موقع عالم اإللكترون... موقع إلكتروني متخصص في علوم الھندسة التكنلوجية واختصاصاتھا المختلفة مكتبة عالم اإللكترون 4electron.com إلى قارئ ھذا الكتاب تحية

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1 موقع عالم اإللكترون... موقع إلكتروني متخصص في علوم الھندسة التكنلوجية واختصاصاتھا المختلفة مكتبة عالم اإللكترون 4eletro.om إلى قارئ ھذا الكتاب تحية طيبة وبعد... لقد أصبحنا نعيش في عالم يعج باألبحاث والكتب والمعلومات وأصبح العلم معيارا حقيقيا لتفاضل األمم والدول والمؤسسات واألشخاص على ح د سواء وقد أمسى بدوره حال شبه وحيد ألكثر مشاكل العالم حدة وخطورة فالبيئة تبحث عن حلول وصحة اإلنسان تبحث عن حلول والموارد التي تشكل حاجة أساسية لإلنسان تبحث عن حلول كذلك والطاقة والغذاء والماء جميعھا تحديات يقف العلم في وجھھا اآلن ويحاول أن يجد الحلول لھا. فأين نحن من ھذا العلم وأين ھو منا نسعى في موقع عالم اإللكترون ألن نوفر بين أيدي كل من حمل على عاتقه مسيرة درب تملؤه التحديات ما نستطيع من أدوات تساعده في ھذا الدرب من مواضيع علمية ومراجع أجنبية بأحدث إصداراتھا وساحات لتبادل اآلراء واألفكار العلمية والمرتبطة بحياتنا الھندسية وشروح ألھم برمجيات الحاسب التي تتداخل مع تطبيقات الحياة األكاديمية والعملية ولكننا نتوقع في نفس الوقت أن نجد بين الطالب والمھندسين والباحثين من يسعى مثلنا لتحقيق النفع والفائدة للجميع ويحلم أن يكون عضوا في مجتمع يساھم بتحقيق بيئة خصبة للمواھب واإلبداعات والتألق فھل تحلم بذلك حاول أن تساھم بفكرة بومضة من خواطر تفكيرك العلمي بفائدة رأيتھا في إحدى المواضيع العلمية بجانب مضيء لمحته خلف ثنايا مفھوم ھندسي ما. تأكد بأنك ستلتمس الفائدة في كل خطوة تخطوھا وترى غيرك يخطوھا معك... أخي القارئ نرجو أن يكون ھذا الكتاب مقدمة لمشاركتك في عالمنا العلمي التعاوني وسيكون موقعكم عالم اإللكترون ww.4eletro.om بكل اإلمكانيات المتوفرة لديه جاھزا على الدوام ألن يحقق البيئة والواقع الذي يبحث عنه كل باحث أو طالب في علوم الھندسة ويسعى فيه لإلفادة كل ساع فأھال وسھال بكم. مع تحيات إدارة الموقع وفريق عمله موقع عالم اإللكترون

2

3 Priiples ad Appliatios of NaoMEMS Physis

4 MICROSYSTEMS Series Editor Stephe D. Seturia Massahusetts Istitute of Tehology Editorial Board Roger T. Howe, Uiversity of Califoria, Berkeley D. Jed Harriso, Uiversity of Alberta Hiroyuki Fujita, Uiversity of Tokyo Ja-Ake Shweitz, Uppsala Uiversity OTHER BOOKS IN THE SERIES: Optial Mirosaers ad Mirospetrometers Usig Thermal Bimorph Atuators Series: Mirosystems, Vol. 4 Lammel, Gerhard, Shweizer, Sadra, Reaud, Philippe, 8 p., Hardover, ISBN: Optimal Sythesis Methods for MEMS Series: Mirosystems, Vol. 3 Aathasuresh, S.G.K. (Ed.) 3, 336 p., Hardover, ISBN: Miromahied Mirrors Series: Mirosystems, Vol. Coat, Robert 3, XVII, 6 p., Hardover, ISBN: Heat Covetio i Miro Duts Series: Mirosystems, Vol. Zohar, Yitshak, 4 p., Hardover, ISBN: Mirofluidis ad BioMEMS Appliatios Series: Mirosystems, Vol. Tay, Frais E.H. (Ed.), 3 p., Hardover, ISBN: Materials & Proess Itegratio for MEMS Series: Mirosystems, Vol. 9 Tay, Frais E.H. (Ed.), 3 p., Hardover, ISBN: Saig Probe Lithography Series: Mirosystems, Vol. 7 Soh, Hyogsok T., Guarii, Kathry Wilder, Quate, Calvi F., 4 p., Hardover ISBN: Mirosale Heat Codutio i Itegrated Ciruits ad Their Costituet Films Series: Mirosystems, Vol. 6 Sugtaek Ju, Y., Goodso, Keeth E. 999, 8 p., Hardover, ISBN: Mirofabriatio i Tissue Egieerig ad Bioartifiial Orgas Series: Mirosystems, Vol. 5 Bhatia, Sageeta 999, 68 p., Hardover, ISBN: Miromahied Ultrasoud-Based Proximity Sesors Series: Mirosystems, Vol. 4 Horug, Mark R., Brad, Oliver 999, 36 p., Hardover, ISBN: X

5 Priiples ad Appliatios of NaoMEMS Physis by Hétor J. De Los Satos NaoMEMS Researh LLC, Irvie, CA, USA

6 A C.I.P. Catalogue reord for this book is available from the Library of Cogress. ISBN (HB) ISBN (HB) ISBN ( e-book) ISBN (e-book) Published by Spriger, P.O. Box 7, 33 AA Dordreht, The Netherlads. Prited o aid-free paper All Rights Reserved 5 Spriger No part of this work may be reprodued, stored i a retrieval system, or trasmitted i ay form or by ay meas, eletroi, mehaial, photoopyig, mirofilmig, reordig or otherwise, without writte permissio from the Publisher, with the exeptio of ay material supplied speifially for the purpose of beig etered ad exeuted o a omputer system, for exlusive use by the purhaser of the work. Prited i the Netherlads.

7 Este libro lo dedio a mis queridos padres y a mis queridos Violeta, Mara, Hetor F. y Joseph. Y sabemos que a los que ama a Dios todas las osas les ayuda a bie, esto es, a los que oforme a su propósito so llamados. Romaos 8:8

8 CONTENTS Prefae Akowledgmets xiii xv. NANOELECTROMECHANICAL SYSTEMS. NaoMEMS Origis. NaoMEMS Fabriatio Tehologies 3.. Covetioal IC Fabriatio Proess 4... Spi-Castig 4... Wafer Patterig 5... Lithography 6... Photoresist Ethig Wet Ethig...3. Dry Ethig...4 Chemial Vapor Depositio Sputterig Evaporatio 6.. MEMS Fabriatio Methods 6... Surfae Miromahiig 7... Bulk Miromahiig Deep Reative Io Ethig...4 Sigle Crystal Silio Reative Eth ad Metal..3 Naoeletrois Fabriatio Elemets..3. Eletro Beam Lithography vii

9 viii Cotets..3. Soft Lithography Moleular Beam Epitaxy Saig Probe Mirosopy Saig Tuelig Mirosope Atomi Fore Mirosopy Carbo Naotubes Naomaipulatio AFM-based Naomaipulatio DIP-Pe Lithography 38.3 Summary 39. NANOMEMS PHYSICS: QUANTUM WAVE-PARTICLE 4 PHENOMENA. Itrodutio 4. Maifestatio of Charge Disreteess 4.. Effets of Charge Disreteess i Trasmissio Lies 4... Idutive Trasmissio Lie Behavior Capaitive Trasmissio Lie Behavior 5.. Effets of Charge Disreteess i Eletrostati Atuatio 5... Fudametal Eletrostati Atuatio 5... Large-sigal Atuatio Swith 5... Small-sigal Atuatio Resoator 5... Coulomb Blokade Sigle Eletro Tuelig Quatum Dots Quatized Eletrostati Atuatio 58.3 Maifestatio of Quatum Eletrodyamial Fores 6.3. va der Waals Fore 6.3. Casimir Fore 6.4 Quatum Iformatio Theory, Computig ad Commuiatios Quatum Etaglemet Eistei-Podolsky-Rose (EPR) State Quatum Gates 7.4. Quatum Teleportatio Deoheree 76.5 Summary NANOMEMS PHYSICS: QUANTUM WAVE PHENOMENA Maifestatio of Wave Nature of Eletros Quatizatio of Eletrial Codutae Ladauer Formula Quatum Poit Cotats 8

10 3.. Quatum Resoae Tuelig Quatum Iterferee Aharoov-Bohm Effet Quatum Trasport Theory Quatized Heat Flow Fermi Liquids ad Lüttiger Liquids Fermi Gas Fermi Liquids Lüttiger Liquids 3. Wave Behavior i Periodi ad Aperiodi Media Eletroi Bad-Gap Crystals Carbo Naotubes Superodutors 3... Superfluidity Superodutivity 3.. Photoi Bad-Gap Crystals Oe-Dimesioal PBC Physis Multi-Dimesioal PBC Physis Geeral Properties of PBCs Advaed PBC Strutures Negative Refratio ad Perfet Leses Cavity Quatum Eletrodyamis Summary NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS Itrodutio NaoMEMS Systems o Chip NaoMEMS SoC Arhitetures NaoMEMS SoC Buildig Bloks Iterfaes Emergig Sigal Proessig Buildig Bloks Charge Detetor Whih-Path Eletro Iterferometer Parametri Amplifiatio i Torsioal MEM Resoator Casimir Effet Osillator Magetomehaially Atuated Beams Systems Futioal Arrays Noise Quatum Squeezig Naomehaial Laser Quatum Etaglemet Geeratio 6 ix

11 x Cotets 4.3. Quatum Computig Paradigms The Io-Trap Qubit The Nulear Mageti Resoae (NMR) Qubit The Semiodutor Solid-State Qubit Superodutig-Based Qubits The Charge Qubit The Flux Qubit The Phase Qubit Summary 9 5 NANOMEMS APPLICATIONS: PHOTONICS Itrodutio Surfae Plasmos Surfae Plasmo Charateristis Naophotois Light-Surfae Plasmo Trasformatio Oe-Dimesioal Surfae Plasmo Propagatio SP Propagatio i Narrow Metal Stripes SP Propagatio i Naowires SP Resoaes i Sigle Metalli Naopartiles SP Couplig of Metalli Naopartiles Plasmoi Waveguides Naophotoi SP-Based Devies Semiodutig Naowires-Based Naophotois Detetio of Surfae Plasmos NSOM/SNOM Summary Appedies A Quatum Mehais Primer 3 A. Itrodutio 3 A. Some Basi Laws Goverig Quatum Systems 3 A.3 Harmoi Osillator ad Quatizatio 5 A.4 Creatio ad Aihilatio Operators 6 A.5 Seod Quatizatio 8 A.5. Field Operators 4 B Bosoizatio 7 B. Itrodutio 7 B. Bosoizatio Rules 7 B.3 Bosoi Field Operators 3 B.4 Bosoizatio Idetity ad Its Appliatio to Hamiltoia with Liear Dispersio 33

12 B.5 Bosoizatio Treatmet of Spiless Eletros i Oe-Dimesioal Wire 39 Referees 4 Idex 53 xi

13 PREFACE This book presets a uified expositio of the physial priiples at the heart of NaoMEMS-based devies ad appliatios. NaoMEMS exploits the overgee betwee aotehology ad miroeletromehaial systems (MEMS) brought about by advaes i the ability to fabriate aometer-sale eletroi ad mehaial devie strutures. I this otext, NaoMEMS-based appliatios will be prediated upo a multitude of physial pheomea, e.g., eletrial, optial, mehaial, mageti, fluidi, quatum effets ad mixed domai. Priiples ad Appliatios of NaoMEMS Physis otais five hapters. Chapter provides a omprehesive presetatio of the fudametals ad limitatios of aotehology ad MEMS fabriatio tehiques. Chapters ad 3 address the physis germae to this dimesioal regime, amely, quatum wave-partile pheomea, iludig, the maifestatio of harge disreteess, quatized eletrostati atuatio, ad the Casimir effet, ad quatum wave pheomea, iludig, quatized eletrial odutae, quatum iterferee, Lüttiger liquids, quatum etaglemet, superodutivity ad avity quatum eletrodyamis. Chapter 4 addresses potetial buildig bloks for NaoMEMS appliatios, iludig, aoeletromehaial quatum iruits ad systems (NEMX) suh as harge detetors, the whih-path eletro iterferometer, ad the Casimir osillator, as well as a umber of quatum omputig implemetatio paradigms, iludig, the io-trap qubit, the NMR-qubit, superodutig qubits, ad a semiodutor qubit. Fially, Chapter 5 presets NaoMEMS appliatios i photois, partiularly fousig o the xiii

14 xiv Prefae geeratio, propagatio, ad detetio of surfae plasmos, ad emergig devies based o them. The book assumes a preparatio at the advaed udergraduate/begiig graduate studet level i Physis, Eletrial Egieerig, Materials Siee, ad Mehaial Egieerig. It was partiularly oeived with the aim of providig ewomers with a muh eeded oheret sietifi base for udertakig study ad researh i the NaoMEMS field. Thus, the book takes great pais i rederig trasparet advaed physial oepts ad tehiques, suh as quatum iformatio, seod quatizatio, Lüttiger liquids, bosoizatio, ad superodutivity. It is also hoped that the book will be useful to faulty developig/teahig ourses emphasizig physis ad appliatios of aotehology, ad to Naotehology researhers egaged i aalyzig, modelig, ad desigig NaoMEMS-based devies, iruits ad systems.

15 ACKNOWLEDGMENTS The idea for this book bega to take shape upo meetig Mr. Mark de Jogh, Seior Publishig Editor of Spriger, at the Europea Mirowave Coferee i Muih, Germay, i Otober, 3. Ubekowst to the author, Dr. Harrie A.C. Tilmas, of IMEC, Belgium, had reommeded him to Mr. de Jogh as a potetial author. Upo a hae eouter Mr. de Jogh itrodued himself ad suggested the writig of a book for (the) Kluwer. The author submitted the book proposal i late November, 3 ad reeived ews of its aeptae soo thereafter, as Spriger s Mirosystems book series editor, Dr. Stephe D. Seturia,. had provided a very positive ad omplemetary report. Therefore, the author is pleased to akowledge Dr. Tilmas, for brigig his ame to Mr. de Jogh s attetio, Dr. Seturia, for his positive reommedatio of the book proposal, ad Mr. de Jogh, for providig him with the opportuity to write the book. Furthermore, the author gratefully akowledges Mr. Mark de Jogh ad Ms. Cidy M. Zitter, his Seior Assistat, for their patiee ad uderstadig durig the ourse of the work. The book ites more tha referees. Aess to these would ot have bee possible without the exellet assistae of Mr. Tim Lee, whom he gratefully akowledges. Fially, the author gratefully akowledges the uderstadig of his wife, Violeta, alog the ourse of the projet, as well as her exellet assistae i preparig the fial ameraready mausript. Hétor J. De Los Satos xv

16 Chapter NANOELECTROMECHANICAL SYSTEMS. NaoMEMS Origis The field of Naotehology, whih aims at exploitig advaes i the fabriatio ad otrolled maipulatio of aosale objets, is attratig worldwide attetio. This attetio is prediated upo the fat that obtaiig early supremay i this field of miiaturizatio may well be the key to domiatig the world eoomy of the st etury ad beyod. NaoMEMS exploits the overgee betwee aotehology ad miroeletromehaial systems (MEMS) brought about by advaes i the ability to fabriate aometer-sale eletroi ad mehaial devie strutures. Ideed, the impat of our ability to make ad otrol objets possessig dimesios dow to atomi sales, perhaps first osidered by the late Rihard Feyma i his 959 talk There is Plety of Room at the Bottom is expeted to be astoudig [], []. I partiular, miiaturizatio, he isiuated, has the potetial to fuel radial paradigm shifts eompassig virtually all areas of siee ad tehology, thus givig rise to a ulimited amout of tehial appliatios. Sie high tehology fuels the prosperity of the world s most developed atios, it is easy to see why the stakes are so high. Progress i the field of miiaturizatio beefited from the advet of the semiodutor idustry i the 96s, ad its rae to irease profits through the dowsalig of iruit dimesios whih, osequetly, ireased the desity ad the yield of iruits fabriated o a give wafer area. This desity, whih derived from progress i photolithographi tools to produe the ever smaller two-dimesioal patters (devie layouts) of a itegrated iruit (IC), has ireased sie by more tha seve orders of magitude ad has ome to be aptured by Moore s law: The umber of ompoets per

17 Chapter hip doubles every 8 moths []. The ulmiatio of suh miiaturizatio program, it is widely believed, is the demise of Moore s law, whose maifestatio is already beomig apparet due to a ireasig predomiae of the quatum mehaial ature of eletros i determiig the behaviour of devies with ritial dimesios (roughly) below m. This lie of developmet is losely related to the field of quatum devies/aoeletrois, whih was prompted by the oeptio of a umber of atomi-level depositio ad maipulatio tehiques, i partiular, moleular beam epitaxy (MBE), origially exploited to ostrut laboratory devies i whih the physis of eletros might be probed ad explored, followig the disovery of eletro tuellig i heavily-doped p-jutios [3]. Naoeletrois did produe iterestig physis, for istae, the disovery of Coulomb blokade pheomea i sigle-eletro trasistors, whih maifested the partile ature of eletros, ad resoat tuellig ad odutae quatizatio i resoat tuellig diodes ad quatum poit otats, respetively, whih maifested the wave ature of eletros [4-6]. These quatum devies, i ojutio with may others based o exploitig quatum pheomea, geerated a lot exitemet durig the late 98s ad early 99s, as they promised to be the geesis for a ew digital eletrois exhibitig the properties of ultra-high speed ad ultra-low power osumptio [7-8]. While efforts to realize these devies helped develop the skills for fabriatig aosale devies, ad efforts to aalyze ad model these devies helped to develop ad mature the field of mesosopi quatum trasport, the sober reality that ryogei temperatures would be eessary to eable their operatio drastially restrited their ommerial importae. A few pratial devies, however, did exert ommerial impat, although oe as muh as that exerted by silio IC tehology, i partiular, heterojutio bipolar trasistors (HBTs), ad high-eletro mobility trasistors (HEMTs), whih exploit the odutio bad disotiuities germae to heterostrutures, ad modulatio dopig to reate -D eletro ofiemet ad quatizatio, respetively, ad reder devies superior to their silio outerparts for GHz-frequey mirowave ad low-trasistorout digital iruit appliatios [9-4]. The ommerial suess of the semiodutor idustry, ad its dowsalig program, motivated emulatio efforts i other disiplies, i partiular, those of optis, fluidis ad mehais, where it was soo realized that, sie ICs were fudametally two-dimesioal etities, tehiques had to be developed to shape the third dimesio, eessary to reate mehaial devies exhibitig motio ad produed i a bath plaar proess [5]. These tehiques, whih iluded surfae miromahiig, bulk miromahiig, ad wafer bodig, beame the soure of what are ow mature devies, suh as aelerometers, used i automobile air bags,

18 . NANOELECTROMECHANICAL SYSTEMS 3 ad pressure sesors, o the oe had, ad a umber of emergig devies, suh as, gyrosopes, flow sesors, miromotors, swithes, ad resoators, o the other. Coiidig, as they do, with the dimesioal features germae to ICs, i.e., miros, these mehaial devies whose behavior was otrolled by eletrial meas, exemplified what has ome to be kow as the field of miroeletromehaial systems (MEMS). Three evets might be ostrued as ospirig to uite aoeletrois ad MEMS, amely, the ivetio of a umber of saig probe mirosopies, i partiular, saig tuelig mirosopy (STM) ad atomi fore mirosopy (AFM), the disovery of arbo aotubes (CNTs), ad the appliatio of MEMS tehology to eable superior RF/Mirowave systems (RF MEMS) [6-8]. STM ad AFM, by eablig our ability to maipulate ad measure idividual atoms, beame ruial agets i the imagig of CNTs ad other 3-D aosale objets so we ould both see what is built ad utilize maipulatio as a ostrutio tehique. CNTs, oeptually, two-dimesioal graphite sheets rolled-up ito yliders, are quitessetial aoeletromehaial (NEMS) devies, as their lose to - m diameter makes them itrisially quatum mehaial -D eletroi systems while, at the same time, exhibitig superb mehaial properties. MEMS, o the other had, due to their iteral mehaial struture, display motioal behavior that may ivade the domai of the Casimir effet, a quatum eletrodyamial pheomeo eliited by a loal hage i the distributio of the modes i the zero-poit flutuatios of the vauum field permeatig spae [9-]. This effet whih, i its most fudametal maifestatio, appears as a attrative fore betwee eutral metalli surfaes, may both pose a limit o the pakig desity of NEMS devies, as well as o the performae of RF MEMS devies []. I the balae of this hapter, we preset the fudametals of the fabriatio tehiques whih form the ore of NaoMEMS devies, iruits ad systems.. NaoMEMS Fabriatio Tehologies NaoMEMS fabriatio tehologies exted the apabilities of ovetioal itegrated iruit (IC) proesses, whih are prediated upo the operatios of formig preise patters of metallizatio ad dopig (the otrolled itrodutio of atomi impurities) oto ad withi the surfae ad bulk regios of a semiodutor wafer, respetively, with the performae of the resultig devies depedig o the fidelity with whih these operatios are effeted. Exellet books o IC fabriatio, givig i-depth overage of the topi, already exist [3] ad the reader iterested i proess developmet

19 4 Chapter is advised to osult these. The expositio udertake here is ursory i ature ad oly aims at providig a uderstadig of the fudametals ad issues of preset ad future NaoMEMS fabriatio tehologies... Covetioal IC Fabriatio Proesses Covetioal IC proesses are based o photolithography ad hemial ethig, ad are sythesized by the iterative appliatio to a wafer of a yli sequee of steps, amely: Spi-astig ad patterig, material depositio, ad ethig. The saliet elemets of these steps are preseted i what follows.... Spi-Castig The first step (after thoroughly leaig the wafer), i defiig a patter o a wafer, is to oat it with a photoresist (PR), Figure -, a visous light- Figure -. Coatig wafer with photoresist. (a) Spi-astig. (b) Soft-bake i ove. () Hard bake i hot plate. sesitive polymer whose hemial ompositio hages upo exposure to ultraviolet (UV) light. The proess of applyig the PR to the wafer i order to ahieve a uiform thikess is alled spi-astig, ad usually ivolves the followig steps: ) Pourig a few drops of the PR at the wafer eter; ) Spiig the wafer for about 3 seods oe it reahes a presribed rotatioal speed of several thousad revolutios-per-miute; ad 3) Bakig it at temperatures of several hudred degrees Celsius to produe a well-adhered

20 . NANOELECTROMECHANICAL SYSTEMS 5 solvet-free dry layer. The resultig PR film thikess is iversely proportioal to the square root of the rotatioal speed, ad diretly proportioal to the peret of solids i it. Determiig these parameters is oe of the first steps i developig a proess.... Wafer Patterig Oe a uiform solid PR layer oats the wafer, this is ready for patterig. This is aomplished by iterposig a glass mask, whih otais both areas that are trasparet ad areas that are opaque, betwee a UV soure ad the PR-oated wafer. As a result, seletive hemial hages are effeted o the PR i aordae with the desired patter, Figure -. Whe it (a) Si Photoresist (PR) SiO (b) Mask Positive PR Si SiO Negative PR () SiO Si Si SiO (d) SiO SiO Si Figure -. Wafer patterig with positive ad egative photoresists. (After [4]). is desired that the reated patter be idetial to that i the glass mask, a positive PR, whih hardes whe exposed to UV light, is employed. Otherwise, whe it is desired that the reated patter be the egative of that i the mask, a egative PR is employed. I the former ase, UV exposure Si

21 6 Chapter hardes the PR, whereas i the latter, UV exposure weakes the PR. Thus, subsequetly, whe the UV-exposed wafer is ethed, the weakeed parts of the PR will be dissolved ad the desired patter revealed. There are two tehiques to dissolve the PR, amely, wet ad dry ethig. These are preseted ext.... Lithography The highest resolutio (miimum size) ad quality of the patter to be defied o a wafer depeds o how well the mask image is trasferred to the PR. Image formatio, i tur, is determied by the lithographi proess ad type of PR employed. The lithographi proess a make use of a optial soure, a eletro beam soure, or a X-ray soure for reatig the desired patter o the wafer. I this setio we deal with the first ad the last approahes. Optial lithography, Figure -3, may be employed i ojutio with (a) (b) () Light Soure Optial System Mask Photoresist Wafer Gap Figure -3. Skethes of ommo approahes to optial lithography. (a) Cotat pritig. (b) Proximity. () Projetio. (After [3]). either, otat pritig, i whih the image is projeted through a mask that is i itimate otat with the wafer, or proximity pritig, i whih the image is projeted through a mask separated by ~ 5µ m from the wafer, or projetio pritig, i whih the mask is separated may

22 . NANOELECTROMECHANICAL SYSTEMS 7 etimeters away from the uderlyig wafer. Beause, the otat ad proximity approahes are proe to suffer from dust partiles preset betwee the mask ad the PR, the projetio approah is preferred for reatig aosale-feature patters. The resolutio of a good projetio optial lithography system is give by.5( λ NA), where λ is the exposure wavelegth ad NA is the umerial aperture of the projetio optis, at a depth of fous apability of ± λ ( NA) [3]. The highest resolutio of optial photolithography appears to be about 5m-m for produtio devies, dow to 7m for laboratory devies, ad is set by diffratio, i.e., at smaller sizes features beome blurred. Overomig these tehial issues, whih ivolves developig smaller wavelegth light soures ad optis, is diffiult. Thus, the ost of optial lithography produtio equipmet apable of reahig resolutios below m, is deemed by idustry as prohibitive [4]. X-ray lithography, see Fig. -4, utilizig the low eergy of soft x-rays at wavelegths betwee 4 ad 5 Å, is relatively impervious to satterig effets. X-ray Soure φ D X-ray Mask Substrate r L g Wafer d δ Figure -4. Sketh of fators eliitig geometrial limitatios i x-ray lithography. Typial values for the geometrial parameters are: φ 3mm, g 4µ m, L 5m, r 63mm. (After [3].) This makes them ameable for use i exposig thik PRs whih, beause of their low absorptio, a peetrate deeply ad produe straight-walled PR images with high fidelity. Beause of diffiulty i reatig optial elemets

23 8 Chapter at these wavelegths, however, the method of image projetio employed is proximity pritig through a mask otaiig x-ray absorbig patters. The mask is separated from the PR-wafer a distae of just about 5 µ m, but sie dust partiles with low atomi umber do ot absorb x-rays, o damage is aused to the patter. Despite the potetial for highest resolutio germae to x-ray lithography, two fators have bee idetified as potetially limitig it. Both fators origiate i geometrial aspets of the illumiatio. I partiular, there is the possibility that a sigifiat peumbral blur δ φ g L be itrodued o the positio of the resist image by the exteded poit soure of diameter φ loated a distae L above a mask separated from the wafer a distae g. Also, a potetial for lateral magifiatio error is preset, due to the divergee of the x-ray from the poit soure ad the fiite mask to wafer separatio. Aordigly, images of the projeted mask are shifted laterally by a amout d r g L. Eve with perfet resolutio, patter formatio quality depeds o how the PR respods to the impigig lightwave or eletro beam. This is addressed ext.... Photoresist The mehaism for image trasfer to the PR ivolves alterig its hemial or physial struture so the exposed area may subsequetly be easily dissolved or ot dissolved. Aordig to the previous setios, patter formatio is effeted o optial resists, eletro beam resists, or x-ray resists. Optial lithography resists may be egative or positive. The fudametal differee, i terms of how they affet the resolutio of the image trasferred, is rooted i their hemial ompositio. I the egative resist, whih ombies a ylized polyisopropee polymer material with a photosesitive ompoud, the latter beomes ativated by the absorptio of eergy with wavelegths i the - to 45-Å rage. The photosesor ats as a aget that auses ross likig of the polymer moleules by trasferrig to them the reeived eergy. As a result of the ross likig, the moleules moleular weight ireases ad this eliits their isolubility i the developig system. The highest resolutio limit of a egative PR derives from the fat that durig developmet the exposed (ross liked) areas swell, whereas the uexposed low moleular weight areas are dissolved. The miimum resolvable feature whe usig a egative resists is typially three times the film thikess [3]. I respose to light the positive resist, whih also otais a polymer ad a photosesitizer, the latter beomes isoluble i the developer ad, thus, prevets the dissolutio of the polymer. Sie the photosesitizer preludes

24 . NANOELECTROMECHANICAL SYSTEMS 9 the developer from permeatig the PR film, o film swellig is produed ad a greater resolutio is possible [3]. Eletro beam lithography also utilizes egative ad positive resists. I a egative resist, the eletro beam prompts ross-likig of the polymer, whih results o ireased moleular weight, ireased resistae to the developer, ad swellig durig developmet. A ommo egative resist used with eletro beam lithography is COP, poly (glyidylmetharylate-o-ethyl arylate), whih reders a resolutio of µ m. I a positive resist, the eletro beam auses hemial bod breakig, redued moleular weight, ad redued resistae to dissolutio durig developmet. Commo positive resists used with eletro beam lithography ilude poly(methyl metharylate) (PMMA) ad poly(butae- ketoe) (PBS), whih reder a resolutio of.µ m. X-ray lithography also utilizes egative ad positive resists, i partiular, COP, PBS ad PMMA with resolutio similar to that stated above is obtaied....3 Ethig Defiig the desired patter o the PR oatig the wafer is ruial. The patter fidelity is defied its seletivity ad aspet ratio, Figure -5. (a) Photoresist Layer Ethed Layer Eth Stop Layer d : Eth Depth Wafer (b) S: Side Eth w: Miimum Width w S Photoresist Layer Ethed Layer Eth Stop Layer d : Eth Depth d : Over Eth Wafer Eth Seletivit y Depth Over Eth Aspet Ratio Eth Depth Miimum Width Figure -5. Patter trasfer defiitio. (a) Ideal. (b) Realisti. (After [5].)

25 Chapter It is see i this figure that the fidelity of the patter trasferred is futio of how preisely the resultig width of the ethed layer resembles that of the PR patter, as quatified by the seletivity ad aspet ratio. Aordigly, four searios may be evisioed, Figure -6, whih reflet the relative stregth with whih the ethat attaks the PR, the ethed material, ad the eth stop. I partiular, it may be surmised from Figure -6(d) that the miimum width of a patter, i.e., how arrow it may be, is limited by the lithography proess to defie the pertiet width i the PR ad the resultig degree of uderut of the PR mask. Thus, ethats produig isotropi profiles (oes i whih the vertial ad horizotal ethig rates are equal), are ot ameable to patter the arrowest features. I geeral, the results deped o a umber of fators otrollig the ethig hemial reatio, suh as temperature ad mixig oditios, whether or ot the ethig aget employed is i the liquid or gaseous state, how well the PR adhered to the wafer durig spi-astig. I the ext setio we address two of the most importat fators, amely, the state of the ethat. (a) (b) () (d) Figure -6. Ethig haraterizatio. (a) Over Eth<<Eth DepthSeletive. (b) Over Eth~Eth DepthNo-seletive. () Side Eth<<Eth Depth. (d) Side Eth~Eth Depth. (After [5].)...3. Wet Ethig I this approah to dissolve the weakeed PR, the pattered wafers are immersed i a liquid hemial ethat, Figure -7. The ethed profile may be isotropi or aisotropi depedig of the wafer orietatio. If this is amorphous, a isotropi profile will result, i.e., the horizotal ad vertial ethig rates are similar. Otherwise, if it is sigle-rystal, a aisotropi profile may result. A umber of hemials employed to effet aisotropi ethig i silio are i use. These ilude tetramethylammoium hydroxide (TMHA), potassium hydroxide (KOH), ad ethylee diamie pyrohateol

26 . NANOELECTROMECHANICAL SYSTEMS (EDP). Detailed experimets to eluidate the mehaism resposible for aisotropi ethig have bee udertake [3]. The fudametal priiple behid aisotropi ethig appears to be this: whe differet rystal plaes possess differet atomi desities, those plaes with greater desity will eth at a slower rate tha those with lower atomi desity. Figure -7. Ethig of wafer immersed i liquid hemial solutio. A exhaustive ompilatio of hemial reatios for pertiet ethig hemials/wafer materials has bee published by Williams ad Muller [9]. Table - below gives some of typial ethed material/ethig solvet pairs. Table -. Wet ethig targets ad solvets Ethed Material Ethig Solvet Silio KOH, TMAH, EDP Silio oxide HF Silio itride H 3 PO 4 Alumium H 3 PO 4 Whe it omes to reatio of free-stadig strutures via surfae miromahiig tehiques (desribed below), wet ethig is aompaied by various drawbaks. For istae, the surfae tesio exerted o the deliate free-stadig strutures by the fluid s hydrodyami fores may prelude their omplete release, or may eve break them. Dry ethig tehiques, irumvet these drawbaks ad are disussed ext Dry Ethig I this approah, show i Figure -8, a gas/vapor or plasma is used as a soure of reative atoms that dissolve the weakeed PR. Typial mathig pairs of ethed material ad ethig gas used i IC fabriatio are show i Table -.

27 Chapter Table -. Ethed material-ethig gas pairs. Silio or Polysilio SF 6, CF 4 Silio dioxide CHF 4 /H Silio itride CF4/O Alumium BCl Two fluorie-otaiig gases have bee reetly adopted for dry ethig proesses, amely, Xeo difluoride, XeF [3] ad Boro Fluoride, BrF 3 [3]. XeF eables a isotropi dry-eth proess for silio, whih is very seletive to alumium, silio dioxide, silio itride ad photoresist. The XeF gas is partiularly useful i the post-proessig of CMOS ICs. It a be sublimated from its solid form at Torr ad room temperature ad, whe applied to solid-phase Si, it obeys the followig reatio: XeF SiXeSiF 4 XeF ethig of Si ahieves high seletivity with a umber of maskig materials, suh as, SiO, Si 3 N 4, Al, PR, ad phosphosiliate glass (PSG), at ethig rates ragig from 3µ m / mi to as high as 4µ m / mi [3], ad is haraterized by the produtio of measurable amouts of heat. Whe i the presee of water or vapor, XeF reats with them to form HF. I terms of its potetial appliatio to aostruture formatio, XeF ethig has the drawbak that the resultig surfaes ted to have a graular fiish with a feature size of about µ m. Eth Gas Wafers Pump Groud Shield Cathode RF Figure -8. Ethig of wafer immersed i plasma

28 . NANOELECTROMECHANICAL SYSTEMS 3 BrF 3 o the other had, eables isotropi ethig of Si with maskig materials suh as Al, Au, Cu, Ni, PR, SiO, ad Si 3 N 4, while ahievig surfae fiish feature size of 4-5m. Dry ethig, it may be oluded, is ot ameable to reatig aostrutures....4 Chemial Vapor Depositio The result of patterig a wafer is to reder some areas of its surfae bare to reeive the depositio of various atomi speies, while prevetig suh depositio i other areas. Chemial vapor depositio (CVD) is oe of the tehiques utilized to itrodue atoms ito the exposed wafer areas ad, for silio wafers, etails the dissoiatio of gasses, suh as silae, SiH 4, arsie (AsH 3 ), phosphie (PH 3 ), ad diborae (B H 6 ), o the wafer surfae at high temperatures, usually i the 45-8 C rage. The hamber otaiig the wafers durig the depositio, Fig. -9, is usually held at pressures betwee. ad Torr, ad the resultig properties of the deposited materials varies. Pressure Sesor Wafers 3-Zoe Furae Pump Load Door Gas Ilet Figure -9. Shemati of hot-wall, redued pressure CVD reator. For istae, uder appropriate parameters of temperature, depositio rate, ad rystalliity of the wafer, the deposited material may grow epitaxially, i.e., maitaiig the same rystallographi ature of the substrate wafer, or beome polyrystallie, i.e., exhibitig a agglomeratio of radomly orieted rystallites. I the otext of silio proesses, typial materials deposited via CVD ilude: polyrystallie silio (polysilio), silio dioxide (SiO ), ad stoihiometri silio itride (Si x N y ), to thikesses ~ µ m. The most ommo reatios for depositig these materials are show i Table -3.

29 4 Chapter Table -3. Commo CVD reatios ad depositio temperatures for pertiet materials. [4] Produt Reatats Depositio temperature ( C) Silio dioxide SiH 4 CO H SiCl H N O SiH 4 N O SiH 4 NO Si(OC H5) 4 SiH 4 O Silio itride SiH 4 NH 3 SiCl H NH Polysilio SiH A alterate method to effet material depositio o a wafer while avoidig the high temperatures required i a CVD reator is to utilize a hotwall plasma depositio reator, Fig. -. I this approah, the wafers are orieted vertially i otat with log alteratig slabs of graphite or alumium eletrodes iside a quartz tube heated by a furae. Pressure Sesor 3-Zoe Furae Graphite Eletrodes Pump Load Door Gas Ilet RF Figure -. Sketh of hot-wall plasma depositio reator. (After [4].) The, oetio of the alterate slabs to a power supply, idues a glow disharge of the gas flowig i the spae betwee eletrodes, whih rus parallel to the wafers. By takig the eergy for the reatio from the glow disharge, the depositio may be ahieved at a wafer temperature i the rage of to 35 C, e.g., Table -4. Table -4. Commo plasma-assisted CVD reatios for depositig pertiet materials [4]. Produt Reatats Depositio temperature ( C) Plasma silio dioxide SiH 4 N O Plasma silio itride SiH 4 NH 3 SiH 4 N

30 . NANOELECTROMECHANICAL SYSTEMS Sputterig While depositio via CVD requires high temperatures to failitate gas dissoiatio, ad migratio oe the atoms/moleules reah the wafer surfae, sputterig ivolves a totally differet mehaism. I sputterig, a plasma is reated by ioizig a iert gas, typially Argo, at low pressures, e.g., ~mtorr. The material oe wats to deposit o the wafer origiates i the bombardmet with high eergy (typially Argo, Ar ) ios, preset i the plasma above the target substrate otaiig the material to be deposited o the wafer. Target (athode) bombardmet auses the ejetio, via mometum trasfer, of its surfae atoms, Fig. -. The ejeted atoms, i tur, fly off from the target ad ome to rest o other surfaes withi the hamber, i partiular, the wafers of iterest. The material trasfer proess is atomi i ature, therefore, its trasfers to the wafer i the same ratio it preset i the target. Substrate Aode Ar Cathode N S Target S N N S Magetro RF Figure -. Sketh of sputterig depositio system. Magetro sputterig is oe of the most versatile sputterig tehiques beause it a be employed to deposit both isulatig ad o-isulatig materials, e.g., Ti, Pt, Au, Mo, W, Ni, Co, Al O 3, SiO, Fe, Cr, Cu, FeNi, TiNi, AlN, SiN, et. The tehique is based o reatig a plasma by iduig the breakdow of a iert gas, e.g., Ar, i the presee of a strog mageti field. The resultig Ar ios are aelerated by the potetial gradiet betwee athode ad aode, impige o the target ad, thus, reate the flux of material towards the substrate to be oated. Typial maximum thikess of deposited materials is ~ 5 µ m.

