Hole Spins in Gaas Quantum Dots

University of New South Wales, Sydney

School of Physics

Hole Spins in GaAs Quantum Dots

Qingwen Wang A thesis submitted in fulﬁllment of the requirement for the degree of Doctor of Philosophy September 2015 2 Acknowledgements

Foremost, my most sincere thanks go to my supervisor Alex Hamilton, for his continuous support during the past four years, for his patience, guidance and encour- agement. Alex always has endless interesting ideas and suggestions about the experiments, without which I would not be able to get the exciting results in this thesis. Even though I sometimes make fun of his hairstyle and ”fashion” taste, I really enjoy talking with Alex, and listening to his ”grandpa” stories. Secondly I wish to thank my co-supervisor Oleh Klochan, without whom I would not be able to make any working devices or know how to run the fridge. Thanks for giving me so many useful suggestions on fabrication, listening to me complaining about everything in the lab, and keeping all the equipments working. I will let beer to express my appreciation when you come back form Ukraine. I would also like to thank everyone else now or formerly in the QED group for making life at UNSW enjoyable: Jason Chen, Andrew See, Jack Cochrane, Ashwin Srinivasan, LaReine Yeoh, Sarah Macleod, Roy Li, Karina Hudson, Scott Liles, Matt Rendell and Elizabeth Marcellina. Special thanks to people who helped me during this project: thanks to Jason Chen and Andrew See for showing me how to do fabrication; thanks to Fay Hudson who helped me with the EBL alignment; thanks to Oleg Sushkov, Dmitry Miserev, Dimitrie Culcer and Elizabeth Marcellina for useful discussions on hole systems. Finally, special thanks go to my parents and friends who are always there for me. Without you, my life would not have been so happy.

3 4 Abstract

In this thesis, we report a new architecture for making lateral hole quantum dots based on shallow accumulation-mode AlGaAs/GaAs heterostructures. Utilizing a double-level-gate architecture, we demonstrate the operation of ultra-small single and double quantum dots in the few-hole regime using electrical measurements. Devices with different dimensions and layouts are tested to reach the single-hole limit. With the flexibility of the double-level-gate architecture, both single and double quantum dots can be formed within the same device with good tunability. With the ability to isolate a single heavy-hole spin, we study the Zeeman splitting of the orbital states in different field orientations via magnetospectroscopy measurements. The extracted value of the hole effective g-factor is found to be strongly dependent on the orbital state, and highly anisotropic with respect to the magnetic field direction. We show that these peculiar behaviours of the heavy-hole spin can be qualitatively explained by the effects of strong spin-orbit coupling and strong Coulomb interactions in hole systems. By varying the dot size in situ, we also demonstrate the tuning of the g-factor anisotropy, and estimate the shape of electrically-defined quantum dot. With the few-hole double quantum dot, we present the first observation of Pauli spin blockade in GaAs hole quantum dots. Utilizing a vector field magnet system, we study the lifting of spin blockade due to the spin-orbit interaction. We found that the effect of spin-orbit coupling on spin blockade to be highly anisotropic in different magnetic field orientations, which agrees with previous theoretical predictions on systems with strong spin-orbit coupling. From the anisotropic lifting of spin blockade, we identify the orientation of the effective spin-orbit field to be along the transport direction, which is very different from experimental results from electron quantum dots and highlights the uniqueness of hole systems. 2 Contents

1 Introduction and Thesis Outline 7 1.1 Introduction ...... 7 1.2 Thesis Outline ...... 8

2 Background Chapter 10 2.1 Introduction ...... 10 2.2 2D systems in GaAs/AlGaAs heterostructures ...... 10 2.2.1 Modulation-doped heterostructures ...... 10 2.2.2 Undoped accumulation-mode heterostructures ...... 12 2.3 Holes in GaAs/AlGaAs heterostructure ...... 14 2.3.1 Valence band structure in GaAs ...... 15 2.3.2 Large hole eﬀective mass ...... 17 2.3.3 Hyperﬁne interaction ...... 18 2.3.4 Spin-orbit interaction ...... 20 2.3.5 The Zeeman splitting in 2D hole systems ...... 21 2.4 Single quantum dots ...... 22 2.4.1 Constant-interaction model ...... 23 2.4.2 Coulomb blockade ...... 25 2.4.3 Bias spectroscopy ...... 26 2.4.4 Spin states in a single dot ...... 29 2.5 Double quantum dots ...... 32 2.5.1 Charge stability diagram ...... 32 2.5.2 Bias triangles ...... 36 2.5.3 Pauli spin blockade ...... 38 2.5.4 The lifting of spin blockade ...... 42

3 Device Fabrication and Measurement Setup 46 3.1 Introduction ...... 46 3.2 Device fabrication ...... 46

3 4 CONTENTS

3.2.1 Device structure and operation principles ...... 47 3.2.2 Dielectric material ...... 49 3.2.3 Shallow wafers ...... 52 3.2.4 Electron Beam Lithography ...... 53 3.3 Measurement setup ...... 55

4 Double-level-gate Architecture for Few-hole GaAs Quantum Dots 58 4.1 Introduction ...... 58 4.2 Literature Review ...... 59 4.3 Nanowire-inspired quantum dot on a planar GaAs heterostructure . 64 4.4 Single dot operation ...... 66 4.4.1 Coulomb blockade and bias spectroscopy ...... 66 4.4.2 Zeeman splitting and anisotropic Land´eg-factor ...... 67 4.5 Double dot operation ...... 70 4.5.1 Tunable interdot coupling ...... 70 4.5.2 Bias triangles and resonant tunnelling ...... 71 4.6 Conclusions and improvements of the design ...... 75

5 Single Hole GaAs Lateral Quantum Dots 79 5.1 Introduction ...... 79 5.2 Literature Review ...... 81 5.3 Single hole quantum dot - Experimental results ...... 87 5.3.1 Device Characterization ...... 87 5.3.2 Bias spectroscopy ...... 89 5.4 Zeeman splitting of the single hole states ...... 94 5.4.1 Suppression of the ground state g-factor ...... 95 5.4.2 Orbital dependence of the hole g-factor ...... 97 5.4.3 Observation of a fourfold degenerate orbital state ...... 99 5.5 Land´eg-factor anisotropy ...... 100 5.6 Conclusions and Future Work ...... 104

6 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots 106 6.1 Introduction ...... 106 6.2 Literature Review ...... 107 6.3 Few-hole double quantum dots - experimental results ...... 114 6.4 Pauli spin blockade ...... 116 6.5 Spin-orbit interaction lifted spin blockade ...... 117 6.6 Anisotropic lifting of spin blockade ...... 119

4 5 CONTENTS

6.7 Conclusion and Future Work ...... 125

7 Conclusions and Future Work 127 7.1 Summary of Results ...... 127 7.2 Future work ...... 128 7.2.1 Theoretical modelling ...... 128 7.2.2 Future experiments ...... 129

A Inﬂuence of surface states on quantum and transport lifetimes in high-quality undoped heterostructures 131 A.1 Introduction ...... 131 A.2 Literature Review ...... 132 A.3 Experimental results ...... 133 A.3.1 Sample and experimental setup ...... 133 A.3.2 Measurements of lifetimes from Shubnikov de Haas oscillations 133 A.4 Comparison with theory and discussions ...... 136 A.5 Predictions from theory ...... 138 A.6 Conclusions and Future work ...... 140

B Magnetospectroscopy of the two-hole states 143

C The eﬀects of barriers on the double dot transport 144

D The (1, 1) → (0, 2) transition 145

5 6 Chapter 1

Introduction and Thesis Outline

1.1 Introduction

Spintronics, i.e. spin-based semiconductor electronics, is an emerging technology which utilizes the spin of charge carriers to store and process information. Grow- ing interests in this field have placed the study of quantum mechanical properties of semiconductors in the centre of research over the past few decades. Thanks to the development of fabrication and experimental techniques, impressive progress, including spin initialization, manipulation and readout [1, 2, 3], as well as all-electric control of a single electron spin [4, 5, 6], has been realized using electron quantum dots, which are one of the most widely used semiconductor systems in quantum information processing. However, fast decoherence of electron spins due to the unavoidable hyperfine interaction with surrounding nuclear spins [7, 8, 6] still remains a major problem for most electron systems with a none-zero nuclear spin. As one possible replacement for electron spins, heavy-hole spins have drawn a lot of attention recently. Unlike conduction band electrons whose wavefunctions are constructed by s-orbitals, holes in the valence band have p-symmetry Bloch wavefunctions with zero density at the site of the nuclei, leading to a vanishing contact hyperfine interaction [8, 9]. Heavy-holes (Jz = ±3/2) consisting of pure p-orbitals are predicted to have negligible spin flip-flops, resulting in an Ising-type hyperfine interaction and potentially long spin coherence times on the scale of µs [10, 11]. Furthermore, holes also have strong spin-orbit coupling due to the non-zero orbital momentum, which provides the possibility of fast all-electric control of hole spins via the spin-orbit interaction [12]. GaAs hole quantum dots have always been a popular candidate for quantum information processing due to the possibility of implementing spin manipulation techniques that have already been developed in their electron counterparts [3, 2]. Up

7 8 Introduction and Thesis Outline to date, experimental work on GaAs hole quantum dots have been mainly conducted via optical measurements of self-assembled quantum dots. Various results of the spin ∗ decoherence time T2 [13, 14, 15] have been obtained and impurity-induced electrical noise in self-assembled quantum dots is considered to be the limiting factor of the ∗ short T2 time [14]. In addition, it is very difficult to incorporate self-assembled quantum dots into complex electric circuits for all-electric control of the hole states. Therefore, electrically isolating and measuring a single hole spin in gate-defined lateral quantum dots is main aim of this thesis. However, due to the large hole ∗ ∗ effective mass in GaAs (mh/me ∼ 6) [16], lateral hole quantum dots need to have much smaller dimensions compared to electron dots to show similar single-particle ∗ energy scales (Eorb ∼ 1/m Adot), which is hard to achieve by simply duplicating the design for electron quantum dots. Even though the operation of lateral GaAs hole quantum dots have been demonstrated [17, 18, 19], electrical isolation of a single heavy-hole spin has not been realized, and the spin properties of three-dimensionally confined heavy-hole states still remain a mystery. In this thesis, we present a new double-level architecture for fabricating ultra- small hole quantum dots on shallow accumulation-mode AlGaAs/GaAs heterostructures. We demonstrate the operation of both single and double hole quantum dots in the few-hole regime using electrical measurements. With the ability to isolate a single hole spin, we study the anisotropic Zeeman splitting of the orbital states and demonstrate the tuning of the hole g-factor anisotropy by changing the dot shape in situ. With a few-hole double quantum dot, we demonstrate the first observed Pauli spin blockade in GaAs hole dots. By studying the anisotropic lifting of the spin blockade due to spin-orbit interaction, we identify the effective spin-orbit field direction in the hole system. The demonstration of electrical isolation of a single hole spin makes a big step towards coherently manipulating heavy-hole spins in GaAs systems. This work paves the way towards all-electric control of a single heavy-hole spin in a GaAs quantum dot.

1.2 Thesis Outline

A brief outline of this thesis is as follows: Chapter 2 gives a brief introduction of background information that is relevant to the experimental work in this thesis, including the advantages of accumulation- mode AlGaAs/GaAs heterostructures, the unique properties of valence band holes in GaAs and basic transport and spin properties of quantum dots.

8 9 Introduction and Thesis Outline

Chapter 3 briefly outlines the fabrication and measurement techniques used in this thesis. Chapter 4 presents the operation principles of the new double-level-gate design. We show electrical measurements of fully gate-tunable single and double hole quantum dots formed using the same device. Chapter 5 studies the Zeeman splitting of the one-hole orbital states via magnetospectroscopy measurements. We observe a hole g-factor which is strongly dependent on the orbital state, and is highly anisotropic with respect to the magnetic field orientation. Chapter 6 studies Pauli spin blockade of holes and the effects of spin-orbit coupling in a double quantum dot. From the anisotropy of the spin blockade, we identify the direction of the effective spin-orbit field. Chapter 7 summarises the main results of this thesis and discusses possible future directions.

9 Chapter 2

Background Chapter

2.1 Introduction

In this section, we present the background information that is relevant to the experimental work in the subsequent chapters. We will first talk about the accumulation- mode heterostructures used in this thesis highlighting the advantages of undoped heterostructures over modulation-doped ones. Following this discussion, we will talk about the complexity and special properties of 2D hole systems. We will start with deriving the valence band structure in bulk GaAs and then add the 2D confinement of the heterostructures in. We will focus on the large effective mass, the hyperfine interaction and the spin-orbit interaction specifically for hole systems which are closely related to our experiments. Finally, we will review the transport and spin properties of single and double quantum dots, including the constant-interaction model, Coulomb oscillations and diamonds, one-electron and two-electron spin states for a single quantum dot, as well as the charge stability diagram, the finite source-drain bias triangles and Pauli spin blockade for a double quantum dot.

2.2 2D systems in GaAs/AlGaAs heterostructures

The focus of this thesis is lateral quantum dots based on two-dimensional hole gases (2DHGs) in GaAs/AlGaAs heterostructures. In this section, we will ﬁrst talk about how to form 2DHGs in heterostructures.

2.2.1 Modulation-doped heterostructures

One way of introducing carriers into the GaAs/AlGaAs heterointerface layer is by doping the AlGaAs region located above the heterointerface, which is called Modula-

10 11 Background Chapter tion doping. Unlike the traditional way of directly doping the region where carriers are needed, modulation doping provides a separation between free electrons or holes and the donors or acceptors. As illustrated in Figure 2.1, for a p-type doping, the introduced holes will travel across into the GaAs layer after being released, where they lose energy and become trapped due to the bandgap mismatch. Those extra charges in both layers result in an electric ﬁeld and thus bending of the valence band which acts as a potential well and conﬁnes the 2DHG. To further prevent the ionized acceptors from scattering the carrier electrons via Coulomb potential, a spacer layer of undoped AlGaAs can be added between the GaAs layer and the doped AlGaAs region.

Figure 2.1: Conduction band of a modulation-doped p-AlGaAs/GaAs heterostructure.

For many years, modulation-doped AlGaAs/GaAs heterostructures have been the centre of research on low-dimensional systems. While extremely high mobility 2D systems [20, 21, 22] have been realized in modulation-doped heterostructures [23, 24], remote ionized impurities, which can act as an additional source of disorder causing both Coulomb scattering and long-range fluctuation of the potential landscape, are introduced to the system ineluctably [25, 26, 27]. Due to these ionized impurities, the closer 2D systems are brought to the surface, the more affected electron transport is [28]. Therefore, although it is possible to attain high electron mobility in deep 2D electron systems, achieving similar mobility in shallower 2D systems, which are desirable for nanostructures with fine lithographic configurations, remains problematic.

11 12 Background Chapter

2.2.2 Undoped accumulation-mode heterostructures

An alternative way of introducing carriers into the GaAs layer is by applying an externally electric ﬁeld, which can be achieved by adding a top-gate to the heterostructure [25]. This is also the structure used in this thesis. With an external electric ﬁeld in the right direction, the conduction band will tilt down and form a triangular potential well together with the discontinuity in the valence band; the valence band holes in the GaAs layer will move towards the heterointerface, where they get trapped and form a 2DHG as shown in Figure 2.2.

Figure 2.2: An induced 2DHG based on undoped GaAs/AlGaAs heterostructure.

Recently undoped heterostructures have drawn attention due to some obvious advantages over modulation-doped heterostructures [25, 26, 27, 29, 30, 31]. Owing to the removal of intentional doping, undoped heterostructures are expected to have fewer ionized impurities, the presence of which is the predominant factor limiting the 2D transport performance of shallow modulation-doped heterostructures [28]. However, induced devices are inherently much more complicated in fabrication than modulation-doped devices, and how to achieve good ohmic contacts without shorting to the top-gate has been a problem for a long time. Kane et al. introduced a way of contacting by self-aligned etching of the GaAs and evaporating of contact metals in 1993 [25]. With this technique, they fabricated devices with variable carrier densities ranging from less than 1010 cm−2 to greater than 5 × 1011 cm−2 with mobilities comparable to high-quality modulation-doped devices [25]. Based on Kane et al.’s method, Harrell et al. demonstrated a direct comparison between the mobility of 2DEGs in induced and modulation-doped heterostructures [32]. As

12 13 Background Chapter shown in Figure 2.3, a higher mobility compared to similar modulation-doped devices, especially at low carrier densities, were achieved in undoped devices.

Figure 2.3: Comparison of mobilities over a range of carrier densities for induced and modulation-doped 2DEGs at 1.4K from Harrell et al.’s result. Figure reproduced from Ref [32].

Since then, considerable efforts have been put into developing fabrication methods for shallow undoped GaAs/AlGaAs heterostructures, which is desirable for pat- terning nanostructures. To transfer fine features from the surface metallic gates to the 2DEG, the heterointerface needs to be as shallow as possible. This is a big problem for modulation-doped devices as bringing the 2DEG closer to the surface means that the thickness of the spacer layer needs to be reduced, which sacrifices the 2DEG mobility. Therefore, induced heterostructures are considered a solution to improve the mobility of shallow modulation-doped heterostructures for nanofabrication. Mak et al. [33] recently demonstrated the fabrication of electron quantum dots on 2DEGs as shallow as 30nm below the surface, in undoped GaAs/AlGaAs heterostructures. As shown in Figure 2.4, an improvement of more than one order of magnitude in mobility(at 2 × 1011 cm−2) with respect to doped heterostructures with similar depths is observed. However, it can also be seen from Figure 2.4 that comparing the two induced electron devices, the 30 nm deep 2DEG showed a smaller mobility than the 60 nm deep one. This is due to the trapped surface charge between the wafer and the dielectric, which affects the 2DEG mobility in a manner similar to ionized dopants in modulation-doped device. As demonstrated by Wang et al. (Appendix A) [34], the effects of surface charge on the 2DEG mobility starts to show and gets significant when the 2DEG is shallower than 50 nm. Based on this calculation, heterostructures with a 2DHG depth of 60 nm are used for the all the quantum dot devices presented in this thesis to achieve the optimal compromise between the requirements of fine nanostructures and high mobility.

13 14 Background Chapter

Figure 2.4: Mobility versus electron density for undoped wafers V625 (60 nm deep) and V627 (30 nm deep), as well as various other doped wafers with shallow 2DEGs. Figure reproduced from Ref [33].

Figure 2.5: Calculated scattering lifetimes as a function of the depth of the 2DEG at ns = 1011 cm−2, evaluated using three scattering mechanisms: background impurity scattering, interface roughness scattering and remote ionized impurity scattering due to surface charge. Dashed lines are the scattering lifetimes calculated with background impurity and interface roughness scattering, while solid lines are the values calculated with scattering from surface states also included. A detailed calculation can be found in Appendix A. Figure reproduced from Ref [34].

2.3 Holes in GaAs/AlGaAs heterostructure

Up to now, we have been mainly talking about electrons in GaAs, which have been intensively studied over the past few decades. Not only the transport and spin properties of GaAs electron systems are well understood, but also the control of a single spin has been realized using GaAs electron quantum dots [1, 2, 3]. However, the short coherent times [1, 35], and the slow rotation rate [3, 4] of electron spins makes GaAs electron systems unsuitable for fast and reliable qubit operations. As one of the candidates to replace electron systems, holes in GaAs have drawn considerable

14 15 Background Chapter attention recently due to their suppressed hyperﬁne interaction and strong spin-orbit coupling. In this section, we will talk about the basic properties of holes in GaAs and the proposed advantages of hole systems over electron systems.

2.3.1 Valence band structure in GaAs

Despite the possible advantages offered by hole systems, holes are much more difficult to work with, particularly due to the complication of the valence band structure: the valence band in GaAs exhibits pronounced nonparabobicity and anisotropy. Since the complexity of the valence band is the origin of the special properties of hole systems, it is important to understand the basics of the valence band structure in GaAs. To model the complicated bandstructure, especially around k = 0, which is the region of interest for most experiments, the k · p method (where k is the wave vector and p =h ¯k is the momentum) is often used. For the Bloch function exp(ik · r)uνk(r) (where r is the position and u is a function with the same periodicity as the crystal lattice), the k · p method involves solving the Schrödingerequation [36]

2 p ik·r ik·r + V0(r) e uνk(r) = Eν(k)e uνk(r) (2.1) 2m0 where m0 is the free electron mass and ν is the band index. Since the Bloch wavefunction in the periodic lattice potential V0(r) can be written as

ik·r ik·r e uνk(r) ≡ e hr|νki, (2.2)

The Schr¨odingerequation can be simpliﬁed to

p2 ¯h2k2 ¯h + V0 + + k · p |νki = Eν(k)|νki (2.3) 2m0 2m0 m0

As can be seen from Equation 2.3, this approach results in a k · p term in the Hamil- tonian. Further extension of the model by including the Pauli spin-orbit interaction term gives the ﬁnal form of the Schr¨odinger equation as [36]

p2 ¯h2k2 ¯h ¯h + V0 + + k · π + 2 2 p · σ × (5V0) |nki = En(k)|nki (2.4) 2m0 2m0 m0 4m0c

0 1 0 −i where Pauli spin matrices σ = (σx, σy, σz) with σx = 1 0 , σy = i 0 , σz = 1 0 0 −1 , and π is deﬁned to be

¯h π = p + 2 σ × 5V0 (2.5) 4m0c

15 16 Background Chapter

Figure 2.6: The band structure of bulk GaAs calculated using a 24×24 k · p model. Figure reproduce from Ref [37]

Index n classifies different bands by the sum of spin and orbital motions, as spin by itself is no longer a good quantum number. Depending on the number of bands being considered, the k · p method requires the solving of a N-dimensional Hamiltonian matrix. Figure 2.6 shows the calculated GaAs bulk band structure using a 24 × 24 k · p model [37]. The Γ points in the graph represent the positions of k = 0 on different bands. We can see from the diagram that the lowest conduction band has a approximate parabolic dispersion around Γ6C , whereas the top valence band is highly anisotropic and non-parabolic near Γ8V . There also exists a valence band

Γ7V , which is isotropic and parabolic near k = 0, similar to the lowest conduction band. This valence band is the so-called spit-off band, which is a direct result of the spin-orbit coupling of GaAs hole systems. In a tight-binding picture, electronic states at the conduction band edge are s-like (angular momentum quantum number l = 0), but electronic states at the valence band edge are p-like (l = 1) [36]. Therefore, the valence band has two states with total angular momentum J = 3/2 and J = 1/2 respectively. (Recall that the total angular momentum J is defined as J = |L ± S| where L is the angular momentum and S is spin.) Due to spin-orbit coupling, these states with different J are split in energy by the spin-orbit gap ∆SO, resulting in a split-off band with J = 1/2 sitting at a lower energy. The bulk value of the spin- orbit gap is as large as ∆SO = 0.341 eV [38], which suggests that the split-off band is usually not populated. After discussing the complication of the valence bandstructure of bulk GaAs, we now add the 2D confinement of the heterostructure into the picture. Figure 2.7 [39] shows a qualitative sketch of the band structure near k = 0 for both bulk GaAs (a) and 2D GaAs systems (b). The 2D confinement lifts the fourfold degeneracy of J = 3/2 states near k = 0, resulting in heavy-hole (HH) states (Jz = ±3/2) and light-hole states (LH) states (Jz = ±1/2), separated in energy by the HH-LH

16 17 Background Chapter

splitting (∆lh−hh). The size of ∆lh−hh is estimated to be ∼ 10 meV in 2D systems, which is usually much larger than the Fermi energy (∼ 1 meV for p = 2 × 1011 cm−2) of the system, which suggests that the LH band is usually empty and does not contribute to electrical transport.

Figure 2.7: (a) Qualitative sketch of the band structure for bulk GaAs; (b) Qualitative sketch of the band structure for a 2D GaAs system where the HH-LH degeneracy is lifted at the band edge due to 2D conﬁnement. Figure reproduced from Ref [39].

Even though analysing the band structure of semiconductors is a task which requires considerable eﬀort, it provides understanding of a lot of fundamental properties of the semiconductor system, which we will talk about speciﬁcally for holes in GaAs in the following sections.

2.3.2 Large hole eﬀective mass

The first property of hole systems directly related to the valence band structure is the large hole effective mass. The effective mass in semiconductor materials is often defined as the curvature of the energy band d2ε(k)/d2k near the band edge. For most semiconductors, if the lowest conduction band and highest valence band can be approximated by a quadratic energy dispersion, the effective mass m∗ is simply a constant. However, since the highest valence band is highly non-parabolic in GaAs, ∗ the situation for holes is more complex: the value of mh depends on the wavevector k, and thus on the hole density. ∗ As an approximation, mh = 0.2−0.4m0 is the most commonly used value for the hole effective mass in GaAs, which is 3 - 6 times larger than the electron effective ∗ mass (me = 0.067m0). Due to the large hole effective mass, GaAs hole systems have been widely used in studying many-body effects [40, 41]. This is because the large effective mass quenches the kinetic energy and enhances interaction effects, which is

17 18 Background Chapter

characterised by the interaction parameter rs

∗ Ep m rs = ∝ √ (2.6) Ek ns where Ep and Ek are the potential and kinetic energies of the systems, and ns is the density of carriers.

The typical value of rs for a GaAs 2DEG is limited to rs ≤ 5. In contrast, for

2D hole systems, rs > 10 is easily achievable. For the hole quantum dots presented in this thesis, we estimate a rs value of ∼ 7 − 10 based on separate measurements of the 2DHG density at similar top-gate voltages. However, since strongly localised quantum dots with only a few holes inside are investigated, the local 2D conﬁnement potential could be strongly distorted to exhibit a hole density very diﬀerent from that in a 2D device. This may result in an even higher value of rs and stronger hole-hole interaction in the hole quantum dot devices measured.

2.3.3 Hyperﬁne interaction

Besides strong hole-hole interaction, another attractive property of hole systems is the weak hyperfine interaction with surrounding nuclear spins, which is the main source of decoherence of electron spins. The dominant hyperfine interaction term for electrons in 2D systems is the contact hyperfine interaction, which can be generally written as [42, 43, 44, 10, 45]:

N X e ~ ~ He = AkS · Ik k N X e x y z = Ak(SxIk + SyIk + SzIk ) (2.7) k N N 1 X X = Ae (I+S + I−S ) + Ae IzS 2 k k − k + k k z k k where S~ is the electron spin, I~k is the k-th nuclear spin (I± = Ix ± iIy and S± =

Sx ± iSy are the electron spin and the nuclear spin raising and lowering operators respectively) and 2µ Ae = 0 γ γ × |ψ(0)|2 (2.8) k 3 e n describes the hyperfine coupling strength between the electron spin to the k-th nuclear spin. (Here µ0 is the vacuum permeability, γe and γn are the gyromagnetic ratios for electrons and nuclei respectively, ψ(0) is the electron wavefunction.) The first term in Eq. 2.7 describes the spin flip-flop process when the electron spin

18 19 Background Chapter exchanges its orientation with the nuclear spin. The second term in Eq. 2.7 describes the interacting between the nuclear spins and the electron spins through the effective e PN e z Overhauser field BN = k AkIk /gµB. As discussed previously, unlike s-type electrons, valence band hole wavefunctions are predominantly p-orbitals, which vanish at the nuclear sites. This eliminates the contact hyperfine interaction leaving only the weak dipole-dipole hyperfine interaction, which has a similar form to that of electrons for light-holes (lh) [42]:

N 1 X H = Ah (I+S + I−S ) + IzS (2.9) lh 3 k k − k + k z k

h 3 e e where Ak = 8π Ak is almost an order of magnitude smaller than Ak. Furthermore, for heavy-hole (hh) states with pure p-symmetry, the spin flip-flop term is also eliminated as pure hh states only couple to the nuclear spin components along the z-axis [44]. Therefore, the hyperfine interaction for hh states has a simple Ising form with only the Overhauser effective field term present [10]:

N 1 X H = AhIzS = Bh gµ S . (2.10) hh 3 k k z N B z k Theoretical calculations have shown that the Ising-form hole hyperfine coupling strength is at least 10 times smaller than the hyperfine coupling for electrons in GaAs [10]. Due to the absence of the spin flip-flop term, the broad frequency distribution of the nuclear spins is the main source of decoherence, which can be suppressed by closed-loop dynamic nuclear polarization (DNP) techniques [8]. Experimental evidence of the hole hyperfine interaction has also been demonstrated, mainly in self-assembled quantum dots based optical measurements [46, 47, 48, 49]. In both Ref [48] and Ref [47], a ratio of ∼ 10% between the heavy-hole and electron hyperfine interactions was shown, which agrees with the Ising-type interaction predicted by theory. On the other hand, electrical measurements on GaAs hole systems so far could not find any evidence of the heavy-hole hyperfine interaction using either resistively detected NMR measurements [50] or Landau level spin diodes [42]. Optical studies on the heavy-hole spin lifetimes have also shown promising results with a long relaxation time T1 ∼ 1 ms [51] and dephasing time ∗ T2 > 100 ns [13], consistent with a reduced hyperfine interaction compared to elec- ∗ trons (T2 ∼ 10 − 20 ns [2, 3]). All these interesting results highlight the advantage of heavy-hole spins over electron spins for coherent spin manipulation and motivate the studies of heavy- hole systems, particularly GaAs hole quantum dots as an alternative to electron spin qubits.

19 20 Background Chapter

2.3.4 Spin-orbit interaction

Besides a weaker hyperfine interaction compared to electrons, the presence of strong spin-orbit coupling in hole systems also makes heavy-hole spins a popular candidate for quantum information processing. As discussed in Sec. 2.3.1, the band structure of GaAs suggests an s-like wavefunction for electrons and a p-like wavefunction for holes. This means that holes in the valence band have a non-zero orbital momentum (l = 1) and thus a strong spin-orbit interaction, whereas for electrons, spin-orbit interaction is only possible via coupling between the conduction band and the valence band. A simple understanding of the spin-orbit interaction is considering it similarly to the interaction between spins and external magnetic field. The only difference is that this magnetic field, i.e. the so-called effective spin-orbit field, is generated internally by the orbital momentum of the spin in an electric field. Depending on the source of the electric field, there are two spin-orbit mechanisms:

• The Dresselhaus term exists in all crystals that exhibit bulk inversion asymmetry (BIA), such as the zinc-blende structure of GaAs. BIA is independent of any external magnetic ﬁelds and is only related to the crystal structure. For 2D electrons in (100) GaAs (i.e. z k [001]), the Dresselhaus term is linear in

the in-plane wavevector kk = {kx, ky, 0}, whereas for 2D holes in (100) GaAs,

the Dresselhaus term is cubic in kk.

• The Rashba term arises from the asymmetry of the potential confining the carriers in the z direction, such as the confinement of 2D gases formed at a GaAs/AlGaAs heterointerface. This asymmetric confining potential is often referred to as structural inversion asymmetry (SIA). The Rashba term can be tuned electrically by varying the confining potential through a gating scheme and can also be minimised in a symmetric quantum well with symmetric dop-

ing. For 2D electrons in (100) GaAs, the Rashba term is linear in kk and the

effective field orientation simply points perpendicular to kk. In contrast, the 3 Rashba term for 2D hole systems is dependent on kk, even though the effective Rashba field direction is still perpendicular to kk

The strong spit-orbit interaction in GaAs hole systems enables spin manipulation via electric ﬁelds, and several theoretical works have proposed heavy-hole spin based EDSR [12] and spin-orbit qubits [52, 53]. The spin-orbit interaction for a heavy-hole conﬁned in a gated (100) GaAs quantum well can be written as [12]

3 3γ0κµB 2 HSO = βk−k+k−σ+ + iαk−σ+ + B−k−σ+ (2.11) m0∆lh−hh

20 21 Background Chapter

2.3.5 The Zeeman splitting in 2D hole systems

The last property of hole systems we would like to talk about is the Zeeman splitting of heavy-hole spins. The Zeeman splitting of holes in isotropic in bulk GaAs but changes drastically when the system is conﬁned to 2D. The Zeeman Hamiltonian term for the valence band in 2D can be expressed as

3 Hν = −2κµBB~ · Jˆ − 2qµBB~ · Jˆ (2.12) ˆ where µB = e¯h/2me is the Bohr magneton, Jˆ = l +s ˆ is the total angular momentum and constants κ and q represent the isotropic and anisotropic contributions respectively. Since the anisotropic coefficient q is usually two orders of magnitude smaller than the isotropic coefficient κ, it is usually omitted for electron systems g∗ ~ (Hc = 2 µBB · ~s). The κ term only couples states different by ±1 in Jz, which 1 1 means it couples the two LH states (Jz = ± 2 ) and the LH states (Jz = ± 2 ) to the 3 3 HH states (Jz = ± 2 ) but does not couple the two HH states (Jz = ± 2 ) to each other. Since 2D confinement of the heterojunction defines the orientation of Jˆ along the growth directionz ˆ, an in-plane magnetic field which favours Jˆ in the 2D plane (xy) will not give any Zeeman splitting for pure HH states, resulting in a zero in-plane effective g-factor. This situation is true for high symmetry growth directions such as the [001] direction used in this thesis as the Zeeman splitting in these directions can be simply described by the spherical part of the Luttinger Hamiltonian and LH and HH states are separate. Therefore, the theoretical calculated 2D effective g-factors for the HH band at k = 0 for (100) GaAs are as shown in Table 2.1

(100) GaAs ∗ ∗ ∗ g[001] g[110] g[110] 7.2 0 0

Table 2.1: Theoretically calculated eﬀective g-factor for the HH band near k = 0 in (100) grown GaAs. Numbers are from Ref. [36]

In addition, when 2D hole systems are further confined in in-plane directions, the situation gets more complicated. A. Srinivasan et al. [54] and F. Nichele et al. [55] ∗ both observed the decrease of g[001] in 1D hole systems when the subband index decreases, which is attributed to an increased heavy hole-light hole (HH-LH) mixing due to the lateral 1D confinement. For the g-factor in GaAs hole quantum dots that we are interested in, hardly any study has been done, particularly experimentally, due to the difficulties in fabricating stable hole quantum dot devices (Chapter 4). Theoretical calculations based on InAs self-assembled quantum dots have shown

21 22 Background Chapter an anisotropic hole g-factor strongly dependent on the dot shape and dot size due to the presence of strong spin-orbit coupling [56]. Measurements based on SiGe self-assembled quantum dots have shown a hole g-factor strongly dependent on the z conﬁnement, which is aﬀecting the g-factor by altering the HH-LH mixing [57]. The literature regarding the g-factor in 0D hole systems will be further discussed in Chapter 5.

