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Miller Thesis - Center for Quantum Devices

1. E 0 5 B mT 0 5 1 A e 2m 0 02 on 20 5 3 icc Are geet L 1000 mV 4 94 ko M53 c 5 24 3 9 A 00 02 i e 250 MV i S Qr a a S aom 7X te i nN an n EN r x M m SEL 4 M et 5 L 0mV N m N d i Ue OE cese E md m a ex tT 0 027 200 mV u ae a F y tern eet 0 6 0 4 0 2 0 0 0 2 0 4 0 6 B mT Figure 5 4 Difference of average conductance g from its value at B 0 as a function of B for various center gate voltages V in the center gated dot squares along with fits to RMT 116 150 Good fits are obtained though the theory assumes homogeneous SO coupling Error bars are the size of the squares Inset Aso and Kj asa function of V extracted from RMT fits see text are shown in the inset in Figure5 4 We note that the 2D samples from the same wafer did not show gate voltage dependent SO parameters 105 However in the 2D case a cubic Dresselhaus term that is not included in the RMT of Ref 116 150 was significant For this reason fits us ing 116 150 might show A V though the 2D case did not Further investigation of the gate dependence of SO coupling in dots will be the subject of future work 43 5 9 Acknowledgements We thank I Aleiner B Altshuler P Brouwer J Cremers V Falko J Folk B Halperin T Jungwirth and Y Lyanda Geller This work was supported in part by DARPA QuIST DARPA SpinS ARO MURI and NSF NSEC Work at UCSB
2. return err XOPEntry This is the entry point from the host application to the XOP for all messages after the INIT message static void XOPEntry void long result 0 switch GetXOPMessage 0 case FUNCTION our external function being invoked result DoFunction 100 break case FUNCADDRS result RegisterFunction break SetXOPResult result main toRecHandle This ts the initial entry point at which the host application calls XOP The message sent by the host must be INIT main does any necessary initialization and then sets the XOPEntry field of the ioRecHandle to the address to be called for future messages HOST_IMPORT void main ioRecHandle IORecHandle ioRecHandle ifdef applec for MPW C for 68K only void _DATAINIT void _DATAINIT for MPW C only UnloadSeg DATAINIT endif ifdef XOP_GLOBALS_ARE_A4_BASED ifdef MWERKS SetCurrentA4 Set up correct 44 This allows globals to work SendXOPA4ToIgor ioRecHandle GetA4 And communicate it to Igor endif endif LoadXOPSegs XOPInit ioRecHandle do standard XOP initialization SetXOPEntry XOPEntry set entry point for future calls if igorVersion 200 SetXOPResult REQUIRES IGOR 200 else SetXOPResult OL if GENERATINGPOWERPC pragma options align reset endif ifdef _WINDOWS_ pragma pack endif 101 All structu
3. Das Sarma Freedman and Nayak 2 have proposed using these braiding rules alone with out taking advantage of the fusion rules explicitly to at least determine whether the v 5 2 state is non Abelian via an interference experiment The idea of the experiment is to localize 4 and 42 on antidots in a three quantum point contact QPC interference device see Figure 2 1 Two tunneling paths at oPC 1 and orc 3 would interfere Assume for now the interference is positive Oxx X f1 it Next 73 would be allowed to controllably tunnel across gpc 2 which should flip the qubit and change the sign of the interference to v3 f4 it If the interference changes as a result of the braid operation then the v 5 2 state must be non Abelian This proposal is exceptionally ambitious It is however the proposal that motivated our own experimental effort at v 5 2 Shortly after Das Sarma Freedman and Nayak published their proposal several other au thors published modifications that make the experiment simpler but still capable of probing non Abelian statistics of the v 5 2 state 3 6 These other proposals all require the use of gates to manipulate the v 5 2 state and typically call for tunneling of quasiparticles between the v 5 2 edge states In the remaining few sections of this introduction the tone will become much more experi mental I will review prior experimental work at v 5 2 discuss our more immediate ex
4. 0 the eigenstates are Landau levels for a charge 2e parti cle in the magnetic field B analogous to the spinless problem 135 For S 1 eigenstates of H and P in general require a numerical solution although analytic solutions exist when either 31 amp 1 OF amp equals zero 132 An analytic solution is found when both a and a2 may be nonzero when g lt Aso where Aso 204 2 m h is the distance over which spin rotates appreciably if gt Aso or dephases if lt Aso due to spin AB like effects Performing a unitary trans formation H H UTHU with U exp ieAs r and expanding in coordinates we find H i k 2eAem Sza where a H gr x 2 212 and Her 2 a5 a2 m eh is the effective SO field P can then be block diagonalized for each m m 0 1 using the Landau basis for particles with charge 2e in the magnetic field B mH g Thus the effect of Op and Ong is to produce spin quantal phases of the AB type 116 123 126 129 Higher expansion terms to H describe spin flip processes and can be taken into account by introducing a spin relaxation time Tso and its corresponding field scale Hs ht 2el Tso The resulting quantum interference contribution takes the form 132 2 Ac B i 2 CCm fim C xoo foo 4 2 where xs B mH g Hir describes the AB dephasing in Ho C x fsm XY N o um Px fsm y fg exp Yfsmt t 2 Ln t dt Ly z are La
5. 20 Das R and Kumar A Applied Physics Letters 89 024107 2006 21 Ballentine L World Scientific New Jersey 1998 22 Wilczek E Phys Rev Lett 49 957 1982 23 Preskill J http www theory caltech edu preskill ph219 topological pdf 2004 24 24 St rmer H L Rev Mod Phys 71 875 1999 103 25 Das Sarma S in S Das Sarma and A Pinczuk eds Perspectives in Quantum Hall Effects Wiley New York 1997 Abrahams E et al Phys Rev Lett 42 673 1979 Halperin B I Phys Rev B 25 2185 1982 Laughlin R B Phys Rev B 23 5632 1981 Wen X G Phys Rev Lett 64 2206 1990 Datta S ed Cambridge University Press Cambridge 2003 Shapere A and Wilczek F World Scientific New Jersey 1989 Laughlin R B Phys Rev Lett 50 1395 1983 Jain J in S Das Sarma and A Pinczuk eds Perspectives in Quantum Hall Effects Wiley New York 1997 Arovas D Schrieffer J R and Wilczek F Phys Rev Lett 53 722 1984 Goldman V J and Su B Science 267 1010 1995 de C Chamon C et al Phys Rev B 55 2331 1997 I do not think Wilczek expected the anyon which he considered essentially a toy mathe matical object to assume a physical form so soon after he published it In the introduction of his anyon paper 22 using a strategy now rarely seen in PRL introductions he stated that practical applications of these phenomena seem remote That was the only point in the paper
6. 211 Chang A M and Cunningham J E Solid State Comm 72 651 1989 212 Chang A M and Cunningham J E Surf Sci 229 216 1990 213 Wang J K and Goldman V J Phys Rev Lett 67 749 1991 214 W rtz A et al Physica E 22 177 2004 109 van Wees B J et al Phys Rev B 38 3625 1988 Alphenaar B W et al Phys Rev Lett 64 677 1990 Alphenaar B W et al Phys Rev B 45 3890 1992 Chung Y C Heiblum M and Umansky V Phys Rev Lett 91 216804 2003 Camino F E Zhou W and Goldman VJ Phys Rev Lett 95 246802 2005 Willett R L et al Phys Rev B 37 8476 1988 Engel L W et al Phys Rev B 45 3418 1992 Gate voltages were restricted to the range 1 9 V depletion to 3 V and allowed to stabilize for several hours at each setpoint Beyond 3 V the conductance was typically hysteretic as a function of gate voltage 223 Wong L W Jiang H W and Schaff W J Phys Rev B 54 R17323 1996 224 Kane C L and Fisher M P A Phys Rev Lett 72 724 1994 225 Lu Y private communication 2006 110
7. R 41b Ops qv ykeky h 4 1c where y arises from the lack of inversion symmetry of the GaAs crystal while a y k2 also depends on the thickness of the wave function in the quantization direction The Rashba coefficient 2 depends on the potential profile of the heterointerface In fitting the data below we assume the effect of gate voltage Vz on Q QJ is through the carrier density n k 27t Good agreement between theory and experiment Figure 4 2 supports this assumption as do previous studies of SO coupling in single interface heterostructures 85 Although 2 can be treated as directly proportional to a uniform electric field 145 the magnitude of 2 in a single interface heterostructure originates mainly from the band offset at the heterointerface which is essentially independent of V 86 140 The symmetry of the linear in k SO terms Op and Or allows these terms to be rep resented as a spin dependent vector potential A that affects the orbital motion and phase of electrons c Opi Og k A 116 125 126 129 That is the linear terms affect electronic interference as a spin dependent AB like effect In contrast the cubic term Eq 1c upon re moving terms with the symmetry of Eq 1a only causes spin relaxation in the diffusive regime although it also can produce AB like effects in the quasi ballistic regime 123 To develop the theory of 2D magnetotransport with SO coupling beyond the d
8. The measurement circuit for the red highlighted orc is drawn schematically with the direction of the edge current flow indicated by the yellow arrows 54 As the temperature increases from 30 mK to 7o mK the plateaus in the Qrc disappear Fractional plateaus are not observed in a 0 5 jm orc and the Ig characteristic is flat for all magnetic fields Together these observations suggest that the 5 2 state is destroyed in the 0 5 ym orc but can survive and exhibit quasiparticle tunneling 66 69 70 202 204 in the larger Qrcs 7 2 Measurement Techniques We measure Ryy Rxx Rp and Ry Fig 7 1 as four wire differential resistances R dV dIac 205 In the IQHE regime these resistances can be readily interpreted in terms of edge channels 206 207 where Npuik is the number of edge channels in the bulk and Nopc lt Nou is the number traversing the apc The bulk Hall resistance Ryy h e 1 Npuk 208 probes the number of edge states in the bulk region The bulk longitudinal resistance Rxx vanishes when Ry shows a plateau The diagonal resistance across a QPC Rp h e 1 Nopc is sensitive only to the number of edge channels traversing the orc and hence provides a orc analog to the bulk Ry The longitudinal resistance across the orc Rg Rp Ryy contains information about both the bulk and the orc region and is not directly analogous to the bulk Rxx On bulk IQHE plateaus the filling fraction is equivalent to the number
9. inset at V 240 mV to essentially pure AL at V 250 mV is observed This crossover demonstrates that a gate can be used to control SO over a wide range as pure WL corresponds to negligible SO rotations within the phase coherence length Lg while AL corresponds to spin rotations 27t The solid curves in Figure 4 1 a are fits of Eq 4 2 with three free parameters Hg H and Heg Ht is fixed at each gate voltage by measured values of density and mobility Figure 4 2 shows extracted parameters H7 and H as a function of 1 Hx is well described by the predicted linear dependence on n with a best fit Figure 4 2 solid line giving y 81 3 eV with zero y intercept see Equation 4 1c The density dependence of H7 is well described by Equation 4 3 Figure 4 2 dotted curve giving fit parameters y 28 4 eV a 4 1 meV and a 5 1 meVA In this way the three SO parameters a1 2 and y are separately obtained from transport measurements by explicitly making use of the density dependence of H24 and H Extracted values of H correspond to dephasing times in the range 33 gl 1100 180 6r E I 7603 MP oo 5 140 2 lt S 120 e Or 30 0 10 20 30 40 50 n 107 m Figure 4 2 Spin orbit effective fields H filled circles and H open squares as extracted using Eq 4 2 plotted as a function of sheet density squared The best fit of Eq 4 3 to Hy
10. 7 1 All quoted temperatures are measured using a RuO resistor mounted on the mixing chamber The bulk mobility of the device measured at base temperature is 2000 m Vs and the electron density is 2 6x 10 m 7 5 Bulk Measurements Bulk Rxx and Rx measurements for the filling fraction range 44 3 to 2 measured in the vicinity of the 1 2 um Orc before the gates are energized are shown in Fig 7 2 Rxx and Rxy are also measured in a region of the Hall bar without gates and found to be virtually indistinguishable showing that the surface gates do not significantly affect the 2DEG Rxx and Ry in an un gated region show no changes caused by energizing gates As temperature is increased Ry near vp 5 2 evolves from a well defined plateau at 56 8 1 2um QPC Rp 0 254 AL T a ao 0 20 0 25 b Bulk Ryy Ry Figure 7 3 Typical IQHE magnetoresistance measured concurrently in the orc a and the bulk b Quantized resistance values are indicated in units of h e The yellow stripe indicates vopc 5 and vy 6 Likewise the blue stripe indicates vopc 4 and vy 5 Ryy 0 4 0 0002 h e to a line consistent with the classical Hall effect for a material with this density There is a stationary point in the middle of the plateau where Rxy is very close to 0 4h e consistent with scaling seen in other quantum Hall transitions 25 223 Activation energies A for the three fractional states vpuk 5 2 21 3 a
11. 7 855nA Each trace is a different B field Traces Not Offset 42 8 4 0 4 8 12 0 3604 Figure 2 3 A series of I4 curves from the o 8 um Qrc each taken at a different magnetic field The The thick red curve that approaches R 0 375 h e at high current is the v 21 3 characteristic which has a dip instead of a peak The thick black curve that approaches R 0 4h e is the 5 2 characteristic which shows a peak At this Qrc gate voltage it was not possible to measure the v 21 3 characteristic because the required magnetic field would take the bulk filling fraction away from v 3 Figure 2 4 A prototype device that could in principle be used to adjust the steepness of the potential in a orc using only some of the gates or in a quantum dot we do not observe any plateau like features for the v 22 3 state in the orc we do observe a hole like zero bias dip in resistance for this state consistent with Roddaro The impact of these findings upon the theories of v 5 2 has yet to be fully explored 16 Future directions The opportunities for continued experimental work on FQHE states in general and the v 5 2 state in particular are myriad More detailed studies of the tunneling properties of the v 5 2 state especially at various temperatures should be carried out The use of extra gates in and around a arc or quantum dot to adjust the steepness of the potential profile or increase the electron density inside
12. I hope to provide just enough back ground to allow me to paint an accurate and intuitive picture of the physics of the v 5 2 state at least as it is understood at this time including the proposed use of 5 2 for quantum computation I then review previous experimental work Finally I discuss in detail the contributions my PhD work has made to this field and I discuss possible future directions Those wishing to come back to the theoretical review later may skip directly to the experimental contributions of my PhD work in Section 2 4 In the last section of the introduction I discuss experimental techniques Prologue Kitaev invented the roc in 1997 7 The only catch is nobody really knows exactly how to build one One possibility is the method published in 2003 by Duan Demler and Lukin 8 which is a general technique that allows one to induce and control strong interaction between spin states of neighboring atoms in an optical lattice including a way to realize experimentally the exotic Abelian and non Abelian anyons that are required for quantum computation Another possible way to realize a TQC was introduced in early 2005 just as I was thinking about what I wanted to work on as one last great project for my PhD Das Sarma Freeman and Nayak proposed using the v 5 2 FQHE state to implement at least elements of a topological quantum computer 2 9 At that time I decided to spend the last four to six months of my PhD working
13. RMT 94 95 was and is an active area of theoretical research even as we conducted the large dot experiments reported in Chapters 5 amp 6 The exact spectrum of energy levels of a large dot may be too complicated to write down but according to RMT the statistical distribution of energy level spacings is universal depending only on the presence or absence of symmetries of the system Dots can be categorized by symmetry properties such as time reversal symmetry spin orbit coupling and spin degeneracy each category possessing a predictable energy level distribution RMT can predict certain measurable features of the chaotic system such as the average and variance of conductance The variance of conductance which arises from the interference of multiple transport paths through the device is completely analogous to ucr in bulk samples However unlike 2D systems the quantum dot is in many ways an ideal place to study quantum chaos Because the overall conductance of the device can easily be set to a few e h the ucr represent a large fractional change in the signal and are easy to measure Quantum dots can be fabricated in any size so many of the relevant scales in the problem can be set by the experimentalist Dots are extremely tunable so large statistically independent samples of the quantum chaos can be measured just by changing a voltage Importantly although dots bestow significant control to the experimentalist they do not trivialize t
14. The papers and appendices After the introductions the second part of the thesis consists of our detailed results in the form of papers that have either been published or submitted for publication The third part consists of appendices which give very technical details of our procedures and techniques that may be of interest to future experimentalists CHAPTER 2 Introduction to the 5 2 experiment 2 1 Overview The work that has occupied the final two years of my PhD the effort to study and eventually manipulate the fractional quantum Hall effect FQHE state at filling fraction v 5 2 in meso scopic devices has been especially challenging but especially rewarding In Chapter 7 I will describe the outcomes of this work in detail Briefly we experimentally observed a plateau near v 5 2 in quantum point contacts QPC and also found evidence that we can use Qrcs to induce quasiparticle tunneling between fractional edge states This is exciting because tunneling of the quasiparticles at v 5 2 is one of the technological capabilities required to test whether these quasiparticles obey non Abelian statistics 2 6 If the v 5 2 state turns out to be non Abelian then needless to say this would be a tremendously exciting discovery aside from being a new unique state of matter a non Abelian state could in principle be used to implement a topological quantum computer TQC In this chapter I first provide a theoretical introduction
15. Transport near vopc 5 2 in the Qrcs is consistent with a picture of chiral Luttinger liquid edge states with inter edge tunneling suggesting that an incompressible state at vorc 5 2 forms in this confined geometry This chapter is similar to an article to appear in Nature Physics August 2007 53 7 1 Introduction The discovery 45 of a fractional quantum Hall effect FQHE at the even denominator filling fraction v 5 2 has sparked a series of experimental 60 62 190 193 and theoretical 51 194 195 studies leading to a prevailing interpretation of the 5 2 state as comprised of paired composite fermions condensed into a BCS like state 46 48 50 Within this picture excitations of the 5 2 ground state possess nonabelian statistics 55 56 196 and associated topological properties The possibility that such a topological state can be accessed in the laboratory has prompted recent theoretical work aimed at experimentally testing the nonabelian character of the 5 2 state 3 6 197 and building topologically protected quantum gates controlled by manipulating the excitations of the 5 2 state 2 198 199 While proposed tests of the statistics of excitations of the 5 2 state make use of confined few micron geometries previous studies of the 5 2 state have been conducted in macro scopic 100 pm 5 mm samples Although experiments using mesoscopic samples with a quantum point contact QPC are now routine the 5 2 st
16. WERT He hoa HY ea he ae ae 6o Si hy a C2 Ohmics i54 ek DAA BAA BRE KE YO ek eo BAS G13 Small cates octave een ek ug s dose oe RA Ba we Ege CTA Connector pates 5 mee hu ee RE et eee eae he ete aes vi D Wafer Data Sheet E Antilocalization Fitting Routines vii 89 91 viii Acknowledgements The Marcus lab is an amazing place to do physics Unlimited helium un limited lock in amplifiers unlimited 50k gigahertz pulse generators no broken equipment to be found and yes tools generally kept where they belong Thanks go to Charlie for invisibly doing the dreary never ending grant writing and note writing that made it possible to do exciting world class research without ever having to worry about a dollar or a missing tool But grants alone do not teach students how to become scientists that requires a role model and a mentor With preter natural creativity irrepressible optimism a keen eye for interesting problems and for craftsman ship and elegance in their solution nor easterly energy levels a willingness to do anything from the menial to the impossible to get the research done and always knowing what to do next one step at a time Charlie is the consummate experimentalist Besides that he is a pleasure to work with quick to make or get the obscure joke and always ready with the unexpected allusion or anecdote Charlie s ability to pair students with the right theorist at the most mutually beneficial
17. Work at UCSB was supported by QUEST an NSF Science and Technology Center 36 CHAPTER 5 Spin Orbit Coupling Antilocalization and Parallel Magnetic Fields in Quantum Dots D M Zumb hl J B Miller C M Marcus Department of Physics Harvard University Cambridge Massachusetts 02138 Division of Engineering and Applied Science Harvard University Cambridge Massachusetts 02138 K Campman A C Gossard Materials Department University of California at Santa Barbara Santa Barbara California 93106 We investigate antilocalization due to spin orbit coupling in ballistic GaAs quantum dots Antilocalization that is prominent in large dots is suppressed in small dots as anticipated theoretically Parallel magnetic fields suppress both antilocalization and also at larger fields weak localization consistent with random matrix theory results once orbital coupling of the parallel field is included In situ control of spin orbit coupling in dots is demonstrated as a gate controlled crossover from weak localization to antilocalization This chapter is published in Phys Rev Lett 89 276803 2002 37 5 1 Introduction The combination of quantumcoherence and electron spin rotation in mesoscopic systems produces a number of interesting transport properties Numerous proposals for potentially revolutionary electronic devices that use spin orbit SO coupling have appeared in recent years including gate controlled spin rotators 89
18. XE Jhtroductiore eu ese em ee ep dare user Ir lo t hee sa Bingen gonial Jat 30 4 2 Previous Theory and Experiments llle 30 43 Theory of Two Dimensional Magnetotransport with Spin Orbit Coupling beyond the Diffusive Approximation e 31 44 Experimental Details en 32 4 5 Crossover from WL to AL and Separation of Spin Orbit Parameters 33 46 Angular Dependence of Spin Precession Rates 1 2 2 0 a 34 427 Comparison with previous Theory lee 34 49 Conclusion e LIA RE RA ES UI eR AUCI e re 35 aig Acknowledegemiernts srir anp acento de ee LAE E acu e feront Selle qna 36 Antilocalization in Quantum Dots 5 Introduction uus ie aE Ge eee S ine ee EA 5 2 Previous Experiments ee 53 Random Matrix Theory 0 0 0 0 0000000008 5 4 Antilocalization and Confinement Suppression of Spin Orbit Effects 5 5 Suppression of Antilocalization by an In Plane Magnetic Field 5 6 Breaking of Time Reversal Symmetry due to an In Plane Magnetic Field 5 7 Effects of Temperature on Antilocalization ss 5 8 In Situ Control of Spin Orbit Coupling with a Center Gate 5 9 Acknowledgements naoa Conductance Fluctuations in Quantum Dots 63 Introduction nk es eee RR ueste emo RU rea 6 2 Previous Work 2 0 0 esane eacee 6 3 Spin Rotation Symmetry Classes 0 000 000 0040 6 4 Experimental Techniques 2 0 000000005000 6 5 Characterization
19. as well as sources and detectors of spin polarized currents 103 146 147 It has also been predicted that the effects of some types of SO coupling will be strongly suppressed in small oD systems i e quantum dots 115 116 148 150 In this Letter we investigate SO effects in ballistic chaotic GaAs AlGaAs quantum dots We identify the signature of SO coupling in ballistic quantum dots to be antilocalization AL leading to characteristic magnetoconductance curves analogous to known cases of disordered 1D and 2D systems 85 98 99 105 122 125 151 AL is found to be prominent in large dots and suppressed in smaller dots as anticipated theoretically 115 116 148 150 Results are generally in excellent agreement with a new random matrix theory RMT that includes SO and Zeeman coupling 116 150 Moderate magnetic fields applied in the plane of the 2D electron gas 2DEC in which the dots are formed cause a crossover from AL to weak localization WL This can be understood as a result of Zeeman splitting consistent with RMT 116 150 At larger parallel fields WL is also suppressed which is not expected within RMT The suppression of WL is explained by orbital coupling of the parallel field which breaks time reversal symmetry 117 118 Finally we demonstrate in situ electrostatic control of the SO coupling by tuning from AL to WL in a dot with a center gate In mesoscopic conductors coherent backscattering of time reversed electron
20. bake at 180 C Second layer 495 PMMA C6 7 bake at 180 C Third layer 950 PMMA Az 10 bake at 180 C The multilayer is for undercut and to get the PMMA thick enough for liftoff Pattern Pattern on Raith using 120pm aperture Develop in 3 1 IPA MIBK for 1min3osec Rinse IPA 15sec 2 min UV Ozone in the Douwe box Evaporate Ammonium hydroxide dip 3 seconds full strength Rinse DI water 15 sec Get into E beam evaporator as quickly as possible Don t trip 86 Figure C 1 An optical micrograph of an annealed ohmic that was measured to have less than 50 resistance at 4 K The white scale bar is 5o um Annealed ohmics that exhibit these cross like structures are virtually guaranteed to have very low resistance 9 10 11 Evaporate the metal stack The wait times are important to keep things from getting too hot For 100nm deep 2Dtc the final three layers can be omitted For 20onm deep 2DEG the final three layers have been shown to decrease the ohmic resistance The thicknesses given are the actual target thickness Multiply by 1 25 for the target in the Marcus lab e beam evaporator A reasonable evaporation rate is about 25 3nm sec Pt 5nm Au 200nm WAIT at least 30 min Ge 100nm Pt 73nm WAIT at least 30 min Au 100nm Ge 5onm Pt 55nm Liftoff in acetone Usually takes 30 seconds Anneal Use the Jipelec RTA in the cleanroom Pyrometer control target 530 C 100s Reduce time
21. e beam lithography which is covered in Appendix A I have tried to include some information about why we do things not just how we do them B 1 Getting Started Obtain supplies from the chemistry stock room or around the lab toolbox safety glasses with anti UV coating pyrex petri dish with lid a few glass microscope slides Carbofib tweezers Obtain material B 2 Cleave Cleave the material To achieve cleave position and chip size precision near 100 jm use the cleaver in the upstairs cleanroom Turn on the nitrogen gas Turn on the pump under the table Turn on the cleave unit power switch left side rear Start the LSD software Adjust the scribe tool pressure and the break force When initialization is finished place your wafer to be cleaved on the blue tacky tape The cleave direction is front to back along the right hand side of the metal bar under the tacky tape Align the wafer cleave axis with this bar as well as possible by hand then be sure that the wafer is sticking to the tape under its entire area or else the scribe may push it aside instead of making a scratch Use the rotation arrow buttons to align the wafer rotation with the on screen cross hair reticle Make sure the angle is correct by checking that an entire side of the wafer is aligned with the reticle move the wafer with the translation arrow buttons Move the wafer so a left or right edge is aligned with the vertical reticle Enter the desired size of the chi
22. for shallower 2DEG C 1 3 Small gates Spin ae 2 4 solvent clean 5 min bake 180 C Spin on PMMA No 5s spin up 4000rpm 45sec 1sec spin up and spin down 950 PMMA A4 10 bake at 180 C 87 Pattern Write the small gates using 20pmaperture on the Raith Develop 1 min in 3 1 15 s IPA rinse 2 min UV Ozone in the Douwe box Evaporate Use thermal evaporator 5nm Cr 15nm Au Liftoff in Acetone overnight 5 sec ultrasound C 1 4 Connector gates Spin 1 2 3 4 solvent clean Bake dry 5 180 C Spin on 3 layer PMMA No 5s spin up 4000rpm 45sec 1sec spin up and spin down First layer 495 PMMA C6 5 bake at 180 C Second layer 495 PMMA C6 7 bake at 180 C Third layer 950 PMMA Az 10 bake at 180 C The multilayer is for undercut and to get the PMMA thick enough for liftoff Pattern Pattern on Raith using 120pm aperture Develop in 3 1 IPA MIBK for 1min3osec Rinse IPA 15sec Evaporate Use thermal evaporator The target thickness is the depth of the mesa etch plus 10 Liftoff in Acetone DONE 88 APPENDIX D Wafer Data Sheet This appendix contains a copy of the wafer data sheet provided by Loren Pfeiffer for the material we used in the experiments reported in Chapter 7 This is the one material we have found so far with good features at v 5 2 and decent gateability 2 25 05 1 act txt 22 EE L Actual Results Substrate wr i