31 6 Chapter...6 Evaporatio I this depositio tehique, the evaporat, the material oe wats to deposit o the wafer, is heated off a ruible. Heatig may be effeted by resistive meas or by diret eletro-beam bombardmet, Fig. -. I the resistive heatig approah, the wafers to be oated ad the ruible otaiig the evaporat, are plaed iside a vauum hamber ad the latter heated util its vapor pressure is greater tha that origially existig i the hamber. Evaporatio results i oatig everythig iside the hamber, i partiular, the wafers of iterest. I the eletro-beam bombardmet approah, lie-of-sight oatig is obtaied. Substrate Vauum B e BEAM - Evaporat Figure -. Sketh of eletro-beam-based evaporatio system. Typial materials deposited by this tehique ilude Al, Cr, Au, Ni, Fe, Ti, Cu, Pt, FeNi, TiNi, SiW, MgO, SiO, Al O 3, AlN, SiN. The depositio rate is a futio of the distae betwee the evaporat ad the substrate, ad its typial maximum thikess is usually ~ 5 µ m... MEMS Fabriatio Methods The reatio of moveable strutures eessitates extedig the -D IC fabriatio proess to ilude shapig of the third dimesio, perpediular to the substrate; this is exemplified, i silio, by four fudametal tehiques, amely, Surfae Miromahiig, Bulk Miromahiig, Deep Reative Io Ethig (DRIE), ad sigle rystal silio reative eth ad metal (SCREAM), whih are preseted ext.

32 . NANOELECTROMECHANICAL SYSTEMS 7... Surfae Miromahiig I surfae miromahiig, 3-D mehaial strutures are ostruted i a layered fashio. Two types of layers, based o their material ompositio/ethig properties, are employed, amely, sarifiial ad strutural layers. The former are ultimately dissolved via a proess step amed release, ad the latter remai, beomig part of the free-stadig movable struture proper. The simplest elemet illustratig the surfae miromahiig tehique is, perhaps, the atilever beam. Figure.3 skethes its formatio. Typial ombiatios of sarifiial ad strutural materials, ad orrespodig ethat are show i Table.5 [7]. SIDE VIEW Sarifiial layer TOP VIEW Wafer Strutural layer Beam Proess sequee Figure -3. Sketh of the formatio of a atilever beam by surfae miromahiig. From top to bottom of the figure, the sarifiial material is deposited ad pattered (top), the the strutural material is deposited ad pattered (middle), the sarifiial layer is released (bottom). Table -5. Strutural/Sarifiial/Ethat Material Systems [7]. Strutural Material Sarifiial Material Ethat Alumium Sigle-rystal silio EDP, TMAH, XeF Alumium Photoresist Oxyge plasma Copper or Nikel Chrome HF Polyimide Alumium Al eth (Phosphori, Aeti, Nitri Aid) Polysilio Silio dioxide HF Photoresist Alumium Al eth (Phosphori, Aeti, Nitri Aid) Silio dioxide Polysilio XeF Silio itride pr Borodoped polysilio Udoped polysilio KOH or TMAH

33 8 Chapter... Bulk Miromahiig As the ame implies, bulk miromahiig sulpts the substrate itself to form the 3-D mehaial struture. The simplest example of this tehique is illustrated by the reatio of a avity, show i Figure.4. As suggested, the aspet ratio of the avity or pit is determied by the ethig properties of the atomi plaes whih, i tur, are futio of the rystallographi properties ad orietatio of the wafer, i partiular, the greater the umber of atoms o a give plae, the slower its ethig rate. To uderstad this statemet we explai the oept of Miller idies [8]. Proess sequee Eth Masks Wafer Eth Mask Wafer Eth Mask θ (a) Wafer () Si () () () () 54.7 o (b) Figure -4. Sketh of bulk miromahied avity. (a) From top to bottom of the figure, a mask is deposited (top), the pattered to expose the wafer (middle), ad the the wafer is exposed to a ethat (bottom). (b) Cavity walls are delimited by the rystallographi plaes of the wafer.

34 . NANOELECTROMECHANICAL SYSTEMS 9 The arragemet ad orietatio of atoms i a rystallie solid is speified with referee to ertai diretios, see Figure -5. Thus, with respet to the origi of a Cartesia set of oordiates, the positio of a atom may be desribed as beig [] Crystal Diretios [ab] [] [] a b Figure -5 Nomelature of rystal diretios. [ab], that is, a uit s alog the diretio [], b uits alog the diretio [], ad uits alog the diretio []. Sie a plae may be desribed by a vetor perpediular to it (its ormal), the diretio [ab] also desribes a plae, whih is deoted the plae (ab), show i Figure -6(a). CrystalPlae [] [] h l (a) (ab) [] k () () () (b) Figure -6. (a) Desriptio of rystallographi plae by its ormal (ab). (b) Desriptio of rystallographi plaes of ubi (atoms oupy the orers ad faes of a ube) rystal by Miller idies. Notie that, sie a plae is desribed by three poits ommo to it, the poits of itersetio betwee a plae ad the three oordiate axes may also be used to deote it. I partiular, see Figure -6(b), the poits h, l, ad k, alog the oordiate axes [], [], ad [], respetively, might be

35 Chapter used for this purpose. However, to aommodate the possibility that the plae might be parallel to oe of the oordiate axes, i whih ase the itersetio would our at ifiity, the reiproals of these poits of itersetio, (/h, /l, /k), are used istead. Figure -6(b) shows examples rystallographi plaes ad their orrespodig of Miller idies [8] for a ubi rystal suh as silio. The fat that the aspet ratio of bulk miromahied strutures is limited by the atural iliatio of the rystallographi plaes makig up the walls, motivated the developmet of tehiques to irease it. The setios below address two of these....3 Deep Reative Io Ethig The idea behid DRIE is to ahieve high-aspet ratio trehes by seletively ehaig the eth rate at the bottom of the treh, while ihibitig the lateral eth rate. This is aomplished by ombiig a sequee of plasma ethig ad polymerizatio steps [3], [3], see Figure -7(a). Eth Oxidize Eth (a) (b) Figure -7. Deep reative io ethig (a) Ethig/polymerizatio sequee. (b) Wall sallopig. Durig the plasma ethig steps, as idiated previously, positive ios resultig from the breakdow disharge of a gas above the silio wafer, bombard the silio surfae as they fall vertially towards the egatively harged wafer. To ahieve vertial seletivity, the sidewalls are proteted by a polymer (PR). Thus, this results i ethig beig primarily effeted at the bottom of the treh. Eah ethig step, whih may result i a lateral eth of

36 . NANOELECTROMECHANICAL SYSTEMS teths of miros, is stopped after the maximum tolerated lateral eth is produed. By repeatig the passivatio/eth sequee, trehes with overall depths of up to several hudred s of miros have bee demostrated. The proess proeeds at room temperature, a produe seletivities of : i stadard PR masks, 3: i hard masks suh as SiO ad Si3N4, ad exhibits ethig rates of 6µ m / se [3]. As a result of this proess, the walls of the ethed trehes exhibit a sallopig struture, see Figure - 7(b). The appliatio of DRIE requires aquirig the DRIE equipmet. A alterative to DRIE for better tha ovetioal bulk miromahiig, but ot as expesive as DRIE, is preseted ext....4 Sigle Crystal Silio Reative Eth ad Metal (SCREAM) Similar to DRIE, the sigle rystal silio reative eth ad metal (SCREAM I) proess effets bulk miromahiig usig plasma ad reative io ethig (RIE) [33], see Fig..8. The proess, however, employs stadard tools, is self-aliged, employs oe mask to defie strutural elemets ad metal otats, ad employs a temperature below 3 C. This low temperature apability makes it ameable for itegratio of MEMS devies with very large sale itegratio (VLSI) tehology [33]. (a) (e) (b) (f) () (g) (d) Figure -8. SCREAM I proess flow. (a) Depositio ad patterig of PECVD maskig oxide. (b) RIE of silio with BCl 3 /Cl. Typially 4- µ m deep. () Depositio of oxide sidewall via PECVD, typially.3 µ m thik. (d) Vertial eth of bottom oxide with CF 4 /O RIE. (e) Eth of silio 3-5 µ m beyod ed of sidewall with Cl RIE. (f) Isotropi RIE release of strutures with SF 6 RIE. (g) Sputterig depositio of alumium metal. The devie show is a beam, free to move left-right, ad its orrespodig parallel-plate apaitor. (After [33].)

37 Chapter..3 Naoetehology Fabriatio Elemets The elemets of aotehology fabriatio rage from tehiques to produe two-dimesioal patters with deep-submiro/aometer-sale widths, to tehiques to produe atomi-thik layers/multi-layers of various material ompositios, to tehiques to preisely maipulate atomi-size partiles. These tehiques, together with those preseted previously, ostitute the arseal at the ore of NaoMEMS...3. Eletro Beam Lithography Eletro beam lithography utilizes eletros, istead of the projetio of a mask image illumiated by photos, to reate diretly the desired patter o the PR, Figure -9. X-Y Mask Data Computer Cotrol Eletro Gu Beam Blakig Defletio Coils Vauum Chamber e - Eletro Resist Metal Film Substrate Table Table Positio Moitor Mehaial Drive Figure -9. Sketh of eletro bea lithography system. (After [3].) Sie the wavelegth of eletro aelerated through a potetial differee V is λ ( Å) 5 V, a eletro beam may be foused to a diameter of..5µm, ad resolutios of m are obtaied. The eletro beam is foused ad saed either i a raster (sequetial) fashio, or i a vetor fashio where the image field osists of idepedetly addressable/exposable pixels, Fig. -.

38 . NANOELECTROMECHANICAL SYSTEMS 3 Eletro-Gu E letro-g u Shutter (a) (b) Figure -. Eletro-beam patters. (a) Raster sa. (b) Vetor sa. The ultimate resolutio of eletro-beam lithography is ot posed by beam spot size, but by the so-alled eletro satterig ad proximity effets, Figures -, -. Eletro Beam Diretio of Sa Desired Lie Desired Lie Resist Substrate Real Resist Image Figure -. Sketh of eletro satterig effets o PR-oated wafer substrate. (After [3].) The former aptures the fat that, i the ourse of peetratig the PR ad uderlyig substrate, the eletro beam satters ad experiees a diretioal hage maifested as a spreadig out of the beam, i.e., irease i its spot size. The latter, i tur, aptures the fat that some of the sattered eletros are absorbed, ot uder the profile of the beam spot, but i areas adjaet to it. Two more effets resultig from beam satterig produe width- ad proximity-depedet patters, Figure -.

39 4 Chapter C B A Iter Proximity Itra Proximity Figure -. Itra- ad iter-proximity effets due to eletro satterig. (After [3].) The itra-proximity effet reflets the fat that the PR area ear the eter of the beam spot reeives more eergy, from adjaet poits, tha the PR earest to the irumferee. Thus orers, like poit A, ted to be uderexposed. The iter-proximity effet, o the other had, reflets the fat that eletros iteded to defie oe patter satter uto adjaet patters, thus extedig the effetive width of the adjaet patter. Refletig all these fators, the highest resolutio of eletro beam lithography as employed for aosale devie fabriatio is about m, however, the slow ature of writig the patters oe at a time, makes this tehique expesive ad ot ameable for mass produtio. Its mai appliatios are i the reatio of masks ad i aotehology researh...3. Soft Lithography The ovetioal IC fabriatio proesses, ad the approahes to MEMS fabriatio derived from them, have as their ore step the photolithographi defiitio of patters o a plaar substrate/wafer. Thus, as idiated previously, their appliatio to reatig aosale devies beomes prohibitively expesive, as the developmet of the oomitat light soures ad tools to reate devies at these legth sales is very expesive. This is of hief import, ot just for researh purposes but, more importatly, for the large sale produtio germae to ommerial appliatios. Soft lithography, the produtio of aosale devies by reatig elasti (soft) polymer masters that a the be used to prit, mold, ad emboss aosale strutures, is a tehique whih has bee the subjet of muh reet researh for the iexpesive reatio of aosale devies. The tehique relies o first makig a elasti stamp, show i Figure -3, ad

40 . NANOELECTROMECHANICAL SYSTEMS 5 LIQUID PRECURSOR TO PDMS MASTER (a) PDMS STAMP PHO TO RES IST (b) () Figure -3. Soft lithography Makig a elasti stamp. (a) A liquid preursor to polydimethylsiloxae (PDMS) is poured over a bas-relief master produed by photolithography or eletro-beam lithography. (b) The liquid is ured ito a rubbery solid that mathes the origial patter. () The PDMS stamp is peeled off the master. (After [34].) appears to have bee advaed by Whitesides [34], who applied it as a extesio of his work o the reatio of haels ad hambers for mirofluidi systems. Pritig is effeted by ikig the elasti stamp with a solutio of orgai moleules alled thiols, ad pressig it agaist a thi film of gold that has bee deposited o a silio wafer, Figure -4(a). Due to the ature of the hemial iteratio betwee the thiol moleules ad the gold, the surfae is wetted with the thiols displayig a preferred orietatio ad reatig a selfassembled moolayer, Figure -4(b), whih delieates the stamp s patter. The feature size or miimum width of the patter is of the order of 5m [34].

41 6 Chapter Figure -4. Mirootat pritig. (a) The elasti stamp (PDMS) is iked i thiols ad the pressed agaist the gold film previously deposited i the wafer. (b) The stamp is retrated, trasferrig a patter of self-assembled thiols. (After [34].) Moldig is effeted by pressig the elasti stamp agaist a liquid polymer o the wafer, show i Figure -5, whih auses the polymer to flow ito Figure -5. Moldig. (a) The elasti stamp is pressed agaist the deposited liquid polymer, whih flows ito the reesses/haels of the mold. (b) Upo urig, the polymer solidifies ito the mold patter. (After [34].)

42 . NANOELECTROMECHANICAL SYSTEMS 7 the stamp s reesses. The, upo urig the polymer, this solidifies aordig to the stamp s patter. The feature size for patters thus reated may be as small as m [34] Moleular Beam Epitaxy The egieerig of moder semiodutor devie strutures relies o the appropriate itrodutio ad distributio of impurities via dopig, together with bad-gap egieerig to effet eletro ofiemet alog the diretio of trasport [34-37]. This latter gives rise to devies i whih tuelig pheomea beomes maifest. The key to these types of strutures is the tehique for depositig dow to moo-atomi-thik layers alled moleular beam epitaxy (MBE). MBE uderwet extesive progress durig the 99s ad is ow a well established produtio tehology [38]. The essetials of MBE for growig a give struture are depited i Figure -6. Cryopaelig Substrate Heatig Blok GaAs Wafer Shutter Ga Ga Al Al A s A s I P P Material Soures Vauum Chamber Wall Liquid Nitroge (a) Vauum (b) Figure -6. (a) Sketh of MBE system. The atomi soures may be either i the solid or the gaseous states. (b) Sketh of layered atomi depositio. (After [38].)

43 8 Chapter The system osists of a steel hamber whih is equipped with pumps, to reate a very low pressure eviromet, typially about Torr, ad a growth hamber otaiig several vauum furaes, alled effusio ells or K-ells, from where a variety of atomi or moleular materials evaporate. The target wafer, o whih growth is to our, is plaed iside the hamber where it is held at a high, otrolled temperature ad uder high vauum, ad rotated to ahieve uiformity over the wafer. Growth ours whe heatig of the K-ells auses the various materials i them to evaporate, thus formig atomi beams that lad o the wafer surfae. The properties of the growig layers are otrolled by a umber of parameters, partiularly, K-ell temperature, whih otrols beam itesity or atomi/moleular flux, ad substrate temperature, whih otrols the dyamis of the atoms oe these reah the wafer surfae, see Figure - 6(b). I partiular, the arrivig atoms evolve aordig to the followig ompetig mehaisms: ) Immediate absorptio to the surfae, i.e., they stik wherever they lad; ) Migratio aross the surfae, i.e., move aroud before omig to a restig plae whih may ot preserve the rystallie struture; 3) Iorporatio ito the rystal lattie; ad 4) Thermal desorptio, i.e., they reevaporate from the surfae. To ahieve good rystal quality, suh a set of flux ad substrate temperature parameters must be disovered that the arrivig atoms have suffiiet eergy to move to the appropriate positio o the surfae, without re-evaporatig, ad be iorporated o the rystal. The MBE tehique is very versatile i that it allows the ompositio of the layers to be fie tued. This is aomplished by equippig the K-ells with shutters whih, through omputer otrol, a tur o or off eah beam aordig to preise timig sequee. The fat that growth is otrolled by omputer, edows MBE with the ability to deliver eve atom-thik layers, of abrupt ompositio, i a reproduible ad reliable fashio. This, i tur, eables badgap egieerig, the use of the material bad gap as a degree of freedom to egieer devie properties. I the IP HBT, a emitter with a bad gap greater tha that of the base, permits high base dopig, without ompromisig urret gai, by virtue of the redutio of hole urret ijetio ito the emitter effeted by the latter s eergy barrier. I the RTD, a lower bad gap regio, a potetial well, sadwihed betwee two large bad gap regios, barriers, allows preferetial urret odutio oly whe the eergy of odutio eletros oiides with the disrete eergy state i the potetial well, thus givig rise to the reatio of a urret-voltage harateristi exhibitig egative differetial resistae. The fat that the path legth of eletro trasport through the devie is very short, leads to these devies exhibitig very high speeds of operatio, e.g., hudreds of GHz i the ase of the HBT, ad lose to a THz i the ase of the RTD.

44 . NANOELECTROMECHANICAL SYSTEMS 9 Figures -7(a) ad (b) show the layer strutures of MBE-grow heterostruture bipolar trasistor (HBT) ad resoat tuelig diodes (RTD), respetively. m GaIAs Cotat x 9 m -3 7 m AlIAs Emitter Cotat x 9 m AlIAs Emitter 8x 7 3 m Compositioal Grade 8x 7 m GaIAs Spaer px 8 6 m GaIAs Base px 9 m GaIAs Spaer p5x 7 5 m GaIAs Spaer x 7 75 m IP Colletor 3x 6 7 m GaIAs Subolletor x 9 m GaIAs Buffer Udoped IP Substrate (a) GaIAs otat (5E8) GaIAs spaer (5E7) GaIAs spaer (ud.) AlAs barrier GaIAs/IAs/G aia s we ll AlAs barrier GaIAs spaer (ud.) GaIAs spaer (E7) GaIAs otat layer (5E8) GaIAs buffer (ud.) IP substrate (semi-isu latig) Å 5Å 5Å 3Å Å/3Å/Å 3Å 5Å 5Å 5Å Å (b) Figure -7. Layer desriptio of MBE-grow devies. (a) IP double heterostruture bipolar trasistor (DHBT) [39]. (b) Resoat tuelig diode (RTD) [4] Saig Probe Mirosopy Progress i Naotehology has bee itimately related to the ivetio of a umber of tehiques for imagig ad maipulatig atoms/aopartiles at aosales. All of these tehiques are based o a very fie tip (with atomi resolutio), ad the ature of what is imaged or maipulated is a futio of the tip itself, i.e., whether it is odutive, isulatig, mageti, o-mageti, et. Exellet review artiles summarizig advaes i saig probe mirosopy has bee published reetly by Giessibl [4] ad Baski [4]. I this setio we fous o two of the mai suh tehiques, amely: ) The saig tuelig mirosope (STM); ) The atomi fore mirosope (AFM).

45 3 Chapter Saig Tuelig Mirosope I STM, a sharp metal tip is brought i very lose proximity to a odutive sample, typially to a distae withi a few Agstroms, see Figure -8 [6]. Figure -8. (a) Sketh of STM system. (b) Probe tip detail. The sample is held i ultra high vauum. (After [6].) The, whe a bias voltage is applied betwee the tip ad the odutive sample, eletros tuel quatum mehaially aross the air gap to eliit a tuelig urret of a magitude ot exeedig several A. Due to the ature of the tuelig urret I t, whih obeys the equatio κ z z ( z) I e I t, where κ z mφ embodies the properties of the tuelig eletro (its mass m), ad the work futio of the tip material Φ, with beig Plak s ostat, the tuelig urret is a very sesitive futio of the tip-sample distae, z. Imagig, therefore, may be produed i two modes: ) Saig the tip i the x-y plae while forig it to remai at a fix z-positio. This, so alled ostat height mode, extrats sample morphology/relief image from modulatio of the tuelig urret magitude as the variatios i the sample relief hage the tip-sample I t, is obtaied; ) Saig the tip i the x-y plae while adjustig the tip positio z to keep the distae. Thus, a image of ( x y, z ostat)

46 . NANOELECTROMECHANICAL SYSTEMS 3 tuelig urret ostat. This is the alled topography mode, ad produes a image of z( x, y, I t ostat). STM tips are fabriated via hemial ethig or mehaial gridig of W, Pt-Ir, or pure Ir [4]. By usig a mageti probe tip the STM a be made sesitive to the spi of the tuelig eletros. Besides the tip sharpess ad material properties, movig the tip with atomi sale preisio, to obtai atomi resolutio image, eessitates the utilizatio of a piezoeletri erami, whose extremely fie deformatio is idued by a applied voltage Atomi Fore Mirosopy I AFM, Figure -9, a sharp tip is also brought very lose to the sample surfae. Figure -9. (a) Sketh of AFM system. (b) Probe detail. The sample may be held at ambiet oditios. (After [4].)

47 3 Chapter However, ulike STM, o voltage is applied betwee the tip ad the sample. Istead of a tuelig urret, the AFM detets the fore eliited betwee the tip ad the sample. The tip is part of a fore-sesig atilever beam so that, whe the latter is raster-saed over the sample, muh like a phoograph, surfae height variatios are deteted by moitorig the iterferee patter produed by a laser beam refletig off the atilever beam whe the latter deflets/deforms. The image of the sample is the extrated by relatig the atilever beam defletio to the fore required to produe it, F TS. F TS i tur, is related to the tip-sample (TS) potetial V TS via its egative gradiet, F V z ad is haraterized by a effetive sprig ostat TS TS kts FTS z. F TS may be attrative or repulsive, as it embodies a variety of fores, eah oe varyig differetly with TS distae z, thus makig it a oliear fore, see Figure -3. F TS z Repulsive Attrative Figure -3. Sketh of AFM tip-sample fore versus their separatio z. For istae, at distaes uder m, short-rage hemial fores are operative whih, for aisotropi hemial bods, are best haraterized by a Stilliger-Weber potetial, V SW V V where both earest eighbor potetial V, give i Eq. (), ad ext earest V potetial V with r σ p r σ q r a () r E A B e for r < aσ, else V () r bod V give i Eq. (), ad (3) are osidered. σ () ( r, r, r ) E [ h( r, r, θ ) h( r, r, θ ) h( r, r, θ )] i j k, () bod ij ik ijk ji jk ijk ki kj ikj

48 . NANOELECTROMECHANICAL SYSTEMS 33 h ( r r, θ ) ij γ rij σ a rik σ a, ik jik λe osθ jik for rij, 3 ik < aσ, else. (3) The optimal parameters, i terms of experimetal agreemet for a silio tip o a silio sample, was foud by Stilliger ad Weber to be as follows: A , p 4,., B , q,., E bo u d aj, a.8,.95 Å, ad σ σ. Similarly, at distaes uder m, log-rage fores, amely, va der Waals, eletrostati, ad mageti fores are operative. The va der Waals fores, are haraterized by a potetial give by Eq. (4) αd V vdw. (4) 6 z For the tip-sample situatio foud i AFM, amely, a spherial tip with radius R separated a distae z from a flat surfae (where z is the effetive distae betwee the plae oetig the eters of the surfae atoms ad the eter of the losest tip atom) the va der Waals potetial is give by [4] Eq. (5) HR V vdw, (5) 6z where H is the Hamaker ostat embodyig the atomi polarizability ad desity of the tip ad sample material pair ad, for the majority of solids ad iteratios aross vauum, has a value of H ev. For tip-sample materials haraterized by this value of Hamaker ostat, ad with a spherial tip of radius R~m separated from flat sample by ~.5m, the respetive va der Waals potetial ad fore are approximately -3eV ad -N, respetively. Whe both the tip ad the sample are odutive ad at separatios of ~m, they may also experiee eletrostati fores, haraterized by the potetial, Eq. (6) [4-45]: F eletrostati ( z) πε RV, (6) z where V is the eletrostati potetial differee. Aordigly, a potetial differee V~Volt, betwee a spherial tip of radius R~m a distae

49 34 Chapter z~.5m from a flat surfae, will experiee a fore ~-5.5N. Based o the method employed to extrat F TS, ad hee the surfae image, AFM operatio is lassified followig three modes: ) Cotat Mode-Stati AFM: I this mode the tip is i repulsio regime ad exerts a large ormal ad lateral fore o the sample. The fore applied to the atilever is kept ostat durig the sa by applyig feedbak, while the z-displaemet is measured yieldig the surfae topography. The mai drawbak of this tehique is that it a oly be applied i ertai ases, amely, at low temperatures, due to the eed to irumvet its lowfrequey oise ad thermal expasio effets o resoae frequey [4]. ) No-Cotat Mode-Dyami AFM: I this mode the atilever is mouted o a atuator whih vibrates ad, thus, exites it with amplitude A drive ad frequey f drive to osillate above the sample. The tip-sample distae is suh that operatio is i the attrative regime. This may avoid the fore ad oise problems of otat mode, but is subjet to jump-tootat if the sprig ostat orrespodig to the tip-sample potetial overomes that of the atilever, i.e., if k < kmaxts. The imagig sigal is derived from the hage i atilever amplitude ad phase that result whe the tip approahes the sample. Sie the exitatio sigal may osist of, either fixed amplitude ad fixed frequey, or fixed amplitude ad varyig frequey, these two modes of operatio are distiguished. The former is alled AM-AFM ad, while this method does provide atomi resolutio, the fat that the time required to apture the tip-surfae iteratio τ AM Q f is proportioal to the quality fator (Q) of the atilever, whih may be tes of thousad, makes it relatively slow. The latter mode, i whih the amplitude is fixed, but the frequey is varied, is alled FM-AFM mode of operatio. This mode also provides atomi resolutio, but it is muh faster tha AM-AFM beause the tipsurfae iteratio time is oly τ FM f. 3) Itermittet Cotat Mode-Dyami AFM : I this mode the tip is exited to osillate above sample, also i the attrative regime, but it is made to otat ( tap ) the sample for a short time durig every yle. Oe of the key aspets of AFM is the desig of the atilever, partiularly, its sprig ostat ad resoae frequey. These are give by Eqs. (7) ad (8), respetively, for a beam of thikess t, width w, legth L, Youg s modulus E, ad mass desity ρ.

50 . NANOELECTROMECHANICAL SYSTEMS 35 3 Ewt k (7) 3 4L t E. 6 (8) L ρ f Aordigly, various aspets, whih deped o the appliatio, must be osidered i desigig the atilever. For example, i the stati AFM mode, the sprig ostat must be hose so that the beam easily deflets i respose to the tip-sample fore. Thus, for k TS betwee N/m ad N/m, the rule is to hoose k betwee.n/m ad ~5N/m, with typial resoae frequeies of khz. O the other had, for the dyami AFM tehiques it has bee foud that, to avoid jump-to-otat, the produt of the atilever sprig ostat ad the vibratio amplitude must exeed the maximum tip-sample attrative fore, i.e., ka >. This meas that there is a trade-off betwee max respose F TS atilever stiffess ad exitatio drive amplitude. I other words, the sprig fore pullig the atilever away from its poit of losest proximity to the sample, must overome the maximum attratio fore. A refied riterio to avoid jump-to-otat ad whih assumes the possibility of a hystereti F TS z relatioship is give by [45]: ( ) ka > E Q TS (9) π ETS is the hysteresis eergy supplied to the atilever beam i eah where vibratio yle. A typial set of k, A values for FM-AFM are k 7 N / m, A 34m. Typially, the AFM atilevers are fabriated via Si or Quartz miromahiig, ad the usual tip materials ilude Si itegrated with beam, W, Diamod, Fe, Co, Sm, CoSm permaet magets, ad Ir.

51 36 Chapter..3.5 Carbo Naotubes Carbo aotubes are, perhaps, the quitessetial elemet of aotehology. Their disovery is the fruit of researh, origially oduted by Kroto ad Smalley i 985, with the aim of studyig the laser vaporizatio of graphite. Suh studies eliited the disovery by them of lusters otaiig 6 arbo atoms (C 6 : Bukmisterfulleree), arraged i a spherial struture, see Figure -3, []. Figure -3. Sketh of the hemial struture of C 6 : Bukmisterfulleree. (After [46].) Cotiued researh to irease the yield of these C 6 lusters led Iijima to disover arbo aotubes (CNT), see Figure -33 [46]. Figure -3. (a) Sketh of the hemial struture of a sigle-wall arbo aotube (SWNT). (After [].) (b) SEM of SWNT ad MWNT. I a multi-walled aotube, a ier SWNT forms the ore of multiple oetri aotubes whih grow aroud it. (Courtesy of Prof. László Forró, Swiss Federal Istitute of Tehology (EPFL), Lausae Switzerlad). CNTs are moleular arbo fibers that osist of graphite yliders losed at eah ed by aps otaiig six petagoal rigs, i.e., eah ap is exatly oe-half of a C 6 moleular luster [46]. They ted to be produed i

52 . NANOELECTROMECHANICAL SYSTEMS 37 three mai modalities, amely, sigle-walled aotubes ( SWNTs), whih rage i diameter from approximately.4m to more tha 3m, multiwalled aotubes (MWNTs), whih rage i diameter from approximately.4m to more tha m, ad ropes, whih are parallel stripes of SWNTs stuk to eah other. Their physial properties are astoudig. With aspet ratios of the order of -, they are several µ m (ropes up to m) log, possess a Youg s modulus, tesile stregth, ad desity of ~TPa (Steel: 3 3.TPa), 45GPa (Steel: GPa), ad.33 ~.4 g / m (Al:.7 g / m ). I additio, their odutivity may be metalli or semiodutig, ad they 3 3 have a urret arryig apability of ~TA / m (Cu: GA / m ). A umber of tehiques are employed to produe CNTs, for istae, the ar disharge, laser ablatio ad hemial vapor depositio methods. These methods usually yield a radom mixture of SWNTs, MWNTs, ad ropes ad researh is uder way to determie tehiques for the otrolled growth of a speifi type of CNT. For istae, Li et al. [47] have reported the developmet of a atalyst-based method that predomiatly yields SWNT. I this method, a silio wafer is pre-pattered with alumia aopartiles, whih serve as atalysts for their CVD growth, produig SWNTs with diameter uder.5m. The arrow diameter of CNTs makes them ideal adidates for appliatios as SPM tips, as well as a umber of devies, suh as haels for field effet trasistors. Figure -33 shows the formatio of CNT tips. Aisotropi eth Catalyst depositio CVD aotube growth Figure -33. Formatio of AFM tips via CNT growth. (After [48].)..3.6 Naomaipulatio The ultimate degree of otrol i aofabriatio, is embodied i the ability to maipulate idividual atoms/aopartiles with preisio. This is

53 38 Chapter aomplished by two tehiques, amely, exploitig AFM to push partiles, ad DIP-Pe lithography AFM-based Naomaipulatio I this tehique, a osillatig AFM tip is brought lose to a partile util, as a result of jump-to-otat, the osillatio amplitude goes to zero. The AFM approahes the aopartile via a fast X-Y saig osillatio, i a plae perpediular to the desired pushig diretio, z, see Figure -34. Oe otat of the AFM with the aopartile is established, motio proeeds i the z-diretio at a slow sa rate. Trajetory of Tip i X,Y Fast Sa Pushig Diretio Slow Sa Figure -34. Pushig a aopartile with AFM. (After [49].) DIP-Pe Lithography I this tehique, developed by Mirki s group [5], see Fig. -35, ad AFM Tip Moleular Trasport Writig Diretio Water Meisus Solid Substrate Figure -35. Close-up of iked AFM tip as moleules flow dow the tip via water meisus. (After [5].)

54 . NANOELECTROMECHANICAL SYSTEMS 39 remiiset of a goose-feather pe, a moleular ik is deposited over a gold surfae aordig to a desired patter. I oe demostratio of the tehique, a AFM tip was oated with a thi film of thiol moleules (the ik ) ad moved two-dimesioally so as to isribe the uderlyig gold surfae. Sie the thiol moleules a oly attah to the gold surfae i oe partiular orietatio, a self-assembled moolayer of them, embodyig the desired writig, results. A variety of iks may be employed ad, i terms of lie width apability, lies a few-aometers wide have bee demostrated.. 3 Summary I this hapter we have itrodued the broad field of aoeletromehaial systems. I partiular, we have traed its origis, motivatio, ad preseted a uified survey of its distitive harateristi, amely, the overgee of fabriatio tehiques, from ovetioal IC fabriatio, to miroeletromehaial systems fabriatio, to aosale fabriatio. I the ext hapter, we address the fudametal physis o whih devies, iruits ad systems exploitig the NaoMEMS fabriatio methods may be prediated.

55 Chapter NANOMEMS PHYSICS: QUANTUM WAVE- PARTICLE PHENOMENA. Itrodutio As disussed i Chapter, NaoMEMS aims at exploitig the overgee betwee aotehology ad miroeletromehaial systems (MEMS) brought about by advaes i the ability to fabriate aometer-sale eletroi ad mehaial devie strutures. This ovel paradigm, i tur, poses a iterestig hallege from the devie physis poit of view. I partiular, the ivetio ad/or disovery of a plethora of ew materials, oepts ad tehiques suh as arbo aotubes (CNTs) [7], photoi bad-gap rystals (PBCs) [5], ad MEMS [5-55], respetively, has opeed up ew possibilities to implemet ovel devies upo whih a ew eletrois tehology, with attributes that are far superior to everythig kow to date, may be prediated. With the simultaeous overgee ad exploitability, at suh small legth sales (e.g., dow to a few aometers), of various types of physial properties ad effets, for istae, eletroi, mehaial, optial, ad mageti ad quatum effets, the ature of the oomitat ew uiverse of devies ad iruits that will fuel this ew eletrois will learly be vast, yet, it is at preset mostly ukow. I this otext, may domais of physis, ot usually ivoked i desribig the behavior of prior-art devies, beome simultaeously pertiet. Suh elemets ilude [56], the maifestatio of harge disreteess, the quatum eletrodyamial (QED) Casimir effet, quatized heat flow, maifestatio of the wave ature of eletros, quatum iformatio theory, omputig ad ommuiatios, wave behavior i periodi ad o-periodi media, ad quatum squeezig. I this hapter, ad the followig, we expose fudametal kowledge required to aalyze devies exploitig these pheomea.

56 4 Chapter. Maifestatio of Charge Disreteess.. Effets of Charge Disreteess i Trasmissio Lies The most fudametal elemet i iruits ad systems is the iteroet or trasmissio lie (TL). TLs play a essetial role i ofigurig iruits ad systems at all legth sales [56]. Ideally, TLs are the medium through whih sigals propagate, from oe poit to aother, with o effet o the sigals, exept a frequey-idepedet delay. Figure - shows a sketh of a mirostrip TL, a ommoly used TL i itegrated iruits. It osists of a metalli stripe of width w ad thikess t s, pattered o a dieletri substrate of thikess h ad dieletri ostat ε r, with the substrate restig o a metalli groud plae. x z y I t s w h E Sigal ε r Figure - Sketh of mirostrip trasmissio lie. From a eletromagetis perspetive, the TL s qualitative operatio is simple [57]. The sigal of iterest is impressed at its iput, by way of its equivalet eletri field E Sigal betwee the metalli stripe ad the groud plae, ad it eliits a propagatig quasi-tem eletromageti wave whih is guided i the dieletri substrate regio betwee the stripe ad the groud plae. A urret I, flowig i oe diretio i the stripe, ad i the opposite diretio i the groud plae, embodies the boudary oditios eessary to sustai the propagatig wave i the substrate, as per Maxwell s equatios [57], ad the magitudes of the mageti ad eletri fields stored alog the lie give rise to a idutae per uit legth, L, ad a apaitae per uit legth, C, whose ratio is aptured i the so-alled harateristi impedae of the lie, give by Z L C. TLs are usually desiged to have Z 5Ω, whih results if, for example, h 635µ m, w 635µ m,

57 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 43 t s µm, ad ε r Uder these oditios of a metal stripe of relatively large dimesios with respet to a Bohr radius, a.59.59m, the urret I may be ostrued as osistig of a esemble of freely-propagatig eletros, eah haraterized by a plae ikz wave-like wave futio ψ ~ e, with otiuous eergy E k m*, where is Plak s ostat, k π λ is the wave vetor, λ the eletro wavelegth, ad m* the effetive mass [58]. Assumig a lossless TL, its iruit behavior may be represeted as a tadem oetio of a umber of fiite-legth ells, eah ell osistig of a legth z of its idutae, L, ad apaitae, C, per uit legth, see Figure -(a) [56]. L L L V C C C i i i R Load z z z (a) m m m k k k q i- q i q i (b) Figure -. (a) Model of ideal trasmissio lie. (b) Model of moatomi liear hai. Thus, the propagatio of a sigal from a soure towards a load, dow a TL, a be visualized as a advaig tide of harge fluid hargig the suessive ells util the load is reahed. Eter aotehology. I oert with exploitig the ability to patter aosale iruits, it is expeted that TLs with stripes of aosale ad subaosale widths ad thikesses will be promiet. I this otext, eletro urrets will be trasported dow very arrow ad thi metalli wires, so arrow ad thi, i fat, that their dimesios may stop at oly tes of Bohr radii. This meas that the eletros ivolved will ot oly experiee quatum mehaial ofiemet, i.e., that their eergy will beome quatized ad give by [58], [59]:

58 44 Chapter E π m* t s ~ x y k z π w () but also, that their disrete ature will be maifest. This latter feature beomes operative whe the system size alog a trasport dimesio beomes of the order of the arrier ielasti oheree legth, ad it implies that, i additio to the quatum mehaial eergy of ofiemet of Eq. (), the Coulomb hargig eergy required for addig or removig a eletro, E q L i where L i is a harateristi legth i diretio i, must be take ito aout [58-6]. Oe must the tur to quatum mehais to properly desribe the TL behavior. The observatio [6]-[63], that the harge q i suessive ells, ad the total eergy, obey equatios () ad (3), d q L dt C i ( qi qi qi ) () H dqi L i i L dt C ( q q ) i (3) whose forms are idetial to the equatios desribig the logitudial vibratio modes i a moatomi liear hai (MLC) [64] (see Appedix A), Figure (b), motivated the appliatio of the quatum mehaial desriptio of the latter to the TL. I partiular, i (3), the first ad seod terms aout for the mageti ad eletri eergies i the TL idutors ad dq apaitors, respetively, ad p L ad q play the roles of mometum dt ad oordiate, respetively. Notie, however, that sie q is harge, p represets eletri urret. The above TL quatizatio assumed the eletri harge q to be a otiuous variable. As has bee observed [59], however, uder appropriate irumstaes, e.g., system size lose to the ielasti oheree legth, the partile (or disrete) ature of eletros beomes evidet. Li [6] osidered the osequees of this possibility ad, aordigly, advaed a theory for TL quatizatio assumig q to be disrete. The possibility of havig the harge adopt exlusively disrete values, was itrodued [6] by imposig the oditio that the eigevalues of the harge operator qˆ be disrete, i.e.,

59 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 45 qˆ q > q q > (4) e I other words, the result of measurig the harge i the TL must be times the fudametal eletro harge, q e, where is a positive iteger. Sie, from a ompariso with the MLC desriptio, harge adopts the role of a oordiate operator i the quatized Hamiltoia, the form of the orrespodig mometum operator pˆ, ad i partiular, pˆ i q q (5) must reflet this ew situatio. This is aomplished by replaig the partial derivative by its fiite-differee approximatio i harge oordiate spae [65], i.e., ψ ψ ( ) ψ ( ) ψ ( ) q q e (6) where q e is the fudametal uit disretizig the harge axis ad ψ is the eletro wavefutio i the harge represetatio. Assumig the lie is drive by a voltage soure V, Shrödiger s for the TL is give by Eq.(7) [6, 6]: qe L qˆ C { ψ ψ ψ } qv ˆ ψ εψ (7) or, usig Eq. (4): qe L q C e { ψ ψ ψ } Vqe ψ εψ. (8) Imposig harge disreteess, thus, turs Shrödiger s equatio for a TL ito a disrete, istead of a partial, differetial equatio. The impliatios of harge disreteess are gauged from the ature of the orrespodig eigevalues ad eigevetors for this equatio. Obtaiig these beomes more trasparet upo developig the quatum theory of mesosopi TLs [6, 6], whih we outlie below followig Li [6].