2.4 Single quantum dots

In this section, we will talk about the transport and spin properties of a single quantum dot, focusing on general principles based on electron dots. A quantum dot is a semiconductor nanostructure which contains three-dimensionally conﬁned electrons or holes. The conﬁnement can be achieve by three means:

• electrostatic potentials (e.g. electrical gating, strain, doping or impurities)

• interface between semiconductors with diﬀerent bandgaps (e.g. heterostructures)

• fabrication-imposed or natural semiconductor surfaces (e.g. etching, nanowire, self-assembled quantum dots)

Nonetheless, in reality, quantum dots are usually defined by a combination of the above methods. The most intensively investigated quantum dot structure, which is also the structure we used, is defined by electrical gating to a two-dimensional electron (hole) gas at a heterointerface. To achieve confinement in all three spatial dimensions, the dot size, and thus the lithographic dimensions of the gates needs to be smaller than the Fermi wavelength of the 2DEG (2DHG), which sets the upper boundary of the dot size to between 10 nm and 1 µm depending on the host material. As shown in Figure 2.8, besides confinement potentials to define a quantum dot, to perform electrical measurements, the quantum dot also needs to couple to source and drain reservoirs via tunnelling barriers, allowing the current through the dot to be measured as a function of the gate voltage which varies the dot occupation, and the source-drain bias voltage which changes the relative potentials of the source and drain reservoirs [43].

22 23 Background Chapter

Figure 2.8: Schematic ﬁgure of a gate-deﬁend lateral quantum dot geometry. The dot is connected to an electric circuit via source and drain reservoirs, which allows the current through the dot to be measured. Figure reproduced from Ref [43].

The electronic transport properties of quantum dots are dominated by two effects. The ﬁrst one is Coulomb interaction between electrons on the dot, which leads to an energy cost to add an extra electron to the dot. The second one is the discrete energy spectrum of the 0D states. To understand the transport through quantum dots governed by these two eﬀects, we will discuss a simple model that describes the electronic states of quantum dots in the following section.

2.4.1 Constant-interaction model

The constant interacting (CI) model is an approximation of the electronic energy levels of a quantum dot based on two assumptions. The ﬁrst one is that the Coulomb interaction between electrons on the dot or tunnelling onto the dot are characterised by a constant capacitance C. Secondly, the discrete energy spectrum of the 0D states is unaﬀected by the interaction between electrons. In the CI model, the ground state energy of an N electron quantum dot is given by [58].

Figure 2.9: Schematic ﬁgure of a gate-deﬁend lateral single quantum dot geometry showing the parameters used in the CI model. Figure reproduced from Ref [59].

23 24 Background Chapter

2 X U(N) = [e(N − N0) − CgVg] /2C + En,l(B) (2.13) N where N0 is the number of electrons on the dot at Vg = 0, C is the total capacitance of the dot, which equals to the sum of the capacitances between the dot and the gate, source and drain C = Cg + CL + CR as illustrated in Figure 2.9. CgVg is a continuous variable and represents the charge induced on the dot by the bias Vg. X The last term En,l(B) is a sum over all the occupied states of the 0D confinement N potential. The electrochemical potential of the ground state of the Nth-electron on the dot is then defined as µ(N) ≡ U(N) − U(N − 1) 1 C (2.14) = (N − N − )E − eV g + E 0 2 c g C N 2 where Ec = e /C is the charging energy for a total capacitance C. From the expression of µ(N) we can see that the electrochemical potential of the N-th electron can be varied linearly by the gate voltage Vg. This means given that Vg does not vary the dot confinement potential (i.e. EN is independent of Vg), the whole ”ladder” of electrochemical potentials for the N electrons on the dot can be shifted up or down by the gate voltage Vg without changing the relative spacings between the levels [43]. Furthermore, from the energy spacing between two consecutive ground state electrochemical potentials, we obtain the so-called addition energy which is the extra energy required to add the N-th electron to the dot

Eadd(N) ≡ ∆µ(N) = µ(N + 1) − µ(N) (2.15) = Ec + ∆E where ∆E = EN+1 − EN is the energy diﬀerence between the two single-particle states. It is worth pointing out that ∆E can be zero if two consecutive electrons are added to the same degenerate level.

The real form of EN depends on the details of the conﬁnement potential. For the ∗ 2 2 simplest parabolic potential V (r) = 1/2m ω0r , the single-particle energy levels can be expressed using two quantum numbers: the radial quantum number n = 0, 1, 2, ... and the angular quantum number l = 0, ±1, ±2, ... as [58] 1 1 E = (2n + |l| + 1)¯h(ω2 + ω2)1/2 − l¯hω (2.16) n,l 0 4 c 2 c ∗ where ωc is the cyclotron frequency ωc = eB/m . This simple single-particle energy level spectrum is known as the Fock-Darwin spectrum, which is a good approximation to energy states of circular symmetric quantum dots with non-interacting

24 25 Background Chapter electrons. Figure 2.10 plots the calculated single-particle energy levels as a function of the magnetic ﬁeld based on Fock-Darwin spectrum for (a) GaAs electrons and

(b) holes with ¯hω0 = 1 meV. Each of the states is twofold degenerate as Zeeman splitting is not included in the calculation. The lowest energy state for both electrons and holes is the first shell with (n, l) = (0, 0), which can contain two electrons (holes). The second shell can hold up to four electrons (holes) as it also has twofold orbital degeneracy at zero-field (l = ±1). Therefore, for a circular symmetric dot, the addition energy of the 3rd and 7th electrons (holes) will be larger than the rest as extra energy ∆E is required to add to the next orbital level. Despite all the similarities between electrons and holes in the Fock-Darwin spectrum, the biggest difference between Figure 2.10(a) and (b) is that the orbitals split much faster for electrons than for holes in a magnetic field. This is caused by the larger hole effective mass in GaAs, which results in a smaller cyclotron frequency and weaker field dependence of l. However, in lateral quantum dots, due to the non-parabolicity of the confinement potential, the real single-particle levels can be very different from the Fock-Darwin spectrum.

Figure 2.10: Calculated energies of the single-particle levels as a function of the magnetic

field for a simple parabolic confinement potential (¯hω0 = 1 meV) for (a) electrons in GaAs ∗ ∗ me = 0.067me and (b) holes in GaAs mh = 0.4mh. Each of the states is double-degenerate as Zeeman splitting in a magnetic field is not included in the calculation.

2.4.2 Coulomb blockade

After discussing the basic theory that describes the energy levels of a quantum dot, we now focus on experimental signatures of transport measurements on quantum dots. First, we will talk about the low-bias regime, where µS − µD = |e|VSD < ∆E, i.e. electron can only tunnel through the dot via one possible transition. As shown in Figure 2.11(b), when the electrochemical potential of a certain ground state transition µ(N) is in the bias window, i.e. µS ≥ µ(N) ≥ µD, one extra electron can tunnel onto the dot so that the number of electrons on the dot increases from N − 1

25 26 Background Chapter to N. When this electron has then tunnelled from the dot to the drain, another electron can tunnel onto the dot again from source, which gives a current through the dot. On the other hand, if the condition µS ≥ µ(N) ≥ µD is not met, i.e. when there is no available ground state transition in the bias window, no current can ﬂow through the quantum dot, which is known as Coulomb blockade. An example of a dot in the Coulomb blockade is illustrated in Figure 2.11(a).

Figure 2.11: Schematics of the quantum dot level alignement for the low-bias regime (a) When the dot is in the Coulomb blockade regime, i.e. no electrochemical potential is inside the bias window. (b) When the ground state transition between N and N − 1 µ(N) is in the bias window and the number of electrons on the dot oscillates between N − 1 and N, which results in a current through the dot. (c) The dot current as a function of the gate voltage indicating the alternating between two cases (a) and (b) and the continuous change of the dot occupation when the gate voltage is swept. Figure reproduced from Ref [43].

As demonstrated in Eq. 2.14, since the electrochemical potential µ(N) can be varied continuously by the gate voltage Vg, sweeping the gate voltage will result in current peaks as illustrated in Figure 2.11(c). As the gate voltage is swept, the electrochemical potentials corresponding to transitions between consecutive ground states are shifted through the bias window. If a certain electrochemical potential is within the bias window, a peak in the dot current is observed. In between current peaks, no transition is available and the number of electrons on the dot is ﬁxed by Coulomb blockade.

2.4.3 Bias spectroscopy

We now look at the high-bias regime when multiple transitions can contribute to the the dot current. The same as for the low-bias regime, current can only ﬂow through the dot when an electrochemical level for a certain transition is within the bias window. As illustrated in Figure 2.12(a), when the source-drain bias VSD is increased so that a transition via an excited state falls within the bias window, there

26 27 Background Chapter are two paths available for the electron to tunnel through the dot. In general, this will result in an increase in the dot current, which enables the bias spectroscopy measurements of the excited states. If VSD is further increased to a point where the bias window is larger than the addition energy, as shown in Figure 2.12, two electrons can tunnel through the dot at the same time, leading to an additional increase of the dot current.

Figure 2.12: Schematic diagram of the electrochemical potentail of a dot in the high bias regime. The level in orange corresponds to the transition via an excited state. (a) The situation when VSD is larger than ∆E but smaller than Eadd. (b) The situation when VSD is further increased to exceed Eadd allowing two-electron tunnelling. Figure reproduced from Ref [43].

In real experiments, current in the high bias regime is often measured to obtain information such as the orbital energy ∆E of the dot. Such a current map is often called the bias spectroscopy diagram. We will now show how to interpret a bias spectroscopy diagram. As illustrated in Figure 2.13(a), we consider four energy states of the dot, two ground states GS(N − 1) and GS(N) and the corresponding excited states ES(N − 1) and ES(N). Between these energy states, we mainly focus on three transitions: the transition between two consecutive ground states GS(N) ↔ GS(N − 1), i.e. the µ(N) in the low-bias regime; the transitions between one ground state and one excited state ES(N − 1) ↔ GS(N) and GS(N − 1) ↔ ES(N). Figure 2.13(b) is a schematic showing the electrochemical potentials of the transitions. As shown in Figure 2.13(c), the current through different transitions are mapped out as a function of the gate voltage Vg and the source-drain bias VSD. The position of a transition on the gate voltage axis at VSD = 0 is determined by its electrochemical potential as illustrated in Figure 2.13(b). For each transition, as the source-drain bias VSD increases, the range of the gate voltage within which current can flow through the dot also increases linearly as µ(N) ∝ Vg. There- fore, a V-shaped region where current can flow through the dot is outlined for each

27 28 Background Chapter

Figure 2.13: Schematic of a high-bias measurement. (a) Energies for N − 1 electrons U(N − 1) and for N electrons U(N). Three possible transitions are considered as indicated by coloured arrows. (b) Electrochemical potentials of transitions in (a) in corresponding colours. (c) Schematic plot of the diﬀerential conductance dIdot/dVSD as a function of

|e|VSD and Vg. The level alignment between the dot and the source (drain) is indicated by schematic diagrams for specific positions. Figure reproduced from Ref [43]. transition. However, electron tunnelling can only occur via the transition between ground states GS(N) ↔ GS(N − 1) in the low-bias regime as demonstrated in Figure 2.11. This indicates that no current can flow through the dot outside the V-shaped region of the ground-state transition GS(N) ↔ GS(N − 1) (solid black line in Figure 2.13(c)) as the dot is in Coulomb blockade. Hence, in Figure 2.13(c), for transitions ES(N − 1) ↔ GS(N) and GS(N − 1) ↔ ES(N), the V-shaped sections in Coulomb blockade region are denoted by dashed lines as no real current can flow in these parts of the map. On the other hand, if the V-shaped region of the two excited state transitions overlaps with the non-blockade region, an increase in the current will be observable as extra tunnelling channels are available. It is worth pointing out that we have been talking about the dc current through the dot so far. However, for real experiments, differential conductance is often plotted instead of dc current for a bias spectroscopy diagram. In this case, only a change in the signal at the solid lines in Figure 2.13(c) will be visible. One important characterization of a quantum dot that is often obtained from the bias spectroscopy diagram is the single-particle level spacing ∆E (or Eorb). Since ∆E is purely determined by the dot confinement potential, it is often used to estimate the dot size. As illustrated in Figure 2.13(c), the excited state transition terminates at the N-electron Coulomb blockade region is the transition via the excited state of the N-electron dot. This statement can be generalized to any value of N. Moreover, this also indicates that at the Coulomb blockade region of N = 0, i.e. an empty dot, no lines correspond to excited-state transitions should terminate. This important signature in the bias spectroscopy diagram is often used in experiments

28 29 Background Chapter

Figure 2.14: Schematic plot of the diﬀerential conductance dIdot/dVSD as a function of

|e|VSD and Vg for a high-bias measurement showing the Coulomb diamonds. Inside the Coulomb diamond, transport through the dot is blocked and the number of electrons is fixed. The level alignment between the dot and the source (drain) is indicated by schematic diagrams for specific positions. to demonstrate the empty state of a single quantum dot. The excited state energy ∆E can be extracted from the position where the excited transition line meets on the Coulomb blockade region as shown in Fig- ure 2.13(c). Since at this point, the size of the bias window exactly equals the single-particle level spacing, the excited state energy can be directly read off the

|e|VSD-axis. So far, we have been mainly talking about a small region of the bias spectroscopy diagram. For a complete map, the gate voltage is usually swept across multiple ground state transitions and for both positive and negative source-drain biases. In this case, the Coulomb blockade regions will grow into diamond-like regions as

V-shaped regions for consecutive ground state transitions join when VSD > Eadd (Figure 2.12(b)) as illustrated by Figure 2.14. The addition energy can be extracted from the height of the diamonds as shown in Figure 2.14. The ratio between the addition energy and the period of the Coulomb blockade peaks in Vg is often called the lever arm, which is the conversion ratio between the gate voltage and energy in a quantum dot.

2.4.4 Spin states in a single dot

So far we have been only talking about the charge of electrons and ignored the electron spin. In this section, we will discuss the properties of single quantum dots when spin is considered.

29 30 Background Chapter

One-electron spin states

A single electron can only have two spin orientations, spin up and spin down. At zero magnetic field, the two spin states are degenerate and have exactly the same energy. When a magnetic field is applied, the spin up and spin down states will ∗ be separated by the Zeeman energy ∆Ez = g µBB. Therefore, if we express the new energy levels of the ground and first excited states of a single electron dot in a magnetic field B, we have four different electrochemical potential levels [43]

µ1,↑,0 = E0,↑ (2.17)

µ1,↓,0 = E0,↓ = E0,↑ + ∆Ez (2.18)

µ1,↑,1 = E1,↑ = E0,↑ + ∆Eorb (2.19)

µ1,↓,1 = E1,↓ = E0,↑ + ∆Eorb + ∆Ez (2.20)

where ∆Eorb denotes the single-particle level spacing. Therefore, if the bias spectroscopy around the N = 0 ↔ 1 transition is measured in a magnetic field, both the ground state and the excited state will show a splitting caused by the Zeeman energy. If a continuous map of the splitting is measured as a function of the magnetic field, the effective Landég-factor can also be extracted. Even though only the levels of a single-electron dot is derived, energy states of a dot with an odd number of electrons is essentially the same presuming all levels are only spin-degenerate at B = 0 and the interaction between electrons is negligible.

Two-electron spin states

Unlike the single-electron dot, two-electron dots are slightly more complicated due to the Pauli exclusion principle, which states that only electrons with opposite spins can be added to the same orbital level. Therefore, if the interaction between electrons is negligible compared to the orbital energy of the dot, at zero magnetic ﬁeld, the ground state of a two-electron dot is always a spin singlet, i.e S = 0, formed by two electrons with antiparallel spins occupying the lowest orbital state [43]: √ |Si = (| ↑↓i − | ↓↑i)/ 2 (2.21)

The ﬁrst excited states of the two-electron dot are the spin triplets S = 1, where the second electron occupies a higher orbital state. The ﬁrst excited state is triply

30 31 Background Chapter

degenerate at B = 0 depending on the z-projection of the total spin (Sz):

|T+i = | ↑↑i,Sz = 1 (2.22)

|T−i = | ↓↓i,Sz = −1 (2.23)

√ |T0i = (| ↑↓i + | ↓↑i)/ 2,Sz = 0 (2.24) Similar to the single-electron case, the energies of the two-electron dot in a magnetic ﬁeld can also be expressed as [43]

US = E↑,0 + E↓,0 + Ec (2.25) = 2E↑,0 + ∆Ez + Ec

UT+ = E↑,0 + E↑,1 − Ek + Ec

= 2E↑,0 + ∆Eorb − Ek + Ec (2.26)

= 2E↑,0 + EST + Ec

UT− = E↓,0 + E↓,1 − Ek + Ec

= 2E↓,0 + 2∆Ez + ∆Eorb − Ek + Ec (2.27)

= 2E↑,0 + 2∆Ez + EST + Ec

UT0 = E↑,0 + E↓,1 − Ek + Ec

= 2E↓,0 + ∆Ez + ∆Eorb − Ek + Ec (2.28)

= 2E↑,0 + ∆Ez + EST + Ec where Ek is the difference in Coulomb energy between the singlet and the triplets and EST = ∆Eorb − Ek is the energy difference between the singlet and the triplets at B = 0. Now if we consider possible transitions between these energy states as illustrated by the schematic in Figure 2.15(a), the electrochemical potential levels of the two-electron dot can take four different values:

µ2,↓,S = E0,↑ + Ec (2.29)

µ2,↑,S = E0,↑ + ∆Ez + Ec (2.30)

µ2,↓,T0 = µ2,↑,T+ = E0,↑ + EST + Ec (2.31)

µ2,↓,T− = µ2,↑,T0 = E0,↑ + EST + ∆Ez + Ec (2.32)

31 32 Background Chapter

Figure 2.15: (a) Schematic energy diagram of the two-electron dot showing the enegy levels and possible transitions when a finite magnetic field is applied. (b) Schematic plot of the differential conductance dIdot/dVSD as a function of |e|VSD and Vg for a high-bias measurement showing the N = 1 ↔ 2 transitions using corresponding colours as in (a). Figure reproduced from Ref [43].

where the notation µ2,↓,S represents a two-electron electrochemical potential of the transition from a spin down (↓) electron to the spin singlet S.

Note that transitions ↑→ T− and ↓→ T+ are neglected as a spin flip of the first electron is required. From the electrochemical potentials, we can see that if a bias spectroscopy of a two-electron dot is measured at zero magnetic field, the energy level of first excited state will be higher than the ground state by EST , which is often called the singlet-triplet splitting of a single dot. In a finite magnetic field, as shown in Figure 2.15(b), only the excited state of a two-electron dot will split into ∗ two levels with an energy difference of ∆Ez = g µBB, which can be used to extract the effective g-factor.

2.5 Double quantum dots

In this section, we will talk about the transport and spin properties of a double quantum dot, also focusing on electron dots.

2.5.1 Charge stability diagram

We consider two dots in series, which means the current can only ﬂow through the system from source to dot 1, then to dot 2 and ﬁnally to the drain. As illustrated by Figure 2.16, each of the two dots is capacitively coupled to a gate and the tunnel barrier between the two dots are characterised by a capacitance Cm. Applying the constant-interaction theory to this double dot system gives the electrochemical potential of dot 1 for the transition between N1 − 1 electrons and N1 electrons on

32 33 Background Chapter

Figure 2.16: Schematic ﬁgure of a gate-deﬁend lateral double quantum dot geometry. The dot is connected to an electric circuit via source and drain reservoirs, which allows the current through the dot to be measured. Figure reproduced from Ref [59] the dot as [59]

µ1(N1,N2) ≡ U(N1,N2) − U(N1 − 1,N2) 1 = (N − )E + N E 1 2 C1 2 Cm (2.33) 1 − (C V E + C V E ) |e| g1 g1 C1 g2 g2 Cm where EC1(2) is the charging energy of the individual quantum dot 1(2) and ECm is the electrostatic coupling energy which represents the energy change in one dot when an electron is added to the other dot. The expressions for these energies are given as [59] ! e2 1 E = (2.34) C1 C2 C1 1 − m C1C2 ! e2 1 E = (2.35) C2 C2 C2 1 − m C1C2 ! e2 1 EC = − 1 (2.36) m C1C2 Cm 2 Cm

Similarly the electrochemical potential of dot 2 for the transition between N2 −1 electrons and N2 electrons on the dot can be expressed as [59]

µ2(N1,N2) ≡ U(N1,N2) − U(N1,N2 − 1) 1 = (N − )E + N E 2 2 C2 1 Cm (2.37) 1 − (C V E + C V E ) |e| g1 g1 Cm g2 g2 C2

The addition energies for the two dots can also be calculated as

33 34 Background Chapter

Eadd1 = µ1(N1 + 1,N2) − µ1(N1,N2) = EC1 (2.38)

Eadd2 = µ2(N1,N2 + 1) − µ2(N1,N2) = EC2 (2.39) We can see from both Eq. 2.33 and Eq. 2.37 that if we consider the case of two completely uncoupled dots, i.e. Cm = 0, we simply get two independent single quantum dots, whose electrochemical potentials are controlled by Vg1 and Vg2 respectively. Therefore, if the number of electrons on the two uncoupled dots are mapped out as a function of the voltages on gates g1 and g2, we get a charge stability diagram like Figure 2.17(a). The lines indicate the gate voltages at which the ground state transitions are aligned with the electrochemical potentials of the source and drain reservoirs and the number of electrons on each dot are as labelled in the boxes sectioned by the transition lines. Note that these lines are either exactly horizontal or vertical as coupling between two dots are not included.

Figure 2.17: Charge stability diagrams of a double quantum dot in the low-bias regime mapping the ground state transitions of the two dots as a function of the voltagees on gates g1 and g2. Numbers in parentheses indicate the occupations of the two dots respectively.

(a) The case when coulping between two dots is omitted, i.e. Cm = 0. (b) The case when a

ﬁnite interdot coupling is included, i.e. 0 < Cm < 1. Figure reproduced from Ref [59].

If the capacitive coupling between two dots is considered (Cm > 0), we can see from Eq. 2.33 and Eq. 2.37 that the electrochemical potential of dot 1 is proportional to both Vg1 and Vg2 and so is dot 2. This means the vertical and horizontal transition lines in Figure 2.17 will now be tilted. Furthermore, for two coupled dots, an additional electron on one dot will change the electrochemical potential of the other dot by ECm . This is suggests that each cross point in the uncoupled case (Fig- ure 2.17(a)) will split into two points, which are called triple points, with a distance determined by the interdot capacitance Cm. Therefore, taking into account all these eﬀects caused by a ﬁnite interdot coupling, we obtain the so-called ”honeycomb”

34 35 Background Chapter pattern of the charge stability diagram as illustrated in Figure 2.17(b). The same as for the uncoupled case, the lines in Figure 2.17(b) only represent the positions of ground state transitions for either dot, and do not indicate a current through the double dot, which requires the electrochemical potentials for both dots to align with the source and drain reservoirs. In fact, at a low source-drain bias, current can only ﬂow through the dot at triple points when the electrochemical potentials of the two dots align with each other within the bias window as shown in the schematic Figure 2.18.

Figure 2.18: Alignment of electrochemical potential lines of a double quantum dot near the triple points of the charge cycles (0, 0) → (1, 0) → (0, 1) → (0, 0) and (1, 1) → (1, 0) → (0, 1) → (1, 1). The solid lines indicate the gate voltages when the electrochemical potential in either dot align with the source or drain level in the low-bias regime. The dashed lines depict the bending caused by a ﬁnite tunnel coupling tc. The inset schematic level diagrams illustrate the alignment of electrochemical potentials in speciﬁc positions of the charge stability diagram. Figure reproduced from Ref [43].

So far, we have assumed that the tunnel coupling tc between the two dots is negligible compared to electrostatic coupling energy ECm , which is often known as the weak-coupling regime. Now we discuss how the tunnel coupling will modify the charge stability diagram when it becomes considerable. If we consider the case when the electron wavefunctions φ1 and φ1 are no longer localised in one dot but span over both dots, depending on whether the interaction between the two wavefunctions is bonding or antibonding, the resultant superposition can take two forms [43]

ψB = αφ1 + βφ2 (2.40) for the bonding orbital and

ψA = αφ1 − βφ2 (2.41) for the anibonding orbital. The interaction between the single-dot states suggests that when the electrochemical potentials in two dots are aligned, the overall energy

35 36 Background Chapter

level will be lowered by |tc| if a bonding orbital is formed, and raised by |tc| if a antibonding orbital is formed. This effect will show as a bending of the straight transition lines near triple points in the charge stability diagram, as depicted by the dashed lines in Figure 2.18. Therefore, in real experiments, the separation between triple points is usually determined by both interdot capacitive coupling and tunnel coupling and the signature revealing different tunnel coupling regimes is the bending near triple points. Note that up to now, we have ignored the single-particle levels of both dots, which will add an extra term ∆E1(2) to the addition energy Eadd1(2). The value of ∆E1(2) is determined by the real confinement potential, which means it can be varying depending on the number of electrons on dot. Moreover, ∆E1(2) can also be zero if consecutive electrons are added to the same spin-degenerate level. Since ∆E1(2) has no qualitative effects on the dot transport at a low-bias (|e|VSD < ∆E1(2)), we will only include it in the next section when we discuss the high-bias regime.

2.5.2 Bias triangles

Up to now, we have been talking about the low-bias regime, where current can only flow through the double dot when electrochemical potentials of two dots are aligned. This kind of tunnelling is usually called elastic tunnelling since the initial and final states have the same energy. As the source-drain bias increases, another type of tunnelling, the inelastic tunnelling can occur. The inelastic tunnelling is the tunnelling process when electrochemical potentials of two dots are misaligned, and energy absorption from or emission to the environment is required to compensate for the mismatch. Since at cryogenic temperatures, absorption of phonons is very rare without external microwave sources, here we focus on the inelastic tunnelling process when dot 1 (next to source) has a higher electrochemical potential than dot 2 (next to drain) and energy is released during the transition. This inelastic tunnelling process dominates the current through the double dot in the high-bias regime when no energy levels are aligned. As illustrated in Figure 2.19, due to inelastic tunnelling, triple points grow into bias triangles when the source-drain bias voltage is increased. It is worth pointing out that for demonstration purposes, the transition cycle (0, 0) → (1, 0) → (0, 1) → (0, 0) will be discussed, but the same processes are valid for any number of electrons. To understand the bias triangle, we first start with the base of the triangle as depicted by the red solid line in Figure 2.19. Remember from the low-bias regime, anywhere on this line corresponds to the case when the electrochemical potentials in two dots are aligned and thus elastic tunnelling is available. Note that since this

36 37 Background Chapter

Figure 2.19: Charge stability diagram near the same charge transitions as in Figure 2.18 in the high-bias regime. At each triple point, a bias triangle, whose size is determined by the source-drain bias VSD applied, is formed. Solid lines depict the gate voltages when the levels in either dot align with the source or the drain level. Yellow areas inside the bias triangles correspond to the gate voltages when transitions involving excited states are available. Green lines in the bias triangle correspond the elastic tunnelling through the dot via excited states. The dashed purple line depicts the detuning axis ε which is perpendicular to the base of the bias triangle. Alignment of electrochemical potentials in two dots with respect to the source and drain levels is depicted by the inset schematic level diagrams, where a black line represents a ground state and a green line represents an excited state, for speciﬁc gate voltages. Figure reproduced from Ref [43]. line is caused by the interdot capacitive coupling, moving anywhere in the charge stability diagram with the same slope as the base of the bias triangle will not change the relative alignment between levels in two dots, but only shift both dot levels together with respect to the source and drain [43]. Therefore if we assume that the drain is grounded and only the source is varied to adjust the bias window, one end of the base of the bias triangle is the triple point, and the other end corresponds to the case when levels in both dots are aligned with the source. Away from the base line, the electrochemical potentials in two dots are no longer aligned, and thus only inelastic tunnelling is possible. Moving upwards along the left edge of the triangle from the base, only the right dot level is varied. As the right dot level is moved down, inelastic tunnelling contributes to the dot current until µ2(0, 1) is shifted out of the bias window. If the source-drain bias is large enough, the excited state of the right dot can enter the bias window. When the excited state of the right dot align with the ground state of the left dot, another elastic tunnelling line parallel to the base of the triangle, as depicted by the green solid line in Figure 2.19, will be visible. Similarly, moving up along the right edge of the triangle from the triple point only shifts the left dot level µ1(1, 0) up. Inelastic tunnelling contributes to the dot current until µ1(1, 0) is higher than the electrochemical potential of the source.

37 38 Background Chapter

Moreover, if both the excited state and the ground state of the left dot fall in the bias window at the same time, a higher dot current is expected in the triangular region near the triple point as two paths are available for transport. It is worth pointing out that in real measurements, the visibility of lines and regions may be very different from the ideal case described here. The dot current is strongly dependent on the relative heights of the tunnel barriers, the relaxation process within the dot and the efficiency of the inelastic tunnelling process [43]. In many real experiments, instead of measuring the whole bias triangle, gate voltages are usually only varied along a line exactly perpendicular to the base of the bias triangle as depicted by the purple dashed line in Figure 2.19. This axis is commonly referred to as the detuning axis ε, which is a measure of the energy difference between ground states in two dots. At the base of the bias triangle, detuning is zero (ε = 0) as grounds states in two dots are aligned. As the gate voltages are varied along the pink dashed line, energy levels in two dots are detuned with respect to each other while keeping the average level at a constant. In our notation, a positive detuning (ε > 0) corresponds to the case when the left dot level is higher than the right dot level (inside the bias triangle in Figure 2.19) while a negative detuning (ε < 0) suggests the opposite.

2.5.3 Pauli spin blockade

Up to now, we have been talking about electron transport in double quantum dots without any spin eﬀects. However, interdot electron transitions require not only aligning of electrochemical potentials but also conservation of the electron spins. This spin selection rule causes a phenomenon called Pauli spin blockade, which is often used to distinguish diﬀerent spin states and realize spin-to-charge conversions in a double quantum dot.

