Electronic Transport in Inas Quantum Devices

Research Collection

Doctoral Thesis

Electronic Transport in InAs Quantum Devices

Author(s): Mittag, Christopher

Publication Date: 2020

Permanent Link: https://doi.org/10.3929/ethz-b-000408571

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ETH Library Diss. ETH No. 26509

Electronic Transport in InAs Quantum Devices

A thesis submitted to attain the degree of

DOCTOR OF SCIENCES of ETH ZURICH (Dr. sc. ETH Zurich)

presented by

Christopher Mittag

M.Sc. in Physics, Technische Universit¨atM¨unchen

born on 27.05.1990 citizen of Germany

accepted on the recommendation of:

Prof. Dr. K. Ensslin, examiner Prof. Dr. T. Ihn, co-examiner Prof. Dr. W. Wegscheider, co-examiner Prof. Dr. D. Loss, co-examiner

2020

Abstract

We experimentally investigate electronic transport in two-dimensional electron gases embedded in a semiconductor heterostructure at cryogenic temperatures. The objective of this thesis is the realization of nanostructures that enable the observation of quantum transport phenomena in quantum wells of the compound III-V semiconductor InAs. This material is characterized by a narrow band gap, a low effective electron mass, and a strong spin-orbit interaction. The latter feature has generated interest in InAs by providing a potential means of manipulating the spin degree of freedom in quantum dot based quantum computing schemes. Additionally, hybrid structures formed by combining spin-orbit materials such as InAs with superconductors have garnered recent attention for realizing Majorana fermions in a solid state system. These hold the promise of providing a platform for fault tolerant topological quantum computation. A peculiarity of InAs is that, in contrast to most other semiconductors, it does not form a Schottky barrier when brought into contact with a metal. At its surface, the Fermi level is pinned in the conduction band, which causes band bending and electron accumulation. A consequence of this is the infeasibility of typical semiconductor microfabrication techniques based on etching, as the carriers accumulated at physical borders cause a parasitic edge conduction that shunts quantum devices formed by electrostatic gates. This has severely limited the availability of InAs two-dimensional electron gases for mesoscopic transport experiments. We study this edge conduction in the context of intending to eliminate it by employing chemical passivation methods during fabrication. However, the application of these procedures did not significantly reverse edge conduction. As a result of this, we introduce a novel sample geometry in which multiple layers of gates electrostatically partition the electron gas, and contact these regions separately, thereby avoid- ing the need of etching steps. This geometry completely circumvents the parasitic edge conduction, thus overcoming the technological challenge of realizing quantum devices in InAs two-dimensional electron gases. Applying this technique, we study one-dimensional transport in a gate-defined quantum point contact that features a full pinch-off of the electron gas. We characterize its energy levels using finite bias spectroscopy and determine the electron g-factor in an applied magnetic field. Furthermore, the magnetoelectric subband structure and the influence of the coupling potential are analyzed. In an additional experiment, these fully controllable tunnel barriers enable the

i formation of a zero-dimensional quantum dot in an InAs quantum well. We observe Coulomb blockade diamonds in the few electron regime and characterize one- and two-electron ground and excited states in a magnetic field. Singlet and triplet states show no avoided crossing, which hints at a vanishing spin-orbit interaction for this particular interplay of crystallographic and magnetic field orientations. Strong coupling of the quantum dot to the leads allows the observation of the Kondo effect, whose dependences on temperature, magnetic field, and bias voltage were found to be in line with theoretical predictions. The g-factor determined from the splitting of the Kondo resonance is in agreement with the value extracted from the excited state spectroscopy. Building on these results, we extend the system by coupling two quantum dots to demonstrate a double quantum dot in an InAs two-dimensional electron gas. By tuning the energy levels in each quantum dot individually, using their respective plunger gates, we map out the charge stability diagram. Finite bias triangles emerge at the transitions between regions of stable occupation when a bias voltage is applied between source and drain contacts. Singlet-triplet spin blockade due to the Pauli exclusion principle rectifies the current flow when both quantum dots are occupied by electrons of the same spin projection. At strong interdot coupling, the spin blockade is pronounced even at zero magnetic field, while at weak interdot coupling it is lifted for the resonant tunneling process. When a magnetic field is applied, this leakage current gives rise to a narrow resonance stemming from hyperfine coupling to the nuclear spins of the crystal.

ii Zusammenfassung

Wir präsentieren experimentelle Untersuchungen des elektronischen Transports in zweidimensionalen Elektronengasen eingebettet in Halbleiterheterostrukturen bei tiefen Temperaturen. Das Ziel dieser Dissertation ist die Realisierung von Nano- strukturen welche es ermöglichen Quantentransportphänomene in Quantentöpfen des III-V Verbindungshalbleiters InAs zu beobachten. Entscheidende Merkmale die- ses Materials sind eine kleine Bandlucke,¨ eine geringe effektive Elektronenmasse, sowie eine starke Spin-Bahn-Wechselwirkung. Besonders die letztere Eigenschaft hat Interesse an InAs hervorgerufen, da sie eine Möglichkeit der Manipulation des Spinfreiheitsgrads von Elektronen in Quantenpunkten bieten könnte, worauf eine potentielle Implementation eines Quantencomputers basiert. Daruber¨ hinaus ha- ben Hybridstrukturen, welche gebildet werden indem man Materialien starker Spin- Bahn-Kopplung wie InAs mit Supraleitern kombiniert, jungst¨ Aufmerksamkeit er- langt um Majorana Fermionen in einem Festkörpersystem zu realisieren. Gemäss aktueller Theorien bilden diese Teilchen eine vielversprechende Plattform fur¨ eine fehlertolerante topologische Quanteninformationsverarbeitung. Eine Besonderheit von InAs im Gegensatz zu den meisten anderen Halbleitern ist, dass es keine Schottkybarriere in Kontakt mit Metallen bildet. Dies lässt sich auf das Ferminiveau zuruckf¨ uhren,¨ welches an der Oberfläche im Leitungsband fest- gehalten ist, was eine Biegung des Leitungsbands und damit eine Anreicherung von Elektronen zur Folge hat. Als Konsequenz daraus sind typische Halbleitermikrofabri- kationstechnologien welche auf Atzverfahren¨ basieren ungeeignet fur¨ InAs, da die an der Oberfläche der physischen Kanten angereicherten Ladungsträger eine parasitäre Randleitfähigkeit verursachen, welche die mit elektrostatischen Kontrollelektroden gebildeten Nanostrukturen uberbr¨ uckt.¨ Infolgedessen ist die Nutzbarkeit von InAs Proben fur¨ mesoskopische Transportexperimente stark limitiert. Im Rahmen des Versuchs der Eliminierung der Randleitfähigkeit durch chemi- sche Passivierungsmethoden während der Fabrikation wird diese untersucht. Kei- ne der getesteten Methoden konnte jedoch eine signifikante Aufhebung der Rand- leitfähigkeit erzielen. Aufgrund dessen fuhren¨ wir eine neue Probengeometrie ein, in welcher das Elektronengas durch mehrere Schichten von Kontrollelektroden elektro- statisch aufgeteilt wird. Die unterschiedlichen Regionen werden separat kontaktiert, wodurch das Anwenden von Atzverfahren¨ redundant wird. Diese Geometrie erlaubt es die parasitäre Randleitfähigkeit komplett zu umgehen, wodurch die technologi- sche Herausforderung der Realisierung von Quantenapparaturen in zweidimensiona-

iii len Elektronengasen von InAs bewältigt wird. Durch die Anwendung dieser Technik realisieren wir das vollständige Abschnuren¨ des Elektronengases im eindimensionalen Elektronentransport durch einen von Kon- trollelektroden definierten Quantenpunktkontakt. Wir spektroskopieren dessen Ener- gieniveaus unter endlicher Vorspannung und determinieren den g-Faktor der Elek- tronen in einem Magnetfeld. Daruber¨ hinaus analysieren wir die magnetoelektrische Subband-Struktur sowie den Einfluss des Kopplungspotentials. In einem weiteren Experiment ermöglichen diese vollständig abstimmbaren Tun- nelbarrieren die Formation einens nulldimensionalen Quantenpunkts in einem InAs Quantentopf. Im Regime weniger Elektronen beobachten wir Coulomb-Blockade- Diamanten und charakterisieren im Magnetfeld die Grund- und angeregten Zustände bei Besetzung mit einem sowie zwei Elektronen. Zwischen den Uberg¨ ängen von Spin-Singlett und Spin-Triplett Zuständen wurde kein vermiedenes Kreuzen festgestellt. Dies könnte auf ein Zusammenspiel der speziellen Orientierung des Ma- gnetfelds und der Kristallrichtung des Transports zuruckgef¨ uhrt¨ werden, welches die Spin-Bahn-Wechselwirkung verschwinden lässt. Eine starke Kopplung des Quanten- punkts an die Zuleitungen ermöglicht die Beobachtung des Kondo-Effekts. Dessen Abhängigkeiten von der Temperatur, dem Magnetfeld sowie der Vorspannung kor- respondieren mit theoretischen Erwartungen. Der aus der Aufspaltung der Kondo- Resonanz bestimmte g-Faktor stimmt mit dem aus der Spektroskopie des angeregten Ein-Elektronen-Zustands extrahierten Wert uberein.¨ Auf diesen Ergebnissen aufbauend koppeln wir zwei Quantenpunkte aneinander und erweitern damit das System zu einem Doppelquantenpunkt in einem zweidimensionalen Elektronengas aus InAs. Das separate Abstimmen der Energieniveaus in beiden Quantenpunkten mithilfe ihrer jeweiligen Kontrollelektroden erlaubt es uns, das Ladungsstabilitätsdiagramm auszumessen. Durch das Anlegen einer Vorspan- nung zwischen Source- und Drain-Kontakt entstehen Dreiecke des erlaubten Strom- flusses aus den stabilen Punkten der Ladungsstabilitätsregionen. Eine Gleichrichtung des Stromflusses durch die Singlett-Triplett-Spinblockade basierend auf dem Pauli- schen Ausschliessungsprinzip wurde beobachtet wenn sich in beiden Quantenpunk- ten ein Elektron mit gleicher Spinprojektion befindet. Bei starker Tunnelkopplung zwischen den beiden Quantenpunkten wurde eine ausgeprägte Spinblockade auch bei null Magnetfeld festgestellt, welche bei schwacher interner Tunnelkopplung fur¨ den resonanten Tunnelprozess aufgehoben ist. Beim Anlegen eines Magnetfelds geht aus diesem Leckstrom in der Spinblockade eine scharfe Resonanz hervor, welche auf die Hyperfeinwechselwirkung mit den Kernspins des Kristalls zuruckzuf¨ uhren¨ ist.

iv Contents

List of Figures viii

List of Tables xii

Constants, Abbreviations and Symbols xiii

1 Introduction1

2 Basic concepts5 2.1 Two-dimensional electron gases...... 5 2.1.1 Band structure...... 5 2.1.2 Diffusive transport within the Drude model...... 7 2.1.3 Spin-orbit interaction...... 9 2.2 Electrons confined to one dimension: quantum point contacts..... 10 2.3 Electrons confined to zero dimensions: quantum dots...... 12

3 Sample fabrication 14 3.1 Introduction...... 14 3.2 Fabrication of Hall bar devices...... 14 3.2.1 Ohmic contact deposition...... 15 3.2.2 Mesa structure definition...... 15 3.2.3 Gate insulator deposition...... 16 3.2.4 Gate metal deposition...... 16 3.2.5 Contact release and bonding...... 17 3.3 Fabrication of devices with an electrostatically defined mesa structure 17 3.3.1 Ohmic contact deposition...... 18 3.3.2 First gate insulator deposition...... 18 3.3.3 Frame gate deposition...... 18 3.3.4 Second gate insulator deposition...... 19 3.3.5 Contact release...... 19 3.3.6 Gate lead deposition...... 19 3.3.7 Electron beam lithography and fine gate deposition...... 20

v 4 Passivation of edge states in etched InAs sidewalls 21 4.1 Introduction...... 22 4.2 Methods of passivating edge states...... 22 4.2.1 Silicon nitride by chemical vapor deposition...... 23 4.2.2 Aluminum oxide by atomic layer deposition...... 25 4.2.3 Oxidation...... 27 4.2.4 Sulfur passivation...... 29 4.2.5 Magnesium borohydride...... 33 4.2.6 PCBM...... 35 4.3 Comparison of diﬀerent passivation methods...... 37 4.4 Conclusion...... 40

5 Circumventing trivial edge conduction by electrostatic mesa deﬁ- nition 41 5.1 Introduction...... 42 5.2 Characterization of the frame gate...... 45 5.3 Proof of principle operation of a quantum point contact...... 47 5.4 Proof of principle operation of a quantum dot...... 50 5.5 Conclusion...... 53

6 Gate-defined quantum point contacts 54 6.1 Introduction...... 55 6.2 Quantized conductance...... 57 6.3 Finite bias spectroscopy...... 58 6.4 Behavior in a parallel magnetic field...... 61 6.5 Behavior in a perpendicular magnetic field...... 64 6.6 Shifting the quantum point contact in real space...... 65 6.7 Influence of the coupling potential...... 66 6.8 Searching for signatures of a 0.7 anomaly...... 68 6.9 Conclusion...... 69

7 Gate-defined lateral quantum dot 71 7.1 Introduction...... 71 7.2 Characteristics of the sample used for defining a quantum dot.... 72 7.3 Tuning the quantum dot...... 73 7.4 Coulomb blockade diamonds at finite bias...... 78 7.5 Excited state spectroscopy...... 80 7.5.1 Behavior in a perpendicular magnetic field...... 83 7.5.2 Behavior in a parallel magnetic field...... 85 7.5.3 Singlet-triplet crossing...... 87 7.6 Kondo effect at strong coupling to the leads...... 89 7.6.1 Tuning into the Kondo regime by opening the dot...... 90 7.6.2 Kondo effect visible as a zero-bias anomaly...... 90

vi 7.6.3 Splitting of the zero-bias anomaly in a magnetic ﬁeld..... 91 7.6.4 Temperature dependence of the zero-bias anomaly...... 93 7.7 Conclusion...... 94

8 Double quantum dots 96 8.1 Introduction...... 96 8.2 Coupling two quantum dots...... 97 8.3 Charge stability diagram...... 100 8.4 Finite bias triangles emerging from triple points...... 102 8.5 Singlet-triplet spin blockade by Pauli exclusion at strong interdot coupling...... 105 8.5.1 Investigating the transition between (0,3) and (1,2) charge occupation...... 107 8.5.2 Investigating the transition between (1,3) and (2,2) charge occupation...... 109 8.5.3 Investigating the transition between (0,4) and (1,3) charge occupation...... 110 8.6 Singlet-triplet spin blockade by Pauli exclusion at weak interdot coupling...... 112 8.6.1 Investigating the transition between (1,3) and (2,2) charge occupation...... 113 8.6.2 Investigating the transition between (0,4) and (1,3) charge occupation...... 117 8.6.3 Investigating the transition between (0,2) and (1,1) charge occupation...... 118 8.6.4 Investigating the transition between (2,0) and (1,1) charge occupation...... 120 8.7 Conclusion...... 121

9 Conclusion and outlook 122 9.1 Conclusion...... 122 9.2 Outlook...... 123

Publications 125

Bibliography 127

Acknowledgements 141

Curriculum Vitae 144

vii List of Figures

2.1 Band lineups of semiconductors with a lattice constant around 6.1 A.˚ 6 2.2 Exemplary layer sequence and band structure of an InAs two-dimensional electron gas...... 7 2.3 Energy level schematic of a quantum dot coupled to two metallic leads. 13

4.1 Optical microscope images of a reference Hall bar sample with a SiNx dielectric layer...... 23 4.2 Longitudinal resistance of a reference Hall bar sample with a SiNx dielectric layer...... 24 4.3 Density and mobility of a reference Hall bar sample with a SiNx dielectric layer...... 25 4.4 Optical microscope images of a Hall bar sample with an Al2O3 dielectric layer...... 26 4.5 Longitudinal resistance of Hall bar samples with an Al2O3 dielectric layer...... 26 4.6 Density and mobility of a Hall bar sample with an Al2O3 dielectric layer...... 27 4.7 Longitudinal resistance of a Hall bar sample with intentionally oxidized edges...... 28 4.8 Density and mobility of a Hall bar sample with intentionally oxidized edges...... 29 4.9 Optical microscope images of Hall bar samples passivated with TAM and ammonium sulfide...... 29 4.10 Longitudinal resistance of a Hall bar sample passivated with TAM.. 30 4.11 Density and mobility of a Hall bar sample passivated with TAM.... 31 4.12 Longitudinal resistance of a Hall bar sample passivated with ammonium sulfide...... 31 4.13 Density and mobility of a Hall bar sample passivated with ammonium sulfide...... 32 4.14 Optical microscope images of a Hall bar sample passivated with Mg(BH4)2. 33 4.15 Longitudinal resistance of a Hall bar sample passivated with Mg(BH4)2. 34 4.16 Density and mobility of a Hall bar sample passivated with Mg(BH4)2. 34 4.17 Optical microscope images of Hall bar samples passivated with PCBM. 35

viii 4.18 Longitudinal resistance of a Hall bar sample passivated with PCBM.. 36 4.19 Density and mobility of a Hall bar sample passivated with PCBM... 36 4.20 Comparison of the longitudinal resistances of Hall bars passivated by different techniques...... 38 4.21 Bar chart comparing specific edge resistivities of the different passivation techniques and a schematic resistor network model...... 39