60 46 Chapter With qˆ as the harge operator, istead of the ovetioal spatial oordiate, the orrespodig ojugate variable is take as pˆ, whih the represets the urret operator, istead of the usual mometum operator. The quatum mehais of the TL the evolves from ( 8) ad the ommutatio relatio: [ q, pˆ ] i ˆ. (9) The fat that the eigestates of qˆ must be speified by a iteger,, allows two oseutive states to be related to oe aother by the appliatio of a ~ pˆ / shift operator, i partiular, Q e iq e. By expadig the expoetial, ad usig ( 4 ) ad ( 9 ), this shift operator may be show to obey the ommutatio relatios: ~ ~ [ qˆ, Q] q Q () ~ ~ [ q, Q ] q Q e ˆ () e ~ ~ ~ ~ Q Q QQ. () The shift operator, whe applied to the umber eigestates defied by, qˆ > q >, produes the followig ew states: e Q ~ > e iα > ~ iα Q > e > (3) (4) where α s are udetermied phases. Therefore, (3) ad (4) lead to the iterpretatio of the shifter operators Q ~ ad Q ~ as ladder operators that irease ad derease the harge of the harge operator i its diagoal represetatio. The quatizatio apparatus is ompleted whe the ompleteess ad orthogoality relatios, ad the ier produt are stipulated, i this ase as give by (5)-(7), respetively, ><, (5)

61 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 47 < m > δ, (6) m Z * < φ ψ > < φ >< ψ > φ ( ) ψ ( ), (7) where belogs to the set of o-egative itegers Z. These relatioships permit obtaiig the fudametal quatum mehaial properties of the TL, amely, the eigefutios of the mometum operator pˆ, i.e., the ature of the urret, ad the eergy spetrum. Assumig the usual relatios [53], pˆ p > p p > ad f ( pˆ) p > f ( p) p >, Li [6] expads the mometum states i terms of the umber states, p > > Z ( p) together with the shiftig operatio ~ iq p Q p > e e ˆ / p >, to obtai the relatioship ( iq p iα ) exp e. This, i tur, yields the mometum expasio i terms of the umber states as, e p Z Z iq p / > κ e > (8) where iα j j κ ad e ( π ) q e iα j j κ e for >. Makig the substitutio p p i the expoetial of (8) yields the same state p >, from where it is determied that the mometum operator pˆ is periodi. Further progress towards obtaiig the eigestates ad dispersio is attaied by otiig that, if oe defies ew disrete derivative operators by: qe ψ ( ) ψ ( ) ψ ( ) q e ( ) ψ ( ) e, (9) ψ q ψ ( ), () e q the Shrödiger s equatio (8), may be expressed as: qˆ { q } Vqψ εψ q L e q e ˆ, () e C

62 48 Chapter from where a mometum operator Pˆ, give by: ~ ~ ( ) ( Q ) Pˆ q q Q e e i iq e, () may be defied. This ew mometum operator is related to pˆ i that pˆ lim Pˆ. q e... Idutive Trasmissio Lie Behavior Idutive behavior is displayed by the so-alled pure L-desig, i whih the TL is osidered to have very arrow width (high impedae). Its mathematial desriptio is give by: Hˆ q L e { } q e q e, (3) where the terms ivolvig the lie apaitae is egleted ad the drivig voltage is set to zero. With this defiitio, ad takig ito aout the ~ iq p relatioship Q p > e e ˆ / p >, the followig relatioships are obtaied: ad q p Pˆ p > si e p >, (4) q e q p Hˆ e p > os p >, (5) qe L These are the desired mometum eigestates ad the eergy spetrum. What is lear from (4) is that the urret i a mesosopi idutive lie, give by I Pˆ L, is periodi, beomes zero wheever p π q e ; qe, ad that it is bouded by ( qel, qel). Similarly, from (5) it is determied that the lowest eergy state is degeerate at p q. e

63 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 49 Aother peuliarity of mesosopi TLs is the ature of their eergy spetrum whe formed ito a rig i the presee of a mageti flux φ. I this ase, Shrödiger s beomes, q L e { D D } ψ εψ q e qe, (6) where D q e ad D q e are disrete derivatives that remai ovariat i the presee of the mageti flux φ ad are defied by Li [6] as, D qe e iq iq q q Q ~ e e φ e e e e φ φ φ ; Dq e e qe qe Qˆ. (7) Applyig the Hamiltoia i (6) to the eigestate p >, the eergy eigevalues are obtaied as, ε q, (8) q e ( p, φ) si ( p φ) e where φ is the mageti flux threadig the TL. Thus, (8) implies that whe the disrete ature of harge is at play, the TL eergy beomes a periodi futio of p or φ, with maximum amplitude ad ulls ourrig wheever p φ qe. Furthermore, it has also bee show that the TL urret is give by, I qe ( φ) φ q L si, (9) e whih implies that it beomes a osillatory futio of the mageti flux. Sie o applied forig futio was assumed, (9) leads to the importat observatio [6] that a TL i the disrete harge regime will, i the presee of a mageti flux, exhibit persistet urrets [59]. These are urrets without dissipatio, suh as the atomi orbital urrets that eliit orbital magetism. q e

64 5 Chapter... Capaitive Trasmissio Lie Behavior I this desig the TL is apaitive (low-impedae) ad the first braketed term i () is egleted ad the Shrödiger equatio is give by, q e qˆ L C Vqˆ ψ εψ, (3) I this ase, the Hamiltoia operator ommutes with the harge operator qˆ, ad osequetly [6], they have simultaeous eigestates. I partiular, the eergy of the state > is give by [67], C ε ( qe CV ) V, (3) C where is the umber of elemetal harges desribig the TL state. Thus, (3) implies that whe the disrete ature of harge is at play i a lowimpedae lie, the TL eergy is a quadrati futio of the state of harges. A iterestig pheomea is predited for the urret flow. I partiular, as the applied voltage ireases, the TL harge a oly irease i disrete steps whih are a multiple of q e. Sie the voltage required to ause this harge to be ijeted ito the TL is q e C, it a be said that the voltage axis is quatized i uits of q e C. Thus, the total harge of a lie i the groud state is give by [67], q q e e q uv k u V k k C C q e, (3) where u(z) is the uit step futio. Cosequetly, by takig the time derivative of (3), oe obtais the orrespodig urret as, I dq dt qe qe dv qe δ V k V k C δ C. (33) dt k Eq. (33) idiates that the urret exhibits a series of delta-futio impulses with periodiity C, osistet with every time a sigle eletro q e

65 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 5 harge is added, ad amplitude proportioal to the slope of the voltage soure. This leads to the importat observatio [67] that a low-impedae ideal TL i the disrete harge regime will exhibit urret flow domiated by Coulomb blokade. Clearly, as limitig ases, typifyig the behavior of ideal high- ad lowimpedae TLs i the disrete harge regime, the pheomea of persistet urrets ad Coulomb blokade-type urret flow, respetively, raise serious questios i the otext of ahievig low-oise aalog ad reliable digital iruits ad systems at aometri-legth sales. As a result, omplete awareess of the possibility that these features might be iadvertetly iluded i the desig spae must be iorporated i TL/iteroet models utilized i the desig ad aalysis of future NaoMEMS... Effets of Charge Disreteess i Eletrostati Atuatio Oe of the distiguishig features of NaoMEMS is the ilusio of futios based o mehaial strutures that a be atuated. For a variety of reasos, i partiular, its ompatibility with IC proesses, eletrostati atuatio is the atuatio mehaism of hoie for these devies [48], ad is the oe o whih we fous our attetio ext.... Fudametal Eletrostati Atuatio Perhaps the most fudametal eletrostatially-atuated elemets/buildig bloks are the sigly-(atilever) ad doubly-ahored beams [5], Figure 3. Figure -3. (a) Catilever beam. (b) Doubly-ahored beam.

66 5 Chapter The devies are essetially parallel-plate apaitors, of omial plate separatio g, i whih the top plate (beam) is free to move i respose to a eletrostati fore developed betwee it ad the rigid bottom plate, as a result of a voltage applied betwee the two.... Large-sigal Atuatio Swith For typial dimesios employed i MEMS [48], e.g., beam gaps, legths, widths, ad thikesses of about µ m, 5µ m, ' s of µ m, ad µm, respetively, the displaemet behavior of the beams, whih maifests itself as otiuous gap redutio versus applied voltage, is ditated by the equilibrium F Coulomb F Sprig established betwee the ε AV quadrati eletrostati fore, F Coulomb, ad the liear sprig ( g ) z fore, FSprig k Beam z, (Hooke s law) whih attempts to brig the beam bak to its udefleted positio. This dyami equilibrium, ad its aompayig smooth displaemet, is maitaied up to about oe-third of the beam-to-substrate distae, at whih poit it is lost ad the beam ollapses oto the bottom plate, abruptly reduig the gap to zero. The voltage demaratig these two regimes is alled pull-i voltage ad is give by [49], V where 3 8k Beamg Pull i, (34) 7Aε k Beam is the sprig ostat of the beam, ad A is the eletrode area.... Small-sigal Atuatio Resoator For appliatio as resoators [54], a AC voltage, together with a soalled DC polarizatio voltage, itrodued to ehae the urret eliited by the variable beam apaitae, are applied. Sie the resoators are iteded for appliatio as stable frequey stadards, with frequey give by [8], E h f r, om. 3κ, (35) ρ L r

67 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 53 where κ is a salig fator that models the effets of surfae topography, iludig for istae, the ahor step-up ad its orrespodig fiite elastiity, E is the Youg s modulus of the beam material, ρ its desity, h its thikess ad L r its legth, the ombied amplitude of AC ad DC voltages is hose to be lower tha pull-i, thus keepig the beam from ollapsig.... Coulomb Blokade The pheomeo of Coulomb blokade [68, 69] refers to the fat that uder ertai oditios, amely, whe jutios are defied whose 5 apaitae is of the order of C ~ F or less, the eergy required to irease the harge by oe eletro is ot egligible with respet to temperature. For example [68], Figure.3 shows that, while a eutral metalli islad, suh as the plates of a apaitor, emits o eletri field ad, thus, allows the uimpeded approah of a eletro, oe this eletro beomes part of the islad it emits a eletri field that may prevet the additio of more eletros. Figure -4. (a) Chargig Coulomb islad. (a) Chargig eergy of small apaitor. At this poit, the islad bloks suh a additio of extra harge. For a 5 jutio apaitae of C ~ F, the miimum voltage required to add a harge q is q C, thus the hargig eergy is E C 3 q C.83 J, whih is lose to the thermal eergy at K. 8 If the apaitae were smaller, e.g., C ~ 6. F, suh as might be

68 54 Chapter typial for aopartiles, the the hargig eergy would be lose to the thermal eergy at 3K. The impliatio of this is that it may be impossible to otiuously ijet harges ito the apaitor whe the hargig eergy exeeds the ambiet temperature. Rather, for a ireasig applied voltage, a hargig evet oly ours every time its magitude exeeds the hargig eergy of a eletro; oe eters the Coulomb blokade regime ad the urret ito the apaitor beomes pulse-like. The situatio is illustrated i Figure -4 with respet to the so-alled sigle-eletro box [69]. Soure Eletrode Tuel Barrier Q-q Small Islad (a) Gate Eletrode V G C Soure - Eletrode - - Tuel Barrier q e -q Gate Eletrode V G Small Islad (b) -q e C V G Qq C J Figure -5. Voltage-otrolled eletro ijetio ito metalli islad. (a) V G. (b) V G >V C. () Ciruit model (After [68], [ 69].) ()

69 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 55 A voltage soure V G is oeted through a small apaitor C, to a small metalli islad that rests over a tuel barrier whih, i tur, is i otat with a eletro reservoir. The apaitae of the tuel barrier is deoted C J, ad the distae betwee the gate eletrode ad the small islad, defiig C, is suh that tuelig is suppressed [69]. With V G, the system is eutral; the small islad otaiig positive harges q, whih are eutralized by a equal amout of egative harges -q, Figure -4(a). Whe the gate voltage ireases, the umber of eletros i the small islad may hage by amouts qe CVG, Figure -4(b). I partiular, the field idued by the gate auses a uompesated harge q to appear o the islad. The apaitae see by the islad is C C J. Therefore, the hargig eergy aompayig the ijetio of a harge C V is, qe G E C ( q qe ) ( C C ) J, (36) It is otied that, while the exteral harge q e is otiuous, the islad harge may oly irease i disrete steps of value q. Therefore, the islad harge is a step-like futio of the gate voltage. As a futio of temperature, the average umber of eletros i the islad is give by [68] (37), Figure -5. EC kbt e (37) EC kbt e Average Charge i Islad (Eletros) - - TK T>K - - Ijeted Exteral Charge, q e C V G (Eletros) Figure -6. Average islad harge versus ijeted harge. (After [69].)

70 56 Chapter..3 Sigle-Eletro Tuelig Upo the islad beig populated by the ijeted harge, the harge tuels through C J ad diffuses to the leads i a harateristi time τ give by the uertaity priiple (38) [69]. E C, (38) τ If the bias V G auses the ijetio of a harge q every τ seods, the a urret of magitude I q / τ is set up, Figure - 7. I C V G - Qq C J Figure -7. Sigle-eletro tuelig shemati. However, if this time is too short, the the urret would appear to be otiuous, as opposed to pulse-like. I this ase, o disrete, sigle-eletro tuelig evet is observed. To observe sigle-eletro tuelig, the harateristi time must exeed the produt of the apaitae times the lead resistae, τ > RC, a oditio whih leads to a miimum value for lead resistae, Eq. (38). R > (38) q Notie that trasport is ourrig through a tuelig jutio...3. Quatum Dots Quatum Dots (QDs) are strutures i whih eletros are ofied i all three dimesios [59]. These strutures ilude both gated layered strutures

71 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 57 grow by MBE, ad metal ad semiodutor aopartiles up to several aometers, e.g., ~-6 m, i size. Beause of their small size, whih is omparable to that of the Bohr exito, a ex ε m exe, eletro eergy levels i QDs are quatized. Eletro trasport through a QD is mediated by tuel barriers, see Fig. -8, ad is effeted via a series of idividual tuelig evets aross the barriers. Eergy Barrier DOT Barrier e- V L (a) C L, Γ L QD C R, Γ R C I - V G - - V L (b) V R Figure -8. (a) Sketh of quatum dot eergy level diagram. The otiuous lie deotes equilibrium, while the dashed lie deotes reflets a applied voltage, V. The dashed arrow deotes suppressed urret due to Coulomb blokade by QD eletros. (b) Equivalet iruit of QD. The tuelig rate aross the barriers is haraterized by the hage i free eergy,, resultig from the tuelig evet, ad the tuel resistae, R t ( R >> h e ), ad is give by [7], [7] Eq. (39). t Γ. (39) e R t exp k BT

72 58 Chapter I geeral, the tuelig rate will deped o the umber of available (empty) states withi the QD. If Γ f is the tuelig rate ito level f i the QD, g f is the degeeray fator, m f is the umber of eletros already oupyig the level, ad F() ε ( exp( ε k BT) ) is the Fermi futio, the the total tuelig rate is give by, Γ FS QD f Γ f QD ( m ) F( ε ) g, (4) f f f FS QD where the iitial ad fial eletro eergies are related by, ε i ε f, FS ε i beig the iitial eletro eergy [3], [33]. Notie that, at small bias voltages, the oupay of QD states preludes tuelig due to Coulomb blokade...4 Quatized Eletrostati Atuatio I otrast to ovetioal eletrostatially-atuated MEM devies, whih exhibit otiuous displaemet versus bias behavior prior to pull-i, the advet of preisio aoeletromehaial fabriatio tehology [7] ad arbo aotube sythesis [7] has eabled aess to beams with dimesioal features (gaps, legths, widths, ad thikesses) of the order of several hudred aometers i whih oditios for the maifestatio of harge disreteess beome also evidet. I fat, reet [73] theoretial studies of suspeded (doubly ahored/lamped) arbo aotubes (CNTs) i whih Coulomb blokade domiates urret trasport have predited that harge quatizatio i the CNTs will result i quatizatio of their displaemet. Speifially, Sapmaz, et al. [73] osidered a sigle-wall aotube (SWNT) modeled as a rod of radius r, ad legth L, ad separated by a gap g over a bottom eletrode, Fig. -9. V Tuel L z g x V G Figure -9. Shemati of suspeded CNT as doubly ahored beam.

73 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 59 They desribed its behavior as follows. As the atuatio voltage, V G, applied betwee the CNT ad the bottom eletrode ireases, the beam beds dowwards ausig the applied eletrostati eergy to be overted ito elasti deformatio eergy, give by, U Elasti [ z( x) ] L EI dx z Tˆ z, (4) 4 where E ad I πr 4 are the CNT Youg s modulus ad momet of iertia, respetively, ad T ˆ T T is total stress, omprised of the residual stress, T, ad the stress idued by V G, whih is give by, T ES L L z dx, S πr. (4) Sie, igorig residual stress, the beam elasti eergy must orrespod to the eletrostati eergy that idued it, the total eergy the state of deformed the beam arrives at is that at whih the sum of elasti ad eletrostati eergies is a miimum. I the Coulomb blokade regime, however, as the bias voltage V is raised, a disrete umber of harges, q, populates the suspeded CNT. Thus, the eletrostati eergy must ilude this otributio, i additio to the atuatio voltage ( V G )-idued deformatio. Takig both eletrostati eergy soures, ito aout, Sapmaz, et al. [73] approximated the total eletrostati eergy by, U Eletrostati ( z( x) ) ( q) C ( z) G qv G ( q) R l L L ( q) L R L z ( x) dx (43) the, miimizig the total eergy with respet to z, the followig equatio for the CNT bedig was obtaied, ( q) IEz Tz F. (44) L R where F is the eletrostati for per uit legth. The bedig of the doublyahored CNT, with the boudary oditios z ( ) z( L) z () z ( L) was give as,

74 6 Chapter sih ξl ( L ) F L osh ξ L osh ξ z ( x), ξ Tξ x sih ξx ξx ξ L T EI. (45) Fially, the effets of harge disreteess are maifest upo examiig the maximum displaemet as a futio of atuatio voltage, ad give by (45) ad (46). ( q ) 5 L EI Er g max.3, ; 4 z T << << (46a) Er g L q L / 3 / 3 5 ( q) L EI Er g z >> max.4, T >> / 3 / 3 L (46b) E r g q L VG L ( ) It δ. (47) r l g / r For a give applied voltage, (47) gives the value of that miimizes the total eergy, where δ is a small orretio. Clearly, (45)-(47) reveal that the beam displaemet is quatized, i.e., its positio hages i disrete steps every time a eletro tuels ito it..3 Maifestatio of Quatum Eletrodyamial Fores Whe the proximity betwee material objets beomes of the order of several aometers, a regime is etered i whih fores that are quatum mehaial i ature [74-76], amely, va der Waals ad Casimir fores, beome operative. These fores supplemet, for istae, the eletrostati fore i outerig Hooke s law to determie the beam atuatio behavior. They also may be resposible for stitio [77], i.e., ausig lose by elemets to adhere together ad, thus, may profoudly hage atuatio dyamis..3. va der Waals Fore va der Waals fores, of eletromageti ad quatum mehaial origi, are resposible for itermoleular attratio ad repulsio. Whe adjaet

75 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 6 materials [78] are separated by distaes R>>r, where r is the atomi radius, the wave futios deay expoetially ad o bodig fores are operative. At these distaes, eah moleule (atom) may be haraterized as a dipole atea emittig a flutuatig field with a frequey distributio haraterized by a average frequey ϖ. For distaes, R, smaller tha the Rϖ average emitted wavelegth, i.e., R < λ or <<, the emitted fields are 3 reative i ature, i.e., they vary with distae as E R. Therefore, with referee to two emittig moleules (atoms), separated a distae R ad edowed with dipole operators < d ˆ ω > αeω, the va der Waals iteratio eergy betwee them derives from the self-osistet field idutio at eah others site. I partiular, atom idues a field at the site of atom give 3 by, Eˆ id ( ) dˆ R, whih, i tur, idues a dipole at the site of atom 3 give by, dˆ id α ( ω) dˆ R, where α ( ω) is the polarizability at the site of atom. Similarly, the idued dipole at atom idues a field at the ˆ ˆ site of atom give by, ˆ d d E id () α ( ω). Thus, the average 3 6 R R groud state dipole eergy of atom is give by [78], ( ) < ˆ * ˆ id α * U > < dˆ dˆ ω R d E > ad is a futio of its average 6 R dipole flutuatio. The sigature of va der Waals fores is the 7 FvdW du vdw dr R distae depedee. For alulatios, Desqueses, Rotki, ad Aluru [79] have modeled the va der Waals eergy by the expressio, U vdw ( R) C dv dv, (48) 6 V 6 V R, ( V V ) where V ad V embody two domais of itegratio of the adjaet materials, ad are the desities of atoms pertaiig to the domais V ad V, R ( V,V ) is the distae betwee ay poit i V ad V, 6 ad C 6, with uits [ ev Å ], is a ostat haraterizig the iteratio betwee atoms i materials ad. While a good first step for modelig purposes, the exlusively pair wise ature of the otributios embodied by (46) may ot be aurate eough for tube geometry sie it is kow [8] that, i exat alulatios, oe eeds to osider three-partile, four-partile, et iteratios, or equivaletly multi-pole iteratios. These multiple

76 6 Chapter iteratios must be iluded to improve modelig results. Nevertheless, applied to a SWNT beam of diameter r ad suspeded by a gap R, they obtaied the va der Waals eergy per uit legth of the CNT as, U vdw L C6σ π r( r R) ( 3r ( R r) ) 7 / ( R r) r ), (49) where σ 38m is the atomi surfae desity, L is the CNT legth. The orrespodig va der Waals fore is give by, F vdw U d L dr vdw ( C6σ π r R( R r) ) ( 8R 3R r 7R r 8Rr 35r ) 5 ( ) 5 R R r. (5) As metioed previously, the va der Waals fore is oe otributor to the pheomeo of stitio. Thus, its promiee must be aouted for i the desig of advaed strutures, e.g., aoeletromehaial frequey tuig systems [54] based o quatum gears [8], as estimates of its magitude are useful i desigig agaist it [8, 8]..3. Casimir Fore The Casimir fore arises from the polarizatio of adjaet material bodies, separated by distaes of less tha a few miros, as a result of quatum-mehaial flutuatios i the eletromageti field permeatig the free spae betwee them [74-77]. It may also arise if vauum flutuatios are a lassial real eletromageti field [83]. The fore may be omputed as retarded va der Waals fores or as due to hages i the boudary oditios of vauum flutuatios; these are equivalet viewpoits as far as it is kow [8]. Whe the material bodies are parallel odutig plates, separated by free spae, the Casimir fore is attrative [74], however, i geeral whether the fore is attrative or repulsive [8], [84] depeds o bot h the boudary oditios, iludig speifi geometrial features, imposed o the field as well as the relatioship amog material properties of the plates ad the iterveig spae. For example, repulsive fores are predited by

77 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 63 Lifshitz formula [75] if the material betwee two plates has properties that are itermediate betwee those of the plates. The startlig aspet of the Casimir fore is that it is a maifestatio of the purely quatum-mehaial preditio of zero-poit vauum flutuatios [74-77] (see Appedix A), i.e., of the fat that, eve i irumstaes i whih the average eletromageti field is zero, its average eergy shows flutuatios with small but o-zero value, i.e., there is virtually ifiite eergy i vauum. Researh efforts aimed at the pratial exploitatio of this extremely large eergy soure, residig i free spae, are uder way [85-87]. Calulatig the Casimir fore etails irumvetig the fat that the zeropoit vauum eergy, E Field ω diverges, ad may tehiques to aomplish this have bee developed [74-77], [88], [89], but iludig these i our presetatio is well beyod the sope of this artile. The essee of may of these alulatios, however, is to ompute the physial eergy as a differee i eergy orrespodig to two differet geometries, e.g., the parallel plates at a distae a apart, ad these at a distae b, where the limit as b teds to ifiity is take. For flat surfaes, the ifiite part of the eergy aels whe the eergy differee of the two ofiguratios is take. The alulated zero-temperature Casimir eergy for the spae betwee two uharged perfetly odutig parallel plates, Figure -, A z Figure -. Casimir effet geometry. is give by, U Casimir π, (5) 7 z ( z) 3 ad, the orrespodig Casimir fore per uit area is give by,

78 64 Chapter F Cas A π. (5) 4 4 z For plaar parallel metalli plates with a area A m ad separated a distae z.5µ m, the Casimir fore is 6 N. May experimets measurig the Casimir fore uder various oditios, suh as effetig ormal displaemet betwee a sphere ad a smooth plaar metal ad betwee parallel metalli surfaes, as well as, effetig lateral displaemet betwee a sphere ad a siusoidally orrugated surfae, have bee performed [89-95]. A good reet review of experimets ad theory for Casimir fores has bee published by Bordag, Mohidee, ad Mostepaeko [89]. Sie the Casimir eergy/fore is a sesitive futio of the boudary oditios, orretios to the ideal expressio (5) have bee itrodued to aout for ertai deviatios. For example, for the sphere-plate geometry, the zero-temperature Casimir fore is give by, F Cas_ Sphere Plate 3 π 36 z ( z) R 3, (53) where R is the radius of urvature of the spherial surfae. To ilude the fiite odutivity of the metalli boudaries, two approahes have bee advaed. I oe, the fore is modified as [96, 97], F, σ Cas ( z) F ( z) Cas _ SpherePlate 4 z ω p 7 5 z ω p, (54) where ω p is the metal plasma frequey [64]. I the other, obtaied by Lifshitz [98], the orretio is igraied i the derivatio of the Casimir fore, ad is give by, F, σ Cas ( z ) R 3 π z dz ' 3 p ξ dpd ξ ( s p ) ( s p ) ( s pε ) ( s pε ) pξz e e pξz, (55)

79 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 65 ( iξ ) where s ε p, ( ) ωε ε i ξ dω is the dieletri π ω ξ ostat of the metal, ε is the imagiary ompoet of ε, ad ξ is the imagiary frequey give by ω iξ. Corretios due to ozero temperature yield [77], F ( z) F ( z) f ( ζ ) 7, (56) π T Cas Cas where ζ k B Tz, k B is Boltzma ostat, T is the absolute temperature, ad f ( ζ ) 3 4 ( ζ π ) ϑ() 3 ( ζ π 45 ), ( ζ 8π ) ϑ( 3) ( π 7 ), for ζ for ζ >, (57) with ϑ () Roy ad Mohidee [9] iluded origially the effets of surfae roughess, whih hages the surfae separatio, by replaig the flat plate with a spatial siusoidal modulatio of period λ, ad the eergy averaged over the size of the plates, L, to obtai, πx π z Asi > λ 7 z < U Casimir 3 m C m A z m, (58) where A is the orrugatio amplitude. The orrespodig Casimir fore is the give by the so-alled, Fore Proximity Theorem [99] relatig the parallel plate geometry ad the sphere-plate geometry, amely, F Cas _ Roughess π R < U Cas _ Rouhess > (59) For λ << L ad z z > A, where z is the average surfae separatio after otat due to stohasti roughess of the metal oatig, they reommed the followig oeffiiets i (58): C, C 3, C 45 8, 4 C A more aurate ad geeral model for stohasti surfae roughess, advaed by Harris, Che, ad Mohidee [88], iludes the

80 66 Chapter effets of surfae roughess, by replaig the flat plate with the mea stohasti roughess amplitude A, to obtai, F A 6 z r Cas ( z) FCas ( z), (6) where A is derived from diret measuremets via a Atomi Fore Mirosope (AFM)..4 Quatum Iformatio Theory, Computig ad Commuiatios The advet of aosale fabriatio tehiques has brought withi our reah the possibility of produig systems whose predomiat behavior is desribed by quatum mehais (QM). While the egieerig of systems based o explotig this ew physis/tehologial paradigm is still i its ifay, this ew paradigm is ultimately expeted to maifest itself i the usherig of a ew eletrois tehology era. Obviously, this ew eletrois is expeted to hage the way i whih systems are implemeted to effet the futios of iformatio proessig, omputig ad ommuiatios [-]. These futios, i tur, will exploit the properties of quatum mehaial wave futios. I this setio we itrodue key aspets of the fudametal physis o whih these futios are prediated, i partiular, we fous o the oepts uderpiig quatum iformatio proessig, amely, quatum bits (qubits), quatum etaglemet, the Eistei-Podolsky-Rose (EPR) State, quatum gates, ad quatum teleportatio. Quatum iformatio is represeted by quatum bits or qubits [3]. Qubits are fudametal physial etities, suh as a two-level atom, whih may adopt two possible quatum (statioary) states (see Appedix A), say the mutually orthogoal states ad. Due to its quatum ature, however, the most geeral state is expressed as, ψ a b, (6) i.e., as a superpositio of both states. Thus, a measuremet of the qubit will ause its wavefutio to ollapse ito the state with probability a, or ito the state with probability b. This meas that durig its time evolutio a qubit may be partly i both the ad state at the same

81 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 67 time, i.e., to the degree that a ad b may adopt a ifiity of values, the qubit has the potetial to be i ay of these. A quatum system possessig qubits is said to have aessible mutually orthogoal quatum states. For example, a system otaiig two oiteratig qubits will have the four states:,,,. States suh as these, whih represet the juxtapositio of idepedet or oiteratig systems (qubits), are alled tesor produt states..4. Quatum Etaglemet I geeral, a tesor produt provides the mathematial desriptio of the state of a system that is ostituted by brigig together oiteratig quatum systems, assumig that they remai without iteratig [6]. Comprehedig this oept is useful to get a lear uderstadig of the defiitio of a etagled state [7-]. I partiular, if assoiated with two quatum systems there are vetor spaes V of dimesio N, i whih resides a vetor φ, ad V of dimesio N, i whih resides a vetor χ, ad where N ad N may be fiite or ifiite, the the tesor produt of V ad V is deoted by the vetor spae V [6], V V V, (6) of dimesio N N, where the vetor, φ χ φ χ, (63) assoiated with the overall spae V, is alled the tesor produt of φ ad χ. If the vetors φ ad χ a be expressed i terms of the respetive v, so that, bases { } i ad u ad { } i φ a i u i, (64) i

82 68 Chapter χ b j v j, (65) j the, the tesor produt may be writte as, φ χ a b u v, (66) i, j i j i j from where it is see that the ompoets of a tesor produt vetor are the produts of the ompoets of the two vetors of the produt. A example will help appreiate the meaig of a tesor produt immediately. Let V x ad V y be two vetor spaes i whih the bases { x } ad { y }, reside. The the tesor produt of the spaes is give by, V xy V V, (67) x y ad the tesor produt of the bases is give by, xy x y. (68) Cosequetly, if X ad Y are operators i V xy, the we have, ( y ) x x ( y ) x x y x xy X xy X x, (69) ( x ) Y y ( x ) y y y x y y xy Y xy. (7) Essetially, the, the operators atig over a tesor produt of spaes operate oly o the vetor spae to whih they belog. Now, assume that the global state of the system is embodied by the wavefutio ψ V V V. The, aordig to the above, ψ ψ ψ, where ψ V ad ψ V. A quatum system is said to be etagled if it is impossible to express its global state as the tesor produt, i.e., ψ ψ ψ. Thus, i a etagled system, it is ot possible to at o oe of its vetor states idepedetly without perturbig the others. It is said the, that the states i a etagled system are orrelated.

83 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea Eistei-Podolsky-Rose (EPR) State I a system with two oiteratig qubits, the global state may be expressed as [8], ψ 3 4, (7) i where ad eah term is the tesor produt of the ompoets of i the orrespodig qubits. Whe 4, ad 3, the resultig state, ( ) ψ EPR, (7) is alled a EPR state [8]. The EPR state is ot a tesor produt of the vetor states, therefore, it represets a etagled state; it does ot belog to ay of the idividual vetor spaes, but is a ombiatio of them. Assoiated with a EPR state is the so-alled Bell-state basis [8], whih embodies the possible states that a result upo measurig two-state quatum systems. I partiular, if, represet the two states of partile, ad, represet two states of partile, the the measuremet of their EPR pair state may result i oe of four state vetors, amely, ad ( ) ± Ψ ±, (73) ( ) ± Φ ±. (74) Oe of the most trasparet demostratios of etaglemet ad its impliatios was the experimet by Kwiat et al. [7], see Figure - below. This experimet exploited the priiple of type-ii parametri dow oversio to produe direted beams of polarizatio etagled photos. I type-ii parametri dow oversio [7] a iidet laser beam pump passes through a rystal, suh as beta barium borate, ad a spotaeously

84 7 Chapter UV-Pump BBO-Crystal Coes Figure -. Etagled photos via type-ii parametri dow oversio. (After [7].) deay ito two photos of lower eergy, oe polarized vertially ad oe polarized horizotally, for istae. I partiular, eah photo a be emitted alog a oe i suh a way that two photos of a pair are foud opposite to eah other o the respetive oes. If it ours that the photos travel alog the oe itersetios, however, the either photo is i a defiite polarizatio state, but their relative polarizatios are omplemetary, i.e., they are etagled. Takig the state of the photos alog the itersetig oes as etagled, i.e., ( H V V ) Φ, (75) H we see that, beause the polarizatio relatioship of omplemetarity must be maitaied, wheever photo is measured ad foud to have vertial polarizatio, the polarizatio of photo will be horizotal, ad vie versa. This meas that o matter the state i whih photo is foud, the state of photo a be predited to be i the orthogoal state whe measured. Etaglemet, therefore, eables a strog orrelatio amog the photos. This is a geeral property amog etagled partiles. By appropriately otrollig the evolutio of aggregates of partiles, it is possible to idued them ito etagled states. The agets that otrol the evolutio of states are alled quatum gates..4.. Quatum Gates Give a qubit prepared i the iitial state ψ ( t ) subsequet time t is give by () t U ( t, t ) ψ ( t ), its state at a ψ, where U is the qubit s trasitio matrix[6] Uitary reversible matries U presribig the evolutio of qubits are alled quatum logi gates [], []. Thus, a

85 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 7 quatum gate trasformig a qubit state suh that ad exp( iωt), would have the form [], U ( θ ) iθ, (76) e where θ ωt. Sie U is a uitary reversible trasformatio, the quatum gate must be reversible. This meas that, give the output, oe must be able to uiquely determie the value of the iput. There are a umber of importat quatum gates of whih quatum iformatio proessig systems are made of, amely, the idetity gate [-],, (77), (78) the NOT gate,, (79), (8) the Z gate,, (8), (8) ad the Hadamard gate,, (83). (84) Quatum gates are represeted graphially, as i Figure - []. I this figure the operatio of the gate is read from left to right usig the followig ovetio. Eah lie represets the propagatio or evolutio of the iput

86 7 Chapter state ad ould, aordigly, represet propagatio via a wire, i time, i spae, or i ay other fashio evolutio may be iteded to take plae. The gate has otrol qubits ad target qubits. A otrol qubit, suh as x, has its lie of propagatio (wire) tapped at a dot. A target qubit, suh as y, has its lie of propagatio (wire) XOR ed with a otrol bit. The gate s purpose is to effet a trasformatio o the target qubit based o the values of the otrol qubit, i partiular, if the otrol qubit is set to oe, the the target qubit is iverted. The realizatio of lassial logi gates, whih are iheretly irreversible, by totally reversible quatum gates may be effeted with the use of the Toffoli gate, see Figure -(b). The Toffoli gate is a irreversible gate that takes three iputs, amely, two otrol qubits ad oe target qubit. By applyig the Toffoli gate twie to its three iput qubits, they are repodued, thus the irreversible gate is made reversible []. z y x z y x y x y x x x x ' x y y ' y x ' x y' y xy z z z y x z y x z y x z y x y x y x y x y x x x x ' x y y ' y x ' x y' y xy z z (a) (b) y x y x x y x ' x y ' y y x y x y x y x x y x ' x y ' y () Figure -. Truth tables ad graphial represetatios of some quatum gates. (a) Cotrol- NOT gate. (b) Cotrol-otrol-NOT (Toffoli) gate. () Bit swappig.

87 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 73 The otrol-not (CNOT) gate, as a be see from Figure -(a), implemets the exlusive-or (XOR) operatio. Thus, the gate iverts y, if x, ad leaves it as is if x. This operatio is expressed as, x y x y ( x ) mod C. (85) Applied to a pair of sigle produt states of two qubits, the CNOT gate produes a set of etagled qubits, i.e., ( ) ( ) C. (86) Similarly, sie the CNOT gate is reversible, whe applied to a etagled state, it produes a set of disetagled states, i.e., ad ( ± ) ( ± ) C, (87) ( ± ) ( ± ) C. (88) These operatios are essetial for quatum teleportatio. Oe may reall that a lassial NOT gate is alled uiversal i the sese that ay other logi gate may be reated by ombiig several NOT gates. Similarly, a uiversal quatum gate should geerate all uitary trasformatios of qubits. It a be show that suh a gate is realized by ombiig a pair of gates, amely, oe that produes a geeral rotatio o a U θ,φ, where, sigle bit, ( ) Uiversal ( θ ) iφ ie si( θ ) si( θ ) os( θ ) os U Uiversal ( θ, φ), (89) iφ ie ad a CNOT gate []..4. Quatum Teleportatio Aordig to Beett et al. [6], quatum teleportatio is a proess that disembodies the exat quatum state of a partile ito lassial data ad EPR orrelatios, ad the uses these igrediets to reiarate the state i

88 74 Chapter aother partile whih has ever bee aywhere ear the first partile. The proess does ot ivolve sedig ay qubits, rather, the seder ad the reeiver must have aess to two other resoures, amely, the ability to sed lassial iformatio, ad a etagled EPR pairs of partiles previously shared betwee them. As per the sketh of Figure -3, teleportatio proeeds as follows. ψ EPR pair 3 S e d e r R e e i v e r ψ Figure -3. Quatum teleportatio of state ψ. (After [8].) There are three partiles ivolved, amely, partile, whose ukow state ψ a b (a ad b are the ukows) is to be teleported by a seder to a reeiver, ad partiles ad 3, whih are prepared by a EPR soure ito a etagled EPR state, for istae, ( ) 3 3 Φ 3. (9) Of these two etagled partiles, oe, amely, partile 3, is set by the EPR soure to the reeiver ad the other, partile, is supplied to the seder. Notie that loally both the seder ad the reeiver possess total kowledge of the states of partiles ad 3, respetively. However, globally, the three states are desribed by tesor produt state, ( a b )( ) 3 3 ψ 3, (9) osistig of the etagled pair, partiles ad 3, ad the ukow state. Now, the speifi atios that effet the teleportatio are as follows. The

89 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 75 seder performs a joit (XOR) measuremet betwee partiles ad. As we saw previously, the outome of measurig a pair of sigle produt states of two qubits, suh as that of partiles ad, has four possible outomes ad ( ± ) Ψ ±, (9) ( ± ) Φ ±. (93) Takig this ito aout, the diret produt state ψ 3 may be expaded i terms of these four outomes ad rewritte as, ψ 3 ( a a b b ) 3 ( a b ) ( a b ) Ψ ( b a ) Φ Ψ ( b a ) Φ (94) The result of performig the XOR betwee partiles ad will be the ollapse or projetio of the global tesor produt state ψ 3 alog oe of ± ± the four vetor states Ψ ad Φ with equal probability, amely, ¼. Notie that this will leave a ew global state osistig of the tesor produt ± ± of oe of the vetors Ψ ad Φ, at the seder, ad a modified qubit 3, at the reeiver. Oe possible result might be, ( b a ) Ψ. (95) 3 3 If these were the ase the, to omplete the teleportatio proess the seder has to ommuiate to the reeiver, usig lassial message, that the global

90 76 Chapter Ψ wave futio ollapsed alog, ad that for its qubit to embody the ukow state ad, hee, omplete the teleportatio, it has to effet the uitary trasformatios: ad o its qubit Deoheree A quatum system is said to deohere whe, i the ourse of its time evolutio, it loses eergy to the eviromet. Uder these irumstaes its trasitio matrix, U, o loger oserves the orm of the states it ats upo. Sie the states hage i a radom maer, the property of superpositio of states is o loger maitaied. From thermodyamis we kow that systems that experiee eergy loss are irreversible, therefore, deoheree preludes the realizatio of quatum gates, e.g., the Toffoli gate, whih must be reversible. The ability of a quatum system to maitai its oheree ad, thus, be apable of maifestig superpositio ad etaglemet, is aptured by the deoheree time. Obviously, the system is useful for quatum iformatio proessig oly durig this period of time. A system made up of may qubits will exhibit a ompouded amout of errors as it approahes its deoheree time., i.e., as it beomes irreversible. The deoheree of a qubit, i partiular, is quatitatively aptured by the quality fator of quatum oheree [], Q πν T, (96) ϕ ϕ where ν is its trasitio frequey ad T ϕ is the oheree time of a superpositio of states. While error-orretig odes tehiques have bee proposed to ombat errors stemmig from deoheree, the eed for a itrisially oheret system to begi with, remais. Therefore, the oeptio of approahes exhibitig log deoheree times, with respet to the iteded omputatioal futio to be implemeted, is ruial, if quatum iformatio proessig is to beome pratial. Vio et al. [] poit out that, give a quatum omputatio with elemetary operatios takig time t op, ative ompesatio of deiheree requires Q ϕ ' s greater 4 tha ν t op. A umber of approahes to the physial implemetatio of qubits, ad their respetive deohereetimes, are disussed i Chapter 4.