23. t where Tay aBy DBs and assume that the combined effects of SO coupling the quantum well We either treat a as a single fit parameter a1 Table 5 3 using b 1 4108 s 1T 6 from device simulations 3 or treat both a and b as fit parameters a5 and b Table 5 3 Fitting both parameters only improves the fit for the unusually shaped center gated dot 5 7 Effects of Temperature on Antilocalization Increased temperature reduces the overall magnitude of g and also suppresses AL compared to WL causing AL at 300 mK to become WL by 1 5K in the 8 m dot Figure 5 3 a Fits of RMT to g B 0 yield Ago values that are roughly independent of temperature Figure 5 3 b consistent with 2D results 151 and T values that decrease with increasing temperature Dephasing is well described by the empirical form 1 ns 7 5 T K 2 5 T K consistent with previous measurements in low SO dots 166 167 As dephasing increases long trajectories that allow large amounts of spin rotations are cut off diminishing the AL feature 5 8 In Situ Control of Spin Orbit Coupling with a Center Gate Finally we demonstrate in situ control of the SO coupling using a center gated dot Figure 5 4 shows the observed crossover from AL to WL as the gate voltage V is tuned from 0 2 V to 1 V At Vg 1 V the region beneath the center gate is fully depleted giving a dot with area 5 8 m that shows WL In the range of Vg gt 0 3 V the amo
24. time is legendary his willingness in the interest of promoting the kind of discussions that lead to advancements in science to suffer gladly even enjoy the occasional eccentricity in both is commendable Thank you It was a tremendous privilege for me to spend most of my graduate career working with another great experimentalist Dominik has experimenter s hands he somehow just gets stuff to work The secret behind those hands is that he never turns any knob or solders any resistor randomly every move he makes in the lab is based on his deep understanding of the physics The same secret is behind his eye which never misses a single wiggle or bump in the data and finds the explanation for each one He is a good friend and significant part of the reason I enjoyed my PhD Thank you Amir s grasp of the fundamentals of physics and how the core concepts show up in the most advanced experiments is an example of how a clear seemingly simple thought process can lead to the most elegant important experiments To work with Amir is to remember that simplex sigillum veri but that only the most accomplished intellect can reduce complicated problems to their fundamental simplicity I am very lucky I was able to work with Amir at Harvard then at the Weizmann Institute and then again at Harvard Thank you The research we do simply cannot be done alone especially a problem as interesting and difficult at the 5 2 experiment As luck would have it the t
25. 0 As By is increased from zero varg B 0 Bj is seen to decrease sharply approaching varg B 7 0 Bj At large Bj the measured variance becomes independent of B within the errorbars solid symbols Figures 6 3 and 6 4 while the RMT predicts that the variance with B 0 is twice the value at B 0 independent of Bj On a comparable B field scale quantum corrections to the average conductance 6g B g B1 0 Bj g Bi1 7 0 Bj are seen to be vanishing upon application of Bj in all devices open symbols insets whereas the RMT calculates a reduced but finite g dashed curves insets Suppression of g in By was 50 0 00 a A 3 Degi c 0 03 8r o 7 2 0 06 i er Se RMT 0 09 4 7 5 RMT FJ ipee M 0 0204 1 2 4 7 de B T R 44 3r E ae B 0 2 B 0 d AT DER w o S gt 10 a O Oo co T T T T T 4 T 0 04177 RMT FJ 0 0204 1 2 4 7 low density n 2 2x 10 m 0 1 2 3 4 5 Bi T Figure 6 4 As Figure 3 but for low density devices Due to effects of By to break TRS the variance for B 0 is seen to be reduced to the variance for B 0 on the same B field scale where WL AL effects are suppressed by B insets previously reported 96 113 and is due to effects of By to break TRS 117 118 Following Ref 117 FJ the suppressions of average and variance can be accounted for by
26. 126 Mathur H and Stone A D Phys Rev Lett 68 2964 1992 127 Lyanda Geller Y B and Mirlin A D Phys Rev Lett 72 1894 1994 128 Iordanskii S V Lyanda Geller Y B and Pikus G E JETP Lett 60 206 1994 129 Lyanda Geller Y B Phys Rev Lett 80 4273 1998 130 Morpurgo A F et al Phys Rev Lett 80 1050 1998 131 Gor kov L P Larkin A I and Khmel nitskii D E JETP Lett 30 228 1979 132 Lyanda Geller Y 2002 unpublished 133 Gasparyan V M and Zyuzin A Y Sov Phys Solid State 27 999 1985 134 Cassam Chenai A and Shapiro B J Phys I France 4 1527 1994 135 Kawabata A J Phys Soc Jpn 53 3540 1984 136 Hansen J E Taboryski R and Lindelof P E Phys Rev B 47 16040 1993 137 Ramvall P Kowalski B and Omling P Phys Rev B 55 7160 1997 138 Nitta J Akazaki T and Takayanagi H Phys Rev Lett 78 1335 1997 139 Sch pers T et al J Appl Phys 83 4324 1998 140 Heida J P et al Phys Rev B 57 11911 1998 141 Schultz M et al Semicond Sci Technol 11 1168 1996 142 Papadakis S J et al Physica E 9 31 2001 143 Winkler R et al Phys Rev B 65 155303 2002 144 Lu J P et al Phys Rev Lett 81 1282 1998 145 145 Ivchenko E L and Pikus G E Superlattices and Other Heterostructures Symmetry and Optical Phenomena volume 110 of Springer Series in Solid State Sciences Springer Verlag 1995 146 Bulgakov
27. AS Temp 351 Mas As Set 1 3le 5 a Comments repeat record DD sample 6 23 00 1 HEAVY Si 626c smoothing Si at 450c label relay rate temp set pt Ga5 3 2 8225 1119 000 1118 000 A14 5 0 8844 1204 000 1203 000 Layer Layer Type Thickness Time Total Time A14 1 chgsi_chgc As110 00 60 00 60 00 2 Ga5As11 7000 00 2480 07 2540 07 3 PauseAsi1 0 00 3600 00 6140 07 qe Ga5Al4As11 100 00 26 98 6167 05 0 239 5 Ga5As11 30 00 10 63 6177 68 sen 6 100 PauseAs1l 0 00 15 00 6192 68 304 Ga5Al4As11 980 00 264 37 11665 00 0 239 305 Al4asil 19 81 22 40 11687 40 306 Ga5As11 5 66 2 01 11689 41 307 t 620 6 0 00 10 00 11699 41 308 chgremp Asll 0 00 100 00 11799 41 309 chgremp Asil 0 00 2 00 11801 41 310 chgsi As11 0 00 15 00 11816 41 sys 311 SiAsll 0 00 28 00 11844 41 312 chgSi As11 0 00 10 00 11854 41 313 GaSAsil 22 64 8 02 11862 43 314 t 432 1 0 00 10 00 11872 43 315 chgremp Asli 0 00 86 00 11958 43 316 startrempAsll 0 00 2 00 11960 43 317 Al4As11 19 81 22 40 11982 83 318 Ga5Al4As11 800 00 215 81 12198 64 0 239 319 PauseAsli 0 00 50 00 12248 64 320 Ga5As11 300 00 106 29 12354 93 7 321 t 615 0 0 00 10 00 12364 93 322 PauseAs11l 0 00 40 00 12404 93 323 Ga5Al4As11 800 00 215 81 12620 74 0 239 324 Al4As11 19 81 22 40 12643 14 nin 325 GaSAsil 5 66 2 01 12645 15 326 chgTemp Asil 0 00 100 00 12745 15 a 327 chgremp As11 0 00 2 00 12747 15 328 chgsi Asil 0 00
28. Bcs like pairing 47 of composite fermions 46 48 50 The Pfaffian picture is supported by numerical studies 51 although experimental confirmation is so far lacking the work in this thesis is a first step towards an experimental verification or refutation of the Pfaffian picture Experimental evidence is of course needed to help determine the nature of the wave function at v 5 2 especially since there are other proposals for the wave function 49 52 Although the Moore Read wave function is only one of many proposals for the v 5 2 state we will study it in some detail for two reasons The first is that the results of numerical studies suggest the Moore Read state has the lowest energy The second is that the Moore Read state leads to non Abelian quasiparticle excitations The Moore Read picture of the v 5 2 state has some similarities to the Chern Simons composite boson picture presented at the end of the previous section in both cases the ground state is a superfluid of bosons that supports vortex quasiparticle excitations However the Moore Read state is a Bcs like condensate of Cooper paired composite fermions whereas the abelian odd denominator FQHE states are condensates of single bose like quasiparticles The Moore Read state also differs from other common scs superfluids in the way the fermions are paired The Moore Read wave function pairs the composite fermions in an angu lar momentum 1 p wave orbital i
29. Control window Focus Correction ood focus is critical not just to achieve small G features but also to get robust repro ducible results in terms of exposure In other words if the spot size of the beam changes from one write session to the next you will not be able to control the dose correctly The method presented below will yield three results First you will be able to correctly set the working dis tance to the surface of the PMMA with an accuracy of better than 1 um By contrast if you try to focus on one of your alignment marks below the PMMA additional 200 nm error or some silver paint unknown thickness your accuracy will probably only be about 50 um or much worse A 50 um error in working distance cor responds to an increase of about 131 nm in the spot di ameter a terrible error for devices with 50 nm feature sizes Second you will be able to see the size and shape of the beam before your write so you know when to stop tweaking the focus and aperture and start writing a time saver and you can verify consistency between write sessions Finally you can reliably correct for the angle of your chip using three point correction so that the focus is correct not just at your alignment marks but anywhere you want to write Choose three non colinear points on your chip to use as the focus locations I typically choose the cen ter of three alignment marks distributed around my target mesas Page 6 6
30. E N et al Phys Rev Lett 83 376 1999 147 Keppeler S and Winkler R Phys Rev Lett 88 46401 2002 148 Khaetskii A V and Nazarov Y V Phys Rev B 61 12639 2000 149 Khaetskii A V and Nazarov Y V Phys Rev B 64 2001 2001 107 150 Cremers J N H J et al Phys Rev B 68 125329 2003 151 Millo O et al Phys Rev Lett 65 1494 1990 152 D yakonov M I and Perel V L Sov Phys JETP 33 1053 1971 153 Papadakis S J et al Science 283 2056 1999 154 Grundler D Phys Rev Lett 84 6074 2000 155 Kurdak C et al Phys Rev B 46 6846 1992 156 Nitta J Meijer F E and Takayanagi H App Phys Lett 75 695 1999 157 Shea H R Martel R and Avouris P Phys Rev Lett 84 4441 2000 158 Engel H A and Loss D Phys Rev B 62 10238 2000 159 Braggio A Sassetti M and Kramer B Phys Rev Lett 87 146802 2001 160 Mireles F and Kirczenow G Phys Rev B 64 24426 2001 161 Hackens B et al Physica E 12 833 2002 162 Buttiker M Phys Rev B 33 3020 1986 163 Baranger H U and Mello P A Phys Rev B 51 4703 1995 164 Brouwer P W and Beenakker C W J Phys Rev B 55 4695 1997 165 Chan LH et al Phys Rev Lett 74 3876 1995 150 151 152 153 154 155 156 157 158 159 160 161 162 165 164 165 166 Huibers A G et al Phys Rev Lett 81 200 1998 167 Huibers A G et al Phys
31. L Il fabs L 1 L L gt 05 The series of Pn MUST be decreasing or we get huge problems Switch to symptotic as soon as this happens flag n P 1 0 sqrt fi fi 2 n 1 2 0 ss ss F 3 if flag gt O P 1 0 sqrt fixfi 2 n 1 2 0 ss ss DONE CALCULATING P Term P P P 1 0 P Sum Term nit while fabs Term Sum gt QUALITY amp amp n maxn Keep adding terms until individual terms are QUALITY smaller than the sum free LL fprintf fp MVtC om 4f f 4f ML f m fi c Sum fprintf fp VtPnINT n 0 WVt 16fVtZzdMVt4dYn Pn ss 0 fi binSize flag n fclose fp p gt result x Sum return 0 XFunc error code static int YuliD struct DOUBLE p4 DOUBLE p3 DOUBLE p2 DOUBLE pil DOUBLE result p double x p gt pl 98 double fi p gt p2 double binSize p gt p3 double QUALITY p gt p4 double P int n 0 double Sum 0 double Term do 1 P Pn sqrt 2 fabs x n fi binSize Term P Pnm sqrt 2 fabs x n 1 fi binSize Pnm sqrt 2 fabs x n 1 fi binSize 2 1 P Sum Term n4 while fabs Term Sum gt QUALITY amp amp n 2000 Keep adding terms until individual terms are QUALITY smaller than the sum p gt result x Sum return 0 The rest of the file is just the interface to Igor struct DPComplexNum DOUBLE real DOUBLE imag F Static int XFUNC1ComplexConju
32. Parameters Aso Tp and x a factor related to trajectory areas are extracted from fits to dot conductance as a function of perpendicular field B The asymmetry parameter Vso is estimated from the dependence of magnetoconductance on parallel field Bj The quantum dots are formed by lateral Cr Au depletion gates defined by electron beam lithography on the surface of a GaAs AlGaAs heterostructure grown in the 001 direction The 2DEG interface is 349 below the wafer surface comprising a 50 GaAs cap layer and a 299 AlGaAs layer with two Si ffi doping layers 143 and 161 from the 2DEG An electron density of n 5 8 x 10 m and bulk mobility u 24 m Vs cooled in the dark gives a transport mean free path le 37m This 2DEG is known to show AL in 2D 105 Measurements were made in a He cryostat at 0 3 K using current bias of 1nA at 338 Hz Shape distorting gates were used to obtain ensembles of statistically independent conductance measurements 165 while the point contacts were actively held at one fully transmitting mode each N 2 5 4 Antilocalization and Confinement Suppression of Spin Orbit Effects Figure 1 shows average conductance g and variance of conductance fluctuations var g as a function of B for the three measured dots a large dot A 8m a variable size dot with an internal gate A 5 8 um or 8pym depending on center gate voltage and a smaller dot 1 2 um2 Each data point represents 200 independent
33. a factor fpy By 1 tay hes where Tay aBi bBy and vl NA h is the escape time The By term reflects interface roughness and dopant inhomogeneities the By term is due to the asymmetry of the well It is assumed that the combined effects of the RMT and flux threading by By can be written as products g Bj dgrur Bi fry By and varg B 0 By vargrur B 0 By 1 frj Bj The coefficient a is obtained from a fit to the experimental g Bj while b is estimated from device simulations Table I The result ing theory curves for both g Bj solid curves insets and var g B 0 By solid curves main panels are in good agreement with the experiment We emphasize that the theoretical variance curves are not fit The coefficients a b estimated from correlation functions 172 are consistent with the values obtained here from g Bj 6 9 Conclusion In summary the variance of conductance fluctuations in open quantum dots in presence of SO coupling and in plane fields Bj is understood in terms of symmetries in the system including V Fal ko and T Jungwirth private communication V Fal ko and T Jungwirth private communication 51 novel spin rotation symmetries as well as time reversal symmetry which can be broken both by perpendicular fields B and parallel fields Bj 6 10 Acknowledgements We thank I Aleiner B Al tshuler P Brouwer J Cremers V Fal ko J Folk B Halperin T Jung wirth and Y Lya
34. alk a k 5 t afk 6 t afk 8 t afk 9 t alk W afk 11 t afk 12 else if w lt 6 9 k int w t w K k 13 k 2 y b k t b k b k 2 t b k 3 b k 5 t b k 6 b k 8 t b k 9 b k 11 t b k 12 y y y yi y ys y 1 y y else y 1 return x lt 0 y y double x a k 1 t t b k t b k t bik 4 4 4 t 7 t 10 t 4 t 7 t 10 t This is the Laguerre Polynomial as in Arfken p 779 double L int j 1 double jj 1 0 double Lag if n 0 return 1 0 93 J L double malloc size_t n 5 sizeof double L 1 0 L 1 L x 1 0 do L 1 L 2 L 1 L 1 x L 1 jj 1 0 0 jj 1 0 L 1 while jj lt n Lag L free L n return Lag static double aLaguerre int n double m double x 1 b This is the Associated Laguerre Polynomial as in Arfken p 780 double L int j 1 double jj 1 0 double Lag if n 0 return 1 0 if n 1 return x 1 0 L double malloc size_t n 5 sizeof double L 1 0 L 1 L x m 1 0 do 1 L 1 L 2 jj m 1 x jj 1 L 1 jj m jj 1 jj 1 0 L 1 while jj lt n Lag L free L n return Lag static double Pn doubl
35. be in quantitative agreement with theory 29 68 71 72 14 2 4 Impact of this work and future directions The effort to probe the statistics of the v 5 2 FQHE state has just begun Prior to our work described in detail in Chapter 7 only bulk experiments on the v 5 2 state had been reported In fact it was not known whether the v 5 2 state could even exist in a confined area since the state exists only by virtue of exceptionally delicate bulk many particle correlations Furthermore it was not known whether the specialized ultra high mobility GaAs AlGaAs samples that support the v 5 2 state which have 6 doping layers both above and below the 2pEG see Appendix D could be processed without destroying the v 5 2 state and whether such material was even gateable using standard top gate depletion techniques Furthermore although the v 5 2 state exhibits all the behaviors of a compressible quantum Hall state there was no experimental evidence that it would definitely even support an edge channel capable of tunneling Although the existence of edge channels has never been in serious doubt experimental confirmation is always important especially since the theoretical proposals to probe the statistics at v 5 2 require interference of edge channels Our experiments have addressed all of these points We found that our standard nanofabrication procedure did degrade the 2pxc but I devel oped an improved procedure to fabrica
36. beam Focus on the alignment cross as well as possible Center the crosshairs on the cross zooming in to minimize error Page 5 Open the Adjust UVW Global window and click on the Origin Correction tab If the button at the lower left reads gt Global click it so that it reads gt Local You want to set the Global UVW Enter the coordinate position of your alignment cross in the U and V set variable area For instance if the center of your design is 0 0 your alignment cross may be located at 0 0 475 Click the Adjust button to adjust the origin of the UV coordinate system Note the XY coordinate system never changes but the UV system should be changed to match your photolithography step as we are now doing Click the Angle Correction tab Click the Read button on line 1 Keep the beam on and move to the exact center of another alignment cross which is supposed to be hor izontally colinear with your origin Click the Read button on line two and click adjust The angle of the UV coordinate system has now been set However thanks to Raith software technology your origin has been screwed up Go back to your first cross center on it and repeat the origin correc tion step NOW the origin and angle of the UV co ordinates are set so it should be possible to move around your sample by entering absolute or relative coordinates in the Stage
37. consists of a GaAs AlGaAs heterostructure grown in the 001 direction with double doping layers set back 143 A and 161 A from the 2psc and a total distance of 349 A from the surface to the 2pEc A 200 um wide Hall bar with 700 um between voltage probes was patterned by wet etching A lithographically defined Cr Au top gate was used to control density and mobility in the Hall bar over the range n 1 4 7 0 x10 m and p 3 6 31 m Vs Measurements were made in a He cryostat at temperature T 300 mK using ac lock in techniques with bias currents ranging from 50 to 500 nA depending on the gate voltage 32 0 200 0 200 200 100 0 100 200 Vg mV V mV Figure 4 1 a Experimental magnetoconductance Ac c B c 0 circles offset for clarity along with three parameter fits to Eq 4 2 solid curves for several gate voltages Inset Experi mental magnetoconductance data for the most negative gate voltage showing pure WL b Den sity and mobility as a function of Vz extracted from longitudinal and Hall voltage measurements c Experimental conductivity showing strong dependence on Vz Note that Ac 10 c At each gate voltage the bias current was experimentally determined not to affect the results 4 5 Crossover from WL to AL and Separation of Spin Orbit Pa rameters Figure 4 1 a shows the longitudinal magnetoconductance as a function of Vg A crossover from pure WL Figure 4 1 a
38. device shapes The large dot shows AL while the small and gated dots show WL Estimates for Aso To and x from RMT fits are listed for each device below the micrographs in Figure 5 1 see Table 5 3 for corresponding e and ej When AL is present i e for the large dot estimates for Aso have small uncertainties 2 576 and give upper and lower bounds when AL is absent i e for the small and gated dots only a lower bound for As 5 can be extracted from fits The value Aso 4 4 um is consistent with all dots and in good agreement with AL measurements made on an unpatterned 2DEG sample from the All measured densities are below the threshold for second subband occupation n 6 6 x 10 m which is known from Shubnikov de Haas measurements and a decreasing mobility with increasing density near the threshold 39 110 OF X Q u 9 6 e Ob X Q u 9 6 re 0 964 c a 10 b za 0 94 f 1 1 1 0 1 0 0 5 0 0 0 5 1 0 B mT 1 00 7 teen 8 Q oss 103 a o 0 90 a Noc nk e Ago 3 2 um lt 0 85 ziii ZBRQRIZG so c e To 0 10 ns 0 80 a ae E mE EN ET 09 Kk 0 33 B mT Figure 5 1 Average conductance g squares and variance of conductance var g triangles calculated from 200 statistically independent samples see text as a function of perpendicular magnetic field B for a 8 0 um dot b 5 8 ym center gated dot and c 1 2 um
39. dotted curve is used to extract y amp and a2 Alternatively the best linear fit to H7 solid line is used to extract y 0 T 0 1 1 0 ns at 300 mK which decrease by more than an order of magnitude as temperature is increased to 2 5 K Within the error bars H5 and H 4 do not depend on temperature over this temperature range Figure 4 3 a displays the magnitudes of the three spin orbit terms as functions of V n and u determined using Eq 4 1 and the extracted values of a1 2 and y Plotted are values along the 110 direction tan k ky where Ops is maximum Error bars indicate uncertainties in the fitting procedure and noise in the data 4 6 Angular Dependence of Spin Precession Rates The total spin precession rate Q is plotted as a function of the direction of the electron momentum in Figure 4 3 b While for most directions Q is an increasing function of density it is seen to decrease with increasing density near sri and Ti The linear Dresselhaus and Rashba terms Op and Og are of comparable magnitude to each other for all densities and T in all directions Near 5 j an integer Ops Opi Og and the SO is controlled by the 2j4 1 x 4 linear terms For near the cubic term becomes comparable to or even exceeds at high densities the linear terms Depending on 6 the linear and cubic terms either add 7 3r or subtract a 71 The extracted values for y 3
40. et al Phys Rev Lett 83 3530 1999 63 Eisenstein J private communication 2006 64 Kouwenhoven L P et al Phys Rev Lett 64 685 1990 65 Camino F E Zhou W and Goldman V J Phys Rev B 72 075342 2005 66 Fendley P Fisher M P A and Nayak C Phys Rev Lett 97 036801 2006 67 Moon K et al Phys Rev Lett 71 4381 1993 68 Fendley P Ludwig A W W and Saleur H Phys Rev B 52 8934 1995 69 Roddaro S et al Phys Rev Lett 93 046801 2004 70 Roddaro S et al Phys Rev Lett 95 156804 2005 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 Wen X G Phys Rev B 44 5708 1991 72 Wen X G Phys Rev B 43 11025 1991 73 Marcus C private communication 2005 74 Stern A private communication 2005 75 Efros A L and Shklovskii B L J Phys C 8 L49 1975 76 Matthews J and M C J Res Natl Inst Stand Technol 1310 497 2005 77 Coleridge P T Stoner R and Fletcher R Phys Rev B 39 1120 1989 78 Coleridge P T Phys Rev B 44 3793 1991 79 Das Sarma S and Pinczuk A eds Wiley New York 1997 80 Goerbig M O Lederer P and Smith C M Phys Rev B 68 241302 2003 81 Eisenstein J P et al Phys Rev Lett 88 076801 2002 82 Jackson J D Classical Electrodynamics Wiley 1975 83 Dresselhaus G Phys Rev 100 580 1955 84 Bychkov Y L and Rashba E I J Ph
41. features so it is possible to at least rank traces in order of electron temperature This is actually very useful since it allows us to check that heating the sample slightly from base temperature actually does make a difference in the electron temperature which shows at least that the temperature has not saturated However since the RIQHE features that are very sensitive from about 3o mK and below are essentially absent at temperatures above about 50 mK and since there is no functional form for the resistance of these features it is impossible to make quantitative estimates of the electron temperature 19 20 CHAPTER 3 Introduction to the spin orbit coupling experiments In this chapter I begin with a brief introduction to the effects of spin orbit coupling upon trans port in a two dimensional electron gas 2DEG followed by a brief introduction to conductance fluctuations and random matrix theory Then I discuss the place of my work in this field of study Finally I record some of our techniques from these experiments 3 31 Theoretical background 3 1 1 Spin orbit coupling Spin Orbit coupling means that an electron s spin can become coupled to its motion as it moves around in the 2pEc The way this happens is quite a fascinating relativistic effect an electric field in the laboratory feels like a magnetic field in the rest frame of the fast moving electron From special relativity 82 the electron feels effective an magneti
42. of Spin Orbit Strength at Zero In Plane Field 6 6 Variance at Zero In Plane Field llle 6 7 Effects of Spin Rotation Symmetry on the Variance 6 8 Orbital effects of Bj on the Variance 5 0 000004 6 9 Conclusion 5 eoe d e Robes Gag be Bele eae bers 6 10 Acknowledgements Experimental observation of the v 5 2 state in a quantum point contact 7 1 7 2 7 3 74 7 5 7 6 7 7 7 8 7 9 Ittod cHon 4g ewe E Ov EE EA EOS E Measurement Techniques llle Previous Experiments eA Experimental Details oe Bulk Measurements 2A Demonstration of the QPC in IQHE and FQHE regimes Observation of plateaus atv 5 2 2 ee ee Temperature data in the QPC ee liedat 34 artus A TEN e dae e ah edt ar e E RR ed ee 7 10 Conclusion 7 11 Acknowledgements Raith Users Guide Detailed Fabrication Procedures B 1 B 2 B 3 B 4 B 5 Getting Started c Beni on ei ee ae BES Sn vmi Beh Mea i Cle ae is i ee ee eue RE EE Be ape em LOT eke E Mesa Photo Tr 5 ume uuu ok ed Ba E Wu Etch Proced r anl isE es Wie A ee ge Bar Summary 24euaesiebeterusgee tem eum b bau B2 3etais 25 us th Mm Gite wh a ru Podio Mod e ek B 4 3 Profilometer Operation eee Th st oS i bs cs Se eee tod ane Ae Dita ESAE ebur Complete Nanofabrication Recipe for 5 2 Devices Cr 9 2 r ady fab recipe s bie bsc aes Apu a be et aS Mesas asart et Es co d otaere ehh
43. of edge states Vpux Nou By analogy in the orc where the filling fraction is not well defined due to nonuniform density we define an effective filling fraction in the QPC vopc h e Rp The edge state interpretation for Ryy Rxx Rp and Ry has been extended to the FQHE 29 64 207 209 214 Within this generalized picture a quantized plateau in Ryy h e Vpulk corresponds to the quantum Hall state at filling fraction vui and a plateau in Rp h e 1 vopc indicates that an incompressible quantum Hall state has formed in the vicinity of the orc with effective filling fraction vopc We associate deviations from precisely quantized values with tun neling which we study below as a function of temperature and bias To simplify the study of quantum states in the vicinity of the orc the perpendicular mag netic field B and gate voltage of the orc Vg are tuned such that 4 is fixed at an integer quantum Hall effect IQHE plateau whenever vopc is at a value of interest With Rxx 0 and Ry quantized to an IQHE plateau features in Rp and Ry measurements can be attributed to the orc region and not the bulk 7 3 Previous Experiments Previously oPcs have been used to selectively transmit integer 215 216 and fractional edge channels 64 217 and to study inter edge tunneling between fractional edge channels 69 203 including in the regime where the bulk is intentionally set to an IQHE plateau 70 204 Compar isons with these
44. quantized value of o 4 1 amp and increases as temperature is decreased Near vopc 5 2 and vopc 21 3 in these orcs the differential resistance exhibits a characteristic shape showing a peak at Ig OnA a minimum near Ig 1 2nA and approaching a constant value at higher currents These observations are consistent with the formation of gapped incompressible FQHE states in the opc with orc induced tunneling between the edge states In a Qrc with 0 5 pm spacing between the gates we do not observe a plateau like feature at any temperature and Ig is flat for the entire range between vopc 3 and vopc 2 This suggests that in our sample no incompressible states form in this Qrc because of either confinement or the effects of decreased electron density All of these measurements have been carried out in a magnetic field range where the bulk filling fraction was on the IQHE tyr 3 plateau while the vopc was tuned to lower values via the gate voltage 7 11 Acknowledgements We gratefully acknowledge helpful discussions with M Fisher B Halperin A Johnson E Kim B Rosenow A Stern X G Wen and A Yacoby Research supported in part by Microsoft Corporation Project Q and HCRP at Harvard University and ARO W911NF 05 1 0062 the NSEC program of the NSF PHY 0117795 and NSF DMR 0353209 at MIT 61 62 APPENDIX A Raith Users Guide I made a lot of devices during the course of my PhD One of the trickiest parts of the devic
45. raith software doesn t work At least this time the focus is Page 8 usually easier After finishing the entire focus correction procedure for the second time the focus correction is usually correct Verify this by moving to several selected lo cations and burning a 5 second spot which should be 40 50 nm in diameter If so focus correction is finished Write Field Alignment inally the beam needs to be aligned to the UV b Open the Microscope Control window and double check that the magnification is 550 and the Field Size is 100 um Make sure the Align Writefield window is visible Zoom to about 550X and position the crosshairs on some easily identifiable feature Zoom in to a few kX to make sure the crosshairs are correctly positioned then zoom back to 550X 10um EHT 30 00 kV SignalA SE2 Date 11 Sep 2004 Photo No 1062 Time 9 5532 KH WD 11mm Click the green SEM button on the Raith computer to freeze the SEM image and assign control of the LEO to the Raith Blank the beam Open a new position list File New Position List and make sure it is selected Click on Filter Align Writefield The Align Writefield Window opens In the Manual tab set the Scan Size to 50x50 um Click the Create amp Scan Marks button The magnification will be automatically set and a slow scan image will be displayed on the Raith computer Page 9 While hol