5.1 Schematic cross section and band edge diagram of an InAs Hall bar.. 42 5.2 Schematic cross-sectional and top view of a sample with an electrostatically defined mesa...... 43 5.3 Optical and scanning electron microscope images of a sample with an electrostatically defined mesa...... 45 5.4 Resistance and conductance traces between the in- and outside of the electrostatically defined mesa...... 46 5.5 Optical and scanning electron microscope images of the fine gate layout for quantum point contact formation...... 47 5.6 Sweep of the voltage applied to a pair of split gates facilitates full pinch-off and disorder-induced resonances...... 48 5.7 Lateral shifting of the constriction by changing the voltages applied to both of the split gates individually...... 49 5.8 Pinch-off and Coulomb resonances visible in the conductance through quantum dot...... 51 5.9 Tunability and finite bias measurements of a quantum dot...... 52

6.1 Schematic representation of the lateral sample geometry and heterostructure employed for the formation of a quantum point contact. 56 6.2 Quantized conductance visible in sweeps of the voltage applied to the split gates...... 57 6.3 I-V characteristics of the Ohmic contacts used for bias correction of the data...... 59 6.4 Finite bias spectroscopy of the quantum point contact...... 60 6.5 Spin-split conductance plateaus emerging in a parallel magnetic ﬁeld. 61 6.6 Schematic band structure used for the estimation of the g-factor.... 62 6.7 Magnetoelectric subbands investigated in a perpendicular magnetic ﬁeld...... 64 6.8 Lateral shifting of the quantum point contact by tuning the voltages on both split gates individally...... 65 6.9 Shaping the coupling potential with gates surrounding the quantum point contact...... 67 6.10 High resolution bias voltage traces of the quantum point contact in search of a 0.7 anomaly...... 69

7.1 Scanning electron micrograph of the gate nanostructure used to deﬁne quantum dots...... 73

ix 7.2 Gate sweep of the voltage applied to the frame gate facilitating pinch- oﬀ at lower voltages due to a thinner dielectric layer...... 74 7.3 Characterization of the ﬁne gates intended to form the quantum dot. 74 7.4 Quantum dot conductance as a function of the barrier voltages.... 76 7.5 Quantum dot conductance as a function of upper and lower plunger gate voltages...... 77 7.6 Finite bias measurement of a Coulomb blockade diamond and corresponding line cut at zero bias...... 78

7.7 Finite bias measurement of Coulomb blockade diamonds up to VDC = ±10 mV...... 79 7.8 Schematic energy spectrum of a quantum dot containing up to two electrons...... 80 7.9 Schematic depiction of a finite bias measurement of a quantum dot containing an excited state with corresponding traces of the current and the conductance...... 82 7.10 Energy level diagrams for transport through a quantum dot taking into account excited states...... 83 7.11 Finite bias measurement revealing a two-electron excited state and spectroscopy of the one- and two-electron ground and excited states in a perpendicular magnetic field...... 84 7.12 Spectroscopy of the one- and two-electron ground and excited states in a parallel magnetic field...... 85 7.13 Singlet and triplet states crossing in a parallel magnetic field leading to a change of the ground state to a spin singlet. Energy level and current schematics explainig resonances of negative differential conductance...... 88 7.14 Kondo effect visible as a valley in the conductance as a function of the plunger gate voltage, and as a zero bias anomaly as a function of the bias voltage...... 91 7.15 Kondo peak visible as a zero bias anomaly within the first Coulomb blockade diamond of a strongly coupled quantum dot...... 92 7.16 Coulomb blockade diamond at finite magnetic field features a split Kondo peak, which in a bias and magnetic field dependent measurement shows a linear Zeeman splitting...... 93 7.17 Temperature scaling of the Kondo peak conductance and fit to a phenomenological model to determine the Kondo temperature..... 93

8.1 First step of double dot tuning procedure: sweeping the middle barrier gates...... 98 8.2 Second step of double dot tuning procedure: separate sweeps of left and right plunger gates...... 98

x 8.3 Third and fourth steps of double dot tuning procedure: recording a map of the dot conductance as a function of left and right plunger gates, and measuring a map of the conductance as a function of the left and right tunnel barriers...... 99 8.4 Schematic network model of a serially coupled double quantum dot.. 100 8.5 Measurement of the charge stability diagram of a double quantum dot.101 8.6 Transport in the vicinity of the triple points around (0,2) and (1,3) charge occupation for increasing source drain bias voltages...... 103 8.7 Energy level schematic visualizing the emergence of ﬁnite bias triangles.104 8.8 Exemplary measurement and energy level schematic illustrating singlet- triplet spin blockade...... 105 8.9 Schematic charge stability diagram with spin blockaded transitions indicated...... 107 8.10 Finite bias triangles at the transition between (0,3) and (1,2) occupation showing no spin blockade...... 108 8.11 Finite bias triangles at the transition between (0,3) and (1,2) occupation at Bk = 100 mT...... 108 8.12 Finite bias triangles at the transition between (1,3) and (2,2) occupation showing transport in forward bias and spin blockade in reverse bias...... 109 8.13 Finite bias triangles at the transition between (0,4) and (1,3) occupation showing spin blockade in forward bias and transport in reverse bias...... 110 8.14 Magnetic ﬁeld dependence of the leakage current in the spin blockaded transition from (1,3) to (0,4) occupation...... 112 8.15 Finite bias triangles at the transition between (1,3) and (2,2) occupation at weak interdot coupling showing transport in forward bias and spin blockade in reverse bias...... 113 8.16 Dependence of the current at the spin blockaded (1,3) to (2,2) transition on Bk showing a narrow resonance at zero detuning...... 114 8.17 Spin blockaded (1,3) to (0,4) transition at weak interdot coupling and its dependence on Bk...... 118 8.18 Spin blockaded (1,1) to (0,2) transition at weak interdot coupling and its dependence on Bk...... 119 8.19 Spin blockaded (1,1) to (2,0) transition at weak interdot coupling and its dependence on Bk...... 120

xi List of Tables

6.1 Energy level spacings of the ﬁrst three modes of the QPC determined by ﬁnite bias spectroscopy...... 60 6.2 Band edge parameters of the semiconductors GaAs, AlAs, InAs, InSb, and InP. Taken from Refs. [1] and [2]...... 63

8.1 Nuclear magnetic resonance data for the relevant isotopes of the elements composing InAs. Taken from Ref. [3]...... 115

xii Constants, Abbreviations and Symbols

Constant Value −23 −1 Boltzmann constant kB = 1.381 × 10 JK = 86.17 µeV K−1 −24 −1 Bohr magneton µB = 9.274 × 10 JT = 57.88 µeV T−1 −11 Bohr radius aB = 5.291 × 10 m −5 conductance quantum G0 = 7.748 × 10 S −31 electron rest mass me = 9.109 × 10 kg elementary charge e = 1.602 × 10−19 C Planck constant h = 6.626 × 10−34 J s = 4.136 × 10−15 meV ps reduced Planck constant ~ = 1.055 × 10−34 J s = 0.6582 meV ps speed of light c = 2.998 × 108 m s−1 −12 −1 vacuum permittivity ε0 = 8.854 × 10 F m −7 −1 vacuum permeability µ0 = 4π × 10 H m

xiii Abbreviation Description 2DEG two-dimensional electron gas AC alternating current ALD atomic layer deposition BOE buffered oxide etch DC direct current DI deionized DMSO dimethyl sulfoxide DQD double quantum dot FET field effect transistor FG frame gate FWHM full width at half maximum IPA isopropyl alcohol LB left barrier MB middle barrier MBE molecular beam epitaxy MIBK methyl isobutyl ketone PCBM phenyl-C61-butyric acid methyl ester PECVD plasma-enhanced chemical vapor deposition PG plunger gate PGL plunger gate left (dot) PGR plunger gate right (dot) PMMA poly(methyl methacrylate) QD quantum dot QPC quantum point contact QSHI quantum spin Hall insulator QW quantum well RB right barrier S spin singlet state SEM scanning electron microscope T+,T0,T− spin triplet states TAM thioacetamide TG top gate THF tetrahydrofuran VTI variable temperature insert

xiv Symbol Explanation Unit A area m2 B magnetic field T B⊥ perpendicular magnetic field T Bk in-plane magnetic field T C capacitance per area F m−2 d layer thickness m E electric field V m−1 E energy J EC conduction band energy J EF Fermi energy J EV valence band energy J f frequency s−1 F force N G conductance S g∗ effective g-factor – I current A j current density A m−1 k wave vector m−1 −1 kF Fermi wave number m le elastic mean free path m L length m m∗ electron effective mass – −2 ns electron sheet density m N electron number – p momentum m−1 r radius m R resistance Ω t time s T temperature K v velocity m s−1 −1 vF Fermi velocity m s V voltage V W width m x, y, z spatial directions m

αgate gate lever arm – −1 αR Rashba coeﬃcient m s −1 βD Dresselhaus coeﬃcient m s δ detuning V ∆SO spin-orbit splitting J

xv εr relative permittivity – λ linear resistivity Ω m−1 λF Fermi wavelength m 2 −1 −1 µe electron Drude mobility m V s µelch electrochemical potential J ρ resistivity Ω σ conductivity Ω−1 σ = (σx, σy, σz) vector of Pauli spin matrices – τe elastic scattering time s −1 ω0 frequency of harmonic potential s −1 ωc cyclotron frequency s

xvi

Chapter 1

Introduction

Semiconductor technology lies at the heart of modern day lives. This is most apparent for ubiquitous devices such as laptops or tablet computers, smart phones and watches. Nowadays however, it even extends to unsuspecting objects like cars, dishwashers, and coffee machines - all these devices require a varying extent of computing power in order to perform the tasks which we have imposed on them. This computing power is centered in a microprocessor unit which uses up to billions of nanometer sized switches that drive computer logic by the presence or absence of current flow, encoding the two logic states of the zero and one. These switches are transistors made in complementary metal-oxide-semiconductor technology, which has only been possible thanks to groundbreaking fundamental physics research per- fomed in the early days of semiconductor physics, like the development of the field- effect transistor. As we are further pushing the boundaries of computing power, certain tasks emerged that lie beyond of what is possible to calculate within reasonable time even using the most advanced supercomputers. These specific tasks for instance include the factorization of large prime numbers [4], and the simula- tion of quantum physical processes or chemical reactions [5]. It was suggested that a quantum computer, where the bits of information can not only assume the values zero and one but also any superposition of these, could be capable of solving the aforementioned problems. Many possible implementations of such a quantum computer have been suggested, but in particular semiconductor quantum dots [6] are considered a promising approach due to their close technological connection to existing semiconductor industry. The purpose of this thesis is neither attempting to directly realize a particular implementation of a quantum computer, nor is it concerned with increasing the speed or decreasing the footprint and energy consumption of classical transistors. We take an approach that is directed more towards fundamental semiconductor device physics and intend to study quantum transport in a material system in which it was thus far not possible to be studied. Insights that are gained in this way are aimed at broadening our understanding of physical systems and opening up the possibilities of realizing certain types of devices, and thus indirectly contribute towards advancing

1 Chapter 1. Introduction both of these computation types in the future. Much like Bardeen, Brattain and Shockley realized the first transistor without immediately visualizing the practical application within the central processing unit of a personal computer, we aim to advance our understanding in a way which may or may not find a direct application in the coming years. The material which we are interested in studying is InAs, a III-V semiconductor composed of indium and arsenic atoms arranged in a zincblende crystal structure. Despite the fact that it was possible to grow this material in superlattices and quantum wells only a few years after GaAs [7], InAs remained mostly unexplored in mesoscopic electron transport physics while GaAs devices thrived to tremendous success, providing a plethora of breakthroughs such as realizing the first quantum point contacts [8,9], and single and double quantum dots [10–13]. Next to heterostructure quality, one of the most significant reason for this disparity in device success can be ascribed to a Fermi level pinning in the conduction band at the surface of InAs. This leads to a bending of the conduction band and thus to the accumulation of electrons at the surface. The resulting lack of a Schottky barrier [14] typically found when semiconductors are put into contact with metals, is very advantageous when trying to establish electrical contact to InAs or when trying to interface it with a superconductor to create hybrid devices [15–17]. When attempting to fabricate devices in a way in which it was pioneered on GaAs and other materials however, severe difficulties are encountered arising from parasitic conductance at the edges of the mesa. We discuss these issues and different methods on how to circumvent them in detail in the course of this thesis. Why should one generally be interested in InAs, if quantum transport has been successfully demonstrated in GaAs already? This question can be answered by considering certain key material properties. InAs is a narrow band gap semiconductor [18] that features a low effective electron mass [19] which is advantageous for reaching high electron mobilities in two-dimensional electron gases and large single- particle energy spacings in quantum dots. The spin-orbit interaction inherent to the material is very strong [20–25], which holds promise for manipulating the spin degree of freedom of electrons trapped in quantum dots [26]. Additionally, this large spin-orbit coupling together with the aforementioned ease in interfacing it with superconductors have caused substantial recent interest for the purpose of investigating Majorana fermions [27–29] for topological quantum computation [30]. Considering all these factors, it is worthwhile to attempt to overcome the technological challenge of realizing nanostructures for quantum transport in an InAs two-dimensional electron gas. We commence this thesis with a brief introduction to important concepts of electronic transport in semiconductor heterostructures in chapter2. Specifically, the influence of the dimensionality of the system under study on the electronic behavior is discussed. The reviewed concepts are applied in the experiments that are presented in the following chapters. In chapter3, the fabrication procedures necessary to manufacture the samples

2 investigated in this thesis are described in detail. We separately outline the fabrication of two fundamentally different types of samples. For Hall bar type devices, a wet etching step is necessary in order to obtain a mesa structure to which the two- dimensional electron gas is confined. The second type of samples is characterized by a mesa structure which is defined completely electrostatically. This requires a different fabrication process in which multiple layers of metallic gates separated by dielectric layers are employed, eschewing the etching step. Chapter4 presents the first experimental efforts undertaken to combat the trivial edge conduction in etched structures of InAs. We investigate the effects of different chemical passivation methods applied during fabrication of Hall bar type devices on the surface accumulation at the etched sidewalls. None of the applied chemicals were successful in reversing the Fermi level pinning at the surface, but an atomic layer deposition of aluminum oxide lead to a significantly higher linear edge resistivity than previously observed. The subsequent chapter5 introduces a fundamentally different device geometry aimed at circumventing the trivial edge conduction. We electrostatically define a mesa structure by using a rectangular gate that partitions the electron gas into an inner and outer part. Both are contacted separately, and a second layer of gates enables the formation of nanostructures in the inner part. This is shown to successfully remove a physical edge of the device and thereby any parasitic conduction. Proof of concept quantum point contact and quantum dot devices are fabricated which show full pinch-off of the electron gas. Building on these results we employ a heterostructure grown on a GaSb substrate, which is of superior quality than the heterostructures used in chapters4 and 5 that were grown on a GaAs substrate, to conduct follow-up experiments in chap- ter6. There, we demonstrate a quantum point contact showing steps of quantized conductance. The energy levels are studied under the application of a finite source drain bias, and a parallel magnetic field enables the extraction of the electron g- factor. The magnetoelectric subband structure is investigated after rotating the field in perpendicularly to the plane of the quantum well growth. Having demonstrated well controlled tunnel barriers, we form a single quantum dot in chapter7. The few electron regime is reached and Coulomb blockade diamonds are analyzed. Excited state spectroscopy at finite bias in a magnetic field reveals the structure of one- and two-electron ground and excited states. A crossover of the ground state from a spin singlet to a spin triplet occurs as a function of magnetic field. The absence of an avoided crossing hints at a vanishing spin-orbit interaction as a consequence of the direction of the parallel magnetic field and the crystal orientation of the sample. The Kondo effect is observed and characterized at strong coupling to the leads and corresponds excellently to theoretical expectations. Chapter8 presents the results of tunnel coupling two single dots to a double quantum dot. We explore the charge stability diagram and the finite bias triangles that result under application of a voltage between source and drain contacts. Due to the Pauli exclusion principle, a singlet-triplet spin blockade of the current is

3 Chapter 1. Introduction

observed. When the interdot coupling is strong, the blockade is pronounced at zero magnetic field. For weak interdot coupling, the blockade is lifted when the energy levels in the dots are resonant. In a magnetic field, this line gives rise to an unexpectedly sharp resonance likely related to hyperfine coupling to the nuclear spins of the crystal. We conclude in chapter9 with a summary of the results of this thesis and provide an outlook on potential future research directions. Follow-up experiments that directly extend the concept of the gate-defined mesa are suggested.