Spin states in a two-electron double quantum dot

To understand spin blockade we ﬁrst discuss the spin states of a two-electron double dot, which involves the transport cycle (0, 1) → (1, 1) → (0, 2) → (0, 1). Since spin blockade occurs at the transition (1, 1) → (0, 2), here we only focus on the spin states of these two cases. For (0,2) state, the spin states are essentially the same as the single dot case as the left dot is empty: √ |S(0, 2)i = (| ↑2↓2i − | ↓2↑2i)/ 2 (2.42)

|T+(0, 2)i = | ↑2↑2i (2.43)

38 39 Background Chapter

|T−(0, 2)i = | ↓2↓2i (2.44) √ |T0(0, 2)i = (| ↑2↓2i + | ↓2↑2i)/ 2 (2.45) where ↓2 denotes a spin-down electron in dot 2. Similar to the single dot case, at zero magnetic ﬁeld, the triplets are separated from the singlet by the singlet-triplet splitting energy EST (Section 2.4.4). For the (1,1) state, the two electron spins in two dots still form singlets and triplets: √ |S(1, 1)i = (| ↑1↓2i − | ↓1↑2i)/ 2 (2.46)

|T+(1, 1)i = | ↑1↑2i (2.47)

|T−(1, 1)i = | ↓1↓2i (2.48) √ |T0(1, 1)i = (| ↑1↓2i + | ↓1↑2i)/ 2 (2.49) Unlike the (0, 2) singlet and triplets, the energy diﬀerence between the (1, 1) states are relatively small as electrons are spatially separated in two dots. Theo- retically, the singlet triplet splitting for the (1, 1) charge conﬁguration depends on the tunnel coupling tc between the two dots as well as the single dot charge energy

Ec [43]. To visually illustrate the charge transition (1, 1) → (0, 2) between diﬀerent spin singlets and triplets, we draw in Figure 2.20 the evolution of the (1, 1) and (0, 2) states as a function of the detuning energy. As shown in Figure 2.20(a), the dot starts in a (1, 1) charge state and the system is in Coulomb blockade. As the levels in two dots are detuned, the (0, 2) states move down. S(0,2) aligns with

S(1, 1) at zero-detuning where the charge transition occurs if tunnel coupling tc is negligible. In the presence of a finite tc, as illustrated by Figure 2.20, S(1, 1) and S(0, 2) hybridize while T(1, 1) and T(0, 2) hybridize since interdot tunnelling √ is spin conserving. This hybridization results in an avoided crossing of 2 2tc at ε = 0 between the two singlet states and the same splitting between the triplet states at a higher detuning ε = EST . This illustrates a very important aspect of the (1, 1) → (0, 2) transition: due to the small energy difference between S(1, 1) and T(1, 1) states, the (1, 1) charge state can be either a triplet or a singlet; however, depending on the spin configuration of the (1, 1) state, the transition (1, 1) → (0, 2) may happen at very different detuning energies as spin is conversed during electron tunnelling. This spin-conserved tunnelling caused current suppression is often called Pauli spin blockade.

39 40 Background Chapter

Figure 2.20: Energy levels of two-electron spin states in a double quantum dot near the transition (1, 1) → (0, 2) as a function of the detuning energy ε for (a) a negligible tunnel coupling between the dots tc = 0, and for (b) a ﬁnite tunnel coupling tc > 0. Black solid lines depict the singlet states and blue solid lines depict the triplet states. Alignment between electrochemical potentials in two dots are illustrated by the inset schematic level diagrams for three diﬀerent detuning energies. Figure reproduced from Ref [43].

Signatures of spin blockade

Pauli spin blockade was ﬁrst observed by K. Ono et al. in experiments on vertically coupled electron quantum dots [60]. As shown in Figure 2.21 [60], current is sup-

Figure 2.21: DC current (I) through a self-assembled quantum dot measured as a function of the source-drain bias (V) showing spin blockade (current suppression for V ∼ 2 − 7 mV) in forward bias and continuous electron transport in reverse bias. Figure reproduced form Ref [60]. pressed due to spin blockade in the range V ∼ 2 − 7 mV. When V > 7 mV, current through the dot resumes as the source-drain bias exceeds the singlet-triplet splitting and charge transitions through triplets become available energetically. In contrast, when the source-drain bias direction is reversed, a non-zero current through the double dot is observed for the whole bias range. This current rectiﬁcation in dc

40 41 Background Chapter transport caused by the conservation of spin during electron tunnelling is the most well-known signature of Pauli spin blockade.

Figure 2.22: Schematic bias triangles illustrating the signature of Pauli spin blockade at transition (1, 1) → (0, 2): (a) Current of the base part of the triangle is suppressed in the blocked region when a positve bias (VSD > 0) is applied. (b) Current is observed in the entire bias triangle when the source-drain bias direction is reversed (VSD < 0). The relative alignement of the singlet and triplet states are illustrated by the schematic level diagrams for three diﬀerent postions (colour-coded) in the bias triangles.

After Ono et al., Pauli spin blockade has since been observed in many systems, particularly in lateral double quantum dots, in which energy levels and tunnelling rates are easily tuned. Bias triangles rather than a single current trace are usually measured to reveal the bias-dependence signature of Pauli spin blockade. As shown in Figure 2.22, the current rectification feature of spin blockade manifests itself as a current-suppressed region at the base of the triangles for only one bias direction. The origin of the bias-dependence of Pauli spin blockade is illustrated by the schematic level diagrams in Figure 2.22. As a positive bias, electrons transfers through the dot via the sequence (0, 1) → (1, 1) → (0, 2) → (0, 1) of the charge configuration. During the transition (0, 1) → (1, 1), the left dot can be filled from the source Fermi sea with either a spin-up or spin-down electron, regardless of the spin of the electron in the right dot. If a singlet S(1, 1) state is formed, the electron then can transfer to the right dot singlet state S(0, 2) and then to the drain. However, if a triplet T (1, 1) is formed, as illustrated by the red-box case in Figure 2.22, the transition T (1, 1) → S(0, 2) is blocked as it violates spin conservation. Therefore, the double dot system will be stuck in the (1,1) charge configuration until the electron spin relaxes. Depending on the spin relaxation time (∼ ms) [43], the resultant current can be either negligible or strongly suppressed, and the dot is said to be spin-blocked. This situation remains until the source-drain bias is large enough to access the triplet

41 42 Background Chapter state T (0, 2), which allows the transition T (1, 1) → T (0, 2) and helps to restore the current through the dot (green-box case in Figure 2.22).

Figure 2.23: (a) Honeycomb pattern for the double quantum dot measured with a charge sensor. (b) Schematic illustrating the transitions where one expects spin blockade to suppress the current in an even-odd level ﬁlling model. (c)-(j) Positive and negative bias triangles for four pairs of triple points demonstrating that spin blockade can only be observed for an (odd, odd) → (even, even) transition. Figure reproduced from Ref [35].

In contrast, when the bias direction is reverse, electron transfers through the double dot via the charge sequence (0, 1) → (0, 2) → (1, 1) → (0, 1). As illustrated by the yellow-box case in Figure 2.19, in this direction, since the left dot is empty to start with, no matter whether a spin singlet S(0, 2) or triplet T (0, 2) is formed in the right, the electron can always transfer to the left dot and then to the left lead. Therefore, a continuous current is observable within the entire bias triangle. Up to now, we have been only talking about spin blockade at the transition (1, 1) → (0, 2). However, for doubly spin-degenerate states, spin blockade is actually observable for any (odd, odd) → (even, even) charge transitions where spin selection happens due to Pauli exclusion principle. As shown in Figure 2.23 [35], spin blockade at different charge transitions is demonstrated by A. C. Johnson et al. through measuring both positive and negative bias triangles at eight different transitions. Spin blockade is confirmed by the current suppression of the base part of the bias triangles at transitions (1, 1) → (2, 0), (1, 3) → (2, 2), (1, 1) → (0, 2) and (1, 3) → (0, 4), all of which involve an (odd, odd) → (even, even) as illustrated by the schematic charge stability diagram in Figure 2.23(b).

2.5.4 The lifting of spin blockade

As spin blockade relies on the conservation of spin in electron tunnelling, any undesirable spin mixing mechanisms will show as a lifting of spin blockade and an

42 43 Background Chapter increase in the current. Since different spin relaxation processes often have distinct magnetic field dependence, the lifting of spin blockade in a magnetic field is often considered as another experimental signature of Pauli spin blockade and used to distinguish different spin mixing mechanisms. Here we will discuss two main spin mixing mechanisms: the hyperfine interaction and the spin-orbit interaction.

Hyperﬁne interaction lifted spin blockade

The hyperfine interaction in a quantum dot can be expressed using an effective Overhauser field created by the nuclear bath [43]

X HHF = AkI~kS~ = gµBB~ N S~ (2.50) k X where Ak is the coupling between electron to the k-th nucleus and B~ N = AkI~k/gµB k is the effective Overhauser field. In a double quantum dot, since electrons in two dots couple to two different sets of nuclei, the effective nuclear fields at two dot sites are slight different. This fluctuation difference in the nuclear field ∆B~ N introduces a coupling between the singlet x y S(1, 1) with the triplets T±,0(1, 1): ∆BN and ∆BN mixes S(1, 1) with T±(1, 1) and z ∆BN mixes S(1, 1) with T0(1, 1) [61]. Figure 2.24(a) and (b) are schematic energy diagrams showing the two-electron singlet and triplet levels as a function of detuning

ε at Bext = 0 and Bext > ∆BN respectively. At Bext = 0, the nuclear field fluctuation ∆BN mixes all (1, 1) triplets with the (1, 1) singlet and spin blockade is lifted as transitions out of the blocked T (1, 1) states are now available. However, this process can only happen when the singlet and triplets are close in energy, i.e. when they are within the energy scale of the nuclear field fluctuation EN = gµB∆BN (depicted by the grey areas in Figure 2.24(a) and (b)). In an external magnetic field Bext, two of the triplets T±(1, 1) will split off from the singlet S(1, 1) by the Zeeman energy gµBBext. When the energy difference between T±(1, 1) and S(1, 1) is larger than

EN , the mixing eﬀect between T±(1, 1) and S(1, 1) states caused by the nuclear ﬁeld

fluctuation becomes negligible. Although T0(1, 1) remains mixed with S(1, 1), the occupation one of the two split-off states T±(1, 1) blocks the current through the dot. This suggests that in the presence of hyperfine interaction, spin blockade is lifted around Bext = 0 and recovers in a finite magnetic field Bext > ∆BN . Therefore, if the current through the dot is plotted as a function of the external magnetic field, a zero-field peak is expected as shown in Figure 2.24(c). Furthermore, if the field- dependence of the dot current is measured for different detuning ε, a colour map like Figure 2.24(d) is often obtained. The zero-field peak in the current is visible all

43 44 Background Chapter

Figure 2.24: (a) and (b) Schematic plots of the energy levels near the transition (1, 1) →

(0, 2) as a function of the detuning energy ε at B = 0 (a) and B > ∆BN (b). Red lines represent singlet states and blue lines represent triplet states. The grey area depicts the energy scale EN = gµB∆BN over which the hyperfine interaction is effective in mixing the spin states. (c) Current through a double quantum dot in the spin-blocked region as a function of the external magnetic field showing the typical signature of a hyperfine- interaction induced lifting of spin blockade: the zero-field peak. (d) Current through a double quantum dot in the spin-blocked region as a function of the magnetic field and the detuning energy. Figure reproduced from Ref [62]. the way from zero to high detuning simply because detuning has no effect on the energy levels of T (1, 1) and S(1, 1) as illustrated by Figure 2.24(a) or (b).

Spin-orbit interaction lifted spin blockade

Besides the hyperfine interaction, another main spin mixing mechanism is the spin- orbit interaction. In a double quantum dot, the interdot tunnel coupling provides a finite overlap between states with different orbital wavefunctions. Due to the spin-orbit interaction, when an electron tunnels from one dot to the other, the variation between the orbital wavefunctions causes the electron spins to change at the same time, which can be simply thought of as an effective magnetic field that the electron experiences during tunnelling. In a simple physical picture, the rotation of the electron spin caused by this effective magnetic field will enable the electron to tunnel from the triplet T (1, 1) states to the singlet S(0, 2) state and lift the spin blockade. However, due to time-reversal symmetry, this process can not happen at zero magnetic field, resulting in a zero-field dip as shown in Figure 2.25(a). If detuning is also considered, a plot like Figure 2.25(b) is expected as the overlap between T (1, 1) and S(0, 2) states decreases as the detuning increases. A complete analysis of the spin-orbit interaction induced lifting of spin blockade

44 45 Background Chapter

Figure 2.25: (a) Current through a double quantum dot in the spin-blocked region as a function of the external magnetic field showing the typical signature of a spin-orbit coupling induced lifting of spin blockade: the zero-field dip. (b) Current through a double quantum dot in the spin-blocked region as a function of the magnetic field and the detuning energy. Figure reproduced from Ref [62]. can be found in Ref [63]. Here we will briefly explain the basic concepts of this model. Focusing on the transition (1, 1) → (0, 2), the transition between T (1, 1) and S(0, 2) caused by spin-orbit interaction is introduced to the tunnelling Hamiltonian as a new non-spin-conserving coupling parameter ~t = {tx, ty, tz} besides the conventional spin-conserving tunnelling parameter t0 [63]:

Hˆt = i~t · |T~ihS02| + t0|S11ihS02| + h.c. (2.51) where |T~i ≡ {|T i, |T i, |T i} in the basis of orthonormal unpolarized triplet states x y z √ 1/2∓1/2 − + 0 |Tx,yi ≡ i {|T11i ∓ |T11i}/ 2 and |Tzi ≡ |T11i. Due to spin-orbit interaction, S(1, 1) couples not only to S(0, 2) with the amplitude t0, but also to the three triplets Tx, Ty and Tz with amplitudes tx, ty and tz, which are real numbers as Tx, Ty and Tz are invariant with respect to time-reversal symmetry [63]. Moreover, since the basis {|Txi, |Tyi, |Tzi} transforms the same as a real space basis, ~t can also be considered as a real vector pointing in a certain direction in space [63]. However, since the wavefunctions of the triplets depend on the detail of the confinement potential, the direction of ~t is hard to predict. One possible way to determine the direction of ~t is by rotating the external magnetic field B~ . The vector nature of the ~t suggests that if it is aligned with the external ± magnetic field, T11 becomes the eigenstates of the magnetic field and is no longer coupled to S02. In this situation, spin blockade could persist to arbitrarily high magnetic fields [63]. More discussion about the effects of spin-orbit coupling on spin blockade can be found in Chapter 6.

45 Chapter 3

Device Fabrication and Measurement Setup

3.1 Introduction

Since the aim of this thesis is to study three-dimensionally confined heavy-hole spins via electrical measurements, one of the key points of the project is fabricating gate- defined lateral quantum dots on GaAs/AlGaAs heterostructures. As discussed in Chapter 1, single hole quantum dots on GaAs have only been realized with self- assembled dots in optical measurements. However, it is very difficult to integrate self-assembled quantum dots into complicated electric circuits, and people has been trying to develop fabrication techniques to achieve gated hole quantum dots for years. Up to date, to the best of our knowledge, electrical transport measurements of GaAs based single lateral quantum dot down to the last hole has never been demonstrated. There are three major difficulties that people have been facing: device stability [17], p-type ohmic contacts and device dimensionality [18, 19]. In this chapter, we will talk about the fabrication of the quantum dot devices used in this thesis and the steps we took to overcome those problems.

3.2 Device fabrication

Based on the previous fabrication method developed in our group [64, 65], there are three main improvements developed in this thesis to achieve a hole quantum dot which is capable of showing transport down to the last hole. We will mainly talk about those fabrication techniques developed in this thesis speciﬁcally for quantum dots. Standard fabrication routines for 2D or 1D hole devices can be found in Ref [64, 65, 66]. Before we go into the details of the improvements, we ﬁrst talk

46 47 Device Fabrication and Measurement Setup about the fabrication key points and operation principles of an induced hole quantum dot device.

3.2.1 Device structure and operation principles

Induced two-dimensional hole gases

Figure 3.1(b) is a sideview schematic of a 2D device fabricated using the same wafer (W641) as the quantum dots measured in this thesis showing the standard layout of an induced hole device. The wafer (W641, Cambridge) used is an MBE grown shallow undoped GaAs/AlGaAs heterostructure with 1 µm of GaAs buffer layer grown on a high-symmetry (100) GaAs substrate (not shown), followed by 50 nm of AlGaAs layer and finally a 10 nm GaAs cap layer. To fabricate p-type ohmic contacts, AuBe is deposited and annealed to make contact to the two-dimensional hole gas (2DHG). As induced devices rely on the applied top gate voltage to electrostatically induce charge carriers, an overall top-gate needs to be deposited as the final step to introduce carriers to the device, which also connects the active measurement region to the metal contacts. To guarantee the connection between the induced 2DHGs and ohmic contacts, the top-gate metal needs to overlap with the contact metal as depicted in Figure 3.1. Therefore, a dielectric is required to insulate those two metal layers. Due to the roughness of annealed AuBe contacts, how to reduce the leakage current between the contact and the top-gate so that high charge carrier density can be obtained is one of the key points of fabricating induced hole devices.

Double-level-gate quantum dot design

To observe clear spaced discrete single-particle energy levels of holes, ultra-small 1 quantum dots are preferred due to the large hole effective mass (Eorb ∝ ∗ ). m Adot To achieve dots with smaller dimensions compared to the conventional design for 2DEG-based lateral quantum dots (Section 4.2), a novel double-level design is used in this thesis. The advantages of the double-level design over the conventional lateral dot design will be discussed in detail in terms of experimental results in Chapter 4. Here we will only talk about the fabrication procedures of the two-level-gate design using Electron Beam Lithography (EBL) and the basic operation principles of the design. Figure 3.2 shows the standard flow of depositing two layers of EBL gates for the double-level design. First narrow parallel finger gates are deposited, then dielectric material AlOx or HfOx are grown over the whole chip using the Atomic Layer De- position (ALD) system, finally a second layer of EBL gates consisting of top-gate

47 48 Device Fabrication and Measurement Setup

Figure 3.1: (a) An optical image of a p-type ohmic contact. Green arrow indicates the overlapping betwen the metal top-gate and the annealed AuBe ohmic contact. (b) Sideview schematic of a 2D device fabricated on wafer W641. channels are deposited. Since the two layers of gates need to be aligned accurately within a few nanometers, small alignment markers (not shown) are written together with the ﬁrst layer of EBL gates, and then used for accurate alignment of the second layer.

Figure 3.2: SEM images showing the steps of fabricating a quantum dot using the double- level design. From left to right: the first layer of EBL patterm consisting of narrow finger gates; deposition of dielectric using the atomic layer deposition (ALD) system; a second EBL step writing the top-gate channel overlapping the finger gates.

To conﬁne a hole quantum dot using the double-level design, a negative bias is applied to the top-gate channel, as depicted in yellow in Figure 3.3(a), to induce holes at the heterointerface. By adjusting the voltages on the three barrier gates,

48 49 Device Fabrication and Measurement Setup a quantum dot can be conﬁned in between two outer barriers (gate 1 and 3) while the middle barrier (gate 2) can be used as a plunger gate. As we will discuss in more detail in the following result chapters, the double-level design has two main advantages:

• First, the quantum dot size is mostly controlled by the width of the plunger gate and the width of the top-gate channel, which means with standard EBL resolution, a lithographic dot size of 50 nm×50 nm is easily achievable;

• Second, since the tunnelling barriers are solely defined by one rectangular thin finger gate, tuning of the barrier is much simplified compared to the conventional lateral dot design where tunnelling barriers are often defined by more than one gate. Moreover, as a width of 30 nm - 50 nm is easily achievable for the finger gates, a sharp tunnelling barrier potential close to the ideal theoretical assumption is also expected.

Figure 3.3: (a) An SEM image of a quantum dot with a double-level-gate design. The ﬁrst layer of gates consist of three narrow ﬁnger gates coloured in grey. Insulated by dielectric

AlOx or HfOx (green), the second layer (yellow) is an inducing top-gate channel perpendicular to the finger gates. The red ellipse indicates the position of the single quantum dot when it is defined and the red arrows indicate the direction of the dot current. (b) A sideview schematic illustrating the cross-section of the device when a quantum dot is confined.

Now that we know the basic fabrication points and operation principles of the induced hole quantum dots, we will discuss the main fabrication techniques we developed to achieve a few-hole quantum dot in the following sections.

3.2.2 Dielectric material

First of all, we implemented the use of atomic layer deposited (ALD) oxides re- placing the previously used polyimide (PI) as the dielectric material. As discussed

49 50 Device Fabrication and Measurement Setup in Section 3.2.1, due to the absence of doping, carriers are only induced at the heterostructure by the potential from the top-gate. To probe the induced 2DHG electrically, metal ohmics are deposited to certain regions to make contact with the 2DHG. Therefore, to guarantee the continuity of carrier, the inducing top-gate needs to overlap with the contact metal, which then requires an insulator between these two pieces of metal to prevent leakage. PI was used previously in our group as the dielectric separating the top-gate from the ohmic contacts [64, 65]. It is generally very easy to pattern as it is UV sensitive. This makes processing with PI very straightforward as only a standard photolithography process is required. However, since PI is spun on, it is very hard to control the thickness of it, which may cause a variation of the 2DHG density across diﬀerent regions of the device. Furthermore, the large thickness and the small dielectric constant of PI means that a considerable bias needs to be applied to the top-gate to induced carriers. After curing, the PI layer is usually around t = 600 nm thick and has a dielectric constant of εr ∼ 3.5. Figure 3.4 plots the 2DEG density versus the top-gate voltage of an induced electron device fabricated on wafer W641 using PI as the dielectric material. Due to a large t and a small εr, VTG = −16 V needs to be applied to achieve a 2DEG density of n = 2.5 × 1015 m−2, which is also expected from a simply parallel capacitor Q εrε0A approximation (C = V = d ).

Figure 3.4: 2DEG density verus top-gate voltage for a standard electron device fabricated on wafer W641 using ∼ 600 nm PI as the dielectric material.

Besides the high top-gate bias required, the main reason why such a thick layer of PI is unsuitable for small quantum dots is because no ﬁne nanostructure fabricated on top of the the PI will be able to survive a thickness of 600 nm due to lateral stray

50 51 Device Fabrication and Measurement Setup of the electric field. Even though for the conventional design of an induced quantum dot, fine finger gates can be deposited directly on top of the wafer surface and only an overall top-gate is insulated by the dielectric, for our double-level design, a nanoscale channel instead of an big top-gate needs to be fabricated above the dielectric. To solve the problem, we employed high-κ ALD oxide as the insulating dielectric instead of PI. The first oxide we tried is AlOx grown in a Savannah 200 ALD system. This is also the dielectric material used in the quantum dot device presented in Chapter 4. Initial calibration of the oxide using a 2D hole device with

30 nm of AlOx is shown in Figure 3.5, which plots the 2DHG density as a function of the top-gate voltage. We can see from the graph that for the same 2DHG density 15 −2 of p = 2.5 × 10 m , a top-gate voltage of only VTG = −1.3 V is required. Using the slope of the trace we can calculate the dielectric constant of the ALD grown

AlOx based on a parallel capacitor approximation as εr = 6.2. Note that this value is slightly smaller than the conventional dielectric constant of AlOx (εr ∼ 9), which is possibly due to chemical processing or uneven surfaces.

Figure 3.5: 2DHG density verus top-gate voltage for a standard electron device fabricated on wafer W641 using 30 nm of AlOx as the dielectric material. Using a parallel capacitor approximation, the dielectric constant of AlOx is calculated to εr = 6.2.

Besides AlOx, another high-k dielectric HfOx is also tested by measuring the 2DHG density as a function of the top-gate voltage as shown in Figure 3.6. Due to the expected high dielectric constant of HfOx, only 10 nm of HfOx is used for the same wafer W641. A 2DHG density of p = 2.5 × 1015 m−2 can be reached at a top-gate voltage of less VTG − 1.2 V. Using the same method, the dielectric constant of HfOx on a real device is calculated to εr = 17.0, which is very close to previous measurement results of ALD grown HfOx [67]. HfOx is used as the dielectric for

51 52 Device Fabrication and Measurement Setup the device presented in Chapter 5 and 6. An ultra-thin oxide layer is especially beneficial for quantum dots with reduced dimensions using the double-level design. As the spacing between parallel finger gates is further reduced to confine a smaller quantum dot, side-growth of the oxide around finger gates may result in regions with extra thick oxide in between the barrier gates. Since quantum dots are also expected to be confined in between barrier gates, a shallow confinement potential caused by the extra thick oxide is undesirable.

Figure 3.6: 2DEG density verus top-gate voltage for a standard electron device fabricated on wafer W641 using 10 nm of HfOx as the dielectric material. Using a parallel capacitor approximation, the dielectric constant of HfOx is extracted as εr = 17.0.

3.2.3 Shallow wafers

For AlGaAs/GaAs based lateral quantum dots, since conﬁnement is achieved electrostatically at the heterointerface by applying bias on gates sitting at the top of the wafer, the thickness of the AlGaAs layer has always been a problem. Wafers with thick AlGaAs layers behave very poorly in transferring ﬁne lithographic patterns to the heterointerface due to broadening of the gate potential. For modulation-doped heterostructures, reducing the thickness of the AlGaAs spacer layer causes a lot of unwanted scattering and switching noise due to ionized impurities. To solve the stability issue, accumulation-mode AlGaAs/GaAs heterostructures are used, but it is much harder to make ohmic contacts to an induced 2DHG as the wafer is undoped and neutral when contacts are deposited and annealed. Previously demonstrated p-type contacts to undoped wafers have always been fabricated on deep heterostructures [65]. To make contact to shallow 2D hole systems, processing parameters need to be optimised for each wafer. During this work,

52 53 Device Fabrication and Measurement Setup we tested various wafers to find the working recipe for fabricating p-type contacts to shallow wafers while gradually reducing the thickness of AlGaAs layer from 300 nm to 150 nm, then to 100 nm and finally 50 nm. Note that even though the fabrication of ohmic contacts is a basic part of the thesis, it actually took more half a year to finally find the working recipe for shallow p-contacts, which is a big portion of the project timeline.

3.2.4 Electron Beam Lithography

Figure 3.7: (a) A image of the device used in Chapter 4. A 10µ m gap on the optical top-gate pattern separates the top and bottom halves. (b) An SEM image of the active dot region. Top and bottom halves of the device is only connected through the EBL top- gates (second-layer). Green arrows indicate the current direction through the device during measurements.

After p-type ohmic contacts can be fabricated on the shallow heterostructure, the finally element we need to achieve a small hole dot is fine lithographic patterns, which gives the lateral confinement of the dot. The first pattern we used is as shown in Figuer 3.7, which is also the dot structure used in Chapter 4. This device is fabricated using an optical top-gate pattern which is generally used for Hall bar measurements. A 10µ m gap in the middle of the optical top-gate pattern separates the top and bottom halves of the device. The quantum dot whose top-gate channel connecting the two halves is written inside the gap. Current flows from one contact in the top half through the dot to another contact in the bottom half as illustrated by the green arrows in Figure 3.7. The first-layer finger gates in this design have a width of 40 nm and a intergate spacing of 80 nm. The second-layer top-gate channels are 150 nm (left) and 250 nm(right) wide respectively. However, as we will discuss in more detail in Chapter 4, the number of finger gates in the first version of design is not enough for a complete control of the dot confinement potential. Therefore, we modified the optical mask and removed the

53 54 Device Fabrication and Measurement Setup long Hallbar-shaped top-gate to accomodate more surface gates as shown in Fig- ure 3.8(a). We also got ride of the big optical top-gate to give more ﬂexibility to the dot design. A total of 5 ﬁnger gates are used in this design with a width of 30 nm and a intergate spacing of 50 nm. A top-gate channel of 50 nm wide is used for the active dot region. The top-gate channel gradually widens up from 50 nm to a few tens of microns where it connects to the optical top-gates.

Figure 3.8: (a) Images of the second version of devices. Optical mask has been modiﬁed to get rid of the big top-gate to accomodate more ﬁnger gates. (b) An SEM image of the active dot region of the second version of the dot design. Green arrows indicate the current direction through the device.

With the new modified optical mask dedicated for quantum dot device, even more complicated quantum dots consisting of ten gates can be fabricated. Figure 3.9 shows the device used in Chapter 5 and 6, which has eight barrier gates and two channel gates. The finger gates are 30 nm wide and 50 nm apart. Both top-gate channels are 50 nm wide and the distance between the two top-gate channels is 150 nm. As we will discuss in detail in Chapter 5 and 6, both few-hole single and double quantum dots can be confined using this flexible design.

54 55 Device Fabrication and Measurement Setup

Figure 3.9: (a) Images of the final version of devices used in Chapter 5 and 6. The optical mask used is same as in Figure 3.8. (b) An SEM image of the active dot region of the final version of the dot design. A total of eight finger gates and two top-gate channels are fabricated. Green arrows indicate the current direction through the device. The unlabelled quantum dot on the left-hand-side operates under exactly the same principles as the labelled one.

3.3 Measurement setup

The typical charging energy of a few-hole quantum dot is on the order of a few meV and the typical orbital energy is on the order of a few hundred µeV. The spin properties such as Zeeman splitting are on an even smaller energy scale. Therefore, to study hole quantum dots our samples are cooled to low temperature down to 30 mK. In addition, to study the anisotropic behaviour of hole spins, magnetic fields in different directions with respect to the device is also required. The measurements presented in Chapter 4 were performed in an Oxford Instru- ments Kelvinox 100 He3/He4 dilution refrigerator with a base temperature of 30 mK. The dilution unit is loaded into a liquid He cryostat with a 10 T superconducting magnet. The Kelvinox 100 refrigerator is equipped with an in situ rotation system [68], which is capable of rotating ∼ 100◦ while maintaining the fridge temperature below 400 mK and enables magnetic fields to be applied in different orientations with respect to the device. The measurements presented in Chapter 5 and Chapter 6 were perform in a vector field fridge, which consists of a cryogen free top loading Leiden refrigerator and a 9 T - 5 T - 1 T three-axis vector magnet system (Cryogenic Ltd.). The base temperature of the vector field fridge is also 30 mK and magnetic fields can be applied in any direction using the vector magnet system. Even though the maximum magnetic field available in the vector field fridge system, particularly along some specific directions (e.g. along the 1 T magnet axis), is smaller compared to the Kelvinox 100 fridge, we still found the vector magnet system preferable compared

55 56 Device Fabrication and Measurement Setup to a rotator for quantum dot measurements. This is mainly because the electrical noise generated during rotating the sample stage can be very dangerous to sensitive quantum dot devices. To measure quantum dots via electrical transport, a standard low-frequency a.c. lock-in technique using Stanford Research SR830 lock-in amplifier was used in the single dot measurements presented in Chapter 4. The d.c. voltages required for biasing the gates were applied using a Keithly 2400 source-measure unit or a YOKOGAWA 7651 programmable dc source. An a.c. excitation voltage of 15 µeV was applied through a typical voltage divider with a ratio of 1:10000 to the source contact and current through the device was detected by connecting the drain contact back to the lock-in amplifier via a variable gain Femto DLPCA-200 current amplifier. For the double dot measurement presented in Chapter 4, standard d.c. measurements were performed instead of a.c. measurements to improve the signal to noise ratio. The d.c. bias is applied using a YOKOGAWA 7651 programmable dc source and current through the device was measured using a Keithley 2000 multimeter via a variable gain Femto DLPCA-200 current amplifier.

Figure 3.10: An image of the Delft IVVI measurement rack used in the measurements presented in Chapter 5 and Chapter 6.

To further improve the noise performance of the measurement setup to measure a pA current through a few-hole quantum dot, the measurements presented in Chapter 5 and Chapter 6 were performed using a Delft IVVI measurement rack. As shown in Figure 3.10, the IVVI rack replaces all the voltage sources in our measurements by its digital to analogue (DAC) outputs as well as the Femto DLPCA-200 current ampliﬁer by its own in-built current pre-amp. Since the IVVI rack is completely powered from batteries, it eliminates any possible noise from the main power supplies to the measurement circuit, which is usually one of the main noise sources in our measurements. Note that a Keithley 2000 multimeter was still used to measure the output from the IVVI pre-amp, but it is completed opto-isolated from the dot

56 57 Device Fabrication and Measurement Setup circuit.

57 Chapter 4

Double-level-gate Architecture for Few-hole GaAs Quantum Dots

We report the fabrication of single and double hole quantum dots using a double-

level-gate design on an undoped accumulation mode AlxGa1−xAs/GaAs heterostructure. Electrical transport measurements of a single quantum dot show varying addition energies and clear excited states. By introducing both in-plane and out-of-plane magnetic ﬁelds, we are able to observe the anisotropic Zeeman splitting of single particle hole states in the GaAs hole quantum dot. In addition, the double-level-gate architecture can also be conﬁgured into a double quantum dot with tunable inter-dot coupling.