46. results are discussed below orcs have also been employed in studies of noise 218 and along with etched trenches interference of quasiparticles 219 in the FQHE regime In all of these studies v 2 where the FQHE gaps are typically much larger 220 221 than those with v gt 2 7 4 Experimental Details The sample is a GaAs AlGaAs heterostructure grown in the 001 direction with electron gas layer 200nm below the surface with Si doping layers 100 nm and 300 nm below the surface A 55 Vbulk 2 9 22 y 21 2 1 0 45 0 35 0 04 R h e 0 02 0 00 Figure 7 2 Bulk transport measurements including temperature dependence The inset is an enlargement of the Ry data near vp 5 2 150 um wide Hall bar is patterned using photolithography and a H20 H2504 H2O 240 8 1 wet etch followed by thermally evaporated Cr Au 5nm 15 nm top gates patterned using electron beam lithography see Fig 7 1 The gates form orcs with lithographic separation between gates of 0 5 0 8 and 1 2 pm 222 Depleting the electron gas beneath only one side of a orc has no effect on transport measurements Measurements are performed in a dilution refrigerator with base temperature 6 mK using standard four wire lock in techniques with an ac current bias excitation Jac ranging from o 2 nA to 0 86 nA and a dc current bias ranging from 0 to 20nA The differential resistances dV dI are measured in four places as shown in Fig
47. since the overall noise between successive points was typically larger than the change in the signal We also used the transformer on the Princeton Applied Research 124 lock in amplifiers Later we spent a month ensuring the linearity of our measurements I would tend to avoid using the transformer in the future unless there is really no other way to take data 3 4 4 Data Analysis The data acquisition for the 2D paper required very careful measurements but the data analysis turned out to be even harder We first tried to fit our data using existing theories 99 122 but these theories did not yield useful results even though the fits looked quite acceptable because they were designed for diffusive samples and were not valid in our ballistic sample Once we arrived with Yuli s help at the correct theory to use things actually become even harder The heart of the theory is the Cooperon function which is the sum of an infinite sum of several integrals None of them converge particularly fast In the end we had the idea to calculate the function only once for a massive 2D parameter space The result is shown in Figure 3 1 The function is at least smooth With such slowly converging integrals we wanted to somehow check that our numerics were working correctly Fortunately Zduniak 121 had calculated the Cooperon function in certain limits and our results matched his see Figure 3 2 so we were able to proceed with confidence that our nume
48. than any where else in the Raith software Check that the beam speed is less than about 4 mm s to avoid breaks and other problems Set up the coordinate system Blank the beam Use the stage control window to drive to clip 1 If you want to watch the stage move you can switch to TV view mode clicking the TV button on the SEM control bar Switch back to SEM view by clicking Zoom to about 250x Make sure the beam is un blanked You should be able to see the upper right hand side of the sample clip which looks like this 68 EHT 30 00 kV WO 10mm Signal A SE2 Date 11 Sep 2004 Time 84124 Photo No 1052 Mag 246X Use the joystick to move along the clip edge until it begins to taper in Soon you will be able to see the corner of your chip Don t linger too long or you could start to expose the PMMA although at this low magnification you have up to 30 seconds before you start to expose SignalA SE2 Date 11 Sep 2004 Photo No 1052 Time 6 42 07 HT 30 00 kV 200ym 10 mm El m WD 10 Zoom in to about 500x As quickly as possible use the joystick to navigate along the edge of your chip to a place where you ex pect to find alignment marks Focus on anything you can find i e the edge of the chip Find your alignment marks trying to stay near the edges of the chip to avoid exposing active areas of the chip If you need to stop and think blank the
49. the orc so that for example the orc can have v 5 2 while vpuk 2 could possibly help stabilize the v 5 2 state within a orc or quantum dot For example see Figure 2 4 It has been suggested 73 that in some cases FQHE states could be stronger in a confined area than in the bulk of certain dirty samples if the filling fraction of interest could be restricted to an area ie in a QPC or dot smaller than the scattering length Devices with more than one tunneling gate could be used to test the theoretical interference predictions Finally high bandwidth studies of the shot noise of the v 5 2 state 74 could turn out to be the best way to probe the statistics of FOHE states including v 5 2 2 5 Techniques In this section I discuss some non obvious experimental details especially the issue of electron temperature in quantum Hall measurements 2 5 1 Refrigerator and wiring We used a Frossati dilution refrigerator with a mixing chamber base temperature of 5 mK as measured using a calibrated RuO resistor mounted on the mixing chamber Coaxial cable was used for electrical wiring from a break out box to the refrigerator and all the way down to Frosatti silver epoxy filters The filters were mounted in the bulkhead of a shielded chamber thermally anchored at the mixing chamber Within this chamber the sample was mounted in a socket on a silver cold finger attached to the mixing chamber The socket was electrically connected to
50. to understand the average conductance and conductance fluctuations in our dots of various sizes and with a top gate in materials with strong and weak spin orbit coupling Our paper along with the theory papers I mentioned above and of course Folk s first observations helped open up an entire field where spin orbit coupling and quantum dots are used to understand and control spin Theoretical and experimental papers continue to report in creasingly sophisticated techniques as exemplified by a recent Letter 120 where single electron resonant tunneling spectroscopy is used in conjunction with the anisotropy of spin splitting to measure the relative strength of Rashba and Dresselhaus spin orbit mechanisms in a quantum dot 26 o 1 2 e2 h 4 2 0 2 4 In x Figure 3 1 This is our calculation of the function C f x the Cooperon term in Equation 4 2 3 4 Techniques 3 441 Data Acquisition The data for the 2D paper Chapter 4 was difficult to acquire because a typical localization fea ture was only about 0 1 of the overall signal In fact it was virtually impossible to see any localization peaks or dips in any single magnetic field trace In order to achieve acceptable signal to noise ratios in finite time we took hundreds of traces and averaged them This turned out to be faster that is we measured more statistically independent points in a given amount of time than turning the time constant on the lock in way up
51. trajectories leads to a conductance minimum WL at B 0 in the spin invariant case and a conductance maximum AL in the case of strong SO coupling 99 122 In semiconductor heterostructures SO coupling results mainly from electric fields 152 appearing as magnetic fields in the electron frame leading to momentum dependent spin precessions due to crystal inversion asymmetry Dresselhaus term 83 and heterointerface asymmetry Rashba term 84 5 2 Previous Experiments SO coupling effects have been previously measured using AL in GaAs 2DEGs 98 105 151 and other 2D heterostructures 85 Other means of measuring SO coupling in heterostructures such as from Shubnikov de Haas oscillations 140 153 154 and Raman scattering 104 are also quite de veloped SO effects have also been reported in mesoscopic systems such as Aharonov Bohm rings wires and carbon nanotubes 123 130 155 160 Recently parallel field effects of SO coupling in quantum dots were measured 96 161 The observed reduction of conductance fluctuations in a parallel field 96 was explained in terms of SO effects 115 116 150 leading to an extension of random matrix theory RMT to include new symmetry classes associated with SO and Zeeman coupling 116 150 5 3 Random Matrix Theory This RMT addresses quantum dots coupled to two reservoirs via N total conducting channels with N gt 1 It assumes y ez Er where y NA 27 is the level broadening du
52. you that you have hit the target remove the photoresist with a little acetone Measure the actual trench depth without the photoresist you ll need that number to set the target evaporation thickness for the gates that connect the pads to the e beam layer That target is the trench depth plus 10 B 5 The rest We have covered the use of all the tools now except the Raith see Appendix A To finish the chip Spin on a single layer PMMA coating downstairs e Pattern the small gates with the Raith using the 20pm aperture except for really tiny fea tures Use a step size of 10 nm Be sure to burn a spot to focus or nothing will be repeatable I no longer use focus correction I think it is defective Instead burn a spot near all your critical patterns Develop for 1 minute in the 3 1 solution rinse in IPA for 15 seconds Clean the residual goop with Douwe s UV Ozone machine in the sample prep room for 2 minutes On the other hand some may advocate raising the tower before removing the chip each time However I say this time wasting paranoia has no place in a safe and efficient fabrication procedure 85 Evaporate 5 nm Cr 15nm Au in the thermal evaporator Liftoff in acetone overnight if you like your chip Be brave use 5 seconds of ultrasound to finish off the liftoff Spin on triple layer PMMA coating downstairs Pattern the large gate pads with the Raith using the 120pm aperture Use a step size of 100nm Don t bo
53. 1 3 eV using Hj 28 4 eV using Hi are in good SO agreement with the value 27 5 eV from band structure calculations 85 86 Estimates for 1 give values for k2 that correspond to a wave function width of 10 nm in the 2 direction which is also reasonable The extracted a2 corresponds to a uniform 145 electric field E 10 MV m using x geE and a value of a 5 33 A from a k p model 85 86 47 Comparison with previous Theory We note that previously existing models for WL AL 85 122 128 provide fits to the data that appear qualitatively reasonable giving values for H that are 5 times higher than those found 34 u m Vs 45 7 10 15 20 25 30 n 1015m 2 15 203 0040 5055 60 65 7 0 a 010 200r 150 100 Q ns 50r 200 100 0 100 200 b 1g Vg mV Figure 4 3 a Magnitudes of isotropic linear Dresselhaus p1 and Rashba Og terms and non isotropic cubic Dresselhaus Opg term as functions of gate voltage V density n and mobility u Insets show theoretical dependence on momentum direction for the three terms indicating that the linear terms are isotropic while the cubic term has a four fold symmetry and is highly anisotropic Maximum magnitude when j Dm is shown for the anisotropic Ops term b Angular variation of Q the magnitude of the total SO precession vector at V 150 mV dotted o mV dashe
54. 15 00 12762 15 re 329 SiAsll 0 00 55 00 12817 15 lt a 330 chgsi As11 0 00 10 00 12827 15 331 Ga5As11 22 64 8 02 12835 17 332 t 433 0 0 00 10 00 12845 17 Se 333 chgTemp As11 0 00 86 00 12931 17 334 startrempAsli 0 00 2 00 12933 17 335 Al4As11 19 81 22 40 12955 57 ans 336 GaSAl4Asil 980 00 7264 37 13219 94 0 239 337 Ga5Asi1 100 00 35 43 13255 37 338 t 620 6 0 00 10 00 13265 37 339 chgremp As1l 0 00 2 00 13267 37 340 chgsi chgc As110 00 2 00 13269 37 z 341 BeepAs11l 0 00 10 00 13279 37 total layers 341 total time 13279 37 sec 2 25 05 1 sum txt total thickness 24095 8 A STRUCTURE 2Dxl MEASURED 3 2 2005 2 20 PM j POSITION A DARK CONTACTS flats MANUAL CONTROL COMMENTS SOURCE CURRENT 100 0uA MAGNET CALIBRATION G A 153 0 MAGNET CURRENT A 3 0 i T 298 7 Ro 2 0249E 3 Rh 1 0336E M 5 1044E 3 bD 6 0470E 11 T 76 9 Ro 9 7669E 1 Rh 1 9010E 7 M 1 9464E 5 D 3 2877E411 i TS 3 7 Ro 2 6503E 0 Rh 2 4664 7 M 9 3061E 6 D 2 5340E 11 AFTER_LIGHT Th 3 7 Ro 2 4969E 0 Rh 2 3467E 7 M 9 3987E 6 D 2 6633E 11 SAMPLE 2 25 05 1 STRUCTURE 20x1 MEASURED 3 2 2005 2 55 PM POSITION A AFTER LIGHT CONTACTS flats MANUAL CONTROL COMMENTS 10 min light SOURCE CURRENT 100 0uA MAGNET CALIBRATION G A 153 0 MAGNET CURRENT A 3 0 vr A Ro 2 49176 0 Rh 2 3466E 7 M 9 4178E 6 D 2 6634E 11 Le d we bl eh 251M T go APPENDIX E Antilocalization Fitting Routines This appendix contain
55. 268661706450773342 11283791670954881569 0 37612638903183748117 12837916709551257377 372510631e 11 4 5493253732e 10 90362766598e 9 6 642090827576e 8 7595634268133e 7 6 21188515924e 6 10388300970969e 5 3 7015410692956173e 4 00233307631218880978 0 0125498847718219221 05657061146827041994 0 2137966477645600658 84270079294971486929 49905026e 12 1 8310229805e 10 39463074e 9 2 721444369609e 8 8045522331686e 7 2 61830022482897e 6 195455056768781e 5 1 6358986921372656e 4 00107052153564110318 0 00608284718113590151 02986978465246258244 0 13055593046562267625 67493323603965504676 82722073e 12 7 421598602e 11 793057408e 10 1 126008898854e 8 1775134830784e 7 1 1199275838265e 6 62023443095201e 6 7 404402135070773e 5 0689993654144881e 4 0 00307553051439272889 01668977892553165586 0 08548534594781312114 56909076642393639985 55296588e 12 3 032205868e 11 0424830707e 10 4 71135111493e 9 011915876293e 8 4 8722516178974e 7 30683284629395e 6 3 445026145385764e 5 4879276133931664e 4 0 00162940941748079288 00988786373932350462 0 05962426839442303805 49766113250947636708 ooN g giH 5 n o000 0 adu0 00oootMttN oooocococtMNvxn ooco static double b 65 2 9734388465e 10 2 69776334046e 9 6 40788827665e 9 1 6678201321e 8 2 1854388148686e 7 2 66246030457984e 6 1 612722157047886e 5 2 5616361025506629e 4 1 5380842432375365e 4 0 00
56. 815533022524927908 0 01402283663896319337 0 19746892495383021487 0 71511720328842845913 1 951073787e 11 3 2302692214e 10 5 22461866919e 9 3 42940918551e 9 3 5772874310272e 7 1 9999935792654e 7 2 687044575042908e 5 1 1843240273775776e 4 8 0991728956032271e 4 0 00661062970502241174 0 00909530922354827295 0 2016007277849101314 0 51169696718727644908 3 147682272e 11 4 8465972408e 10 6 3675740242e 10 3 377623323271e 8 1 5451139637086e 7 2 03340624738438e 6 1 947204525295057e 5 2 854147231653228e 5 0 00101565063152200272 0 00271187003520095655 0 02328095035422810727 0 16725021123116877197 0 32490054966649436974 92 static double Laguerre int n 1 2 31936337e 11 2 64888267434e 9 1 1371857327578e 7 3 68797328322935e 6 6 5860243990455368e 4 0 02585434424202960464 6 303206648e 11 2 050708040581e 8 2 11211337219663e 6 9 823686253424796e 5 7 5285814895230877e 4 0 11637092784486193258 0 18267336775296612024 3 67789363e 12 1 93319027226e 9 1 8006992266137e 7 6 75407647949153e 6 1 7604388937031815e 4 0 0206412902387602297 0 09084526782065478489 S4 OB c Noct EOM Ilana A O07 x xj 2 2 Ww W int t k 13 CCCCCCCCCCCCa Ek t afk 11 t 2 0876046746e 10 4 35953392472e 9 7 8441223763969e 7 8 428418334440096e 5 0 0023972961143507161 0 06905562880005864105 afk 2 t a k 3 t
57. 9 Enter the u v values for these three points into the U and V columns of the Adjust UVW Global window Make sure you see the message Focus Correction displayed in red on the Adjust UVW Global win dow If not turn focus correction on under Project Options Make sure the beam is blanked then click the UV lightning bold on line 1 to move to your first point If your point is an alignment cross then first focus on the cross as well as possible up to a zoom of about 50 kX Use the joystick to move a bit away from the cross and zoom in to about 300 kX Center click the Short Spot button to stop the beam from rastering and to hard bake a spot into the PMMA at the location of the crosshairs Wait for 30 seconds then center click the button again to return to normal SEM mode Because the working distance is probably off by up to 0 1 mm the beam will not be very concentrated on the spot and it takes a long time 30 s to burn a visible spot After returning to normal SEM mode it may or may not be possible to see a faint ghostly blob You may need to zoom out to about 100 200 kX if the spot is quite large Focus on the ghostly blob When the focus is correct the ghostly blob looks more like a ghostly doughnut that is brighter around the edges and darker in the middle 70 Signal A SE2 Date 14 Sep 2004 Photo No 1062 Time 6 32 19 This ghostly blob is the result of a 30 secon
58. Electron Transport in GaAs Heterostructures at Various Magnetic Field Strengths a dissertation presented by Jeffrey Burnham Miller to the Division of Engineering and Applied Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Applied Physics Harvard University Cambridge Massachusetts January 2007 2007 Jeffrey Burnham Miller All rights reserved Dissertation Advisor Professor Charles M Marcus Author Jeffrey B Miller Electron Transport in GaAs Heterostructures at Various Magnetic Field Strengths Abstract This thesis describes two sets of experiments which explore transport in a two dimensional elec tron gas in the presence of a magnetic field We used nanofabrication techniques to make samples on GaAs AlGaAs heterostructures and measured the samples at cryogenic temperatures using ac lock in techniques In the first set of experiments the low field experiments we studied the effect of spin orbit coupling We tuned the strength of spin orbit coupling from the weak localization regime to the antilocalization regime using in situ gate control Using a new theory we separately extracted the values for the three material dependent spin orbit constants We also measured the average and variance of conductance in assorted quantum dots with and without strong spin orbit cou pling and found quantitative agreement with recent random matrix theory predictions as l
59. If no waste containers are available find empty solvent bottle left for this purpose on the shelves near the sliding doors Use a glass bottle for TCE waste Always fill out the waste label properly write the full name of the waste Isopropyl alcohol not IPA check the right boxes flamable for Acetone and IPA Toxic Poison and Flamable for TEC write Marcus on the advisor line Cleanroom for room 6 5546 for the phone number Don t fill in the date that is to be filled in on the day the bottle is full The DI water can go down the drain While you are cleaning the chip set up the spinner Only the Headway spinners are able to achieve repeatable results It happens that upstairs the right hand spinner is a Headway but it is also the spinner that is designated for epoxy SU 8 and other thick disgusting resists Some types 78 of SU 8 resist can form a residue on the spinner if not cleaned by the previous user that will spin a voluminous spiderweb like into the air when you use the spinner which lands on your chip ruining it However since SU 8 is just as frequently used on the left hand spinner even though it is forbidden you are better off using the Headway To prevent contamination line the spinner tray with foil Make a tidy hold for the spindle so the foil does not rub any moving parts Next select a chuck to hold your chip For chips less than 5mm squre the best chuck is the one with a single 1mm pillar in the middle with just on
60. In fact you are probably carrying some GaAs right now its in your cell phone 3 We means humanity in general but Loren Pfeiffer and Art Gossard the creators of the GaAs heterostructures I measured for this thesis in particular This small laboratory has been housed within a bigger laboratory provided by Prof Charlie Marcus see Acknowl edgements 5The chapters in my thesis are not arranged in the same order that we actually conducted the experiments 6The materials were the same GaAs AlGaAs heterostructures but the details were a bit different The wafers we used for the 5 2 experiments had electron mobility up to 2000 vs whereas the best mobility in the spin orbit samples was only 30 m vs the problem and the current status of the problem The introductions also discuss some of the non obvious experimental techniques we used for each set of experiments In the case of the 5 2 experiment the introduction also is intended to provide a fairly complete physically oriented theoretical background at a level that is comfortably accessible to the interested experimentalist As far as I know there is no such review in the literature which has not made my life easy over the past several years Hopefully the 5 2 introduction can serve as a starting point for other experimentalists who are new to topological quantum computation new to non Abelian anyons or new to incompressible weakly coupled Cooper pairs of composite fermions
61. Rev Lett 83 5090 1999 168 Brouwer P W Cremers J N H J and Halperin B I Phys Rev B 65 R81302 2002 169 Fujisawa T et al Nature 419 278 2002 170 Elzerman J M et al Nature 430 431 2004 171 Golovach V Khaetskii A and Loss D Phys Rev Lett 93 16601 2004 172 Zumb hl D M et al Phys Rev B 69 121305R 2004 173 Al tshuler B L JETP Lett 41 648 1985 174 Lee P A and Stone A D Phys Rev Lett 55 1622 1985 175 Meir Y Gefen Y and Entin Wohlman O Phys Rev Lett 63 798 1989 176 Al tshuler B L and Shklovskii B I Sov Phys JETP 64 127 1986 177 Stone A D Phys Rev B 39 10736 1989 178 Feng S Phys Rev B 39 8722 1989 179 Imry Y Europhys Lett 1 249 1986 180 Debray P et al Phys Rev Lett 63 2264 1989 181 Moon J S Birge N O and Golding B Phys Rev B 56 15124 1997 182 182 Fukai Y K et al Solid State Comm 94 757 1995 108 183 Birge N O Golding B and Haemmerle W H Phys Rev Lett 62 195 1989 184 Mailly D et al Europhys Lett 8 471 1989 185 Gustin C et al Physica E 17 154 2003 186 Gustin C et al Phys Rev B 68 241305R 2003 187 Hackens B et al Physica E 17 156 2003 188 Huibers A G et al Phys Rev Lett 81 1917 1998 189 Alves E R P and Lewenkopf C H Phys Rev Lett 88 256805 2002 190 Pan W et al Phys Rev Lett 83 3530 1999 191 Pan W et al Phys Rev Le
62. Zutic L Fabian J and Sarma S D Reviews of Modern Physics 76 323 2004 107 Rudolph J et al Applied Physics Letters 82 4516 2003 108 Duckheim M and Loss D Nature Physics 2 195 2006 109 Schliemann J Loss D and Westervelt R M Phys Rev B 73 085323 2006 110 Serebrennikov Y A Phys Rev B 73 195317 2006 111 Citro R Romeo F and Marinaro M Phys Rev B 74 115329 2006 112 Trushin M P and Chudnovskiy A L Physica E 34 397 2006 113 Zumb hl D M et al Phys Rev Lett 89 276803 2002 114 Zumb hl D M et al to be published 2004 115 Halperin B I et al Phys Rev Lett 86 2106 2001 116 Aleiner LL and Fal ko V L Phys Rev Lett 87 256801 2001 117 Fal ko V I and Jungwirth T Phys Rev B 65 81306 2002 118 Meyer J S Altland A and Al tshuler B L Phys Rev Lett 89 206601 2002 106 119 Meyer J S Fal ko VI and Al tshuler B L in NATO Science Series II edited by I V Lerner B L Altshuler V I Fal ko and T Giamarchi Kluwer Academic Publishers Dordrecht Vol 72 117 2002 120 Konemann J et al Phys Rev Lett 94 226404 2005 121 Zduniak A Dyakonov M I and Knap W Phys Rev B 56 1996 1997 122 Al tshuler B L et al Sov Phys JETP 54 411 1981 123 Aronov A G and Lyanda Geller Y B Phys Rev Lett 70 343 1993 124 Koga T et al Phys Rev Lett 89 046801 2002 125 Bergmann G Phys Rep 107 1 1984
63. a With this theory the Rashba contribution and linear and cubic Dresselhaus contributions to spin orbit coupling are separately estimated allowing the angular dependence of spin orbit precession to be extracted at various gate voltages This chapter is published in Phys Rev Lett 90 76807 2003 29 4 1 Introduction An important component along the path toward realizing quantum spintronic devices 100 101 is a structure that allows manipulation of electron spin without destroying phase coherence Spin orbit SO coupling has been the focus of recent studies because of its potentially useful role in coherent spin rotators 89 spin interference devices 123 and spin filters 103 124 The mecha nisms by which SO coupling affects transport 98 99 122 125 have recently been considered in the context of Aharonov Bohm AB phase and Berry phase 113 116 125 126 129 underscor ing the richness of the underlying physics The results in this and other recent experiments 130 cannot be explained without considering these AB like effects The conductivity of low dimensional systems shows signatures of quantum interference that depend on magnetic field and SO coupling 26 90 99 122 131 In particular constructive destructive backscattering associated with pairs of time reversed closed loop electron trajectories in the absence presence of significant SO interaction leads to negative positive magnetoresis tance effects
64. a distinct peak at zero bias and dips at intermediate biases resembles the Ig curves in that work In Ref 7o it is convincingly argued that the Ig curves are a signature of quasiparticle tunneling between the FQHE edge states based on quantitative comparison to applicable theory That theory states that the characteristic for tunneling between FQHE edge states 29 72 224 is expected to have a peak at zero bias and a minimum at intermediate biases 67 68 whereas tunneling between IQHE edge channels is expected to yield a flat ohmic curve The data we present for vopc 5 2 both the temperature dependence and the Ig curves are consistent with the formation of a FQHE state with tunneling related backscattering We interpret that a mechanism for the deviation of Rp from 0 4h near 5 2 and 21 3 as well as the peak and dip behavior of the Ig data could be tunneling between edge channels on opposite sides of Hall bar in the vicinity of the orc We do not believe the data can be explained by transport via thermally excited particles through the small bulk region of the Qrc since this process would be expected to have the opposite temperature dependence 7 10 Conclusion In conclusion we have observed plateau like features near vopc 5 2 and vopc 21 3 in QPcs with 1 2m and 0 8 um spacings between the gates The plateaus disappear between 30 70 mK At lower temperature the resistance of the plateau like feature is higher than the bulk
65. ab art project thank you for everything especially the soda sculpture Reilly brought with him tremendous knowledge about all aspects of low temperature experiments along with a selfless willingness to help others including me apply them He also brought his Australian friendliness and good humor and helped make work fun back in the good new days Jimmy tea connoisseur a cleanroom craftsman and vegetarian thanks for showing us how to put on boxing gloves and smash the bag I learn something every time I talk to any member of the lab Andrew Christian Doug Edward Hugh Jason Jennifer Josh Kate Leo Lily Mike Nadya Nathaniel Reinier Ron Sarah Slaven Susan and Yiming Thank you Yuli who seems never to have forgotten any number was kind enough to dial our number in lab over and over again until we all understood spin orbit coupling Bert Bernd and Xiao Gang were always happy to help us think about the 5 2 data while Loren was always glad to discuss tea or wafers All the members of Microsoft s Project Q team have been a tremendous help especially Ady and Steve David G G has been a source of advice experience knowledge and kindness Thank you So many people work out of everyday sight to keep a research group productive James Jim James Ralph Joan Susan thanks for keeping everything running smoothly One of my first projects was to help watch while Jim designed the BiasDAC a tool that has been invaluable for all o
66. abytes or more of data only to finish the calculation with a probabilistic projection However Bell s theorem specifically prevents local probabilistic algorithms from reproducing the quantum mechanical result In summary it is the exceedingly complex non local correlations of the quantum state along with the enormous size of the vector space Hilbert space of a quantum system that make it useful for efficient computation of certain types of problems At the end of this introduction we will see how the physics of the 5 2 state could meet these criteria Lots of schemes little coherence There are quite a few ideas for how to actually implement qubits including optically trapped atoms 8 or ions 12 quantum optics 13 cavity QED 14 NMR 15 and solid state implemen tations 16 17 In fact few qubit Qcs have already been demonstrated in some of these systems 18 20 but nobody has been able to build a oc with anywhere near enough qubits to be useful The difficulties including gate accuracy and noise reduction can largely be traced back to the fact that qubits are so good at forming non local correlations that they do not really know when to stop the qubits become correlated the more graphic word is entangled with the environment This entanglement effectively measures the system and that measurement collapses the qubits onto a set of basis states thereby destroying coherence This decoherence problem is extremely hard to over
67. ach ohmic behavior at high Ig fields 60 61 and variable density samples 62 have led to the conclusion that the v 5 2 state is probably spin polarized which tentatively rules out some competing theoretical explanations Ongoing experimental work 63 is likely to clarify the spin polarization properties of the bulk 5 2 state even further Prior ront tunneling experiments and some theory The use of a orc to selectively backscatter fractional edge channels for v 1 was experimen tally demonstrated as long ago as 1990 by Kouwenhoven and coworkers 64 Camino Zhou and Goldman have observed the v 11 3 plateau in an etched device with self aligned gates 65 In addition to selectively backscattering edge states a orc can bring edge states into close enough proximity to allow tunneling between them For FQHE edge states including 5 2 66 this is predicted 67 to cause the longitudinal resistance to diverge at zero temperature and zero voltage when all of the current tunnels into the counterpropagating edge Of particular relevance to our results the theoretical I V characteristic for tunneling between FQHE edge channels has a very distinct shape 68 illustrated in Figure 2 2 The peak at I4 0 and the minimum at intermediate Iac are understood to be signatures of tunneling between FQHE edge states 68 For IQHE edges states the tunneling behavior is ohmic This behavior has been observed experimentally 69 70 and found to
