Ballistic Transport in One-Dimensional P-Type Gaas Devices

The University of New South Wales, Sydney School of Physics

Ballistic transport in one-dimensional p-type GaAs devices

Oleh V. Klochan A thesis submitted in fulﬁllment of the requirements for the degree of Doctor of Philosophy July 2007 Acknowledgements

Firstly, I am deeply grateful to my supervisors Alex Hamilton and Adam Micolich who were trying methodically to teach me how to do science during last three and a half years. It was not that easy for all of us. Secondly, I am indebted to Romain Danneau and Warrick Clarke who helped me a lot with processing, measurements, analysis of the data and writing this thesis. Thirdly, I have to mention Ulrich Z¨ulicke and Oleg Sushkov, who helped immensely with understanding the physics behind the experimental results. Special thanks to Richard Newbury, who gave me a chance to do my PhD in UNSW and Jack Cochrane, whose technical expertise was always required. I would also like to thank everyone with whom I shared all this time in the Quantum Electronic Devices group and outside: Neil Kemp, Ali Rashid, Martin Aagesen, Lap- Hang Ho, Andrew See, Jason Chen, Kurt Erlich, Ted Martin, Ivan Terekhov, and Lasse Taskinen. I am thankful to personnel of the Centre for Quantum Computer Technology for the access and assistance in the Semiconductor Nanofabrication Facility and National Magnet Laboratory. Needless to say it would be impossible for me to go through this years without the invaluable support of my family and friends back in Ukraine and here in Australia. Abstract

In this thesis we study GaAs one dimensional hole systems with strong spin-orbit interaction effects. The primary focus is the Zeeman splitting of 1D subbands in the two orthogonal in-plane magnetic field directions. We study two types of 1D hole systems based on different (311)A grown heterostructures: a modulation doped GaAs/AlGaAs square quantum well and an undoped induced GaAs/AlGaAs triangular quantum well. The results from the modulation doped 1D wire show enhanced anisotropy of the effective Lande g-factor for the two in-plane field directions (parallel and perpendicular to the wire), compared to that in 2D hole systems. This enhancement is explained by the confinement induced reorientation of the total angular momentum Jˆ from perpendicular to the 2D plane to in-plane and parallel to the wire. We use the intrinsic anisotropy of the in-plane g-factors to probe the 0.7 structure and the zero bias anomaly in 1D hole wires. We find that the behaviour of the 0.7 structure and the ZBA are correlated and depend strongly on the orientation of the in-plane field. This result proves the connection between the 0.7 and the ZBA and their relation to spin. We fabricate the first induced hole 1D wire with extremely stable gate characteristics and characterize this device. We also fabricate devices with two orthogonal induced hole wires on one chip, to study the interplay between the confinement, crystallographic anisotropy and spin-orbit coupling and their effect on the Zeeman splitting. We find that the ratios of the g-factors in the two orthogonal field directions for the two wires show opposite behaviour. We compare absolute values of the g-factors relative to the magnetic field direction. For B [011] the g-factor is large for the wire along [011] and small for the wire along [233]. Whereas for B [233], the g-factors are large irrespective of the wire direction. The former result can be explained by reorientation of Jˆ along the wire, and the latter by an additional off-diagonal Zeeman term, which leads to the out-of-plane component of Jˆ when B [233], and as a result, to enhanced g-factors via increased exchange interactions. Publications arising from this work

1. O. Klochan, W. R. Clarke, R. Danneau, A. P. Micolich, L. H. Ho, A. R. Hamil- ton, K. Muraki and Y. Hirayama, Conductance quantization in induced one- dimensional hole systems, AIP Conference Proceedings 893, 681 (2007).

2. R. Danneau, O. Klochan, W. R. Clarke, L. H. Ho, A. P. Micolich, M. Y. Simmons, A. R. Hamilton, M. Pepper, D. A. Ritchie and U. Z¨ulicke, Anisotropic Zeeman splitting in ballistic one-dimensional hole systems, AIP Conference Proceedings 893, 699 (2007).

3. O. Klochan, W. R. Clarke, R. Danneau, A. P. Micolich, L.H. Ho, A. R. Hamilton, K. Muraki and Y. Hirayama, Ballistic transport in induced one-dimensional hole systems, Appl. Phys. Lett. 89, 092105 (2006).

4. R. Danneau, O. Klochan, W. R. Clarke, L.H. Ho, A. P. Micolich, A. R. Hamilton, M. Y. Simmons, M. Pepper, D. A. Ritchie and U. Z¨ulicke, Zeeman splitting in ballistic hole quantum wires, Phys. Rev. Lett. 97, 026403 (2006).

5. R. Danneau, W. R. Clarke, O. Klochan, L.H. Ho, A. P. Micolich, A. R. Hamil- ton, M. Y. Simmons, M. Pepper and D. A. Ritchie, Ballistic transport in one- dimensional bilayer hole systems, Physica E 34, 550 (2006).

6. R. Danneau, W. R. Clarke, O. Klochan, A. P. Micolich, A. R. Hamilton, M. Y. Simmons, M. Pepper and D. A. Ritchie, Conductance quantization and the 0.7 × 2e2/h conductance anomaly in one-dimensional hole systems, Appl. Phys. Lett. 88, 012107 (2006).

Manuscripts Currently Under Review

1. R. Danneau, O. Klochan, W. R. Clarke, L.H. Ho, A. P. Micolich, M. Y. Simmons, A. R. Hamilton, M. Pepper and D. A. Ritchie, Enhanced g-factor near the 0.7 structure in ballistic hole quantum wires, submitted to Phys. Rev. Lett.

2. A.R. Hamilton, O. Klochan, R. Danneau, W.R. Clarke, L.H. Ho, A.P. Micol- ich, M.Y. Simmons, M. Pepper, D. A. Ritchie, K. Muraki and Y. Hirayama, Quantum transport in one-dimensional GaAs hole systems, submitted to Int. J. Nanotechnology. ii

3. F. J. Rueß, B. Weber, K. E. J. Goh, O. Klochan, A. R. Hamilton and M. Y. Sim- mons, 1D conduction properties of highly phosphorous-doped, planar nanowires patterned by scanning probe microscopy, submitted to Phys. Rev. B.

International Conference Contributions

X R. Danneau, O. Klochan, W. R. Clarke, L.H. Ho, A. P. Micolich, M. Y. Sim- mons, A. R. Hamilton, M. Pepper and D. A. Ritchie, 0.7 structure and zero bias anomaly in ballistic hole quantum wires,17th International Conference on the Electronic Properties of Two-Dimensional Systems, in Genova, Italy, accepted for Poster Presentation, (2007).

X L. H. Ho, W.R. Clarke, R. Danneau, O. Klochan, A. P. Micolich, A. R. Hamil- ton, M. Y. Simmons, M. Pepper, D. A. Ritchie, Eﬀect of screening long-range Coulomb interactions in a dilute 2D system using a bilayer heterostructure,17th International Conference on the Electronic Properties of Two-Dimensional Sys- tems, in Genova, Italy, accepted for Poster Presentation, (2007).

X O. Klochan, W. R. Clarke, R. Danneau, A. P. Micolich, L.H. Ho, A. R. Hamilton, K. Muraki and Y. Hirayama,Conductance quantisation in an induced hole quantum wire, International Conference on Physics of Semiconductors, in Vienna, Austria, Oral presentation, (2006).

X O. Klochan, W. R. Clarke, R. Danneau, A. P. Micolich, L.H. Ho, A. R. Hamil- ton, K. Muraki and Y. Hirayama, Conductance quantisation in induced one- dimensional hole systems, Quantum nanoscience conference in Noosa, Australia, Poster presentation, (2006).

X R. Danneau, W.R. Clarke, O. Klochan, L.H. Ho, A.P. Micolich, A.R. Hamil- ton, M.Y. Simmons, M. Pepper and D.A. Ritchie, Ballistic transport in one- dimensional bilayer hole systems,16th International Conference on the Electronic Properties of Two-Dimensional Systems, in New Mexico, USA, Poster Presenta- tion, (2005). Contents

1 Introduction and Thesis Overview 1 1.1 Introduction ...... 1 1.2 Spin-orbit coupling eﬀects for holes in GaAs ...... 2 1.3 Large eﬀective mass of holes in GaAs ...... 3 1.4 Previous work on 1D hole systems ...... 4 1.5 Thesis overview ...... 8

2 Background Chapter 14 2.1 Introduction ...... 14 2.2 Confining carriers to 2D in GaAs/AlGaAs heterostructures ...... 15 2.3 Confining carriers to 1D in GaAs/AlGaAs heterostructures ...... 17 2.3.1 The conductance of ideal 1D systems ...... 18 2.3.2 Source-drain bias technique ...... 18 2.4 The Kondo effect ...... 21 2.4.1 The Kondo effect in quantum dots ...... 22 2.5 Spin-orbit coupling interactions ...... 25 2.5.1 Ballistic transport in the presence of SO interactions ...... 26 2.6 The structure of the valence band in GaAs ...... 29 2.6.1 The band diagram of the GaAs valence band in 2D ...... 33 2.6.2 The Zeeman splitting in 2D hole systems ...... 34 2.6.3 Spin-orbit coupling effects ...... 38 2.6.4 The band diagram of the GaAs valence band in 1D ...... 38

3 Device Fabrication and Measurement Setup 45 3.1 Introduction ...... 45 3.2 Devices for studies in Chapter 4 and 5 ...... 46 3.3 Devices for studies in Chapter 6 and 7 ...... 47 3.4 Fabrication routine ...... 50 3.4.1 2D device fabrication (device DQPC25) ...... 50

iii iv CONTENTS

3.4.2 2D device fabrication (device CQPC17) ...... 53 3.4.3 Modiﬁcation of the fabrication procedure for KQPC19 device . 55 3.4.4 1D device fabrication ...... 57 3.5 Measurement setup ...... 59

4 Anisotropic Zeeman splitting in doped hole 1D wires 65 4.1 Introduction ...... 65 4.2 Literature overview ...... 66 4.3 Device structure and the principles of its operation ...... 68 4.4 Determining the g-factors of the 1D subbands ...... 74 4.4.1 Source-drain bias spectroscopy ...... 75 4.4.2 Magnetic ﬁeld measurements ...... 78 4.4.3 Calculations of g-factors and discussion ...... 83 4.5 Theoretical interpretation of the results ...... 86 4.6 Conclusions and future work ...... 89

5 In-plane magnetic ﬁeld studies of the 0.7 structure and the zero-bias anomaly in doped hole 1D wires 92 5.1 Introduction ...... 92 5.2 Literature overview ...... 93 5.2.1 Studies of the 0.7 structure and the zero-bias anomaly ..... 95 5.2.2 The 0.7 structure in hole systems ...... 97 5.2.3 Theoretical results on the 0.7 structure ...... 98 5.3 Device structure and initial characterization ...... 99 5.4 Testing the 0.7 structure ...... 102 5.4.1 Temperature dependence of the 0.7 structure and the ZBA . . 102 5.4.2 In-plane magnetic ﬁeld measurements ...... 104 5.5 Discussion of the results ...... 108 5.6 Conclusions and future work ...... 110

6 Conductance quantization in induced GaAs/AlGaAs hole 1D wires 116 6.1 Introduction ...... 116 6.2 Literature overview ...... 117 6.3 Device structure and the principles of its operation ...... 119 6.4 Contact resistance and conductance quantization of the wire ...... 121 6.5 The stability of the device ...... 125 6.6 Density dependence of conductance quantization ...... 127 6.7 Temperature dependence of conductance quantization ...... 128 6.8 Source-drain bias measurements ...... 129 CONTENTS v

6.9 Magnetic ﬁeld measurements ...... 132 6.10 Conclusions and future work ...... 134

7 Anisotropic Zeeman splitting in orthogonal induced hole 1D wires 138 7.1 Introduction ...... 138 7.2 Device structure and initial characterization ...... 140 7.3 Determining the g-factors of the 1D subbands ...... 145 7.3.1 Source-drain bias spectroscopy ...... 146 7.3.2 Magnetic ﬁeld measurements ...... 148 7.3.3 Results for QPC233 ...... 149 7.3.4 Results for QPC011 ...... 153 7.3.5 Calculations of the g-factors and discussion of the results . . . 158 7.4 Conclusions and future work ...... 161

8 Thesis summary and future work 165 8.1 Thesis summary ...... 165 8.2 Future work ...... 166 Chapter 1

Introduction and Thesis Overview

Chapter Outline In this chapter we present our motivation for the research done in this thesis, highlighting the advantages of hole systems compared to well studied electron systems. These advantages include strong spin-orbit coupling eﬀects and enhanced particle-particle interactions. We also explain why hole one- dimensional systems attract less attention than their electron counterparts. Fi- nally we conclude with an overview of the content of this thesis.

1.1 Introduction

In this thesis we study ballistic transport in hole GaAs one dimensional systems. There are three main aspects, which relate to the material system in this research - GaAs :

1. One dimensional systems. The method we use to fabricate such structures is based on two powerful technologies: molecular beam epitaxy (MBE) and electron beam lithography (EBL). The MBE technique allows to grow high purity GaAs/AlGaAs heterostructures with a single layer control over the growth. In the heterostructures the carrier transport is restricted to a single plane. The EBL technique is then used to conﬁne the system further to one dimension (1D) by nano-patterning the surface of the heterostructure using either deposited metal depletion gates or wet/dry etching to form a narrow 1D constriction.

2. Ballistic transport. To study ballistic transport in a 1D system, its length should be shorter than the mean free path of the charge carriers. Therefore the host two-dimensional (2D) system should have a large mobility (i.e. few scattering centers). That is why high purity, lattice matched GaAs/AlGaAs heterostructures are ideal for these studies, as they allow us to achieve mean free paths

1 2 1. Introduction and Thesis Overview

exceeding 100 μm in electron 2D systems [1] and 10 μm in hole 2D systems [2, 3].

3. Hole systems. In hole systems the current is carried by holes rather than electrons. Holes in GaAs devices have two major advantages over electrons which we will elaborate on in the following sections:

X The p-type valence band enhances spin-orbit coupling eﬀects for holes. X ∗ · ∗ · Holes have a much higher eﬀective mass mp (0.2 m0

1.2 Spin-orbit coupling eﬀects for holes in GaAs

Recently the spin-orbit (SO) coupling effects in semiconductors have attracted a lot of attention, stimulated by a rapidly growing field of spintronics [4]. Electrical control over the spin degree of freedom in the semiconductor 2D systems via the Rashba effect [5] has been used for the theoretical proposals for new spintronic devices (e.g. the Datta-Das transistor [6]). The hole systems from this point of view are more promising than the electron systems. This is due to the band structure of a typical semiconductor, which has a s-type conductance band and p-type valence band. The orbital angular momentum for electrons in conductance band is equal to zero, and the spin-orbit coupling is only possible due to coupling to the valence band. This coupling depends on the energy gap between conduction and valence bands and therefore SO is strong for electrons in narrow band semiconductors. However narrow band semiconductors are really difficult to gate due to the absence of the Schottky barrier between the metal and the narrow band semiconductor. In contrast, for holes in the p-type valence band, the orbital momentum is non zero and the SO interactions are strong. We discuss the spin-orbit interactions in more detail in Chapter 2. The spin-orbit coupling effects in 2D hole systems have been used to study spin transitions between the states with different spin polarization in the fractional quantum 4 Hall (FQH) regime at ν = 3 [7]. In 2D electron systems these transitions can be stimulated by changing the carrier density or tilting the sample in a magnetic field. Muraki et al. have shown that in 2D hole systems the spin transitions can be triggered by changing the asymmetry of the 2D confining potential (by tuning both the front and the back gates) while keeping the hole density constant. The authors have calculated the effect of the asymmetry of the quantum well on the Landau levels and showed that it is substantial and can change the spin configuration of the FQH state. Shayegan’s group studied how the 2D confinement dramatically alters the properties of the hole system and particularly the Zeeman splitting in a magnetic field [8, 9]. 1. Introduction and Thesis Overview 3

The isotropic splitting for holes in bulk GaAs becomes highly anisotropic in 2D systems, due to drastic changes in the valence band structure. The large effective g-factor along the growth direction is contrasted by the zero in-plane g-factor. However for low symmetry growth directions, the spin-orbit interaction enhances the in-plane g-factors by coupling the spin to the highly anisotropic crystal lattice. Additionally, it was shown that the in-plane Zeeman splitting in 2D hole systems is very sensitive to the asymmetry of the confinement potential [10], which can have potential for spintronics applications. More details on the anisotropic Zeeman splitting in 2D hole systems are presented in Chapter 2. On the other hand, the effect of the 1D confinement on anisotropy of the Zeeman splitting remains mainly unexplored. In this thesis we study 1D hole systems with strong SO coupling. In particular we focus on how the 1D confinement affects the anisotropy of the in-plane 2D g-factors. We extract the effective g-factor directly by comparing source-drain bias measurements with the Zeeman splitting of the 1D subbands in an in-plane magnetic field oriented both parallel and perpendicular to the wire. Moreover, we fabricate and characterize a new type of 1D hole wires, based on a GaAs/AlGaAs SISFET heterostructure. This type of device overcomes limitations of doped systems and exhibits extremely stable gate characteristics. Using these new 1D hole wires we investigate the interplay between confinement, crystallographic anisotropy and spin-orbit coupling and its effect on the anisotropic Zeeman splitting of the 1D subbands.

1.3 Large eﬀective mass of holes in GaAs

The effective mass m∗ is very often defined as a curvature of the energy band d2ε(k)/d2k. The conductance band of GaAs can be approximated by a parabolic energy dispersion, ∗ resulting in a constant effective mass me. The situation for holes is more complicated ∗ because of a pronounced nonparabolicity of the valence band. Therefore mp depends on the value of k, and hence on the hole density. Moreover, the Rashba and Dressel- haus effects cause zero field spin-splitting of the valence band resulting in two branches for different k and hence in two values of the hole effective mass (the details on the band structure of the GaAs and spin-orbit coupling effects are presented in Chapter ∗ · 2). The most commonly used value for hole effective mass is me =0.38 m0 which is larger by a factor of 5 than the electron effective mass.

The high effective mass of holes enters the interaction parameter rs, which is defined as the ratio of the potential energy EP to the kinetic energy EK , and is proportional to the effective mass: EP a ∗ rs = = ∝ m , (1.1) EK aB 4 1. Introduction and Thesis Overview

where a is the carrier-carrier distance, aB is the eﬀective Bohr radius.

The typical range of rs accessible in 2D electron systems varies from 1 to 5. In contrast, 2D hole systems enable much higher rs ≈ 40 to be routinely achieved [11].

The high effective mass of holes in GaAs has been used to access high values of rs for studies of the metal-insulator transition phenomenon [12] and to prevent tunneling between the two closely spaced 2D layers in studies of electron-hole condensates [13] in 2D hole systems. Additionally, 2D hole systems are good candidates to probe the crystalline phase of charge particles, known as the Wigner crystal [14], and can exhibit a richer phase diagram than electron systems [10]. Interaction effects are very important in 1D systems. For any interaction strength the Fermi liquid theory breaks down in 1D and has to be replaced by a model such as the Tomonaga-Luttinger liquid, which is expected to represent many types of 1D systems [15]. However, the conductance of electron quantum point contacts (very short 1D wires) exhibited quantized plateaus in units of 2e2/h and this was explained by non-interacting picture [16, 17]. The discovery of the 0.7 structure [18], a small shoulder below the first plateau, highlighted the importance of the interactions in 1D wires. Many experimental and theoretical attempts have been directed towards understanding the origin of the structure but it remains unknown. The evolution of the 0.7 structure to 0.5 × 2e2/h in an in-plane magnetic field suggests that it is due to a spin polarized state at B =0 T, contradicting the Lieb-Mattis theorem [19] that the ground state of the 1D system can not be ferromagnetic. On another hand, the temperature dependence of the 0.7 structure and the distinct peak around zero bias in source-drain bias measurements (called the zero-bias anomaly), shows striking similarity to the signatures of the Kondo effect in quantum dots [20]. Details on the experiments and theory of the 0.7 structure are presented in Chapter 5. In this thesis we use 1D hole wires to study the behaviour of the 0.7 structure and the zero-bias anomaly (ZBA) in an in-plane magnetic field. We probe the 0.7 structure and the ZBA in an in-plane magnetic field oriented along the wire and perpendicular to the wire. Owing to the highly anisotropic g-factor we are able to test directly the connection between the 0.7 structure and the ZBA and their relation to spin.

1.4 Previous work on 1D hole systems

Despite the advantages of hole over electron systems, the number of studies on 1D hole wires is very limited compared to their electron counterparts. This is mainly due to the difficulties in fabrication, combined with the poor stability of their electrical characteristics. Thus 1D hole systems have remained relatively unexplored. Only 1. Introduction and Thesis Overview 5 recently have devices using modulation doped AlGaAs/GaAs heterostructures been sufficiently stable to observe the 0.7 structure in 1D wires and single hole transport in quantum dots. We will now review the previous works concentrating on problems with device fabrication and stability. Zailer et al. [21] used a modulation Si-doped GaAs/AlGaAs heterostructure, which was grown on a (311)A oriented substrate. Two narrow quantum point contacts (QPCs) in parallel, separated by a square region, were etched and metal gates were deposited on top of the etched regions. The side gates were then electrically connected to external circuitry, whereas the central dot remained unconnected. This particular geometry was chosen to study Aharonov-Bohm oscillations in a quantum Hall regime. By pinching off one of the constrictions it was possible to observe the transport through a single QPC only. Figure 1.1a shows the conductance through the QPC as a function of the side gate voltage at B = 0 T and at B = 3 T (in the quantum Hall regime). The conductance suffered from random telegraph noise and therefore presented traces were averaged over 40 side gate sweeps. The trace at B = 0 shows small inflections which are not quantized in units of 2e2/h. The application of the large perpendicular magnetic field (B = 3 T), spin polarizes the system and leads to conductance plateaus quantized in steps of e2/h - characteristic signature of the quantum Hall regime. Additionally, the estimation of the subband spacings were done only qualitatively from the temperature measurements because the direct measurements using a source-drain bias were prevented by the telegraph noise. Daneshvar et al. [22] used a symmetrically Si-doped quantum well grown on a (311)A GaAs substrate. The QPC was patterned into three gates: two split gates and the middle gate, using a standard EBL technique with subsequent deposition of the metal gates. The wire was oriented diagonally between the [233] and [011] crystallographic axes. Figure 1.1b shows the conductance of the QPC vs the side gate voltage at B = 0 T (top trace) and several values of the perpendicular magnetic field

B⊥. The zero field trace shows weak plateau-like features around quantized values of conductance (in units of 2e2/h). However Daneshvar et al. stated that in order to achieve the quantization a voltage-dependent contact resistance was subtracted, which was determined by fitting the plateaus to their quantized positions. Moreover, the authors emphasized that the side gate voltage at which the plateaus appeared was found to vary significantly between voltage sweeps. They ascribed this gate instability to problems with the fabrication technology. The sample was used to study the Zeeman splitting of 1D subbands in the in-plane magnetic field B perpendicular to the wire direction. Poor stability of the devices resulted in significant experimental error in the estimated effective g-factors, which exceeded 30 %. We will discuss these results in more detail in Chapter 4. 6 1. Introduction and Thesis Overview

(a) (b)

Figure 1.1: (a) The conductance of the one QPC as a function of a gate voltage at B = 0 and 3 T. The figure is taken from Zailer et al., Ref. [21]; (b) The conductance of the one QPC in units of e2/h as a function of the side-gate voltage for different B perpendicular to the 2D plane. The inset shows a schematic of the device. The figure is taken from Daneshvar et al., Ref. [22].

Rokhinson et al. [23] tried to avoid metal gating by using a AFM local anodic oxidation nanolithography (LAO), to pattern short constrictions over shallow Si-doped quantum well grown on a (311)A wafer. Figure 1.2a shows the conductance of the QPC vs the side gate voltage. The plateau-like features around the quantized value are clearly present for up to 9 × 2e2/h with the exception of 2e2/h, where no feature is visible. However, the plateaus are strongly perturbed by resonances. This may indicate that the ballistic transport in these type of QPC is severely affected by disorder, which can be due to the shallowness of the quantum well. Later on, Rokhinson et al. [24] performed magnetic focusing experiments using a pair of similar QPCs as the injector and detector of holes. The authors argued that they were able to observe two focusing peaks corresponding to spin polarized hole states due to intrinsic spin-orbit coupling. This technique has been used further to test the spin properties of the 0.7 structure present in these devices [25]. We should note however that the conductance of the QPCs studied in [25] exhibits a rich variety of additional features below and above the 2e2/h and the unambiguous interpretation of the results is hindered. We will look on this work in more detail in Chapter 6. All of the previously discussed works have been done on Si-doped (311)A grown GaAs heterostructures. (311) grown heterostructures have more complicated valence 1. Introduction and Thesis Overview 7 band and large crystallographic mobility anisotropy compared to their (100) counterparts [26]. For (311)A heterostructures, Si can be used as a p-dopant, in contrast to (100) heterostructures, where Be is a p-dopant. Be is much more diffusive than Si, which results in reduced hole mobilities in (100) grown heterostructures [27]. That is why ballistic transport has been predominantly studied in high mobility (311)A grown GaAs heterostructures. Only recently, a development in MBE technology has allowed the usage of carbon as a p-type dopant in (100) grown heterostuctures, producing very high hole mobilities [2, 3]. Pfeiffer et al. [28] used the cleaved edge overgrowth technique [29] combined with C doping to fabricate long hole wires (2 μm) in (100) oriented heterostructures. The conductance of the hole wires exhibits up to three plateaus, which became weaker as the subband index was increased (see Fig. 1.2b). Resonances in the conductance trace are evident, especially on and below the first plateau. Note, that the conductance is not quantized in units of 2e2/h due to abrupt 2D to 1D wave function mismatch in this type of 1D wires [30].

(a) (b)

Figure 1.2: (a) The conductance of the QPC as a function of a gate voltage. The inset shows the conductance at 4.2 K for several values of a constant gate offset. The figure is taken from Rokhinson et al., Ref. [23]; (b) The conductance of the 1D wire as a function of the top gate voltage. The inset shows the conductance of three different devices. The figure is taken from Pfeiffer et al., Ref. [28].

Finally, Ensslin’s group [31, 32] has fabricated the hole QPCs using the AFM LAO technique on C-doped (100) heterostructure. Figure 1.3a shows the conductance of the QPC vs the top gate voltage. The plateaus are clearly present but with additional features superimposed on top of it. The authors also mention the existence of time dependent conductance fluctuations that happen at certain gate voltages. Again, these 8 1. Introduction and Thesis Overview resonances could be due to the shallowness of the heterostructure used for AFM LAO fabrication technique. Later on, Grbić et al. presented measurements of single hole transport through quantum dots [33] and Aharonov-Bohm effect in ring structures [34] based on the same heterostructure. Recently Danneau et al. [35] fabricated QPCs on the Si-doped, (311)A grown bilayer quantum well heterostructure, which exhibited exceptionally clean quantized conductance in units of 2e2/h (see Fig. 1.3b) and were sufficiently stable to perform source-drain bias measurements and observe for the first time a reproducible 0.7 structure.

(a) (b)

Figure 1.3: (a) The conductance of the QPC as a function of the top gate voltage. The figure is taken from ETH annual report, Ref. [31]; (b) The conductance of the top wire as a function of the side gate voltage. The inset shows a four stage pinch-off trace of the bilayer system. The figure is taken from Danneau et al., Ref. [35].

1.5 Thesis overview

In this thesis we present four experimental chapters related to studies of ballistic transport in 1D hole systems, particularly in an in-plane magnetic field, and also the development of a new type of 1D wires with extremely stable gate characteristics. The next chapter will review the background information relevant to the research work in this thesis. This includes the formation of the 2D systems, linear and nonlinear ballistic transport in ideal 1D wires, the Kondo effect in quantum dots and spin- orbit interactions in electron systems. Then we discuss hole systems and show the band structure of the valence band in GaAs and its modification with the spatial confinement. We also present theoretical and experimental results on the Zeeman effect and spin-orbit interactions in 2D hole systems. 1. Introduction and Thesis Overview 9

In Chapter 3 we present the devices we used for the studies in this thesis as well as the heterostructures details. We will also outline the fabrication of this devices with special attention to the fabrication of a new type of 1D wires and modifications on previously developed fabrication processes. We then describe the measurement setup and modifications we made to reduce electrical noise in the setup. In Chapter 4 we study how the 1D confinement alters the 2D anisotropic Zee- man splitting. We perform transport measurements in ballistic hole quantum wires fabricated on a (311)A bilayer heterostructure with the application of an in-plane magnetic field and d.c. source-drain bias voltage. Extracted g-factors for 1D subbands parallel and perpendicular to the wire show an anisotropy that is significantly higher than in the 2D case. This anisotropy is explained in terms of interaction between 1D confinement and spin-orbit coupling. In Chapter 5 we investigate the behaviour of the 0.7 structure and the zero bias anomaly (ZBA) in ballistic hole wires fabricated on a (311)A bilayer heterostructure with the application of an in-plane magnetic field. We find that the magnetic field dependencies of both phenomena are distinctly anisotropic, owing to the highly anisotropic g-factor. The data also show a strong correlation between the anisotropy of the 0.7 structure and the ZBA suggesting that the two phenomena are intimately related and have the same origin. These results link unambiguously the 0.7 structure and the ZBA to spin. In Chapter 6 we present the characterization of a new type of 1D wires fabricated using a GaAs/AlGaAs SISFET structure. The device exhibits clear conductance quantization, including the 0.7 structure, and has extremely stable gate characteristics. The hole density in devices of this type can be varied over a wide range, while the mobility of holes remains high enough to observe conductance quantization. In Chapter 7 we study the Zeeman splitting in a new type of 1D wires fabricated along two orthogonal crystallographic directions. By comparing the results between the two orthogonal wires and the results in this chapter to those in Chapter 4 for the same wire direction, we investigate the interplay between 1D confinement, crystallographic anisotropy and spin-orbit coupling and its effect on the anisotropic Zeeman splitting of the 1D subbands. In Chapter 8 we summarize our results and conclusions and address the possibilities for further work that this project has generated. Bibliography

[1] L. Pfeiﬀer, K. W. West, H. L. Stormer and K. W. Baldwin, Electron mobilities exceeding 107 cm2/Vs in modulation doped GaAs, Appl. Phys. Lett., 55(18), 1888 (1989).

[2] M. J. Manfra, L. N. Pfeiﬀer, K. W. West, R. de Picciotto and K. W. Baldwin, High mobility two-dimensional hole system in GaAs/AlGaAs quantum wells grown on (100) GaAs substrates, Appl. Phys. Lett., 86, 162106 (2005).

[3] C. Gerl, S. Schmult, H.-P. Tranitz, C. Mitzkus and W. Wegscheider, Carbon-doped symmetric GaAs/AlGaAs quantum wells with hole mobilities beyond 106 cm2/Vs, Appl. Phys. Lett., 86, 252105 (2005).

[4] I. Zuti´c,J.ˇ Fabian and S. D. Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys, 76(2), 323 (2004).

[5] Y. A. Bychkov and E. I. Rashba, Oscillatory eﬀects and the magnetic susceptibility of carriers in inversion layers, J. Phys. C, 17, 6039 (1984).

[6] S. Datta and B. Das, Electronic analog of the electrooptic modulator, Appl. Phys. Lett., 56(7), 665 (1990).

[7] K. Muraki and Y. Hirayama, Spin transition of a two-dimensional hole system in the fractional quantum Hall eﬀect, Phys. Rev. B, 59(4), R2502 (1999).

[8] S. J. Papadakis, E. P. de Poortere, M. Shayegan and R. Winkler, Anisotropic magnetoresistance of two-dimensional holes in GaAs, Phys. Rev. Lett., 84(24), 5592 (2000).

[9] R. Winkler, S. J. Papadakis, E. P. de Poortere and M. Shayegan, Highly anisotropic g-factor of two-dimensional hole systems, Phys. Rev. Lett., 85(21), 4574 (2000).

10 BIBLIOGRAPHY 11

[10] R. Winkler, E. Tutuc, S. J. Papadakis, S. Melinte, M. Shayegan, D. Wasserman and S. A. Lyon, Anomalous spin polarization of GaAs two-dimensional hole systems, Phys. Rev. B, 72(19), 195321 (2005).

[11] H. Noh, M. P. Lilly, D. C. Tsui, J. A. Simmons, E. H. Hwang, S. D. Sarma, L. N. Pfeiﬀer and K. W. West, Interaction corrections to two-dimensional hole

transport in the large-rs limit, Phys. Rev. B, 68(16), 165308 (2003).

[12] E. Abrahams, S. Kravchenko and M. Sarachik, Colloquium: Metallic behavior and related phenomena in two dimensions, Rev. Mod. Phys, 73(2), 251 (2001).

[13] E. Tutuc, M. Shayegan and D. A. Huse, Counterﬂow measurements in strongly correlated GaAs hole bilayers: evidence for electron-hole pairing, Phys. Rev. Lett., 93(3), 036802 (2004).

[14] W. Pan, G. A. Csáthy, D. C. Tsui, L. N. Pfeiffer and K. W. West, Transition from a fractional quantum Hall liquid to an electron solid at Landau level filling 1 ν = 3 in tilted magnetic fields, Phys. Rev. B, 71, 035302 (2005).

[15] J. Voit, One-dimensional Fermi liquids, Rep. Prog. Phys., 57, 977 (1994).

[16] D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A. Ritchie and G. A. C. Jones, One- dimenstional transport and the quantization of the ballistic resistance, J. Phys. C: Solid State Physics, 21, L209 (1988).

[17] B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, D. van der Marel and C. T. Foxon, Quantized conductance of point contacts in a two-dimensional electron gas, Phys. Rev. Lett., 60(9), 848 (1988).

[18] K. J. Thomas, J. T. Nicholls, M. Y. Simmons, M. Pepper, D. R. Mace and D. A. Ritchie, Possible spin polarization in a one-dimensional electron gas, Phys. Rev. Lett., 77(1), 135 (1996).

[19] E. Lieb and D. Mattis, Theory of ferromagnetism and ordering of electronic energy levels, Phys. Rev, 125(1), 164 (1962).

[20] S. M. Cronenwett, H. J. Lynch, D. Goldhaber-Gordon, L. P. Kouwenhoven, C. M. Marcus, K. Hirose, N. S. Wingreen and V. Umansky, Low-temperature fate of the 0.7 structure in a point contact: a Kondo-like correlated state in an open system, Phys. Rev. Lett, 88(22), 226805 (2002). 12 BIBLIOGRAPHY

[21] I. Zailer, J. E. F. Frost, C. J. B. Ford, M. Pepper, M. Y. Simmons, D. A. Ritchie, J. T. Nicholls and G. A. C. Jones, Phase coherence, interference and conductance quantization in a conﬁned two-dimensional hole gas, Phys. Rev. B, 49(7), 5101 (1994).

[22] A. J. Daneshvar, C. J. B. Ford, A. R. Hamilton, M. Y. Simmons, M. Pepper and D. A. Ritchie, Enhanced g factor of a one-dimensional hole gas with quantized conductance, Phys. Rev. B, 55(20), R13409 (1997).

[23] L. P. Rokhinson, D. C. Tsui, L. N. Pfeiﬀer and K. W. West, AFM local oxidation nanopatterning of a high mobility shallow 2D hole gas, Superlatt. and Microstruct., 32(2-3), 99 (2002).

[24] L. P. Rokhinson, V. Larkina, Y. B. Lyanda-Geller, L. N. Pfeiﬀer and K. W. West, Spin separation in cyclotron motion, Phys. Rev. Lett., 93(14), 146601 (2004).

[25] L. P. Rokhinson, L. N. Pfeiﬀer and K. West, Spontaneous spin polarization in quantum point contacts, Phys. Rev. Lett., 96, 156602 (2006).

[26] R. Winkler, Spin-orbit coupling eﬀects in two-dimensional electron and hole systems (Springer-Verlag, Berlin, Germany, 2003), 1st ed.

[27] F. Fischer, D. Schuh, M. Bichler, G. Abstreiter, M. Grayson and K. Neumaier, Modulating the growth conditions: Si as an acceptor in (110) GaAs for high mobility p-type heterostructures, Appl. Phys. Lett., 86, 192106 (2005).

[28] L. N. Pfeiﬀer, R. de Picciotto, K. W. West, K. W. Baldwin and C. H. L. Quay, Bal- listic hole transport in a quantum wire, Appl. Phys. Lett., 87(7), 073111 (2005).

[29] A. Yacoby, H. L. Stormer, K. W. Baldwin, L. N. Pfeiﬀer and K. W. West, Magneto-transport spectroscopy on a quantum wire, Solid State Commun., 101, 77 (1997).

[30] R. de Picciotto, H. L. Stormer, A. Yacoby, L. N. Pfeiﬀer, K. W. Baldwin and K. W. West, 2D-1D coupling in cleaved edge overgrowth, Phys. Rev. Lett., 85(8), 1730 (2000).

[31] B. Grbi´c,C. Ellenberger, T. Ihn, K. Ensslin, D. Reuter and A. D. Wieck, Nanos- tructures on p-type C-doped AlGaAs heterostructures, Tech. rep., ETH Zurich (2004).

[32] K. Ensslin, Quantum point contacts: pinning down the last spin, Nature Phys., 2, 587 (2006). BIBLIOGRAPHY 13

[33] B. Grbi´c,R. Leturcq, K. Ensslin, D. Reuter and A. D. Wieck, Single-hole transistor in p-type GaAs/AlGaAs heterostructures, Appl. Phys. Lett., 87, 232108 (2005).

[34] B. Grbi´c, R. Leturcq, T. Ihn, K. Ensslin, D. Reuter and A. D. Wieck, Aharonov-Bohm oscillations in the presence of strong spin-orbit interaction, arXiv:0704.1264v1.

[35] R. Danneau, W. R. Clarke, O. Klochan, A. P. Micolich, A. R. Hamilton, M. Y. Simmons, M. Pepper and D. A. Ritchie, Conductance quantization and the 0.7 × 2e2/h conductance anomaly in one-dimensional hole systems, Appl. Phys. Lett., 88, 012107 (2006). Chapter 2

Background Chapter

Chapter Outline In this chapter we present the background information that is relevant to the experimental work in the subsequent chapters. We first consider electron systems and discuss the formation of the two dimensional systems, ballistic transport in one dimensional systems and the Kondo effect in zero-dimensional systems. Then we introduce spin-orbit coupling interactions in electron systems. After that, we consider hole systems. We present the evolution of the valence band in GaAs as we reduce the dimensionality of the system. We also discuss the mechanisms and show experimental results on the Zeeman splitting and the spin-orbit coupling effects in 2D hole systems.

2.1 Introduction

In this section we start from a simple description of how two dimensional (2D) systems are formed in the semiconductor heterostructures, highlighting the differences between modulation doped and induced systems. These two types of the semiconductor heterostructures are the basis of the 1D devices used throughout the thesis. Then, we reduce the dimensionality of the system from 2D to one dimension (1D), and show the conductance of a ballistic 1D conductor in both the linear and nonlinear regimes. We also note the presence of a feature below the first quantized plateau, called the 0.7 structure, and two of its possible origins: spontaneous spin polarization at zero magnetic field [1] and the Kondo mechanism [2]. We then look at the origin of the Kondo effect, and its signatures in the quantum dots, which also have been seen in the 1D wires. Following this discussion, we introduce a simple picture of the spin-orbit (SO) interactions in electron systems and show as an example, the effect of the SO interactions on ballistic transport in 1D electron systems. Finally, we turn our attention to hole

14 2. Background Chapter 15 systems and show that the physics of the hole systems is defined by the complexity of the valence band. We describe the methods for calculating the band diagrams of the semiconductors and trace the evolution of the GaAs valence band from the bulk (three dimensions) to the plane (two dimensions) and finally to the wire (one dimension). We also analyze the origins of the anisotropic Zeeman spin splitting, describe the effect of spin-orbit coupling effects and show previous experimental data on those effects in 2D GaAs hole systems.