4 Chapter 2

Basic concepts

This chapter serves as a basic introduction to concepts of electronic transport in semiconductor heterostructures that are made use of in later chapters of this work. We begin in section 2.1 by introducing the band structure of an InAs two-dimensional electron gas, outlining important parameters of such a system, and by describing diﬀusive transport in two dimensions within the Drude model. In addition, we detail the eﬀect of spin-orbit-interaction on the Hamiltonian of the system. Sub- sequently, we present how the formation of a narrow constriction makes transport one-dimensional and introduces conductance quantization in section 2.2. Ultimately, we turn to systems where the motion of electrons is quantized in all three spatial directions, which are known as quantum dots, in section 2.3. The overview given in this chapter is not intended to be comprehensive, and for more detail Refs. [1, 31, 32] can be consulted.

2.1 Two-dimensional electron gases

2.1.1 Band structure Two-dimensional electron gases are an area of solid state phyics research that has brought to light intriguing effects over the past years and is still actively investigated. Quantum mechanics has taught us that reducing the size of a system in a given direction leads to the formation of quantized energy levels in this direction. This can be achieved either by utilizing a material that is intrinsically two-dimensional, like an exfoliated sheet of graphene or of a transition metal dichalcogenide, or by confining charge carriers to a very thin conductive layer which is surrounded by an insulating material. The latter can be achieved in semiconductor heterostructures by combining semiconductors of different band gaps. Fig. 2.1 shows the band gaps and their respective alignments when put into contact of semiconductors of the so- called 6.1 A˚ family [18], to which InAs belongs with its lattice constant of 6.06 A.˚ On the right hand side of the image we can see that sandwiching a thin InAs layer in between two layers of AlSb, which has a much larger band gap, creates a quantum

5 Chapter 2. Basic concepts

Figure 2.1: Band lineups of semiconductors with a lattice constant around 6.1 A.˚ The grey boxes represent the band gaps and the arrows show the oﬀsets between the bands, which are all labeled in units of eV. Adapted from Ref. [18].

well of depth 1.35 eV in the conduction band. If the Fermi energy is positioned such that it lies above the top of the conduction band of InAs and within the band gap of AlSb, the quantum well is populated with charge carriers and a two-dimensional electron gas is created. Samples used for the experiments in chapter4 and5 feature exactly this material composition, while for the samples used in the experiments in chapters6,7, and8 the barriers are composed of a slightly different material in Al0.8Ga0.2Sb. We show the band structure corresponding to such a system on the right hand side of Fig. 2.2 with the sequence of layers depicted on the left hand side. The blue dashed line specifies the position of the Fermi energy, which allows for the population of one subband of the electronic system. The corresponding wave function within the quantum well is drawn in red. At the bottom of the conduction band in InAs we assume a parabolic dispersion relation with an isotropic effective mass m∗ with respect to the two in-plane directions x and y, and n quantized subbands populated in growth direction z. Therefore, we can write the total dispersion relation of the system as the sum of their energies

2k2 E = E + ~ , (2.1) nk n 2m∗

with the wave vector k = (kx, ky) pointing in the plane of the two-dimensional electron gas. The wave function of a given state with subband index n and wave vector k can then be written as

i(kxx+kyy) ψnk(x, y, z) = e χn(z), (2.2)

6 2.1. Two-dimensional electron gases

which is a combination of a plane wave in x and y directions and a wave function χn corresponding to the quantized state in growth direction z.

Figure 2.2: Exemplary layer sequence and associated band structure of a heterostructure used for the experiments in this thesis. The blue dashed line speciﬁes the position of the Fermi energy and the red curve shows the wave function of the two-dimensional electron system. Adapted from Ref. [33].

The density of states for a parabolic dispersion calculated with the free electron model for two dimensions is constant within each subband and is given by g g m∗ D(E) = s v . (2.3) 2π~2

The degeneracy factors take into account the spin (gs) and valley (gv) degrees of freedom. In InAs, we only have one occupied valley which means that gv = 1, unlike in graphene or Si. At zero magnetic ﬁeld, gs = 2, as both spin species are degenerate in energy. Knowing the density of states and the Fermi energy, we can simply calculate the electron sheet density within the two-dimensional electron gas as ns = D· EF. (2.4)

We denote the Fermi energy EF as the electrochemical potential of the system measured from the quantized energy of the ground state subband.

2.1.2 Diﬀusive transport within the Drude model All experiments in this thesis were performed on two-dimensional electron gases, and therefore it is instructive to give an overview of their basic transport characterstics. The fundamental law governing electron transport states that the voltage drop U

7 Chapter 2. Basic concepts

between two longitudinal contacts is proportional to the current I and the resistance R according to U = R · I, (2.5) which is known as Ohm’s law. This can be rewritten as

j(r) = σE(r) (2.6) with the current density j(r) and the electric field E(r) which depend on the position vector r. The electrical conductivity σ does not, for a homogeneous material, depend on r. In the three dimensional case, j and E are three-component vectors, and σ is a three by three tensor that can be reduced depending on material isotropy. For the two-dimensional case that we are concerned with, they get reduced to two- component vectors and a two by two tensor, respectively. We consider diffusive transport within the Drude model, which means that we introduce a mean elastic scattering time τe after which a scattering event random- izing the momentum of the electron happens. Furthermore, we assume that the corresponding scattering length is small compared to the sample dimensions. Let the motion of charge carriers of a two-dimensional electron gas be free in x and y directions and quantized in z direction. We impose an electric field E = (Ex, 0, 0) and apply a magnetic field perpendicular to the quantum well, B = (0, 0,Bz). Con- sidering the Lorentz force F = −e(E + v × B) (2.7) acting on the electrons of charge −e moving with a velocity v, we can write two equations of motion. These are

dv m∗ x = −|e|(E + v B ) and (2.8) dt x y z dv m∗ y = +|e|v B , (2.9) dt x z with the effective electron mass m∗. Solving these equations, we find the electron mobility µe as |e|τ µ = e , (2.10) e m∗ which corresponds to the ratio of drift velocity and applied electric field at zero ∗ magnetic field. Introducing the cyclotron frequency ωc = |e|Bz/m , we can define the components of the conductivity tensor as

2 nse τe 1 σxx = ∗ 2 2 (2.11) m 1 + ωc τe 2 nse τe ωcτe σxy = ∗ 2 2 . (2.12) m 1 + ωc τe

8 2.1. Two-dimensional electron gases

The relations σxx = σyy and σyx = −σxy are obtained from solving equation 2.9 for an electric ﬁeld applied in the y-direction. This allows us to rewrite equation 2.6 for a two-dimensional system as

j σ −σ E x = xx xy x . (2.13) jy σxy σxx Ey

Matrix inversion of the conductivity tensors gives us the speciﬁc resistivities ρxx and ρxy, which link the electric ﬁeld to the current density as

E ρ ρ j x = xx xy x . (2.14) Ey −ρxy ρxx jy

The explicit results of the matrix inversion of equation 2.12 are

∗ σxx m 1 ρxx = 2 2 = 2 = (2.15) σxx + σxy nse τe |e|nsµe σxy Bz ρxy = 2 2 = , (2.16) σxx + σxy |e|ns from which it becomes apparent that we can extract both the mobility as well as the density of a two-dimensional electron gas by measuring the longitudinal and transverse resistivities. This corresponds to the classical Hall effect, where a magnetic field perpendicular to a conductive sheet is used to extract its carrier density. Samples in a Hall bar geometry are investigated in chapter4, and such magnetic field dependent measurements are used in order to extract the sheet density and mobility.

2.1.3 Spin-orbit interaction In the discussion of the two-dimensional electron gas we have thus far assumed that the two spin species are degenerate and do not play a role in the transport behavior. However, in certain materials a relativistic effect called spin-orbit interaction can change this picture. If we assume that the potential that the electrons are moving in lacks inversion symmetry, they are subject to an electric field E. In the reference frame of the electrons a magnetic field B that is perpendicular to their velocity v and the aforementioned electric field, 1 B = − v × E, (2.17) c2 is present. This magnetic field causes a Zeeman splitting of the two spin species and thus enters the Hamiltonian of the system as an additional term gµ H = − B (v × E)σ. (2.18) SO 2c2

9 Chapter 2. Basic concepts

Here, σ denotes the vector of Pauli matrices that describes the spin. This effect is more pronounced in materials that consist of heavy elements, as their nuclei contain more charges thus contributing to a larger electric field. It has been found that in crystals which are not symmetric under inversion, such as diamond and zinc-blende structures, this electric field exists. This so-called Dresselhaus contribution due to bulk inversion asymmetry [34] gives rise to a contribution to the Hamiltonian of the form

HD = βD(σxkx − σyky). (2.19)

Here, σx and σy are the Pauli matrices and are not to be confused with the longitudinal conductivity discussed in section 2.1.2. The Dresselhaus coeﬃcient βD depends on the material and on the thickness of the quantum well. Narrow wells contribute to a large Dresselhaus coeﬃcient, whereas wide quantum wells result in a smaller one. A second way to create a spatial asymmetry in the potential is by structural inversion asymmetry, which depends on the layer sequence in which the heterostructure is grown. This adds the Rashba term [35, 36]

HR = αR(σxky − σykx) (2.20) to the Hamiltonian. The Rashba coefficient αR is composed of a material specific proportionality constant and the average electric field in z-direction. This means that the strength of the Rashba contribution to the spin-orbit interaction can be influenced by applying electric fields perpendicular to the plane of the quantum well, for instance by electrostatic gating from the top [37, 38]. Combining both contributions we receive a single-particle Hamiltonian

H = H0 + αR(σxky − σykx) + βD(σxkx − σyky), (2.21) where H0 is the kinetic energy of the electrons without spin-orbit interaction. This Hamiltonian can lead to a splitting of the spin degeneracy even at zero magnetic ﬁeld, resulting in two branches E(k), one for spin up and one for spin down.

2.2 Electrons conﬁned to one dimension: quantum point contacts

In a two-dimensional electron gas, the charge carriers are free to move in x and y directions, and exhibit quantized energy levels in the z-direction. Using electrostatic gating from the top, we can deplete electrons in such a way that a narrow constriction is formed in the electron gas. This introduces quantized energy levels in one of the in-plane directions and thereby creates a one-dimensional channel coupling the two reservoir contacts. Assuming a channel extending into the x-direction that is much longer than its cross section, we can separate the wave function of the system into a

10 2.2. Electrons conﬁned to one dimension: quantum point contacts

plane wave in x-direction and quantized states χn(y, z) perpendicular to the direction of the channel. Therefore, we write the wave function of the states in the wire as

1 ikxx ψnk = χn(y, z) · √ e . (2.22) L Here we have introduced a normalization length L that is large compared to the wavelength of the electrons. The mode index n serves as a quantum number labeling the subbands or modes of the wire. Assuming a parabolic potential and calculating the energy En(kx) of the modes as a function of the wave vector kx along the wire, we get 2k2 E (k ) = E + ~ x , (2.23) n x n 2m∗ with the subband energies En of the quantized modes in y and z direction. In order to observe transport through the one-dimensional wire, we need to leave the thermodynamic equilibrium and apply a source-drain voltage between the left and right reservoirs. The current is then given by

|e| X Z ∞ I = −g dE[f (E) − f (E)], (2.24) s h L R n En with the spin degeneracy factor gs and the Fermi-Dirac distributions of the left and right leads, fL(E) and fR(E) respectively. The applied bias voltage VSD corresponds to the diﬀerence in Fermi levels in the left and right leads according to |e|VSD = µL − µR. If this is small compared to kBT , we are able to rewrite the term in the integrand of equation 2.24 as

∂fL(E) ∂µL(E) µL(E) − µR(E) = (µL − µR) = − |e|VSD, (2.25) ∂µL ∂E which makes the integration of the energy in equation 2.24 straightforward and thus lets us write it as e2 X I = g f (E )V . (2.26) s h L n SD n Dividing this expression for the current by the applied voltage yields an expression for the linear conductance G of a quantum point contact,

I e2 X G = = g f (E ). (2.27) V s h L n SD n From this equation, the quantization of conductance that is observed in quantum point contacts becomes apparent. If the energy En of a mode is below the electrochemical potential in the source lead µL, its Fermi-Dirac distribution fL is equal 2 to one, and thus this mode adds a conductance of gse /h to the total conductance of the constriction. If on the other hand the energy En of another mode is above

11 Chapter 2. Basic concepts

the electrochemical potential, the Fermi-Dirac distribution is equal to zero, and the modes do not contribute to transport. The consequence of this is that for an en- ergetic mode spacing that is larger than the thermal broadening, we increase the 2 conductance stepwise by gse /h when changing the Fermi level within the constriction. If no magnetic field is applied, the spin degeneracy factor gs = 2 and we observe quantized steps in 2e2/h which upon the application of a magnetic field give rise to steps in e2/h as spin degeneracy is broken due to the Zeeman splitting. In chapter6 this is investigated in a gate-defined constriction in an InAs quantum well.

2.3 Electrons conﬁned to zero dimensions: quantum dots

In the previous section we treated a quantum point contact, in which the transport of electrons is restricted to one spatial dimension and is quantized in the other two dimensions. By utilizing a different geometry of the nanostructured gates which locally deplete the electron gas, we can confine the electrons in all three spatial directions, thus completely quantizing their energy levels. Such a small metallic island is called a quantum dot. Electronic transport happens only by weak tunnel coupling to the electron reservoirs on either side of the dot. A consequence of this is that the number n of electrons on the dot is well defined and thus the total charge of the dot is quantized. The electrostatic energy of the quantum dot is given by e2n2 E = , (2.28) electrostat 2C where C is the capacitance of the island. Its magnitude depends on the shape of the quantum dot, and for an island defined by gating of a two-dimensional electron gas a flat disk is a valid assumption. This leads to C = 8εrε0r where r is the radius of the disk and εr is the relative dielectric constant of the host material. The charging energy, which is equal to the change in the electrostatic potential of the dot when an extra electron is added, is e2 E = . (2.29) c C In addition to the charging energy arising due to the Coulomb interaction we also need to take into account the quantization energy of the levels in the dot. This additional mean spacing between energy levels can be determined to be 2 ∆E = ~ . (2.30) m∗r2 This single particle energy spacing is inversely proportional to the effective electron mass and the square of the radius of the island. InAs has a small effective mass, which means that we receive a relatively large single particle energy spacing compared to GaAs. For a dot of radius r = 50 nm it amounts to ∆E ≈ 900 µeV which is approximately a third of the Coulomb energy.

12 2.3. Electrons conﬁned to zero dimensions: quantum dots

The energy spectrum of a quantum dot consists of fully quantized levels while the leads that it is coupled to are metallic and ﬁlled up to their respective Fermi energies µL and µR. This situation is schematically depicted in Fig. 2.3. At zero

(a) (b) (c) μn+1 μn+1 μn+1 μL μL μR μL μn μR μn μn μR

Figure 2.3: Schematic of the energy levels of a quantum dot coupled to two metallic leads with Fermi levels µL and µR. (a) No current flows due to Coulomb blockade while no state within the dot is in resonance with the Fermi levels in the leads at zero applied bias. (b) Current flows when a level in the dot is in resonance with the leads. (c) At finite source drain bias, current flows while a level in the quantum dot is within the bias window. temperature and applied bias, current flow can only occur when a state in the dot is on resonance with the electrochemical potentials of the left and right leads. In Fig. 2.3 (a) this is not the case and no current is allowed to flow, which is known as the Coulomb blockade effect. When a plunger gate is used that capacitively couples to the quantum dot, the energy levels of the dot can be shifted and brought in resonance with the Fermi level in the leads. We show this situation in Fig. 2.3 (b) where the arrows mark that current flow is now possible with an electron hopping from the left lead onto the empty state of the quantum dot and then tunneling out into the right lead. Upon application of a source drain voltage, a bias window of size µL − µR = −eVSD is opened. This allows current to flow while a state is within this window, as can be seen in Fig. 2.3 (c). The addition energy, required to increase the number of electrons in the dot by one at fixed gate voltages, is given by

e2 µ − µ = ∆E + , (2.31) n+1 n C the sum of single particle energy spacing and the charging energy.

13 Chapter 3

Sample fabrication

3.1 Introduction

This chapter provides a detailed description the fabrication processes which have been used to manufacture the different types of samples with which the experiments shown in this thesis were carried out. We utilized semiconductor nano- and microfabrication techniques within the shared cleanroom facility FIRST located at ETH Hönggerberg. The semiconductor heterostructures that host the two-dimensional electron gases in which we realize our nanostructures are grown by our collaborators by molecular beam epitaxy (MBE). This process starts from a circular slab of crystal, in our case typically of a thickness of 500 µm and of a diameter of 2 inches, which is the substrate wafer. In the ultra high vacuum environment of the MBE machine, the heterostructure is deposited atomic monolayer by atomic monolayer onto the substrate wafer [39]. The samples used for the passivation experiments of chapter4 and for the proof of concept experiments of electrostatic mesa definition in chapter5 were grown on a GaAs substrate in the group of Werner Wegscheider at ETH Zurich. The samples for the experiments on quantum point contacts in chapter6, on single quantum dots in chapter7, and on double quantum dots in chapter8 were grown on a GaSb substrate wafer by the group of Michael Manfra at Purdue University. Details on the heterostructure compositions and layer sequences are given in the respective chapters. At the start of the fabrication process, the wafer is cleaved along the (011) and (011) crystallographic orientations into 6 mm×5.6 mm dies using a diamond scriber.