68. all the solvents and can itself be completely removed from the surface of the chip through evaporation Organic solvents form a monolayer on the chip surface that will not evaporate even at 180 degrees C DI water is available from the plastic gun like dispenser in the fume hood Always check that the resistance on the DI water filter is 18 2MOhms or higher Upon removing the chip from the water blow it off with compressed nitrogen immediately While blowing hold the chip on a cleanroom wipe so any liquid that is tempted by surface tension to cling to the sides will be wicked away IF any liquid of any kind is ever allowed to dry on the chip it WILL be carrying some solid impurites which will precipitate onto the surface and potentially ruin the chip Drive away the water by heating the chip on a hotplate at 180 degrees C for five minutes When using a hotplate be sure to put the chip on an area that is clear of hard baked photoresist and other crap that inconsiderate cleanroom users have left for you Also be sure that in the unlikely event that you contaminate the hot plate to clean it off with an appropriate solvent generally acetone or PGMEA if you are using LOR 3A Only jerks ruin the cleanroom for other users and you are not a jerk Empty the cleaning solvents into waster containers found in the yellow cabinets The TCE goes into a separate container in the toxics cabinet The aceton and IPA go into the general solvent waste container
69. ample loading port Do not drop the sample holder as some people have previously done If there is any silver paint on the front of the sam ple holder carefully clean it off with acetone and a cleanroom wipe or a cleanroom Q tip Liberally blow off the front and back of the sample holder using compressed nitrogen Be sure to remove all dust and cat hair from the three holes on the back of the holder anything sitting in these holes will lead to drift E Back of the sample holder Be sure to blow out the indicated holes The Raith computer is on the left the Leo SEM computer is on the right The joystick far left is used to move the SEM stage Start the Raith software and activate your user ac count BAS Front of the sample holder showing the Faraday cup and clip i EA 1 Secure your chip with clip 1 oriented correctly as indicat If needed activate the Navigator Exchange win ed dow by clicking on the stop light near the Roman nu Wearing gloves to protect the Raith from your greasy meral I on the command bar Click the Load Sample Page2 65 button mand tab select Faraday cup on holder and click go This drives the stage to the Faraday cup If all goes well your sample will be loaded in about 10 minutes If wearing a bunny suit or snowpants Turn on the beam by clicking the beam blank un causes a psychological need for you to pee now blank button on the Raith computer
70. ance is correct for this u v location 71 If the spot is not circular you will need to adjust the aperture until it is circular I always click the marker button to put two measurement markers on the SEM image to check the diameter in the two 45 degree direc tions otherwise the optimistic eye can be fooled into believing that an oval is a circle Once the working distance and the aperture are cor rectly set these values need to be fed to the Raith computer In the Adjust UVW Global window click the Read button on line 1 to get the x y coordinates and the working distance of the current location Type the current u v coordinates into the U and V columns These values can be found in the Coordinates window It is important to enter the u v values correctly or the entire UV coordinate system could become stretched rotated or shifted Click the checkbox on line 1 to tell the Raith that everything is set Blank the beam click the UV lightning bolt on line 2 to move to your next focus location and repeat the entire focus procedure Repeat for line 3 When all three checkboxes are clicked click the Adjust button which causes the Raith to use the new values for focus correction Think your done No way The Raith software now requires you to repeat the ENTIRE procedure Un check all the boxes and repeat the entire focus pro cedure again Seriously Don t ask me why the
71. ate is exceptionally fragile only the highest quality GaAs AlGaAs heterostructures exhibit a 5 2 state even in bulk samples Experimental investiga tion of the statistics of the 5 2 ground state is crucial especially since alternative models have been proposed to explain the 5 2 state in confined geometries 200 and in the bulk 49 52 In this paper we study the 5 2 state in the vicinity of a quantum point contact Near a QPC the electron density is not uniform so the notion of a orc filling fraction is not well defined However based on transport measurements it is possible to define an effective filling fraction in the vicinity of the orc vopc as discussed below Below 30 mK a plateau like feature with diagonal resistance also defined below near but above the bulk quantized value of 0 4h e is evident at vopc 5 2 in orcs with 1 2 jm and 0 8gym spacings between the gates On this plateau we find a peak in the differential resistance at dc current bias I4 0 and a dip around lac 1 2nA a characteristic shape that is consistent with oprc induced quasiparticle tunneling between fractional edge states 68 We also observe a zero bias peak at vopc 21 3 whereas we find a zero bias dip near vopc 22 3 201 consistent with previous orc studies for vopc lt 1 70 Figure 7 1 a SEM micrograph of the 0 5 ym orc b Optical micrograph of the entire device the outline of the wet etched Hall bar has been enhanced for clarity
72. b servations are consistent with the formation of a gapped incompressible FQHE state in the orc at these filling fractions with orc induced tunneling between the edge states In a opc with 0 5 um spacing between the gates we do not observe a plateau like feature at any temperature and the lac characteristic is flat for the entire range between v 3 and v 2 This suggests that in our sample no incompressible states form in this opc probably due either to confinement or the ef fects of decreased electron density All of these measurements were carried out in a magnetic field range where the bulk filling fraction was on the IQHE v 3 plateau while the filling fraction in the orc was tuned to lower values via the gate voltage Interestingly we find that there is a peak in resistance at zero Ig for the v 5 2 state not a dip see Figure 2 3 In the language of an interesting paper by Roddaro 70 this means that the 5 2 state behaves like a particle tunneling state not a hole tunneling state Furthermore although All irreversible gate behavior was reversible upon warming and re cooling the device We did find consistent with the literature that illumination improved the bulk ronz properties of the material 15 0 4304 0 4201 0 4104 0 4004 h e 0 3904 R 0 380 3704 d 775 nm point contact i
73. bottles are not appropriate as solvents will explode when mixed with acid Have a fluoroware beaker full of DI water avialable Use a pipet to suck 1 ml of sulfuric acid directly out of the bottle and measure it 81 into the small graduated cylinder Empty any excess acid out of the pipet into the waste container and rinse the pipet a few times with the clean DI water from your nearby beaker Dispose of the pipet in the sharps waste not the trash otherwise an innocent person is likely to be stabbed while handling the trash Add the acid to the mixture and mix well The acid is added before the H202 so that the heat liberated by the reaction does not degrate the H202 Finally use a pipet to measure H202 into the mixture and stir well Clean all the measurement tools with everything with DI water ensuring that all chemicals end up in waste containers and finishing the job by rinsing well with DI water in the sink Allow the etch solution to sit and equilibrate while you measure the thickness of the photoresist B 4 3 Profilometer Operation Log into the profilometer computer user PBC pass wismad Start the profilometer software Place a chip on the sample holder Click the view sample button the screen will be black since the camera is still raised Click the button to lower the tower As the tower descends use the x and y translation knobs to position the chip when the tip lands it must hit your chip not the sample holder in ord
74. c field Bso 7 x E 1 v 8 3 1 where v is the velocity of the electron which causes the spin to Larmor precess via s Bgo So we see a magnetic field or at least an effective magnetic field creeping into the system even when we did not explicitly apply one What is really interesting though is that this effective field does not lift Kramer s degeneracy the way a normal magnetic field would because spin orbit coupling does not break time reversal symmetry For an electron in a GaAs AlGaAs heterostructure spin orbit coupling can arise via three different mechanisms The first two mechanism are due to the GaAs itself the polar bond between the gallium and arsenic atoms is asymmetric and produces an electric field which produces a spin orbit contri bution known as the Dresselhaus term Silicon by contrast is inversion symmetric it only has one kind of atom so does not exhibit Dresselhaus type spin orbit coupling Dresselhaus 83 wrote down the full Hamiltonian for 3p conduction band electrons with this type of interaction but in 2D only two terms survive One term is linear in the momentum k of the electron note that k 27r is related to the electron density which we can control experimentally via a top gate Ht 04 Oxky oyky 3 2 and one term is cubic in k HY y oxkxk oykyk2 D Oxrkx y FyKy s 3 3 The third source of spin orbit coupling in a GaAs AlGaAs heterointerface is the electric field f
75. ch less likely to cause confusion Finally for example say you want to write your 2 2 mm by 1 mm pattern with a working area of U 1150 um to 1150 um V 550 um to 550 um at the center of the mesa The correct position to specify is 1 1 mm 0 5 mm Set the dose Factor For example if you specified the default dose to be 100 pAs cm using the dose calculator window but you want to write the pat tern at 300 uAs cm enter 3 0 into the Dose Factor Page 11 box click the Exposure Parameter button to dis play the Dose information Usually if you set up the dose when you measured the beam current all the boxes can be clicked Default Create more positions if desired One way is to shift drag existing positions to copy them then modify some of the parameters as needed Another way par ticularly useful for exposure matrices is to select the position then click Filter Matrix Copy The Create Position Matrix window opens and you can specify a step size and a dose factor The selected position will be duplicated as many times as you specify off set by the step size and with the specified dose factor increments When you have all the positions you want click Scan All The Raith moves the stage to the correct location and exposes the PMMA On screen you will see a graphical representation of the order in which the gates are written Incidentally this representa tion is the size of the working area so if yo
76. come since it is very hard to isolate the physical qubits ie little atoms or electrons from the entire rest of the universe Each proposed oc implementation has schemes to reduce the decoherence But the implementation I am gearing up to describe the topological oc has uniquely among known implementations a built in resistance to decoherence However before I can describe a topological Qc I need to introduce anyons 2 2 2 Introduction to anyons Topological quantum computers roc depend on a mathematical construction called the anyon Amazingly it turns out that this mathematical construction is also a physical reality the quasi particles that form in the regime of FQHE plateaus are anyons In this section I introduce these amazing little particles I begin by discussing the quantum mechanical concept of identical parti cles Systems of identical particles are a fundamental topic in quantum mechanics 21 Identical is a technical term the theory of quantum mechanics states that it is not possible even in principle to distinguish two electrons two protons two neutrons etc As a result if we swap two electrons there may not be any observable differences between the two states More mathematically we can define a permutation operator P in 3D according to P x1 x2 x2 x1 2 3 It is easy to confirm that PP I the identity matrix and so the only eigenvalues of P are 1 This result is a mathematical deduction fro
77. command bar would be a good time because the next few steps take a while On the Leo SEM computer the right computer turn on the SEM control window by clicking on the red and green circular dial looking button on the command bar To display the Navigator Exchange window click the I stop light Load and unload sample buttons are available in the Navigator Exchange window Initial Setup After sample is loaded Raith asks several ques tions Reset coordinate system Yes Switch on beam Yes Select accel Voltage 30 kV Ap erture size 10 um No values for astigmatism and aperture align in database OK scum 1 3e 005 Tor Gun Vacuum 832010 Torr The beam current ramps up over about 1 minute or less Note the time in your lab book it is nice to know how long the beam has been warming up before you begin writing While the beam is ramping up open the Find home Click on the Detectors tab and set Signal A to window and click the lightning bold to find home in SE2 Set the brightness to about 52 the contrast the x direction then the y direction I do this opera to about 4396 tion every time I load the Raith because I think it re duces drift problems caused when the sample holder is poorly seated on its three pin suspension system To set the zoom of the Leo first click on the zoom fo cus button on the Leo control bar the l
78. d and 250 mV solid corresponding to densities of 2 3 5 0 and 7 0 x 10 m respectively using Eq 4 2 However these fits also lead to the unphysical result that Tso lt T Such unphysical results are not surprising given that for V gt 50 mV the SO length vr Q is less than while theory 85 122 128 assumes diffusive spin evolution amp Ago Ly Finally we note that a theory for arbitrarily strong SO coupling 129 may also be used to fit this data by including B via Lg yielding values for Op3 and Op for all V s which agree with our estimates using Eq 4 2 to within a factor of 3 However the theory in 129 does not separate Op and Og terms 4 8 Conclusion In conclusion we have realized an in situ gate controlled crossover from weak localization to antilocalization in a GaAs AlGaAs 2DEG experimentally demonstrating that spin rotation can be strongly modulated in a clean phase coherent system New theory addresses spin orbit effects 35 in the quasi ballistic regime and allows separate measurement of the Rashba linear Dresselhaus and cubic Dresselhaus terms 49 Acknowledgements We thank I Aleiner H Bruus and S Studenikin for illuminating discussions and F Mancoff for device fabrication This work was supported in part by DARPA QUuIST DARPA SpinS ARO MURI and NSF NSEC We also acknowledge support from ONR and NSA Y L G NDSEG J B M and the Harvard Society of Fellows D G G
79. d burn after focus ing as well as possible on the nearby alignment mark A cre ative eye is required and it doesn t hurt to believe in ghosts r2 EHT 30 00 kV SignalA SE2 Date 14 Sep 2004 Mag 301 96 K X WD 11mm Photo No 1063 Time 26 33 30 Here is the same 30 second spot in focus You can tell the spot is in focus because it looks like a ring or a doughnut Notice the diameter is about 120 nm After focusing tap the joystick to move away from your 30 second spot and burn another spot for about 10 seconds This spot should have a much smaller diameter because the beam is much better focused which is also why a shorter burn time can be used Focus on the 10 second spot move away from it and burn a 5 second spot Page 7 Signal A7 SE2 Date 11 Sep 2004 Photo No 1059 Time 9 21 33 Mag 303 73 K X A spot burned for 10 s after focusing on a 30 s spot EHT 30 00 kv SignalA SE2 Date 11 Sep 2004 WD 11mm 48 Photo No 1058 Time 9 20 20am I4 Mag 307 97 KX The same 10 s spot after focusing EHT 30 00 kv wo SignalA SE2 Date 11 Sep 2004 11mm 20am Mag 642 09 K X m Photo No 1060 Time 9 25 43 Le BFR STII EL CANIS FLEES OE EOE ET ODD SB EET A 5s spot in focus You will know that the working distance is correct when the diameter is 40 50 nm no more focusing is required Note that the zoom here is 650 kX When the spot size is 40 50 nm the working dist
80. d loosening the two screws on the sides Install the correct chuck Lower the mask frame and place a piece of glass not your mask agains the pegs Turn on the mask vacuum Hold in the leveler buton on the aligner on the left front of the aligner and raise the cuck by turning the z knob If the knob rotates without raising the chuck tighten the idiotically long screw on the side of the knob If the knob won t turn this rediculously long screw is probably hitting the edge move the stage back with the x micrometer dial until there is enough clearance Incidentally the top of the z knob is supposed to slip when the pressure of the chip chuck against the mask is appropriate and this slip tension can be adjusted by the screw in the top on the knob hidden under a plastic cap Unfortunately people often adjust this tension often I think people mistakenly radically overtighten the tension screw when they really wanted to tighten the rediculously long screw so this feature cannot be relied upon Continue raising the chuck and holding the level button until the top of the chuck is pressed into the glass and has been leveled by it Release the level button and lower the chuck Replace the glass with your mask Check that all the micrometers are apromimately cen tered so you have maximum adjustment capability in both directions The chromimum side brown looking side of the mask should be face down Put the chip in the center of the chuck in the desired o
81. d to cover the chip surface on drop usually and quickly start the spinner Some people recommend applying the resist during a 5 second 500 RPM pre spin but it is not necesary for our small chips and is likely to increase problems with bubbles When all your chips are finished spinning bake them on the hotplate at 115 C for 2 minutes If you are working with more than one chip you can keep them clean safe and identifiable lining them up on a glass slide inside a covered glass petri dish To time more than one chip place the first chip on the hotplate and start the watch Then place the other chips in a line When the alarm sounds remove them in the same order it will take the same amound of time to remove them as 79 it took to place them so they will all get the same bake time Dispose of any excess photoresist waste in a correctly labelled waste container Throw the empty beaker into the covered trashcan If the photomask is not clean you can assume it is not unless you cleaned it hold it over 1 2 wipes and squirt it with a strong jet of acetone Blow it dry then rinse blow with IPA It is very important to blow the mask completely dry This should remove almost any dirt but if it dosn t ask the CNS staff about photomask cleaning products Move to the AB mask aligner Raise the mask frame and make sure that the small squre chuck with a single vacuum hole is installed If not remove the chuck by pulling the hoses off the aligner an
82. ding the Ctrl key drag the center of the green cross to match the crosshairs on the SEM im age the cross turns blue Click continue Two more slow scans will be pre sented align the cross in each one The Raith asks if you want to accept the changes You can examine the changes to Zoom Shift and Rotation in the Align Writefield window Accept the changes Unblank the beam reactivate user control of the SEM click the button Move just off the edge of your alignment mark zoom in to 300 kX and burn a new spot Zoom out enough to see the spot and a good portion of the alignment mark If you do not move the joystick and there is no drift the crosshairs will be exactly centered on the burn spot Click sent and blank the beam repeat align write field procedure with a scan size of 10 um After the align writefield procedure is complete again go back to the SEM and make sure the spot is exactly in the crosshairs Repeat again with a scan size of 5 um then again with 1 um Repeat the align writefield procedure until the changes are acceptably small Specifically the change in zoom should be a factor between 0 99975 and 1 00025 which for a 100 um writefield yields a zoom error of 25 nm or less per writefield which should not result in stitching errors The shift error should be just a few nm and the rotation error should be a few 0 001 s of a degree Write these values down in your lab b
83. dot at T 0 3K along with fits to RMT solid curves In b the center gate is fully depleted Vertical lines indicate the fitting range error bars of 2 are about the size of the squares same wafer 105 Comparing Figures 5 1 a and 1 c and recalling that all dots are fabricated on the same wafer one sees that AL is suppressed in smaller dots even though Aso is sufficient to produce AL in the larger dot We note that these dots do not strongly satisfy the inequalities L Aso 1 N 7 1 having N 2 and L Ago 0 64 0 34 for the large small dot Nevertheless Figure5 1 shows the very good agreement between experiment and the new RMT 5 5 Suppression of Antilocalization by an In Plane Magnetic Field We next consider the influence of Bj on g In order to apply tesla scale Bj while maintaining subgauss control of B we mount the sample with the 2DEG aligned to the axis of the primary solenoid accurate to 1 and use an independent split coil magnet attached to the cryostat to provide B as well as to compensate for sample misalignment 96 Figure 5 2 shows shape averaged magnetoconductance relative to B gt gt 9 A i e fully broken time reversal symmetry g B Bj g B By 8 B1 gt o A Bj as a function of B at several values of Bj along with fits of RMT 116 150 with parameters Aso Tp and x set by a single fit to the By 0 data The low field dependence of 52 0 Bj on By Figure 5 2 b allo
84. e fabrication process is electron beam lithography A small part of the trickiness is that making arbitrarily shaped 3o nm gold lines that is lines of gold only about 200 atoms wide is near the edge of technological capability It is near the edge but it is not over the edge At Harvard the main part of the trickiness is the Raith a slightly rickety pile of passable hardware and defective software Over the course of time the Raith users at Harvard developed methods to outsmart the defective software and push the hardware to the white knuckle edge of functionality ie we got the thing to work to spec I wrote these methods down in Jeff s Fairly Comprehensive Raith Usage Notes a document that became the de facto Raith user s manual at Harvard 225 I m rather proud of this little document which I have included in its original form in this Appendix Jeff s Fairly Comprehensive Raith Usage Notes start on the next page 63 Jeff s Fairly Comprehensive Raith Usage Notes Jeff Miller Marcus roup Harvard University Te document is intended to be a friendly fairly comprehensive guide to using the Raith Elec tron Beam Lithography system including tricks to get small dots to write nicely even if the Raith is not being nice Written primarily as an appendix to Jeff s thesis it should be updated whenever somebody learns a new trick Pre Raith Preparation Design the Device Use DesignCAD to make your device He
85. e s int n double f double binSize 1 This procedure integrates the curve as described in Pn binSize is the w size of the trapezoids for integration double b double c Laguerre n 0 0 double i 1 0 double Sum 0 double t double t2 double nSmall O0 double SMALL 1e 25 l1e 50 double ce do 94 binSize i t2 tet exp s f t t2 2 c ce Laguerre n t2 if fabs c lt SMALL ce lt 1e 150 nSmall We are looking for enough small numbers in a row to quit else nS8mall 0 Sum binSize 2 c b Add the area of the new trapezoid to the overall integral a oO ll itt while nSmall lt 6 Quit if we find 5 small numbers in a row return s Sum static double Poo double x double f double binSize This procedure return Po either exactly if erf 1 0 or by numerical integration double A sqrt 2 x f double ErfResult derf A sqrt 2 double Pint double exact sqrt 2 x exp 0 5 A A sqrt 3 141592653589793238 2 1 ErfResult if ErfResult gt 9999999999990 1 Pint Pn sqrt 2 x 0 f binSize return Pint else return exact Fs static double Pnm double s int n double m double f double binSize This procedure integrates the curve as described in Pnm binSize is the w size of the trapezoids for integration double b double c Laguerre n 0 0 double i 1 0 double Sum 0 double t double t2 double nSma
86. e to escape A is the mean level spacing z gugB is the Zeeman energy and Er is the Thouless energy Ta ble 5 5 Decoherence is included as a fictitious voltage probe 116 150 162 164 with dimension 38 A A T Er A ev ep A ay az by m eV ns ns 1T ns T 8 2 6 0 0 35 33 0 15 0 04 6 6 6 6 0 24 12 17 73 0 32 0 33 3 2 0 140 8 0 9 2 3 86 3 6 3 1 1 4 0 9 3 7 Table 5 1 Dot area A LL 130nm edge depletion spin degenerate mean level spacing A 27h m A m 0 067me dwell time T4 h NA Thouless energy Er fivr VA e and ey for the fits in Figure 5 1 B coefficients 4 and a from one and two parameter fits B coefficient b from two parameter fit see text less dephasing rate N h AT where T is the phase coherence time SO lengths 1 2 along respective principal axes 110 and 110 are assumed within the RMT to be large compared to the dot dimensions L4 along these axes We define the mean SO length Aso 1 A5 and SO anisotropy Vso A1 As SO coupling introduces two energy scales e x Er L1L5 A2 representing a spin dependent Aharonov Bohm like effect and er L3 M L2 A2 e9 SO providing spin flips AL appears in the regime of strong SO coupling ee gt 4 where 4 y h 1 is the total level broadening Note that large dots reach the strong SO regime at relatively weaker SO coupling than small dots
87. e vacuum hole and no additional vacuum channels The vacuum hole is there to hold the chip while it spins so make sure it is clear of photoresist Otherwise the interlock will detect good vacuum allow the spinner to start and fling your chip at high speed into the spinner tray If you need to clean the hole use either a needle tool which you carry just for such a purpose and or acetone followed with an IPA rinse to avoide damaging the O ring too much Some photoresists are better removed with PGMEA Blow the chuck dry and firmly press onto the spindle Program the spinner Press Recipe o recipe o is the only recipe we are allowed to edit the others are set to a fixed program Press Step 1 Press Speed then type 5000 enter to set the spin rate in RPM Press Speed again choose 5000 enter to set the ramp rate in RPM sec Press StepTerminate and type 45 enter to set the spin time Press Step 2 StepTerminate o enter the recipe ends at the first o time step Press Step o to end the programming sequence and return to run mode Always spin a junk chip Balance the chip on the chuck This is often easier if you turn on the vacuum by pressing the VacuumOnAuto button to toggle the vacuum ON Don t leave vacuum on for highly extended periods it isn t that great for the pump Center the chip carefully rotating the chuck by hand to be sure Toggle vacuum back to Auto Press the green foot switch to start the spinner Check the speed on the display a
88. e wave function of the system is exactly Eq 2 6 But when the field is detuned a localized quasihole is created The wave function of the ground state plus a quasihole localized at position zo is obtained by acting on the ground state pr LG zo 2 7 1 It is clear from the z zo term that all the electrons in the ground state feel a barrier at zo and are pushed away The range of the repulsion is short far away from zo the wave function is essentially unaffected Less immediately obvious is the fact that the quasihole has fractional charge e m However it can be shown 32 that the spatial extent of the bubble in the electron fluid excludes exactly 1 11 of an electron so the quasihole carries the fractional charge e m It can also be shown using topological Berry phase arguments similar in spirit to our introduction to anyons in Section 2 2 2 that the quantum statistics of the quasiholes are fractional although to do so correctly is extremely non trivial Quasiparticles are described by a slightly more complicated wave function but they also have fractional charge and fractional statistics The upshot is that these FQHE quasiholes and quasiparticles are anyons 27 34 Fractional charge has been observed experimentally 35 but to observe the anyonic statistics remains an experimental challenge 2 36 5Depending on the sign of the detuning either a quasiparticle or a quasihole can be created Quasiholes are technicall
89. eam expanded to meet the problem Iuliana has a sharp eye and a physical way of thinking about things that helped us keep making progress even when nothing seemed to be working She is also dependable with a capital D She stuck with the project through almost intolerable difficulties and kept things running without a minute s interruption even when it was time for me to go have a baby Marc welcomed me into his lab and became a soothing source of valuable advice and encouragement I always looked forward to spending a few minutes or a few hours looking at data in Marc s office he knows that every little piece of data is telling part of a story and he knows how to go from noisy messy working data to a neat tidy story by reading with a careful eye and asking a few perceptive questions Eli learned a lot of details amazingly fast and helped keep a seemingly endless experiment running 1x when everyone else was just too exhausted Thank you For all practical purposes I learned quantum mechanics statistical mechanics and con densed matter physics from Heather Thank you for your patience and help with all those prob lem sets and for helping to make work fun back in the good old days Alex was always interested and able to help with any problem from the most obscure detail of mathematical physics to Raith alignment difficulties to helping move a sofa up the stairs He also in a lab full of creative folks wins the prize for the most memorable l