2.2 Conﬁning carriers to 2D in GaAs/AlGaAs heterostructures

In this thesis we study 1D hole systems fabricated on two substantially different 2D heterostructures: the modulation doped square quantum well and the undoped triangular quantum well. Therefore in this section we will describe these two types of heterostructures, highlighting two key differences: the shape of the confinement potential and the different methods of generating carriers in these heterostructures. The first step towards studying low-dimensional systems is to confine the carriers to a 2D plane. This can be done in GaAs/AlGaAs heterostructures using band gap engineering. When an AlGaAs layer is grown on top of a GaAs layer, discontinuities in the conduction and valence bands, due to different energy gaps in GaAs and AlGaAs, form. It is possible to trap carriers at one of these discontinuities and therefore obtain a 2D sheet of carriers at the GaAs/AlGaAs interface.

Figure 2.1: (a) Generic symmetrically doped heterostructure wafer layout; (b) The band diagram of the heterostructure shown in Fig. 2.1a.

Figure 2.1a shows a heterostructure layer diagram of the active region of the modulation doped quantum well heterostructure: an undoped GaAs layer is sandwiched between the two undoped AlGaAs to form a quantum well (QW). The carriers in the well are provided from the n-doped AlGaAs layers on either side of the quantum 16 2. Background Chapter well. The schematic band diagram of this heterostructure is presented on Fig. 2.1b.

Doping an AlGaAs layer with n-type dopants shifts the Fermi energy EF towards the conduction band. The EF in the heterostructure remains constant throughout and therefore the energy bands in the undoped layers bend to shift EF closer to its intrinsic position in the center of the band gap. As a result of this band bending, two triangular quantum wells are formed at the GaAs/AlGaAs interfaces on each side of a GaAs layer. If the doping in the n-AlGaAs layer, the thicknesses of the undoped

AlGaAs and GaAs layers are chosen correctly, EF rises above the conductance band edge in the entire GaAs layer, forming a single QW. Electrons therefore accumulate in the GaAs QW creating a 2D sheet with a width determined by the thickness of the GaAs QW (typically 10 − 20) nm. By optimizing the heterostructure parameters the square quantum well confinement potential can be achieved in this type of quantum well. Modulation doped heterostructures have revealed exceptionally high mobilities at cryogenic temperatures [3] and are the basis for HEMT devices. However, the scattering of the carriers in these structures is rather complex. It consists of a long-range disorder potential scattering caused by the presence of the ionized dopants in the modulation doped layer, and a short-range disorder potential caused by roughness of the GaAs/AlGaAs interface and background impurities in the heterostructure. Usually, in clean systems at low carrier densities, the long-range scattering mechanism dominates over the short-range mechanism, thereby limiting mobility of the carriers. While it is possible to reduce the effect of remote ionized dopant scattering by increasing the separation between the ionized dopants and the 2D carrier layer (i.e. increasing the thickness of the undoped AlGaAs layer), this also limits the minimum size of devices that can be defined.

Figure 2.2: (a) Generic n+ SISFET wafer layout; The band diagram of the n+ SISFET structure with (b) no bias applied to the top gate and (c) positive bias applied to the top gate VB >Vthreshold.

An alternative approach that eliminates the effect of remote ionized dopant scat- 2. Background Chapter 17 tering entirely resulted in the development of SISFET devices. The structure and the principle of operation of a SISFET is similar to that of a silicon MOSFET. An undoped GaAs/AlGaAs heterojunction is followed by a thin undoped GaAs layer and a heavily doped GaAs cap layer, as shown in schematic in Fig. 2.2a. The doping in the cap layer pins EF close to the conduction band and therefore the bands in the undoped layer have to bend to maintain a constant EF throughout the system. A wide well at GaAs/AlGaAs interface is situated above EF and therefore is empty (see Fig. 2.2b). To populate the well, a voltage has to be applied to a heavily doped top layer, which is used as an in-situ top gate. When a positive voltage VB is applied to the top gate, the bands in the undoped layers bend upwards. Moreover, EF exhibits a discontinuity eVB, which effectively raises EF at the GaAs/AlGaAs interface, so that the bottom of the well is below EF , and hence the well is populated (see Fig. 2.2c). The shape of the confinement potential in this type of well of these type is triangular. The fact that the top layer is metallic also means that free carriers in this layer screen the random potential fluctuations from the ionized dopants. Thus, when the 2D system is induced, it does not “feel” the random potential fluctuations from the ionized dopants, and remote ionized impurity scattering is eliminated. As we show in Chapter 6, the absence of the doping layer in the heterostructure is very important for the fabrication of 1D hole systems with stable gate characteristics.

2.3 Conﬁning carriers to 1D in GaAs/AlGaAs heterostructures

To conﬁne the 2D system further, we use electron beam lithography technique to pattern nanoscale features on the surface of a heterostructure with either the subsequent deposition of metal depletion gates or wet/dry etching to form a narrow 1D constriction. By ballistic we mean that the conductor has dimensions smaller than the typical distance between two scattering events for a charge carrier moving along the conductor. If the conductor is ballistic, its conductivity cannot be described by the Drude formula: ne2τ σ = enμ = e , (2.1) e m∗ where e is the electron charge, n is the carrier density, μe is the electron mobility, and τe is the average time between scattering events. Clearly this formula cannot be applied to the ballistic transport because quantities like μe and τe are not applicable anymore (no scattering). Moreover, the usual scaling relation between conductivity and conductance also breaks down: W G = σ, (2.2) L 18 2. Background Chapter where W and L are the width and the length of the conductor. For example, in a ballistic conductor decreasing the length L by a half would not lead to a doubled conductance. The conductivity being an average quantity loses its meaning for the ballistic transport.

2.3.1 The conductance of ideal 1D systems

The conductance of an ideal 1D wire connected to source and drain reservoirs can be calculated using a common equation for the current through a conductor:

dn I = eΔnυ = e( )dEυ = e(1/hυ)eV υ =(e2/h)V, (2.3) dE where I is the current, V is voltage, Δn is the gradient of electron density, υ is the dn group velocity of the electrons and dE is the 1D density of states. As can be seen from Eq. 2.3, the terms that are dependent on the energy and the subband index (i.e. the group velocity and the density of states) cancel each other, e2 resulting in a constant conductance h per subband. This is known as equipartition rule, which says that each 1D subband contributes equally to the current through the wire. Moreover, the 1D transport does not depend on the actual material of the system. Hence, for the N subbands, the total conductance G appears as:

I 2e2 G = = N, (2.4) VSD h where the prefactor of 2 corresponds to the twofold spin degeneracy of the 1D subbands. This result has been confirmed by a series of experiments [4, 5], which showed the step-like conductance of the quantum point contacts fabricated on GaAs/AlGaAs 2DEG. With the application of an in-plane magnetic field the twofold degeneracy is lifted, which leads to the conductance in steps of e2/h. However, the natural question appears: Why does a ballistic conductor, which has no scattering, have a finite resistance? Imry [6] has pointed out that the 1D conductance depends on where the electrochemical potentials of the source and drain are chosen. He showed that the quantized resistance of the 1D wire is actually the contact resistance between the wire and the reservoirs.

2.3.2 Source-drain bias technique

Equation 2.4 in the previous section assumes that the applied voltage VSD between source and drain is much less than the separation between the two 1D subbands. In this section we will consider a VSD that is equal or larger than the separation between the two 1D subbands, which introduces strong nonlinearity into the system. 2. Background Chapter 19

The conductance of a 1D wire with the application of a large VSD has been calculated by Glazman and Khaetskii [7] and was later used to determine the subband spacings of the 1D subbands in GaAs/AlGaAs 1D wires [8]. The model used by

Glazman and Khaetskii considers an adiabatic constriction with VSD dropping symmetrically on each side of the constriction. Because the wire is ballistic, it has no resistance and VSD drops on the interface regions between the reservoirs and the 1D wire. With no d.c. bias applied the electrochemical potentials of the source and drain are closely spaced μS ≈ μD ≈ EF , as shown in the inset in Fig. 2.3a. Depending on the

Fermi energy, μS and μD can be between the 1D subbands (plateau in conductance) or aligned with one of the subbands (riser between the conductance plateaus). The conductance G is due to the transmission of N occupied 1D subbands and described by Eq. 2.4 (see Fig. 2.3a). We will begin from the situation where both the source and drain electrochemical potentials are located between the N and N − 1 subbands. When a d.c. voltage is added, μS and μD oﬀset symmetrically with respect to the Fermi energy in the 1D wire.

Increasing the d.c. voltage moves μS up and μD down, until for example μD drops below the N th subband (see Fig. 2.3b). In this case, the number of occupied subbands on the source side will be larger by one than the number of occupied subbands on the drain side. For all subbands up to (N − 1) there are electrons moving in both directions from source to drain (forward) and from drain to source (backward). This results in a known conductance:

2e2 G = (N − 1) (2.5) h

For the N th subband there are only electrons moving from the source to drain and none moving backward. Therefore the current for the N th subband will not depend on the electrochemical potential of the drain but only on the diﬀerence (μS − EF ), hence:

eVSD EF + 2 2e 2 2e eVSD I = dE = (EF + − EN ), (2.6) h EN h 2 which gives a conductance: 2e2 1 G = ( ). (2.7) h 2 Then, the total conductance of the 1D wire will be:

2e2 1 G = (N − ). (2.8) h 2 This results in half plateaus start to develop in the conductance. 20 2. Background Chapter

5 5 N N µ E µ µS S F D 4 4 EF µD N-1 N-1 3 3 /h) 2 /h) 2 2 2 G (2e G (2e

1 1

0 0 0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 VSG (a.u.) VSG (a.u.) (a) (b) 5 5 µS µS N N+1 E E 4 F µD 4 F N N-1 N-1 µ 3 D /h) 3 2 /h) 2 2 G (2e 2 G (2e

1 1

0 0.0 1.0 2.0 3.0 0 0.0 1.0 2.0 3.0 VSG (a.u.) VSG (a.u.) (c) (d)

Figure 2.3: Conductance of the QPC vs VSG for diﬀerent values of d.c. oﬀset VSD. The insets show the corresponding energy levels diagrams. 2. Background Chapter 21

When EF sits exactly in the middle between the subbands N and N + 1, when at a certain d.c. voltage VC , μS and μD align with the two subband levels. In this case, only half plateaus will be present in the conductance (Fig. 2.3c). This value of the th source-drain voltage VC is a direct measure of the subband spacing between N and N +1th subbands

EN,N+1 = eVC . (2.9)

th By increasing the VSD further, μS rises above the N +1 subband and μD falls below N −1th subband. Using the same argument as before, it is only N −1 subbands, which have electrons moving in both directions. The subbands N and N + 1 will have only forward moving electrons which will result in aconductance:

2e2 G = N, (2.10) h which means that the integer plateaus reappear and the half-plateaus vanish (Fig. 2.3d). By comparing the evolution of 1D subbands in the source-drain bias and magnetic ﬁeld, it is possible to extract the eﬀective g-factors of 1D subbands.

2.4 The Kondo eﬀect

In previous sections we have considered the conductance of an ideal 1D wire in linear and nonlinear regimes and found that the 1D quantized conductance can be obtained considering no interactions between the electrons. On the other hand, as was mentioned in Chapter 1, for any interaction strength the Fermi liquid theory breaks down in 1D and has to be replaced by a model such as Tomonaga-Luttinger model [9]. The discovery of the 0.7 structure [1], a small shoulder below the first plateau, highlighted the importance of interactions in 1D wires. The review of experimental results and the relevant theoretical explanations of the 0.7 structure are presented in Chapter 5. Yet, the origin of the 0.7 structure remains unknown. The evolution of the 0.7 structure in an in-plane magnetic field suggests that it is due to a spin polarized state [1]. On another hand, the temperature dependence of the 0.7 structure and the distinct peak around zero bias in source-drain bias measurements (the zero-bias anomaly), shows striking similarity to the signatures of the Kondo effect in quantum dots [2]. To understand the origin of the zero-bias anomaly, in this section we review the Kondo effect, focusing on the results in quantum dots. We will start from the description of the conventional Kondo effect in metals. The resistance of metals is mainly determined by phonon scattering at high temperatures. Hence, when the temperature is decreased the resistance of the metal drops. At very low temperatures ≈ 10 K, the resistance of a pure metal saturates due to the presence 22 2. Background Chapter of defects. This saturation resistance depends on the concentration of the defects. However if magnetic impurities are introduced, the situation changes dramatically. The resistance of such a system will first decrease and then, at a certain temperature, start to increase with decreasing temperature. This phenomenon was explained by Kondo [10] in terms of the interaction between the spin of the magnetic impurity and the spin of electrons, which results in a logarithmic increase in the resistance with decreasing temperature. Further modifications were made by Anderson [11] to account for temperatures close to absolute zero. The mechanism of the Kondo effect can be described by two Fermi seas of electrons with energy EF separated by an atom with a discrete energy spectrum (for simplicity we will consider only one level ε0 with one electron). This electron can tunnel out from the ε0 level to the left or right Fermi sea only if ε0 >EF , otherwise it remains on the ε0 level. Quantum mechanically, the tunneling is possible due to exchange processes that can effectively change the direction of the spin on the impurity. These processes are possible due to Heisenberg uncertainty, which allows energetically unfavorable configurations for very short periods of time. Thus the electron from the ε0 level can tunnel out of the impurity to the Fermi sea and another electron from the Fermi sea can tunnel onto the impurity, but the spin of that electron can be different. When many of these flipping processes are taken together, the energy spectrum of the system exhibits an additional peak at EF . Because at low temperatures only the electrons close to

EF take part in the conductance, this additional peak causes enhanced scattering and hence the resistance increases. This eﬀect happens below a certain temperature known as the Kondo temperature (TK ). Theoretically the Kondo temperature was related to the energy scales of the model via the following relation: √ ΓU πε0(ε0 + U) T = exp , (2.11) K 2 ΓU where Γ is the width of the impurity level ε0, which is broadened due to tunneling, and U is the Coulomb energy required to add a second electron on the level ε0. Moreover, the ratio of the resistance at a ﬁnite temperature to the zero temperature resistance

(RT /R0) is a function of a scaled temperature (T/TK ). This function is the same for 1 all materials with a spin- 2 impurity.

2.4.1 The Kondo eﬀect in quantum dots

Quantum dots are essentially artiﬁcial atoms, which can be fabricated by further conﬁnement of a 1D wire using either surface gate or etching techniques (see Chapter 1). An example of the quantum dot and corresponding energy scales are shown on 2. Background Chapter 23

Fig. 2.4. The quantum dot consists of two tunnel barriers (defined by the gates SG) and a plunger gate (P) in between, which allows control of the number of the electrons on the dot. Depending on the coupling between the reservoirs and the dot, the dot can operate in different regimes. In the limit of the weak coupling, the number of the electrons on the dot is well defined (i.e. the energy spectrum is discrete) and the dot is described by single electron transport (the Coulomb blockade regime). On the other hand, in the limit of strong coupling the number of the electrons on the dot is undefined (the energy spectrum is continuous) as they can travel freely to/from source/drain reservoirs. To observe the Kondo effect, a quantum dot should be tuned

Figure 2.4: (a) The schematic of the quantum dot consisting of the tunnel barriers, deﬁned by side gates (SG) and a plunger gate (P) in between. The source and drain ohmic contacts are marked as S and D correspondingly. (b) The schematic diagram the energy scales in the dot taken from Ref. [12]. between the above two limits. This is because the following requirements for the observation of the Kondo eﬀect in quantum dots should be obeyed:

1. The dot should have a discrete spectrum (the spacing between the levels Δε is large).

2. The coupling between the dot and source/drain reservoirs should be relatively large (the Γ is large).

3. The measurement temperature should be below TK . The first criterium is satisfied when the dot dimensions are small and the second criterium is satisfied by adjusting the source and drain tunneling barriers (adjusting the QPCs voltages). The third criterium is satisfied when TFermi level of the source/drain. When the level

N with energy εN lines up with EF the electrons can transport from the source to the drain through the dot, otherwise there is no transport. This is known as Coulomb blockade [13].

We now consider the situation when εN

(a) (b) (c)

Figure 2.5: The model of a magnetic impurity with a single level ε0 below the Fermi level EF . (a) Initial state; (b) virtual state; (c) ﬁnal state. Figures are taken from Ref. [14].

Unlike in a metal, in a quantum dot this many body state enhances the conductance through the dot. This happens due to reduced dimensionality of the dot compared to a metal. In metals the Kondo mechanism effectively increases the radius of the scatterer (magnetic impurity), hence the resistance increases. In the quantum dots, there is no path around the dot, and the Kondo correlated state promotes the transport through the dot, hence the resistance decreases (actually the conductance can reach the unitary limit of 2e2/h, implying the transmission probability T = 1 through the dot [15]). The first signature of the Kondo effect is the temperature dependence. It can be clearly seen from the temperature dependence of the Coulomb blockade traces, providing the measurement temperature is below TK [16]. The peaks on the traces bunch up into pairs (separated by energy U) corresponding to the filling of each energy level with two electrons. Different pairs are separated by energy U + ε0. Between the peaks inside the pair the number of electrons in the dot is odd and between the pairs 2. Background Chapter 25 the number is even. The temperature dependence of the Coulomb blockade traces exhibit two distinctive features:

1. the conductance between the peaks inside one pair (number of electrons is odd) increases with decreasing temperature;

2. the conductance between the pairs (number of electrons is even) decreases with decreasing temperature.

Clearly, there are two opposite trends in the temperature evolution of conductance depending on whether the number of electrons on the dot is even or odd. The enhancement of conductance for odd configurations with decreasing temperature strongly resembles the Kondo effect in metals. The second signature of the Kondo effect is a peak in the density of states at

EF . This peak enhances the conductance through the dot. When the bias voltage

VSD is applied between the source and the drain, the Kondo peak splits into two peaks corresponding to the positions of the electrochemical potentials of the source

(μS) and drain (μD). When the separation of μS and μD is large, an electron in the source cannot resonantly tunnel to the drain and the conductance is suppressed [16]. It is possible to visualize the Kondo peak in the density of states by measuring the differential conductance through the dot vs the applied bias VSD. The differential conductance exhibits a peak at zero VSD rapidly falling as a bias increased. This peak is known as the zero bias anomaly (ZBA) and is directly related to the Kondo resonance in the density of states. With the application of an in-plane magnetic field the spin degeneracy of the levels on the dot are lifted and the ZBA splits into two peaks following the linear Zeeman relation [14]. The signatures of the Kondo effect discussed above have been seen in electron 1D wires [2] and have been used to explain the origin of the 0.7 structure in 1D wires. Recently the ZBA has been seen in hole 1D wires [17]. To the best of my knowledge, no theoretical work on the Kondo effect in hole systems has been done so far. Moreover, 1 it is unclear if the Kondo mechanism can be simply mapped from electron (J = 2 ) 3 into hole (J = 2 ) systems. Therefore further studies of the ZBA and its relation to the 0.7 structure have to be done in hole systems to understand the origin of both phenomena and in order to compare the results obtained in hole systems to those obtained in electron systems.

2.5 Spin-orbit coupling interactions

As was explained in Chapter 1, spin-orbit coupling effects in the semiconductors have attracted a lot of attention recently due to possible spintronic applications [18]. For 26 2. Background Chapter electrons, SO interaction effects are possible only via coupling of the conductance band to the valence band, whereas for holes SO interactions are an intrinsic property. Before approaching hole systems, we first describe the spin-orbit interaction effects in electron systems and how they can affect the ballistic transport of quasi-one dimensional wires. The spin-orbit interaction can be viewed similarly to the interaction of spin and the external magnetic field (the Zeeman effect): the energies of the electrons will be different depending on the spin orientation relative to the field direction. Spin-orbit coupling interaction can be viewed in a similar way but instead of the external magnetic field, the internal magnetic field is generated due to electron orbital motion. There are two SO mechanisms which can remove the spin degeneracy of the energy levels in the absence of magnetic field. These mechanisms are related to certain asymmetries present or induced in the system:

X The Dresselhaus term arises from the fact that the zinc-blende structure of GaAs crystal lacks inversion symmetry [19]. This bulk inversion asymmetry is independent of any external ﬁelds and depends only on the crystallographic direction. In the bulk, the Dresselhaus term for electrons in the conductance band is proportional to k3, which means that it can only be signiﬁcant at k>0. By reducing dimensionality of the system to 2D, the Dresselhaus term becomes linear in k near k ≈ 0.

X The Rashba term is specific to 2D systems and is related to the asymmetry of confinement potential [20]. This asymmetry, often referred to as a structural inversion asymmetry, causes an electric field E perpendicular to the 2D plane, which can be tuned using a gating scheme. Qualitatively the effect of field E on the moving elecrons/holes can be viewed as an effective in-plane magnetic field acting on electrons via the Lorentz force. The Rashba term is linear in E and k for electrons in the conductance band:

ˆ ERashba = βRJ · k × E, (2.12)

where the prefactor βR is inversely proportional to the energy gap between the subbands. Note also that the Rashba eﬀect for electrons in the conduction band is only possible via coupling to the valence band.

2.5.1 Ballistic transport in the presence of SO interactions

Ballistic transport in electron quasi one dimensional wires in the presence of a finite spin-orbit coupling was theoretically considered by many authors (see Ref. [21] and references therein). An example of the electron quasi 1D wire with perfect transmission 2. Background Chapter 27 and the effect of the SO interaction on the 1D conductance has been considered ana- lytically by Pershin et al. [22]. The authors pointed out that the Rashba SO coupling is controlled by the asymmetry of the well and is typically larger than the Dresselhaus SO term. Therefore only the Rashba term in the Hamiltonian was considered: ∂ ∂ H = β (σ p − σ p )=iβ (σ − σ ), (2.13) R R x y y x R x ∂y y ∂x where βR is the Rashba coefficient, σx and σy are the components of the Pauli spin matrix and px and py are the components of momentum operator. Unlike the Zeeman spin splitting which splits spin degenerate subbands in energy, the Rashba splitting is in the vector k direction [23], as shown in schematic in Fig. 2.6. The combined effect of the Zeeman and Rashba splitting for different orientations of the in-plane magnetic field B is shown in Fig. 2.7a and 2.7b. For B perpendicular to the wire the subbands simply shifts along k and E (see Fig. 2.7a). However, for B parallel to the wire the energy bands exhibit anticrossings resulting in local maxima at k = 0 (see Fig. 2.7b).

E E E

↑

↑ ↑↓ k ↓ k ↓ k

Figure 2.6: The twofold degenerate conductance band (left) and the eﬀect of Zeeman splitting (middle) and the Rashba splitting (right).

The conductance of the ideal 1D wire exhibits steps corresponding to the chemical potential μ moving through the bottom of the 1D subbands. When the subbands are spin degenerate they contribute 2e2/h into the total conductance. With the application of an in-plane magnetic field, the subbands split and each of the subband contribute only e2/h. On the energy diagram shown on Fig. 2.7a we can see that by sweeping μ from zero energy up through the subbands, μ will cross the spin-split subbands one by one and each of them will contribute e2/h. The conductance in this case is presented on Fig. 2.7c, which is not different from the spin-split conductance that occurs due to the Zeeman effect. Therefore the conductance is insensitive to the Rashba contribution for this direction of the magnetic field. 28 2. Background Chapter

The situation changes dramatically when we consider the energy diagram for B parallel to the wire (see Fig. 2.7b). Again, each subband minimum will contribute e2/h and the maximum will reduce the conductance by e2/h. By sweeping μ from zero energy up through the subbands, it crosses two minima which have the same energy and contribute 2e2/h but then it crosses the maximum and the conductance reduces by e2/h, ﬁnally μ crosses the second subband and the conductance restores back to 2e2/h (see Fig. 2.7d).

(a) (b)

Figure 2.7: Dispersion relation E vs p for the direction of magnetic ﬁeld B (a) perpendicular to the wire and (b) parallel to the wire. Conductance G as a function of the chemical potential μ corresponding to the energy dispersion in (c) Fig. 2.7a and (d) Fig. 2.7b. Figures are taken from Ref. [22].

Pershin et al. also noted that temperature washes out a sharp transition in the conductance resulting in a peak-like feature below each quantized plateau. Note that in the results in this thesis, we have not observed repetitive resonances below each quantized plateau in conductance of hole 1D wires. Only in Chapter 7, we have observed resonant structure below the 2e2/h that does not appear at higher plateaus. This suggests that the resonant structure may be related to a defect around the 1D 2. Background Chapter 29 constriction.

2.6 The structure of the valence band in GaAs

As was briefly mentioned in Chapter 1, holes are much more difficult to work with, particularly due to the pronounced nonparabolicity and anisotropy of the valence band. In this thesis we mostly focus on studies of the Zeeman splitting in different 1D hole systems. The primary question is: “How does 1D confinement alter the 2D in-plane g-factor anisotropy?”. To answer this question, we need to understand the origin of the in-plane g-factor anisotropy in hole systems by tracing the evolution of the valence band in GaAs from the bulk (three dimensions) to the plane (two dimensions) and finally to the wire (one dimension). We also describe the Rashba and Dresselhaus effects and show experimental results in GaAs 2D hole systems to understand how important these effects are compared to the Zeeman spin splitting. Figure 2.8 shows the typical band diagram of a bulk semiconductor. The bands are considered to be parabolic (i.e. the effective mass m∗ is constant). The conductance band is separated from the valence band by an energy gap Eg (Eg =1.43 eV for GaAs at T = 300 K, [24]). The valence band consists of the two branches with different curvature and therefore different hole effective mass: the heavy hole (HH) band and the light hole (LH) band, which are degenerate at k = 0. Note also the split-off band below the HH and LH bands, that we will discuss later in the section. However, this is just a qualitative picture, which becomes significantly modified for real semiconductors. In particular Fig. 2.8 is not valid around the top of the valence band (when the wave vector k = 0), where most of the transport phenomena take place. In order to calculate the band diagrams around k = 0 several methods had been developed. The k·p theory [25] solves the Schrödingerequation for periodic parts of the Bloch functions, eliminating the plane wave prefactor. We can write the Schrödinger equation for electrons in state n with a wave function ψn,k: p2 + V (r) ψn,k = εn(k)ψn,k, (2.14) 2m0 where p = −i∇ is the momentum operator, m0 is the free electron mass, V (r)is the potential of the crystal lattice and k is a wave vector. We now present the wave function in Bloch representation:

ψn,k = Un,k(r)exp(ik · r), (2.15) where Un,k(r) is a periodic part of the wavefunction and exp(ik · r) is a plane wave. 30 2. Background Chapter

Figure 2.8: The simple band structure of the semiconductors. The bands are considered to have a parabolic dispersion. The top s-like conductance band is separated by the energy gap from the double degenerate p-like valence band.

Substituting Eq. 2.15 into Eq. 2.14 we obtain the Schr¨odingerequation for the periodic functions U(n, k): p2 2k2 + V (r) + k · p + Un,k(r)=εn(k)Un,k(r), (2.16) 2m0 m0 2m0

As can be seen from Eq. 2.16, this approach results in a k · p term in the Hamilto- nian. Because k ≈ 0, this additional term can be treated as a perturbation, typically to the second order. This theory describes the shape of the bands and predicts the effective mass of electrons in the semiconductors extremely well, despite the fact that it is not applicable for k>0. Additionally, this theory has a very limited application to the valence band at k = 0, where the “light” and the “heavy” hole bands become degenerate. Therefore more sophisticated models are required for valence band calculations. The Kane model is an extension of the k ·p theory, which overcomes the limitations mentioned above and solves the Schrödingerequation using a set of eigenfunctions at k = 0. Being an exact model rather than an approximation, it allows the band diagrams of the semiconductor to be calculated with an arbitrary degree of precision, providing a complete set of eigenfunctions is given. Usually the Kane model is limited to a certain subset of the eigenfunctions, depending on the accuracy of the band calculations required [26]. The modification of the Kane model includes the spin-orbit (SO) coupling interaction. SO interaction follows from nonrelativistic Dirac equation, that results in a 2. Background Chapter 31

Pauli term: − · × ∇ HSO = 2 2 σˆ pˆ ( V0), (2.17) 4m0c where is a Planck’s constant, c is the speed of light, p is the momentum operator, σ is a Pauli spin vector and V0 is the Coulomb potential of the atomic core. In a simpliﬁed picture the Pauli term can be considered as an energy of the magnetic moment, proportional to a product of the orbital momentum ˆl and the spins ˆ:

ˆ HSO ∝ (l · sˆ). (2.18)

For electrons in conductance band l = 0 and for holes in the valence bands ˆl has a magnitude l = 1. Because l = 0, the valence band has two states with different total ˆ ˆ 3 1 orbital momentum J = l +ˆs: J = 2 and J = 2 . Typically the spin-orbit interaction 3 1 causes a splitting Δ between J = 2 and J = 2 states (see Fig. 2.8), which depends on the energy gaps and for bulk GaAs is Δ = 0.341 eV [24]. Because of the large energy 1 gap, J = 2 states are usually empty and do not take part in electrical transport. This band is called the split-off band. 3 3 1 The J = 2 state has four projections on the z axis: Jz = ± 2 ; ± 2 and therefore 3 is fourfold degenerate at k = 0. The doubly degenerate state with Jz = ± 2 is called 1 the heavy-hole (HH) band and the doubly degenerate state with Jz = 2 is called the light-hole (LH) band (see Fig. 2.8). In contrast, for the conductance band the magnitude of orbital momentum l =0 and therefore there is no direct spin-orbit interactions (c.f. Eq. 2.18). For electrons in conductance band spin-orbit effects arise only due to the coupling of the conductance band to the more remote bands and inversely proportional to the energy gap. Therefore the spin-orbit effects in electron systems are only large in the narrow gap semiconductors. As was mentioned above, the Kane models are usually limited to a certain subset of eigenfunctions. The extended Kane model [27, 28] takes into account k · p and SO interactions between the five top-most bands: split-off Γ7v, fourfold degenerate HH-LH band Γ8v, and the conduction bands Γ6c,Γ7c,Γ8c. It results in a 14 × 14 Hamiltonian matrix, which can be reduced to six non zero matrix elements that describe the gaps and couplings between the bands. To simplify the calculations, the Kane model with a smaller Hamiltonian matrix can be considered:

X The 8 × 8 Kane model, which includes only three bands (Γ6c,Γ8v,Γ7v).

X The 4 × 4 Kane model called the Luttinger Hamiltonian, which only takes into

account one band Γ8v and the interactions with the remote subbands are repre- sented by three Luttinger parameters 32 2. Background Chapter

All Kane models give the same result at k = 0, but the higher order Kane models are more accurate for k>0. An example of the calculated GaAs band diagram using a 24×24 Kane model is shown on Fig. 2.9. The point Γ on wave vector axis corresponds to k = 0. As can be seen from the diagram the lowest conductance band Γ6c is isotropic and has approximately a quadratic dispersion at small k. In contrast, the top valence band Γ8v is highly anisotropic and nonparabolic. The split-oﬀ valence band Γ7v is isotropic and has a quadratic dispersion at small k similar to the conductance band

Γ6c.

Figure 2.9: The band structure of bulk GaAs calculated using a 24 × 24 Kane model. The ﬁgure is taken from Ref. [29].

A powerful method of analysis of the Hamiltonian is the theory of invariants [30], which treats the Hamiltonian based on symmetry considerations. Using this theory, it is possible to predict which terms can enter the Hamiltonian and which must cancel. The symmetry of the Hamiltonian reﬂects the symmetry of the crystal. For zinc-blende compounds the Hamiltonian can be separated into the three parts: spherical, cubic and tetrahedral.

H = Hspherical + Hcubic + Htetrahedral. (2.19)

It is very common that only the spherical part is used. Note however, that the spherical part does not distinguish between electrons and holes, as it depends only on the 2. Background Chapter 33

coupling between the conduction Γ6c band and the valence Γ8v,Γ7v bands. The spherical part accounts for the nonparabolicity of the bands but not their anisotropy. The anisotropy corrections for holes comes from the cubic part of the Hamiltonian which contains the coupling to more remote bands. The tetrahedral term contains corrections of higher orders and usually is omitted. In the 2D case the symmetry of the Hamiltonian is reduced, and this results in an axial term, with the symmetry axis perpendicular to the 2D plane [31]. By reducing the dimensionality of the systems we reduce its symmetry, therefore the spatial directions become nonequivalent. In the following discussion we will always assign z direction to the direction of the vector Jˆ and x and y to the remaining two directions. For example, in the 2D case Jˆ points in the direction perpendicular to the 2D plane, hence the z direction is the direction perpendicular to the plane, and x and y are the in-plane directions.

2.6.1 The band diagram of the GaAs valence band in 2D

By confining the 3D system to two dimensions, the 3D energy levels of the system are 3 transformed into a series of 2D subbands. Holes in Γ8v (HH-LH) band have J = 2 and are strongly affected by size quantization. The fourfold degenerate HH-LH subband 3 splits into a twofold degenerate HH band with Jz = ± 2 and a twofold degenerate LH 1 ˆ band with Jz = ± 2 (see Fig. 2.10). The confinement sets the quantization axis of J along the growth direction (perpendicular to the 2D plane).

Figure 2.10: A schematic of the valence band in the 2D case. The solid lines correspond to the shape of the bands while the dashed lines mark the anticrossings. The ﬁgure is taken from Ref. [31].

The terms “heavy” and “light” holes are not strictly applicable in the 2D case. For the bound states along the quantization axis (z), the mass of the holes in the HH subband is lighter than the hole mass in the LH subband. Therefore the curvature of 34 2. Background Chapter the HH band around k = 0 is larger than that of LH band as shown on Fig. 2.10. But for in-plane (xy) motion, the HH band has a higher eﬀective mass than the LH band.

This results in HH-LH anticrossings at kxy > 0 (see Fig. 2.10) due to the oﬀ-diagonal terms in the Hamiltonian, which causes mixing of HH and LH states.

Figure 2.11: Band diagram for 15 nm GaAs/AlGaAs quantum wells with diﬀerent growth directions (left) (100) and (right) (311), calculated using the 8×8 Kane model. Dotted lines correspond to the axial approximation. The ﬁgure is taken from Ref. [31].

Figure 2.11 shows the calculated band structure of a 15 nm GaAs/AlGaAs quantum well for different growth directions, using the 8 × 8 Kane model with the terms in the Hamiltonian up to the cubic correction (solid lines). The dotted lines on the graphs show calculations using only the axial approximation. Both approximations give very close results near k = 0, but as k increases the differences are clearly visible. Additionally, as can be seen from the graphs the topmost 2D band is always heavy hole (HH1), independent of the growth directions. Whereas the second band can be either heavy hole (HH2 for [113] growth direction) or light hole (LH1 for [001] growth direction). Typically in GaAs heterostructures, only the first subband is occupied, which means that the transport in 2D hole systems is via the heavy holes. Note also the anticrossings that are obvious for [001] growth direction.

2.6.2 The Zeeman splitting in 2D hole systems

The Zeeman splitting of holes is isotropic in bulk GaAs. However the situation changes drastically when the system is conﬁned to 2D. The Zeeman term for the valence band includes isotropic and anisotropic contributions with the proportionality constants κ 2. Background Chapter 35 and q respectively: ˆ3 Hv = −2κμBB · Jˆ − 2qμBB · J , (2.20)

The anisotropic coefficient q is two orders of magnitude smaller than κ and usually is omitted. The κ term couples states different by ±1inJz. In other words, the 1 magnetic field B acts a ladder operator, for example, it couples LH (Jz = − 2 ) states 1 3 1 to LH (Jz = 2 ) states. It is also couples HH states (Jz = ± 2 ) to LH states (Jz = ± 2 ), 3 3 but no direct coupling between HH and HH states(Jz =+2 and Jz = − 2 ) exists. As discussed earlier, size quantization defines the orientation of the vector Jˆ along the growth direction (z). An in-plane magnetic field favors Jˆ to be in the 2D plane (xy), but Jˆ is already predefined to be along z. Therefore the g-factors for HH states in an in-plane magnetic field should be zero. This is true for high symmetry growth directions ([100] and [111]) (see Table 2.1). These directions can be described in a simple way by the spherical part of the Luttinger Hamiltonian. The spherical part can be diagonalized, which means that the HH and LH states can be separated.

(100) (311) B[001] B[100] B[010] B[311] B[233] B[011] 7.2 0 0 7.2 0.6 -0.2

Table 2.1: Theoretically calculated eﬀective g-factors for the HH band in 20 nm (011) and (311) grown GaAs/AlGaAs quantum wells. The numbers are taken from Refs. [32, 33].

In contrast, for low symmetry directions (such as [311]), the spin couples to the highly anisotropic orbital motion and the in-plane g-factors become finite and anisotropic due to additional HH-LH coupling (see Table 2.1). In this case, the spherical term of the Luttinger Hamiltonian does not describe the system and the addition of the cubic term is needed. Therefore the Luttinger Hamiltonian cannot be diagonalized, which means that the states become a mixture of HH and LH states even at k = 0. The off-diagonal terms depend on the anisotropy of the orbital motion and cause a finite and anisotropic g-factor in the in-plane directions. But still, the in-plane g-factors are due to higher order terms in the Luttinger Hamiltonian, therefore being an order of magnitude smaller than the g-factor along z direction. For an arbitrary low-symmetry growth direction, Winkler [32] calculated the HH g- factor using the Luttinger Hamiltonian (see Fig. 2.12). The HH g-factor is essentially an oscillating function of the angle θ between the arbitrary crystallographic direction and the [011] direction. As discussed above for high symmetry directions [001] and

[111], the g-factors are equal to zero. For [311] growth direction, g[233] ≥ 0.6 and g[011] = −0.2, and therefore the ratio of the g-factors g[233]/g[011] ∼ 4. As can be 36 2. Background Chapter

Figure 2.12: (a) Coordinate system for quantum wells grown in the [mmn] direction. m and n are integers, θ is the angle between [001] and the [mmn] direction; (b) Effective g-factor g∗ of the 20 nm GaAs/AlGaAs quantum well vs the angle θ. The figure is taken from Ref. [32]. seen from Fig. 2.12 and Table 2.1, the in-plane g-factor for high symmetry growth directions is equal to zero. In order to calculate this g-factor, higher order (in B) terms in the Zeeman Hamiltonian have to be considered. Essentially, for the high symmetry growth direction, the in-plane Zeeman splitting becomes cubic in B. Experimentally, the anisotropy of the in-plane g-factor has been studied by Pa- padakis at al. [34] in a 20 nm thick (311)A grown GaAs/AlGaAs quantum well. The authors measured the field required to polarize the 2D system in the two orthogonal in-plane crystallographic directions. The ratio of these fields is equal to the ratio of the effective g-factors. The sample was patterned into an L-shaped Hall bar allowing Pa- padakis et al. to perform measurements along the [233] and [011] direction in the one device. The measured magnetoresistance ρ(B) in an in-plane magnetic field B[233] and

B[011] exhibits a distinct feature - a small region with a reduced slope (see Fig. 2.13). The center of the region was marked by B∗ and was associated with the depopulation of the minor spin-split subband. This assumption was confirmed by self-consistent calculations. Additional confirmation came from magnetoresistance measurements for different hole densities (see Fig. 2.13). The field B∗ decreases with decreasing hole density which is again consistent with the subband depopulation. It is apparent from the data that B∗ for each direction of the field B is independent of the current (I) direction. Moreover, B∗ along [233] is consistently smaller than B∗ along [011] for all measured hole densities. This is explained by a highly anisotropic in-plane g-factor for the (311)A grown GaAs/AlGaAs quantum well. The anisotropy of the g-factor depends only on the orientation of magnetic field relative to crystallographic axes and is independent of the current direction. 2. Background Chapter 37

Figure 2.13: Normalized magnetoresistance ρ(B)/ρ(B = 0) vs an in-plane magnetic ﬁeld B. Left column correspond to B [233] and right column to B [011]. Each raw corresponds to a certain 2D hole density. Field B∗ is marked by an arrow in each case. The ﬁgure is taken from Ref. [34].