3.2 Fabrication of Hall bar devices

For the fabrication of Hall bar devices, we immediately cleave the dies into four 3 mm × 2.8 mm pieces which are subsequently processed. The freshly cleaved pieces are ﬁrst cleaned in 50 ◦C acetone for 10 minutes, and then in 50 ◦C isopropyl alcohol

14 3.2. Fabrication of Hall bar devices

(IPA) for 10 minutes before being blow dryed with nitrogen. Sonication can be applied during the cleaning process, but it has to be done carefully in short pulses especially when handling material grown on a GaSb substrate, as it is relatively brittle and pieces of the chip can be accidentally cleaved oﬀ during sonication.

3.2.1 Ohmic contact deposition The first fabrication step of a Hall bar device is the deposition of Ohmic contacts using optical lithography and electron beam evaporation. For the optical lithography, the sample has to be spin-coated with photoresist. We perform a dehydration bake of the sample on a hot plate for 120 s at 120 ◦C to remove any water that might have condensed on the surface. Subsequently, the sample is spin coated with AZ5214E, an image reversal photoresist that is used in order to get a stronger undercut for the lift-off process. The resist is spun at 3000 rpm for 3 s with a 3 s ramp and then at 5000 rpm for 60 s with a 5 s ramp. As the pieces are rectangular and fairly small, resist inhomogeneities and edge beads are severe issues during optical lithography processes. The spin-off process at low speed and an off-centered spinning using a custom made sample holder help to contain these problems. After spinning, the sample is pre-baked at 110 ◦C for 60 s. In a Karl SüssMJB3 mask aligner, the pattern of the Ohmic contacts is exposed with an intensity of 35 mJ cm−2. To reverse the image, we perform a reversal bake for 120 s at 120 ◦C on a hot plate with an ensuing flood exposure in the MJB3 of 200 mJ cm−2. The resist is developed for 30 s in Microposit MF-319 developer and then rinsed for 20 s in deionized (DI) water. The pattern is checked in an optical microscope and an additional, shorter MF-319 step can be performed in case it has not been fully developed yet. After developing, the metal for the Ohmic contacts has to be deposited. For this, the sample is introduced into a Plassys electron beam evaporator and a sequence of the metals Ge/Au/Ni/Au of thicknesses 18/50/40/100 nm is deposited on the samples that are grown on a GaAs substrate. For samples grown on a GaSb substrate wafer, we deposit the metals Ti/Ni/Au to thicknesses of 20/100/100 nm. Subse- quently, the residual metal deposited on the unexposed resist is lifted off by leaving the sample in dimethyl sulfoxide (DMSO) at 80 ◦C for 1 hour, then in acetone for 5 minutes at room temperature, and then in IPA for 3 minutes at room temperature.

3.2.2 Mesa structure definition The next step is the definition of a mesa structure by optical lithography and wet chemical etching. A dehydration bake on a hot plate for 120 s at 120 ◦C is performed after which the positive photoresist AZ1505 is spun on the sample in an off-centered fashion. The spinning speeds are 3 s at 3000 rpm with a 3 s ramp and 60 s at 5000 rpm with a 5 s ramp. The spinning is followed by a pre-bake at 100 ◦C for 60 s. For the definition of the mesa, good resolution of the optical lithography is critical. For this reason, we conduct an edge bead removal before the actual exposure. This

15 Chapter 3. Sample fabrication

process consists of exposing the area around the edge of the chip at 100 mJ cm−2 using a rectangular mask and then developing away the edge bead for 20 s in MF-319 followed by a 20 s DI water rinse. Having removed the edge bead, we can bring the sample in close contact to the mask and expose the actual structure at a dose of 35 mJ cm−2. The development is done using MF-319 for 20 s with a 20 s DI rinse afterwards. For the wet etching step we prepare an etch solution consisting of H20, C6H807 (50%), H2PO4 (85%) and H2O2 (30%) mixed in a ratio 220 : 110 : 3 : 5 by volume [40–43]. The solution is left for 10 minutes while being agitated by a magnetic stirrer before the sample is introduced. While the stirring is maintained, the sample is etched to a depth that surpasses the second barrier, which is veriﬁed with a proﬁlometer. After the etching is complete, the sample is rinsed for 20 s in DI water and then the resist is removed using acetone and IPA for 3 minutes each at room temperature.

3.2.3 Gate insulator deposition

After etching of the mesa, a gate insulator has to be deposited such that an opera- tional gate can be deposited onto the sample and also to protect the etched crystal surface from degradation and oxidation at ambient conditions. It is therefore crucial that the sample is introduced into the vacuum of a Picosun Sunale R-150B atomic layer deposition (ALD) machine immediately after the etching process and resist ◦ removal. We deposit aluminum oxide (Al2O3) at a temperature of 120 C with a typical thickness of 30 nm. This temperature is suﬃcient to anneal the metal deposited for the Ohmic contacts into the crystal and thus enables good electrical contact.

3.2.4 Gate metal deposition

In the next step, we deposit a metallic gate on top of our structure by optical lithography and electron beam evaporation. The optical lithography is performed using AZ5214E image reversal resist following the same parameters given for the lithography of the Ohmic contacts in section 3.2.1. The top gate is deposited using electron beam evaporation of Ti/Au/Au with thicknesses of 10/50/20 nm. The first two layers are deposited under straight incidence, whereas the last gold layer is deposited under an angle of 20◦. This is to ensure that the the metal covers part of the sidewall created by etching of the mesa, and thus connects the actual gate on top of the Hall bar with its lead on the etched part of the chip. The lift-off process after deposition of the top gate is again executed anaologously to the lift-off of the Ohmic contacts in section 3.2.1.

16 3.3. Fabrication of devices with an electrostatically deﬁned mesa structure

3.2.5 Contact release and bonding

The deposition of the gate insulator described in section 3.2.3 by ALD provides conformal coverage of the whole sample. This means that the metal deposited for the Ohmic contacts is at this point covered with an Al2O3 layer which needs to be removed in order to be able to contact the sample with bond wires. To release the contacts, the sample is spin-coated with image reversal resist AZ5214E as described previously in section 3.2.1. Using optical lithography according to the parameters given in the same chapter, we pattern holes on top of the Ohmic contacts. After development, the resist is hard baked for 60 s at 120 ◦C in order to better withstand the next step. We etch the Al2O3 layer using buﬀered oxide etch (BOE), a diluted hydroﬂuoric acid based etch solution, for 60 s followed by two 30 s rinsing steps in DI water. This step concludes the fabrication process of the sample itself. In order to be able to insert the device into a measurement setup it requires a packaging. For this, we glue it into a ceramic chip carrier using poly(methyl methacrylate) (PMMA) glue which is baked for 10 minutes at 120 ◦C on a hot plate. In the next step, the contact pads on the chip carrier need to be electrically connected to the Ohmic contac pads on the surface of the sample. This is done with 25 µm gold wires using a Westbond 747677E gold wire bonder. In order to prevent leakage to the substrate by physical damage from the bond head of the wire bonder, we do not perform a regular bonding process however. Instead, a bond is only performed on the chip carrier, after which the wire is cut and carefully bent in such a way that it forms an arc and gently rests on the bond pads of the sample. The wires are subsequently glued into place using EPO-TEK H20E electrically conductive silver epoxy which is cured in an oven for 20 minutes at 120 ◦C.

3.3 Fabrication of devices with an electrostatically deﬁned mesa structure

A second type of samples explored within the scope of this thesis are samples featuring an electrostatically deﬁned mesa structure by a frame gate geometry. The operating principle of such a type of device is thoroughly explained in chapter5. The development of this device and the fabrication process pertaining to it constitutes a major result of this thesis. We therefore describe this process and its intricacies in the following. We start with a cleaved 6 mm × 5.6 mm die, which will at a later step be cleaved again and thus allows us to fabricate four samples simultaneously. The die is cleaned in acetone for 10 minutes at 50 ◦C, and in IPA at 50 ◦C for 10 minutes after which it is blow dryed with nitrogen.

17 Chapter 3. Sample fabrication

3.3.1 Ohmic contact deposition

In the first step, we fabricate the Ohmic contacts by first patterning the sample with optical lithography and then depositing the metal by evaporation. The cleaned sample is subjected to a dehydration bake on a hot plate at 120 ◦C for 120 s. After the bake, the image reversal photoresist AZ5124E is spun onto the chip at at 3000 rpm for 3 s with a 3 s ramp and at 5000 rpm for 60 s with a 5 s ramp. We subsequently perform a pre-bake for 60 s at 110 ◦C on a hot plate. Off-centered spinning is not needed in this case as we are processing a larger 6 mm × 5.6 mm piece of semiconductor and thus edge beads are not as pronounced as for the chips that are a quarter of this size. The resist is then exposed with an intensity of 35 mJ cm−2 in a Karl SüssMJB3 mask aligner. We reverse the image employing a 120 s reversal bake on a hot plate at 120 ◦C and a subsequent flood exposure of 200 mJ cm−2. After completion of the exposure, the resist is developed for 30 s in MF-319 and then rinsed for another 30 s in DI water. The sample is subsequently introduced into a Plassys electron beam evaporator and the Ohmic contact metal consisting of 20/100/100 nm of Ti/Ni/Au is deposited without tilt. The residual metal is lifted off in DMSO at 80 ◦C for 1 hour, and then acetone and IPA at room temperature for 5 and 3 minutes respectively.

3.3.2 First gate insulator deposition

Once the Ohmic contacts have been fabricated, the whole sample is covered by a dielectric in order to prevent leakage to the gate which will be deposited in the next step. The sample is therefore introduced into the Picosun Sunale R-150B ALD ◦ machine and a layer of Al2O3 is deposited at a temperature of 120 C. The layer thickness was typically chosen between 15 − 30 nm and is provided in the later chapters whenever a speciﬁc sample is discussed. A thinner dielectric proved to be advantageous as it allows operation of the whole device at lower voltages, which can prevent charging of layers between the quantum well and the gates. Especially for samples with a wider GaSb capping layer, which can be populated by holes, operation voltages as low as possible are desired.

3.3.3 Frame gate deposition

For depositing a first gate layer we employ optical lithography for patterning and electron beam evaporation of the gate metal. The resist AZ5214E is used in an image reversal process with the same parameters as given in section 3.3.1. A metal layer of Ti/Au of thickness 5/45 nm which forms the frame gate that electrostatically defines a mesa is evaporated at straight incidence. Thereafter, a lift-off process as described in section 3.3.1 is conducted to remove the excess metal and resist.

18 3.3. Fabrication of devices with an electrostatically deﬁned mesa structure

3.3.4 Second gate insulator deposition

In order to fabricate a nanostructure on the inside of the electrostatically defined mesa, a second layer of gates is required which needs to traverse the first gate layer and hence must be insulated from this layer by a dielectric. We employ an ALD process analogous to section 3.3.2 and deposit a layer of Al2O3. The thickness was again chosen between 15 − 30 nm, starting from the higher value and was reduced as the process was more refined. A thinner layer brings the fine gates, which are deposited in the following, closer to the quantum well and thus allows for a sharper confinement potential in addition to lower operation voltages.

3.3.5 Contact release

At this point in the fabrication process, the Ohmic contact metal is covered by two layers of Al2O3 and the bond pads of the frame gate are covered by one layer. Therefore, we need to release these areas by wet etching of the aluminum oxide with a hydroﬂuoric acid based process. To pattern the area, optical lithgraphy with the image reversal resist AZ5214E according to the parameters of section 3.3.1 is performed. After development, a post bake of the resist at 120 ◦C for 60 s on a hot plate is necessary to harden the photoresist such that it better withstands the etching process. The sample is then dipped into a BOE solution for 50 s to completely etch both Al2O3 layers above the Ohmic contacts. This is followed by two 30 s DI water rinse steps and resist removal in DMSO at 80 ◦C for 15 minutes, then in acetone and IPA at room temperature for 5 and 3 minutes respectively.

3.3.6 Gate lead deposition

The next fabrication step is the deposition of bond pads and gate leads to the fine gates which will form the nanostructure in the center of the device. This is done using an image reversal process in optical lithography. The sample is prepared, coated with AZ5214E and exposed according to the parameters given in section 3.3.1. A metal layer forming the optical gate leads of 10/50 nm of Ti/Au is deposited at straight incidence using electron beam evaporation. The ensuing lift-off process following the steps in section 3.3.1 finishes the deposition of the gate leads. Before continuing fabrication, the 6 mm × 5.6 mm chip that we have used thus far has to be cleaved into four smaller pieces, such that we can use the subsequent electron beam lithography step to pattern distinct nanostructures on these. To protect the sensitive surface of the sample from mechanical damage during cleaving, we spin coat it with AZ1518 photoresist at 3000 rpm for 60 s and bake it at 120 ◦C for 60 s. Then, the sample is cleaved into four 3 mm × 2.8 mm pieces and the thick protective resist layer is removed in 50 ◦C acetone for 5 minutes and in 50 ◦C IPA for 5 minutes.

19 Chapter 3. Sample fabrication

3.3.7 Electron beam lithography and fine gate deposition In this last and most important fabrication step, the nanostructure determining the functionality of the device is produced. For quantum dots and quantum point contacts, the size of the gates needs to be on the nanometer scale. Therefore, the resolution of optical lithography is not sufficient any more and electron beam lithography is used. This process is similar to optical lithography in that it requires a resist to be spin coated on to the sample which is then exposed. In the case of electron beam lithography, the exposure is a direct write with a beam of focused electrons. During development, this exposed area is then washed away by a chemical used as developer. The sample is first subjected to a dehydration bake for 120 s at 120 ◦C on a hot plate and then spin-coated with PMMA of 950k molecular weight dissolved in anisole at 4.5% solid content. The sample is spun at 1000 rpm for 1 s with a 1 s ramp and then at 6000 rpm for 45 s with a 5 s ramp. This should result in a PMMA thickness of 190 nm. The sample is spun centered, as for the ebeam write any edge bead is irrelevant since the write only happens in the very center of the sample where the thickness of the resist is homogeneous. After spinning, the sample is baked on a hot plate for 240 s at 180 ◦C. The exposure is done in a Raith 150 electron beam lithography system at an acceleration voltage of 30 keV. The sample is subsequently developed in a 1 : 3 mixture of methyl isobutyl ketone (MIBK) and IPA for 60 s and then rinsed in IPA for 30 s. The metal forming the fine gates is now deposited in an electron beam evaporator. To reach a better vacuum which is beneficial for the film quality in this step, we employ a titanium pre-pump step in which 10 nm of Ti are evaporated onto the closed shutter. Following this, 5/20 nm of Ti/Au are evaporated at straight incidence and at low evaporation speed to guarantee that the thin film is homogeneous and devoid of large grains. Following evaporation, lift-off is performed as outlined in section 3.3.1 and completes the microfabrication of a sample with an electrostatically defined mesa. The last step is composed of glueing the sample into a chip carrier and establishing electrical connections from the sample to the chip carrier via careful bending and glueing of bond wires, as described for Hall bar samples in section 3.2.5.

20 Chapter 4

Passivation of edge states in etched InAs sidewalls

In this chapter, we explore the possibility of removing trivial edge conduction inherent to InAs by reversing the band bending and Fermi level pinning at the surface through a passivation process performed during fabrication. For this we are testing a plethora of different approaches. To begin with we investigate a reference process in section 4.2.1, where no chemical passivation was performed and a silicon nitride layer was deposited by plasma-enhanced chemical vapor deposition (PECVD) as a gate dielectric. A different method of growing a dielectric layer in atomic layer deposition (ALD) of aluminum oxide was examined in section 4.2.2, where also no additional chemical passivation occurred. In order to investigate how oxidized edges of a mesa behave, we measure purposefully oxidized samples in section 4.2.3. Throughout section 4.2.4 we explore ammonium sulfide and thioacetamide, two different methods of sulfur passivation. Two other chemicals which so far have not been analyzed in the context of transport experiments in literature are tested in section 4.2.5 in magnesium borohydride, and in section 4.2.6 in phenyl-C61-butyric acid methyl ester (PCBM). Ultimately, the performance of all of these different methods with respect to linear edge resistance is compared in section 4.3. Out of all methods tested, it was found that atomic layer deposition of aluminum oxdie with no further chemical treatment yielded the best results. However, this method is not sufficient to completely remove trivial edge conduction and we therefore investigate a different, edgeless sample geometry capable of completely circumventing this issue in the next chapter.