90. eau like behavior in the 1 2 jm Qrc and somewhat less well developed plateaus in in the 0 8 ym orc although vy is not on a plateau when vopc 21 3 but again these features are suppressed in the 0 5 um orc We do not observe any plateaus near Vopc 22 3 in any of the arcs The reentrant integer quantum Hall effect features 80 which are clearly visible in the bulk do not survive at all in the arcs We interpret the plateau like features in the two larger orcs as indicating that the incom pressible states at vopc 5 2 and vopc 21 3 are not destroyed by the confinement The linear plateau less behavior in the 0 5 um orc is reminiscent of a classical Hall line suggesting that no incompressible states survive in this QPC 7 8 Temperature data in the orc Temperature dependence for a representative V setting of the 1 2 ym orc is shown in Fig 7 6 Below 30mK a distinct plateau like feature is evident This plateau disappears between 30 to 58 0 5F a QPC Figure 7 5 Typical magnetoresistance from v 3 to v 2 measured concurrently in the QPC a and the bulk b In a the Rp curves are from three different Qrcs of lithographic size 0 5 gm black 0 8 um red and 1 2 um blue The colored stripes highlight regions in field where the resistance in the 1 2 yim and o 8 yim Qrcs forms a plateau like feature near vopc 5 2 with Vpulk 3 The applied gate voltages V are 2 2 2 0 and 1 9 V for the 1 2 0 8 and 0 5 ym QPcs a
91. eft mouse button of the SEM mouse and move the mouse left or right to zoom out or in Zoom all the way out Look for the Faraday Cup Use the joystick to move center the left hand edge of the Faraday cup under the green crosshairs If the green crosshairs are not visible they can be toggled using the indicated button In the Stage control window under the com Page 3 66 Signal A SE2 Date 11 Sep 2004 Photo No 1052 Time 82932 Focus on the metal burr on the left hand side of the Faraday cup To focus hold the center mouse button on the Leo mouse and drag left or right 20um EHT 30 00 kV g 282X T WO 11mm Click the Apertures tab on the SEM control panel Signal A SE2 Date 11 Sep 2004 Photo No 1052 Time 83118 67 Zoom to about 50 kXClick the Focus Wobble box then the Aperture Align button After clicking the Aperture Align button the current values for the aperture alignment appear in a green box at the bot tom of the SEM screen Double click this box to en ter the alignment values from your previous Raith session as a starting point Fine tune the aperture alignment by either dragging the red dot in the SEM control panel or by moving the slider bars or by after clicking the Ap Align button left click dragging up down or left right on the SEM image The alignment is correct when the image does not move due to focus wobble but in stead onl
92. en symbols of conductance as a function of magnetic field B perpendicular to the 2DEG at a temperature T 300 mK and zero magnetic field Bj 0 in the plane of the 2DEG measured in the devices on high density 2DEG Insets show device micrographs AL due to SO coupling is seen in the big 8 pm dot a WL is seen in the smaller 1 2 um dot b fabricated on the same material demonstrating confinement suppression of SO effects Both dots show a larger variance at Bj 0 when TRS is not broken Fits of the RMT 116 150 to g B are shown as dashed curves Solid curves are the RMT for var g B with the same parameters as obtained from fits to g times an overall correction factor see text n A To Aso Vso KL fear a b m pm ns pm nsf ms Tr 2 0 3 0 0 18 8 5 1 0 0 15 1 0 2 8 0 5 L0 1 0 028 2 0 8 0 0 21 8 5 1 0 0 25 0 6 3 0 0 37 0 07 0 028 5 8 1 2 0 10 3 2 1 4 0 33 1 9 1 0 6 641 0 14 5 8 8 0 0 39 4 4 1 4 0 23 0 7 0 45 1 40 4 0 14 Table 6 1 Carrier density n dot area A L coherence time Ty spin orbit parameters As and vso RMT parameters x foar and and FJ parameters a and b see text is estimated It is noted that a Aso 8 5 gm noticeably reduces the WL correction amplitude at the lowest temperatures T 50 mK The resulting low temperature saturation of coherence times extracted using theory neglecting SO effect
93. er to correctly calibrate the tip height The tip should not damage photoresist or GaAs but if you are paranoid use a junkchip Meanwhile NEVER move the sample when the tip is down as it could damage the profilometer After the tower lowers and the tip hits your sample the tip will automatically raise slightly Use the two rings on the camera to adjust the zoom and focus Generally maximum zoom is appropriate for our devices Use the lightbulb icons to adjust the light level After setting the zoom click the tip down button Right click the point where the tip hits its shadow and choose to adjust the crosshair calibration This readjusted crosshair is your best guide for positioning the scan Since the tip is not touching the sample use the knobs to move the feature you want to measure to a position about 2 3 crosshair divisions to the right of your calibrated crosshair The scan will move from left to right starting at the crosshair on the screen Physically this is from front to back on your sample Rotate the stage to get the correct angle Generally it is best to start and end the scan on a surface that is known to be level like the surface of the chip and arrange for a bump a narrowish region of photoresist to be in the middle of the scan This allows you to confidently level the scan on the two side regions Before starting the scan click the recipe set up button Choose a scan length 100 200 microns a scan time 30 60s a
94. esired For instance on a particu larly thin gate you may wish to boost the dose to 1 2 to give that gate 20 more dose Page 10 Close the GDSII Editor but be sure to save the new GDSII data Click File New Position List Drag your pattern from the GDSII database onto the position list Right click Properties on the pattern in the posi tion list to open the Exposure Properties window Enter the layers you would like to write Typically for an exposure matrix you will enter just the 1 3 layers with the smallest features When writing the real device I typically write just these small features i e everything that fits into a single working area and then specify a second position in the position list for the larger gates Enter the working area and position These steps are actually a bit confusing The working area is the por tion of your design that you would like to write For instance if you are writing an exposure matrix you may only want to write the central 10 um of your device You could therefore specify the working area to be 5 um to 5 um in both U and V However you may instead want to specify a working area that is an integer multiple of the writefield size writefield size is 100 um because it simplifies the calculation of the position special Raith feature In that case you would design your layers in such a way that ex posure matrix sized chunks are defined by layer If t
95. ething unexpected about ucr in a quantum dot Folk s experiment was designed 9To protect the privacy of the sender I have removed names and places 25 to try to determine whether electron electron interactions break spin degeneracy in a quantum dot by applying an in plane field In Folk s words 96 The original concept for the measurement was that if the system were spin degen erate at low field then a large in plane field would lift the degeneracy via Zeeman splitting with associated changes in the amplitude of conductance fluctuation If on the other hand spin degeneracy at low field were already lifted by interactions then a large parallel field would not alter spectral statistics and hence conductance fluc tuation amplitude Surprisingly we find that the conductance fluctuations are indeed suppressed by a strong parallel field suggesting degeneracy at low field but in many cases by a significantly greater factor than can be understood in terms of a simple breaking of spin degeneracy At the end of the paper we suggest a possible explana tion for this large suppression as resulting from field dependent spin orbit scattering Something unexpect was going on and the cause was probably spin orbit coupling Halperin and coworkers published a theory paper the very next letter after Folk supporting spin orbit as the explanation 115 Meanwhile Dominik and I had a chip with strong well studied spin orbit coupling in the fr
96. f our research Thank you To my parents for teaching by example for having exuberant confidence in me and for all your love thank you To Neer who has only just started helping you have been a constant delight while I ve been doing a lot of typing you even helped a little at the keyboard and you would have liked to do more thank you Experiments take a long time sometimes it seems like I never came home But when I did come home without fail it was to a loving reception Following Manjari to Harvard turns out to have been a fabulous decision here I ve had a chance to work with all the wonderful people I mentioned learn from them how to be a scientist push my perfectionism problem to almost absurd levels also following some world class examples help out with really interesting research but most rewardingly to love Manjari Thank you CHAPTER I Introduction T SEEMS INEVITABLE THAT ARSENIC the garlicky king of poisons and poison of kings and gallium an element named after France would combine to form a compound of some intrigue Indeed the compound gallium arsenide GaAs has played host to some very intriguing physics over the years One feature that makes GaAs so scientifically useful is that we can grow a nearly perfect interface between GaAs and AlGaAs where electrons become trapped in a two dimensional 2D universe When these electrons are under the influence of a strong magnetic field there is almost no tellin
97. g use the knobs to move the chip out from under the tip When the chip is well clear of the tip use a tweezers to remove it from the stage Never try to pull a chip out from under the tip with the tweezers due to the risk of hitting the tip Etch the chip A typical etch rate is 5 nm s The etch depth target should be set to remove all the dopants plus about 10 if the sample is single side doped you do not have to etch all the way to the 2pzc if the sample is double side dope you need to etch all the way to the under side doping layer or you ll have leaky gates The reason not to etch too far is that eventually you will have to evaporate continuous gates over the mesa edge and you don t want to have to make them too thick because liftoff gets hard Calculate an etch time that should etch about half way to the target That gives enough time to calculate the etch rate but does not risk going way to far the etch rate shouldn t be 6 nm s Use the profilometer to measure the depth calculate the etch rate and iterate towards the target It should be possible to hit the target within 576 Occasionally some materials will have some oxide or something on the surface that takes a long time to etch Be patient cowboy Once the oxide is gone the etch rate goes back to the expected rate you don t want to etch all the way to China just to get through the oxide with fewer profliometer iterations Remove the photoresist When the profilometer tells
98. g what wild physics can happen nobody predicted that 2p electrons and some magnetic flux would condense into an entirely new state of matter 1 the anyonic fractional quantum Hall fluid right at the GaAs AlGaAs boundary During my PhD I have studied how electrons confined to 2D behave when I apply a mag netic field The laboratory for these studies has been the GaAs AlGaAs heterostructure The results of my research are contained in this thesis Organization of the thesis This thesis is organized into three general parts the introduction the papers and the appendices The introductions My PhD work can be divided into two different magnetic field ranges which I introduce sepa rately in two different chapters Chronologically I first studied the low field transport properties of electrons in a two dimensional electron gas 2DEG in particular the effects of spin orbit cou pling and magnetic field applied both parallel and perpendicular to the 2pxEc The last two years of my PhD I studied the high field properties of the same system in particular the fractional quan tum Hall effect state at filling fraction v 5 2 In both introduction chapters the primary goal is to lay out the problems I have studied during my PhD why the problem is interesting the status of the problem before my work the specific contributions my work has made to understanding 1France Gallia Other features of GaAs also make it practically useful
99. gate struct struct DPComplexNum pl complex parameter struct DPComplexNum result complex result p 1 p gt result real p gt pi real p gt result imag p gt pl imag return O0 h static long RegisterFunction 1 int funcIndex NOTE Some X Ps should return a result of NIL in response to the FUNCADDRS message See XOP manual Restrictions on Direct XFUNCs section funcIndex GetXOPItem 0 which function invoked switch funcIndex case O0 XFUNC1Add p1 p2 return long YuliC break case 1 XFUNCi1Divu pi p2 99 return long YuliD break case 2 XFUNC1ComplexConjugate p1 return long XFUNCiComplexConjugate break return NIL DoFunction This will actually never be called because all of the functions use the direct method It would be called if a function used the message method See the XOP manual for a discussion of direct versus message XFUNCs static int DoFunction int funcIndex void p pointer to structure containing function parameters and result int err funcIndex GetXOPItem 0 which function invoked p void GetXOPItem 1 get pointer to params result switch funcIndex case 0 XFUNC1Add p1 p2 err YuliC p break case 1 XFUNCiDivu pi p2 err YuliD p break case 2 XFUNC1ComplexConjugate p1 err XFUNCiComplexConjugate p break
100. gets stuck at v over finite regions in B as the density n is held constant that is plateaus develop in the Hall resistance Integer Ouantum Hall Effect To explain the IQHE we need to remember that in a magnetic field the continuous energy spec trum of the 2pEG breaks up into discretely spaced highly degenerate allowed energy levels En n 1 2 ho called Landau levels At finite temperature the Landau levels broaden slightly into a very narrow energy band 25 When the chemical potential lies within one of these bands the material is metallic that is the electron wave functions are not localized and transport can occur throughout the sample with some finite conductivity Away from these extended Landau level states any real material will have localized states due to slight local variations in electron density caused by tiny local variations in the 2D potential landscape When the chemical poten tial is in the region of the localized states varying the number of electrons only adds or subtracts localized states which carry no current so the current remains fixed at the full Landau level value Therefore when the chemical potential is between Landau levels the system is incompressible In a semiclassical picture the Landau levels correspond to electrons moving in circular orbits of quantized size due to the Lorentz force In a bulk region of the 2Dpzc the circular orbits cause the electrons to be localized But within a magnetic le
101. gner light source powersupply is turned on This powersupply is on a shelf under the table If it is on it will be possible to see a blue glow around the light source housing If it is not on flip on the circuit breakers wait about 10 minutes then press and hold the fire button until the starter fires with a loud click The light should now be visible but if not try firing again Incidentally the lamp should cool for 3o minutes after the last person turned it off before you fire so make sure the last user didn t just turn it off Typically you do not need to turn the lamp off unless you are the last user of the night how would you ever know this To turn it off switch of the breakers Clean the chip Find four fluoroware beakers and label them TCE ACE IPA and DI H20 you can keep these beakers and use them over and over Pour a little TCE trichloroethylene into a fluoroware beaker Put the chip in the TCE and put the beaker in the ultrasound bath on off buttons are on the panel above the fume hood shield for 5 minutes TCE removes grease If the level in the bath is too low add a little deionized DI water If the beaker floats add more TCE or try to remove some water from the bath After 5 minutes transfer the chip to another beaker of Acetone for 5 minutes Acetone removes photoresist and other organic impurities Next transfer to IPA for 5 minutes IPA removes the acetone Next transfer to DI water for 5 minutes Water removes
102. guerre polynomials y Aes and Hj h 2e The dephasing factors fsm are given by fii 1 Hy Hso Hi foo 1 Ho Hi fio 1 Hg 2Hs0 Hir where Hy h 4eL5 and Lg is the phase breaking length Equation 4 2 does not include all B dependent interference terms notably excluding Cooper channel contributions due to electron electron interactions 90 and a reduction of WL due to electron diffraction effects 133 Also in an attempt to capture the effects of cubic terms on H and Hso we introduce an effective vector potential Ag As v5 ky k ws As Ti k which leads to an effective SO field Hg 2 03 af 2nay r yh m Jeff 4 3 Equation 4 2 is applicable when B gt H which corresponds in the present experiment to B between 20 100 p T depending on V see Figure 5 2 We have confirmed that fitting only to data where B gt H gives within error bars the same results as fitting over the entire measured range of B Modification of the commutator k 2eA2 r by A induces spin flipping terms y 4 in the transformed Hamiltonian H The corresponding Hz 3 n m n eh using its expression in the diffusive regime 4 4 Experimental Details We now turn to a discussion of the experiment Three similarly fabricated samples made on three separate heterostructure materials were measured all showing qualitatively similar behavior The sample for which data is presented
103. h fo RENE E gi edt EAERIO ve he oh ene oerte hv be o 3 2 2 Theoretical Background lees 4 2 2 1 General introduction to quantum computers a 4 2 2 2 Introduction toanyons s ne netes me nbe e aa ua aD E 5 2 2 3 Integer and fractional quantum Hall effects aoaaa aa 7 2 214 Topological Quantum Computation 00000000 ee eee 11 2 3 Prior exp rimental work 5 2 ce eom rodeo s og y RE E ds 13 2 4 Impact of this work and future directions 2 eee 15 Bib Techniques cac dera heen ETE 8 ee ae Mined faa hes que ee ae IR 17 2 5 1 Refrigerator and wiring 2 ee 17 2352 OE Dawa bae hel AN hear eas Ma oi Mod LA e ALB ins eh ud dom dr ate E 17 2 5 Electron Temperature een 18 3 Introduction to the spin orbit coupling experiments 21 31 Theoretical background llle 21 3 11 OSpin orbitcoupling eee 21 3 1 2 Weak localization and antilocalization llle 22 3 1 3 Conductance fluctuations e 23 3 1 4 Physics in the dot and random matrix theory sss 23 3 2 Perspectives on the 2p spin orbit experiment 2 2 llle 24 3 3 Perspectives on the quantum dot experiments 2 2 0 0 0 0 0005 25 3 4 Vlechiiquesits sss co gees be ee cee seals eases Be ee Sots Be a 2 Be hd eh A oak 27 341 Data Acquisition sop re UE Ses Aloe hate RIA eom EDS OS em A 27 34 2 Data Analysis eu eere ea Qe a ee desee eacus e e E 27 4 Spin Orbit in Two Dimensions 29
104. he cryostat to provide B 96 The Hall effect measured in a GaAs Hall bar as well as the location of weak anti localization extrema in transport through the dot itself visible Bj S 2T were used to determine B 0 Statistics of conductance fluctuations were gathered using two shape distorting gates 165 while the point contacts were actively held at one fully transmitting mode each N 2 Based on about 400 200 statistically independent samples for the low density high density dots the average and variance of conductance were obtained Measurements were taken at various fixed By as a function of B with high resolution around B 0 increasing the number of statistically independent samples for B 0 by about a factor of 5 6 5 Characterization of Spin Orbit Strength at Zero In Plane Field The average conductance g B is used to characterize the strength of SO coupling The large dot on high density material shows AL due to SO coupling Fig 6 1 a while the smaller dot on the same material displays WL Fig 6 1 b showing that SO effects in the small dot are suppressed due to the extra confinement as previously reported 113 Fits of g B to the RMT 116 150 give the average SO length Aso AjA2 where A15 are the SO lengths along the main crystal axes the phase coherence time T and a parameter related to typical trajectory area The SO inhomogeneity Vso y 1 A2 can be extracted from g B
105. he problem In fact the combined influence of coherence confinement electron electron interactions spin and spin orbit coupling 96 makes the physics an extremely rich ground for experimental and theoretical studies Because the interference is for all electron paths the spin rotation needs to be averaged so interference term is 1 2 not 1 92 23 3 2 Perspectives on the 2D spin orbit experiment In this and the following section I describe the specific contributions of my PhD research to understanding the problems introduced in this chapter This section deals with the 2D spin orbit work reported fully in Chapter 4 The experiment described in Chapter 4 took place during somewhat of an awakening about the importance of spin orbit coupling in GaAs AlGaAs heterostructures Shortly prior to our experiment the prevailing view was that there essentially was no significant spin orbit coupling in GaAs Quoting directly from Beenakker and van Houten 97 We will not discuss the effects of spin orbit scattering or of superconducting fluc tuations since these may be neglected in GaAs AlGaAs heterostructures The beginning of the end to that attitude was an important paper by Paul Dresselhaus and cowork ers 98 showing weak antilocalization in GaAs AlGaAs heterostructures that was tunable via a top gate Clearly spin orbit is not negligible in GaAs Even before Dresselhaus experiment it was known that spin orbit coupling a
106. he total size of your pattern is larger than a single writefield size I urge you to specify the working area in enough integer multiples of writefields to contain the pattern For example if your pattern is 2 2 mm by 1 mm specify a working area of U 1150 um to 1150 um V 550 uum to 550 um The result is a working area that is large enough to contain the en tire pattern built out of integer multiples of 100 um Also note that the critical area around 0 0 is not di vided up between different write fields so you don t have to worry about the Raith stitching your de vice incorrectly 74 Now specify the position The position you enter into the exposure properties window specifies the loca tion of the center of lower left most writefield By the way if the working area is smaller than one write field then the lower left corners of the writefield and the working area are coincident For instance say you want to write a 10 um square portion of your device at the center of your mesa centered at position 0 0 If you specify a working area in both U and V of 5 um to 5 um with a writefield size of 100 um then you need to enter the values 0 045 mm 0 045 mm for the position This is a little confusing and can lead to errors so an alternative method would be to spec ify an working area of 50 um to 50 um Restrict the size of the pattern by choosing only select layers In this case the position would be 0 0 which is mu
107. idge We pulled the chip Dominik made quantum dots and we got to work trying to sort out what was going on As we were measuring Aleiner and Fal ko published a paper that expanded upon Folk and Halperin s theory that spin orbit coupling was involved in fact they identified all possible symmetry classes which arise from the interplay between SO coupling and Zeeman splitting in a disordered or chaotic semiconductor quantum dot and describe all the physically achievable parametric dependencies treated as crossovers between distinct symmetry classes 116 In other words they massively expanded the applicability of random matrix theory Our work 113 provided quantitative confirmation of these theories which represents quite an advance in the understanding of a very complicated many body problem in physics The only small detail is that our results did not agree quantitatively with the theory in fairly high parallel magnetic fields It turns out that this did not represent a failure of the theory just an opportunity to add one more detail The same year we were measuring Fal ko and Jungwirth 117 and Meyer and Alt shuler 118 119 published papers correcting the naive view that in plane fields influence only the Zeeman energy without affecting the orbital physics In fact the parallel field can break time reversal symmetry and couple to the orbital motion of electrons Adding the Fal ko Jungwirth terms to the Aleiner and Fal ko theory allowed us
108. iffusive approximation 132 we follow Refs 133 135 which treat the quasi ballistic case fg lt p Vh 2eB is the magnetic length and is the transport mean free path without spin orbit coupling The approach is to introduce an operator P GR o r rz 01 G2 ri 12 2 27vT for the probability of an electron to propagate both forward and backward along a path segment from r to r2 where G G are single electron retarded advanced Green functions 7 2 are the Pauli spin operators for particle moving forward backward v is the density of states per spin and c is the scattering time The interference contribution from the n traversal of a closed path is given by the trace of P In the presence of SO coupling of the form in Eq 4 1 formulas in 134 remain valid once a summation over spins is included in the trace Introducing the total spin of interfering partial waves S 0 02 we write Tr P 5Tr P Po where operators Po and P describe singlet S 0 and triplet S 1 contribu tions To calculate Tr Po 1 we diagonalize Py We find that when Op and Og are taken into account Py has the same eigenfunctions as the Hamiltonian H for particles with charge 2e spin S and spin frequency 20 H E k 2eAem 4 2eAs where A is the vector potential associ ated with the applied perpendicular magnetic field B and As zu Sx N2Sy 5S4 amp 1Sy is the SO vector potential For 5
109. in a vortex hence at rational values of v 1 21 the energy of the system is minimized by forming composite fermions which are new particles comprised of an electron that has captured 2n vortices The vortex number has to be even to preserve the Fermi statistics of the crs At v 1 2 the entire external magnetic field is used up making the crs and the Coulomb interaction is essentially screened the system becomes mathematically equivalent to a system of fermions moving in a zero magnetic field 40 As the magnetic field is varied away from this effective zero point at v 1 2 the crs experience Shubnikov de Hass oscillations as effective Landau levels form As the deviation from zero field zo 2gp41 when p Landau levels are occupied p is a positive or negative integer g is an integer We have just shown that the FQHE can be conceptualized as an IQHE of crs 33 increases the crs undergo an effective integer quantum Hall effect with plateaus at v Fractionally charged excitations can occur near v oa on the FQHE plateaus For sim plicity take the case of v 1 3 If the magnetic field deviates from exactly v 1 3 for fixed density by one flux quantum then one vortex will be formed this vortex is a quasihole Because each electron normally carries three vortices at v 1 3 a single vortex which causes the local electron density to go to zero looks like a deficit of 1 3 electron so the quasihole charge is 3 Thi
110. in the presence of AL and is taken as Vso 1 4 1 0 for the high low density devices An additional parameter x of order one in the RMT relevant in the strong SO limit is taken as x 1 for all devices For fit details see Ref 113 parameters are listed in Table I In absence of AL only a lower bound on A can be found The extracted coherence times are comparable for all devices and consistent with previous experiments 167 Note that the SO length Aso is comparable to the device diameter L VA of the big dot On the low density material both devices show WL see Figure 6 2 indicating that for both dots Aso gt gt L the regime of confinement suppressed SO coupling Note that while both 8 pm dots have nominally identical geometries only the device on the high density 2DEG shows AL Constrained by experiments observing WL rather than AL in identical devices made on this wafer 167 down to the lowest dilution refrigerator temperatures a lower bound As 2 85 ym 47 a oo m n 58x10 m op 00 P8 i var g e hY x 10 uj 9 6 Y s o x T me S dg I 10 97 40r S S n od oH I To 10 94 20r gt 5 10 91 oL 12 um um 2 1 1 3 4 B mT Figure 6 1 Average g B solid dots and variance var g B op
111. ing time reversal symmetry s is the Kramers degeneracy parameter and X characterizes mixing of different spins when Kramers degeneracy is already broken Spin rotation symmetry is classified as either not broken s 2 2 1 partially broken s 1 3 1 or completely broken s 1 2 2 The 46 variance is reduced by a factor of two when a crossover into the class with next lower symmetry occurs The Kramers degeneracy can be lifted by a Zeeman field as well as SO coupling if B 4 0 Once Kramers degeneracy is broken s 1 mixing of spins X 2 is due to SO coupling and can be possible already at B 0 due to SO coupling or can be revived by Bj when SO coupling is con finement suppressed 115 at Bj 0 Finite temperatures and decoherence strongly reduce var g 116 150 but the relative reduction factor R varg B 4 0 Bj 0 varg B 4 0 By gt 0 is affected only weakly 6 4 Experimental Techniques Four quantum dots of various sizes were measured made on two different 2DEG s with electron densities n 2 x 10cm and n 5 8 x 10 cm 7 see Ref 113 172 for details Figures 1 and 2 show device micrographs insets Measurements were made in a He cryostat at 0 3K using current bias of 1nA at 338 Hz In order to apply tesla scale Bj while maintaining sub gauss control of B we mount the sample with the 2DEG aligned to the axis of the primary solenoid accurate to 1 and use an independent split coil magnet attached to t