Note however, that the traces in Fig. 2.13 cannot be explained by a single value of the g-factor because of its dependence on the value of the wave vector k. Therefore the measured g-factor is an average over k up to the Fermi level. Moreover, these measurements provide only the ratio of the g-factors but not their absolute values. Furthermore, it is not possible to verify if the Zeeman splitting is linear in B with this technique.

Finally, Winkler et al. [35] pointed out that the Zeeman splitting depends significantly on the asymmetry of the confinement potential. This is explained by the fact that the values of B∗ are inversely proportional to the HH-LH splitting, and are much smaller for wide quantum wells (with small HH-LH splitting) than for narrow quantum wells (with large HH-LH splitting). Similarly B∗ increases with increasing electric field E due to confinement asymmetry. 38 2. Background Chapter

2.6.3 Spin-orbit coupling eﬀects

The Zeeman splitting (discussed above) is not the only mechanism that lifts the spin degeneracy of the bands. Because SO effects are an intrinsic property of hole systems, we have to understand how strong these effects are compared to the Zeeman splitting. In the bulk, the Dresselhaus term for electrons in the conductance band and holes in the LH band is proportional to k3. In contrast, holes in the HH band have a linear and cubic Rashba term, which have a similar magnitude and enter the Hamiltonian with opposite signs. This causes, for example, the spin splitting to change its sign at small k along [110] direction [31]. The 2D confinement results in a Dresselhaus term that is linear in k for holes in the LH and HH bands. Moreover, the HH-LH splitting reduces prefactors for the Dresselhaus term in the 2D compared to the bulk. The Rashba term is linear in E and k for electrons in the conductance band and holes in the LH band (see Eq. 2.13). For holes in the HH band the situation is more complicated: the Rashba term has a leading term proportional to k3 [36]. Experimentally, SO coupling effects have been studied in quantum wells and heterojunctions grown on (311)A oriented GaAs by Shayegan’s group [37]. The results for a 15 nm square quantum well showed that the Rashba contribution had a nonmonotonous density dependence, compared to the case of a triangular well, where the Rashba contribution decreased monotonously with decreasing hole density. Lu et al. confirmed their results by self-consistent calculations, concluding that the main contribution to the spin splitting is due to the Rashba effect rather than the Dresselhaus effect. Later on Winkler et al. [36] showed both theoretically and experimentally, that in accumulation-layer heterojunctions the Rashba spin splitting increases with decreasing hole concentration and electric field E. This is due to the fact that the Rashba coefficient βR of the HH states is inversely proportional to the 2D subband spacings. By increasing the electric field E, the subband spacings in a triangular quantum well also increase, thereby reducing the Rashba coefficient. This result has been verified for (100) grown GaAs/AlGaAs heterojunctions by Habib et al. [38]. These results indicate that the Dresselhaus effect is small compared to the Rashba effect. Additionally, the Rashba effect can result in noticeable spin splitting in triangular wells in contrast to the square wells, where the Rashba effect is small.

2.6.4 The band diagram of the GaAs valence band in 1D

The 1D confinement modifies the 2D subbands further, resulting in a staircase of 1D subbands. The treatment of the 1D bands is much more complicated than in electron systems due to the nonparabolicity of the bands, strong spin-orbit coupling 2. Background Chapter 39 and the spin splitting between heavy and light holes. The calculations of the 1D band diagram were performed by Zülicke et al. [33] for a hard-wall 1D confinement potential characterized by Wx and Wy, the widths of the confinement in x and y directions. x is the growth direction (perpendicular to the 2D plane) and y is the confinement direction perpendicular to the wire. The calculations were based on the Luttinger Hamiltonian, which included spherical, axial and cubic parts. The qualitative evolution of the 2D subbands when the wire is narrowed can be obtained by considering the spherical part of the Hamiltonian. The spherical Hamil- tonian can be separated into the following parts based on the symmetry:

wire sb hl 1D mix Hspherical = H + H + H + H , (2.21) where Hsb corresponds to the 1D subband energies, Hhl is the HH-LH coupling due to the asymmetric confinement in x and y directions, H1D is the quadratic energy dispersion for holes in the wire, and Hmix is the intersubband mixing between HH and LH. In the limit of only one 1D subband, the term Hmix disappears, and additionally in hl wire a symmetric 1D confinement the H term vanishes. Hspherical can be then diagonalized providing that the quantization axis (the axis of Jˆ) is set along the 1D wire. Then, the Hamiltonian is diagonalized in this basis and the HH and LH subbands are not mixed and separated by HH-LH splitting. However, when the asymmetry of the 1D confinement is present, the Hhl term introduces coupling between the HH and LH subbands, and the subbands become a mixture of HH and LH states. This type of coupling has no analogue in the 3D case and in the 2D case the coupling arises only by introduction of a cubic correction. Note, that a cubic correction to the 1D Hamiltonian also contains confinement induced couplings between the HH and LH subbands, dominating over the spherical part. Figure 2.14 shows the calculated evolution of the 2D HH1 and LH1 subbands into the first five 1D subbands as the aspect ratio Wx/Wy changes from 0 (pure 2D case) to 1 (pure 1D case). The figure was produced using the spherical part of the Luttinger Hamiltonian described above. As the wire starts to form, the lowest 2D state (HH band) splits onto quasi 1D subbands, which rise quickly in energy and approach quasi 1D subbands split from the higher energy 2D state (LH band). As can be seen from the Fig. 2.14 the quasi one dimensional subbands that originate from HH and LH 2D bands exhibit numerous crossovers at different values of Wx/Wy. Note however, that Fig. 2.14 does not reflect the HH-LH nature of the subbands. In the 2D limit (Wx/Wy = 0) using a spherical approximation the states of the system can be separated onto HH and LH along the growth direction x. As the Wx/Wy increases the spherical part of the Hamiltonian does not describe the system properly and cubic 40 2. Background Chapter

Figure 2.14: Evolution of the 2D bands vs the ratio of the confinement in the xy plane (Wx/Wy). The diagrams calculated using the spherical part of the Luttinger Hamiltonian. The figure is adapted from Ref. [33]. term should be included. As a result, the Hamiltonian can not be diagonalized, which means that off-diagonal elements cause HH and LH states to mix. Finally, in the limit of Wx/Wy = 1 the LH and HH states can be separated again but only along the wire direction z. In this direction the LH 1D subband is lower than the HH 1D subband. Therefore in the 1D case the ground state should be LH-type. Similarly to the 2D case, the “heavy” and “light” holes change the effective mass such as in the confinement direction y the LH effective mass appear heavier than the HH mass. Whereas, along the wire direction z the LH effective mass is still lighter than the HH effective mass. Bibliography

[1] K. J. Thomas, J. T. Nicholls, M. Y. Simmons, M. Pepper, D. R. Mace and D. A. Ritchie, Possible spin polarization in a one-dimensional electron gas, Phys. Rev. Lett., 77(1), 135 (1996).

[2] S. M. Cronenwett, H. J. Lynch, D. Goldhaber-Gordon, L. P. Kouwenhoven, C. M. Marcus, K. Hirose, N. S. Wingreen and V. Umansky, Low-temperature fate of the 0.7 structure in a point contact: a Kondo-like correlated state in an open system, Phys. Rev. Lett, 88(22), 226805 (2002).

[3] L. Pfeiﬀer, K. W. West, H. L. Stormer and K. W. Baldwin, Electron mobilities exceeding 107 cm2/Vs in modulation doped GaAs, Appl. Phys. Lett., 55(18), 1888 (1989).

[4] D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A. Ritchie and G. A. C. Jones, One- dimensional transport and the quantization of the ballistic resistance, J. Phys. C: Solid State Physics, 21, L209 (1988).

[5] B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, D. van der Marel and C. T. Foxon, Quantized conductance of point contacts in a two-dimensional electron gas, Phys. Rev. Lett., 60(9), 848 (1988).

[6] Y. Imry, Introduction to mesoscopic physics (Oxford Univ. Press, Oxford, UK, 1997).

[7] L. I. Glazman and A. V. Khaetskii, Non-linear quantum conductance of a lateral microconstraint in a heterostructure, Europhys. Lett., 9(3), 263 (1989).

[8] N. K. Patel, J. T. Nicholls, L. Martin-Moreno, M. Pepper, J. E. F. Frost, D. A. Ritchie and G. A. C. Jones, Evolution of half plateaus as a function of electric ﬁeld in a ballistic quasi-one-dimensional constriction, Phys. Rev. B, 44(24), 13549 (1991).

41 42 BIBLIOGRAPHY

[9] J. Voit, One-dimensional Fermi liquids, Rep. Prog. Phys., 57, 977 (1994).

[10] J. Kondo, Resistance minimum in dilute magnetic alloys, Prog. Theor. Phys., 32(1), 37 (1964).

[11] P. W. Anderson, Ground state of a magnetic impurity in a metal, Phys. Rev., 164(2), 352 (1967).

[12] D. Goldhaber-Gordon, J. G¨ores,M. A. Kastner, H. Shtrikman, D. Mahalu and U. Meirav, From the Kondo regime to the mixed-valence regime in a single-electron transistor, Phys. Rev. Lett., 81(23), 5225 (1998).

[13] L. Kouwenhoven, C. Marcus, P. McEuen, S. Tarucha, R. Westervelt and N. Wingreen, in L. Sohn, L. Kouwenhoven and G. Schon (eds.), Proceedings of the NATO Advanced Study Institute on Mesoscopic Electron Transport, 105 (Kluwer Series E345, 1997).

[14] S. M. Cronenwett, T. H. Oosterkamp and L. P. Kouwenhoven, A tunable Kondo eﬀect in quantum dots, Science, 281, 540 (1998).

[15] W. G. van der Wiel, S. D. Franceschi, T. Fujisawa, J. Elzerman, S. Tarucha and L. P. Kouwenhoven, The Kondo eﬀect in the unitary limit, Science, 289, 2105 (2000).

[16] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav and M. A. Kastner, Kondo eﬀect in a single-electron transistor, Nature, 391(1), 156 (1998).

[17] L. P. Rokhinson, L. N. Pfeiﬀer and K. W. West, Spontaneous spin polarization in quantum point contacts, Phys. Rev. Lett., 96, 156602 (2006).

[18] I. Zuti´c,J.ˇ Fabian and S. D. Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys, 76(2), 323 (2004).

[19] G. Dresselhaus, Spin-orbit coupling eﬀects in zinc-blende structures, Phys. Rev., 100(2), 580 (1955).

[20] Y. A. Bychkov and E. I. Rashba, Oscillatory eﬀects and the magnetic susceptibility of carriers in inversion layers, J. Phys. C, 17, 6039 (1984).

[21] D. S´anchez, L. Serra and M.-S. Choi, Strongly modulated transmission of a spin-split quantum wire with local Rashba interaction (2007), cond- mat/arXiv:0706.0107v1. BIBLIOGRAPHY 43

[22] Y. V. Pershin, J. A. Nesteroff and V. Privman, Effect of spin-orbit interaction and in-plane magnetic field on the conductance of a quasi-one-dimensional system, Phys. Rev. B, 69, 121306(R) (2004).

[23] M. Governale and U. Z¨ulicke, Rashba spin splitting in quantum wires, Solid State Commun., 131(2004), 581 (2004).

[24] I. Vurgaftman, J. R. Meyer and L. R. Ram-Mohan, Band parameters for III-V compound semiconductors and their alloys, J. Appl. Phys., 89(11), 5815 (2001).

[25] E. O. Kane, Handbook on Semiconductors, chap. Energy band theory, 193 (North- Holland, Amsterdam, 1982).

[26] J. H. Davies, The physics of low-dimensional semiconductors: an introduction (Cambridge University Press, Cambridge, UK, 1997), 1st ed.

[27] U. R¨ossler, Nonparabolicity and warping in the conduction band of GaAs, Solid State Commun., 49, 943 (1984).

[28] H. Mayer and U. R¨ossler, Spin splitting and anisotropy of cyclotron resonance in the conduction band of GaAs, Phys. Rev. B, 44(16), 9048 (1991).

[29] O. Zitouni, K. Boujdaria and H. Bouchriha, Band parameters for GaAs and Si in the 24 − k · p model, Semicond. Sci. Technol., 20, 908 (2005).

[30] G. L. Bir and G. E. Pikus, Symmetry and strain-induced eﬀects in semiconductors (Wiley, New York, 1974).

[31] R. Winkler, Spin-orbit coupling eﬀects in two-dimensional electron and hole systems (Springer-Verlag, Berlin, Germany, 2003), 1st ed.

[32] R. Winkler, S. J. Papadakis, E. P. de Poortere and M. Shayegan, Highly anisotropic g-factor of two-dimensional hole systems, Phys. Rev. Lett., 85(21), 4574 (2000).

[33] U. Z¨ulicke, Electronic and spin properties of hole point contacts, Phys. Stat. Sol C, 3(12), 4354 (2006).

[34] S. J. Papadakis, E. P. de Poortere, M. Shayegan and R. Winkler, Anisotropic magnetoresistance of two-dimensional holes in GaAs, Phys. Rev. Lett., 84(24), 5592 (2000).

[35] R. Winkler, E. Tutuc, S. J. Papadakis, S. Melinte, M. Shayegan, D. Wasserman and S. A. Lyon, Anomalous spin polarization of GaAs two-dimensional hole systems, Phys. Rev. B, 72(19), 195321 (2005). 44 BIBLIOGRAPHY

[36] R. Winkler, H. Noh, E. Tutuc and M. Shayegan, Anomalous Rashba spin splitting in two-dimensional hole systems, Phys. Rev. B, 65, 155303 (2002).

[37] J. P. Lu, J. B. Yau, S. P. Shukla, M. Shayegan, L. Wissinger, U. R¨ossler and R. Winkler, Tunable spin-splitting and spin-resolved ballistic transport in GaAs/AlGaAs two-dimensional holes, Phys. Rev. Lett., 81(6), 1282 (1998).

[38] B. Habib, E. Tutuc, S. Melinte, M. Shayegan, D. Wasserman, S. A. Lyon and R. Winkler, Negative diﬀerential Rashba eﬀect in two-dimensional hole systems, Appl. Phys. Lett., 85(15), 3151 (2004). Chapter 3

Device Fabrication and Measurement Setup

Chapter Outline In this chapter we present the devices we used for the studies in this thesis as well as the heterostructure details. We will also outline the fabrication routine with special attention to the fabrication of a new type of 1D wires in undoped heterostructures and modiﬁcations of previously developed fabrication processes. We then describe the electrical measurement setup and modiﬁcations I made to reduce electrical noise in the setup.

3.1 Introduction

The objects of this thesis are GaAs/AlGaAs p-type 1D wires. In our studies we use two types of wires, each of which are fabricated on substantially different heterostructures. These heterostructures have differences in the method of producing the 2D hole system and the 2D confinement potential. However, the common feature of the two heterostructures is the (311)A growth direction. As was mentioned in the Chapter 1, the major advantage of (311) heterostructures for studies of hole ballistic transport, is their much higher mobility compared to (100) wafers. This is due to the fact that Si can be used as a p-type dopant in (311) but not in (100) heterostructures. For (100) heterostructures Be is used as a p-type dopant, however it is more diffusive than Si and therefore its diffusion to the 2D hole layer limits the hole mobility [1]. Recent development of MBE technology allowed carbon to be used as a p-type dopant in (100) grown heterostuctures, producing very high hole mobilities [2, 3]. However, for studies in this thesis the use of (311) wafers is imperative beyond the reasons of mobility given above. As was outlined in Chapter 1, in this thesis we pri- marily concentrate on the behaviour of 1D hole systems in an in-plane magnetic field.

45 46 3. Device Fabrication and Measurement Setup

In the 2D hole systems the in-plane Zeeman splitting is significantly suppressed [4]. Moreover for high symmetry growth directions ((100) and (111)), the linear Zeeman term is equal to zero and only higher order effects contribute to a finite splitting at very large magnetic fields [5]. On the other hand, for low symmetry growth directions such as (311), the suppression of the Zeeman splitting is lifted due to coupling of the hole spin to the highly anisotropic crystal lattice, which allows the Zeeman splitting to be probed at moderate magnetic fields. A common argument against the use of (311) heterostructures is large crystallographic anisotropy of the hole mobility [6]. Previous experimental studies of transport in 2D hole systems [7, 8] found the mobility along the [011] direction to be approximately 3 times lower than the mobility along [233] at high hole densities. This anisotropy of the hole mobility is due to interface roughness of the (311) surfaces [9, 10], which causes additional scattering along [011]. A large mobility anisotropy could prevent the observation of the ballistic transport in 1D wires along the low mobility [011] direction and therefore make the study of the Zeeman splitting impossible. Note however that the devices used Chapter 7 were oriented along [011] and [233] crystallographic axes and both showed ballistic 1D conductance quantized in units of 2e2/h. In the following sections we will firstly present the devices used for the studies in this thesis and then the required fabrication procedures for making these devices. Finally, we will discuss the measurement setup and modifications I made to reduce the electrical noise in the measurements.

3.2 Devices for studies in Chapter 4 and 5

For the studies in Chapter 4 and 5 we use the sample DQPC25, which was fabricated by Romain Danneau. The sample was fabricated on the (311)A oriented modulation doped heterostructure T483 grown by Michelle Simmons at the Cavendish Laboratory in Cambridge. The wafer layout is shown in Fig. 3.1a. The heterostructure is grown on a (311)A n+ GaAs substrate which is used as an in-situ back gate. The active region consists of 20 nm GaAs bottom QW, 30 nm AlGaAs barrier, 20 nm GaAs top QW. The carriers are provided by 200 nm Si-doped AlGaAs layers on each side of the QWs and are separated from the QWs by undoped AlGaAs layers. The hole density 11 −2 2 and mobility are ns =1.2 × 10 cm and μ =92m/Vs for the top layer. The 11 −2 2 numbers for the bottom layer are smaller: ns =1.0 × 10 cm and μ =87m /Vs. The optical micrograph of the sample is shown of Fig. 3.1b and the sample active region is schematically shown on Fig. 3.1c. The 1D wire consists of three metal gates: two side gates (SG) and the middle gate (MG), which has a width of 600 nm and is 3. Device Fabrication and Measurement Setup 47

Figure 3.1: (a) The layout of T483 heterostructure; (b) the optical micrograph of DQPC25; (c) the schematic of the active region. S and D mark the source and drain ohmic contacts, SG, MG, TG and DG stand for “side gates”, “middle gate”, “top gate” and “depletion gate” respectively. separated from the SG by 100 nm on each side. There are also depletion gates (DG) which run across the mesa and an overall back gate (BG). The functions of the gates are the following:

1. The SG deﬁnes the wires in both layers and subsequently pinch them oﬀ.

2. The MG is used to vary the hole density in the top wire.

3. The BG controls the hole density in the bottom wire.

4. The DG cuts the top layer forcing the source-drain current to ﬂow through the bottom wire only.

By tuning the voltages on all four gates it is possible control the hole density in each of the layers and observe conductance quantization in each of the wires or both wires simultaneously.

3.3 Devices for studies in Chapter 6 and 7

For studies in Chapter 6 and 7 we use the sample CQPC17 fabricated by Warrick Clarke and KQPC19 fabricated by myself. The samples are fabricated on the (311)A oriented heterostructure R165 grown by Koji Muraki at NTT Basic Research Labs, Japan. The schematic layout of the R165 wafer is shown in Fig. 3.2a. The heterostructure is grown on a (311) GaAs substrate and consists of a 1.5 μm GaAs layer, a 175nm AlGaAs layer and a 25nm GaAs layer, all of which are undoped. The ﬁnal layer in the structure is a 75 nm heavily doped (5 × 1018 cm−3 Si) p+-GaAs layer, which is used 48 3. Device Fabrication and Measurement Setup

Figure 3.2: (a) The layout of R165 heterostructure layout; (b) the optical micrograph of CQPC17. The schematic of the active regions of (c) CQPC17 and (d) KQPC19. S and D mark the source and drain ohmic contacts, SG and TG stand for “side gate” and “top gate” respectively; V1 and V2 mark the voltage probe ohmic contacts. as an in-situ top gate. The holes are induced electrostatically at GaAs/AlGaAs heterointerface by negatively biasing the top gate. The degenerate p+ doping in the cap layer ensures that the Fermi level in the heterostructure is pinned close to the valence band and therefore only small negative threshold voltages (≈−0.1 V) are required to induce the holes. The optical micrograph of CQPC17 is shown in Fig. 3.2b. The 1D wire is oriented along the high mobility [233] crystallographic axis and fabricated by patterning the conducting cap layer onto three separate gates: two side-gates (SG) and a top gate (TG) (see Fig. 3.2c). The functions of the gates are the following:

1. The top gate is used to induce and subsequently vary the hole density. Typically to observe clean conductance quantization with a large number of plateaus, high hole densities (≥ 1 × 1011 cm−2) were required.

2. The side gates are used to squeeze the 1D wire and eventually pinch it oﬀ. Note 3. Device Fabrication and Measurement Setup 49

that because the wire is etched there is not 2D to 1D transition (i.e. the deﬁnition of 1D wire is absent) and the wire is deﬁned at zero side gate voltage.

The sample KQPC19 consists of two orthogonal 1D wires oriented along the [233] and [001] crystallographic axes (see Fig. 3.2d for a schematic). The wires were designed to be of the same dimensions - 400 nm both in width and length; hence it was possible to perform all measurements for the same value of the top gate voltage in both wires (the hole density in both wires was the same).

2.0 8 7 6

5 1.5 4 ) -2 3 cm / Vs) 2

11 1.0

µ (cm 2 p (×10

0.5

5 10 9 0.0 8 2 3 4 5 6 -0.5 -0.4 -0.3 -0.2 -0.1 10 11 10 -2 10 VTG (V) p (cm ) (a) (b)

Figure 3.3: (a) The hole density p vs the top gate voltage VTG for the 2D device fabricated on R165 wafer. The triangles correspond to the experimental data points and the straight line is the least squares ﬁt; (b) A log-log plot of hole mobility μ vs density p for the same device. The ﬁgures are adapted from Ref. [11].

The hole density and mobility for the R165 wafer have been extracted from the Hall and magnetoresistance measurements for diﬀerent values of the top gate voltage for a 2D device oriented along the high mobility [233] direction [12]. Figure 3.3a shows a linear dependence of the hole density vs the top gate voltage. The maximum top gate voltage and therefore the maximum hole density is limited by the current leakage between the top gate and the ohmic contacts. The minimum top gate voltage is limited by the 2D threshold voltage. The hole density can be typically varied over an order of magnitude 1.7×1010

3.4 Fabrication routine

The fabrication process for our 1D devices consists of two basic stages:

1. The fabrication of the host 2D device using the standard ultra-violet UV photolithography techniques.

2. The fabrication of the 1D device using the electron beam lithography with subsequent deposition of the metal surface gates (devices from Chapters 4 and 5) or wet etching of the doped cap layer (devices from Chapters 6 and 7).

3.4.1 2D device fabrication (device DQPC25)

This stage of the fabrication process consists of the following steps:

1. Cleaving the wafer and initial cleaning. Typically, for a single device a rectangular chip 3.5mm×4 mm is cleaved from the wafer T483. The size of the chip is determined by the photolithography mask pattern. Special care was taken to keep track of the orientation of sides of the chip relative to the crystallographic axes. For the (311) wafers it is relatively easy to determine the orientation as the edge of the chip cleaved along ([233]) is perpendicular to the surface, whereas the edge along ([011]) is at the angle to the surface. We always cleave the chips such that the longer side runs along [233], this direction will become the long direction of the Hall bar. After cleaving, the chip is cleaned in acetone in the ultrasonic bath for 15 min to remove any contamination from the surface. When the chip is removed from acetone, it is rinsed immediately with isopropanol and dried under N2 gas. The rinsing in isopropanol after the acetone is important to prevent the appearance of dry stains on the wafer surface (see Fig. 3.4a).

2. Photolithography process for a mesa. The chip is placed on the spinner and pre-cleaned by spraying with acetone followed by isopropanol and then the chip is thoroughly dried under N2 gas. The positive resist AZ6112 is then spun at a rate of 5000 rpm for 60 s. This results in a thickness of the resist around 1.1 μm (see Fig. 3.4b). The chip is then baked at 110◦ C for 2 minutes. 3. Device Fabrication and Measurement Setup 51

After that, the chip is placed into the Quintel 600 mask aligner and aligned to the mesa pattern on the photolithography mask. The chip is then exposed to UV light (intensity 20 mW/cm2) for 2.3 seconds (see Fig. 3.4c). This is followed by development of the photoresist in MIF300 developer for 35 s, a short rinse in DI water and ﬁnally the chip is dried under N2 gas. The development removes all of the exposed resist and there is only a layer of the photoresist in the shape of a mesa left on the surface of the chip (see Fig. 3.4d).

After the development, the chip is dipped into the HF buﬀered etch (HF/NH3F: H2O:H2O2, 6 : 60 : 3) and the unprotected area is etched to a depth below the bottom QW (≈ 300 nm below the surface) (see Fig. 3.4e). When the etching time elapses the chip is rinsed immediately in DI water for 2 min and then thoroughly dried under N2 gas. The left photoresist is then removed by soaking the chip in the acetone followed by a rinse in isopropanol and drying under N2 gas.

3. Photolithography process for ohmic contacts. The same photolithography process used for a mesa is repeated for the ohmic contacts.

X initial pre-clean; X spinning of the resist (see Fig. 3.4f); X baking of the resist; X aligning and exposing (see Fig. 3.4g) the ohmics pattern; X development of the exposed resist (see Fig. 3.4h).

After development, the sample is placed into the Edwards evaporator and the chamber is pumped out to P ≈ 5 × 10−7 mbar. The ohmic contacts alloy is then thermally evaporated at a rate of 1.2 nm/s until the required thickness of 160 nm is achieved (see Fig. 3.4i). For contacting p-type devices we use AuBe alloy (99 % of Au and 1 % of Be by weight). After evaporation the sample is soaked in acetone for 10 min. Acetone dissolves the resist underneath the metal and therefore removes metal from the areas covered with the resist. The metal remains only on developed areas, were it is in direct contact with the surface (see Fig. 3.4j). The chip is then rinsed in isopropanol and dried under N2 gas. The chip is then placed into the rapid temperature annealer (RTA) and annealed ◦ in the ﬂow of forming gas (mixture of N2 and H2 gases) at 490 C for 2 min. The 52 3. Device Fabrication and Measurement Setup

(a) Cleave and clean (b) Spin photoresist (c) Align and expose wafer mesa pattern

(d) Develop mesa (e) Etch mesa (f) Spin new photore- pattern sist

(g) Align and expose (h) Develop ohmic (i) Evaporate AuBe ohmics pattern pattern

(j) Liftoﬀ AuBe (k) Anneal ohmics (l) Chlorobenzene soak

(m) Develop gates (n) Evaporate Ti/Au (o) Liftoﬀ Ti/Au pattern

Figure 3.4: Process ﬂow for the fabrication of bilayer p-type devices. The ﬁgure is adapted from Ref. [11]. 3. Device Fabrication and Measurement Setup 53

alloy diﬀuses trough the quantum wells and makes the ohmic contact to both 2D hole systems (see Fig. 3.4k).

4. Photolithography process for gates and contact pads. Again, the photolithography process for the gate and contact pads pattern was repeated.

X initial pre-clean; X spinning of the resist; X baking of the resist; X aligning and exposing the gate and contact pads pattern.

Before the development the chip is placed in chlorobenzene for 3 min to harden the resist surface and then dried under N2 gas (see Fig. 3.4l). This is done to make the photoresist profile undercut after the development. After the hardening the photoresist is developed (see Fig. 3.4m) and the chip is placed into the Edwards evaporator. The chamber is pumped down and the gate metal (20 nm of Ti and 80 nm of Au) are evaporated sequentially at rates of 0.3 and 1 nm respectively (see Fig. 3.4n). Then, the chip is soaked in acetone to lift-off the metal from the areas protected by photoresist (see Fig. 3.4o). The undercut profile of the resist helps the closely spaced gate pattern to lift-off easily.

5. The EBL stage for the device DQPC25. The fabrication of the 1D wire for DQPC25 is discussed in a separate section below.

6. Packaging and bonding. After all photolithography processes are complete the sample is mounted with conducting Ag paste on the surface of LCC20 chip carrier and baked at 120◦ C for 10 min for the paste to set hard. After that, the chip is bonded using the K&S gold ball bonder or Al wedge bonder.

3.4.2 2D device fabrication (device CQPC17)

The same routine was used for the devices fabricated on the R165 wafer with two exceptions:

X The EBL stage for induced devices is done after the initial cleaving and cleaning of the wafer for the reasons, discussed later in the chapter. 54 3. Device Fabrication and Measurement Setup

X The ohmics stage for induced devices is diﬀerent to that for doped devices. Because the top layer of R165 is a heavily doped layer which is used as in-situ top gate, contacting the 2D hole system becomes a formidable challenge. We use the modiﬁed Kane’s technique [13] (so called self-aligned ohmics), which was transferred to ohmics processing on R165 wafer by Warrick Clarke [12]. The following fabrication routine has been used to fabricate the sample CQPC17 for studies in Chapter 6.

The photolithography for the ohmics is the same as previously:

X initial pre-clean;

X spinning of the resist;

X baking of the resist;

X aligning and exposing the ohmic contacts pattern;

X developing the resist.

However, after development the ohmics pattern is etched using a H2SO4 etch (H2SO4 :H2O:H2O2, 1 : 80 : 8) to about 15 nm above the heterointerface, rinsed in DI water for 2 min and dried under N2 gas. The etch was selected based on its etch proﬁle on (311) GaAs surfaces. For the self-aligned ohmics scheme, the overhang of the top conducting layer over the ohmic contacts is essential for producing working ohmics. The sample is then placed into the Edwards evaporator, such that the normal sample surface makes a small angle (15 − 20◦) to the direct line of sight to the evaporation source. This is done in order to evaporate the ohmics metal under the overhanging top gate layer. If the angle is to large, all the ohmics short to the top layer but if the angle is small, there is a large lateral separation between the gate and the ohmics and therefore the ohmics will not make electrical contact to the induced 2D hole layer. The thickness of evaporated AuBe metal is 60 nm, which is much smaller than for DQPC25. This is done to increase the vertical separation between the ohmic and the gate and therefore to reduce the leakage current between the top gate and the ohmic contacts. After the evaporation, the sample is removed from the evaporator and soaked in acetone to lift-oﬀ the metal from the areas protected by the photoresist. Special care is taken to ensure that no small pieces of the ohmic metal due to roughness of the ohmics edges run over the top of the mesa, as they will short the ohmics to the top gate. Then the chip is annealed in the RTA at 490◦ C for 100 s. The subsequent photolithography for gates and contact pads, followed by the bonding and packaging stages are the same as previously described. 3. Device Fabrication and Measurement Setup 55

3.4.3 Modiﬁcation of the fabrication procedure for KQPC19 device

The optimized ohmic contacts fabrication procedure, discussed below in this section, has been used to fabricate the sample KQPC19 for studies in Chapter 7. The processing of self-aligned ohmics is a difficult task. During past 4 years since our group started working with the R165 heterostructure, we have had a number of difficulties with the fabrication of the ohmics. The main difficulty was the unreliable annealing of the ohmic contacts. We initially used a home-made rapid thermal annealer (RTA) (shown on Fig. 3.5a), which consisted of a resistive heating system where a metal foil strip was heated by passing a current through it. The sample was placed on the foil strip, and its temperature was read-out using a K-type thermocouple which had to be mounted directly on the foil strip, but clearly could not be in an electrical contact with the boat. The size of the foil strip was small and therefore clamping the thermocouple to the boat was not possible due to an excessive thermal load, which reduced significantly the warm-up time which was not tolerable. As a solution, the thermocouple was glued to the metal foil strip with a special high temperature cement. The repetitive thermal cycling degrades cement’s properties and eventually the thermocouple stops reading the right values of the foil strip temperature, and the foil strip is heated uncontrollably. Then, the foil strip and the thermocouple are usually replaced and RTA recalibrated. Another problem with the fabrication of the self-aligned ohmics is the angle of the ohmic metal evaporation. I found that the number of the working ohmics is very sensitive to the evaporation angle, with the yield falling off dramatically within 2◦ on either side of the optimum value. The original Edwards Rotatilt system shown on Fig. 3.5b, did not provide a suitable scale to set the angle precisely and reproducibly. Above problems has reduced significantly the yield of the working devices. More- over, as described in Chapter 6, at low-temperatures the ohmics become nonlinear and very resistive. This significantly limited our studies of the sample CQPC17. To overcome these problems two major modifications have been done and implemented by myself. Firstly, the commercial RTA Ulvac-Riko MILA 3000 was purchased, installed and calibrated to replace the home-made system used previously. The RTA uses an near infra-red quartz lamp to heat a carbon susceptor on which the sample is placed (see Fig. 3.5c). The in-build Omron digital controller is used in this annealer to set the required annealing temperature profile and minimize temperature fluctuations. Secondly, a new tilting system for a sample holder in the Edwards evaporator was designed and implemented allowing us to control the evaporation angle within 0.5◦ (see Fig. 3.5d). Subsequently, most of the parameters of the ohmic contacts processing stage have 56 3. Device Fabrication and Measurement Setup

Figure 3.5: (a) A photograph of the home-made annealer. Note that the thermocouple is mounted at the bottom of the metal strip; (b) A photograph of the Ulvac-Riko MILA 3000 RTA; (c) A photograph of the original Edwards Rotatilt system; (d) A photograph of the home-made tilting system installed on the Edwards Rotatilt system. been reassessed:

X The optimum angle of evaporation was found to be 17◦.

X The ohmic metal thickness 160 nm.

X The annealing temperature 470◦ C.

X The annealing time 30 s.

This optimization has led to the following results:

X The number of working ohmic contacts per sample increased substantially. Pre- viously from the total of 8 ohmics only 4−5 worked, after the optimization 7−8 ohmics worked.

X The contact resistance of the ohmics decreased by more than an order of mag-

nitude from RC ≈ 50 kΩ to RC = 3 kΩ at T =4.2K. 3. Device Fabrication and Measurement Setup 57

X The ohmic contacts became more uniform, that is all working ohmic contacts on a sample have a similar contact resistance.

X Previously, to “connect” the ohmic contacts to the 2D hole systems the sample had to be brieﬂy illuminated by a red LED at low temperatures. Otherwise no current could be passed through the ohmics. After the optimization, ohmics worked both with and without the illumination which indicates that the ohmics are well connected to the 2D hole system.

There is still a room for improvement of the ohmic contacts. Usually we test the electrical characteristics of the devices at T =4.2 K, and if the ohmics worked well at 4.2 K it was a good indication that they would work equally well at millikelvin temperatures. However, with the devices used in Chapter 7, we noticed that although the optimized ohmics may show excellent characteristics at T =4.2 K, they often degrade noticeably as the temperature lowers to Tbase = 20 mK, and subsequently when a magnetic ﬁeld is applied.

3.4.4 1D device fabrication

The 1D devices in this chapter were made using the electron beam lithography (EBL). The process is very similar to photolithography, but instead of UV light an electron beam is used to expose the PMMA resist, resulting in two orders of magnitude higher resolution. For our 1D samples, we use two diﬀerent methods of conﬁning a 2D system to form a 1D wire:

X For the sample DQPC25 we use metal surface gates (two side gates and the middle gate) to deﬁne the 1D wire.

X For the sample CQPC17 we use wet etching to deﬁne a narrow 1D constriction by separating the conducting cap onto three separate gates: a top gate and two side gates.

The EBL processing routine is structured as follows:

1. The chip is placed on the spinner and pre-cleaned by spraying with acetone followed by isopropanol and is then thoroughly dried under N2 gas.

2. The PMMA EBL resist A3 is spun at a rate of 5000 rpm for 60 s. The resulting thickness of the resist is around 100 nm.

3. The chip is then baked at 180◦ C for 3 minutes on a hot plate. 58 3. Device Fabrication and Measurement Setup

4. The chip is placed into the FEI Sirion EBL system and the pattern is written. The pattern has the smallest feature size of around 80 nm. The electron beam changes the chemical bonding between the polymer chains in the PMMA resist, which makes the exposed resist more soluble in the MIBK developer (MIBK : IPA, 1 : 3).

5. The development is done in the MIBK developer for 30 s followed by a rinse in isopropanol and drying under N2 gas.

After development, sample DQPC25 was placed into the Edwards evaporator. Thin films of Ti (10 nm) and Au (30 nm) are deposited sequentially. The sample is then soaked in acetone for several hours to lift-off the metal from the areas protected by PMMA resist. The lift-off of EBL patterns is much more problematic than in the photolithography process due to the small size of the features and lower solubility of the EBL resist in acetone. The central area of the 1D wire is shown on a SEM micrograph (see Fig. 3.6a). The middle gate of the wire has width and length of approximately 600 nm and is separated from the side gates by 100 nm on each side. After development, sample CQPC17 was dipped into the H2SO4 etch (the same as for ohmics photolithography stage) to etch through the cap layer, followed by a rinse in DI water and a drying under N2 gas. The central area of the wire is shown on a SEM micrograph (see Fig. 3.6b). The 1D constriction has a width and length of around 400 nm and is separated from the side gates by 250 nm on each side. This large separation occurs because the etch removes GaAs both vertically and laterally. To reduce the width of the etched features, a different heterostructrure with a thinner conducting top layer would be required. The two orthogonal 1D wires on KQPC19 were etched using a HCl etch (HCl : H2O :H2O2, 1 : 100 : 4). This is because the etch rate of the HCl etch is approximately 6 times slower than that of the H2SO4 etch, allowing more precise control in stopping the etch at the target depth. Figures 3.6c and 3.6d show the central regions of the two orthogonal 1D wires. Both wires have a central region that is approximately 400 nm by 400 nm. In contrast to the symmetric 1D region of the wire along the Hall bar (Fig. 3.6d), the 1D region of the wire across the Hall bar (Fig. 3.6d) is not symmetric: the side gate on the left is sharper at the end and has a larger separation from the top gate than the side gate on the right. We believe that the difference in the patterns is due to the anisotropy of the etch in the different crystallographic directions, because the parameters for the EBL writing (magnification, dose and electron beam scan direction) for both wires were identical. It is important to note that the EBL process is performed at different stages in the fabrication for devices DQPC25 and CQPC17/KQPC19. For DQPC25, the wire metal 3. Device Fabrication and Measurement Setup 59

(a) (b)

Figure 3.6: The SEM micrographs of the central regions of the 1D wires used in this thesis: (a) sample DQPC25, (b) sample CQPC17, (c) sample KQPC19 (1D wire along the Hall bar) and (d) sample KQPC19 (1D wire across the Hall bar). SG, MG and TG stand for “side gates”, “middle gate” and “top gate” respectively. gates are deposited after the photolithography, so that the pattern can be accurately aligned to the photolithographical metal gates. For CQPC17 and KQPC19 devices, the EBL stage is done immediately after the initial cleaving and cleaning of the wafer and before all photolithography processes. Large alignment markers along with the wire pattern are deﬁned by EBL and subsequently etched. These markers are then used to align the optical patterns to the etched wires. The EBL stage is done before the photolithography because after the ohmic contacts are annealed, the surface of the wafer becomes resistant to the etchant and the 1D wires could not longer be deﬁned. We think this is due to chemical reconstruction of the surface.