Parts of this chapter have been published in:

Passivation of Edge States in Etched InAs Sidewalls C. Mittag, M. Karalic, S. Mueller, T. Tschirky, W. Wegscheider, O. Nazarenko, M. V. Kovalenko, T. Ihn, K. Ensslin Appl. Phys. Lett. 111, 082101 (2017)

21 Chapter 4. Passivation of edge states in etched InAs sidewalls

4.1 Introduction

In contrast to the most commonly used materials in semiconductor physics, GaAs and Si, which feature a Fermi level pinned at mid-gap in the case of GaAs or a well-insulating native oxide layer in the case of Si, InAs shows a peculiar specialty. At the surface, its conduction band is bending downward and its Fermi level is pinned above the conduction band minimum, which leads to an electron accumulation [44, 45]. Recently, a double quantum well structure containing InAs and GaSb has been suggested as a quantum spin Hall insulator (QSHI) [46–48] and edge transport in etched Hall bars has been verified [49–51]. However, edge transport of a similar magnitude was also found in InAs/GaSb samples which are not in the topological phase [52–54]. The presence of this trivial edge transport and the aforementioned accumulation of charge carriers at InAs surfaces suggests that it originates from the InAs layer and is verified by observing edge transport in single InAs QWs [54]. This unwanted edge conduction can not only obscure data in the intricate InAs/GaSb material system, but is also detrimental to realizing quantum devices in InAs heterostructures, as mesa edges and structures defined by etching introduce a parallel conducting channel, and InAs nanowires, where surface accumulation is believed to drastically lower the mobility at low temperatures [55, 56]. In recent work, these problems could be controlled for field effect transistors (FETs) working in a drastically different regime, displaying promising room temperature operation of devices in industry dimensions [57–59].

4.2 Methods of passivating edge states

During the course this chapter, we investigate InAs Hall bars which display edge conduction after the bulk is pinched off by top-gating and apply different passivation techniques during fabrication with the goal of alleviating this edge conduction. The heterostructure used for this work is grown by molecular beam epitaxy (MBE) on a GaAs substrate and consists of 200 nm GaAs, followed by 6 nm AlAs, a 32 nm AlSb layer, a 1100 nm AlxGa1−xSb layer (x = 65%), a 500 nm GaSb layer, a 50 nm super- lattice of 10 iterations of 2.5 nm AlSb and 2.5 nm GaSb layers and then the 15 nm InAs quantum well sandwiched between two 50 nm AlSb barriers and a 3 nm GaSb capping layer to prevent oxidation of the barrier. Ohmic contacts (Ge/Au/Ni/Au) have been evaporated and 4 × 8 µm Hall bars were defined using standard wet etch recipes [43, 60]. Following the etching process and resist removal in acetone and isopropanol, different passivation techniques have been applied, which consisted either of utilizing a chemical and depositing a gate dielectric, or directly depositing a dielectric layer. The dielectric is always necessary to insulate the structure from the subsequently evaporated Ti/Au top gate, and can thus not be omitted. As InAs/GaSb heterostructures have widespread use in infrared detection where surface accumulation can lead to unwanted signals, different passivation techniques have been explored for optoelectronic devices [61]. A range of these are applied to

22 4.2. Methods of passivating edge states

transport measurements in the following. In the subsequent section we will explore the deposition of silicon nitride layer by PECVD, the deposition of an aluminum oxide layer by ALD, natural oxidation of the surface, two kinds of sulfur passivation, magnesium borohydride, and finally PCBM. In the following subsections we present optical microscope images and electronic transport data in order to probe the aforementioned passivation methods with respect to their success at suppression the edge conduction, their effect on the general transport behavior, and their influence on the fabrication process ensuing the passivation step. The transport data have been collected in liquid Helium variable temperature insert (VTI) at a temperature of T = 1.3 K. We measured with AC lock-in techniques at a frequency of fAC = 31.41 Hz, and chose a time constant of 100 ms, a wait time of 1 s between data points and used a filter slope of 24 db.

4.2.1 Silicon nitride by chemical vapor deposition In order to be able to properly gauge the eﬀect of any passivation method, one needs to set a reference with which to compare. For this, we fabricate a Hall bar with the parameters given in the beginning of this section and deposit 150 nm of ◦ SiNx by PECVD at a temperature of 300 C after etching the mesa and before depositing the top gate. No method of chemical passivation has been performed. This type fabrication has been perennially used as the standard gate dielectric in previous works both in our group as well as within other research groups working on heterostructures containing either pure InAs quantum wells or the very similar composite InAs/GaSb double quantum wells [49, 51, 54, 60, 62–67].

(a) (b) (c)

400 μm 100 μm 20 μm

Figure 4.1: Optical microscope images of a reference Hall bar sample with a SiNx dielectric grown by PECVD after etching and no additional chemical passivation. Images were taken at magniﬁcations of 50x (a), 200x (b), and 1000x (c).

Fig. 4.1 shows three images at diﬀerent magniﬁcations of a typical reference sample that features a silicon nitride dielectric. Due to the relatively thick dielectric layer, the wafer surface acquires a yellow hue. Throughout all three images, one can see the granularity of the etched wafer surface resulting from inhomogeneities in the wet etch process. We now investigate electronic transport behavior through the Hall bar in order to characterize its behavior. In Fig. 4.2 we plot the longitudinal resistance Rxx

23 Chapter 4. Passivation of edge states in etched InAs sidewalls

Figure 4.2: Longitudinal resistance Rxx of a Hall bar sample with a SiNx dielectric layer and no chemical passivation as a function of the top gate voltage VTG. Past the kink at VTG = −3 V, the bulk is depleted and conduction happens via trivial egde states.

measured as function of the voltage VTG applied to the top gate. For positive VTG, we continuously populate the quantum well with more electrons, thereby increasing its mobility which in turn leads to a decreasing Rxx. When charging the top gate with negative voltages, we push electrons out of the quantum well, depleting it of carriers and thus increasing its resistance, which is reflected in the strongly increasing Rxx in the range from VTG = −1 V to VTG = −3 V. It is expected that once the quantum well is fully devoid of charge carriers, it enters a completely insulating regime. Reflecting this behavior, the resistance should assumes a value in the MΩ to GΩ range. In stark contrast to this expectation, one can see in Fig. 4.2 that past VTG = −3 V, the curve of Rxx shows a kink and saturates at Rxx = 7 kΩ, after which it is only very weakly affected by the gate. A resistance of a few kΩ still corresponds to a reasonably good conductor and signals that there must be a different path along which the current can flow once the quantum well is fully depleted. We know from Corbino disk measurements [68] on the same material that the quantum well can be pinched off fully. Therefore, only the bulk is probed and no edges are involved, which means that this residual conductance must stem from the edges. A behavior as it can be seen in Fig 4.2 is the hallmark of trivial edge conduction stemming from charge carrier accumulation at the edges due to Fermi level pinning and band bending. As expected, depositing silicon nitride on top of the sample after etching does not influence the edges in any way other than presenting a physical protection layer preventing further oxidation at ambient conditions and acting as a gate dielectric, thus preventing leakage. To check whether the passivation treatments which will be applied in the following have any effect on the bulk transport characteristics of our InAs quantum well we calculate the sheet electron density ns and the electron mobility µe and plot these as functions of VTG in Figs. 4.3 (a) and (b). The density decreases with decreasing VTG, as is expected. Around VTG = −3 V, which is exactly the point where Rxx starts

24 4.2. Methods of passivating edge states

(a)(a) (b)

Figure 4.3: (a) Density ns and (b) mobility µe of a Hall bar sample with a SiNx dielectric layer and no chemical passivation as a function of the top gate voltage VTG. to saturate in Fig. 4.2, the density shows an unphysical bend upwards with another bend and a plateau afterwards. This behavior is caused by the fact that the density is obtained from Hall measurements, that means from the longitudinal and transverse resistances of the Hall bar in low magnetic fields, where it is assumed that the carriers are moving in the bulk of the sample and are subject to a Lorentz force due to the magnetic field. It fails when the transport happens via one-dimensional edge modes and is reflected by abnormalities in the Hall density, such as kinks, bends, or oscillations. From a fit to the regime in which the density linearly depends on VTG, an estimate for the point at which the quantum well is expected to be depleted can be determined, and it corresponds to the value at which VTG kinks and begins to saturate in Fig. 4.2. One should also note that due to the high thickness of 150 nm of the silicon nitride film, relatively large voltages are required to achieve meaningful physical changes in the device, for instance one needs to apply 4 V to the top gate −12 −2 −12 −2 in order to change the density from ns = 0.5 × 10 cm to ns = 1 × 10 cm . The electron mobility in Fig. 4.3 (b) continuously decreases as a function of VTG and reaches zero at the point at which the quantum well is expected to be fully depleted.

4.2.2 Aluminum oxide by atomic layer deposition In this section, we again do not apply any chemical passivation technique, but instead investigate the inﬂuence of changing the dielectric layer employed after wet etching. We employ atomic layer deposition of a 40 nm Al2O3 layer at a temperature of 150 ◦C. Fig. 4.4 shows optical microscope images of a sample featuring such a dielectric layer. Integration into the fabrication process is straightforward, as the deposition is isotropic and shows high uniformity. Again, roughness in the etched surface of the sample is clearly seen, and due to the thinner layer as compared to silicon nitride, the turquoise and grey surface of the wafer retains its original color.

25 Chapter 4. Passivation of edge states in etched InAs sidewalls

(a) (b) (c)

400 μm 100 μm 20 μm

Figure 4.4: Optical microscope images of a Hall bar sample with an Al2O3 dielectric grown by ALD after etching and no additional chemical passivation. Images were taken at magniﬁcations of 50x (a), 200x (b), and 1000x (c).

Figure 4.5: Longitudinal resistance Rxx of Hall bar samples with an Al2O3 dielectric layer and no chemical passivation as a function of the top gate voltage VTG. The resistance of the edge conduction past pinch-oﬀ of the bulk is orders of magnitude higher than for the reference samples with a silicon nitride dielectric.

Measuring electronic transport through this Hall bar sample, we depict Rxx as function of VTG in Fig. 4.5. There, we show traces for two types of samples passivated with ALD Al2O3. Sample B, which was left in air for approximately one hour during chemical passivation of other samples and was intended as a reference sample for comparison, and sample A, where the same process was repeated without any waiting time. Both theses samples display a longitudinal resistance increasing to MΩ after depletion of the quantum well, which is surprising as one would naively expect the water precursor of the ALD process to further edge oxidation. After pinch-off, the resistance slightly increases with decreasing gate voltage and develops more noise. These fluctuations do not reproduce exactly in subsequent measurements. They lead us to believe that the carrier accumulation at the edge is sufficiently reduced such that transport is close to breaking down. The behaviors of sample A and B are almost identical and from this data, we therefore recognize that the oxidation process must be happening on timescales slower than one hour, as the oxides forming on the surface of etched InAs structures are expected to be conductive [61].

26 4.2. Methods of passivating edge states

(a)(a) (b)

Figure 4.6: (a) Density ns and (b) mobility µe of a Hall bar sample with an Al2O3 dielectric layer and no chemical passivation as a function of the top gate voltage VTG.

In Fig. 4.6 (a) and (b) we determine and plot the electron density and mobility of sample with an Al2O3 dielectric layer. As the behavior of samples A and B is very similar, we decide to just show the data corresponding to device A for clarity. The density linearly decreases as a function of VTG and shows unphysical spikes past the point at which one expects the quantum well to be depleted from a linear ﬁt of the density for positive gate voltage. This point corresponds to the kink in Rxx in Fig. 4.5. Due to the thinner layer of the aluminum oxide layer as compared to the silicon nitride layer in the previous section, the capacitance is much higher and one can operate the devices at lower voltages, for instance the voltage needed to change −12 −2 −12 −2 the density from ns = 0.5 × 10 cm to ns = 1 × 10 cm is 1 V as compared to 4 V in the case of silicon nitride. Lower voltages means less risk of breakthrough and also less danger of being able to charge any layer inbetween the quantum well and the gate and should thus always be desireable. An additional advantage of the Al2O3 layer can be seen in the mobility presented in Fig. 4.6 (b). It shows the expected behavior, decreasing with more negative top gate and reaches zero for the case of a fully depleted quantum well. The mobility at zero gate voltage however, is slightly higher than for the sample with a silicon nitride gate dielectrict, and due to the more eﬀective gate one can reach higher densities and in turn also higher mobilities.

4.2.3 Oxidation Once a sample of a heterostructure containing a quantum well is etched, independently whether a wet chemical etching process or a dry reactive ion etching process was chosen, the sidewalls are exposed and not protected by the surrounding crystal material. From this point on they are subject to atmospheric conditions and an oxidation process is bound to start, as it is physically impossible during fabrication to go from etching to covering the etched surfaces without any exposure to air or

27 Chapter 4. Passivation of edge states in etched InAs sidewalls

chemicals. Even the solvents needed to remove the photoresist mask after successful etching contain a ﬁnite amount of oxygen. It is susupected that oxidation of the edges could be a possible reason for trivial edge conduction, yet there are no reports where oxidized edges have been speciﬁcally investigated. We therefore etched a sample and purposefully oxidized it at cleanroom air atmosphere during the course of two weeks before continuing fabrication in order to investigate oxidized edges. Due to the advantages of the thinner Al2O3 gate dielectric grown by ALD mentioned in the preceding section, we chose aluminum oxide as a gate insulator. The appearance of the samples is similar to the ones seen in Fig. 4.4.

Figure 4.7: Longitudinal resistance Rxx of a Hall bar sample with intentionally oxidized edges covered by an Al2O3 dielectric layer as a function of the top gate voltage VTG. As expected, past bulk pinch-oﬀ transport is enabled by the conductive edge states.

In Fig. 4.7 the longitudinal resistance Rxx is depicted as a function of VTG. We again recognize the behavior of trivial edge conduction, apparent from the sudden kink in previously monotonically increasing Rxx while decreasing VTG. For this oxidized sample, the saturation occurs around Rxx = 15 kΩ, which is a factor of two higher than for the sample covered by silicon nitride, but orders of magnitude lower than the samples that were passivated by aluminum oxide directly after etching. We can therefore confidently state that oxidized edges display trivial edge conduction. Additionally, from the results of the previous section, we recognize that the oxidation process is happening on timescales slower than one hour, since samples A and B in section 4.2.2 showed almost identical behavior, but faster than two weeks. The shapes of the electron density and mobility curves in Fig. 4.8 as a function of VTG are qualitatively very similar to the curves of the samples passivated with aluminum oxide immediately after the wet etching process. There are minor quani- tative differences, such as a generally slightly lower mobility, but it is not clear what the influence of oxidation of the eges on this bulk quantity is. This difference could very well be a sample specific variation, and the systematics of it have not been investigated in the course of this work, nor do we deem it a particularly interesting topic for further studies.

28 4.2. Methods of passivating edge states

(a) (b)

Figure 4.8: (a) Density ns and (b) mobility µe of a Hall bar sample with intentionally oxidized edges covered by an Al2O3 dielectric layer as a function of the top gate voltage VTG.

4.2.4 Sulfur passivation The most widespread technique in the literature that has been used to wet chemically passivate InAs surfaces is sulfur passivation, using either ammonium sulﬁde [69], thioacetamide (TAM) [70] or 1-octadecanethiol (ODT) [71, 72]. The general idea of this method is the removal of any oxide present on the surface and the ensuing creation of covalent bonds to a layer of sulfur adatoms, thereby reducing the density of surface states. This is supposed to make the surfaces chemically more stable, preventing re-oxidation at the surface [72]. These methods have so far mostly been applied to nanowires [71, 72], well-deﬁned crystal surfaces [69, 70], and InAs/GaSb infrared photodetectors [61].

(a) (b)

100 μm 100 μm Figure 4.9: (a) Optical microscope image at a magnification of 200x of a Hall bar sample passivated with TAM and an Al2O3 gate dielectric. (b) Optical microscope image at a magnification of 200x of a Hall bar sample passivated with ammonium sulfide and an Al2O3 gate dielectric.

In this section, we explore two of the aforementioned sulfur passivation methods - TAM and ammonium sulﬁde - with respect to their inﬂuence on etched Hall bars and the trivial edge conduction present in these. For the TAM passivation, we dissolved ◦ TAM in H2O at a concentration of 0.2 mol/l, at a temperature of 60 C and at a

29 Chapter 4. Passivation of edge states in etched InAs sidewalls

pH of 2 which was adjusted by adding acetic acid, following the recipe given by Petrovykh et al. [70]. After mesa etch and solvent-based resist removal, the samples were immersed in the TAM solution. The time of the dip was varied up to 15 minutes in a multitude of runs and was found to neither inﬂuence the resulting data nor the visual appearance of the sample. An image of a typical sample passivated with TAM and subsequently covered by an Al2O3 gate dielectric can be seen in Fig. 4.9 (a). The sample looks completely regular, has not been damaged or compromised in any way by the TAM dip, and no residues of the solution are visible on the surface. The thin threads or strands that vary from being almost colorless to dark blue in color are not caused by the TAM dip but are photoresist residues that appared during fabrication of the top gate.

Figure 4.10: Longitudinal resistance Rxx of a Hall bar sample passivated with TAM and an Al2O3 dielectric layer as a function of the top gate voltage VTG.