4.1 Introduction

Long-lived heavy-hole spins have drawn significant attention recently due to the suppressed hyperfine interaction with surrounding nuclei [8, 9, 10, 11, 50, 42], which is the main mechanism leading to fast decoherence (∼ 10 ns) of electron spins in ∗ GaAs quantum dots [2, 3]. Theory predicts the dephasing time T2 for holes to be on the scale of µs, with an Ising-like hyperfine interaction [10, 11]. Optical ∗ measurements of self-assembled quantum dots have shown varied results, with T2 > ∗ 100 ns from coherent population measurements [13], and T2 up to 20 ns from Ramsey ∗ fringes [14, 15]. The short T2 was shown not to be limited by nuclear spins, but by charge noise [14]. The promising optical measurements described above indicate that hole spins in gate-defined quantum dots should be investigated. However, due to the large

58 59 Double-level-gate Architecture for Few-hole GaAs Quantum Dots

∗ ∗ eﬀective mass (mh/me ∼ 3 − 13) [16], hole quantum dots need to have much smaller dimensions compared to electron dots to show similar single-particle energy scales ∗ (Eorb ∼ 1/m Adot), which is hard to achieve by simply duplicating the design for electron quantum dots. Even though the operation of GaAs single hole transistors has been demonstrated [17, 18, 19], to date it has not been possible to observe the Zeeman splitting of the single-particle levels, which is a prerequisite for measure- ∗ ments of the T1 and T2 spin lifetimes, and for coherent spin manipulation. In this chapter, we present a double-level-gate design that allows formation of both single and double hole quantum dots. Electrical transport measurements through the dot show orbital energies comparable to those of GaA electron quantum dots [69]. We observe Zeeman splitting of the excited orbital states in an applied magnetic ﬁeld, and show that the Zeeman splitting is highly anisotropic, due to the strong spin-orbit interaction of holes in GaAs.

4.2 Literature Review

Few-electron quantum dots used to study atomic behaviour are first realized in double-barrier GaAs heterostructures [70]. Those so-called vertical quantum dots are etched pillars of ∼ 10 nm thick with conducting contacts separate from the dot by tunnelling barriers. They are usually fabricated to be circular with a diameter of a few hundred nanometers to reproduce the symmetry of real atoms. Even though vertical quantum dots work very well as an artificial atom for studying atomic properties [58], it is hard to have a complete control over a vertical quantum dot for spin manipulation, or integrate vertical dots into complex electronic circuits. Therefore, gate-defined lateral quantum dots based on GaAs heterostructures have since been widely investigated [71]. For a lateral quantum dot, metal gates are deposited on top of a GaAs/AlGaAs heterostructure with a high-mobility 2-dimensional electron gas (2DEG). Negative voltages are applied to the metal gates to deplete the 2DEG beneath to form quantum dots. However, control over the electron number down to zero is more difficult compared to vertical dots. This is because decreasing the electron number in lateral dots requires driving the gate voltages to more negative values, which also reduces the tunnel coupling between the dot and the leads, resulting in an unmeasurable current in the few-electron regime. The first measurement of lateral quantum dots down to the last electron was report by M. Ciorga et al. in 2000. The device geometry is shown in Figure 4.1 [72]: big gates enclosing the quantum dot are used to define the dot confinement; the upper thin gate is used to set the tunnel barriers and the lower thin gate is used as the plunger gate to vary the dot occupation. This surface gate geometry allows the number of electrons on the

59 60 Double-level-gate Architecture for Few-hole GaAs Quantum Dots dot to be varied over a wide range while maintaining a measurable current through the dot.

Figure 4.1: (a) A scanning electron microscope (SEM) image of the lateral electron quatnum dot design by M. Ciorga et al.. (b) An SEM image of a double quantum dot device with charge sensor quantum point contacts (QPCs) incorporated by J. M. Elzerman et al.. Ohmic contacts (crosses) and current directions are drawn on the images. Figure from Ref [43].

Applying very similar design principles, few-electron lateral double quantum dots shown as Figure 4.1 were demonstrated by J. M. Elzerman three years later in 2003. In addition, two quantum point contacts (QPCs) were also incorporated in the device acting as charge sensors, which allows detecting of interdot charge transitions without transport through the dot. This has since become the conventional design for lateral quantum dots, and only minor adjustments are made by various research groups. It is worth pointing out that the same device (Figure 4.1(b)) can also be used as a single quantum dot if only half of the device is defined. However, so far, research based on electrical measurements of single and double quantum dots has been focused on electron systems. Heavy-hole systems, as a new emerging candidate for spintronic devices due to its suppressed hyperfine interaction [8, 9, 10, 11, 50, 42] and strong spin-orbit coupling [36], remain relatively unexplored due to difficulty in device fabrication and stability. Some progress has recently been made in developing gated hole quantum dots based on GaAs/AlGaAs heterostructures. Y. Komijani et al. [17] demonstrated fabrication of hole quantum dots based on p-doped GaAs/AlGaAs heterostructures. As shown in Figure 4.2 [17], charging energies up to 2 meV and single-particle level spacings of ∼ 100 µeV were observed in these quantum dot. However, due to charge rearrangement in the doping layer, low-frequency switching noise is also visible in Figure 4.2 at certain plunger gate voltages, indicated by the red arrows. The presence of these switching events suggests that not only will any long time-

60 61 Double-level-gate Architecture for Few-hole GaAs Quantum Dots

Figure 4.2: Top: Coulomb diamonds of the p-doped hole dot by Y. Komijani et al.. Inset: an SEM image of the device. Bottom: Zoomed-in Coulomb diamonds showing the positions of the excited states. Figure reproduced from Ref [17]. scale measurements be challenging but also reproducing any experimental results be diﬃcult. To improve the stability of hole quantum dots, induced quantum dots,

Figure 4.3: (a) Conductance G through the hole dot as a fucntion of the plunger gate voltage for one Coulomb blockade peak measured from an induced hole quantum dot by Klochan et al.. (b) Time sweeps of the conductance G at diﬀerent locations on the Coulomb blockade peak as indicated by arrows. Figure from Ref [18]. in which carriers are introduced electrostatically by a biased gate, are proposed. Klochan et al. [18] showed the operation of an induced GaAs single hole transistor in an accumulation-mode GaAs/AlGaAs heterostructure. The dot is deﬁned by wet etching the p+ degenerately doped cap which is used as the metallic gate. Owing to the removal of modulation doping, the hole quantum dot is proven to be very stable compared to previous measured modulation-doped hole quantum dots. Figure 4.3(b) shows the conductance through the induced dot as a function of time for several

61 62 Double-level-gate Architecture for Few-hole GaAs Quantum Dots set values of the plunger gate voltage corresponding to diﬀerent locations on the Coulomb blockade peak. It can be seen from the conductance ﬂuctuations that even at half-height of the Coulomb blockade peak, where the conductance through the dot is most sensitive to noise, the maximum variation is only 2.4% of an electron charge. However, even though device stability has been greatly improved in induced dots, as shown in Figure 4.4 [18], due to the large dimensions of the etched structure, charging energies of the dot are only a few hundred micro-electronvolts, and no clear excited states could be observed.

Figure 4.4: Coulomb diamonds of the etched induced hole quantum dot by Klochan et al.. Inset: an SEM image of the etched quantum dot. Figure reproduced from Ref [18].

Recently, Tracy et al. [19] demonstrated hole quantum dots based an completely undoped GaAs/AlGaAs heterostructure using the conventional gate architecture for electron quantum dots (see Figure 4.1) as shown in Figure 4.5(a). Since the dimensions of the dot are smaller compared to the previous work by Klochan et al., transport through the hole dot shows much clearer Coulomb diamonds as well as visible excited states (Figure 4.5(b)). However, since the dot design is no different from that used for most electron dots, the scale of the energy states observed in the hole dot is smaller compared to similar electron dots [69] due to the large hole ∗ ∗ effective mass (mh/me ∼ 3 − 13) [16]. This means measurements like Zeeman splitting of the energy states, which is the basic of many advanced experiments, remain hard with hole quantum dots. Therefore, to study the properties of single heavy-hole spin in GaAs, the current focus is how to reduced the size of the dot to achieve orbital energies comparable to those of electron dots. In this chapter, we show a double-level-gate architecture for hole quantum dots, which is capable of confining dots with much smaller dimensions than the conventional lateral dot design.

62 63 Double-level-gate Architecture for Few-hole GaAs Quantum Dots

Figure 4.5: (a) An SEM image of the accumulation mode gated hole quantum dot by Tracy et al.. (b) Coulomb diamonds of the induced hole dot. Figure reproduced from Ref [19].

Figure 4.6: (a) An SEM image of a nanowire on thin parallel gold gates. (b) A schematic diagram illustrating the bias measurement setup and the five gates used for dot tuning. (c) Coulomb diamonds of a single quantum dot defined with the left three barriers (d) Charge stability diagram of a double quantum dot defined with all five barrier gates. Insets: schematic diagrams illustating the position of the two dots. Figure reproduced from Ref [73].

The inspiration for this double-level design comes from another type of widely used quantum dot geometry besides lateral dots: nanowire quantum dots. As shown in Figure 4.6(a), nanowire quantum dots are often fabricated by depositing nanowires on top of a processed substrate with patterned parallel thin finger gates insulated by a dielectric. Quantum dots are then defined by locally depleting the carriers in sections of the nanowire right above the finger gates. Due to the small dot size, nanowire quantum dots often exhibit large orbital energies, which are beneficial for further field-dependence measurements. Besides the strong confinement potential, another advantage of the nanowire dot design is the simplicity of incorporating extra dots. As shown in Figure 4.6(c) and (d), depending the number of finger gates used,

63 64 Double-level-gate Architecture for Few-hole GaAs Quantum Dots the device can either work as a single dot if half of the gates are biased, or as a double dot if all gates are biased. To adopt the nanowire design on undoped heterostructures, instead of the big overall inducing top-gate used in Ref [19], we only deposit a 1D inducing top-gate channel to imitate the function of the nanowire, which will be discussed in detail in Section 4.3. Similar designs have been used for silicon quantum dots in metal-oxide- semiconductor (MOS) structures and proven to work well down to the few-electron regime. As shown in Figure 4.7(a) and (b), in the silicon MOS device, a narrow metal top gate used to induced the dot quantum, as well as vary the dot occupation, is deposited on top of two parallel ﬁnger gates, which are biased in the depleting mode to act as tunnel barriers. The bias spectroscopy diagram of the dot shown in Figure 4.7(c) demonstrates stable Coulomb diamonds with charging energies up to 6 meV and clear excited states. Since our system is very similar to the MOS structure, it is expected that the double-level design may perform better for hole quantum dots compared to the conventional design used in Ref [19].

Figure 4.7: (a) An SEM image and (b) a sideview schematic of a Si MOS quantum dot. (c) A bias spectroscopy diagram of the dot in the many-electron regime showing stable Coulomb diamonds and clear excited states. Figure reproduced from Ref [74].

4.3 Nanowire-inspired quantum dot on a planar GaAs heterostructure

The hole quantum dot presented in this chapter is fabricated on a shallow undoped (100) GaAs/AlGaAs heterostructure (wafer W641) comprising a 10 nm GaAs cap and a 50 nm AlGaAs layer on a GaAs buﬀer layer. There are two main reasons why we picked this wafer for fabricating hole quantum dots. First of all, the undoped heterostructure is expected to improve the stability of quantum dot devices [17, 18,

64 65 Double-level-gate Architecture for Few-hole GaAs Quantum Dots

19]. Secondly, the heterointerface is only 60 nm deep, which means that very ﬁne lithographic features can be transferred from surface metallic gates to the 2DHG. Separate measurements of a 2D Hall bar device from the same wafer show that the 2D holes have a mobility of 600, 000 cm2/Vs at p = 2.5 × 1011 cm−2 and T = 250 mK.

The quantum dot architecture is a double-level-gate design: ﬁrstly, three parallel barrier gates are deposited directly on top of the wafer, then a 30 nm dielectric

AlOx is deposited, and finally a 150 nm wide top-gate is deposited over the barrier gates. Figure 4.8(a) shows an SEM image of the device geometry, and Figure 4.8(b) shows a schematic cross-section. When the top-gate (TG) is negatively biased, holes accumulate at the heterointerface forming a 1D channel. Tuning the voltages on the barrier gates (1-3) confines the 1D hole channel into isolated islands, i.e. quantum dots. With the flexibility of three parallel barrier gates, the device can be tuned into either a single quantum dot using any two consecutive barriers while lifting the third, or a double quantum dot using all three barriers. The device was measured in a dilution refrigerator (Kelvinox 100) with a base hole temperature of 80 mK.

Figure 4.8: (a) A Scanning Electron Microscope (SEM) image of the device showing the double-level-gate design. Three barrier gates are on the bottom layer (labelled 1, 2, 3) and have a width of 40 nm and a inter-gate spacing of 80 nm. Another ﬁnger gate (labelled 4) of the same width was used as the plunger gate. The top-gate channel (labelled TG) is on the top layer and has a width of 150 nm. (b) A side-view schematic of the device. The undoped

GaAs/AlxGa1−xAs heterostructure used has a 10 nm GaAs cap and a 50 nm AlxGa1−xAs layer; 30 nm of AlOx is used as the insulator between the two layers of Ti/Au gates. (c) Diﬀerential conductance g of the dot measured as a function of the voltages on gate 2 and

3 when gate 1 is lifted to V1 = −0.9 V, showing the formation of a single quantum dot in between the two barrier gates 2 and 3. (d) A linecut of (c) at V3 = −0.703 V showing the diﬀerential conductance peaks as a function of gate 2 bias V2.

65 66 Double-level-gate Architecture for Few-hole GaAs Quantum Dots

4.4 Single dot operation

4.4.1 Coulomb blockade and bias spectroscopy

Figure 4.9: (a) Coulomb blockade peaks: diﬀerential conductance g of the single quantum dot versus the plunger gate bias V4. (b) Bias spectroscopy diagram of the single quantum dot: diﬀerential conductance g of the quantum dot is plotted as a function of the source- drain bias and the plunger gate bias V4. Dashed lines are guides to the eye highlighting features corresponding to excited states in the dot. Dotted lines show the ground states.

We first present results in the single quantum dot configuration. For the single dot measurement, the device was measured using standard lock-in techniques and an AC excitation of 15µ V. Figure 4.8(c) shows the differential conductance g = dI/dVSD of the dot as a function of the voltages on gate 2 and 3 while VTG = −3.1

V, and gate 1 is lifted to V1 = −0.9 V and not used as a barrier. The dark lines represents conductance peaks due to Coulomb blockade. The straight unbroken lines indicate the formation of a single quantum dot that does not break into two as number of holes goes down. The lines in Figure 4.8(c) are almost at 45◦, indicating formation of a single quantum dot which has almost the same capacitive coupling to the two gates 2 and 3, and thus must be located half way between the two barriers, as depicted in Figure 4.8(b). Figure 4.8(d) plots a linecut of Figure 4.8(c) at V3 = −0.703 V showing the Coulomb blockade peaks as a function of the gate 2 bias V2. Figure 4.9(a) shows the Coulomb blockade peaks of the same dot when gate 2 and 3 are set to V2 = −0.7529 V and V3 = −0.6648 V (on the last visible peak in Figure 4.8(c)) and gate 4 is used as a plunger gate to change the number of holes in the dot. The suppression of the peak height when gate 4 is more positive is due to opaque barriers caused by the crosstalk between gate 2/3 and gate 4. Figure 4.9(b) shows the diﬀerential conductance of the dot as a function of both

66 67 Double-level-gate Architecture for Few-hole GaAs Quantum Dots the source-drain bias and the plunger gate voltage. This bias spectroscopy diagram shows an increasing addition energy from 1.8 meV to 2.7 meV, which indicates the dot is operating in the few-hole regime where the varying orbital energies are no longer negligible compared to the near constant charging energy. The dashed lines in Figure 4.9(b) run parallel to the edges of the Coulomb diamonds, marking excited orbital states within the dot. From Figure 4.9(b), the orbital level spacing is 300-400 µeV. This orbital energy scale is comparable to that of 2D GaAs electron quantum dots [69], in which single electron spin properties have been extensively studied. It is also worth pointing out that a large orbital energy is only achievable when the hole dot size is significantly reduced compared to its electron counterpart given the big mismatch between the heavy hole and electron ∗ ∗ 2 ∗ effective masses in GaAs (mh/me ∼ 3 − 13 [16]). Using Adot = 2π¯h /Eorbmh, we −15 2 ∗ estimate the dot size to be around 4 × 10 m (mh = 0.4m0), consistent with the lithographic dimensions of the dot (∼ 150 nm×30 nm, note that due to the side-growth of the AlOx, the actual lithographic dimension of the dot is expected to be smaller than just the inter-gate spacing of the finger gates). Therefore, from the orbital energy, we show that the double-level design can effectively reduce the size of the dot from the conventional lateral dot structure [69, 17, 19], and we are able to achieve single hole energy level spacings similar to those of electrons in GaAs [69], which is necessary to study hole spin states in lateral quantum dots.

4.4.2 Zeeman splitting and anisotropic Land´eg-factor

Figure 4.10: Zoomed-in bias spectroscopy diagrams focusing on the splitting of energy states at (a) B⊥ = 0 T and (b) at B⊥ = 1 T. Both ground and excited states show a clearly resolvable splitting. (c) and (d) are line cuts along the red dotted lines in (a) and (b) showing corresponding resonance peaks in diﬀerential conductance.

Since the ability to resolve Zeeman splitting is essential to distinguish and ma-

67 68 Double-level-gate Architecture for Few-hole GaAs Quantum Dots nipulate spin states, we now focus on investigating the behaviour of the orbital states in magnetic fields. In GaAs electron systems, the Landég-factor is isotropic, so the magnetic field is usually applied in the plane of the 2D system to cause Zeeman splitting without coupling to the orbital motion. However, for holes, the g-factor ∗ is highly anisotropic. For 2D hole systems, the out-of-plane g-factor is g⊥ = 7.2, ∗ whereas the in-plane g-factor gk is negligible [36]. For 1D hole systems, the g-factor ∗ is also highly anisotropic [75, 76], although g⊥ decreases as the 1D subband index goes down [54, 55]. Therefore, to Zeeman split the energy states of the dot, we first introduce a magnetic field B⊥ perpendicular to the heterostructure. Figure 4.10 shows the zoomed-in bias spectroscopy diagrams at (a) B⊥ = 0 T and (b) B⊥ = 1 T focusing on the splitting of single particle levels. Linecuts along the red dashed lines in (a) and (b) are further plotted in Figure 4.10(c) and (d) illustrating the resonance peaks in the differential conductance corresponding to transport through different energy states. As indicated by the green and blue dashed lines, both the ground state and the excited state split in an out-of-plane magnetic field. The ground state splits slightly slower than the excited state with Zeeman energies of 80 µeV and

140 µeV at B⊥ = 1 T respectively. It is worth pointing out that the same splitting is visible in both positive and negative bias edges of the Coulomb blockaded region which conﬁrms that the observed splitting comes from the energy levels of the dot but not from states in the leads [43].

Figure 4.11: (a) Diﬀerential conductance through the dot is measured as a function of

VSD and B⊥ showing the continuous Zeeman splitting of the energy states. Gate 4 is set at V4 = −0.45 V. Dotted lines are guide to the eye highlighting the ground state splitting which is used to extract the g-factor. (b) Zeeman energy of the ground state splitting is plotted as a function of the perpendicular magnetic ﬁeld. Dotted line is a ﬁt to the linear part of the splitting which gives an out-of-plane g-factor of 2.6.

A continuous map of the Zeeman splitting is shown in Figure 4.11(a): diﬀerential

68 69 Double-level-gate Architecture for Few-hole GaAs Quantum Dots

conductance at V4 = −0.45 V is swept for negative source-drain bias up to VSD = −2 mV while stepping the perpendicular magnetic ﬁeld from 0 to 1 T. We focus on the splitting of the ground state as highlighted by the dotted lines. A plot of the Zeeman splitting of the ground state as a function of the magnetic ﬁeld strength is shown as

Figure 4.11(b). The Zeeman energy increases linearly when B⊥ < 0.7 T, which gives ∗ ∗ an effective out-of-plane g-factor of g⊥ = 2.6 (dashed line, ∆E = g⊥µBB). When B⊥ > 0.7 T, the Zeeman energy starts to deviate from a linear splitting and even decreases when the field goes above 0.9 T. This can also been seem in the colour map where the split line deviates from the linear dotted guide line. We attribute this non-linear behaviour of Zeeman energy at large fields to the repulsion between levels with opposite spins [43], which is a result of the strong spin-orbit interaction of heavy-holes in GaAs [36]. ∗ ∗ The measured out-of-plane g-factor g⊥ = 2.6 is smaller than the 2D value of g⊥ = ∗ 7.2, but is consistent with the suppression of g⊥ observed in 1D hole systems [54, 55], and is possibly a result of the strong confinement potential of the dot. Strong confinement can change the degree of heavy hole-light hole (HH-LH) mixing in GaAs hole systems [77] and thus modify the magnitude and anisotropy of the g-factor. The ∗ observed suppression of g⊥ is a typical signature of increased HH-LH mixing, which ∗ is also predicted to result in a non-negligible gk [77]. Therefore, it is very interesting to explore how the energy states of the dot behave in an in-plane magnetic field.

Figure 4.12: Zoomed-in bias spectroscopy diagrams focusing on the splitting of energy states at Bk = 0 T (a); at Bk = 5 T (b). The differences compared to Figure 4.10 are caused by a slightly different dot configuration after rotation. No clear splitting is visible from any energy states. (c) and (d) Line cuts along the red dotted lines in (a) and (b) showing the differential conductance traces before and after splitting.

After rotating the sample stage by 90 degrees, while keeping the temperature below 400 mK using a in-situ rotation system [68], we show in Figure 4.12(a) and

69 70 Double-level-gate Architecture for Few-hole GaAs Quantum Dots

(b) the same bias spectroscopy diagram with magnetic fields Bk = 0 T and Bk = 5 T applied along the current direction, as well as in Figure 4.12(c) and (d) the corresponding linecuts along the red dashed lines. It is worth pointing out that compared to Figure 4.10, all the differential conductance peaks are broader in Figure 4.12, which is caused by a small variation of the dot configuration. The quantum dot was grounded to avoid charge noise from the rotator between the two measurements. Even though the dot was re-tuned to the same hole number, the configuration of the barriers used in Figure 4.12 is slightly more transparent than that used for the perpendicular-field measurement previously, resulting in broader features in the differential conductance. Nonetheless, this should not affect the qualitatively comparison between the in-plane and out-of-plane g-factors as only a upper limit for the in-plane g-factor taking into account the level broadening will be estimated. As shown in Figure 4.12(b) and (d), no clear splitting can be seen except broadening of all energy levels and similar broadening persists to Bk = 10 T which is the maximum field we could apply. From the energy level broadening, we estimate the ∗ ∗ in-plane effective g-factor gk to be less than 0.4. This small gk is consistent with ∗ the characteristics of 2D heavy holes in GaAs which have negligible gk. However, ∗ ∗ it is difficult to reconcile this small gk with the suppression of g⊥, since if there is ∗ HH-LH mixing it might be expected to increase gk. This surprising result makes ∗ it unclear what is the mechanism of the suppression of g⊥ and raises the possibility that there might be processes other than HH-LH mixing affecting the Zeeman anisotropy. However, since the orbital energy of the quantum dot (∼ 300−400 µeV) is more than 10 times smaller than the separation of the heavy and light hole 2D subbands (∼ 5 meV), the degree of HH-LH mixing may not be sufficient to lead to a ∗ measurable gk. Therefore, it will be very interesting to repeat these measurements on quantum dots with even stronger confinement potentials and larger orbital energies to disentangle the effects of HH-LH mixing from other possible mechanisms ∗ through a measurable gk.

4.5 Double dot operation

4.5.1 Tunable interdot coupling

In this section, we present the measurements in the double dot configuration. Double quantum dots are of great interest since they allow spin-to-charge conversion, which enables spin readout in coherent spin manipulation [2, 3, 4, 5]. The tunability of interdot coupling is one of the fundamental features required for double quantum dot devices [59]. It helps to disentangle effects from different spin mixing mechanisms

70 71 Double-level-gate Architecture for Few-hole GaAs Quantum Dots such as the hyperﬁne interaction and the spin-orbit interaction [62].

Figure 4.13: (a) A side-view schematic of the device showing the double dot conﬁguration. (b) Diﬀerential conductance g as a function of voltages on barrier gate 1 and 3 for strongly coupled double dot (V2 = −0.80 V); (c) for intermediately coupled double dot (V2 = −0.79

V); (c) and for weakly coupled double dot (V2 = −0.78 V).

An advantage of the double-level-gate design is the straightforward incorporation of an additional dot to form a double quantum dot and the simple interchanging between single and double dot configurations. With the same device, a double quantum dot can be easily constructed by bringing down the voltage on gate 1, which was lifted to V1 = −0.9 V for the single dot measurement, and using it as a third barrier. Similar to the operation of gate 2 and 3, this will also confine another dot in between gate 1 and 2 as shown in Figure 4.13(a). The differential conductance of the double dot as a function of voltages on gate 1 and 3 is shown in Figure 4.13(b)-

(d) with V2 = −0.80 V, V2 = −0.79 V and V2 = −0.78 V respectively. The coupling between the two dots is controlled by the bias V2 on the middle barrier gate. In Figure 4.13, we show three regimes of inter-dot coupling: (b) strongly coupled double dot with curving diagonal parallel lines resembling Figure 4.8(c), indicating the two dots are merging into one; (c) intermediate coupled double dot with standard honeycomb pattern; and (d) weakly coupled double dot when leakage current is greatly suppressed and only transport through triple points is visible. The ability to conﬁgure double hole quantum dots with tunable interdot coupling suggests that the device design also allows more complicated double dot measurements and spin- dependent transport.

4.5.2 Bias triangles and resonant tunnelling

Compared to a single dot, where 0D states are probed by the reservoir, with a thermal distribution of kBT , a double dot allows the tunnelling from one dot into the

71 72 Double-level-gate Architecture for Few-hole GaAs Quantum Dots other, and thus probing 0D states using 0D states which have an energy broadening limited by the lifetime of the 0D state, not the temperature. Therefore, resonant tunnelling through 0D states in a double quantum dot removes the broadening of energy levels in single dot experiments due to the finite temperature of the reservoir [78]. As shown in Figure 4.14(a), DC current through the double dot in the weakly coupled regime is measured as a function of the voltages on left and right barriers with a source-drain bias of VSD = 0.8 mV. With a finite bias, triple points grow into bias triangles whose size directly reflects the bias applied. A detailed discussion of the features of bias triangles can be found in the background chapter (Section 2.5.2). Note that from now on, we switch to the DC measurement configuration (Section 3.3). This is because for bias spectroscopy measurements of a double quantum dot, if only the change in current is monitored (AC measurement configuration), one would not be able to tell whether the current through the dot is suppressed (e.g. spin blockade) or simply not changing. Therefore, DC current instead of AC current is often monitored when measuring a double quantum dot.

Figure 4.14: (a) Stability diagram of one pair of bias triangles in the weakly coupled regime with a source-drain bias of VSD = 0.8 mV. The blue arrow indicates the detuning axis. (b) DC current through the double dot alone the red dashed line in (a) showing the conductance resonance peaks when the energy levels of two dots are aligned. Corresponding peaks are denoted by the same symbol in the colour map. (c) Simple schematics showing the alignment of levels in two dots for positions in the bias triangle denoted by the star, triangle and square. In notation µm,n, m represents the dot number (dot 1 or dot 2) and n represents the energy state, i.e. 0 for a ground state and 1 for the ﬁrst excited state.

Resonant tunnelling of the two dot levels can be seen as the current peaks parallel to the base of the triangle as indicated by the arrows in Figure 4.14(a). Here we focus on the base part of the bias triangle. The current through the dot along the red dashed line is plotted as a function of the voltage on gate 1 in Figure 4.14(b), illustrating the three resonant tunnelling peaks (star, triangle and square) seen in

72 73 Double-level-gate Architecture for Few-hole GaAs Quantum Dots the bias triangle. One possible level alignment for those resonant tunnelling peaks is illustrated by the simple schematics in Figure 4.14(c). Since the experiment is performed in the weak-coupling regime, we ignore the mixing between quantum states in one dot with states in the other dot and only consider the alignment of two separate sets of electrochemical potential levels in two dots. The detuning axis ε is defined to be perpendicular to the base of the triangle as depicted by the dark blue arrow in Figure 4.14(a). As discussed in Section 2.5.2, the detuning energy is a measurement of the ground state energy difference between the two dots, and a positive detuning (ε > 0) suggests that the energy level of the left dot is detuned to be higher than the energy level of the right dot. The first peak (star) corresponds to the base of the bias triangle when the ground states of the two dots align. Going up the bias triangle to a small positive detuning, the energy levels of the left dot move up in energy and a second resonant tunnelling peak (triangle) is seen when the excited states of two dots align. When the levels of the left dot move further up, the ground state of the left dot eventually aligns with the excited state of right dot resulting in a third peak (square). Note that the second resonant peak has a smaller current compared to the other two peaks, which is possibly caused by relaxation within the left dot. If the relaxation rate in the left dot is comparable to the interdot tunnelling rate, holes on the excited state may relax to the ground state before it tunnels to the right dot, which reduces the visibility of the corresponding resonant peak (blue triangle). From the proposed scenario of the level alignment in Figure 4.14(c) and the corresponding peak positions, the single-particle level spacings of the two dots can be extracted as ∆E1 ∼ 130 µeV for the left dot and ∆E2 ∼ 200 µeV for the right dot, which are consistent with previous single dot measurements when only half of the device is defined. It is worth pointing out that the energy difference between the first and second resonant peak is only only ∼ 70 µeV, which confirms the fact that resonant tunnelling in a double dot have excellent resolution of energy states. To examine the energy evolution of the levels as a function of the magnetic field to gain more information about the spin states in the double dot, a perpendicular

field B⊥ is applied while measuring the current through the dot. As shown in Figure 4.15(a), gate 1 is swept along the red dashed line in Figure 4.14(a) while the perpendicular magnetic field B⊥ is stepped from 0 to 1 T. A few interesting features can be seen in Figure 4.15(a). First, we observe a splitting of the base of the bias triangle (green star) as indicated by the red and green dots. To explain this splitting, we draw in Figure 4.15(b) the evolution of the energy levels in left and right dot respectively (the two upper diagrams). We assume that spin is conserved in the tunnelling process and the ground states of both dots split in a perpendicular field,

73 74 Double-level-gate Architecture for Few-hole GaAs Quantum Dots

Figure 4.15: Magnetic field spectroscopy of the double dot. (a) Current through the double dot along the red dashed line in Figure 4.14(a) plotted as a function of the voltage on gate 1 and the magnetic field. (b) Schematics showing how levels evolve in a magnetic field for the left dot, the right dot and the resultant peak positions for the double dot when the levels from two dots are combined. Up and down arrows indicate the spin of a specific energy level. (c) Schematics illustrating the level alignment between the two dots for three different positions in the magnetic field spectroscopy (red, green and pink).

but with different rates. If the ground state of the left dot splits much slower than the right dot so that when the left dot levels are used to probe the right dot levels (Figure 4.15(c) red and green panels), the difference in the splitting rate between the two dots will result in a splitting-like feature in the current through the double dot (also as illustrated schematically by the lowest diagram in Figure 4.15(b)). The energy of the splitting measured from the double dot current thus corresponds to the difference of Zeeman splitting between the two dots. Second, we observe parallel resonant tunnelling lines which have the same B-dependence as indicated by the green and pink dots. If we follow the previous assumption of the level alignment (Figure 4.14(c)), the line indicated by the pink dot corresponds to the alignment of excited states between two dots. Since no splitting-like feature is observed for this resonant peak, at least one of the excited states is spin-polarised and has the same spin configuration as the resonant peak it is parallel to. According to this feature, we label the spins of different levels in two dots as shown in Figure 4.15(b) and (c). Note that in all the schematics, we assume excited states for both dots are spin- polarised for simplicity of the drawings. The existence of a spin-polarised excited state is consistent with previous experiments on single dots with closely spaced energy levels, where a parallel spin filling is demonstrated to be favourable [79]. However, due to the unknown number of holes on the dot, the level configuration (Figure 4.14(c)) we have been using to explain different resonant tunnelling events

74 75 Double-level-gate Architecture for Few-hole GaAs Quantum Dots is not exclusive. Even though the base of the bias triangle always corresponds to the alignment of the ground states in two dots, the first resonant peak above the base (blue triangle) could come from other cases of level alignment. Figure 4.16 shows the schematics of two examples of possible level alignment between two dots at B = 0 for the first resonant peak above the bias triangle base (blue triangle): (a) it comes from the alignment of the excited state of a lower hole occupation in the left dot with the ground state of the right dot; and (b) it comes from the alignment of the left dot ground state with the right dot excited state. Despite the complexity of the energy evolution when a magnetic field is further applied, both cases can be argued to explain the same features in the magnetic field spectroscopy discussed previously. The fact that we are not able to rule out the possibility of different dot level configurations based on the magnetic field spectroscopy measurements suggests that even smaller dots with well defined hole occupation will be necessary to understand more about the hole spins in a double quantum dot.