91 . NANOMEMS PHYSICS: Quatum Wave-Partile Pheomea 77.5 Summary This hapter has dealt with physial pheomea exploitig wave-partile duality. We bega by addressig oditios that maifest harge disreteess, ad its osequees o the performae of trasmissio lies, amely, persistet urrets ad urret exhibitig Coulomb blokade (pulsatig) behavior. The, after itroduig the oepts of sigle-eletro tuelig, the effet of harge disreteess i eletrostati atuatio was preseted. I this otext, we saw that hargig domiated by Coulomb blokade may lead to quatized eletrostati atuatio. Followig this, we addressed the maifestatio of quatum eletrodyamial fores, i partiular, va der Waals ad Casimir fores ad their substatial ifluee i movig ao- ad miro-meter-sale devies. The hapter oluded with a expositio of the saliet poits of quatum iformatio theory, omputig ad ommuiatios. I partiular, we foused o the oepts of quatum bits, quatum etaglemet, the Eistei-Podolsky-Rose (EPR) state, quatum gates, ad quatum teleportatio. Lastly, the ruial issue of deoheree was disussed.

92 Chapter 3 NANOMEMS PHYSICS: QUANTUM WAVE PHENOMENA 3. Maifestatio of Wave Nature of Eletros The priiples of aosale devies are based o the physis domiatig this dimesioal regime. I partiular, as the devie size is redued below about m, the eletro behavior stops obeyig lassial physis, i whih its mometum ad eergy are otiuous, ad starts obeyig quatum mehais, i whih it behaves as waves with quatized eergy, Figure 3-. Fig.3- Cotiuous Eergy p p mv E ~ m L z L x L y Size RedutioQuatized Eergy h p λ (,, ) E x y z E hω h xπ yπ zπ m Lx Ly Lz L < ~ i m x p h x > Figure 3-. Size-depedet behavior of eletros. ~

93 8 Chapter 3 The, depedig o the partiular devie struture, behavior suh as iterferee, diffratio, et., harateristi of waves, or Coulomb iteratio, harateristi of partiles, may be promiet. The various types of behavior are preseted ext. 3.. Quatizatio of Eletrial Codutae The oept of eletrial odutae quatizatio emerges from osiderig eletro trasport i short, arrow (quatum) wires, Figure 3-. E F E F d e V - Figure 3-. Eletro trasport dow short, arrow wire betwee eletro reservoirs with Fermi levels E F ad E F, uder the ifluee of applied voltage V. Here we have a short, arrow wire oeted betwee two eletro reservoirs haraterized by Fermi seas (otats) filled up to eergy levels E F ad E F, Uder the ifluee of a applied voltage V, whih misaligs the Fermi levels, eletros travel from reservoir E F towards reservoir E F, i a effort to equalize the Fermi levels ad, as a result, establish a urret. Sie the wire is very short, trasport evolves without satterig, i.e., ballistially. However, sie the wire is very arrow, the uertaity priiple fores its trasverse mometum (ad osequetly, its eergy) to be quatized, i.e., p ~ d, where is a iteger represetig the bad i whih trasport is ourrig Ladauer Formula The questio before us is: What is the odutae of this system? The aswer was determied by Ladauer [3], ad may be arrived at as follows [76]. The urret is the balae betwee the umber of eletros beig lauhed from the left-had reservoir ito the wire, ad the umber of eletros beig lauhed from the right-had reservoir ito the wire. I

94 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 8 partiular, sie i the mometum iterval dp, this umber equals dp π, the orrespodig urret is dj evdp π. Therefore, the total left-right urret, assumig a sigle bad, ad takig ito aout two spis, is give by: J F e e vdp de π π E ee F π. () A similar result is obtaied for the right-left urret, e e J vdp de π π µ ee F π, () so, the et urret from left to right is: J J J e π ( E F E ) F. (3) The, width the substitutio EF EF ev, we obtai, J e V. (4) π The proportioality fator betwee urret ad voltage is the quatized odutae for a sigle bad: e g. (5) π Assumig trasport is ourrig i N bads (haels) uder the Fermi level, the total odutae is, g N. (6) g This expressio learly reveals that the odutae is quatized i uit s of g. I reality, there is a fiite probability that i goig from the reservoir ito the wire, ad vie versa, some eletros may be baksattered, i whih ase the umber of bads through whih trasport is operative is less tha N. I that ase the effetive value for N is odutae is give by:

95 8 Chapter 3 N Effetive N i T ( E ) F, (7) where T is the trasmissio oeffiiet of bad. Clearly, astig the odutae i terms of the trasmissio oeffiiet uovers its depedee o the wave ature of the eletro Quatum Poit Cotats I derivig the quatized eletrial odutae of a quatum wire above it was poited out that it is proportioal to N, the umber of bads through whih trasport is operative. The quatum poit otat (QPC), Fig. 3-3, represets a virtually zero-legth quatum wire, i whih the details of T domiate trasport ad are made patetly maifest i the odutae. Top View E F DEG DEG E F V (a) Cross Setio I Split-Gate 5 m -AlGaAs m AlGaAs Cofied DEG 3 (b) GaAs Codutae (e /h).6k.3k Gate Voltage (V) Figure 3-3. Quatum poit otat. (a) Top view. (b) Cross-setio. () Codutae versus gate voltage. (After [4].) I the QPC a ostritio is formed by modulatig via, e.g., depletio regios, the width of the hael betwee two two-dimesioal eletro gas

96 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 83 (DEG) regios, Figure 3-3 (a), (b). A reditio of the first experimetal demostratio of the effet is show i Figure 3-3(). It is observed that the odutae dereases approximately liearly as the gate voltage is ireased egatively, i.e., as the ostritio or hael width arrows. I partiular, at V G -.V, the hael is pihed-off ad the odutae is zero. Notie also, that the odutae dereases i disrete steps of e h. A explaatio of the observed quatized odutae was attributed to the resistae of the ostritio upo ompariso with the semi-lassial formula for the odutae of a ostritio i a DEG, deoted G S, after Sharvi who derived it [68]. G S is give by, G S D e dn vfw, (8) π de * where dn D de m π is the quatum mehaial desity of states, * iludig a fator of two for spi, v k m is the Fermi veloity, with F kf π λf πs beig the Fermi vetor ad S the DEG eletro desity, ad W is the width of the ostritio. Rewritig (65) so that the quatized odutae beomes expliit, oe obtais, F G S e h k W π e h W λ F. (9) F The fat that this equatio iludes the ratio W λ F suggested that, experimetally, there should be deviatios due to the maifestatio of the wave ature of eletros wheever λ ~ F W. I partiular, it was determied that the plateau values of odutae are obtaied wheever W is a itegral multiple of λ F /. Therefore, the quatized odutae is a maifestatio of the wave ature of eletros i that as the voltage is ireased from pih-off, a ew mode (bad) for trasport beomes available every time the ostritio wides by λ F /. The trasmissio oeffiiet of the ostritio aptures this [5]. The deviatios from flatess of the odutae plateaus were attributed to satterig or to the abruptess of the ostritio. Fially, as the temperature ireases, the odutae steps smear out util at high temperature they disappear. This is due to the o-mooeergeti, wider, distributio of eletros lauhed by the reservoirs ito the ostritio [68] ad exposes oe of the pratial limitatios of QPCs, amely, that their utilizatio requires extremely low temperatures.

97 84 Chapter Quatum Resoat Tuelig Oe of the fudametal devies exploitig the wave ature of eletros, ad whih fids pratial appliatio at room temperature, is the resoat tuelig diode (RTD) [6], [7], see Figure 3-4. B W B E E E FL E Cotat E Cotat E FR z E E FL E E V Peak Curret p E FR E E FL E VV (a) Valley e Curret E FR I RTD I p V RTD IV Vp VV (b) Figure 3.4. Resoat tuelig diode. (a) Eergy bad diagram ad operatio. (b) Curretvoltage harateristi. The RTD osists of a double barrier sadwihig a potetial well, ad i tur lad by two eletro reservoirs (otats). The potetial well dimesios are of the order of tes of Agstrom, suh that eletros i it are ofied ad, thus, a oly exist i quatized eergy levels. The barrier legths are of the order of a few Agstroms, so that eletros a tuel through them.

98 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 85 Resoat tuelig devies are implemeted i a variety well/barrier materials systems [6], iludig, Type-I heterostrutures (trasport ours exlusively i the odutio bad) suh as GaAs/Al x Ga -x As, IAs/AlSb, I.53 Ga.47 As/AlAs, ad Type-II heterostrutures (trasport ivolves odutio ad valee bads) suh as GaSb/AlSb [8] The ideal RTD urret-voltage harateristi is show i Fig. 3-4(b) ad, with respet to Fig. 3-4(a), a aepted plausible explaatio of it is as follows [6[, [7]. With o voltage applied, the system is i equilibrium as o fores are experieed by the eletros i the otats ad o urret flows: () As the voltage is ireased eletros tuel the left-had barrier, propagate through the well ad tuel through the right-had barrier, ad a ireasigly large urret flow; () Whe the voltage is suh that the eergy of the iomig eletro distributio overlaps the first quatized eergy of the well, E, maximum urret trasmissio is ahieved, this is the resoat tuelig oditio; (3) Whe the overlap dereases, at higher applied voltages, the trasmissio, ad thus urret, rapidly dereases, thus the egative resistae regio is produed. This explaatio assumes the eletro mometum trasverse to the well is oserved. Sie the itrisi time it takes a eletro to traverse the struture is related to Heiseberg s uertaity priiple, τ Γ, where Γ is the eergy width of the quatized level, the proess is very fast, i.e., ~ps, so the devies are ideal for THz appliatios [8, 9]. The simulatio ad modelig of RTDs is a relatively mature subjet [6-3] ad iludes a variety of approahes ragig from those egletig satterig ad harge effets to those iludig them to a variety of degrees. These models typially reprodue features of the I-V urve related to eergy levels i the devie, suh as the voltages at whih peak ad valley urrets our, but ot the magitudes of these urrets. A typial approah is the two-bad tight-bidig model, exposed by Shulma [4] for modelig a GaAs-GaAlAs RTD. I partiular, by egletig satterig ad harge effets it fouses o alulatig the trasmissio oeffiiet of the struture by employig a atom-to-atom trasfer matrix tehique that builds up the eletro wave futio as it propagates through the devie layers. The model divides the struture as show i Figure 3-5, assumes that the wave futio is a ombiatio of s-like orbitals o eah atio (Ga, Al) ad a p-like orbital o eah aio (As), of the form, Csφ s C p φ p Ψ, () ad sets up a tight-bidig Hamiltoia of the form,

99 86 Chapter 3 H I E s U ( eika / ) U ( eika / ), () E p H I Ψ I EΨI, () Left Claddig B W B Right Claddig III II I E () z INCOMING REFLECTED TRANSMITTED E ( z )... GaAsGa AsAl... AlAsGaAs.. GaAsAlAs.. AlAsGaAsGaAs Figure 3-5. RTD struture for two-bad tight-bidig modelig. where E S ad E P are orbital eergies prior to ouplig to ext eighbors, ad a is the lattie ostat. Next, solutios are formulated for the three regios as follows. For regio I, we have (3) ad (4). L i k ( a / 4) Ψ ( E E i E E e E p φ I E E s s φ (3) p) s p R i k ( a / 4) Ψ ( E E i E E e E p φ I E E s s φ (4) p) s p For regio II we have the trasfer matrix (5). C s( ) C p( ) U(, ) U(, ) C ( ) s (5) [ E E ( ) ] U(, ) U(, ) [ E E ( ) ][ E ( ) E] C p( ) s s p U(, ) U(, ) U(, ) E ( ) E( ) p U(, ) U(, ) U(, ) For ouplig regios II ad III we have (6) ad (7).

100 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 87 Ψ III C L C R s φ s C p φ p C Ψ Ψ (6) L III R III C C L R E E s E p E E ika p ( e / ) E E e ika / p ( ) i E E e ika ika s ( /4 e /4 ) C s i C p E E e ika eika/4 s ( ) (7) The dispersio relatio, veloity, ad overlap itegral defiig the tightbidig are give by (8), (9), ad (). 4 k ( E ) ± arsi ( E E s )( E E p a U (8) au si( ka / ) v ± (E E s E p ) (9) U ( E s E p ) m * a () Fially, the urret is give by (), where x E kt. J ( V ) e( kt ) 4π dx C L v I v III k III v III ( E / kt x) e F l ( E ev ) / kt x e F () This formulatio, though ot fully preditive, is a useful tool for the aalysis ad desig of RTDs ad related devies. A typial I-V urve produed usig this formalism is show i Figure 3-6. Curret (A/m) 4 4 IGaAs/AlAs RTD V oltage (V olts) Figure 3-6. Curret-voltage urve alulated via two-bad tight-bidig formalism.

101 88 Chapter Quatum Iterferee While the RTD I-V harateristis are the result of ostrutive ad destrutive iterferee betwee the barriers, as a oe-dimesioal devie these are really futio of the degree of resoae with the eergy levels i the well. Whe trasport ours i two dimesios, we may have ostrutive ad destrutive iterferee as a result of waves travelig thorough differet paths that overge at oe poit Aharoov-Bohm Effet The quitessetial example of this type of iterferee, whih also exposes the wave ature of eletros, is the Aharoov-Bohm (AB) effet [5], Figure 3-7. The essee of the AB effet, see Fig. 3-7, is that a eletro beam, with wavefutio ψ i, split at poit A ito two waves, ψ ad ψ, whih subsequetly follow paths ABF ad ACF, aroud a soleoid establishig a mageti flux φ stritly i its iterior, will gai respetive phases S ad S so that at F the wavefutio is, F is e is e ψ ψ ψ, (5) or, i other words, there is a phase differee ( S S ) betwee them. I partiular, the phase shift is give by, B Eletro Beam A Shadow Soleoid Radius, R F Iterferee Regio Metal Foil C (a) t ~e ik L I φ Out t ~e ik L (b) Figure 3-7. (a) Aharoov-Bohm-effet eletro wave iterferee setup. (After [5].) (b) Sketh of metalli rig implemetatio.

102 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 89 S e ABF A e A e ϕ dt dl ϕdt dl A dl, (6) ACF where ϕ is the salar potetial ad A is the vetor potetial, whih is related to the mageti field iside the soleoid by (7). φ A dl H ds, (7) The remarkable aspet of this effet is that, beause of (7), it predits, ad has bee ofirmed, that a vetor potetial exists eve where o mageti field is existet, amely, outside the soleoid i this ase, ad this vetor potetial edows the wave futios with a phase shift differee whih establishes that the eletros may exhibit iterferee. I partiular, the phase shift may be expressed as, e χ φ, (8) so that whe χ π there is ostrutive iterferee, ad whe χ π there is destrutive iterferee. ( ) 3..4 Quatum Trasport Theory The wave ature of eletros is resposible for a umber of pheomea, suh as quatized eletrial odutae, resoat tuelig, ad quatum iterferee, whih fid their geesis i the quatum ature of eletros. Sie, i fat, at dimesios approahig m feature sizes, these effets are already begiig to domiate the harateristis of pratial devies, the questio of how to simulate the behavior of these quatum devies has reeived muh attetio. I this setio, we fous o the priiples of typial theoretial approahes to the quatum trasport of heat ad eletros Quatized Heat Flow I bulk devies, the rate of heat odutio per uit area is proportioal to the temperature gradiet, i.e., Fourier s law, Q A κ T, where κ is the bulk oeffiiet of thermal odutivity. This expressio assumes κ γcvl p [6], where γ is a umerial fator, C is the speifi heat per

103 9 Chapter 3 uit volume, v is the veloity of soud, ad l p is the phoo mea free path, i.e., the typial devie dimesio L >> l p. At aosale dimesios, however, L < l p ad the phoos propagate ballistially. I this ase, theory developed by Rego ad Kirzeow [7], ad experimets performed by Shwab, Herikse, Worlok, ad Roukes [8], have show that the thermal odutae betwee isolated right ad left temperature reservoirs, whih are oly iteroeted through the devie, is give by Ladauer s theory as, ( ω) ( ) Nα N α R L ω κ dω ω ζ α( ω) π α T α ω () α where ω ( k) ad ( k) α α dω ω R ( ω) ( ) T L ω ζ α ( ω), (9) ζ are the frequey ad phoo trasmissio ω probability of ormal mode α, respetively, ad ( ) ( ) k B T i ω e i represets the thermal distributio of phoos i reservoir with temperature T i. While, it has bee demostrated i the works of Agelesu, Cross, ad Roukes [9], ad of Rego ad Kirzeow [7], that the trasmissio probability is sesitive to the geometrial features of the aosopi systems, i partiular, to phoo satterig due to surfae roughess ad trasitios (o-adiabati mode ouplig), the mai olusio from ( 9) was that at low temperatures heat trasport is mediated by a uiversal ostat, amely, the quatum of thermal odutae due to phoos, k B π 3h [8]. This has serious impliatios pertaiig to the maximum rate at whih power a be dissipated i NaoMEMS, ad ideed aosale thermal trasport is a very ative area of urret researh [3] Fermi Liquids ad Lüttiger Liquids As suggested at the begiig of this hapter, trasmissio lies (TLs) are ubiquitous i iruits ad systems at all legth sales. Sie TLs should simply trasfer or guide sigals from oe loatio to aother, without dereasig their amplitude or power, it is imperative that they exhibit the lowest possible loss. This is the reaso why metals, due to their lowest resistivity, are preferably utilized to implemet iteroets (TLs). The resistivity of ovetioal (large-dimesio) TLs reflets the dimesioality of eletro motio. For istae, i TLs of retagular rosssetioal area A, as dimesios shrik eletro motio may beome quatized i ertai diretios, thus givig rise the to the reatio of eergy

104 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 9 sub-bads or haels i whih trasport a oly our oe the eletros aquire the orrespodig eessary eergy, i other words, eletros behave as waves with disrete (quatized) wave vetors. The quatized eletrial odutae is a maifestatio of this. I otrast, eletros i TLs of relatively large dimesios may exist at virtually all eergies ad, if there were o iteratio amog eletros, they would behave as free partiles. The theory of eletro behavior i a metal, whe eletro-eletro iteratios are take ito aout, is due to Ladau [3] ad is deoted Fermi liquid theory. A Fermi liquid is osidered to be made up of quasipartiles, whih are fititious etities that, while beig physially differet from eletros, behave similarly to eletros, but with a differet mass ad dispersio relatioship. Whe eletro trasport is ofied alog oe dimesio, a behavior differet to that of free eletros ad that of a Fermi liquid is observed. The ew aggregate of etities is said to osist of aother fititious quasipartile, amely, the plasmo, ad is referred to as a Lüttiger liquid (LL). The distitio betwee Fermi liquid ad Lüttiger liquid behaviors is importat to the realizatio of aosale iruits ad systems, ot oly from the poit of view of TL properties, but also beause their differet behavior eliits ew issues whe oetig a Fermi liquid TL to a Lüttiger liquid TL. The fudametal aspets of Fermi ad Lüttiger liquids are addressed ext Fermi Gas The Fermi liquid theory explais the suess of the free-eletro approximatio i the alulatio of trasport problems, eve i the otext of eletro-eletro iteratios. The usual poit of departure for desribig the Fermi liquid is the Fermi gas. This is the oeptual situatio i whih the metal is modeled as a solid of volume V ad legth L o a side 3 ( V L ), whih otais movig o-iteratig eletros i muh the same way as atoms ad moleules move iside a gas otaier. Sie the eletros are assumed to be idepedet, i.e., do ot iterat, they eah obey a Shrödiger equatio of the form [3], p H ψ U ( r ) ψ Eψ, (3) m where the potetial eergy is take to be U ( r ). The solutio of this equatio is the obtaied by assumig that all spae is filled by ubes of side

105 9 Chapter 3 L, ad that the wavefutio fulfills periodi boudary oditios at eah of its faes, amely, ψ ˆ, (3) ( r Lx) ψ ( r Lyˆ ) ψ ( r Lzˆ ) ψ ( r ) These assumptios yield solutios of the form φ kσ ik r e, (3) V ( r ) χ σ where σ ± represets the two values of eletro spi ad χ σ represets the two spi futios, χ, χ. (33) Beause of the periodi boudary oditio, the wave vetor is defied by, π π π kx x, k y y, kz z, (36) L L L where,,, ±, ±,..., x y eigevalues of (3) are give by, z k k k k. The eergy x y z k E k E k. (37) σ m The saliet properties of the eletro gas as a whole are aptured by its wave futio, its total eergy, ad various quatities suh as its speifi heat, ad its mageti suseptibility. The wave futio is give by the Slater determiat [3], ψ... ν (,,3,... N ) ν ν ν 3 N φ ν φν N!... φ ν N () φν ( )... φν ( N ) () φ ( )... φ ( N ) ν... () φ ( )... φ ( N ) ν N... ν ν N..., (38) whih esures that the Pauli exlusio priiple is obeyed, i.e., if two of the oe-partile states ν i are the same, the ψ ν ν.... With U ( r ), the ν N

106 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 93 lowest eergy eigevalue (groud state eergy) is give by the sum of the oe-eletro eergies up to a maximum eergy level deoted by E F ad alled Fermi eergy. This is obtaied whe N eletro states with eergy less tha E F are oupied, ad all states with eergy greater tha E F are uoupied. To obtai a expressio for E F, oe pitures the states i (38) as a grid of poit i k xk ykz spae where they form a fie three-dimesioal grid of spaig π L, suh that a sphere etered at k would otai 3 3 4π 3 Vk Vk k poits of the grid whe its radius is k, 3 3 π 6π 3π L iludig spi. Sie eah poit i the grid represets oe eletro, the umber of grid poits otaied i a sphere with the largest radius, k F, orrespodig to E F must equal N, 3 Vk F N. (39) 3π Thus, the largest eletro mometum is, k F 3π N V ( 3π ) / 3 / 3, (4) where is the eletro desity i the metal, ad the Fermi eergy is, E F / 3 ( 3π ) / 3 3π N m V. (4) m At absolute zero, all levels are filled up to E F. For a arbitrary eergy E, less tha E F, the total umber of eletros with eergy less tha E is give by, N V 3π m 3 /, (4) from where the desity of states is give by,

107 94 Chapter 3 D dn de V π m / ( E ) E 3. (43) Exitatio of the groud state of a Fermi gas requires, due to Pauli exlusio priiple ostraits, the additio of partiles with mometum greater k > k F, or the destrutio of a partiles (reatio of holes) with k < kf. However, if these partiles ame from outside the system, the the total umber of partiles N would hage ad we would have a differet system. Whe oe isists o iduig exitatios that oserve the umber of partiles, the reatig a partile with k > kf k, is aompaied by reatig a hole with k < kf k, i.e., partile-hole exitatios whih a be idetified by two quatum umbers k, k are reated. These exitatios may be aused by a umber of ifluees, i partiular, a rise i temperature or the appliatio of a mageti field k F. Sie, uder o iteratio, all states are oupied up to k F, eletros losest to E F will require the miimum eergy to exite. Thus, the eergy eessary to exite ( kf k ) a eletro of mometum k, for istae, is EExitatio. m Temperature-idued exitatios of the Fermi gas are aptured by the speifi heat, give by [8], C el E T π 3 D ( E ) k T γt F B, (44) where k B is Boltzma s ostat, ad mageti field-idued exitatios are aptured by the mageti suseptibility give by, M χ D ( E F ) µ, (45) B B where µ B is the Bohr mageto. Clearly, these quatities ivolve the desity of states evaluated at oe poit, amely, the Fermi eergy. This fat, oupled to the irumstae that, as log as oe is dealig with a oiteratig free eletro gas D ( EF ) will have the same value, suggests that solvig both (44) ad (45) for D ( EF ) ad takig the ratio of the resultig quatity must be equal to oe. This ratio, alled the Wilso ratio, is give by [33],

108 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 95 π k B χ R W, (46) µ γ 3 B ad aptures the degree to whih there are eletro iteratio effets. I partiular, deviatios from RW sigal the presee of iteratio. Disussios o the Fermi liquid oept, whih embodies pheomea due to eletroeletro iteratio, usually make use of this idex as a haraterizatio parameter Fermi Liquids Fermi liquid theory assumes that as the eletro-eletro iteratio is tured o, from its zero value i the Fermi gas, the states i the ow iteratig system evolve diretly from those of the oiteratig system, i suh a way that the exited partiles may also be labeled by mometum pairs k, k, just as i the oiteratig eletro ase [34]. This irumstae is exemplified by the evolutio of states i a oiteratig eletro gas situated i a ifiite-wall potetial well as the iteratio betwee them is tured o very slowly (adiabatially), see Figure 3-8 [34]. Havig idetial quatum labels for oiteratig eletros ad quasipartiles implies that quatities that deped o these labels, suh as the ofiguratioal etropy ad the eergy distributio, remai uhaged after the iteratio is tured o [34]. Suh is ot ase with the total eergy beause the eergy of iteratio modifies its value from the simple sum of that of the free partiles. d ψ V ( x ) ψ E ψ dx Eergy V ( x ) λ x x < x N4 π π N4 N3 N N N λ N3 N N N Figure 3-8. Adiabati otiuity explais how the labels of the eergy states i a oiteratig eletro gas may otiue to be used as the iteratio λ is tured o. Notie that, as the eergy levels ad their orrespodig eigefutios evolve, the quatum labels (N) of the origial oiteratig problem remai. After [34].

109 96 Chapter 3 O the other had, the exited partiles, while fidig themselves at the same k, k as the free eleros, exhibit a differet mass ad a differet E vs. k relatioship tha these, i partiular, see Figure 3-9, iteratios amog the partiles with states below E F, ad betwee these ad the exited eletros with eergy above E F, are resposible for this. Thus, the dyamial properties of quasi-partiles differ from those of free eletros. Uder these irumstaes, the theory assumes that for low-eergy exitatios, the quasi-partile distributio evolves i suh a way that, if [33], ( k ) if if k k < < k k F F (47) (a) - (b) () Figure 3-9. Fermi liquid represetatio. (a) Groud state. (b) Exited state. () The quasipartile exhibits a ew effetive mass, m*, whih derives from its iteratio with groud state eletros as it moves through them. This effetive mass is i additio to the mass derived from its iteratio with the rystal lattie (aptured by the eergy bad urvature), i.e., the dispersio relatio E vs. k. - the the distributio of the oiteratig gas is ( k), ad, upo exitatio ( k) ( k) δ( k), where δ ( k) whe a quasi-partile is exited, ad ( ) δ k whe a quasi-hole is exited. Here, k ( k,σ ),

110 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 97 ad (, ) σ represets spi. Similarly, the orrespodig eergy hage is assumed to be give by, δ E Ek δ( k ) f ( k, k ) δ( k ) δ( k ), (48) Ω k kk where the first term represets the eergy of a idividual quasi-partile, defied as, E k k F ( k k F ) m *, (49) with m * represetig its effetive mass, ad the seod term, i partiular, f ( k, k ) apturig the iteratio eergy betwee quasi-partiles. Further, i aalogy with the the ase of oiteratig states, the probability of a quasipartile oupig a state k obeys Fermi statistis, ( k ) βe k e, (5) where β / k B T ad E k is give by (49). I the ase of the Fermi liquid, it has bee foud that alulatios may be simplified by expressig the iteratio futio as the sum of symmetri ad ati-symmettri terms, amely, ad f f a ( k k ) f ( k k ) f ( k k ) S,,,, (5a) a ( k k ) f ( k k ) f ( k k ) S,,,. (5b) The, assumig that these iteratio futios exhibit rotatioal symmetry, ad vary slowly with k, the approximatio k k k F is made, whih permits a Legedre expasio of the form [3], s, a s, a k k f ( k, k ) f L PL ( os θ ), os θ, (5) L k F where P L are the Legedre polyomials. Iversio of the expasio gives the oeffiiets,

111 98 Chapter 3 f s, a L ( os s a θ ) f ( k, ) L, d ΩP k L, (53) 4π whih i ormalized form are rewritte as, F a, s k F m * L a, s f. (54) L π Followig the osideratios i the disussio of the oiteratig eletro gas, exitatios of the Fermi liquid are also aptured by the speifi heat ad the mageti suseptibility. These alulatios assume that, for low eergies, E ad m m *, ad yield [33], ad k E k m * k F k γ B, (55) 3 µ B k F m * χ F (56) π a from where the Wilso ratio is give by (57) i terms of the Ladau a parameter F. R W. (57) F a For the quitessetial example of a Fermi liquid, amely, liquid helium 3 3 a ( He ), a oeffiiet of F. 7 [33] was obtaied experimetally, resultig i a Wilso ratio R W 3. 33, whih deotes strog iteratio. Ladau s Fermi liquid theory sueeds i apturig the pheomeology of ear equilibrium properties, as show above, however, i situatios whe it is ot possible to write a simple expasio for f, as is the ase i highly aisotropi metals, the appliatio of the theory to obtai quatitative results beomes impossible [33], [34]. A more fudametal limitatio of the theory derives from the irumstaes uder whih the oept of quasi-partiles is valid, amely, whe their lifetime is loger tha the time it takes to tur o the iteratio [33], [34]. I partiular, if the Hamiltoia for the iteratig system as a

112 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 99 futio of the iteratio parameter V ad tur-o time τ V is give by [35], H H Ve t /τ V, (58) the the time it takes a quasipartile of exitatio eergy ε to deay, τ ε, must be muh greater tha the iteratio tur-o time, τ V, ad also muh greater tha the time it takes the quasipartile of to absorb the exitatio eergy, give by Heiseberg s uertaity priiple ε form, τ ε >> τ V >>. (59) ε Obviously, at large exitatio eergies E ε, the assoiated time durig whih this eergy is absorbed ε may beome muh smaller tha the lifetime τ ε, whih meas that o quasipartile has a hae to form ad, thus, the quasipartile oept breaks dow. A estimate of this lifetime is give i [34] by alulatig the deay rate of a quasi-partile with eergy ε above the Fermi eergy E F, at absolute zero. Usig Fermi's golde rule, whih desribes the trasitio betwee iitial states i ad fial states f eliited by a satterig potetial V, if τ ε π f V if δ ( ε ε ) f, (6) assumig V if is ostat ad eforig oservatio of eergy ad Pauli exlusio priiples, see Figure 3-, oe obtais, loses ω k,ε p,ε q,ω kq,ε p-q,ε-ω ε quasi-partile ε gais ω filled Fermi sea Figure 3-. Eergy relatioship of quasi-partile satterig proess. The eergyω lost i a satterig evet by the quasi-partile must be lower tha its iitial eergy ε, ad there must be a eletro state at a eergy ε apable of absorbig at this eergy ω.

113 Chapter 3 τ ε π ~ V π ~ V D ε 3 D ( E F ) dω D( E F ) dε δ ( ε ω ε ε ) D( E F ) ( E ) ε F ω dε. (6) This results suggests that, the smaller the quasi-partile (exitatio) eergy ε, the loger will its lifetime be, i partiular, as ε, the lifetime teds to ifiity. A iterestig result that relates the validity of the quasipartile oept to the dimesioality d of the system was derived by Shofield [34], by makig a hage of variables to express (6) i terms of the mometum ad eergy trasferred. His result was the expressio, τ ε π ε D ( E ) F ωdω k ω v F F q d dq ( π L) d D( q, ω) ( v q) F. (6) The itegral (6) is iterpreted as follows [36]: ) The itegral over ω aouts for the umber of possible hole exitatios that a be reated; ) The lower limit of the mometum itegral, over q, sigifies that a miimum mometum must be trasferred to give a hage i eergy of ω ; 3) The deomiator ( v ) F q i the itegrad embodies the fat of already havig performed itegratio over the diretio of the mometum ad it reflets that there is a ireased time available for small defletios; 4) The umerator, D( q,ω) is the matrix elemet for the satterig proess. Examiatio of the impat of settig the dimesio to d reveals that, if oe assumes D ( q,ω) to be ostat, the due to the sigularity of the q itegral, the projeted quasipartile lifetime τ ε, is ot muh greater tha, but i fat is it lose to, ε. Therefore, (6) is violated as the quasipartile, i priiple, a ever have eough time to form. The importae of this result is that Fermi liquid theory breaks dow whe applied to oe-dimesioal metalli systems, suh as are typial at aosales. The ew situatio is desribed by the oept of the Lüttiger liquid Lüttiger Liquids The term Lüttiger liquid is used to deote the behavior of iteratig eletros ofied to oe-dimesioal trasport [37]. Suh behavior is

114 3. NANOMEMS PHYSICS: Quatum Wave Pheomea uraveled by solvig the iteratig eletro problem. The Hamiltoia i questio, give by (see Appedix B), H π L v g g g Nˆ g L v L, R Nˆ R q Nˆ v q qv ( b b b b ) qr q ql q b b qr qv ql, (63) must be diagoalized to determie the pertiet types of solutios holdig i oe dimesio. This Hamiltoia diagoalizatio is failitated by the proedure of bosoizatio [37]-[39] disussed i detail i Appedix B. I essee, oe-dimesioal bosoizatio trasforms a odiagoal fermioi Hamiltoia ito a diagoal bosoi oe, with the assumptio that the oedimesioal dispersio relatoship is liear, ad give by E( k) v F k k F [34]. The ature of this dispersio relatio gives rise to the trasport haraterizatio i terms of spiless left- ad right-movig eletros with respetive eletro desities N L ad N R, the parameter g, whih aptures the eletro-eletro iteratio stregth i the problem, ad the Fermi veloity v F. Kae ad Fisher [4] have aptured this pheomeology with he followig set of expressios. The Hamiltoia (63) is rewritte as [4], [4], [ N N λ ] H πv N N R L R L, (64) with v ( g g) ( g ), v λ, (65) g with λ as the iteratio stregth parameter betwee the left- ad rightmovig eletro speies, ad g, alled the Lüttiger parameter. For g the iteratio is zero, ad the Hamiltoia the aptures the behavior of a oiteratig eletro gas with veloity equal to the Fermi veloity v v F. From (65) it is see that repulsive iteratios, whih per (64) imply λ >, lead to g <, ad the opposite is true for attrative iteratios. I terms of the two-partile iteratio potetials, V ad V 4, betwee fermios movig i opposite diretios, amely, left ad right, ad either both left- or both right-movig, respetively, v ad g are give by,

115 Chapter v 4 v v v F F F V V V π π, (66) ad F F F F V V V V g v v v v 4 4 π π π π. (67) Kae ad Fisher [4] iterpret v i the ase 4 > V V, as the plasmo veloity, whih ireases above F v whe the repulsive iteratios redue the ompressibility of the eletro gas. Whe the eletro spi is iluded i the Hamiltoia, the iteratio beomes, ( ) ( ) ( ) ( ) ( ) ( ) ' ~ ~ ' ' ~ ~ ' ' ' x x x x V x x x x V σ σ σσ ρ ρ ρ ρ. (68) I this ase, the kieti eergy part of the Hamiltoia may be writte as follows [33]. ( ) ( ) ( ) ( ) ( ) q q L k k k k H s s q s F s k s k s k F s k s k F F ki ± >,,,,,,,,,,,,, v v α α ρ α ρ π, (69) where the substitutio, ( ) ± k s k s k s q,,,,, ρ, (7) represetig desity operators for spi projetios, s has bee made. The potetial eergy, i tur, otais two types of iteratio, amely, bakward satterig ad ward satterig. The bakward satterig Hamiltoia is give by, s q k k t q k p t k s k t s q p k F F g L H,,,,,,,,,,,, it_, (7) for

116 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 3 whih aptures satterig evets i whih ( k, s; k, t) ( k, s; k t) for F F F F, s t. The forward satterig Hamiltoia is give by, H it_ ( g ( q) ρ α, s ( q) ρ α, t ( q) g 4 ( q) ρα, s ( q) ρα, t ( q) ). (7) L The full bosoized Hamiltoia has bee show by Shulz [33] to take the form, H H H ki ρ H H σ it_ H g ( πa) it_ dx os ( 8φ ) σ, (73) where a is a short-distae utoff, ad for ν ρ, σ, π u ν K ν u ν ( ) H ν dx Π ν φ, (74) x ν π K ν with, g 4, ν g πv ν F g 4, ν gν u v ν F, Kν, (75) π π πv F g 4, ν gν ad g ρ g g, g σ g, g 4, σ. Shulz [33] has exposed a umber of situatios by examiig (75). For istae, he poits out that a oiteratig system, for whih uν v F ad, thus exhibits equal harge ad spi veloities, is obtaied by settig K ν. That if g, the there is o baksatterig ad (75) desribes uoupled harge ad spi desity osillatios with a dispersio relatio ω ( k) u k ad the system is odutig. The Hamiltoia (75) offers, as oe of its osequees, the possibility of omplete separatio i the dyamis of spi ad harge. I partiular, if u ρ u σ, the spi ad harge waves propagate with differet veloities. The eletro, i this ase, is said to dissolve ito two partiles, amely, a spi partile, alled a spio, ad a harge partile, alled a holo [34]. A ν ν

117 4 Chapter 3 simple piture for visualizig spi-harge separatio is show i Figure 3-. holo spio Figure 3-. Illustratio of spi harge separatio. If a photo impiges o a atiferromageti Mott isulator a removes a eletro, the disruptio left behid hages both the spi ad harge order. Eletro motio ito the vaat site results i spi ad harge separatio, givig rise to two distit partiles, amely, a holo ad a spio. (After [34].) Qualitatively, the pertiet physis of the Lüttiger liquid follow from the dispersio relatio ad may be surmised from Fig. 3- [34]. A examiatio of this figure idiates that, due to the liear dispersio relatio, hages i mometum determie eergy hages. I partiular, a mometu exitatio q imposed o the D eletro system, will ause a ompressio ad rarefatio of the eletro desity with a wavelegth π q. The degrees of ompressio ad rarefatio embody a desity wave, ad has two osequees. First, beause q determies the kieti eergy E i a uique way, the desity wave has a well-defied kieti eergy. Seod, the oomitat desity will deped o both the spi iteratio ad the Coulomb iteratio amogst eletros whih, beig futios of distae, embody the potetial eergy of the system. Therefore, the total eergy of the system may be speified by the properties of a desity wave. This desity wave, i tur, otais a spi desity ad a harge desity. This spi-harge separatio ad oexistee is the hallmark of the Lüttiger liquid. Eergy k F δ E k δ q Figure 3-. Exitatio of eletro-hole pairs i oe-dimesioal struture. After [34].