Measuring transport through a Hall bar structure passivated with TAM, we find Rxx to depend on VTG as depicted in Fig. 4.10. Normal 2DEG behavior is observed for positive gate voltages, and for negative gate voltages one can see that Rxx exhibits two kinks where the slope changes before finally saturating at a value of around 5 kΩ. This is the lowes resistance observed thus far, already hinting at the fact that this type of passivation might not be very effective. It is also noteworthy that despite the thin Al2O3 gate insulator which was used, a high absolute magnitude of gate voltage was needed in order to reach bulk depletion, setting in at VTG = −5 V. A likely interpretation of this experimental observation is that there are now more charge carriers available which need to be depleted before the bulk is devoid of carriers and transport occurs via trivial edge states. Validating this hypothesis, we turn to Fig. 4.11, where the density and mobility in the sample passivated with TAM can be seen as a function of VTG. In Fig. 4.11 (a) one can see that the density at zero applied top gate voltage is now above 1 × 1012 cm−2 whereas in an untreated quantum well the nominal density was determined to be −12 −2 ns = 0.55 × 10 cm . This confirms the hypothesis that there are more electrons present in a Hall bar which has been passivated with TAM than in an unpassivated

30 4.2. Methods of passivating edge states

(a) (b)

Figure 4.11: (a) Density ns and (b) mobility µe of a Hall bar sample passivated with TAM and an Al2O3 dielectric layer as a function of the top gate voltage VTG. An enhanced density at zero top gate voltage is visible. one, postulated based on the behavior of the longitudinal resistance. The electron mobility at zero applied gate voltage has not been compromised strongly when compared to an unpassivated sample, yet its behavior as a function of VTG is strongly irregular. The mobility decreases with increasing gate voltage, despite the fact that the density increases, and the mobility shows a peak around VTG = −1 V. One possible reason for such a behavior could be that at these high densities, a second subband within the quantum well becomes populated, thereby posing an additional scattering channel. This was the subject of further investigation, as it is falls outside of the goal of this chapter, which is removing the trivial edge conduction by means of chemical passivation. From the present data, it becomes clear that TAM is not a suitable method to achieve this goal.

Figure 4.12: Longitudinal resistance Rxx of a Hall bar sample passivated with ammonium sulﬁde and an Al2O3 dielectric layer as a function of the top gate voltage VTG.

For the passivation with ammonium sulﬁde, samples were dipped in a 20 % (NH4)2S

31 Chapter 4. Passivation of edge states in etched InAs sidewalls

solution at room temperature for times up to 15 minutes. This chemical is extremely aggressive, etching away the Hall bar and parts of the heterostructure at this concentration and immersion time. Reducing the dip time and diluting the solution further (up to 1 : 4) in H2O did alleviate this problem slightly, but could not remove it fully. This is visible in the bottom half of the optical microscope image in Fig. 4.9 (b) where a part of the heterostructure (dark blue) has been lifted off and shifted downwards by a few µm. It could not be determined which part precisely has been shifted around, but it is known that the AlSb barriers are soluble by water and we therefore assume that they have been dissolved in the ammonium sulfide solution. Hall bars in which parts of the mesa structure was compromised resulted to be nonfunctional, as their transport behavior did not depend on gate voltages and magnetic fields in a way characteristic for two-dimensional electron gases. Fig. 4.12 depicts data from one of the few sampels subject to the lowest concentration of ammonium sulfide that could be electrically measured, and we plot the longitudinal resistance as a function of VTG. Over the whole range in gate voltage, which again is strongly increased for a sample with an Al2O3 gate dielectric, the resistance was found to only vary from 350 Ω to 550 Ω. This indicative of the fact that the sample never reaches a regime where the bulk is pinched off.

(a) (b)

Figure 4.13: (a) Density ns and (b) mobility µe of a Hall bar sample passivated with ammonium sulﬁde and an Al2O3 dielectric layer as a function of the top gate voltage VTG.

Taking a look at the density as a function of VTG, presented in Fig. 4.13 (a) we −12 −2 conﬁrm this. The overall density is strongly enhanced, surpassing ns = 5 × 10 cm for almost the whole gate voltage range, which is an order of magnitude above the nominal quantum well density. This asserts the behavior seen in the Hall bars passivated with TAM and shows that sulfur passivation introduces a strong doping to the bulk, strongly increasing its electron density. The electron mobility, depicted in Fig. 4.13 (b), stays mostly constant as a function of VTG and is strongly reduced compared to the value reached in samples that have not been treated with ammonium sulﬁde. Considering all these pieces of evidence, we can safely say that the sulfur pas-

32 4.2. Methods of passivating edge states

sivation, as tested here, was unsuitable for our purpose. A feature common to both methods of sulfur passivation was the strong doping eﬀect, leading to a highly increased bulk density. We assume that this is related to the fact that during passivation, sulfur atoms can covalently bond to indium atoms of the quantum well, or even to aluminum atoms in the barriers. They thereby replace arsenic or antimony atoms, but in comparison contain one additional valence electron. If these additional electrons are now mobile enough, they can move into the quantum well and become trapped by the conﬁnement potential in growth direction, acting as a dopant from the edge.

4.2.5 Magnesium borohydride Motivated by the failure of the sulfur passivation techniques, we explore two other methods of passivation intended to act in a similar way, but taking carriers away from the edge instead of supplying a surplus. The ﬁrst of these methods, which were suggested by our collaborators from the inorganic chemistry department at ETH, Prof. Maksym Kovalenko and Olga Nazarenko, is passivation with magnesium borohydride (Mg(BH4)2).

(a) (b) (c)

400 μm 100 μm 20 μm

Figure 4.14: Optical microscope images of a Hall bar sample passivated with Mg(BH4)2 with an Al2O3 dielectric grown by ALD after passivation. Images were taken at magniﬁ- cations of 50x (a), 200x (b), and 1000x (c).

This chemical is intended to reduce native and other oxides after wet etching and to then terminate the surface with Mg2+, thereby acting as a compensation doping for the surplus of negative charge carriers at the surface. The Mg(BH4)2 was dissolved in tetrahydrofuran (THF) in a glovebox at argon atmosphere, into which the samples had been introduced before. The samples were then dipped into the solution for two minutes at room temperature and at 60 ◦C respectively. No inﬂuence of the temperature could be determined between these two runs. After passivation, the samples were taken out of the glovebox and transferred back into the cleanroom, where they were immediately introduced into the ALD machine in order to grow the Al2O3 gate dielectric. This treatment also preserved the sample quality, as can be seen on the optical microscope pictures of a sample passivated with Mg(BH4)2 in Fig. 4.14. The only visual change of the treatment is a slight red and purple hue on the surface of the sample.

33 Chapter 4. Passivation of edge states in etched InAs sidewalls

Figure 4.15: Longitudinal resistance Rxx of a Hall bar sample passivated with Mg(BH4)2 and an Al2O3 dielectric layer as a function of the top gate voltage VTG.

In Fig. 4.15 we plot the longitudinal resistance as a function of VTG for a sample passivated with magnesium borohydride. For positive gate voltages we observe regular behavior of a two-dimensional electron gas being accumulated with carriers, and for negative gate voltages the resistance starts to increase strongly, but then shows a pronounced kink and saturates at a value of Rxx = 8 kΩ. This behavior has been seen before and looks very similar to unpassivated samples with a silicon nitride dielectric. This leads us to believe that with Mg(BH4)2, we were not able to achieve the desired eﬀect of compensation doping of the edge, thereby stripping it of carriers that could contribute to trivial edge conduction.

(a) (b)

Figure 4.16: (a) Density ns and (b) mobility µe of a Hall bar sample passivated with Mg(BH4)2 and an Al2O3 dielectric layer as a function of the top gate voltage VTG.

Characterizing the bulk properties of this sample further, we show the sheet density in Fig. 4.16 (a) and the mobility in Fig. 4.16 (b) as a function of VTG. Similar to the longitudinal resistance, there are no diﬀerences to an unpassivated sample. In contrast to sulfur passivation, which seemed to be capable of enriching the system

34 4.2. Methods of passivating edge states

with carriers that we presume to be stemming from the extra electron which sulfur carries with respect to the group V elements within the crystal, our Mg(BH4)2 treatment did not have an eﬀect on the bulk carrier density. Consequently, the mobility also corresponds to the nominal value for the wafer used and depends on the gate voltage in a regular fashion for this type of gated Hall bar device.

4.2.6 PCBM Another chemical which was chosen to supply extra holes at the surface and thus trap charge carriers responsible for trivial edge conduction is PCBM. This chemical, which is a fullerene derivate and contains a C60 buckyball bound to an organic molecule, is a known electron acceptor in organic photovoltaics, and could therefore also act as an acceptor for our types of charges.

(a) (b) (c)

400 μm 100 μm 20 μm (d) (e) (f)

Figure 4.17: Optical microscope images of a Hall bar sample passivated with 5 µL PCBM in (a-c) and with 30 µL PCBM in (d-f). An Al2O3 dielectric was grown by ALD after passivation. Images were taken at magniﬁcations of 50x (a,d), 200x (b,e), and 1000x (c,f).

Performing a dip was not possible for PCBM. Therefore, using a pipette, a toluene solution of PCBM was dropped onto the sample lying on a hotplate in the glovebox. This technique unfortunately resulted in very uneven spatial distribution and the formation of small droplets, which can be seen in Fig. 4.17. On the sample shown in the top row (a-c), an amount of (5 µL) was dropped. On the sample shown in the bottom row (d-f), an amount of (30 µL) was dropped. As becomes apparent immediately, the PCBM is not distributed equally, but rather aggregate in large clusters. They made subsequent cleanroom fabrication unfeasible, as they can be in the way of the gate leads, making those break apart as for the lower Hall bar in Fig. 4.17 (b). There was only one sample containing the lowest amount of PCBM dropped on which a working gate could be fabricated, albeit with very limited tuning range.

35 Chapter 4. Passivation of edge states in etched InAs sidewalls

Figure 4.18: Longitudinal resistance Rxx of a Hall bar sample passivated with 5 µL PCBM and an Al2O3 dielectric layer as a function of the top gate voltage VTG. Due to diﬃculties in the fabrication process, the range in gate voltage is strongly reduced.

The longitudinal resistance of this sample can be seen in Fig. 4.18 as a function of VTG. Probably due to fabrication difficulties that are a consequence of large PCBM aggregates close to the top gate, it could only be operated without leakage in a range from VTG = −1.5 V to VTG = 0.6 V. The longitudinal resistance in this range just slightly increased as a function of the gate voltage from Rxx = 200 Ω to Rxx = 600 Ω. This is a sign that the gate modulation is not enough to fully deplete the quantum well, and thus the effect of PCBM on the trivial edge states can not be fully understood. However, the severe fabrication difficulties by themselves make PCBM in this form an unsuitable method of passivation for this type of experiment.

(a)(a) (b)

Figure 4.19: (a) Density ns and (b) mobility µe of a Hall bar sample passivated with 5 µL PCBM and an Al2O3 dielectric layer as a function of the top gate voltage VTG. Both quantities seem to be unaﬀected by the PCBM passivation in the gate voltage range that could be investigated.

In Fig. 4.19 (a) we show the density and in (b) the mobility as a function of VTG.

36 4.3. Comparison of diﬀerent passivation methods

Both are very limited in tuning range but correspond to the nominal values around zero gate voltage for the wafer from which the samples were taken. From this we cannot see any eﬀect of the PCBM treatment on the bulk transport properties of our Hall bar sample.

4.3 Comparison of diﬀerent passivation methods

In Fig. 4.20 we aggregate the previously collected transport data in order to compare the corresponding methods of passivation. The longitudinal resistance Rxx of Hall bars passivated with the passivation methods explained in detail in the previous sections, is plotted as a function of effective density neff . This quantity has been calculated by determining the relation between electron density ns in the quantum well and the top gate voltage, VTG, from Hall effect measurements, and extending the scale linearly to negative gate voltages. We want to emphasize that only neff ≥ 0 has a physical meaning corresponding to the density in the quantum well and negative values merely provide a renormalized gate voltages axis which is independent of different capacitances and charge redistributions. From Corbino disk measurements in InAs it is known that the quantum well can be completely depleted by a top gate, driving the resistance in the bulk of the device to arbitrarily high values, whereas for Hall bars with etched edges it saturates in the kΩ range [54]. A successful passivation is therefore expected to either completely suppress the edge conduction, facilitating insulating behavior after quantum well pinch-off, or at least increase their resistance to a value high enough to unambiguously observe ballistic h transport phenomena in InAs-based material systems, i.e., Redge e2 for typical edge lengths. Turning to Fig. 4.20, we can immediately distinguish two different families of curves. For samples passivated with TAM, Mg(BH4)2, SiNx and the oxidized sample, Rxx saturates between 3 − 14 kΩ. This allows us to draw the following conclusions: firstly, our measurements are consistent with the results of Ref. [54], where edge conductance of the same order of magnitude was measured in PECVD SiNx overgrown Hall bars. Secondly, the presumption that oxidized samples display edge conduction is correct, and its magnitude is similar to samples which have been overgrown by PECVD directly after etching, and therefore have not been purposely oxidized. Here it is important to mention that the exact dynamics of the oxidation process are not well understood, which means that edges could already start to ox- idize during resist removal necessary after wet etching and the transfer time to the vacuum chamber for dielectric deposition. Thirdly, neither the sulfur passivation with TAM nor the Mg(BH4)2 dip led to a significant change in transport behavior after pinch-off. The two types of samples fabricated without chemical passivation and ALD Al2O3 display a resistance increasing to MΩ after depletion of the quantum well, which is surprising as one would naively expect the water precursor of the ALD process to further edge oxidation. After pinch-off, the resistance slightly increases

37 Chapter 4. Passivation of edge states in etched InAs sidewalls

Figure 4.20: Longitudinal resistances Rxx of Hall bars passivated by different techniques as a function of effective density neff , obtained by extending a linear fit of the gate voltage to density characteristic to negative voltages. Positive values of this quantity correspond to the density in the quantum well, while negative values provide a renormalized gate voltage axis allowing comparison of devices with different gate dielectrics and thicknesses. ALD Al2O3 samples shown are passivated directly after wet etching (A) and with one hour of waiting time (B). Adapted from Ref. [73]. with decreasing gate voltage and develops more noise. These fluctuations do not reproduce exactly in subsequent measurements. They lead us to believe that the carrier accumulation at the edge is sufficiently reduced such that transport is close to breaking down. Sample B, which was left in air for approximately one hour during chemical passivation of other samples and was intended as a reference sample, surprisingly first showed enhanced resistance. After this observation, the process was repeated without waiting time (sample A), displaying a further increase in resistance. Therefore, as mentioned in section 4.2.3 we recognize that the oxidation process is happening on timescales slower than one hour, but faster than two weeks. The different transport behavior can be seen by comparing the oxidized sample, represented by the black curve in Fig. 4.20 with the blue and green curves corresponding to the samples immediately overgrown with aluminum oxide after etching. Finally, the data of the PCBM passivated sample (cyan curve in Fig. 4.20) shows a tuning range limited by gate leakage which occured due to the severe fabrication difficulties described in section 4.2.6. Out of four devices, only one functioned, rendering PCBM unsuitable as a method of passivation. While the data in Fig. 4.20 show the resistance across the whole device, it has been verified by nonlocal transport measurements that the current flows along the edge for gate voltages below pinch-off, and that there are no other bulk leakage paths that the current can take after the quantum well is pinched off. Using a resistor network model shown in Fig. 4.21 (b), it is possible to calculate a specific edge resistivity λedge. There are three assumptions on which this model is

38 4.3. Comparison of diﬀerent passivation methods

(a) (b)

V2 V3

λL12 λL23 λL34

V1 V4

λL61 λL56 λL45

V6 V5 Al2O3 A SiNx TAM

Al2O3 B Mg(BH4)2 oxidized

Figure 4.21: (a) Bar chart comparing specific edge resistivities λedge of different passivation −1 techniques. All methods investigated in this paper display λedge < 2.5 kΩ µm , with the −1 exception of ALD Al2O3, where λedge reaches 72.8 kΩ µm . (b) Resistor network model used to calculate the specific edge resistivity. Adapted from Ref. [73]. based. Firstly, there needs to be a complete quantum well pinch-off. We are certain that this can be achieved for sufficiently negative voltages as it has been checked independently in Corbino disk measurements, as mentioned previously. Secondly, there is current flowing entirely along the edges. This assumption has been verified in nonlocal measurements and is also the case for our sample. The third assumption is that the edge resistance scales linearly with edge length, which is expected for a diffusive conduction path along the edge and is confirmed in our measurements. The specific value of λedge depends on the point chosen for analysis in the region where the quantum well is pinched off. This is because the edges can be gated weakly, resulting in a finite slope after depletion of the quantum well, which can be seen in Fig. 4.20. For consistency we have evaluated the edge resistivity at the point neff = 0 for all samples. The comparison of λedge for all feasible passivation techniques used is displayed in Fig. 4.21 (a), where λedge of the ALD Al2O3 passivation exceeds all other passivation techniques by more than an order of magnitude. We acknowledge this experimental observation, but have thus far lacked a straightforward physical explanation, which is why we take a closer look at the chemistry taking place at the surface during the ALD process in order to gain insight into the surprising outperformance of this method compared to dedicated passivation techniques.