112. ious experiments in which SO rates are measured using WL AL in a gated GaAs het erostructure have not reported in situ gate control 98 136 137 Very recently Koga et al 102 demonstrated gate controlled SO coupling in InGaAs heterostructures using WL AL but did not report a full crossover from WL to AL in any single sample We know of no previous study in which an in situ crossover from WL to AL is demonstrated Modification of Rashba SO coupling using gated quantum wells has been observed using beating patterns in Shubnikov de Haas os cillations in InGaAs 138 139 InAs AlSb 140 and HgTe 141 Gate controlled SO coupling in GaAs 2D hole systems 142 144 has also been investigated using beating of Shubnikov de Haas oscillations The angular variation of SO coupling in GaAs quantum wells has been measured using Raman scattering 104 but to our knowledge has not been extracted from transport data 30 43 Theory of Two Dimensional Magnetotransport with Spin Orbit Coupling beyond the Diffusive Approximation The Hamiltonian for conduction band electrons in a 001 2DEG is H ru e Q where m is the effective mass k k k kx ky is the in plane wave vector 7 Cx 0y is the Pauli spin operator and Q Qx Qy is the total SO frequency Q Op Ops Og can be written as the vector sum of linear Op and cubic Ops Dresselhaus terms and the Rashba term Og Op amp k ky h 4 1a OR M Kky 9k
113. known as weak localization antilocalization Antilocalization is the paradigmatic experimental signature of SO coupling in phase coherent electronic systems 125 In this Letter we demonstrate in situ control of SO coupling in a moderately high mobility GaAs AlGaAs two dimensional electron gas 2DEG inducing a crossover from weak localization WL to antilocalization AL as a function of an applied top gate voltage see Figure 4 1 Theory beyond the diffusive approximation must be used to extract gate voltage dependent SO parame ters from magnetotransport when the SO precession frequency becomes comparable to the inverse transport scattering time T71 as occurs here and when the magnetic length becomes comparable to the mean free path Such a theory which also takes into account AB like spin quantal phases and spin relaxation 132 is developed here and used to estimate separately the various SO terms Rashba linear and cubic Dresselhaus defined below over a range of gate voltages ranging from WL to AL 4 2 Previous Theory and Experiments Conventional WL theories assume SO times much longer than transport scattering times 99 122 128 and so cannot be applied to clean materials such as high mobility 2pEcs Previous theories that go beyond the diffusive approximation do not treat SO 133 134 or treat it only as spin relaxation 121 135 without accounting for Berry phase effects which play a crucial role as we show here Prev
114. l Additionally the samples are of relatively high mobility gt 100 000 cm Vs compared to non GaAs systems Some of our samples are in the ballistic regime where the mean free path exceeds the magnetic length We have been trying to fit our data with the ballistic theory introduced in a PRL on which you a co author PRL 90 76807 2003 We have not found this to be easy David tells me that you were the fitting expert for the previous gated GaAs results I was wondering if you could share some insights with us Thanks in advance This email simultaneously illustrates a feature of our paper and a weakness The feature is that ours is still essentially the only theory that is applicable in the ballistic regime for fitting antilocalization data The weakness is that the formulas really are a bear to work with although I guess they are what they are based on physics 3 3 Perspectives on the quantum dot experiments In this section I discuss the specific contributions to the literature of the quantum dot experiments described in Chapters 5 amp 6 These two chapters and two papers 113 114 present results on a single overarching topic the study of an extremely rich many body physics problem The dis tinction between the two chapters is that Chapter 5 deals mainly with average conductance while Chapter 6 is primarily about variance Just before we measured the spin orbit properties in our 2D sample Folk and coworkers 96 had noticed som
115. ll O0 double SMALL 1e 50 double ce do b t loa binSize i t2 t t 95 ce exp s f t t2 2 c ce aLaguerre n m t2 if fabs c lt SMALL ce lt 1e 150 nSmall We are looking for enough small numbers in a row to quit else nSmall1 0 Sum binSize 2 c b Add the area of the new trapezoid to the overall integral itt printf inZf nSmall 4f t X4f Sum A4f i nSmall t Sum while n8mall lt 6 Quit if we find 5 small numbers in a row return s sqrt n 1 0 Sum static double Dfunc double x double fi double binSize double QUALITY double P int n 0 double Sum 0 double Term do 1 P Pn sqrt 2 fabs x n fi binSize Term P Pnm sqrt 2 fabs x n 1 fi binSize Pnm sqrt 2 fabs x n 1 fi binSize 2 1 P Sum Term n4 if n 4 100 0 printf nTerm fe n 4d P he Sum 4f Term n P Sum while fabs Term Sum QUALITY Keep adding terms until individual terms are QUALITY smaller than the sum return x Sum void Go IORecHandle ioRecHandle 1 HOST IMPORT void main IORecHandle main ioRecHandle The function YuliC returns the cooperon for a given x and f static int YuliC struct 1 DOUBLE p4 DOUBLE p3 DOUBLE p2 DOUBLE pi DOUBLE result p double x p gt pi double fi p gt p2 96 double binSize p gt p3 double QUALITY p gt p4 START OF PROCEDURE FUNCTIONALITY double P Pa
116. ll traces are measured with Vg 2 7 V and an ac lock in excitation of 0 86 nA Vpyi 3 for the entire B range of the main panel but not the full range of the inset 1 2 um QPC 0 43 0428 f 0 42 Rp h e 0 41 3 5 3 6 3 7 Figure 7 7 Dependence upon dc current bias of the 5 2 and 21 3 states in the 1 2 jim orc The main panel a shows the Rp data as a function of magnetic field each trace represents a different lacfrom OnA to 3nA Rp as a function of Ig for selected magnetic fields indicated by the color coded arrows are shown in b and c The dotted grey lines in the insets indicate resistance values of 3 7h e b and 2 sh e c All traces are measured with V 2 4V and an ac lock in excitation of 0 2nA vy 3 for all fields shown in this figure nounced dip at la 0 a peak at intermediate values and high current saturation In the 0 5 um arc the Ig traces are flat for all filling fractions between vopc 3 and vopc 2 All the traces in Fig 7 7 are measured with an ac lock in excitation Iac 0 2 nA while the data in all other figures 60 have been measured with Iac 0 86 nA Fig 7 7 provides a key point of comparison to previous experimental and theoretical work on the FQHE In a recent experiment 70 a oPc is used to measure tunneling differential resistance characteristics Ia curves for vopc lt 1 while vp is fixed on an IQHE plateau Our Ig data for 2 lt Vopc lt 3 and vga 3 with
117. long n 0 double Sum 0 double jj 71 0 double Term double ss sqrt 2 x int flag 0 long maxn 1000000 This is the maximum number of terms to include in the sum fp fopen 0906file tat a double L double LL double Po Poo x fi binSize LL double malloc size t maxn 5 sizeof double L LL do CALCULATE P if na 0 P Po F if n 1 P fi ss ss fi fi ss ss Po if P gt Po fabs Po P Po gt 05 flag 1 P 1 0 sqrt fi fi 2 n 1 2 0 ss s8s F J if n 2 double Pi fi ss ss fi fi ss ss Po P 1 f 1i fi ss ss Po P1 2 0 if P gt P1 fabs P1 P P1 gt 05 flag 2 P 1 0 sqrt fi fi 2 n 1 2 0 ss s8s F F if n 3 L Po n 2 L 1 L fi ss ss fi fi ss ss Po n 1 L 1 1 fi fi ss ss Po L 2 0 n 0 L 2 jj i fi fi ss ss L L 1 jj L 1 2 Cjj 2 5 jj 1 0 L 1 P L 1 97 if L 1 gt L II C fabs L 1 L L gt 05 S The series of Pn MUST be decreasing or we get huge problems Switch to symptotic as soon as this happens flag 3 P 1 0 sqrt fi fi 2 n 1 2 0 ss s8s u if n gt 3 amp amp flag 0 L 2 jjt it fi fixss ss L L 1 jj L 1 jj 2 jj 1 0 L 1 Pos L 1 Pa 1 0 sqrt fi fit 2 n 1 2 0 ss s8s if L 1 gt
118. m the principles of quantum mechanics 21 The postulates of quantum mechanics go on to identify particles with the eigenvalue 1 as bosons and 1 as fermions All of this is a standard and fundamental topic of quantum mechanics known as quantum statistics but is only strictly true for three or more dimensions In two dimensions the situation becomes even more interesting Kitaev elegantly intro duces anyons 7 Anyons are particles with unusual statistics neither Bose nor Fermi which can only occur in two dimensions Quantum statistics may be understood as a special kind of in teraction when two particles interchange along some specified trajectories the overall quantum state is multiplied by e In three dimensions there is only one topologically distinct way to swap two particles Two swaps are equivalent to the identity transfor mation hence e 1 On the contrary in two dimensions the double swap corre sponds to one particle making a full turn around the other this process is topologically nontrivial Therefore the exchange phase can in principle have any value hence the name anyon Here is a concrete example of an anyon which I have adapted from the Physical Review Letter where Wilczek first introduced and named anyons 22 Suppose we have some 2D fluid comprised of particles with charge q Suppose we apply a magnetic field perpendicular to the 2D plane and for some reason the field forms point like tubes ca
119. me pairs of particles that can fuse in more than one way and there is a Hilbert space of two or more dimensions spanned by these distinguishable states For the non Abelian Moore Read anyon model we can write the fusion rules 53 yxy lL o0xo I y pxo o 2 9 where I is the ground state the superfluid condensate y is a single composite fermion that is half of a cooper pair and c is a charge 4 half vortex The fusion rules can be understood physically when two ys fuse they form a Cooper pair and condense into the ground state I When two cs fuse they can either reveal that the core was empty II or that it contained a fermion y This is the fusion rule that makes the v 5 2 Moore Read state non Abelian The third rule comes from the associativity of the other rules The braiding rules for the Moore Read state are best illustrated with an example 2 As sume we have four charge 4 half vortices labeled 41 42 43 and 744 Let y and y2 form the qubit if this pair of half vortices has a fermion in the core then we will call the state of the qubit 1 otherwise it is 0 If we move say 73 around both 14 and rp the state acquires some phase if the core is empty but it acquires that phase plus an extra phase factor of 1 if the core is occupied by a fermion If we instead move 73 around only one of 11 or 42 then the state of the qubit flips that is an empty vortex acquires a fermion or a full one loses its fermion
120. mples was degrading by about 1 or 2 million vs during the fabrication That was a small percent change in mobility but it was enough to degrade the 5 2 features We traced the problem to the solvents used for liftoff in the LOR photoresist process My first attempt at a solution was to switch to another photolithography process AZ 5200E that used acetone as the liftoff solvent but I found the process to be unstable although may be it is not more unstable than the Marcus lab standard LOR 3A recipe somebody should look in to the AZ process more carefully someday The solution I went with in the end was to use multilayer PMMA patterned with e beam lithography 85 C 1 5 2 ready fab recipe C 1 1 Mesas Photo d 2 4 solvent clean TCE Acetone IPA DI water 5 each in ultrasound Bake dry 180 C 5 Spin on 1813 photoresist No 5s slow spin 5000rpm 45s 1s spin up and spin down Bake 115 C 2min Expose 4s Develop CD 26 45s Rinse DI 15s UV Ozone in upstairs cleanroom 30s Do not use plasma Etch Etch in 240 8 1 H20 H25O04 H2 O Rinse in DI water for 15s for each etch Etch rate is approx 3nm sec Target depth is 330nm the depth of the second doping layer 10 Remove photoresist with acetone C 1 2 Ohmics Spin di 2 4 solvent clean Bake dry 5 180 C Spin on 3 layer PMMA No 5s spin up 4000rpm 45sec 1sec spin up and spin down First layer 495 PMMA C6 5
121. n effects 10 We already established that anyons do not exist in 3p and higher So 2D is truly a special situation for quantum statistics The second comment is that even in this 3D universe 2D is not just a mathematical figment but a physical reality thanks to the GaAs AlGaAs heterointerface Finally the inherently topological origin all that matters is the Aharonov Bohm winding number not the precise path of anyons is interesting in the context of oc If a qubit could be encoded topologically that is by using the winding number then the information would be intrinsically robust against decoherence small local interactions with the environment would not change the number of times particles have been moved in complete loops around each other 2 2 3 Integer and fractional quantum Hall effects Having met anyons we now turn to a less abstract concept the Hall effect The treatment of the integer and fractional Hall effects I present here is designed to lead straight to the 5 2 state St rmer s Nobel lecture 24 is a masterpiece and I recommend it as a more general introduction to both the rou and the FQHE Classically the Hall effect predicts a simple linear relationship between the Hall resistance Rxy and the magnetic field Ry B ne where n is the electron density and e is the charge of an electron The basic observation of the quantum Hall effects both integer and fractional is that at certain rational values of v 2h Rxy
122. nd the ac lock in excitation is 0 86nA 70 mK consistent with the disappearance of the plateaus in the bulk However unlike the bulk where the 5 2 plateau disappears symmetrically around a stationary point at Ryy 0 4h e as temperature increases in the orc there is an additional resistance Rp exceeds the quantized value of 0 4h e by 26 Q 5 Q We also note that the the extra resistance on the plateau decreases as the temperature increases behavior consistently observed in both the 0 8 um and 1 2 um Qrcs We interpret this as indicating that the temperature dependence comes not only from the thermal excitation of quasiparticles but also from the temperature dependence of their backscattering 7 9 la data The dependence of the differential resistance on dc source drain bias Ig Fig 7 7 provides ad ditional insight into this excess resistance At base temperature the resistances Rp vs Ig near vopc 5 2 and vopc 21 3 in the 1 2pm Fig 7 7c and o 8 gm not shown arcs show pro nounced peaks at I4 0 a dip at intermediate values and saturation to a constant value at high currents In these arcs the Ig behavior near vopc 22 3 not shown is inverted with a pro 59 0 41 1 2 um QPC 0 39 3 55 3 60 265 3 70 3 75 B T Figure 7 6 Temperature dependence of the 5 2 state in the 1 2 ym orc The inset shows an ex panded range of the 8 mK trace with the grey box indicating the range of the data in the main panel A
123. nd 22 3 are extracted from the linear portion of the data in a plot of In Ryx vs 1 T using the minimum Rxx for each FQHE state and Rxx e M 2T giving A54 60mK As 130mK and A 110 mK consistent with previous measured values 45 59 81 7 6 Demonstration of the QPC in IQHE and FQHE regimes We now focus on measurements with one orc formed as shown in Fig 1 Low field Rp and R data from the 1 2 um orc along with concurrently measured Rxy and Rxx show regions where one IQHE state forms in the bulk with a lower IQHE state in the arc see Fig 7 3 Figure 7 3 also shows the appearance of a plateau like feature in the orc between vopc 5 and vopc 4 in both the 1 2 gm and 0o 8 um opcs which remains unexplained At higher magnetic fields Fig 7 4 Rp and Ry show FQHE plateaus while the bulk is quantized at the IQHE value vpulk 2 57 a 1 2um QPC Rp h e 9 4 H A DV fh AN 0 0 5 0 6 0 7 0 8 0 B T Figure 7 4 Typical FQHE magnetoresistance measured concurrently in the orc a and the bulk b Quantized resistance values are indicated in units of h e The colored bands indicate field regions where selected FQHE states form in the orc and vy 2 7 7 Observation of plateaus at v 5 2 We now concentrate on the range vopc 3 to vopc 2 with puk 3 Fig 7 5 Plateau like structure near vopc 5 2 is evident in the 1 2 pm and 0 8m arcs but is not seen in the 0 5 um arc Near vopc 21 3 we also see plat
124. nd a tip force 10 mg Click the view Sample button and when ready click the start scan button The profilometer violently drags the tip to the left front then starts again slowly to the right back When the tip returns to the starting position it begins taking data as shown on the screen The violent action of the tip at the start of a scan can sometimes move the chip If this happens try sandwiching your chip between microscope slides in front and in back to add weight It is also possible to reduce the tip force although too little force may result in excessively noisy data After the scan the screen switches to view the data Drag the cursors to two spots ideally as far away from each other as possible to maximize the lever arm that should be level and click the level button The trace will be adjusted Drag the cursors to the two levels you wish to measure and pull them out to get regions to average over The height difference between the cursor and the average height difference between the regions is indicated in the box on the left Switch back to the sample view and repeat the scan as often as necessary to convince yourself it is accurate 82 Incidentally there will be variations in photoresist height across the chip but the etch rate is typically uniform across the chip Therefore the most accurate mesa height can be achieved by carefully measuring the same place or places on the chip When you have finished measurin
125. nd antilocal ization could be important especially in other materials such as InGaAs which have very strong spin orbit coupling Hikami Larkin and Nagaoka 99 had included a spin orbit term in a formula for the shape of weak localization in a magnetic field Knap and coworkers extended the theory to include different spin orbit mechanisms 85 but all of these theories used a diffusive approxima tion and were not actually applicable to the very clean ballistic GaAs samples that Dreselhaus ourselves and others were measuring Meanwhile spintronics 100 101 the manipulation of electron spin without destroying phase coherence was becoming a hot topic in condensed matter physics Instead of neglecting spin orbit coupling people were employing it as a key feature in devices such as the Datta and Das coherent spin rotator 89 and spin orbit based spin filters 102 103 This was the state of the field when we were doing our experiment People wanted to use spin orbit coupling to make spin transport devices but there was no applicable theory to explain antilocalization one of the most easily observed effects of spin orbit coupling in high mobility GaAs heterostructures Moreover although the three materials specific spin orbit constants had been calculated 85 86 and measured using Raman spectroscopy 104 they had never been separately measured using transport Knowledge of these three constants see Section 3 1 1 is especially impor
126. nd make sure the chip dosn t fly away After the spin cycle ends inspect the chip to make sure the spin cycle didn t contaminate the surface After your chip is done drying for 5 minutes put it on the spinner and spin if for 45 seconds This will enusre it is cool and that it dosn t fly away If it does fly away you ll need to clean it again But first check the back for bumps that prevent a good vacuum and try to get it to work If the spinner refuses to work it is probably clogged with photoresist or may need to have the interlock adjusted seek professional assistance or jam a clothes hanger down the hole to clear it Never dump solvents down the vacuum hole Find Shipley 1813 in the yellow cabinet Pour a little into a fluoroware beaker Prepare a little aluminum foil mat with one end folded enough times to form a little ridge The idea is to be able to lay the dropper on the ridge so that the tip hovers above the foil mat without touching it but no so high that all the liquid rushes to the bulb Get a dropper and fill it with enough liquid to spin Avoid air bubbles by allowing the bulb to fully fill before pulling the tip out of the liquid With the chip on the chuck and your foot poised above the green switch drip a few drops of photoresist into the spinner pan to clear away phantom dust and to make sure no air bubbles are near the tip With the tip near the chip surface to avoide excess air bubbles and turbulance drip JUST enough liqui
127. nda Geller This work was supported in part by DARPA QuIST DARPA SpinS ARO MURI and NSF NSEC We also acknowledge support from NDSEG J B M and the Har vard Society of Fellows D G G Work at UCSB was supported by QUEST an NSF Science and Technology Center 52 CHAPTER 7 Experimental observation of the v 5 2 state in a quantum point contact J B Miller I Radut D M Zumb hl E Levenson Falk M A Kastner C M Marcus L N PfeifferT K W West Division of Engineering and Applied Science Harvard University Cambridge Massachusetts 02138 T Department of Physics Massachusetts Institute of Technology Cambridge Massachusetts 02139 Institut f r Physik Universit t Basel Klingelbergstr 82 CH 4056 Basel Switzerland Department of Physics Harvard University Cambridge Massachusetts 02138 TBell Labs Lucent Technologies Murray Hill New Jersey 07974 We study the transport properties of quantum point contacts OPcs fabricated on a GaAs AlGaAs two dimensional electron gas that exhibits well developed fractional quantum Hall effect including at bulk filling fraction vp 5 2 We find that a plateau at effective filling factor vopc 5 2 is identifiable in point contacts with lithographic widths of 1 2 um and 0 8 ym but not 0 5 um We study the temperature and dc current bias dependence of the vopc 5 2 plateau as well as neighboring fractional and integer plateaus in the Qrc while keeping the bulk at Vuk 3
128. ng the quasiparticles Small local interactions with the environment cannot change the occupation of a single vortex Thus the entangled quantum state is protected from decoherence 2 2 4 Topological Quantum Computation We have introduced the general concept of quantum computation argued that quantum comput ers would be useful and discussed decoherence as a main pitfall in actually building a oc We have introduced anyonic particles which acquire phase in nontrivial fractions of 27t and identified a physical system FQHE where anyons exist We then outlined how the Moore Read quantum Hall wave function which may describe the FQHE state at v 5 2 would lead to non Abelian anyons due to the quantum entanglement of vortex zero modes and their possible occupation by a fermion So finally we are in a position to discuss topological quantum computation and how it could be done using the v 5 2 state A nice introduction to topological quantum computation is available online John Preskill s sixty eight page lecture notes 23 on the topic give quite a physically grounded introduction In the next few paragraphs I motivate the fundamental concepts and in the process I outline how the Moore Read 5 2 state could be used to implement a Troc Topological quantum computation depends on the existence of particles in 2p that acquire non trivial phase when they are braided roc requires an anyon system An anyon system is char acterized by two sets
129. ngth 0g A eB of the edge the orbits will skip off the edge potential and form quasi 1D channels One channel forms for each occupied Landau level A fully quantum mechanical treatment yields the same result 27 28 Either way when the quantum Hall fluid is incompressible all the current will flow around the edges of the sample in 1D edge channels in a direction clockwise or counterclockwise set by the magnetic field Since each edge can only support current flowing in one direction and since the edges are spatially well separated backscattering is not possible and the longitudinal The existence of extended states in 2D is not allowed by scaling localization at zero magnetic field but is now the generally accepted picture for the quantum Hall effects 25 27 which occur at substantial fields The exact fate of the extended states as the field decreases towards zero does not seem to be completely settled in the literature 4In the hierarchical picture of fractional edge channels 29 the picture is more complicated but the result is the same resistance along an edge channel is vanishing The resistance of each edge state is just the 1D contact resistance h e 30 The Hall resistance is quantized because only the edge states carry current and their resistance is quantized FQHE and quasiparticles The FQHE unlike the IQHE 28 arises due to interactions between electrons Because of the com plete degeneracy of states there i
130. nstead of the more common 0 s wave orbital 53 54 This non zero angular momentum results in the breaking of spin rotation spatial rotation parity and time reversal symmetries 49 These broken symmetries can lead to textures in the order parameters for the pairing and quasiparticle excitations of vanishing excitation energy on these textures called zero modes 49 which I describe below Another property of the Moore Read state a property which is not unlike typical Bcs superfluids but which is different from other FQHE states is that the vortices are actually half vortices with a charge of 4 instead of e 2 Physically one can associate this halving with the pairing of crs into Cooper pairs Furthermore each vortex is associated with an intra vortex zero mode To gain a physical sense of the zero mode we consider that each vortex is a small circular edge like the edge of the Hall bar with vanishing electron density at the center Thus each vortex has a domain wall to separate the vacuum at the center from the Cooper paired phase outside 49 These domain walls must satisfy the same boundary conditions as the edges of the sample The result is that due to the p wave pairing of the crs in addition to positive and negative energy chiral modes on opposite edges the edge states there is a zero energy mode that is shared between the edges Applied to the vortices these boundary conditions endow each vortex with one zero energy m
131. o Rxx which is thought to have a power law temperature dependence 75 The temperature where hopping becomes dominant is not even necessarily that low for IQHE it can already be dominant as high as 4K 76 So using the center of the quantum Hall dips as a thermometer is not effective At lower magnetic fields it is possible to find a region in field where the Rxx minima of the Shubnikov de Hass oscillations are just approaching Rxx 0 at base temperature As the temperature increases the resistance of these minima is expected to be activated which in this case is a reasonably good assumtion Similarly at low fields the amplitude of the Shubnikov de Hass oscillations can be analyzed using Dingle plots 77 78 to estimate a temperature However due to the significant difference in Hall resistance between low and high field there is no guarantee that the low field electron temperature is applicable to the high field regime 18 Another option is to observe changes in the most delicate quantum Hall features as tem perature changes In our samples the feature most sensitive to small changes in temperature near base temperature was the reentrant integer quantum Hall effect RIQHE There is no accepted theoretical functional form for the development of this feature with temperature 79 but qual itatively the development of this feature is well known 59 80 81 Very small changes in the temperature lead to quite dramatic changes in the RIQHE
132. o the position of scatterers in the sample because the interference depends in detail on the path lengths between scattering events Slightly changing the fermi energy of the sample or slightly moving some of the scatters can completely change the interference leading to fluctuations in the sam ple conductance Similarly to weak localization these fluctuations have a universal magnitude of about e h regardless of the overall conductance of the sample as long as the overall conductance is higher than about e h Datta s book offers a very clear explanation of this universality 89 3 14 Physics in the dot and random matrix theory I turn now turn to the quantum dot 93 Using nanoscale depletion gates it is possible to create a zero dimensional confinement potential for electrons connected to the 2D reservoirs by quasi 1D quantum point contact QPC channels The confinement causes the continuous dispersion of the free electron gas to break up into discrete allowed energy levels la the particle in a box problem from introductory quantum mechanics In this thesis I only deal with large quantum dots 1 um 8pm which contain roughly 100 1000 s of electrons where the picture is more complicated In fact the situation is chaotic The study of quantum chaos is new only 30 years old compared to the study of quantum mechanics in general about 100 years old A technique to study the quantum chaos in quantum dots random matrix theory
133. ode that is entangled with all the other vortex zero modes leading for 2n vortices to 2 1 distinct degenerate states 55 56 This large degeneracy of ground states due to the zero modes which arise due to the p wave pairing of crs is what allows the Moore Read state to be non Abelian 23 49 7The Pfaffian is the square root of the determinant of an antisymmetric matrix 9Briefly the Pfaffian is treated using a scs mean field theory which is diagonalized using Bogoliubov de Gennes equations The solutions to these equations are allowed excitations including the zero mode 49 10 Furthermore in the presence of more than one vortex a Cooper pair may be broken such that one or two of its constituents are localized within the correlated zero mode of the vortex cores A ground state is a superposition which has equal probability for the vortex core to be empty or occupied by one of these fermions 56 If one vortex encircles another vortex the phase it acquires will differ by zt depending on whether the stationary vortex is occupied or not Finally since the ground state is a superposition with equal weights for the two possibilities this relative 7t phase shift could transform the system from one ground state to another 56 Importantly the information about vortex occupation is stored topologically not locally only pairs of vortices may be occupied or unoccupied and this occupation can only be changed topologically by braidi