3.5 Measurement setup

All measurements presented in this thesis were performed in an Oxford Instruments Kelvinox 100 He3/He4 dilution refrigerator with the base temperature of 20 mK. The dilution unit is loaded into a liquid helium cryostat with a 10/12 T superconducting NbTi/Nb3Sn magnet at the bottom of the cryostat. To measure the transport properties of the 1D wires we use a standard low fre- 60 3. Device Fabrication and Measurement Setup quency a.c. lock-in technique using Stanford Research SR830 and EG&G Instruments 5210 lock-in amplifiers. For for the d.c. voltages required for biasing the gates we use the digital to analogue (DAC) outputs of the SR830 lock-in amplifiers or a Keithly 2400 source-measure unit. Two typical measurement configurations are two-terminal (2T) and four-terminal (4T). In the 2T configuration we usually work in a constant voltage mode. In this mode a voltage divider with a typical dividing ratio 1 : 10000 is connected in series with the a.c. voltage source of a lock-in amplifier. The small a.c. signal after the divider is then applied to the source ohmic contact of the device and the current from the drain ohmic contact is detected by the current pre-amplifier of the lock-in amplifier. In the 4T configuration we again use the constant voltage mode but in addition to the 2T configuration, we monitor the two voltage drop across the voltage probe ohmic contacts. The devices DQPC25 and CQPC17 were measured in the 2T configuration whereas KQPC19 was measured in the 4T configuration.

Figure 3.7: (a) The schematic diagram of the RCR cold ﬁlters; (b) The photograph of the cold ﬁlters installed on the 1k pot stage of the dilution refrigerator.

Significant effort was put into modifying the measurement setup in order to reduce electrical noise, both at low and high frequencies. To reduce high frequency noise we use commercial low-pass filters Mini-Circuits BLP-1.9 with a cut-off frequency of 1.9 MHz. These filters were connected in series with all a.c. wiring that goes into the cryostat. The d.c. gate lines were filtered using external home-made RCR filters with a 0.1 Hz cut-off frequency. To reduce the high frequency noise further, we have designed and installed a set of passive internal RCR cold filters for all electrical lines going into the fridge (see schematic in Fig. 3.7a). These filters are placed on the 1K pot stage of the dilution fridge and therefore operate at ≈ 1.7 K (see Fig. 3.7b). We have used the ceramic multi-layer surface mount capacitors with a nominal capacitance of 100 nF at room temperature and as a resistance we have used the resistance of the wiring ≈ 110 3. Device Fabrication and Measurement Setup 61

Ω on each side of the capacitors. A test of the capacitors at liquid helium temperature showed that the capacitance drops to 30 nF but the resistance of the wiring remains almost unchanged.

(a)

) 10 Ω 105 104 103 102 101 Measured resistance ( 100 100 101 102 103 104 105 106 107 Frequency (Hz) (b)

Figure 3.8: (a) The diagram of the 4T circuit used for PSpice modeling; (b) The frequency response of the 4T circuit in a constant voltage mode. Diﬀerent curves correspond to diﬀerent resistances of the device (R5): 10, 100, 103,104,105 and 106 Ω.

We have used the PSpice software to model the simplified 4T measurement circuit in constant current and constant voltage modes for a number of device resistances. The schematic of the circuit and the results of the modeling for constant voltage mode are shown on Fig. 3.8a and 3.8b. The source voltage is applied through the 1 : 10000 voltage divider (R1:R14) via the RCR filter (R2:C4:R3) to the device (R5). The current from the device, goes to ground via the RCR filter (R6:C3:R7). The voltage from the device is measured via the RCR filters (R8:C1:R9 and R10:C2:R11) by the lock-in amplifier with 10 MΩ voltage input resistance (R12) and 25 pA capacitance (C5). From Fig. 3.8b it is obvious that the cut-off frequency depends on the resistance 62 3. Device Fabrication and Measurement Setup of the sample, but even for 1 MΩ, the cut-off frequency is ≈ 10 kHz. However, the base temperature test of the filters showed that for device resistances exceeding 1 MΩ the lock-in amplifier frequency should not exceed 19 Hz, which is probably due to the simplicity of the circuit diagram used for the modeling. Finally, 50 Hz noise has been found to affect the quality of the data dramatically. Typically large 50 Hz noise is an indication of ground loops in the measurement circuit. Therefore, optimization of the grounding arrangement has to be done prior to the measurements. All electronics is usually connected to the same ground point and the GPIB interface is isolated from the computer via an opto-isolator bus. This ground connection is the only ground for the fridge insert. Additionally, the He3 and He4 pumps are also electrically disconnected from the electronics ground by using the plastic KF clamps and o’rings on all fridge pumping lines. Bibliography

[1] F. Fischer, D. Schuh, M. Bichler, G. Abstreiter, M. Grayson and K. Neumaier, Modulating the growth conditions: Si as an acceptor in (110) GaAs for high mobility p-type heterostructures, Appl. Phys. Lett., 86, 192106 (2005).

[3] C. Gerl, S. Schmult, H.-P. Tranitz, C. Mitzkus and W. Wegscheider, Carbon-doped symmetric GaAs/AlGaAs quantum wells with hole mobilities beyond 106 cm2/Vs, Appl. Phys. Lett., 86, 252105 (2005).

[4] R. Winkler, Spin-orbit coupling eﬀects in two-dimensional electron and hole systems (Springer-Verlag, Berlin, Germany, 2003), 1st ed.

[5] R. Winkler, E. Tutuc, S. J. Papadakis, S. Melinte, M. Shayegan, D. Wasserman and S. A. Lyon, Anomalous spin polarization of GaAs two-dimensional hole systems, Phys. Rev. B, 72(19), 195321 (2005).

[6] B. Habib, E. Tutuc, S. Melinte, M. Shayegan, D. Wasserman, S. A. Lyon and R. Winkler, Negative diﬀerential Rashba eﬀect in two-dimensional hole systems, Appl. Phys. Lett., 85(15), 3151 (2004).

[7] Y. Hanein, H. Shtrikman and U. Meirav, Very low density two-dimensional hole gas in an inverted GaAs/AlAs interface, Appl. Phys. Lett., 70(11), 1426 (1997).

[8] M. Y. Simmons, A. R. Hamilton, S. J. Stevens, D. A. Ritchie, M. Pepper and A. Kurobe, Fabrication of high mobility in-situ back-gated (311)A hole gas heterojunctions, Appl. Phys. Lett., 70(20), 2750 (1997).

[9] M. Wassermeir, J. Sudijono, M. Johnson, K. Leung, B. Orr, L. D¨aweritz and K. Ploog, Reconstruction of the GaAs (311)A Surface, Phys. Rev. B, 51(20), 14721 (1995).

63 64 BIBLIOGRAPHY

[10] W. Braun, Applied RHEED: Reﬂection high-energy electron diﬀraction during crystal growth, vol. 154 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, Germany, 1999).

[11] W. R. Clarke, Quantum interaction phenomena in p-GaAs microelectronic devices, Ph.D. thesis, School of Physics, University of New South Wales, Australia (2005).

[12] W. R. Clarke, A. P. Micolich, A. R. Hamilton, M. Y. Simmons, K. Muraki and Y. Hirayama, Fabrication of induced two-dimensional hole systems on (311)A GaAs, J. Appl. Phys., 99, 023707 (2006).

[13] B. E. Kane, L. N. Pfeiﬀer, K. W. West and C. K. Harnett, Variable density high

mobility of two-dimensional electron and hole gases in a gated GaAs/AlxGa1−xAs heterostructure, Appl. Phys. Lett., 63(15), 2132 (1993). Chapter 4

Anisotropic Zeeman splitting in doped hole 1D wires

Chapter Outline This chapter presents a study of anisotropic Zeeman splitting in ballistic hole quantum wires fabricated on a (311)A bilayer heterostructure. Transport measurements show spin splitting of 1D subbands when the direction of the in-plane magnetic field B is parallel to the wire. In contrast, when B is perpendicular to the wire, no spin splitting is detected up to B = 9 T. Extracted g-factors for 1D subbands show an anisotropy that is significantly higher than in the two-dimensional case. This anisotropy is explained in terms of interaction of one dimensional confinement and spin-orbit coupling.

4.1 Introduction

The question we ask in this chapter is how confining holes to a 1D wire affects their spin properties. We answer this question by performing transport measurements with the magnetic field applied in the two-dimensional (2D) plane and oriented along (B) or perpendicular (B⊥) to the one-dimensional wire. These measurements allow us to extract the effective g-factors of 1D subbands for both orientations of in-plane magnetic field. As was discussed in Chapter 2, the confinement of the 3D hole system into 2D changes dramatically the behavior of the system in magnetic field: the isotropic 3D Lande g-factor becomes highly anisotropic in the 2D case. The in-plane anisotropy of g-factor has been measured and theoretically explained in (311)A grown 2D hole systems [1, 2]. By confining the 2D system into 1D wire, the energy spectrum is expected to change, which can modify the behavior of the system in magnetic field further. Moreover, in contrast to the 2D hole systems, where the anisotropic ratio of

65 66 4. Anisotropic Zeeman splitting in doped hole 1D wires the g-factors was estimated by the ratio of the depopulation magnetic fields [2], 1D systems allow us to study the evolution of the Zeeman splitting in a magnetic field, as well as a direct measurement of the 1D g-factors. However, no systematic work has been done in 1D hole systems so far. We have used the conductance quantization to measure directly the evolution of the 1D subbands in a magnetic field and the effective g-factors for in-plane magnetic field B oriented either along or perpendicular to the 1D wire direction. The data shows that the lower subbands split linearly in B, whereas no visible splitting exists for B⊥. Moreover, the anisotropy of the effective g-factors for two orthogonal field directions ∗ ∗ is significantly larger (g/g⊥ > 4.5) than that in 2D GaAs hole systems.

4.2 Literature overview

The Zeeman splitting in low-dimensional systems with spin-orbit coupling has attracted a lot of attention as a possible mechanism for spintronic applications. For example Björk et al. studied the Zeeman splitting in InAs nanowire-based quantum dots of the different length [3]. InAs nanowires with a diameter up to 70 nm were grown in the MBE chamber from the Au nanoparticles deposited on the substrate. The dots were defined by MBE growth of InP layers which define the barriers of the dot. The length of InAs wire sandwiched between the barriers was varying from 7 to 20 nm, hence the dots are strongly confined in the transport direction. The transport characteristics of the dots show a Coulomb blockade pattern with an apparent shell structure. With application of a magnetic field perpendicular to the wire, the Coulomb peaks corresponding to one energy level move apart linearly as a consequence of the Zeeman splitting. The authors showed that by increasing the length of the quantum dots the g-factor can be varied from 2.3 to 8 (the bulk value for InAs g∗ =14.7) and explained it by the confinement dependent energy gap between the levels of the InAs dot. Haendel et al. [4] used SiGe systems to study the tunneling of holes between two SiGe quantum wells separated by a thick Si barrier with embedded Ge well only 4 monolayer thick. The holes to SiGe wells are provided by boron-doped Si layers situated on each side of the SiGe double well structure. During the growth, the p- dopant (boron) diffuses into the Ge well in the barrier and facilitates tunneling between the Si/Ge quantum wells. A current-voltage characteristic of the device has a step-like dependence reflecting the resonant tunneling of heavy holes through the levels of boron atoms. Note that these measurements were done for the voltages below the onset of the resonant tunneling through the 2D subbands of the Ge well. With the application of magnetic field parallel to the growth direction, the tunneling levels of boron in the 4. Anisotropic Zeeman splitting in doped hole 1D wires 67

Ge well split and the differential conductance of the device shows a characteristic linear evolution of the conductance steps in magnetic field. On the other hand, if the field is in the 2D plane, there is no splitting visible up to the highest measured field (18 T). The explanation of these results is very similar to that described in Chapter 2. The confinement and strain (due to significant lattice mismatch of the layers) cause a very large HH-LH splitting and therefore the in-plane g-factor is suppressed, whereas the g-factor in the growth direction is large. We will now review previous work done on Zeeman splitting in GaAs 1D systems featuring both electron and holes. The application of an in-plane magnetic field B changes the energy spectrum of one-dimensional wires. The leading term linear in B, called the Zeeman term, lifts the spin degeneracy of the two-fold degenerate 1D subbands with the application of an external magnetic field. Historically, the Zeeman effect in 1D wires has been investigated in GaAs electron systems. Initial studies [5] found the g-factor to be enhanced (g∗ =1.1) from the bulk value of 0.44 and independent of the subband index. Additionally, some anisotropy of the g-factor measured for two in-plane directions had been suggested. More detailed studies [6] showed three main results: a) the in-plane anisotropy of the g-factor is small; b) when the 1D constriction is very wide both g-factors approach the bulk value; c)both g-factors increase as the subband number becomes smaller. The enhancement of g-factor at low subband indexes was explained by pronounced electron-electron interactions at low electron densities. Because of the difficulties with fabrication and stability, the number of experiments in 1D hole systems is very limited. Daneshvar et al. [7] used a modulation doped heterostructure grown on a (311)A GaAs substrate. The 1D wire was formed by a surface gate technique using two side gates and a middle gate. The wire was not aligned with the [233] or [110] crystallographic directions. An in-plane magnetic field perpendicular to the current direction (B⊥)was applied to probe the Zeeman splitting. The calculated g-factors for B⊥ are found to be between 1.7to0.6, increasing with decreasing subband index (similar to electron systems). However, due to gate instabilities of the wire, the average error in g-factor calculations was approximately 30%. Daneshvar et al. suggested that the enhancement of the g-factors was due to exchange interaction as in electron systems. The similarity to electron systems allowed authors to speculate that the exchange interaction might dominate the 1D behavior irrespective of the charge carrier. Rokhinson et al [8] studied QPCs formed by local anodic oxidation on a modulation doped shallow quantum well grown on a (311)A GaAs substrate. Two orthogonal QPCs were oriented along the [233] and [110] crystallographic directions and the magnetic field B was applied parallel and perpendicular to the current I through the QPC. 68 4. Anisotropic Zeeman splitting in doped hole 1D wires

∗ ∗ ∼ The main findings were that the ratios g233/g110 1.8 and > 4 for QPCs oriented in [110] and [233] directions respectively. However, the absolute values of the g-factors were not presented, which limited the information that can be extracted from these measurements. The authors claimed that they observed the intrinsic crystallographic anisotropy in (311)A GaAs but also indicated that the ratios are significantly different from the values obtained for 2D hole systems [1, 2]. Unfortunately, this data exists only in their abstract for the 2002 EP2DS conference and details of the measurements could not be found elsewhere.

4.3 Device structure and the principles of its operation

This section presents the fabrication and characterization of the doped bilayer 1D system. The device used for the studies in this chapter has been fabricated by Romain Danneau and measured by the author and Romain Danneau. The wafer T483 was grown by Michelle Simmons at Cavendish Laboratory, University of Cambridge. The growth structure of the wafer is illustrated on Fig. 4.1a. The heterostructure was grown on a conducting (311)An+-GaAs substrate. The active region consists of a 20 nm GaAs bottom quantum well (QW), a 30 nm AlGaAs barrier,and a 20 nm GaAs top 11 −2 2 QW. The hole 2D density and mobility are ns =1.2 × 10 cm and μ =92m/Vs 11 for the top layer. These numbers are smaller for the bottom layer, ns =1.0 × 10 cm−2 and μ =87m2/Vs.

(a) T483 wafer layout (b) EBL micrograph of the device

Figure 4.1: (a) Schematic digram of T483 wafer structure, which was used for the studies in this chapter; (b) EBL micrograph of the active region of the device. SG stands for “side gates”, MG - for “middle gate” and DG - for “depletion gates”. 4. Anisotropic Zeeman splitting in doped hole 1D wires 69

The use of a bilayer system originally was motivated by the instabilities found in previous 1D wires [9, 7], and which was attributed to doping layer. The original idea was that to measure the conductance quantization in one layer, while using the other layer as a ground plane to screen one of the dopant layers. It turned out that the bilayer system exhibited clean conductance quantization [10] in both layers, and was stable enough to perform source-drain bias and magnetic field measurements. The Hall bar mesa was formed by wet etching and was aligned along the high mobility [233] direction. The AuBe ohmics are evaporated and annealed in order to contact the 2D hole gas in both QWs. Ti/Au gate metallization is deposited using optical lithography. The surface gates of the QPC are defined using standard EBL techniques followed by Ti/Au metal evaporation, which overlaps the optical gates. The details of the process flow for this device can be found in Chapter 3. An SEM micrograph of the active region of the device is shown in Fig. 4.1b. The 1D wire is defined using three gates: two side gates (SG) and the middle gate (MG). The middle gate has a width of 600 nm and separated by 100 nm from the side gates on each side. There are also depletion gates (DG), which run across the mesa in order to deplete locally the top QW (the typical depletion voltage used in the measurements is ∼ +0.7 V). Depending in which layer (top or bottom) we wanted to measure conductance quantization, certain values of VMG, VBG and VDG were set prior to sweeping the side gates. Therefore it is necessary to map the middle gate, back gate and depletion gate parameter space in order to find the optimum values for the observation of conductance quantization in the top and bottom layers. Before the mapping, we need to understand the function of each gate. The purpose of DG is to stop the current flow through the top layer thereby forcing the current to pass through the bottom layer only. In Fig. 4.2a we show the characteristic two step depletion of the bilayer system biasing the depletion gate. By increasing VDG, the top layer beneath the depletion gate starts to deplete, and at a voltage ∼ 0.55 V, the top layer is pinched off ( marked by a dashed line in Fig. 4.2a).

Increasing VDG further depletes the bottom layer, and at ∼ 1.2 V no current ﬂows through the ohmics (conductance G drops to zero). Because the quantum wells are separated only by 30 nm barrier, the ohmics contact both layers in parallel. Note that there is no interlayer tunneling. The MG is used to vary the density in the top QW and can be biased between +2.0 and −0.5 V with current leakage to ohmic contacts below 10 pA, as shown in

Fig. 4.2b. The middle gate sweep trace also has a two stage pinch-oﬀ at VMG ≈ 0.6V and 1.2 V corresponding to the pinch-oﬀ of the top and the bottom layers beneath the MG. Because the MG runs only across a half of the mesa, the current I ( conductance 70 4. Anisotropic Zeeman splitting in doped hole 1D wires

600 600

500 500

400 400

300 300 G (µS) G (µS)

200 200 bottom TL definition BL definition top + bottom 100 100

0 0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 VDG (V) VMG (V) (a) Depletion gates characterization (b) Middle gate characterization

600 600

500 500

400 400 BL pinch-off

300 300 G (µS) G (µS)

200 200

100 100 TL pinch-off BL definition TL definition

0 0 0.5 1.0 1.5 2.0 2.5 0.0 1.0 2.0 3.0

VBG (V) VSG (V) (c) Back gate characterization (d) Side gates characterization

Figure 4.2: (a) Conductance of the bilayer system vs VDG at VSG = VMG = VBG =0 V; (b) Conductance of the bilayer system vs VMG at VSG = VDG = VBG =0V; (c) Conductance of the bilayer system vs VBG at VSG = VMG = VDG = 0 V;(d) Conductance of the bilayer system vs VSG at VMG = VDG = VBG = 0 V (top trace) and VMG = VBG = 0 V and VDG =+0.75 V (bottom trace). 4. Anisotropic Zeeman splitting in doped hole 1D wires 71

G) do not drop to zero and saturates. The conducting substrate is used as an in-situ back gate (BG) to control the density in the bottom QW with an operational range from 0 to +2.5 V, as shown in Fig. 4.2c.

The BG range is enough to deplete the bottom layer at VMG = 0 V and slightly reduce the density in the top layer for VMG > 1.2V. Positively biasing the side gates depletes the 2D hole gas underneath and defines the 1D channel in both QWs. By increasing the bias on the side gates further, the width of the 1D channels decreases and finally the channels pinch off (Fig. 4.2d).

Figure 4.2d shows two traces measured at VMG = VBG = 0 V and two diﬀerent

VDG = 0 V and +0.75 V. Because the VMG is set to zero, the density the top layer is very low and the top wire can be pinched-off before the bottom wire. This results in four stage pinch-off top trace at VDG = 0 V and two stage bottom trace at VDG =0.75 V having the same pinch-off voltage (this confirms that the bottom layer pinches off after the top layer).

Figure 4.3a illustrates the mapping of G vs VSG for traces at diﬀerent VMG at a constant value of VBG =0.8 V. Each trace in this ﬁgure corresponds to a certain

VMG varying from +0.5 V (the rightmost trace) to −0.3 V (the leftmost trace). As previously seen in electron systems [11], by setting the voltage on the middle gate it is possible to observe conductance quantization in the top or in the bottom wire. In general, sweeping the SG from zero to pinch-off the conductance of the system goes through the following stages: a) top wire definition (first inflection point at ∼ 0.6 V); b) bottom wire definition (second inflection point at ∼ 1.6 V); c) bottom wire pinch-off and d) top wire pinch-off. By increasing the VMG, stage d) weakens (dashed arrow line) and when VMG ≥ 0 V (last four traces), the top wire is fully depleted, which results in the absence of stage d).

Figure 4.3b shows the same data taken at VDG =0.8 V, when the current flows only through the bottom layer. The traces exhibit a two-stage pinch-off with the inflection point at ∼ 1.6 V (the same as on Fig. 4.3a, confirming the definition of the bottom layer after the top layer. Note, that the leftmost four traces on Fig. 4.3a and 4.3b do have identical pinch-off voltages, indicating that for VMG ≥ 0 V the top wire is fully depleted and only the bottom 1D wire is measured.

Additionally, for traces with VMG > 0 V on Fig. 4.3b, there is a strange feature close to the pinch-off voltage: the current seems not to drop to zero but remains constant until a certain voltage which is approximately the same as the pinch-off voltage for the corresponding trace on Fig. 4.3a. This suggests that somehow the current flows through the top layer after the bottom layer is pinched-off, despite VDG being applied. Several values of VDG have been tested but the residual current still exists even in the limit when DG fully depletes the top layer and strongly depletes the 72 4. Anisotropic Zeeman splitting in doped hole 1D wires

500 V =0V (a) DG VBG = 0.8 V -0.5V

(µS) 300

200 top+bottom G 100

0 (b) 400 VDG = 0.8 V

300 (µS) 200 bottom G 100 TL definition BL definition

0 1 2 3 4 VSG (V)

Figure 4.3: (a) Conductance of the 1D wire vs VSG for −0.5

wire is fully depleted by the middle gate), VBG = +1 V (the bottom layer is slightly depleted by the back gate) and VDG =+0.7 V (current runs through the bottom layer only). Figure 4.4 shows the conductance of the top and bottom wires for the optimized values of VMG and VBG.

VSG (V) 2.5 3.0 3.5 4.0

10 TOP

8 /h)

2 6

G (2e 4

2 BOTTOM 0 2.6 2.8 3.0 3.2 3.4

VSG (V)

2 Figure 4.4: Conductance of the 1D wire in units of 2e /h vs VSG. Top trace shows 1D conductance in the top layer with VBG =2.5V,VMG = −0.5V,VDG = 0 V. Bottom trace shows 1D conductance in the bottom layer with VBG =1V,VMG =+0.2V, VDG =+0.75 V

All transport measurements in this chapter were performed using two terminal conﬁguration. Therefore in order to convert raw conductance into conductance in 2 units of 2e /h the constant series resistance RC should be subtracted from the raw data. This series resistance comprises of contact resistance of the ohmics, resistance of 2D regions contacting the wire and the measurement circuitry. Because the resistance of the 2D regions varies with the hole density in both layers, the value of subtracted series resistance will depend on VBG and VMG, but RC is typically less than 2.5 kΩ. 2 Consequently, the value of RC was chosen to align the ﬁrst quantized plateau to 2e /h, this procedure automatically adjusts all plateaus to their quantized positions. As can be seen in Fig. 4.4, up to 10 quantized plateaus are achievable in each of the layers. An additional resonant feature below 1 × 2e2/h is apparent only on the bottom layer trace. We will address this issue in more detail in the next chapter. 74 4. Anisotropic Zeeman splitting in doped hole 1D wires

4.4 Determining the g-factors of the 1D subbands

The application of an in-plane magnetic ﬁeld lifts the spin degeneracy of the two-fold degenerate 1D subbands and causes the subbands to split. Taking into account only th the linear term, the energy splitting EN of the N level can be written as

∗ EN = gN μBB (4.1) ∗ th where gN is the eﬀective g-factor of the N level, and μB is the Bohr magneton. There are two common methods to obtain g-factors from magnetic ﬁeld measurements and source-drain bias measurements:

1. The ﬁrst method [5] is to extract the splitting rates of the transconductance

peaks in magnetic ﬁeld (∂VSG/∂B) and in source-drain voltage (∂VSG/∂VSD). Taking derivative of Eq.4.1 we then obtain

∗ ∂EN ∂EN × ∂VSG ∂eVSD × ∂VSG gN μB = = = (4.2) ∂B ∂VSG ∂B ∂VSG ∂B

2. The second method is to compare crossings of 1D subband edges from the source-

drain bias measurements (ΔEN,N+1) with crossings of the 1D subband edges

from the magnetic ﬁeld measurements (BC ). Using Eq. 4.1 for two adjacent levels we can then get the following relations:

± ∗ EN = EN (B =0) gN μBB (4.3)

and ± ∗ EN+1 = EN+1(B =0) gN+1μBB (4.4) By taking a diﬀerence between the two we will obtain:

− ± ∗ ∗ EN,N+1 = EN (B =0) EN+1(B =0) gN gN+1 μBB (4.5) ∗ ∗ where gN gN+1 is the average eﬀective g-factor for the N and N + 1 subbands. When the levels cross at some magnetic ﬁeld BC , EN,N+1 = 0 and therefore:

− ∗ ∗ ΔEN,N+1 = EN (B =0) EN+1(B =0)= gN gN+1 μBBC (4.6)

The first method allows us to calculate the g-factor for each individual subband. Additionally it is possible to monitor the subband splitting and verify the linearity of the Zeeman splitting. On the contrary, the second method assumes linearity and takes the average of the g-factors for two adjacent spin-split subbands. In this chapter we aim to measure the effective g-factors for the in-plane magnetic field B oriented either along or perpendicular to the 1D wire direction. Firstly, the 4. Anisotropic Zeeman splitting in doped hole 1D wires 75 sample was placed in the dilution fridge insert in such a way that B was in-plane and parallel to the wire (B [233]).¯ We denote this orientation of magnetic field by the symbol . Then, the system was cooled down to the base temperature. Prior to magnetic field measurements, a full characterization of the gates on the sample was performed (see Fig. 4.2), followed by the source-drain bias measurements. Sub- sequently, the system was warmed up and the sample reoriented such that B was in-plane but perpendicular to the wire (B ⊥ [233]).¯ We denote this orientation by the symbol ⊥. All the measurements (characterization, source-drain bias and magnetic

field) were repeated for the B⊥ orientation. The consistency of the results relies on consistency of the gate characterization and source-drain bias measurements. Note that because of the modulation doping some characteristics of the device, such as pinch-off voltages, can vary moderately between the cool-downs. However these small changes do not alter the measured 1D subband spacings and therefore the calculations of the effective g-factors.

4.4.1 Source-drain bias spectroscopy

The source-drain biasing (SDB) technique [12, 13] was used to determine the subband spacings of 1D subbands in the top and bottom layers. This technique is described in Chapter 2. Typically, SDB measurements are performed by sweeping the d.c. voltage

VSD and monitoring the diﬀerential conductance g(VSD) at diﬀerent values of VSG.

The plateau in conductance will appear as accumulation of g(VSD) traces at VSD =0. With application of a d.c. bias the traces evolve to form accumulations at ﬁnite

VSD, which corresponds to half integer plateaus, as shown in schematic in Fig. 4.5a. To highlight the diamond shape pattern, the data is presented as a colour map of transconductance dG/dVSG plotted vs VSG and VSD coordinates (see Fig. 4.5b). The plateau in conductance will produce zero transconductance and the riser between the conductance plateaus will produce large negative transconductance (note that on the riser, conductance drops with increasing VSG for hole systems). Fig. 4.6a and Fig. 4.6b show measured differential conductance g as a function of VSD at various set VSG for top and bottom layers respectively. The data has been adjusted for a constant series resistance RC to align the first accumulation of the conductance curves (the first plateau) with 2e2/h. The ohmic resistance acts as a voltage divider for the applied VSD: at low wire conductances, most of VSD drops across the wire, but as conductance of the wire increases, more voltage drops across the ohmics and less voltage drops across the wire. The SDB colour maps of transconductance of the top and bottom layers (Fig. 4.6c and Fig. 4.6d) show the diamond shape patterns (c.f. Fig. 4.5b). Yellow areas (small transconductance) correspond to plateaus in 76 4. Anisotropic Zeeman splitting in doped hole 1D wires

5/2 5/2

(V) Δ 3/2 V3/2SD SG V 1

1/2 1/2

0 VC VSD (mV) (a) (b)

Figure 4.5: (a) Differential conductance of a 1D wire vs VSD for different values of VSG; (b) Map of transconductance vs VSD on x axis and VSG on y axis. The lines represents the risers in conductance, the white areas inside the lines correspond to the plateaus in conductance. the conductance, while black areas (large negative transconductance) correspond to the risers between conductance plateaus. The yellow areas around VSD = 0 V are the integer plateaus; there are also yellow areas situated between adjacent integer plateaus at finite VSD representing half-plateaus. The center of those regions are marked by VC which is a direct measure of the subband spacings ΔE between the pair of adjacent integer plateaus.

Table 4.1 shows the extracted values of ΔE = eVC for the first five subbands for each of the wires. In addition, the same data is presented for the second cooldown showing a maximum error of 10 μeV for the estimation of the subband spacings. As can be seen from Table 4.1, the subband spacing decreases with the increasing subband index for both wires. There is also a noticeable difference between top and bottom subband spacings reflecting a difference in confining potential. Moreover, the values of subband spacings for hole wires are much smaller than in electron wires, which have a typical scale of subband spacings around 2.5 meV [14]. This is because the subband spacing is inversely proportional to the effective mass of ∗ · the carriers and the effective mass of holes in GaAs (mp =0.38 m0) is much higher ∗ · than that of electrons (me =0.067 m0). With the application of a d.c. voltage, the risers between the plateaus split linearly as shown in the schematic in 4.5b. The dashed white lines on Fig. 4.6c highlight the position of the risers. We have extracted the difference ΔVSG between the risers as a 4. Anisotropic Zeeman splitting in doped hole 1D wires 77

6 5 5 4 4 /h) /h)

2 3 2 3 (2e g (2e g 2 2

1 1

0 0 -0.5 0.0 0.5 -0.5 0.0 0.5

VSD (mV) VSD (mV) (a) VBG =2.5V,VMG = −0.5V,VDG =0V (b) VBG =1V,VMG =0.2V,VDG =0.75 V

2.8 2.4 0 0

-5 -5 3.0 2.6

-10 -10

3.2 -15 2.8 -15 (V) (V) SG SG V V

3.4 3.0

3.6 3.2

-0.5 0.0 0.5 -0.5 0.0 0.5 VSD (mV) VSD (mV) (c) Transconductance colour map (top layer) (d) Transconductance colour map (bottom layer)

2 Figure 4.6: (a) Differential conductance of the top wire in units of 2e /h vs VSD for different VSG (step size 0.01 V); (b) Differential conductance of the bottom wire 2 in units of 2e /h vs VSD for different VSG (step size 0.01 V); (c) Colour map of transconductance vs VSD on x axis and VSG on y axis. Data obtained by numerical differentiation of the data in Fig. 4.6a; (d) Colour map of transconductance vs VSD on x axis and VSG on y axis. Data obtained by numerical differentiation of the data in Fig. 4.6b. 78 4. Anisotropic Zeeman splitting in doped hole 1D wires

Levels ΔE (μV) top ΔE (μV) bottom 1,2 378 / 387 288 / 279 2,3 281 / 288 172 / 162 3,4 239 / 255 152 / 151 4,5 196 / 188 136 / 131

Table 4.1: Subband separation of the ﬁrst ﬁve subbands for top and bottom wires. Two values separated by “/” are extracted from the two separate cool-downs.

function of VSD and plotted it on Fig. 4.7 for the three subbands (2, 3, and 4) of the top wire. The experimental points can be approximated by the straight line.

N=4 0.1

0.0 N=3

(V) 0.1 SG V

0.0 N=2 0.1

0.0 0 100 200 300 400 μ VSD ( eV)

Figure 4.7: Splitting of the subband edges ΔVSG as a function of VSD for the ﬁrst three subbands of the top wire. Straight lines show the least squares ﬁt through the data points.

4.4.2 Magnetic ﬁeld measurements

Figure 4.8 shows a schematic of linear evolution of 1D subbands in an in-plane magnetic field B. Two adjacent subbands, the spin-up lower subband N(↑) and the next higher ↓ ∂VSG spin-down subband N +1( ) move towards each other in magnetic field at a rate ∂B and cross at some value of magnetic field BC . These values can be used to calculate the effective g-factors of the 1D subbands. Figure 4.9a shows the conductance of the top layer for different values of magnetic

ﬁeld B in the range between 0 and 9 T. The leftmost trace corresponds to B =0T 4. Anisotropic Zeeman splitting in doped hole 1D wires 79

5/2

(V) Δ VSD 3/2 SG V 1

1/2

0 B C B (T)

Figure 4.8: Schematic diagram of the Zeeman splitting of 1D subbands. BC labels the ﬁeld at which two adjacent spin-split subbands cross.

and has no offset (leftmost blue trace). Other traces are offset sequentially along VSG axis by 0.05 V. The magnetic field B lifts the spin degeneracy of the 1D subbands and the conductance evolves from being quantized in units of 2e2/h at B =0Ttoe2/h for B =3.6 T, corresponding to complete spin resolved 1D levels (middle blue trace). At B =7.8 T the subbands cross, resulting in half plateaus only being observed in conductance (rightmost blue trace). The bottom wire shows similar behavior (Fig. 4.9b) to the top layer. However the values of B that correspond to fully spin resolved levels and level crossings are considerably smaller (2.2 and 3.6 T respectively). Additionally, the conductance quantization in the bottom layer is much weaker than in the top layer resulting in a poor conductance quantization at high magnetic fields. Therefore, in this chapter results from the bottom wire are shown just for qualitative comparison with the data from the top wire. In order to observe the diamond-like pattern, as shown in a schematic in Fig. 4.8, we calculated the transconductance (dG/dVSG) for each magnetic field B and plotted the data as a colour map against VSG and B coordinates, as shown in Figs. 4.10a and 4.10b for top and bottom layers respectively. Additionally, it is possible to plot transconductance as a colour map against G and B coordinates in order to highlight the evolution of the 1D subbands in magnetic

field and extract BC (shown in Figs. 4.10c and 4.10d for top and bottom layers correspondingly). The fields BC can be extracted from these colour maps as the field 80 4. Anisotropic Zeeman splitting in doped hole 1D wires

/h) 3 2

2 G (2e 1

0 4.0 4.5 5.0 5.5 6.0 6.5

VSG (V)

(a) VBG =2.5V,VMG = −0.5V,VDG =0V

/h) 3 2

2 G (2e 1

0 2.5 3.0 3.5 4.0

VSG(V)

(b) VBG =1V,VMG =0.2V,VDG =0.75 V

Figure 4.9: (a) Conductance of the top wire vs VSG for different values of B (step size 0.2 T). Traces offset from the left to the right in multiples of 0.05 V; (b) Conductance of the bottom wire vs VSG for different values of B (step size 0.2 T). Traces offset from the left to the right in multiples of 0.04 V. 4. Anisotropic Zeeman splitting in doped hole 1D wires 81

3.0 2.0

3.2

2.2 3.4

3.6 (V) (V) SG

SG 2.4 V 3.8 V

4.0 2.6 4.2

0 2 4 6 8 0 2 4 6 8 B (T) B (T) (a) VBG =2.5V,VMG = −0.5V,VDG =0V (b) VBG =1V,VMG =0.2V,VDG =0.75 V

6 6

5 5

4 4 /h) /h) 2 2 3 3 G(2e G (2e 2 2

1 1

0 0 0 2 4 6 8 0 2 4 6 8 B (T) B (T) (c) VBG =2.5V,VMG = −0.5V,VDG =0V (d) VBG =1V,VMG =0.2V,VDG =0.75 V

Figure 4.10: (a) Transconductance of the top wire vs magnetic field B on x axis and side gate voltage VSG on y axis; (b) Transconductance of the bottom wire vs B on x axis and VSG on y axis; (c) Transconductance of the top QPC vs B on x axis and conductance G on y axis; (d) Transconductance of the bottom QPC vs B on x axis and conductance G on y axis. Blue colour corresponds to plateaus in conductance and red to the risers between conductance plateaus. 82 4. Anisotropic Zeeman splitting in doped hole 1D wires at which two adjacent integer plateaus disappear and only the half plateau in between them remains. The fields BC for first five subbands are labeled by black squares in Figs. 4.10c and 4.10d.

0.4

0.2 N=5 0.0 N=4 0.2

0.0 N=3

(V) 0.2 SG V

Δ 0.0 N=2 0.2

0.0 N=1 0.2

0.0 0 2 4 6 B|| (T)

Figure 4.11: Splitting of the subband edges ΔVSG as a function of B for the first five subbands of the top wire. Straight lines show the least squares fit through the data points.

In order to quantify the evolution of the subbands in Fig. 4.10a, Fig. 4.11 shows the difference in VSG (ΔVSG) between the two spin-split subbands (N ↑ and N ↓) as a function of magnetic field for the first five 1D subbands of the top wire. As can be seen from Fig. 4.11, as B increases, the spin-split subbands diverge and

ΔVSG increases approximately linearly for the lower subbands. However, the data for subband 5 exhibit nonlinear behavior and can be fitted with two lines of different slope that cross at around B =4.4 T. Note that this change in the slope could be due to the fact that around 4.4 T the subbands 5 ↓ and 4 ↑ cross. For lower plateaus the change in slope is not visible because the subband crossings appear at higher fields and we can only extract the data reliably up to the crossings but not after. Figure 4.12a shows the data taken for the top wire in the same way as that in Fig.

4.9a, but for B⊥ instead. The plateaus remain at their quantized positions up to the highest measured field of 9 T and no sign of Zeeman splitting is observed. The colour map of transconductance looks completely different to the case of B. Instead of the diamond pattern there are lines curving upwards as the magnetic field B increases (Fig. 4.12b). This behavior is probably due to a quadratic term B2 in the 1D Hamiltonian, 4. Anisotropic Zeeman splitting in doped hole 1D wires 83 known as the diamagnetic shift as seen previously in electron systems [6]. Plotting the transconductance as a function of G and VSG confirms that the conductance plateaus are not affected by the magnetic field (Fig. 4.12c).