The role of the ALD Al2O3 process in controlling the surface states of etched InAs heterostructures can be rationalized as follows. The trimethylaluminum, Al(CH3)3, precursor used in the ALD process can act as a powerful reducing agent, which is known as “self-cleaning” [74]. As an example, Al(CH3)3 has been shown to remove native oxides from the surface of GaAs [75–77]. There are also reports on surface reduction of InAs and InAs/GaSb heterostructures with Al(CH3)3 [78–81]. Reduction of native oxides is essentially already complete after the ﬁrst Al(CH3)3 pulse in the

39 Chapter 4. Passivation of edge states in etched InAs sidewalls

ALD process [82]. After the native oxides are reduced, only Al2O3 forms as it has a much lower Gibbs free energy of formation (−377.9 kcal/mol) [83] as compared to all other possible oxides on the surface of InAs (the Gibbs free energies of In2O3, As2O3, As2O5 are −198.6 kcal/mol, −137.7 kcal/mol, −187.0 kcal/mol) [79]. This strong chemical reducing capability of Al(CH3)3 under ALD-conditions has been reported also for other systems [84, 85]. A possible reason for the ineffectiveness of sulfur passivation could be the fact that, as a group VI element, it provides an additional electron and can act as a donor, which in a different context has been reported to increase band bending at the surface [86, 87]. This is also true for the related group VI element selenium [88]. For transport applications, this effect seems to outweigh the advantages of termi- nating etched surfaces reported in optoelectronics [61]. This motivated the use of Mg(BH4)2 and PCBM to counter-dope at the surface, which, disregarding fabrication difficulties for PCBM, was not effective. Possible reasons could be too low or too high concentrations or usage of an unsuitable chemical, and would need to the subject of further, more detailed, study.

4.4 Conclusion

In conclusion, different methods to passivate edge states in transport through etched InAs quantum well structures have been investigated, of which deposition of Al2O3 by means of ALD directly after etching without any additional chemical passivation has been determined as the most effective technique. This method was capable of achieving specific edge resistivities of up to λedge = 72.8 kΩ/µm. While edge conductance could not be removed completely, we hope that our findings are sufficient to stimulate research and applications in InAs and InAs-based materials and contribute to discussions on edge states in these systems. The maximum edge resistivity achieved here is not sufficient for applications such as quantum dots or quantum point contacts, where narrow gates would need cross a mesa structure with a triv- ially conducting edge. For applications however, where the edge resistivity does not 2 need to be much larger than h/e , simple overgrowth with ALD Al2O3 could be a feasible method of containing trivial edge conduction. As an outlook for further efforts in passivation, finding a suitable chemical is still an open question. Another feasible option could be the passivation by MBE overgrowth with a related material of larger band gap, such as a quaternary alloy [89]. In order to realize quantum devices and nanostructures, a different approach is needed and will be presented in the next chapter: the fabrication of a mesa structure without a physical edge and without any etching steps relying purely on electrostatic definition of the whole structure.

40 Chapter 5

Circumventing trivial edge conduction by electrostatic mesa deﬁnition

Surface Fermi level pinning in InAs does not only lead to the accumulation of charge carriers at the surface of the sample [45], but also to their accumulation at the edges. Such edges are a result of standard semiconductor fabrication processes used to define mesa structures in many different semiconductor materials, most prominently in GaAs for instance. In InAs however, such fabrication techniques are not feasible for realizing nanostructures, as the electrons accumulated at the edges lead to trivial edge conduction which in turn is responsible for a parasitic leakage channel. In the previous chapter, the possibility of reversing this charge carrier accumulation and band bending by means of chemical passivation methods during fabrication was explored. However, the resulting linear resistance of the edge is not sufficient to enable the implementation of standard mesa etching techniques for realizing nanostructures in InAs. In this chapter, we therefore introduce a new approach of defining a mesa purely by electrostatic gating instead of relying on a physical etching process. In the first section 6.1, the gating architecture and general working principle of the frame gate are introduced. Thereafter, the frame gate itself is characterized in section 5.2. The proof-of-principle demonstration of a quantum point contact is provided in section 5.3, which is followed by the proof-of-principle operation of a quantum dot in section 5.4.

Parts of this chapter have been published in:

Edgeless and Purely Gate-Deﬁned Nanostructures in InAs Quantum Wells C. Mittag, M. Karalic, Z. Lei, T. Tschirky, W. Wegscheider, T. Ihn, K. Ensslin Appl. Phys. Lett. 113, 262103 (2018)

41 Chapter 5. Circumventing trivial edge conduction by electrostatic mesa deﬁnition

5.1 Introduction

In chapter1 InAs was introduced as a semiconductor material of strong spin-orbit interaction, low effective mass and large g-factor, for which it has gained recent interest. It arises, for instance, from proposals for investigating topological phenomena. One-dimensional InAs nanowires combined with superconductors are expected to be a host system for Majorana physics [27, 28, 90–92], anticipated to be a platform for fault-tolerant topological quantum computation [30, 93]. Composite quantum wells of InAs and GaSb were proposed to show the quantum spin Hall effect [46, 48– 51, 66], thus offering another avenue towards achieving this goal, which is however currently complicated by technological difficulties. For potential future applications of topological quantum computation which require upscaling to a large number of qubits, it would be advantageous to start from a two-dimensional structure [94–96], which would severely simplify integration. Therefore, developing functional nanostructures made from InAs two-dimensional electron gases is a natural starting point. Achieving the fabrication of reliable Coulomb islands or nanostructures which do not suffer from any kind of trivial leakage channels is paramount for these kinds of experiments.

(a) (b) 2DEG

E E EF EF E E C x C x Figure 5.1: Schematic cross sections and band edge diagrams of an etched InAs Hall bar. (a) InAs two-dimensional electron gas portrayed by the red dashed line and conﬁned by the edges of an etched mesa (top), and the corresponding band edge diagram (bottom). (b) Same structures as in (a), but with a gate (yellow) on top charged with a negative voltage, thus depleting the two-dimensional electron gas. Due to the pinned Fermi level at the surface, electrons accumulate in the triangular potential wells at the edges, which are shaded red in the band edge diagrams. This leads to trivial edge states depicted by the red dots in top panel. Adapted from Ref. [97].

Yet, up to date, convincing nanostructures in InAs two-dimensional electron gases have not been realized with the exception of a few attempts with trench-etched quantum point contacts [98–100]. These kind of structures usually do not enable a complete pinch oﬀ, and as explicitly mentioned by Koester et al. in Ref. [98] residual conduction probably stemming from surface charges is generally subtracted from the data. We believe that the reason why such nanostructures have not been realized so far lies in the trivial edge conduction recently investigated in more detail in InAs [73, 101] and InAs/GaSb quantum wells [52–54]. This conduction persists after

42 5.1. Introduction

the bulk of the two-dimensional electron gas has been depleted and can neither be eliminated by more negative gate voltages [54] nor by means of chemical passivation during sample fabrication, as demonstrated in chapter4. In InAs nanowires, exquisite control over single and multi-dot systems has been reported [102, 103]. The nanowires are grown using vapor-liquid-solid growth and have very clean surfaces with particular crystallographic orientations [55]. This is likely to result in favorable surface chemistry when compared to surfaces created by mesa-etching of two-dimensional InAs quantum wells, which are known to be etched isotropically, suﬀering from high surface roughness as well as redeposition of etching residues as well as potential partial oxidation [40, 41, 43]. For InAs two- dimensional electron gases, the Fermi level is pinned in the conduction band at the surface [44, 45]. The origin of this Fermi level pinning is not clear yet, but possible reasons are discussed in Refs. [101] and [52].

(a) (b) 30 nm Al2O3 30 nm Al2O3 3 nm GaSb 50 nm AlSb 15 nm InAs 50 nm AlSb

Figure 5.2: (a) Schematic view of the cross-section of the heterostructure and the fabricated layer sequence. Ohmic contacts are colored in dark brown, the frame gate in bright yellow, and the fine gates in dark yellow. (b) Schematic top view of the sample showing the lateral structure consisting of Ohmic contacts in an exemplary 4-point configuration, the rectangular frame gate, and fine gates defining a quantum dot. The color code is the same as in (a). Adapted from Ref. [97].

The eﬀect of this Fermi-level pinning is illustrated in Fig. 5.1 (a) and (b). The top panel of Fig. 5.1 (a) shows a schematic cross-section of a sample with an etched mesa hosting an InAs two-dimensional electron gas, and in the bottom panel one can see its corresponding band edge diagram as a function of the real space coordinate x. The edges of the etched mesa now constitute a new surface, where the Fermi level is pinned in the conduction band. When the bulk of the two-dimensional electron gas is populated by carriers, this does not cause any problems as the sample is conductive everywhere in the plane of the two-dimensional electron gas. However, if one now adds a gate onto the structure as depicted by the yellow stripe covering the top of the sample in Fig. 5.1 (b) and energizes the gate with a negative voltage, thus pushing the Fermi level into the band gap in the bulk, the situation changes.

43 Chapter 5. Circumventing trivial edge conduction by electrostatic mesa deﬁnition

While the bulk of the sample is now depleted, as resembled by the absence of the dashed red line representing the two-dimensional electron gas in the upper panel of Fig. 5.1 (b), the Fermi level remains pinned in the conduction band at the edges. This causes electron accumulation [44, 45] in the triangular potential well created at the edges [indicated in red in Fig. 5.1 (b)]. When measuring transport through such a structure (into the plane of Fig. 5.1), these accumulated electrons form a parasitic channel which conducts in parallel. As the edges are not or only weakly affected by the gate [52, 54, 104], this leakage channel will persist even if the gate drives the fabricated nanostructure, such as a quantum dot or quantum point contact for instance, into pinch-off. The standard semiconductor nanofabrication approach is therefore not feasible in this case. In this chapter, we propose a gate geometry that circumvents the parasitic edge conduction. We use a heterostructure containing a 15 nm InAs quantum well with 11 −2 5 2 an electron density ns = 5.5 × 10 cm and a mobility of µe = 1 × 10 cm /Vs in-between two 50 nm AlSb barriers with a 3 nm GaSb capping layer. The heterostructure was grown on a GaAs substrate and it corresponds to the same wafer as used for the passivation experiments in chapter4, thus the material composition is identical. The layer sequence explained in the following is depicted in the schematic cross- section of the device shown in Fig. 5.2 (a) and in the schematic top view displayed in Fig. 5.2 (b). While a detailed description of the fabrication methods and recipes is given in chapter3, we shall give a brief summary of the necessary steps to fabricate a frame gated sample. This is aimed to help the reader in understanding this chapter by itself, without constantly needing to consult another chapter for fabrication details. In a first step, Ohmic contacts of (Ge/Au/Ni/Au) were deposited. Then, a 30 nm dielectric layer of Al2O3 was deposited by atomic layer deposition at a temperature of 150 ◦C. At the same time this temperature anneals the metal pads deposited on the wafer surface before the atomic layer deposition step, resulting in Ohmic contact to the two-dimensional electron gas. In the next step we deposited a frame-shaped gate of Ti/Au (10/70 nm) on the outside of the Ohmic contacts, which will be referred to as frame gate in the following. The frame gate is paramount to circumventing the trivial edge conduction. Upon depletion of the electron gas underneath, the inner part of the sample containing the Ohmic contacts without a physical edge or surface at which electrons could accumulate, is decoupled from the outside part.

The deposition of a second 30 nm Al2O3 film by atomic layer deposition allows for depositing fine Ti/Au (5/25 nm) gates by electron beam lithography and metal evaporation. These gates form the desired nanostructure on top of the sample. In Fig. 5.3, the finished sample is shown. Figure 5.3 (a) is an optical microscope image of the whole sample at a magnification of 50x, where one can see the frame gate with two bond pads on the left and right side, and 10 Ohmic contacts on its inside. On the outside of the frame gate are the bond pads for the nanostructure gates, from which thin leads meander over the frame gate towards the center of the structure.

44 5.2. Characterization of the frame gate

(a) (b) (c)

500 μm 20 μm 400 nm Figure 5.3: (a) Optical microscope image taken at a magniﬁcation of 50x of a sample similar to the quantum dot sample used in this chapter. (b) Optical microscope image at a magniﬁcation of 1000x showing a zoom into the red rectangle in (a). (c) Scanning electron micrograph of the nanostructure forming the quantum dot. The area roughly corresponds to a zoom into the red rectangle in (b). Adapted from Ref. [97].

A zoom into the red box taken with an optical microscope at a magnification of 1000x in Fig. 5.3 (b) shows the gate leads to the nanostructure, which can be seen on the scanning electron micrograph of Fig. 5.3 (c). The nanostructures measured in this chapter are a quantum point contact consisting of a split gate with 200 nm separation, described in section 5.3 and a quantum dot formed by a semicircular top gate and three finger gates forming a right and left barrier and a plunger gate, described in section 5.4. The fabrication process proved to be reliable across multiple runs, and two samples with different nanostructures are shown in the following.

5.2 Characterization of the frame gate

The purpose of the frame gate is to completely partition the electron gas into an inner and an outer part, which are electrically decoupled and separated by a region in which the quantum well is completely depopulated by carriers and thus turns insulating. Thereby, this is similar to a mesa etch in a sense that it defines a precise geometric area which gets contacted and on which experiments, either in the bulk of the two-dimensional electron gas or on additionally defined structures, are performed. The crucial difference to a mesa etch however is that the wafer surface stays pristine, and no additional surface states leading to trivial edge conduction are created. The necessary precondition for this enterprise to be successful is the knowledge that the material can be fully depleted and turned insulating when no edges are involved. Otherwise, if no full depletion can be reached electrostatically, a leakage current between out- and inside of the frame gate would persist, and the efforts were in vain. A measurement technique which can elucidate whether this is the case or not is the Corbino disk geometry [68], in which the current between two contacts in the shape of concentric rings with an overlapping gate inbetween is measured. This configuration is therefore able to only probe the bulk contribution, with no contribution from edges. Previous results from our group on samples fabricated from the same wafer have demonstrated that indeed full pinch-off of the bulk

45 Chapter 5. Circumventing trivial edge conduction by electrostatic mesa deﬁnition

can be achieved [54]. We can therefore fulfill the necessary conditions to operate a frame gate sample as designed. One should note that our sample structure in principle is an adaption of a Corbino disk geometry, where the gate is very thin and rectangular, and there are multiple smaller contacts on the in- and out-side of the gate, instead of having the whole in- and outside be a singular contact, thus leaving room for additional gates on top. Additionally, it is necessary that the outside of the frame gates features at least one Ohmic contact, because the unetched electron gas there always needs to be at a defined potential. Otherwise, if it were floating it could charge up over time and lead to frame gate leakage or other undesired effects. During intended normal operation of the device, the outside of the frame gate should be kept on ground.

(a) (b)

Figure 5.4: (a) Conductance G between two contacts on the in- and outside of the electrostatically deﬁned mesa measured as a function of the voltage VFG applied to the frame gate. (b) Two-terminal resistance R measured between the same two contacts on the in- and outside of the frame gate. Adapted from Ref. [97].

We ﬁrst characterize the frame gate in order to show that it completely discon- nects the two-dimensional electron gas bulk from the boundaries. These measurements have been done at a temperature of T = 1.3 K using AC lock-in techniques at a frequency of fAC = 31.41 Hz, with a time constant of 100 ms, a wait time of 1 s between data points and at 24 db ﬁlter slope. We apply a bias voltage between a contact on the outside part of the two-dimensional electron gas [which is not shown in the optical microscope image in Fig. 5.3 (a)] and the contacts on the inside. The results can be seen in Fig. 5.4 (a) where we plot the conductance and in Fig. 5.4 (b) where we plot the resistance as a function of VFG, the voltage applied to the frame gate. At VFG = −0.8 V, the conductance drops to zero as the two-dimensional electron gas underneath the frame gate is fully depleted. This voltage agrees with the expected depletion voltage taking into account the capacitance and electron density of the structure. At the same voltage, the resistance increases rapidly up to 107 Ω, which was the maximum detectable resistance in our measurement setup. From this result we deduce that the regions of the two-dimensional electron gas in- and outside

46 5.3. Proof of principle operation of a quantum point contact

the frame gate are suﬃciently decoupled from each other to realize nanostructures only using the inner part of the frame gate, as if it were an etched mesa in standard semiconductor fabrication schemes. Together with a full pinch-oﬀ in a quantum point contact measurement which is provided in the next section this proves that our gate geometry circumvents the parasitic trivial edge conduction.

5.3 Proof of principle operation of a quantum point contact

In order to investigate whether our eﬀorts on circumventing the trivial edge conduction come to fruition, we conduct a proof of concept experiment by testing whether a split gate structure aimed at realizing a quantum point contact, pinches oﬀ the electron gas completely. The sample used for this measurement can be seen in Fig. 5.5.

(a) (b)

20 μm 500 nm Figure 5.5: (a) Optical microscope image of a sample featuring seven sets of split gates in series. The separations range from 100 nm to 400 nm. The image was taken at a magniﬁcation of 1000x. (b) Scanning electron micrograph of a split gate with a separation of 400 nm.

