The University of New South Wales

School of Physics

Many-body related phenomena in quantum wires

David John Reilly A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy December 2001 U N S W 1 1 SEP 2003

LIBRARY Abstract

There is controversy as to whether a one-dimensional (ID) gas can spin polarise in the absence of an applied magnetic field. Together with a phe­ nomenological model, this thesis reports experimental results supportive of a spin polarisation at B = 0, driven by many-body interactions. A spin energy gap is indicated by the presence of a feature in the range 0.5 — 0.7 x 2e2/h in conductance data. Importantly, it appears that the degree of polarisation is not static but a function of both the one- and two-dimensional electron density. Previous work on the 0.7 x 2e2/h conductance feature has established that it is due to many-body interactions involving spin. Here, the 0.7 feature is studied in GaAs/AlGaAs quantum wires of exceptionally high quality, free from the disorder associated with modulation doping. These novel structures have facilitated investigation of the 0.7 x 2e2/h conductance feature as a function of density, temperature, magnetic field, bias and length of the ID region. In a direct comparison in zero magnetic field, conductance structure is observed near 0.7 x 2e2/h for a zero length contact, where the quantum wire of length l = 2pm shows structure evolving with increasing confinement to 0.5 x 2e2/h, the value expected for an ideal spin-split sub-band. For intermediate lengths (l = 0.5/im and 1.0/cm) the conductance feature at 0.7 x 2e2/h evolves below 0.6x2e2/h with increasing confinement. These results suggest that the dominant mechanism through which interact can be strongly affected by both the length and confinement of the ID region. Investigations using a source - drain bias technique have lead to the develop­ ment of a new phenomenological model, consistent with all of the key published results on the subject. In the context of this model, a density dependent spin polarisation produces quarter plateaus at 0.75 x 2e2/h with the application of a source - drain bias. Features at 0.25 x 2ei2/h remain absent however, since the spin energy gap remains closed at low densities. In linear response conductance a quasi-plateau near 0.7 x 2e2/h results from the simultaneous increase of the Fermi energy and spin energy gap. Complementing these investigations, perturbed conductance data taken on a device is consistent with spin dependent scattering, reminiscent of Kondo type physics seen in metals and recently in quantum dots. Although no clear connection can be established, taken together with the stud­ ies made here on open quantum dots this result suggests that spin is important in a complete description of quantum transport in low dimensional mesoscopic systems. This thesis also describes research related to the fabrication and mea­ surement of micro-magnetometers designed for use in explosive flux compression experiments. These novel devices were used to measure the de Haas - van Alphen effect in the heavy Fermion compounds LaB6 and CeB6. For Armanda iii

Acknowledgments

I must admit that I have looked forward to writing the acknowledgments section of my thesis in order to have the opportunity to thank the many people who have truly made a difference. Of course, I am grateful first of all to my academic supervisor, Prof. Bob Clark, for his enthusiasm and encouragement, as well as providing the excellent facilities that make up the Centre for Quantum Computer Technology. I am enormously indebted to Alex Hamilton, who has spent so much time talking physics and patiently teaching me over these past years. Equally to Andrew Dzurak, - there were times when I didn’t think I would have made it without your encouragement and good humor. I’m deeply grateful to Bruce Kane who taught me a lot. Thanks to Michelle Simmons, Neil Curson and Richard Newbury for so many helpful chats. I couldn’t imagine working with a better group of friends than Jeremy O’Brien, Tilo Buehler, Rolf Brenner, Steven Schofield, Geoff Facer, Rachel Heron, Carlin Yasin, Pradeep Sriganesh and Rita McKinnon. In particular, thanks to Tilo for all your help and encour­ agement over the past years and to Carlin for proof reading this thesis. None of this work would have been possible without the help of Bob Starrett, Dave Barber, Karen Jury, Renata Jones, Gavin Hicks, Venus Lim, Martin Brauhart, Mark Gross, Nancy Lumpkin, Jack Sandal, Andrei Skougarevsky, Eric Gauja, Albert McMaster, Sophie Kinna, Lars Oberbeck, Fred Green, Linda Macks and Fay Stanley. Thanks also to Oleg Sushkov and Ross McKenzie for patiently discussing physics with me. I am very grateful for the opportunity to work at Los Alamos National Laboratories with Chuck Mielke, Dwight Rickell, Greg Boebinger, Chris Hammel, Marilyn Hawley, Geoff Brown and Dennis Pelekov. I must also thank Loren Pfeiffer and Ken West of Bell Laboratories who supplied the magnificent material used in this work. Finally I could never have attempted a Ph.D without the support and en­ couragement of my family; Mum, Dad, Jeff, Nat and Jen - thankyou. Thank you also to my parents in law - Armando and Mary. Lastly, to my wife Armanda, thanks for more than I can say. I could never have done it without you. iv

Publications arising from this work Journal Publications

• Density dependent spin polarisation in ultra low-disorder quantum wires D. J. Reilly, T. M. Buehler, J. L. O’Brien, A. R. Hamilton, A. S. Dzurak, R. G. Clark, B. E. Kane, L. N. Pfeiffer and K. W. West Submitted to Physical Review Letters 2001

• Many-body spin related phenomena in ultra low-disorder quantum wires D. J. Reilly, G. R. Facer, A. S. Dzurak, B. E. Kane, R. G. Clark, P. J. Stiles, J. L. O’Brien, N. E. Lumpkin, L. N. Pfeiffer and K. W. West. Physical Review B 63, Rapid Communications. R121311 (2001).

• Many-body spin interactions in semiconductor quantum wires D. J. Reilly, G. R. Facer, A. S. Dzurak, B. E. Kane, R. G. Clark, P. J. Stiles, J. L. O’Brien, N. E. Lumpkin, L. N. Pfeiffer and K. W. West. Australian Journal of Physics, 53, 543 (2000).

• The 0.7 structure in one-dimensional constrictions with tunable potential landscapes J. L. O’Brien, D. J. Reilly, A. S. Dzurak, T. M. Buehler, R. Brenner, R. G. Clark, L. N. Pfeiffer and K. W. West To be submitted to Physical Review B

• AC magnetic susceptibility measurements of the superconducting transi­ tion in single crystal cr-uranium using co-planar micro-magetometers J. L. O’Brien, A. R. Hamilton, R, G. Clark, C. H. Mielke, J. L. Smith, J. C. Cooley, D. G. Rickel, R. P. Starrett, D. J. Reilly, N. E. Lumpkin, R. J. Hanrahan Jr. , and W. L. Hults Submitted to Physical Review B Rapid Communications.

• Construction of a Silicon-Based Solid-State quantum Computer A. S. Dzurak, M. Y. Simmons, A. R. Hamilton, R. G. Clark, R. Brenner, T. M. Buehler, N. J. Curson, E. Gauja, R. P. McKinnon, L. D. Macks, M. Mitic, J. L. O’Brien, L. Oberbeck, D. J. Reilly, S. R. Schofield, F. Stanley, D. Jamieson, S. Prawer, C. Yang and G. J. Milburn. To appear in Journal of Quantum Information and Computing. Self-aligned fabrication process for silicon quantum computer devices T. M. Buehler, R. P. McKinnon, N. E. Lumpkin and R. Brenner, D. J. Reilly, L. D. Macks and A. R. Hamilton and A. S. Dzurak and R. G. Clark Submitted to Applied Physics Letters.

Conference Proceedings Correlated electron phenomena in ultra-low-disorder quantum point con­ tacts and quantum wires D. J. Reilly, G. R. Facer, A. S. Dzurak, B. E. Kane, R. G. Clark, N. E. Lumpkin, L. N. Pfeiffer and K. W. West 1998 Conference on Optoelectronic and Microelectronic Materials and De­ vices. Proceedings (Cat. No.98EX140). IEEE. 1999, pp.486-8. Piscat- away, NJ, USA.

Construction of a Silicon-Based Quantum Computer M. Y. Simmons, S. R. Schofield, J. L. O’Brien, N. J. Curson, R. G. Clark, T. M. Buehler, R. P. McKinnon, R. Brenner, D. J. Reilly, A. S. Dzurak and A. R. Hamilton Proceedings of the 2001 1st IEEE conference on nanotechnology

Contributed Abstracts Many-body spin related phenomena in ultra low-disorder quantum wires D. J. Reilly, G. R. Facer, A. S. Dzurak, B. E. Kane, R. G. Clark, P. J. Stiles, J. L. O’Brien, N. E. Lumpkin, L. N. Pfeiffer and K. W. West. Annual American Physical Society March Meeting, 2001, Seattle Washington USA Bulletin of the American Physical Society (2001).

Many-body spin interactions in ultra low-disorder quantum wires D. J. Reilly, G. R. Facer, A. S. Dzurak, B. E. Kane, R. G. Clark, P. J. Stiles, J. L. O’Brien, N. E. Lumpkin, L. N. Pfeiffer and K. W. West. Annual American Physical Society March Meeting, 1999.

Effect of a Kondo impurity in the conductance of a quantum point contact G. R. Facer, D. J. Reilly, B. E. Kane, A. S. Dzurak, R. G. Clark, P. J. Stiles, J. L. O’Brien, N. E. Lumpkin, L. N. Pieiffer and K. W. West. Annual American Physical Society March Meeting, 1999. vi

• Many-body spin related phenomena in ultra-low-disorder quantum wires D. J. Reilly, G. R. Facer, A. S. Dzurak, B. E. Kane, R. G. Clark, P. J. Stiles, J. L. O’Brien, N. E. Lumpkin, L. N. Pfeiffer and K. W. West 14th National Congress of the Australian Institute of Physics Conference handbook TF 145, (2000). • Many-body spin related phenomena in ultra-low-disorder quantum wires D. J. Reilly, G. R. Facer, A. S. Dzurak, B. E. Kane, R. G. Clark, P. J. Stiles, J. L. O’Brien, N. E. Lumpkin, L. N. Pfeiffer and K. W. West 24th ANZIP Condensed Matter Physics Meeting, Wagga, Australia Feb 2000 Conference Handbook TP8. • Electrical transport of quantum dots fabricated in undoped GaAs/AlGaAs heterostructures J. L. O’Brien, A. S. Dzurak, T. M. Buehler, R. Brenner, D. J. Reilly, R. G. Clark, N. E. Lumpkin, L. N. Pfieffer and K. W. West 24th ANZIP Condensed Matter Physics Meeting, Wagga, Australia Feb 2000 Conference Handbook WM8. • Correlated electron phenomena in ultra-low-disorder quantum wires D. J. Reilly, G. R. Facer, A. S. Dzurak, B. E. Kane, R. G. Clark, N. E. Lumpkin, J. L. O’Brien, L. N. Pfeiffer and K. W. West 23th ANZIP Condensed Matter Physics Meeting, Wagga, Australia Feb 1999 Conference Handbook WP13. • Fabrication of aluminum single-electron- using bilayer electron beam resist and shadow evaporation T. M. Buehler, R. Brenner, N. E. Lumpkin, A. S. Dzurak, R. G. Clark, M. Gross, E. Gauja, R. P. Starrett, A. R. Hamilton, M. Y. Simmons, J. L. O’Brien and D. J. Reilly 24th ANZIP Condensed Matter Physics Meeting, Wagga, Australia Feb

2000. • The Dirac Experiments - Results and Challenges R. G. Clark, J. L. O’Brien, A. S. Dzurak, B. E. Kane, N. E. Lumpkin, D. J. Reilly, R. P. Starrett, D. G. Rickell, J. D. Goetee, L. J. Campbell, C. M. Fowler, C. Mielke, N. Harrison, W. D. Zerwekh, D. Clark, B. D. Bartram, J. C. King, D. Parkin, H. Nakagawa and N. Miura Proceedings of the Eighth International Conference on Megagauss Mag­ netic Field Generation and Related Topics, Tallahassee, 1998 edited by H. Schneider-Muntau (World Scientific, Singapore.) vii

• Single-electron architectures for simulation of solid state quan­ tum computer read-out R. Brenner, T. M. Buehler, R. P. McKinnon, D. J. Reilly, A. R. Hamilton, A. S. Dzurak, R. G. Clark, N. E. Lumpkin and G. J. Milburn Poster Presentation, 14th National Congress of the Australian Institute of Physics Conference 2000.

Seminars • Electron interactions in quantum wires Dept, of Applied Physics & Physics, Yale University, New Haven, CT, USA (August 2000). • Electron spin interactions in quantum wires Dept, of Physics, Princeton University, Princeton NJ, USA (September 2000). • Many-bodv electron phenomena in quantum wires Los Alamos National Laboratory, NM USA (September 2000). • Electron spin interactions in quantum wires Centre for High Technology Materials, University of New Mexico, NM USA (September 2000). • Many-body electron phenomena in quantum point contacts CSIRO, Division of Telecommunications ond Industrial Physics, Lindfield Australia. (1998). • Electron spin interactions in quantum wires Department of Applied Physics, University of Technology, Sydney. (1999). • Australian Institute of Physics, Award for Postgraduate Excellence presentation, University of New South Wales, (October 2000). • Many-body spin interactions in quantum wires at UNSW, Coogee, Australia, (December 2000). • Towards a phenomenological understanding of the 0.7 conductance feature Centre for Quantum Computer Technology, UNSW, (July 2001). • Density dependent spin polarisaton in ultra low-disorder quantum wires Mesoscopic Physics at UNSW, Coogee, Australia, (December 2001). Contents

1 Introduction 1 1.1 Perspectives...... 1 1.2 The 0.7 Conductance Feature...... 4 1.3 Micro-Magnetometers...... 8 1.4 Thesis Outline ...... 9

2 Background Theory 11 2.0. 1 Introduction...... 11 2.0. 2 Ballistic Electron Transport...... 12 2.1 Preliminary Concepts ...... 13 2.1.1 The Two Dimensional Electron Gas ...... 14 2.1.2 The and the Fermi-Dirac Distribution . . 16 2.1.3 Electrons Confined to One Dimension ...... 17 2.2 Landauer-Biittiker Formalism...... 19 2.2.1 Quantised transmission of a saddle-point constriction ... 22 2.3 Interacting Electrons in ID ...... 23 2.3.1 Correlation and the Exchange Interaction ...... 24 2.3.2 The ...... 27 2.4 The Spin Polarised Electron Gas ...... 29 2.5 The Origin of the 0.7 Feature: Theory and Models ...... 29 2.5.1 The Lieb-Mattis Theorem...... 30 2.5.2 Self-Consistent Calculations...... 31 2.5.3 Wigner Crystallisation...... 34 2.5.4 Electron Bound States...... 35 2.5.5 Spin-Orbit Interactions...... 36 2.5.6 Phenomenological Descriptions ...... 37

3 Experimental Methods 39 3.1 Introduction ...... 39 3.2 Sample Fabrication...... 39 3.2.1 Molecular Beam Epitaxy...... 41 3.2.2 Optical Lithography and Electron Beam Lithography ... 42

viii CONTENTS ix

3.3 Measurement ...... 46 3.3.1 Cryostats and Dilution Refrigerators...... 46 3.3.2 Measurement Electronics ...... 50

4 Dependence of the 0.7 feature 54 4.1 Introduction...... 54 4.2 l = 0 Quantum Wires ...... 60 4.2.1 QPC:A...... 61 4.2.2 QPC:B...... 63 4.3 l = 0.5fim Quantum Wires ...... 69 4.4 l = 1.0iim Quantum Wires ...... 72 4.4.1 Quantum Wire: 1/rra : A ...... 75 4.4.2 Quantum Wire l/im : B...... 78 4.5 / = 2.0/rm Quantum Wires ...... 81 4.6 Discussion of Results...... 82 4.7 Conclusion...... 90

5 Confinement Potential: Bias Spectroscopy 92 5.1 Introduction...... 92 5.2 ID Sub-band Energies: DC Source-Drain Dependence...... 93 5.2.1 l = 0.5/im quantum wire...... 95 5.2.2 l/im quantum wires...... 102 5.3 Adiabatic Coupling: Different Geometries ...... 108 5.3.1 Curved and Rectangular l/im Quantum Wires ...... 108 5.4 Asymmetric Confinement Potentials...... 114 5.5 Further exploration of the confinement potential...... 116 5.5.1 Charge traps: Potential Perturbations ...... 119 5.6 Discussion of Results and Conclusions...... 120

6 A Density Dependent Spin Polarisation in ID 122 6.1 Introduction...... 122 6.2 Conclusion...... 135

7 Non-Quantised Structure 136 7.1 Introduction...... 136 7.2 Kondo-like Conductance Perturbations...... 137 7.3 Electron interferometry in wires with finger gates...... 147 7.4 Summary of Results ...... 154

8 Micro-Magnetometers 157 8.1 Introduction...... 157 8.2 Technology ...... 158 8.3 Fabrication of the Magnetometers ...... 164 CONTENTS x

8.4 The de Haas - Van Alphen Effect: Mapping the Fermi Surface . . 164 8.5 Experimental Results ...... 169 8.5.1 Lanthinum - and Cerium - Hexaboride...... 170 8.6 Micro-Magnetometers: Conclusion...... 171

9 Conclusions and Future Work 175 9.1 Magnetic Resonance Force Microscopy: Detection of possible spin polarisation in ID?...... 175 9.1.1 MRFM...... 176 9.1.2 Using MRFM to detect an electron spin polarisation . . .178 9.2 Other future work ...... 179 9.3 Thesis Summary and Final Conclusions ...... 180 List of Figures

1.1 Data taken from some of the first experiments to measure quan­ tised conductance. Note the presence of the strong conductance anomaly near 0.7 x 2e2/h. (Taken from Van Wees et al. [1]) ... 5 1.2 Data taken from Thomas et al. [2] showing the 0.7 x 2e2/h conduc­ tance feature evolving smoothly into the Zeeman spin-split level at 0.5 x 2e2/h with the application of a in-plane magnetic field. . 6

2.1 The quantised conductance of a Quantum Point Contact. (Taken from Chapter 4.) ...... 13 2.2 A schematic of the bottom of the conduction band as it varies through the heterojunction of an enhancement mode FET...... 16 2.3 Schematic diagram depicting the formation of a quasi-one dimen­ sional region, connected to 2D reservoirs...... 18 2.4 A schematic depicting the dispersion relation E(k) for electrons occupying distinct ID sub-bands...... 20 2.5 The calculated transmission for a saddle potential, dashed line is for ujx/ujy = 2 and solid line is for ujx/ujy = 3...... 23

3.1 Photographs showing processing facilities within the class 350 (a) and class 3.5 (b) clean-rooms at UNSW’s Semiconductor Nanofabrication Facility (SNF)...... 40 3.2 Schematic of the wafer used for the experiments reported in this dissertation. Layers were grown using Molecular Beam Epitaxy at Bell Laboratories, Lucent technologies New Jersey USA. ... 42 3.3 Optical lithography processing steps required to fabricate the ohmic contacts and mesa structure. See text for details...... 43 3.4 Illustration showing MBE grown layer structure with ohmic metal touching the AlGaAs/GaAs heterojunction on the side of the mesa and making contact with the 2DEG (under the application of a bias)...... 44 LIST OF FIGURES xii

3.5 Photographs showing (a) the UV optical aligner used to fabri­ cate the ohmic contacts and metal interconnects, and (b) Leica EBL 100 Electron Beam Lithography System used to fabricate the nanoscale quantum wires...... 45 3.6 Optical microscope photographs showing the device structure af­ ter optical lithography processing. (1) Entire device layout, in­ cluding bond pads. (2) Increased magnification showing 4 ohmic contacts (total of 8 for both devices). (3) Top gate FET struc­ ture prior to EBL. (4-6) Device after subsequent EBL, and wet etch. The ID quantum wire is located at the center of the cross structure in (6)...... 47 3.7 Atomic Force Microscope images of two quantum wires (length / = 1.0/am) prior to wet etching. The patterns are defined in the PMMA. Left image shows a narrow wire with rounded edges and the right image shows a wire with the standard rectangularly defined gate structure...... 48 3.8 Atomic Force Microscope images of a quantum wire of length l — 0.5//m. The images were taken after the device had been thermally cycled many times between room and milliKelvin tem­ peratures...... 48 3.9 Photographs showing the two dilution refrigerators used in the quantum wire experiments, (a) shows an older OI Kelvinox sys­ tem and electronics racks used to study the point contact and / = 2/mi quantum wire devices, (b) Experimental setup pho­ tographed from above, (c) shows the newer OI Kelvinox 100 sys­ tem, used for all other low temperature measurements reported. . 49 3.10 Photograph showing extensive cryogenic measurement platforms housed within the National Magnetic Laboratory at UNSW. Two dilution refrigerators can be seen at the back (OI K100) and left (OI Plastic) of the room. The foreground shows characterisation stations operating at liquid Helium temperatures. Close observa­ tion reveals a student teleporting back to the mothership...... 51 3.11 Circuit diagram for the most standard circuit used in the conduc­ tance measurements...... 52 3.12 Photographs showing electronic equipment used in the low noise transport experiments. Left: Setup for measurements at pumped He4 temperatures. Right: Measurements at milli-Kelvin temper­ atures...... 53 LIST OF FIGURES xm

4.1 Schematic of the quantum wires studied in this thesis. I defines the length of the ID region. The green shaded regions represent the side gates, (biased negative). The blue region represents the top gate, (biased positive). The two dimensional reservoirs are also indicated...... 57 4.2 Schematic of the MBE grown layer structure (see Chapter 3 for detail). The bias applied to the top gate (Vr) and side gates (Vs) control the shape of the ID confinement potential (shown in yellow). A large bias applied to all gates steepens the potential and increases the ID sub-band spacing...... 58 4.3 Schematic showing how a larger ID sub-band energy spacing leads to a larger ID density, (and larger Fermi level) for the same con­ ductance or occupancy...... 59 4.4 QPC:A (/ = 0). Conductance as a function of side gate bias Vs for top gate voltages Vr ranging: 170mV - 306mV in steps of 4mV. T = 50mK...... 61 4.5 QPC:A (/ = 0). Transconductance (dG/dVs), calculated numeri­ cally from the data in figure 4.4. Traces have been geometrically scaled to align the main transconductance peaks...... 62 4.6 Temperature dependence of QPC:A. Top graph: G as a function of Vs for top gate bias, Vr=180mV: T = 4000mK (red), 1500mK (orange), 1200mK (pink), 960mK (lblue), 760mK (dblue). Bot­ tom graph: G as a function of Vs for top gate bias, VA=300mV: T = 3500mK (red), 1500mK (orange), 1200mK (pink), 950mK (lblue), 800mK (dblue)...... 64 4.7 QPC:B (/ = 0). Conductance as a function of side gate bias Vs for top gates Vr ranging: 170mV - 294mV in steps of 4mV. T = 50inK ...... 65 4.8 QPC:B (/ = 0). Transconductance (dG/dVs) calculated numeri­ cally from the data in figure 4.7. Traces have been geometrically scaled to align the main transconductance peaks...... 66 4.9 Temperature dependence of QPC:B. Top graph: G as a function of Vs for a top gate bias of, Vr = 170mV: T = 2730mK (red), 1650mK (orange), 960mK (pink), 510mK (lblue), 50mK (dblue). Bottom graph: G as a function of Vs for a top gate bias of, VT=300mV: T = 2850mK (red), 1500mK (orange), 970mK (pink), 500mK (lblue), 50mK (dblue)...... 67 4.10 Plot showing the activated temperature dependence of QPC:B. . 68 4.11 QPC:B (/=0). Conductance as a function of side gate bias Vs, for large top gate and side gate bias, Vp = 300mV - 680mV. T = 50mK...... 69 LIST OF FIGURES xiv

4.12 The conductance G of a l = 0.5//ra quantum wire as a function of side gate voltage, Vs- T = 50mK. Vr = 560mV - 1500mV...... 70 4.13 l = 0.5//77i quantum wire. Transconductance (dG/dVs) calculated numerically from the data in figure 4.12. Traces have been geo­ metrically scaled to align the main transconductance peaks. ... 71 4.14 Conductance as a function of side gate bias Vs for a / = 0.5//777 Quantum wire at T = 1.6K...... 72 4.15 Conductance G as a function of side gate bias Vs for a l = 0.5//m Quantum wire at T = 4.2K. Top plot shows a dense number of curves from Vs' 0 to -700mV, and the bottom plot shows data taken out to Vs = -2.0V...... 73 4.16 Temperature dependent conductance of / = 0.5i±m quantum wire. Lattice temperature is indicated in the legend. TOP: Conduc­ tance G as a function of side gate voltage, Vs for Vr = 500mV. BOTTOM: Same as Top only traces for 4.2K, 1.0K and 30rriK are shown for easy comparison...... 74 4.17 Quantum wire: l//m : A. Conductance G as a function of side gate bias Vs for Vr = 300mV - 800mV (bottom trace). T = 4.2K. 76 4.18 TOP: Conductance of 1.0fim quantum wire (Device A) at T = 50mK. Vr = 270mV - 630mV. BOTTOM: Zoom of data for 1st two integer plateaus...... 77 4.19 Transconductance (dG/dVs) of l = 1 /im quantum wire (Device:A) at T — 50mK. Traces have been geometrically scaled to align the main transconductance peaks...... 78 4.20 Conductance of 1.0//m quantum wire (Device B) at T = 4.2K. Vr = 440mV - 990mV in lOmV steps...... 79 4.21 Conductance of 1.0[im quantum wire (Device B) at temperatures T « 1.6K (blue) and T = 4.2K (red). VT = 0.44V - 0.79V in 50mV steps...... 80 4.22 Conductance of 1.0fim quantum wire (Device B) at T « 50mK. Vt = 0.44V - 0.79V in 50mV steps. A small dc offset bias was present (approximately: 0.4mV). Inset: A zoom of the region below the first plateau (n=l)...... 81 4.23 Conductance G as a function of side gate voltage, Vs for a quan­ tum wire of length l = 2/xra. TOP: T = 50mK. BOTTOM: T = IK...... 83 4.24 Feature position as a function of side gate voltage, Vs- The posi­ tion of the corresponding 11= 1 plateau is also shown. Red: / = 0, Green: / = 0.5/im, Blue: / = 1.0nm, Black: l = 2.0/im...... 84 LIST OF FIGURES xv

4.25 Conductance histograms for l = 0,0.5,1.0, 2\im quantum wires. The high number of counts in the shorter devices reflects the flat­ ness of the integer plateaus. The count number n is reduced as the quality of the plateaus decreases with increasing length of the ID region. The 0.5 — 0.7 x 2e2/h structure is indicated by the red arrows. The blue arrows indicate the presence of structure near 1.7 x 2e2/h...... 86 4.26 Schematic depicting electron density as a function of position through a quantum wire. Note how the contact region, (Con) is larger for the longer quantum wire...... 89

5.1 Schematic illustration depicting the differential conductance (di/dv) as a function of source - drain dc bias. Note how the curves bunch at the integer and half - integer plateaus. The green shaded re­ gions represent the effective chemical potential of the source (S) and drain (D), with the pink lines indicating the ID sub-band edges. The bias required to move from one half-plateau (or in­ teger plateau) to the next, is the ID sub-band energy spacing, indicated by the brown lines...... 95 5.2 Differential Conductance (di/dv) of a quantum wire of length l = 0.5fim. Vt = 1500mV. T = 50mK...... 96 5.3 Differential Conductance (di/dv) of a / = 0.5/mi quantum wire. Vt = 630mV. T = 50mK...... 97 5.4 Differential Conductance (di/dv) of a / = 0.5/mn quantum wire. Vt = 550mV. T = 50mK...... 98 5.5 Differential Conductance (di/dv) of a l = 0.5/im quantum wire. VT = 470mV. T = 50mK...... 100 5.6 Differential Conductance (di/dv) of a l = 0.5/mi quantum wire. VT = 720mV - 820mV. T = 4.2K. = 0V...... 101 5.7 Differential Conductance (di/dv) of a l = 0.5/im quantum wire. VT = 700mV - 808mV. T = 4.2K. = 0V...... 101 5.8 Differential Conductance (di/dv) of a / = 0.5/mi quantum wire. VT = 700mV - 808mV. T = 4.2K...... 102 5.9 Differential Conductance (di/dv) of a l = 1.0fim quantum wire. Vt = 240mV - 360mV. T = 50mK ...... 103 5.10 Differential Conductance (di/dv) of a / = 1.0/im quantum wire. Vt = 385mV. T = 50mK...... 104 5.11 Differential Conductance (di/dv) of a l = 1.0/im quantum wire. Vt = 600mV. T = 50mK, 450mK, 700mK (top to bottom). . . . 106 5.12 Differential Conductance (di/dv) of a / = 1.0/rra quantum wire. VT = 700mV. T = 50mK...... 107 LIST OF FIGURES xvi

5.13 Conductance G as a function of side gate bias Vs- TOP: De- vice:A. BOTTOM: Device:B. Insets illustrate the geometrical form of the device, / defines the effective length of the ID region. T = 50mK for both devices...... 109 5.14 Differential conductance of 1.0/im (Device B). T = 50mK, Vs = -800mV. VT = 600mV - 740mV...... 110 5.15 Conductance G of 1.0/im quantum wire as a function of top gate bias. (Device B) T = 50mK. Vsd = 0 to 1.4mV in steps of O.linV. 112 5.16 Differential conductance of a 1.0fim (Device B) T = 4.2K. TOP: Vs = 0, Vt = 320mV to 431mV lmV steps. BOTTOM: Vs = -800mV. Vt = 600mV to 708mV lmV steps...... 113 5.17 Conductance of a l = 0.5/im quantum wire, with an antisymmet­ ric side gate bias. Vt = 800mV. T = 50mK. TOP: left side gate sweeps bias, right side gate steps from 0V -2V. BOTTOM: left side gate steps bias 0V to -2V, right side gate sweeps...... 115 5.18 Conductance of a l = 0.5fim quantum wire with a antisymmetric side gate bias. Vt = 500mV, T = 4.2K...... 117 5.19 TOP: Conductance of a l = 0.5nm quantum wire, as a function of top gate voltage (Vt) for different side gates, (Vs = 0, -0.03mV, -O.O6111V, -0.09rnV, -O.llOmV. T = 50mK. BOTTOM: Conduc­ tance of a / = 1.0\im quantum wire (Device B), as a function of top gate voltage (Vt) for different side gates, (Vs = 0 and -SOOrriV. )...... 118 5.20 Conductance of a l = 0.5(im quantum wire in two distinct states. The different states arise due to surface charge traps that modify the potential landscape. T = 50rnK...... 119