134. of rules the braiding rules that describe what happens when two anyons are exchanged and the fusion rules that describe what happens when two anyons are combined To implement a roc the following physical capabilities are required 23 Pair creation the ability to create pairs of anyons The simplest possible system would contain chargeons with no effective flux and fluxons with no charge but particles with both charge and flux can also be used e Pair annihilation the ability to bring pairs together and observe whether the pair annihilates completely or if the pair was carrying some other particle leaves some detectable particle behind Braiding the quantum gates are performed by exchanging particles to create different mem bers of the topological braid group A computation proceeds as follows 23 Many pairs of anyons are prepared the pairs are manipulated to form a particular braid and pairs of anyons are fused to see whether they anni hilate completely or not The braiding operations act on a system with quantum entanglement which provides the large Hilbert space for quantum computation The system is then measured by fusion which is a non deterministic measurement 11 The property of an anyon system that makes it non Abelian 23 is if for at least some anyon pairs the fusion can occur in two or more different ways In an Abelian model any two particles fuse in a unique way In a non Abelian model there are so
135. om matrix theories 94 95 176 179 offer a universal classification of statistical properties such as the average and variance of conductance in terms of the fundamental symmetry classes These theories were widely confirmed by experiments in diffusive 2D and 1D systems in both metals and GaAs 2DEG s including observed reductions in variance due to Zeeman splitting 180 181 SO coupling 151 182 and breaking of TRS both in the presence 183 and absence of SO coupling 184 In open quantum dots an observed large reduction of conductance fluctuations in By 96 161 185 186 has been explained by SO effects that increase upon application of Bj while SO effects at Bj 0 are confinement suppressed 115 This has led to an extended random matrix theory RMT 116 150 including a classification of transport properties in terms of spin rotation symmetries Subsequent experiments found AL 113 187 in high density dots due to strong SO coupling at Bj 0 allowing the SO length Aso to be extracted Orbital effects of Bj were ob served via a suppression of weak anti localization 113 as well as in correlations of conductance fluctuations 172 In this study we report on effects of Bj on the variance in dots of various SO strength 6 3 Spin Rotation Symmetry Classes The RMT 116 150 gives the variance at zero temperature T 0 in terms of symmetry param eters varg s BX 116 150 where f is the conventional parameter describ
136. on this interesting problem Two years later having answered some significant scientific and technological questions the problem looks even more interesting if more challenging than we had initially realized 2 2 Anexperimentalist s theoretical introduction to the fractional quantum Hall effect the state at v 5 2 and topological quantum computation To motivate our experimental interest in the FQHE state at v 5 2 provide the prevailing theoreti cal picture of how this unusual state of matter forms and describe how to use this matter to build a computer requires a fairly intricate arc of reasoning In this section I will start this arc with a very brief introduction to the general principles of quantum computation admittedly a long way from the experiments that comprise my PhD research However this starting point allows the intellectual arc to curve gently and hopefully illuminatingly through a list of tricky topics on the way to our goal So here we go 2 2 1 General introduction to quantum computers In this section I introduce the idea of quantum computation For a more detailed treatment I recommend John Preskill s Cal tech lecture notes on the topic 10 which are available on the internet He calls them lecture notes but they run to hundreds of pages and I think he is getting ready to turn them into a book My treatment borrows heavily from his but is of course much shorter and is tuned towards topological quantum comp
137. ong as we also properly included the effects of parallel magnetic field In the second set of experiments the high field experiments we studied the transport properties of quantum point contacts QPc fabricated on a GaAs AlGaAs two dimensional elec tron gas that exhibits excellent bulk fractional quantum Hall effect including a strong plateau in the Hall resistance at Landau level filling fraction v 5 2 We demonstrate that the v 5 2 state can survive in oPCs with 1 2 um and o 8 um spacings between the gates However in our sample all signatures of the 5 2 state are completely gone in a 0 5 im orc We study the temperature de pendence at v 5 2 in the orc and find two distinct regimes at temperatures below 19 mK a we find a plateau like feature with resistance near but above the bulk quantized value of 0 4h e while at higher temperatures this plateau does not form We study the dc current bias I4 de pendence of the plateau like feature and find a peak in the differential resistance at Ig 0 and a dip around Ig 1 2nA consistent with quasiparticle tunneling between fractional edge states In a orc with 0 5 um spacing between the gates we do not observe a plateau like feature at any temperature and the Ig characteristic is flat for the entire range between v 3 and v 2 iii iv Contents Abstract iii Acknowledgements ix 1 Introduction 1 2 Introduction to the 5 2 experiment 3 LE OVERVIEW E falar gohan We v
138. ongly on the SO properties The open symbols in the main panels of Figures 6 3 and 6 4 show that the variance is reduced upon application of Bj and saturates at large Bj giving reduction factors R varg B 7 0 Bj 0 varg B 0 Bj gt 0 between R 1 6 for the dot showing pronounced SO effects at Bj 0 and R 4 for the low density dots showing WL at Bj 0 Reduction factors as small as R 1 3 are seen in center gated devices with stronger SO coupling not shown Within the RMT these new experimental results are explained in terms of spin rotation symmetries in dots showing AL SO coupling breaks Kramers degeneracy s 1 and mixes up and down spins to some extent at already By 0 if B 0 resulting in small reduction factors 1 lt R lt 2 In dots showing WL on the other hand spin rotation symmetry is intact at By 0 s 2 4 1 but can be broken upon application of Bj resulting in reduction factors R 4 low density dots Breaking of spin rotation symmetry besides the Zeeman effect ez gugB g 0 44 which breaks Kramers degeneracy is caused by SO coupling combined with Bj introducing a new energy scale 115 116 150 e C e2 2Er A A2 A is the device area is a geometry 49 0 00 a RMT RMT FJ Ti 0002 0103 1 3 7 B T anl 0 10 9e Pd 24307 353 R 2 2 1 5 zT ne B 0 RMT uo B40 RMT FJ LM se D b E
139. ook Interest ingly the values for zoom have been about 1 37767 73 and 1 40704 for years At this point you could re measure the current us ing the Faraday Cup if you desire If you do be sure to double check your origin when you return to the sample as large stage movements have occasionally resulted in measurable shifts If necessary you can re origin the UV coordinate system without ruining all the work you have already done Writing the pattern t long last it is time to write the pattern Writ As an exposure matrix is the first step in any Raith fabrication before writing any device on valuable 2DEG an exposure matrix should be written on junk be sure the junk is the same material as the 2DEG as different materials can have different electron backscattering properties Once the device is shown to repeatably work at some exposure it is time to write the real device on the real 2DEG Open the GDSII Database window and make sure the GDSII window is active Click on File Open Pattern Click Edit Hit ctrl a to select all your gates etc Click Modify Dose Set and enter a dose value of 1 0 This ensures that all gates are assigned a dose If you forget this step the Raith won t bother to write any of your gates Click the E button to view the editing tool palette Zoom in if desired Double click on gates to view more information and set alternative dose levels to certain gates if d
140. ot sized gates features sizes down to 50 nm or smaller it is best to use a single layer of PMMA The recipe here results in a PMMA thickness around 200 nm which is thick enough to result in easy lift off but thin enough that narrow towers of resist be tween nearby gates do not tend to fall over 3 solvent clean TCE Acetone Methanol 5 min each in ultrasound Bake 2 min at 180 C to drive off water Spin on PMMA 950K 3 in Chlorobenzene Set up spinner with a 5 s spin up at 500 RPM followed by a 40 s spin at 4000 RPM Use aramprate of 2000 RPM Use a glass pipet to place a few drops of PMMA on the sample during the 5 s spin up Bake 10 min at 180 C to harden PMMA Raith Procedures Copy GDS file to Raith Computer Insert the disk with your GDS file into the Raith com puter the CPU is located directly under the loading port Copy the file into an appropriate directory on the Raith computer hard drive The Raith computer monitor as opposed to the Leo computer is the one on the left Page 1 64 hairy dandruffy hands replace the sample holder into the loading port Install your chip as shown cor rectly oriented Use only soft tweezers around sample holder Note the numerous scratches caused by inconsiderate peo ple who have used metal tweezers Close the port and fasten the two clamps Jm i Sample loading port on the Raith Load Sample Carefully remove the sample holder from the s
141. p into the x translation variable entry box by clicking on the units to the right of the entrybox Send the dimension to the x axis Move exactly this distance by clicking the blocky left or right triangular arrow button Make sure that the mode is Scribe and Break and that the stop hook above the threaded rod used to stop the cleave carriage in peck mode has been lifted Click S1 to scratch the wafer the scribe will scratch along your wafer from front to back along the vertical reticle You may choose to break the wafer by hand If not place a clean Kimwipe on your sample to protect the surface from accumulated evil crap on the roller wheel and press the B1 button to break the wafer along the scrbe line It is also possible to measure x or y dimensions For x measurements move the vertical reticle to one side of your object to be measured click the X1 Ruler button Move to the other side click the button again The x dimension will be displayed in the x translation variable entry box Good tweezers for handling chips are the anti Magnetic anti acid stainless steel ones with carbofib tips available from techni tool com They are ESD safe soft and non scratching 77 When finished exit the LSD program click the file cabinet button turn off the scriber the gas and the pump Sign the log book Clean off the break wheel wiht IPA and a Kimwipe B 3 Mesa Photo In the upstairs cleanroom make sure that the AB mask ali
142. perimen tal goals and results and provide a brief outlook for future experiments 9The truly goal oriented researcher would call this operation a Nor gate 1 As far as I know nobody has proposed a way to take advantage of the fusion rules Presumably a scheme with fusion would eliminate the need to measure the system using interference 12 LAL TITITIT Figure 2 1 Artist s rendering of the device proposed by Das Sarma Freedman and Nayak The current flows along the edges as indicated by arrows Tunneling occurs at the orcs labeled 1 and 3 The two tunneling paths will interfere either constructively or destructively influencing the conductance The two half flux quasiparticles of a qubit are localized on the two stars Another half flux may tunnel at orc 2 If the v 5 2 state is non Abelian this braiding operation will switch the interference from constructive to destructive or vice versa 2 3 Prior experimental work In 2005 when Das Sarma Freedman and Nayak 2 published their method to experimentally study the non Abelian statistics of the v 5 2 state it immediately prompted several experimental groups including our group of course to begin studying the manipulation and measurement of the v 5 2 state in mesoscopic devices In terms of studying 5 2 with gates or in etched structures small enough to observe tunneling between edge states I am not aware of any published prior experimental work However a tremendo
143. re are a few basic design tips I ll add more later if I feel like it In general the depletion length is approximately the same as the depth of the 2DEG and the wavelength of the electron is determined by the density of the mate rial For example say you have a material with a wave length of 50 nm and the 2DEG is 100 nm below the surface and you want to make a point contact that is will have at least 3 conductance plateaus You need to space the gates at least 350 nm apart 100 nm each for depletion length 50 nm each per plateau Once the design is finished it is useful to specify the write order of the gates Gates will be written in the order that they were added to the DesignCAD file Of course you were not likely to have designed the device in the same order you would like it to write so you will have to make a new file and paste the gates into it in the correct order Alternatively I have written a little DesignCAD macro that you can use to specify the write order of the gates I usually write my smallest gates Print this out on cleanr Version 20040929 first and I am careful to write the gates that would E most sensitive to Raith stage drift consecutively m paper Make a GDS file From DesignCAD export your design as a DXF file Using LinkCAD installed on the design computer convert to a GDS file Save the GDS file on a diskette for transfer to the Raith computer Spin On PMMA For quantum d
144. res are 2 byte aligned 102 Bibliography 1 Sethna J in L Nagel and D Stein eds Order Parameters Broken Symmetry and Topology 1991 Lectures in Complex Systems Santa Fe Institute Studies in the Sciences of Complexity Proceedings Addison Wesley New York 1992 2 Das Sarma S Freedman M and Nayak C Phys Rev Lett 94 166802 2005 3 Stern A and Halperin B I Phys Rev Lett 96 016802 2006 4 Bonderson P Kitaev A and Shtengel K Phys Rev Lett 96 016803 2006 5 Hou C Y and Chamon C Phys Rev Lett 97 146802 2006 6 Feldman D E and Kitaev A Phys Rev Lett 97 186803 2006 7 Kitaev A quant ph 9707021 2006 8 Duan L M Demler E and Lukin M D Phys Rev Lett 91 090402 2003 9 Das Sarma S private communication 2005 10 Preskill J http www theory caltech edu preskill ph229 2004 11 BELL J S Rev Mod Phys 38 447 1966 12 Garcia Ripoll J J Zoller P and Cirac J I Phys Rev A Atomic Molecular and Optical Physics 71 062309 2005 13 Dell Anno F De Siena S and Illuminati F Physics Reports Review Section Of Physics Letters 428 53 2006 14 Mabuchi H and Doherty A Science 298 1372 2002 15 Schulman L and Vazirani U 16 Loss D and DiVincenzo D P Phys Rev A 57 120 1998 17 Taylor J M et al Nature Physics 18 Schmidt Kaler F et al Nature 422 408 2003 19 Leibfried D et al nature 422 412 2003
145. rical methods were correct The full source code used for our Cooperon calculation 27 Agreement between Zduniak et al and our XOP function YuliC x f BS 1e 5 Qu 1e 12 C x f 4n3h e2 0 005 B Qu cutoff for sum Max number of terms in sum 1000000 In x Figure 3 2 We were able to check our numerical methods colored points against previously published calculations 121 black lines is listed in Appendix E 28 CHAPTER 4 Gate Controlled Spin Orbit Quantum Interference Effects in Lateral lransport J B Miller D M Zumb hl C M Marcus Department of Physics Harvard University Cambridge Massachusetts 02138 Division of Engineering and Applied Science Harvard University Cambridge Massachusetts 02138 Y B Lyanda Geller Naval Research Laboratory Washington D C 20375 D Goldhaber Gordon Department of Physics Harvard University Cambridge Massachusetts 02138 Department of Physics Stanford University Stanford California 94305 K Campman A C Gossard Materials Department University of California at Santa Barbara Santa Barbara California 93106 In situ control of spin orbit coupling in coherent transport using a clean GaAs AlGaAs 2DEG is realized leading to a gate tunable crossover from weak localization to antilocalization The necessary theory of 2D magnetotransport in the presence of spin orbit coupling beyond the diffusive approximation is developed and used to analyze experimental dat
146. rientation squarely aligned with the mask as much as possible Lower the mask it shouldn t touch the chip yet Look through the mask and move the chip into place with the x and y micrometer screws in front and on the right and rotate the chip with the angle micrometer on the left Switch on the sample vacuum Move the alignment microscope use the align switch above the mask Use the lowest magnification to start Use the lowest light level to minimize unwanted exposure The two buttons on the microscope movement handle can be pressed to move the scope in the x or y direction Focus on the mask and raise the chip enough that it is ALMOST in focus You don t want the chip to touch the mask until you have finished aligning When ready slowly raise the chip into the mask In fact the first part of the chip to touch the mask will be the corner beads the thicker bumps of photoresist that form in the corners of the chip If the chips moves or rotates due to edgebeads then lower the chip slightly and realign Holding in the level button to allow the chip to self adjust for uneven corner bead heights raise the mask until the shadows of the mesas disappear ie until the mask is touching the photoresist Switch to the next higher magnification focus on the mask As you raise the chip the mask will be pushed up and out of focus You want to get the contact between the chip and mask as close as possible without popping the mask vacuum or breaking
147. ring off clouds 88 and electrons scattering off disorder in the 2pEc When the wave scatters it follows all kinds of random paths and on average the interference cancels out However all closed loop scattering paths that start and end at the same point are special because there are always two ways to scatter around a loop forward and backwards or forward and time reversed that unless time reversal symmetry is broken have exactly the same phase The interference for these two paths e 91 92 e 1 will always be constructive So the process for a wave to backscatter right back to where it started is enhanced In phase coherent weakly disordered conductors this enhanced backscattering process causes the conductance to decrease by approximately e h no matter what the overall conduc tance is This phenomenon is known as weak localization 26 90 91 When at least one magnetic flux quantum threads a closed electron trajectory then time reversal symmetry is broken The phase acquired over time reversed paths will differ by the Aharonov Bohm phase the interfer ence will no longer be fully constructive and weak localization will be lifted This all only works if the electrons remember their phase so any path longer than the phase coherence length will also not contribute to the localization So far we have assumed that the spin of the electrons does not change much over any tra jectory If the sample has negligible spin orbit co
148. rom the potential which confines the electrons to the 2D boundary This term called the 1Spin orbit coupling is not exclusively a 2D effect the way anyons are but I will only talk about the 2D version P pling y y any y 21 Rashba term 84 will not play a role in symmetrically doped square wells where there is no net field but does contribute linearly in k for the triangular potential in a single side doped sample Hg a2 exky Oykx 3 4 The constants in these Hamiltonians 1 2 and y have been calculated by adding the spin orbit interaction as a perturbation to band structure calculations 85 86 but the answers vary In Chapter 4 we report how we separately measured these three constants using transport To explain this measurement however I first need to introduce weak antilocalization 3 1 2 Weak localization and antilocalization Ohm s law feels like an old friend Like an old friend it sometimes needs a few corrections Quantum mechanically the wave function for a particle such as an electron in a 2DEG includes a phase that changes as the particle follows any given trajectory If the electron scatters changes momentum several times without losing phase coherence then interference corrections need to be applied to Ohm s law Enhanced backscattering is quite a general phenomenon when coherent waves meet a random array of scatters it is observed for laser light scattering in a cloudy liquid 87 radar scatte
149. rrying flux Finally suppose the charge q merges onto the edge of the flux tube to form a new type of particle Now lets rotate this composite particle counterclockwise by exactly 27 During this rotation the charge makes a complete loop around the flux which results in an Aharonov Bohm phase of e 1 But this unitary rotation can also be represented via the angular momentum e 27J h eit f 2 4 and so the angular momentum can take eigenvalues J m 1 27 m 8 2n 2 5 where m is an integer and we have defined the angle 0 We will call e the topological spin of the anyon 23 As we already mentioned in 3p the only allowed values of 0 are 0 and 7t because in the language of groups a 47t rotation in the the 3p rotation group SO 3 can be contracted smoothly to a trivial path 23 However the 2D group SO 2 allows in principle any value of 0 Hence anyons By a similar argument moving one anyon around another results in the same nontrivial Aharonov Bohm phase as rotating one anyon We will see this braiding again At this point I pause to make several comments The first is that the situation in one dimension is less interesting or at least more ambiguous because in order to swap two particles they must pass through one another so it becomes hard to separate quantum statistics from At this point were are doing a math problem not worrying about whether such a particle would ever really form interactio
150. s 0 00 555 gt o D 10 0 01 9 RMT OE E 8 RMT FJ ee 7 00 02 0 103 1 3 7 6 irr By T ST gum R 1 6 4 high density n 5 8x 10 m 0 1 2 3 4 5 6 7 Bi T Figure 6 3 Variance of conductance fluctuations through high density devices as a function of in plane field Bj with B 0 solid symbols and B 0 open symbols sufficiently large to break TRS It is seen that the big dot with strong SO effects at Bj 0 shows a smaller reduction of the variance in B than the small dot Insets show 0g Bj g B 0 Bj g B 40 Bj open symbols Dashed curves show RMT the solid curves are RMT F see text and B direction dependent coefficient and Er is the conventional Thouless energy The associated field scale given by et y where is the level broadening due to escape and decoherence 113 becomes large in small dots and in the weak SO limit and is inaccessible in the smallest dot giving R 2 due to breaking of Kramers degeneracy only In the bigger low density dots where this field scale is one to two Tesla the SO strength As cannot be independently extracted from a varg Bj measurement because of the extra coefficient Using as the only fit parameter the dashed RMT curves in Figures 6 5 and 6 4 are obtained giving good agreement for all devices 6 8 Orbital effects of B on the Variance Finally we turn to orbital effects of Bj on the variance measured when TRS is not externally broken B
151. s then is another way to think about anyons Composite bosons By extension of the composite fermion picture 41 44 it is possible to describe odd denominator FQHE States in terms of composite bosons At odd denominator filling fractions it is possible to describe composite particles that are composed of an electron and an odd number of effective flux quanta instead of the even number in the composite fermion picture and hence the composite particles have bose statistics These bosons then condense into a superfluid The superfluid ex hibits a Meisner effect like other superfluids expelling the Chern Simons effective flux quanta and forming vortices As before these vortices constitute the fractionally charged anyonic quasi particles This minor extension of the composite fermion picture is useful to have in mind as we move to the v 5 2 state The v 5 2 state So far we have only discussed the origin of the odd denominator FQHE plateaus The experimental observation of an even denominator plateau at v 5 2 45 cannot be explained in any of the pictures of the FQHE or IQHE we have discussed In fact at this time the actual quantum mechanical wave function of v 5 2 state is under considerable debate One proposal by Moore and Read 46 is to multiply the Laughlin FQHE wave fuction by a Pfaffian factor 2 1 Pf z Zj z1 2n i ze n El 2 8 I lt where the Pfaffian is a way of describing a
152. s effectively no kinetic energy associated with the electrons in the Landau level yet the Coulomb repulsion is still effective One configuration that may have minimized the energy in this picture would be if the electrons were equally spaced as far apart from each other as possible forming a Wigner crystal 31 However the position of the electrons is uncertain to within a magnetic length In the range of density and magnetic field where the uncertainty broadened electron wave functions overlap any Wigner crystal that may have formed could melt and the lowest energy wave function is not necessarily a crystal In fact the ground state is described by Laughlin s 32 wave function 2z1 2n Gi zj e a rar I lt 2 6 where z x iy is the position of a particle and rri which corresponds to filling fraction v 1 m is an odd number to preserve Fermi statistics The exponential term comes from the Landau level wave function 33 and the z z keeps the particles far apart to reduce Coulomb energy 31 Importantly this wave function leads to an incompressible quantum liquid Incompressibility just as in the IQHE case implies plateaus in Rxy It also implies that to keep the macroscopic filling fraction pegged at the favorable value small changes in B or n must cause only well localized deviations from v What this means is that when the magnetic field is tuned to exactly the right value for filling fraction 1 m th
153. s is consistent with the large dot results of Ref 167 6 6 Variance at Zero In Plane Field The variance of conductance fluctuations varg B at By 0 is seen to be reduced upon ap plication of a small perpendicular field B Figures 6 1 and 6 2 This is due to breaking of TRS by B and is well known 165 188 Using the parameters obtained from fits to g B and an additional overall factor fvar Table I to match the RMT variance at B 0 with the experi 48 varg eh x10 varg e hy x 10 B mT i Figure 6 2 Average g B solid dots and variance var g B open symbols at a temperature 300mK and B 0 measured in the devices on low density 2DEG Both devices display WL indicating that SO effects are weak Note that while both 8 jim dots have nominally identical geometry only the high density device shows AL RMT is shown as dashed and solid curves as described in the caption of Figure 1 ment the solid RMT curves in Figures 6 1 and 6 2 are obtained from Eq 37 of Ref 116 150 which includes effects of thermal smearing and decoherence The RMT applicable for N gt 1 in chaotic dots calculates a ratio var g B 0 B varg B 7 0 B of two independent of By see below Theories valid for N 2 are not currently including SO effects 189 6 7 Effects of Spin Rotation Symmetry on the Variance The variance in an in plane field Bj when TRS is broken by B 0 depends str
154. s the full listing of the routines we used to fit the data in the 2D spin orbit coupling paper Chapter 4 This listing is a C program I wrote that generates a detailed 2D matrix of values for the Cooperon function in Equation 4 2 The Cooperon function is the sum of the sum of two integrals none of which converge very fast By calculating all possible values once we were actually able to do data fitting XFUNCi c This is the procedure used for calculating the Yuli C function the cooperon for weak Localization antilocalization including Spin Orbit include XOPStandardHeaders h Include ANSI headers Mac headers IgorX0P h XOP h and XOPSupport h include XFUNC1 h include lt stdio h gt include lt stdlib h gt include lt math h gt All structures are 2 byte aligned if GENERATINGPOWERPC pragma options align mac68k endif ifdef _WINDOWS_ pragma pack 2 endif FILE fp File can be used for diagnostic purposes to write output error function in double precision double derf double x 1 Downloaded from http momonga t u tokyo ac jp ooura gamerf html Searched for Algorithm on http www mathtools net C Mathematics int k double w t y static double a 65 1 5 958930743e 11 1 13739022964e 9 1 466005199839e 8 1 635035446196e 7 1 6461004480962e 6 1 492559551950604e 5 1 2055331122299265e 4 8 548326981129666e 4 91 is 00522397762482322257 0 0
155. scientific progress spurred by spintronic research is ongoing For example Schliemann Loss and Westervelt 109 have pro posed a method to study zitterbeweung a long standing prediction of relativistic quantum me chanics that leads to an oscillatory term in the Hamiltonian using spin orbit scattering in GaAs heterostructures Another interesting active avenue of research has to do with the importance of Berry s phase in spin orbit systems 110 112 a point we also highlighted in our paper These papers 106 109 110 demonstrate one of the two main ways our paper has had an impact on the field of spintronics Our result is often cited in the theoretical literature when quan titative values of the three spin orbit materials constants are required to make predictions The sec ond impact our paper has had is to provide a valid theory to fit other spin orbit measurements mostly it seems in other materials systems such as InSb InA Sb Every so often I receive an email like this one which arrived on 19 Dec 2006 Jeff I don t believe we have met however David Goldhaber Gordon recommended that I contact you in regards to a physics problem I have a group at the University of somewhere doing transport measurements on InSb InAlSb quantum wells for the last few years Recently we have been doing some low field measurements InSb has enormous spin orbit coupling so as you might suspect we get a significant anti weak localization signa