Similarly, Fig. 4.13a presents data for the bottom wire for B⊥. The plateaus in conductance do not move to the half-integer values in magnetic field. However, it is difficult to analyze the data at high magnetic fields because the plateaus become weak. Additionally, as can be seen in colour maps in Figs. 4.13b and 4.13c, the pinch-off voltage shifts significantly with applied magnetic field, which can again attributed to diamagnetic shift.

Because the subbands do not split up to the highest measured field B⊥,itisnot possible to extract directly fields BC at which the adjacent spin-split subbands cross or the splitting rates of the subbands in magnetic field. Thus, we can only estimate ∗ BC by comparing the field B when the subbands just start to split for both parallel and perpendicular field. In the parallel direction the spin splitting in the top wire is evident at B∗ ≈ 2 T for all subbands and in the perpendicular direction the spin ∗ ∗ splitting is not present up to 9 T. Hence the ratio B /B⊥ is at least 4.5 for the top ∗ ∗ layer. Assuming a linear Zeeman splitting, the ratio B /B⊥ = BC ()/BC (⊥). Note that the ratio BC ()/BC (⊥) is equal to the ratio of the g/g⊥ as the subband spacings remain the same (c.f. Eq. 4.6). For the bottom layer, the spin splitting in B direction is observed at 1.2T,howeverinB⊥ direction the conductance plateaus become very ∗ weak and B⊥ can not be estimated.

4.4.3 Calculations of g-factors and discussion

We have performed calculations of the 1D g-factors using both methods described previously:

X From the splitting of the subband edges in source-drain voltage ∂VSG and in ∂VSD ∂VSG magnetic ﬁeld ∂B . Using this method we estimated the eﬀective g-factor for the three subbands (2, 3 and 4) in the top layer. The bottom layer shows much weaker conductance quantization and therefore the reliable extraction of the subband edge slopes is hindered.

X From the crossings of the subband edges in source-drain voltage VC and magnetic

ﬁeld BC we have calculated the average g-factors between the subbands 1 & 2, 2 & 3, 3 & 4 and 4 & 5 for both wires.

We found the g-factors calculated by two methods to be consistent with each other within the accuracy of calculations. 84 4. Anisotropic Zeeman splitting in doped hole 1D wires

6 /h) 2 4 G (2e

0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

VSG (V) (a) VBG =2.5V,VMG = −0.5V,VDG =0V

2.5 8

3.0 6 /h) 2

(V) 3.5

SG 4 V G (2e

4.0 2

4.5 0 0 2 4 6 8 0 2 4 6 8 B (T) B (T) (b) VBG =2.5V,VMG = −0.5V,VDG =0V (c) VBG =2.5V,VMG = −0.5V,VDG =0V

Figure 4.12: (a) Conductance of the top wire vs VSG for different values of B⊥ (step size 0.2 T). Traces offset from the left to the right in multiples of 0.05 V; (b) Transcon- ductance of the top wire vs magnetic field B on x axis and side gate voltage VSG on y axis. Blue colour corresponds to plateaus in conductance and red to the risers between the conductance plateaus. Data for differentiation taken from Fig. 4.12a; (c) Transconductance of the top wire vs B on x axis and conductance G on y axis. The colours are the same as in Fig. 4.12b. 4. Anisotropic Zeeman splitting in doped hole 1D wires 85

4 /h)

2 3

2 G (2e

0 3.0 3.5 4.0 4.5 5.0 5.5 VSG (V) (a) VBG =1V,VMG =0.2V,VDG =0.75 V

2.6 8

2.8 6 /h) 2

(V) 3.0 SG V G (2e 4

3.2 2

3.4 0 0 2 4 6 8 10 0 2 4 6 8 10 B (T) B (T) (b) VBG =1V,VMG =0.2V,VDG =0.75 V (c) VBG =1V,VMG =0.2V,VDG =0.75 V

Figure 4.13: (a) Conductance of the bottom wire vs VSG for different values of B⊥ (step size 0.2 T). Traces offset from the left to the right in multiples of 0.05 V; (b) Transconductance of the bottom wire vs magnetic field B on x axis and side gate voltage VSG on y axis. Blue colour corresponds to the plateaus in conductance and red to the risers between the conductance plateaus. Data for differentiation taken from Fig. 4.13a; (c) Transconductance of the bottom wire vs B on x axis and conductance G on y axis. The colours are the same as on Fig. 4.13b. 86 4. Anisotropic Zeeman splitting in doped hole 1D wires

Table 4.2 shows g-factor values calculated using the crossings of the subband edges for both the top and bottom wires. The maximum error for the g-factors is ±0.04 and ±0.14 for the top and bottom wires respectively. This error includes the uncertainty in the estimation of the magnetic field BC ± 0.2 T and variation of the subband spacings for different cool-downs ±10 μeV. Table 4.3 shows g-factor values calculated using the splittings of the subband edges for the top wire. The maximum error for the g-factors is ±0.07, which reflects the uncertainty in the slope of a linear fit to the experimental points (see Fig. 4.11).

top bot top bot ∗ ∗ Levels ΔE (μV) ΔE (μV) BC (T) BC (T) gtop gbot 1,2 382.5 283.5 6.8 3.4 0.97 1.44 2,3 284.5 167 6.2 2.4 0.79 1.21 3,4 147 151.5 5 1.8 0.85 1.45 4,5 192 133.5 4.4 1.4 0.75 1.64

Table 4.2: g-factor values for the top and bottom wires calculated using subband spacings ΔE and the magnetic ﬁeld at subband crossings BC .

∗ Level ∂VSG/∂eVSD ∂VSG/∂B gtop 2 312 16.073 × 10−3 0.89 3 441 18.379 × 10−3 0.72 4 517 22.145 × 10−3 0.74

Table 4.3: g-factor value for the top wire calculated using the splitting rates of the subbands in source-drain voltage ∂VSG and magnetic ﬁeld ∂VSG . ∂VSD ∂B

Figure 4.14 summarizes the main results of this chapter. The solid circles represent g-factor values for the top layer for the B direction calculated using both methods. The shaded area at the bottom of the graph shows the possible values for g-factors in the top wire for B⊥ (open circles are the maximum estimated values). The anisotropic value of the g-factors in the 2D case are indicated by the two dashed lines. As can be seen from Fig. 4.14, g exhibits a monotonous decrease as the subband index increases, approaching a 2D value g[233]. On the other hand, g⊥ is estimated from the ratio BC ()/BC (⊥) to be at least 4.5 times smaller than g. This anisotropy of the 1D ∗ ∗ g-factors in parallel and perpendicular in-plane directions (g/g⊥ > 4.5) is in marked ∗ ∗ contrast to 2D hole systems where (g/g⊥ ≤ 4) [1, 2].

4.5 Theoretical interpretation of the results

In this section we will discuss our results for the 1D hole wire in comparison with existing data for 2D hole systems. In particular we show that the two results can 4. Anisotropic Zeeman splitting in doped hole 1D wires 87

1.2

g|| 1.0

0.8

top g 0.6 [233] g*

0.4

g2D 0.2 [011]

g⊥ 0.0 0 1 2 3 4 5 Subband index

Figure 4.14: Calculated g-factor for the lowest 1D subbands. Solid blue circles correspond to g calculated from the subband edge splittings. Solid red circles correspond to g calculated from crossings and open circles to the maximum values of g⊥. The shaded area below the open circles shows the possible values of g⊥. be understood under a common framework providing that the axis of total orbital momentum Jˆ is appropriately chosen for each system. As was discussed in Chapter 2, the 2D conﬁnement drastically changes the properties of hole systems:

X It lifts the degeneracy of the top four-fold degenerate valence band, so that HH 3 (j = 2 ) subband is the highest in energy. Therefore in most cases 2D electrical transport in holes is through the HH subband.

X It deﬁnes the direction of total angular momentum Jˆ to be parallel to the growth direction z (Jˆ z), as indicated in Fig. 4.15a.

X ˆ ∗ When B J the Zeeman splitting (gz ) is large for both HH and LH subbands. On the contrary, when B ⊥ Jˆ (B is in the 2D plane) there will be no Zeeman ∗ splitting (gx,y = 0) for HH subbands.

X For low symmetry growth directions, such as (311), cubic term in the Luttinger 88 4. Anisotropic Zeeman splitting in doped hole 1D wires

Hamiltonian introduces HH-LH coupling, but it is reduced by the energy gap between HH and LH due to 2D conﬁnement. This cubic correction results in a ﬁnite and anisotropic in-plane g-factor.

X Both the calculations and the experimental data for the (311) grown 20 nm QW ∗ ∗ show that g/g⊥ ≤ 4 [1, 2]. This anisotropy depends only on the direction of the magnetic ﬁeld B and has no dependance on the current I direction.

Figure 4.15: Schematic of the device showing the orientation of total angular momentum Jˆ in (a) a 2D hole system and (b) in a 1D hole wire.

Our results in 1D case show much stronger anisotropy and can not be explained by the 2D in-plane anisotropy. In order to interpret our data we need to take into account the fact that the 1D wire is formed by biasing the side gates and that the holes are then conﬁned to move only in one direction. This 1D conﬁnement realigns the vector Jˆ from being perpendicular to the plane to in plane and parallel to the wire (marked as z in the schematic in Fig. 4.15b). From this point, qualitatively the same argument as in the 2D case can be applied for the 1D wire. When B is parallel to the wire direction the resulting g is enhanced. On the other hand, for B⊥ no direct coupling between the HH states exists and the

ﬁnite g⊥ arises from the cubic HH-LH coupling. This coupling depends strongly on the 1D conﬁnement potential (which controls the gap between the LH and HH states) in the x and y directions.

The monotonous decrease of g with increasing subband index can be explained qualitatively by comparing the strength of the confinement in the x and y directions. For higher subbands the confinement in the growth (311) direction is stronger than the electrostatic in-plane confinement in the [011] direction and therefore g tends to the 2D value of 0.6 (vector Jˆ is aligned along the growth direction). As the number of 4. Anisotropic Zeeman splitting in doped hole 1D wires 89 occupied subbands decreases, the in-plane confinement becomes comparable with the growth confinement and the vector Jˆ realigns along the wire, causing increase in g and enhanced ratio g/g⊥. It is also possible to explain the enhancement of the g-factor in 1D hole systems by exchange interaction effects [5, 6] despite recent findings by Winkler et al. [15], who showed that the spin susceptibility enhanced in diluted 2D electron systems due to exchange interactions remains unaffected in the case of 2D hole systems. This argument can only be strictly applied to (100) grown quantum wells and not for (311), particularly when the 1D wire is formed and the vector Jˆ becomes oriented along the wire. It is expected that the suppression of the exchange enhancement is lifted along the wire but still remains in the perpendicular ([011]) direction [16]. Therefore ∗ ∗ the enhanced ratio of the g-factors g/g⊥ can be due to both the Zeeman term and exchange enhancement. Finally, there has been recent theoretical work [17] showing that by constricting the system from 2D into a 1D wire, the HH ground state of the 2D system becomes a mixture of HH and LH states when the width of the channel narrows, and finally in the 1D case the ground state is the LH state. It is also known [1] that the in-plane ∗ ∗ anisotropy for the LH state in the 2D case is g/g⊥ = 2. On the contrary, our data shows much stronger anisotropy and can only be explained assuming the transport through the HH subbands. We will address the transition from the HH state in 2D to the LH state in 1D in more detail in Chapter 5.

4.6 Conclusions and future work

In this chapter we presented measurements of the anisotropic Zeeman splitting in ballistic hole quantum wires fabricated on a (311)A oriented bilayer heterostructure. From the transport measurements, spin splitting of the 1D subbands is evident when the direction of the in-plane magnetic field B is parallel to the wire. On the other hand, when B is perpendicular to the wire no sign of spin splitting is detected up to the ∗ ∗ highest measured field of B = 9 T. The ratio g/g⊥ > 4.5 is significantly higher than in the two-dimensional case. This anisotropy is explained by a model which considers the reorientation of the vector Jˆ from being perpendicular to the plane to in-plane and parallel to the wire. From the discussion in the previous section we assume that the g-factor anisotropy in our model depends on the wire direction, which is in marked contrast with the 2D case [1, 2], where the g-factor anisotropy depends on the magnetic field direction. In order to verify these assumptions, the comparison between 1D wires in two orthogonal directions [011] and [233] should be made. Bibliography

[1] R. Winkler, S. J. Papadakis, E. P. de Poortere and M. Shayegan, Highly anisotropic g-factor of two-dimensional hole systems, Phys. Rev. Lett., 85(21), 4574 (2000).

[2] S. J. Papadakis, E. P. de Poortere, M. Shayegan and R. Winkler, Anisotropic magnetoresistance of two-dimensional holes in GaAs, Phys. Rev. Lett., 84(24), 5592 (2000).

[3] M. T. Björk, A. Fuhrer, A. E. Hansen, M. W. Larsson, L. E. Fröberg and L. Samuelson, Tunable effective g factor in InAs nanowire quantum dots,Phys. Rev. B, 72, 201307R (2005).

[4] K.-M. Haendel, R. Winkler, U. Denker, O. G. Schmidt and R. J. Haug, Giant Anisotropy of Zeeman Splitting of Quantum Conﬁned Acceptors in Si/Ge, Phys. Rev. Lett., 96, 086403 (2006).

[5] N. K. Patel, J. T. Nicholls, L. Martin-Moreno, M. Pepper, J. E. F. Frost, D. A. Ritchie and G. A. C. Jones, Properties of a ballistic quasi-one-dimensional constriction in a parallel high magnetic-ﬁeld, Phys. Rev. B, 44(19), 10973 (1991).

[6] K. J. Thomas, J. T. Nicholls, M. Y. Simmons, M. Pepper, D. R. Mace and D. A. Ritchie, Possible spin polarization in a one-dimensional electron gas, Phys. Rev. Lett., 77(1), 135 (1996).

[7] A. J. Daneshvar, C. J. B. Ford, A. R. Hamilton, M. Y. Simmons, M. Pepper and D. A. Ritchie, Enhanced g factor of a one-dimensional hole gas with quantized conductance, Phys. Rev. B, 55(20), R13409 (1997).

[8] L. P. Rokhinson, D. C. Tsui, L. N. Pfeiffer and K. W. West, Modification of the effective g-factor by lateral confinement, Abstract of EP2DS 2002 Conference (2002).

90 BIBLIOGRAPHY 91

[9] I. Zailer, J. E. F. Frost, C. J. B. Ford, M. Pepper, M. Y. Simmons, D. A. Ritchie, J. T. Nicholls and G. A. C. Jones, Phase coherence, interference and conductance quantization in a conﬁned two-dimensional hole gas, Phys. Rev. B, 49(7), 5101 (1994).

[10] R. Danneau, W. R. Clarke, O. Klochan, A. P. Micolich, A. R. Hamilton, M. Y. Simmons, M. Pepper and D. A. Ritchie, Conductance quantization and the 0.7 × 2e2/h conductance anomaly in one-dimensional hole systems, Appl. Phys. Lett., 88, 012107 (2006).

[11] I. M. Castleton, A. G. Davies, A. R. Hamilton, J. E. F. Frost, M. Y. Simmons, D. A. Ritchie and M. Pepper, Closely separated one-dimensional wires: coupled ballistic conduction, wave function hybridization and compressibility investigations, Physica B, 249-245, 157 (1998).

[12] L. I. Glazman and A. V. Khaetskii, Non-linear quantum conductance of a lateral microconstraint in a heterostructure, Europhys. Lett., 9(3), 263 (1989).

[13] N. K. Patel, J. T. Nicholls, L. Martin-Moreno, M. Pepper, J. E. F. Frost, D. A. Ritchie and G. A. C. Jones, Evolution of half plateaus as a function of electric ﬁeld in a ballistic quasi-one-dimensional constriction, Phys. Rev. B, 44(24), 13549 (1991).

[14] K. J. Thomas, M. Y. Simmons, J. T. Nicholls, D. R. Mace, M. Pepper and D. A. Ritchie, Ballistic transport in one-dimensional constrictions formed in deep two- dimensional electron gases, Appl. Phys. Lett., 67(1), 109 (1995).

[15] R. Winkler, E. Tutuc, S. J. Papadakis, S. Melinte, M. Shayegan, D. Wasserman and S. A. Lyon, Anomalous spin polarization of GaAs two-dimensional hole systems, Phys. Rev. B, 72(19), 195321 (2005).

[16] U. Z¨ulicke, private communications.

[17] U. Z¨ulicke, Electronic and spin properties of hole point contacts, Phys. Stat. Sol C, 3(12), 4354 (2006). Chapter 5

In-plane magnetic ﬁeld studies of the 0.7 structure and the zero-bias anomaly in doped hole 1D wires

Chapter Outline In this chapter we study the magnetic field dependence of the 0.7 structure and the zero bias anomaly (ZBA) in a ballistic hole wire fabricated on a (311)A bilayer heterostructure. We analyze the behaviour of the 0.7 structure and the ZBA with the application of an in-plane magnetic field B oriented parallel or perpendicular to the wire. We find that the magnetic field dependencies of both phenomena are distinctly anisotropic owing to the highly anisotropic g-factor. The data also show a strong correlation between the anisotropy of the 0.7 structure and the ZBA suggesting that the two phenomena are intimately related and have the same origin. These results link unambiguously the 0.7 structure and ZBA to spin.

5.1 Introduction

The quantization of conductance in ballistic one-dimensional (1D) wires observed by Van Wees et al. [1] and Wharam et al. [2] can be explained using the Landauer- Buttiker formalism [3, 4] in a single particle framework. An elementary explanation of the quantization views the 1D constriction as an electron wave guide, through which a small integer number of transverse modes can propagate at the Fermi level. The wide two-dimensional (2D) regions at opposite sides of the constriction are reservoirs of electrons in a local equilibrium. A voltage diﬀerence V between the reservoirs induces a current I through the constriction, equally distributed among the N modes. Summing

92 5. In-plane magnetic field studies of the 0.7 structure and the zero-bias anomaly in doped hole 1D wires 93 over all modes in the wave guide, one obtains the conductance G = I/V = Ne2/h. The experimental step size at zero magnetic field is 2 × e2/h because spin-up and spin-down modes are degenerate. Even though the transport is ballistic, the electron wave guide has a non-zero resistance because of the reflections that occur when a small number of propagating modes in the wave guide are matched to a larger number of modes in the reservoirs. In 1996, Thomas et al. [5] pointed out an anomalous plateau near 0.7 × 2e2/h that had been seen previously (see for example [6]) but was generally neglected or ascribed to a resonance caused by an impurity inside the wire. This plateau, commonly referred as the 0.7 structure, could not be explained using noninteracting theory and therefore is believed to be a many-body phenomena. In 2002, Cronenwett et al. [7] presented source-drain bias data featuring a characteristic peak at zero source-drain voltage called the zero-bias anomaly (ZBA). The authors observed a correlation between this peak in the source-drain bias and the 0.7 structure. Following the observation of the Kondo effect in quantum dots, the Kondo mechanism was also proposed to explain the 0.7 structure in 1D wires. However, the origin of the 0.7 structure is still highly debated. Studying the 0.7 structure in new 1D systems can reveal important information about its origin and also provide constraints on existing theories of the 0.7 structure. In this chapter we study the behaviour of the 0.7 structure in a 1D hole wire in response to two orthogonal directions of the in-plane magnetic field, parallel (B) and perpendicular B⊥ to the wire. Unlike electrons [8], holes have a strong anisotropy of the effective g-factor [9], which allows us to directly test the relation of the 0.7 structure and the spin. Additionally, we performed studies of the ZBA in the source- drain bias measurements and its anisotropy in an in-plane magnetic field in order to unambiguously verify the connection between the ZBA and the 0.7 structure.

5.2 Literature overview

In 1996, Thomas et al. published a paper [5] on the 0.7 structure, emphasizing that it is not a conductance resonance. Two main distinctive features of the 0.7 structure have been investigated.

1. The temperature dependence of the 0.7 structure: With increasing temperature T , the integer subbands smear out while the 0.7 structure becomes enhanced. This is in a marked contrast to resonances which disappear rapidly with increasing T .

2. The behaviour in an in-plane magnetic ﬁeld: The 0.7 structure smoothly shifts 5. In-plane magnetic ﬁeld studies of the 0.7 structure and the zero-bias 94 anomaly in doped hole 1D wires

from its normal conductance value at B = 0 T to a spin-polarized value of 0.5 × 2e2/h with increasing B. From this behaviour the authors predicted a spin-polarized ground state at B = 0 induced by electron-electron interactions.

Additional evidence that the 0.7 structure is not a resonance came from studying many devices and from laterally shifting the channel by applying an offset to the gates. The 0.7 structure remained unaffected, which suggests that it is not caused by a defect in the vicinity of the channel. It was also shown that the 1D g-factor of the integer subbands is isotropic and increases with decreasing subband number, indicating an enhancement of electron- electron interactions in the wire at low conductances. The fact that the 0.7 structure at B = 0 T evolves to a fully spin-split value of 0.5 × 2e2/h at finite B, allowed the authors to argue that the 0.7 structure is due to a spin-polarized state of the 1D system at B = 0 T, possibly caused by enhanced electron-electron interactions at low densities in the wire. Since then the Cavendish group have produced a series of papers studying the 0.7 structure in 1D wires with different configurations and disorder [10, 11, 12]. Particular attention was paid to the density dependence of the 0.7 structure. An additional gate between the two side gates or an overall back gate were used to vary the electron density in the channel, however the values of the 1D density were only estimated based on the directly measured 2D density. The results on the density dependence of the 0.7 structure are controversial. Firstly, it was shown by Thomas et al. [10] that the 0.7 structure moved to 0.5 with decreasing density. On contrary, Reilly (from UNSW) et al. [13] argued that the 0.7 structure moves to 0.5 with increasing density. Later, Pyshkin et al. [11] evidenced that the 0.7 structure remains at 0.7 × 2e2/h at intermediate densities and shifts to 0.5 with decreasing or increasing density. Additionally, studies in high magnetic fields at the crossing of the first two spin- split subbands, revealed the existence of an additional structure at 1.5 × 2e2/h (the 0.7 analogue) which has the same properties as the 0.7 structure. The results were explained by the spontaneous spin polarization that occurs when the two subbands with opposite spins cross. The Copenhagen group concentrated mainly on the temperature and d.c. source- drain bias dependence of the 0.7 structure [14]. They were able to perform measurements over a wide temperature range due to the very large 1D subband spacings (up to 20 meV) of their shallow etched QPCs. These structures allowed the authors to observe the 0.7 structure up to T = 20 K [15]. Their results showed that the temperature dependence of the 0.7 structure has an activated behaviour with a characteristic temperature Ta(VSG). Additionally, source-drain bias data revealed the evolution of 5. In-plane magnetic field studies of the 0.7 structure and the zero-bias anomaly in doped hole 1D wires 95 the 0.7 structure at zero source-drain bias voltage into a symmetrical feature at finite d.c. bias at G =0.85 × 2e2/h. By comparing the source-drain bias and the temperature measurements, the authors established the link between the 0.7 structure and the feature at finite bias at G =0.85 × 2e2/h. In order to reduce the effect of disorder on the 0.7 structure, Reilly et al. [13] investigated ultra-high mobility material without modulation doping. However, their wire is defined by chemical etching, which can potentially introduce disorder that is not present in Schottky gate devices [16]. The 1D structures were induced electrostatically and had lengths varying from 0 to 5 μm. The results showed that by increasing the length of 1D wire from 0 to 2 μm, the 0.7 structure shifts from its initial value to 0.5 × 2e2/h. In the case of short wires, the 0.7 structure moves to 0.5 × 2e2/h with increasing electron density in the wire. The electron density however is measured in the 2D regions of the samples and is not necessarily correspondent to the 1D density in the wire. The authors explained the length dependence of the 0.7 structure in terms of a critical length beyond which 1D system can have a spin-split ground state and ascribed the density dependence to the effect of 1D density on the confining potential. Later on, the authors proposed a model based on density dependent dynamic spin splitting, which describes a variety of experimental data [17]. Finally, shot noise experiments in the region of the 0.7 structure [18, 19] indicated the existence of two conduction channels.

5.2.1 Studies of the 0.7 structure and the zero-bias anomaly

The Harvard group [7] performed temperature and in-plane magnetic field measurements of the 0.7 structure. Their basic measurements of the 0.7 structure (temperature and in-plane magnetic field measurements) confirmed the well known signatures of the 0.7 structure:

1. The 0.7 structure is enhanced with increasing temperature.

2. The 0.7 structure shifts to the spin-resolved value of 0.5 × 2e2/h with the application of an in-plane magnetic ﬁeld.

Additionally, these authors performed source-drain measurements at different temperatures and magnetic fields. Figure 5.1a shows their source-drain bias data at high magnetic field (B = 8 T). The system is fully spin-polarized and the conductance exhibits half-plateaus, which evolve into quarter plateaus with the application of a 2 source-drain voltage VSD. The lowest spin-resolved half plateau at 0.5 × 2e /h shifts 2 to ∼ 0.8 × 2e /h at finite VSD (highlighted by dashed circle). On the other hand, Fig.

5.1b presents source-drain bias measurements at B =0T.AtVSD = 0 V there are 5. In-plane magnetic ﬁeld studies of the 0.7 structure and the zero-bias 96 anomaly in doped hole 1D wires

Figure 5.1: (a) Differential conductance of a QPC vs VSD measured for different VSG at T =80mKandB = 8 T. Dashed circle highlights the position of ∼ 0.8 × 2e2/h at finite VSD; (b) Differential conductance of a QPC vs VSD measured for different VSG at T =80mKandB = 0 T; (c) Differential conductance of a QPC vs VSD measured for different VSG at T = 600 mK and B = 0 T. The figures are adapted from the Ref. [7].

integer plateaus, which evolve to half plateaus with the application of VSD. Note that below the first integer plateau at VSD = 0 there there is no strong 0.7 structure but 2 at finite VSD there is still a plateau near 0.8 × 2e /h (highlighted by dashed circle), resembling the high field data when the plateau at 0.5 × 2e2/h is present. Figure 5.1b and 5.1c present a comparison of source-drain bias measurements at different temperatures. The low temperature data (Fig. 5.1b) exhibits an additional peak at zero-bias (highlighted by dashed rectangle) in a number of traces below 2e2/h. When this peak, known as the zero-bias anomaly (ZBA), is present, there is no 0.7 structure and the conductance is enhanced to the unitary limit (i.e. G =2e2/h). At higher temperatures (Fig. 5.1c) there is no ZBA (highlighted by dashed rectangule) and the conductance shows a clear 0.7 structure. Similar behaviour is observed in the magnetic field dependence of the ZBA: at B = 0 T, the 0.7 structure is weak and we see the ZBA, while at B = 8 T, the system is fully spin-polarized (half integer plateaus are present) and the ZBA is suppressed (highlighted by dashed rectangle). As was explained in Chapter 2, the enhancement of the conductance to the unitary limit at low temperatures and the presence of the ZBA in the density of states at the Fermi energy (which can be directly seen in source-drain bias measurements) are the distinct signatures of the Kondo effect in the quantum dots [20, 21]. To verify this hypothesis in 1D electron wires, Cronenwett et al. [7] showed the following features of 5. In-plane magnetic field studies of the 0.7 structure and the zero-bias anomaly in doped hole 1D wires 97 their data:

2 X The temperature dependence of conductance at diﬀerent VSG below 2e /h col- lapse onto a single curve - an empirical Kondo-like form using a single scaling

factor TK (the Kondo temperature). Note however that the Kondo equation shown in Chapter 2, was modiﬁed based on empirical considerations. The authors used the modiﬁed Kondo formula so that the high temperature limit of the conductance is e2/h. Therefore, the position of the 0.7 structure above e2/h is

due to the Kondo correlated state with high TK .

X Extracted Kondo temperatures correspond to the measured widths of the ZBA

(at diﬀerent VSG);

X The ZBA peak splits in an in-plane magnetic field. Note that the splitting is not uniform for different traces below the first plateau. The reason for that is the different Kondo temperature for different traces. When the measurement temperature is lower or higher than the Kondo temperature the splitting can not be seen. Therefore only few traces around 0.7 × 2e2/h exhibit splitting in magnetic field.

As was discussed in Chapter 2, the Kondo formalism considers a dynamic spin [7], which can be eﬀectively ﬂipped, in contradiction with the static spin polarization picture proposed in Ref. [5].

5.2.2 The 0.7 structure in hole systems

The first observation of the 0.7 structure in hole systems was made by Danneau et al. [22], and subsequently Rokhinson et al. [23] attempted the first study of the 0.7 structure in a 1D hole system using a magnetic focusing technique [24]. Rokhinson et el. fabricated two QPCs in parallel using the AFM local anodic oxidation (LAO) technique in an ultra shallow (311)A grown QW, designed such that at certain perpendicular magnetic field, the holes emerging from the injector QPC will focus in the detector QPC. In a system with spin-orbit interaction, carriers with the same energy but different spin states will acquire different momenta and therefore, in perpendicular magnetic field have different cyclotron radii. The authors used this technique to detect the spin-polarization in the region G<2e2/h. Prior to the focusing measurements, the authors characterized the 0.7 structure by performing temperature, source-drain bias and in-plane magnetic field measurements of the 1D conductance. The behaviour of 1D conductance in temperature and in-plane magnetic field reproduce the signatures of the 0.7 structure in electron systems [5]. 5. In-plane magnetic field studies of the 0.7 structure and the zero-bias 98 anomaly in doped hole 1D wires

Additionally, source-drain bias data exhibits the ZBA, which evolves with temperature and an in-plane magnetic field, in agreement with data in Ref. [7]. The first step in the magnetic focusing experiments involved setting the conductance of both injector and detector QPCs to 2e2/h. As the perpendicular magnetic field is swept, two additional peaks are observed in the magnetoresistance of the detector QPC, corresponding to two spin-split states of the holes emerging from the injector QPC. As one would expect for G =2e2/h, the amplitudes of the peaks were roughly the same. When the conductance of injector is reduced below 2e2/h, the relative height of the two peaks is observed to change: one of the peaks is enhanced while the other one is suppressed. These results were explained by a finite spin-polarization of injected holes, which is enhanced in samples where the 0.7 structure is present. From this Rokhinson et al. proposed that the origin of the 0.7 structure is due to a static spin-polarization around 0.7 × 2e2/h, which is incompatible with the Kondo picture of the 0.7 structure.

5.2.3 Theoretical results on the 0.7 structure

Many theories have been developed that explained particular properties of the 0.7 structure in electron systems, some of which we will review below. The magnetic field dependence of the 0.7 structure presumes a spin-polarized ground state at zero magnetic field. Unfortunately, this contradicts a Lieb-Mattis [25] theorem, which states that the ground state of an ideal 1D system can not be magnetized. However, many authors [26, 27, 28] argue that the 1D wires used in experiments are not ideal (e.g., finite dimensions, connections to the 2D reservoirs) which can result in violation of the theorem. The proposed mechanisms for the 0.7 structure, which implement the spin-polarized ground state, are ferromagnetic Wigner crystal- lization [28], spontaneous spin-polarization due to exchange interactions [26] and spin density waves [27]. A particular difficulty is the temperature dependence of the 0.7 structure. The fact that the 0.7 structure becomes more developed with increasing temperature indicates that it is not a ground state property. To overcome this, Starikov et al. [29] used a realistic model of a split gate 1D wire and found that the ground of the system is a magnetized state below a normal nonmagnetized state. As the temperature increases the magnetized state splits into a metastable state, which is still below the normal state. Another approach to explain the temperature dependence was developed by Seelig et al. [30], who proposed electron backscattering by acoustic phonons is the origin of the 0.7 structure. Following the observation of the zero bias anomaly by Cronenwett et al. [7], theo- 5. In-plane magnetic field studies of the 0.7 structure and the zero-bias anomaly in doped hole 1D wires 99 ries were developed justifying the formation of a dynamic spin inside or in the vicinity the 1D wire [31, 32, 33, 34]. A bound electron in the QPC acts as a scatterer for the incoming electrons with the same spin, hence reducing the conductance.As the temperature is lowered, the Kondo effect - screening of a localized spin by electrons in the leads, restores the conduction to 2e2/h. In contrast to previous theories Sushkov [35] performing Hartree-Fock calculations of the 1D wire conductance found both the spin-polarized ground state and the Kondo model to be improbable. Instead, the calculations suggest the existence of a few electron bound state with zero total spin. This result is similar to the theory of Flambaum et al. [36], which considers the formation of a two-electron bound state in the QPC. In this model, plateaus at 0.75×2e2/h and 0.25×2e2/h are the consequence of a single-triplet distribution. The authors note that bound states of several electrons are possible, but would result in plateaus at different fractions of 2e2/h. Finally, a recent theory [37] proposes a very simple origin for the 0.7 structure. When the wire is near the pinch-off, the Fermi level is close to the top barrier. Electrons effectively slow down near the top of the barrier, which enhances the electron-electron interactions just above transmission coefficient 0.5. This leads to a correction to the conductance at finite temperatures. This theory claims to agree with both magnetic field and temperature measurements.

5.3 Device structure and initial characterization

The device used for studies in this chapter is the same device (DQPC25) used in the previous chapter. The details on the device fabrication are contained in Chapter 3 and the device operation and electrical characterization are discussed in Chapter 4. The wafer T483 used for this device is a bilayer, symmetrically doped (311)A heterostructure consisting of two 20 nm quantum wells separated by a thick 30 nm barrier. The wire is formed by conventional EBL technique followed by metallization and oriented along the high mobility [233] direction. The active area of the device consists of two Schottky side gates (SG) separated by 100 nm from the 600 nm wide top gate (TG). There are also depletion gates (DG) on each side of the wire and the conducting substrate is used as an overall back gate (BG). The wire is oriented along the high mobility [233] crystallographic axis and designed to be 400 nm in length and width. By tuning the voltages on all four gates it is possible to observe conductance quantization in both wires simultaneously or just in one of the wires separately. The function of the gates are the following: 5. In-plane magnetic ﬁeld studies of the 0.7 structure and the zero-bias 100 anomaly in doped hole 1D wires

1. The side gates deﬁne the wires in both layers and subsequently pinch them oﬀ.

2. The middle gate is used to vary the hole density in the top wire.

3. The back gate controls the hole density in the bottom wire.

4. The depletion gate cuts the top layer between the wire and the end of the Hall bar, forcing the source-drain current to ﬂow through the bottom wire only.

All transport measurements in this chapter were performed using a standard low frequency lock-in technique in a dilution fridge with a base temperature of 20 mK. Two terminal measurements of the wire conductance were performed by applying a 20 μV a.c. voltage at f = 17 Hz to a source ohmic contact, and the resulting current through the drain ohmic contact was measured by the lock-in. Temperature measurements were done by heating the mixing chamber and providing enough time (typically 1 hour) for the sample to thermalize. On the initial cool down, the sample was placed in the dilution fridge insert in such a way that B was in-plane and parallel to the wire (B [233]).¯ After magnetotransport and source-drain bias measurements were obtained, the system was warmed up and the sample reoriented such that B was in- plane but perpendicular to the wire (B ⊥ [233])¯ and the measurements were repeated.

VMG = -0.5 V 2 /h) 2

G (2e 1

VMG = -0.15 V 0 3.2 3.4 3.6 3.8 4.0 4.2

VSG (V) 2 Figure 5.2: Conductance in units of 2e /h of the top QPC vs VSG for −0.5

In the previous chapter we showed that conductance quantization in the bottom 2 wire exhibits an additional feature below 2e /h at VMG =0.2 V and VBG =1V.

However, the conductance quantization in the top wire at high hole density (VMG = 5. In-plane magnetic ﬁeld studies of the 0.7 structure and the zero-bias anomaly in doped hole 1D wires 101

−0.5 V and VBG =2.5 V) did not show any additional structure. Knowing that the 0.7 structure depends significantly on the carrier density in the 1D wire (as well as the confinement potential) [5, 17], we have measured the conductance of the top wire at a constant value of VBG =2.5 V for different VMG, in order see if we can observe an 2 additional feature below 2e /h for some value of VMG. Figure 5.2 presents the conductance of the top wire corrected for a constant contact resistance for different values of VMG. As shown in Fig. 5.2 there are number of traces where a weak structure below the first plateau is visible. For further studies, we have chosen the trace at VMG = −0.225 V (blue bold trace), where the additional structure is the strongest. Figure 5.3 shows two traces of conductance quantization in the top and the bottom wires exhibiting an additional feature below 2e2/h. Note that the number of quantized steps in the top wire is smaller compared to the traces in the previous chapter due to reduced density in the wire (set by VMG).

VSG (V) 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

8 Bottom wire 6 /h) 2 4 Top wire G (2e 2

0 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 VSG (V)

2 Figure 5.3: Conductance in units of 2e /h of the QPC vs VSG. Red trace (top x axis) presents 1D conductance in the top layer - VBG =2.5V,VMG = −0.225 V, VDG =0 V. Blue trace (bottom x axis) shows 1D conductance in the bottom layer - VBG =1 V, VMG =0.2V,VDG =0.75 V.

In the following sections we show that an additional feature observed below 2e2/h (see Fig. 5.3) is the 0.7 structure. However, further in the chapter only data from the top wire be presented for two reasons. Firstly, the 0.7 structure in the bottom wire has a peak-like shape, resembling a resonance. In contrast, the structure in the top wire seems to be more plateau like. Secondly, the strong 0.7 structure in the bottom wire 5. In-plane magnetic ﬁeld studies of the 0.7 structure and the zero-bias 102 anomaly in doped hole 1D wires suppresses the zero bias anomaly in the source-drain bias measurements (as discussed earlier). Therefore to observe the 0.7 structure and the ZBA simultaneously, the 0.7 structure should be relatively weak.

5.4 Testing the 0.7 structure

Although the strength and the shape of the feature changes between the cool-downs, there are deﬁnitive tests which allow to distinguish the 0.7 structure from a resonance. As discussed earlier the two distinctive signatures of the 0.7 structure, which we will describe in the following sections:

1. The temperature dependence of the 0.7 structure.

2. The in-plane magnetic ﬁeld dependence of the 0.7 structure.

5.4.1 Temperature dependence of the 0.7 structure and the ZBA

Unlike resonances, which rapidly disappear with increasing temperature, the 0.7 structure remains at the highest measured temperatures even after the integer plateaus are completely thermally smeared and is enhanced with increasing temperature.

2.0 T = 20 mK T = 200 mK T = 320 mK 1.5 T = 550 mK /h)

2 T = 650 mK 1.0 G (2e 0.5

0.0 3.5 3.6 3.7 3.8 VSG (V)

Figure 5.4: (a) Conductance of the quantum wire G vs VSG for different temperatures T =20, 200, 320, 550, 650 mK. The traces are offset sequentially by −0.02 V, from right (20 mK, no offset) to left (650 mK, offset by −0.08 V).