Figure 4.16: Schematics showing the other two possible level alignemnt between two dots that could result in the same resonant peak (blue triangle) in Figure 4.14 at B = 0. Black lines indicate ground states and blue lines indicate excited states.

Nonetheless, comparing to the Zeeman splitting of the single dot conﬁguration (Figure 4.11(a)), the lines in Figure 4.15(a) are much sharper (line width 25 µeV ∼

290 mK) and clear splitting is visible even at B⊥ = 0.2 T with better resolution. This diﬀerence between the single and double dot measurements highlights the advantage of double quantum dots, especially in measuring intrinsic properties of 0D states.

4.6 Conclusions and improvements of the design

In conclusion, we fabricated both single and double few-hole quantum dots on an undoped AlxGa1−xAs/GaAs heterostructure using a new double-level-gate design.

75 76 Double-level-gate Architecture for Few-hole GaAs Quantum Dots

Electrical transport measurements through the single quantum dot show orbital energies comparable to those of electron quantum dots in GaAs. By applying both

B⊥ and Bk, we observe the Zeeman splitting of the orbital states, from which the anisotropic Landég-factor of holes in a quantum dot is estimated. Finally, we show the device can also be easily configured into a double quantum dot with tunable inter-dot coupling, which illustrates the flexibility of the double-layer architecture. Even though the dot is in the few-hole regime and performs quite well compared to the conventional design, we still could not completely empty the dot, and we estimate there are roughly 10 holes on the dot. This essentially means that the single-particle level spacing we observed is still smaller than what one would expect for a dot containing only one or two holes. This is also the reason why the Zeeman splitting we observed is not linear when the perpendicular magnetic field is larger than 0.8 T (Figure 4.11), as it is masked by the level repulsion between closely spaced energy levels, which leaves limited field range to extract the g-factor, particularly for the single dot. This problem suggests even smaller dot dimensions will be desirable to fully investigate the properties of 0D hole states without any artificial effects.

Figure 4.17: SEM images showing the improvement of the dot design: (a) First version: the design presented in this chapter. Finger gates are 40 nm wide with an intergate spacing of 80 nm. Top-gate is 150 nm wide. (b) Second version: Finger gates are 30 nm wide with an intergate spacing of 50 nm. Top-gate channel is 50 nm wide. (c) Third version: the ﬁnal design used in the next two chapter. Gate demensions are the same as in (b) but more ﬁngers are deposited.

On the other hand, in the single dot conﬁguration, the plunger gate used is too far away from the dot and thus couples rather weakly to the dot. This is the reason why we only see a few Coulomb blockade peaks over the full range of the plunger gate bias without leakage (Figure 4.9). Therefore, a dedicate plunger gate, ideally right above the dot, will be preferable for a complete control of the dot occupation

76 77 Double-level-gate Architecture for Few-hole GaAs Quantum Dots within the tunable range of the plunger gate voltage. Based on these two requirement, we improved our design to a second version as shown in Figure 4.17(b). In the second version of the double-level design, we reduced the width of the ﬁnger gates from 40 nm to 30 nm, the intergate spacing from 80 nm to 50 nm and the top-gate channel width from 150 nm to 50 nm. Due to the reduced intergate spacing, we also decreased the dielectric AlOx between the two layers of gates from 30 nm to 10 nm, even though 30 nm of AlOx is still used outside the eﬀective dot region as the insulator for the ohmic contacts.

Figure 4.18: Coulomb oscillations measured with the second version of the dot design with

VTG = −3.1 V, VL = −0.817 V and VR = −0.83 V.

As shown in Figure 4.18, a basic characterization of the new design shows many more Coulomb oscillations compared to the previous device (Figure 4.9(a)), indicating a much wider tunable range of the dot occupation. Further bias spectroscopy measurement reveals a considerably larger orbital energy of ∼ 800 µeV as shown in Figure 4.19, which confirms a more than twofold reduction of the dot size. However, the visibility of the last Coulomb diamond is still very poor due to opaque barriers, and we could not see the formation of a double quantum dot even when the top-gate voltage is raised to VTG = −4 V. To solve these remaining problems, we further improved the dot design to a third version as shown in Fig- ure 4.17(c), which is also the final version with which we could see the empty state of both the single and double quantum dots. There are two major modifications implemented compared to the second version. The first one is the number of finger gates: we added two more parallel finger gates in the first layer which will give the capacity to form a double dot as well as a single dot within the same device depending on which gates are used as tunnelling barriers. Secondly, we replaced the dielectric AlOx (κ ∼ 8) insulating two layers of gates by HfOx (κ ∼ 18) [67], whose higher κ allows a stronger potential from the top-gate for the same oxide thickness

77 78 Double-level-gate Architecture for Few-hole GaAs Quantum Dots

Figure 4.19: Derivative of the current versus source-drain bias and plunger gate voltage measured using the second version of the dot design. Dot configuration is the same as in Figure 4.18. Dashed lines are guide to the eye indicating the edges of the last few Coulomb diamonds, and the first orbital state of the last visible transition. without leakage. We will show the interesting results obtained using the final design in the following chapters: Chapter 5 will focus on the single dot configuration and Chapter 6 will focus on the double dot configuration.

78 Chapter 5

Single Hole GaAs Lateral Quantum Dots

We present experiments with a gated GaAs hole quantum dot, which allows transport measurements down to the single-hole limit. Zeeman splitting of the orbital states is studied via magnetospectroscopy measurements. The extracted value of the hole g-factor strongly depends on the orbital state, and is also highly anisotropic with respect to magnetic field orientations. We show that these peculiar behaviours of the hole spin can be qualitatively explained by the effects of strong spin-orbit coupling and strong Coulomb interaction in a zero-dimensional (0D) hole system. By varying the dot size in situ, we also demonstrate the tuning of the g-factor anisotropy and estimate the electrically- defined dot shape.

5.1 Introduction

Semiconductor quantum dots have been intensively studied as one of the main candidates for spintronic and quantum information processing devices [80]. Impressive progress, including spin initialization, manipulation and readout [1, 2, 3, 4], has been made using GaAs/AlGaAs heterostructure electron quantum dots, which are one of the most used and well characterized semiconductor systems. However, fast decoherence of electron spins due to the unavoidable hyperﬁne interaction with surrounding nuclear spins [7, 8] still remains a major problem in GaAs electron systems. As one possible replacement for electrons, holes in GaAs have drawn a lot of attention recently due to their long spin lifetimes and potential applications in quantum information processing [8, 9, 10, 11]. Optical experiments on self assembled quantum dots have demonstrated promising properties of hole spins in GaAs [13, 51, 14, 15].

79 80 Single Hole GaAs Lateral Quantum Dots

However, it is diﬃcult to incorporate self assembled quantum dots into complex circuits, and gate-deﬁned quantum dots are better candidates for the ultimate control of the performance of the device. Even though the operation of gated hole quantum dots based on GaAs/AlGaAs heterostructures have been shown before [17, 18, 19], to the best of our knowledge, few-hole quantum dots with hole occupations down to zero have not been demonstrated in transport measurements. Besides the interest in a long-lived spin states, as a system with strong spin-orbit coupling, heavy-holes in GaAs quantum dots are proposed to be suitable for all-electric and ultrafast spin manipulations for new spintronic devices [53, 12].

Furthermore, GaAs hole quantum dots are excellent systems to study spin-orbit interaction. 2D and 1D GaAs hole systems have been intensively studied over the past years especially focusing on the anisotropic Zeeman splitting in magnetic fields [36, 55, 54] caused by strong spin-orbit coupling. 2D GaAs hole systems such as 2D hole gases (2DHGs) in GaAs/AlGaAs heterostructures have been shown to have a highly anisotropic g-factor as a result of the change in the valence band structure caused by the 2D confinement potential. 1D GaAs hole systems such as quantum point contacts (QPCs) achieved by adding extra confinement in the plane of 2DHGs, have also been demonstrated to exhibit an anisotropic g-factor which is sensitive to the in-plane confinement potential [54, 55]. The g-factor in 0D GaAs hole systems, on the other hand, remains unexplored experimentally so far mainly due to difficulties in isolating a single hole spin in electrical measurements.

In addition, due to a large eﬀective mass, the interaction parameter rs for holes is usually larger than 10 and a value of 40 is generally achievable. This means holes in GaAs are strongly correlated, which makes GaAs hole quantum dots a good system to study interaction eﬀects such as Wigner molecules [40].

In this chapter, we present experiments with a GaAs hole quantum dot, which allows transport measurements down to the single-hole limit. Zeeman splitting of different orbital states is studied via magnetospectroscopy measurements. The hole g-factor is observed to be strongly dependent on the orbital state, as well as highly anisotropic with the magnetic field orientation. These peculiar behaviours of the hole spins can be qualitatively explained by the effects of strong spin-orbit coupling and strong hole-hole interaction in a 0D confinement potential. By varying the dot size in situ, we also show the tuning of the g-factor anisotropy, which is useful in both applications such as selective spin control, and basic characterisations including identifying the shape of the dot and the strength of heavy hole-light hole (HH-LH) mixing.

80 81 Single Hole GaAs Lateral Quantum Dots

5.2 Literature Review

In this section, we review the current progress in the study of quantum dots, mainly focusing on three aspects that are closely related to our experiments:

• The irregular shell ﬁlling due to the Coulomb interaction;

• The anisotropic Land´eg-factor due to the spin-orbit interaction;

• The eﬀects of HH-LH mixing in hole quantum dots.

Figure 5.1: (a) Coulomb oscillations in the current as a function of the gate voltage measured using vertical GaAs quantum dots. Inset: Addition energy versus electron number calculated from the distance between adjacent peaks. (b) Schematics showing the addition of electrons to circular orbits. The ﬁrst shell can hold two electrons and the second shell can contain up to four electrons, which means the addition of the third and seventh electron will cause extra energy. Figure from Ref [58].

Electron quantum dots have been widely used as an effective means to study the atomic-like properties of artificial atoms. S. Tarucha et al. [81] observed shell filling effects in vertical GaAs quantum dots, which are defined as pillars etched in a GaAs/AlGaAs double-barrier heterostructure. As shown in Figure 5.1 [58], low-bias Coulomb blockade oscillations were measured down to the last electron on the dot to reveal the dependence of the oscillation period on the dot occupation. The energy to add one more electron to the dot was directly extracted from the spacing of the Coulomb blockade peaks. It was observed that the addition energy is unusually large for electron numbers N = 2, 6 and 12, which agrees with the complete shell filling numbers for a 2D harmonic potential in a disk-like quantum dot. ∗ ∗ Holes in GaAs, on the other hand, are much heavier than electrons (mh/me ∼ 6), ∗ rs ∝ m for holes is generally larger than 4, which places GaAs hole systems in the strong-interaction regime [40]. This suggests that the Coulomb interaction plays

81 82 Single Hole GaAs Lateral Quantum Dots

Figure 5.2: The energy position of the charging peaks versus the magnetic field measured an InAs hole quantum dot by D. Reuter et al.. The insets show the occupation of the single-particle levels for each peak before and after level crossing. Figure reproduced from Ref [82]. an important role in hole quantum dots, in contrast to the conventional filling of single-particle orbitals demonstrated for electron dots. D. Reuter et al. [82] studied the hole charging spectra of self-assembled InAs quantum dots in perpendicular magnetic fields by capacitive-voltage spectroscopy. As shown in Figure 5.2, from the magnetic-field dependence of the peaks, only the s-orbital was found to be filled with two holes with opposite spins while higher orbitals p−(l = −1), p+(l = +1) d−(l = −2), d+(l = +2) were all found to be only half occupied by a single hole with

Jz = −3/2. The resulting ssp−p+d−d+ shell filling suggests a strongly spin-polarised six-hole ground state (S = −6), with five out of six holes having Jz = −3/2. This incomplete shell filling is very different from electron quantum dots system, in which the six-electron ground state has a ssp−p−p+p+ configuration and is unpolarised (S = 0). This result confirms the strong correlation in hole quantum dots as it suggests that the reduction of total Coulomb energy through exchange interaction is large enough to overcome the energy difference between d and p shells and leads to the filling of the d shell before the p shell is full [82]. However, since a self-assembled quantum dot was used in that study, lack of control over the dot confinement potential limited a thorough investigation of the observed hole-hole interaction. A gated quantum dot structure is usually preferred in exploring the interplay between the classical Coulomb interaction and the filling of quantised single-particle levels, for the ultimate control of the dot potential and thus the changeover between non-interacting and interacting regimes [83, 84, 85]. Since the GaAs hole quantum dot studied in this thesis is completely defined by metallic gates on a heterostructure, it is considered a promising candidate for investigating

82 83 Single Hole GaAs Lateral Quantum Dots the interesting many-body effects. Another interest in studying hole quantum dots focuses on the effects of strong spin-orbit coupling and the anisotropic Landég-factor, which are intrinsic to hole systems. S. Takahashi et al. [86] studied the anisotropy of the spin-orbit interaction in an InAs self-assembled electron quantum dot. An external magnetic field was applied in different directions to map out the angular dependence of the electron g-factor. From the splitting of the 1st, 3rd and 5th Coulomb blockade peaks, the g-factors for the 1s, 2p− and 3d− orbital states were extracted. As shown in Fig- ure 5.3 [86], the electron g-factor exhibits a strongly anisotropic behaviour for the ∗ 2p− and 3d− orbitals with a maximum g along the out-of-plane direction and a minimum g∗ in-plane, whereas the g-factor for the 1s state barely changes over the same field orientation range. This qualitatively reflects that electrons with larger angular momentum experience stronger magnetic confinement, resulting in a stronger angular dependence of the g-factor [86]. Moreover, we can see from S. Takahashi et al.’s result that the electron g-factor exhibits a strong dependence on the orbital state: the higher the orbital, the smaller the g-factor. Even though this interesting behaviour of the g-factor is not explained in the paper, we suspect that it can be a result of the different orbital wavefunctions.

Figure 5.3: (a) Schematic illustrates the orientation of the magnetic ﬁeld with respect to the self-assembled InAs quantum dot by Takahashi et al.. (b) Electron g-factor as a function of

θ (see (a)) for 1s, 2p− and 3d− orbital states respectively. Figure reproduced from Ref [86].

The experimental results by Takahashi et al. agree very well with an earlier theoretical calculation for InAs quantum dots by C. E. Pryor and M. E. Flatté[56]. C. E. Pryor and M. E. Flattécalculated the g-factor for self-assembled InAs/GaAs quantum dots using 8-band strain dependent k · p theory. The magnetic field was included by coupling to the envelop function and the spin. Figure 5.4 [56] shows the main results from the model for both electrons and holes. The calculated g-factor

83 84 Single Hole GaAs Lateral Quantum Dots

is plotted as a function of Eg, which can be considered as a parameter inversely proportional to the dot size or a parameter characterising the energy difference between discrete single-particle levels in the quantum dot. Due to spin-orbit coupling, which preferentially aligns the spin antiparallel to the orbital momentum, the bare g = 2 is modified resulting in an anisotropic effective g-factor. Since the spin-orbit coupling is much stronger for the valence band, the theory predicts a huge correction ∗ of the hole g-factor. As shown in Figure 5.4, the effective hole g-factor gh is highly anisotropic with the out-of-plane values (Figure 5.4(b)) ∼ 8 times larger than the in- plane values (Figure 5.4(c) and (d)). As a function of the in-plane dot size (∝ 1/Eg), ∗ |gh| is predicted to increase as the dot gets smaller for all directions (dashed lines in Figure 5.4(b)-(d)). More details of this theoretical calculations will be discussed in Section 5.4.2 and Section 5.5 in comparison to our experimental results.

Figure 5.4: (a) and (b) Calculated InAs/GaAs quantum dot g-factor as a function of the dot size (parameterized by Eg) in the out-of-plane [001] direction. (c) and (d) InAs/GaAs quantum dot g-factor vs the dot size parameter Eg in the two in-plane directions [110] and [110].¯ The g-factors are calculated for diﬀerent dot shapes. Blue traces are the results for a ﬂat dot (h=1.7 nm) while red traces are the results for a tall dot (h=2.8 nm). Two in-plane dot shapes are also calculated: crosses are the g-factors calculated for a circular dot (e = d[110]/d[110]¯ =1) and dots are g-factors calculated for an elongated dot in the [110] direction (e=1.4). Only the calculated electron g-factors are plotted in (a) whereas in (b)- (d), both electron and hole g-factors are plotted with dashed lines representing holes and solid lines representing electrons. Figure reproduced from Ref [56].

Even though the electron g-factor in III-V semiconductor quantum dots predicted by the theory for self-assembled InAs dots [56] has been verified [86], the Zeeman splitting of the hole counterparts, which have a much stronger spin-orbit interaction, remains mainly unexplored experimentally. Therefore, a qualitative comparison between our experimental results on a single hole quantum dot and the theoretical calculations by C. E. Pryor and M. E. Flatté[56] will be beneficial to understand

84 85 Single Hole GaAs Lateral Quantum Dots the anisotropic behaviour of hole systems. It is worth pointing out that as shown in Figure 5.3(a), a self-assembled quantum dot has a pyramidal shape, which, at first glance, may seem to have a very different confinement potential compared to the lateral quantum dot structure we are using. However, as shown in schematic Figure 5.7(b), due to the strong confinement potential and the small dot size, our hole quantum dot may look very much like an inverted pyramid. This similarity in the actual dot shape between the self-assembled quantum dot and our gated quantum dot suggests that the theoretical calculation by C. E. Pryor and M. E. Flatté[56] could be applicable to our experimental results.

Figure 5.5: Isosurface of first three electron states and first three hole states for a flat (Dot F, 25.2 nm base and 3.5 nm height) and a tall (Dot T, 25.2 nm base and 5 nm height) InGaAs/GaAs self-assembled quatnum dot. Figure from Ref [87].

Besides strong spin-orbit coupling, another important eﬀect in hole quantum dots that people often overlook is the HH-LH mixing. In GaAs/AlGaAs heterostructures, the HH (Jz = ±3/2) and LH (Jz = ±1/2) states are separated by a energy gap

∆hh−lh, which is caused by the 2D confinement potential (a detailed discussion can be found in Section 2.3.1. For a 2DHG in high-symmetry directions such as (100), HH-LH mixing is usually negligible at normal carrier densities [36]. However, in a quantum dot, the degree of HH-LH mixing can be strongly altered by the dot confinement potential, significantly influencing the spin splitting of quantised energy levels [77]. As shown by G. Bester et al. [87], who calculated the wavefunctions of a self-assembled quantum dot by solving the Schrödingerequation with a confining potential, the mix of light-holes (Jz = ±1/2) in the ground state rapidly increases from 10% to 27% when the height of the dot is only varied modestly from 3.5 nm to 5 nm (the base of the dot stays 25.2 nm). It is worth pointing out that this variation in the dot height only corresponds to a 4% change in the height to base (25.2 nm) ratio and the overall shape of the dot remains alike. As illustrated in

85 86 Single Hole GaAs Lateral Quantum Dots

Figure 5.5, unlike the pure orbital character shown by the electrons, the hole states are strongly mixed and show p-shell (doughnut-like) characters. For the tall dot, the mix of p-shell is significant even for the first hole (41% s-like and 28% p-like) [87]. This predicted strong HH-LH mixing and a p-like ground state in hole quantum dots can possibly add unwanted dephasing mechanisms to the heavy-hole spin, which was originally thought to be long-lived [44]. Therefore, it is important to investigate the strength of the HH-LH mixing in gated hole quantum dots, which is one of the main candidates for spin manipulation. To experimentally demonstrate the HH-LH mixing, the change in the Landég- factor is often measured as the local confinement potential is varied by metallic gates. A strong modification on the out-of-plane g-factor by the HH-LH mixing has been observed recently by N. Ares et al. in SiGe self-assembled hole quantum dots. As shown in Figure 5.6, the out-of-plane g-factor g⊥ is plotted as a function of F , which characterizes the symmetry of the z confinement potential and was experimentally varied by the biases on the top and back gates of the quantum dot. Illustrated by the inset schematics in Figure 5.6, as the symmetry of the z confinement potential changes, the relative positions of the HH and LH wavefunctions vary, giving rise to a change in the mixing between the two hole states and thus a variation in the out-of-plane g-factor g⊥. In GaAs hole systems, similar effects have only been demonstrated using 1D quantum point contacts [54, 55, 77]. In this chapter, by tuning the size of the dot in situ, we demonstrate the effects of the HH-LH mixing on the anisotropy of the hole g-factor in a quantum dot for the first time.

Figure 5.6: g⊥ measured from a SiGe hole quantum dot as a function of F , which is a parameter characterising the symmetry of the quantum well potential. Red solid and black dashed lines are ﬁttings obtained from the oﬀ-diagonal elements of a standard 4×4 Luttinger Hamiltonian [88]. Figure reproduced from Ref [57].

86 87 Single Hole GaAs Lateral Quantum Dots

5.3 Single hole quantum dot - Experimental results

5.3.1 Device Characterization

Figure 5.7: (a) A Scanning Electron Microscope (SEM) image of the device and a diagram of the confinement potential for a single quantum dot. First layer of gates consists of five finger gates (labelled 1-5) with a width of 30 nm and a inter-gate spacing of 50 nm. A top-gate channel (labelled TG) on the top layer has a width of 50 nm and is running along the y-axis of the vector magnet system. (b) A side-view schematic of the device and a rough plot of the dot potential assuming parabolic confining potentials in x and y directions. The undoped

GaAs/AlxGa1−xAs heterostructure used has 10nm GaAs cap and 50 nm AlxGa1−xAs layer.

10 nm of HfOx is used as the insulator between the two layers of Ti/Au gates. (c) Low-bias

(VSD = 50 µeV) charge stability diagram of a single quantum dot: current through the dot measured as a function of the voltages on left/right barriers (gate 2/4) and plunger gate (gate 3). Labelling of the number of holes on the dot will be demonstrated in Section 5.3.2. The double lines in between N = 3 and N = 4 are caused by a trap state near the dot, which will be discussed in Section 5.3.2.

The single hole quantum dot device is fabricated on a shallow undoped GaAs/AlGaAs heterostructure comprising a 10 nm of GaAs cap and a 50 nm of AlxGa1−xAs layer on a GaAs buffer layer as shown in Figure 5.7(b). The wafer used is the same as in Chapter 4. A separate quantum Hall measurement with the same heterostructure has shown a 2D hole mobility of 600, 000 cm2/Vs at p = 2.5 × 1011 cm−2 and T = 250 mK. The device also has the double-level-gate design as presented in Chap- ter 4 but with a different layout and dimension: 5 barrier gates are deposited first directly on top of the wafer with a width of 30 nm and a inter-gate spacing of 50 nm, then 10 nm of HfOx is deposited as the dielectric between two layers of gates, and finally a top-gate channel of 50 nm is deposited as the top layer. Figure 5.7(a) is an SEM image of the device showing the dimension and layout of the double-level gates. Figure 5.7(b) shows a schematic side-view of the device showing the position

87 88 Single Hole GaAs Lateral Quantum Dots of the dot when all the gates are properly biased. We also show in Figure 5.7(b) a rough estimation of the dot potential assuming parabolic confining potentials in x and y directions from the dot confinement, and a triangular confining potential in the z direction from the single heterojunction. The dot was measured in a dilution refrigerator with a vector magnet system at a base hole temperature of T ∼ 100 mK.

To conﬁne a single quantum dot with the 5-gate double-level design, the top-gate (TG) is negatively biased to -1.05 V to induce holes at the heterointerface. Finger gates 2 and 4 are used as the left and right tunnel barriers of the dot while gate 1 and 5 are unbiased to -0.73 V and form parts of the source or drain leads as shown in the potential energy diagram Figure 5.7(a). Gate 3 is used as the plunger gate which ﬁne tunes the dot potential and changes the number of holes in the dot. Figure 5.7(c) shows the low-bias charge stability diagram of the dot: the plunger gate bias V3 is swept from V3 = −0.53 V to V3 = −0.65 V while the two barrier gates 2 and 4 are stepped together from V2/V4 = −0.50 V to V2/V4 = −0.62 V. Straight parallel lines of current are observed indicating the formation of a single quantum dot. As the potential of the two barriers are lowered (more negative biases), a continuous shift of the last current peak to lower plunger gate voltages is observed without causing the dot to break into multiple dots. Note that in between the third and fourth holes, double lines are visible when V2/V4 > −0.56 V and V3 < −0.62 V, which is caused by a trap state near the dot and will be discussed in detail in Section 5.3.2.

Figure 5.8 shows the current through the dot as a function of the voltage on the plunger gate 3 at a low bias while the left and right barriers are set to V2 = −0.5347

V and V4 = −0.5348 V (along the red dashed line in Figure 5.7(c)). The inset of Figure 5.8 shows the zoom-in of the Coulomb blockade peaks in the low current regime when the plunger gate is less than -0.66 V and the barriers are opaque.

No more Coulomb blockade peaks are seen when V3 < −0.60 V, and the labelling indicates the number of holes on the dot. Note that here we ignore the small peak in between N=3 and N=4 which corresponds to the double-line feature in Figure 5.7(c), and is caused by a trap state as we will discuss in Section 5.3.2.

88 89 Single Hole GaAs Lateral Quantum Dots

Figure 5.8: Coulomb blockade peaks of the dot: current at a low bias (VSD ∼ 50 µeV) plotted as a function of the plunger gate voltage at V2 = −0.5347 V and V4 = −0.5348 V. Inset: zoom-in of the low current regime. Labelling of the number of holes on the dot will be demonstrated in Section 5.3.2.

5.3.2 Bias spectroscopy

Coulomb diamonds

To prove the last seen Coulomb blockade peak corresponds to the addition of the first hole on the dot, Figure 5.9 shows the bias spectroscopy of the device in the same configuration: the derivative of the current through the dot is plotted as a function of source-drain bias and plunger gate voltage V3. From Figure 5.9, we can see that the size of the Coulomb diamonds increases as the hole occupation goes down, and the last diamond opens up to 12 mV and does not close again. The fact that the last diamond opens up to more than 3 times larger than the second last diamond and does not close again without losing the current through the dot strongly suggests that no more holes are in the dot. The emptying of the dot is also supported by Figure 5.7(c), which is the charge stability diagram plotting the current through the dot at a low bias while stepping the voltages on gate 2 and 4 together and sweeping the plunger gate bias. In Fig- ure 5.7(c), the parallel lines show the shifting of the Coulomb blockade peaks while the left and right tunnelling barriers are lowered. The fact that we do not observe more transport lines but only a continuous shift of peak positions due to crosstalk between the barriers, indicates that there are no lower energy states available in the quantum dot. This rules out the possibility that the absence of transport is a result of opaque barriers, and verifies the hole occupation of the dot as labelled in the figures. It is worth pointing out that, similar to Figure 5.7(c) and Figure 5.8, in Figure 5.9, the small diamond in between the third and fourth holes is ignored in

89 90 Single Hole GaAs Lateral Quantum Dots the labelling as it is caused by the trap state (Section 5.3.2).

Figure 5.9: Coulomb diamonds of the single dot: derivative of the current is plotted as a function of the source-drain bias and the plunger gate voltage V3. The last diamond opens up to 12 mV and does do not close any more, which strongly suggests that there are no more holes in the dot. The Coulomb blockade diamonds are labelled according to the number of holes on the dot. The small diamond in between N = 3 and N = 4 is not counted as it is caused by the trap state. The yellow box highlights the region used in Figure 5.11. The red line indicates the size of the orbital energy of the one hole state and the green line indicates the corresponding change in the source-drain bias which is used to calculate the bias to energy conversion ratio α1 = Eorb/∆VSD = 1 meV/1.32 mV=0.76 meV/mV. The blue lines indicate the energy to plunger gate bias conversion ratio α2 = Eadd/∆Vg =2.24 meV/0.01 V=22.4 meV/mV which is used to extract the addition energy from the Coulomb blockade peak spacing in Figure 5.8.

Hole shell ﬁlling

As illustrated by the blue lines in Figure 5.9, using the energy to gate voltage conversion ration α2 = Eadd/∆Vg = 2.24 meV/0.01 V=22.4 meV/mV, we extract the addition energy from 1 to 15 holes from the Coulomb blockade peak spacing in Figure 5.8. The extracted addition energy is plotted as a function of the number of holes on the dot as shown in Figure 5.10(a). For electrically defined quantum dots, the dot capacitance and thus the the charging energy can change dramatically for the first few charge carriers on the dot [89]. Therefore to illustrate the shell filling effects, an exponential background (black dashed line) as a rough estimation of the varying charging energy is extracted from the addition energy, and the difference is plotted in Figure 5.10(b). As shown in Figure 5.10(b), we see a clear odd-even fluctuation in the energy between N = 6 and N = 13 indicating the consecutive filling of doubly degenerate orbitals. When N > 13, the addition energy is predominantly the charging energy of the dot and stays a constant. For N < 6, we could not

90 91 Single Hole GaAs Lateral Quantum Dots see any clear evidence of shell ﬁlling, which, in relation to D. Reuter’s results [82] (Figure 5.2), could be an indication that the ﬁrst six holes on the dot are spin- polarised.

Figure 5.10: (a): Addition energy as a function of the number of holes on the dot calculated from the distance between adjacent Coulomb blockade peaks in Figure 5.8. The dashed black line is an exponential fit to the addition energy as a rough estimation of the varying charging energy of the dot. (b) The difference between the addition energy and the exponential background plotted as a function of the number of holes on the dot illustrating the shell filling effects.

Orbital energy of the quantum dot

Figure 5.11: (a) Bias spectroscopy of the one hole states: same as the highlighted region in Figure 5.9 but the DC current rather than the derivative of the current is plotted versus

VSD and V3; (b) DC current along the red line cut in (a) as a function of the source-drain bias.

Single-particle energy levels of the dot can also be extracted from the bias spectroscopy diagram. For a single quantum dot, transport through excited states shows

91 92 Single Hole GaAs Lateral Quantum Dots as conductance resonance lines running parallel to the edge of the Coulomb diamonds. However, due to the low dimensionality of the leads, unwanted resonances from 1D states in the lead will also show as peaks in diﬀerential conductance, which looks the same as those from the orbital states in bias spectroscopy diagrams. There- fore, to distinguish the orbital states of the dot from other complications, we plot in Figure 5.11(a) the DC current through the dot of the highlighted region in Figure 5.9, and in Figure 5.11(b) a line cut of the DC current as a function of the source-drain bias at V3 = −0.605 V. As shown in Figure 5.11, in the region of interest, all the conductance resonance lines correspond to a step in the DC current through the dot.

This agrees with the integral of 0D density of states (g0D(E) ∼ δ(E)) rather than 1D −1/2 density of states (g1D(E) ∼ E ) and thus indicates that those resonances come from transitions via the energy states of the quantum dot rather than the energy states of the 1D lead. We obtain the orbital energy of the dot to be Eorb = 1 meV 2 ∗ as indicated by the red line in Figure 5.9. Using equation Adot = 2π¯h /Eorbmh, the dot size is estimated around 1.2 × 10−15 m−2 using a hole eﬀective mass of

0.4m0 [16]. This calculated dot size agrees with the lithographic dimensions of the dot (30 nm×50 nm) and also suggests that the dot is mainly underneath the plunger gate.