6.1 Illustration depicting a density dependent spin gap opening lin­ early with increasing density (n) or gate voltage (Vs). The Fermi level is non-linear with gate voltage because of the non-linear den­ sity of states with energy. The energy gap between spin up and down is assumed to increase linearly with density...... 124 6.2 Illustration depicting a density dependent spin gap opening lin­ early with increasing density (n) or gate voltage (Vs). El, E2, E3 indicate the ID sub-band edges. The Fermi level indicated by the dashed line, is non-linear with density n, due to the singularity in the ID density of states...... 125 LIST OF FIGURES xvii

6.3 Schematic showing the three main scenarios that lead to features near 0.5 x 2e2/h and 0.7 x 2e2/h in the conductance at both zero (left) and finite source - drain bias (right). The shaded regions below each graph represent the Fermi distribution. Scenario (I) occurs if the spin gap is large in comparison to kT. Scenario (II) occurs at high temperature when kT is close to the size of the spin gap. Finally scenario (III) occurs for the case of weak spin splitting. Shown on the left is real data taken from Chapters 4 and 5 to illustrate the three scenarios...... 126 6.4 Conductance of a / = 0.5/mi quantum wire as a function of side gate voltage for top gate voltages in the range; Vp = 420mV- 1104mV (right to left). The red curves are for T = 4.2K and the black curves are for T = 50mK. For the left most curves the position of the feature moves upward from 0.5 x 2e2/h to 0.7 as the temperature is raised. (From Chapter 4.) ...... 129 6.5 Differential conductance of a / = 1.0fim quantum wire as a func­ tion of source - drain bias Vsd, T=50mK. (a) Vr=385mV. (b)VT=600mV. (c)Vr=700mV. (d)VT=700mV, B=2.75T. (From Chapter 5.) . . 130 6.6 TOP: Conductance of a l = 1.0/xm quantum wire as a function of Vs for VT in the range, 270mV - 800mV (right to left). The dashed curves are for a constant offset bias of Vsd = 0.5mV and the solid curves for Vsd = 0. BOTTOM: The application of a constant bias produces the non-monotonic feature in the conductance. Left side is conductance data, right side is a schematic relating the potentials of the source (S) and drain (D) to the feature...... 132 6.7 Schematic illustrating how the density dependent spin gap leads to a cusp feature below the first integer plateau. As the ID density is increased, the source - drain bias required to move above the second spin band edge also increases. Bottom right shows a zoom of S-D data...... 133 6.8 Schematic illustrating the anticipated spin -gap dependence with 2D density...... 134

7.1 Conductance G of a quantum point contact containing an impu­ rity close to the ID region. Each trace is for a different side gate bias Vs, ranging 0 to -1.5V in steps of -lOOmV, left to right. T = 50mK. Inset: zoom of region of interest and comparing traces Vs = 0 to V;s- = -1.5V...... 138 7.2 Conductance G of a quantum point contact containing an impu­ rity close to the ID region. Each trace is for a different side gate bias Vs, ranging 0 -1.5V in steps of -lOOmV, left to right. T = 3 75 m K...... 139 LIST OF FIGURES xvm

7.3 Conductance G as a function of top gate bias Vt. Side gates are grounded. Each trace corresponds to a different temperature, as indicated by the graph legend. Inset: A zoom of the region of interest...... 140 7.4 Schematic comparison between electron scattering trajectories for the case of B = 0 and finite perpendicular magnetic field. The for­ mation of edge states in the Quantum Hall regime steer electrons away from the influence of the scatterer...... 141 7.5 Conductance G of a quantum point contact containing an impu­ rity close to the ID region. Each trace is for a different side gate bias Vs, ranging 0 to -1.3V in steps of-lOOmV, left to right. B = 0.8T. T = 50mK...... 142 7.6 Conductance G of a quantum point contact containing an impu­ rity close to the ID region. Each trace is for a different side gate bias Vs, ranging 0 to -600mV in steps of - lOOmV, left to right. B = 1.5T. T = 50mK...... 143 7.7 Conductance G of a quantum point contact containing an impu­ rity close to the ID region. Each trace is for a different side gate bias Vs, ranging 0 -1.4V in steps of -lOOmV, left to right. B = 2.70T. T = 50mK...... 144 7.8 Conductance G as a function of top gate bias Vt for a quan­ tum point contact containing an impurity close to the ID region. Each trace is for a different parallel magnetic held, ranging B = 0 (black) 0.25T (pink) 0.50T (yellow) 0.75T (red) LOOT (green) 1.25T (purple) and 1.5T (blue). T = 50mK...... 145 7.9 TOP: AFM images showing the two devices with huger gates, (Taken from O’Brien et al. [3] ). BOTTOM: The correspond­ ing conductance G as a function of top gate bias Vt, with Vs grounded. T = 50mK...... 148 7.10 Device A. Conductance G as a function of the bias applied to side huger gate A. T = IK. Vt = 540mV - 623mV. Left, middle an right most traces are shown in red to facilitate comparison. . . .149 7.11 Device A. Conductance G as a function of the bias applied to side huger gate B. T = IK. Vt — 540mV - 594mV...... 151 7.12 Device A. Conductance G as a function of the bias applied to both huger gates. T = IK. Vt = 540mV - 656mV. Inset: zoom of the region of interest...... 152 7.13 Device A. Conductance G as a function of bias applied to both hnger gates together. Vt — 580mV, 590mV and 600mV. Blue trace: T = lOOmK, Black trace: T = 2.IK, Red trace: T = 4.2K. 153 LIST OF FIGURES xix

7.14 Device B. (a) Conductance G as a function of side gate bias V$. T = 50mK. Vt = 630mV - 900mV. (b) Shows a zoom of the central region with an isolated trace (red) to aid observation, (c) What is presumably Coulomb charging peaks can be seen predominately below the 0.7 feature, (d) Gate geometry and device layout. . . . 155

8.1 Illustration of a micro-magnotometer connected to a set of co- planar transmission lines. The magnetic field B is perpendicular to the plane of the coils as indicated. The magnetisation signal threading the coils is entirely due to the sample. (Illustration by R. P. Starrett)...... 159 8.2 Result of calculations simulating the effective coil pickup. The vast majority of the magnetisation flux threads the compensated coils. (Calculation and illustration by R. P. Starrett)...... 160 8.3 Magnotomter sample mounting procedure. The usual sequence is to mount the sample on the surface of the micro-magnotometer using Duco cement (a). After the glue is dry, the magnetometer can be mounted and bonded to the CTL printed circuit board with the sample fitting in the small aperture shown in (b). A custom made mounting jig is used to keep the magnotometer in contact with the CTLs until the glue is drv (c). (Illustration bv A. Wang)...... 162 8.4 Photographs showing the G10 fibre-glass probe used in the de Haas - van Alphen experiments to 60T. (a) shows the tail with a loop of plastic used to minimise vibrations in the pulse, (b) is a close-up of the CTL printed circuit boards, and (c) is a spliced photograph of the entire insert...... 163 8.5 Optical microscope photographs showing the micro-magnetometers. (a-c) show the coaxial EBL written coils (generation 1) at increas­ ing magnifications, (d-f) show side by side coils (generation 2) fabricated using standard optical lithography techniques...... 165 8.6 Schematic of the process used to fabricate the micro-magnetometers (Part 1). Standard optical lithography techniques are used to de­ fine the metal coils and SiN layers that provide electrical isolation between layers. Selective etching of SiN enables electrical inter­ connects between the metal layers. The top row shows the results of the steps outlined in the columns below...... 166 8.7 Schematic of the process to fabricate the micro-magnetometers (Part 2) See caption of Figure 8.6 for details...... 167 8.8 Raw signal obtained from a 40 turn side by side micro-magnetometer. Gain = 5000. T = 4.2K ...... 171 LIST OF FIGURES xx

8.9 Top:LaB6 de Haas - van Alphen signal plotted as a function of l/B. Clear periodic oscillations can be seen. Bottom: Fourier transform of the data showing clear peaks corresponding to the various a k - space cross sections...... 172 8.10 CeB6 dHvA signal from a 80 turn micro-magnetometer at T = 450mK. (a) Raw signal with preamplifier gain of 10000. (b) Fourier transform indicating clear dHvA oscillations. Data taken by C. H. Mielke (NHMFL)...... 173

9.1 Basic configuration of a magnetic resonance force microscope: the force on the sample magnetisation in the gradient of a proximate magnetic field source causes a deflection of the cantilever which is sensed optically. The RF coil is used to excite spin resonance. . . 177 Chapter 1

Introduction

1.1 Perspectives: The Electronics of Tomorrow

The dwindling size of circuits in electronic chips drives much of the interest in mesoscopic and nanoscale physics. The electronics industry is deeply interested in developing new nanofabrication methods to continue in the long term trend of building smaller, faster and less expensive devices. Before these developments can happen however, we must understand the unique physics that governs matter at these dimensions. describes t he behavior of the universe at all scales however, in the realm of nanoscale devices classical and quantum physics depart and common sense is left behind. One of the most striking ex­ amples of this departure can be seen in the electrical resistance of mesoscopic devices. Feynman anticipated, in part, such odd behavior: “/ have thought about some of the problems of building electric circuits on a small scale, and the prob­ lem of resistance is serious... ” [4]. Landauer [5] also envisaged that quantum mechanics would have direct consequences for small scale devices in which the electrons flow coherently from one side to the other. In 1987 two groups [6, 1] confirmed these expectations in a remarkable way. The electrical conductance of a mesoscopic device was found to be quantised in units of ~ 12kQ, corresponding

1 1. Introduction 2 to the number of occupied quantum modes. In this Thesis electrical conduction at the quantum level is further explored. Experimental evidence is shown to be consistence with the formation of a new and novel state in which the electron interactions give rise to magnetic ordering. Electron correlations in clean semiconductor systems create some truly re­ markable phenomena. For the case of two dimensional systems, the fractional quantum Hall effect provides an apt and fascinating example of where the be­ havior is governed not by single electrons interacting with an external field, but by the complex interplay of some 1011 electrons in collaboration [7]. In one dimensional systems (ID) such as the quantum wires considered in this thesis, interacting electrons are expected to create a strongly correlated state know as a Luttinger Liquid. Luttinger liquids are without a Fermi surface, even at zero temperature and lack single particle excitations at low energies. Collective exci­ tations are separable in terms of spin and charge, giving rise to properties with no analog in Fermi liquid descriptions. In addition to Luttinger Liquids, low density quasi-one dimensional systems are expected to exhibit correlated properties in connection with the electron spin. This Thesis reports on a number of examples where this is the case. The first concerns the behavior of an anomalous conductance feature near 0.7 x 2e2/h that cannot be explained within the context of a non-interacting picture. Evidence is reported in Chapters 4 and 5 linking the conductance feature to the existence of a spontaneous spin polarisation in, or near the one dimensional electron gas. Consistent with the higher dimensional cases, strong exchange coupling between the electrons in ID is expected to drive magnetic ordering in the absence of an applied magnetic held. The inclusion of the electron spin in open systems [8] connects tunneling to a magnetic-exchange process. In this case, the physics of a quantum dot connected to leads becomes similar to the physics of magnetic impurities 1. Introduction 3 coupled to the conduction electrons in a metal host, that is the Kondo effect [9]. So far however, the Kondo effect has not been seen in one-dimensional systems. In spite of this, the experimental data reported in Chapter 7 is consistent with spin dependent scattering effects. The data suggests that the magnetic moment of a single defect can perturb the electron transport in a complex way, destroying conductance quantisation. The experimental study of many-body interactions requires systems of re­ markable purity. Modulation doping is the technique most commonly used to create a two dimensional electron gas in GaAs/AlGaAs semiconductor systems. Although the doping provides the necessary electrons, it also introduces impu­ rities which strongly scatter the carriers. This random impurity potential can ultimately limit the electron mobility at very low temperatures. For the work conducted in this Thesis, a novel technique was exploited in order to avoid the doping impurity potential. All of the data was taken on quantum wires that were fabricated from undoped enhancement-mode Field Effect Transistor (FET) devices, where the electron gas is induced by an applied gate bias. These de­ vices are extremely clean and have exhibited electron mobilities in excess of 10‘cm2/VS with mean free paths greater than 160//m. These novel structures are ideal systems in which to study fundamental electron correlation effects, since the electron wave function remains coherent over great distances. Of further im­ portance, these devices allow the carrier density and potential confinement to be continuously and independently varied during the course of an experiment. This versatility greatly facilitates investigations into many-body physics in mesoscopic systems. This Thesis includes investigations of many-body electron interactions in ID clean semiconductor quantum wires. The prime focus is on an experimental study of the anomalous 0.7 x 2e2/h conductance feature, and how it relates to the notion of a spin polarisation. Initial studies included mapping the behavior 1. Introduction 4 of the 0.7 feature as a function of electron density and the length of the ID region. It was found that both the strength and position of the anomalous feature increase with increasing ID length and the 2D density [10]. For high 2D densities (or strongly negative side gate bias) the feature appears not at 0.7 x 2e2//i, but close to 0.5 x 2e2/h the value expected for an ideal spin-split Zeeman level. These results suggest that the energy gap between spin up and spin down electrons increases with 2D density (or side gate bias) and length of the ID region. The application of a source - drain bias across the ID constriction allows for spectroscopy of the ID sub-band energies. This technique was used to char­ acterise the relationship between gate bias and the sub-band energy spacing or confinement potential. Following these studies, bias spectroscopy was used to in­ vestigate the 0.7 conductance feature. Under the application of a bias the feature appears as an anomalous band edge, similar to the single particle ID sub-bands. For the case of the 0.7 feature however, the band-edge energy appears to be a function of the ID density. These observations lead to the development of a simple phenomenological model to explain the 0.7 feature in terms of a den­ sity - dependent spin polarisation, as discussed in Chapter 6 and reference [11]. Within the context of this model, a conductance feature near 0.7 is produced by the simultaneous increase of both the Fermi level and the spin energy gap. A quasi-plateau is observed since the difference between the Fermi energy and the upper spin band-edge remains constant for a time.

1.2 The 0.7 Conductance Feature

The following section introduces the 0.7 conductance anomaly via a brief chrono­ logical review of the essential results to date. The emphasis is on the experi­ mental results since theoretical explanations for the conductance anomaly are 1. Introduction 5

1.6 K

0.6 K

0.3 K

0.7 feature

-2 -i.B GATE VOLTAGE (V)

Figure 1.1: Data taken from some of the first experiments to measure quantised conductance. Note the presence of the strong conductance anomaly near 0.7 x 2e1 /h. (Taken from Van Wees et al. [1]) reviewed in Chapter 2. The presence of a conductance feature near 0.7 x 2e2/h can be seen in the very first ballistic experiments on Quantum Point Contacts (QPCs) by van Wees et al and Wharam et al [6, 1] in 1987. Figure 1.1 shows data taken by van Wees et al that exhibits a strong 0.7 feature. Although subsequent experiments con­ tinued to observe a robust conductance anomaly, no comment was made in the literature until the brief discussion by Patel et al. [12] in 1991 in the context of source - drain bias experiments. A seminal work by Thomas et al. [2] (1996) created a strong interest in the field with the suggestion that the feature was in some way related to many-body interactions involving the electron spin. Two important results were uncovered by Thomas and co-workers. The first was that the conductance structure evolves continuously into the Zeeman spin-split level at 0.5 x 2e2/h with the application of a magnetic held in the plane of the current 1. Introduction 6

Gate Voltage VK (V)

Figure 1.2: Data taken from Thomas et al. [2] showing the 0.7 x 2e2/h conduc­ tance feature evolving smoothly into the Zeeman spin-split level at 0.5 x 2e2/h with the application of a in-plane magnetic field.

(Fig. 1.2). This behavior suggests that the feature is due to a ferromagnetic or spin polarised state, occurring at zero magnetic field. Secondly, Thomas et al. revealed an enhancement of the electron (/-factor for the lower ID sub-bands. In accordance with well established electron correlation theory, an enhanced g- factor is suggestive of strong many-body interactions. This work was followed by a detailed investigation of the temperature and density dependence of the conductance anomaly [13] (1998). By making use of a back gate technique [14] the strength or width of the quasi-plateau appeared to increase with decreasing carrier density, as expected for exchange and correlation type interactions. In­ consistent with this picture however, the feature also broadens and increases in strength with increasing temperature and often remains at temperatures com­ parable to the ID sub-band spacing. In 1996 Tscheuschner and Wieck [15] presented conductance measurements on in-plane-gate transistors with mobilities near 6.6 x 105cm^2V~ls~l. Conduc- 1. Introduction 7 tance quantisation in units of 0.5 x 2e2/h was seen, however due to the effects associated with a series resistance, interpretation is made difficult in these lower mobility structures. 1998 saw the emergence of a number of important papers signifying the universal nature of the conductance feature. Ramvall et al. [16] presented data with features near 0.2, 0.7 and 1.5 x 2e2//i taken on lower mobility samples, fabricated from GalnAs/InP heterostructure. In addition, the first in­ vestigations made on the induced FET structures considered here were reported by Kane et al. [17]. At about this time Kristensen et al. [18, 19] also reported important measurements on the temperature and bias dependence. These re­ sults indicate that the feature behaves like a ID sub-band edge and exhibits an activated temperature dependence. Ballistic transport measurements in constrictions of n-PbTe free from modu­ lation doping also revealed quantised structure near 0.5 x 2e2/h [20]. Studies by Liang et al. [21] provided further evidence that the conductance structure was not caused by an electron resonance. More recent work has concentrated on the dependence of the feature with electron density, length, temperature, magnetic held, thermopower and source-drain bias [22, 23, 24, 25, 10, 26]. These inves­ tigations (discussed in detail later) re-iterate the earlier studies and provided further evidence that the feature is an intrinsic property of one-dimensional semiconductor systems. In fact, there is some evidence to suggest that the effect is not limited to devices based on . Experiments on carbon nan­ otube quantum resistors and gold also show conductance structure of striking similarity to the semiconductor results discussed above, (see for exam­ ple Fig. 2c in reference [27]). Recent experiments on metallic nano-constrictions under electro-chemical control [28] also indicate fractional conductance quanti­ sation near 0.5 and 1.5 x 2e2/h. An explanation for these stable and well defined conductance steps is yet to be uncovered. Recently a new held termed ‘spintronics’ has arisen to exploit the electron 1. Introduction 8 spin as a basis for new technology. Possible applications of spintronics includes high speed magnetic filters, sensors, quantum transistors and ultimately spin qubits for quantum computers [29, 30, 31]. The notion that the carriers in a ferromagnetic metal such as Fe, Co or Ni, should themselves be magnetically polarised, dates from the earliest realisations that ferromagnetism is essentially a quantum mechanical effect arising from the spin of the electron. The presence of magnetic moments in these metals reflect an imbalance between up and down spins. If the 0.7 conductance feature is due to a spin energy gap, then ID quantum wires maybe ideal structures for use in spintronic applications. The work presented here takes this concept further, speculating that the degree of polarisation is a function of the density or gate bias. The ability to control the spin of electrons with a surface gate, opens a window to novel spintronic applications.

1.3 Micro-Magnetometers

This thesis also reports on the use of micro-magnetometers for measurement of quantum oscillations in microscopic samples. Primarily designed to measure magnetisation in association with the de Haas - van Alphen effect, the devices may also potentially find application in spintronic type circuits. Here the concern was fabrication and characterisation of the devices for use in micro-second pulsed magnetic fields. Chapter 8 is essentially self contained with all of the work relating to magnetometer devices reported there. Micro-magnetometers of various geometries were fabricated using both elec­ tron beam and optical lithography by myself and N. Lumpkin. These devices consist of an inductive coil arrangement, defined precisely using lithographic techniques to be close to perfectly compensated. In comparison to conventional hand-wound devices these micro-magnetometers have several advantages. Firstly 1. Introduction 9 because of the near perfect compensation, the magnetometers are expected to exhibit high sensitivity. Secondly, due to high packing density of lithographically defined coils, the devices have a large sample filling factor and couple strongly to the magnetisation flux. Measurements at the National High Magnetic Field Laboratory at Los Alamos NM, USA on heavy Fermion compounds provided a ideal testing ground for the sensitivity and practicality of these magnetometers. These results are reported in Chapter 8.

1.4 Thesis Outline

The Thesis is outlined as follows. Following this brief introductory Chapter, a more detailed discussion ensues. Chapter 2 provides general background theory relating to mesoscopic transport and electron correlations in one dimensional systems. Theoretical models proposed to explain the 0.7 conductance structure are briefly reviewed towards the end of the chapter. Chapter 3 reports on the experimental methods used in the fabrication of the quantum wire devices stud­ ied in this dissertation. Accompanying this discussion is a brief summary of the low noise electrical techniques used in the low temperature transport measure­ ments. Chapter 4 reports on the length, density and temperature dependence of the 0.7 x 2e2/h conductance feature. A total of six samples were characterised in these studies, with consistent results obtained across each of them. Chap­ ter 5 reports on the results obtained using a source - drain bias spectroscopy technique. The technique and results obtained are discussed, with emphasis on the implications for the 0.7 conductance structure. Chapter 5 also examines the effects of an asymmetric confinement potential and surface charge traps on the 0.7 structure. Chapter 6 presents the phenomenological model developed as part of this work to explain the 0.7 feature in terms of a density dependent spin polarisation. Some of the data from the earlier chapters is also included here to 1. Introduction 10 show consistency with the proposed model. Chapter 7 reports on data taken on three devices in which novel conductance perturbations were present. The data taken on a quantum point contact indicate the presence of a single magnetic impurity close to the ID region. These results are consistent with spin depen­ dent scattering, reminiscent of the Kondo effect seen in metals and recently in quantum dots. The second half of Chapter 7 reports measurements made on two novel devices, fabricated by O’Brien. These measurements explore the interest­ ing regime between OD and ID confinement. Evidence for the co-existence of the 0.7 conductance feature and Coulomb charging effects is seen, in addition to geometrical resonance phenomena. Chapter 8 is self contained and discusses the fabrication of micro-magnetometer devices. Data taken using these magnetome­ ters at Los Alamos is also presented. Finally Chapter 9 ends the dissertation by means of a conclusion. Future experiments are also briefly proposed. Chapter 2

Background Theory

2.0.1 Introduction

In 1988 investigators uncovered one of the most striking examples of quantum behavior in mesoscopic systems; quantisation of the ballistic conductance [1, 6]. This result is well explained by quantum transport theory, without the need to include the interactions between electrons [32, 33]. Despite the success of this non-interacting picture, several exceptions have been found that cannot be explained without appealing to electron correlation effects. This dissertation primarily investigates one such example: the anomalous conductance feature seen near 0.7 x 2e2/h in semiconductor quantum wires. The conductance fea­ ture can be seen in the earliest ID transport experiments and serves as a prime example for the need to include inter-particle interactions in a complete descrip­ tion. Despite strong evidence that the 0.7 feature is a result of electron-electron interactions, a complete microscopic understanding is still lacking. The main purpose of this chapter is to provide a formal context for the ex­ perimental work reported in the later chapters. Firstly, non-interacting quantum transport is treated with particular reference to ballistic transport in one dimen­ sion (ID). Following this, the general concepts of electron-electron interaction

11 2. Background Theory 12 theory are briefly discussed, since an appreciation of this vast topic is vitally im­ portant to the interpretation of results presented in chapters 4-7. In addition, the chapter also summarises some specific models proposed to explain the origin of the 0.7 x 2e2/h conductance feature.

2.0.2 Ballistic Electron Transport

With the development of modern nanofabrication techniques, electronic devices with dimensions comparable to the Fermi wavelength can be produced. Often, as in the case of this thesis, devices are designed and fabricated specifically for the study of quantum transport. These devices are usually based on material systems of unprecedented quality, fabricated using advanced techniques such as Molecular Beam Epitaxy (MBE) where precise control enables a structure to be ‘grown’ atomic layer by atomic layer. In parallel with these advances in mate­ rial quality, Electron Beam Lithography (EBL) has allowed feature sizes to be reduced far below what is possible using optical lithography. Limited only by the wavelength of the electron and the performance limits of the resist, EBL has fast become an essential tool in nano-electronics. Together these technological advances have opened a window into the physics of ballistic transport, where the electron mean free path (both elastic and inelastic) exceeds the dimensions of the conductor. It is in this regime that the wave-nature of the electron becomes prominent, resulting in behavior that strongly deviates from a classical descrip­ tion. Ultimately the fabrication of small, low-disorder conductors has allowed significant progress in our understanding of the meaning of electrical resistance at the quantum mechanical level. Although early pioneering experiments were performed using metallic con­ ductors [34], the relatively large Fermi wavelength in semiconductors (typically 30nm) has seen them used as the prime system for transport investigations. Ballistic electron transport in ID was first demonstrated in 1988 by two groups, 2. Background Theory 13

Side Gate Voltage, Vs (V)

Figure 2.1: The quantised conductance of a Quantum Point Contact. (Taken from Chapter 4.)

(Wharam et al. [6] Van Wees et al. ) [1] using high quality AlGaAs/GaAs het­ erostructures. These results provide a striking demonstration of how quantum transport departs from classical behavior when conduction occurs ballistically. Figure 2.1 shows the ballistic conductance of a quantum point contact, studied in this thesis. The conductance is quantised to units of 2e‘2/h ~ (12.9K fl)-1, and directly reflects the number of occupied quantum modes in the constriction.

2.1 Preliminary Concepts

The fundamental properties of electron transport are best studied in the ballistic regime, where the elastic and inelastic mean free paths exceed the dimensions of the conductor. In particular the observation of many-body interactions gen­ erally requires devices with reduced dimensionality and low-disorder, for several 2. Background Theory 14 reasons. Firstly, correlation effects are often difficult to detect and study in higher dimensional systems since they are generally manifested as weak correc­ tions to the non-interacting result (with some important exceptions). In fact, the successfulness of Landau’s formulation of Fermi liquid theory, in terms of quasi-particles is a direct consequence of this. Secondly, clean devices reduce spurious phenomena associated with disorder and facilitate the detection and isolation of correlation effects. In recent years the quest for samples of higher mobility and lower disorder has resulted in the development of novel device architectures and fabrication techniques. In the field of ID quantum wires, investigators have made use of various material systems [20, 16] and designs including ID systems based on high electron mobility transistors (HEMTs) [1, 6], Cleave Edge Over Growth wires [35], V-groove quantum wires [36], in plane gated transistors (IPGT)[15], enhancement mode FETs [37, 38, 39, 17] and other novel structures [40]. Disor­ der in the form of impurities and defects causes a randomisation of the potential which leads ultimately to momentum relaxation [41]. The mobility is a pa­ rameter that provides a direct measure of the disorder, since it relates the drift

—t t velocity (V(i) to the applied electric held (E):

(2.1)

[e] rm (2.2) m where m is the electron mass, e is the electron charge and rm is the momentum relaxation time.

2.1.1 The Two Dimensional Electron Gas

The work reported in this thesis is based on the gallium arsenide (GaAs)/aluminum gallium arsenide (AlGaAs) material system. Work on mesoscopic conductors 2. Background Theory 15 has been primarily concerned with this system due to the very high quality two-dimensional conducting channel formed at the interface between GaAs and AlGaAs. The small lattice mis-match between the two materials facilitates epi­ taxial growth, but leads to a difference in band gap. When the two materials are brought together (grown in a MBE chamber) this difference in the band gap between the two materials causes the bands to bend and forms a potential-well at the interface. In the case of doped samples, positively charged donors in the AlGaAs give rise to an electrostatic potential that attracts free electrons in the GaAs into the . Alternatively, for the case of the induced structures considered in this work, an external potential is applied to a gate electrode above the heterojunction to provide the electrostatic potential (See figure 2.2). In either case an extremely thin conducting layer of electrons is formed at the interface: the two Dimensional Electron Gas (2DEG). The system is two-dimensional (2D) in the sense that the electrons (or holes) do not have a continuous spectrum of states but are quantised to eigenstates associated with some potential U(z) in the z-direction, (the direction of growth). The carriers are free to propagate in the x-y plane with a wave-function of the following form:

'I'M = cj)n(z)exp(ikxx)exp(ikyy) (2.3)

Where the index n refers to the different sub-bands, each having a different wave-function (f)Tl(z). In the work considered here, only the lowest sub-band with n = 1 is occupied. Consequently the z-dimension can be ignored and the system can be considered exactly two-dimensional. In addition, the effect of the lattice potential can be adequately incorporated into the concept of an effective mass. The single-band effective mass Schrodinger equation may then be solved to yield the eigenfunctions (normalised to an area A) of form:

®(x,y) = —=zexp(ikxx)exp(ikyy) (2.4) v A with Eigenvalues given by the dispersion relation: 2. Background Theory 16

Fermi Level (2DEG)

n+ GaAs GaAs AlGaAs GaAs Gate Spacer

Figure 2.2: A schematic of the bottom of the conduction band as it varies through the heterojunction of an enhancement mode FET.

E En + 2^* {kx + ky ) (2-5)

Where En is the eigenvalue of the associated standing wave in the confining direction (z) and m* is the effective mass (typically 0.067me in GaAs/AlGaAs).