Figure 5.4 shows the temperature dependance of the conductance quantization in the top wire. The traces are offset from the rightmost curve (T = 20 mK) in multiples of −0.02 V. The integer plateaus are smeared out with increasing temperature and at T ≈ 500 mK the second plateau at 2 × 2e2/h disappears. On the contrary, the 5. In-plane magnetic field studies of the 0.7 structure and the zero-bias anomaly in doped hole 1D wires 103 structure below the first plateau becomes significantly more pronounced, confirming that it is not due to a resonance. To complete the temperature characterization of the 0.7 structure, we performed source-drain bias measurements for different temperatures. Figure 5.5a shows source- drain bias measurements in the region below 2e2/h at T = 20 mK. The differential 2 conductance traces exhibit an anomalous peak at VSD = 0 mV below 2e /h, which rapidly diminishes with applied d.c. voltage (see for example the bottom bold trace). On the contrary, the high temperature data presented in Fig. 5.5b do not reveal any enhancement at zero source-drain voltage (see for example the bottom bold trace).

The peak in diﬀerential conductance at VSD = 0 mV (the ZBA) and its temperature dependence, are similar to the previous results in electron [7] and hole [23] systems and resemble the ZBA observed in quantum dots [20] due to the Kondo eﬀect.

1.0 1.0 /h) /h) 2 2 (2e g (2e g 0.5 0.5

T = 20 mK T = 320 mK 0.0 0.0 -0.5 0.0 0.5 -0.5 0.0 0.5 VSD (mV) VSD (mV) (a) (b)

Figure 5.5: Diﬀerential conductance of the quantum wire g vs VSD for diﬀerent values of VSG at (a) T = 20 mK, (b) T = 320 mK.

Before discussing measurements in in-plane magnetic fields, we should mention the behaviour of the 0.7 structure in a magnetic field B perpendicular to the 2D plane. Applying a small magnetic field B perpendicular to the 2D plane (of order of 100 mT) is a common way of masking/eliminating the resonances in the 1D systems [7]. In contrast, the 0.7 structure remains mostly unaffected by the small perpendicular magnetic field. Therefore to ensure that the observed 0.7 structure is not due to a resonance measurements in perpendicular magnetic field up to 0.7 T were performed, showing that the structure below 2e2/h survives and does not change significantly [38]. 5. In-plane magnetic field studies of the 0.7 structure and the zero-bias 104 anomaly in doped hole 1D wires

5.4.2 In-plane magnetic ﬁeld measurements

The smooth evolution of the 0.7 structure from 0.7 × 2e2/h to 0.5 × 2e2/h with application of an in-plane magnetic field is a characteristic signature of the 0.7 structure. 1D hole systems offer a powerful tool of studying the origins of this structure. As was shown previously [9] and discussed in Chapter 3, the effective g-factor of the 1D subbands is highly anisotropic depending on the orientation of the in-plane magnetic field. If the 0.7 structure and the ZBA relate to spin, then because of the 1D g- factor anisotropy, the evolution of the 0.7 structure and the ZBA should also be highly anisotropic depending on the orientation of an in-plane magnetic field. Figure 5.6a shows conductance traces for the top wire with the application of a magnetic field parallel to the wire direction. Below the first plateau, the relatively weak 0.7 structure at B = 0 T becomes more evident with application of the magnetic field, smoothly shifting to 0.5 × 2e2/h at B =3.6 T, and then remaining unaltered up to 9 T. This behaviour is similar to the behaviour of the 0.7 structure in electron systems [5]. In order to see the evolution of the integer plateaus and the 0.7 structure more clearly, Fig. 5.6b shows a colour map of the transconductance dG/dVSG vs conductance G and magnetic field B. The transconductance has been obtained by numerical differentiation of the conductance data presented on Fig. 5.6a. High transconductance (corresponding to the risers in conductance) is displayed as a red colour, and small transconductance (plateaus in conductance) appear blue. From Fig. 5.6a and 5.6b it can be seen that a half-plateau at 1.5 × 2e2/h starts to develop around B =1.8 T. Increasing the field above ≈ 6 T causes the half-plateau to evolve back to the integer value. Unlike the electron systems [7], much smaller fields B are required to achieve the full spin polarization. This is due to the difference in ∗ mh the subband spacing, reflecting the ratio of the effective masses ∗ ≈ 5. me We also observe a strong correlation between the behaviour of the 0.7 structure and the ZBA in magnetic field. Source-drain bias measurements have been performed at B = 0 T (see Fig. 5.7a) and at B =3.6 T (see Fig. 5.7b). At zero field, the 0.7 structure is weak and the ZBA is present. At high field (B =3.6 T), the 0.7 structure evolves into a spin polarized plateau at 0.5×2e2/h and the ZBA is completely suppressed. This result is in agreement with that previously reported in electron [7] and hole systems [23]. After thermally cycling to room temperature, the sample was reoriented and the same data was taken for the in-plane magnetic field perpendicular to the wire. Con- ductance traces in Fig. 5.8a show that the integer plateaus at 1 × 2e2/h and 2 × 2e2/h do not evolve with magnetic field and that no spin splitting is visible up to B =10 5. In-plane magnetic field studies of the 0.7 structure and the zero-bias anomaly in doped hole 1D wires 105

2 B|| = 9 T /h) 2 1 G (2e B|| = 0 T

0 3.5 4.0 4.5 5.0 5.5 6.0

VSG (V) (a)

2.0

1.5 /h) 2 1.0 G (2e

0.5

0.0 0 2 4 6 8 B|| (T) (b)

Figure 5.6: (a) Conductance of the top wire G vs VSG for different magnetic fields B ranging from0Tto9Tinsteps of 0.2 T. For clarity the traces are offset in multiples of 0.05 V, from left (0 T, no offset) to right (9 T, offset by 2.25 V). The bold trace corresponds to the conductance at B =3.6 T , when the 1D subbands are fully spin-polarized and the 0.7 structure reaches 0.5 × 2e2/h; (b) Transconductance of the top wire dG/dVSG vs G and B coordinates. Blue colour corresponds to the plateaus in conductance whereas red colour corresponds to the risers between conductance plateaus. 5. In-plane magnetic field studies of the 0.7 structure and the zero-bias 106 anomaly in doped hole 1D wires

1.0 1.0 /h) /h) 2 2 (2e g 0.5 g (2e 0.5 T = 20 mK T = 20 mK B|| = 3.6 T B = 0 T 0.0 0.0 -0.5 0.0 0.5 -0.5 0.0 0.5 VSD (mV) VSD (mV) (a) (b)

Figure 5.7: Diﬀerential conductance of the quantum wire g vs VSD for diﬀerent values of VSG at (a) B = 0 T (same as Fig. 5.5a), (b) B =3.6T.

T. However, below the first plateau the situation is different. A weak 0.7 structure at B = 0 T remains unchanged up to ≈ 5 T and and then shifts smoothly towards 0.5 × 2e2/h at B>9 T. The structure remains weak and therefore its evolution is more clearly seen on the transconductance colour map shown in Fig. 5.8b. Note that there is an additional feature splitting off the first plateau at B>7T, the origin of this feature is unclear. It is very unlikely that this additional plateau is due to the Zeeman splitting of the first plateau, because the double degenerate plateau at 2e2/h cannot split into three plateaus. By performing temperature and magnetic field measurements, we have proved that the 0.7 structure at B = 0 T is not a resonance. Moreover the simultaneous observation of the 0.7 structure and the ZBA at B = 0 T reinforces our statement that the observed 0.7 structure is not a resonance. Additionally, the data in the previous chapter did not exhibit an additional feature at high fields for the same orientation of magnetic field. Note also, that a second repetitive cool-down had been set up to verify the data and the same results were obtained.

Figures 5.9a and 5.9b present source-drain bias measurements at different B⊥. First, Fig. 5.9a shows data at B⊥ =3.6 T to compare with that previously shown 2 in Fig. 5.7b. There, for B =3.6 T, the 0.7 structure reaches 0.5 × 2e /h and the ZBA is completely suppressed. In contrast, for B⊥ =3.6 T, the 0.7 structure has not moved noticeably from its zero field position and the ZBA is still strong (see Fig. 5.9a). By increasing the field further to B = 10 T (see Fig. 5.9b), the 0.7 structure 5. In-plane magnetic field studies of the 0.7 structure and the zero-bias anomaly in doped hole 1D wires 107

2 B⊥ = 10 T /h) 2 1 G (2e

B⊥ = 0 T 0 3.5 4.0 4.5 5.0 5.5 6.0 VSG (V) (a)

2.0

1.5 /h) 2 1.0 G (2e

0.5

0.0 0 2 4 6 8 10 B⊥ (T) (b)

Figure 5.8: (a) Conductance of the top wire G vs VSG for different magnetic fields B⊥ ranging from0Tto10Tinsteps of 0.2 T. For clarity the traces are offset in multiples of 0.05 V, from left (0 T, no offset) to right (10 T, offset by 2.5 V); (b) Transconductance dG/dVSG of the top wire vs G and B coordinates. 5. In-plane magnetic field studies of the 0.7 structure and the zero-bias 108 anomaly in doped hole 1D wires shifts towards its fully spin-polarized position of 0.5 × 2e2/h and the ZBA is strongly suppressed.

1.0 1.0 /h) /h) 2 2 (2e g (2e 0.5 g 0.5

T = 20 mK T = 20 mK B⊥ = 3.6 T B⊥ = 10 T 0.0 0.0 -0.5 0.0 0.5 -0.5 0.0 0.5 VSD (mV) VSD (mV) (a) (b)

Figure 5.9: Diﬀerential conductance of the quantum wire G vs VSD for diﬀerent values of VSG at (a) B⊥ =3.6 T, (b) B⊥ =10T.

5.5 Discussion of the results

There are three main results that are evident from the data presented in the previous section:

1. The evolution of the integer subbands in an in-plane magnetic ﬁeld is highly anisotropic depending on the orientation of the ﬁeld, either parallel or perpendicular to the wire, which is aligned along the [233] direction. This result is similar to the results from Chapter 4, where we showed that the g-factor of the g integer subbands exhibits an anisotropy with a ratio > 4.5. This anisotropy g⊥ exceeds the theoretical and experimental value found in 2D hole systems [39, 40].

2. The main result of this chapter is that the anisotropic behaviour is also evident in the evolution of the 0.7 structure with application of in-plane parallel and perpendicular fields. This anisotropy correlates with the anisotropy of the g- factors of the integer subbands, giving direct and unambiguous evidence that the 0.7 structure is related to spin. Moreover, the ZBA observed in the source- drain bias measurements has the same behaviour in magnetic field as the 0.7 5. In-plane magnetic field studies of the 0.7 structure and the zero-bias anomaly in doped hole 1D wires 109

structure, conﬁrming a direct connection between them and reinforcing their relation to spin.

3. The last result is that the anisotropy of the 0.7 structure is smaller than that of the integer subbands. In the in-plane magnetic field parallel to the wire, both the integer subbands and the 0.7 structure quickly evolve into half-plateaus by B =3.6 T. In contrast, when the in-plane field is perpendicular to the wire direction, the integer subbands remain unchanged, but the 0.7 structure clearly shifts to the 0.5 × 2e2/h value at B>9 T. This can indicate the enhanced g- factor around 0.7 × 2e2/h in a perpendicular magnetic field compared to integer g subbands. This results in a effective g-factor ratio < 4. g⊥

To explain the reduced anisotropy of the g-factors around 0.7 × 2e2/h we refer to recent theoretical calculations by Zülicke [41], who considered the qualitive evolution of the 2D subbands into 1D subbands as a 1D hole system is narrowed. The confinement is characterized by the ratio of Wx/Wy, where Wx is the width of confinement in the direction orthogonal to 2D plane and Wy is the width of the wire in the 2D plane. These calculations are summarized in Chapter 2. Zülicke showed that by narrowing the 1D constriction from the 2D to 1D system, the quasi 1D levels, which originate from different 2D levels, cross at certain values of the ratio Wx/Wy. Moreover, the 2D

HH-like ground state (Wx/Wy = 0) becomes a HH-LH mixture as the wire narrows, and ﬁnally in the 1D limit (Wx/Wy = 1), the ground state of the system is purely LH-like. Fig. 5.10 shows g-factors calculated for a 20 nm quantum well grown on (311) GaAs substrate. The peculiar evolution of the ground state of the system causes the eﬀective g-factors to evolve as a function of Wx/Wy. In the 2D limit (Wx/Wy = 0), g[233] >g[011] and g[233]/g[011] ≈ 0.6/0.2 = 3; as Wx/Wy increases, g[233] remains almost constant while g[011] increases noticeably. At Wx/Wy ≈ 0.4 there is a crossover between g[233] and g[011], and after that g[011] >g[233].

It is possible to estimate the maximum achievable Wx/Wy in our device, knowing the quantum well width (Wx = 20 nm) and the width of wire (Wy), when the conduc- 2 2 tance is below 2e /h (0.7 × 2e /h). Wy can be approximated as a half of the Fermi wave length (λF ), which is directly related to the 2D hole density. Note however, that as the 2D system is conﬁned to 1D, the 1D density will be smaller than the 2D value.

Experimentally it was shown [42] that λF in 1D is approximately four times larger than in 2D. In our case the 2D λF ≈ 70 nm, thus the 1D λF /2 will be ≈ 140 nm and the maximum Wx/Wy =20/140 ≈ 0.14. Hence in our experiment we can access the range of Wx/Wy between 0 and 0.14 which is well before the predicted crossover in Ref. [41]. 5. In-plane magnetic ﬁeld studies of the 0.7 structure and the zero-bias 110 anomaly in doped hole 1D wires

As can be seen from Fig. 5.10, by changing Wx/Wy from 0 (2D case) to 0.14, g[233] g[233] does not change signiﬁcantly, but g[011] does increase. Therefore the ratio g[011] decreases when the 1D wire narrows. This is in very good qualitative agreement with our experimental results, which show no spin splitting for the integer subbands

(Wx/Wy closer to zero) in B⊥ but ﬁnite visible splitting of the 0.7 structure.

Figure 5.10: Calculated g-factor for B [233] (dotted line) and B [011] (dashed line) vs Wx/Wy ratio. The shaded area shows the range 0

Note however, that the theory [41] is developed for hard-wall confinement potential. In our samples, the confinement is produced by electrostatic depletion of the 2D square quantum well using a surface gate technique. The electrostatic confinement is often described by a parabolic potential [43]. As was pointed out by Zülicke [41], 1D confinement plays a crucial role in the HH-LH subband splitting. Therefore, the presented theory can only be used for qualitative comparison. The effect of the confinement on transport characteristics of the 1D hole wires is discussed in more detail in Chapter 6.

5.6 Conclusions and future work

In this chapter we presented a study of the 0.7 structure and the ZBA in 1D hole system with application of in-plane magnetic fields parallel and perpendicular to the wire. When B is parallel to the wire, integer subbands as well as the 0.7 structure 5. In-plane magnetic field studies of the 0.7 structure and the zero-bias anomaly in doped hole 1D wires 111 evolve to half-plateaus by B =3.6 T. In contrast, when B is perpendicular to the wire, the integer plateaus remain unchanged up to the maximum measured field of 10 T. 2 However, the 0.7 structure evolves to 0.5×2e /h at B⊥ > 9 T, which indicates reduced g-factor anisotropy below 2e2/h compared to the integer subbands. The smaller ratio g < 4 around 0.7×2e2/h is in agreement with theoretical calculations, which consider g⊥ a crossing between g[233] and g[011] as the wire narrows [41]. The anisotropic behaviour of the 0.7 structure and the ZBA exhibit the same trend as the integer subbands. These results show directly and unambiguously the connection between the 0.7 structure and ZBA and their relation to spin. Therefore the theories which do not consider spin (e.g. Ref. [30]) appear to be improbable. Additionally, in Ref. [7], the Kondo mechanism was contrasted with the static spin polarization. On contrary to Ref. [7], our data shows that the ZBA is related to spin splitting. Moreover, the Kondo effect in hole systems has not been investigated and therefore it is not clear if the Kondo mechanism will be the same as in electron systems.

In future work, it would be interesting to study 1D hole wires with larger Wx/Wy. This would allow a study of the evolution of the g-factor anisotropy as the wire narrows from the 2D to 1D limit. Bibliography

[1] B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, D. van der Marel and C. T. Foxon, Quantized conductance of point contacts in a two-dimensional electron gas, Phys. Rev. Lett., 60(9), 848 (1988).

[2] D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A. Ritchie and G. A. C. Jones, One- dimensional transport and the quantization of the ballistic resistance, J. Phys. C: Solid State Physics, 21, L209 (1988).

[3] R. Landauer, Electrical resistance of disordered one-dimensional lattices, IBM J. Res. Dev., 1, 223 (1957).

[4] M. Buttiker, Symmetry of electrical conduction, IBM J. Res. Dev., 32, 317 (1988).

[5] K. J. Thomas, J. T. Nicholls, M. Y. Simmons, M. Pepper, D. R. Mace and D. A. Ritchie, Possible spin polarization in a one-dimensional electron gas, Phys. Rev. Lett., 77(1), 135 (1996).

[6] H. van Houten and C. Beenakker, Quantum point contacts, Physics Today, (7), 22 (1996).

[7] S. M. Cronenwett, H. J. Lynch, D. Goldhaber-Gordon, L. P. Kouwenhoven, C. M. Marcus, K. Hirose, N. S. Wingreen and V. Umansky, Low-temperature fate of the 0.7 structure in a point contact: a Kondo-like correlated state in an open system, Phys. Rev. Lett, 88(22), 226805 (2002).

[8] K. J. Thomas, J. T. Nicholls, N. J. Appleyard, M. Y. Simmons, M. Pepper, D. R. Mace, W. R. Tribe and D. A. Ritchie, Interaction eﬀects in a one-dimensional constriction, Phys. Rev. B, 58(8), 4846 (1998).

[9] R. Danneau, O. Klochan, W. R. Clarke, L. H. Ho, A. P. Micolich, M. Y. Simmons, A. R. Hamilton, M. Pepper, D. A. Ritchie and U. Z¨ulicke, Zeeman splitting in ballistic hole quantum wires, Phys. Rev. Lett., 97, 026403 (2006).

112 BIBLIOGRAPHY 113

[10] K. J. Thomas, J. T. Nicholls, M. Pepper, W. R. Tribe, M. Y. Simmons and D. A. Ritchie, Spin properties of low-density one-dimensional wires, Phys. Rev. B, 61(20), R13365 (2000).

[11] K. S. Pyshkin, C. J. B. Ford, R. H. Harrell, M. Pepper, E. H. Linﬁeld and D. A. Ritchie, Spin splitting of one-dimensional subbands in high quality quantum wires at zero magnetic ﬁeld, Phys. Rev. B, 62(23), 15842 (2000).

[12] C.-T. Liang, M. Pepper, M. Y. Simmons, C. G. Smith and D. A. Ritchie, Spin- dependent transport in a quasiballistic quantum wire, Phys. Rev. B, 61(15), 9952 (2000).

[13] D. J. Reilly, G. R. Facer, A. S. Dzurak, B. E. Kane, R. G. Clark, P. J. Stiles, A. R. Hamilton, J. L. O’Brien, N. E. Lumpkin, L. N. Pfeiﬀer and K. W. West, Many-body spin related phenomena in ultra low-disorder quantum wires, Phys. Rev. B, 63, 121311(R) (2001).

[14] A. Kristensen, H. Bruus, A. E. Hensen, J. B. Jensen, P. E. Lindelof, C. J. Mar- ckmann, J. Nyg˚ard,C. B. Sørensen, F. Beuscher, A. Forchel and M. Michel, Bias and temperature dependence of the 0.7 conductance anomaly in quantum point contacts, Phys. Rev. B, 62(16), 10950 (2000).

[15] A. Kristensen, J. B. Jensen, M. Zaﬀalon, C. B. Sørensen, S. M. Reimann, P. E. Lindelof, M. Michel and A. Forchel, Conductance quantization above 30 K in GaAlAs shallow quantum point contacts smoothly joined to the background 2DEG, Phys. Rev, 83(1), 607 (1998).

[16] D. Csontos and H. Q. Xu, Eﬀects of boundary roughness on the conductance of quantum wires, Appl. Phys. Lett., 77(15), 2364 (2000).

[17] D. J. Reilly, T. M. Buehler, J. L. O’Brien, A. R. Hamilton, A. S. Dzurak, R. G. Clark, B. E. Kane and L. N. Pfeiﬀer, Density-dependent spin polarization in ultra- low disorder quantum wires, Phys. Rev. Lett., 89(24), 246801 (2002).

[18] P. Roche, J. S´egala,D. C. Glattli, J. Nicholls, M. Pepper, A. C. Graham, K. J. Thomas, M. Simmons and D. A. Ritchie, Fano factor reduction on the 0.7 conductance structure of a ballistic one-dimensional wire, Phys. Rev. Lett., 93(11), 116602 (2004).

[19] L. DiCarlo, Y. Zhang, D. T. McClure, D. J. Reilly, C. M. Marcus, L. N. Pfeiﬀer and K. W. West, Shot-noise signatures of 0.7 structure and spin in a quantum point contact, Phys. Rev. Lett., 97(3), 036810 (2006). 114 BIBLIOGRAPHY

[20] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav and M. A. Kastner, Kondo eﬀect in a single-electron transistor, Nature, 391(1), 156 (1998).

[21] S. M. Cronenwett, T. H. Oosterkamp and L. P. Kouwenhoven, A tunable Kondo eﬀect in quantum dots, Science, 281, 540 (1998).

[22] R. Danneau, W. R. Clarke, O. Klochan, A. P. Micolich, A. R. Hamilton, M. Y. Simmons, M. Pepper and D. A. Ritchie, Conductance quantization and the 0.7 × 2e2/h conductance anomaly in one-dimensional hole systems, Appl. Phys. Lett., 88, 012107 (2006).

[23] L. P. Rokhinson, L. N. Pfeiﬀer and K. West, Spontaneous spin polarization in quantum point contacts, Phys. Rev. Lett., 96, 156602 (2006).

[24] L. P. Rokhinson, V. Larkina, Y. B. Lyanda-Geller, L. N. Pfeiﬀer and K. W. West, Spin separation in cyclotron motion, Phys. Rev. Lett., 93(14), 146601 (2004).

[25] E. Lieb and D. Mattis, Theory of ferromagnetism and ordering of electronic energy levels, Phys. Rev, 125(1), 164 (1962).

[26] C. K. Wang and K. F. Berggren, Local spin polarization in ballistic quantum point contacts, Phys. Rev. B, 57(8), 4552 (1998).

[27] S. M. Reimann, M. Koskinen and M. Manninen, End states due to a spin-Peierls transition in quantum wires, Phys. Rev. B, 59(3), 1613 (1999).

[28] B. Spivak and F. Zhou, Ferromagnetic correlations in quasi-one-dimensional conducting channels, Phys. Rev. B, 61(24), 16730 (2000).

[29] A. A. Starikov, I. I. Yakimenko and K. F. Berggren, Scenario for the 0.7- conductance anomaly in quantum point contacts, Phys. Rev. B, 67(23), 235319 (2003).

[30] G. Seelig and K. A. Matveev, Electron-phonon scattering in quantum point contacts, Phys. Rev. Lett., 90(17), 176804 (2003).

[31] Y. Meir, K. Hirose and N. S. Wingreen, Kondo model for the “0.7 Anomaly” in transport through a quantum point contact, Phys. Rev. Lett., 89(19), 196802 (2002).

[32] K. Hirose, Y. Meir and N. S. Wingreen, Local moment formation in quantum point contacts, Phys. Rev. Lett., 90(2), 026804 (2003). BIBLIOGRAPHY 115

[33] T. Rejec, A. Ramˇsakand J. H. Jeﬀerson, Spin-dependent resonances in the conduction edge of the quantum wires, Phys. Rev. B, 62(19), 12985 (2000).

[34] T. Rejec and Y. Meir, Magnetic impurity formation in quantum point contacts, Nature, 442, 900 (2006).

[35] O. P. Sushkov, Restricted and unresticted Hartree-Fock calculations of conductance for a quantum point contact, Phys. Rev. B, 67(19), 195318 (2003).

[36] V. V. Flambaum and M. Y. Kuchiev, Possible mechanism of the fractional- conductance quantization in a one-dimensional constriction, Phys. Rev. B, 61(12), R7869 (2000).

[37] C. Sloggett, A. I. Milstein and O. P. Sushkov, Correlated electron current and temperature dependence of the conductance of a quantum point contact, arXiv:cond- mat/0606649v2.

[38] W. R. Clarke, Quantum interaction phenomena in p-GaAs microelectronic devices, Ph.D. thesis, School of Physics, University of New South Wales, Australia (2005).

[39] R. Winkler, S. J. Papadakis, E. P. de Poortere and M. Shayegan, Highly anisotropic g-factor of two-dimensional hole systems, Phys. Rev. Lett., 85(21), 4574 (2000).

[40] S. J. Papadakis, E. P. de Poortere, M. Shayegan and R. Winkler, Anisotropic magnetoresistance of two-dimensional holes in GaAs, Phys. Rev. Lett., 84(24), 5592 (2000).

[41] U. Z¨ulicke, Electronic and spin properties of hole point contacts (2007), to be published in Phys. Stat. Sol.

[42] R. Crook, C. G. Smith, M. Y. Simmons and D. A. Ritchie, One-dimensional probability density observed using scanned gate microscopy, J. Phys.: Condens. Matter, 12, L735 (2000).

[43] M. B¨uttiker, Quantized transmission of a saddle-point constriction, Phys. Rev. B, 41(11), 7906 (1989). Chapter 6

Conductance quantization in induced GaAs/AlGaAs hole 1D wires

Chapter Outline This chapter presents the characterization of a new type of 1D hole wire using a GaAs/AlGaAs SISFET structure (i.e. without modulation doping). The device exhibits clear conductance quantization, including the 0.7 structure, and has extremely stable gate characteristics. The hole density in devices of this type can be varied over an order of magnitude, while the mobility of holes remains high enough to observe conductance quantization. We present initial measurements of the dependence of the 1D conductance on density, temperature, source-drain bias and magnetic ﬁeld.

6.1 Introduction

A common feature of all 1D hole devices studied so far is the presence of a modulation doping layer in the heterostructure [1]. However, as was described in Chapter 2, in modulation doped structures, carriers in the 2D system are scattered by random potential fluctuations due to ionized dopants in the modulation doped layer. In contrast, in the SISFET structures without modulation doping, remote ionized impurity scattering is eliminated. The cap layer is heavily doped that it is effectively metallic and this means there are sufficient carriers in this layer to screen the 2D system from the random potential fluctuations from the ionized dopants in the cap layer. Moreover, in modulation doped 1D hole systems [2, 3], the doping layer is believed to be the cause of gate instabilities. Therefore the development of undoped 1D hole wires is essential to overcome these instabilities. The absence of the modulation doping layer will also

116 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires 117 be advantageous for one-dimensional devices in the following ways: a) the conﬁnement potential in the device is more uniform, which makes the observation of ballistic transport in long wires much easier than in modulation doped devices [4]; b) the mobility of carriers can be much higher for induced devices at low densities, where the carrier-carrier interaction eﬀects are strongest [5].

Furthermore, using holes rather than electrons will enable studies of carrier-carrier interaction phenomena in limit of high interaction parameter rs. This is because the ∗ interaction parameter rs is proportional to the eﬀective mass of a charge carrier m :

EP a ∗ rs = = ∝ m (6.1) EK aB where EP and EK is potential and kinetic energies of the system, a is the carrier- carrier distance, aB - effective Bohr radius. The most commonly used value of the ∗ holes effective mass in GaAs is m =0.38 · m0, where m0 is the mass of a free electron in a vacuum [6, 7, 8]. It is approximately five times larger than the effective mass for ∗ electrons in GaAs (m =0.067 · m0). Thus at the same carrier density, the rs for holes is five times larger than for electrons, allowing rs ≥ 30. Hence, the goal of the studies in this chapter is to develop and characterize 1D hole devices without modulation doping.

6.2 Literature overview

In this section we will review previous work done on 1D electron wires fabricated using SISFET structures without modulation doping concentrating on fabrication and the range of accessible electron densities and mobilities. The elimination of remote ionized impurity scattering in SISFET structures allowed ballistic transport to be studied in a wide density range, particularly at low densities. A special interest was paid to the 0.7, structure which is believed to be a many-body phenomenon [9]. Kane [12] et al. fabricated long 1D wires using a heterostructure shown in Fig. 10 −2 11 6.1a. The range of accessible density was very wide 1 × 10 cm

(a) n+ SISFET (b) ISIS (c) MISFET

Figure 6.1: (a) Heterostructure wafer layout used by (a) Kane et al. [10], (b) Hirayama [11], (c) Harrell et al. [5]. “G1” and “G2” label the gates and “I” denotes the insulator layer. the 1D conductance. The 0.7 structure was also observed along with its satellite at 1.7 × e2/h. These studies have been followed by a series of papers that used the same type of devices to explore the 0.7 structure [13, 14] in ultra-low disorder systems. Hirayama et al. [11] used a specially designed ISIS heterostructure, in which a heavily doped back gate was used to induce electrons (Fig. 6.1b). The range of 10 −2 11 −2 accessible density was 1 × 10 cm

6.3 Device structure and the principles of its operation

This section presents the fabrication and characterization of the ﬁrst induced 1D hole system. The device used for studies in this chapter was fabricated by Warrick Clarke and characterized by the author. The heterostructure R165 used for fabrication of this device was kindly provided by Koji Muraki from NTT Basic Research Labs (Japan). The schematic layout of the wafer is presented on Fig. 6.2a. The GaAs/AlGaAs heterojunction is grown on a (311) GaAs substrate and consists of a 1.5 μm GaAs layer, a 175nm AlGaAs layer and a 25nm GaAs layer, all of which are undoped. The ﬁnal layer in the structure is a 75 nm heavily doped (5×1018 cm−3 Si) p+-GaAs layer, which is used as a metallic gate.

Figure 6.2: (a) Schematic diagram of the heterostructure used for studies in this chapter; (b) SEM micrograph of the central region of the device. S and D denote source and drain ohmic contacts, SG and TG stand for “side gate” and “top gate”. The darker regions are a 75 nm deep trench etched to electrically separate SG and TG.

The use of a p+ gate [18] as opposed to an n+ gate [10] allows the accumulation of holes at much lower gate bias due to the Fermi level pinning, thereby reducing the chance of current leakage between the gate and source/drain contacts. These low threshold biases allow the deposition of metallization gates directly onto the mesa 120 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires without the dielectric layer between them, simplifying the processing procedure for these devices and allowing greater yield. Electrical contact to the 2DHS was achieved using annealed AuBe ohmic contacts, which were deposited using the modiﬁed self- aligned process [18]. The details of the processing routine can be found in Chapter 3.

In order to induce holes in the 2DHS, the top layer should be negatively biased to a voltage below the threshold (typically ∼−0.1 V). The 2DHS is induced directly beneath the top gate, reproducing the shape of the gate. This property allows fabrication of the fine scale features by patterning the top gate. We have used electron beam lithography and wet etching to pattern and etch the p+-GaAs cap (etch depth ∼ 75 nm) into three separate, independently biasable gates - two side gates (SG) and a top gate (TG) as shown in the SEM micrograph in Fig. 6.2b. The wire is oriented along the high mobility [233] direction. The TG has a ‘bow-tie‘ shape, and when the TG is negatively biased, it induces two large 2DHS regions connected by a narrow 1D channel. For induced devices the number of occupied 1D subbands in the wire is controlled by VTG and the etched width of the channel at VSG = 0 V. For a given VTG a positive VSG can then be applied to squeeze the 1D channel, reducing the number of occupied subbands and eventually, pinching off the channel. On contrary, in modulation doped devices, the 1D system at VSG = 0 V is not defined, and therefore should be first defined and then squeezed by the side gates.

The hole density p (controlled by VTG) is determined by measuring the Hall slope in the 2D reservoirs adjacent to the wire. However it should be mentioned, that this density is that of the 2D region and the actual 1D density is lower when a positive side gate voltage is applied. Nonetheless the 2D density can still be used as a good estimate value for the density in the 1D channel at zero side gate voltage. The top gate for the presented device operates over the bias range from the threshold voltage of VTG =

−0.22 V to VTG = −0.41 V where the top gate starts to leak to the ohmic contacts (current leakage ≥ 1 nA). This top gate range corresponds to the hole density changing from 5.84 × 1010 cm−2 to 1.38 × 1011 cm−2. This particular device has signiﬁcantly greater threshold voltage than previously measured 2D systems (VTG = −0.12 V; p =1.64 × 1010cm−2) [18], which limits the lowest density attainable. Additionally, for ballistic transport studies we need to ensure that the mobility of the holes at low densities is suﬃciently high to observe conductance quantization. The hole mobility in the presented device ranges from μ =5.06 × 105 cm2/Vs for p =3.3 × 1010 cm−2 to μ =7.1 × 105 cm2/Vs for p =1.35 × 1011 cm−2 providing the possibility of observing ballistic transport in the low density limit. 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires 121

6.4 Contact resistance and conductance quantization of the wire

In this section we discuss the eﬀect of the ohmic contact resistance on the measurements of the 1D wire conductance. We observe that the ohmic contacts in the device used in this study has a very big ohmic contact resistance and moreover, the ohmics show nonlinear behaviour. Hence we discuss the methods we used to determine the ohmic contact resistance and how we account for it in the measurements of the 1D wire conductance.

10 (a)

S) 10 μ ( 5 9 uncorr g 8 0.0 0.4 0.8 1.2 0 Ω 6 RC=97.5 k (b) R =96.5 kΩ 5 C Ω RC=95.5 k /h)

2 4 VTG=-0.36V 3 g (2e 2 1 0 0.2 0.4 0.6 0.8 1.0 1.2 VSG (V)

Figure 6.3: (a) Raw conductance guncorr of 1D quantum wire vs side gate voltage; (b) Conductance of 1D quantum wire corrected for diﬀerent values of constant contact resistance as a function of VSG.

The conductance through the 1D wire is typically a two-terminal measurement and it is necessary to subtract the contact resistance of the ohmic contacts and measurement circuitry in order to obtain quantized conductance of the 1D wire. Figure

6.3a shows a typical example of a SG voltage sweep at set value of VTG = −0.36 V 10 −2 (p =1.18 × 10 cm ). Applying a positive bias VSG to the side gates causes the 1D channel to narrow, reducing the conductance as shown in Fig. 6.3a. The conductance ultimately drops to zero (i.e., the 1D channel pinches oﬀ ) at VSG ∼ 1.2 V. Conduc- tance plateaus are evident in the data (see the inset in Fig. 6.3a, conﬁrming that the device operates as a 1D ballistic hole system. Note that at VSG = 0 V in Fig. 6.3a, the conductance is ∼ 10 μS, much less than 2e2/h (77 μS). This low conductance is due to 122 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires

the high resistance RC of the ohmic contacts added in series to quantized resistance of the quantum wire in the measurements. The common way to convert the raw conductance into conductance in units of 2e2/h in doped systems is to subtract the resistance of 2D region and the measurement circuitry R(VSG = 0 V). However for the induced systems, a 1D wire is already formed at VSG = 0 and certain number of 1D subbands is occupied, hence RC = R(VSG = 0), but this still can be used as a good estimate because RC R1Dwire. Alternatively, it 2 is possible to find the value of RC that aligns the first plateau to 1×2e /h. Subtracting the RC found in this way should also automatically align the higher plateaus. Figure 6.3b presents three traces of corrected conductance where we tried both methods of conversion. The top red curve corresponds to the value of resistance RC = R(VSG =0 V) = 97.5 kΩ. The first plateau is well aligned and needs only a small adjustment, but at the same time the second plateau appears at 3 × 2e2/h. The green middle curve corresponds to RC =96.5 kΩ which is approximately 1% smaller than the initial resistance R(VSG = 0). This small change lowers the first plateau making it exactly quantized, but the second plateau moves from 3 × 2e2/h to 2.5 × 2e2/h. Decreasing the RC further (blue bottom trace), it is possible to align the second plateau to its value of 2 × 2e2/h but now the first plateau will be below 1 × 2e2/h. An even more pronounced “misalignment” effect is visible for the third plateau.

The fact that it is not possible to align the plateaus with the constant value of RC , and its high value, can potentially mean that the ohmic contact resistance depends on the current through the wire and can actually inﬂuence the measurements. There were few other indications of the poor ohmics. Firstly, the threshold voltage at which

2DHS starts to conduct is signiﬁcantly lower in this device (VTG = −0.22) to that previously found in similar 2D devices (VTG = −0.12) [18]. Secondly, we found that

RC is strongly dependant on VTG ranging from RC ∼ 80 kΩ at VTG = −0.28Vto

RC ∼ 150 kΩ at VTG = −0.40 V. To understand and eliminate the effect of the ohmics, d.c. I-V characteristics have been measured for different values of VTG at VSG =0V as shown in Fig. 6.4a. It is evident that I-V traces are not linear, thus, the ohmic contact resistance will vary as the current through the wire changes. Another feature of the traces in Fig. 6.4a is their asymmetry: for the same value of negative and positive d.c. voltage applied to the ohmics the measured current is different. It is also apparent that the nonlinearity and asymmetry reduce with increasing VTG. The asymmetry of the I-V characteristics can be attributed to self- gating effects seen in induced electron wires [19]. To check the origin of the asymmetry, we plot RC vs VSD for different values of VTG in Fig. 6.4b. The traces are asymmetric which supports the self-gating explanation for the asymmetry of the ohmic contacts.

To emphasize the change of the contact resistance, Fig. 6.5 presents RC vs I traces, 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires 123

3 V =-0.36 V TG 160 VTG=-0.38 V 2 VTG=-0.40 V 140

1 120

Ω) 100

(nA) 0 (k C DC I R 80 -1 60 -2 40 VTG=-0.40 V VTG=-0.38 V V =-0.36 V -3 20 TG -0.4 -0.2 0.0 0.2 0.4 -2 -1 0 1 2 V (mV) DC VDC (mV) (a) (b)

Figure 6.4: (a) d.c. I-V characteristics of the device at diﬀerent values of VTG at VSG = 0 V; (b) Resistance of the device at VSG = 0 V vs applied voltage VSD. which have been calculated from the I-V characteristics. Three traces are shown for three diﬀerent values of VTG. The resistance RC (I) has its maximum at I =0nA and then decreases with increasing current. As in I-V characteristics, the higher VTG results in bigger change in RC (I) for the same value of the current. The asymmetry of the curves for positive and negative current is also apparent.

It is important to emphasize that this dependence of RC on current is due to the poor ohmics and not the nonlinearity of the wire. Indeed, it is possible to speculate that in induced 1D wires with biasing the top gate, the 1D wire is formed simultaneously, hence the ohmic contacts and the wire are measured in series. Therefore, it is possible to argue that the nonlinear resistance is caused by the changing resistance of the 1D wire when a d.c. source-drain voltage is applied. However, we have shown that the resistance of the ohmics is much larger than the resistance of the 1D wire and therefore the d.c voltage drop across the wire is signiﬁcantly smaller than the subband spacing of the 1D subbands. Thus the resistance of the wire remains approximately constant, and hence, the dependence of RC on I at VSG = 0 V is caused by nonlinear ohmic contacts. This dependence means that in order to convert raw conductance into the 2 conductance of the 1D wire in units of 2e /h, we need to subtract a function RC (I) rather than a constant value of RC . As can be seen from Fig. 6.4a the function is 124 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires

diﬀerent for diﬀerent values of VTG.

180

160

VTG=-0.40V

140 Ω) (k C VTG=-0.38V R 120

100 VTG=-0.36V

80 -2 -1 0 1 2 I (nA)

Figure 6.5: Resistance of the device at VSG = 0 V vs current I through the device (red lines). Blue triangles adjacent to each line show the values of a contact resistance at which the plateaus are quantized. Blue traces are quadratic polynomial ﬁts through the triangles.