Observation of a trapped state

Finally, we will discuss some anomalous features observed in the measurements, including the double-line feature in Figure 5.7(c), the ignored Coulomb blockade peak in Figure 5.8, the small Coulomb diamond in between N = 3 and N = 4 in Figure 5.9, the zig-zag on the edge of the N = 2 Coulomb diamond in Figure 5.9, as well as the faint shadows above the conductance resonance peaks which will be shown in Figure 5.13. All those abnormal structures are caused by a trap state near the quantum dot [90]. As the trap charges or discharges, the dot potential is slightly varied, which shows as a discontinuity and duplication of features in the transport measurement at a slightly oﬀset gate bias. Since the trap must couple diﬀerently to the barrier gates from the dot itself, to verify our hypothesis, crosstalk maps which show the relative capacitive coupling of two pairs of gates to the dot, and to the trap state, are shown in Figure 5.12. The crosstalk map is done by monitoring the current through the dot while sweeping the voltages on left (gate 2) or right (gate 4) barriers and stepping the bias on plunger gate 3 at the same time. As highlighted by the red circle in Figure 5.12 (a), we see double lines which is a measurement of the average occupation of the trap when it is switching. When the trap state charges and discharges much more rapidly compared to the bandwidth of the DC measurement,

92 93 Single Hole GaAs Lateral Quantum Dots the current through the dot senses the change in the trap state as a time-average occupation of the trap. The relative change in the current strength of the two peaks (one goes up while the other one goes down) indicates the gradual change of the trap state from mostly unoccupied (occupied) to mostly occupied (unoccupied) as the gate biases V4 and V3 are varied. Similarly, the blue circle in the crosstalk map between the left barrier (V2) and the plunger gate (V3) (Figure 5.3.2 (b)) also illustrates the region when the dot senses the average occupation of the switching trap state. However, the change of the occupation of the trap happens in a much smaller gate bias range in Figure 5.12(b) than in Figure 5.12(a). This suggests that there exists a trap state, which is more strongly coupled to the left barrier compared to the right barrier. The rapid charge and discharge of this trap state causes duplication of features such as the double lines in Figure 5.7(c), the double peaks in Figure 5.8 and the extra Coulomb diamond in Figure 5.9. Moreover, as demonstrated in Ref [90], the small offset in the gate bias caused by the trap state will also show as zig-zags on the edge of the Coulomb diamonds, which is a result of the superposition of two sets of slightly offset Coulomb diamonds. Therefore, in transport measurements, we also see features like zig-zags and shallows when bias spectroscopy is measured. However, it should be pointed out that this trap state only causes offsets in the gate bias and duplication of features, but does not change any features or affect our results from the measurement.

Figure 5.12: Current through the dot at a low bias as a function of (a) the voltages on right barrier V4 and plunger gate V3 while bias on gate 2 is ﬁxed at V2 = −0.5347 V and (b) the voltages on left barrier V2 and plunger gate V3 while bias on gate 4 is ﬁxed at V4 = −0.5348 V. Red and blue circles highlight the region when the dot senses the average occupation of the switching trap.

93 94 Single Hole GaAs Lateral Quantum Dots

5.4 Zeeman splitting of the single hole states

After characterizing of the quantum dot’s hole occupation and orbital energy, we move on to exploring the behaviour of the energy states in magnetic ﬁelds. In this section we investigate the response of a single hole spin to a magnetic ﬁeld using a vector magnet system, which is capable of applying a maximum of Bz = 9 T ◦ out-of-plane, By(Bx) = 1 T and Bx(By) = 5 T in-plane, and a full 360 rotation of

Btot = 1 T. The relative orientations of device with respect to the three magnets are indicated by the x (Bx), y (By) and z-axis (Bz) in Figure 5.7(a) and (b) (as well as in Figure 5.13). We focus on the configuration when the dot only has one single hole spin inside. In this configuration, the quantum dot can be considered as an artificial hydrogen atom which contains only one positive charge. The excited states of the dot can be thought of as the higher orbitals of the artificial hydrogen atom. We focus on three main properties of the single hole states observed in the magnetospectroscopy measurement:

• the suppressed g-factor for the ground state

• varying g-factors for diﬀerent orbital states

• a fourfold degenerate excited state

Figure 5.13: Derivative of the current plotted as a function of energy and magnetic ﬁeld

(a) Bx, (b) By and (c) Bz. Dashed lines are guides to the eye showing the splitting of energy states. The very faint shadows above some of the conductance resonance lines are caused by the trap state as discussed in Section 5.3.2.

As shown in Figure 5.13, we plot the derivative of the current through the dot as a function of energy ξ and magnetic ﬁeld (a) Bx, (b) By and (c) Bz. The relative orientation of the dot with respect to the vector magnets is illustrated by

94 95 Single Hole GaAs Lateral Quantum Dots

the schematics in Figure 5.13(a)-(c): Bx is in-plane perpendicular to the transport channel, By is in-plane parallel to the current direction and Bz is pointing out-of- plane, i.e. perpendicular to the heterointerface. The measurements are done by sweeping the source-drain bias at V3 = −0.607 V and stepping the magnetic ﬁeld in three diﬀerent directions while monitoring the current through the dot. The source- drain bias is then converted into energy using the bias to energy conversion ratio

α1 = 0.76 meV/mV. α1 is calculated from the ratio of the orbital energy to the corresponding change in the source-drain bias as illustrated by the red and green lines in Figure 5.9. It is worth pointing out that the lever arm α1 is not found to be B field dependent within the measurement error for 0 < |B| < 4 T. Magnetic fields up to 4 T were applied in x and y directions, and only up to 1.5 T was applied in z direction due to a strong suppression of current caused by the out-of-plane magnetic field. In all three plots Figure 5.13(a), (b) and (c), the bottom trace at the lowest energy corresponds to the one hole ground state, followed by the first and higher excited states as the energy goes up. It is worth pointing out that the faint broad lines above the conductance resonance lines are caused by a trap state near the dot as discussed in Section 5.3.2. Those lines behave exactly like ”shadows” of the resonance peaks: they follow the corresponding peaks but do not interact with any of them. Since those shadows do not cause any change to the main features that we are interested in, we will ignore those faint lines in the following discussions.

5.4.1 Suppression of the ground state g-factor

There are a couple of peculiar features we observed from this measurement. First of all, there is a strong suppression of the g-factor of the one hole ground state, especially in the z direction. As shown in Figure 5.13(c), the ground state barely splits up to 0.5 T and exhibits a very small Zeeman splitting when the field further increases. From the linear Zeeman splitting, we extract the out-of-plane g-factor of ∗ ∗ the ground state gz,ground ' 1.5 assuming g = ∆Ez/µBB. In contrast, the first ∗ excited state splits more than three times faster which gives a g-factor of gz,excited ' 5.5. The same effect is also observed in both in-plane directions. In the y direction, as shown in Figure 5.13(b), the ground state does not show any clear splitting but only a broadening of the resonance peak up to By = 4 T , which gives a upper limit ∗ of the g-factor gy,ground < 0.3. The first excited shows a linear Zeeman splitting ∗ ∗ with gy,excited = 1.2, which is at least four times larger than gy,ground. In the x direction, things are slightly more complicated. As shown in Figure 5.13(a), the

ﬁrst excited state is fourfold degenerate in Bx, which we will discuss in more detail in Section 5.4.3. Here for extracting the g-factor, the splitting highlighted by the

95 96 Single Hole GaAs Lateral Quantum Dots red dashed lines, which shows a linear splitting rate as shown in Figure 5.14, is used.

The g-factors in x direction are extracted to be gx,ground = 0.4 and gx,excited = 0.5.

Figure 5.14: (a) Derivative of the current as a function of energy and magnetic field Bx (Figure 5.13(b)) after aligning the resonance peak of the ground state, which removes the curvature of all the traces to illustrate the linear Zeeman splitting of all the orbital states. Dashed lines are guides to the eye. (b) Extracted Zeeman splitting from the first excited state ∗ in (a) (red dashed lines) as a function of magnetic field Bx. A linear fit using ∆Ez = g µBB ∗ gives a g-factor of gx,excited ∼ 0.5.

It is not clear what is the origin of this strong suppression of the g-factor for the one hole ground state. We propose three possible causes for the vanishing Zeeman splitting, but a dedicated calculation for GaAs hole quantum dots will be required to fully explain the effect. First of all, it is suspected that the correction of the g-factor may come from a mixing of higher orbital states into the conventional s-shell ground state [91]. As shown in Figure 5.5, theoretical calculations have shown that depending on the dot shape, even the ground state of the first hole in a quantum dot can have a considerate contribution from the p-shell orbitals [87]. Moreover, in self-assembled hole quantum dots, the conventional p-like valence band states were found to exhibit properties of d-orbitals, which reduces the symmetry of hole wavefunctions and contributes to the hyperfine interaction [49]. All the evidence suggests that the one- hole ground state can be very different from the pure s-like electron ground state in a lateral quantum dot. This difference in the orbital wavefunctions can lead to a distinct Zeeman splitting for holes, possibly resulting in the suppressed g-factor for the ground state. Even though theoretical calculations of the orbital wavefunctions in self-aseembled hole quantum dots have been carried out [87, 56], to our best knowledge, no dedicated modelling for lateral hole quantum dots has been done up to date. Since wavefunctions are very sensitive to the confinement potential,

96 97 Single Hole GaAs Lateral Quantum Dots the wavefunction in a lateral hole quantum dot like the one we measured can be very different from that in a self-assembled quantum dot. Therefore, theoretical modelling specifically for a GaAs heterostructure based lateral hole quantum dot will be necessary to know the exact form of the orbital wavefunctions and the variation caused in the effective g-factor.

Secondly, due to the large value of rs ∼ 7−10, Coulomb interaction can play a sig- niﬁcant role in hole quantum dots [40]. In similar quantum dot systems with strong interaction, a suppression in the g-factor has been observed previously. Jarillo- Herrero et al. [92] measured the Zeeman splitting of the one-hole orbital states in a carbon nanotube and demonstrated a reduced value of the g-factor g ≈ 1.1 from the known value g = 2 in carbon nanotubes. The reduction was also shown to disappear when the number of holes on the dot increases. Since spin-orbit coupling is weak in carbon, the lower g-factor was attributed to a strong hole-hole interaction in the 1D nanotube. The similarities between Jarillo-Herrero et al.’s results and our observations suggest the possibility that the suppression of the g-factor could be related to the strong Coulomb interaction among holes in our quantum dot. Finally, the suppression of the g-factor has also been demonstrated as a result of HH-LH mixing in hole quantum dots. As reviewed previously (Figure 5.6), Ares et al. showed that HH-LH mixing adds an extra correction to the out-of-plane g-factor g⊥ of SiGe quantum dots [57]. For GaAs/AlGaAs heterostructures, LH-HH mixing is shown to be non-negligible even in 1D systems [55, 54, 77]. Hence, we expect that the HH-LH mixing in our hole quantum dots can be strong enough to modify the g-factor if the same mechanism of g correction holds for GaAs qauntum dots. However, without rigorous theoretical modelling and more experimental proof, we could not rule out any of the above possibility or make any conclusions about the dominant eﬀect that leads to the suppression of the ground state g-factor. Never- theless, all three possible causes of the suppressed g-factor point towards the unique properties of holes and highlight the intriguing behaviour of hole spins.

5.4.2 Orbital dependence of the hole g-factor

∗ Besides the suppressed gground, another strange behaviour of the hole g-factor observed is the dependence of g∗ on the orbital state. Figure 5.15 shows the extracted g-factor from the Zeeman splitting (Figure 5.13) for all three ﬁeld orientations. The g-factor is plotted as a function of the single-particle level, with single-particle index of 0 presenting the ground state, 1 presenting the ﬁrst excited state and so on. As shown in Figure 5.15, the in-plane and out-of-plane g-factors exhibit contrast- ing dependence on the orbital state. In the z direction, if we ignore the suppressed

97 98 Single Hole GaAs Lateral Quantum Dots

∗ Figure 5.15: The extracted g-factor using g = ∆Ez/µBB from Figure 5.13 plotted versus the single-particle level index (an index of 0 represents the ground state, 1 represents the ﬁrst excited state and so on) for x, y and z directions respectively. In the y direction, the Zeeman splitting of some energy states is not resolvable and an upper bound of the g-factor g∗ = 0.3 is plotted instead.

∗ ground state g-factor which is discussed in Section 5.4.1 (single-particle level 0), gz ∗ ∗ decreases for higher orbital states. In contrast, gx and gy stay quite constant for all the orbital states measured. The same effect has also been observed in InAs electron quantum dots by S. Takahashi et al. (Figure 5.3) [86]. Theoretically, the calculation presented by C. E. Pryor and M. E. Flatté[56] qualitatively agrees with our result. As shown in Figure 5.4(b)-(d) [56], the calculated hole g-factor (dashed lines) are plotted as a function of Eg, which is a parameter characterizing the dot size: Eg increases as the dot size decreases. Since the single-particle level spacing of a dot is directly related to the dot size (Eorb ∝ 1/Adot), we can also consider Eg as a parameter directly proportional to the orbital level spacing of the dot. In an electrically defined quantum dot, due to the complexity and non-parabolicity of the real confinement potential, the orbital level spacing is often varying. As shown in Figure 5.13, the spacing between different single-particle levels decreases for higher orbitals. Therefore, in relation to our measurement, here we simply consider Eg as a parameter qualitatively charactering the orbital level spacing with a smaller Eg indicating a higher orbital level. Now if we have a look at Figure 5.4(b) [56], we can conclude that the calculated out-of-plane g-factor decreases for higher orbital states

(i.e. smaller Eg), which agrees with the change in gz we observed in Figure 5.15. Furthermore, as illustrated in Figure 5.4(c) and (d) [56], depending on the real in- plane shape of the dot (crosses: circular, dots: elliptic), gx and gy could have very weak dependence on Eg, particularly in comparison to gz. As many-body eﬀects and HH-LH mixing are not included in Pryor and Flatt´e’s

98 99 Single Hole GaAs Lateral Quantum Dots theory, this qualitative agreement between our experimental results and their theoretical calculations suggests that the observed orbital-dependent g-factor is most likely a result of the strong spin-orbit interaction in hole systems. Spin-orbit interaction prefers antiparallel alignment of the spin to the orbital angular momentum and thus modifies the bare g-factor g = 2 to an effective g-factor, whose value depends on both the strength of spin-orbit coupling and the spacing between discrete energy levels in a quantum dot [56]. Therefore, for higher orbital levels, gz rapidly decreases as the energy levels become closer spaced. Moreover, the difference between gz and gx or gy also illustrates the angular dependence of the spin-orbit interaction, whose effects on the hole g-factor are maximised in the out-of-plane direction.

5.4.3 Observation of a fourfold degenerate orbital state

Finally, we observed a very special fourfold degenerate orbital state, whose degeneracy can only be lifted by the magnetic field in the x direction. As shown in Figure 5.13(a), the first excited state splits into four resonance peaks in an in-plane magnetic field Bx, which is perpendicular to the current direction. However, the same excited state only splits into two peaks in either By or Bz as shown in Fig- ure 5.13(b) and (c). To confirm the observed fourfold degenerate state is real, an out-of-plane magnetic field of Bz = 0.4 T was first applied to Zeeman split the excited state, and then the in-plane field Bx is stepped from 1.5 T to 4 T while sweeping the source-drain bias across the split levels and monitoring the current through the dot. The derivative of the current is then plotted as a function of energy and Bz/Bx as shown in Figure 5.16. It can be clearly seen that both Bz-split levels further split into two more levels when a large enough Bx is applied. This verifies that each of the split levels in Bz (or By) are still twofold degenerate and the first excited state of the hole quantum dot is actually fourfold degenerate. Even though fourfold degeneracy has been observed in quantum dots before [58], it is extremely uncommon in a lithographically non-circular quantum dot based on single heterostructures. Conventional understanding of a fourfold degenerate states suggests a double-degenerate orbital state plus spin states [58], which is only considered possible in spherically symmetric quantum dots. However, considering the fact that the dot is fabricated on a GaAs/AlGaAs heterostructure, as well as the double-level design of the dot (Figure 5.7(a)), the dot is very unlikely to be spherically symmetrical. Moreover, the fact that only Bx can lift the fourfold degeneracy indicates that the quantization axis of the orbitals is pointing along x direction and magnetic fields applied in y or z directions only couple to the spin degree of freedom of the holes. This is intriguing given that the orbital momentum of a 2DHG formed

99 100 Single Hole GaAs Lateral Quantum Dots in a heterostructure is usually along z-axis, i.e. perpendicular to the heterointerface. This suggests that either the dot conﬁnement is so tight that the quantum dot no longer looks like a ”pancake” but more like a ”pyramid”, or the mixing of diﬀerent shells is causing complicated behaviour of the orbital states that are not well understood. In either case, rigorous modelling will be necessary to gain more insights into the problem.

Figure 5.16: Derivative of the current plotted as a function of energy and B ﬁeld.

An out-of-plane magnetic field of Bz = 0.4 T is first applied to split the first orbital state into two levels, and then an in-plane field Bx up to 4 T is further applied.

5.5 Land´eg-factor anisotropy

Another important feature in the magnetospectroscopy measurements in Section 5.4 that we have not discussed yet is the anisotropic splitting along different field directions. In this section, we show measurements using a rotated magnetic field and focus on the anisotropic behaviour of the g-factor. We also show the tuning of the g-factor anisotropy by varying the dot confinement potential in situ, from which we analyse the shape of the quantum dot. For GaAs, 2D heavy-hole systems are highly anisotropic with a large out-of-plane ∗ ∗ g-factor g⊥ = 7.2 and a negligible in-plane g-factor gk ∼ 0 [36]. Similarly for 1D hole ∗ ∗ systems, a large g⊥ and a very small gk < 1 have also been experimentally demon- ∗ strated, and the out-of-plane g-factor g⊥ has been shown to decrease as the subband index goes down [54, 55]. For 0D hole systems, theoretical calculations show that the g-factor is still anisotropic [56], which is also demonstrated in Chapter 4. However, as predicted by theory [56], the anisotropic g-factor is also a function of the dot size, which we could not verify using the quantum dot device presented in Chapter 4 due to a limited control of the confinement potential as well as the large dot size

100 101 Single Hole GaAs Lateral Quantum Dots and unknown number of holes inside. As this prediction points towards electric-field control of g-factors in quantum dots, which can be used to selectively control spin procession and drive spin resonance [56], it is very interesting to investigate the effects of the confinement potential on the g-factor anisotropy using the energy states of the single-hole dot. Two different dot configurations are measured to determine the influence of confinement on the g-factor anisotropy: one is the dot configuration presented in Sec- tion 5.3 with barriers set at V2 = −0.5347 V and V4 = −0.5348 V; the other one is a dot configuration with more transparent barriers V2 = V4 = −0.57 V, resulting in a shallower confinement potential and a larger dot size. Figure 5.17 shows the Coulomb diamonds for the big dot configuration, labelled according to the number of holes inside the dot. Compared to the small dot shown in Figure 5.9, the big dot has smaller charging energy for the same hole occupation and broader conductance resonance lines. The orbital energy has also decreased by 10% to Eorb = 0.9 meV, which confirms a larger dot size of 1.35 × 10−15 cm−2 as expected. Therefore, by varying the voltages on the left and right barriers, we are able to change the dot size in situ. It is worth pointing out that on the negative-bias edge of the N = 0 diamond (Figure 5.17), some double-line features are visible. Those lines are caused by the trap state in the left barrier as discussed in Section 5.3.2. Since the biases on the barriers are varied for different dot configurations, the energy level of the trap state also changes between the two dot configurations. Depending on the energy level of the trap state, distinct changes in the dot current are caused by the trap state as the source-drain bias is swept. Nonetheless, those small features observed in the dot current caused by the trap state should not affect the Zeeman splitting of the energy levels on the dot and thus are ignored in following measurements. For both dots, the energy states with a single hole on the dot are investigated. Figure 5.18 shows the Zeeman splitting of the energy states and the extracted g-factor for both configurations. The experiment is done by sweeping the plunger gate bias V3 for different field orientations at a fixed source-drain bias of VSD = −4 mV for both the small dot (Figure 5.18(a)) and the big dot (Figure 5.18(b)). A magnetic field of 0.9 T was rotated in the yz plane for a full 360◦. The Zeeman splitting of the first excited state, as indicated by the red arrows, is used to extract the g-factor for both dot configurations. It is worth pointing out that there are a couple of reasons why the splitting of the first excited state is used for calibrating the g-factor anisotropy. First of all, the ground state g-factor is strongly suppressed and no clear splitting could be observed at the magnetic field used for the rotation. Secondly, for higher excited states, the energy levels are closer spaced and Zeeman splitting becomes non-linear at a very small magnetic field due to the level repulsion

101 102 Single Hole GaAs Lateral Quantum Dots

Figure 5.17: Bias spectroscopy of the large dot conﬁguration (V2 = V4 = −0.57 V): the colour scale represents the derivative of the current through the dot, plotted as a function of the source-drain bias and the plunger gate voltage V3.

Figure 5.18: Derivative of the current as a function of the plunger gate bias V3 and θ, which denotes the angle between the rotated magnetic field and z-axis, for the small dot (a) and the big dot (b) with one hole on the dot. Red arrows indicate the Zeeman split states used to extract the g-factor anisotropy; (c) the measured g-factor anisotropy for the small dot; (d) the measured g-factor anisotropy for the ∗ large dot. In both cases, g-factor is extracted assuming ∆Ez = g µBB. The line fit ∗ p ∗ 2 ∗ 2 used is gHH = (gmax cos(θ + φ)) + (gmin sin(θ + φ)) . caused by spin-orbit interaction, which makes them unsuitable for extracting g- factors. Last of all, in the yz plane, the splitting of the first excited state is purely Zeeman as demonstrated in Section 5.4.3, in contrast to the possible orbital splitting observed in the x direction. The extracted g-factors are plotted in Figure 5.18(c) and (d) as a function of the angle between the magnetic field and z-axis, with error bars given by the line-

102 103 Single Hole GaAs Lateral Quantum Dots width of the excited state. The extracted g-factors show a strong anisotropy in field orientations with a maximum along z-axis and a minimum along y-axis for both configurations, which agrees with the characteristics of 2D heavy-holes. Fitting with ∗ p ∗ 2 ∗ 2 ∗ equation gHH = (gmax cos(θ + φ)) + (gmin sin(θ + φ)) yields gmax = 5.5 ± 0.1 ∗ ∗ ∗ and gmin = 1.3 ± 0.2 for the small dot and gmax = 5.2 ± 0.1 and gmin = 0.3 ± 0.2 for ∗ ∗ the large dot. Fitting parameters gmax and gmin can be simply understood as the ∗ ∗ conventional out-of-plane g-factor g⊥ and in-plane g-factor gk for the hole system. ∗ Comparing the two dot configurations, we see that they have similar g⊥ whereas the ∗ small dot exhibits a three times larger in-plane g-factor gk than the large dot. Two- dimensional heavy-hole systems in GaAs are known to have a negligible in-plane g-factor [36]. Theory predicts that strong confinement potentials can cause HH-LH mixing, which will then result in a measurable in-plane g-factor depending on the degree of the mixing [77]. The tighter the confinement, the stronger the HH-LH ∗ mixing and the larger the in-plane g-factor gk. Our results qualitatively agree with ∗ this prediction in that the smaller quantum dot has a larger value of gk. However, the ∗ same theory also predicts a decreasing g⊥ as the degree of HH-LH mixing increases, which we did not observe. This suggests that the existing theory based on GaAs hole systems is insufficient to explain the behaviour of strongly localized holes in a quantum dot and more specific modelling of 0D hole systems is necessary. Even though there is no dedicated theoretical calculation for hole dots based on GaAs/AlGaAs heterostructures, our experimental results qualitatively agree with the theoretical prediction for the hole g-factor in InAs/GaAs self-assembled quantum ∗ dots very well. As shown by the dashed lines in Figure 5.4(b)-(d) [56], g⊥ increases as the dot size reduces (parameter Eg is inversely proportional the dot size), and ∗ |gk| also increases as the dot size decreases in both in-plane directions. This is exactly what we observed in our hole dot. The fact that the observed change in the g-factor can qualitatively explained by Ref [56] indicates that the main cause of the confinement-dependent g-factor anisotropy is the spin-orbit interaction rather than HH-LH mixing, similar to the observation presented in Section 5.4.2. A small HH-LH mixing also agrees with the observed strongly anisotropic g-factor, which suggests a predominantly heavy-hole like state. Furthermore, using the theoretical prediction of Ref [56], we can also estimate the shape of the quantum dot. As shown in Figure 5.4 (c), depending on the parameter e = d[110]/d[110]¯ , which is ratio of the dot diameter in the two orthogonal in-plane ¯ ∗ crystalline orientations [110] and [110], gk shows different dependence on Eg. For a ∗ circular dot (e = 1, dots), gk stays roughly constant as Eg changes, whereas for an ∗ elongated (in [110]) egg-shaped dot (e > 1, crosses), |gk| increases as Eg increases. Since an increase in the in-plane g-factor as the dot size reduces is observed in our

103 104 Single Hole GaAs Lateral Quantum Dots hole dot, we argue that the in-plane shape of the measured dot is elliptic rather than circular. To ﬁgure out in which in-plane direction the dot is elongated, a comparison ∗ between Figure 5.4(c) and (d) shows that the calculated |gk| is larger in the elongated direction [110]. Therefore, since for our dot, gy > gx as demonstrated in Section 5.4, we conclude that the dot is elongated in y-direction (i.e. along the current direction), which agrees with the lithographic dimensions of the dot as shown in Figure 5.7(a).

5.6 Conclusions and Future Work

We have studied the spin properties of a gated GaAs hole quantum dot, which allows transport measurements of the single-hole limit. Zeeman splitting of orbital states is measured via magnetospectroscopy measurements. The extracted value of the hole g-factor strongly depends on the orbital state, and is also highly anisotropic with magnetic field orientation. The behaviour of the hole spins can be qualitatively explained by the effects of strong spin-orbit coupling in a 0D confinement potential. By varying the dot size in situ, we also demonstrate the tuning of the g-factor anisotropy, from which we also estimate the shape of the dot. However, some of the peculiar behaviour observed from the single hole, particularly the suppressed Zeeman splitting of the one-hole ground state can not be fully explained by existing theory. Progress in modelling of hole wavefunctions in gated quantum dots on heterostructures will be necessary to understand the observed properties of a single heavy-hole spin in quantum dots. In particular, the mixing of different shells for the single hole orbital states, and the effects of shell mixing on the effective g-factor and the degeneracy of the orbital levels need to be examined. Experimentally, based on the current result, a few measurements will be interesting to carry out in the future. First, the shell filling we observed was inconclusive for the first six holes which could be an indication of spin polarization as suggested in Ref. [82]. However, since the charging energy is also included in the addition energy, an increasing charging energy in the few-hole regime could disguise the change in the orbital energy and make shell filling effects hard to identify. Therefore, a complete measurement of the magnetic field dependence of the addition energy (the Fock-Darwin spectrum) needs to be carried out to verify the shell filling and the spin states of the holes. Due to the suppression of the current caused by the magnetic field, we did not have a large enough field range to conduct those experiments. It will be very interesting to measure the Fock-Darwin spectrum on a similar quantum dot device which allows transport measurements up to a larger magnetic field. Second, as we will show in Appendix B, the measurement on the N = 2 hole states was also inconclusive. We did not observe the conventional singlet and triplet

104 105 Single Hole GaAs Lateral Quantum Dots signatures of the two-hole states, which could possibly be related to the strong Coulomb interaction between the holes. However, due to the limited understanding of 0D GaAs hole systems either theoretically or experimentally, we could not make any solid conclusions solely based on our observations from a single device. There- fore, it will be very beneﬁcial to repeat the magnetospectroscopy measurements on multiple devices to rule out any eﬀects originated from device artefacts. Last, due to lack of a non-invasive charge sensor, we could not measure the spin relaxation time that many people are interested in for heavy-hole systems. There- fore, incorporating a charge sensing circuit should be a priority in future quantum dot designs to carry out electrical measurements of the T1 time using Elzerman’s method [1].

105 Chapter 6

Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots

We present the observation of spin blockade in a GaAs hole double quantum dot and the lifting of spin blockade due to spin orbit interaction. We found that the effect of spin-orbit coupling on spin blockade to be highly anisotropic in different magnetic field orientations, which agrees with previous theoretical predictions. From the anisotropic lifting of spin blockade, we identify the orientation of the effective spin-orbit field, which is very different from previous experimental results on electron quantum dots and highlights the uniqueness of hole systems.

6.1 Introduction

Electrically defined semiconductor quantum dots are attractive systems for spin manipulation and quantum information processing [43]. Recently, all-electrical control of single electron spins has been demonstrated in electron systems with strong spin- orbit coupling using electric dipole spin resonance (EDSR) technique [5, 6]. Utilizing the coupling between spin and orbital states, an oscillating electric field can effectively rotate the electron spin coherently [93]. However, in most electron systems with strong spin-orbit coupling, a large nuclear spin is also expected [94, 95, 96]. Consequently, the hyperfine interaction between electron spins and nuclear spins will lead to unavoidable spin relaxation and phase randomization, which is the dominant limiting factor of the spin lifetimes in those systems [61]. Holes, on the other hand, are predicted to have much weaker hyperfine interaction with nuclear spins due to the p-orbital symmetry of their Bloch wavefunc-

106 107 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots tion [9, 10, 11, 8]. Therefore, as a possible solution to improve the spin lifetimes in quantum dots, holes have drawn significant attention recently. Nonetheless, understanding of the complicated hole quantum dot system is still limited. Theoretically, J. Fisher et al. [10] showed that heavy holes confined in two-dimensional quantum dots have an Ising-type hyperfine interaction, which suggests a ten times smaller hyperfine constant compared to its electron counterpart. Experimentally, various optical measurements based on self-assembled GaAs quantum dots by B. D. Ger- ardot et al. [51], D. Brunner et al. [13], A. Greilich et al. [14] and K. De Greve et al. [15] have also demonstrated the predicted long hole spin coherence times and reduced hyperfine interaction. As a system with both strong spin-orbit interaction and weak hyperfine interaction, heavy-holes in GaAs are proposed to be one of the promising candidates for all-electrical spin manipulation [53, 12]. However, GaAs hole quantum dots are much less studied electrically [17, 18, 19], and the spin mixing mechanisms due to spin-orbit coupling in hole quantum dots still remains unclear [97]. In this chapter, we show our measurements with a GaAs double hole quantum dot where we observe clear spin blockade. By applying a magnetic field in different orientations, we demonstrate the anisotropic behaviour of the spin-orbit induced lifting of spin blockade, and identify the direction of the effective spin-orbit field. We show that the observed effective spin-orbit field orientation is very different from previous experimental results on electron quantum dots, which is possibly due to the complicated spin-orbit coupling mechanisms in hole systems. By aligning the external magnetic field with the effective spin-orbit field, we can extend the field range in which spin blockade is observed. This result provides important implications for experiments such as Rabi oscillations where spin blockade in a finite magnetic field is necessary for the readout of spin states.

6.2 Literature Review

Pauli spin blockade is the one of the most widely used methods to realize spin-to- charge conversion for readout of the spin state in double quantum dots. It was first observed by K. Ono et al. in 2002 as a current rectification effect when the source- drain bias direction on a vertically coupled electron quantum dot was reversed [60]. Since then, Pauli spin blockade has been observed in many electron systems, particularly in lateral double quantum dots and nanowire quantum dots [35, 43, 98, 94, 5]. Positive and negative bias triangles are usually measured to demonstrate the observation of Pauli spin blockade. A detailed discussion of the signatures of spin blockade can be found in Section 2.5.3.

107 108 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots

For hole systems which we are interested in, to the best of our knowledge, spin blockade has only been observed in InSb nanowire quantum dots [97] and carbon nanotubes [99, 100], but not yet been demonstrated in lateral quantum dots based on GaAs/AlGaAs heterostructures. As shown in Figure 6.1(a), the origin of spin blockade for holes are exactly the same as for electrons: if holes in two dots form a spin triplet, the hole in the right dot is not able to tunnel to the left dot because the only energetically allowed state is a spin singlet. Therefore, the current through the double dot will be suppressed until the source-drain bias is large enough to access the higher triplet level in the left dot as illustrated by the bias triangle in Figure 6.1(b). It is worth pointing out that spin blockade could be much more complicated for holes due to the complexity of the valence bandstructure. One may remember from Chapter 5 that a fourfold degenerate state was observed in our single dot, which, if assumed true for a double dot, suggests that the suppression of the dot current could be caused by either spin blockade or orbital blockade as transport is only blocked when holes in two dots have not only the same spin but also the same orbital. However, since the degeneracy of the states highly depends on the shape of the dot, which could be completely different for a double dot, and understanding of the origin of the degeneracy is still missing, no definite conclusion could be made based on the current knowledge of hole systems. Therefore, it will be very interesting to further investigate the behaviour of hole dots in other devices with different dot geometries.

Figure 6.1: (a) Schematic illustrating the origin of spin blockade for a InSb nanowire double hole quantum dot. (b) Charge stability diagram near a spin-blocked transition at B = 0. The suppression of the current through the base part of the bias triangle is a typical signature of spin blockade. Figure reproduced from Ref [97].

As a spin-to-charge conversion process, spin blockade relies on slow spin relaxation within the dot as well as spin conservation during tunnelling events. Any undesirable spin mixing mechanisms may lift the spin blockade and cause an in-

108 109 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots crease in the leakage current, which reduces the accuracy and reliability of spin blockade based spin readout techniques. Therefore, understanding diﬀerent spin- blockade lifting processes in double quantum dots and thus suggesting possible ways to preserve spin blockade are very important for real applications in quantum information processing. As a popular candidate for spintronics, heavy-holes remain relatively unexplored in this ﬁeld, particularly experimentally. Due to similarities in the origin of spin blockade between electrons and holes, the lifting of hole spin blockade is expected to resemble the characteristics of similar processes in electron quantum dots. Here we review a few experimental work on electron systems that are closely related to our experiments.