2.1.2 The Density of States and the Fermi-Dirac Distri­ bution

The electron density N(E)dE between energies E and E + dE is the product of two independent quantities, the density of states D(E)dE between E and E + dF, and the probability of occupation of a quantum state:

N(E)dE = P(E)D(E)dE (2.6) 2. Background Theory 17

In the low temperature limit, the Fermi energy corresponds to the energy of the top-most filled level. In this case P(E) = 1 for states below the Fermi level and N(E) is simply equal to the density of states. In ID the Density of States is:

(2.7)

The total number of electrons is given by:

(2.8)

The variation with E~1//2 being characteristic of a one-dimensional system. The factor of 2 comes from the spin degeneracy of the states in zero magnetic held. In 2D the density of states is a constant, independent of E. The total number of electrons is given by: (2.9)

Where A refers to the total area and spin degeneracy is again included. At non-zero temperature, some excitation to higher, unoccupied states takes place. Only electrons close to the Fermi surface however, have available empty states within an energy kBT. The effect of the thermal energy can be to blur the Fermi level, and this is of vital importance when considering models explain­ ing the 0.7 conductance feature. The probability of a state being occupied is described by the Fermi-Dirac distribution:

P(E) = ([exp(E - EF)/kBT) + l)-1 (2.10)

2.1.3 Electrons Confined to One Dimension

The 2DEG at the GaAs/AlGaAs heterojunction is strongly confined to the in­ terface. If only the first 2D sub-band (\(z)) is populated the system can be treated exactly two dimensional. The dimensionality can be lowered further by lateral confinement of the 2DEG to form a narrow channel of electrons: the one 2. Background Theory 18

Surface Gates f

2DEG

2DEG

1DEG

Figure 2.3: Schematic diagram depicting the formation of a quasi-one dimen­ sional region, connected to 2D reservoirs. dimensional electron gas (1DEG). This lateral confinement is typically achieved electrostatically by applying a bias to surface gates and selectively depleting the 2DEG in spatially separated regions (see Figure 2.3). Other novel techniques have included selective removal of the 2DEG via wet and reactive-ion etching, to form a narrow conducting channel of electrons [42]. The devices studied here combine both electrostatic and lithographic confinement to produce a 1DEG where the potential function V(x,y) and the ID density can be separately var­ ied. One-dimensional electron systems are ideal for studying quantum transport, as discussed above. Although the systems considered here confine the electrons to ID using electrostatic techniques, it is important to mention that ID electrons have been studied in a number of other systems. For instance, a ID electron system is formed at the edge of a 2D electron gas subject to a strong magnetic held [43]. Other examples include carbon nano-tubes [27], polymer and molec­ ular systems [44]. The quantum wire devices studied here consist of a narrow channel in a 2DEG, produced by electrostatic and lithographic confinement. The confinement potential V(x,y) is generally agreed to be a saddle point [32]. If the mean-free path of the electrons exceeds the channel length and the width approaches the Fermi wavelength (Af), then the electron wave-function can be separated into its components in an analogous way to the 2D case considered 2. Background Theory 19 above: y(x,y,t) = [n(y)exp(iEnt/h)][exp{ikx)][exp(iEkt/h)]. (2.11)

n(y) represents a standing wave in the confinement or transverse direction (y) with eigenvalue En. The other two terms represent a traveling wave in the x direction. The total energy includes the standing mode eigenvalue (En) in addition to the dispersion relation for the transport (x) direction:

h2k2 E = En + -o4 (2.12) 2m*

The dispersion relation is shown in figure 2.4. Many of the principal phenomena in ballistic transport are exhibited in the cleanest and most extreme way by ID quantum wires. The lateral constriction of the 2DEG produces a conducting channel that acts as an electron waveguide. Each mode of this waveguide contributes a conductance of 2e2//i, with the factor of 2 due to spin degeneracy. This remarkable result arises naturally by consid-

% ering that quantum transport is a transmission problem and that in ID, the density of states {pm) directly cancels the group velocity {V). The relationship between conductance and quantum mechanical transmission probabilities is the subject of the next section.

2.2 Landauer-Biittiker Formalism

Ballistic electron transport was shown by Landauer [45] to be a transmission problem, in the quantum mechanical sense (also see also references [46, 47]). The current through a conductor is expressed in terms of the probability that an electron can transmit through it. Although ID systems display quantised conductance due to the cancellation of the group velocity with the ID density of states, the notion of a contact resistance is fundamental to ballistic transport in any dimension. Where does the contact resistance come from, if transport is 2. Background Theory 20

Figure 2.4: A schematic depicting the dispersion relation E(k) for electrons occupying distinct ID sub-bands. ballistic with no scattering? The origin of a contact resistance can be traced to the redistribution of the current as it makes its way from a near infinite number of modes in the contact reservoirs, into the few modes inside the quantum wire. DePiccotto et al [48] have studied this redistribution of current by using non- evasive voltage contacts to probe the chemical potential inside a quantum wire. The results show that indeed a ballistic conductor does have zero resistance, if the potential ‘behind’ the contact region can be probed, consistent with the notion of a contact resistance, as proposed by Irnry [46]. Each transverse mode carries a finite amount of current per unit energy. The total current is easily calculated by finding the current per mode, and counting the number of modes occupied below the Fermi level. For a uniform electron gas with n electrons per unit length, moving with a velocity u, the current is defined as J = enV (2.13)

Since the electron density of a single A;-state in a conductor of length L is just (1/L), the dispersion relation yields the group velocity and the current can be 2. Background Theory 21 written as: (2.14) k k With the inclusion of a function f{E) to account for how the k states are occupied, and converting to an integral (assuming periodic boundary conditions for k) the current is: I=jJf(E)dE (2.15) with a factor of 2 coming the from spin degeneracy. The total current is now just equation 2.15 multiplied by the number of occupied transverse modes or sub-bands. This is a general result independent of the dispersion relation and the dimensionality: the current carried per mode per unit energy by an occupied state is equal to 2e/h (approximately 80nA/meV). The contact resistance can now be written in terms of the current:

2 / = 2e/h(n\ — ^2) ~^ — 2—[ ~ ]-/V0 (2.16)

G = NQ2e2/h -> (2.17)

Where No is the number of occupied transverse; modes or sub-bands and n\ and //2 are the chemical potentials either side of the ID quantum wire. The above treatment assumes an ideal electron waveguide, however it is straight forward to also consider the non-ideal case. Incorporation of a transmis­ sion probability accounts for transmission and reflection of electron waves into (and out of) the conductor. Further, this quantum mechanical description [5] facilitates calculation of the associated expectation values, via solution of the Schroedinger equation for a given confining potential. The Landauer formula is:

e2 G = 2—N0T (2.18) /?.

The factor T represents the average probability that an electron injected at one end of the conductor will transmit to the other end. If the transmission 2. Background Theory 22 coefficient is unity, then the conductor is an ideal waveguide and the conductance is quantised in units of the contact resistance, (2e2/h). Biittiker [49, 32] extended Landauer’s general result to include multi-terminal devices. This extended treatment makes strong contact with real devices and has facilitated interpretation of experimental results since its inception. If transport is coherent across the conductor, then the current per unit energy is given by

oP __ HE) = - £ T„mfp(E) - f,(E)] (2.19) <7

Tpq(E) is the total transmission from terminal p to terminal q at the energy E and fp{E) is the Fermi function for terminal p (like wise for fq(E)).

2.2.1 Quantised transmission of a saddle-point constric­

tion

Surface gates electrostatically define a potential \ (x,y) via selective depletion of the 2DEG. The form of this potential is generally agreed to be a saddle-point:

V(x, y) = V0 - irrfulx2 + ~rn*u2y y2 (2.20)

Where Vo is the electrostatic potential and ux,uy describe the curvature of the potential. Calculation of the the conductance via the Landauer transmission formula (2.18) for this saddle potential is due to Biittiker [32]. Quantum me­ chanically, transmission and reflection at the saddle allows for sub-bands that are neither completely open nor completely closed, but are allowed to transmit with probability T. The transmission is then

T = —---- 7------r (2.21) 1 + exp{— 7T£n)

With en equal to

en = 2[E - huy(n + i) - Vo]/hujx (2.22) 2. Background Theory 23

(E'V0)/hwx

Figure 2.5: The calculated transmission for a saddle potential, dashed line is for = 2 and solid line is for ux/ujy = 3.

Only transmission probabilities for which the incident channel and the out-going channel are the same are nonzero. Equation 2.21 predicts that when the Fermi level Ep is exactly aligned with a ID sub-band edge, then the transmission co­ efficient is 0.5, and the resulting conductance 0.5 x 2e2/h. Figure 2.5 shows the calculated total transmission probability (conductance) as a function of Fermi level, for two different confining potentials (ratios of ujx/ujy.)

2.3 Interacting Electrons in ID

Although many of the principles of mesoscopic physics (and quantum mechanics in general) can be adequately illustrated by considering systems that consist of only one particle subject to external forces, there are a number of important phenomena that require the inclusion of two or more particles. Electrons inter­ act via their charge and spin in a complicated way. An accurate description of 2. Background Theory 24 such interactions requires that the inter-particle potentials be included in the many-particle Schroedinger equation. However, as this approach is often im­ practical, it is therefore generally necessary to resort to other techniques and approximations. Theories addressing electron interactions naturally make use of such techniques, and an abundance of literature exists on this subject. The reader is referred to reviews by Al’tshuler and Aronov [50], Fukuyama [51] and by Lee and Ramakrishnan [52] for a comprehensive treatment of electron-electron interactions in mesoscopic physics. As discussed in the introduction to this Chapter, electron-electron interac­ tions are predicted to form the basis for a microscopic description of the 0.7 conductance anomaly. In fact, various recent explanations for the 0.7 feature have made good use of standard techniques used in the treatment of many-body interactions. These include the Hartree-Fock approximation [53, 54], Random Phase Approximation (RPA) [55] and ultimately (Spin) Density Functional The­ ory [56, 57]. Although a complete review of such methods is beyond the scope of this dissertation, attempts have been made to briefly characterise the most elementary forms of electronic interaction and to a lesser extent the techniques used in their treatment. Importantly, the subject is only treated with reference to proposed explanations for the 0.7 conductance feature.

2.3.1 Correlation and the Exchange Interaction

The 0.7 conductance feature observed in ID mesoscopic transport experiments may be due to magnetic ordering in the region of the ID constriction. In par- ticnlar the notion of a spontaneous spin polarisation in zero magnetic held has been suggested initially by Thomas et al [2] and expanded upon by Wang and Berggren [58] and also by Gold and Calmels [53]. In general, magnetic ordering is a consequence of the exchange interaction between electrons. Although ex­ change can be difficult to treat without approximation techniques, the essence 2. Background Theory 25 of the interaction is easily communicated. To begin, note that the interaction is a consequence of both the electrostatic Coulomb interaction between the electron charges and the fact that electrons are identical and obey the Pauli exclusion principle (ie the overall wave-function must be antisymmetric). As an example, consider the hydrogen molecule. The Schroedinger equation for two non-interacting electrons moving in similar potentials is:

h2 (Vi + Vi) + V(qi) + V(q2) 'ijj = Elf) (2.23) 2m’

where q\ and q2 are generalised coordinates of electrons 1 and 2. Possible solutions are ^(1)^(2) or 'ipb(l)ipa(2) with

E = Ea + Eb. (2.24)

The term t/ja( 1) is the single electron wave-function when electron 1 is in state a and it is a solution of the one electron Schroedinger equation; ^(2) is the solution for electron 2 in state b. Similarly, ^(l) is the wave-function for electron 1 in state b and 'ipa(2) for electron 2 in state a. As the electrons are indistinguishable, the total wave-function ^(1,2) must be such that

\'ip(l,2)\2dqldq2 = |^(2,1)| 2dqldq2 (2.25) and thus either V>( 1,2) = +0(2,1) (2.26) or V>(1,2) = -V>(2,1). (2.27)

The first total wave-function is symmetric, the second is antisymmetric. Linear combinations of the single particle wave-functions satisfying these requirements are ipsym(11 2) = \M 1)^(2) + „(2)] /2'/2 (2.28) 2. Background Theory 26

1,2) = [V>a(l)^(2) -*(l)^a(2)]/21/2 (2.29)

The total eigenfunctions depend on both the space variables and the spin vari­ ables of the two electrons. Separating the variables the total eigenfunction can be written as: (total eigenfunction) = (space eigenfunction) x (spin eigenfunction) or

0 = (r)x (2-30) where (r) is the solution for an electron without spin and x ls the spin com­ ponent. Since the overall wave-function must be antisymmetric, the total wave- function is either ■0(1,2) = sym(T 2)Xonfi(l, 2) (2.31) or 0(l,2) = 0anti(l,2)XsJ/m(l,2) (2.32)

Returning now to the Hydrogen molecule, the Hamiltonian for the non­ interacting electron case is

H = e2 [(l/rab)(l/r12) - (l/ra2) - (l/rw)] (2.33) where rab is the distance between nuclei, r\2 is the distance between electrons and ra2 and rb\ are the distances between electrons and nuclei of the other atom. The energy is obviously E = J ip'HipdT (2.34) and the additional energy due to the electron interactions can be found using perturbation theory. The additional energy for the singlet state (S = 0) is

E\ = A2(Cl2 + Ju) (2-35) and for the triplet state (5=1)

E2 = B2(Cn - Jn) (2.36) 2. Background Theory 27

Where A and B are normalising factors and

If 0*(l)<^(2)//12

0o(l)'W(2)tfi2

The term J12 is the exchange integral. For the example of the Hydrogen molecule, J is negative and hence the ground state is anti-parallel. Importantly, J is proportional to the amount of wave-function overlap and thus is a function of density. The integral C is referred to as the ‘Coulomb energy’ because it is equivalent to that of the classical electrostatic interaction between two contin­ uous charge distributions. Refer to section 2.6.2 for further discussion of the coulomb energy and correlation.

2.3.2 The Luttinger Liquid

The finite resistance of a perfect non-interacting quantum wire is entirely due to the contact resistance (2e2/h) and non-interacting Fermi liquid theory has been the starting point for most mesoscopic electron transport theories. In ID however, there have been a number of theories that suggest under certain con­ ditions a Fermi liquid will destabilise as a result of Coulomb interactions, and form a new correlated fluid known as a Luttinger liquid [59, 60, 61]. This novel correlated state is predicted to give rise to a renormalisation of the conductance

(2.39)

Where the parameter K characterises the size and the strength of the inter­ actions: K < 1 for repulsion; K > 1 for attraction; K = 1 in the absence of interactions. To see why this is possible, recall the reasoning for the quantisation of conductance in ID: the exact cancellation of the Fermi velocity with the ID density of states. For interacting electrons Kane and Fisher [62] showed that 2. Background Theory 28 both the density of states and the velocity are renormalized, and this cancel­ lation need no longer precisely occur. Experiments to observe Luttinger liquid behavior are difficult due to the fact that any residual disorder or non-ideality greatly obscures the predicted power-law characteristics for conductance as a function of temperature and length. Thus far there has been no clear exper­ imental evidence reported that supports the existence of the Luttinger Liquid state in ID semiconductor systems of the kind studied in this dissertation (at least in an unambiguously accepted form). Further uncertainty arises when con­ sidering the role of the Fermi liquid reservoirs in real devices. There exists a number of theoretical arguments [63, 64, 65, 66] intended to explain the ob­ served non-renormalisation of the conductance in a ID electron gas, however agreement has still not been realised. The excitations of an interacting ID system are bosonic in nature. Luttinger liquids are without a Fermi surface, even at zero temperature and lack single particle excitations at low energies. Collective excitations are separable in terms of spin and charge, giving rise to properties with no analogue in Fermi liquid descriptions. This remarkable phenomenon of spin-charge separation has in fact been postulated by many to underly superconductivity in the cuprate materials. Interestingly, connections between Luttinger Liquid type interactions and the anomalous conductance feature near 0.7 were proposed initially in some form by Thomas et al [2]. Balents and Eger [67] have also discussed spin polarised transport and effects associated with spin polarisation as providing a means of probing spin-charge separation. However, despite these sentiments no strong connection between the behavior expected for a Luttinger liquid state and the 0.7 conductance feature has been established. 2. Background Theory 29

2.4 The Spin Polarised Electron Gas

Iii low dimensional systems, electron-electron interactions can play an increas­ ingly important role. The ground state of the two-dimensional electron gas has been studied extensively both theoretically and experimentally using a va­ riety of techniques. Early Monte Carlo calculations [68] revealed three distinct phases of the 2DEG: an unpolarised fluid (rs < 13), a fully spin polarised fluid (13 < rs < 33) and Wigner crystallisation {r8 > 33) 1 Recent sophisticated cal­ culations affirm the notion of a spin polarised ground state in the low density 2DEG [69]. In ID interactions become especially crucial as the density becomes increasingly lower. A quantum wire, with only a few sub-bands occupied may contain oidy 2-40 electrons. At these densities exchange interactions may be particularly strong. Wang and Berggren [58, 70] have used the density functional theory of Kohn and Sham [71] to model both infinite single mode quantum wires, and devices with more realistic geometries. Their results detail the possibility of a spontaneous spin polarisation occurring in the region of the saddle point, as a result of strong exchange coupling between electrons. For low sub-band fillings exchange is found to dominate over the kinetic energy, leading to a fully spin polarised ground state.

2.5 The Origin of the 0.7 Feature: Theory and Models

Electron-electron interactions are thought to give rise the anomalous conduc­ tance feature seen near 0.7 x 2e2/h in quantum wires. The following section re­ views some of the relevant theoretical work essential to an understanding of the 1 The dimensionless interaction strength rs is the inter-particle spacing in units of the effec­ tive Bohr radius, or, equivalently, the ratio of the typical interaction energy between electrons to the Fermi energy (PE/KE). 2. Background Theory 30

0.7 conductance feature. Self-Consistent approximation techniques are briefly reviewed, with an emphasis on how these methods have been used in explaining aspects of the conductance feature. A number of scenarios including Wigner crys­ tallisation, electron bound-states and spin-orbit coupling are considered in some detail before some recent phenomenological descriptions are reviewed. Firstly though, the rigorous theorem of Leib and Mattis must take precedence in the framework of understanding.

2.5.1 The Lieb-Mattis Theorem

The Lieb-Mattis theorem [72] states that there can be no ferromagnetism unless one postulates explicitly spin- or velocity-dependent forces. The ground state must always have 5 = 0, with the only proviso being that no spin or velocity dependent forces are present. This rigorous enunciation is not limited to one- dimensional cases but applies to interacting fermions in all dimensions, so long as the potential is not pathological and remains separately symmetric. The authors show that under no circumstances can a ID electron system be ferromagnetic with only space-dependent forces. The question that now requires immediate attention is whether the notion of ferromagnetic ordering in quantum wire devices is in violation of the Lieb- Mattis result. This question remains somewhat open, however attempts have been made to address the issue. Although the Lieb-Mattis theorem remains a stern warning that the criterion for ferromagnetism must be rather detailed, it may not apply under the following conditions. Firstly, coupling of large 2D reservoirs to the ends of a quasi-ID constriction may constitute a pathological potential. Further still, the ID region is finite in width and length with often more than a single channel occupied. Secondly, perhaps the 0.7 feature arises from an excited system. In this case the ground state could remain unpolarised without any violation of the 2. Background Theory 31

Lieb-Mattis theorem. Other considerations may apply in addition to these ar­ guments. The formation of transient or partially bound 5=1 states is one possibility. While such ‘states’ remain forbidden by the Lieb-Mattis theorem, the uncertainty principle permits an apparent violation of energy conservation for a short time. Clearly any theoretical argument that is based on these ideas must necessarily also include the rigorous detail required to validate such claims. One final possibility consistent with the Leib-Mattis theorem is that the 0.7 fea­ ture is due to a polarisation in the 2D reservoirs and not the ID quantum wire. The probability for transmission would then depend on how the electron spin was aligned with the polarisation in the reservoirs. This somewhat glib proposal may have some merit and is considered in more detail at the end of Chapter 4.

2.5.2 Self-Consistent Calculations

The wave function of a many-electron system must be antisymmetric under ex­ change of any two electrons because the electrons are fermions. This antisymme­ try produces a spatial separation between electrons that have the same spin and thus reduces the Coulomb energy of the electron system. As discussed in section 2.4.1., the reduction in the energy of the electronic system due to the antisym­ metry of the wave-function is called the exchange energy. It is straightforward to include the exchange in a total energy calculation using the Hartree-Fock approximation. The Coulomb energy of the system is below its Hartree-Fock value, if electrons that have opposite spins are also spatially separated. In this case the Coulomb energy is reduced at the cost of increasing the kinetic energy. The difference between the many-body energy and the energy calculated in the Hartree-Fock approximation is called the correlation energy [73]. It is extremely difficult to calculate the correlation energy of a complex system and alternative methods are required to describe the effects of electron-electron interaction. Density-functional theory [74, 71] allows one, in principle, to map exactly 2. Background Theory 32 the problem of a strongly interacting electron gas onto that of a single particle moving in an effective local potential. Although this potential is not known precisely, local approximations to it work remarkably well. At present there are no a priori arguments to explain why these approximations work. Nevertheless density-functional theory has been shown countless times to reproduce a variety of ground state properties to within a few percent of experiments. Hohenberg and Kohn [74] proved that the total energy, (including exchange and correlation), of an electron gas (even in the presence of a static potential) is a unique functional of the electron density. The minimum value of the total- energy functional is the ground-state energy of the system, and the density that yields this minimum value is the exact single-particle ground-state density. Kohn and Sham [71] showed how it is possible, formally, to replace the many-electron problem by an exactly equivalent set of self-consistent one-electron equations. Wang and Berggren [58] present a self-consistent theoretical calculation of the electronic structure of a quasi-one-dimensional, infinite, straight electron channel. In the presence of a weak magnetic field, spontaneous spin polarization is found at low electron densities. Of particular interest to the work presented in this dissertation, their calculations show that spin polarisation is also pre­ dicted when the Fermi level EF evolves through sub-band threshold energies with increasing density. This work is elaborated upon in a second paper [70]. Here the author’s purpose is to demonstrate that spin polarisation may indeed also occur in devices with more realistic geometries, (i.e. not an infinite channel) with B —» 0. Again methods based on the Kohn-Sham equations [71] are used with self-consistency being reached after the Fermi energies in successive interactions are identical to within a given numerical accuracy (« 10~AmeV). The effective Schroedinger equation is taken as

rPl + Py + Vconf(x,y) + V°xch(x,y) = Ea(f)(T{x,y), (2.40) 2 m* 2. Background Theory 33 where a = ±1/2 refers to spin and Vconf is the usual saddle point confinement potential defined in section 2.3.1. In the Kohn-Sham local-density approximation (LDA), the exchange potential is [75]:

v?xch(x,y) =----- (2.41.) where e is the dielectric constant and na(x, y) is the spatial distribution for cr-spin electrons. The results of these calculations suggest that due to the exchange coupling between electrons in the saddle region, the effective potential barriers for spin up and spin down electrons is different. This difference in barrier potential is highly dependent on the electron density and the effective electrostatic potential Vo defined by surface gates. One scenario is that the spontaneous polarisation leads to the barrier for one spin direction increasing above the Fermi level. Conduction occurs only through tunneling for one spin species which leads to a feature in the conductance between 0.5 x 2e2/h and 1.0 x 2e2/h. How such a picture might apply to the experimental results is discussed in Chapters 4-6. Further self consistent calculations have also been undertaken by Gold and Calmels [53, 76] for the case of an infinite cylindrical wire and Zabola for 3D metal nanowires [77]. Gold and Calmels have addressed the possibility of fer­ romagnetism in ID using a variety of techniques including the random-phase approximation (RPA), taking into account the local-field correction [55]. In analogy with three-dimensional spin polarised electron gases, their calculations have hinted at the occurrence of a Bloch instability in a quasi-ID system with cylindrical confinement [76]. Other considerations have focused on the form of electron-electron interac­ tions in a context of screening and effects associated with image charges. The possibility of a paramagnetic-ferromagnetic phase transition in the ground state of a quantum wire is discussed by Byczuk and Dietl [78] using these techniques. Recently, very interesting results were published by Hirose, Li and Wingreen 2. Background Theory 34

[79, 57]. Spin density functional theory is again employed in conductance calcula­ tions. The authors suggest that due to electron-electron interactions, additional features in the conductance, near (n + 1/2)2e1 /h may occur due to spin po­ larised electrons or band-edge pinning effects. The singularity in the ID density of states leads to pinning of the Fermi energy near the band edge. Further dis­ cussion about this important topic is deferred to Chapter 6, where it is treated in some detail.

2.5.3 Wigner Crystallisation

The phenomena of Wigner crystallisation is well known in 2 and 3 dimensions. However little is known about the one dimensional case. The possibility of crystallisation in ID has been considered in the context of transport by Glazman, Ruzin and Shklovskii [80]. Of further interest, Wigner crystallisation and waves (CDWs) have been proposed as mechanisms behind the observed 0.7 conductance anomaly by Sushkov [54, 81] and also Spivak and Zhou [82]. In the later work the authors consider the melting of the Wiger crystal, (a second order Lifshitz phase transition) as the origin of the conductance feature. Such phenomena was previously studied in the context of quantum melting of bulk Helium crystals [83]. In the case of semiconductor systems it is proposed that the crystal is pinned by interstitial electrons or vacancies in the ID Wigner crystal. Conductance occurs via hopping events since the crystal is frozen. Melting of the Wigner crystal leads to the development of a conductance feature seen near 0.7 x 2e2/h. The model predicts negative magneto-resistance in the pinch-off region, where G « 2e2//?,, due to the propagation of a polaron with increasing magnetic field. The prediction was tested experimentally as part of the work reported in this thesis. No negative magneto-resistance was detected, although only one sample was examined. The pinning of the Wigner crystal is predicted to give rise to conductance 2. Background Theory 35 anomalies. Sushkov [54] has calculated the transmission expected for a pinned Wigner crystal using the Hartree-Fock method. By considering the effect of an impurity on the current flowing in a ID ring with electron-electron interactions, the model predicts the formation of charge density waves (CDWs) that extend shallowly into the 2D reservoirs. The author argues that these CDWs are a precursor to Wigner cryallisation. A second paper calculates the transmission when a parallel magnetic field is applied [81]. A difference in spin population is assumed and the Zeeman energy E = ggBS is considered. The model pre­ dicts conductance anomalies occurring at values below the conductance quantum (2e2/h) as a function of ID channel length and width.

2.5.4 Electron Bound States

The formation of singlet and triplet bound states in a ratio 1:3 is proposed as an explanation for the 0.7 feature by Flambaum and Kuchiev [84] and also, in connection with blockade type resonances by Rejec, Ramsak and Jefferson [85]. Flambaum and Kuchiev rely on the formation of two-electron bound states, perhaps via some phonon exchange mechanism reminiscent of superconductivity. The attraction between electrons near the saddle point potential leads to the formation of triplet states in a ratio 3:1, with singlet bound states. The value 0.75 x 2e2/h follows from the singlet-triplet statistical weight ratio. The crux of the argument is that triplet bound-states with S = 1 are the ground state of the system, however this seems in direct contradiction to the exact theory of Lieb and Mattis (section 2.6.1). The binding of the triplet state is then supposed to provide enough energy to overcome the barrier (fiujx), while the singlet binding is insufficient. Further theories of this type are presented by Rejec et al. [85]. The authors consider the case of an open quantum dot, perhaps of the type considered in Chapter 7. The open dot is like a straight quantum wire with a bulge near the 2. Background Theory 36 center. This bulge can catch an electron and trap it there. The interactions be­ tween current carrying electrons transversing the wire and the trapped or bound electron give rise to Coulomb blockade type resonances that are spin depen­ dent. The authors propose that the bound electron may form singlet and triplet states with transport electrons and that this process could produce conductance anomalies. A feature near 0.25 x 2e1 /h is predicted for singlet resonances with one near 0.75 x 2e2//i, related to a triplet resonance. Substitution of the trapped electron for a magnetic impurity leads to Kondo type interactions. Such a sce­ nario is treated in more detail in Chapter 7, where data reminiscent of Kondo type resonances is presented.

2.5.5 Spin-Orbit Interactions

An electron in an atom possesses two kinds of angular momenta: orbital angu­ lar momentum associated with its motion, and intrinsic angular momentum or spin. The coupling of these two quantities (known as the spin-orbit interaction) can produce spin splitting, due to the difference in energy between spins aligned and anti-aligned with the orbital angular momentum. Photoconductivity mea­ surements on GaAs-AlGaAs 2DEGs by Stein, von-Klitzing and Weimann [86] have shown that the interface potential due to the dopants (or surface gates) is without inversion symmetry. Analogous to spin-orbit splitting, an electric field transformed into the reference frame of a moving electron acts as a mag­ netic field which lifts the spin degeneracy. The estimated splitting, determined from linear extrapolation of electron spin-resonance energies, is proportional to the Fermi wave vector and the effective electric field at the interface channel. For a density n = 4.6 x 1015m~2 the splitting is 32/ieK. Lomrner et al [87] have studied the apparent spin-splitting at B = 0 and ascribe its origin to the band-structure (nonlinear k3 term) in bulk material and spin-orbit coupling at the interface (Rashba term). Other evidence of zero field spin-splitting can be 2. Background Theory 37 found in spin relaxation [88], spin precession [89], Raman scattering [90] and magneto-conductance measurements [91]. Schmeltzer et al. [92] suggests that the lack of inversion symmetry and the presence of an interface electric field in GaAs/AlGaAs heterostructures, induces zero field spin splitting as discussed above. The splitting leads to the lifting of the spin degeneracy and creates spin-polarised sub-bands in a zero magnetic field. The conductance feature at 0.7 x 2e2/h results from a hybridisation between spin up and spin down electrons in the ID region. Despite these predictions, experiments (see chapters 4,5 and 6) suggest that the observed effective spin­ splitting is at the very least an order of magnitude larger than that predicted on the basis of spin-orbit or interface electric field effects. Nevertheless, these type of interactions might provide an initial driving field or seed, that when coupled to strong exchange forces can lead to larger spin-splitting in one-dimensional systems.

2.5.6 Phenomenological Descriptions

Despite the premise that a spin polarisation can occur in ID semiconductor quantum point contacts, many questions still remain. In particular, the reason­ ing behind the exact position of the feature near 0.7 x 2e2/h, the temperature, magnetic field and DC bias dependence remains largely unexplained. Several at­ tempts have been made to explain this behavior, including the model presented in this thesis, (see Chapter 6). Bruus, Cheianov and Flensberg [93] presented a simple phenomenological model interpreting the experimental observations on the 0.7 conductance anomaly. Their model assumes the presence of a spin polarisation in the ID region. The Leib-Mattis exact theorem [72] is addressed by emphasizing some of the points made in section 2.6.1. The essence of the model is based on a quadratic de­ pendence on the parameter A(/i), which measures the difference between the 2. Background Theory 38 chemical potential and the bottom of the uppermost spin-split sub-band. The parameter A(/z) remains linear below the band-edge, but becomes parabolic after the potential passes through the band-edge bottom, due to interaction induced pinning. The feature near 0.75 x 2e2/h arises due to thermal de-population of the upper spin-split sub-band, reducing it’s contribution to the conductance by 0.5. Conductance calculations based on this model show remarkable agreement with the experimental data, including the temperature and magnetic field de­ pendence. Some inconsistencies remain however, particularly in relation to the DC source-drain bias spectroscopy, the details of which are discussed in chapter

6. Bruus and Flensberg have also considered [94] the presence of localized plas- mons in the region of the point contact. Other ideas by Lindelof [95] have included the formation of an Isomer state and Kondo like resonances resulting from interaction with this Isomer. Lindelof suggests that the 0.7 conductance feature is caused by thermal population between the isomer state, which is an isolated (5 = 1) bound state in the middle of the constriction, and the usual (5 = 0) ground state. Chapter 3

Experimental Methods

3.1 Introduction

All of the experimental work described in this dissertation involved the use of low- noise techniques to measure small currents and voltages in nanoscale devices at low temperatures. The following Chapter reviews the experimental methods used to fabricate the quantum wire samples studied in this dissertation and discusses the measurement techniques employed in this work, together with a brief review of the primary apparatus. The experimental details relating to magnotometer samples are discussed in the relevant sections of Chapter 8, which is essentially self contained.