118 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 5 The behavior of the Lüttiger liquid at low eergy exitatios is aptured by the speifi heat ad the mageti suseptibility. The speifi heat is give by, γ γ v F v F u ρ u σ, (76) where γ is the speifi heat oeffiiet for oiteratig eletros at Fermi veloity v F, ad the spi suseptibility is give by, χ χ v F. The Wilso ratio is give by [33], u σ χ γ u ρ R W γ χ u u ρ σ. (77) The presetatio i this setio has exposed the fat that i oedimesioal trasport, the quasi-partiles of a Fermi liquid morph ito two ew etities, amely, spios ad holos, whih, idividually, trasport spi ad harge, respetively, ad haraterize the Lüttiger liquid. It will be see i the ext setio, that the maifestatio of spi-harge separatio is resposible for a quatitative hage i the behavior of D TLs. 3. Wave Behavior i Periodi ad Aperiodi Media The ability to reate patters of very high preisio, made available by NaoMEMS fabriatio tehology, will edow egieers with the ability to effet sigal proessig o a variety of wave pheomea, e.g., eletroi, eletromageti, aousti, et. Muh of this futioality will exploit the pheomeo of bad gaps; typially, domais of eergies or frequeies i whih wave propagatio is forbidde. I what follows, the topis of eletroi [8] ad photoi badgaps [5, 4], are addressed. 3.. Eletroi Bad-Gap Crystals 3... Carbo Naotubes Carbo aotubes (CNTs) were already itrodued i Chapter. They are a relatively ew type of material ad are osidered by may to be the

119 6 Chapter 3 quitessetial aotehology devie. Their properties are related to those of a D perodi graphite sheet, see Figs y x a a C Figure 3-3. Sketh of a graphee lattie, a sigle sheet of arbo atoms arraged i the hoeyomb struture, showig vetors utilized i desribig the lattie. I this ase, the vetor C is defied by the pair 4, m4, i.e., (4, 4). The graphee lattie is defied by a vetor C of the form C a ma, where a ad a are the uit ell base vetors of the graphee sheet, Fig. 3-3, with a a.46m. The pair of itegers (, m), where m, is used to represet a possible CNT struture [46]. Three types of CNT strutures are typially idetified aordig to how the oeptual graphee rollig ito a ylider is effeted, amely, the armhair, the zigzag, ad the hiral CNT strutures, see Fig. 3-4 [43]. The hiral agle, θ, of the wrappig vetors desribig these CNTs are related to the idies ad m by the equatio [46], θ si (78) 3m m m with θ for the Zigzag CNT, θ 3 for the Armhair CNT, ad < θ < 3 for the Chiral CNT. The orrespodig CNT diameter is give by, ( Å).783 m m d CNT. (79)

120 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 7 Carbo Naotube Axis uit (a) Carbo Naotube Axis uit (b) Carbo Naotube Axis Carbo Naotube Axis Chiral Vetor θ () Figure 3-4. Carbo aotube strutures aordig to how the grapheme sheet is wrapped. (a) Armhair. (b) Zig-zag. () Chiral. After [43]. As with ovetioal rystals, (eletro) wave propagatio is a futio of the atomi (diret lattie) periodiity ad its reiproal lattie, ad is aptured by the dispersio relatio, E vs. k. I the ase of graphee, the diret lattie is of the hoeyomb type, Fig. 3-5(a) ad applyig the tightbidig or liear ombiatio of atomi orbitals (LCAO) method [64], the graphee bad struture is obtaied as, E 3ak ( ) y ak x ak x k ± γ 3 4 os os 4 os, (8) where a is the lattie ostat, i.e., a 3a. A plot of this futio is show i Figure 3-5(b). It may be otied from this figure that at the K-

121 8 Chapter 3 poits (the orers of the first Brilloi zoe) there is zero gap betwee odutio ad valee bads i graphee. y a r a r b r b r x K r Reiproal lattie poits st Brilloui zoe poit Figure 3-5 (a) Reiproal lattie of graphee with the st Brilloi zoe (shaded). b ad b are the primitive lattie vetors. The K poit lies at the edge of the BZ. D grapheme sheets rolled aroud the y axis, will give rise to armhair CNTs. (b) LCAO badstruture of grapheme. The Fermi level lies at E. Courtesy of Prof. Christia Shöeberger, Uiversity of Basel, Switzerlad]. The effet of rollig the graphee sheet to form the CNT maifests itself i the bad struture as follows. O the oe had, the mometum of eletros alog the irumferee of the ylider beomes quatized. O the other, propagatio is ow oly possible alog the ylider axis, i.e., i oe dimesio, thus the oomitat CNT bad struture orrespods to slies of the D graphee struture. Whe the slie passes through a K-poit, the CNT is metalli sie, at these poits, the gap is zero; whe it does t, it is semiodutig. I partiular, CNT struture type ad its eletroi properties are related as follows [46]. For armhair CNTs, the irumferetial mometum vetor is quatized aordig to,

122 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 9 ν ν π k x, (8) N 3a x for ν,..., N x, where N x is the umber of uit ells spaig the irumferee. Thus, it a be show that a armhair CNT rolled suh that its irumferee lies alog k x ad the trasport logitudial axis is alog k y, would have logitudial D bad strutures at eah of the disrete values of k x give by (8), see Figure 3-8. Similarly, a zigzag CNT has its irumferetial mometum vetor quatized aordig to, k ν ν y N y π, (8) a for ν,..., N y. I this ase, the resultig CNT may be either metalli or semiodutig. Metalli, whe its idex is divisible by three, i whih ase a slie passes through a K-poit ad the tube behaves as a D metal 5 with Fermi veloity v F 8 m / s [44], ad otherwise, semiodutig. I the otext o f ballisti CNTs, their odutae is give by Ladauer s formula, G ( Ne h)t, where N, the umber of oedimesioal haels is four, due to eletro spi degeeray ad the two bads at K- ad K -poits, see Fig. 3-7(a). This works out to G ( 4e h) / 6.5kΩ, assumig T. The eergy gap of semiodutig CNTs is related to their diameter by [44], [45], 4v F.9eV E GAP. (83) 3d CNT d CNT [ m ] I the geeral ase of a hiral CNT, Dresselhaus et al. [46], [47] have show that a metalli CNT is obtaied wheever, m 3q, (84) where q is a iteger. I summary, the urret kowledge of eletroistrutural properties of SWNTs is as follows [46]: all armhair tubes are expeted to be metalli, oe-third of zigzag ad hiral tubes are expeted to be metalli, ad the rest are expeted to be semiodutig [46].

123 Chapter 3 K M K Γ k y (a) K k x M Γ K k x k y (b) Figure 3-6 (a) A D bad struture lies at eah of the disrete values of k x for a (5, 5) armhair CNT, ditated by the irumferetial quatizatio i this diretio. The armhair CNT is metalli. (b) A D bad struture lies at eah of the disrete values of k y for a (9, ) zigzag CNT, ditated by the irumferetial quatizatio i this diretio. The zigzag CNT is semiodutig. (After [46].) It may be surmised from the slie passig through the K, K poits, see Fig. 3.7, that eah hael is four-fold degeerate, o aout of spi degeeray ad the sublattie degeeray of eletros i graphee [44]. E(k) K K E F k Figure 3-7. Eergy bad diagram of metalli CNT for slie through Fermi poits K, K.

124 3. NANOMEMS PHYSICS: Quatum Wave Pheomea This fat has bee utilized by Burke [48] to propose a AC iruit model for CNTs, iludig eletro-eletro iteratio, see Fig Spi L K Spi L K C Q Spi L K C Q Spi L K C Q C Q C ES C ES C ES C ES Figure 3-8. AC iruit model for iteratig eletros i CNT. The four-fold degeeray is aptured by four haels. (After [48].) The iruit model is iterpreted by Burke [48] as follows. The iruit aptures the existee of four modes, amely, three spi modes, whih orrespods to a differetial exitatio, ad oe harge mode, whih orrespods to ommo mode exitatio. I the latter ase (harge mode), all four trasmissio lies appear i parallel, ad they are haraterized by a effetive lie possessig a harge-mode propagatio veloity ad harateristi impedae give by [48], v p 4 4C Q v F v F, (85) L K C Q C ES C ES g ad, 4L L h, (86) g e K K Z,CM C ES C Q where, L K h e v F (h is Plak s ostat) is the kieti idutae per C Q e hv is the quatum apaitae,ad uit legth, F ( h d) C ES πε osh (h here is the CNT-to-groud distae) is the eletrostati apaitae (the CNT-to-groud apaitae). Typial values for these parameters are: L K 6H / µ m, C ES 5aF / µ m, ad C Q af / µ m. The harateristi impedae for the three spi modes is give by,

125 Chapter 3 L h, (87) e K Z,DM C Q whih Burke iterprets as defiig the ratio of exitatio voltage to eliited urret whe the spi wave is exited. With diameters of the order of approximately m, CNTs are ideal systems where the harateristis of Lüttiger liquids, amely, strog eletro-eletro iteratio ad spi-harge separatio, should be maifest. Aordigly, efforts have bee expeded to develop ways of haraterizig ad asertaiig suh behavior. Notieable amog these, is experimetal work by Bokrath et al. [49] who dedued, from the measured 3D-D α tuelig odutae di dv V, CNT Lüttiger parameters g with values betwee. ad.3. These were extrated from ompariso of measuremet to the theoretial relatios α ( g ) / 4 ( g g ) / 8 Ed or α Bulk, for 3D-D otats loated at the ed or at the bulk, respetively, of the CNT [5], see Fig Bulk Cotat Ed Cotat e - e - L>>L E Luttiger Liquid Figure D-D otat to arbo aotube. (After [5].) Similarly, efforts have bee expeded, ad are beig vigorously pursued, to uover the predited spi-harge separatio. These ilude proposals to diretly exite Lüttiger liquid behavior i CNTs by impressig mirowave voltage waves i CNTs atig as trasmissio lies [49] Superodutors The pheomeo of superodutivity maifests itself as the drop i the eletrial resistae of metals ad alloys at suffiietly low temperatures, aompaied by the ihibitio of mageti fields from peetratig iside of them [8]. Coversely, a material i the superodutig state loses this property whe its temperature is raised past a ritial temperature, T, or it

126 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 3 is exposed to a high ritial mageti field H [8]. We disuss the priiples of superodutivity here, maily beause of the importae of superodutors (materials exhibitig superodutivity) as a alterative meas of implemetig quatum bits (qubits). Our poit of departure i disussig superodutivity is the oept of superfluidity, from whih it may be uderstood i a ituitive fashio Superfluidity Superfluidity refers to the property exhibited by a superfluid, i.e., a liquid that flows without fritio. A suessful explaatio of superfluidity was put forth by Ladau [53], [54]. Ladau s reasoig was as follows [3]. If oe assumes that the Bose quatum fluid of mass M is i its groud state at absolute zero, ad flowig withi a apillary tube with veloity v, ad eergy Mv, the, i a oordiate system ahored i the fluid, the fluid would be at rest ad the apillary would appear to be movig at a veloity v. If fritio emerges betwee the apillary ad the fluid, the the part of the latter i otat with the tube would o loger be at rest, but would begi to be arried alog by the apillary wall. However, sie this part of the fluid would o loger be at rest, the at of it beig arried alog by the tube wall must idue exitatios from its groud state. These exitatios, i tur, would maifest as hages i its eergy ad mometum, E ad p, so that the fluid s total eergy would ow be E p v Mv. Upo exitatio, the fluid itself would lose eergy. Therefore, eergy hage must be egative, i.e., E p v <. (88) Sie the fluid is a quatum system of Bose partiles, its eergy is quatized ad must hage disretely. The smallest eergy exitatio, therefore, is that for whih E p v is a miimum, whih ours whe p ad v are opposite. This meas that oe must have, E pv < or E v >. (89) p This equatio sets the miimum veloity at whih exitatios would begi, as the ritial veloity,

127 4 Chapter 3 E v > mi. (9) p I partiular, if v, the it is possible for the fluid to flow free of exitatios, i.e., without fritio/dissipatio, as log as v < v. This is the so-alled Ladau s riterio for superfluidity. This oditio is maitaied as log as v is less tha the speed of soud. This isight, led Ladau to propose that the low eergy exitatios of the superfluid groud state should osist of two types of partiles, amely, phoos ad rotos. Phoos beig quatized soud waves, with a eergy dispersio, E Sp, (9) where S is the speed of soud ad p the mometum, ad rotos beig quatized rotatioal motio (vorties), with a eergy dispersio, E ( p p ) m eff. (9) At temperatures above absolute zero, the fluid will be exited by thermal eergy. Therefore, it will be possible for some of the thermally exited fluid partiles to ahieve veloities greater tha v ad will, osequetly, experiee fritio. Uder these irumstaes, the fluid will be omposed of these ormal partiles ad superfluid partiles, resultig i a mass urret give by, j ρ v ρ v, (93) where ρ s ad ρ ad with veloity s s v are the mass desity ad veloity of the ormal fluid, ad v those of the superfluid. If oe assumes that the whole fluid flows s v ( ρ ρ ) v ρv v v, the the total mass urret may be writte as, s j. (94) s Oe of the fudametal properties of a superfluid derives from the fat that, sie it possesses o exited partiles, its mometum does t hage ad, osequetly, it a t exert a fore o a body immersed i it. Flow with this property, deoted potetial flow, is mathematially haraterized by the equatio,

128 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 5 v s. (95) Eq.(95) sigifies that a superfluid is irrotatioal, i.e., it exhibits o vortiity. The quitessetial example of a superfluid is embodied by a Bose liquid, whih osists of atoms of itegral-value spis, i partiular, liquid helium (He 4 ), whih does ot solidify at absolute zero ad flows through apillaries without dissipatio. Ladau s argumets, preseted above, while suessfully explaiig liquid helium behavior, were of a ituitive ad pheomeologial ature. Elemets for a first-priiples theory to explai superfluid behavior bega takig shape with observatios by Fritz Lodo [55], to the effet that the ostitutio of He atoms, whih are omposed of a eve umber of elemetary partiles ( protos, eutros, ad eletros) suggested that they should be desribed by a symmetri wavefutio ad, osequetly, should obey Bose statistis, together with the earlier observatio by Eistei that, at appropriately low temperatures ad mass ad desity oditios, a gas of o-iteratig Bose partiles odeses with the remarkable property that a ozero fratio of the odesed atoms oupies a sigle oe-partile state. Suh a state, i partiular, is a oheret state ad has ome to be kow as a Bose-Eistei odesate (BEC) [55]. A fudametal theory apturig this behavior is the Gross-Pitaevskii (GP) model. The GP equatio models the geeral Bose gas by the equatio [78], ψ i ψ U mf ψ, (96) t m where m is partile mass, U mf e ( x ) ψ d x x x, (97) is the mea field for Coulomb iteratio betwee atoms, ad may be expressed as, U mf ( x x ) ψ( x ) dx gδ( x x ) ψ( x ) dx g ψ( x ) V. (98) Substitutig (98) ito (96) oe obtais a oliear Shrödiger equatio, ψ i t m ψ g ψ m * ( x) ψ( x) ψ gψ ψ. (99)

129 6 Chapter 3 Codesatio is aptured whe i (99) oe imposes the oditios for obtaiig the lowest possible state, ψ, amely, that the wave futio be homogeeous, i.e., ψ. () This leads to the relatio, N, () V ψ where N is the umber of atoms ad V is the volume. I tur, substitutio of () ad () ito ( 99) leads to a simplified equatio of motio, amely, ψ i g, () t ψ with a solutio of the form, ψ Ce. (3) gt The dispersio relatio for low-level exitatios are obtaied by liearizig (99), i partiular, writig ψ ψ χ, where χ << ψ, ad substitutig ito (99), oe obtais, i χ t m * χ gψ χ gψ χ. (4) Sie this equatio otais the two ukows χ ad seod equatio by takig its omplex ojugate, * χ, we geerate a * χ * * - i χ gψ χ gψ χ. (5) t m The, postulatig solutios of the form, χ iet ipx ~ ξe, (6)

130 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 7 ad χ * ~ ηe iet ipx, (7) substitutig them ito (4) ad (5), together with (), ad solvig the resultig system of equatios for E, oe obtais the result, 4 p E ( p) S p. (8) 4m This is the dispersio relatio of a superfluid. Expressig the fluid veloity i terms of it, we obtai, v E p p 4m S. (9) This equatio has a positive miimum, ourrig at p, ad give by the ostat veloity S. Sie this veloity is idepedet of mometum, E(p) must otai a eergy gap. A eergy gap i its spetrum, thus, is aother maifestatio of superfluidi behavior. The zero-vortiity property of a superfluid is derived from first priiples as follows. From (3) it may be see that the wave futio for the Bose odesate i its lowest eergy state is a oe-partile omplex wave. Geeralizig this expressio to, i ( x ) ( ) ψ e χ ψ x, () oe a express the mass desity as are related, as usual, by, ρ m ψ, where ( x) ψ ad the urret i j * * ( ψ ψ ψ ψ ). () It the follows that, isertig () ito () oe obtais, j ψ χ ρ χ, () m whih, upo ompariso with (94) yields,

131 8 Chapter 3 v χ, m (3) that is, the veloity is related to the phase, χ, of the wave futio, so oe a rewrite (3) as, v φ, (4) whih learly expresses that the flow is a potetial flow, sie the url of ay gradiet is zero, ad the potetial is give by, φ m χ. (5) A further pheomeo aomplayig superfluidity, ad eluidated by first-priiples osideratios, pertais to the dyamis of superfluids whe plaed i a rotatig otaier. I partiular, it is experimetally foud, Fig. 3-9, i a vessel otaiig a mixture of ormal ad superfluid ompoets, ad rotatig at a agular veloity Ω, that the dyami behavior of the two ompoets is quite differet. O the oe had, as is expeted from lassial hydrodyamis, the ormal ompoet rotates with the vessel (i.e., it is arried alog with the vessel due to fritio), so that it aquires a eddy urret v Ω r, ad this veloity, i tur, gives rise to a aompayig vortex, sie v Ω, see Fig The superfluid ompoet, o the other had, beomes populated by a distributio of vorties. This appearae of vorties i the superfluid ompoet would appear to otradit the fudametal oditio for superfluidity of zero vortiity, see Eq.(95). The lue to this behavior was to be foud i the reogitio that potetial flow, haraterized by (95), may also be obtaied wheever the equivalet form, based o Stokes theorem, v s dr, (6) is satisfied. I partiular, if the potetial of the rotatig fluid is proportioal to the agle, see Fig. 3-, so that oe has, φ Γ π α, (7)

132 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 9 z Ω Fluid (a) r α v Ω α r (b) Figure 3-. (a) Normal fluid i rotatig vessel aquires meisus with shape depedig oly o agular veloity Ω. Top view of fluid-otaiig vessel rotatig with agular veloity Ω. The ormal fluid aquires a eddy urret with veloity v α. the the fluid veloity may be alulated as, Γ v () r φ, (8) α r α πr ad, sie the veloity deays with distae, this is the profile of a vortex. Now, alulatio of the irulatio of this vortex gives, α v dl φ Γ. (9) Examiatio of Eq. (9) reveals that if the irulatio (potetial hage) is zero, oe still has the oflit betwee the mathematial violatio of vortiity ad the experimetal observatio of vorties. However, if the agle α is ot uiquely defied, exept up to modulus π, the it would be possible to reoile the two if the potetial φ were ot sigle-valued. This, i tur, would be the ase if the phase of the wavefutio was ot uique, but also defied modulo π, so that χ πn. I this ase, the irulatio (9) would be expressed as,

133 Chapter 3 Γ φ π m N, () that is, it would be quatized. Thus, a hage i potetial of π m would brig it to the poit of departure, due to its o-sigle-valuedess, yet would allow a o-zero vortiity due to its fiiteess. The quatum ature of a superfluid otaied i a rotatig vessel maifests, therefore, i that its irulatio beomes quatized. Oe remarkable aspet of a rotatig vessel otaiig a superfluid pertais to the shape of its meisus. I partiular, from the fat that a ormal fluid i a vessel of area A rotatig at a agular veloity Ω has a irulatio Ω A, ad that a superfluid o the same vessel would have a irulatio Γ νa, where ν is the desity of vorties per uit area, oe fids, equatig irulatios, that the Ω Γν. This sigifies, that although the superfluid would ot eessarily be rotatig, due to the appearae of vorties, the shape of its meisus will be the same as that of a ormal fluid rotatig at a agular veloity Ω. I other words, oe a simulate the effet of rotatio o a ormal fluid by a populatio of vorties. The fat that the irulatio of a superfluid otaied i a rotatig vessel is quatized meas that the vessel must reah a ertai miimum agular veloity, the ritial agular veloity, Ω, ad rotatioal eergy before the vorties begi to be reated. From the ratio of vortex eergy to vortex agular mometum it a be show that, Ω, () mr where R is the vessel radius. Figure 3- shows a piture of vorties i a superfluid. Figure 3-. Observatio of vortex latties. The examples show otais approximately 8, vorties. The vorties have rystallized i a triagular patter. Reprited with permissio from [56]. Copyright AAAS.

134 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 3... Superodutivity Our uderstadig of superfluidity, gaied i the previous setio, failitates that of superodutivity. Superodutivity, the absee of eletrial resistae to eletro trasport, may be oeptually visualized as the superfluidity of eletros. A qualitative aalogy betwee these two pheomea may be summarized as follows. Whereas a superfluid embodies a boso odesate of, e.g., helium atoms, a superodutor, o the other had, embodies boso odesates of, e.g., boud eletro pairs. Eletros, as is kow, due to the Coulomb fore of repulsio betwee them, do ot, stritly speakig, odese. However, uder ertai irumstaes, a effetive bidig fore may be preset that overomes the fore of repulsio betwee eletro pairs ad turs these pairs, effetively, ito bosos. These eletro pairs, whih behave as bosos, are alled Cooper pairs ad have zero spi (just as the helium atoms). Thus, while a boso odesate of helium atoms may behave as a superfluid, uder appropriate irumstaes, ad whe it does so it exhibits trasport without fritio, so too a odesate of a aggregate of Cooper pairs, behaves as a superodutor. Cotiuig with the aalogy, while superfluid trasport exists for veloities less tha a ~ mi E p, so too superodutive trasport exists ritial veloity, v ( ) below a ritial veloity ~ ( ) v p, where i this ase is the bidig eergy of a Cooper pair. Fially, while dissipatio ad fluid vorties (rotos) appear above v i the superfluid, so too ohmi dissipatio ad so-alled vortex states, i.e., irulatio of superodutig urrets i vorties throughout the system, appear beyod v i the superodutor. With these qualitative prelimiaries, we ext address the saliet aspets of superodutivity, amely, the riterio for superodutivity i light of its oeptual relatioship to superfluidity, the bidig eergy of Cooper pairs, the ihibitio of a mageti field iside superodutig materials, the oditios for the extitio of superodutivity. I aalogy with (5), the equatio for a sigle eletro movig i a superodutor may be writte as, ( x, t ) ψ σ * ψ σ gψ σ i ψ σ ψ σ, () t m where g represets harge, σ or represets the spi state, ad ψ * σψ σ is a -idex summatio that embodies the desity from all spis. I this otext, the wave futio of a pair of eletros is a produt give by,

135 Chapter 3 σσ ( x x ) ψ ( x ) ψ ( ) Ψ. 3), σ σ x Ψ σσ x, x must satisfy Pauli s exlusio priiple, wheraby it must be ati-symmetri. Furthermore, sie spis ad spatial oordiates operate i differet (tesor) spaes, the wave futio must be a produt of a spi-depedet fator, ad a oordiatedepedet fator, i.e., Beig the wave futio of a boso, ( ) Ψ, σσ Eσσ, (4) g ( x x ) E f ( x x ) σσ, where E σσ is the ati-symmetri spi-depedet fator. With this defiitio, oe a rewrite (4) as, ( x,t) * ψ σ i ψ σ E σσψ σ. (5) t m Followig the same proedure as i the previous setio, the dispersio relatio is obtaied from the set of equatios, ( x,t) * ψ σ i ψ σ E F ψ σ E σσψ σ, (6a) t m ad ψ ( x, t ) * σ * - i ψ σ E F t m ψ * σ * E σ σ ψ σ, (6b) where the eergy is ow referred to the Fermi eergy. The, postulatig solutios of the form, ad ψ ψ iet/ ipx/ σ ~ η σ e, (7a) * σ ~ ζ σ e iet/ ipx/, (7b) it a be show, upo substitutio o (6), that the set of equatios,

136 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 3 E η σ p m E F η σ E σ σ ζ σ, (8a) ad - E ζ σ p m E F ζ σ * E σ σ η σ, (8b) is obtaied. Solvig (8) for E oe obtais ( ) E ± v p p. This is the dispersio relatio for superodutig eletros. It represets a parabola with a miimum at p p F, orrespodig eergy, ad eergy gap. Therefore, appliatio of the Ladau riterio for superfluidity, to the preset ase of superodutivity, yields the ritial veloity, v ( p F ), below whih eletro trasport experiees o eletrial resistae, i.e., is superodutive. Next, we address the formatio of Cooper pairs. I exploitig the superfluid physis aalogy to desribe superodutivity, oe must ofrot the issue of explaiig how eletros, whih would ordiarily be preluded from bidig, due to Coulomb s repulsio fore, would bod/odese to form bosos. The lue to this possibility was advaed by the disovery that [57], [58] i superodutig elemets, the produt of the square root of their isotopi mass ad the ritial temperature, M / T, is a ostat. This experimetal fat, i tur, was iterpreted by Fröhlih [54] to mea that the properties of the zero-poit or thermal lattie phoos, were ivolved i superodutivity ad, i partiular, that eletros residig withi the rystal lattie were apable, via iteratios mediated by these phoos, of attratig oe aother. This pheomeo is demostrated ext. To determie the ature of the phoo-mediated eletro-eletro iteratio, we assume the oexistee of phoos ad eletros is desribed by a Hamiltoia osistig of three terms, amely, the eergy of the eletros, the eergy of the phoos, ad the eergy of iteratio betwee eletros ad phoos, respetively. The first two terms are aptured by the uperturbed Hamiltoia: H E ω k, σ k, σ k, σ qa qa q. (9) k, σ q F F

137 4 Chapter 3 The third term is the familiar eletro-phoo iteratio [59], i whih a aousti phoo distorts the lattie ad, as a osequee, produes a gratig i the bad edges whih, i tur, auses eletros to satter off of it. This iteratio is aptured by the iteratio potetial for aousti phoos give by, U AP, (3) ( r, t) D u( r, t) where D is the deformatio potetial ad, u ( r,t) [ i q r ( q) t ] * i[ r ( q ) t] ( a e a e q ω ω ), (3) ρvω ( q) q the lattie displaemet. The pertiet eergy of iteratio is, H where, Ψ ep d r Ψ () r U () r Ψ () r AP ik r () r e φ () r k k k q, (3), (33) is the uperturbed oe-eletro Blok state. With these defiitios, the firstorder eletro-phoo iteratio may be writte as, H ' id k, σq * ( a a ) k q, σ k, σ q q. (34) The Hamiltoia desribig the eletro-phoo system, the, is give by, H k, σ * ( a ) E ωaa id a k, σ k, σ k, σ q q q k q k q q. (35) q kq Now, to determie the ature of the eletro-eletro iteratio, we have to trasform (35) ito a Hamiltoia that does ot otai the O(D) term, i.e., i whih the phoo oordiates are elimiated ad oly eletro-eletro iteratio terms are preset. This is aomplished by trasformig (35) ito a ew Hamiltoia give by H ~ e S He S, ad so hoosig S that H ~

138 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 5 otais o off-diagoal terms of O(D). I partiular, if we take ' H H H, where H gives rise to the solutios whe H ', so that H E, the H ~ may be expaded as, [3], H ~ e S H H ' S S ' ( H H ) e S... ( H H ) [ H,S] [ H,S ],S] ' H... ' [,S,S] [ H,S]... ' [ H,S] H [ H ] S S.... (36) ' If we selet [ H,S] H, the the seod ad third terms i (36) vaish ad we have a presriptio for S, amely, ' ' H ' ' ' ' ( E ' E ) S S, (37) E ' E H whih yields the desired Hamiltoia as, ' [ H,S] O( ) H ~ H S. (38) Now, i this diagoal formulatio, effetive eletro-eletro iteratio is ' eluidated by osiderig the ase i whih the perturbatio H auses the followig trasitios: Either the eletro i state k emits a phoo q ad this is absorbed by the eletro i state k, or the eletro i state k emits a phoo q ad this is absorbed by the eletro i state k. These trasitios may be mathematially represeted as ourrig from a iitial state i to a fial state f via a virtual state m, i terms of whih the expetatio value of the ommutator i (38) may be expressed as, ' ' ' [ S, H ] i ( f S m m H i f H m m S i ) f. (39) m Followig [54], osideratio of the phoo system at absolute zero, so that oe of the phoo states refers to the vauum, the matrix elemet alulatio (34) over the phoo operators yields, without loss of geerality,

139 6 Chapter 3 ω k q q k k k q k q E E D i S, (4a) ad ω ' k q k ' q k ' k ' q k ' q E E D i S. (4b) Substitutig (4) ito (39) oe obtais, [ ] ω ω kk' q q k' q k' q q k k k' q k' k' q k' ' E E E E D i S,H f. (4) Realizig that, due to eergy oservatio, k q k q k' ' k E E E E, (4) may be simplified to yield, ( ) ω ω q ' kk q q k k k ' q k k ' q k ' q '' E E D H. (43) Equatio (4) reveals that i irumstaes whe ( ) q q k E k E ω <, this term is egative, thus embodyig a eletro-eletro iteratio that is attrative, ad that gives rise to the bosoi behavior metioed previously. Havig show that it is physially possible for a pair of eletros to attrat oe aother i the presee of a phoo, the ext questio before us is to determie the bidig eergy of the pair. As usual, this is obtaied from the eergy eigevalues of Shrödiger equatio, ψ ψ E H. Towards this ed, we begi by expressig the Hamiltoia, ( ) r r V p p H m m, (44) where the potetial ( ) r r V models the iteratio (43), i the eterof-mass ad relative-motio oordiates, i.e., ( ) r V p 4 P H m m, (45)

140 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 7 with ( ) r r R, r r r, p p P, ad / ) p (p p. The, expressig the solutio as, ψ k r ik k ik R e e h, (46) ad takig ito osideratio the symmetry properties of the problem, i partiular, upo iterhage of r ad r, R R, r r, ad k k h h, ad i the frame of referee i whih the system is at rest K, substitutio of (46) ito the Shrödiger equatio, ψ ψ E H, yields, ( ) ( ) ( ) Ω Ω ' ' ' ' ' ' ' ' ' k k kk k k r ik r ik k k r ik k k k r ik k r ik k V e r V e k E e E e r V e k h h dr h m h h m h. (47) Sie the eletro-eletro iteratio is mediated by phoos, ad the phoo eergies lie betwee ad D ω, where D ω is the Debye eergy, the eletros will be uder the ifluee of the bidig potetial as log as the their exitatio eergy of the pair is lower tha the Debye eergy, i.e., D k k ' ω < ε ε, m k k ε. I this otext, we have, V V ' k k (48) ad we a write (47) as, ' k k k ' ' V k E h h m, (49) whih, may be expressed as, ' k ' k k ' k k ' ' ' k E V m h h, (5) whih may be fatored as,

141 8 Chapter 3 ' ' V ( h ) k k ' k k E m, (5) from where we get, V k E m ' ' k. (5) Replaig summatio by itegratio we obtai, V EF ωd E ω EF F D dε dεn () ε VN( ) ε E, (53) ε E EF where N() is the desity of eletroi states for a sigle spi populatio i the ormal metal [64]. Upo arryig out the itegratio we get, E VN() l ωd E E E F F whih may be solved by the eergy of the pair, E F E e ω VN() D, (54). (55) Clearly, (55) deotes a system eergy that is below the Fermi eergy, therefore, we have a boud state. Observig that the redued mass m ad the eletro mass m are related by m m, effetig the orrespodig substitutio k m k m εk, ad repeatig the operatios of (53)-(54) oe obtais the result, E E F ωd. (56) VN() e

142 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 9 The zero-temperature bidig eergy (gap) is give by, D e ω VN() E b. (57) The bidig eergy (57) determies how far apart the eletros formig a Cooper pair may separate while still atig as boud. I this otext, the radius R of a Cooper pair has bee estimated as [6], R ~ k me F b, (58) whih, umerially, is of the order of µ m. The impliatio of the bidig eergy is as follows. At absolute zero, a eergy greater tha the bidig eergy is required to separate Cooper pairs ad, thus, reate exited eletros whih are geerated i pairs. At eergies lose to this threshold, E, the urret will osist of both Cooper pairs ad sigle (ormal) eletros resultig from the breakig of the pairs, givig rise to a two-fluid model trasport. Abrikosov has show that as the temperature ireases E dereases util it reahes zero a the ritial temperature, T. This is temperature depedee is give by, b b E b ( T T) 3.6 T. (59) Next, we osider the pheomeo of mageti field exlusio from a superodutor. We examie the superurret i a superodutor otaiig a desity of s eletros movig with veloity v s ad, thus, give by J e v, i the presee of a vetor potetial field A '. I geeral, the s s s partile veloity i a vetor potetial is give by, q ' v p A. (6) M I the ase of the superodutor, M m, ad q e. If we let Ψ be the wavefutio of the eletro pair (boso), the we a express (6) as, e

143 3 Chapter 3 Ψ v M m e * q iψ Ψ Ψ * e iψ Ψ Ψ ' A. (6) ' A iχ Now, writig the omplex wave futio as Ψ Ψ e, where χ is a spae-depedet phase, ad substitutig ito (6) we obtai, v s m ' χ A. (6) e e m e This equatio reveals that, eve if χ, urret flow may be exited by ' the vetor potetial. I fat, sie B A, we may redefie A to ilude the phase, without hagig B, i.e., ' A A χ, (63) e from where we get, v s e m ' A. (64) e The superuret, the, is give by, J s e s ' A. (65) m e The effets of a superodutor o a mageti field iside its bulk follow from from substitutig (64) ito the equatio (65), 4π B J s, (66) ad takig its url, i.e., 4π 4πe s B J s B. (67) m e

144 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 3 Sie B, (67) beomes, 4πe s B m e B, (68) whih may be rewritte as, B δ L B, (69) with the Lodo peetratio depth give by, δ L m e 4πe s ω p, (7) where ω p is the plasma frequey i the material. Take alog oe diretio, say z, (7) beomes, d Bx Bz, (7) dz δ L B x is the mageti field at the surfae of the superodutor. The solutio stipulates that the mageti field deays iside the superodutor with a harateristi legth δ L. Assumig a plasma frequey of 5 / s, the approximate value of the Lodo peetratio depth is 3Å. This meas that at distaes greater tha ~3Å from the surfae, the mageti field ad, per (65), the urret, vaish iside a superodutor, see Fig. 3-. where ( ) Vauum B x x Superodutor B x / L () z B () e z δ x z δ L Figure 3-. Deayig mageti field i superodutor.

145 3 Chapter 3 The vaishig of the mageti field iside a superodutor is alled the Meisser effet, ad has ertai pratial osequees. For istae, if a superodutig wire is tured ito a rig, the the fat that its bulk mageti field ad urret are zero implies that, e e ' ' χ A χ χ χ A dl dl. (7) C Therefore, the Cooper pair wave futio may be writte as, C Ψ A Ψ e iχ e e i φ i A dl Ψ e Ψ e, (73) where φ is the mageti flux iside the hollow part of the rig. Sie the phase must equal a iteger multiple of π, however, we have, e φ π, (74) or, φ π e πh eπ φ. (75) Thus, the mageti flux ofied by the superodutig rig is quatized i uits of flux φ h e, alled a fluxoid. The phase of the Cooper pair wave futio ad the fluxoid are at the heart of two effets of fudametal import for appliatios, amely, the Josephso effet ad the oliear Josephso idutae. The Josephso effet refers to the fat that, wheever two superodutors at the same temperature are brought i proximity to oe aother, separated by a thi isulatig layer (so thi that tuelig of Cooper pairs may our), Fig. 3-3, a superurret I J flows, whih depeds o the V I J ψ,χ ψ,χ S I S Figure 3-3. Shemati of Josephso jutio.

146 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 33 phase differee δ χ χ of the respetive wave futios i the superodutors. Sie the veloity of a Cooper pair is proportioal to the phase gradiet of its wave futio, i.e., v ~ χ, ad sie the phase has a period of π, it is ot diffiult to aept that the superurret be periodi. Ideed, it a be show [8] that the Josephso jutio urret is give by, I where, J I si δ, (76) φ dδ V, (77) π dt is the voltage aross the jutio. The Josephso idutae, i tur, derives from substitutig (76) ad (77) i the defiitio of idutae voltage, amely, V di J L. (78) J dt Thus, di dt J dδ π I os δ I os δ V, (79) dt φ ad, from (78) we obtai, L J φ. (79) πi os δ Clearly, the deomiator, os δ makes the idutae oliear, beomig large as δ π, ad egative i the rage π < δ < 3π. The oliearity of the Josephso idutae gives rise to the formatio of the Josephso qubit, whih is a oliear LC resoator osistig of the Josephso jutio s idutae, L J, ad apaitae. To olude our expositio o superodutivity, we poit out that there are two types of superodutors aordig to how the Meisser effet maifests i them [8]. I partiular, type I superodutors are haraterized by a magetizatio versus applied mageti field urve that ireases up to a ritial field, H, where it drops to zero ad, ourretly,

147 34 Chapter 3 the superodutig state disappears (it beomes ormal). Type II superodutors, o the other had, are haraterized by two ritial fields, amely, a lower ritial field H, below whih the superodutig state exists exlusively, ad above whih the superodutor is threaded by flux lies that give rise to a lattie of vorties, ad a upper ritial field H, beyod whih superodutivity disappears. The vorties are irulatig superodutig urrets aroud ormal regios, ad are suh that the oset of a vortex ours whe the orrespodig flux is that of a sigle fluxoid. Quatitatively, H φ, (8) πδ L ad H φ, (8) πξ where δ L is the mageti field peetratio depth, ad ξ v F [8] is the oheree legth, whih aptures the lattie ostat of vortex lattie. 3.. Photoi Bad-Gap Crystals Cotiuig with the topi of wave pheomea i periodi strutures, we ow briefly take o the subjet of eletromageti wave propagatio ad maipulatio i periodi dieletri strutures or photoi bad-gap rystals (PBCs) [5]. PBCs are -, -, or 3-dimesioally periodially pattered materials whose dispersio relatio, i.e., propagatio ostat versus frequey respose, exhibits rages i whih wave propagatio is forbidde (bad gaps) ad rages i whih it is allowed Oe-dimesioal PBC Physis The fudametal physis of a PBC are easily grasped from osideratios of a -D PBC, whih is of fiite extet ad osists of alteratig regios of dieletri ostat, ε ad ε, respetively, see Fig. 3.4.