156. se to rich physics including novel spin rotation symmetries 116 150 a suppression of SO effects due to confinement 96 113 115 116 148 150 leading to very long spin life times 148 149 169 171 and lifting of the SO suppression by an in plane field 96 115 116 150 as well as by a spatial dependence of the SO parameters 168 Further mag netic fields Bj applied in the plane of the 2D electron gas 2DEG change the electron dispersion and in particular can break time reversal symmetry TRS 113 117 118 172 adding additional complexity to this system In this communication we present an experimental study of the variance of conductance fluctuations var g through open quantum dots defined by lateral gates on a GaAs AlGaAs 2DEG The By dependence of the variance varg B 0 By with TRS broken by a perpendicular field B 0 is seen to depend strongly on the SO strength and can be characterized by novel spin rotation symmetries found in Ref 116 150 which gives good fits to our data Further var g B Bj is seen to become independent of B at large B due to effects of By breaking TRS This is in good agreement with theory 117 118 as well as experiments on average 113 and correlations 172 of conductance fluctuations 6 2 Previous Work Theory of low dimensional diffusive systems has long predicted conductance fluctuations 173 174 to be reduced by both SO coupling 127 175 as well as Zeeman effects 176 178 Rand
157. t to a portion of the data and the slope of this line is used to esti mate the activation energy A via Rxx e 4 2T_ The deviation from the line at low temperatures is probably due to variable range hopping The arrows indicate an ill advised method to estimate temperature which requires that the deviation is due to poor cooling of the electrons instead of variable range hopping 2 5 3 Electron Temperature With quantum Hall effect measurements it is difficult to measure the temperature of the electrons because unlike Coulomb blockade peak width in quantum dots there is no convenient absolute thermometry One could attempt to handle this problem using an Arrhenius plot see Figure 2 5 with In Rxx where Rxx measured in the center of a quantum Hall minimum plotted against 1 Tmc where Tyc is the mixing chamber temperature Assuming the resistance is activated Rxx x e T the plot of In Rxx vs 1 T should be a straight line Assuming that at reasonably high temperatures the electrons and the mixing chamber are the same temperature and assuming any deviations from the line at low temperature are due only to deviations between the mixing chamber and electron temperature it would be possible to estimate the actual electron tempera ture using the linear fit see Figure 2 5 However these assumptions are quite dubious and not likely to be valid In fact at low temperatures variable range hopping is likely to be the dom inant contributor t
158. t under vso r v o where r L1 L and gives an extremal value of 6 0 Bj at Vso yT As a consequence fits to g 0 Bj cannot distin guish between Yso and r Vso As shown in Figure 5 2 b data for the 8m dot r 2 are consistent with 1 vs X 2 and appear best fit to the extremal value vs 1 4 Values of vso that differ from one indicate that both Rashba and Dresselhaus terms are significant which is consistent with 2D data taken on the same material 105 2 2 i seri Zg ZA The symmetry is precise if one takes f kz zf A See Ref 116 150 41 5 6 Breaking of Time Reversal Symmetry due to an In Plane Magnetic Field Using Vso 1 4 and values of Aso Tp and x from the By 0 fit RMT predictions for g B By agree well with experiment up to about Bj 0 2T Figure 5 2 a showing a crossover from AL to WL For higher parallel fields however experimental 59 s are suppressed relative to RMT predictions By By 2T WL has vanished in all dots Figure 5 2 c while RMT predicts significant remaining WL at large B One would expect WL AL to vanish once orbital effects of Bj break time reversal sym metry Following Ref 117 118 FJ we account for this with a suppression factor frj Bj B and flux threading by B can be written as a product g 0 Bj 6grmr 0 Bj frj Bj The Bj term reflects surface roughness or dopant inhomogeneities the By term reflects the asymmetry of 1 Tay Tz
159. tant for spintronic applications because they affect the spin precession for elec trons traveling along specific crystallographic directions in fact at certain densities there could be certain directions where there is no precession As far as I know our experiment 105 was the first time that the spin orbit strength was tuned in situ with a gate voltage from complete weak localization to antilocalization Moreover our paper included a new theory of antilocalization that was applicable to clean ballistic meaning the spin orbit length was longer than the mean free path samples The combination of the theory and our gate voltage dependent data allowed us to separately measure the contributions of the three spin orbit mechanisms Rashba linear Dresselhaus and cubic Dresselhaus In the years since we published our experiment 105 the interest in spintronics has con tinued 106 and there have been many interesting experimental advances such as a spin laser 107 Theoretical proposals for new spintronic devices such as Electric dipole induced spin reso nance in disordered semiconductors published this year in Nature Physics 108 appear regularly 7 t is surprising but apparently true that writing important papers about spin orbit coupling is a hereditary trait 24 Although it seems like practical applications of spintronic devices may need to wait for advances in materials 106 especially perhaps magnetic semiconductors the
160. te Hall bars and nanoscale devices without affecting the wafer mobility or the quality of v 5 2 features My complete nanofabrication recipe is printed in Appendix C Another difficulty was that the growth parameters that produce these remarkable bulk ma terials are not necessarily compatible with easy gating We tested quite a few wafers with good v 5 2 features that were ungateable due to switching noise giant gate drift unmanageable hys teresis and irreversibility of applied gate voltage Eventually we found a wafer that happened to have both manageable gates and good bulk v 5 2 features Unfortunately at this time there is no clear correlation between growth parameters and useable gates although some pattern could emerge as we test even more wafers Incidentally we also found that all materials were utterly ungateable after illuminating the cold sample with an infrared LED We have observed plateau like features near v 5 2 and v 21 3 in Qrcs with 1 2 ym and 0 8 um spacings between the gates At temperatures above about 18 mK the plateaus disappear Below this temperature the resistance of the plateau like feature is higher than the bulk quantized value of o 4 1 amp and increases as temperature is decreased The Ig traces near v 5 2 and v 21 3 in these Qrcs exhibit a characteristic shape showing a peak in resistance at l4 OnA a minimum near Ig 1 2nA and approaching a constant value at higher currents These o
161. that turned out to be incorrect 38 Arovas D 39 Jain J K Phys Rev Lett 63 199 1989 40 Halperin B I Lee P A and Read N Phys Rev B 47 7312 1993 41 Read N Bull Am Phys Soc 32 923 1987 42 Read N Phys Rev Lett 62 86 1989 43 Zhang S C Hansson T H and Kivelson S Phys Rev Lett 62 82 1989 44 Zhang S 1992 45 Willett R et al Phys Rev Lett 59 1776 1987 38 39 40 41 42 43 44 45 46 Moore G and Read N Nucl Phys B 360 362 1991 47 Bardeen J Cooper L N and Schrieffer J R Phys Rev 106 162 1957 48 Greiter M Wen X G and Wilczek F Phys Rev Lett 66 3205 1991 49 Read N and Green D Phys Rev B 61 10267 2000 50 Scarola V W Park K and Jain J K Nature 406 863 2000 51 Morf R H Phys Rev Lett 80 1505 1998 52 T ke C and Jain J K Phys Rev Lett 96 246805 2006 53 Stone M and Chung S B Phys Rev B 73 014505 2006 54 54 Tewari S et al quant ph 0606101 2006 104 55 Nayak C and Wilczek E Nulcear Physics B 479 529 1996 56 Stern A von Oppen F and Mariani E Phys Rev B 70 205338 2004 57 Halperin B I Helv Phys Acta 56 75 1983 58 Gammel P L et al Phys Rev B 38 10128 1988 59 Xia J S et al Phys Rev Lett 93 176809 2004 60 Eisenstein J P et al Phys Rev Lett 61 997 1988 61 Eisenstein J P et al Surf Sci 229 31 1990 62 Pan W
162. the filters with copper wire 2 5 2 LED Within this chamber we mounted an infra red light emitting diode We used digikey part number 516 1262 ND Emitter IR 5mm 875nm We were able to flash this LED at base temperature al though the length of time of the flash was limited to about 5 minutes to avoid crashing the mixing chamber due to excess heat Attempts to mount the LED at the 1vc bulkhead effectively at the liquid helium bath bringing the light to the sample with a fiber optic were unsuccessful The fiber optic cable was difficult to thermally anchor at the mixing chamber and therefore heated the sample Also the efficiency of the fiber optic coupling even though we physically mounted the fiber optic into the LED was poor and the desired effect of improving the 2DEG mobility was never achieved using this method With the LED mounted directly shining on the sample we used a Keithley 2400 to source 3 0 to 3 6 V with the current compliance limited Once the LED reached its steady state temperature the current at 3 0 V was about 3 mA We were able to achieve significant improvements in the sample mobility using the LED although it made the gates unusable 17 e Center of 5 2 dip bulk 5 64 e In R Linear fit 6 41 6 84 In Rxx 7 24 0 025 0 050 0 075 0 100 1 T Figure 2 5 An Arrhenius plot of In Rxx from the center of a bulk 5 2 dip against mixing chamber temperature The red line is a fi
163. the chip or mask or else light will leak around the patterns and cause an overexposure Send the alignment microscope back to the right Make sure the On and Auto buttons are pressed on the exposre control to the left and set the time to 4 seconds Since the auto button is pressed when you flip the switch on the right to move the exposure unit into position it will atuomatically turn on the lightsource when it is in place Other than checking to ensure that the 80 light switches itself on don t stare at the light during the exposure the system is supposed to be safe but look at it there is no way all the UV is contained correctly After the light switches off you can hear the switch flip the switch to move the lightsource out of the way Lower the chuck to make sure the chip is not stuck to the mask raise the mask remove the chip Fill 2 labeled fluoroware beakers about 1 2 full with CD 26 developer Fill the DI Water beaker with DI water Swirl the chip in the first CD 26 for 20 seconds quickly switch to the second beaker for 25 more seconds then rinse in DI for 15 seconds Blow the chip off Inspect the results under a microscope There will be invisible redidual photoresist on the surface of the chip This must be removed with UV Ozone Place the chip in the Uv Ozone machine Close the lid Open both valves on the oxygen cylinder under the table The main valve is tricky make sure you have opened it at least a few turns Set
164. the time to 30 seconds Don t use heat Turn on the main power the UV the Ozone Make sure the vent is closed button not pressed and hit start Make sure there the flow indicator bead is floating near the top of the scale Afterwards vent the chamber for a minute or two unless you like inhaling ozone and getting asthma The chip is now ready to etch B 4 Etch Procedure B 4 1 Summary DI Water Sulfuric Acid Hydrogen Peroxide 240 1 8 approx rate 3 5 nm sec Measure photoresist thickness Etch Measure photoresist trench Remove photoresist with acetone ultrasound Measure actual trench depth for future reference B 4 2 Details Gather materials Get the hydrogen peroxide bottle out of the small fridge in the Westervelt sample prep room If the door is locked it is possible to break into the room by removing the ventilation grating with a screwdriver Also take a calculator a few glass pipets and bulbs from the sample prep room It may be wise to take an empty glass bottle for waste the cleanroom does not reliably provide clean empty bottles Proceed to the downstairs cleanroom In the gowning area you will find a red Marcus toolbox with glass beakers take it in with you Mix the solution Use the large graduated cylinder to measure 240 ml of DI water from the filter on the back wall the DI water in the fume hood downstairs is not to be trusted Make sure an appropriate waste bottle is available and ready Empty solvent
165. ther to burn a spot to focus it won t matter with this aperture Develop in 3 1 solution for 9o sec Do the 2 minute UV Ozone Evaporate 15nm Cr and xnm of Au where x is the mesa height 10 Liftoff overnight in acetone This is the liftoff step that is most likely to fail as could have been predicted by Murphy s law DONE 84 APPENDIX C Complete Nanofabrication Recipe for 5 2 Devices In this Appendix I list the complete nanofabrication recipe I developed to process the ultra high mobility wafers for the 5 2 experiment In fact this recipe is only a small tweak to the standard Marcus lab recipe that has been developed and improved by many group members over many years We had two general difficulties in processing these wafers The first difficulty was making ohmic contacts I think the problem was that the doping in the ultra high mobility wafers is designed to such tight tolerances that there is not much room for error in the ohmic contact doping The ohmic contact recipe in this Appendix routinely yields contacts with 50 Q or less resistance I think one important improvement over previous Marcus lab ohmic recipes is that the Ge and Au are evaporated separately Trying to evaporate a eutectic alloy is just asking for trouble Another improvement is the thickness of the metal This makes the ohmics a pain to evaporate but not as much of a pain as cooling ohmics that do not work The second difficulty was that the mobility of the sa
166. tt 83 820 1999 192 Lilly M P et al Phys Rev Lett 83 824 1999 193 Pan W et al Physica E 6 14 2000 194 Haldane ED M and Rezayi E H Phys Rev Lett 60 956 1988 183 184 185 186 187 188 189 190 191 192 193 194 195 Rezayi E H and Haldane F D M Phys Rev Lett 84 4685 2000 196 Tserkovnyak Y and Simon S H Phys Rev Lett 90 016802 2003 197 Chung S B and Stone M Phys Rev B 73 245311 2006 198 Kitaev A Ann Phys N Y 303 2003 199 Bonesteel N E et al Phys Rev Lett 95 140503 2005 200 Harju A Saarikoski H and R s nen E Phys Rev Lett 96 126805 2006 201 Miller J B et al unpublished 202 D Agosta R Vignale G and Raimondi R Phys Rev Lett 94 086801 2005 203 Roddaro S et al Phys Rev Lett go 046805 2003 204 Lal S condmat o611218 2006 205 All resistances are differential with zero dc current bias unless otherwise noted 206 B ttiker M Phys Rev Lett 57 1761 1986 207 207 Beenakker C W J and van Houten H Solid State Physics 44 1 1991 see discussions in Secs IV B and IV C Our Rp corresponds to R in this reference 208 In the absence of tunneling across the Hall bar the derivations for Rxy and other resistances yield an equality e g Rxy h e 1 Nyan 209 Beenakker C W J Phys Rev Lett 64 216 1990 210 MacDonald A H Phys Rev Lett 64 220 1990
167. u have specified a smaller area you can track the progress much more clearly If you want to be sure something is actually happening you can double check that the red LEDs on the High Speed Pattern Processor are moving and the current measured when the beam is blanked is fluctuating on the picoammeter Finishing When you are finished writing all your devices click Unload Sample in the Navigator Exchange win dow and wait 10 minutes Remove your sample close the door Post Raith Processing Develop Develop the sample in 3 1 Isopropyl Alcohol Methyl Isobutyl Ketone solution for 1 minute Rinse in Iso propyl Alcohol and blow off 75 You should be able to see your pattern in the PMMA under the microscope Evaporate Evaporate Liftoff Liftoff in Acetone Let the sample sit for at least an hour preferably overnight Be bold give it a good shot of ultrasound I ve never seen a gate lift off due to ultrasound although sev eral reviewers of this document tend to disagree View or use If you made an exposure matrix view it in the SEM If you made a device finish your last photo steps and measure publish a paper and graduate Page 12 76 APPENDIX B Detailed Fabrication Procedures This Appendix written primarily for Eli s benefit but likely to be of use to any new users of the McKay cleanrooms covers all aspects of fabrication except evaporation which is trivial and
168. unt of AL is controlled by modifying the density under the gate For V gt 0V the AL peak is larger than in the ungated 87m dot We interpret this enhancement not as a removal of the SO suppression due to an inhomogeneous SO coupling 168 which would enhance AL in dots with L Aso lt 1 not the case for the 8 m dot but rather as the result of increased SO coupling in the higher density region under the gate when Ve gt OV One may wish to use the evolution of WL AL as a function of V to extract SO parameters for the region under the gate To do so the dependence may be ascribed to either a gate dependent Aso or to a gate dependence of a new parameter xj ep a My L2 A2 e Both op tions give equally good agreement with the data fits in Figure 5 4 assume As V including the parallel field dependence not shown Resulting values for Aso or assuming the other fixed 3V Falko T Jungwirth private communication 42 s o o 2 e 100 p3 Aso bw id 5 SX ww zx TN 5s 3 05 7 4 44 0 3 04 05 06 0 0809 1 1 5 Temperature K Figure 5 3 a Difference of average conductance from its value at large B 6 B 0 for various temperatures with B 0 for the 8 0 7m dot squares along with RMT fits solid curves b Spin orbit lengths As circles and phase coherence times Ty triangles as a function of temperature from data in a
169. upling this is often a good assumption However if the spin does rotate significantly as an electron traverses a closed loop then we need to include this in the calculation In Section 2 2 2 we learned that the wave function of a fermion picks up a phase of 1 when rotated by 27 Spin orbit coupling can cause such a spin rotation without For example the FQHE samples discussed in Chapter 7 are double doped square well samples that would be expected to exhibit negligible Rashba effect Most of the wafers we have used for top gate controlled mesoscopic physics have had triangular potentials 4The disorder is considered weak if the conductance is about e h or higher 5To see that the magnitude of the correction is independent of the overall conductance requires a subtle argument For a very clear explanation including the way this universal correction scales for larger coherent conductors of various geometry I recommend Datta s book 89 Chapter 5 22 breaking time reversal symmetry and the resulting minus sign shows up in the interference In samples with strong enough spin orbit coupling this phase shows up in the transport the weak localization dip is transformed into an antilocalization peak in conductance 3 1 3 Conductance fluctuations Another manifestation of quantum interference which also has a universal magnitude is univer sal conductance fluctuations vcr 90 The conductance of a sample is sensitive in detail t
170. us amount of experimental work has been done to study the v 5 2 state in the bulk and to study other FQHE states using QPCs Prior v 5 2 experiments The first quantized Ry plateau at v 5 2 was observed by Willett and coworkers in 1987 45 At that time the discovery of an even denominator FOHE state was somewhat of a surprise although Halperin four years earlier had already proposed the possibility of boson like bound electron pairs 57 and there was certainly no consensus then or even now about the physics of the state Over the two decades since then the quality of available 2DEG GaAs AlGaAs heterostructures has improved tremendously The measured 5 2 energy gap A has increased from A 52mK in 1988 58 to more than 500 mK today 2 At the present time the quality of the 2pEG has become so good that the v 5 2 state is just one relatively stable phase out of many exotic phases that can be observed between v 3 and v 2 59 Important works using tilted magnetic 13 Rp at v 5 2 0 503 Rp at v 2 Offset 0 1 h e 15 10 5 0 5 10 15 Figure 2 2 Comparison of the I V characteristic for an IQHE v 2 and FQHE v 5 2 plateau Rp is in fact dV dI the differential resistance The IQHE state shows ohmic behavior while the FQHE behavior is highly nonlinear due to a complicated tunneling density of states at very low temperature and voltage The FQHE curve is seen to appro
171. user can then easily check If the answer is wrong the algorithm can be re run This probabilistic behavior makes Qcs seem worse than classical computers even when everything is working perfectly acs don t always get the answer right To see the advantage of a QC consider 50 qubits instead of just one A 50 qubit oc can be represented as 1Even macs 49 ly L Ay X 2 2 x 0 where Y ax 1 and the basis vectors x are either 0 or 1 To perform a computation we prepare in some input state perform unitary operations on selected qubits these operations are known as quantum gates and project the result onto the 0 1 basis For each x the proba bility of measuring 0 is a And that s it As promised this procedure can be done either with a QC or a classical computer The trick is how long it would take to run the computation on a classical computer We only have 50 qubits but to model with a classical computer we would need to keep track of 2 complex numbers that is more than 2000 terabytes single precision Now imagine trying to compute rotations of 2000 terabyte matrices it cannot be done Bell s theorem 11 prevents the use of the following shortcut to classically simulate a Qc 10 It might have been possible since the output of the oc is probabilistic to use probability distributions and a random number generator along each step of the computation instead of performing exact vector math on ter
172. utation and the v 5 2 rou state Why go quantum All personal computers are universal 10 which means any computation that could be done using a quantum computer Qc could also be done on a home PC The advantage of the Qc is that certain types of computation could be done much faster We will now see how this works The unit of quantum information is the qubit We can model the qubit as a vector in a two dimensional complex vector space with inner product We can call the qubit basis vectors 0 and 1 reminiscent of a classical bit and then we can write down Ip al0 bj1 2 1 where a and b are complex numbers normalized a b 1 When we measure the qubit the state is projected onto the basis The probability of measuring 0 is a and of course the probability of measuring 1 is b This brings up the interesting fact that the output of a quantum computer is not deterministic repeated measurements with exactly the same inputs will yield a probability distribution not an answer in the sense of a classical computer This probabilistic behavior is inherent to Qcs Part of the art in developing a good quantum algorithm is to find a way to get the desired output with very high probability This also helps explain why the types of algorithms that have already been developed for Qcs involve problems that are hard to solve but easy to check to factor a large number is hard but a Qc can find an answer fast which the
173. was supported by QUEST an NSF Science and Technology Center JBM acknowledges partial support from NDSEG 44 CHAPTER 6 Conductance Fluctuations in Open Quantum Dots with Spin Orbit Coupling and Zeeman Fields D M Zumb hl J B Miller C M Marcus Department of Physics Harvard University Cambridge Massachusetts 02138 Division of Engineering and Applied Science Harvard University Cambridge Massachusetts 02138 D Goldhaber Gordon Department of Physics Harvard University Cambridge Massachusetts 02138 Department of Physics Stanford University Stanford California 94305 J S Harris Jr Departement of Electrical Engineering Stanford University Stanford California 94305 K Campman A C Gossard Materials Department University of California at Santa Barbara Santa Barbara California 93106 Conductance fluctuations in GaAs quantum dots with spin orbit and Zeeman coupling are investigated experimentally and compared to a random matrix theory formulation that defines a number of regimes of spin symmetry depending on experimental parameters Accounting for orbital coupling of the in plane magnetic field which can break time reversal symmetry yields excellent overall agreement between experiment and theory This chapter is published in Phys Rev B 72 081305 2005 45 6 1 Introduction The combination of quantum confinement spin orbit SO coupling and Zeeman effects in lateral semiconductor quantum dots gives ri
174. ws the remaining parameter vso to be estimated as described below Besides Zeeman energy ez calculated using g 0 44 rather than fit parallel field com i M v z A NE bined with SO coupling introduces an additional new energy scale ef ES Lij 12 x X where 40 OF 4 RMT ic 0 um 4 0 01 RMT FJ1 2j qo RMT FJ2 c oc TUE tr Et o a 5 8 um 1 0 02 oe eee d LH Otel 7 S 0 04 Oe A A F 1 2 um 0 05 0 1 4 AeA S Su ES J 0 0 02 0 05 0 1 0 5 1 5 10 B T Figure 5 2 a Difference of average conductance from its value at large B g B 1 By asa function of B for several By for the 8 0 m dot at T 0 3K squares with RMT fits curves b Sensitivity of g 0 By to Vso for the 8 07m dot 1 lt vso lt 2 shaded vso 1 4 solid line and vs 0 8 dashed line c 6g 0 Bj markers with RMT predictions dashed curves and one parameter solid curves or two parameter fits dotted curves using RMT including a suppression factor due to orbital coupling of By see text Kz is a dot dependent constant and l 7 are the components of a unit vector along By 116 150 Be cause orbital effects of By on g B Bj dominate at large By e must instead be estimated from RMT fits of var 2 with already broken time reversal symmetry which is unaffected by orbital coupling The RMT formulation 116 150 is invarian
175. y simpler quasiparticles are treated in the literature 34 An interesting historical note papers were published showing the statistics of the quasiholes are Fermi Bose and Other The correct answer published by Halperin 27 and later expanded and confirmed by Arovas Schrieffer and Wilczek 34 is Other the quasiholes are anyons with fractional statistics Before moving on it is important to be clear about flux tubes Following Wilczek 37 I introduced anyons as a charge magnetic flux composite Section 2 2 2 However real FQHE quasiparticles do not actually carry magnetic flux 38 the true origin of the fractional statistics is a complicated many body effect Having emphasized this detail I will now move on to the composite fermion picture of the quasiparticles Composite fermions Jain 39 realized that the essential features of the FQHE can be understood intuitively in terms of a new kind of particle the composite fermion cr The composite fermion theory is based on the single hypothesis 33 that a system of electrons can reduce its energy if each electron captures an even number 2n of vortices Conceptually a vortex forms when one magnetic flux quantum pierces the 2DEG Mathematically the vortices are singular points in a Chern Simons gauge representation 40 The electron density at the center of a vortex is zero increasing to the bulk value at the perimeter Electrons can reduce their Coulomb interaction by sitting
176. y changes focus Click on the magnifying glass button on the control bar to return to zoom focus mode Otherwise when you try to zoom focus you will accidentally adjust the ap erture align Refocus zoom in repeat Click the Stigmation button Enter the previous good stigmation values in the green box Unless somebody turns the filament off or otherwise crashes the system the stigmation values are fairly stable from day to day so write your values down to use next time Zoom to about 100 kX or higher Fine tune the stig mation The idea is to first focus then try to improve the focus and eliminate directionally preferential fo cusing by adjusting the stigmation Again refocus zoom in repeat A sufficiently long time after ramping up the beam I usually think 20 30 minutes is long enough center the crosshairs over the Faraday cup and zoom all the way in Open the Current window and click the Measure button to measure the beam current Fife x ELT This is a good time to set up the default exposure parameters Open the Exposure window Page4 Click on the Calculator button to open the Expo sure Parameter Calculator window Enter the Area Step Size 0 006 um and the Area Dose 100 pAs cm Click on the calculator icon on the Area Dwell Time line to calculate the dwell time aprox 0 00134 ms Note that the Beam Current is shown to higher precision in this window
177. ys C 17 6093 1983 85 Knap W et al Phys Rev B 53 3912 1996 86 Pfeffer P Phys Rev B 59 15902 1999 87 87 Akkermans E Wolf P and Maynard R Phys Rev Lett 1986 105 88 Ishimaru K volume 2 Academic Press New York 1978 89 Datta S and Das B Appl Phys Lett 56 665 1990 90 Altshuler B L and Aronov A G Electron Electron Interactions in Disordered Systems edited by A L Efros and M Pollak p 11 North Holland Amsterdam 1985 91 Lee P A and Ramakrishnan T V Rev Mod Phys 57 287 1975 92 Bergmann G Solid State Comm 42 815 1982 93 Kouwenhoven L and Marcus C M Physics World 11 35 1998 94 Beenakker C W J Rev Mod Phys 69 731 1997 95 Alhassid Y Rev Mod Phys 72 895 2000 96 Folk J A et al Phys Rev Lett 86 2102 2001 97 Beenakker C W J and van Houten H Solid State Physics 44 1 98 Dresselhaus P D et al Phys Rev Lett 68 106 1992 99 Hikami S Larkin A I and Nagaoka Y Prog Theor Phys 63 707 1980 100 Wolf S A et al Science 294 1488 2001 101 Awschalom D Loss D and Samarth N Semiconductor Spintronics and Quantum Computation Springer Verlag 2002 102 Koga T et al Phys Rev Lett 88 126601 2002 103 Kiselev A A and Kim K W Appl Phys Lett 78 775 2001 104 Jusserand B et al Phys Rev B 51 4707 1995 105 Miller J B et al Phys Rev Lett go 76807 2003 106

Miller Thesis - Center for Quantum Devices

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