3.2 Sample Fabrication

In modulation-doped single interface heterostructures, [96] carriers are confined at the GaAs/AlxGai_xAs interface by an electric field generated by dopants lo­ cated in the AlxGa!_xAs. While modulation doping has lead to the fabrication of material of unprecedented quality in the last decade, [97] the technique is not without its deficiencies: disorder is inevitably introduced to the material by the

39 3. Experimental Methods 40

Figure 3.1: Photographs showing processing facilities within the class 350 (a) and class 3.5 (b) clean-rooms at UNSW’s Semiconductor Nanofabrication Facil­ ity (SNF). presence of the remote ionized dopants [37]. This disorder can be an important factor limiting the mobility in high quality modulation-doped samples [98]. Ad­ ditionally, both the carrier type and density in modulation-doped structures is fixed when the material is grown. An alternative to modulation doping is to confine carriers to the GaAs/Al xGai_xAs interface with an externally applied electric held [37, 38, 39, 17]. Using this technique both two-dimensional (2D) electron and hole gases may be pro­ duced with adjustable densities limited only by tunneling across the AlxGai_xAs barrier. The mobility of such devices has exceeded 6 x 106 cm2V-1s-1 at 4.2K, and increases further at lower temperatures (recent measurements indicate mo­ bilities in excess of 2.4 x 10' cm2V_1s_1 at milli-Kelvin temperatures). Further, because the random impurity potential associated with doping is avoided the electron mean free path typically exceeds 160/im [99] which is greater than the sample dimensions. These high mobility GaAs heterostructures form the ba­ sis of the quantum wires studied here and were grown using Molecular Beam Epitaxy (MBE) by Loren Pfeiffer and Ken West of Lucent Technologies. The 3. Experimental Methods 41 material was then subsequently processed using a combination of both electron beam and optical lithography to fabricate the mesa structure, ohmic contacts and nanoscale elements. Each of these fabrication processing steps are briefly reviewed in the following sections.

3.2.1 Molecular Beam Epitaxy

Molecular beam epitaxy (MBE) was developed in the early 1970s as a means of growing high-purity epitaxial layers of compound semiconductors [100, 101]. Since that time it has evolved into a popular technique for growing III-V com­ pound semiconductors as well as several other materials. MBE can produce high-quality layers with very abrupt interfaces and provides good control over the thickness, doping, and composition. Figure 3.2 shows the layer structure used in this work. Although a number of other heterostructures were studied in relation to this work, the data presented in this Thesis was taken on samples fabricated from a single wafer. This was done ill order to facilitate comparison between samples of different geometrical design, without having to account for additional effects associated with the heterostructure parameters. The layer structure consists of firstly a super-lattice of thin GaAs and AlGaAs layers deposited onto a GaAs (100) substrate. The supper-lattice minimises prop­ agation of growth defects from both the substrate and epitaxial growth boundary, and captures impurities such as carbon [102] which may otherwise contaminate the GaAs/AlGaAs heterojunction. Secondly a thick GaAs layer followed by Al­ GaAs forms the confining potential well. Finally a GaAs spacer separates the conducting (heavily doped % 3 x 1018cm-3) gate from the AlGaAs. The spacer separates any local disorder associated with the gate from the heterojunction. The heavy carrier concentration in the gate ensures that the gate behaves as a metal with the application of a voltage bias. 3. Experimental Methods 42

350A GaAs (delta - dopped) 250 A GaAs 750 A AlGaAs

5000A GaAs

1800A (30 periods of 30A GaAs, 30A AlGaAs) 1000A GaAs buffer

Substrate Wafer: 10.25.96.2 Figure 3.2: Schematic of the wafer used for the experiments reported in this dissertation. Layers were grown using Molecular Beam Epitaxy at Bell Labora­ tories, Lucent technologies New Jersey USA.

3.2.2 Optical Lithography and Electron Beam Lithogra-

phy

Figure 3.3 schematically details the fabrication steps required for the formation of self-aligned ohmic contacts and metal interconnections. Common to each of the three optical processing phases are the following procedures: spinning on photo­ resist (shown for (a)); exposure through a metal/glass mask (b), (e), (j)); and development of the pattern in the resist ((c), (f), (1)). Following development, the pattern is transferred to the chip either by etching (steps (d), (g)) or by depositing a layer of evaporated metal ((h), (ni)). Upon removal of the resist layer, any excess metal is also removed (lift-off process); (steps (i) and (n)). The resist profile for clean metal lift-off is improved using a pre-develop soak in chlorobenzene (k). Finally after deposition of the ohmic contact metal (step i), the metals are alloyed into the semiconductor bv a rapid thermal anneal at reduced pressure (H2/N2). The final result of all the optical lithographic 3. Experimental Methods 43

bond pad formation

'•> j j_U i • tun

(i)

Figure 3.3: Optical lithography processing steps required to fabricate the ohmic contacts and mesa structure. See text for details. 3. Experimental Methods 44

n+ GaAs GaAs Spacer Ohmic Contacls AlGaAs

GaAs 2D Electrons

Figure 3.4: Illustration showing MBE grown layer structure with ohmic metal touching the AlGaAs/GaAs heterojunction on the side of the mesa and making contact with the 2DEG (under the application of a bias). processing is illustrated in Figure 3.4. The reader is referred to references [37, 38, 39, 17] for a detailed discussion of the ohmic contact fabrication techniques used in the fabrication of these undopped samples. The development of electron beam technology was in fact the impetus for the first experiments on mesoscopic quantum wires [1, 6]. Since that time electron beam lithography has further progressed to the point where highly complex pat­ terns can be written with line-widths of less than lOnm. Although the smallest feature size required for the quantum wire devices is ^ 50nm, the high resolution of EBL ensures that the patterns are sharply defined without spurious disorder. All the devices studied here were fabricated using the Leica Ltd EBL 100 system, shown in Figure 3.5(b). Patterns were exposed at an accelerating voltage of 50kV and defined in 60nm of poly (methyl-methacrylate) (PMMA). Following exposure, the patterns were developed in a 3:1 mixture of IPA and MIB (methyl- isobutyl-ketone) for 30 seconds. Figure 3.6 shows optical microscope photographs of the virgin device struc­ ture following optical lithography processing (1-3). Figure 3.8 (4-6) shows EBL written quantum wires, where the one dimensional channel can be seen at the 3. Experimental Methods 45

Figure 3.5: Photographs showing (a) the UV optical aligner used to fabricate the ohmic contacts and metal interconnects, and (b) Leica EBL 100 Electron Beam Lithography System used to fabricate the nanoscale quantum wires. 3. Experimental Methods 46 center of the cross feature. Following the electron beam lithography the top gate structure is sectioned into three separately controllable gates using wet-etching. Etching of the GaAs top gate is performed using an acid/oxidant solution (HC1:H202:H20). The GaAs is oxidized by the H202 and the resulting oxide is continuously removed by the HC1 in solution. The process is generally performed in the dark to minimise photo-electric effects that can alter the etching properties. Figure 3.7 shows Atomic Force Microscope (AFM) images of two quantum wires of length / = 1.0fim defined in PMMA, prior to etching. Figure 3.8 shows a quantum wire of length / = 0.5/xra, after a series of experiments. The majority of apparent dis-colouration is caused by repetitive thermal cycling between milli-Kelvin and room temperature.

3.3 Measurement

Distinguishing quantum from classical phenomena in transport measurements generally requires very low temperatures approaching T = OK and low noise electrical measurement techniques. Nearly all of the data reported in this Thesis was obtained in a dilution refrigerator capable of base temperatures of the order T = lOmK. The following section discusses the cryogenic platforms used for detailed investigation and characterisation of the quantum wire devices. A brief summary of the electrical measurement setup then follows in section 3.3.2.

3.3.1 Cryostats and Dilution Refrigerators

A quick consideration of the energy scales characterising ballistic transport in mesoscopic devices reveals the requirement of low temperatures. In fact, the re­ sults presented in Chapters 4, 5, 6 and 7 suggest that some particular phenomena are so fragile that they are characterised by energy scales less than 10fieV or T 3. Experimental Methods 47

Figure 3.6: Optical microscope photographs showing the device structure after optical lithography processing. (1) Entire device layout, including bond pads. (2) Increased magnification showing 4 ohmic contacts (total of 8 for both devices). (3) Top gate FET structure prior to EBL. (4-6) Device after subsequent EBL, and wet etch. The ID quantum wire is located at the center of the cross structure in (6). 3. Experimental Methods 48

Figure 3.7: Atomic Force Microscope images of two quantum wires (length / = 1.0fim) prior to wet etching. The patterns are defined in the PMMA. Left image shows a narrow wire with rounded edges and the right image shows a wire with the standard rectangularly defined gate structure.

Figure 3.8: Atomic Force Microscope images of a quantum wire of length l = 0.5/rra. The images were taken after the device had been thermally cycled many times between room and milliKelvin temperatures. 3. Experimental Methods 49

Figure 3.9: Photographs showing the two dilution refrigerators used in the quan­ tum wire experiments, (a) shows an older 01 Kelvinox system and electronics racks used to study the point contact and / = 2/im quantum wire devices, (b) Experimental setup photographed from above, (c) shows the newer 01 Kelvinox 100 system, used for all other low temperature measurements reported.

= lOOrnK. Initially proposed by London [103], the dilution refrigerator is capable of continuous operation at base temperatures of around T = lOmK. Two separate dilution refrigerator systems were used in the experiments reported here. Figures 3.9 (a) and (b) shows the first system (original Oxford Instruments Kelvinox), used for the measurements made on quantum point contacts and a quantum wire of length / = 2/rra. Figure 3.9 (c) shows the newer system (Oxford Instruments Kelvinox 100) used for all other experiments below T = IK. For intermediate temperatures a 1.5K cryostat based on pumped He4 was used. This system, con­ structed in house by R. Starrett (NML) permits measurements at temperatures 3. Experimental Methods 50 approaching T = 1.4K for a limited time. Figure 3.10 is a photograph of the National Magnet Laboratory where all the quantum wire experiments were performed. The primary system (Blue dewar - Kelvinox 100) can be seen at the back of the room. In the left of the photograph is a second dilution refrigerator with a plastic tail set (White dewar). The foreground of the photo shows the two ‘dipping’ stations also used to characterise the devices at liquid helium temperatures.

3.3.2 Measurement Electronics

At temperatures below T = lOOrnK the applied voltage across a device should not exceed ~ 10/iV in order to remain within the linear response regime. Applying larger voltages will generally have a similar effect to increasing the temperature in terms of smearing between energy levels. Consequently in order to have a good signal to noise ratio, special techniques must be employed to ensure in­ terference and noise is kept to a minimum. AC phase sensitive methods based on lockin amplifiers are typically employed to pick out very small signals from a background noise spectrum. However, although these techniques for fill the requirements for good signal to noise there is a second issue of vital importance to be considered. At very low temperatures the heat capacity of a degenerate electron gas becomes exceedingly small [104] and when energy coupling between the electrons and surroundings is weak, electron heating can occur [105]. Inter­ fering signals from such sources as mains power, pump motors, ground loops, radio frequency emitters, etc can couple down the dilution refrigerator and heat the electrons. Importantly, phase sensitive lockin techniques generally do not reveal any information about the magnitude of interference signals heating the electrons and so other methods must also be applied to ensure this does not occur. For the measurements made here special care was taken in every instance to ensure that electron heating was keep to a minimum by characterising the 3. Experimental Methods 51

Figure 3.10: Photograph showing extensive cryogenic measurement platforms housed within the National Magnetic Laboratory at UNSW. Two dilution refrig­ erators can be seen at the back (01 K100) and left (01 Plastic) of the room. The foreground shows characterisation stations operating at liquid Helium tempera­ tures. Close observation reveals a student teleporting back to the mothership. 3. Experimental Methods 52

GPIB

Lockin Amp 5210 lOMohm

RuO (Therm) 5210 Lockin Amp lOKohms IMohm

lOOohms

QWIRE

lOkohms

Preamp

5210 Lockin Amp Preamp

Figure 3.11: Circuit diagram for the most standard circuit used in the conduc­ tance measurements. interference coupling. Precautions were taken to ensure proper shielding and grounding and mains line power conditioners were generally used. In every in­ stance the signals going to and coming from the sample were examined with an oscilloscope and spectrum analyzer. Cryogenic and room temperature electrical filters were also used in the low temperature sensitive measurements. Figure 3.11 shows the general setup used for most of the experiments. Considerations were made to decouple the sensitive analog electronics (and the grounding) from the high signal (TTL) digital circuits. Figure 3.11 shows that a analog to digital converter (IOTECH) was used to avoid GPIB connections to the EG&G analog lockin amplifiers. In addition, optical-isolation decoupled 3. Experimental Methods 53

Figure 3.12: Photographs showing electronic equipment used in the low noise transport experiments. Left: Setup for measurements at pumped He4 temper­ atures. Right: Measurements at milli-Kelvin temperatures. the computer from the measurement circuits. All data was gathered using a PC running LabVIEW (National Instruments), with virtual instrument software designed specifically for the studies undertaken in this Thesis. Chapter 4

Length, Density, and Temperature Dependence of the 0.7 Feature

4.1 Introduction

Zero length quantum wires (or point contacts) exhibit unexplained conductance structure close to 0.7 x 2e2/h in the absence of an applied magnetic field. Both the position and the strength of this conductance feature is found to depend on the density, temperature, and length of the ID region. This chapter investigates these dependencies via measurements on quantum wires with nominal lengths /=0, 0.5, 1.0 and 2gm, fabricated from structures free of the disorder associated with modulation doping. In a direct comparison in zero magnetic field, structure is observed near 0.7 x 2e2/h for /=0, whereas the /=2/im wire shows structure evolving with increasing electron density to 0.5 x 2e2//i, the value expected for an ideal spin-split sub-band. For intermediate lengths (/ = 0.5\im and 1.0\im) the feature at 0.7 x 2e2/h evolves below 0.6 x 2e2/h with increasing density. These results suggest the dominant mechanism through which electrons interact can

54 4. Dependence of the 0.7 feature 55 be strongly affected by both the electron density and length of the ID region. Quantum wires have been used extensively to study ballistic transport in one dimension (ID) where the conductance is quantised in units of 2e1 /h [6, 1]. This result is well explained by considering the allowed energies of a non-interacting electron gas confined to ID, where the factor of 2 is due to spin degeneracy (see Chapter 2 for a detailed discussion). Electron interaction effects in ID have been considered for some time, involving models [59] which go beyond the conventional Fermi liquid picture. Such correlated electron models have been applied to quantum wire systems [106] and recent experimental studies [40, 35] have investigated their predictions. Although recent theories have considered the effect of weak disorder on correlation effects [107], it is generally accepted that low-disorder nanostructures are necessary for such investigations. Low-disorder quantum point contacts (which are quantum wires of length /=0) formed in GaAs/AlGaAs heterostructures exhibit unexplained conductance structure close to 0.7 x 2e2/h in the absence of a magnetic held [2, 13, 17, 18, 15, 24, 23, 10, 21, 25, 26]. These studies suggest that the structure is a manifestation of electron-electron interactions involving spin. In particular, Thomas et al. [2] studied the continuous evolution of the 0.7 x 2e2/h structure into a Zeeman spin- split conductance plateau with the application of an in-plane magnetic held. In addition, an enhancement of the electron (/-factor for lower ID channels was also observed, consistent with an interpretation [2] involving electron-electron interactions. In this chapter transport data taken on ID systems free from the disorder associated with modulation doped heterostructures is reported. This data in­ cludes evidence for spin related many-body effects in long ID regions. Conduc­ tance structure comparable to Thomas’ is found in the zero length wires, while the 2 //m quantum wire exhibits plateau-like structure near 0.5 x 2e2/h in zero magnetic held, the value expected for an ideal spin-split level. Further, for wires 4. Dependence of the 0.7 feature 56 of intermediate length (/=0.5//ra, 1.0/xm) a pronounced feature at 0.7 x 2e1/h is found that evolves continuously to 0.55 x 2e2/h with increasing ID electron density. Theories involving electron-correlation effects have been developed recently to explain why experiments predominantly show a feature at 0.7 x 2e2//i, rather than at 0.5 x 2e2//i, the value expected for simple spin-splitting. Chapter 2 provided a discussion of these models which included two-electron spin sin­ glet/triplet pairing [84, 85], Fermi-pinning at a spontaneously spin-split sub-band [93], Wigner crystallisation [82, 54] and spin-orbit coupling [92]. In addition to these explanations, Chapter 6 presents a new phenomenological model [11] that is consistent with all of the key data published on the subject. The data pre­ sented here, together with recent results obtained by Thomas et al. [22] and Pyshkin et al. [23], indicate that the ideal value of 0.5 x 2e2/h can also oc­ cur, either above a certain length scale, or for some critical carrier density or potential profile, thereby lending strength to an interpretation in terms of spin polarisation (see Chapter 6.). The study of correlated electron states requires devices with ultra low-disorder since such states are expected to be easily destroyed by disorder and may be masked by other effects associated with localisation. The samples studied in this thesis comprise of a novel GaAs/AlGaAs layer structure which avoids the major random potential present in conventional HEMT devices by using epitax­ ially grown gates to produce an enhancement mode FET [39]. These devices are advantageous for the study of ID interacting systems because they eliminate the need for a dopant layer in the AlGaAs adjacent to the 2DEG, thus greatly reducing disorder while allowing the electron density in the 2DEG to be varied over a large range. The electron mobility in the 2DEG is typically 4 — 6 x 106 cm2V_1s_1 at 4.2K and increases further at lower temperatures. At lOOmK the 2D ballistic mean free paths exceed 160//m [99] which is greater than our sample 4. Dependence of the 0.7 feature 57

Figure 4.1: Schematic of the quantum wires studied in this thesis. / defines the length of the ID region. The green shaded regions represent the side gates, (biased negative). The blue region represents the top gate, (biased positive). The two dimensional reservoirs are also indicated. dimensions. These devices are comparable with the highest mobility electron systems yet produced. Ballistic conductance plateaus have been demonstrated in quantum wires up to 5[inn in length, with the data exhibiting more than 15 plateaus [17]. To investigate the sensitivity of many-body effects to the length of the ID region, measurements were made of the conductance of quantum wires of nominal length /=0, 0.5/im, 1.0gm, and 2fim. I is defined in the device schematic shown in figure 4.1. The devices were patterned from ultra-high-mobility heterostructures, com­ prising a 75 nm layer of Alo.3Gao.7As on top of GaAs to produce the 2DEG interface. A 25 11m GaAs spacer separated the epitaxial conducting top gate from the AlGaAs. NiAuGe ohmic contacts were made to the 2DEG using a self-aligned technique. Electron beam lithography and shallow wet etching were used to selectively remove the top gate to form the quantum wires. This device structure is described in detail in Chapter 3 and reference [17]. The epitaxial top 4. Dependence of the 0.7 feature 58

Vs (-) Vt (+) Vs (-)

GaAs (Spacer)

AlGaAs

GaAs

GaAs (Superlattice)

V> (-) Vt (+) V> (-)

GaAs (Spacer)

AlGaAs

GaAs

GaAs (Superlattice)

Figure 4.2: Schematic of the MBE grown layer structure (see Chapter 3 for detail). The bias applied to the top gate (Vt) and side gates (Vs) control the shape of the ID confinement potential (shown in yellow). A large bias applied to all gates steepens the potential and increases the ID sub-band spacing. layer was sectioned into three separately controllable gates (see Figure 4.1). The center (top gate) was biased positively relative to the contacts to induce a 2DEG at the GaAs/AlGaAs interface. This positive bias Vt determined the carrier den­ sity in the 2DEG reservoirs which was typically tunable from 0.6 — 6 x 10ncm~2. A negative voltage Vs was then applied to the side gates to produce electrostatic ID confinement in addition to the geometric confinement already present. Low frequency four-terminal conductance measurements were made with an excitation voltage below 10[iV using two lock-in amplifiers to monitor both cur­ rent and voltage, as described in Chapter 3. The results presented here are raw data as no equivalent series resistance has been subtracted and no attempt has been made to adjust the plateau heights to fit with quantised units of 2e2/h. The ID electron density n\o may be controlled using both the top and side gates to vary the shape of the potential well perpendicular to the channel (See occupancy. Figure to 4.

a Dependence

larger

4.3:

Schematic ID

density, DOS

of n=

the 1

showing

(and 0.7 n=2

feature larger

how

a Fermi

larger Large Small

level)

ID

G G ID ID

sub-band

= = sub-band sub-band for

2x 2 the

x

same 2e/h

2e/h energy spacing spacing

conductance

spacing

leads

59 or

4. Dependence of the 0.7 feature 60

Figure 4.2). When both the top and side gates are strongly (weakly) biased posi­ tive and negative respectively the confining potential is presumed steep (shallow), leading to a larger (smaller) ID sub-band spacing and a corresponding high (low) ID electron density. In this way it is possible to maintain a constant ID occu­ pancy, and hence conductance, while varying n\D, as shown in Figure 4.3. This is confirmed by measuring the ID sub-band energy spacing using a DC source - drain technique [12], and the details are the subject of Chapter 5. It is im­ portant to note however, that because the ID density of states (DOS) varies as p « E1-1/2, the electron density is NOT linear with the sub-band spacing. The effect of the ID sub-band spacing on the ID density is discussed in more detail in Chapters 5 and 6.

4.2 l — 0 Quantum Wires

Two identical Quantum Point Contacts (or l = 0 Quantum Wires) were fab­ ricated by Facer et al. [108] using electron beam lithography (EBL) and wet etching (as discussed in Chapter 3). These devices were made from the standard heterostructure (wafer: 25.10.96.2) used throughout this thesis, and were pro­ cessed identically on the same chip. The 0.7 conductance structure was studied as a function of density and temperature, for both devices on the chip and the following sections report the results of these studies. Although the devices may differ slightly due to the presence of random impurities in the heterostructure, they are in every other sense identical. The opportunity to study identical de­ vices simultaneously in the same cool down, provides an important and crucial test for the reproducibility of the 0.7 conductance structure. 4. Dependence of the 0.7 feature 61

Side Gate Voltage, Vs (V)

Figure 4.4: QPC:A (/ = 0). Conductance as a function of side gate bias Vs for top gate voltages Vt ranging: 170rnV - 306mV in steps of 4mV. T = 50rnK

4.2.1 QPC:A

Figure 4.4 shows conductance measurements for QPC:A as a function of side gate voltage, Vs at a temperature T = 50mK. Data was taken at a series of top gate voltages corresponding to different densities and potential profiles. Both the 2D density and the potential profile increase (from shallow to steep) when compar­ ing curves from right to left. Five clean plateaus are shown, free from conduc­ tance perturbations associated with weak-disorder or localisation effects. Below the first plateau (G < 2e2/h) a very slight additional feature near 0.7 x 2e2/h is observed on some traces. The presence of this anomaly is better detected by studying the derivative data (dG/dVs). Figure 4.5 shows the derivative of conductance G with respect to side gate voltage, Vs. Each trace has been geo­ metrically adjusted so that the peaks in the transconductance are aligned. These peaks correspond to regions where the conductance is rapidly changing, between 4. Dependence of the 0.7 feature 62

Side Gate Voltage, Vs

Figure 4.5: QPC:A (/ = 0). Transconductance (dG/dVs), calculated numerically from the data in figure 4.4. Traces have been geometrically scaled to align the main transconductance peaks. the plateaus at 2e2/h. To the right of the first transconductance peak (left most peak), a weak shoulder feature can be seen. In comparison to other devices re­ ported (later), the 0.7 conductance feature appears to be extremely weak, and can only be detected as an asymmetric broadening of the first transconductance peak. The behavior of the conductance feature was investigated further as a func­ tion of temperature in the range, T = 50mK - 4000mK. Figure 4.6 shows the conductance of the last plateau (1 x 2e2/h) as a function of side gate voltage, Vs for different temperatures. The conductance was found to be temperature independent until T >500mK, and therefore only traces above this temperature 4. Dependence of the 0.7 feature 63 are shown. Measurements were made at a high and low top gate bias (Vt) to observe the effect of density on the behavior. Comparing the temperature de­ pendence data with the data presented in the initial work by Thomas et al [2], confirms that the extremely weak conductance structure below the first plateau is indeed the 0.7 feature. A discussion of this unusual dependence is reserved for the next section, where the conductance feature appears substantially stronger.

4.2.2 QPC:B

Figure 4.7 shows the conductance of a second identical quantum wire of length /=0. The conductance was studied as a function of side gate voltage Vs, for a range of top gate voltages Vt, at a temperature T = 50mK. Consistent with the data presented in the previous section, a conductance anomaly near 0.7 x 2e1 /h is revealed below the first plateau. In contrast to QPC:A however, the 0.7 feature here (QPC:B) is particularly strong. Although the trend is not fully monotonic, a definite strengthening of the feature is seen when comparing the furthest left and right most traces. This behavior is strongly evident in the transconductance

(idG/dVs) data. Figure 4.8 presents the transconductance dG/dVs of QPC:B. Again traces have been geometrically scaled in order to align the main transcon­ ductance peaks. In contrast to the data taken on the other identical device on the chip (QPC:A), the 0.7 feature in this device (QPC:B) is revealed as a satellite peak, completely separate from the first main transconductance peak. The height and strength of this satellite peak steadily increases as the top and side gate bias is increased (Vt and Vs). Further examination also reveals an asymmetric broad­ ening of the second main transconductance peak, corresponding to the presence of an extremely weak 1.7 x 2e2/h conductance anomaly. Figure 4.9 presents the temperature dependence of the 0.7 feature seen in QPC:B. Conductance is plotted as a function of side gate bias (Vs), for temper- 4. Dependence of the 0.7 feature 64

a 0.5

Side Gate Voltage, Vs (V)

a o.5

Side Gate Voltage, Vs (V)

Figure 4.6: Temperature dependence of QPC:A. Top graph: G as a function of Vs for top gate bias, Vr=180mV: T = 4000mK (red), 1500rnK (orange), 1200mK (pink), 960mK (lblue), 760mK (dblue). Bottom graph: G as a function of Vs for top gate bias, Vr=300mV: T — 3500mK (red), 1500mK (orange), 1200mK (pink), 950mK (lblue), 800mK (dblue). 4. Dependence of the 0.7 feature 65

« 3.0

8 2.0

-1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 Side Gate Voltage, Vs (V)

Figure 4.7: QPC:B (/ = 0). Conductance as a function of side gate bias Vs for top gates VT ranging: 170mV - 294mV in steps of 4mV. T = 50mK atures T = 50mK - 2850mK. Measurements were made at two different top gate voltages, (Vt = 170mV and Vp = 300mV) to observe the effect of density on the temperature dependence. Although the same characteristic behavior can be identified for both the high and low density data, the feature appears stronger at high gate bias and facilitates observation. The temperature dependence of the 0.7 feature is characterised by two ef­ fects. Firstly the position (in conductance) where the slope (transconductance) begins to decrease, moves slightly lower. Secondly the feature becomes flatter and broader with increasing temperature. Such an unusual temperature de­ pendence is opposite to the behavior generally expected for many-body effects, which usually decrease in strength with increasing temperature. Never the less, the temperature data reported here is entirely consistent with measurements made by others, [2, 18, 24]. Following the work of Kristensen et al. [18, 24], Figure 4.10 plots the de- 4. Dependence of the 0.7 feature 66

Side Gate Voltage, Vs

Figure 4.8: QPC:B (/ = 0). Transconductance (dG/dVs) calculated numerically from the data in figure 4.7. Traces have been geometrically scaled to align the main transconductance peaks. viation from ideal conductance as a function of the inverse temperature, for four different side gate voltages where the 0.7 feature occurs. The linear depen­ dence with 1/T in this Arhenius plot confirm, in accordance with Kristensen et al. [24], that the temperature dependence of the 0.7 feature is indeed ‘acti­ vated’. Although a discussion of the activated behavior is deferred to Chapter 6, the reader should note that behavior of the conductance with temperature in a non-interacting single particle picture can also be described as ‘activated’. This follows directly from the Fermi-Dirac distribution where the occupation probability depends exponentially on temperature and the difference between the Fermi energy and the band-edge. Confirmation that the 0.7 feature also has 4. Dependence of the 0.7 feature 67

C 0.5

Side Gate Voltage, Vs (V)

Side Gate Voltage, Vs (V)

Figure 4.9: Temperature dependence of QPC:B. Top graph: G as a function of Vs for a top gate bias of, Vp = 170mV: T = 2730mK (red), 1650mK (orange), 960mK (pink), 510mK (lblue), 50mK (dblue). Bottom graph: G as a function of Vs for a top gate bias of, VT=300mV: T = 2850mK (red), 1500mK (orange), 970mK (pink), 500mK (lblue), 50mK (dblue). 4. Dependence of the 0.7 feature 68

Ta = 0.34

© -2

Ta=I.19 j= -2.5

Vs =-1.60 V Vs =-1.64 V Vs = -1.68 V Vs = - 1.71V

Figure 4.10: Plot showing the activated temperature dependence of QPC:B. an activated dependence with temperature, is evidence that the feature results from an anomalous band-edge. Figure 4.11 shows again the conductance of QPC:I3 as a function of side gate bias. In this case very high biases were applied to the gates, in a hope to observe an increased strengthening of the 0.7 feature. No such enhancement is seen and further, the 0.7 feature appears even weaker for very strong bias. Such behavior may be explained by considering that the potential defining the ID channel has likely become strongly perturbed, and transport somewhat non-adiabatic. This is indicated by the decreasing quality of the quantisation on the plateaus. Of further consideration, a reasonable leakage current was present between the top gate and ohmic contacts for the very large gate voltages. Comparison of the two devices QPC:A and QPC:B reveals a large difference in the strength of the 0.7 feature. At T = 50mK QPC:A bearly shows any deviation from an ideal, single particle conductance. Although the feature can be clearly identified at higher temperatures (T >1K) or, by examination of the transconductance (clG/dVs) the 0.7 feature characterising QPC:B is clearly stronger. Given that the two 4. Dependence of the 0.7 feature 69

O 3.0

o 1.0

-3.0 -2.0 -1.0 Side Gate Voltage, Vs (V)

Figure 4.11: QPC:B (/=0). Conductance as a function of side gate bias Vs, for large top gate and side gate bias, Vt = 300mV - 680mV. T = 50mK. devices are nominally identical such a striking difference between them begs fur­ ther analysis and explanation. One possibility is that the devices are slightly different due to the presence of random impurities. These ‘background’ impuri­ ties however, do not effect the quality of the integer plateaus (at n x 2e2/h) and therefore must be far from the ID region. An alternative explanation, consistent with other recent results presented in Chapter 5 suggests that surface charge traps exist that can modify the potential landscape. Charge traps however, are known to re-distribute with thermal cycling.