148 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 35 d d d d d d d ε ε d... Iomig Wave ε ε ε ε ε ε Figure 3-4 Sketh of oe-dimesioal PBC. ε Exitig Wave Fousig o a uit ell, see Fig. 3-5, we otie that if a wave impiges from the left o this uit ell, it will i geeral, udergo multiple refletios ad trasmissios at two plaes, amely, t, r at the first ε / ε, disotiuity, ad r, t at the ε / ε disotiuity. e ik z r ik z Total e d d ε ε Uit Cell (a) d ε ik z t Total e t Total ε ε ε d... ε z ik d ' tt e (b) Figure 3-5 (a) PBC uit ell. (b) Trasmissio/refletio aalysis. wave vetor i regio i. tt e ik ' d r e ik ' d tt e ik ' d r e ik ' d... t Total k µε 4 i ω i is the The, the amplitude of the trasmitted wave will be give by the sum of the followig terms [58]: () The fratio that is trasmitted through the ε / ε iterfae, is phase-shifted while traversig (left-to-right) the regio ε of legth d, ad the is trasmitted through the ε / ε disotiuity, amely, ik d ' te t. This is the amplitude for diret trasmissio through two disotiuities.

149 36 Chapter 3 () The fratio that is refleted from ε / ε, is phase-shifted while traversig (right-to-left) ε of legth d, ad the is refleted agai at ε / ε, phase-shifted left-to-right the regio ε of legth d, ad so o. This is the amplitude for trasmissio after two refletios, ad so o. The frequey seletivity origiates as follows [58]. At frequeies where kd is a eve multiple of π, we have, 4 ( r' r'...) π t Total k d eve umber tt', (8) that is, every term iside the parethesis is exatly i phase ad there is ostrutive iterferee; this results i maximum trasmissio. O the other had, if kd is a odd multiple of π, we have, 4 ( r' r'...) π t Total k d odd umber tt', (83) that is, every term iside the parethesis alterates i sig ad there is destrutive iterferee, whih results i miimum trasmissio. With i Z i µ ε i represetig the harateristi impedae of regio ε i, we obtai the omplex refletio ad trasmissio oeffiiets as follows, r' Z Z ε ε, (84) Z Z ε ε t ' Z ε Z Z ε ε. (85) The real refletio ad trasmissio oeffiiets are give by, R r', (86) ad

150 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 37 Z T t' Z. (87) The overall trasmissio oeffiiet for the ε / ε / ε of Fig. 3-5(b) is give by, T Total T ttotal R Rosk d. (88) This expressio a be used to ompute the trasmissio oeffiiet of the uit ell, whih iludes fiite ε regios of legth d /, by replaig kd ( kd kd ). Figure 3-6 shows a plot of the trasmissio oeffiiet of suh a uit ell, Eq.(88). Trasmissio Coeffiiet U it C ell Phase Shift (k d k d )/(π /) Figure 3-6. Trasmissio oeffiiet versus phase shift for uit ell for PBC i Fig Parameters: d.6i, d.4i, ε, ε 8.9. At odd multiples of π / oe fids valleys, whereas at eve multiples of π / oe fids peaks of the trasmissio oeffiiet. The destrutive iterferee, of a sigle uit ell i this example, is resposible for a valley trasmissio amplitude of oly ~.36. As the umber of oseutive uit ells, N, makig up the rystal ireases, the umulative effet of the uit ell s atteuatio drives the overall rystal atteuatio from ~.36, for a sigle uit ell, to arbitrarily low values, depedig o N. [6]. Whe multiple layers of uit ells are asaded, the total trasmissio is drastially redued ad a photoi badgap is formed at the frequey i questio. The -D PBC, beig most ofte foud i its embodimet as a multilayer film i dieletri mirrors ad i optial filters, is already a extesively

151 38 Chapter 3 studied struture. From these appliatios it is kow that PBCs a at as perfet mirrors for light whose frequey lies withi a well-defied rage, amely, whe kd (where d is the lattie ostat) is a odd multiple of π /, ad that they may loalize modes whe edowed with defets [6]. The appliatio of PBCs i the otext of routig ad otrollig the propagatio of light waves, for example, requires their realizatio i, at least, -D. Next, we deal with multi-dimesioal PBCs Multi-dimesioal PBC Physis The properties of - ad 3-D PBCs may be formulated i terms of the oheret satterig properties of - ad 3-D latties [64]. Fig. 3-7(a) typifies a -D triagular-lattie PBC osistig of yliders of dieletri ostat ε embedded i a host of dieletri ostat ε. Top View a Host ε Lattie ε (a) aosθ θ θ' a k k aosθ' (b) Figure 3-7. (a) Sketh of -D PBC with lattie ostats a ad a osistig of yliders of dieletri ostat ε embedded i a host of dieletri ostat ε. (b) Detail of a iomig wave with wave vetor k impigig o two objets separated a distae a, ad sattered alog wave vetor k '.

152 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 39 Speifially, the properties of these strutures are give by argumets advaed by Bragg [64], whose essee (for a -D periodi lattie) is that the path differee (phase shift) betwee iomig ad sattered rays, a ( k k ), see Fig. 3-7(b), determies whether the trasmissio of the struture exhibits a maximum or a miimum; a maximum whe is a iteger multiple of π, ad a miimum whe it is a odd multiple of π. For a 3-D PBC, o the other had, ( k k ) R must be valid simultaeously for all vetors R that are Bravais lattie vetors [64]. A large umber of omputatioal tehiques to obtai the properties of geeral PBCs have bee developed, most of whih derive from the solid state physis literature o omputig bad strutures [6]-[66]. Obviously, it would be impossible to egage i detailig these tehiques here, thus we istead provide a umber of aalytial results derived by Joaopoulos et al. [6] that apture some geeral properties of PBCs ad failitate oe s ituitio whe thikig about them Geeral Properties of PBCs Iitially, tehiques for omputig the properties of dieletri PBCs exploited previously itrodued methods for omputig the bad strutures of semiodutors. Ideed, a ompariso betwee the equatios of quatum mehais (QM), used to desribe semiodutors, ad eletromagetis (EM), used to desribe dieletri PBCs, shows may similarities, Table 3-. Table 3-. Compariso betwee quatum mehais ad eletromagetis formulatios. [59]. Field iωt iωt Ψ( r, t) Ψ( r) e H( r,t) H( r) e Eigevalue problem H Ψ EΨ ΞH ( ω ) H Hermitia operator ( ) H V() r m Ξ ε () A key differee, however, whih restrits the geeral appliability of the QM formulatio to solve PBC problems is the salar ature of the QM problem ompared to the vetor ature of the EM problem. Fortuately, however, ulike the QM semiodutor bad struture problem, i whih the Bohr radius itrodues a fudametal legth sale ad, as a result, similar latties with differig dimesios give rise to differet behaviors, the EM problem possesses o fudametal legth sale ostat. This meas that the properties of PBCs whih differ oly via a legth expasio or otratio of all distaes, are related by simple expressios. I partiular, give a EM eigemode obeyig the equatio, r

153 4 Chapter 3 ω H() r H() r, (89) ε() r if the dieletri profile defiig a PBC is saled as follows, ε () r ε' () r ε()s r, where s is the salig fator, the it a be show that the saled PBC will obey the equatio, ω ' H( r' /s) H( r' /s), (9) ε' ( r' ) s from where oe derives that the properties orrespodig to the saled PBC are derived from those of the usaled oe as follows: H' ( r' ) H( r' s) ad ω ' ω/s. Thus, oe the PBC solutios are kow at oe legth sale, they are automatially kow at all others. As a pratial appliatio, mirowavelegth-sale PBCs may be exploited as vehiles to study to optial-sale PBC oepts. Similarly, there is o fudametal value of dieletri ostat, therefore, it may be show that wheever the dieletri ostat is uiformly saled throughout a PBC as follows: ε () r ε' () r ε() r s, where s is the salig fator, the the saled PBC will obey the equatio, sω H() r H() r. (9) ε' () r This meas that, upo salig the dieletri ostat, the mode geometry remais uhaged, but the frequey sales as: ω ω ' sω. Thus, multiplyig the dieletri ostat by a fator of /9 will result i multiplyig the frequey of their modes by three. Lastly, the properties of PBCs deped o parameters suh as fillig fratio, the otrast betwee host ad lattie dieletri ostats, ad the umber of layers employed. Fig. 3-8 shows the omputed trasmissio oeffiiet for a eleve-layer PBC as the idex of refratio ε is ireased from. to.98.

154 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 4..8 Trasmissio oeffiiet Normalized Frequey (/a) Figure 3-8. Eleve-layer -D PBC trasmissio oeffiiet with idex of refratio as a parameter. The bad gap atteuatio ireases from a few db for. to lose to 8dB at Advaed PBC Strutures The iitial ivestigatios i the field of PBCs foused o dieletri materials-based PBCs, whose struture osisted of either periodi arrays of suitably shaped holes i a dieletri slab, thus formig a otiuous dieletri host matrix, or a periodi array of suitably shaped ad isolated dieletri objets. The former PBC is exemplified by a slab pattered with a array of ylidrial air holes, whereas the latter PBC is exemplified by a array of isolated yliders embedded i air. These PBCs permitted the reatio of bad gaps at fiite frequeies, but did ot produe them at DC. Further ivestigatios o metal-based PBCs, suh as a wire mesh, soo followed, whih demostrated the existee of bad gaps dow to DC [67], [68]. While eablig the maipulatio of eletromageti waves, i partiular, ahievig diffratioless guidae of light aroud sharp beds [6], the overall propagatio behavior i dieletris ad metalli meshes followed the usual right-had (RH) rule, i whih the diretios of the eletri ad mageti fields, E ad H, ad the propagatio vetor k form a righthaded system with oiidee of the diretio of eergy flow ad k. Further work, aimed at maipulatig the properties of the PBC medium, led Pedry to propose two shemes, amely, a omposite medium made up of a array of metal posts whih reated a frequey regio with egative

155 4 Chapter 3 permittivity, ε eff <, ad a array of iterspersed split-rig resoators whih reated a frequey regio with egative permeability, µ eff <. These materials have beome kow as metamaterials ad, whe implemeted so that both the permittivity ad the permeability are simultaeously egative, they exhibit a egative refrative idex ( ω) εeff ( ω ) µ eff ( ω), whih is real ad gives rise to the existee of propagatig modes with the remarkable property that they follow a lefthad (LH) rule. I this ase the vetors E, H, k form a left-haded system, i.e., the diretio of propagatio is reversed with respet to the diretio of eergy flow [69]. Left-haded materials have bee the subjet of muh attetio beause they exhibit uusual propagatio properties. For istae, they exatly reverse the propagatio paths of rays withi them, whih may be exploited to implemet low refletae surfaes by exatly aelig the satterig properties of other materials. Aother appliatio, exploits their potetial to produe perfet leses Negative Refratio ad Perfet Leses The oept of a perfet les was itrodued by Pedry [7], upo further examiig the earlier aalysis of Veselago [69] o the osequees of egative refrative idex materials. Veselago [69], i partiular, had idiated that refletio ad refratio betwee vauum ad a egative refratio material, would follow the situatio depited i Fig φ φ Vauum ψ ψ Refrative Idex 3 4 Figure 3-9. Cosequees of egative refrative idex o refratio properties. Iidet beam. Refleted beam. 3 Refrated beam for <. 4 Refrated beam for >. (After [69].) Fig. 3-9 shows, that otrary to the usual ase of a positive idex, whe the refratio idex is egative the agle of refratio is also egative with respet to the surfae ormal. As a result, whe suh a medium is used as a

156 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 43 les, Fig. 3.3, it auses light origially divergig from a poit soure S to be reversed ad to overge bak to a poit S i the medium. S S S - Figure 3-3. Parallel-sided medium with egative refrative idex refouses light. (After [7].) The speial feature otributed by a egative refratio les was eluidated by Pedry [7]. It osists i that, by beig apable of amplifyig evaeset waves, all wave ompoets emaatig from the soure are preset at the overgig fous; this eables the perfet reostrutio of the soure image. This property was prove by Pedry [7] by assumig a iidet wave with eletri field give by, E ( ik z ik x i t) ŷexp z x, (9) S ω where, sie k x k y > ω, the wave vetor, k z ω i k x k y, (93) implies a expoetially deayig (evaeset) wave, a refleted wave give by, E ( ik z ik x i t) rŷexp z x, (94) S ω ad trasmitted wave give by, E where, ( ik' z ik x i t) tŷexp z x, (95) S ω

157 44 Chapter 3 εµω εµω k' z i k x k y,k x k y >. (96) The, usig the formula for the trasmissio oeffiiet of a slab of width d, i.e., T S where, ( ik' z d) ( ik' d) tt'exp, (97) r' exp z t ad t are the vauum/medium ad medium/vauum trasmissio oeffiiets ad r ad r the orrespodig refletio oeffiiets, give by, t ad µ k µ k k ' z z z,r, (98) ' ' µ k z k z µ k z k z t' k' k' µ k ' z z z,r'. (99) k' z µ k z k' z µ k z If both the permeability ad permittivity approah egative uity, the the trasmissio oeffiiet beomes, lim µ ε TS lim µ ε tt ' exp r' exp ( ik ' z d ) ( ik ' d ) z exp ( ik ' d ) exp ( ik d ) z z. () Sie k z is imagiary, see (96), () is a growig expoetial ad the wave is amplified. By otrast, i a ormal les the large trasverse wave vetor of propagatig waves are evaeset ad deay prior to reahig the fous, thus the iomplete spetral otets makes it impossible to idetially reostrut the image.

158 3. NANOMEMS PHYSICS: Quatum Wave Pheomea Cavity Quatum Eletrodyamis The field of Cavity Quatum Eletrodyamis or, avity QED, deals with the effet of the surroudig eviromet o the spotaeous emissio rate of atoms [7]. The oept was itrodued by Purell i 946 [7] i the otext of ulear mageti momet trasitios. He observed that at oditios of temperature, radio frequey, ad ulear mageto give by 3 K, ν 7 se, ad µ, respetively, the orrespodig rate of spotaeous emissio, give by, A ν 8πν 8π µ 3h 3 h ν se, () adopts a value of se. So small is, ideed, this value, that it implies the virtual impossibility of the spi system beig able to ahieve thermal equilibrium with its surroudigs. This expressio, Eq. ( ), for the spotaeous emissio rate A betwee iitial ad fial states i ad f, assumes the atom is i free spae ad derives from Fermi s golde rule [7], amely, A f H i ρ( ν), () where the iitial state i, represets a atom i the absee of ay photos, ad the fial state f, represets the atom with a sigle photo. The Hamiltoia H represets the atom-field iteratio, ad ρ ( ν) represets the desity of photo states or umber of radiatio osillators per uit volume, i a uit frequey rage whih, for free spae, adopts the value of, S 3 ( 8πν ) ρ. (3) I other words, ρ S embodies the umber of eletromageti modes ito whih photos may be emittted at the loatio of the emitter [73]. Whe the atom is elosed by a mirowave avity of quality fator Q, however, the umber of radiatio osillators per uit volume is limited to those oupyig the frequey rage ν Q, whih is, i fat, exatly oe. If oe assumes the avity volume ad the wavelegth to be related by

159 46 Chapter 3 V ( λ ) 3 ( ν) 3 per uit volume, ( mode ) /( νv) Q as follows,, the the desity of photo states per uit frequey,, may be expressed i terms of the avity mod e V ( ν)( ) mod e 3 ν Q ν 8ν Q 3 ρ. (4) Comparig (3) ad (4) it is see that they are related by, ρ Q ρs. (5) π Thus, a avity elosure of quality fator Q ireases the effetive desity of photo states i free spae by the fator of ( Q π). I tur, sie the spotaeous emissio rate is proportioal to this desity of photo states, this rate is ireased, i partiular, to [7], A QA. (6) The larger issue eliited by Purell s observatio was that the spotaeous emissio rate of a atom may be modified aordig to the properties of the surroudigs. I partiular, as Klepper [7] poited out, the spotaeous emissio of a atom i a avity may be ihibited if the avity has dimesios smaller tha the radiato wavelegth, but it may be ehaed (ireased), as i (6), if the avity resoates at this wavelegth. This realizatio that the spotaeous emissio rate of a atom may be suppressed or ehaed by modifyig the properties of the radiatio field i the surroudigs, has may pratial appliatios. For istae, i solid-state eletrois it is well kow that spotaeous emissio is fudametally resposible for o-radiative reombiatio proesses, whih limit the performae of semiodutor lasers, heterojutio bipolar trasistors, ad solar ells [5]. How would oe apply the avity QED oept to ihibit the spotaeous emissio i these situatios, where oe is dealig ot with sigle atoms, but with etire devies, is ot at all obvious. The aswer to this questio was advaed by Yabloovith i 987 [5] with his photoi bad-gap rystal (PBC) idea. Ideed, by surroudig the devies i questio with a PBC exhibitig a bad gap whih overlaps the eletroi bad edge (aross whih the o-radiative trasitios would our) the spotaeous

160 3. NANOMEMS PHYSICS: Quatum Wave Pheomea 47 emissio a be forbidde, thus potetially elimiatig o-radiative trasitios. This is so beause, i the bad gap of a PBC, the desity of photos states, ρ. The first experimetal demostratio of the use of PBC three-dimesioal PBCs to otrol the dyamis of spotaeous emissio from quatum dots has bee reetly published [73]. I this ase, Fig. 3-3, Figure 3-3. (a) Saig eletro mirosope image of the () fae of a titaia iverse opal with lattie parameter a46 m. Reprited with permissio from[7] Copyright 4 Nature. (b) Lumiesee deay urves of quatum dots iside three differet photoi rystals. The data are reorded at frequeies 5,67 m - (a 37 m) ad 5, m - (a 4 m, ad a5 m). The urves have bee overlapped after 5 s. The first part of the deay urve is iflueed by emissio of titaia (reorded at 5,4 m - ). After 5 s this otributio is egligible. [73]. the spetral distributio ad time-depedet deay of light emitted from exitos ofied i the CdSe quatum dots are show to be otrolled by the host PBC. I partiular, the fat that lifetimes of 9.6 ±.s ad 9.3 ±.s for quatum dots embedded i PBCs of lattie ostats a4 m ad a5 m, respetively, are obtaied, demostrate a fator of

161 48 Chapter 3 variatio produed by the PBCs. This orroborates the strog role the PBC plays i otrollig the radiative lifetime of the emitters. 3.3 Summary This hapter has dealt with the physis of waves that is of relevae to quatum pheomea ourrig i NaoMEMS. It bega with typial pheomea that maifest ad exploit the wave ature of eletros, i partiular, the quatizatio of eletrial odutae, its alulatio with Ladauer s formula, ad its maifestatio i quatum wires, quatum poit otats, resoat tuelig ad quatum iterferee (Aharoov-Bohm effet). The, the topi of quatum trasport theory was take up, with partiular emphasis o dealig with pheomea domiatig i oedimesioal trasport, suh as the Lüttiger liquid. Fially, wave behavior i both periodi ad o-periodi media was addressed, i partiular, arbo aotubes, superodutors, photoi badgap rystals, ad avity quatum eletrodyamis. I ext hapter fouses o the appliatio of the material preseted thus far to egieer a variety of iruits ad systems that typify elemets to be foud i NaoMEMS.

162 Chapter 4 NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 4. Itrodutio The ew eletrois, eabled by NaoMEMS, will exploit the oexistee of mesosopi ad mehaial devies operatig i the quatum mehaial regime. Thus, a plethora of pheomea, suh as tuelig, harge quatizatio, the Casimir effet, motio quatizatio, etaglemet, et., are at our disposal to be exploited i reatig powerful omputig ad ommuiatios hardware. This hapter exposes a variety of emergig devies that embody potetial aoeletromehaial quatum iruits ad systems (NEMX) devie-iruit paradigms []. 4. NaoMEMS Systems o Chip NaoMEMS Systems-o-Chip (SoC) may be prediated upo a multitude of physial pheomea, e.g., eletrial, optial, mehaial, mageti, fluidi, quatum effets ad mixed domai. Therefore, their uiverse of possible implemetatios ad appliatios is vast ad oly limited by our imagiatio. Possible areas of edeavor, already uder researh, ilude: Naoeletrois, Naoomputatio, Naomehais, Naoegieerig, Naobiotehology, Naomediie, Naohemistry, ad RF MEMS. I priiple, the, there is the potetial for oeivig ew devies that might spark a revolutio as importat ad wide-ragig as that egedered by the ivetio of the trasistor ad ICs. Ultimately, however, the suess of the

163 5 Chapter 4 tehology may well lie o its ability to deliver improved performae at low ost o tehology-blid appliatios, Figure 4-, as well as i eablig ew appliatios (some of whih are right ow oly limited by our imagiatio). For the purposes of this book, we fous o NaoMEMS SoCs i terms of implemetatio ad appliatios. Iput Iform atio NaoMEMS SoC Output Iform atio Eletrom ageti Heat Soud Biomoleules Fore Piture Voie Computig Diagosti Data Figure 4-. Coeptual reditio of a NaoMEMS System-o-Chip. 4.. NaoMEMS SoC Arhitetures Regardless of the tehology of implemetatio utilized, a system must perform a defiite futio ad is haraterized by how lose it omes to meetig ertai tehology-blid speifiatios (spes). Typially, the desig proess begis with a blok diagram of the system i questio, whih displays a arhiteture or high-level topologial diagram showig how the ostituet buildig bloks are iteroeted to trasform or proess oe or more iput sigals ito oe or more output sigals, see Figure 3-. Followig this, overall systems aalysis assigs or flows dow the overall system spes to the idividual buildig bloks, whih are the desiged. I the ase of NaoMEMS SoCs this is diffiult to do beause the field is so premature that, usig a iruit aalogy, the equivalets of passive ompoets (resistors, idutors, apaitors, diodes) ad ative ompoets (trasistors) is ot yet available to the degree of ompleteess that would allow a omplete osistet system implemetatio. Our ourse of atio, therefore, is to expose a variety of potetial NaoMEMS SoC buildig bloks.

164 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS NaoMEMS SoC Buildig Bloks 4... Iterfaes The idea behid NaoMEMS is that of reatig a system that, i order to aomplish a give futio, avails itself of devies ad tehiques spaig the rage from the miro- dow to the ao-sale ad beyod. I the most geeral ase, the iput sigal to a NaoMEMS SoC will be aalog, i.e., will exhibit otiuous amplitude ad will exist at all times, see Figure 3-. Proessig this sigal, therefore, will etail deidig whether it is feasible to at o it as reeived/deteted, or to trasform it to a more oveiet state. The ature of the iterfae sesor, i partiular, its sesitivity, badwidth, ad dyami rage, will ome ito play here ad will ditate the eed for trasdutio, amplifiatio, digitizatio, filterig, et., thus determiig the rest of the arhiteture. I this otext, the doubly-ahored Si beam has bee osidered as a potetial mehaial sesig elemet i future NaoMEMS SoCs, ad impressive estimates for its itrisi fore sesitivity (S F ), dyami rage (DR), mass sesitivity (M), ad badwidth (BW) have bee obtaied by Roukes [74]. For istae, a beam of legth, width, ad thikess. x. x. miros ad ative mass ag would exhibit / 7 S F ( ω ) 3 N / Hz, DR 35dB, M.7 g, ad BW 7. 7GHz, assumig a temperature of 3K ad a Q of,. Ufortuately, it is ulear whether the full extet of these parameters will be aessible due to various pratial diffiulties suh as mass variatio due to upreditable adsorbates, ad the impossibility of realizig a oiseless read-out. This latter theme is also ommo to eletrostati- ad optiallybased sesig iterfaes as well. I the former ase, whih aordig to 8 Roukes [74] may attai a miimum apaitae of F, the parasiti apaitae would prelude resolvig it. I the latter ase, the fat that the spot size of the light delivered by the optial fiber used i AFM displaemet-sesig is muh greater tha aosale dimesios, preludes its resolutio ad, hee, proper detetio. I systems with a eletroi iput sigal sesig sheme, however, the sesor may take the form of a quatum superlattie-based aalog-to-digital overter, Fig. 4- [75]. Here, the pulsatig ature of the superlattie s urret-voltage harateristi diretly samples/quatizes the voltage axis. The resultig urret is used to geerate pulses that drive a outer whose output is a digital represetatio of the iput voltage. For highest resolutio, the superlattie may be realized with moleular devies.

165 5 Chapter 4 (a) I SL I SL () I SL (V) () I SL (t) (4) V SL (t) V SL di SL /dt (3) t t t (b) V i Iput Devie Differetiatig Devie Swithig Devie Itegratig Devie Dτ Coutig Devie N-bit Lath.. Digital Output Code - () Figure 4-. Superlattie-based aalog-to-digital overter arhiteture. (a) Superlattie bad diagram. (b) SL A/D oversio priiple. () ADC arhiteture [75] Emergig Sigal Proessig Buildig Bloks While the speifi struture of a NaoMEMS SoC is still the subjet of muh researh, a umber of potetial buildig bloks for NaoMEMS-based sigal proessig have bee proposed. I what follows, we preset a umber of these [], amely, a harge detetor, a whih-path eletro iterferometer, a parametri amplifier usig a torsioal MEM resoator, a

166 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 53 Casimir effet-based osillator, a magetomehaially atuated beam, ad array-based futios. We olude with a example of exploitig quatum squeezig to redue oise i mehaial strutures Charge Detetor This devie was experimetally demostrated by Krömmer et al. 76]. I this devie a low-power RF sigal propagates through a suspeded resoator, Figure 4-3, ad sets it ito vibratio. w TOP VIEW Resoator END VIEW RF Sigal Gate SIDE VIEW B g L Substrate Trasmissio Lies Figure 4-3. Shemati of harge detetio resoator system []. With a i-plae mageti field applied perpediular to beam, a Loretz fore perpediular to the substrate surfae is developed. Appliatio of a voltage, V, betwee the gate ad the beam, idues a harge, Q, o the beam via the relatio, Q CV, ad essetially, modifies its stiffess (sprig ostat). This results i a mehaial resoae frequey shift of Q C z δ f, where C is the gate-beam ouplig apaitae, ad C C C represets the seod derivative of the apaitae with respet to beam elogatio amplitude, z(t), evaluated at z. Optimum harge detetio (maximum frequey shift) is obtaied whe RF power drives the beam to the verge of oliear amplitude vibratio. For a gate bias of V ± 4V, a mageti field of T, ad a RF power of -5.8dBm at 37.9MHz, a harge

167 54 Chapter 4 detetio resolutio of about 7 q / Hz. This devie has the potetial to exploit harge disreteess effet Whih-Path Eletro Iterferometer Armour ad Bleowe [77], [78] preseted a theoretial aalysis for this oept. A atilever resoator operatig at radio frequeies is disposed over oe of the arms of a Aharoov-Bohm (AB) [5] rig otaiig a quatum dot (QD), Figure 3-4. Eletrostati ouplig of the vibratig beam with V DC V AC - Bottom Eletrode Aharoov-Bohm Rig Beam Resoator Φ Quatum Dot SIDE VIEW E Φ Quatum Dot Substrate Figure 4-4. Shemati of mehaial whih-path eletro iterferometer []. eletros hoppig i/out of the QD modulates the iterferee friges, aordig to vibratio frequey ( ω )-eletro dwell time, τ d Ei, produt, where Ei is the eletro eergy spread. For ω τ d <<, short dwell time, iterferee friges are destroyed if qe x th > Ei., where xth is the thermal positio uertaity of the atilever ad E the eletri field. This sigals eletro dephasig ad detetio i QD arm. For ω τ d ~, the beam-qd behaves as a oheret quatum system, beam vibratio ad QD exhage virtual eergy quata i resoae, ad iterferee friges are modulated at beam vibratig frequey. For the largest dwell times, the eviromet idues lost of oheree. This devie has the potetial to exploit harge disreteess effet.

168 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS Parametri Amplifiatio i Torsioal MEM Resoator This devie was experimetally demostrated by Carr, Evoy, Sekari, Craighead ad Parpia [79]. A torsioal resoator of quality fator Q, Figure 3-5, is exited at a fudametal drivig frequey, ω, whih applies Torsioal Resoator TOP VIEW V V V DC AC_ ω AC_ ω END VIEW w Paddle SIDE VIEW ϕ g L Trasmissio Lies Substrate Figure 4-5. Shemati of torsioal parametri amplifier []. a torque τ ( ωt). If the devie is drive at resoae, with a applied torque give by τ () t τ si( ωt θ ), where θ is the phase agle betwee exitatios at ω ad ω, the the torsioal sprig ostat exhibits a modulatio, κ () t κ os( ω t). Uder these irumstaes, the agular amplitude respose, ϕ, adopts the form τ Q ϕ κ os ( Qκ κ ) ( Qκ κ ) θ si θ /. () Aordigly, with zero sigal amplitude at ω, κ () t, ad the agular respose is τ Q κ. Otherwise, the square-root fator ats as a phase-depedet gai ad, beomig ifiity whe θ π, ad κ κ Q. For < θ < π, the agular respose may be approximated by,

169 56 Chapter 4 τ Q ϕ κ os θ ( V V ) ( V V ) AC _ ω si θ AC _ ω /, () where V is a struture-depedet parameter, showig that the gai ireases with the pump sigal amplitude. The devie has the potetial to exploit the Casimir effet Casimir Effet Osillator This devie, whih was proposed ad aalyzed by Serry, Walliser, ad Malay [8] i 995, Figure 4-6, ad experimetally realized by Cha, Aksyuk, Kleima, Bishop ad Capasso [8] i, represets the first lear demostratio of the impat of the Casimir fore i the performae of NEMX. 3 π R F Sprig kz F Casimir 4 z V (z).. z z max z. Sprig SprigCasimir..5.5 z z V ( z ) C Sprig V z Casimir ( z) 3 max z z max z max Figure 4-6. Summary of oliear Casimir effet MEM resoator physis []. The experimet etailed hagig the proximity of a vibratig rotatioal resoator to a metalli sphere, Figure 4-7(a), to measure its behavior i the absee/presee of the Casimir fore. After determiig the drive for liear respose, the proximity of the osillator to a metalli sphere was varied ad the resoae frequey measured exhibited a behavior as depited i Figure 4-7(b). For sphere-osillator distaes greater tha 3.3µ m, the osillator resoae frequey was equal to the drive frequey, 748Hz, ad the agular amplitude frequey respose was symmetri ad etered aroud the drive frequey, ω k I, where k is the sprig ostat ad I the momet of iertia, osistet with mass-sprig fore osillator behavior.

170 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 57 b ϕ - V z Piezo Stage (a) Agular Amplitude D ereasig osillator-sphere separatio Frequey Respose (b) Figure 4-7. Shemati of torsioal MEMS osillator ad sketh of Casimir effet o resoae respose []. However, as the sphere-osillator distae was dereased, i partiular, at 4m, 6.5m, ad 98m, the resoae frequey shifted, aordig to, ω ω [ b F ( z) Iω ], where F ( z) is the first derivative of the exteral fore evaluated at z, ad the agular amplitude frequey respose asymmetri ad hystereti. This behavior was show to be osistet with the dyamis of a mass-sprig-casimir fore system. The ramifiatios of this beautiful experimet are eormous, i partiular, it may be oluded that the Casimir fore will be oe of the fators limitig the itegratio level or desity of NEMX Magetomehaially Atuated Beams This idea was proposed ad theoretially aalyzed by Blom [8]. I additio to their futio as mehaial elemets (atuators), arrow metaloated aosale beams also embody mesosopi wires. If suh a beam is elogated due to, say, eletrostati atuatio, this results i a redutio i its ross-setioal area, ad i partiular, that of the urret-arryig metallizatio/wire, ad as a osequee, the odutae of the latter hages as trasverse quatized modes are pushed above the Fermi level. The hage i thermodyami potetial as the wire elogates, i tur produes a fore alog the legth of the wire, whih is give by,

171 58 Chapter 4 F m π 4 3 ( E E ) 3 / ( E E ) F F / E, (3) where E F is the Fermi eergy, m the eletro mass, ad E is the eergy of the trasverse modes. This fore maifests as fore ad beam defletio flutuatios. O the other had, if the beam is ot eletrostatially atuated, but a mageti field is applied alog its legth, it will also ause odutae hages as the Ladau levels [6] push the eergy above the Fermi level. Thus, the beam is magetomehaially atuated. This devies has the potetial to exploit harge disreteess effet Systems Futioal Arrays The dyami properties of the olletive modes i a MEMS resoator array were studied experimetally by Buks ad Roukes [83], ad theoretially by Lifshitz ad Cross [84]. I this oept, the lateral eletrostati ouplig of a array of doubly-ahored beams leads to olletive modes that resemble phoos. Adjustmet of the ouplig serves to tue the diffratio properties of the mehaial lattie the array embodies. I a related oept, De Los Satos [85] uveiled the idea of populatig a rigid photoi bad-gap rystal lattie with a sub-array of MEMS swithes. The, by exploitig the oivasive properties of these, i.e., their ideal ON/OFF states, loalized states modes ould be formed that eabled the ON/OFF swithig of pass bads withi the photoi bad-gap, thus makig the system programmable Noise Quatum Squeezig Ultimately, the purity of resoator vibratio is determied by its zeropoit flutuatios. I this otext, quatum squeezig tehiques [86] may be applied to redue the flutuatios i flexural motio. Appliatio of quatum squeezig to mehaial resoators has bee studied theoretially by Bleowe ad Wyboure [87]. Aordigly, by exitig the resoator with a pumpig voltage of the form, V p () t V os( ω pt φ), its sprig ostat beomes, k mω k, where k CV g, ad k t k os ω t φ. Whe the effetive resoator sprig ostat, p () ( ) () t p k k k p, ireases, the urvature of the effetive potetial arrows [87] ad this squeezes the wavefutio. I partiular, for a phase φ π 4, the quatum uertaity i the flexural displaemet beomes,

172 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 59 ( ) T Q k Z, (4) m ω m ω T e. The, with mω defiig the zero-poit uertaity, the squeezig fator R Z mω beomes, ω k T where ( ) B R T Q k / mω <, (5) whih, for R <, deotes the ourree of quatum squeezig. Bleowe ad Wyboure [74] foud that usig typial resoator values, e.g., desity, 3 3 ρ 3.99 kg / m, Youg s modulus, E 3.7 N / m, beam to substrate distae, g 5m, beam thikess, t m, ad legth, L 7m, the squeezig fator is R. 5, whih sigals the realizatio of quatum squeezig, i.e., oise redutio below that of zeropoit flutuatios i the flexural displaemet mode Naomehaial Laser This devie oept was proposed by Bargati ad Roukes [88]. The fudametal idea is to egieer a laser-like devie i whih the resoator is realized by a aomehaial beam, whose tip has bee futioalized with a ferromageti material, ad whose vibratio iterats with a adjaet ative medium otaiig ulear spis biased by a exteral mageti field, B. With the appropriate geometrial ofiguratio, see Fig. 4-8, C atilever Z X Y Ferromageti Tip Sesitive Slide B Preessig Spis M irowave Pumpig Figure 4-8. Sketh of mehaial laser. (After [88].)

173 6 Chapter 4 vibratio of the aomehaial beam auses superpositio of the field produed by its ferromageti tip with the exteral mageti field. This results i a modulatio of the mageti field pereived by the ulear spis ad, as a osequee, a stimulate trasitios i the Larmor frequey of ulear spis (Zeema effet). I tur, a dipolar iteratio ouples the rotatig trasverse ompoet of the ulear magetizatio of the ulear spis with the ferromageti tip, resultig i a fore that drives the beam osillatios. This proess, uder resoae betwee Larmor frequey ad beam vibratio, leads to self-sustaied oillatios, i.e., to laser behavior. The proposed devie was alled atilaser. Typial parameters are as follows: Fudametal frequey of beam, MHz, effetive sprig ostat,. 5 N/m, quality fator,, trasverse mageti field gradiet due to ferromageti tip, 6 T / m, trasverse relaxatio time of ulear spis, 5µ s, ulear gyromageti ratio, π MHz / T, exteral mageti field, Tesla Quatum Etaglemet Geeratio As disussed i Chapter 3, quatum etaglemet is a fudametal igrediet for effetig quatum iformatio proessig. Most shemes for quatum etaglemet, however, were demostrated i the otext of optial experimets, where the objet of etaglemet was photo polarizatio. While the realm of implemetatio of NaoMEMS SoCs iludes variats that exploit optial sigal proessig, i.e., the proessig ad maipulatio of photos, eletros ad, thus, eletroi sigal proessig i solid-state systems remai a importat paradigm. It is ot surprisig, therefore, that a umber of efforts have bee aimed at fidig ways to ahieve the eletro pair etaglemet ad trasport over log distaes. The superodutorarbo aotube jutio, proposed by Bea, Vishveshwara, Balets, ad Fisher [89] is a lever idea alog these lies, see Fig SC I A δ A SW NT B V A I B δ V B Figure 4-9. Quatum etaglemet jutio. A setup of two aotubes A ad B otatig a superodutor. Voltage drops V A ad V B may be preferetially applied aross tubes A ad B respetively, ad urrets through eah of them may be measured. [89].

174 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 6 The oept osists i exploitig the iheret etaglemet of superodutig Cooper pair together with eletro-eletro iteratios i oe dimesio to eable the sequetial ijetio of etagled pairs from a superodutor ito two aotubes loated ext to eah other at a distae well below the Cooper pair oheree legth. The key to the Cooper pair ijetio ad separatio ito etagled eletros relies o the Lüttiger liquid behavior exhibited by CNTs haraterized by a iteratio fator g ad subbad spaig ε. I partiular [89], the tuelig rate, ( ev h)( kt ) g ΓAA ~, at whih Cooper pairs tuel from the superodutor ito the ed of a CNT, beig proportioal to evρ e, turs out to be muh smaller tha the tuelig rate ( )( ) g Γ AB ~ h kt ev, at whih split etagled pairs are ijeted ito both CNTs. This differee, is rooted i the fat that Lüttiger liquid behavior, maifested as the oheret arragemet of all eletros i the CNT bulk, auses the sigle-eletro tuelig desity of states, ( ) ( ) 4 g ρ e E ~ ε E ε to domiate the Cooper pair tuelig desity of states, ρe( E) ~ ε ( E ε )g. With Γ AA << ΓAB, virtually all the harge tuelig that ours ivolves split etagled pairs. Oe split, the etagled eletros may propagate for log distaes due to the ballisti property that haraterizes trasport i CNTs of Fermi veloity v F ad legth L at low temperatures T < T φ vf k BL at whih loss of oheree due to thermal effets are oexistet Quatum Computig Paradigms As idiated i Chapter, the fudametal buildig blok o whih quatum iformatio proessig systems are based is the qubit, a two-state quatum system. Qubits may take o may physial forms, however, to be useful i realizig real, pratial, systems, they must be edowed with three key properties [9]: ) They must be deoupled from the eviromet to avoid disturbaes whih may deviate their time evolutio from uitarity; ) They must be able to respod, i a otrolled fashio, to purposeful maipulatio, i order to eable the formatio of quatum logi gates ad etagled states, whih rely o suh iteratios; 3) They must withstad the mometary, but strog, ouplig to the eviromet itrodued by a measurig devie. I this setio, we preset the priiples of various qubit implemetatios, i partiular, io-trap-, ulear-mageti resoae-, solidstate-, ad superodutig-based qubits.