The measured RC (I) includes both the changing resistance of the ohmics and the constant resistance of the wire. Therefore to align the conductance plateaus, the values of RC should be reduced by a constant resistance of the 1D wire at the corresponding value of VTG at VSG = 0 V. For example, for VTG = −0.40 V and VSG = 0 V the wire 1 × h ∼ contributes approximately 5 2e2 5 kΩ (for five occupied subbands). We have calculated the values of RC required to align the plateaus to their quantized positions at each value of VTG (blue triangles on Fig. 6.5). The triangles are plotted symmetrically both for positive and negative current, and follow the same trend as the measured RC , but differ from RC by a small resistance, consistent with a constant contribution of the 1D wire. The discrepancies for the negative currents are related to the asymmetry of I-V characteristics. The good correspondence between the measured and calculated RC values proves that we can use the calculated values of RC to convert the measured conductance into conductance of the 1D wire in units of 2e2/h. Blue traces on Fig. 6.5 represent the quadratic polynomial fit through the calculated points for different VTG. This fit was used as RC (I) to convert the raw data 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires 125

10 (a)

10 (µS) 5 9 uncorr g

8 0.0 0.4 0.8 1.2 0 3 (b) VTG=-0.36 V

/h) 2 2 g (2e 1

0 0.0 0.2 0.4 0.6 0.8 1.0 1.2

VSG (V)

Figure 6.6: a) Raw conductance guncorr of the 1D wire vs side gate voltage VSG prior to subtraction of the contact resistance RC ; (b) Contact-corrected conductance g as a function of VSG. To demonstrate the device stability we show three consecutive sweeps for the same VTG. Traces are oﬀset horizontally by −0.12 V for clarity. into conductance of the 1D wire as shown in Fig. 6.6 and also throughout this chapter unless stated overwise. Although poor ohmics limits the use of this device, it is still possible to perform measurements of the dependence of the 1D conductance on density, temperature, source-drain bias and magnetic ﬁeld.

6.5 The stability of the device

An on-going challenge in the development of p-type quantum wires has been instabilities in the electrical characteristics of these devices [2, 3, 20, 21]. This typically results in noisy, poorly quantized conductance plateaus, and for a fixed gate bias configuration, large fluctuations in the device conductance over long time scales (of order minutes to hours). To demonstrate the reproducibility and stability of our device, we

ﬁrstly show three consecutively obtained traces in Figure 6.6 at VTG = −0.36 V. The traces share the same features and have the same pinch-oﬀ voltage to within 0.01 V

(the increment of the VSG). The characteristic conductance suggests that the device operates as a ballistic 1D wire with extremely stable gate characteristics. Most sig- niﬁcantly, and in contrast to previous studies [2, 3, 20, 21], the lower plateaus are 126 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires

(a) (b) VTG=-0.36V

2 /h) 2

g (2e 1

0 0.0 0.4 0.8 1.2 0 50 100 150

VSG (V) t (s)

Figure 6.7: (a) Conductance g vs VSG at VTG = −0.36 V; (b) Conductance g as a function of time t for several set values of VSG

clear and well defined with no resonances. Furthermore, we observe the presence of a reproducible feature around 0.7 × 2e2/h, commonly called the 0.7 structure, which has been extensively studied in electron systems. To explore the stability of the device further, the time evolution of conductance at various fixed VSG has been monitored. Figure 6.7 presents the data for VTG = −0.36 V. At lower conductance, the device is remarkably stable with less than 0.02 × 2e2/h average conductance drift over several minutes. The stability of the present device is in marked contrast to previous studies of modulation doped 1D hole systems [2, 3, 20, 21] and is likely due to the absence of remote doping which introduces charge traps with long recombination times [5]. However, the conductance at the higher plateaus in Fig. 6.7 appears to be more noisy than that of the first plateau. It should be emphasized that the noise that appears in each of the traces is the constant measurement noise with a magnitude I ∼ 10 pA, and not the noise caused by temporal instabilities of the device. Because of the very high contact resistance RC , higher plateaus in raw conductance appear at almost the same value of conductance (c.f. Fig. 6.6) and subtracting the RC effectively “stretches” the conductance and the noise, highlighting its presence at higher plateaus. On the other hand, at low conductances around 2e2/h the resistance of the 1D wire becomes comparable with RC , and therefore changes in RC will not affect the conductance strongly and the ‘noise‘ appears to be small. 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires 127

6.6 Density dependence of conductance quantization

Induced systems are ideal for particle-particle phenomena studies at low density because of the absence of remote ionized impurity scattering, which is the dominant scattering mechanism at low carrier densities.

In Fig. 6.8 we plot the conductance g as a function of VSG, for different hole 10 −2 11 −2 densities p between 8.33 × 10 cm and 1.35 × 10 cm . The value of VSG at which the quantum wire pinches off, as well as the strength and number of plateaus decreases with decreasing density. This is consistent with the fact that the top gate controls the hole density in the wire and less voltage on the side gates is required to pinch-off the channel with smaller hole density. The number of plateaus observed in the data in Fig. 6.8 (up to five plateaus depending on p) is consistent with the 400 √2π nm width of the 1D channel and a Fermi wavelength of 70 nm (λF = 2πp). As the hole density is reduced, the Fermi wavelength becomes larger and at a certain density exceeds the width of the 1D channel, and therefore the channel cannot transmit the holes. For our device, the conductance at p =8.33 × 1010 cm−2 (the leftmost curve in Fig. 6.8) shows only one plateau, and the 1D wire is pinched off at lower densities. This pinch-off density could be lowered by increasing the lithographic width of the wire, as the 2DHS is measurable down to p =1.64 × 1010 cm−2 as was seen in the devices fabricated previously from the same heterostructure [18]. From the general relation (6.1) we obtain: e2m∗ r = √ (6.2) s 2 πp where is the permittivity of GaAs (∼ 12.80), p is the density of holes in the 2D system ∗ and m =0.38 · m0 is the effective mass of the holes. The highest measured density 11 −2 in this sample p =1.35 × 10 cm corresponds to rs =10.8, the lowest density of 10 −2 p =8.33×10 cm corresponds to rs =13.7, the minimum possible density measured 10 −2 in the 2D devices of the same type p =1.64 × 10 cm [18] corresponds to rs =30.9.

Thus, a very wide range of rs can be accessed to study hole-hole interaction eﬀects in this system. A noteworthy feature of the data in Fig. 6.8 is that the 0.7 structure appears to evolve as a function of the hole density. As the number and strength of the integer plateaus decreases with decreasing density, the 0.7 structure, on the contrary, broadens and becomes more pronounced. It is diﬃcult however at this stage to discuss the behaviour of the 0.7 structure as a function of hole density for two reasons. Firstly, our device has a limited density range. Second and most important, is the large contact resistance RC that changes with density and can potentially introduce a small shift (maximum of 0.06 × 2e2/h) into the position of the 0.7 structure. However, the fact 128 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires

-2 VTG (V) p (cm ) rs -0.40 1.35·1011 10.8 4 -0.38 1.27·1011 11.1 -0.36 1.18·1011 11.5 -0.34 1.10·1011 12.0 10 3 -0.32 9.97·10 12.5 -0.30 9.92·1010 13.0 /h) 2 -0.28 8.33·1010 13.7

g (2e 2

0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

VSG (V)

Figure 6.8: Conductance g vs VSG at different values of the top gate voltage VTG. VTG values and their corresponding p and rs values are indicated in the legend. that the 0.7 structure is present at all densities, opens a way for further exploration in the low density regime, where particle-particle interactions are the strongest and not disturbed by the ionized donor potential. Studying the density dependence of conductance quantization is complicated by the fact that VTG controls both the density and confinement potential in the 1D wire. For example, this complicated behaviour may be responsible for controversial results on the density dependence of the 0.7 structure [13, 22, 17, 15]. Induced structures provides the unique possibility of changing the confinement potential at constant density by appropriate tuning of VTG and VSG over a wide range.

6.7 Temperature dependence of conductance quantization

One of the key methods for distinguishing the 0.7 structure from resonances caused by an impurity in or near the 1D channel is the temperature dependence of 1D conductance (Fig. 6.9). The conductance traces in Fig. 6.9 have been offset horizontally for clarity, with low T on the left and high T on the right. The plateaus become thermally smeared with increasing temperature, with the 2e2/h plateau disappearing at T ∼ 600 mK, while higher plateaus weaken at lower T . This can be compared with electron systems [23], where plateaus typically persist to higher temperatures of order ∼ 4 K. This indicates that the 1D subband spacing is much smaller than that 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires 129 found in electrons; which is due to the larger effective mass of holes compared to that of electrons. In contrast, the plateau at 0.7 × 2e2/h becomes significantly stronger with increasing temperature, which is a signature of the 0.7 structure in 1D electron systems [9]. This suggests that the 0.7 structure observed here has a similar origin to that observed in electron quantum wires.

VTG=-0.40 V

25 mK 715 mK /h) 2 g (2e 1

0 0.0 0.4 0.8 1.2 1.6

VSG (V)

Figure 6.9: Conductance of the quantum wire g vs VSG for different temperatures T =25, 175, 370, 515, 620, 715 mK. For clarity the traces are offset sequentially by 0.12 V, from right (715 mK, no offset) to left (25 mK, offset by −0.60 V).

6.8 Source-drain bias measurements

In order to directly measure the subband spacings, a d.c. source-drain bias technique has been used [24, 25]. This technique is described in detail in Chapter 2. There are two ways of taking the data. The most commonly used method is to measure a series of d.c. voltage sweeps at various values of VSG, as we used in Chapter 4. The second method is to measure SG voltage sweeps at diﬀerent d.c voltages. Because of the asymmetry and high resistance of the ohmics, we preferred to use the second method.

The measured conductance is converted by subtracting a constant RC , and then the data is numerically diﬀerentiated to produce the transconductance (dg/dVSG), which is plotted vs VSG and VSD coordinates.

The full data set is presented for the source-drain bias measurements at VTG = −0.40 V in Fig. 6.10. Conductance of the 1D wire presented in Fig. 6.10a was calculated by subtraction of the constant RC from the raw data. The plateaus on the traces evolve with applied VSD, however the traces are very asymmetric relative to 130 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires

5 V = 3 mV 4 SD

/h) 3 2

2 g (2e

V = -3 mV 1 SD

0 0 1 2 3 4 VSG (V) (a)

0.2 VTG=-0.40 V

3 0.4 150x10

0.6 Ω)

( 0.8 100 (V) =0 V) SG

SG 1.0 V R(V 1.2 50 1.4

1.6 0 -2 -1 0 1 2 -2 -1 0 1 2 V (mV) SD VSD (mV) (b) (c)

Figure 6.10: (a)Conductance of the 1D wire corrected for a constant contact resistance RC vs VSG at VTG = −0.40 V. The leftmost curve corresponds to VSD = −3 mV and the rightmost to VSD = +3 mV (with a step size of 0.2 mV); (b) Resistance R(VSG =0) V as a function of VSD; (c) 3D colour map of transconductance vs VSG on y axis and VSD on x axis. Data obtained by numerical diﬀerentiation of the data shown on Fig. 6.10a. Yellow colour corresponds to plateaus, black to risers between plateaus. 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires 131

VSD = 0 V and are aﬀected by the large contact resistance.

To see the evolution of RC with application of source-drain bias, we have plotted the measured resistance R(VSG =0V )vsVSD in Fig. 6.10b. Applied VSD strongly aﬀects RC , reducing it dramatically from > 150 kΩ at VSD = 0 mV to ≈ 50 kΩ at

VSD = 1 mV. Note that this dramatic change can not be explained by nonlinearity of 1 × h ≈ the wire, which has a resistance 5 2e2 5kΩatVSG =0V.

To make the evolution of plateaus clearer, in Fig. 6.10c we show a 3D colour map of transconductance calculated from the data in Fig. 6.10a. Yellow represents plateaus and black the risers between the plateaus. The colour maps are very asymmetric, however half plateaus for subbands 1 and 2 are clearly present. To eliminate asymmetry due to self-gating eﬀects, the conductance was symmetrized with regards to VSD =0

V by taking an average of the two traces at opposite values of VSD as in Ref. [26]. Figure 6.11a shows the symmetrized 3D colour map. The blue crosses on the 3D map ∗ highlight the centers of the half-plateaus at finite source-drain voltage VSD, which is ∗ a direct measure of the subband spacings. For subbands 1 and 2, VSD = 2 mV, but only a small part of that voltage drops across the 1D wire due to large RC . The 2 resistance of the 1D wire is R1D = e /3h =8.6 kΩ, in contrast to RC ∼ 48 kΩ, hence the voltage drop across the wire is 0.35 mV. For the subband spacing between 2nd rd 2 and 3 subbands, the resistance of the 1D wire R1D = e /5h =5.16 kΩ and there- ∗ fore VSD =0.21 mV. These results are comparable with those previously measured in bilayer 1D hole systems [27], but a rapid change of the subband spacing suggests the noticeable difference in confinement potential between the modulation doped bilayer and induced devices.

The conﬁnement potential of the 1D wire is directly related to the subband spacing. In induced structures, changing the bias on the top gate not only changes the density but also the conﬁnement potential [12]. To check this, we analyzed the source- drain bias data for three values of the top gate voltage: VTG = −0.36 V, VTG = −0.38 V and VTG = −0.40 V. The results are shown on Fig. 6.11b. The subband spacings

ΔE1,2 have a linear dependence on VTG in agreement with recent studies in electron triple-gate structures [28]. We ﬁt a straight line through the experimental data points and extrapolated it to the range of the top gate voltages accessible by this type of induced device. If the trend presented on Fig. 6.11b is correct, then achieving low hole densities may result in much smaller subband spacings, which may prevent the observation of quantized 1D conductance at low hole densities. 132 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires

0.2 700 x 0.4 x x 600

x 0.6 500

x x 0.8 400 (V) x (µV) SG 1.0 1,2 V 300 ΔΕ

x x 1.2 200 x 1.4 x x 100

1.6 0 0.1 0.2 0.3 0.4 0.5 0.6 -2 -1 0 1 2 VSD (mV) VTG (V) (a) (b)

Figure 6.11: (a) The symmetrized 3D colour map of transconductance vs VSG on y axis and VSD on x axis. Data obtained by numerical diﬀerentiation of the data shown on Fig. 6.10a. Yellow corresponds to plateaus, black to the risers between plateaus; (b) The subband spacing between ﬁrst and second subbands calculated from source-drain bias data as a function of the top gate voltage VTG.

6.9 Magnetic ﬁeld measurements

In this section we study the behaviour of the device in responce to a magnetic field B oriented parallel to the wire. With this field orientation we expect to observe the Zeeman splitting of 1D subbands at moderate fields, similar to the results presented in Chapter 4. The effect of an in-plane magnetic field is shown schematically on Fig. 4.8. When the magnetic field is applied, it lifts the spin degeneracy of the 1D subbands and spin polarized subbands separate in energy as a linear function of magnetic field. As was explained in Chapter 4, from the magnetic field measurements it is possible to extract the effective g-factors of 1D subbands. As discussed previously, the ohmics in this device are non ideal, and therefore they strongly affect the evolution of the 1D wire conductance as a function of magnetic field. The magnetic field data is analyzed in the similar way to d.c. source-drain bias data. Conductance of the 1D wire is measured for different values of magnetic field B. The full data set is presented in Fig. 6.12 for VTG = −0.40 V. The conductance corrected for a constant contact resistance RC is presented on Fig. 6.12a for B between 0 and 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires 133

4 /h)

2 3

g (2e 2 B=0T B= 1.5 T

0 0.0 0.5 1.0 1.5 2.0 2.5

VSG (V) (a)

6 5 VTG = -0.40 V 4 0.2 3

2 0.4

0.6 (Ω) 106

7 (V)

=0 V) 6 0.8 SG SG 5 4 V

R(V 1.0 3

2 1.2

1.4 105

7 0.0 1.0 2.0 3.0 0.0 0.4 0.8 1.2 B (T) VSD (mV) (b) (c)

Figure 6.12: (a) Conductance of the 1D wire corrected by a constant RC . Traces are offset horizontally in multiples of VSG =0.1 V from left (no offset, B = 0 T) to right (offset by +1.5V,B =1.5 T); (b) The contact resistance RC as a function of magnetic field B; (c) 3D plot of transconductance vs side gate voltage on y axis and magnetic field B on x axis. Data obtained by numerical differentiation from the data on Fig. 6.12a. Yellow colour corresponds to plateaus, black - to risers between plateaus. 134 6. Conductance quantization in induced GaAs/AlGaAs hole 1D wires

1.5 T. The ohmic contacts degrade very quickly in magnetic ﬁeld and therefore the conductance appears very noisy. To see the evolution of RC vs B, Fig. 6.12b shows a plot of R(VSG = 0 V) as a function of B. The ohmics start to be more resistive around

B = 1 T and by B =1.5T,RC reaches 200 kΩ. For higher fields, the measured current through the device becomes too small to distinguish conductance plateaus. However, similar to the source-drain bias measurements, when the transconductance is calculated by numerical differentiation, the evolution of the subbands in magnetic field is clearer, as shown on Fig. 6.12c. Unfortunately the accessible range of magnetic field is not enough to see the splitting of the 1D subbands. Note that we have performed magnetic measurements for other values of the VTG = −0.38 and −0.36 V, which show the same problem with high resistance of the ohmic contacts.

6.10 Conclusions and future work

In this chapter we presented the first p-type 1D quantum wire fabricated using an AlGaAs/GaAs SISFET structure without modulation doping. This device exhibits very stable, clear conductance quantization, including the 0.7 structure due to absence of modulation doping. We have also presented characterization measurements such as density and temperature dependence of the 1D conductance, followed by source-drain bias and magnetic field measurements. This device stability and tunable density will enable studies in the highly interacting regime (10 ≤ rs ≤ 30), where very low densities and high mobility can be achieved simultaneously. The main limitation of the current device is poor ohmic contacts, which affect the measured data, and do not allow high accuracy source-drain bias and magnetic field measurements. Therefore it is necessary to optimize the fabrication procedure for ohmic contacts in order to use these devices for further research. Additionally, the 1D wire should be wide enough in order to observe conductance quantization at low densities, which will enable, for example, a study of the 0.7 structure over a very wide range of hole densities. Nonetheless, the measurements in this chapter prove that induced 1D hole systems exhibit high stability and clean conductance quantization. This opens a new way for studying the effects of spin-orbit coupling and strong carrier-carrier interactions in mesoscopic hole systems. Bibliography

[1] R. Dingle, H. L. Stormer, A. C. Gossard and W. Wiegmann, Electron mobilities in modulation-doped semiconductor heterojunction superlattices, Appl. Phys. Lett., 33(7), 665 (1978).

[2] I. Zailer, J. E. F. Frost, C. J. B. Ford, M. Pepper, M. Y. Simmons, D. A. Ritchie, J. T. Nicholls and G. A. C. Jones, Phase coherence, interference and conductance quantization in a conﬁned two-dimensional hole gas, Phys. Rev. B, 49(7), 5101 (1994).

[3] A. J. Daneshvar, C. J. B. Ford, A. R. Hamilton, M. Y. Simmons, M. Pepper and D. A. Ritchie, Enhanced g factor of a one-dimensional hole gas with quantized conductance, Phys. Rev. B, 55(20), R13409 (1997).

[4] J. A. Nixon, J. H. Davies and H. U. Baranger, Interaction eﬀects in a one- dimensional constriction, Phys. Rev. B, 43, 12638 (1991).

[5] R. H. Harrell, K. S. Pyshkin, M. Y. Simmons, D. A. Ritchie, C. J. B. Ford, G. A. C. Jones and M. Pepper, Fabrication of high-quality one- and two-dimensional electron gases in undoped GaAs/AlGaAs heterostructures, Appl. Phys. Lett., 74(16), 2328 (1999).

[6] K. Hirakawa, Y. Zhao, M. B. Santos, M. Shayegan and D. C. Tsui, Anoma- lous cyclotron-resonance line splitting of two-dimensional holes in (311)A

AlxGa1−xAs/GaAs heterostructures, Phys. Rev. B, 47(7), 4076 (1993).

[7] Y. Y. Proskuryakov, A. K. Savchenko, S. S. Safonov, M. Pepper, M. Y. Sim- mons and D. A. Ritchie, Hole-hole interaction eﬀect in the conductance of the two-dimensional hole gas in the ballistic regime, Phys. Rev. Lett., 89(7), 076406 (2002).

[8] B. E. Cole, W. Batty, Y. Imanaka, Y. Shimamoto, J. Singleton, J. M. Cham- berlain, N. Miura, M. Henini and T. Cheng, Eﬀective-mas anisotropy in GaAs-

135 136 BIBLIOGRAPHY

(Ga,Al)As two-dimensional hole systems: A comparison of theory and very high- ﬁeld cyclotron resonance experiments, J. Phys. : Condens. Matter, 7, L675 (1995).

[9] K. J. Thomas, J. T. Nicholls, M. Y. Simmons, M. Pepper, D. R. Mace and D. A. Ritchie, Possible spin polarization in a one-dimensional electron gas, Phys. Rev. Lett., 77(1), 135 (1996).

[10] B. E. Kane, L. N. Pfeiﬀer, K. W. West and C. K. Harnett, Variable density high

mobility two-dimensional electron and hole gases in a gated GaAs/AlxGa1−xAs heterostructure, Appl. Phys. Lett., 63(15), 2132 (1993).

[11] Y. Hirayama, K. Muraki and T. Saku, Two-dimensional electron gas formed in a back-gated undoped heterostructure, Appl. Phys. Lett., 72(14), 1745 (1998).

[12] B. E. Kane, G. R. Facer, A. S. Dzurak, N. E. Lumpkin, R. G. Clark, L. N. Pfeiﬀer and K. W. West, Quantized conductance in quantum wires with gate-controlled width and electron density, Appl. Phys. Lett., 72(26), 3506 (1998).

[14] D. J. Reilly, T. M. Buehler, J. L. O’Brien, A. R. Hamilton, A. S. Dzurak, R. G. Clark, B. E. Kane and L. N. Pfeiﬀer, Density-dependent spin polarization in ultra- low disorder quantum wires, Phys. Rev. Lett., 89(24), 246801 (2002).

[15] S. Nuttinck, K. Hashimoto, T. S. S. Miyashita, Y. Yamamoto and Y. Hirayama, Quantum point contacts in density-tunable two-dimensional electron gas, Jpn. J. Appl. Phys, 39(7A), L655 (2000).

[16] K. Hashimoto, S. Miyashita, T. Saku and Y. Hirayama, Back-gated point contact, Jpn. J. Appl. Phys, 40(4B), 3000 (2001).

[17] K. S. Pyshkin, C. J. B. Ford, R. H. Harrell, M. Pepper, E. H. Linﬁeld and D. A. Ritchie, Spin splitting of one-dimensional subbands in high quality quantum wires at zero magnetic ﬁeld, Phys. Rev. B, 62(23), 15842 (2000).

[18] W. R. Clarke, A. P. Micolich, A. R. Hamilton, M. Y. Simmons, K. Muraki and Y. Hirayama, Fabrication of induced two-dimensional hole systems on (311)A GaAs, J. Appl. Phys., 99, 023707 (2006).

[19] D. J. Reilly, Many-body spin related phenomena in semiconductor quantum wires, Ph.D. thesis, School of Physics, University of New South Wales, Australia (2001). BIBLIOGRAPHY 137

[20] L. P. Rokhinson, D. C. Tsui, L. N. Pfeiﬀer and K. W. West, AFM local oxidation nanopatterning of a high mobility shallow 2D hole gas, Superlattices and Microstructures, 32(2-3), 99 (2002).

[21] L. N. Pfeiﬀer, R. de Picciotto, K. W. West, K. W. Baldwin and C. H. L. Quay, Bal- listic hole transport in a quantum wire, Appl. Phys. Lett., 87(7), 073111 (2005).

[22] K. J. Thomas, J. T. Nicholls, M. Pepper, W. R. Tribe, M. Y. Simmons and D. A. Ritchie, Spin properties of low-density one-dimensional wires, Phys. Rev. B, 61(20), R13365 (2000).

[23] B. J. van Wees, L. P. Kouwenhoven, E. M. M. Willems, C. J. P. M. Harmans, J. E. Mooij, H. van Houten, C. W. J. Beenakker, J. G. Williamson and C. T. Foxon, Interacion eﬀects in a one-dimensional constriction, Phys. Rev. B, 43(15), 12431 (1991).

[24] L. I. Glazman and A. V. Khaetskii, Non-linear quantum conductance of a lateral microconstraint in a heterostructure, Europhys. Lett., 9(3), 263 (1989).

[25] N. K. Patel, J. T. Nicholls, L. Martin-Moreno, M. Pepper, J. E. F. Frost, D. A. Ritchie and G. A. C. Jones, Evolution of half plateaus as a function of electric ﬁeld in a ballistic quasi-one-dimensional constriction, Phys. Rev. B, 44(24), 13549 (1991).

[26] A. Kristensen, H. Bruus, A. E. Hensen, J. B. Jensen, P. E. Lindelof, C. J. Mar- ckmann, J. Nyg˚ard,C. B. Sørensen, F. Beuscher, A. Forchel and M. Michel, Bias an temperature dependence of the 0.7 conductance anomaly in quantum point contacts, Phys. Rev. B, 62(16), 10950 (2000).

[27] R. Danneau, W. R. Clarke, O. Klochan, A. P. Micolich, A. R. Hamilton, M. Y. Simmons, M. Pepper and D. A. Ritchie, Conductance quantization and the 0.7 × 2e2/h conductance anomaly in one-dimensional hole systems, Appl. Phys. Lett., 88, 012107 (2006).

[28] H.-M. Lee, K. Muraki, E. Y. Chang and Y. Hirayama, Electronic transport characteristics in a one-dimensional constriction deﬁned by a triple-gate structure,J. Appl. Phys., 100, 043701 (2006). Chapter 7

Anisotropic Zeeman splitting in orthogonal induced hole 1D wires

Chapter Outline This chapter presents a study of Zeeman splitting in induced ballistic 1D wires fabricated along two orthogonal crystallographic directions. The data for the wire oriented along the [233] direction exhibits non-

monotonous anisotropy with the ratio g/g⊥ > 5.2 for lower subbands. The data

for the 1D wire along [011] shows a reversed anisotropy with the ratio g⊥/g ≥ 2, which is in contrast with our results for the wire oriented along the [233] direction. To explain these two different results we consider the anisotropy of the g-factors relative to the magnetic field direction. We show that for B [011] the g-factor anisotropy can be explained by reorientation of the vector Jˆ from perpendicular to the 2D plane to in-plane parallel to the wire, whereas for B [233] the additional off-diagonal Zeeman term leads to the out-of-plane component of Jˆ, and as a result, to enhanced g-factors irrespective of the wire direction due to increased exchange interactions.

7.1 Introduction

In this chapter we aim to study the Zeeman splitting in induced hole 1D wires fabricated along two orthogonal directions. This study would allow us to answer the following question:

1. What is the eﬀect of the crystal anisotropy on Zeeman splitting in 1D wires?

2. How does the conﬁnement potential (e.g. the single heterojunction used in this chapter vs the quantum well used in Chapter 4) aﬀect the Zeeman splitting in 1D hole systems?

138 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires 139

These experimental results would allow to answer important theoretical question raised in Chapter 4: “What determines the 1D g-factor: the crystallographic anisotropy [1] or the 1D confinement?”. As described in Chapter 2, the existence of an in-plane g-factor in 2D hole systems that is both finite and anisotropic can be explained by the way in which holes couple with the (311)A GaAs crystal lattice. It is possible to show that the total angular momentum vector Jˆ aligns to the crystal growth direction and hence we expect the in-plane g-factor to be zero [2]. However, for low symmetry growth directions, because of the spin-orbit coupling the spin of the holes couples to highly anisotropic orbital motion (due to the bulk inversion asymmetry of the GaAs crystal). This lowers the symmetry of the system, which results in a finite and anisotropic in-plane g-factor. The anisotropy can also be explained from a band diagram perspective. The 2D confinement lifts the heavy hole-light hole (HH-LH) degeneracy at k = 0. As a result, the carrier transport is via the HH subband. An in-plane field B couples states differing by Jz = ±1 (HH states to LH states, and LH states to LH states) but no direct coupling between HH states is generated. Thus, the in-plane g-factor for HH states is zero. However, due to crystal cubic anisotropy for the (311)A low symmetry growth direction, a dynamic coupling is produced, resulting in a finite and anisotropic in- plane g-factor for HH states. This dynamic coupling is suppressed by HH-LH energy splitting due to the 2D confinement, which is why the in-plane g-factors are finite but an order of magnitude smaller than the out-of-plane g-factor. Finally, the 1D confinement further modifies the band diagram [3, 4], thereby altering the HH-LH coupling. This modifies the value of the in-plane g-factors. Experimentally, the in-plane g-factor anisotropy has been studied in (311) GaAs 2D hole systems [5] and found to be independent of the current direction with a ratio of the g-factors for orthogonal in-plane fields less than 3. This is in a reasonable agreement with g/g⊥ = 4 predicted by theoretical calculations [6]. The effect of the 1D confinement has been studied by us [7], presented in Chapter 4. The data on the 1D hole wire oriented along [233] show enhanced anisotropy of the in-plane g-factor ∗ ∗ (g/g⊥ > 4.5) which is significantly larger than the in-plane anisotropy in the 2D case. These results were explained by reorientation of the vector Jˆ from perpendicular to the 2D plane to in-plane and parallel to the wire. It is possible to apply the same argument as in the 2D case, that the g-factor along Jˆ should be much larger than the g-factor perpendicular to Jˆ, to the 1D wire. Therefore the enhanced ratio of the g-factors becomes straightforward. On the other hand, Rokhinson et al [1] studied QPCs formed by local anodic oxidation on a modulation doped shallow quantum well grown on (311)A GaAs substrate. Two orthogonal QPCs were oriented along [233] and [110] crystallographic 140 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires directions and magnetic field B was applied parallel and perpendicular to the current ∗ ∗ ∼ I through the QPC. The main findings were the ratios g233/g110 1.8 and > 4 for the QPCs oriented in [110] and [233] directions respectively and the absolute values of the g-factros have not been presented. The authors claimed that they observed the intrinsic crystallographic anisotropy in (311)A GaAs but also indicated that the ratios are significantly different from the values obtained for 2D hole systems [6, 5]. Note that Rokhinson’s data for [233] wire is in a good agreement with our results. In this chapter we study the Zeeman splitting in induced ballistic 1D wires fabricated along two orthogonal crystallographic directions. The data for the wire oriented along the [233] direction exhibits nonmonotonous anisotropy with the ratio g/g⊥ > 5.2 for lower subbands. The data for the 1D wire along [011] shows a reversed anisotropy with the ratio g⊥/g ≥ 2, which is in direct disagreement with our results for the wire oriented along the [233] direction. To explain these two different results we analyze absolute values of the in-plane g-factors relative to the magnetic field direction. We show that for B [011], the g-factor anisotropy can be explained by reorientation of vector Jˆ from the perpendicular to the 2D plane to the in-plane and parallel to the wire, whereas for B [233] the additional off-diagonal Zeeman term leads to the out-of-plane component of Jˆ, and as a result to enhanced g-factors irrespective of the wire direction due to increased exchange interactions.

7.2 Device structure and initial characterization

This section presents the fabrication and characterization of the orthogonal induced hole 1D wires. The device used for studies in this chapter was fabricated, tested and measured by the author. The heterostructure R165 was grown by Koji Muraki from NTT Basic Research Labs (Japan). The same wafer was used for the studies in Chapter 6. The GaAs/AlGaAs heterojunction is grown on a (311)A GaAs substrate and resides 200 nm below degenerately doped (5×1018 cm−3 Si) p+-GaAs layer, which is used as a metallic gate. The details on the fabrication and electrical characterization of the sample can be found in the Chapter 3. To induce the hole system at the GaAs/AlGaAs interface, the top gate (TG) is negatively biased below a certain threshold voltage, which is defined by the voltage at which the current starts to flow through the ohmic contacts. The typical threshold voltage is ≈−0.1 V. The sample used in Chapter 6 had a higher threshold voltage due to the poor ohmics. Poor ohmics limited the studies in source-drain bias (see Fig. 6.10) and in-plane magnetic field (see Fig. 6.12). To perform the studies in this chapter, we needed to improve the ohmic contacts to induced 1D wires. For the current sample, the procedure of ohmic contact formation has been improved, which reduced 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires 141 the ohmic contact resistance by an order of magnitude. Moreover, the improved ohmic fabrication procedure allowed a wider top gate range to be accessed with no measurable leakage current up to VTG = −0.49 V compared with VTG = −0.40 V for the device measured in Chapter 6. For the studies in this chapter two orthogonal 1D wires have been fabricated on the same Hall bar: one parallel to the [233] crystal direction (shown on the Fig. 7.1a) and the other parallel to [011] direction (Fig. 7.1b). The wires are designed to be 400 nm both in length and width after etching. In the following discussion we will refer to these 1D wires as QPC233 and QPC011 respectively.

(a) (b)

Figure 7.1: SEM micrograph of (a) QPC233, (b) QPC011. Note that both wires are fabricated on the same Hall bar. The dark grey area is the etched part dividing the top conducting layer into the top gate and the side gates, the light rectangles at the top and the bottom of the picture are the optical gates contacting to the two side gates.

As discussed in the previous chapter, by patterning the top heavily doped layer using EBL and wet etching, it is possible to fabricate 1D wires with three independently biasable gates : two side gates (SG) and a top gate (TG). The top gate has a “bow- tie” shape, connecting two large 2D regions via a narrow 1D channel. Therefore, by applying a voltage to the top gate which is beyond the threshold, a 1D system is established between two large 2D reservoirs. The hole density is controlled by the voltage on the top gate and has a linear dependence (see Fig. 3.3a) [8]. The extended gate voltage range in this device (−0.12

The VTG is varied from −0.48 V (rightmost trace) to −0.27 V (leftmost trace), which 10 11 −2 corresponds to a density range 7.96×10 rs > 9.62). As observed in Chapter 6, the number and strength of the plateaus decreases with decreasing density. The leftmost trace on both graphs exhibits no plateaus, because 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires 143

80 6 6 V = -0.48 V TG VTG = -0.48 V QPC233 400 QPC233 5 5 60 4 300 4 /h) /h) 2 2 S) S) μ μ ( 40 3 ( 3 G (2e G (2e 2T 4T 200 G G 2 2 20 100 1 1

0 0 0 0 0.0 0.5 1.0 0.0 0.5 1.0 VSG (V) VSG (V) (a) (b)

100 6 400 6 V = -0.48 V TG VTG = -0.48 V QPC011 QPC011 80 5 5 300 4 4

60 /h) /h) 2 2 S) S) μ μ ( 3 ( 200 3 2T 4T G (2e G (2e

G 40 G 2 2 100 20 1 1

0 0 0 0 0.0 0.5 1.0 0.0 0.5 1.0

VSG (V) VSG (V) (c) (d)

Figure 7.2: Two terminal conductance vs VSG with no adjustments G2T (red trace / left axis) and corrected for a constant series resistance G (blue trace / right axis) for (a) QPC233, (c) QPC011. Four terminal conductance vs VSG with no adjustments G4T (red trace / left axis) and corrected for a constant series resistance G (blue trace / right axis) for (b) QPC233, (d) QPC011. The axes are coloured appropriately to the colours of the conductance traces. The series resistances used to align the conductance plateaus are: RC (2T ) = 1160 Ω and RC (4T ) = 370 Ω for QPC233 and RC (2T ) = 8550 Ω and RC (4T ) = 1003 Ω for QPC011. 144 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires

QPC || [233] 6 10 11 -2 7.96×10 < p < 1.69×10 cm /h) 2 4 (2e 4T

G 2

0 0.0 0.2 0.4 0.6 0.8 1.0

VTG (V)

(a)

QPC || [011] 6 10 11 -2 7.96×10 < p < 1.69×10 cm /h) 2 4 (2e 4T

G 2

0 0.0 0.2 0.4 0.6 0.8 1.0 1.2

VTG (V) (b)

Figure 7.3: Four terminal conductance G4T vs VSG for values of VTG from −0.27 to −0.48 V ( VTG step size −0.03 V) for (a) QPC233, (b) QPC011. The corresponding 10 11 −2 density and rs ranges are: 7.96 × 10 rs > 9.6. 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires 145

2 the 1D wire conductance is below 2e /h.AboveVTG = −0.27 V the wire is pinched-off, and no current runs through the ohmic contacts. This 1D threshold voltage is much higher than the 2D threshold voltage and is due to the narrow width of the fabricated wires. We have stressed in Chapter 6 that it is possible to reduce the wire pinch-off voltage by fabricating a wider wire, which will allow to study ballistic transport in hole 1D systems at low densities (high rs). Pursuing this aim, induced 1D wires have been fabricated with widths of 1000 and 800 nm. The pinch-off voltages for these wires shifted towards the 2D threshold voltage, being −0.14 and −0.18 V respectively, providing the possibility to access low hole densities. Unfortunately, we have found no conductance quantization in these samples (even at high density), which we ascribe to a very small subband spacing between 1D subbands due to less sharp confinement (the side gates are far apart). Therefore the studies in this chapter were done on narrow wires (400 nm) in the high density limit. All subsequent measurements in this chapter were performed at dilution fridge base temperature in a constant voltage four terminal configuration for

VTG = −0.48 V. The 4T scheme eliminates the eﬀect of the ohmic contacts on the 1D wire behaviour.

7.3 Determining the g-factors of the 1D subbands

In this section we calculate the effective g-factors of 1D subbands in the two orthogonal induced hole wires for two orthogonal in-plane field directions. The methods of calculating effective g-factors of 1D subbands were presented in Chapter 4. g-factors can be calculated from either the splitting rates of the subband edges in magnetic field and source-drain voltage, or from the crossings of the subband edges in magnetic field and source-drain voltage. The first method allows us to calculate g-factors for each individual subband. Additionally it is possible to monitor the subband splitting and verify the linearity of the Zeeman splitting. In contrast, the second method assumes linearity and takes the average of the g-factors for two adjacent spin-split subbands. It is difficult to extract the g-factor if the subband does not split and in that case we can only place the upper bound on g∗ (as was done in Chapter 4). This will be described in more detail as the data is presented. In order to collect all the necessary data, we need to take measurements in two subsequent cool downs. On the first cooldown the sample is oriented such that the magnetic field is parallel to [233] direction. Source-drain bias measurements of the QPC233 and QPC011 are taken first at B = 0 T. Then, the measurements of four- terminal conductance in magnetic field B [233] for the QPC233 and QPC011 are 146 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires obtained. The system is then was warmed-up and the sample was reoriented such that B [011]. On the second cooldown the source-drain bias and magnetic field measurements were repeated for both wires. To avoid confusion with the orientations of the wires and magnetic fields relative to the crystallographic axes, we will present the data for each wire and refer to the orientation of the in-plane magnetic field as parallel (), when the direction of the field is in-plane and parallel to the wire, and perpendicular (⊥), when the direction of the field is in-plane and perpendicular to the wire. In cases where it is necessary, we also provide a corresponding crystallographic direction. Note that because the conductance traces in both wires have resonant features on and below the first plateau, we will consider the Zeeman splitting of the higher subbands only. The investigations of the origin and properties of those resonances we set aside for future work. However, the fact that the resonances are pronounced only around the first plateau indicates that they may be related to a defect around the 1D constriction, which has a significant effect only at low densities (less screening). Another confirmation of this is the evolution of the resonances with hole density (c.f. Fig. 7.3b and 7.3b).