4.3 l = 0.5fim Quantum Wires

The following section reports conductance data for a l = 0.5fim quantum wire. Importantly, this data together with the results in the next section link the re­ sults of the l = 0 wires (QPC:A, QPC:B) to the data taken on a / = 2fim device, where a full lifting of the spin degeneracy is observed at B = 0. Figure 4.12 shows conductance data for a / = 0.5/im quantum wire at T — 50mK. Anal- 4. Dependence of the 0.7 feature 70

-1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 Side Gate Voltage, Vg (V)

Figure 4.12: The conductance G of a l = 0.5/rm quantum wire as a function of side gate voltage, Vs- T = 50mK. Vt = 560mV - 1500mV. ogous to the results reported in the previous sections, conductance is plotted as a function of side gate bias (V$) for a range of top gate voltages [Vt) corre­ sponding to an increasing density (right to left). The right most traces reveal a strong conductance feature near 0.75 x 2c2//?, that evolves continuously down­ ward toward 0.5 x 2e2/h with increasing gate bias. The left most trace reveals a plateau-like feature at 0.52 x 2e2//?, close to the result expected for an ideal Zeeman spin-split sub-band. The evolution of the feature towards 0.5 x 2e2/h can also be seen in the transconductance data {dG/dVs) shown in Figure 4.13. The anomalous structure appears as a completely separate peak in the transconductance, sharpening very slightly with increasing gate bias. Further, for the n — 2 transconductance peak, a strong shoulder-like feature can be identified to the right of the main peak. This 1.7 x 2e2/h feature also increases with strength and moves downwards toward 1.5 x 2e2/h with increasing bias. 4. Dependence of the 0.7 feature 71

Side Gate Voltage, (Arb. Units)

Figure 4.13: / = 0.5/.im quantum wire. Transconductance (dG/dVs) calculated numerically from the data in figure 4.12. Traces have been geometrically scaled to align the main transconductance peaks.

Turning now to the temperature dependence of the feature, new and unex­ pected results are seen. Figures 4.14 and 4.15 show conductance as a function of side gate voltage, for a range of top gate bias at T = 1.6K and 4.2K respectively. Comparing this data with the conductance data taken at T — 50mK reveals that the position of the 0.7 feature is strongly temperature dependent. At low gate bias the feature behaves similarly to the / = 0 devices studied in the previous section. The position where an initial change of slope occurs moves downward, and the feature becomes flatter and broader consistent with the l = 0 devices and measurements made by others [2, 24]. This behavior is shown in figure 4.16, for a temperature range TLattice — lOmK - 4200mK. At high gate bias (left most curves in figures 4.12 4.14 and 4.15) new, unexpected behavior is observed. At low temperatures (T < 500mK) the position of the feature is around 0.55 x 2e2/h for the left most traces. However, as the temperature is raised, the position of 4. Dependence of the 0.7 feature 72

-0.4 -0.3 -0.2 Side Gate Voltage, Vs (V)

Figure 4.14: Conductance as a function of side gate bias Vs for a l = 0.5pun Quantum wire at T — 1.6K. the feature moves upward to 0.7 x 2e2/h and becomes broader. At temperatures close to T — 4K the position of the conductance anomaly remains constant, as the top gate bias (Vr) is varied. A comparison of Figures 4.12 and 4.15 suggests that the strong evolution of the feature from 0.7 to 0.5 x 2e2/h at T = 50mK with increasing density, is a low temperature effect. At higher temperatures above T « 2K the density dependence is destroyed.

4.4 l = 1.0fim Quantum Wires

The potential landscape in the ID - 2D contact region (see Figure 4.1) is believed to play an important role in adiabatic ballistic transport. Abrupt, sharp changes in the potential caused by the surface gates can lead to reflections of the electron wave. In order to investigate these effects, two devices of length l = 1.0pirn (l/mi: A and l/mi:B) were fabricated from the same herterostructure and subject 0 Quantum Figure 4. to

Dependence

-700mV,

4.15:

wire Conductance, G (2e7h) Conductance, G (2e /h)

Conductance and

at

T of the

—

the

bottom 4.2K. -

1.6 0.7

G Side Side

Top

as - feature plot 1.4

a

Gate

Gate

plot

function shows - 1.2

shows

Voltage, Voltage,

- data 1.0

of

a

side

dense taken - 0.8

gate Vs V

- out number 0.6 s

(V) (V) bias

to - 0.4

Vs Vs

of

= for - curves 0.2

-2.0V.

a

/

0.0

= from

0.5

iim Vs 73 ’.

4. Dependence of the 0.7 feature 74

— 0.01 K 0.20 K 0.30 K 0.40 K v - 0.50 K e 1.0 0.60 K w« 0.70 K o 3 0.80 K HD 1.00 K C 1.50 K O 1.80 K U 2.35 K 4.20 K

-0.3 -0.2 -0.1 Side Gate Voltage, Vs (V)

— T = .03 K T = 1.0 K T = 4.2 K

Side Gate Voltage, Vs (V)

Figure 4.16: Temperature dependent conductance of l = 0.5fim quantum wire. Lattice temperature is indicated in the legend. TOP: Conductance G as a function of side gate voltage, Vs for Vr = 500mV. BOTTOM: Same as Top only traces for 4.2K, 1.0K and 30mK are shown for easy comparison. 4. Dependence of the 0.7 feature 75 to identical processing. One device (l/rm:A) was fabricated according to the standard rectangular geometry used for all other devices studied in this thesis (refer to Chapter 3). The other device (l/im:B) was designed to have smooth, curved contact regions joining the 2D reservoirs to the ID quantum wire. Close comparison of the two quantum wires reveals that device is in fact slightly longer due to the curved contact regions. Chapter 5 discusses in some detail the effect of the different geometries, but here the only concern is the effect of the slightly different lengths.

4.4.1 Quantum Wire: 1 fim : A

Figure 4.17 shows the conductance of a / = l fim quantum wire (device A) at T = 4.2K. Due to the smaller ID sub-band spacing (in comparison to the / = 0.5fim device) the plateaus appear smeared and washed out at this temperature. None the less, a slight feature near 0.7 x 2e2/h can be identified. Note that the position of this feature remains close to 0.7 as the top gate bias is varied (Vt = 300mV - 850mV) consistent with the temperature data presented in the previous section for the / = 0.5/im device. Of further interest, the leftmost green trace on the bottom figure shows the plateaus becoming more resolved as the side gate voltage is increased, consistent with an increasing confining potential. Figure 4.18 shows the conductance of a / = 1 fim quantum wire (device A) at T = 50rriK. Comparison with the data taken at T — 4.2K reveals a strong, broad 0.7 feature that evolves downwards towards 0.5 x 2e1 /h with increasing gate bias. Measurements were only made to a side gate bias of Vs ~-1000mV, for fear of irreversible damage due to leakage currents. Despite this limited range a clear evolution towards 0.5 is observed. Of further interest is the strong 1.7 x 2e2/h feature seen evolving downwards towards 1.5 x 2e2/h with increasing gate bias. Figure 4.19 shows the transconductance (dG/dVs) at T = 50mK for a / = 1 \im quantum wire (device A). Consistent with the transconductance data reported 4. Dependence of the 0.7 feature 76

-0.2 Side Gate Voltage, Vs (V)

O 6.0

-1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 Side Gate Voltage, Vg (V)

Figure 4.17: Quantum wire: lfim : A. Conductance G as a function of side gate bias Vs for VT — 300mV - 800mV (bottom trace). T = 4.2K. 4. Dependence of the 0.7 feature 77

Side Gate Voltage, Vs (V)

-0.8 -0.6 -0.4 Side Gate Voltage, V§ (V)

Figure 4.18: TOP: Conductance of 1.0/rm quantum wire (Device A) at T = 50mK. Vt — 270mV - 630mV. BOTTOM: Zoom of data for 1st two integer plateaus. 4. Dependence of the 0.7 feature 78

in > 5 O T3

0.0

Side Gate Voltage, (Arb. Units)

Figure 4.19: Transconductance (dG/dVs) of I = Ifim quantum wire (Device:A) at T — 50mK. Traces have been geometrically scaled to align the main transcon­ ductance peaks. in previous sections, the data has been geometrically scaled in order to align the main integer transconductance peaks. The 0.7 feature can been seen as a distinct separate peak to the right of the n=l transconductance peak. As the gate bias

(Vt and Vs) are increased the satellite peak splits further away from the n— 1 transconductance peak and grows in strength. An extremely strong shoulder peak can also be identified to the right of the n=2 peak corresponding to the 1.7 x 2e2/h feature. The 1.7 transconductance peak also becomes more defined with increasing gate bias. Of further importance, close examination to the right of the n=3 transconductance peak reveals an extremely weak 2.7 x 2e2/h feature.

4.4.2 Quantum Wire 1 fim : B

Quantum wire l/rm:B has rounded, adiabatic 2D - ID contact regions as dis­ cussed above. A consequence of making the contact regions smooth however, is 4. Dependence of the 0.7 feature 79

O 6.0 8 5.0

O 2.0

//////////■/////

Side Gate Voltage, Vs (V)

Figure 4.20: Conductance of 1.0fim quantum wire (Device B) at T = 4.2K. Vt = 440mV - 990mV in lOmV steps. to make the effective length of the ID region slightly longer. Figure 4.20 shows the conductance of device B as a function of side gate bias, for T = 4.2K. Com­ paring this data with data taken on Device:A reveals that the ID confinement potential (Tiujy) is greater in this case. As a result, clean integer plateaus can be seen together with a significant 0.7 and 1.7 feature at an elevated temperature of T = 4.2K. Figure 4.21 compares the conductance data taken at T = 4.2K with data taken close to T « 1.6 K, in a pumped He4 cryostat. The 0.7 conductance feature is resolved in the T ~ 1.6K data as a small plateau. For the high density traces shown left most (Figure 4.21), the 0.7 feature exhibits a non-monotonic ‘dip’ in conductance. Such behavior has been observed for other devices, most notably the l = 2iim quantum wire considered in the next section, and is consistent with the phenomenological model proposed in Chapter 6. Upon cooling to T = 50mK, Device B exhibits a pronounced plateau at 4. Dependence of the 0.7 feature 80

T ~ 1.6 K T = 4.2 K

0 -0.8 -0.6 -0.4 Side Gate Voltage, Vs (V)

Figure 4.21: Conductance of 1.0//m quantum wire (Device B) at temperatures T % 1.6K (blue) and T = 4.2K (red). Vp = 0.44V - 0.79V in 50mV steps.

0.7 x 2e2/h, that is at least half the width of the plateau at 1 x 2e1 /h. Once again a strong evolution is seen with increasing gate bias {Vp)- In this case the position of the feature oidy changes by a small amount, however the strength and width of the plateau increases with increasing density. Close analysis of the experimental conditions reveals the presence of a small dc offset bias (VsD=0.4mV), between the source and drain contact terminals. In accordance with the phenomenological model proposed in Chapter 6 to explain the 0.7 feature, such an offset bias will suppress the evolution towards 0.5 x 2e1/h with density, (see Chapter 6 for details). Close observation of the data in Figure 4.22 also reveals the presence of additional conductance structure, just below the first plateau. These features are most probably due to weak disorder associated with the enhanced length of Device B. The interplay between the 0.7 conductance feature and disorder potentials has been explored by O’Brien, Reilly et al.[3] and some of these results are reported in Chapter 7. 4. Dependence of the 0.7 feature 81

Side Gate Voltage, V. (V)

Figure 4.22: Conductance of 1.0nm quantum wire (Device B) at T ~ 50mK. Vt = 0.44V - 0.79V in 50mV steps. A small dc offset bias was present (approxi­ mately: 0.4mV). Inset: A zoom of the region below the first plateau (n=l).

4.5 l = 2.0jim Quantum Wires

Figure 4.23 shows the conductance G of a quantum wire with l — 2//m. Data were obtained at temperatures T = IK and T = 50mK. Clear conductance quan­ tisation is seen near integer multiples of 2e2/h with up to 15 plateaus evident, indicating ballistic transport along the full length of the 2fim wire. The data collected at T = IK show a clear plateau-like feature below 2e2/h which be­ comes more pronounced and evolves downwards in G towards 0.5 x 2e2/h as n\B is increased. A much weaker inflection is also present near 0.7 x 2e2/h on some traces. The presence of a feature near 1.7 x 2e2/h is also observed, consistent with data taken on shorter length devices. As the 2gm wire is cooled to T = 50mK the feature near 0.5 x 2e2/h re­ mains however, rich evolving structure is also revealed. Conductance inflections occur below each of the integer plateaus (within e2/h) and predominantly evolve downwards in G with increasing Vg. One explanation within a single-particle 4. Dependence of the 0.7 feature 82 picture is that these inflections result from weak disorder, leading to interference of electron waves along the quantum wire. However, against this, remnants of the strongest features survive at T=1K, in particular the feature near 1.7 x 2e1 /h is reminiscent of the 0.7 x 2e2/h feature seen in low-disorder /=0 wires, such as QPC:A and QPC:B. Of further interest, the plateau at 1 x 2e2/h is suppressed for both data sets (T = IK and T = 50mK). For the shorter length devices considered in this thesis the plateau at 2e2/h remains almost constant, but in the case of the data taken on the / = 2pm the plateau falls in G (by up to 8%) as the density is increased. Suppression of plateaus below the ideal quantised values has been observed in previous studies on quantum wires [40, 35, 17], and considered theoretically in a number of many-body treatments [106, 107]. In the case presented here the suppression cannot be explained by a simple increase of the effective series resistance associated with the 2D contact regions, since the 2D sheet resistance decreases with increasing Vp- Abrupt coupling of the 2D reservoirs to the low density ID region could result in a reduction of the transmission coefficient as the 2D electron density is increased. This is consistent with the density mismatch being larger for the longer wire, since the top-gate voltage threshold for conduction is almost twice as large for l = 2pm as for / = 0 devices.

4.6 Discussion of Results

A detailed study of the 0.7 x 2e2/h conductance anomaly has been undertaken on ultra low-disorder quantum wires of nominal length l = 0,0.5,1.0 and 2pm. Figure 4.24 details the evolution in position of the conductance features with varying Vs- The position of the feature is defined as the conductance G at which the transconductance dG/dVs is a local minimum. The feature near 0.7x2e2/h in / = 0 devices becomes slightly more pronounced 4. Dependence of the 0.7 feature 83

S 2.0

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 Side Gate Voltage, V§ (V)

Hill

! ! i I i ! ! i / 1 I

i ./ / /

Side Gate Voltage, Vs (V)

Figure 4.23: Conductance G as a function of side gate voltage, Vs for a quantum wire of length / = 2/rm. TOP: T — 50mK. BOTTOM: T — IK. 4. Dependence of the 0.7 feature 84

1.1 n = 1 Position 1.0

0.9

CJ 0.7 ** —___x ^ s__ C 3 U 0.6 3 'O C O 0.5 — 2.0 micron QW U — • 0.5 micron QW 0.4 — • 0.0 micron QW — • 1.0 micron QW 0.3. -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 Side Gate Voltage, (V)

Figure 4.24: Feature position as a function of side gate voltage, Vs. The position of the corresponding n=l plateau is also shown. Red: / = 0, Green: / = 0.5//m, Blue: / = 1.0//m, Black: l = 2.0(inn. 4. Dependence of the 0.7 feature 85 with increasing Vs (Figure 4.7) but the variation in conductance is small. This is in contrast to the plateau-like features seen in the l = 0.5\im, 1.0[am and l=2^m wire data, which evolves downward towards 0.5 x 2e2/h with increasing U\d■ In the case of the / = 2/rra wire, note that if the n = 1 plateau is normalised to equal 2e2/h, then this feature still evolves downwards in G but never falls below 0.5 x 2e2//i, the position expected for a spin-split ID plateau. Figure 4.25 presents a conductance histogram for each device. The histogram is generated by dividing each conductance trace into small segments correspond­ ing to the histogram bin-widths. The ?/-axis counts the number of times a trace has a corresponding conductance that falls into the range defined by the bin- width. Clear peaks are seen near 0.5 — 0.7 x 2e2/h and also near (1.7 x 2e2/h) for each device. A clear evolution in the size and position of the feature toward 0.5 x 2e2/h is seen as the length of the quantum wire is increased. Conductance data suggestive of many-body effects in ID have now been ob­ served in a variety of high mobility structures including split-gated HEMTs [2], gate metallised structures [18] and the undoped enhancement mode FETs con­ sidered here. Some evidence for this effect has also been seen in low mobility quantum wires based on ion-beam defined GaAs transistors [15] and other ma­ terial systems such as GalnAs/InP [16] and n-PbTe [20]. The diverse number of experimental systems that have been examined would seem to establish the feature as an intrinsic property of a ID correlated system. In particular the temperature dependence, described as activated by [18] and confirmed here for the / = 0 and l — 0.5^m wires, remains consistent between devices of different design. Some important exceptions do exist, however, as in measurements of narrow wires by Yacoby et al. [35] and Tarucha et al. [40] there appears to be no strong feature present even though clear quantisation is seen. The absence of the feature in reference [35] may be associated with a large ID sub-band spacing made possible in that case due to a novel epitaxial confinement technique. 4. Dependence of the 0.7 feature 86

Conductance, G (2e /h)

Figure 4.25: Conductance histograms for l = 0,0.5,1.0, 2/im quantum wires. The high number of counts in the shorter devices reflects the flatness of the integer plateaus. The count number n is reduced as the quality of the plateaus decreases with increasing length of the ID region. The 0.5 —0.7 x 2e2/h structure is indicated by the red arrows. The blue arrows indicate the presence of structure near 1.7 x 2e2/h. 4. Dependence of the 0.7 feature 87

The most commonly invoked explanation for additional conductance struc­ ture near 0.7 x 2e2/h has been some form of spontaneous spin polarisation medi­ ated through the exchange interaction [53, 58] as discussed in Chapter 2. Chapter 6 describes a phenomenological model, formulated by Reilly [11] that is consis­ tent with all of the data presented in this chapter, however a microscopic theory of the 0.7 conductance feature is still lacking. The fact that structure is seen near 0.7 x 2e2/h in / = 0 wires and structure evolving towards 0.5 x 2e2/h in longer wires (with l = 0.5/im, 1.0gm, 2/xm and 5fim [17]) leads to a possible scenario in which the spin-splitting is only fully resolved in wires above some critical length scale or ID density. The additional structure observed in 1=2 [im devices near 1.7 x 2e2/h, and in higher sub-bands below IK, also suggest that many-body effects become enhanced in longer ID regions. Recently Thomas et al. [22] have reported ID conductance measurements in double structures. Their results show that for l = 0.4/irn devices the feature at 0.7 x 2e2/h evolves to 0.5 x 2e2/h with decreasing ID density. These results further indicate that the 0.7 feature may evolve as a function of n.\o or potential profile, although the dependence is the opposite to what is reported in this work. Note that the devices studied in [22] may share a similar confining potential with the devices reported here, since strong gate biases are required in both cases to force the conductance feature to evolve downwards to 0.5 x 2e2//i. Of further interest, very recent results have been reported by Pyshkin et al. [23] on induced electron systems similar to the devices discussed in this work. Again a strong evolution towards 0.5 x 2e2/h is seen for both increasing and decreasing density. Although these results confirm that the 0.7 feature can indeed evolve downwards toward 0.5, consistent with increasing polarisation, a microscopic theory remains to be uncovered. Of further importance, the amount by which the surface gates can vary the ID density should be considered. Even under optimistic conditions the non-linear ID density of states (see Figure 4.2) 4. Dependence of the 0.7 feature 88 requires a very large change in sub-banc! spacing to affect the ID density by any significant amount. Finally, considering the possibility of an exchange driven spin polarisation, the question of how the length and density dependence is produced still remains. At least three possibilities exist. Firstly, as discussed in Chapter 2 the pres­ ence of an interface electric field from the surface gates can lead to spin splitting from effects associated with spin-orbit coupling. The degree of splitting depends on the Fermi wavelength, Fermi energy and the interface electric field, all of which are tunable via the surface gates. Schmeltzer et al. [92] have considered a scenario where such mechanisms lead to zero-field spin splitting in the type of heterostructures considered here. Generally spin-orbit splitting is considered far too small in GaAs/AlGaAs structures to produce such effects, although in the presence of strong exchange coupling a polarisation seeded by spin-orbit type interactions remains a possibility. Secondly of notable importance, various calculations (see Chapter 2 section 2.5) predict magnetic ordering in the 2DEG at low densities. For rs > 13 a spin polarised ground state is predicted, before the formation of a Wigner crystal. However, the 2D densities where such polarised phases may exist are far lower than what occurs in the devices considered here and further, the 0.7 feature increases in strength with increasing 2D density. Such a low density scenario might occur however, just before the ID region, as the electron density decreases massively. Figure 4.26 plots (schematically) the electron density as a function of position through the contact reservoirs and ID region. Perhaps it is the side gate bias (Vs) that reduces the density in the region of the contact reservoirs, while keeping the conductance constant. Figure 4.26 sketches the expected density dependence with position x for both a short and long quantum wire. Note how the density of the contact reservoirs pass through the density range predicted to give rise to a spin polarlised 2DEG. longer through Figure 4.

Dependence

quantum 4.26: density density a

quantum

Schematic wire.

of

wire. the

depicting 0.7 Note

feature

how

electron

the

contact

density

region,

as

a (Con)

function

is

larger

of

position

for

the 89

4. Dependence of the 0.7 feature 90

Perhaps it is the large exchange at these low densities that drives the polarisation in ID. Maslov [64] has shown that interactions strictly within the ID region do not modify the conductance. The development of a spin polarisation in the low density 2D - ID contact regions however, could perhaps produce a conductance of 0.5 x 2e2/h via a lifting of the spin degeneracy in that region. Such a scenario would be somewhat consistent with the evolution of the conductance towards 0.5 x 2e2/h with increasing length and/or negative side gate bias. Thirdly, perhaps the observed length dependence is related to tunneling. Assuming the presence of some static spin polarisation, a conductance of 0.5 x 2e2/h would occur when the Fermi level lies between the spin-split band edges. A feature near 0.7 may occur due to the weighted average of current between the lower spin band and some fraction of electrons that are able to tunnel into the higher spin-band. Increasing the length of the ID region reduces this tunnel current and the conductance tends towards 0.5 x 2e2//?,, since fewer electrons are occupying the upper spin band. Despite the simplicity of this argument, a brief consideration of the Landauer Buttiker formalism (described in Chapter 2) requires that the energy range (position of the Fermi level) for which such a scenario might occur is extremely narrow. It is also difficult to see how a quasi­ plateau would result in this static picture. Further possible explanations for the length dependence reported in this chapter are discussed in Chapter 6, in the context of the phenomenological model presented there.

4.7 Conclusion

In conclusion, this chapter reports studies of ultra low-disorder quantum wires utilising a novel GaAs/AlGaAs layer structure which avoids the random impu­ rity potential associated with modulation doping. These devices are ideal for the study of electron correlation effects in ID, in particular the conductance 4. Dependence of the 0.7 feature 91 anomaly near 0.7 x 2e2/h. In common with other workers structure is found near 0.7 x 2e2/h in wires with l = 0, whereas in longer wires the dominant structure evolves towards 0.5 x 2e2/h with increasing side gate bias. Without further investigations on many samples it is not possible to definitively rule out disorder-related backscattering as leading to the length dependent results ob­ served. In spite of this, the data shows a consistent trend with varying length on six different samples and, taken together with the recent results in reference [22], indicate that both the length over which interactions occur and the density play an important role in determining the effect of correlation mechanisms upon electrical transport. Chapter 5

Confinement Potential: Bias Spectroscopy

5.1 Introduction

When an electron gas is confined to one dimension (ID) quantised energy sub­ hands are revealed. Using a method developed by Patel et al. [12], these ID sub-band energies can be measured via the application of a dc source-drain bias. This Chapter reports non-linear differential conductance measurements on ultra low-disorder quantum wires with lengths / = 0.5urn and l = 1.0[im. The sub­ band energies are studied as a function of top gate bias Vr and side gate bias Vs in an effort to characterise the ID confinement potential. Of particular interest, strong evidence for the existence of an anomalous sub-band edge is reported near 0.7 x 2e2/h in conductance measurements. Adiabatic coupling is explored in section 5.3, where a comparison between curved and rectangularly defined / = 1 /im quantum wires is made. In section 5.4 data taken using an asymmetric side gate bias are reported. Although the results are largely unexplained, they perhaps uncover an important aspect of the 0.7 x 2e2/h feature, namely the requirement for a symmetric potential. The

92 5. Confinement Potential: Bias Spectroscopy 93

Chapter concludes with a study of surface charge traps, and their effect on the 0.7 x 2e2/h conductance feature.

5.2 ID Sub-band Energies: DC Source-Drain Dependence

When a dc source - drain voltage bias (eVso) comparable to the ID sub-band spacing, is applied between the reservoirs either side of a quantum wire, the conductance exhibits non-linear behavior and additional plateaus (called half plateaus) are observed at conductance values midway between those observed in the linear regime [12]. For the case where the source - drain bias (Vsd) fore fills the condition eVso < Ef and transport remains ballistic, then the Landauer formula can be extended to finite bias. Following a derivation by Martin-Moreno [109] the current is given by:

(5.1) via extension of the linear response Landauer formula (see chapter 2 section 2.2). The differential conductance is then given by

(5.2)

Where © represents the classical step function (©(a;) = 1 for x > 0 and 0(rr) = 0 for x < 0). If the voltage dropped at the ‘bottle neck’ of the constriction is linear in Vsd, then Eu(Vsd) — En — /3eVsD, and equation 5.2 predicts that new plateaus will be introduced with increasing Vsd- Equation 5.2 then simplifies to

G = (2e2/h)[PN+ + (1 - /3)JV_) (5.3) 5. Confinement Potential: Bias Spectroscopy 94

Where N+ is the sub-band number such that E^+ < Ep -I- (3eVsD < En++1, and 7V_ is the sub-band number such that En_ < Ep — (1 — /3)eVsD < EW-+1- Equation 5.3 predicts new plateaus in the differential conductance at values of G = (N + (3)2e2/h, (N 4- 2(3)2e2/h, (N + 3/3)2e2/h..., where N is an integer, depending on whether the number of conducting sub-bands in the forward and backward polarity direction differ by 1,2,3 and so-on, respectively. The physical origin of the half-plateaus can be more clearly understood if an energy reference frame is used, in which the sub-band energies En remain constant as a function of Vsd■ In this reference frame, when a voltage difference is applied, the electro-chemical potential at the source reservoir is = Ep+(3eVsD, and the electro chemical potential of the drain reservoir is = Ep — (1 — (3)eVsD- The total current, / is due to electrons in the energy interval between ^ and /zs, whereas the differential conductance is related to electrons with energies close to the source and drain electro-chemical potentials. Therefore, the differential conductance at finite Vsd is a weighted average of two zero - Vsd conductances, one for a Fermi energy of Ep + (3eVsD, and the other for a Fermi energy of Ep — (1 — (3)eVsD■ When the number of occupied sub-bands is different for these two ‘Fermi energies’, the average differential conductance is not necessarily an integer multiple of 2e2/h. For most of the measurements reported in this Chapter it is assumed that (3 ~ 1/2, in accordance with calculations by Glazman and Khaetskii [110]. However, this is certainly not the case in all instances, particularly when the bias is dropped asymmetrically at pinch off. Figure 5.1 illustrates schematically the differential conductance (di/dv) as a function of source - drain bias. Note how the curves bunch at the integer and half - integer plateaus (indicated by the blue regions). The green shaded regions represent the effective chemical potential of the source (S) and drain (D), with the pink lines indicating the ID sub-band edges (an energy reference frame in 5. Confinement Potential: Bias Spectroscopy 95

di/dv

s D

s D

s D D Bias

ID sub-band energy

Figure 5.1: Schematic illustration depicting the differential conductance (di/dv) as a function of source - drain dc bias. Note how the curves bunch at the integer and half - integer plateaus. The green shaded regions represent the effective chemical potential of the source (S) and drain (D), with the pink lines indicating the ID sub-band edges. The bias required to move from one half-plateau (or integer plateau) to the next, is the ID sub-band energy spacing, indicated by the brown lines. which the band-edges remain constant with S-D bias is used, as described above). The bias required to move from one half-plateau (or integer plateau) to the next, is the ID sub-band energy spacing indicated by the brown lines.