175 6 Chapter The Io-Trap Qubit The io-trap qubit was proposed by Cira ad Zoller [9]. It is embodied by atomi ios ofied by a eletrode struture desiged i suh a way that a 3- dimesioal harmoi potetial well (trap) is produed [9]. Coolig the ios lowers their eergy ad, were it ot beause of Coulomb s fore of repulsio, whih maitais them apart, they would desed to, ad meet at, the bottom of the well. Istead, the olletive state of the ios is the result of a balae betwee the potetial well eergy profile ad the fores of repulsio betwee ios, whih maifests i their assumig a liear array, see Figure 4-. Figure 4-. Sketh of io trap qubit. The eletrodes reate a 3-D harmoi potetial well that ofies the ios. The io trap simultaeously implemets two types of qubit, Fig. 4-. I oe Figure 4-. Qubit realizatios with io trap. istae, the two states of the qubit are embodied by the diretio of the io s mageti momet, whih is parallel or atiparallel to a exterally applied mageti field. I the seod istae, the olletive motio of a array of ios forms the qubit. I partiular, whe expressed i terms of ormal modes, the two states of the motioal qubit are the oe i whih the ios move simultaeously i the same diretio, ommo mode (CM) ad the oe i whih adjaet ios move i opposite diretios, strethed mode. I the motioal ase, the qubit is ot assoiated with ay idividual io,

176 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 63 but rather, with the array as a whole. Sie the io trap produes two qubits, a otrolled iteratio betwee them allows the realizatio of quatum gates. I the ase of the spi-orietatio qubit, the iteral spi state of the io may be set ito the dow ( ) or up ( ) states, by appliatio of a uiform mageti field. Alteratively, it may be prepared ito superpositio states by varyig the time duratio of applied RF fields. Further futioality is obtaied out of the io-trap system by ouplig its spi-orietatio qubit to its motioal qubit. I partiular, superpositio of a spatially o-uiform mageti field alog the motioal qubit, for istae, of magitude B at the io s left most positio ad B at its rightmost positio, auses the io to experiee a field of amplitude B ad frequey equal to the motioal osillatio frequey. Uder these irumstaes, a exhage of eergy betwee the spi ad the motioal states,, esues if the mageti field frequey oiides with the eergy differee betwee the two spi states. More geerally, if the spi qubit is i a superpositio state, e.g., the, osistet with oservatio of eergy, the eergy exhage produes the trasitio. As depited i Fig. 4-, ( ) ( )... Cofiig Eletrode Laser Beam (a) Phooi Motio e e ii) i) iii) (b) Figure 4-. (a) Cira-Zoller io-trap qubit. (b) Qubit states g ad e a eergy ω. g, are separated by

177 64 Chapter 4 the iteratio may be partiularized, to the state of oe of the ios i the motioal qubit, by ausig the mageti field gradiet to exist o it. This is aomplished by fousig a laser beam o the io i questio, see Figure 4-. Aalytially, the ier workigs of the io-trap qubit were desribed i detail by Cira ad Zoller [9] as follows. The two states of a partiular io, amely, its groud ad exited states, are deoted by g ad e, respetively. The 3-dimesioal motio ofiemet of the ios is desribed by a aisotropi harmoi potetial haraterized by frequeies x << y, z. The typial eergy level sheme otemplated for the io trap is show i Fig. 4-(b). Whe the extet of the io s motio is muh less tha the iverse wavevetor of the laser field, the so-alled Lamb- Dike limit (LDL), the osillatios of the groud state beome ormal modes. Uder these irumstaes, a laser beam with frequey ω ω ν, or detuig equal to mius the CM mode frequey, L x x, will exite the ommo mode exlusively. This is the situatio i whih trasitios lead to motioal mode (phoo umber) trasitios. O the other had, if ω L ω ν x, the the trasitio leads to trasitios. Fially, whe ω L ω the idued trasitios leave uhaged. Thus, the relatioship betwee laser detuig, ad motioal frequey, ad the fat that the frequeies of the differet ormal modes are well separated i the exitatio spetrum, allows the otrol of iteratio betwee ios via the CM motio ad, i fat ostitutes the ouplig of two qubits whih is eessary to produe quatum gates. After the quatum qubits are maipulated to effet a quatum omputatio, the result must be read. I the ase of the io trap this is aomplished by measurig the spi-depedet sattered light whe a laser beam impiges upo a io. Exploitig the fat that satterig is substatially greater for the spi tha for the spi, the state of the spi is iferred. The maipulatio of the state of a N-io-trap qubit by a laser beam is drive by the iteratio betwee a io ad the eletri field of the laser. Startig with the Hamiltoia for the -th io, H, i the groud state ad i the absee of ay laser field, ad hoosig the laser frequey as above, i.e., x, ad the io positio to oiide with a ode of the laser stadig wave, the system is desribed by,

178 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 65 H i i [ eq g ae g eq a ], (6) N, q e where a ad a are the reatio a aihilatio operators of CM phoos, respetively, Ω is the Rabi frequey, φ is the phase of the laser field at the mea positio of the io, q, levels ivolved i the eergy trasitio exited by the laser, ad k M << is the LDL parameter, with x k kos, k the laser wavevetor, θ the agle betwee the diretio of propagatio of the laser ad the x-axis of motio of the qubit, ad M the io mass. The Rabi frequey, Ω E d ˆ ε L, haraterizes the trasitio frequey betwee the groud ad metastable states produed by a laser with eletri field amplitude E ad polarizatio vetor ˆ L i a io of eletri dipole operator d. The evolutio of the system upo beig impiged by a laser beam pulse of time duratio t k ( N ) o the -th io is desribed by the uitary operator, U k,q i i ( ) exp ik ( e g ae g e a e ) q q. (7) Appliatio of this uitary operator o the various states of the -th qubit yields the results of Table 4-. ad represet the populatio of the CM mode with zero ad oe phoo, respetively. Table 4-. Effet of Io-Trap Uitary Operator o State Evolutio Operator Iitial State Fial State k,q U g g k,q U k,q U i g os( k ) g ie si( k ) e i e os( k ) e ie si( k ) g The above iteratio is ameable to the implemetatio of a two-bit gate. I partiular, Cira ad Zoller [9] have show that this is aomplished by followig three steps: ) Apply a π laser pulse with polarizatio q ad,, phase φ to the m-th io to reate the evolutio Û m Û m ( ) ; ) Tur o the laser direted to the -th io for a time duratio π ad polarizatio q q

179 66 Chapter 4 ad phase q ad phase φ, respetively. This reates the evolutio, operator Û, whih exlusively hages the sig of the sate g via a rotatio through the state e ; 3) Apply agai a π laser pulse with polarizatio q ad phase φ to the m-th io to reate the evolutio,, Û m Û m ( ). Sie these operators at o o-iteratig ios, the,,, overall effet is give by the produt Û m, Û m Û Û m i Eq. (8) below. Compariso of the first ad last olums reveals that the effet of the omposite operatio is to hage the sig of the state oly whe both ios are iitially exited, thus, Eq. (8) embodies a C-NOT gate. e e g g m m m m e g e g, Û m g g i g i g m m m m e g e g, Û g g i g i g m m m m e g e g, Û m e e g g m m m m e g e g. (8) May suessful implemetatios of io-trap qubits have bee experimetally demostrated [9]. Key to these experimetal demostratios are tehiques to address a variety of issues, most otably: ) Mitigatig the deoheree of the io trap, whih is due to the spotaeous deay of the iteral atomi states ad the motio dampig; ) Suppressig spotaeous emissio; 3) Obtaiig highly effiiet read-out shemes. A thorough disussio of problems ad solutios regardig io-trap qubits is give by Wielad et al [9] The Nulear Mageti Resoae (NMR) Qubit As is well kow, some atoms exhibit a itrisi ulear mageti momet µ ad a agular mometum I, ad these are related through the gyromageti ratio γ by [8], µ γi. (9) Sie the agular mometum is quatized [6], with values m I I,I,..., I, a uleus with a itrisi agular mometum of half a uit, i.e., I, will have the allowed values of m I ±. Thus, i the presee of a mageti field B B z ˆ, the eergy of iteratio betwee the mageti momet ad the field,

180 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 67 H µ B B I, () I z will split ito two eergy levels, see Fig. 4-3 at the top of ext page. These two eergy levels i a o-zero field embody a two-state quatum system that a be used as a qubit. The otrolled maipulatio of these qubits to effet quatum omputatios is the goal of NMR-based quatum omputig (QC). The origis, developmet, progress ad status of NMRbased QC has bee addressed reetly i extesive review artiles by Laflamme, Kill, Cory et al. [93], ad by Vadersype ad Chuag [94]. Our presetatio, therefore, will follow these losely. B BB m I µ ω B m I Figure 4-3. Eergy level splittig whe a uleus of itrisi agular mometum I is exposed to a ostat mageti field B. I pratie, limits germae to urretly available tehiques prelude detetig the eergy absorbed by a sigle uleus. Therefore, a substae otaiig a multitude of ulei, whose otributios add, must be employed [93]. The system of hoie for NMR-based QC osists of the very large umber of ulei belogig to atoms formig a moleule i a liquid, soalled liquid-state NMR. Fig. 4-4 depits a typial moleule used to form Cl Cl 3 C 3 C H Cl Figure 4-4. Trihloroethylee moleule for liquid-state NMR-based QC. The proto (H), ad the two arbos ( 3 C) are employed to realize qubits. The 3 C uleus has spi ½. [93]. qubits is the trihloroethylee (TCE) moleule, whih otais a hydroge uleus possessig a strog mageti momet. As a result, whe the moleule is exposed to a ostat strog mageti field, B, eah hydroge s

181 68 Chapter 4 spi oriets itself i the diretio of the field. If, i additio, a RF field is applied i a pulsed fashio, the spis are made to tip off-axis, while preessig about the diretio of the ostat field. The preessio frequey is the so-alled Larmor frequey ad is give by µb. For the hydroge atom (proto), the mageti momet is 4.7MHz/T ad, at a typial field of B.7T, its preessio frequey is 5MHz. Sample examiatio is aomplished plaig a oil aroud it, tued to the preessio frequey, whih piks up the osillatig urrets idued as a osequee of the mageti field produed by the preessig protos. The devie that applies the stati mageti field ad the RF otrol pulses, ad the detets the mageti idutio is alled a NMR spetrometer, see Fig Sample Tube Mixer C Diretioal Coupler RF Computer Osillator RF Coil B Amplifier Stati Field Coil Figure 4-5. Sketh of experimetal NMR spetrometer. (After [95].) NMR pheomea, whih were first observed i 946 [96], [97], beame the basis for a multitude of aalytial studies of materials, i partiular, the determiatio of moleular strutures [98], ad mageti resoae imagig [99]. I these otexts, the tehology of NMR spetrosopy is rather mature. The appliatio of NMR to QC was advaed Cory, Fahmy ad Havel [], ad Gershefeld ad Chuag [] i 997. To overome the diffiulty i detetig the spi of idividual, adoptio was made of qubits implemeted i the liquid state, where additive effets ould be exploited to yield a reasoably large sigal amplitude. Also adopted were methods to diser the fratio of ulear spis poitig i the exteral field diretio, despite the effets of temperature-idued radom spi orietatio. The twostate quatum system was realized by hoosig moleules possessig spi- / ulei, whih i the presee of the exteral mageti field adopts two

182 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 69 states, amely, a low eergy state deoted by, ad a higher eergy state deoted by. Aalytially, a NMR-based QC system is desribed i terms of two Hamiltoias, amely, the system Hamiltoia, whih aptures the eergy of sigle ad oupled spis i the presee of a mageti field, ad the otrol Hamiltoia, whih aptures the effets of applied RF pulses otrollig the operatios with qubits. The system Hamiltoia for sigle spis is give by, H / B I z I z, () / where I z is the z-ompoet of the agular mometum I xi ˆ x yi ˆ y zˆ I z. I geeral, the three ompoets of the agular mometum are related to the Pauli spi matries as follows [6], x I x, y I y ; z I z, () where, i x, y ; z i. (3) - H embodies the time evolutio give by the e ih t/ U, whih represets the preessio of the overall state vetor (the so-alled Bloh vetor) with respet to the axis B, defied by the stati mageti field, see Fig. 4-6 [94]. Z X Y i Figure 4-6. Preessio of a spi-/ about the axis of a stati mageti field. (After [94].) Vadersype ad Chuag [94] idiate that i the most geeral ase, the system Hamiltoia for a moleule possessig N isolated ulei is give by, H N N i i i ( i ) i B I z I z i i, (4)

183 7 Chapter 4 where i labels the ulei, ad i deotes the so-alled hemial shifts, whih haraterize the fat that distitly differet preessio frequeies are exhibited by idetial atomi speies withi a give moleule, whe the shieldig eviromet produed by their surroudig eletros results i a differet mageti field, B. They also poit out [94] that typial hemial shifts rage i the order of a few kilohertz, ompared to the preessio frequeies, whih rage i the MHz. I additio to isolated spi ulei, liquid-state NMR iludes the presee of oupled spis. These are haraterized by either a diret or a idiret ouplig mehaism. The diret ouplig is of the mageti dipole-dipole ature, similar to the iteratio betwee two adjaet bar magets ad, for ulei i ad j, separated by a distae r ij, is give by [94], µ ( )( ) < i j i j 3 i j H D I I I rij I r 3 ij, (5) i j 4 r ij rij where µ is the free spae mageti permeability, ad I i is the mageti momet vetor of spi i. Uder ertai oditios, Eq. (5) may be simplified. For istae, for large preessig frequeies it redues to [94], H D i< j µ i j 3 8 r ij i j i ( 3os )[ 3I I ( I I )] ij j, (6) where ij is the agle betwee B ad r i j ij, whereas if ω ω is muh greater tha the ouplig stregth it redues to [9], H D i< j µ i j 3 4 r ij ( 3os ) ij I i z I j z. (7) The idiret ouplig is haraterized by a stregth J, whih aptures the overlap of eletroi wavefutios betwee two atomi ulei, ad has values ragig from several Hz, for three- to four-bod oupligs, to several KHz for oe-bod ouplig. The idiret ouplig Hamiltoia takes the form [94], i j i j i j i j H I I J I I I I I I, (8) J i< j ij i< j ij ( ) x where J ij haraterizes the ouplig betwee spis i ad j. Simplifiatio of this expressio is also possible i ertai irumstaes, i partiular, whe x y y z z

184 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 7 ω i ω j >> π J ij, whih may be obtaied whe dealig with heteroulear spis or with small homoulear moleules, it redues to [94], H N i j π I I z. (9) J J ij i< j z Eq. (9) aptures the irumstae that, i additio to a ostat exterally applied mageti field, B, the atual field at a give spi loatio iludes a stati field alog ± ẑ, whih is eliited by spis i its eighborhood. The osequee of this additioal field is to shift the spi s eergy levels ad maifests as a hage i the Larmor frequey. For istae, a eighborig spi j i state will shift the frequey of spi i by J ij, whereas if spi j is i state, it will shift the frequey of spi i by J ij J ij. I geeral, it turs out that, whe i the presee of eighborig spis, the spetrum of a give spi would show, istead of a sigle lie at its Larmor frequey, two lies for every eighborig spi, the lies beig separated by the ouplig stregth J ij ad loated equidistatly above ad below the Larmor frequey. I the majority of NMR-based QC experimets, the system Hamiltoia realized is desribed by the simplified Hamiltoias [94], i.e., H sys i i i i j I J I I, () z i< j ij z where the first term arises from the eergy of isolated spis, ad the seod from the eergy of iteratig (oupled) spis. To effet the maipulatio of qubits i NMR-based QC [94], it is eessary to apply a mageti field that will rotate the state of the spi-/ ulei, see Fig This is aomplished by addig to the stati ẑ -direted mageti field, B, a time-varyig (RF) eletromageti field orieted i the xˆ ŷ plae, of a frequey ω RF lose to the spi preessio frequey ω. This RF field gives rise to the otrol Hamiltoia whih, for a sigle spi, is give by [94], H RF [ os( t ) I si( t φ) I ] RF x z B, () where B is the applied RF field amplitude ad φ its phase. For liquid-state NMR, γ B 5KHz ω. I the presee of N spis, the total otrol Hamiltoia is the sum of the terms suh as Eq. () of eah spi. The implemetatio of quatum gates i NMR-based QC exploits the ability to RF y

185 7 Chapter 4 idue a ertai time evolutio of a spi state by the fie perturbatio that varyig the amplitude, frequey, ad phase of the otrol Hamiltoia affords. The aalysis of spi rotatios is failitated by desribig the motio with respet to the so-alled rotatig frame [93], [94]. This is a oordiate system that rotat es with respet to the ẑ axis at a frequey ω RF. A give rot state i the rotatig frame ψ ad the orrespodig state ψ i the laboratory (o-rotatig) frame are related by [9], ψ rot exp ( iω ti )ψ. () RF z It a be show by substitutio of () ito Shödiger s equatio, that i the rotatig frame ad i the presee of may, e.g., r, applied RF fields, the system ad otrol Hamiltoias adopt the forms [94], ad H i j H π I z I, (3) otrol sys J ij i< j i,r ω r z r i r i r i r i [ os( ω ω ) t φ ) I si( ω ω ) t φ ) I ] RF x RF y. (4) The effet of the otrol Hamiltoia is most easily visualized with referee to the Bloh sphere, see Fig. 4-7, whose surfae otais the Qubit Tip Trajetories Figure 4-7. Bloh sphere surfae: Dashed lies delieate the trajetories of the tip of a qubit as a futio of the RF pulse stregth ad duratio. Whe the RF frequey equals the Larmor frequey, i.e., at resoae, the pulse produes a 9 rotatio. As ωrf ω ireases, the rotatio dereases, i partiular, at large offsets the trajetory remais lose to. [94].

186 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 73 lous of the tip of a qubit vetor as a futio of ω RF ω, for a give RF pulse duratio ad the parameter ω. NMR-based quatum gates are geerated by tuig the parameters i the otrol Hamiltoia to ahieve a desired qubit rotatio. Sie ay quatum gate may be ostruted from sigle-qubit rotatios ad the C-NOT gate, the problem of NMR-based quatum omputig redues to determiig the otrol Hamiltoia that will implemet these. I this otext, we ote that the most geeral qubit rotatio is defied by [94], iˆ R ˆ exp, (5) where ˆ deotes the 3-dimesioal axis of rotatio, θ is the agle of rotatio, ad x ˆ x y ˆ y zˆ z is a vetor of Pauli matries. Furthermore, it a be show that ay qubit trasformatio may be implemeted as a sequee of rotatios about oly two axes. I partiular, Bloh s theorem stipulates suh a trasformatio as [94], ( β) R ( γ) R ( δ) i e α R x y x U. (6) Therefore, i terms of the otrol Hamiltoia parameters, implemetig a sigle-qubit gate may be aomplished i the rotatig frame usig RF pulses. Speifially, if a RF field of amplitude ω ad frequey is ω RF ω is applied to a sigle spi, this will evolve aordig to [94], [ ω ( os φi si φi ) ] U exp i, (7) x y t pulse where the RF pulse duratio is give by t pulse. I the otext of the Bloh sphere, this trasformatio would rotate the qubit by a agle θ ~ ω t pulse, with respet to a axis i the xˆ ŷ plae give by the phase φ. For istae, the parameters: φ π ad 9 ω effet the ( ) t pulse π rotatio about xˆ, whereas doublig the pulse duratio implemets R x ( 8), ad hagig the phase to φ π effets the rotatio about ŷ. I geeral, the phase of the RF pulse determies the utatio axis i the rotatig frame, so that to perform xˆ ad ŷ rotatios it is ot eessary to oriet the RF field alog these axes; hagig the phase suffies. A rotatio about the ẑ axis i terms of rotatios about xˆ ad ŷ is give by [94], ( θ) XR ( θ) X YR ( θ)y U R, (8) z y x R x

187 74 Chapter 4 where the bar over X ad Y deotes a rotatio of -9 degrees with respet to X or Y. The NMR-based implemetatio of the C-NOT quatum gate ivolves a series of two-qubit rotatios, amely []:,,, ad. Addressig a partiular qubit, without affetig the eighborig oe, is aomplished by exploitig the fat that differet atoms possess differet resoae frequeies, ω, or that the same type of atoms with a differet hemial shift also possess differet ω. Takig two oupled spi-/ atoms with resoae frequeies ω ad ω, ad ouplig J, the C-NOT gate is implemeted if applyig a arrowbad 8-degree pulse at a frequey ω J /, auses spi to be iverted oly if spi is i state. I this ase, spi is the otrol qubit ad spi the target qubit. Pitorially, the C-NOT gate may be visualized followig the ostrutio of Steffe, Vadersype ad Chuag [], see Fig The sequee of rotatios is produed as follows: ) A RF pulse at a frequey ω, of a badwidth suh overig the frequey rage ω ± J, but that does ot overlap with ω, rotates spi from Z to Y; ) The spi system is allowed to evolve freely for a duratio of / J seods; 3) Durig the free evolutio period, the preessio frequey of spi will be shifted by ± / aordig to whether spi is i the J or state. This will result i the rotatio of spi to either X or X by the ed of this period, depedig o the state of spi ; 4) A 9-degree pulse applied to spi about the Y axis rotates spi to Z if spi is i state, or to Z if spi is i state. Z 9 X Delay(/J ab ) 9 -Y X Y Figure 4-8. Left-to-right: Sequee of qubit rotatios for implemetig the C-NOT quatum gate i NMR-based QC. The oordiate system rotates aroud the ẑ axis at a frequey whe spi is (solid lie), ad (dashed lie). (After [].) ω

188 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 75 While the maturity of NMR spetrosopy has eabled the suessful proof-of-oept implemetatio of various QC algorithms, the fat that the tehique must rely o measurig esembles of spis to obtai a detetable read-out sigal is a limitig aspet of it, sie this implies that oe must begi with the highly-mixed iitial esemble state; this is the result of there beig a very small eergy differee betwee up ad dow spis at room temperature, maifestig itself as a early radom equilibrium distributio [93]. A highly-mixed state possesses equally likely spi-up ad spi-dow states, for example [93], ( ε) ε I/, (9) 5 ε ~, whih is a almost radom state with a small exess of the state [93]. This expressio for the equilibrium state follows from the H/kT desity matrix thermal whih, beig proportioal to e (where the ulear spis i a moleule posses the iteral Hamiltoia H, T is temperature ad k is the Boltzma ostat), admits a expasio [9], ( ) ( /kt ) z z /kt H/kT e e e..., (3) whih with, e ( ) /kt ( I - ) z ε /kt..., (3) σ z may be writte as, H/kT ( ) ( e I /kt - ) /kt..., (3) z z where I is the idetity matrix ad, for spi i, the parameter i represets the eergy differee betwee up ad dow states. While the desired iitial state is a pure oe, i whih all spis are i the same state, e.g.,, the atual radomess of the iitial esemble state may be overome by a tehique to trasform it ito a almost pure state. A almost pure state is oe that produes a sigal that is proportioal to that of a pure-state sigal. It is geerated by exploitig three fats [93], amely: ) That the magetizatio is determied by the traeless part of the desity matrix; ) That the ompletely mixed state I/ is preserved uder both uitary ad o-uitary trasformatios; ad 3) That all sales are relative, i partiular, that oly the ratio of two magetizatios determies the fial aswer of a quatum omputatio, i.e., the deidig fator i a measuremet is, ot the absolute magetizatio, but its relative value ompared to the oise [93].

189 76 Chapter 4 Costrutig a pseudo-pure state makes use of the oept of deviatio desity matrix. This is the arbitrary matrix δ for whih δ ρ λi for some ostat λ. From this defiitio, ad ispetio of Eq. (9), it is lear that the matrix ε is i fat a deviatio matrix from the equilibrium state of oe ulear spi. A iterestig property of the deviatio matrix is that, if mˆ is a traeless operator, the [93], tr ( δmˆ ) tr( ( ρ λi) mˆ ) tr( ρmˆ ) tr( mˆ ) tr( ρmˆ ). (33) Thus, the expetatio value (the measuremet) of a traeless observable may be obtaied either from the desity matrix or from the deviatio matrix, as presribed i Eq. (33). A pseudo-pure state, i fat, is defied as oe whose equilibrium state has the deviatio δ ε. Its sigifiae is as follows. If we are iterested i the probability of p of measurig state, give that the iitial state was, the this is give by [93], p U U tr( U U ) tr( U U ( I σ z ))/ ( tr( U U ) tr( U U σ ) ( tr( U U σ )/ z z /, (34) Where U is the total uitary operator assoiated with a omputatio. Therefore, Eq. (34) idiates that by measurig the iitial ad fial expetatio values of z, a tr( z ), ad a tr( z ) tr( U U σ z ), respetively, oe a determie p. I fat, p ( ( a a )), idepedet of the sale. Most importatly, the tehique may be exteded to the ase i whih oe desires to determie the probability p of measurig the state, i the ase i whih this state refers to the first qubi resultig from applyig a

190 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 77 quatum gate to a iitial state... [93]. The result is the same, amely, ( ( a a )) p, exept that ow the deviatio i questio takes the formδ ε I geeral, if a state has a deviatio proportioal to a pure state ψ ψ, i partiular, δ ε ψ ψ, it is alled a pseudo-pure state. Physially, Cory et al. [] stated that the justifiatio for ostrutig a pseudo-pure state derives from the fat that the spis i the differet moleules of a liquid are virtually idepedet of oe aother ad that, as a result, they may be ostrued as a large umber of opies of a sigle type of moleule, thus permittig the liquid to be approximated by a Gibbs esemble. Beause of N this, istead of dealig with a desity matrix of size, whih is the total umber of moleules, oe a deal with a redued desity matrix of size, where is the umber of spis i a sigle moleule. Aalytially, istead of the desity matrix [], Ψ ( ψ) p ψ ψ dψ, (35) { ψ} where p ( ψ) is the probability desity of the pure state desribed by the spior ψ ad { ψ } deotes the set of all uit orm spiors, oe uses the approximatio [], ( α ) I α ψ ψ Ψ ( α ), (36) ( α ) α where ψ is a uit spior. Thus, sie the esemble average of a observable O is obtaied by takig the trae of its produt by the desity matrix, tr ( OΨ), a simplifiatio is obtaied from usig the pseudo-state, sie the esemble average is ow give by, tr ( OΨ) ( α) tr( O) α ψ O ψ, where tr ( O) is kow. While the pseudo-pure state otiues to be made up of a statistial mixture of moleules, sie by Eq. (36), eah spior determies a uique psudo-pure desity matrix, ad eah pseudo-pure desity matrix determies a spior that is uique to withi a overall phase fator (assumig the polarizatio is α kow), eah additio of the magetizatios of all the moleules reveals the predomiae of oe partiular state preset, i effet apturig eah moleule s state for the fial spetrum without the eessity of wavefutio ollapse []. The prie paid as a result of usig pseudo-pure states is the loss of a fator of the order of oe millio i the effetive umber of

191 78 Chapter 4 moleules per state beause the et polarizatio of spis is oly about oe part i oe millio. Herei lies oe of the mai limitatios of NMR-based QC [93], []: The fat that the pseudo-pure state sigal dereases expoetially with the umber of qubits prepared, while the oise level remais ostat, preludes the methods for extratig pseudo-pure states from workig for more tha about ulear spis. Thus, the use of pseudo-pure states eables oe to obtai a result despite the highly radom ature of the iitial state. The questio the beomes, how does oe trasform a iitial radom state ito a pseudo-pure state with deviatio......? A tehique, amog various, that is employed applies mageti field gradiet to the sample i order to make the frequey of the preessig spis positio-depedet ad, thus, make it possible to distiguish differet parts of the sample. I partiular, the gradiet field idues a positio-depedet phase hage alog the sample. This is the basis of NMR imagig [93]. Aother issue that derives from the esemble ature of the sample, is that are must be take to redue uiteded ouplig betwee qubits [93]. The established tehique to aomplish this is alled refousig [93], [94]. The fudametal idea is to apply a pulse at the midpoit of the evolutio period to a give spi, of suh a phase (typially 8 ) as to udue the evolutio it has experieed over the time period due to the ifluee of the udesired ouplig [93]. Oe ommo issue with QC is the effet of deoheree. I the ase of NMR-based QC deoheree is haraterized i terms of two parameters, amely, the eergy relaxatio rate, T, ad the phase radomizatio rate, T [94]. T aptures the eergy lost by preessig spis to various mehaisms suh as oupligs to other spis, ad to phoos ad paramageti ios, ad hemial reatios suh as ios exhages with the solvet. This soure of deoheree may, by properly hoosig the moleules ad liquid samples, be exteded to tes of seods. T aptures eergy losses due to short- ad log-rage spi-spi oupligs, the effets of flutuatig mageti fields due to the spatial aisotropy of the hemial shifts, loal paramageti ios, or ustable laboratory fields. These fators, by properly hoosig the quality of the samples ad laboratory equipmet allow a deoheree time of oe seod or more for moleules i solutio [94] The Semiodutor Solid-State Qubit Give the predomiae of solid state silio eletrois tehology, there is a strog motivatio to disover ad develop paradigms for quatum omputig that exploit qubits embedded i silio wafers. A early example of this is the sheme for a silio-based ulear spi quatum omputer

192 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 79 itrodued by Kae [], see Fig I this setio this example is reexamied. B J-Gates A-Gates B AC Barrier e - e - Silio 3 P Qubits Figure 4-9. Sketh of ulear spi QC oept. Illustrated are two ells i a oe-dimesioal 3 array otaiig P doors ad eletros i a Si host wafer, separated by a barrier from 3 metal gates o the surfae. B ~ a Tesla, ad B ~ Tesla. (After [].) I this sheme the qubits are embodied i the ulear spis of door atoms loated udereath biasig metalli gates i doped silio strutures, ad the ouplig betwee qubits is eabled by the hyperfie iteratio, whih ouples eletro ad ulear spis. I partiular, with the wave futio of the door eletro beig oetrated at the uleus, a large hyperfie eergy, ad thus ouplig, betwee eletro ad ulear spis is guarateed whih, i tur, may be ommuiated to adjaet qubits by the extesio/overlap of the eletro wave futios of the orrespodig door eletros. Modulatio of the ouplig betwee eletroi wave futios, ad thus betwee qubits, is failitated by the harge ature of eletros, whih eables their maipulatio via applied eletri fields. Quatum omputatio, therefore, may be effeted by applyig voltages through biasig gates loated o the wafer surfae, i partiular, A gates, whih otrol the resoae frequey of the ulear spi qubits, ad J gates, whih otrol the eletro-mediated ouplig betwee eighborig ulear spis. I additio, two other biasig mageti fields are eessary, amely, a global field B a, to eable flippig of the ulear spi at resoae, ad a loal mageti field, B, to break the two-fold spi degeeray of germae to eletros oupyig the lowest eergy-boud state at the door, whih maifests itself at low temperatures. The detailed physis of the silio-based ulear spi quatum omputer is aptured by the parameters goverig the magitude of the spi iteratios, whih determies the time required for maipulatig qubits ad

193 8 Chapter 4 the separatio required betwee adjaet doors. I the presee of a mageti field B z, ad assumig a door uleus with I / embedded i a silio host, the iteratio i questio, amely, the ulear-spi iteratio, is give by the Hamiltoia [], e N e N H e N µ B B z g N µ N B z A, (37) where µ N is the ulear mageto, σ are the Pauli spi matries, g N is the 8 ulear g-fator, ad A µ ( ) B g N µ N is the otat hyperfie 3 iteratio eergy whe the probability desity of the eletro wavefutio, ( ) is evaluated at the uleus. Clearly, examiatio of Eq. (37) idiates that the iteratio eergy is a diretly proportioal to the mageti field ad is a strog futio of the wave futio probability desity at the uleus. A trade-off exists, however, beause for eletros i their groud state the frequey separatio betwee ulear levels is [], A h A g N µ N B A, (38) µ B B whih, for fields B < 3.5T is domiated by the seod term. Thus, i this regime the magitudes of the ulear mageto ad the wavefutio probability desity at the uleus take o a domiat harater. To perform arbitrary rotatios o the ulear spi, Kae idiates that it is eessary to alter its preessio frequey i ompariso with that resultig from the applied mageti field B a []. This is aomplished by exploitig the fat that the proximity of the door-ulear spi system to the A gate allows the hyperfie iteratio to be redued by shiftig the evelope of the eletro-door wavefutio away from the uleus, i.e., by reduig ( ). I essee, suh a shiftig alters the frequey, Eq. (35), ad auses the ulear spi-door system to behave as a voltage-otrolled osillator produig, for a door plaed Å uder the gate, a tuig parameter of the order of 3 MHz/V []. I additio to the sigle-qubit rotatio, the two-qubit C-NOT operatio must be implemeted i order to eable geeral quatum omputatios. I the otext of the ulear spi-door system, aomplishig this requires developig the ability to idue ulear-spi exhages betwee two uleus-eletro spi systems. The iteratio betwee two suh systems is aptured by the Hamiltoia [], H N e N e e e ( B) A A J H, (39)

194 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 8 where ( B) H represets the mageti field iteratio terms betwee spis, the respetive hyperfie iteratio eergies of the uleus-eletro systems is give by A ad A, respetively, ad 4J is the exhage eergy, whih is a futio of the eletroi wavefutio overlap ad, for doors i a host semiodutor of dieletri ostat, ad Bohr radius a B, ad separated by a distae r of about - Å, is give by [3], 5 e r r () 4J r.6 exp. ( 4) a B a B a B The wavefutio overlap, to whih J is proportioal, is aptured by this exhage eergy. Thus, varyig the voltage applied via the J-gate oe a modulate ouplig betwee separated qubits. Oe qubits have bee maipulated to effet a quatum omputatio, the result of the omputatio must be read off. I the silio-based ulear spi QC, this is aomplished by measurig the urret that results from the oversio of ulear spis ito eletro polarizatio, i respose to a bias voltage, see Fig. 4- below. I partiular, this oversio of the ulear spi ito a eletro polarizatio is prompted by the ouplig of the states ad, whih is produed by the hyperfie iteratio betwee the ulei ad the eletroi states as the exhage eergy J is ireased adiabatially from J < µ BB/ to J > µ B B/, see Fig. 4-(a) [99].

195 8 Chapter 4 J A >A Eergy Levels 4J µ B B J (a) A-Gates J-G ates Barrier Si e - P P (b) Figure 4-. (a) Eergy levels for eletros (solid lies) ad lowest eergy-oupled eletroulear (dashed lies) systems as a futio of exhage eergy, J. Whe J < µ BB/, it is possible to perform two-qubit omputatios by exerisig otrol over the level splittig with the J-gate. Above J µ BB/, the states of the oupled system evolve ito states with differig eletro spi polarizatio. Whe J the state of the uleus with the larger eergy splittig, whih is otrollable by the A-gate, determies the fial eletro spi state after a adiabati irease i J. (b) Oly eletros i state a make trasitios ito states i whih eletros are boud to the same door (D - states). These trasitios eliit a eletro urret that is measurable by apaitive meas, thus eablig the uderlyig spi states of the eletros ad ulei to be determied. []. This implies a hage i wavefutio symmetry, i.e., from that of to that of. Two eletros with the latter symmetry, however, are apable of oupyig the same door. I the Si:P the door takes the form of a D state, whih is always a siglet state with a seod eletro bidig eergy of.7mev. Uder these irumstaes, it will be possible, with the appropriate bias betwee the A-gates, to idue eletros from oe door to move the adjaet, already oupied oe i order to establish the D state i it. This harge motio, i tur, is detetable utilizig sigle-eletro

196 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 83 apaitae tehiques ad produes a sigal that remais observable util the spi relaxes; for Si:P this time may be of the order of hours []. Kae poits out that a umber of pratial osideratios must be addressed to make this sheme workable []. For istae, before begiig a omputatio, iitializatio will require the idividual determiatio of gate biases to aout for flutuatios due to the variatio with positio of both doors ad gate sizes. These voltages, i tur, will have to be stored to effet the alibratio as eeded. Also, gate voltage flutuatios i essee ouple the eviromet to the qubits, thus otributig to spi deoheree. This deoheree is eliited by the idutio of differee spi preessio frequeies i pertiet qubits, ad maifests i that two spis i phase at a give time, will be 8 out of phase a time t φ later. It a be show that [], t φ, ( 4) π α ( V) S ( ν ) V st where α d dv is the tuig parameter of the A-gates, with the flutuatig differetial preessio frequey of the spis, S V is the spetral desity of the frequey flutuatios, ad ν st is the frequey differee betwee the ad states. Estimates, assumig the use of 6 low-temperature eltrois to bias the gates, suggest t φ se, whih 5 implies the ability of the ulear spi QC to perform betwee logial operatios durig t φ. Fially, measures have to be take to reder a predomiae of ertai polarizatio of eletros spis, e.g., 6 ( < ), so that they a effetively mediate ulear spi iteratios. This, i tur, requires the eletros to oupy the lowest eergy levels, whih ours whe µ B B >> kt. With B T, this sets the operatig temperature at mk Superodutig-Based Qubits I the searh for two-level quatum systems upo whih qubits might be based, Josephso jutio-based superodutig qubits are urretly the most advaed. I otrast to the previously disussed qubits, whih are based o mirosopi quatum effets of idividual partiles, suh as ios, eletros, or ulei, superodutig-based qubits are based o marosopi quatum oheree effets [4], [5]. These are effets i whih the qubit

197 84 Chapter 4 state is embodied, ot i the wavefutio of elemetal partiles, but o the oheret olletive behavior of may partiles, e.g., a superfluid. Thus, the qubit states are defied by marosopially observed quatities, suh as the harge or the urret of partile odesates. The key to superodutig qubits is the oliear ature of the resoat LC iruit embodied i the Josephso jutio [6]. The quatum mehaial behavior of a liear LC iruit is aptured by the flux Φ through the idutor, whih plays the role of positio oordiate, ad the harge Q o the apaitor, whih plays the role of ojugate mometum, thus eablig the ommutatio relatio [, Q] i. With the Hamiltoia give by, H Φ L Q C, the usual eigeeergy states are give by ( ) E, where ω LC is the resoae frequey. Refletig the quadrati ature of the potetial, the eergy states are equally spaed. Thus, it is diffiult to defie the two lowest states as the qubit states, sie trasitios betwee higher-lyig states are as equally likely [6]. The LC resoator may be made useful as a qubit if its eergy spetrum is aused to exhibit two lowest-lyig states separated from the higher-lyig states. This is aomplished if a oliear idutae is itrodued [6]. I partiular, the oliear Josephso idutae, L J Φ / πi os δ, where δ φ L φ R, φ L, R is the phase of the wavefutio o either side of the jutio, ad I is the ritial urret, itrodues a oliear potetial i whih the two lowest-lyig states are well separated from the higher-lyig states. These variables afford haraterizatio of the Josephso jutio i terms of its eergy, E J ( Φ ext ) Φ I os δ / π E J os δ. I this otext, the ojugate variables of the quatum mehaial desriptio of the LC resoator beome the flux, ow give by Φ ϕ θ, where ϕ Φ π, ad θ δ mod π represets a poit i the uit irle (a agle module π ), ad the harge, ow give by Q en, whih represets the harge that has tueled through the jutio, ad N a operator with iteger eigevalues apturig the umber of Cooper pairs that have tueled. The ommutatio relatio ow is give by [ θ, N] i [6]. The Hamiltoia is give by, ( N Q / e) E θ H E CJ r J os, ( 4) where E CJ ( e) C J embodies the Coulomb eergy for addig oe Cooper pair worth of harge to the jutio apaitae C J, ad Q r embodies a residual radom harge apturig a iitial harge existig o

198 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 85 the apaitor before it was oeted to the idutor [6]. Q r origiates from the ievitable work futio differee ad/or the presee of exess harged impurities o the apaitor eletrodes of the jutio. I the ourse of developig approahes to miimize the effet of Q r, while retaiig the oliearity of the resoator, three fudametal types of Josephso-based superodutig qubits have bee developed, amely, the harge qubit, the flux qubit, ad the phase qubit, see Fig. 4-. Φ ext V g C g L I b (a) (b) () <δ > large L~L J E E E II (d) (e) (f) Figure 4-. Fudametal types of superodutig qubits. (a) Charge qubit. (b) Flux qubit. () Phase qubit. (d), (e), (f) Potetial (dotted lie), showig qualitatively differet shapes for these three respetive qubit types. I (e) the oliearity of the first levels omes about from the aellatio betwee the superodutig loop idutae ad the jutio idutae ear Φ ext Φ /. No losed-form expressios exist for the eigevalues ad eigefutios of the potetial, but its features are aptured by two aspet ratios, amely, E / ad λ L J / L J E CJ. Groud-state wavefutio is also idiated (dasheddouble-dot lie). The x represets a Josephso jutio. (After [6] ad [7].) The ature of the Josephso-based qubit is a futio of the relatioship betwee the relative magitudes of the Josephso eergy, E J, whih reflets the stregth of the ouplig aross the jutio, ad the Coulomb hargig eergy, E CJ, whih reflets the eergy eeded to irease the harge o the jutio by a Cooper pair, e [8].