7.3.1 Source-drain bias spectroscopy

We use the source-drain biasing (SDB) technique to directly measure the 1D subband spacings of the two wires (QPC233 and QPC011). The details of the technique are presented in Chapters 2 and 4. Figures 7.4a and 7.4b show the four-terminal differential conductance g as a function of d.c. voltage VSD at different values of VSG for QPC233 and QPC011 correspondingly. The four terminal differential conductances have been corrected for the constant series resistance contribution of the 2D hole system resistance adjacent to each of the wires. These additional resistances were chosen to align the first plateau to 2e2/h in conductance and were calculated to be 370 and 1003 Ω for QPC233 and QPC011 respectively. The d.c. voltage drop across each wire has been corrected for a constant ohmic contact resistance of 12 and 7.5 kΩ for QPC233 and QPC011 (the ohmic contact resistance acts as a voltage divider).

The accumulations of diﬀerential conductance traces at VSD = 0 V on Fig. 7.4a and 7.4b correspond to quantized plateaus in the conductance. Additional accumulations can be seen around ﬁnite VSD ≈ 350 μV corresponding to half integer plateaus. In order to see the evolution of the plateaus with applied VSD more clearly, Fig. 7.4c and 7.4d present the colour map of the transconductance (dg/dVSG) plotted vs VSG and VSD coordinates. The transconductance has been numerically calculated from the 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires 147

12 QPC||011 10 QPC||233 10

8 8 /h) 2 /h) 2 6 6 g (2e g (2e 4 4

2 2

0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 V (mV) VSD (mV) SD (a) (b)

QPC||233 QPC||011 0.0 0.0

0.2 0.2 0.4 0.4 (V) (V) SG

SG 0.6 V 0.6 V 0.8 0.8 1.0 1.0 1.2 -0.4 0.0 0.4 -0.4 0.0 0.4 V (mV) VSD (mV) SD (c) (d)

2 Figure 7.4: Four terminal differential conductance in units of 2e /h vs VSD for different VSG (step size 0.02 V) and VTG = −0.48 V for (a) QPC233, (b) QPC011. Colour map of the transconductance vs VSD on x axis and VSG on y axis. Data obtained by numerical differentiation of the data in (c) Fig. 7.4a, (d) Fig. 7.4b. 148 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires four terminal data. The yellow regions correspond to high transconductance (risers in conductance) and black areas to low transconductance (plateaus in conductance). The white lines are shown as a guide and connect the centers of the yellow regions and highlighting the diamond pattern (see Fig. 4.5b). The distance between the corners of each diamond is a direct measure of the subband spacing between the two adjacent subbands. We have also extracted splitting rates ( ∂VSG ) for the 1D subbands. ∂eVSD Table 7.1 summarizes the results of this section by showing the extracted 1D subband spacings (ΔE) for QPC233 and QPC011. Note that the source-drain bias measurements have been repeated on the second cool-down. The difference in measured subband spacings was less than 10 μV confirming the stability and reproducibility of these devices. As can be seen from Table 7.1 there is a noticeable difference in subband spacings between the QPC233 and QPC011. At first sight it could be attributed to the difference in widths of the two wires due to anisotropic wet etch used to form both wires (see Chapter 3). However the comparable 1D conductance of the wires at the same value of VTG and pinch-off characteristics suggests that the dimensions of the wires are very similar. Hence we suspect that the differing subband spacings between QPC233 and QPC011 is instead due to the in-plane anisotropy of the hole effective mass m∗ since ΔE ∝ 1/m∗.

Levels ΔE (μV) ΔE (μV) ∂VSG/∂eVSD ∂VSG/∂eVSD QPC233 QPC011 QPC233 QPC011 1,2 365 282 430 528 2,3 222 186 636 870 3,4 188 149 799 1194 4,5 168 109 906 1428 5,6 149 115 1096 - 6,7 140 98 13268 - 7,8 133 - - -

Table 7.1: Subband spacings and the splitting rates ∂VSG/∂eVSD of the 1D subbands for QPC233 and QPC011.

7.3.2 Magnetic ﬁeld measurements

With the application of an in-plane magnetic field B the spin degeneracy of the 1D subbands is lifted, allowing us to observe spin split subbands at a finite B. Increasing the field further forces the adjacent spin up and spin down subbands to cross at some

field BC (see Fig. 4.8). Before we discuss the magnetic field measurements we have to mention that all the measurements have been done in the range from 0 to 4.2 T with a step of 0.2 T. The 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires 149 limited range of magnetic field is because the ohmic contacts degrade quickly with increasing magnetic field, with a resistance exceeding 100 kΩ at B =4.2 T. Although it is still possible to pass current through these ohmics, the voltage probes stop func- tioning and as a result the traces become very noisy, especially at high conductances, when the resistance of the wire becomes much smaller than the ohmic resistance. That is why we have limited the range of the field to 4.2 T, where we were able to perform four terminal measurements, minimizing the effect of the ohmic contact resistance.

7.3.3 Results for QPC233

Figure 7.5a shows the four terminal conductance of QPC233 adjusted for the constant resistance of the hole system at each value of parallel magnetic field B (B [233]). The traces are offset in multiples of 0.05 V from the leftmost (B = 0 T, no offset) to the rightmost (B =4.2 T, offset by 1.05 V). As can be seen from Fig. 7.5a, some plateaus spin-split as the magnetic field increases. The noise at high conductances and high fields is related to the high ohmic contact resistance. As a result, only a small fraction of the applied a.c. voltage drops across the wire, leading to a small signal to noise ratio. In order to analyze the evolution of the plateaus more clearly, the transconductance dG4T /dVSG has been numerically calculated from the data presented in Fig. 7.5a.

Figure 7.5b shows the transconductance vs VSG and B coordinates. Blue colour (low transconductance) corresponds to plateaus in conductance, while red (high transconductance) to the risers between the plateaus. To highlight the evolution of the plateaus,

Fig. 7.5c presents the transconductance vs G4T and B coordinates. In contrast to our previous results in Chapter 4, the plateaus split very nonuniformly in magnetic field. For example, subband 2 to starts to split at around 0.8 T, subband 4 at 3 T, whereas subband 5 does not show any splitting up to the highest measured field of 4.2T. As can be seen in the Fig. 7.5b, subbands 1 − 7 spin-split, while subband 5 is unaffected by magnetic field. However, only one crossing field BC can be directly measured between subbands 6 and 7. For the other subbands, which do not exhibit crossings before 4.2 T but clearly have moving risers, we have extracted the field B∗, when the particular subband starts to split in magnetic field. Finally, for subband ∗ 5, which shows no splitting we have assumed the minimum value of Bmin =4.2T. ∂VSG Additionally we have extracted the subband splitting rates ( ∂B ) for subbands 2, 3 and 6. Figure 7.6a shows the evolution of the four-terminal conductance of QPC233 in the perpendicular magnetic field B⊥ (B [011]). The measured conductance is not quantized in units of 2e2/h and cannot be adjusted by a constant contact resistance. 150 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires

QPC|| [233] 6 /h) 2 4 B = 4.2 T (2e B|| =0T || 4T

G 2

0 0.0 0.5 1.0 1.5 2.0 VSG (V) (a)

0.0 7

0.2 6

5 0.4

/h) 4 2 (V) (2e

SG 0.6

V 3 4T G 0.8 2

1 1.0 0 0 1 2 3 4 0 1 2 3 4 B|| [233] B|| [233] (b) (c)

Figure 7.5: (a) Conductance of QPC233 vs VSG for different values of B [233] (step size 0.2 T). Traces offset by multiples of 0.05 V from the left to the right; (b) Transconductance of QPC233 vs magnetic field B on the x axis and side gate voltage VSG on the y axis. Blue colour corresponds to plateaus in conductance and red to risers between conductance plateaus; (c) Transconductance of QPC233 vs B on the x axis and conductance G4T on the y axis. The colours are the same as in Fig. 7.5a. 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires 151

This is because one of the voltage probes stopped working on the second cool-down, therefore the actual measurement is three-terminal. Moreover, the ohmics became slightly nonlinear and degraded in magnetic field faster than during the first cool- down. This would pose a problem if we would require absolute values of conductance, but we are interested only in whether the subbands split in magnetic field. As can be seen from the raw data in Fig. 7.6a, the plateaus do not split up to the highest measured field, with the possible exception of subband 6. As before, Figs. 7.6b and 7.6c present colour maps of the transconductance. No splitting of the subbands is detectable with the possible exception of subband 6. How- ever due to poor ohmics on the second cool-down, high conductance data at high fields is too noisy (see Fig. 7.6a) to enable us to resolve the spin splitting for subband 6. Therefore we assume that subband 6 does not split. Because all subbands do not split in magnetic field, the values of BC as well as the splitting rates cannot be directly ∗ measured and the minimum value of Bmin =4.2 T is taken for all subbands. Table 7.2 summarizes the results for QPC233 in the two orthogonal directions of magnetic field. Analogous to the 2D case, assuming that Zeeman splitting is linear, ∗ ∗ B ()/B (⊥)=BC ()/BC (⊥)=g⊥/g, and therefore by plotting this ratios we can observe how anisotropic the in-plane g-factor is. As discussed in Chapter 4, the g- factor anisotropy for the 1D wire fabricated on modulation doped heterostructure and oriented along [233] was found to be g/g⊥ > 4.5 and this ratio was independent of the subband index. In contrast, as can be seen from Table 7.2, the data for the induced 1D wire oriented along the same crystallographic axis, exhibits nonmonotonous behaviour.

∗ ∗ Levels B () BC () B (⊥) BC (⊥) ∂VSG/∂B 2 0.8 >4.2 > 4.2 >4.2 19.787 × 10−3 3 1.6 >4.2 >4.2 >4.2 14.974 × 10−3 4 3 >4.2 >4.2 >4.2 - 5 >4.2 >4.2 >4.2 >4.2 - 6 1.8 >4.2 >4.2 >4.2 26.453 × 10−3 7 - 3.2 - >4.2 -

Table 7.2: The fields at which the subbands start to split B∗, the crossing fields BC and splitting rates ∂VSG/∂B for 1D subbands in QPC233 for the two orthogonal in-plane field directions: B (B [233]) and B⊥ (B [011]).

∗ ∗ Figure 7.7 presents the ratio g⊥/g estimated from both B ()/B (⊥) and BC ( ∗ ∗ )/BC (⊥) as a function of subband index for QPC233. The ratio B ()/B (⊥) (red trace) is nonmonotonous, reﬂecting the nonuniform splitting of diﬀerent subbands in

B (B [233]) because no splitting is seen in B⊥. For example, subband 2 starts 152 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires

QPC||[233] 6 /h) 2 4 (2e B⊥ = 4.2 T 4T B⊥ = 0 T G 2

0 0.0 0.5 1.0 1.5 2.0

VTG (V) (a)

0.0 7

0.2 6

5 0.4

/h) 4 2

(V) 0.6 (2e TG 3 4T V

0.8 G 2 1.0 1

1.2 0 0 1 2 3 4 0 1 2 3 4 B|| [011] B|| [011] (b) (c)

Figure 7.6: (a) Conductance of QPC233 vs VSG for different values of B [011] (step size 0.2 T). Traces offset by multiples of 0.05 V from the left to the right; (b) Transconductance of QPC233 vs magnetic field B on the x axis and side gate voltage VSG on the y axis. Blue colour corresponds to plateaus in conductance and red to risers between conductance plateaus; (c) Transconductance of QPC233 vs B on the x axis and conductance G4T on the y axis. The colours are the same as in Fig. 7.6a. 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires 153

to split in B at 0.8 T but no splitting is seen for B⊥ up to 4.2 T. Therefore the ratio g⊥/g is below 0.2. On contrary, subband 5 does not split up to 4.2 T for both orientations of magnetic ﬁeld, therefore the ratio g⊥/g can not be estimated and only the upper bound can be placed to be 1. The ratio BC ()/BC (⊥) (blue trace) has little information (BC ()/BC (⊥) = 1), as the crossings of the subbands for both directions of magnetic ﬁeld cannot be observed before 4.2 T, with the only exception of the crossing between subbands 6 & 7.

QPC233 B*(||) / B*(⊥) 1.2 ⊥ BC(||) / BC( )

|| 0.8 / g ⊥ g 0.4

0.0 1 2 3 4 5 6 7 Subband Index

∗ ∗ Figure 7.7: The ratio of eﬀective g-factors g⊥/g estimated from both B ()/B (⊥) (red trace, open squares) and BC ()/BC (⊥) (blue trace, open triangles) as a function of subband index for QPC233. The markers correspond to the maximum estimated ratio of g⊥/g and the error bars to zero indicate possible values of the ratio.

To explain this nonuniform splitting, we refer to the theoretical work of Zülicke [4], who indicated that confinement alters the position of HH and LH levels in the 1D system, thereby modifying the coupling between the HH and LH bands. This leads to a renormalization of the effective g-factor. Moreover, recently Csontos et al. [13] showed that in a cylindrical quantum wire the g-factor fluctuates dramatically as a function of the subband index. This behaviour is due to the confinement potential, which is different from the square well potential used in previous calculations [4]. Therefore it is possible that the difference between our data in this chapter and the data in Chapter 4 is due to a different confinement potential in the 1D wires under study.

7.3.4 Results for QPC011

Figure 7.8a shows the four terminal conductance for QPC011 for B (B [011]).

Traces are offset from right (B = 0 T) to left (B =4.2 T) in multiples of 0.05 V. In contrast to QPC233, where in B most of the subbands split before 2 T, for QPC11 154 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires the splitting is seen at higher magnetic fields, where it is hidden by the noise. Note that this data was taken on the second cool-down and the ohmics degraded at lower magnetic fields than on the first cool-down, reducing signal to noise ratio. The colour maps of the transconductance give more information on the evolution of the subbands, as shown in Figs. 7.8b and 7.8c. Subbands 2−6 split in the magnetic field, whereas for subband 1 it is unclear if it splits because it is masked by resonant features, which also evolve in magnetic field. Values of BC cannot be extracted directly for both lower and higher subbands even though the colour maps suggest that subbands 4&5and5&6cross before 4.2 T. The raw data at high conductance in high magnetic field is too noisy to distinguish splitting. Therefore only values of B∗, when the subbands start to split are extracted from the colour maps in B.

Figure 7.9a presents the four terminal conductance of QPC011 for B⊥ (B [233]).

In contrast to the B case and opposite to QPC233, spin splitting appears at much smaller ﬁelds for all subbands in B⊥. As before, Figs. 7.9b and 7.9c show the colour maps of transconductance, highlighting the evolution of the plateaus. The crossings

BC of the subbands 2 & 3, 3 & 4, 4 & 5 and 5 & 6 were measured directly from the colour maps as well as the values of B∗ for each of the subbands. Additionally we have ∂VSG − extracted the subbands splitting rates ( ∂B ) for subbands 2 5. Finally, Table 7.3 summarizes the magnetic field measurements for QPC011 by showing extracted values of the magnetic fields when the subbands start to split B∗ and the crossing fields BC for QPC011 in two orthogonal field directions. As can be seen from the table, the values of B∗() are consistently smaller than B∗(⊥) for most of the subbands. The same trend is seen for the values of BC . This means that the g-factor anisotropy is opposite to that found in QPC233.

∗ ∗ Levels B () BC () B (⊥) BC (⊥) ∂VSG/∂B 2 1.6 >4.2 0.8 >4.2 24.286 × 10−3 3 2.8 >4.2 1.0 3.6 37.425 × 10−3 4 2.8 >4.2 1.2 3.4 56.425 × 10−3 5 2.4 >4.2 1.2 3.2 40.925 × 10−3 6 2 >4.2 2 3.6 -

∗ Table 7.3: The fields at which the subbands start to split B , the crossing fields BC and the splitting rates ∂VSG/∂B for 1D subbands in QPC011 for the two orthogonal field directions: B (B [011]) and B⊥ (B [233]).

∗ ∗ Figure 7.10 presents the ratio g/g⊥ estimated from both B (⊥)/B () and BC (⊥ ∗ ∗ )/BC () as a function of a subband index for QPC011. The ratio B (⊥)/B () (red trace) generally below 0.5 and only subband 6 splits similarly in the two orthogonal

ﬁeld directions. The ratio BC (⊥)/BC () (blue trace) again has little information on 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires 155

8 QPC || [011] 6 /h) 2 B = 4.2 T 4 || (2e B|| =0T 4T G 2

0 0.0 0.5 1.0 1.5 2.0

VTG (V) (a)

0.0 7

0.2 6

5 0.4

/h) 4 0.6 2 (V) (2e

SG 3 4T V 0.8 G 2 1.0 1 1.2 0 0 1 2 3 4 0 1 2 3 4 B|| [011] B|| [011] (b) (c)

Figure 7.8: (a) Conductance of QPC011 vs VSG for different values of B [011] (step size 0.2 T). Traces offset by multiples of 0.05 V from the left to the right; (b) Transconductance of QPC011 vs magnetic field B on the x axis and side gate voltage VSG on the y axis. Blue colour corresponds to plateaus in conductance and red to risers between conductance plateaus; (c) Transconductance of QPC011 vs B on the x axis and conductance G4T on the y axis. The colours are the same as in Fig. 7.8a. 156 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires

QPC||[011] 6 /h) 2 4 B⊥ = 4.2 T (2e B⊥ = 0 T 4T

G 2

0 0.0 0.5 1.0 1.5 2.0 VSG (V) (a)

0.0 7

0.2 6

0.4 5

/h) 4 0.6 2 (V) (2e SG

V 3 0.8 4T G 2 1.0 1 1.2 0 0 1 2 3 4 0 1 2 3 4 B|| [233] B|| [233] (b) (c)

Figure 7.9: (a) Conductance of QPC011 vs VSG for different values of B [233] (step size 0.2 T). Traces offset by multiples of 0.05 V from the left to the right; (b) Transconductance of QPC011 vs magnetic field B on the x axis and side gate voltage VSG on the y axis. Blue colour corresponds to plateaus in conductance and red to risers between conductance plateaus; (c) Transconductance of QPC011 vs B on the x axis and conductance G4T on the y axis. The colours are the same as in Fig. 7.9a. 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires 157

the spin splitting as BC (⊥)/BC () ≈ 1. Note that the ratio of the g-factors for QPC011 is in a good agreement with Rokhinson’s unpublished data [1].

QPC011 B*(⊥) / B*(||) 1.2 ⊥ BC( ) / BC(||)

0.8 ⊥ / g || g 0.4

0.0 1 2 3 4 5 6 7 Subband Index

∗ ∗ Figure 7.10: The ratio of eﬀective g-factors g/g⊥ estimated from both B (⊥)/B () (red trace, open squares) and BC (⊥)/BC () (blue trace, open triangles) as a function of a subband index for QPC011. The markers correspond to the maximum estimated ratio of g⊥/g and the error bars to zero indicate possible values of the ratio.

Summarizing the results for the two wires QPC233 and QPC011, we showed that for QPC233 the ratio g/g⊥ is non monotonous exceeding 5.2 for the second subband, which is in a good agreement with our previous data in Chapter 4. The different trend in the g-factor ratio can be explained by the difference in confinement potential between the two different types of 1D wires. In marked contrast to our previous data in Chapter 4 and the data for QPC233, QPC011 shows the opposite g-factor anisotropy with g⊥/g ≥ 2.

As discussed in Chapters 2 and 4, the enhanced anisotropy of the g-factors for the [233] oriented 1D wire (i.e. QPC233) can be theoretically explained by reorientation of the total orbital momentum vector Jˆ from perpendicular to the 2D plane to in-plane and parallel to the wire, when the wire is defined. Hence, the g-factor along Jˆ will be large and the g-factor perpendicular to Jˆ will be small. That implies that when the wire is oriented along the [011] direction (i.e. QPC011) g should be much larger than g⊥. Our data for QPC011 shows opposite behaviour. This poses the question of what mechanism defines the spin splitting in the 1D hole wires along the [011]: the 1D confinement or the crystal anisotropy? To answer this question, we have to analyze absolute values of g-factors, which has not been done in Ref. [1]. 158 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires

7.3.5 Calculations of the g-factors and discussion of the results

Using the methods described in Chapter 4 we have calculated the eﬀective g-factors for QPC233 (data in Tables 7.1 and 7.2) and QPC011 (data in Tables 7.1 and 7.3) in two orthogonal directions of an in-plane magnetic ﬁeld (see Fig. 7.11).

Figure 7.11a shows the effective g-factor for QPC233 in B (B [233]). g-factors for subbands 2, 3 and 6 have been calculated from the splitting rates of the subbands in magnetic field and source-drain voltage. The maximum error, obtained from the linear fit to the splittings is calculated to be ±0.12. The average g-factor for subbands 6 & 7 is calculated from the crossings of the subbands in magnetic field and source- drain voltage. The maximum error in the calculation of the g-factors, consists of both the uncertainty in measured subband spacings (±10 μV) and in BC (±0.2 T), and is found to be ±0.07. The g-factors for subbands 4 and 5 have been calculated separately as follows. Because subband 5 does not exhibit any splitting up to the maximum measured field of 4.2 T in both field directions, we assume it to be between zero and maximum possible value, which we estimated and showed as an error bar. For all subbands ∗ ∗ which exhibit splittings we have calculated the smallest energy ΔE = gμBB , which we able to resolve from the colour maps. We found ΔE∗ to be in the range 35−40 μeV depending on the subband. Now we use this value to estimate the maximum possible g-factor for the subbands which do not show spin splitting. The minimum value of B∗ for subband 5 is 4.2 T. Using the smallest ΔE∗ =35μeV and B∗ =4.2T,wehave calculated the maximum possible g-factor for subband 5 to be 0.15. Using the same considerations we have calculated g-factor for subband 4 to be 0.21. The error in the estimation of the g-factor for subband 4, which rises due to varying ΔE∗, is smaller than the previously calculated error of ±0.12.

Figure 7.11b shows the eﬀective g-factors for QPC233 in B⊥ (B [011]). Because ∗ all of the subbands do not split up to the highest ﬁeld of 4.2TinB⊥, we set g⊥ =0 for all subbands. However, as discussed above, we can estimate the maximum possible ∗ B (⊥) g values from the ratio ∗ = . We have marked those maximum values by error B () g⊥ bars at the corresponding subband indexes.

Figure 7.11c shows the effective g-factors for QPC011 in B (B [011]). Because the subbands split only at higher fields, values of BC cannot be extracted. Moreover, the accurate determination of the splitting rates is also hindered. Therefore we can ∗ ∗ ∗ only extract fields B and estimate the g-factors from the ratio B ()/B (⊥)=g⊥/g. The error bars correspond to the maximum error for calculation of g⊥.

Figure 7.11d shows the effective g-factors for QPC011 in B⊥ (B [233]). g-factors for subbands 2 − 5 have been calculated from the splitting rates of the subbands in 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires 159 magnetic field and source-drain voltage. The maximum error, obtained from the linear fit to the splittings is calculated to be ±0.12. The average g-factor for subbands 2 & 3,3&4,4&5and5&6iscalculated from the crossings of the subbands in magnetic field and source-drain voltage. The maximum error in calculation of g-factors, which consists of both the uncertainty in the measured subband spacings (±10 μV) and in

BC (±0.2 T), is found to be ±0.07. By looking at Fig. 7.11, we can see two distinctive features of the data:

X The eﬀective g-factors with B [011] are anisotropic relative to the wire direc-

tion, with a minimum possible ratio gQP C011/gQP C233 ∼ 2.

X The eﬀective g-factors with B [233] are generally isotropic relative to the wire direction and much higher than with B [011]. They also exceed the 2D value of 0.6 for a number of subbands.

The behaviour of the g-factors in B [011] can be explained by the same argument we used in Chapter 4. Indeed we can see that the values of the g-factor for QPC011 are consistently higher than for QPC233. This suggests that the g-factor is enhanced when the field direction is aligned with the wire and suppressed when the field direction is perpendicular to the wire. This is in a good agreement with the reorientation of the total orbital momentum vector Jˆ from perpendicular to the 2D plane to in-plane and parallel to the wire. So when B Jˆ, the g-factor is large, and when B ⊥ Jˆ, the g-factor is small. The g-factors in B [233] are much higher than those in B [011] irrespective of the wire direction. Clearly, this isotropic behaviour of the g-factor cannot be explained by the reorientation of the total orbital momentum vector Jˆ from perpendicular to the 2D plane to the parallel along the wire. The high values of the g-factors in 1D electron systems were previously explained in terms of enhanced exchange interaction at low carrier densities [14]. In contrast, the recent work of Winkler [3] argues that the exchange interaction is suppressed for 2D heavy holes in (100) GaAs/AlGaAs systems. Previously, Winkler [15] showed that 3 the 4 × 4 spin density matrix of J = 2 holes can be transformed into a sum of the multipoles, where the monopole is the total hole density, the dipole corresponds to spin polarization, quadrupole quantifies the HH-LH splitting and the octupole has no equivalent in electron systems. Analyzing the data for (100) grown GaAs/AlGaAs 2D hole systems Winkler et al. [3] found that the dipole moment is much smaller than 1 in an in-plane magnetic field, even when one of the spin split subbands is completely depopulated. However the authors noticed that for low symmetry growth directions, like (311), a large dipole moment exists. This result can also be traced to the fact that 160 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires

QPC233 (a) B||233 0.5

0.0 QPC233 (b) B||011 0.5

g* 0.0 QPC011 (c) B||011 0.5

0.0 QPC011 (d) B||233 0.5

0.0 1 2 3 4 5 6 7 Subband Index

Figure 7.11: Calculated effective g-factors for (a) QPC233 in B [233], (b) QPC233 in B [011], (c) for QPC011 in B [011], (d) for QPC011 in B [233]. Solid circles correspond to calculated values whereas open circles represent estimated values. 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires 161 the in-plane magnetic field B [233] results in an off-diagonal Zeeman term, which tries to align the spin perpendicular to the 2D plane direction [2]. The existence of the dipole moment removes the suppression of the exchange interaction and therefore the g-factor for B [233] becomes large. In contrast, for B [011], there is no off-diagonal Zeeman term and no dipole moment, therefore exchange interactions are suppressed. Hence the Zeeman splitting is only defined by the orientation of Jˆ. To summarize the discussion in this chapter we present a simple picture of the Zeeman splitting in 1D hole wires fabricated on (311) grown heterostructures. In 2D hole systems, the 2D confinement defines the direction of total orbital momentum Jˆ perpendicular to the 2D plane. When the magnetic field is parallel to Jˆ, the g-factor is large, and when the magnetic field is in the 2D plane, the g-factor is small. By defining the wire, the vector Jˆ is forced to align along the wire. For the case of B [011], when the wire (and Jˆ) is oriented along the field, the g-factor will be enhanced compared to the g-factor for the wire perpendicular to the field direction. For the case of B [233] again the 1D confinement rotates Jˆ along the wire, but the off-diagonal Zeeman term tries to keep Jˆ along the direction perpendicular to the 2D plane. Our data confirms that the second mechanism is stronger and therefore Jˆ has a component along the direction perpendicular to the 2D plane independent of the wire direction. Moreover, the suppression of the exchange interaction is removed and therefore the g-factor is large.

7.4 Conclusions and future work

In this chapter we presented measurements of the Zeeman splitting in two orthogonal induced 1D wires. The anisotropy of the g-factors for the [233] wire exhibits nonmonotonous behaviour with subband index, with the ratio g/g⊥ > 5.2 for lower subbands. The nonmonotonous trend may be explained by the different confinement potential of the induced wire compared to the sample used in Chapter 4. The data for the 1D wire along [011] shows a reversed anisotropy with the ratio g⊥/g ≥ 2, which is in contrast to our results for the wire oriented along the [233] direction. To explain these two different results we consider the absolute values of the g-factors relative to the magnetic field direction. We show that for B [011] the g-factor anisotropy can be explained by reorientation of vector Jˆ from the perpendicular to the 2D plane to in-plane and parallel to the wire, whereas for B [233] the additional off-diagonal Zeeman term leads to the out-of-plane component of Jˆ, and as a result, to enhanced g-factors irrespective of the wire direction due to increased exchange interactions. As a future work it would be useful to improve the ohmic contacts further, so that the higher magnetic fields would be accessible, which will allow better estimates of the 162 7. Anisotropic Zeeman splitting in orthogonal induced hole 1D wires minimum values of the g-factors to be found. It would also be interesting to study the Zeeman splitting in 1D hole wires with different confinement potential, for example, 1D wires grown by the cleaved edge overgrown technique [16]. Bibliography

[1] L. P. Rokhinson, D. C. Tsui, L. N. Pfeiffer and K. W. West, Modification of the effective g-factor by lateral confinement, Abstract of EP2DS 2002 Conference (2002).

[2] R. Winkler, Spin-orbit coupling eﬀects in two-dimensional electron and hole systems (Springer-Verlag, Berlin, Germany, 2003), 1st ed.

[3] R. Winkler, E. Tutuc, S. J. Papadakis, S. Melinte, M. Shayegan, D. Wasserman and S. A. Lyon, Anomalous spin polarization of GaAs two-dimensional hole systems, Phys. Rev. B, 72(19), 195321 (2005).

[4] U. Z¨ulicke, Electronic and spin properties of hole point contacts, Phys. Stat. Sol C, 3(12), 4354 (2006).

[5] S. J. Papadakis, E. P. de Poortere, M. Shayegan and R. Winkler, Anisotropic magnetoresistance of two-dimensional holes in GaAs, Phys. Rev. Lett., 84(24), 5592 (2000).

[6] R. Winkler, S. J. Papadakis, E. P. de Poortere and M. Shayegan, Highly anisotropic g-factor of two-dimensional hole systems, Phys. Rev. Lett., 85(21), 4574 (2000).

[7] R. Danneau, O. Klochan, W. R. Clarke, L. H. Ho, A. P. Micolich, M. Y. Simmons, A. R. Hamilton, M. Pepper, D. A. Ritchie and U. Z¨ulicke, Zeeman splitting in ballistic hole quantum wires, Phys. Rev. Lett., 97, 026403 (2006).

[8] W. R. Clarke, A. P. Micolich, A. R. Hamilton, M. Y. Simmons, K. Muraki and Y. Hirayama, Fabrication of induced two-dimensional hole systems on (311)A GaAs, J. Appl. Phys., 99, 023707 (2006).

[9] Y. Hanein, H. Shtrikman and U. Meirav, Very low density two-dimensional hole gas in an inverted GaAs/AlAs interface, Appl. Phys. Lett., 70(11), 1426 (1997).

163 164 BIBLIOGRAPHY

[10] M. Y. Simmons, A. R. Hamilton, S. J. Stevens, D. A. Ritchie, M. Pepper and A. Kurobe, Fabrication of high mobility in situ back-gated (311)A hole gas heterojunctions, Appl. Phys. Lett., 70(20), 2750 (1997).

[11] M. Wassermeir, J. Sudijono, M. Johnson, K. Leung, B. Orr, L. D¨aweritz and K. Ploog, Reconstruction of the GaAs (311)A Surface, Phys. Rev. B, 51(20), 14721 (1995).

[12] W. Braun, Applied RHEED: Reﬂection high-energy electron diﬀraction during crystal growth, vol. 154 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, Germany, 1999).

[13] D. Csontos and U. Z¨ulicke, Anatomy of anomalous spin splitting in hole quantum wires, cond-mat/0703797v1.

[14] K. J. Thomas, J. T. Nicholls, M. Y. Simmons, M. Pepper, D. R. Mace and D. A. Ritchie, Possible spin polarization in a one-dimensional electron gas, Phys. Rev. Lett., 77(1), 135 (1996).

3 [15] R. Winkler, Spin-density matrix of spin- 2 hole systems, Phys. Rev. B, 70, 125301 (2004).

[16] L. N. Pfeiﬀer, R. de Picciotto, K. W. West, K. W. Baldwin and C. H. L. Quay, Bal- listic hole transport in a quantum wire, Appl. Phys. Lett., 87(7), 073111 (2005). Chapter 8

Thesis summary and future work

8.1 Thesis summary

In this thesis we have studied ballistic transport in GaAs one-dimensional hole systems and in particular their behavior in an in-plane magnetic field. Here we present the general and overall conclusions of the research. In Chapter 4 we studied how the 1D confinement alters the 2D anisotropic Zee- man splitting. We performed transport measurements in ballistic hole quantum wires fabricated on a (311)A modulation doped heterostructure and oriented along the [233] crystallographic axis. Directly measured effective g-factors for the two orthogonal ∗ ∗ in-plane field directions show pronounced anisotropy. The ratio g/g⊥ > 4.5 is significantly higher than in the two-dimensional case [1, 2]. This enhanced anisotropy can be explained by the reorientation of the total angular momentum Jˆ from perpendicular to the 2D plane to in-plane and parallel to the wire. This implies that in contrast to the 2D case, where the in-plane g-factor depends only on the orientation of magnetic field, in a 1D wire the in-plane g-factor depends only on the wire direction. In Chapter 5 we investigated the behavior of the 0.7 structure and the zero bias anomaly (ZBA) in ballistic hole wires fabricated on (311)A modulation doped heterostructure with the application of an in-plane magnetic field. Using the highly anisotropic in-plane g-factor in 1D hole systems, we found that the magnetic field dependencies of the 0.7 structure and the ZBA are distinctly anisotropic and correlated to each other. This indicates that the two phenomena are intimately related and have the same spin related origin. The anisotropy of the evolution of the 0.7 structure in the two orthogonal directions of the in-plane field is reduced compared to the integer subbands. This result is in a good agreement with recent theoretical calculations of the evolution of the 2D bands into 1D bands as the wire is narrowed [3]. The g-factors for the two orthogonal directions are expected to cross due to mixing of the heavy hole

165 166 8. Thesis summary and future work and light hole quasi 1D bands, which is caused by the increasing 1D confinement. In Chapter 6 we presented the characterization of a new type of 1D wire fabricated using a GaAs/AlGaAs SISFET structure with no modulation doping. The device exhibits clear conductance quantization, including the 0.7 structure, and has extremely stable gate characteristics. We present initial measurements of the dependence of the 1D conductance on density, temperature, source-drain bias and magnetic field. The hole density in devices of this type can be varied over a wide range, while the mobility of holes remains high enough to observe conductance quantization. Finally, in Chapter 7 we investigated the Zeeman splitting in two orthogonal undoped 1D wires, which has been fabricated on the same sample, with the application of magnetic field in the two orthogonal directions. This study allowed us to investigate the effect of the 1D confinement and crystallographic anisotropy on the Zeeman splitting of the 1D subbands. The ratio of the in-plane g-factors for the wire oriented along [233] is in a general agreement with the results from Chapter 4, whereas the difference in the absolute values may be due to different confinement potential between the samples used in Chapter 4 (square quantum well) and Chapter 7 (triangular quantum well). In contrast, the ratio of the in-plane g-factors for the wire oriented along [011] is reversed. To explain this discrepancy the analysis of the absolute values of the g-factors was performed. We show that for B [011], that the g-factor is large for the wire along [011] and small for the wire along [233], whereas for B [233], the g-factor is large irrespective of the wire direction. The former result can be explained by the reorientation of the total angular momentum vector Jˆ from perpendicular to the 2D plane to in-plane and parallel to the wire, and the latter by the additional off-diagonal Zeeman term [4], which causes a spin polarization perpendicular to the 2D plane when B [233], hence exchange interactions are allowed [5]. This results in an enhancement of the g-factors irrespective of the wire direction for B [233].

8.2 Future work

Although the results in Chapters 4 and 7 signiﬁcantly clariﬁed the mechanisms of the Zeeman splitting in 1D hole wires, there are still some issues to be addressed:

1. The effect of the confinement potential on the 1D effective g-factors. This work can be done by comparing the hole systems with different confinement potential or by using a combination of the top and back gates to change the confinement while keeping the hole density constant.

2. Additionally, the eﬀects of 2D to 1D heavy hole- light hole band mixing can be studied in 1D hole wires with a stronger 1D conﬁnement potential. This would 8. Thesis summary and future work 167

allow for example, the ratios of the g-factors in two orthogonal directions of an in-plane magnetic ﬁeld for two 1D subbands to be reversed. This is possible if

one of these two subbands corresponds to Wx/Wy before the crossover point and

the other subband corresponds to Wx/Wy after the crossover point [3].

3. The Zeeman splitting in 1D hole wires grown on (100) heterostructures remains unexplored. Although we have mentioned that in the 2D case for high symmetry growth directions such as (100) the in-plane g-factor is suppressed, the reorientation of the vector Jˆ along the wire in 1D, suggests that the g- factors along the wire should be measurable within the experimentally accessible range of magnetic ﬁeld. Moreover, the in-plane g-factor along the wire should always be large irrespective of the wire direction. This work is also possible due to the carbon doping technique [6, 7], which allows high mobility (100) oriented GaAs/AlGaAs heterostructures to be grown.

Finally, the work on the 0.7 structure in 1D hole systems should be continued, especially on the origins of the zero bias anomaly. In contrast to Ref. [8], our result shows that the zero-bias anomaly is related to spin splitting and therefore its origin is not necessarily related to the Kondo eﬀect. Additional work on zero-dimensional hole systems should be done to see if the Kondo eﬀect can be observed in hole systems. Bibliography

[1] R. Winkler, S. J. Papadakis, E. P. de Poortere and M. Shayegan, Highly anisotropic g-factor of two-dimensional hole systems, Phys. Rev. Lett., 85(21), 4574 (2000).

[2] S. J. Papadakis, E. P. de Poortere, M. Shayegan and R. Winkler, Anisotropic magnetoresistance of two-dimensional holes in GaAs, Phys. Rev. Lett., 84(24), 5592 (2000).

[3] U. Z¨ulicke, Electronic and spin properties of hole point contacts, Phys. Stat. Sol C, 3(12), 4354 (2006).

[4] R. Winkler, Spin-orbit coupling eﬀects in two-dimensional electron and hole systems (Springer-Verlag, Berlin, Germany, 2003), 1st ed.

[5] R. Winkler, E. Tutuc, S. J. Papadakis, S. Melinte, M. Shayegan, D. Wasserman and S. A. Lyon, Anomalous spin polarization of GaAs two-dimensional hole systems, Phys. Rev. B, 72(19), 195321 (2005).

[6] M. J. Manfra, L. N. Pfeiﬀer, K. W. West, R. de Picciotto and K. W. Baldwin, High mobility two-dimensional hole system in GaAs/AlGaAs quantum wells grown on (100) GaAs substrates, Appl. Phys. Lett., 86, 162106 (2005).

[7] C. Gerl, S. Schmult, H.-P. Tranitz, C. Mitzkus and W. Wegscheider, Carbon-doped symmetric GaAs/AlGaAs quantum wells with hole mobilities beyond 106 cm2/Vs, Appl. Phys. Lett., 86, 252105 (2005).

[8] S. M. Cronenwett, H. J. Lynch, D. Goldhaber-Gordon, L. P. Kouwenhoven, C. M. Marcus, K. Hirose, N. S. Wingreen and V. Umansky, Low-temperature fate of the 0.7 structure in a point contact: a Kondo-like correlated state in an open system, Phys. Rev. Lett, 88(22), 226805 (2002).

168