5.2.1 / = 0.5gm quantum wire

Turning now to the experimental data, Figure 5.2 shows the differential conduc­ tance (di/dv) of a quantum wire of length / = 0.5/im. The top gate bias Vr was fixed at 1500mV and each trace is for a different side gate bias, (0 to -2V, in -20mV steps). The linear response integer plateaus are indicated at zero source

- drain bias by the bunching of traces at n x 2e2/h. As Vsd is increased the in- 5. Confinement Potential: Bias Spectroscopy 96

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 Source - Drain Bias, V (mV)

Figure 5.2: Differential Conductance (di/dv) of a quantum wire of length l = 0.5/cm. VT = 1500rnV. T = 50mK. teger plateaus become weaker, while new structure develops in the conductance at half-integer values of 2e2/h; these are the half-plateaus. Close observation reveals a slight increase in the ID sub-band spacing (the horizontal distance in S-D bias between half-plateaus ) as the side gate bias Vs becomes more strongly negative. That is, the half-plateaus at the top of the graph near 6.5 x 2e1 /h are more closely spaced in S-D bias than the half plateaus near say, 1.5 x 2e1 /h. The increase in sub-band energy is slight and at the very most is an increase of 1 meV (from near 7meV to 8meV). However, the data in Figure 5.2 is raw and has not been corrected for effects associated with the ohmic contact resistance. Assuming that the ohmic contact resistance remains constant for a fixed S-D bias, the potential drop across the quantum wire decreases with increasing conductance as the side gate bias is reduced and more of the voltage is dropped across the contacts. This behavior effectively stretches the differential conductance traces outwards, with increasing bias. Although Figure 5.2 still in- 5. Confinement Potential: Bias Spectroscopy 97

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 Source - Drain Bias, V (mV)

Figure 5.3: Differential Conductance (di/dv) of a / = 0.5/xm quantum wire. Vr = 630mV. T = 50mK. dicates a small increase in the ID sub-band spacing with increasing confinement from the side gate bias Vs, the effect due to ohmic contact resistance (which moves in the opposite sense) may somewhat cancel this effect in the data. It is therefore reasonable to expect a slightly larger change in sub-band spacing with gate bias than what is indicated in the data. Careful inspection of Figure 5.2 reveals that below the last plateau at 2e2/h the differential conductance deviates from the ideal case. Two distinctly differ­ ent effects are seen. Firstly, the last half-plateau near 0.5 x 2e2/h is strongly asymmetric. The left side (negative source) appears relatively normal, but the right side (positive source) appears suppressed. Such behavior has been seen in numerous devices of various geometries and is attributed to the non-linear voltage drop across the constriction near pinch-off [12, 111, 109, 24]. This effect is often further enhanced in these FET devices, since the effective gate bias (Vt for example) is with respect to the average potential of the ohmic contacts. Such 5. Confinement Potential: Bias Spectroscopy 98

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 Source - Drain Bias, V (mV)

Figure 5.4: Differential Conductance (di/dv) of a / = 0.5/im quantum wire. Vp = 550mV. T = 50mK. a ‘self-gating' effect can be corrected for such as in reference [24]. Secondly, just below the first plateau a feature near 0.85 x 2e2/h is observed which was first commented upon by Patel et al. [12]. Latter works, [13, 24] revealed this structure to be the anomalous 0.7 x 2e2/h conductance feature, discussed in the previous chapters of this thesis. Figure 5.3 shows the differential conductance of the / = 0.5/rra quantum wire for a top gate bias of Vp = 630mV. Comparison with the Vp = 1500mV data (Fig­ ure 5.2) indicates that the ID sub-band spacing has decreased slightly (perhaps 8meV to 6ineV) with decreasing top gate voltage, as expected (see discussion in section 4.1 and Figure 4.2). Note that this reduction cannot be explained by effects associated with the ohmic contacts, since the contact resistance increases with decreasing top gate bias. Figure 5.4 shows differential conductance data for a top gate voltage of Vp = 550inV. Again the sub-band spacing is just slightly smaller than the Vp = 630mV case. Three additional features appear in the 5. Confinement Potential: Bias Spectroscopy 99 conductance that cannot be explained within the context of the spin degenerate model [109] reviewed above. Firstly a strong 0.7 x 2e2/h feature appears just below the first plateau, oc­ curring near 0.85 x 2e2/h at high source - drain bias. At zero bias the feature dips down toward 0.5 x 2e2/h forming a characteristic cusp in the differential conductance. In addition, very weak conductance structure can be seen just below the second plateau; the 1.7 x 2e2/h conductance feature. This feature seems to appear in a stronger form for moderate top gate voltages (or densities). As confirmed (later) in a second device (/ = 1.0pan wire), the 1.7 x 2e2/h first becomes more defined with increasing top gate bias, but then disappears as Vp is increased further. Perhaps such puzzling behavior is consistent with measure­ ments made by Pyshkin et al [23] and Thomas et al [22]. In those cases, it was not the 1.7 feature but the 0.7 x 2e2/h feature that was found to strengthen at low and high densities but weaken in the moderate case. Finally, in addition to these features, a weak feature near 1.25 x 2e2/h can be seen at a large source - drain bias (7mV). A feature in this vicinity has been confirmed by measurements made on other devices (reported later), and also by Kristensen et al. [24, 112]. The origin of this feature is explained in Chapter 6. Figure 5.5 shows differential conductance data for the l = 0.5/im quantum wire with the top gate bias reduced further to Vp = 470mV. The low top gate bias induces only one sub-band in the quantum wire even without an applied side gate bias, Vs. Estimation of the ID sub-band spacing is difficult in this regime, without the presence of any half-plateaus. Note however, that the integer plateau at 1 x 2e2/h begins to weaken at approximately 2mV, corresponding to a sub­ band spacing near 4meV. Figures 5.6 and 5.7 show the differential conductance of the / = 0.5/am quan­ tum wire at a temperature T = 4.2K. The side gate bias Vs was kept constant at ground potential (OmV) and the top gate bias was swept over the range Vp 5. Confinement Potential: Bias Spectroscopy 100

H3 0.6

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 Source - Drain Bias, V (mV)

Figure 5.5: Differential Conductance (di/dv) of a l = 0.5/am quantum wire. Vr = 470mV. T — 50mK.

= 700mV - 820mV. The large sub-band energy spacing in this device facilitates measurements at elevated temperatures, and allows the 0.7 x 2e2/h feature to be studied over a large range of thermal energy, kT. The presence of the anoma­ lous feature below the first plateau can be easily identified, despite the thermal smearing. Consistent with the measurements reported in Chapter 4, the zero bias 0.7 feature rises upward from 0.6 to 0.75 x 2e2/h with increasing tempera­ ture for a large top gate bias, (Vt). This behavior can be seen at zero source - drain bias, together with a smearing of the plateau due to the increased temper­ ature. Chapter 6 presents a phenomenological model consistent with this data and provides an explanation for the temperature dependence of the feature. Figure 5.8 extends the measurements of figures 5.6 and 5.7 to include the high bias regime. At around Vs£>=10mV the differential conductance undergoes a cross-over to the strong non-linear regime, where a negative differential resistance is observed. Such behavior has been predicted by Kelly [113] due to a saturation 5. Confinement Potential: Bias Spectroscopy 101

Source - Drain Bias, VSD (mV)

Figure 5.6: Differential Conductance (di/dv) of a / = 0.5/rm quantum wire. Vt = 720mV - 820inV. T = 4.2K. = 0V.

-10.0 0.0 10.0 20.0 Source - Drain Bias, Vcn (mV)

Figure 5.7: Differential Conductance (di/dv) of a l — 0.5/im quantum wire. Vt = 700mV - 808mV. T = 4.2K. Ks' = 0V. 5. Confinement Potential: Bias Spectroscopy 102

5 1.0

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 Source - Drain Bias, V (mV)

Figure 5.8: Differential Conductance (di/dv) of a / = 0.5pm quantum wire. VT = 700mV - 808mV. T = 4.2K. of the number of carriers and a decrease of the transmission probabilities through the constriction at high source - drain bias bias.

5.2.2 1 fim quantum wires

As previously discussed in Chapter 4, the potential landscape in the ID - 2D contact region (see Figure 4.1) is believed to play an important role in adiabatic ballistic transport. Abrupt, sharp changes in the potential caused by the surface gates can lead to reflections of the electron wave. In order to investigate these effects, two devices of length l = 1.0pm (lpm:B and lpm:A) were fabricated from the same herterostructure and subject to identical processing. One device (l/ira:B) was designed to have smooth, curved contact regions joining the 2D reservoirs to the ID quantum wire. The other device (1pm:A) was fabricated according to the standard rectangular geometry used for all other devices studied in this thesis (refer to Chapter 3). Close comparison of the two quantum wires 5. Confinement Potential: Bias Spectroscopy 103

_8 -7 -6 -5 -4 -3 -2 -1 0 12 3 4 5 6 7 8 Source - Drain Bias (mV)

Figure 5.9: Differential Conductance (di/dv) of a / = 1.0/zra quantum wire. Vp = 240mV - 360mV. T = 50rnK reveals that device l/rra:B is in fact slightly longer due to the curved contact regions. This section explores the effects of the different coupling geometries on both the conductance in linear response and on the ID sub-band spacing. The investigation begins with device: 1.0/ira:A, where a comparison with the data taken on the 0.5/im quantum wire (reported in the previous section) reveals a small enhancement of the 0.7 x 2e2/h conductance feature. Figure 5.9 shows the differential conductance (di/dv) of a l = 1.0/ira quantum wire (device:A) as a function of dc source - drain bias. The effect of a non­ linear voltage drop between the source and the ID constriction can be modeled by adding a quadratic term to the dependence of the electrostatic potential at the constriction with source - drain bias [109]. The potential then becomes

Uq(Vsd) = U0- peVsD + ieVso/Z- In this case the half - plateaus drift to higher values of conductance (or lower values if 7 is negative) as a function of Vsd- If 7 becomes large, the potential drop is highly non-linear and half-plateaus will 5. Confinement Potential: Bias Spectroscopy 104

Source - Drain Bias (mV)

Figure 5.10: Differential Conductance (di/dv) of a / = 1.0fim quantum wire. Vr = 385mV. T = 50mK. not occur. Inspection of all of the differential conductance data presented in this thesis reveals that 7 is positive and non-zero, but not too large since clear half plateaus are seen throughout. For the case of low top gate voltages Vr, studies of the I — V characteristic reveal strong non-linearities. Returning to Figure

5.9, note that the top gate bias Vr was stepped in the range from Vt = 240mY - 360mV, covering the threshold voltage to an occupancy of n = 2. At these low top gate voltages the contact resistance can be several 10s of Mls, and the potential drop between the source and the ID constriction is highly non-linear. These non-linearities are observed in Figure 5.9 primarily in the form of curved, rather than flat integer and half-plateaus. Of further interest, the data appears asymmetric about zero bias with the slope of the half-plateaus being different for the source and drain. This behavior is likely to be related to a difference in resistance between the source ohmic contact and that of the drain at these low top gate voltages. 5. Confinement Potential: Bias Spectroscopy 105

Figure 5.10 shows the differential conductance (di/dv) of an / = 1.0 fim quan­ tum wire (device:A) at temperature T = 50mK. In this case the top gate bias was held constant at Vp = 385mV and the side gate bias was stepped in the range Vs = -200mV to -320mV. Consistent with the data presented for the l = 0.5fim quantum wire in the previous section, a characteristic cusp is seen below the first integer plateau near 0.7 x 2e2/h at zero bias. As the source - drain bias voltage is increased the feature rises upward toward 0.9 x 2e2/h consistent with data taken by others [23, 112]. Figure 5.5 shows data taken on the l = 0.5fim quantum wire for a top gate bias of Vp = 470mV. From an estimation of the width of 1st integer plateau, a ID sub-band spacing of approximately 4meV was suggested (section 5.2.1). A similar estimate of approximately 4meV is made here for the l = 1.0fim quantum wire (device:A), with Vp set to 385mV. This estimate is consistent with the temperature dependence of the linear response conductance. Increasing the top gate bias has a number of effects, as discussed in the pre­ vious section for the case of the / = 0.5//m quantum wire. Figure 5.11 displays the differential conductance of the / = 1.0/im quantum wire (device:A), with Vp set to 600mV for three different temperatures, T = 50mK, 450mK and 700mK. Firstly, the increased top gate bias produces a strengthening in the electron con­ finement and the sub-band spacing increases slightly in comparison to the data taken with Vp = 385mV. In addition to this minor perturbation, the 0.7 feature seen below the first integer plateau can be observed to increase in strength. Im­ portantly, (see Chapter 6 for a detailed discussion) the cusp feature near zero bias becomes more defined as the top gate bias is increased from Vp = 385mV to 600mV. Increasing the temperature from T = 50mK to T = 700mK has very little effect, both on the single particle plateaus and on the anomalous conductance structure, as shown by Figure 5.11. At T = 700mK the effective thermal en- 5. Confinement Potential: Bias Spectroscopy 106

Source - Drain Bias (mV)

Source - Drain Bias (mV)

r> I , I ■ I , I ■ I I---T------, I---- . , I -5 -4 -3 -2 -1 012 3 4 5 Source - Drain Bias (mV)

Figure 5.11: Differential Conductance (di/dv) of a / = 1.0gm quantum wire. Vr = 600mV. T = 50mK, 450mK, 700mK (top to bottom). 5. Confinement Potential: Bias Spectroscopy 107

Source - Drain Bias (mV)

Figure 5.12: Differential Conductance (di/dv) of a l = 1.0gm quantum wire. Vr = 700mV. T = 50mK. ergy (4kT) is less than 0.3meV and therefore for the case of the integer and half plateaus, this result is hardly surprising given a sub-band spacing of «6meV. This result is important for the case of the anomalous 0.7 feature, since any proposed explanation of the conductance structure should also explain this char­ acteristic energy scale. For example, an explanation based on electron-electron interactions (see Chapter 2), should incorporate a mechanism where by the en­ ergy scale exceeds at least 0.3meV in order not to be destroyed by the thermal energy. Turning now to Figure 5.12, where the top gate bias is increased to Vr = 700mV, (T = 50mK) close observation reveals the presence of a strong 1.7x2e2/h below the n = 2 integer plateau, in addition to the structure near 0.7. Following the behavior of the ‘cusp’ feature near 0.7 x 2e2/h with increasing top gate bias (through Figures: 5.10, 5.11, 5.12), a clear dependence is observed. The position of the feature at zero bias pulls down toward 0.5 x 2e2/h with increasing 5. Confinement Potential: Bias Spectroscopy 108 top gate bias, consistent with the results reported in Chapter 4. In addition to this behavior, the strength and width in source - drain bias can also be seen to increase with increasing gate bias. These results, taken together with the data presented in Chapter 4 reinforce the notion of a spin polarisation occurring in the region of the quantum wire. A detailed explanation for the dependence of the differential conductance with source - drain bias is given in Chapter 6, where a phenomenological model of a density dependent spin polarisation is proposed.

5.3 Adiabatic Coupling: Different Geometries

Recent calculations [114] have reinforced the importance of the 2D:ID coupling region on the nature of electron interactions in quantum wires. Although the region where the 2DEG makes contact with the quantum wire has been known for some time [41, 115, 62] to be significant in adiabatic transport, the implica­ tions for the 0.7 conductance feature are yet to be explored. The initial impetus for fabricating two l = 1 fim quantum wires with curved and rectangular cou­ pling was to investigate how these different geometries affect if at all, the 0.7 conductance feature. However, despite the attempts to keep all other variables constant between the two devices, they in fact also differ in the effective length and width of the lithographically defined ID region.

5.3.1 Curved and Rectangular l/im Quantum Wires

Figure 5.13 shows the conductance G of the two devices as a function of side gate bias Vs for a temperature T = 50mK. The inset to each figure illustrates the coupling geometry. A comparison between the two figures reveals that in fact the data for the device with the curved coupling geometry exhibits resonance­ like conductance perturbations. These predominantly occurs below the first and second integer plateaus, although some additional perturbation can be seen at defines 5. BOTTOM: Figure

Confinement

5.13: the Conductance, G (2e /h) Conductance, G (2e /h) effective

Conductance Device:B.

Potential: length

Insets

G of Side

as

the

Bias illustrate a

Gate function ID

Spectroscopy region.

Voltage, the

of

side T geometrical

=

gate

50mK V s

bias (V)

for

form Vs- both

TOP: of

devices. the

Device:A. device, 109

l

5. Confinement Potential: Bias Spectroscopy 110

0.5

-8 -7 -6 -5-4-3-2-101234 Source - Drain Bias (mV)

Figure 5.14: Differential conductance of 1.0/im (Device B). T = 50mK, Vs = -800mV. Vt = 600mV - 740rnV. low side gate voltages Vs for the higher plateaus. A number of suggestions can be made to explain the presence of these resonance features in the conductance of device:B. Firstly as discussed above, making the 2D:ID coupling region smooth and curved has in fact also increased the effective length / of the ID region. Quantitatively it is difficult to estimate the amount of extra length added by the curved geometry due to the fact that the width is also changing. Nixon and Davies [41, 115] have shown that due to the background random potential, conductance quantisation is highly sensitive to the length of the ID region. The devices studied here were fabricated from an ultra low-disorder heterostructure, free from the disorder associated with modulation doping and the ballistic mean free path exceeds 160\im [99]. Despite this, fabrication of surface gate structures (in particular wet etching in acid) can alter the intrinsic background potential. Disorder associated with the gates themselves can also produce rapid inter-mode scattering together with consecutive backscatter events 5. Confinement Potential: Bias Spectroscopy 111 that ultimately breakdown the conductance quantisation entirely. Of further consequence, the smooth, adiabatic coupling of device:B may in fact promote phase coherent reflection of electron waves, leading to conductance features [116, 117]. In addition to the increase in effective length, device:B is also narrower in width. This is confirmed by a comparison of the data taken on each device at T = 4.2K, shown in Chapter 4 section 4.4. The curved geometry has slightly reduced the width and there by increased the ID sub-band energy spacing. A larger sub-band spacing accounts for the well defined conductance plateaus at elevated temperatures, in comparison to device:A. Figure 5.14 shows the differential conductance (di/dv) of device:B as a func­ tion of source - drain bias for top gates Vt = 600mV to 740mV, with Vs fixed at -800mV. The data exhibit clear integer and half - plateaus consistent with the differential conductance data presented for device:A in Figures 5.11 and 5.12, but here the top gate Vt is stepped as opposed to the side gate bias Vs- Again a cusp is observed for the 0.7 x 2e1 /h feature in keeping with the data presented previously, together with a particularly strong 1.7 feature occurring just below the second integer plateau. Figure 5.15 displays a series of slices through the differential conductance plot, for constant source - drain bias Vsd■ Starting with an initial offset bias of approximately 0.5mV, the position in conductance of the 0.7 feature can be observed to initially decrease as the offset bias is counteracted, then return to higher positions as the bias is increased. In terms of the source - drain data shown in Figure 5.14, Figure 5.15 corresponds to beginning on the left side of the cusp, and moving through zero to the right side. Turning now to a comparison of the 0.7 x 2e2/h feature for the two / = 1 pm quantum wires, an important result is uncovered. Taking both the differential conductance and linear response data together, no clear difference is seen in ei­ ther the strength or position of the 0.7 feature between each device. This is 5. Confinement Potential: Bias Spectroscopy 112

o QJ — VSD = 00mV e VSD = 0.2mv u - VsD = 0-4mV 3 VSD = 0.6mV TD C ---- VSD = 0.8mV o VSD=1.0mv U VSD = ' -2mV

~ VSD='4mV

U.60 0.62 0.65 0.68 0.70 0.73 0.75 Top Gate Voltage, V (V)

Figure 5.15: Conductance G of 1.0gm quantum wire as a function of top gate bias. (Device B) T = 50mK. Vsd — 0 to 1.4mV in steps of O.lmV. a key result and has important implications for our understanding. Consistent with data presented in Chapter 7 and in reference [3], the existence and ap­ parent strength of the 0.7 feature appears to depend only very weakly on the residual background disorder. Of further interest, it is found that single particle interference phenomena such as conductance resonances can co-exist with the 0.7 feature. Again taken with the results presented in Chapter 7 and reference [3], these results suggest that the feature may not explicitly require ultra low- disorder systems to occur and more profoundly, may in fact be an intrinsic effect of the ID electron gas. Finally Figure 5.16 presents differential conductance data for device:B at a temperature T = 4.2K. The enhanced confinement in this quantum wire is due to the lithographically defined curved coupling geometry. The larger sub-band spacing allows the 0.7 x 2e2/h feature to be studied over a large temperature range. Note the behavior of the feature below the first plateau. In comparison 5. 600nrV Figure Vs

Confinement —

0,

5.16: to Vf

di/dv (2e7h) di/dv (2e7h) 708mV

= Differential -9

320mV

-8

lmV Potential: -7

to -6 steps.

conductance 431mV -5

Source Source -4

Bias

-3 lmV

-2

Spectroscopy - -

of steps. Drain Drain -1

a

0 1.0

BOTTOM: fim Bias 1 Bias

2

(Device

(mV) (mV) 3

4

B) 5 Vs

T 6 =

= -800mV. 7

4.2K. 8

9

TOP:

V t 113

=

5. Confinement Potential: Bias Spectroscopy 114 to the data taken at T = 50rnK, the cusp feature is observed to flatten with increasing temperature. Consistent with the linear response data presented in Chapter 4, the feature remains close to 0.7 x 2e2/h for temperatures above approximately 2K depending on the sub-band spacing. Importantly the feature is not destroyed at elevated temperatures but behaves in much the same manner as a single particle ID sub-band edge. Comparing the data taken with no side gate bias Vs = 0 (TOP) to the data taken with Vs = -800mV, a clear difference can be seen in the quality of the data. Perhaps the bias from the side gates effectively smoothes the electric field, potential and frees up any trapped charges. The occurrence of random telegraph signals (RTSs) [118] can be identified on some traces for Vs = 0, consistent with this interpretation.

5.4 Asymmetric Confinement Potentials

The following section presents data taken on a / = 0.5fim quantum wire, where the side gates were biased unevenly. Measurements on split-gated HEMTs have long made use of this technique to shift the ID channel laterally. For the case of the 0.7 x 2e2/h structure, the technique was used initially by Thomas et al. to rule out explanations based on an impurity trapped in the ID region [13]. Contrasting these results, measurements made on the l = 0.5/im quantum wire with an asymmetrically gate bias suggest profound implications for the 0.7 feature. Figure 5.17 shows linear response conductance G as a function of side gate bias Vs. The top gate bias was fixed at Vp = 800mV and the temperature T = 50mK. The data in the top of the Figure was obtained by sweeping the left side gate (voltage indicated on x axis (Vs)), and then stepping the right side gate from 0 V on the left to -2 V on the right. The data in the bottom of the Figure was obtained in the same manner, with the role of each gate reversed, i.e. 5. Confinement Potential: Bias Spectroscopy 115

5 2.0

B 10

-1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 Side Gate Voltage, Vs (V)

fj 2.0

e l.o

-1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 Side Gate Voltage, Vs (V)

Figure 5.17: Conductance of a l = 0.5fim, quantum wire, with an antisymmetric side gate bias. Vr = 800mV. T = 50mK. TOP: left side gate sweeps bias, right side gate steps from OV -2V. BOTTOM: left side gate steps bias OV to -2V, right side gate sweeps. with the right gate sweeping and the left gate now stepping. Close examination of the top graph reveals a 0.7 x 2e2/h feature that dramatically increases in strength as the right gate is stepped from OV to -2V. Such behavior may be consistent with the 0.7 conductance structure arising from a spatially constant impurity. In the case of split gate devices the channel may be shifted laterally from left to right by an asymmetric gate bias since the bias moves the depletion region. In the case of the enhancement mode devices considered here however, the situation is further complicated by the presence of both the lithographically 5. Confinement Potential: Bias Spectroscopy 116 defined quantum wire and the top gate bias Vr. Never the less, the very strong dependence with stepped gate bias requires some explanation. If the apparent behavior of the 0.7 feature was related to an impurity in or near the ID region then by shifting the channel in the other direction the dependence should weaken. What is in fact observed, as shown in the bottom graph of Figure 5.17, is the exact opposite to this expected result. At least two suggestions can be made to explain the behavior. Firstly, micro-graphs of the surface gates indicate the presence of a fair degree of roughness associated with the etching process. How this roughness affects the electrostatically defined potential is unknown, but it is not too hard to image that the bare lithographically defined potential landscape contains a fair degree of disorder. Perhaps the side gate bias shields or smoothes this apparent roughness as indicated by the schematic illustrations of the ID potential either side of the graphs in Figure 5.17. An alternative explanation is that the asymmetric side gate bias has little effect on the ID potential landscape, but greatly modifies the 2D:ID coupling. Such a notion has been suggested in Chapter 4, section 4.6 but clear evidence is still lacking. Figure 5.18 shows the conductance G as a function of asymmetric side gate bias, at at temperature T = 4.2K. Even at elevated temperatures, a clear de­ pendence is seen in the strength of the 0.7 x 2e2/h feature with stepped side gate bias. These puzzling results require further investigations. In particular, confirmation of this behavior on a second, different sample would be desirable.

5.5 Further exploration of the confinement po­ tential

Throughout the course of this dissertation linear response conductance data has been presented as either a function of the side gate bias Vs or the top gate bias Vp. Is there any specific difference between these? In order to answer this 5. Confinement Potential: Bias Spectroscopy 117

-1.0 -0.8 -0.6 -0.4 Side Gate Voltage, Vs (V)

Figure 5.18: Conductance of a / = 0.5/ura quantum wire with a antisymmetric side gate bias. Vp = 500mV, T = 4.2K. question a careful study of the entire data set should be carried out, however the results seem to depend on both the details of the sample and the specific voltage applied to the gates. Previously as reported in section 5.3, the presence of a side gate bias was found to cleanup the data and reduce the occurrence of RTSs and other spurious sources that may degrade the conductance data. In contrast to this, Figure 5.19 shows conductance G as a function of top gate bias Vp for two different quantum wires (/ = 0.5/im and / = 1.0fim). In the TOP plot clean data is obtained despite the application of only a small side gate bias Vs. This behavior is reinforced in the BOTTOM plot where two traces for Vs = 0 and Vs = -800mV are compared. In this case the side gate bias can be seen to introduce conductance perturbations on the plateaus, contrasting the apparent behavior reported in section 5.3. 5. Confinement Potential: Bias Spectroscopy 118

0.50 0.55 0.60 Top Gate Voltage, VT (V)

Side Gates = -800mV

Side Gates = 0V

Top Gate Voltage, VT (V)

Figure 5.19: TOP: Conductance of a l = 0.5/im quantum wire, as a function of top gate voltage (Vr) for different side gates, (Vs = 0, -0.03mV, -0.06mV, -0.09mV, -O.llOmV. T = 50mK. BOTTOM: Conductance of a / = 1.0//m quantum wire (Device B), as a function of top gate voltage (Vt) for different side gates, (Vs = 0 and -800mV. ) 5. Confinement Potential: Bias Spectroscopy 119

— State A — State B

-1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 Side Gate Voltage, V§ (V)

Figure 5.20: Conductance of a / = 0.5\±m quantum wire in two distinct states. The different states arise due to surface charge traps that modify the potential landscape. T = 50mK.

5.5.1 Charge traps: Potential Perturbations

As a heterostructure is cooled from room temperature to below T = 50mK, car­ riers return to localised states at donor locations. Generally however, defects (particularly on the surface) contain dangling bonds that have potential to cap­ ture and trap carriers. Of further importance, the rate at which a sample is cooled can also determine weather the free carriers have enough thermal energy and time to escape capture from defect dangling bonds. Trapped charge can distort the potential landscape and interfere with phenomena that rely on low disorder structures. In order to study these effects on the strength of the 0.7 con­ ductance feature, the / = 0.5^m quantum wire was cooled to T — 50mK at two different rates. Figure 5.20 shows data from these two conditions. ‘State A’ was obtained via rapid cooling to base temperature, while ‘State B‘ was produced by slow cooling. The inset to the Figure shows the transconductance (dG/dVs) of 5. Confinement Potential: Bias Spectroscopy 120 the two traces, where the traces have been offset to facilitate comparison. Clearly for the case of State B, the anomalous conductance structure near 0.7 x 2e1 /h ap­ pears stronger. Accompanying this difference in strength of the 0.7 feature, the two states also were characterised by different threshold voltages Vthres required to induce an electron gas under the top gate. This behavior is consistent with the notion that trapped charges can strongly modify the effective electrostatic potential near the ID region.

5.6 Discussion of Results and Conclusions

Throughout the course of this chapter the focus has remained on the anomalous conductance feature near 0.7 x 2e2/h. Sections 5.21 and 5.22 have shown how the anomalous feature appears to behave in a very similar fashion to a single particle sub-band edge. The characteristic ‘cusp' shape that occurs below the first integer plateau has been shown to be reproducible in the three different samples reported here and in the data of others [23, 13]. The region in bias voltage for which the differential conductance pulls down toward 0.5 x 2e2/h increases with increasing top gate bias Vt consistent with the results presented in Chapter 4. Yet important questions remain, both from a microscopic point of view and in terms of the phenomenological variables such as density, sub-band energy and temperature. The following Chapter (Chapter 6) presents a simple phenomenological model that is consistent with the key results presented in both Chapter 4 and this Chapter. However despite this, some of the fundamentally important observations made in this chapter remain unexplained. In particular the results of section 5.3, where the hardness of the lithographic potential was shown to weaken the 0.7 structure in a systematic way. Further, the role played by the 2D:ID coupling region remains an important question, particularly in the light of recent calculations and predictions for correlated phenomena [114]. 5. Confinement Potential: Bias Spectroscopy 121

Section 5.5 presented evidence that the strength of the 0.7 feature is sensitive to the cooling rate and presumably charge trap density, although this behavior has only been confirmed in one sample. In contrast to this sensitivity, a com­ parison between the two l = 1.0/zm wires with different geometries reveals that single particle interference phenomena can co-exist with the 0.7 conductance structure. Perhaps these contradictory results can be reconciled with a more detailed study of how the 0.7 structure depends on the details of the potential landscape. Chapter 7 investigates some of these effects further. In conclusion, using a source - drain bias technique developed by Patel et al. [Ill] the ID sub-band energies were studied as a function of top gate bias

Vt, side gate bias Vs and temperature T. The Chapter has focused on the anomalous conductance structure occurring below the first integer plateau near 0.7 x 2e2/h. The apparent strength of the feature is observed to increase with increasing top gate bias, consistent with the results presented in Chapter 4. Studies of the potential landscape indicated that the 0.7 conductance structure is both sensitive and insensitive to perturbations of the potential depending on the scale and nature of the disturbance. Chapter 6

A Density Dependent Spin Polarisation in ID

6.1 Introduction

In the presence of strong exchange coupling, electrons can spin polarise in the absence of an applied magnetic field. Such a scenario is predicted for a variety of different systems, including three dimensional (3D) metal nanowires [77] the two dimensional electron gas (2DEG) [69], one dimensional (ID) ballistic quantum wires [58, 70, 53, 76, 55] and circular quantum dots [119]. In the case of ID, interactions become increasingly important at low densities and models such as the Tomanaga-Luttinger liquid theory [59, 64] are required to describe them. Despite the large exchange energy present in low-density ID systems, there are strict theoretical arguments against magnetic ordering [72]. The notion of a ID spin polarised ground state remains the subject of wide debate, in particular since the important experimental results of Thomas et al [2]. Together with a phenomenological model, this chapter focuses on the ex­ perimental results of semiconductor quantum wires in zero magnetic field that provide strong evidence in favor of a spin energy gap developing in the ID region.