199 86 Chapter The Charge Qubit The harge qubit, see Fig. 4-, also kow as the Cooper pair box, aims at ompesatig the residual offset harge Q r by biasig the Josephso jutio with a voltage soure V g i series with a gate apaitor C g. I this ase it a be show that the Hamiltoia, with potetial show i Fig. 4-(d), is give by, H E ( N N g ) E J os θ where E ( e) ( ( C C ) C, (43) C J g represets the eergy required for hargig the islad of the box ad N g Q r C gvg / e. To futio as a harge qubit, E > E, i whih ase the iruit favors fixig the umbers CJ J of Cooper pairs. I the absee of tuelig, this state of affairs yields a eergy versus gate voltage as give by the dashed lies i Fig. 4-(b), that is, as the gate voltage ireases, the eergy of the zero state ireases ad that of the oe state dereases. C g V g E/E C E J Cooper- Pair Box (a).5 C g V g e (b) Figure 4-. Charge qubit. (a) A qubit is reated by the superpositio of the two lassial states embodied by the presee of zero ad oe extra Cooper pair i the box. (b) Eergy levels as a futio of otrollig gate voltage. However, i the presee of tuelig, ouplig auses the eergy levels to split ad avoid rossig, thus refletig the reatio of two ew quatum states (solid lies), amely, oe materialized as the symmetri superpositio, ad the other as their of the lassial zero ad oe states ( ) atisymmetri superpositio ( ), both separated by a eergy gap of magitude E J [8]. The dyami behavior of the harge qubit is otrolled by applyig timevaryig sigals to the voltage gate. Iitial demostratio of the oheret

200 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 87 otrol of marosopi quatum states i a sigle-cooper-pair box was reported by Nakamura, et al. [9]. I these experimets, the superpositio of two harge states (i.e. states with differet umber of Cooper pairs N) was deteted by a tuelig urret through a probe jutio. I partiular, a ormal eletro esaped through the probe jutio every time the system adopted the oe state. Cotrol of the state of the qubit was effeted by varyig the legth of the voltage pulse, with the probability of the system returig to the zero or oe state osillatig i proportio to it. The major soure of deoheree was foud to be the probe jutio itself, whih limited the oheree time to s [6]. Nakamura et al. s [6] approah was improved by the quatroium devie demostrated by Vio et al. s [] see Fig I this devie, the Josephso jutio of the Cooper pair is split ito two small parallel Josephso jutios whih are haraterized by their eergy E J os( δ / ), where δ is the superodutig phase differee aross the series ombiatio of the two jutios. These jutios, i tur, are shuted by a larger Josephso jutio, haraterized by a eergy E J E J ad by a phase γ, thus formig a loop. A urret I φ applied to a adjaet oil produes a flux Φ that passes through the loop, with the osequee that it idues a phase φ that ow liks the loop phases as follows, δ γ φ, where φ eφ /. This atio etagles the state of the box, N, via δ, with the phase γ, see Fig. 4-3(a). The quatum state of the qubit is maipulated by applyig a mirowave pulse of frequey ν ν ~ 6.5GHz, the trasitio frequey betwee harge levels i the box orrespodig to the zero ad oe states. Depedig o the pulse duratio, ay state Ψ α β a be prepared. Readig the state exploits the fat that a urret pulse I b () t, see Fig. 4-3(b), of peak amplitude slightly below the ritial urret of the large jutio, I ee J /, auses a superurret to develop i the loop that is proportioal to N. I partiular, whe there is o extra harge i the box, this superurret eliits a lokwise urret i the loop formed by the two jutios, whereas whe there is a extra harge i the box, the urret is outerlokwise. I the former ase, the urret adds to the bias urret i the large jutio with the result that, for preisely adjusted amplitude ad duratio of the I b () t pulse, it swithes to a fiite voltage for a state oe ad

201 88 Chapter 4 preparatio quatroium iruit readout U(t) E J / C g E J / N δ φ γ E J C C I b (t) V tuig I φ V(t) U(t) (a) I b (t) t d τ R I P V th V(t) (b) Figure 4-3. Quatroium iruit. (a) The iruit osists of a Cooper pair box islad (ode N), to whih two small Josephso jutio brahes are oeted. These, together with a larger Josephso jutio, that is shuted by a apaitae C (to redue phase flutuatios), form a loop. The state of the iruit is embodied by the umber of Cooper pairs, N, ad the phases δ ad γ. To tue the quatum eergy levels, a DC voltage V is applied to the gate apaitae, C g, ad a DC urret iruit loop. (b) To prepare arbitrary quatum states, mirowave pulses () t I φ is fored through the oil to produe a flux φ i the U are applied to the gate. To read out the state a urret pulse I b () t is applied to the large jutio ad the resultig voltage V () t aross it is measured. A typial write/read timig sequee is show. (After [].) it does ot swith for a state zero. I essee, the quatroium uses a phase iruit to measure urret, istead of the harge, thus avoidig the probeidued deoheree problem of Nakamura et al s. A deoheree time of.5µ s was measured [] The Flux Qubit The flux qubit, see Fig. 4-(b) above, is osidered as the dual of the harge qubit [6]. It osists of a jutio that is oupled to a urret soure via a trasformer, istead of a gate apaitor, with the jutio itself beig oeted i series with a idutae L, ad the system beig

202 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 89 biased by a exteral flux Φ ext through a auxiliary oil. I the flux qubit the approah to ompesatig the detrimetal effet of Q r relies o shutig the jutio with the superodutig wire of the loop ad hoosig the oditio E < E. This results i makig the quatum flutuatios of q CJ muh larger tha those of Fig. 4-(e), is give by, J Q r q φ e H E J os ext C L J. The Hamiltoia, with potetial show i ( φ φ ), (44) where φ is the itegral of the voltage aross the idutor L, whih gives the flux through the superodutig loop, ad q is its ojugate variable, whih represets the harge o the jutio apaitae C J. Both obey the ommutatio relatio [ φ, q] i. The prototypial flux qubit osists of three Josephso jutios formig a loop ad beig otrolled by a applied mageti field perpediular to the loop to otrol the phase, see Fig E/E J E Φ Φ (a).5 Φ Φ (b) Figure 4-4. Flux qubit. (a) A qubit is reated by the superpositio of the two lassial states embodied by the loop phase of zero ad π. While oe or two jutios would be suffiiet, three jutios allow greater otrol over the behavior of the system. (b) Eergy levels as a futio of otrollig mageti flux. The eergy gap, E Φ ζ( Φ / L)( N Φ / ), plays the same role as E J. ζ is a umerially determied parameter ad N Φ Φ ext / Φ. [7], [8]. I this ase the two qubit states ad are embodied i trasitios i phase from loop phases of to π, whih are assoiated with urrets irulatig aroud the loop i lokwise ad ati-lokwise diretios. I partiular, states of zero ad π phase differee aroud the loop, are oupled whe the flux through the loop equals half the quatum mageti flux i the superodutor, i.e., whe Φ Φ /. Uder this state of

203 9 Chapter 4 affairs, two ew states, ( ) ad ( ), that are quatum superpositios, are formed, with the eergy betwee them ow give by the tuelig stregth. Cotrol of the qubit, suh as to hage its state, is effeted by ouplig to the flux φ, whih is aomplished by sedig urret pulses o the trasformer primary. Measuremets of the states, made with a superodutig quatum iterferee devie (SQUID), a devie whih osists of two Josephso jutio i parallel, to detet the mageti flux, reveals that the urrets are arried by a billio Cooper pairs, with tuelig beig the mehaism by whih the diretios of all of these partiles is reversed simultaeously [8]. The deoheree times, whih are limited by defets i the jutio are i the rage of 5 s to 4µ s The Phase Qubit The phase qubit, see Fig. 4-(), utilizes oly oe Josephso jutio, ad the two quatum states are embodied i the quatum osillatios of the phase differee betwee jutio eletrodes [7]. I this ase the approah to ompesatig the detrimetal effet of Q r relies o usig large ratios of E J / E CJ. A large oliearity i the Josephso idutae is ahieved by biasig the jutio at a urret I ~ I. The Hamiltoia, with potetial show i Fig. 4-(f), is give by, H E CJ p Iϕ δ I ϕ os δ. (45) The ojugate variables, give by the phase differee operator δ, whih is proportioal to the flux aross C, ad the harge o the apaitae ep, obey the ommutatio relatio [, p] i approximated by the ubi form, J δ [7]. The potetial is I ϕ V( δ ) ϕ ( )( ) ( ) 3 I I δ π / δ π /, (46) 6 from where it a be show that the lassial frequey of osillatio at the bottom of the well is give by, J J [ ( I I ) ] / 4 ω p, (47) L C ad the first two levels that a be used for the qubit states have the trasitio frequey ω. ω [7]. 95 p

204 4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 9 Read out of the qubit state is aomplished by exploitig tuelig through the barrier separatig the potetial well from the otiuum, ad subsequet self-amplifiatio due to the egative slope potetial, see Fig. 4- (f). I partiular, sie the barrier beomes thier at higher eergies, ad those higher eergy states have a ireasig probability of esape, the oe state is measured by sedig a probe sigal to idue a partile i the oe state to tuel out of the well. Upo tuelig out of the well, the dowward aeleratio of the potetial leads to the appearae of a voltage / e aross the jutio. This voltage is assoiated with readig a oe state for the qubit; zero voltage is assoiated with readig a zero state. I terms of operatig temperature, it is lear that superodutig qubits must be operated at temperatures suh that kt << ω <<, where ω is the trasitio frequey betwee the eergy levels represetig states ad, ad is the eergy gap of the superodutig material. This eessitates oolig to temperatures of the order of mk. 4.4 Summary This hapter has dealt with a umber of aspets surroudig the atual implemetatio of NaoMEMS iruits ad systems. We bega disussig arhitetural issues, as this is the first step i defiig a NaoMEMS system o hip (SoC). The, emergig adidate buildig bloks, iteded for appliatios ragig from iterfaes to sigal proessig futios, were desribed. These iluded a harge detetor, whih-path eletro iterferometer, torsioal MEM resoator for parametri amplifiatio, Casimir effet osillator, magetomehaially atuated beam, futioal arrays, ad a quatum etaglemet geerator. These buildig bloks represeted aoeletromehaial quatum iruits ad systems (NEMX), as they exploited the oexistee of eletroi ad mehaial strutures. The hapter oluded with a presetatio of physial implemetatios of quatum bits (qubits), suh as the io-trap, the ulear mageti resoae, the semiodutor solid-state, ad superodutig qubits, upo whih quatum omputig paradigms might be prediated.

205 Chapter 5 NANOMEMS APPLICATIONS: PHOTONICS 5. Itrodutio The ability to fabriate aometer-sale strutures has give ew impetus to the field of miiaturizatio of optial devies, whose ultimate goal might be artiulated as that of itegratig optis ad eletrois i the otext of a moolithi tehology. While there are o fudametal limits to the miiaturizatio of eletroi futios dow to ao- ad sub-aometer sales, the miimum size of devies maipulatig optial sigals is limited by diffratio to about half the wavelegth ( λ ) [], whih i pratial terms eompasses dimesios i the several hudreds of aometer []. Two approahes have bee devised to overome these limitatios, amely, the desig of optial elemets based o very high refrative idex materials [], whih is aompaied by high losses i the sub-3 m size regime [3], ad the oversio of photos ito eletromageti modes whose size is determied by the size of the waveguide rather tha by the wavelegth of the optial field [4]. The latter approah is based o surfae plasmos (SPs), olletive osillatios of free eletros resultig from the iteratio of eletromageti waves with free eletros at a dieletri-metal iterfae [5]. I partiular, Dikso ad Lyo [] poit out that, by employig SPs to trasport light, the miimum waveguide size beomes oly limited by a ombiatio of the Thomas-Fermi sreeig legth, whih is ~. m i Au, ad size effets affetig the dieletri ostat, whih have a oset at dimesios less tha 5 m i Au. While we will fous o SP-based approahes, a third approah to sub-wavelegth photoi iruit elemets,

206 94 Chapter 5 proposed by Barrelet, Greytak, ad Lieber [6], employs semiodutig aowires ad will be touhed upo briefly. I this hapter, we deal with the fudametal priiples of aophotois, the proessig of light by aometer-sale devies. I partiular, we address the topis of geeratio, propagatio, ad detetio of surfae plasmos, ad emergig devies based o them. 5. Surfae Plasmos The oept of plasmos emerges from osiderig the motio of a oetratio ( r, t) of free eletros, i a positive bakgroud, as a result of a applied eletri field E. I partiular, assumig the eletros to behave as a fluid of veloity v( r, t), their motio is presribed by the osistet solutio of Newto s ad the otiuity equatios [3], dv m m( v ) v ee, () dt ad t ( v). () As a first step towards the solutio, after egletig the seod term i () due to its quadrati ature i v, oe postulates that the effet of the eletri field is to ause the loal eletro desity to deviate from the ostat bakgroud desity by δ. I this otext, the extet of this deviatio is related to the eletri field by Poisso s equatio, ( ) 4πe E 4πe δ, (3) ad, beause of eletro iertia ad the restorig fore supplied by Coulomb attratio to regai equilibrium, i.e., δ, osillatios esue. These olletive bulk eletro osillatios are deoted as volume plasmos, ad their frequey of osillatio is obtaied by substitutio of δ ito (), resultig i, δ v, (4) t whih, upo differetiatig with respet to time, beomes,

207 5. NANOMEMS APPLICATIONS: PHOTONICS 95 δ δ ee v t t t m, (5) ad whih, i tur, upo substitutig (3) ito (5) beomes, δ 4πe δ. (6) t m Eq. (6), beig aalogous to that of a harmoi osillator, presribes the frequey of plasmo osillatio as, 4π e ω p. (7) m Of partiular iterest i this hapter, is the oept of surfae plasmos, (SPs), Fig. 5-, thoroughly reviewed by Raether [5]. These are eliited by the iteratio of exteral eletromageti fields with surfae eletros, ad are haraterized by a dispersio relatio, a spatial extesio, ad a propagatio legth or lifetime. 5.. Surfae Plasmo Charateristis The dispersio relatio for SPs at the iterfae betwee a dieletri haraterized by ε, deposited o the plae surfae of a semi-ifiite metal ' '' haraterized by ε ε iε, is give by [5], Dieletri, ε Evaeset Wave x z Metal, ε Figure 5-. Sketh of surfae plasmo. The field aompayig a surfae plasmo peaks at the dieletri-metal iterfae ad dimiishes expoetially away from the iterfae. k ω ε i k x, i,, (8) zi

208 96 Chapter 5 where the wave vetor k x, is give by, k x ω. (9) ε ε ε ε Substitutig the omplex dieletri ostat expressio ito (9), the wave ' '' vetor beomes k k ik, with ompoets, k ' x ad k ω x ε ε x x ', () ' ε ε ' 3 / '' '' ω εε ε x '. () ' ε ( ) ε ε Sie ω < k x, see (9), ad ε ' <, (harateristi of the metal), both k z ad k z are imagiary. As a result, the SP field beomes evaeset. The orrespodig spatial deay of the field, away from the iterfae, is thus proportioal to exp( k zi z ) [5], ad is haraterized by the distae at whih it has dereased ito either medium by /e [5]. Thus is give by, z ε ε ' λ, () π ε ito the medium with ε, ad, z ' λ ε ε, (3) π ε ' ito the medium with ε. The propagatio legth L i for SPs propagatig alog a smooth surfae is defied as the distae, away from the iterfae, at whih their itesity, '' exp k x, has dereased by /e, amely, whih is proportioal to ( ) '' x x L i. (4) k

209 5. NANOMEMS APPLICATIONS: PHOTONICS 97 Raether [5] has poited out that at visible wavelegths i silver, L i may be as high as µ m at λ 545Å, ad 5µ m at λ,6å. I additio to haraterizig the SP deay by a distae, it may also be haraterized by its lifetime. This is related to the SP group veloity by, ' '' T i L i v g ad, is a omplex frequey ω ω iω ad real k ' x are '' assumed, may be expressed as T π ω, where from (9), oe obtais, ε ε ε i '' ' '' ' ω k x. (5) ' ' ( ε ) ε ε Sie SPs are assoiated with both a field ad eletro motio, their lifetime is iflueed by mehaisms givig rise to atteuatio. These ilude, radiatio dampig (oversio of the SP ito light due to satterig), eletro satterig proesses givig rise to ohmi losses, ad hemial iterfae dampig due to high iterfae state desities [7]. Two steps are essetial, therefore, i the miiaturizatio of optis by exploitig SPs, amely, the proesses of exitig the SPs by light, ad of trasportig SPs with miimum loss. These subjets are take up by aophotois. 5.3 Naophotois Naophotois deals with the realizatio of aometer-sale optial ompoets ad sigal proessig futios. While the goal is to produe miiaturized optial ompoets, it is oeivable that ompoets i the SP domai, while performig equivalet optial futios, might take differet forms ot derivable from a diret dowsalig of their optial outerparts. Nevertheless, futios suh as light-to-sp oversio, SP wave guidig, ad SP-to-light oversio are expeted to be fudametal to these pursuits Light-Surfae Plasmo Trasformatio Shemes for overtig light ito SPs, ad vie versa, derive from irumvetig the iompatibility of their dispersio relatios, whih do ot iterset, see Fig. 5- below, ad the eessity to oserve mometum. Aordigly, there are two fudametal elemets to supply the additioal mometum, amely, the gratig oupler, ad the ATR prism.

210 98 Chapter 5 ω SP Light C D A Light B SP k AB k CD k x k x Figure 5-. Sketh of dispersio relatios for light, k x ω, ad SPs, ' k x ω εε ε ε. A iomig light wave with wave vetor k x, eessitates ad added mometum k AB to overt to a SP. Coversely, a SP Neessitates losig a mometum k CD to trasform to a light wave. (After [5].) I the gratig oupler tehique, the wave vetor of light impigig upo the gratig-metal iterfae at a agle θ is resolved ito oe ompoet perpediular to the gratig-metal iterfae, ad oe ompoet alog the iterfae, see Fig I partiular, for a gratig of period a, the wave vetors alog the iterfae are give by ω si θ ± g, where is a iteger ad g π a is the reiproal lattie vetor of the gratig. Couplig betwee the light ad the SPs is ahieved whe the oditio, ω ω ε k x si θ ± k x k SP, (6) ε k ω/ k z θ k x k x Fig. 5-3 Coept of gratig oupler to trasform light ito SPs. (After [5].)

211 5. NANOMEMS APPLICATIONS: PHOTONICS 99 that is, whe a iidee agle θ exists at whih the sum or differee of the ompoet of the light wave vetor ad a multiple of the gratig reiproal lattie vetor equal a SP wave vetor. Redutio of a SP vetor by k x trasforms it ito light, whereas additio of k x to the light s wave vetor trasforms it ito a SP. I the ATR method,, see Fig. 5-4, the wave vetor of light impigig upo a hemispherial prism of dieletri ostat ε ad the metal iterfae at a agle θ resolves its wave vetor ito ompoets that are perpediular ad parallel to the prism-metal iterfae. I this ase, ouplig betwee light ad SPs ours whe the ompoet of the light s wave vetor alog the iterfae, ε ωsi θ, equals the SP wave vetor, k x k SP ω εε ε ε. If the metal thikess is fiite, e.g., of extet d, there exists the possibility that for a ertai value of d, the evaeset field at the ε / ε iterfae may ouple to the lower ε / ε iterfae, where it ould also exite SPs [5], see Fig ω ω ε Normal ε θ d ε k x ε Figure 5-4. Coept of ATR oupler. A metal layer of thikess d ad dieletri ostat ε is sadwihed betwee a prism of dieletri ostatε ad a dieletri ε. (After [5].) 5.3. Oe-Dimesioal Surfae Plasmo Propagatio Oe light has bee overted ito SPs, the ext questio is how to provide effiiet eergy guidae. To eluidate the issues ivolved, a umber of studies o surfae plasmo propagatio, utilizig various forms of waveguide, have bee udertake.

212 Chapter SP Propagatio i Narrow Metal Stripes Lampreht et al. [4] oduted studies of SP propagatio i mirosale Au ad Ag metal stripes of widths i the mirometer rage, ad determied the effet of film width o SP propagatio legth, see Fig Propagatig Surfae Plasmos CCD High N.A. Objetive Metal Struture 5 m SiO 5 m Al E θ Glass Substrate Exitatio Regio Measuremet Regio Figure 5-5. Sketh of setup for spatially ofied SP exitatio ad measuremet. (After [4].) I partiular, they fabriated 7 m-thik gold ad silver stripes with widths i the 54 µ m rage. Their experimetal sheme, see Fig. 5-5, ivolved loalized light-sp ouplig by a prism arragemet utilizig a opaque alumium sree to ahieve well demarated exitatio ad propagatio regios. The propagatio legths were observed by detetig SP stray light with a CDD amera upo exitatio with three differet wavelegths, amely, 54, 633, ad 785 m. The experimet oluded that the SP propagatio legth dereased with dereasig lateral stripe width, the rate of derease beig very dramati below µ m, ad ireased with wavelegth. At a wavelegth of 633 m, the propagatio legth i a silver stripe was about 58µ m ad a few miros, for stripe widths of 54µ m ad µ m, respetively SP Propagatio i Naowires Dikso ad Lyo [], oduted studies of SP propagatio i highaspet-ratio metal aostrutures ad m diameter, 5µ m -log Au ad Ag rods, observig propagatio over distaes greater tha µ m for light wavelegths of 53 m ad 8 m. I partiular, they reported that oe the SP propagatio is iitiated, the SPs are guided dow the legth of the wire ad reemerge from the ed as photos via plasmo satterig. I additio, for speifi iidet exitatio wavelegth ad waveguide ompositio, they were able to demostrate uidiretioal SP propagatio.

213 5. NANOMEMS APPLICATIONS: PHOTONICS SP Resoaes i Sigle Metalli Naopartiles Further efforts were made to study the ofiemet of SPs to metalli aopartiles. Amog these, Klar et al. [7] reported the measuremet of SP resoaes i sigle metalli aopartiles, ad of the homogeeous lie shape of their resoae, via photo saig tuelig mirosopy (PSTM) (PSTM detets a sigal at the exit of a optial fiber tip that is proportioal to the ear field.) These SP resoaes are kow to be determied by the dieletri properties of the medium i whih the partiles are embedded, ad by the size ad shape of the partiles, ad are aompaied by a large resoat ehaemet of the loal field both iside ad ear the partile, see Fig. 5-6 [8]. Light Ioi Cluster Surfae Charges Eletri Field Eletroi Cluster Time t Time tt/ Figure 5-6. Sketh illustratig the exitatio of the dipole surfae plasmo osillatio. The eletri field of a iomig light wave idues a polarizatio of the free eletros with respet to the muh heavier ioi ore of a spherial metalli aopartile. The et harge differee is oly felt at the aopartile surfae whih, i tur, ats as a restorig fore. I this way a dipolar osillatio of the eletros is reated with period T. (After [8].) The setup utilized by Klar et al. [7], see Fig. 5-7, osisted of a tuable otiuous wave (CW) laser illumiatig the sample via a tapered Al-oated fiber tip. The aopartiles were gold spheres with a typial diameter of 4 m, ad oupyig a volume fill fratio of 3 %, embedded i a mthik dieletri sol-gel TiO matrix with a refrative idex.9. The experimet proeeded to positio the fiber tip 7 m from the sample ad to shie laser light of various photo eergies, i partiular,, ev, ev,.94 ev, ad.9 ev. Detetio was effeted by a silio photodetetor ad plots of the trasmitted light itesity, saed aross a surfae area of 75 x 75 m were made. Three key results were obtaied i the experimet, amely, a ehaed trasmissio by a maximum fator of, with respet to the bakgroud itesity, for a aopartile loated ear the eter of the sa area, a typial resoae width of ~6meV, orrespodig to a dephasig

214 Chapter 5 time of 7fs, ad a double-peak resoae struture. The field ehaemet was explaied as aused by the exitatio of the SP resoae by the evaeset field of the fiber aperture ad subsequet radiatio, by the partile, of propagatig modes ito the far field, muh like a atea. The Laser Al-Coatig Fiber Tip Au- Naopartile TiO Film Glass Substrate Photodiode Figure 5-7. Sketh of setup for measurig surfae plasmo resoaes i sigle metalli aopartiles. The fiber tip has a aperture diameter of about 8 m ad positioed 7 m away from the m thiktio film, whih is supported by a mm-thik glass substrate. (After [7].) double-peak feature was explaied as deotig the eletromageti ouplig of two lose-lyig partiles SP Couplig of Metalli Naopartiles The properties of SP ouplig betwee lose-lyig metalli aopartiles were studied by Kre et al. [9] ad Kottma ad Marti []. Kre et al. [9] utilized PSTM to eluidate the evolutio of the optial ear-field patter whe a large umber of idetial partiles are arraged i a liear hai. Compariso with theoretial alulatios lead them to ofirm the uexpeted squeezig of the optial ear field due to SP ouplig above a hai of half oblate Au spheroids aopartiles with sizes averagig x m i setio, by 4 m height. Kottma ad Marti [] oduted a theoretial ivestigatio of the plasmo resoaes of iteratig silver ylidrial aopartiles with 5 m diameter at various separatios, e.g., see Fig This figure shows that at a separatio of 5 m ad iidee alog the major axis (i.e., alog the horizotal arrow) a sigle ylider exhibits a resoae (dotted lie) at λ 344m. This resoae has the same magitude, although shifted dow to λ 34m, for two yliders (dashed lie). I additio, a extra resoae at about 37 m is observed (dashed lie) for this latter ase, showig the ouplig of the two yliders. I this ase, a ehaemet i gap field amplitude, with respet to the iidet field amplitude, by a fator

215 5. NANOMEMS APPLICATIONS: PHOTONICS 3 of 8 is observed. Whe the wave is iidet ormal to the major axis (as idiated by the dashed arrow), a broad resoae is observed at λ 38m, with a gap field ehaemet of 4 with respet to the iidet illumiatio. Satterig Cross Setio(m) Wavelegth(m) Figure 5-8. Satterig ross setio (SCS) alulatio of 5 m diameter yliders with 5 m separatio. Illumiatio is i two differet diretios, as idiated by the arrows i the iset. The iidet field polarizatio is i-plae, perpediular to the arrows. The dotted urve orrespods to a sigle ylider. [] Plasmoi Waveguides The oept of exploitig the ouplig of resoat SP fields betwee adjaet metal aopartiles to realize plasmo waveguides was studied by Maier et al. [] via fiite-differee time-domai (FDTD) simulatios ad experimetally. The FDTD simulatios ivolved exitig a liear array of 5 m Au spheres with a eter-to-eter spaig d 75m, ad drive by a soure dipole plaed before the first partile. The drivig pulse was etered at.4 ev, the resoae eergy of a idividual partile ad orrespodig to k π d, the highest group veloity waveguide mode. The pulse had a width of 3 fs, equivalet to 95% of the badwidth of the dispersio relatio for eah polarizatio, ad 4% of the total simulatio time. For a liear hai of ie aopartiles, the FDTD simulatios predited group veloities of m / s ad 5.7 m / s for field exitatios of trasverse ad logitudial polarizatio, respetively. Similarly, eergy deay legths, estimated by moitorig the maximum field amplitudes at the eter of eah partile ad at the logitudially polarized soure, of 6dB/8m ad

216 4 Chapter 5 6dB/86m were determied. The FDTD study oluded that by optimizig partile geometry it should be possible to ahieve eergy trabsport at a veloity of. ( is the speed of light). The diret experimetal evidee of eergy trasport a waveguide osistig of liear arrays of 9 m x 3 m x 3 m rod-shaped Ag aopartiles with a iter-partile spaig of 5 m ad havig the log axis of the rods orieted perpediular to the propagatio diretio to irease the ear-field ouplig was fabriated. To probe eergy trasport, the fluoresee of Moleular Probes Fluorspheres F-88, polystyree aospheres with a diameter of ± 8m, plaed radomly alog the waveguide, see Fig. 5-9, was deteted. Itesity Fiber Dye Laser Light Field Couplig Naopartiles Far Field Detetio Figure 5-9. Sketh of SP propagatio detetio alog waveguide by fluoreset moleules. (After [].) The proedure etailed exitatio of the first partile i the waveguide by ouplig laser light at a wavelegth of 57 m, the sigle partile resoae wavelegth, via the tip of a optial fiber, ad moitorig its propagatio dow the guide by measurig the positio-depedet itesity of the light emitted by the fluoreset moleules. The presee of plasmo trasport was sigaled by a broader full width at half maximum of the fluoreset ao spheres whe a sa is doe alog the waveguide tha perpediular to it. The results of the experimet were a deay legth of 6 db /95 ± 8m, orrespodig to a eergy propagatio distae of.5µ m Naophotoi SP-Based Devies While still i its ifay, a umber of SP-based devies have bee proposed [], []. For istae, Bozhevolyi et al. [] advaed SP-

217 5. NANOMEMS APPLICATIONS: PHOTONICS 5 based waveguidig strutures ispired by photoi badgap rystal (PBC)- based desigs. I partiular, the propagatio of SPs i the rage of 78-8 m lauhed ito aostrutured gold film surfaes with areas of -mwide satterers arraged i a 4-m period triagular lattie otaiig lie defets was demostrated, see Fig. 5-. (a) (b) Figure 5-. Sketh of SP-PBC devies. (a) Lie defet waveguide. (b) Lie defet jutio. The white irles represet 45-m-thik gold posts. The periodiity of the metalli satterers was arraged to ihibit SP propagatio iside these areas, thus reatig a plasmoi bad gap at a ertai rage of wavelegths, i partiular, at 85 m. Guidae of SPs ourred at 78 m alog the lie defets. This was the first observatio of SP bad-gaps ad SP guidig alog lie defets i SP-PBC strutures. Figure 5- shows skethes of the SP-PBCs. Kre et al. [], o the other had, demostrated two-dimesioal optis based o SPs, i partiular, loal SP soures, Bragg mirrors, ad beam iterferometer. The goal of the SP soure was to lauh laterally a SP beam, ad was based o the gratig approah. I partiular, it osisted of periodi aosale protrusios o a metalli film with geometries providig the mathig betwee the light ad SP wave vetors. SPs were lauhed by

218 6 Chapter 5 fousig a 75 m, 5mW laser beam o a silver aopartile of m diameter ad 6 m height. Bragg mirrors, see Fig. 5-(a), osisted of five Bragg Mirror SP Lauher (a) (b) SP Lauher SP Lauher Figure 5-. Skethes of SP-based devies. (a) Bragg mirror. (b) Beam iterferometer. The irle represets the fous of the impigig laser. The dashed arrows represet propagatig SPs. lies of gold 4 m diameter, 7 m height gold ilied at a 3 agle with respet to the aowire used for lauhig the SPs. Withi eah lie, the eter-to-eter partile distae was m ad, to fulfill the Bragg oditio at a SPP wavelegth of 6 m, the iter-lie distae was 35 m. A refletio oeffiiet of ~9% was estimated. Sie the trasmitted itesity was foud to be egligible, this was take to mea that % of the SP itesity was overted to light. A beam iterferometer was ofigured

219 5. NANOMEMS APPLICATIONS: PHOTONICS 7 by ombiig two Bragg mirrors symmetrially with respet to a aowire used for lauhig SPs, see Fig Semiodutig Naowire-Based Naophotois I additio to the SP-based aophotois approah, a approah based o usig ative aowire waveguides has bee advaed by Lieber s group [6]. This approah is motivated by a attempt to irumvet the loss limitatios exhibited by passive waveguides, suh as SP-based devies, whih may hider their appliability for maipulatig light over the extet of itegrated photoi systems. Early examples of semiodutig aowires ilude aosale lasers [3], i whih a sub-wavelegth diameter aoavity is reated by exploitig the high refrative idex otrast betwee a aowire ad its surroudigs. The ative waveguide oept pursued by Lieber s group [6] ivolves utilizig avities suh as these as waveguides. The feasibility of the oept was ivestigated by quatitatively haraterizig the losses through straight ad sharply bet CdS aowires, of sub-wavelegth ( m) diameter, by saig optial mirosopy. I partiular, the experimets reorded spatial maps of the itesity of light emitted from oe ed of the aowire, as a futio of the positio of a diffratio-limited laser spot with eergy greater tha the CdS bad gap. I this otext, the laser eergy absorbed by the CdS aowire was re-emitted via photolumiesee ad subsequetly guided by it. The experimet idiated that ative CdS aowires are apable of effiiet guidig over straight ad sharp ad aute agle beds, with typial losses of about -db i a abrupt bed. I additio, by studyig the harateristis of jutios betwee two aowires it was foud that light may be oupled effiietly through sub-wavelegth beds defied by them. Fially, by applyig a variable eletri field aross a aowire, it was demostrated that it is possible to modulate the itesity of 5 the light exitig the aowire ~5% at a field of ~.4 V / m. 5.4 Detetio of Surfae Plasmos The detetio of SPs relies o their oversio to light, ad the subsequet detetio of this light. I this otext, oe a metio two detetio shemes. I oe sheme, detetio is effeted by moitorig the light emitted by fluoreset moleules overig the etire devie; suh was the approah employed i Setio to show diret evidee of SP propagatio i a plasmo waveguide []. This approah is more of a diagosti tool ad does ot seem ameable to utilizatio i atual sigal proessig systems where oe is iterested i detetig the output at the exit

220 8 Chapter 5 of, e.g., a aowire. I a seod approah, a ear-field saig optial mirosope (NSOM), whih allows sub-wavelegth resolutio [4], is utilized. I this setio we provide the fudametal priiples of operatio of the NSOM NSOM/SNOM Near-field saig optial mirosopy (NSOM), also alled saig ear-field optial mirosopy (SNOM), is a super-resolutio optial mirosopy tehique that eables the ability to view samples at spatial resolutios beyod those attaiable with ovetioal optial tehiques [4], [5]. Covetioal optial tehiques are limited by the diffratio of light. This is haraterized by the size of the spot to whih a light beam a be foused. The spot is part of a family of oetri rigs, kow as the Airy disk patter, ad its size is defied as the distae d from the poit of highest itesity, loated at the middle of the eter spot, to the first ode i itesity (demaratig the begiig of the first rig), ad it is give by, λ d.6, (7) si θ where λ is the free-spae wavelegth, is the idex of refratio o the medium i whih the light propagates, ad θ is the agle desribig the light overgee for the fousig elemet [5]. With the value of the deomiator, deoted as umerial aperture (NA), for the objetive, beig typially as high as.3-.4, (7) is usually simplified to d λ. This is take as the distae two objets may be approahed to oe aother other while still beig distiguishable. To irumvet this limit, Syge [6], [7] proposed the sheme show i Fig. 5-. Iidet Light Opaque Sree Near Field Far Field Sample Surfae Wavelegth of Light Figure 5-. Sketh of Syge s oept for overomig diffratio limit. (After [5].)

221 5. NANOMEMS APPLICATIONS: PHOTONICS 9 Here a opaque sree otaiig a aperture of dimesio muh smaller tha the optial wavelegth is iterposed i the light path, i frot the sample surfae, thus irumsribig the passig light to diffrat from this small aperture. Fig. 5-3 shows a sketh of a typial SNOM imagig system. Normal Fore Sesor Laser Laser Laser PSD Optial Fiber Laser Tapered Optial Fiber Probe Tip Sample o Saig Stage Figure 5-3. Sketh of typial SNOM system. The probe-sample distae is otrolled via ormal fore feedbak. (After [8].) By plaig the sample surfae i the immediate viiity of the aperture, the light emergig from it would be made to iterat with the sample before diffratig out, thus allowig a higher resolutio image to be formed. I pratie, the sample is illumiated via a 5- m-diameter hole i a tapered optial fiber probe tip [8]. The system may be operated i at least four modes, Fig. 5-4, aordig to whether the probe tip is used for illumiatio, for light olletio, or for both [8]. I the trasmissio mode, Fig. 5-4(a), the probe tip illumiates the sample ad the trasmitted light is olleted ad proessed. I the refletio mode, the probe tip illumiates the sample, ad the refleted sample is olleted ad proessed. I the olletio mode, Fig. 5-4(), a exteral light soure illumiates the sample, ad the probe tip ollets the light refleted from the surfae. I the illumiatio ad olletio mode, Fig. 5-4(d), the probe tip is employed to both illumiate the sample ad ollet the refleted light.

222 Chapter 5 Figure 5-4. Modes of operatio of SMON system. (a) Trasmissio mode imagig. (b) Refletio mode imagig. () Colletio mode imagig. (d) Illumiatio/olletio mode imagig. (After [8].) The theory of diffratio by small holes was origially treated by Bethe [9] ad orreted by Bouwkamp [3], [3]. The proper expressios for the field ompoets i the ear-field regio i the immediate viiity of the aperture are give by [3], ikau x y E x ikz v artav, (8) π 3 u v 3a ( u v )( v ) E E y z 4ikxyu, (9) 3πa ( u v )( v ) 4ikxv, () 3πa ( u v )( v ) where a is the aperture radius, k is the wave umber, ad x, y, ad z are related to the oblate-spheroidal oordiates u, v, ad ϕ via the equatios, / [( u )( v )] ϕ / [( u )( v )] ϕ x a os y a si, (), () z auv. (3) 5.5 Summary This hapter has dealt with the appliatio of NaoMEMS tehiques to photois. After poitig out the limitatios of ovetioal optis to reder miiaturized devies at sub-wavelegth sizes, we wet o to osider the