Figure 6.2: (a) Current ISD as a function of voltages on two barrier gates with VSD = 3 mV near a spin-blocked transition for electrons in an InAs nanowire double quantum dot. Spin blockade suppresses the current in the base region of the triangle until the spin triplet is energetically accessible as indicated by the arrow. White dashed line indicates the detuning axis used in (c)-(d). (b) Same plot for the reverse bias, continuous current is observed for the whole triangle. (c) Current ISD as a function of the magnetic field and the detuning energy for weakly coupled dots showing a zero-field peak. (d) Same plot for strongly coupled dots showing a zero-field dip. Figure reproduced from Ref [94].

A. Pfund et al. [94] investigated the lifting of spin blockade in InAs nanowire double quantum dots. Triplet-singlet relaxation was observed to be mainly caused by two mechanisms: hyperfine interaction with nuclear spins in the weak interdot coupling regime, and spin-orbit interaction in the strong interdot coupling regime. As shown in Figure 6.2(c), for weakly coupled dots, when the current through the system was measured as a function of the magnetic field and the detuning energy, a current peak around B = 0 mT was observed. This indicates a lifting of spin blockade at zero magnetic field which is then suppressed in a finite field. This lifting of spin blockade at zero-field in the weakly coupled regime is found to be caused by the mixing of triplet (1, 1) and singlet (1, 1) states due to uncorrelated local nuclear

109 110 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots

fields experienced by electrons in two dots. The width of the zero-field peak directly ∗ reflects the RMS fluctuations of the intrinsic hyperfine field BN = EN /g µB in the system, which is usually around a few mT for electrons. This mixing between the singlet and triplet (1, 1) states is then suppressed when the external magnetic field exceeds BN as two of the triplets are Zeeman-split away from the singlet in the energy spectrum (|ET±(1,1) − ES(1,1)| > EN ). In contrast, as shown in Figure 6.2(b), a different effect was observed for strongly coupled dots: spin blockade is only lifted in a finite magnetic field resulting in a current dip around B = 0 mT. This zero- field dip in the strong coupling regime is attributed to another spin mixing effects: the spin-orbit interaction. In the presence of strong spin-orbit coupling, which is expected in InAs, there exists no fixed spin quantization axis and thus singlet and triplet states are mixed. This effect can not cause spin relaxation at zero-field due to time reversal symmetry, resulting in a zero-field dip of the spin blockade current. For GaAs hole systems that we are studying, a strong spin-orbit interaction as well as a very weak hyperfine interaction are expected from previous works [36, 8]. Therefore, spin-orbit coupling caused spin relaxation is anticipated to be the main effect causing the lifting of spin blockade. Based on the work by A. Pfund et al., a zero-field dip in the strong-coupling regime will be the main signature to look for in our experiments.

Figure 6.3: Dot current I as a function of the magnetic field at detuning ε = 0 for a strongly coupled InAs nanowire double quantum dot showing the zero-field dip due to spin- orbit interaction. The solid line shows the fitting with theory presented in Ref [63]. Figure reproduced from Ref [62].

After A. Pfund et al., S. Nadj-Perge et al. [62] also observed the same interplay between spin-orbit and hyperﬁne mechanisms in lifting the Pauli spin blockade using electron double quantum dots based on InAs nanowires. One step ahead from A. Pfund et al.’s work, S. Nadj-Perge et al. also showed a complete quantitative analysis

110 111 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots of the leakage current as a function of the magnetic field at different detuning energies. Using the transport model proposed by J. Danon and Yu V. Nazarov [63], S. Nadj-Perge et al. fully reproduced the measurement results over a wide range of interdot coupling. The transport model considered electron transport in the Pauli spin blockade regime in the presence of strong spin-orbit coupling. Spin relaxation processes due to the spin-orbit interaction as well as the hyperfine interaction are both accounted for in the model, but here specifically in relation to our experiments, we only focus on the spin-orbit part of the model. A detailed theoretical analysis of the leakage current caused by hyperfine interaction can be found in Ref [61]. To model spin-orbit coupling induced lifting of spin blockade, J. Danon and Yu V. Nazarov introduced the effect of spin-orbit interaction as a non-spin-conserving interdot tunnel coupling element, which essentially describes the hybridization between (1, 1) triplets and the (0, 2) singlet caused by spin-orbit coupling. By solving the effective evolution equation for the electron density in the presence of non-spin-conserving tunnelling, a analytical expression of the leakage current is obtained. As shown in Figure 6.3, the model numerically reproduces the experimental observed Lorentzian- shaped zero-field dip, which is often used to identify a spin-orbit interaction induced lifting of spin blockade. Besides the excellent agreement with experimental results, a very important prediction J. Danon and Yu V. Nazarov made based on their theory is that if the external magnetic field is applied along the direction of the effective spin-orbit field, the range of magnetic fields in which spin blockade can be observed will largely increase. We will give a detailed explanation of this mechanism later in this chapter, but it is not hard to see the significance of this prediction for experiments such as Rabi oscillations, where spin-blockade based spin readout needs to be performed in a magnetic field. In addition, this prediction also suggests a method to measure the direction of the effective spin-orbit field B~ SO. The prediction by J. Danon and Yu V. Nazarov was also verified experimentally by S. Nadj-Perge et al. [96] using the EDSR spectrum of a InSb nanowire double DD quantum dot. As shown in Figure 6.4, the singlet-triplet repulsion gap ∆SO between the T−(1, 1) state and the S(0, 2) state, which directly reflects the degree of hybridization between (1, 1) triplets and the (0, 2) singlet due to spin-orbit coupling, were measured for different magnetic field orientations. The leakage current through the dot was also monitored simultaneously for comparison. Indicated by DD the arrows in Figure 6.4(b)-(d), ∆SO vanishes when the external magnetic field is applied perpendicular to the nanowire, which indicates that the direction of the effective spin-orbit field of the system is also perpendicular to the nanowire. This result agrees with the expected effective spin-orbit field direction for electrons in

111 112 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots

Figure 6.4: Experimental results by S. Nadj-Perge et al. on an InSb nanowire electron double dot. (a) Schematic illustrating the orientations of the external magnetic field and the effective spin-orbit field with respect to the nanowire. (b)-(d) The avoided crossing in the EDSR spectrum for different magnetic field orientaions as illustrated by the schematic in the top-left corner of each plot. (e) Schematic illustrating the spin-orbit induced coupling between (1, 1) triplets and the (0, 2) singlet. The strength of the coupling scales as |B~ SO ×B~ | for small B~ . (f)-(h) Current I as a function of the magnetic field B and the detuning energy ε for field orientations as in (b)-(d) correspondingly. Figure reproduced from Ref [96]. a structure inversion asymmetry (SIA) dominated [111] InSb nanowire. The same effect was also observed in the leakage current (Figure 6.4(f)-(h)) as predicted by J. Danon and Yu V. Nazarov [63]: when the external magnetic field is applied parallel to the effective spin-orbit field (i.e. perpendicular to the nanowire), spin blockade preserves and leakage current remains low for the entire field range (B = 0−75 mT) (Figure 6.4(h)); in contrast, when the external magnetic field is applied perpendicular to the effective spin-orbit field (i.e. parallel to the nanowire), a shape increase in the leakage current at a small field (B ∼ 20 mT) is observed (Figure 6.4(f)). There- fore, if a freely rotatable magnetic field, such as the vector field magnet system, is available, one would be able to find the effective spin-orbit field orientation by monitoring the leakage current through a double quantum dot in the spin blockade regime. Since the spin-orbit interaction of holes in GaAs heterostructures can be very different from electrons due to the complexity of the valence band structure [36], this experiment is also of interest from the point of view of fundamental physics. For hole systems, similar experiments have only been demonstrated recently by V. S. Pribiag et al. [97] also using InSb nanowire quantum dots. Taking advantage of the small bandgap of InSb, hole transport is achieved by strong band bending induced by local finger gates, which forms an attractive potential for holes between two p-n junctions [97]. This technique allows measurements of hole quantum dots with n-type contacts, but is only possible for narrow bandgap semiconductors.

112 113 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots

Figure 6.5: Experimental results by V. S. Pribiag et al. on an InSb nanowire hole double dot. (a) Bias triangle of the (1, 1) → (0, 2) transition in the strong interdot coupling regime. (b)-(d) Current I versus magnetic field and detuning energy in the strong-coupling regime for three different field orientations. Zero-field dips with similar widths are observed in all three directions. Figure reproduced from Ref [97].

As shown in Figure 6.5, the magnetic field dependence of the leakage current in the spin blockade regime is measured for strongly-coupled dots. Even though zero- field dips indicating a spin-orbit induced lifting of spin blockade were observed, the expected anisotropy of the leakage current as a function of the magnetic field orientation is absent. Since the expected anisotropy of spin blockade is not related to the dot shape or size, but only determined by the fixed direction of the Rashba spin-orbit field, the observed isotropic lifting of spin blockade only for holes is very intriguing, especially given the fact that hole Bloch wavefunctions are already anisotropic [36]. Despite limited understanding of the real mechanism causing the isotropic lifting of spin blockade, this result still suggests that the method proposed by J. Danon and Yu V. Nazarov [63] will not be able to extend the spin blockade region if the isotropy is universal for hole systems. This could potentially undermine the interests in hole systems as most spin manipulation processes based on double quantum dots require spin-to-charge conversion in the presence of a finite magnetic field. There- fore, studying the field-dependence of the lifting of spin blockade in a different and conventional hole system is necessary to gain more understanding of this strange behaviour. In this chapter, we show that gate-tunable double quantum dots can be formed in GaAs hole systems. With electrical measurements of the current through the double dot, we show Pauli spin blockade of holes and an anisotropic lifting of spin blockade due to spin-orbit interaction. From the anisotropy of the spin blockade, we identify the direction of the effective spin-orbit field in GaAs hole systems, which provides

113 114 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots important implications for further experiments on GaAs hole quantum dots.

6.3 Few-hole double quantum dots - experimental results

Figure 6.6: (a) A Scanning Electron Microscope (SEM) image of the device. First layer of gates consists of ﬁve barriers (labelled 1-5) with a width of 30 nm and a inter-gate spacing of 50 nm. A top-gate channel (labelled TG) on the top layer has a width of 50 nm; (b) A side-view schematic of the device. The undoped GaAs/AlxGa1−xAs heterostructure used has 10 nm GaAs cap and 50 nm AlxGa1−xAs layer. 10 nm of HfOx is used as the insulator between the two layers of Ti/Au gates; (c) Charge stability diagram of the double quantum dot: current through the dot measured as a function of the voltages on the left plunger

(gate 3) and the right plunger (gate 4) with VSD = 0.5 mV showing the typical honeycomb pattern. Inset: Charge stability diagram of the last pair of bias triangles with VSD = 2 mV.

The double hole quantum dot device is fabricated on a shallow undoped GaAs/AsGaAs heterostructure comprising a 10 nm GaAs cap and a 50 nm AsGaAs layer on a GaAs buffer layer as shown in Figure 6.6(b). A separate quantum Hall measurement with the same heterostructure has shown a 2D hole mobility of 600, 000 cm2/V s at p = 2.5 × 1011 cm−2 and T = 250 mK. The device is the same one used in Chapter 5 for the single dot measurement, but with a different gate configuration. It has a double-level-gate design: 5 barrier gates are deposited first directly on top of the wafer with a width of 30 nm and a inter-gate spacing of 50 nm, then 10 nm of HfOx is deposited as the dielectric between two layers of gates, and finally a top-gate channel of 50 nm is deposited as the top layer. Figure 6.6(a) is an SEM image of

114 115 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots the device showing the dimension and layout of the two-level gates. The double dot was measured in a dilution refrigerator with a vector magnet system at a base hole temperature of T ∼ 100 mK.

To tune the device into a double quantum dot, the top-gate channel is negatively biased to VTG = −1.05 V to induce holes at the heterointerface. Gates 2 and 5 are used as the left and right barriers of the dot while gate 1 is not used as a barrier and lifted to V1 = −0.73 V as depicted in Figure 6.6(b). This conﬁguration ensures the double dot can be emptied before the barriers are too opaque to allow a measurable current through. Two inner barrier gates 3 and 4 are used as plunger gates for left and right dots respectively, and control the inter-dot coupling at the same time. The operation of the double dot is demonstrated by the charge stability diagram as shown in Figure 6.6(c), which plots the current through the double dot with a source-drain bias of VSD = 0.5 mV versus the voltages on gate 3 and gate 4. Dashed lines are guides to the eye outlining the typical honeycomb pattern for a double quantum dot. The absence of current in the top right corner of the stability diagram suggests that no more holes are left in both dots.

To verify that there really are no holes in the dots in the (0, 0) state, current through the dot is monitored with VSD = 4 mV, which is larger than the largest addition energy of the single hole quantum dot presented in Chapter 5. No lower charge states are observed in either dot for a wide range of plunger gate voltages as shown in Figure 6.7. Figure 6.7(a) shows the high-bias stability diagram while V4 is lowered and V3 is opened up, conﬁrming that no more charge states are available in the right dot. Similarly, Figure 6.7(b) shows that no current is observed when gate

3 is lowered to V3 = −0.55 V, indicating an empty state for the left dot.

Moreover, varying addition energies are observed in both dots as shown in Figure 6.6(c), which also supports the few-hole regime. An addition energy of

Eadd = 3 ∼ 4 meV for the second hole in both dots is also comparable to previous single dot measurements as presented in Chapter 5. It is worth pointing out that the current through the last row of bias triangles in Figure 6.6(c) is minimal due to a suppression of both elastic and inelastic tunnelling events at small source- drain biases, which is caused by imbalanced barriers (Appendix C). Therefore, we plot in the inset of Figure 6.6(c) the last pair of bias triangles (highlighted by the red rectangle) with a larger source-drain bias of VSD = 2 mV. The last pair of bias triangle is clearly visible at this bias but has a current suppressed base, which in turn explains why it is hardly visible in the big stability diagram with a source-drain bias of VSD = 0.5 mV.

115 116 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots

Figure 6.7: Charge stability diagram with a large source-drain bias VSD = 4 mV showing no more transport observable after the last row of triangles in both dots when (a) opening up gate 3 and pinching oﬀ gate 4, and (b) opening up gate 4 and pinching oﬀ gate 3.

6.4 Pauli spin blockade

Even though transitions like (1, 1) to (2, 0) or (1, 3) to (2, 2) are often studied to probe the spin states of the dots, we could not see any clear evidence of spin blockade in transitions with low hole occupations (Appendix D). This might be an indication that the ﬁrst few holes are spin polarized or strongly interacting due to the large value of rs ∼ 7 − 10 of the hole system [40].

Figure 6.8: Stability map focusing on the transition (5, 5) to (4, 6): (a) with VSD = +0.5 mV, and (b) with VSD = −0.5 mV. (c) and (d) are Schematics showing the charge transitions of the double dot with positive and negative biases respectively.

Instead, we focus on the transition (5, 5) to (4, 6) where we observe typical spin blockade. Figure 6.8(a) and (b) show the region of the charge stability diagram of interest with source-drain biases of VSD = +0.5 mV and VSD = −0.5 mV respectively.

116 117 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots both figures are plotted with the same colour scale except an overall change in the sign of the current. Comparing Figure 6.8(a) with (b), it can be seen that the top and bottom pairs of bias triangles in two figures look very similar to each other, whereas the current through the base part of the middle pair of bias triangles (pointed by the red arrow) is strongly suppressed in the positive-bias direction but flows freely in the negative-bias direction. This is a typical signature of Pauli spin blockade as illustrated in the schematics Figure 6.8(c) and (d). When a negative bias is applied, current flows through the device via states (1, 0) → (2, 0) → (1, 1) → (1, 0) (Note that here we only talk about the holes that are relevant to the transport and ignore the ones that are already in the dot but not involved in the transport cycle), during which the right dot has only one unpaired hole. However, when a positive bias is applied, the charge transport sequence reverses into (1, 0) → (1, 1) → (2, 0) → (1, 0). Assuming the existing hole in the left dot is a spin-down, then during the transition (1, 1) → (2, 0), the left dot can only accept a hole with spin-up from the right dot due to Pauli exclusion principle. Hence, if a spin-down is loaded into the right dot from the right lead, transition (1, 1) → (2, 0) is forbidden as the energetically allowed ground state of the (2, 0) configuration (i.e. singlet S(2, 0)) is not spin conserved. This suppression of transport persists till the source-drain bias is large enough to allow transitions into the (2, 0) excited states (T (2, 0)) at higher energies. The bias at which the suppression of current disappears also gives the energy gap of so-called singlet-triplet splitting ∆ST . From Figure 6.8(a), the singlet triplet splitting ∆ST is ∼ 150 µeV for the hole occupation studied. The fact the suppression is only observed for the middle pair of triangles but not the ones above or below in Figure 6.8 also agrees with the mechanism of spin blockade. For doubly degenerate energy levels, spin blockade only happens at transitions with occupations of (odd, odd) → (even, even), which guarantees that the second dot in the transport direction always has an unpaired hole so that only the opposite spin is allowed to go through. This means that spin blockade will not happen in consecutive bias triangles, whose hole number is only different by one.

6.5 Spin-orbit interaction lifted spin blockade

Besides the strong dependence on bias direction, another signature of spin blockade is its response to magnetic ﬁelds. Figure 6.9(a) and (b) shows the same pair of bias triangles at (a) B = 0 T and (b) Bz = 200 mT. As shown in Figure 6.9(b), in a small perpendicular magnetic ﬁeld of Bz = 200 mT, the current through the spin blocked region is almost fully recovered, which indicates a lifting of the forbidden (1, 1) → (2, 0) transition. This lifting of spin blockade is a result of the spin-orbit

117 118 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots interaction mediated transition. In GaAs hole systems, strong spin-orbit coupling leads to the hybridization between the triplet T (1, 1) and the singlet S(2, 0), which allows the transition (1, 1) → (2, 0) and lifts the spin blockade. However, this effect of the spin-orbit interaction is cancelled at zero magnetic field due to time-reversal symmetry. Therefore, spin blockade is robust at B = 0 T but lifted at a finite field when the spin-orbit mechanism is activated.

Figure 6.9: Stability map of the spin blocked pair of bias triangles at (a) B = 0 T, and

(b) B = 200 mT with a source-drain bias of VSD = 0.5 mV. The green arrow indicates the direction of the detuning axis ε. (c) Schematic showing the unblock of the forbidden transition (1, 1) → (2, 0) under the eﬀects of spin-orbit interaction.

This distinct behaviour of the spin-orbit induced lifting of spin blockade is further illustrated in Figure 6.10(a), which plots the current through the dot as a function of the perpendicular magnetic ﬁeld Bz and the detuning energy ε. The colour map is done by sweeping the right plunger gate voltage V4 along a line cut at V3 = −0.5676

V as indicated by the red arrow in Figure 6.8(a) with VSD = 0.5 mV, while stepping the magnetic field Bz. The current through the dot shows a dip around zero field in the spin-blocked region. As a comparison, Figure 6.10(b) plots the same colour map with the opposite source-drain bias VSD = −0.5 mV, where no spin blockade exists. The absolute value of the current is plotted as there is an overall change in the sign of the current when the bias direction is reversed. As expected, no change in current is observed as the magnetic field varies. The current at zero detuning for both bias directions is further plotted as a function of the magnetic field in Figure 6.10(c) for comparison. As shown in Figure 6.10(c), the current in the positive bias direction increases monotonically as the field goes up and saturates around B ∼ 200 mT, whereas the current in the negative bias hardly varies in this range. Therefore, from both bias dependence and magnetic-field dependence of the suppressed current, we demonstrated the first observation of hole spin blockade based on

118 119 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots lateral GaAs quantum dots. Up to now, the magnetic field has only been applied in the z (out-of-plane) direction. From next section on, we will start to vary the magnetic field orientation and study the anisotropic effects of the spin-orbit interaction on the lifting of spin blockade.

Figure 6.10: Current through the dot plotted as a function of the perpendicular magnetic

ﬁeld Bz and the detuning energy ε with a positive bias VSD = +0.5 mV (a) and a negative bias VSD = −0.5 mV (b). Dashed lines indicate the position of the zero detuning. (c)

Current though the dot at zero detuning (ε = 0 meV) as a function of Bz for positive (red) and negative (green) biases respectively.

6.6 Anisotropic lifting of spin blockade

As predicted by theory [63], the hybridization of T (1, 1) and S(2, 0) states also strongly depends on the orientation of the magnetic field. To investigate this anisotropic lifting of spin blockade, magnetic field dependence of the dot current is measured in different magnetic field orientations. As shown in Figure 6.11, the positive-bias current through the dot is monitored as a function of the detuning energy and the magnetic field (a) Bx, (b) By and (c) Bz. In Figure 6.11(d)-(f), the current at zero detuning is plotted as a function of the magnetic field for the three field orientations respectively. The relative orientation of the x, y, z axes with respect to the device is as illustrated in Figure 6.6 (also in Figure 6.11). It can be seen from Figure 6.11 that in all three directions, the leakage current in the blocked region increases monotonically as the field increases until it reaches value of the non- blocked case. Even though a zero-field dip is observed for all three orientations, the widths of the dip are dramatically different. As shown in Figure 6.11(f), a full recov- ery of the spin-blocked current occurs at Bz ∼ 200 mT, whereas in Figure 6.11(d), the current saturates at a much higher field Bx ∼ 800 mT. Along the other in-plane direction y, at the maximum field By = 1 T we could apply, no saturation but only a

119 120 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots linear increase of the leakage current was observed as shown in Figure 6.11(e). Nev- ertheless, this suggests the largest width of the zero-ﬁeld dip amongst three cases, and highlights the anisotropic magnetic ﬁeld dependence of the leakage current.

Figure 6.11: Current through the dot plotted as a function of the detuning energy and the magnetic field (a) Bx, (b) By and (c) Bz. Dashed lines indicate the line cut at zero detuning. (d)-(f) current through the dot at zero detuning as a function of magnetic field for x, y and z directions respectively. The black lines are fittings with equation 2 2 2 I = Imax(1 − 8Bc /(9B + 9Bc )) and the values of the fitting parameter Bc are shown on the plots correspondingly.

To qualitatively characterize this anisotropic lifting of spin blockade, the zero- detuning currents for three cases, i.e. the traces in Figure 6.11(d)-(f), are ﬁtted with a simple theory from Ref [63]. By analytically solving the evolution equation for the charge density matrix in the presence of non-spin-conserving tunnelling events caused by strong spin-orbit coupling, the theory states that for any quantum dot systems with a strong spin-orbit interaction, the leakage current I through the dot at zero detuning ε = 0 is given by

2 8 Bc I = Imax(1 − 2 2 ), (6.1) 9 B + Bc √ 2 2 2 −1/2 p where Bc = 2 2(1 + |~η| )(ηx + ηy) t0 Γrel/Γ and Imax = 4eΓrel. Here ~η is a dimensionless parameter given by ~η = ~tSO/t0, where ~tSO = {tx, ty, tz} characterises the non-spin-conserving due to spin-orbit interaction and t0 characterises the usual spin-conserving tunnelling events. It can be seen from the form of Equation 6.1 that the current exhibits a Lorentzian-shaped dip at B = 0 T. The fits with Equation 6.1 are plotted as the black lines in Figure 6.11(d)-(f) respectively. From the fits, we can extract the fitting parameter Bc, which is called the characteristic width of the zero-field dip [98]. The value of Bc is determined by parameters ~tSO, t0 and

Γrel. Since the interdot coupling t0 and the spin relaxation rate Γrel is not varied

120 121 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots

during the experiment, the value of Bc directly reflects the strength of spin-orbit interaction induced tunnelling in the corresponding direction. Similar to the real width of the zero-field dip, the larger the value of the characteristic width Bc, the weaker spin-orbit interaction induced tunnelling is in this direction. The values of the Bc used in the fits of Figure 6.11(d)-(f) are 0.38 T, 0.86 T and 0.10 T respectively, which again suggests a strong anisotropic response of the spin-orbit induced leakage current to the magnetic field direction. It is worth pointing out that an overall background offset of Ib ∼ 3.4 pA is included in the fitting as the current is not completely suppressed at B = 0. This leakage current in the spin-blocked region could be caused by other spin flip mechanisms such as the interaction between holes on the dot and holes in the lead [101]. From the maximum leakage current Imax, we estimate the singlet-triplet spin relaxation rate Γrel to be around 10 MHz. The fact that a similar maximum current is observed in the x (Figure 6.11(d)) and z

(Figure 6.11(f)) directions conﬁrms the assumption that the relaxation rate Γrel is not B orientation dependent.

Figure 6.12: Current through the dot as a function of θ (the angle between B~ and z-axis) and ϕ (the angle between B~ and x-axis in the xy plane) while rotating a fixed magnetic field (a) B = 0.2 T in xz plane, (b) B = 0.2 T in yz plane and (c) B = 0.5 T in xy plane. The black lines are fitting with equations I = I0|cos(θ − θ0)| + Ib for (a) and (b), p 2 2 2 I = I0 cos (ϕ − ϕ0)cos (θ0) + sin (θ0) for (c).

To systematically demonstrate the anisotropy of the lifting of spin blockade, a fixed magnetic field is rotated in the xz, yz and xy planes respectively, while monitoring the current through the dot. As shown in Figure 6.12(a) and (b), a fixed magnetic field B = 200 mT is rotated in xz and yz planes for a full 360◦ and the current through the dot is plotted as a function of the angle between the magnetic field and positive z-axis. Both plots show a strong anisotropic leakage current with maxima along z-axis and minima in xy plane. The traces in Figure 6.12(a) and (b)

121 122 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots

are fitted with equation I = I0| cos(θ −θ0)|+Ib [96] using similar fitting parameters: ◦ I0 = 4.67 pA, Ib = 4.45 pA and θ0 = 1.27 for Figure 6.12(a) and I0 = 4.57 pA, ◦ ◦ Ib = 4.31 pA and θ0 = 3.23 for Figure 6.12(b). The fact that θ0 is very close to 0 for both cases confirms that the leakage current reaches minima when the magnetic field points in the plane of the heterointerface. To further illustrate the anisotropy of spin blockade in this plane, Figure 6.12(c) shows the leakage current through the double dot while rotating a fixed magnetic field B = 0.5 T in the xy plane. The leakage current exhibits a sinusoidal relationship to the angle ϕ between the magnetic field and the x-axis, and is fitted with equation p 2 2 2 ◦ I = I0 cos (ϕ − ϕ0)cos (θ0) + sin (θ0) [96], where I0 = 7.72 pA, θ0 = 40.5 , ◦ ϕ0 = 14.5 . The value of ϕ0 suggests a minimum of the leakage current when the external magnetic field B is pointing ∼ 14◦ to the y-magnet in-plane. Taking into account the misalignment between the magnets and the sample holder in our fridge, we conclude that the leakage current is minimised when the external field is applied in-plane parallel to the direction of transport. Note that for the simplicity of describing the magnetic field orientation, we ignore this misalignment between the magnet and the sample and simply take the direction of the current as the y-axis in following explanations, which makes no qualitative difference in all the analysis. The anisotropic lifting of spin blockade with respect to the external magnetic field can be understood as follows: as shown in Equation 6.1, the leakage current due to spin-orbit interaction is characterised by the parameter ~tSO = {tx, ty, tz}, where the amplitudes of t , t and t reflects the coupling of the singlet S(2, 0) to x y z √ √ the three orthonormal (1, 1) triplets Tx = (T− − T+)/ 2, Ty = i(T− + T+)/ 2 and

Tz = T0 [63]. In the new real-space basis T~ = {Tx,Ty,Tz} with unpolarized triplet states, ~tSO can be considered as a real vector in space and tx, ty and tz are real numbers due to time-reversal symmetry [63]. At a finite magnetic field, Tx and Ty q q 2 2 2 2 decay to S(2, 0) with an amplitude ∼ B tx + ty/ t0 + tSO. If decay due to spin- orbit interaction is the limiting mechanism of the transport, the leakage current is 2 2 2 2 2 2 2 ~ 2 2 2 proportional to B (tx + ty)/(t0 + tSO) = B sin α|tSO| /(t0 + tSO), where α is the angle between the external field B~ and ~tSO. If we consider the extreme case when

B~ and ~tSO both point along the z direction, tx and ty will be zero. This means that triplets Tx, Ty and thus T± are no longer coupled to singlet S(2, 0), and the spin blockade can not be lifted by those states. Therefore, we argue that the direction of

B~ when the leakage current reaches a minimum is the direction of ~tSO and conclude that the effective spin-orbit field B~ SO of the hole quantum dot system is pointing along the direction of transport (y-axis). From the same theory [63], the width of zero-field dip also scales as p 2 ~ ~ t0 + tSO2 /(|tSO|| sin α|), which means that if the magnetic field B is swept along

122 123 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots

different directions with respect to ~tSO, a continuous change of the width of the zero- field dip should also be observed. In addition, when the magnetic field is aligned with ~tSO, an infinite width, i.e. no change in the leakage current is expected. To confirm the prediction of the width of the zero-field dip, magnetic field sweeps along different directions are performed while monitoring the current through the double dot as shown in Figure 6.13(a). A smooth change in the width of blocked region is observed as the magnetic field is gradually changed from being along z-axis to in- plane along x-axis, and then further rotated in-plane from perpendicular to parallel to the transport channel. A great increase of the zero-field dip is observed as the angle between B~ and ~tSO is reduced, and finally when B~ and ~tSO are aligned, i.e. when B~ is pointing along the current direction, spin blockade persists up to fields larger than B = 200 mT. The evolution of the zero-field dip can also be reproduced using Equation 6.1, which is shown as the black lines in Figure 6.13(a). The extracted values of the characteristic field Bc are plotted in Figure 6.13(b) and (c) as functions of θ and ϕ respectively, which are angles as illustrated by the corresponding inset schematics.

For both in-plane and out-of-plane magnetic fields, strong dependence of Bc on the field orientation is observed. A stronger angular dependence when the magnetic field is rotated out-of-plane is also noticeable from the steeper slope of Bc in Figure 6.13(b) than in Figure 6.13(c). This is possibly an effect caused by the g-factor anisotropy of the 0D hole states. Since the Zeeman splitting between the triplet states T± for a certain magnetic field scales with the g-factor in the same orientation, the decay from these states and thus the value of Bc or the width of the zero-field dip will be modified by the g-factor anisotropy correspondingly. As presented in Chapter 5, a highly anisotropic g-factor was observed in a single hole quantum dot confined within the same device, with a maximum g∗ ∼ 5 along z-axis, which explains the more rapid change of Bc when the magnetic field is orientated out-of-plane. However, we also argue that the observed change in the dip width, especially when the field orientation is varied in-plane, can not be purely caused by the g-factor anisotropy. This is because the observed in-plane hole g-factor exhibits a maximum along the y-axis, which suggests that if the g-factor anisotropy is dominating the value of

Bc, Bc should reach a in-plane minimum along the transport direction. This is exactly the opposite to our observation as illustrated by Figure 6.13(c). Therefore, we rule out the possibility that the observed anisotropic lifting of spin blockade is an effect caused by the anisotropic hole g-factor. This further consolidates our previous argument that the extension of spin blockade region is a measurement of the intrinsic effective spin-orbit field direction. Similar measurements on electron quantum dots by S. Nadj-Perge [96] have

123 124 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots demonstrated an effective spin-orbit field perpendicular to the current direction (Fig- ure 6.4), which is very different from our results on hole quantum dots. For structure inversion asymmetry (SIA) dominated electron systems, the effective Rashba spin- orbit field simply points perpendicular to the momentum ~k, which agrees very well with S. Nadj-Perge’s results. However, due to the complexity of the valence band structure, the spin-orbit interaction for holes is much more complicated than for electrons and is not yet well understood, particularly for quantum dots. For a 2D hole system at a single heterojunction, preliminary calculations1 suggest that the Rashba term is much larger than the Desselhaus term and dominates the spin-orbit interaction. Since the Rashba term for holes has a |~k|2~k dependence, a simple theory thus predicts the effective spin-orbit field of 2D hole systems to be perpendicular to the transport direction, in contrast to our observation of a spin-orbit field parallel to the transport direction. This intriguing result suggests that either the dominant spin-orbit interaction for hole quantum dots is different from that for 2D hole systems, or spin-orbit coupling induced spin relaxation mechanisms for holes are not the same as for electrons. A vigorous theoretical calculation on hole quantum dot systems is necessary to fully explain the origin of the observed spin-orbit field direction.