122 6. A Density Dependent Spin Polarisation in ID 123

This density-dependent energy gap between spin-up and spin-down electrons is revealed in transport measurements as an anomalous conductance feature in the range 0.7 — 0.5 x 2e2/h. A feature near 0.7 x 2e2/h can be seen in some of the earliest ID ballistic transport measurements on quantum point contacts [1]. In 1996, Thomas et al [2] revealed that the anomalous feature was related to spin, by showing that it evolves smoothly into the Zeeman spin-split level at 0.5 x 2e2/h with an in­ plane magnetic field. Since that time, experimental studies have concentrated on the behavior of the anomaly as a function of temperature, source-drain bias, magnetic field, thermopower, wire length and density [13, 17, 24, 10, 22, 23, 25]. Together with these investigations, numerous mechanisms to explain the origin of the conductance feature have been proposed [92, 84, 82, 79, 54, 93, 57]. Amongst the most compelling of these models is the notion of Fermi-level pinning [93, 57, 79], in the presence of a static spin energy gap. In this Chapter an alternative simple phenomenological model is proposed that appears to explain all of the characteristic details of the 0.7 feature by means of a density dependent spin gap arising in the region of the quantum wire. Starting from full spin degeneracy at pinch off, the spin gap begins to open as the ID channel is populated, as illustrated in Figure 6.2. The non-linear dependence of Fermi level (Ep) with density or gate voltage can be traced to the singularity in the ID density of states; p ~ e-1/2 [79]. Figure 6.1 illustrates schematically how the non-linear Fermi level arises. The polarisation is assumed to open linearly with increasing electron density. Important key differences exist between this simple model and other expla­ nations based on Fermi level pinning. This model explains why a feature often occurs close to 0.5 x 2e2/h, without the co-existence of a feature near 0.7 x 2e2/h. Consistent with bias spectroscopy data, this model also predicts that the spin gap will continue to open as the Fermi level rises above the spin-split levels. 6. A Density Dependent Spin Polarisation in ID 124

Figure 6.1: Illustration depicting a density dependent spin gap opening linearly with increasing density (n) or gate voltage (V$). The Fermi level is non-linear with gate voltage because of the non-linear density of states with energy. The energy gap between spin up and down is assumed to increase linearly with den­ sity. 6. A Density Dependent Spin Polarisation in ID 125

* Fermi level

Vs ~ n

Figure 6.2: Illustration depicting a density dependent spin gap opening linearly with increasing density (n) or gate voltage (Vs). El, E2, E3 indicate the ID sub-band edges. The Fermi level indicated by the dashed line, is non-linear with density n, due to the singularity in the ID density of states.

In contrast to the case of a static spin polarisation the sub-bands remain spin degenerate until they are populated, after which the spin gap opens linearly with increasing density. Such behavior is suggestive of interaction effects and is consistent with a spin polarisation driven by exchange as predicted by Wang and Berggren [58, 70]. In that case the polarisation weakens with increasing kinetic energy as higher sub-bands are populated, (see Figure 6.2). The proposed model is consistent with conductance data obtained on ultra- low-disorder GaAs/AlGaAs quantum wires free from the disorder associated with modulation doping. Although the model is illustrated via comparison with this data (reported in Chapters 4 and 5), the model is not limited to these samples but appears to be consistent with all of the key published data on the conductance feature. Turning now to the conductance measurements, note that the presence and shape of conductance anomalies depends on the details of the system parameters and the rate at which the spin gap opens with ID density. Figure 6.3 depicts 6. A Density Dependent Spin Polarisation in ID 126

1.75 _ / = 0.5 gm

V = 1.5 V

l = 0.5 Jim

0.75 —

0.75 =,

Figure 6.3: Schematic showing the three main scenarios that lead to features near 0.5 x 2e2/h and 0.7 x 2e2/h in the conductance at both zero (left) and finite source - drain bias (right). The shaded regions below each graph represent the Fermi distribution. Scenario (I) occurs if the spin gap is large in comparison to kT. Scenario (II) occurs at high temperature when kT is close to the size of the spin gap. Finally scenario (III) occurs for the case of weak spin splitting. Shown on the left is real data taken from Chapters 4 and 5 to illustrate the three scenarios. 6. A Density Dependent Spin Polarisation in ID 127 the three main scenarios that may occur. Scenario (I) occurs if the spin splitting opens quickly with increasing density, so that an appreciable energy gap develops in comparison to the thermal energy kT. In this case a fully resolved spin-split plateau at 0.5 x 2e2/h is seen in linear response conductance G and no feature near 0.7 x 2e2//i, as shown on the left side of Figure 6.3. The right side of the figure shows the dependence of the differential conductance (di/dv) with finite source-drain bias, where the thick lines represent conductance plateaus. Due to an averaging of the conductance at the chemical potential of the source S and drain D, half-plateaus at 0.5, 1.5 x2e2/h occur at finite bias when the two potentials differ by one sub-band as discussed in detail in Chapter 5 and references [12, 109]. In a large applied magnetic field experiments by Patel [111] indicate that with the lifting of the spin degeneracy a finite source-drain bias produces quarter plateaus at 0.25, 0.75, 1.25, 1.75, x2e2//i, in addition to the half plateaus already present for the unpolarised case. One of the key differences between a density dependent spin polarisation driven by interactions and the static case of an applied field is the following. For the case of where the polarisation results from interactions, the quarter plateaus at 0.25 & 1.25 x2e2/h will not occur if the spin gap has not yet opened by more than kT as Ep crosses over the band- edges. As the density increases however, the spin gap opens further than kT and a differential conductance plateau near 0.75 x 2e2/h is observed due to source and drain chemical potentials differing by one spin sub-band. Now consider Scenario (II) in figure 6.3. At elevated temperatures the ther­ mal energy kT is comparable to the spin gap as Ep crosses the band edges and a feature near 0.5 x 2e2/h will not be resolved (left side of scenario (II)). Instead, as Ep moves towards the upper spin band-edge the ‘tail' of the Fermi function crosses it, and a conductance between 0.5 and 1 x 2e2/h occurs depending on the number of electrons thermally excited into the upper spin band. A quasi-plateau 6. A Density Dependent Spin Polarisation in ID 128 is seen near 0.7 x 2e2/h because as Ep increases, so does the spin gap and the number of electrons thermally excited into the upper spin band remains approx­ imately constant. Figure 6.2 shows that Ep and the upper spin band edge move together for a time, before Ep crosses it. The right side of Scenario (II) (Figure 6.3) illustrates the behavior of the differential conductance at elevated temperatures. In contrast to the low tem­ perature case (Scenario (I)), the feature remains close to 0.7 x 2e2/h even when the source - drain bias is close to zero. Scenario (III) (Figure 6.3) illustrates the case for weak spin splitting. At low temperatures a feature near 0.5 x 2e2/h is absent if the spin gap remains very small in comparison to kT. With increasing density however, the spin gap opens by more than kT and a feature near 0.75 x 2e2/h can be seen in the differential conductance due to the source and drain potentials differing by one spin band. Note that the spin gap remains open with the application of a source - drain bias since, to a first approximation the bias does not change the ID density. Conductance data taken on two different samples is now reviewed to illustrate the consistency of this simple model with the experimental results (data taken from Chapter 4 and 5). Figure 6.4 shows data taken on a / = 0.5^m quantum wire at T = 4.2K (dashed curves) and T = 50mK (solid curves). The data taken at T — 50mK show a characteristic evolution towards the spin-split level at 0.5 x 2e2/h, with increasing 2D density (right to left). At T=4.2K the position of the anomalous feature does not evolve with 2D density but remains close to 0.7 x 2e2/h as Vp is varied. Very similar behavior is seen for two other devices of length l = 1/rm, as discussed in Chapter 4. Comparison with the proposed model suggests that the low temperature data (solid curves) are consistent with scenario (I) of figure 6.3. As the temperature is raised from T = 50mK to T = 4.2K, scenario (II) relates the conductance feature to the model of a density dependent spin gap. Consistent with this model, the feature in Figure 6.4 only 6. A Density Dependent Spin Polarisation in ID 129

Figure 6.4: Conductance of a / = 0.5[im quantum wire as a function of side gate voltage for top gate voltages in the range; Vt = 420rnV- 1104mV (right to left). The red curves are for T = 4.2K and the black curves are for T = 50mK. For the left most curves the position of the feature moves upward from 0.5 x 2e2/h to 0.7 as the temperature is raised. (From Chapter 4.) tion Figure 6. (c)V

A rT

of =700mV. Density

6.5: source

Differential di/dv (2e7h) di/dv (2e7h) di/dv (2e /h) di/dv (2e /h)

(d)Vr=700mV, - Dependent

drain

bias conductance

Spin V

sd B=2.75T. ,

Polarisation T=50mK.

of

a

(From l

=

(a) 1.0

Chapter

in fim Vr=385mV.

ID

quantum

5.)

wire (b)Vj

as ’ =600mV.

a

func­ 130

6. A Density Dependent Spin Polarisation in ID 131 evolves downwards towards 0.5x2e2//i for low temperatures but becomes ‘flatter’ at high temperatures. The broadness of the quasi-plateau near 0.7 x 2e2/h results from the small slope of the Fermi function at high temperatures where it overlaps the upper spin band edge. Figure 6.5 shows the differential conductance (di/dv) of a l — lfim quantum wire as a function of dc source-drain bias (Vsd), for different side gate voltages (Vs) at T = 50mK (data taken from Chapter 5). Comparing the differential conductance at three different 2D densities (corresponding to Vt = 385mV (a), 600mV (b) and 700mV (c)). Clear half-plateaus at 0.5,1.5 x 2e2/h can be seen developing near ±1.5mV, however the last half-plateau on the right side of each graph is suppressed due to the asymmetric bias across the constriction at pinch- off [109]. Note that quarter plateaus near 0.25 x 2e2/h are absent but strong features near 0.9 — 0.75 x 2e2/h can been easily identified, consistent with the model of a dynamic polarisation. The 0.75 feature often occurs closer to 0.85 x 2e2/h since the upper spin band contributes to the current when the Fermi function overlaps it significantly. We now focus on the regions close to zero bias where a characteristic ‘cusp’ feature is seen below the first and (to a less extent) second integer plateaus. In the context of our simple model, increasing the spin-gap is comparable to reducing the thermal energy, kT by moving from scenario (II) to scenario (I). Our data indicates that with increasing VT, the spin gap opens further and the zero bias conductance tends towards 0.5 x 2e2/h. Moving from scenario (II) to scenario (I) of our model, we observe a characteristic cusp in the dc-bias data developing with decreasing temperature or alternatively, increasing top gate Vt as shown in Figure 4. With the application of a small external magnetic field the gap opens a little further and the cusp feature widens (Fig 4. (d)) however, interpretation is made difficult due to effects associated with the magneto-resistance of the ohmic contacts. The characteristic shape of the cusp results from the spin gap opening 6. A Density Dependent Spin Polarisation in ID 132

Figure 6.6: TOP: Conductance of a l = 1.0gm quantum wire as a function of Vs for VT in the range, 270mV - 800mV (right to left). The dashed curves are for a constant offset bias of Vsd — 0.5mV and the solid curves for Vsd = 0. BOTTOM: The application of a constant bias produces the non-monotonic feature in the conductance. Left side is conductance data, right side is a schematic relating the potentials of the source (S) and drain (D) to the feature. further with side gate bias (Vs) for each trace. Lastly the effect of a constant dc offset bias on both the shape and position of the conductance feature is explored. Figure 6.6a shows conductance data for a / = 1.0/xm quantum wire at T = 50mK. The solid lines are measurements made with the excitation voltage < 10fiV and the dashed lines are measurements made with a constant dc offset bias of Vsd — 0-5 mV. A non-monotonic feature is seen in the differential conductance when the dc offset bias is present. Analogous to the case of thermal excitation (Figure 6.4), the offset bias distributes the current between both spin-bands, as shown in the schematic illustration of Figure 6.6c. This non-monotonic feature has been seen for at least five samples of 6. A Density Dependent Spin Polarisation in ID 133

di/dv

^------m S - D Bias

Source - Drain Bias (mV) s D 0.5 0.75 (di/dv)

Figure 6.7: Schematic illustrating how the density dependent spin gap leads to a cusp feature below the first integer plateau. As the ID density is increased, the source - drain bias required to move above the second spin band edge also increases. Bottom right shows a zoom of S-D data. various lengths reported in this dissertation, and also in the data of others [22]. Note that such a feature may occur even without an offset bias if the spin-gap remains closed initially as Ep passes, but then rapidly opens with the upper spin band moving above the Fermi-level. It is unclear how models evoking a static polarisation or Fermi-level pinning could explain the non-monotonic feature. A recent work [112] points out a weak differential conductance feature oc­ curring near 1.25 x 2e2/h for a source - drain bias greater than the sub-band spacing. Close examination of the data on the l = 0.5//m wire (see Figure 5.4) also reveals the presence of this feature. In the context of the proposed model, 6. A Density Dependent Spin Polarisation in ID 134

Increasing 2D density (Vt)

1D density, n

Figure 6.8: Schematic illustrating the anticipated spin -gap dependence with 2D density a feature near 1.25 x 2e2/h would result from (say) the source potential S being above the spin degenerate second sub-band (2 x 2e2/h) and the potential of the drain being between the two spin-split, levels of the first sub-band (0.5 x 2e2/h). As discussed in Chapter 4 and reference [10] and confirmed by others [22, 23] the 0.7 feature evolves towards 0.5 x 2e2/h with increasing top gate bias Vt or 2D density. In terms of the proposed model this behavior is consistent with the spin gap opening at a greater rate with ID density n as Vr is increased, as shown in Figure 6.8. Although the exact mechanism behind the dependence on Vt is unknown, note that the ID saddle potential (ujx/u>y) appears to remain essentially constant throughout, as Vt is varied. As also discussed in the previous Chapters, future investigations might focus on examining spin instabilities in the low density 2D reservoirs just away from the ID region. 6. A Density Dependent Spin Polarisation in ID 135

6.2 Conclusion

In conclusion, a simple model to explain the 0.7 conductance feature is presented in terms of a density dependent spin gap arising in the ID region. Experimental results on ID ultra-low-disorder induced electron systems are consistent with this interpretation. The ability to control the degree of spin polarisation with a surface gate may have important applications for spintronic devices. Chapter 7

Spin Related Non-Quantised Structure in the Characteristics of Quantum Wires

7.1 Introduction

The influence of electron spin on electrical transport measurements in semicon­ ductor nanostructures is presently a subject of great interest. The following Chapter presents detailed measurements made on three devices, in which per­ turbed forms of conductance structure were present in addition to the usual quantisation. In each situation the role of spin seems implicitly linked to a complete understanding of the phenomena. The first device is a quantum point contact (QPC) studied earlier in Chapter 4 where the 0.7 x 2e2/h conductance feature was examined. Here the interest is focused on the higher plateaus, where the behavior of non-quantised conductance structure is found to depend very strongly on temperature and parallel magnetic field. There is some evidence to suggest that this result is related to spin dependent scattering from a single impurity atom, in a similar sense to the Kondo effect seen in metallic systems

136 7. Non-Quantised Structure 137 and quantum dots [8]. The second half of the Chapter considers experiments performed on two quan­ tum wires (/ = 2/im) in which sharp finger gates were used to strongly perturb the potential landscape and in some instances form a (OD) quantum dot. Such resonators are capable of producing rich conductance structure associated with single particle electron interference and resonant tunneling phenomena. Here the co-existence of the 0.7 feature with both resonance conductance structure and Coulomb charging effects is explored. Some evidence is presented to suggest that the 0.7 feature depends only weakly on the details of the ID potential landscape.

7.2 Kondo-like Conductance Perturbations

In 1964 Kondo [9] presented a detailed explanation of electron scattering from localised magnetic moments due to transition metal impurities in alloys. On the basis of an s — d exchange model, Kondo showed that an impurity atom may retain its magnetic moment in a metal and in such a case scattering may depend on the relative orientation of the spins of the impurity atom and the conduction electrons. Since that time scattering due to single impurities has also been considered [120, 121, 117] for the case of quasi-one-dimensional semiconductor wires. In these treatments however, the electrons are assumed not to interact and the role of spin is neglected. Recent work in quantum dots [8, 122] has re-ignited interest in the Kondo effect and researches are eagerly searching for its signature in other dimensional systems. The following section presents data taken on a quantum point contact (QPC) where the presence of an impurity perturbs the quantised conductance. The impurity is shown to be located very close to the ID region and depends strongly on temperature and the application of a parallel magnetic field. Figure 7.1 shows the conductance G of a quantum point contact at T = 50mK. 7. Non-Quantised Structure 138

T = 50mK

Top Gate Voltage, VT (V)

Figure 7.1: Conductance G of a quantum point contact containing an impurity close to the ID region. Each trace is for a different side gate bias Vs, ranging 0 to -1.5V in steps of-lOOmV, left to right. T = 50mK. Inset: zoom of region of interest and comparing traces Vs = 0 to Vs = -1.5V.

The conductance was measured as a function of top gate bias Vp, for side gates ranging from 0 to -1.5V in steps of -lOOmV. For the left most traces the usual conductance quantisation is disturbed with conductance structure predominately occurring between 6 and 8 x 2e2/h. Unlike resonance structure seen in low- mobility devices, the structure present in this QPC occurs only for the higher sub­ bands, with the plateaus above and below the disturbance remaining essentially quantised. Of further significance, the suppressed conductance is restored with the application of a moderate side gate bias Vs- This indicates that the cause of the conductance structure is likely to be an impurity scatterer located close to the ID region, since in general Vs only modifies the local potential near the ID region. Bagwell [121] has examined the non-propagating evanescent modes 7. Non-Quantised Structure 139

Top Gate Voltage, VT (V)

Figure 7.2: Conductance G of a quantum point contact containing an impurity close to the ID region. Each trace is for a different side gate bias Vs, ranging 0 -1.5V in steps of-lOOmV, left to right. T = 375mK. developing around defects in a ID constriction. In this approach the presence of a scatterer can lead to quasi-bound-states that split off from one of the higher-lying confinement sub-bands. A bound state associated with the lowest sub-band is analogous to a donor level below the conduction band minima of a semiconductor. This leads to unusual scattering properties when the Fermi energy approaches either a sub-band minima or a quasi-bound-state splitting off of a higher-lying confinement sub-band. Thus in the presence of scattering defects, the states available are not only those associated with the confining potential but also include evanescent modes that can greatly perturb the usual quantisation of conductance. Figure 7.2 repeats the conductance characteristics at an elevated tempera- 7. Non-Quantised Structure 140

— T = 50m K T = 135mK T = 200m K T = 310mK T = 378mK - T = 520mK

Top Gate Voltage, VT (V)

Figure 7.3: Conductance G as a function of top gate bias VT. Side gates are grounded. Each trace corresponds to a different temperature, as indicated by the grapli legend. Inset: A zoom of the region of interest. ture of T — 375mK. At very low temperatures the heat capacity of a degenerate electron gas becomes exceedingly small [104] and when energy coupling between the electrons and surroundings is weak, electron heating can occur [105]. Under these considerations a lattice temperature T = 375mK likely represents only a slight increase in electron temperature from the T = 50mK data presented in Figure 7.1. Despite the increase in thermal energy being small, the conductance structure undergoes a near complete restoration. Figure 7.3 presents the con­ ductance G as a function of top gate bias Ur, with the side gates grounded. The lattice temperature is increased from T = 50mK to T = 520mK, however the conductance characteristics do not evolve smoothly. Instead, the perturbed conductance structure jumps discontinuously when the temperature is above T 7. Non-Quantised Structure 141

Figure 7.4: Schematic comparison between electron scattering trajectories for the case of B = 0 and finite perpendicular magnetic field. The formation of edge states in the Quantum Hall regime steer electrons away from the influence of the scatterer.

= 200mK. At T — 400mK a complete restoration of the conductance plateaus is observed. Studies by van Wees [123] indicate that conductance structure arising from localised impurities is strongly affected by a perpendicular magnetic field. This result is explained in terms of the Quantum Hall Effect (QHE) where the for­ mation of edge channels can guide the current away from the influence of an impurity. Figure 7.4 schematically illustrates this concept by considering the trajectory of an electron in zero and finite magnetic held. In contrast to the orbital effects associated with a magnetic held oriented perpendicular to the x-y plane, a parallel held only serves to lift the spin degen­ eracy. In this case the Lorentz force remains zero and edge states do not occur. 7. Non-Quantised Structure 142

B = 0.8T

mum MMmJ T = 50mK

Top Gate Voltage, VT (V)

Figure 7.5: Conductance G of a quantum point contact containing an impurity close to the ID region. Each trace is for a different side gate bias Vs, ranging 0 to -1.3V in steps of -lOOrnV, left to right. B = 0.8T. T — 50mK.

Figure 7.5 shows the conductance characteristics in the presence of a B = 0.8T parallel magnetic held. Although the held is slight, some restoration of the per­ turbed conductance is observed. Interestingly as the held is increased to B = 1.5 T (Figure 7.6), quantisation is restored to the conductance characteristic despite the fact that the magnetic energy (g/j,BBS) of the electrons spins is not suffi­ cient to exceed thermal fluctuations (kT). This intriguing result suggests that the strong held dependence seen in this sample is related not to the conduction electron spins but the magnetic moment associated with an impurity atom. In any case the impurity potential seems very shallow given the small amount of energy (magnetic or thermal) required to restore the conductance. Close examination of Figure 7.6 reveals that although the conductance quan- 7. Non-Quantised Structure 143

T = 50mK

Top Gate Voltage, VT (V)

Figure 7.6: Conductance G of a quantum point contact containing an impurity close to the ID region. Each trace is for a different side gate bias Vs, ranging 0 to -600mV in steps of-lOOmV, left to right. B = 1.5T. T = 50mK. tisation has been restored near 9x2e2/h by the application of a B = 1.5T parallel field, a strong conductance perturbation (suppression) is now seen for the left most traces near the first plateau. Increasing the confinement with a side gate bias Vs restores quantisation, consistent with the behavior seen for the higher plateaus at B = 0. Figure 7.7 shows the conductance characteristics for B = 2.70T. Comparing this result to data taken at lower magnetic fields reveals that the conductance perturbation occurring for the lower plateaus has increased in magnitude. In addition, strong oscillating structure is observed that decreases with increasing top gate bias. Further, the application of a side gate bias has less of an effect on the conductance characteristics than for the case of smaller fields. The origin of the periodic structure is puzzling. Perhaps the simplest 7. Non-Quantised Structure 144

B = 2.70 T

T = 50mK

Top Gate Voltage, VT (V)

Figure 7.7: Conductance G of a quantum point contact containing an impurity close to the ID region. Each trace is for a different side gate bias Vs, ranging 0 -1.4V in steps of -lOOmV, left to right. D = 2.70T. T = 50mK. explanation is that the sample is not perfectly aligned with the parallel field and the periodic magneto-conductance is related to Shubnikov - de Haas oscillations associated with the small perpendicular component of the field [124]. A more exotic explanation in accordance with calculations by van der Marel [125] is that the oscillations are related to weak diffraction of electron waves associated with reflections from the impurity, although this is very unlikely. Finally Figure 7.8 presents the left most conductance characteristic (Vs = 0) for parallel magnetic fields of magnitude B = 0, 0.25T, 0.50T, 0.75T, 1.0T, 1.25T and 1.50T. Little variation in the conductance structure is observed between 0 and B = 0.25T, however above this field the characteristics are steadily improved and complete restoration of the usual quantisation is observed for B = IT. 7. Non-Quantised Structure 145

B = 1.5 T

T = 50mK

Top Gate Voltage, VT (V)

Figure 7.8: Conductance G as a function of top gate bias Vr for a quantum point contact containing an impurity close to the ID region. Each trace is for a different parallel magnetic field, ranging B — 0 (black) 0.25T (pink) 0.50T (yellow) 0.75T (red) LOOT (green) 1.25T (purple) and 1.5T (blue). T = 50mK.

Although impurity states have been rigorously explored in lower mobility samples [126], the results presented here are of interest for several reasons. Firstly the quantised conductance is disturbed only in a very narrow range of gate bias, occupancy, temperature and magnetic field. The impurity state has no effect on the quantised conductance below 4 x 2e2/h or above 10 x 2e2/h, but significantly alters the conductance staircase when the top gate bias is tuned to Vr ~ 350mV. This behavior supports the notion that the effect is related to a single impurity atom or defect, and not the weak disorder associated with the ID potential land­ scape which would be expected to modify the conductance over a larger range. Secondly, the strong dependence with parallel magnetic field and temperature is particularly intriguing. Generally speaking, resonance states broaden contin- 7. Non-Quantised Structure 146 uously with increasing temperature, and eventually disappear. However for the case studied here, a discontinuous jump is observed in the conductance structure indicative of a characteristic temperature - Tc « 250mK, again consistent with Kondo type physics [8]. A similar discontinuous evolution is seen with increasing parallel magnetic field so that by B « = 0.5T the magnetic field is observed to restore the conductance quantisation. This last result can be explained by evok­ ing arguments involving the cyclotron magnetic length (see Figure 7.4) however, the parallel direction of the applied field suggests an alternative explanation. Although Figure 7.7 (B = 2.70T) shows Shubnikov - de Haas oscillations modulated by the conductance plateaus, there is good reason to believe that orbital effects associated with a perpendicular magnetic field are not responsible for the observed behavior of the anomalous conductance structure. Firstly, very similar results were obtained in a previous cool down [108]. In that case consis­ tent behavior was seen with the application of a parallel magnetic field despite the absence of conductance oscillations - even for larger fields. This would tend to suggest that the sample was mis-aligned in the D field for the data reported here. Secondly of further importance, the size of the perpendicular field com­ ponent (obtained from the period of the oscillations in Figure 7.7), is generally not sufficient to produce the tight edge states illustrated in Figure 7.4. Taken together this reasoning suggests that spin effects are responsible for the strong parallel field dependence, although in-plane orbital phenomena cannot be ruled out. One possible explanation is that strong electron backscattering occurs when the magnetic moment of the impurity interacts with the electron spins. Increas­ ing the temperature or magnetic field re-orients the impurity spin and degree of backscattering. Without a more detailed study however, (possibly making use of the bias spectroscopy technique outlined in Chapter 5) it is difficult to rule out alternative simpler explanations for the conductance structure. 7. Non-Quantised Structure 147

7.3 Electron interferometry in wires with finger gates.

Open quantum dots are mesoscopic systems in which the tunnel coupling to the leads is increased above the usual values required to observe single electron tunneling. In this regime electrons become less localised and the number of electrons on the dot less well defined. In this open regime, theories of non­ interacting electrons usually give a proper description of transport. The theory is more complicated in the intermediate regime where the tunnel coupling is relatively strong but the discreteness of charge still plays an import role. For this case, the transport description needs to incorporate higher order tunneling processes through virtual, intermediate states. When spin is neglected, these processes are known as co-tunneling [127]. When one keeps track of the spin, it can be convenient to view tunneling as a magnetic-exchange coupling [8]. Here data taken on devices fabricated to explore the open quantum dot regime are reported. In particular, the influence of spin polarisation on the conductance of an open-dot is explored via the 0.7 feature, making contact with the rich variety of new physics emerging in zero dimensional systems [128]. The following section reports data taken on two quantum wires of length l = 2nm. In addition to the usual ID wire region, these devices have sharp finger gates that can be used to strongly perturb the potential landscape of the ID constriction (see AFM images in Figure 7.9). For moderate gate voltages these devices produce rich conductance structure associated with the interference of electron waves due to the geometrical arrangement. With the application of strong bias to the finger gates, a zero dimensional quantum dot is produced and transport is dominated by Coulomb Charging and 0D resonances. Smith et al. [129] investigated Fabry-Perot type interference phenomena in devices where both the width and length of the ballistic channel were separately variable. Their 7. Non-Quantised Structure 148

Z 3.0 Z 3.0

0.6 0.7 0.8 0.9 Top Gate Bias \ Top Gate Bias \

Figure 7.9: TOP: AFM images showing the two devices with finger gates, (Taken from O’Brien et al. [3] ). BOTTOM: The corresponding conductance G as a function of top gate bias Vt, with Vs grounded. T = 50mK. 7. Non-Quantised Structure 149

Side Gate A, Vs (V)

Figure 7.10: Device A. Conductance G as a function of the bias applied to side finger gate A. T = IK. VT = 540mV - 623mV. Left, middle an right most traces are shown in red to facilitate comparison. results indicate that periodic oscillations in the conductance are observed as the channel length is varied. Analogous to the case of an electromagnetic waveguide, standing waves are produced when an integer number of Fermi wavelengths make uj) the cavity length. Some evidence for this type of interference is presented for the devices studied here. When an electron gas is spatially confined in all three dimensions the result­ ing system is termed a quantum dot, since the energy spectrum is comprised of discrete zero dimensional (0D) eigenstates [130]. For the case of strong finger gate confinement electron transport becomes dominated by two principle mech­ anisms. At low temperatures Coulomb Blockade (CB) oscillations are observed for the case where conduction occurs via tunneling, (ie the barrier resistances are greater than e2/h). For the devices considered here the oscillations are mod- 7. Non-Quantised Structure 150 ulated by the usual ID conductance plateaus [131]. Depending on the energy state spacing in the zero dimensional region and the details of the tunnel barri­ ers, an alternative regime exists where OD resonance effects produce sharp dips in the conductance as a function of gate bias [132]. In the devices studied here the strong conductance perturbations potentially result from a variety of these effects and it is generally difficult to identify and separate each mechanism. Im­ portantly however, the 0.7 x 2e2/h conductance feature can be seen in most instances to co-exist with these other effects, indicating that perhaps the oc­ currence of a spin polarisation depends only weakly on the details of the ID potential landscape. Figure 7.9 shows AFM images of the two devices fabricated by O’Brien [3]. The corresponding conductance traces are also shown in the bottom of Figure 7.9, where the top gate bias was varied with the finger side gates grounded. Device B exhibits strong noil-quantised structure throughout the conduc­ tance staircase and oscillating structure below the first plateau. Focusing on Device A, reasonable conductance quantisation is observed with an additional non-monotonic feature occurring below the first plateau. Comparing this data with the data taken on a third quantum wire device of length .1 = 2\im reported is Chapter 4, reveals that this feature is most likely the 0.7 conductance structure. The particular non-monotonic behavior can be explained within the context of a density dependent spin polarisation, as presented and discussed in Chapter 6 (see Figure 6.6). Figure 7.10 shows the conductance G of Device A as a function of the left side finger gate bias (side gate A). The presence of additional structure between 0.5 and 1.0 x 2e2/h can be seen on all traces as the top gate bias is varied. Given that the additional structure predominately occurs above the first spin-split plateau, it is likely to be due to a complex interplay between geometric resonant effects and features associated with a density dependent spin polarisation (the 0.7 fea- 7. Non-Quantised Structure 151

Side Gate B, Vs (V)

Figure 7.11: Device A. Conductance G as a function of the bias applied to side finger gate B. T = IK. Vt = 540mV - 594mV. ture). Within the context of the model presented in Chapter 6, a spin gap that develops with increasing density is likely to produce rich conductance structure when the Fermi wavelength of the minority spin band-edge corresponds to an integer multiple of the channel length. A clear and strong 1.7 x 2e1 /h feature can also be identified, consistent with the data taken on other devices (Chapter 4). The data in Figure 7.10 was taken at a temperature T = IK, in order to sup­ press smaller conductance perturbations associated with weak disorder apparent in these longer devices. Figure 7.11 shows the conductance G of Device A as a function of the right side finger gate bias (side gate B). Two strong dips are seen in the conductance near 0.25 x 2e2/h and 0.75 x 2e1 /h with some smaller additional structure just below the first plateau. The origin of these dip structures is essentially unknown. 7. Non-Quantised Structure 152

5.0 wmmmm 1.2 1.0 4.0 0-8 ^ 0.6 C ^ssmmmmm 0.4 O O 3.0 PmffSM 000196610.2 0J U c -0.8 -0.6 2.0 A & B Vs (V) u 3 3 o 1.0 U m WmmgjmWum HfflwMMi 0.0 1 m B wKBiw 1.0 -0.8 -0.6 -0.4 -0.2 0.0 Side Gate A & B, Vs (V)

Figure 7.12: Device A. Conductance Casa function of the bias applied to both finger gates. T = IK. Vp = 540mV - 656mV. Inset: zoom of the region of interest.

One possible explanation is that they result from single particle electron inter­ ference, as investigated by Smith et al. [129, 132]. A more exotic explanation involving electron spin singlet and triplet bound states has been proposed by Re- jec et al. [85] and also by Flambaum and Kuchiev [84], as reviewed in Chapter

2. When the same potential is applied to both finger gates the conductance exhibits a hybridisation of the features seen when each gate was varied separately. Two strong features stand out in the family of curves. The dipping structure never ‘peaks’ above 0.5 x 2e2/h for the first instance, and then never ‘dips’ below 0.5 x 2e2/h for the second occurrence. A clear line at 0.5 x 2e2/h can be identified 7. Non-Quantised Structure 153

T = 2.1K T = 3.1K T = 0.1K

Side Gate Voltage, V§ (V)

Figure 7.13: Device A. Conductance G as a function of bias applied to both finger gates together. VT = 580mV, 590mV and 600mV. Blue trace: T = lOOrnK, Black trace: T = 2.IK, Red trace: T = 4.2K. where the conductance structures meet. Secondly, just below the first plateau, near 0.8 x 2e2/h each trace exhibits what is most likely a 0.7 feature. Figure 7.13 Shows the conductance G as a function of side finger gate bias, with the same potential applied to the two gates, as in Figure 7.12. Here the conductance at three different temperatures is compared. Little variation is seen above T = 2.IK, however the T = lOOrnK data exhibits strong periodic oscillations. These are likely due to either Coulomb charging effects, or resonant transmission as states in the open quantum dot line up with the Fermi level of the 2D reservoirs. Turning now to Device B, Figure 7.14 shows the conductance G as a function of side gate bias Vs, with the same potential applied to both gates. The geometric arrangement of the gates, indicated in Figure 7.14 (d) produces a QD quantum 7. Non-Quantised Structure 154 dot when the finger gates are strongly biased. For intermediate bias however, the conductance characteristics exhibit Coulomb charging peaks modulated by the usual conductance staircase. Figure 7.14 shows data taken over a range of top gate bias Vr = 630mV - 900mV at T — 50mK. Complex conductance structure is observed, most probably due to an interplay of interference, Coulomb charging, resonant tunneling and spin polarisation effects. Figure 7.14 (b) shows a zoomed region of the curve family and illustrates the complex behavior with an isolated red trace. The conductance oscillations are closely periodic until the conductance approaches 0.7 x 2e2/h. As discussed in Chapter 6, near G = 0.7 x 2e2/h the tail of the Fermi function begins to cross over into the upper spin-split sub-band, and in this case destroys the Coulomb blockade oscillations, (Figure 7.14 (c)). The rich interplay between Coulomb charging events and spin depend phenomena is currently of great interest. In fact, as discussed in the previous section, the Kondo effect seen recently in open quantum dot systems [8] essentially describes the physics of such interactions.

7.4 Summary of Results

This Chapter has explored non-quantised conductance structure seen in the char­ acteristics of undoped, induced quantum wire devices. The first section investi­ gated a peculiar suppression of the conductance staircase, and presented some evidence to suggest the effect was related to spin dependent scattering from a magnetic impurity near the ID region. Although the study is not conclusive, the data presented appears very unlike results reported for geometrical conductance resonances and a more thorough examination is required. The second section of the Chapter presented data taken on two quantum wires with finger gates. Rich conductance structure associated with Coulomb charging, geometrical res­ onances, resonant tunneling and spin polarisation was observed. Distinguish- 7. Non-Quantised Structure 155

Side Gate Voltage, Vs (V)

Figure 7.14: Device B. (a) Conductance G as a function of side gate bias Vs. T = 50mK. Vt = 630mV - 900mV. (b) Shows a zoom of the central region with an isolated trace (red) to aid observation, (c) What is presumably Coulomb charg­ ing peaks can be seen predominately below the 0.7 feature, (d) Gate geometry and device layout. ing between each mechanism remains a difficult task. The coexistence of the 0.7 x 2e2/h conductance feature in the presence of these other effects makes an important point concerning the nature of the 0.7 feature. Presuming that the feature is due to some form of spin polarisation, these results indicate that the mechanism driving the polarisation is only weakly dependent on the details of the potential landscape or alternatively, that the polarisation occurs slightly away from the ID region, where the effects of the side gate bias is small. The notion that the polarisation is insensitive to the details of the potential landscape is consistent with the fact that the 0.7 feature has been observed now in a variety 7. Non-Quantised Structure 156 of different samples comprising of different material systems and geometries. Chapter 8 de Haas - van Alphen measurements using Micro-Magnetometers

8.1 Introduction

The Dirac Series of experiments was instituted in 1996 at Los Alamos National Laboratory so that a variety of experimental projects could access magnetic fields approaching D = 1000T for the study of new physical phenomena in condensed matter systems. These ambitious experiments require advanced innovative tech­ niques to eliminate Faraday pick-up [133] and allow useful measurements to be made during the short time of the pulsed magnetic field. The fabrication and testing of micro-magnotometers for use in such extreme environments formed a major part of the work undertaken in this Thesis. Ultimately, the many advan­ tages of these novel devices may see them used in de Haas - van Alphen fis pulsed field experiments to map the Fermi surface of the important high temperature superconducting material YBq^Cu^Ot^s (YBCO). The outline of the Chapter is as follows. Section 8.2 summarises some of

157 8. Micro-Magnetometers 158 the immense technological challenges undertaken in high magnetic pulsed held experiments and the techniques used to overcome them. Section 8.3 reports on the fabrication of the micro-magnetometer coils using both optical and electron beam lithography. Finally sections 8.4 and 8.5 discuss the de Haas - van Alphen effect and report on some of the data taken using the micro-magnotometers at the National High Magnetic Field Laboratory, Los Alamos NM.

8.2 Technology

Russian-developed MCl-class [134] hux-compression generators are capable of producing pulsed magnetic fields near B = 1000T. The conditions during such a pulse are extreme with the generator, cryostat and samples all destroyed after the pulse. During the pulse, dB/dt can reach l()9T/s, creating voltages up to 1KV in a conducting loop of area of 1 mm2. Minimisation of Faraday pick-up is therefore critical and specially designed [133] coplanar transmission lines (CTLs) patterned on a printed circuit board (PCB) substrate were used to achieve this. Together with these extreme conditions, low temperature measurements need to be made within a time frame of a few /is, all the while keeping noise and interference to a minimum. Such challenges have lead to the development of novel technologies [133, 135] and paved the way to the development of the micro-magnotometer devices considered here. Traditional magnetometers are made of wound copper wire and are used in a variety of experiments including magnetisation, susceptibility and de Haas van Alphen measurements in pulsed fields. The coil windings are usually wound to be compensated, so that no signal results from dB/dt pick-up of the applied field and only the magnetisation M of the sample contributes to the signal. Hand - wound coils however, apart from being extremely tedious to make, usually do not obtain complete compensation and this degrades their sensitivity. In contrast 8. Micro-Magnetometers 159

Figure 8.1: Illustration of a micro-magnotometer connected to a set of co-planar transmission lines. The magnetic field B is perpendicular to the plane of the coils as indicated. The magnetisation signal threading the coils is entirely due to the sample. (Illustration by R. P. Starrett). 8. Micro-Magnetometers 160

Figure 8.2: Result of calculations simulating the effective coil pickup. The vast majority of the magnetisation flux threads the compensated coils. (Calculation and illustration by R. P. Starrett). 8. Micro-Magnetometers 161 to traditional coils the micro-magnotometers considered here are compensated inductive devices of millimeter scale size. They consist of lithographically defined planar coils where precise placement of the coil wires ensures almost perfect compensation. Of further importance, the close packing of coil windings on a mm scale opens a window into the physics of very small pure single crystals. Figure 8.1 shows an illustration of a micro-magnotometer coupled to the co-planar transmission lines that minimize Faraday pick-up. These are side by side coils, counter wound to cancel out the background field. The coil axis is perpendicular to the applied magnetic field B, resulting in the electron orbit planes also being perpendicular to the magnotometer plane. The coils are designed for use with mm sized samples that fit within the two coils taken together (Figure. 8.1). Under these conditions the efficiency of the magnotometer is at a maximum with nearly all of the flux threading through the magnetometer coils, as shown in Figure 8.2. In addition to the high sensitivity achievable with these devices, care was taken to ensure that the technique is practical and without tedious sample preparation. The devices are entirely compatible with the co-planar transmission lines (CTLs) used in the explosive flux-compression experiments. By making use of a novel ‘flip-chip’ arrangement, the devices make contact with the CTLs without the need for soldering. Instead, contact is made by bonding the magnotometer chip to the printed circuit board using Duco hobby cement. After the experiment is complete the sample and magnotometer can be recovered using a suitable glue solvent. The sample can be mounted prior to bonding the magnetometer to the CTLs (Figure 8.3 (a)) or, after bonding by making use of a small aperture in the printed circuit board as shown in Figure. 8.3 (b). A custom made jig keeps the chip in contact with the CTLs until the glue is dry, as shown in Figure 8.3 (c). Explosive flux compression experiments destroy the cryostat. Consequently the magnotometer experiments were designed to work with a cheaply constructed 8. Micro-Magnetometers 162

Figure 8.3: Magnotomter sample mounting procedure. The usual sequence is to mount the sample on the surface of the micro-magnotometer using Duco cement (a) . After the glue is dry, the magnetometer can be mounted and bonded to the CTL printed circuit board with the sample fitting in the small aperture shown in (b) . A custom made mounting jig is used to keep the magnotometer in contact with the CTLs until the glue is dry (c). (Illustration by A. Wang). 8. Micro-Magnetometers 163

Figure 8.4: Photographs showing the G10 fibre-glass probe used in the de Haas - van Alphen experiments to GOT. (a) shows the tail with a loop of plastic used to minimise vibrations in the pulse, (b) is a close-up of the CTL printed circuit boards, and (c) is a spliced photograph of the entire insert. probe and cryostat. The probe was constructed from GlO fibre-glass to alleviate effects associated with magnetisation and eddy current heating in metal appa­ ratus. The probe used for non-destructive de Haas - van Alphen experiments is shown in Figure 8.4. A small loop of plastic at the end of the probe (Fig 8.4a) ensures a tight fit into the tail of the cryostat and reduces vibrations as­ sociated with the field pulse. The CTL printed circuit boards (Fig 8.4b) make contact to triaxial screening cable to minimise spurious interference and the steel boxes at the top of the probe (right of Fig 8.4c) allow for the incorporation of preamplifiers directly into the insert. 8. Micro-Magnetometers 164

8.3 Fabrication of the Magnetometers

The ability to fabricate micro-magnetometers by lithographic means is advan­ tageous in producing perfectly compensated devices. Further, because of the nature of lithography many devices can be produced simultaneously. The first generation magnotometer devices were fabricated using electron beam lithog­ raphy (EBL) and consisted of co-centric coaxial coils with 250nm metal line widths and 250nm pitch. Large quantities of these devices were produced, since because of the large area associated with the coil windings they were especially sensitive to contamination. EBL is essentially a slow process in comparison to optical lithography and 20 devices took several days to expose. To over come these limitations the devices were rescaled and fabricated using conventional UV photo-lithography. Figure 8.5 shows optical microscope photographs of the mag­ netometers in various stages of processing. The devices fabricated using optical lithography were made by N. Lumpkin. The original coaxial magnotometer design (see Figure 8.5(b)) also made it difficult to achieve close to perfect compensation. Although each loop obviously consists of a slightly different effective area, the total area for the inner and outer coils must be exactly equal. Achieving this requirement is non-trivial and software was written to assist in this task. Switching to the optically defined side by side geometry effectively removed this problem and provided a means of fabricating near perfectly compensated coils.

8.4 The de Haas - Van Alphen Effect: Mapping the Fermi Surface fn 1930, in the course of measurements on the magnetic susceptibility of bismuth single crystals at low temperatures, de Haas and van Alphen [136] found that the 8. Micro-Magnetometers 165

Figure 8.5: Optical microscope photographs showing the micro-magnetometers, (a-c) show the coaxial EBL written coils (generation 1) at increasing magnifi­ cations. (d-f) show side by side coils (generation 2) fabricated using standard optical lithography techniques. 8. Micro-Magnetometers 166

Diagram Keys:

Gallium Arsenide Substrate Photoresist mmmmt Gold 4KKB Silicon Nitride

Figure 8.6: Schematic of the process used to fabricate the micro-magnetometers (Part 1). Standard optical lithography techniques are used to define the metal coils and SiN layers that provide electrical isolation between layers. Selective etching of SiN enables electrical interconnects between the metal layers. The top row shows the results of the steps outlined in the columns below. 8. Micro-Magnetometers 167

Figure 8.7: Schematic of the process to fabricate the micro-magnetometers (Part 2) See caption of Figure 8.6 for details. 8. Micro-Magnetometers 168 susceptibility (defined as the ratio of magnetisation to field) was not constant, as is usual for feebly magnetic materials, but varied in an oscillatory manner with the field. The amplitude of the oscillations diminishes as the temperature is raised and the effect disappears altogether above 30 or 40K. Quite soon af­ terwards Peierls [137] laid the basis of the theory of the effect by showing that the magnetisation of a free electron gas should oscillate as the field was varied because of the quantisation of the free electron orbits in a magnetic field. The following summary provides an insight into the origin of the de Haas - van Alphen effect and is based on a derivation given by Shoenberg [138]. Consider the energy levels of a two-dimensional (2D) metal in a perpendicular magnetic field. In a magnetic field electrons move in curved paths as a results of the Lorentz force. Quantum theory restricts the possible motions to only certain orbits, namely those that satisfy the Bohr-Sommerfeld condition J (p — eA/c) • ds = (r + y)/i (8.1) where p is the electron momentum, hk] A is the vector potential of the field and the integration is round the orbit in real space; r is an integer and 7 is a phase constant, which is 1/2 for E(p) = p2/2m (circular orbits). Since p is just eH/c multiplied by the radius vector of the real space orbit and by making use of the relation J A ■ ds = OH (8.2) where O is the area of the orbit, the quantisation condition (eq. 8.1) becomes

eHO/c = (r + 7 )h (8-3) or, since

a(E) = 0(eH/c)2 (8.4) where a(E) is the area of the curve of constant energy E in momentum space,

a(E) = (r + i)ehHjc (8.5) 8. Micro-Magnetometers 169

Equation 8.5, which can be regarded as an implicit equation specifying the energy levels E\(H), E2(H), ... Er(H)... of the system, shows that the permitted curves of constant energy are separated by equal areas ehH/c. In a magnetic field these (Landau levels) are the only states permitted. As the field H is increased, the curves of constant energy and also the energy levels will become more and more separated; moreover since each will be able to accommodate more and more electrons, fewer and fewer energy (Landau) levels will be occupied. As a Landau level de-populates, the total (free) energy [139] of the system U must undergo a sudden discontinuity with H. A discontinuity must also be accompanied in M since M = dU/dH. These discontinuities must occur at equal intervals eh/ca(Eo) of 1 /H, (E0 is the Fermi energy), where the Landau levels cross the Fermi surface. Although this simple derivation was made for the 2D case, extension of the argument to a three-dimensional (3D) metal is permissible. In 3D the amplitude of the oscillations is generally reduced. From the arguments given above, clearly the period of the oscillations maps directly to the volume contained within the Fermi surface. Careful measurement of the 1/B frequency spectrum for different field directions allows for the Fermi surface to be entirely reconstructed. Energy surfaces in momentum space are generally assumed to be ellipsoidal in shape. Although this assumption is physically plausible in the immediate neighborhood of a zone boundary, there is no reason why it should continue to hold further from the boundary and the de Haas - van Alphen effect offers plenty of evidence to show that the Fermi surface is not usually ellipsoidal at all [138].

8.5 Experimental Results

The de Haas - van Alphen measurements reported here were made on small sin­ gle - crystals of Lanthinum Hexaboride (LaB6) and Cerium Hexaboride (CeB6) 8. Micro-Magnetometers 170 at the National High Magnetic Field Laboratory (NHMFL), Los Alamos USA. Magnetisation studies were also carried out on Ca^R^O-?, but these results are not shown here. The primary motivation for performing these particular experiments was to characterise the sensitivity of the micro-magnetometers in comparison to traditional coils. Further, given the rich physics recently uncov­ ered [140, 141] on these Heavy Fermion compounds, additional investigations using sensitive micro-magetometers may provide further insight.

8.5.1 Lanthinum - and Cerium - Hexaboride

Recent measurements on LaB6 [141] indicate the presence of quantum interfer­ ence effects, a rare phenomenon in metals. Of particular interest, one of the interference frequencies can be shown to correspond to the Brillouin zone where the effective mass is zero. The samples measured here were grown at Los Alamos using the method outlined in [142]. The samples were « 300/rm in length and « 30/ira in height. Figure 8.8 shows the raw signal obtained from a side bv side micro-magnetometer with 40 turns at a temperature T = 4.2K. A Standford Research Systems preamplifier was used with a gain of 5000. Figure 8.9a shows the same data plotted as a function of 1/B. Clear periodic oscillations can be seen corresponding to the de Haas - van Alphen effect. The Fourier transform of the dHvA signal is shown in Fig 8.9(b) for the falling magnetic field. The experiments on Cerium Hexaboride (CeB6) proved difficult for several reasons. Firstly the heavier effective mass [143] reduces the signal significantly in comparison to LaB6. Therefore in order to increase the dHvA signal, the sample was initially cooled to T % 2.IK, by reducing the LHe4 vapor pressure with a strong pump. Below the A point (near T = 2K) however, He4 undergoes a super-fluid transition and vibrations associated with the pump and field pulse are easily coupled to the sample. Attempts to over come microphonic effects included using a damping loop as shown in Figure 8.4(a). Ultimately however, 8. Micro-Magnetometers 171

B (T) Figure 8.8: Raw signal obtained from a 40 turn side by side micro-magnetometer. Gain = 5000. T = 4.2K

the experiment was performed in a custom built LHe,3 cryostat, where vibration effects were negligible. Figure 8.10 shows data taken at T = 450mK on CeB6. The signal shown in Figure 8.10(a) was obtained from a 80 turn, side by side micro-magnetometer with a preamp gain of 10000. The Fourier transform shown in Figure 8.10 (b) indicates strong oscillations associated with the de Haas - van Alphen effect (Data taken by C. H. Mielke of NHMFL).

8.6 Micro-Magnetometers: Conclusion

Micro-magnetometers have been fabricated using electron beam and optical lithography specifically for the purpose of making sensitive measurements on small samples. These novel devices were demonstrated via measurements on the heavy Fermion compounds, Lanthinum Hexaboride (LaB6) and Cerium Hex- aboride (CeB6). Clear oscillations associated with the de Haas - van Alphen showing Clear Figure 8.

Micro-Magnetometers

periodic

8.9: clear

Top:LaB6 FFT Amplitude (arb.) dH/dt (arb)

peaks o s c i l l a t i o n s corresponding

de

Haas can Inverse

be - Frequency

van

seen. to

Alphen the

Field Bottom: various

(Tesla) signal

(

T*

ok- Fourier

frequency plotted ) space

(tesla)

transform as

a cross

function sections.

of

the

of

1 data

172 /B.

8. Micro-Magnetometers 173

46 48 b) Field (tesla)

i

3 O. I £

Frequency (tesla)

Figure 8.10: CeB6 dHvA signal from a 80 turn micro-magnetometer at T = 450mK. (a) Raw signal with preamplifier gain of 10000. (b) Fourier transform indicating clear dHvA oscillations. Data taken by C. H. Mielke (NHMFL). 8. Micro-Magnetometers 174 effect were observed, indicating the sensitivity and versatility of these devices. Future applications will hopefully include mapping the Fermi surface topology of the high temperature superconductor YBCO, in micro-second explosive flux compression experiments. Chapter 9

Electron Correlations in ID: Conclusions and Future Work

The conductance of a one-dimensional electron gas coupled to reservoirs is quan­ tised in units of 2x2e2/h. At low densities however, electron-electron interactions become important and these can modify this result. This Thesis has focused on one particular instance where a conductance anomaly, occurring near 0.7 x 2e2/h cannot be explained without evoking many-body electron interaction theory. In particular, the behavior of the feature is consistent with a density dependent spin polarisation. In this final chapter, the main results of this Thesis are reit­ erated by way of a conclusion and some suggestions for future experiments are proposed.

9.1 Magnetic Resonance Force Microscopy: De­

tection of possible spin polarisation in ID?

To date, a detailed microscopic understanding of the 0.7 conductance feature is still to be uncovered. Given that all of the experiments reported so far have focused on using transport as an experimental probe, the next (big) logical step

175 9. Conclusions and Future Work 176 might be to utilise some other means of probing the electronic configuration of a quantum wire. Despite these sentiments, probing the very small number of elec­ trons in the ID channel is a daunting task. One novel possibility however, might be to make use of a recent technique known as Magnetic Resonance Force Mi­ croscopy (MRFM). As part of the work for this Thesis, an experimental method was developed to make use of this technique in the detection of a possible spin polarisation in or near the quantum wire. The following section briefly reviews the technique of MRFM and is followed by a discussion of how it could be used to detect a spin polarised quantum wire.

9.1.1 MRFM

Nuclear magnetic resonance (NMR.) is not only the most detailed probe of molec­ ular structure and dynamics with the highest spectral resolution but is also the basis for magnetic resonance imaging [144]. Nonetheless, an ever present draw­ back of NMR, due to the low transition frequency, is a requirement for large amounts of sample, containing at least ~ 1015 — 1016 spins. Therefore, the sug­ gestion by Sidles [145] that it might be possible to use force to detect and image the nuclear resonance of a single proton was remarkable. His idea was to effect three-way resonant coupling among (1) a precessing proton mounted on (2) a vibrating substrate and (3) a mechanical oscillator holding a magnetic particle which produces both a dc field and a strong magnetic field gradient. The proton exerts a force on the mechanical oscillator through interaction with the mag­ netic field gradient generated by the particle. Energy transfer into the oscillator, mediated via the magnetic force between the proton and oscillator, is sufficient that single-proton detection is possible by sensing excitation of the mechanical oscillator. Sidles and Garbini [146] have reviewed the current status of magnetic reso­ nance force microscopy, and have concluded that the use of a sub-micron size 9. Conclusions and Future Work 177

Cantilever UMJ

Magntic field Sample gradient source

Figure 9.1: Basic configuration of a magnetic resonance force microscope: the force on the sample magnetisation in the gradient of a proximate magnetic held source causes a deflection of the cantilever which is sensed optically. The RF coil is used to excite spin resonance. gradient source, coupled with interferometric detection methods, [147] make sin­ gle spin detection with MRFM a realistic possibility. Figure 9.1 shows the conventional setup for MRFM. In this configuration the sample is placed on the end of the cantilever. By modulating the radio frequency signal to the coil at the natural resonance frequency of the cantilever, the interferometry can detect extremely small changes in magnetic force, due to the excited spins in the sample. A modified approach due to Berman and Tsifrinovich [148] makes use of the same technique, but with a magnetic particle located on the cantilever and the sample separately fixed. 9. Conclusions and Future Work 178

9.1.2 Using MRFM to detect an electron spin polarisa­

tion

The measurements reported in this thesis suggest that the 0.7 conductance struc­ ture is due to a spin polarisation that is a function of the electron density. From the source - drain bias spectroscopy data reported in Chapter 5, with the Fermi energy well above the second spin band edge the energy difference between spin up and down is likely to be a few meV. A quick consideration of the electron g factors of Ga and As, reveal that the frequency of radiation required to do elec­ tron spin resonance is essentially too large (THz), though perhaps using laser techniques it is possible. In contrast to this, nuclear magnetic resonance (NMR) experiments have previously been performed on the material system at a few tens of Megahertz [149]. An experiment to simultaneously measure the conductance and hyperfine shift in Ga and As nuclear resonance is proposed. Using the modified technique of MRFM with a scanning cantilever (AFM) the 1D wire can be precisely located under the tip of the cantilever. By performing spin resonance on the nuclei of the host atoms (Ga and As) a modulated magnetic force can be detected at the reso­ nance frequency of the cantilever. The next step is to apply a bias to the surface gates and simultaneously monitor the conductance and the resonance frequency of the nuclei. If the electrons spin polarise in the vicinity of the quantum wire, the effective local field will modify the nuclear resonance frequency of nearby Ga and As atoms via the hyperfine interaction. MRFM enables precise control over the depth of the sensitive slice and so could detect changes in resonance frequencies of host nuclei very close to the ID electron gas. Correlation with simultaneous conductance measurements could perhaps shed some light on the microscopic nature of the 0.7 conductance feature. In particular, because of the relevance of this type of experiment to , plans are made to perform initial investigations at Los Alamos National Laboratories in the near 9. Conclusions and Future Work 179 future.

9.2 Other future work

Despite the sentiments expressed throughout this Thesis, a large amount of work is yet to be done before an effective microscopic theory of many-body spin inter­ actions in quantum wires can be developed. Future efforts may concentrate in a number of important areas, both theoretically and with experiments. Initial ex­ periments are planned to focus on the 2D-ID coupling region, where the quantum wire makes contact with the reservoirs. Based on some of the results reported in this Thesis, the importance of the coupling region seems apparent. Fabrication of gated structures designed to vary the electrostatic potential just away from the ID quantum wire would be an interesting beginning to understanding more about this important region. Following on from the work reported in Chapter 5, more adiabatic coupling and asymmetric bias type experiments need to be performed. Based on the experimental data presently available, it is difficult to rule out a relation between geometrical resonance phenomena associated with the ID electron waveguide and the 0.7 feature. Although the feature is unlikely a resonance conductance structure, there is evidence to suggest that it may be related to electron interference effects, (see Chapter 5 section 5.4 for example). More work is needed in this area. Another very interesting experiment would be to fabricate ID hole systems. Do they exhibit a 0.7 feature? Even if these systems are found not to exhibit the conductance structure, important information would be gained. For the case of the enhancement mode FET structures considered in this work, there are possibilities to combine ID electron and hole systems. Such experiments would open a window into the physics of ID electron-hole correlations. 9. Conclusions and Future Work 180

9.3 Thesis Summary and Final Conclusions

One Dimensional semiconductor quantum wires are ideal systems to study the fundamental properties of quantum transport. In particular these devices are test beds for a range of predicted phenomena involving correlated electrons. This Thesis has presented experimental and phenomenological results that show the relevance of the electron spin in many-body interactions. In particular the work has focused on the anomalous 0.7 x 2e2/h conductance feature shown to be due to electron interactions involving spin. Chapters 4 and 5 presented experimental data taken on ultra low-disorder quantum wires fabricated from heterostructures free from the disorder associated with modulation doping. These novel devices provided a means of studying the 0.7 feature as a function of length, ID & 2D density, potential profile, temperature and magnetic held. The results uncovered are consistent with a phenomenological model presented in Chapter 6 to explain the 0.7 feature in terms of a density dependent spin polarisation. This model is not limited to the data presented here, but applies to all of the key data published on the subject. Chapter 7 examined perturbed conductance effects, and their relation to spin. Reminiscent of the Kondo effect seen in metals and recently in quantum dots, a conductance perturbation in a QPC was shown to be consistent with spin dependent scattering from a single magnetic impurity. Complementing these studies, the second part of Chapter 7 studied the regime were electron transport is subject to the effects associated with open quantum dots. Here evidence was reported for the simultaneous existence of Coulomb charging events, resonant tunneling phenomena, geometrical electron interference and the 0.7 conductance feature. The essential implication of this result is that the 0.7 feature may not implicitly require a clean potential landscape, but can continue to exist in a strongly perturbed confinement. Chapter 8 reported the work done in relation to the fabrication and measure- 9. Conclusions and Future Work 181 merit of micro-magnetometer devices for use in micro-second explosive flux com­ pression experiments. The lithographically defined inductive coils were tested at the National High Magnetic Field Laboratory at Los Alamos NM, USA. Chapter 8 presented data, taken on the heavy Fermion compounds LaB6 and CeB6. Clear evidence of the de Haas - van Alphen effect was seen. In conclusion, conductance experiments on one-dimensional quantum wires reveal an anomalous feature near 0.7 x 2e2/h. This Thesis has primarily focused on the nature of this feature and its association with many-body spin related phenomena. The 0.7 conductance feature has been shown to be consistent with the notion of a density dependent spin polarisation occurring in or near the ID region. Bibliography

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