Figure 6.13: (a) Current through the dot plotted as a function of the external magnetic field. All traces are offset for clarity. From top to bottom: magnetic field is oriented in zx plane from z-axis to x-axis in steps of 15◦and then in xy plane from x-axis to y-axis. (b) and (c): extracted fitting parameter Bc plotted as a function of the angle between the magnetic field and x/y-axis respectively.

It is also worth pointing out that even though a strong anisotropic ﬁeld-dependence of the width of the zero-ﬁeld dip is demonstrated, we did not observe the predicted

1Done by E. Marcellina and D. Culcer

124 125 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots persistence of spin blockade up arbitrarily high field [63]. Here we propose two possible reasons for this. First, in our vector-field magnet system, the vector magnetic field is realized by three superconducting coils with different diameters. Sweeping one magnet may generate a magnetization in another coil, resulting in a small magnetic field in the perpendicular direction. Therefore, even though the magnetic field is set to sweep along the y-direction, the device may still experience a non-negligible magnetic field in the x-direction, which causes an increase in the leakage current. Second, there may exist other spin relaxation mechanisms, such as the Coulomb interaction between holes on the dot, that could also lift the spin blockade [102]. However, it is unclear if such processes will also exhibit field dependence in a manner similar to spin-orbit coupling and thus more theoretical modelling will be needed to fully explain our experimental observations.

6.7 Conclusion and Future Work

In conclusion, we present the measurement of a double hole quantum dot where we observe spin blockade. We obtain a zero-field dip in the leakage when an external magnetic field is applied, which suggests the lifting of spin blockade due to the spin-orbit interaction. By varying the magnetic field orientation, we demonstrate the anisotropic behaviour of the lifting of spin blockade as predicted by theory [63]. From the anisotropy of spin blockade, we identify the direction of the effective spin- orbit field in our system to be along the transport direction. By aligning the external magnetic field with the effective spin-orbit field, we also show the extension of the magnetic field range in which spin blockade is preserved. This experiment not only enables the measurement of the effective spin-orbit field direction, but also provides important implication for future experiments, such as EDSR with a hole double dot system. In an EDSR experiment, an external magnetic field is required to split the different spin states, between which spin resonance is driven. For the GaAs/AlGaAs heterostructure we are studying, the conventional idea is to apply the external magnetic field along the z-axis due to the large out- of-plane hole g-factor. However, based on the results of this chapter, the external magnetic field actually needs to be applied in-plane along the transport direction. By applying the external magnetic field along the effective spin-orbit field direction, the unwanted spin mixing processes caused by spin-orbit coupling can be greatly reduced, which directly influences the accuracy of readout of the spin state and thus the outcome of the EDSR experiment. On the other hand, despite the successful demonstration of anisotropic lifting of spin blockade, many experimental results we obtained with the double dot system

125 126 Anisotropic Pauli Spin Blockade in GaAs Double Hole Quantum Dots still remain mysterious. For example, we could not observe clear evidence of spin blockade in transitions like (1, 1) → (2, 0), where spin blockade is often studied in double quantum dots. The exact reason for this problem is unknown but we suspect the strong spin-orbit interaction and strong hole-hole interaction in the system (rs ∼ 7−10) may play an important role. Therefore, it will be very interesting to conduct more experiments on these transitions, especially during different cool downs and using different devices to confirm the reproducibility of odd behaviours that are unique to hole systems. Due to the limitation of the current measurement setup, we were not able to do any fast gate measurements to investigate the hole spin lifetimes in comparison to optical measurements. As we are currently building a new probe with fast lines, measuring the hole spin lifetimes and determining the main spin relaxation or phase randomization mechanisms will also be an interesting experiment to conduct in the near future. Moreover, unlike the conventional lateral quantum dots, we do not have a working charge sensor in the current version of the double-level design. Since a charge sensor is not only useful in ambiguously determining the dot occupation, but also favourable in spin lifetime measurements, incorporating a charge detecting circuit to the current dot design should also be a high priority in future works.

126 Chapter 7

Conclusions and Future Work

Electrically isolating a single hole spin using lateral quantum dots on AlGaAs/GaAs heterostructures has drawn considerable attention recently due to predicted slow spin decoherence and strong spin-orbit coupling of heavy-hole spins. Main experimental diﬃculties of achieving a few-hole lateral quantum dot are the instability of p-type devices and the small single-particle level spacing of existing hole dots due to the large hole eﬀective mass. In this thesis, we demonstrate both single and double hole quantum dots on an accumulation-mode AlGaAs/GaAs heterostructure with electrical transport measurements down to the single-hole limit. By exploring the spin properties of the heavy-hole states, we obtain interesting results that are unique to hole systems.

7.1 Summary of Results

In this section, we brieﬂy summarise the main experimental work presented in this thesis:

• We fabricated both single and double few-hole quantum dots on an undoped accumulation-mode AlGaAs/GaAs heterostructure using a new double-level- gate architecture. Electrical transport measurements down to the single-hole limit are demonstrated.

• Both the single and double quantum dots are fully gate-tunable, including the dot potential, dot size for a single dot and the interdot tunnel coupling for a double dot.

• With the few-hole single quantum dot, an odd-even shell ﬁlling for hole occupations N > 6 is demonstrated.

127 128 Conclusions and Future Work

• Zeeman splitting of the one-hole orbital states in a single dot is measured via magnetospectroscopy measurements. The extracted value of the hole g-factor shows strongly dependence on the orbital state, and is also highly anisotropic with magnetic ﬁeld orientation.

• By varying the dot size in situ, we demonstrate the tuning of the g-factor anisotropy, and estimate the shape of the dot.

• We demonstrate the ﬁrst observed Pauli spin blockade in a double hole quantum dot in GaAs.

• We demonstrate the anisotropic behaviour of the lifting of spin blockade due to spin-orbit coupling, from which we identify the direction of the effective spin-orbit field. By aligning the external magnetic field with the effective spin- orbit field, we also show the extension of the magnetic field range in which spin blockade is preserved.

7.2 Future work

In this section, we talk about possible future work based on the current experimental progress presented in this thesis, including necessary theoretical calculations on hole quantum dots to explain some of our experimental ﬁndings, and interesting following measurements towards the goal of all-electric control of heavy-hole spins.

7.2.1 Theoretical modelling

Some peculiar spin properties of the 0D hole states, especially the suppressed Zeeman splitting for the one-hole ground state (Chapter 5) can not be explained by existing theory. Progress in modelling of hole wavefunctions in gated quantum dots on heterostructures will be necessary to fully explain the observed properties of a single heavy-hole spin in quantum dots. In particular, the mixing of different shells for the single hole orbital states, and the effects of shell mixing on the effective g-factor and the degeneracy of the orbital levels need to be examined. The observed effective spin-orbit field direction (Chapter 6) is very different from electron systems and does not agree with a simple prediction of the Rashba spin- orbit field direction in a 2D hole system. This intriguing result suggests that either the dominant spin-orbit interaction for hole quantum dots is not Rashba-type, or spin-orbit coupling induced spin relaxation mechanisms for holes are not the same as for electrons. A vigorous theoretical calculation on hole quantum dot systems will be very beneficial to fully explain the origin of the observed spin-orbit field direction.

128 129 Conclusions and Future Work

7.2.2 Future experiments

Experimentally, based on the current result, a few measurements will be interesting to carry out in the future.

Hole shell ﬁlling

In the single dot measurement, we did not see a larger addition energy at magic numbers N = 2, 6, 12 as observed previously in electron quantum dots. Instead, we observe an odd-even shell filling only from the seventh hole on the dot (Chapter 4). Due to the suppression of the current caused by the magnetic field, we did not have a large enough field range to see a complete Fock-Darwin spectrum, which is necessary to confirm the shell filling effects as well as verify the spin states of the holes on the dot. Therefore, it will be very interesting to measure the Fock-Darwin spectrum on a similar quantum dot device which allows transport measurements in larger magnetic fields.

Two-hole spin states and hole-hole interaction

In the measurement of the single dot, we did not observe the conventional singlet and triplet behaviours of the two-hole spin states, which could possibly be related to the strong Coulomb interaction between holes in the system. However, due to limited understanding of 0D GaAs hole systems either theoretically or experimentally, we could not make any solid conclusions solely based on our observations from a single device. Similarly, many experimental results we obtained in the double dot measurement also remain mysterious. For example, we could not observe clear evidence of spin blockade in transitions like (1, 1) → (2, 0), where spin blockade is often studied in double quantum dots. The exact reason for this problem is unknown but we also suspect the strong spin-orbit interaction and strong hole-hole interaction in the system (rs ∼ 7 − 10) may play an important role. Therefore, it will be very interesting to conduct more experiments to explore the odd behaviour of the first two holes in both single and double dot measurements, especially during different cool downs and using different devices to confirm the reproducibility of special properties that are unique to hole systems.

Incorporation of charge sensor and fast gate measurements

Unlike conventional lateral quantum dots, we do not have a working charge sensor in the current version of the double-level design. Since a charge sensor is not only

129 130 Conclusions and Future Work useful in ambiguously determining the dot occupation, but also favourable in spin lifetime measurements, incorporating a charge detecting circuit to the current dot design should also be a high priority in future works. In addition, due to the limitation of the current measurement setup, we were not able to do any fast gate measurements to investigate the hole spin lifetimes in comparison to optical measurements. As we are currently building a new probe with ∗ fast lines, measuring the hole spin lifetimes T1 and T2 , demonstrating Rabi oscillations of holes using electric dipole spin resonance (EDSR), and determining the main spin relaxation or phase randomization mechanisms will be exciting experiments to conduct in the near future.

130 Appendix A

Inﬂuence of surface states on quantum and transport lifetimes in high-quality undoped heterostructures

In this appendix, we present some work on the 2D scattering theory that has been conducted during the Ph.D. study but is not particularly related to the main interest of this thesis. This work has been publish in Phys. Rev. B 87, 195313 (2013).

A.1 Introduction

For many years, modulation-doped AlGaAs/GaAs heterostructures have been the centre of research on low-dimensional systems. While extremely high mobility 2D systems [20, 21, 22] have been realized in modulation-doped heterostructures [23, 24], remote ionized impurities, which can act as an additional source of disorder causing both Coulomb scattering and long-range fluctuation of the potential landscape, are introduced to the system ineluctably [25, 26, 27]. Due to these ionized impurities, the closer 2D systems are brought to the surface, the more affected electron transport is [28]. Therefore, although it is possible to attain high electron mobility in deep 2D electron systems, achieving similar mobility in shallower 2D systems which are desirable for nanostructures with fine lithographic configurations remains problematic. Recently undoped heterostructures have drawn attention due to some obvious advantages over modulation-doped heterostructures [25, 26, 27, 29, 30, 31]. Owing

131 Influence of surface states on quantum and transport lifetimes in high-quality 132 undoped heterostructures to the removal of intentional doping, undoped heterostructures are expected to have fewer ionized impurities, the presence of which is the predominant factor limiting the 2D transport performance of shallow modulation-doped heterostructures [28]. Nanostructures fabricated using shallow undoped AlGaAs/GaAs heterostructures have been shown to have much improved electron mobility [33] compared to similarly shallow modulation-doped ones. However, there still exists one possible drawback for metal-gated undoped heterostructures: unavoidable surface charge may affect the carriers in a manner similarly to remote ionized impurity scattering, adding unwanted disorder to undoped devices. Although it has been shown that surface charge reduces the 2DEG mobility of very shallow systems [103], there is little understanding of its effects on 2DEGs of deep undoped heterostructures, or on the stability of devices based on shallow 2D systems. In this chapter, we present both experimental and theoretical analysis of undoped heterostructures showing that even though the carrier mobility is not affected by surface states when the 2DEG is deep, the electron quantum lifetime τq is significantly reduced by surface charge in high-quality undoped heterostructures.

A.2 Literature Review

The scattering mechanisms in modulation-doped AlGaAs/GaAs heterostructures have been intensively studied over the past decades. Shallow modulation-doped heterostructures, which are more preferable to deep ones for pattering nanostructures, have also been investigated. D. Laroche et al. [28] showed that scattering by remote dopants, rather than background charged impurities, is the limiting factor of the mobility in shallow modulation-doped two-dimensional electron gases (2DEGs). Gated heterostructures, in which intentional doping is removed, have drawn considerable attention recently as one of the possible means to improve the mobility of shallow 2DEGs. W. Y. Mak et al. [103] investigated the scattering mechanisms of shallow undoped GaAs/AlGaAs heterostructures of various 2DEG depths. Using conventional scattering theory, eﬀects of background impurities, interface roughness and surface states were demonstrated. Besides the unavoidable background impurities and interface roughness, scattering due to the charge in surface states were shown to aﬀect the mobility of shallow 2DEGs similarly to remote ionized dopants in modulation-doped devices. Therefore, to improve the mobility of shallow 2DEGs for nanostructure fabrication, it is very important to understand the role of surface charge in undoped heterostructures. In this chapter, we present both experimental work and theoretical calculations of transport and quantum lifetimes in undoped heterostructures. We demonstrate

132 Influence of surface states on quantum and transport lifetimes in high-quality 133 undoped heterostructures that depending on the depth of the 2DEG, surface states can have different effects on the transport and quantum lifetimes of the system. This result provides indication in choosing heterostructures with the right depth for future experiments which require high-mobility systems.

A.3 Experimental results

A.3.1 Sample and experimental setup

The undoped device used in the experiment is a single heterojunction Hall bar fabricated on wafer B13520, which has a 17nm GaAs cap followed by 300nm undoped

Al0.34Ga0.66As and then a 1µm GaAs buﬀer. Polyimide is used as the insulator between the ohmics and the top-gate metal. Detailed fabrication methods and schematics of the device can be found in Ref [104]. Magnetotransport measurements were performed in a dilution fridge with a constant excitation voltage of 100µV using standard lock-in techniques. The base temperature of the dilution fridge was 25mK and the base electron temperature was 80mK determined from measurements of variable range hopping [105] in a similar sample with the same measurement setup. By changing the voltage applied to the top-gate, we varied the 2DEG density over an order of magnitude from 1.3 × 1010cm−2 to 1.5 × 1011cm−2 as shown in Figure A.1(a). The 2DEG density is linear with the top-gate bias and its gradient is determined by the separation between the top-gate and the 2DEG. The 2DEG mobility is plotted as a function of density in Figure A.1(b). The 2DEG mobility increases monotonically with the carrier density and reaches 5.6 × 106cm2/V s 11 −2 at ns = 1.5×10 cm , the highest density measured at VTG = 7V . Top-gate biases larger than 7V were not applied to avoid breakdown of the polyimide gate dielectric.

For VTG ≤ 7V , the gate leakage was below the measurement resolution of 10pA.

A.3.2 Measurements of lifetimes from Shubnikov de Haas oscillations

Figure A.2 shows the low-ﬁeld Shubnikov de Haas (SdH) oscillations and Hall resistance for three diﬀerent densities. The SdH oscillations show clear zeros and no parallel conduction. The amplitude of the SdH oscillations can be written as [106]

∆Rxx = 4R0X(T ) exp(−π/ωcτq) (A.1) where R0 is the zero-ﬁeld resistance, ωc is the cyclotron frequency, and X(T ) is a thermal damping factor given by

2 2 X(T ) = (2π kT/¯hωc)/ sinh(2π kT/¯hωc). (A.2)

133 Inﬂuence of surface states on quantum and transport lifetimes in high-quality 134 undoped heterostructures

Figure A.1: Device characterization: (a) 2DEG density as a function of applied top gate bias obtained from the low-ﬁeld Hall resistance. (b) 2DEG mobility as a function of density.

The quantum lifetime τq can be obtained from a Dingle plot of the logarithm of

Figure A.2: Shubnikov de Haas oscillations and Hall resistance for three diﬀerent 2DEG densities.

134 Inﬂuence of surface states on quantum and transport lifetimes in high-quality 135 undoped heterostructures

∆Rxx/R0X(T ) versus 1/B. In a “good” Dingle plot, the intercept at 1/B = 0 is

4 and the slope of the resultant straight line gives τq [106]. Figure A.3 shows the

Dingle plots used to extract τq. The data are well described by straight lines with a fixed intercept of 4, leaving the slope as the only fitting parameter. All Dingle plots show excellent agreement between the experimental data and fitting lines, which suggests that reliable values of τq have been obtained.

Figure A.3: Dingle plots for three different densities corresponding to the ρxx data shown in Figure A.2, using an electron temperature of 80mK from variable range hopping measurements [105] on a similar sample with the same experimental setup. It is worth noting that Dingle plots are insensitive to the carrier temperature used - a 10mK difference in the temperature only results in a 1% change in τq with our fitting method where the 1/B = 0 intercept is fixed 4. Moreover, if the intercept is left as a fitting parameter, the thermal correction term X(T ) which contains the temperature only affects the intercept of Dingle plots, but does nothing to the slope or the value of τq.

The extracted τq is then plotted as a function of density, together with the transport scattering lifetime τt obtained from the 2DEG mobility, as shown by the triangles in Figure A.4. Solid down-pointing triangles show the measured τq with a ﬁxed 1/B = 0 intercept of 4 for Dingle plots while the open triangles show the

τq obtained with the intercept used as a second fitting parameter. The two fitting methods give only a 15% difference in τq on average. Both τq and τt increase as the density increases while τt increases faster with ns than τq. To understand the role of different scattering mechanisms we compared our experimental results with

135 Inﬂuence of surface states on quantum and transport lifetimes in high-quality 136 undoped heterostructures numerical calculations.

Figure A.4: Experimental and theoretical scattering lifetimes plotted as a function of density with the right axis showing corresponding mobility calculated using µq,t = eτq,t/m.

The symbols show the measured τt and τq. Dashed lines show τt and τq calculated with background impurity (BI) scattering only. Solid lines show τt and τq calculated with both

BI and interface roughness (IR) scattering. Dotted lines show τt and τq calculated with BI,

IR and surface charge (SC) scattering. The calculated τt including SC scattering (dotted line) is very close to the values of τt calculated with BI and IR scattering only (solid line) and can hardly be distinguished. The calculated τq with (solid line) or without (dashed line)

IR scattering lie on top of each other as the addition of IR scattering does not alter τq.

A.4 Comparison with theory and discussions

At T = 0, the transport scattering lifetime τt can be calculated as [107]

1 m∗ Z 2kF | U(q) |2 q2 = dq (A.3) 3 2 2 q τt π¯h k 0 (q) 2 2 F 4kF − q where |U(q)|2 is the scattering potential and (q) is the screening function. The transport lifetime τt shows a diﬀerent density dependence for diﬀerent scattering mechanisms, which can be used to identify the various scattering process in the system. The dominant scattering mechanism in deep undoped heterostructures is

136 Inﬂuence of surface states on quantum and transport lifetimes in high-quality 137 undoped heterostructures

Coulomb scattering from background impurities [107, 103], for which |U(q)|2 is given by [107]

2 2 1 e 2 |U(q)|BG = ( ) [NbAl Ga As FAl0.34Ga0.66As(q) + NbGaAs FGaAs(q)]. (A.4) 2q 20q 0.34 0.66 where F (q) is the form factor. The experimental τt is well described with background impurities as the sole scattering mechanism, as shown by the dashed line in 14 −3 13 −3 Figure A.4 with NbAlGaAs = 1.65 × 10 cm and NbGaAs = 5.5 × 10 cm . The higher impurity density in AlGaAs is consistent with previous studies [104, 107, 103]. The only discrepancy between the calculation and measured mobility occurs at densities above 8 × 1010cm−2, when interface roughness scattering starts to take over and limit the mobility. Interface roughness scattering can be modeled using the following scattering potential:

2 2 2 2 2 n2De − 1 Λ2q2 |U(q)|IR = πΛ ∆ exp 4 (A.5) 20 where we use ∆ = 2A˚ as the amplitude and Λ = 4nm as the correlation length of the roughness. Including interface roughness scattering gives excellent agreement between experimental τt and the calculations over the whole density range as shown by the solid line in Figure A.4.

The single-particle lifetime, τq, which counts both small-angle and large-angle scattering processes and diﬀers from τt by the weighting factor 1 − cos(θ), is more complex than calculating τt since the ﬁrst order approximation

1 2m∗ Z 2kF |U(q)|2 1 = 3 2 q dq (A.6) τq π¯h 0 (q) 2 2 4kF − q has a logarithmic divergence [108]. To solve this problem we used multiple-scattering theory [108] to calculate the renormalized τq following the method in Ref [107]. The calculated τq is plotted as the solid line in Figure A.4. Surprisingly, the values of

τq calculated with the same background impurity density and interface roughness used to ﬁt the τt data are more than double the values of the measured τq. This discrepancy can be explained by introducing another scattering process - scattering from remote surface charges. Since polyimide is used as the insulator between the top-gate and the ohmic contacts [104], charge can be trapped at the interface between polyimide and the GaAs cap layer [109]. This surface charge acts like remote ionized impurities, scattering electrons in the 2DEG. [103] To model the scattering caused by this extra disorder, we considered a sheet of charged impurities with an areal density of Nsc at a distance d from the 2DEG, for which the scattering

137 Inﬂuence of surface states on quantum and transport lifetimes in high-quality 138 undoped heterostructures potential is given by

2 2 2 e |U(q)|SURF = Nsc exp(−2qd)F (q) (A.7) 20

11 −2 With Nsc = 0.9×10 cm and d = 300nm, we greatly improved the fitting between theoretical and experimental values of τq as shown by the dotted line in Figure A.4. It is important to note that adding this extra disorder makes little difference to the calculated values of τt. This is expected as the surface charge is far from the 2DEG, and only causes small-angle scattering events which hardly affect τt. However, τq is sensitive to all scattering events, so is strongly affected by the surface charge.

We note that although the fitting of τq is significantly improved by including surface charge scattering, there still exists a small discrepancy between the theoret- 10 −2 ical and experimental τq at densities above 7 × 10 cm which cannot be explained using our model. It is tempting to ascrive this discrepancy to interface roughness scattering, but unlike τt including interface roughness scattering does not improve the agreement between experiment and modelling in the high-density regime. This is because the quantum scattering rate from interface roughness is negligible compared to that from charged impurities (interface roughness scattering only has a very small effect on τt, yet since it is large-angle it will affect both τt and τq equally).

A.5 Predictions from theory

We now investigate how this unavoidable surface charge will aﬀect shallow 2D systems. We have calculated the single particle and transport lifetimes as a function of the depth of the 2DEG at a density of 1 × 1011cm−2 using the same model and impurity densities as in Figure A.4. From the calculation result plotted in Fig- 11 −2 ure A.5, we can see that at ns = 1 × 10 cm , the transport lifetime τt remains insensitive to small-angle scattering caused by surface states if the 2DEG is deeper than 60nm, whereas the quantum lifetime τq is strongly aﬀected at similar or even greater depths. For shallow systems where the 2DEG depth is usually less than

60nm, both τt and τq are affected by surface states, although the reduction in τt stays lower than that of τq. Moreover, the calculation also shows that even by if all backgroun imprities could be removed, ultra high electron mobilities can only be achieved in insulated gate devices if the 2DEG is deeper than 200nm. To furthur explore the effects of surface states in shallow systems, we have also repeated the calculation in Figure A.5 for different carrier densities. The calculations show that as the 2DEG density increases, the depth at which the 2DEG mobility starts to be affected by surface states gets shallower. This result can be explained by the matrix

138 Inﬂuence of surface states on quantum and transport lifetimes in high-quality 139 undoped heterostructures

Figure A.5: Calculated scattering lifetimes as a function of the depth of the 2DEG at ns = 1011cm−2, evaluated using three scattering mechanisms: background impurity scattering, interface roughness scattering and remote ionized impurity scattering due to surface charge. Dashed lines are the scattering lifetimes calculated with background impurity and interface roughness scattering, while solid lines are the values calculated with scattering from surface states also included. All parameters used are the same as in Figure A.4. element of surface charge scattering. As shown by Equation A.7, the term exp(−2qd) is always less than 1 since both the wavevector q and the distance d are positive. Therefore, the greater the product qd is, the less electrons are scattered by surface charge. Since the mobility is strongly affected by q ' 2kF backscattering process, if the depth of the 2DEG d is decreased, the density must be increased to increase kF to keep the same amount of surface charge scattering. Accordingly, the shallower the 2DEG is, the higher the carrier density must be to attain high mobility. Our results also show that even though high mobility can be achieved in undoped shallow wafers at a large carrier density [31], scattering from surface charge will affect device performance through the reduction of τq. As a result, all quantum lifetime or phase related measurements based on insulated-gate heterostructures, such as the Aharonov-Bohm effect [110] and the observation of the ν = 5/2 fractional quantum Hall effect(FQHE) states [31, 111], will be significantly influenced even though the electron mobility can be high and modulation doping has been removed. In addition, it can be hard to make high-mobility nanostructures on very shallow undoped wafers in the presence of surface charge scattering. A possible way to eliminate the

139 Influence of surface states on quantum and transport lifetimes in high-quality 140 undoped heterostructures effects of surface charge is by using a semiconductor-insulator-semiconductor field- effect transistor (SISFET) structure, in which a degenerately doped cap acts as the top-gate [26, 112] and screens the 2DEG from surface charge, allowing mobilies over 107cm2/V s. However, while extremely useful for large area 2D devices, the SISFET structure has drawbacks for making nanostructures. To define small features the 2DEG should be shallow, but in the SISFET it is hard to form ohmic contacts to a shallow 2DEG without shorting to the doped cap. Secondly the degenerately doped n+ or p+ GaAs cap used in the SISFET is not suitable for extremely small gate patterns. This is because the gates are defined by etching the cap, and surface state pinning means that gates narrower than ∼ 70nm are no longer metallic [113].

A.6 Conclusions and Future work

Even though the measurements and calculations were carried out for electrons in AlGaAs/GaAs heterostructure, the same theory predicts identical effects from surface charges to the tansport and lifetimes of holes in GaAs which is the focus of the rest of the thesis. 2DHG mobility of three wafers with different 2DHG depth are measured to confirm our prediction. Table A.1 shows the peak 2D mobility of the 2DHGs measured at 230mK. As the 2DHG gets shallower, the peak mobility obtainable decreases. However, since wafer B13520 is grown in a different MBE system (Bochum) from wafers W639 and W641 (Cambridge), the background impurity levels of B13520 can be very different from the other two wafers. Nonetheless, comparison between W639 and W640 can still show the effects of surface states as a function of the 2DHG depth. Overall, the hole mobility measured from wafers with different 2DHG depths agrees with the theory qualitatively. More measurements of heterostructures with different AlGaAs thickness ideally from the same MBE system need to be carried out if one wants to compare the effects of surface charge with theory quantitatively.

2 Wafer tAlGaAs (nm) tcap(nm) d2DHG(nm) Peak µ(cm /V s) B13520 300 17 317 88,200 W639 150 10 160 68,600 W641 50 10 60 60,300

Table A.1: Peak 2DHG mobility at T = 230mK for three wafers with diﬀerent AlGaAs layer thickness.

140 Inﬂuence of surface states on quantum and transport lifetimes in high-quality 141 undoped heterostructures

1.6

1.4

1.2

1.0 ) W 0.8 (k xx

R 0.6

0.4

0.2

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 B (T)

Figure A.6: SdH oscillations of 2DHG measured on wafer W641 at a temperature of 230mK. The 2DHG density is 2.5 × 1011cm−2 from a standard Hall measurement. The SdH oscillations show the typical beating behaviour due to the existence of two heavy hole masses, which is a result of the asymmetric z conﬁnement in single heterojections [16].

It will also be very interesting to compare the quantum and transport lifetimes of holes to verify if scattering mechanisms due to surface states are the same for holes. One complication to extract quantum lifetimes for 2DHG in heterojunctions is the presence of two effective masses, which results in a beating pattern [16] of SdH oscillations as shown in figure A.6. Figure A.6 is a typical measurement of hole SdH oscillations using a hall bar device on wafer W641. To separate the oscillations due to two HH subbands with different effective masses, Fourier and inverse Fourier transform needs to be applied to the original SdH oscillations first before any further calculations can be performed to extract the quantum lifetimes. This complexity in data analysis and the sensitivity of quantum lifetimes to the quality of oscillations after inverse Fourier transform may suggest that it would be better to conduct similar measurements on symmetric quantum well rather than single heterostructure. Therefore, possible work in the future will be conducting similar measurements and calculations for holes in quantum wells with different 2DHG depth. It will be very interesting to see if holes show the same trend in mobility as a function of 2DHG depth, the relationship between quantum and transport lifetimes for holes as well as whether different scattering mechanisms affect electrons and holes alike or not [104]. In conclusion, we have measured the transport and quantum lifetimes of an induced 2DEG at a single Al0.34Ga0.66As/GaAs heterojunction and compared the results with theoretical calculations. From the comparison, we have detected the existence of surface charge and demonstrated its effects on the scattering lifetimes

141 Inﬂuence of surface states on quantum and transport lifetimes in high-quality 142 undoped heterostructures

for deep undoped heterostructures. We found that the quantum lifetime τq is significantly reduced by surface charge even though the 2DEG is 300nm deep and its transport mobility is unaffected. These findings will be important for the development of high quality induced nanostructures.

142 Appendix B

Magnetospectroscopy of the two-hole states

Figure B.1 shows the magnetospectroscopy of the two-hole states measured using the single dot presented in Chapter 5. The source-drain bias is swept at a plunger gate bias of V3 = −0.623V while stepping the magnetic field in (a) the z direction and (b) the x direction. Both the ground state and the excited state seem to split in a perpendicular magnetic field, whereas only the excited state splits in a parallel magnetic field. This magnetic field dependence of the two-hole states is very different from the case for electrons as discussed in section 2.4.4. We suspect that this odd behaviour of the two-hole spin states could be caused by the strong Coulomb interaction in the hole system. However, due to the lack of theoretical calculations on hole quantum dots, we could not make any solid conclusions solely based on this measurement.

Figure B.1: Magnetospectroscopy measurements of the two-hole states. Derivative of the current through the dot plotted as a function of energy and magnetic ﬁeld (a) Bz and (b)

Bx.

143 Appendix C

The eﬀects of barriers on the double dot transport

Figure C.1 shows the charge stability diagram of the double dot presented in Chapter

6 near the transition (1, 0) → (0, 1) with a source-drain bias of VSD = 2mV . Left (gate 2) and right (gate 5) barriers are set to slightly diﬀerent conﬁgurations to illustrate how the relative height of the barriers change the transport through the dot. As the biases on the two barriers are varied, the visibility of the base of the bias triangle and the sharpness of the resonant tunnelling lines in the bias triangle change.

Figure C.1: Charge stability diagram near the transition (1, 0) → (0, 1) with a source- drain bias of VSD = 2mV when the left (gate 2) and right (gate 5) barriers are set at (a)

V2 = −0.5364V , V5 = −0.6295V ; (b)V2 = −0.5364V , V5 = −0.628V ; (c)V2 = −0.5355V ,

V5 = −0.628V .

144 Appendix D

The (1, 1) → (0, 2) transition

Figure D.1 shows the charge stability diagram of the double dot presented in Chapter 6 near transitions (1, 1) → (0, 2) ((a) and (b)) and (1, 1) → (2, 0) ((c) and (d)) where spin blockade is usually observed [35]. However, comparing the forward and reverse bias triangles, we could not see any signatures of spin blockade.

Figure D.1: Bias triangles of the double dot presented in Chapter 6 near conventional spin- blockade transitions. (a) and (b) are reverse-bias and forward-bias trangles of the transition (1, 1) → (0, 2). (c) and (d) are reverse-bias and forward-bias trangles of the transition (1, 1) → (2, 0).

To further look for signatures of spin blockade near the conventional (1, 1) →

(0, 2) transition, current through the dot is monitor while sweeping V3 along the pink dashed line in Figure D.2(a) at different magnetic fields. As shown in Figure D.1(c), the base of the (1, 1) → (0, 2) bias triangle splits in a perpendicular magnetic field, indicating a transition involving a spin triplet rather than a spin singlet, which is very different from observations in electron quantum dots [114]. The same measurement is repeated in an in-plane field Bx as shown in Figure D.2(d), but no splitting of

145 146 The (1, 1) → (0, 2) transition any states is observed.

Figure D.2: (a) Forward-bias and (b) reverse-bias trangles of the transition (1, 1) → (0, 2) with more balanced barriers compared to Figure D.1. (c) Current along the pink dashed line in (a) is plotted as a function of the bias on gate 3 and the magnetic ﬁeld (c) Bz and

(d) Bx. Black dashed lines are guides to the eye.

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