The optical microscope image in Fig. 5.5 (a) shows seven sets of split gates of varying separations between the two gates ranging from 100 nm up to 400 nm, as designed in the electron beam lithography mask. The width of the gates is 1 µm, as thinner gates have been found to not completely pinch off the electron gas by themselves in these samples. It is worth noting that the gates studied in chapters6, 7, and8 are a lot thinner and nonetheless are capable of pinching off completely. We attribute this to a few potential causes: firstly, the wafer material that was used in these upcoming chapters features a much shallower quantum well (21.5 nm below the surface as compared to 53 nm here), and the Al2O3 dielectric layers are also thinner (2x 15 nm for the sample in chapter8 compared to 2x 30 nm used here). This means that the gates are a lot closer to the quantum well and therefore have a larger capacitance, allowing them to have a stronger influence on the quantum well at similar voltages. Secondly, the material used in these later chapter is also of a higher mobility at a slightly lower density. One can expect the lower density

47 Chapter 5. Circumventing trivial edge conduction by electrostatic mesa deﬁnition

material to be depleted more easily, and the higher mobility means there are less defects in the material which could cause a hopping-like conductance through an otherwise depleted region. Additionally, especially when operating a single and double quantum dot, multiple gates which are spatially very close are energized at the same time, leading to an eﬀectively wider gate and therefore enabling a pinch- oﬀ at more positive voltages. In Fig. 5.5 (b), a scanning electron micrograph of one of split gates with a designed separation of 400 nm and an actual separation after fabrication of 370 nm can be seen. All split gates are designed in the same geometry of a wide rectangular base with two sharp triangular tips facing each other.

Figure 5.6: Conductance G through a quantum point contact with a width of 200 nm as a function of the voltage VQPC applied symmetrically to the split gates. Full pinch-oﬀ can be reached, but plateaus or steps of quantized conductance are absent while a multitude of resonances induced by disorder is visble. Adapted from Ref. [97].

We measure the quantum point contact formed by split gates with a separation of 200 nm in a 4-terminal geometry using AC lock-in techniques at T = 1.3 K. The lock-in frequency, time constant, wait time and filter slope are the same as in the measurements in the previous section 5.2. In Fig. 5.6 we show the conductance G as a function of the voltage VQPC applied to both quantum point contact gates simultaneously. Full pinch-off can be reached at VQPC = −1.95 V, which demonstrates that there is no parasitically conducting channel present underneath the gates and thereby validates the achievement of our primary goal for this type of structure. Going one step further than a pure proof of concept demonstration, we acknowledge that neither quantized conductance steps nor any reminiscence of a 0.7-anomaly are visible. The data shows a plethora of resonances which are caused by disorder in the channel and spoil the observation of quantum transport phenomena. It is worth noting that the relatively high temperature of T = 1.3 K should thermally smoothen the background disorder potential and is therefore known to help achieve better plateaus, as compared to dilution refridgerator temperatures. In the case present here though, this thermal averaging does not seem to be sufficient for the strength of the disorder potential in the channel.

48 5.3. Proof of principle operation of a quantum point contact

Figure 5.7: Color plot of the conductance G through a pair of split gates with a separation of 200 nm as a function of the voltages VQPCL and VQPCR applied to the two gates forming the constriction. By shifting the quantum point contact in real space, the disorder can not be circumvented.

To further characterize this pair of split gates and the disorder present in the channel, we laterally shift the location of the constriction in real space by changing the voltages applied to the two gates forming this constriction, VQPCL and VQPCR. The resulting color plot of the conductance as a function of these two voltages is shown in Fig. 5.7. There has been a shift in the magnitude of the applied voltages of around −0.7 V between this measurement and the line trace in Fig. 5.6. The general behavior however is unchanged, as one can see in conspicuous features such as the reentrant feature shortly before pinch-off around VQPC = −1.85 V in Fig. 5.6 now present at VQPCL = −2.6 V and VQPCR = −2.6 V in Fig. 5.7. This is likely due to charging of one of the layers between the quantum well and the gates, and this effect appears combined with strong hysteresis between up- and downsweep and temporal instability in these devices. The most likely candidate for a layer that can be charged is the 3 nm wide GaSb capping layer, as it is wide enough that the quantized energy states for holes can be populated at sufficiently high gate voltages, thus acting as a narrow hole quantum well. The holes trapped in this well would then screen the effect of the gates, requiring more negative gate voltages in order to sufficiently deplete the InAs quantum well below. As more charge accumulates over time, this effect aggravates to the point that one risks a breakthrough of the dielectric by further increasing the voltage applied to the gates. This effect can be reset by thermal cycling the sample, but it is only a matter of time until its reappearance. Other than the charging effect and shift in the gate voltage range, one can see that the resonances do appear in both gates and are also affected by both gates as they wiggle around in Fig. 5.7 with varying slopes with respect to the left and right gates. One can therefore assume that resonances and thus localized defects are present all throughout the channel as well as underneath both split gates and are not solely caused by one gate featuring an unlucky clustering of defects. We expand upon

49 Chapter 5. Circumventing trivial edge conduction by electrostatic mesa deﬁnition

this type of experiments in chapter6, demonstrating full quantum point contact behavior utilizing a sample containing a strongly diﬀerent heterostructure but the same frame gate approach, proving that the aforementioned issues such as lack of quantized steps, high disorder, and instabilities, do not arise from the approach used here, but from within the heterostructure.

5.4 Proof of principle operation of a quantum dot

In this section we investigate another type of sample with a frame gate, containing fine gates for forming a quantum dot as depicted in the scanning electron micrograph in Fig. 5.3(c). In this image one can see the semicircular top gate and three gates on the lower half: the left barrier gate, the plunger gate in the middle, and the right barrier gate. In between these, a metallic island is formed. The two barrier gates are intended to tune the tunnel coupling to the reservoirs on the left and right side, and with the plunger gate one can change the electron occupation on the dot. The subsequent measurements were performed in a dilution refrigerator with a base temperature of T = 60 mK using AC lock-in techniques in a 2-terminal configuration. The measurements in this section were performed at a frequency of fAC = 77 Hz, using a time constant of 30 ms, a wait time of 50 ms between data points and 12 db filter slope In a first step, we demonstrate that also the narrower quantum dot gates fully pinch off the two-dimensional electron gas. For this purpose, we apply a voltage Vall to all four quantum dot gates at the same time and operate the device as a quantum point contact. As seen in Fig. 5.8 (a) the electron gas can be pinched off in this geometry. No plateaus of quantzied conductance are visible, which is to be expected given the geometry of the nanostructure. As was the case for the split gate measurements in the previous section, strong resonances caused by disorder appear on the conductance curve. We now tune the gates such that we form a small metallic island in the region between the gates. We apply a voltage VPG to the plunger gate to change the occupation of the quantum dot while measuring its conductance G. The result of this measurement is displayed in Fig. 5.8 (b). Sharp, evenly spaced conductance resonances indicate charging the quantum dot with single electrons. On top of these resonances, there is a smooth background of the order of 0.05 × e2/h that decreases with more negative VPG. This means that there is still a finite current flowing through the dot when it is in the Coulomb blockade regime, indicating very open tunnel barriers. Past VPG = −1.75 V, the evenly spaced resonances disappear and more irregular peaks emerge. These are likely resonances through localized states and defects visible when the barriers are almost closed due to the cross-capacitance of the plunger gate. Further characterizing the dot we show in Fig. 5.9 (a), its conductance as a function of the voltages VLB and VRB applied to the left and the right barrier, respectively,

50 5.4. Proof of principle operation of a quantum dot

(a) (b)

Figure 5.8: (a) Conductance G through the quantum dot as a function of the voltage Vall applied to all four quantum dot gates simultaneously demonstrating quantum point contact like operation of the gates leading to full pinch-off. (b) Coulomb resonances visible in the conductance G through the quantum dot as a function of VPG. Each peak corresponds to the charging of an individual electron onto the dot. Adapted from Ref. [97]. in order to elucidate the tunability of the system. Charging both gates to more negative voltages we pass multiple Coulomb resonances represented by stripes of high conductance in between two regions of near zero conductance, indicating that electrons are being expelled from the dot. This is due to cross-capacitance, as with more negative voltages the tunnel barriers are not only changing the coupling but also have a lever arm on the energy levels in the dot. These resonances do not have a uniform slope but wiggle around and show pronounced bends, indicating charge rearrangements as well as temporal instabilities, as has already been observed in the previous chapter. The overall conductance decreases when both voltages are more negative, which is a sign of closing tunnel barriers and therefore showcases tunable operation of the quantum dot. In Fig. 5.9 (b) we show a color plot of the quantum dot differential conductance while varying the applied DC source-drain bias voltage VDC on the x-axis in addition to VPG on the y-axis. Coulomb blockade diamonds are faintly visible as regions of zero conductance periodically spreading from VDC = 0. From their extent in VDC we can determine a charging energy Ec ≈ 1 meV. This is an approximate value, as the outlines of the Coulomb diamonds are not very sharp and they increase in size for more negative VPG. To confirm whether this agrees with estimations based on the self-capacitance C of the island, we calculate

e2 e2 ∆Ec = = , (5.1) C 8εrε0r where r is the radius of the metallic disc within a medium of relative permittivity εr that constitutes the quantum dot [1]. Assuming a radius of our dot of r = 150 nm we determine from equation 5.1 a charging energy of ∆Ec = 0.99 meV, corresponding

51 Chapter 5. Circumventing trivial edge conduction by electrostatic mesa deﬁnition

(a) (b)

Figure 5.9: (a) Quantum dot tunability shown by varying the voltage applied to both tunnel barriers in a color plot of the conductance G as a function of VLB and VRB. (b) Color plot of a ﬁnite bias measurement displaying the conductance G through the quantum dot as a function of the source drain bias VDC and VPG. Coulomb blockade diamonds emerge from zero bias. Adapted from Ref. [97].

to the value extracted from the data presented in Fig. 5.9 (b). From the size of the quantum dot and the electron density in the wafer we estimate the number of electrons in the dot according to

N = ns · A, (5.2) with A = πr2 the area of the dot. Applying equation 5.2 we estimate the number of electrons on the dot to be N = 389. Neither excited states nor signatures of single particle levels are visible, which is related to wafer stability and independent of the technique of gate-deﬁning a mesa. Combining the measurements of Figures 5.9 (a) and (b) we determine the lever arms of the gates constituting the quantum dot to be αPG = 0.05 for the plunger gate, αLB = 0.08 for the left barrier, and αRB = 0.06 for the right barrier, respectively. These values are within expectations considering the distance of the two- dimensional electron gas from the gates and the geometry of the structure. This indicates that the quantum dot is formed in the center of the lithographically deﬁned area, and not purely by disorder in one of the barriers, which is also supported by a mean free path le = 1 µm larger than the size of the dot. While size-quantized excited states and other features typical of quantum dots in technologically more advanced two-dimensional electron gas systems such as GaAs or Si are not evident from the data, a proof-of-principle operation of such a quantum dot structure in an InAs quantum well has not been demonstrated before.

52 5.5. Conclusion

5.5 Conclusion

In conclusion, we demonstrated in this chapter a technique which circumvents parasitic trivial edge conduction in InAs quantum wells and enabled proof of concept operation of nanostructures such as quantum point contacts and quantum dots. This was achieved by separating the sample edge from the bulk of the two-dimensional electron gas electrostatically with a continuous rectangular frame gate partitioning the electron gas in an inner and outer part instead of applying physical mesa etching techniques. Quantitative analysis of the nanostructure conductance was however still severely limited by disorder in the material system. This issue can be overcome by improving material quality [33], which is independent of the device geometry introduced here and will be demonstrated in the ensuing chapters. There, the same gating geometry is used to deﬁne and operate quantum point contacts in chapter6, to form single quantum dots in chapter7, and to couple two of such dots to double quantum dots in chapter8. The frame gate could in principle also be applied to other narrow-band material systems that suﬀer from similar undesired edge conduction or Fermi-level pinning issues.

53 Chapter 6

Gate-deﬁned quantum point contacts

In InAs quantum wells, the observation of quantized conductance in point contacts has been hampered by trivial edge conduction at etched edges of InAs devices, leading to parallel conductive channels. In the previous chapter, a new sample geometry was introduced which circumvents this issue by employing purely electrostatic confinement without the need for physical etching. A sample using such a frame gate was studied in various proof of principle experiments throughout chapter5. How- ever, due to limitations in sample quality, it was not possible to observe quantized conductance. In this chapter, we employ the same geometry on samples grown on a GaSb substrate, which exhibit cleaner transport behavior with respect to mobility and elastic mean free path. We show three steps of quantized conductance and study the energy level spacings at a finite bias in sections 6.2 and 6.3. In a parallel magnetic field as discussed in section 6.4, the previously spin-degenerate modes show a Zeeman splitting into conductance steps of integer multiples of e2/h. Under the influence of a parallel magnetic field, orbital effects come into play in addition to pure spin splitting and we characterize the resulting magnetoelectric subbands in section 6.5. Furthermore we analyze the influence of shifting the constriction in real space and the influence of changes in the surrounding potential landscape in sections 6.6 and 6.7 before briefly commenting on the absence of the 0.7 anomaly in our sample in section 6.8.

Parts of this chapter have been published in:

Gate-Deﬁned Quantum Point Contact in an InAs Two-Dimensional Electron Gas C. Mittag, M. Karalic, Z. Lei, C. Thomas, A. Tuaz, A. T. Hatke, G. C. Gardner, M. J. Manfra, T. Ihn, K. Ensslin Phys. Rev. B. 100, 075422 (2019)

54 6.1. Introduction

6.1 Introduction

When a ballistic two-dimensional charge carrier system is confined to a narrow constriction, a striking quantum mechanical phenomenon can be observed: the conductance is quantized in integer multiples of 2e2/h [8,9]. Such a structure, called a quantum point contact (QPC), has since its inception been demonstrated in a plethora of different materials[105–112]. InAs is a material system featuring a low effective mass and a high spin-orbit interaction, and can be contacted with ease with superconductors due to Fermi level pinning at the surface. This provoked a substantial amount of recent interest in this material for the purposes of topological quantum computation [90–92, 113]. An architecture based on a two-dimensional material system would be beneficial for scaling and integration [94, 114–116], which is why full control and understanding of nanostructures in InAs quantum wells has profound implications for future research. However, experiments on QPCs in InAs have been scarce, performed in trench-etched structures[99, 100, 117, 118], and were limited by parallel conduction [98] due to surface charge carrier accumulation inducing trivial edge conduction [44, 45]. In the following we study a completely electrostatically defined QPC in InAs without any background conductance in order to provide a detailed understanding of its energy levels, g-factor and magnetoelectric subband structure. The device was fabricated on a heterostructure grown by molecular beam epitaxy on an undoped GaSb substrate in (100) crystal orientation, as described by Thomas et al. [33]. The schematic cross-section in Fig. 6.1 shows the layer sequence, which in growth direction consists of a 25 nm GaSb layer, a 800 nm Al0.8Ga0.2Sb0.93As0.07 quaternary buffer layer, two 20 nm Al0.8Ga0.2Sb barriers below and above the 24 nm wide InAs quantum well, and a 1.5 nm InAs cap layer. The mobility of the two- 6 2 dimensional electron gas in the quantum well is µe = 1.4 × 10 cm /Vs at an electron 11 −2 density ns = 5.1 × 10 cm as measured in a Hall bar geometry at T = 1.5 K. We determine an elastic mean free path le according to µ √ l = v · τ = ~ e 2πn , (6.1) e F e e s where vF is the Fermi velocity and τe is the elastic scattering time.√ In equation 6.1 we have related the Fermi velocity to the Fermi wave vector kF = 2πns according to ∗ ∗ vF = ~kF/m and deduced the scattering time from the mobility via τe = µem /e. The resulting mean free path le = 16.5 µm is much larger than the size of the constriction we investigate in the following. In order to avoid the trivial edge conduction inherent to InAs [44, 45] we re- frain from any etching steps and employ a fully gate-defined sample geometry as introduced in chapter5 and published previously [97]. The fabrication process of a sample with a frame gate geometry is given in detail in section 3.3 of chapter3. In the schematic of the sample in Fig. 6.1, the black crossed squares represent Ohmic contacts to the two-dimensional electron gas. Of the four available Ohmic contacts,

55 Chapter 6. Gate-deﬁned quantum point contacts

two proved to be disconnected from the sample. This is likely the result of an imper- fect etching process of the Al2O3 layers during sample fabrication. The remaining two contacts were used as the source (S) and drain(D) contacts for the measurements presented in this chapter and are labeled accordingly. The frame gate (FG), which is separated from the wafer surface by an Al2O3 layer is represented by the yellow rectangular frame. Another Al2O3 layer separates it from the fine gates forming the nanostructure, which are represented by thin black lines. The nanostructure consists of 5 pairs of opposing gates with separations of 100 nm and 150 nm and can be seen in the scanning electron micrograph of a sample similar to the one used in this chapter in the inset of Fig. 6.1. The two gates colored in red are vertically offset by 150 nm and horizontally offset by 100 nm with respect to the plane of the image and were used to define the QPC investigated in this study. We refer to them as gates QPC1 and QPC2 and they are labeled accordingly in the inset of Fig. 6.1. This configuration showed the best transport data out of all combinations of gates, likely due to a locally favorable potential landscape. The surrounding gates used for investigating the influence of changes in the coupling potential are colored in yellow and are kept on ground except for the measurements in section 6.9. All subsequent measurements were performed in a dilution refrigerator

FG D S

InAs (1.5 nm) 1 AlGaSb (20 nm) B InAs (24 nm) 200 nm AlGaSb (20 nm) 2 AlGaSbAs (800 nm) GaSb (25 nm) GaSb (100)

Figure 6.1: Schematic representation of the lateral sample geometry and layer sequence of the heterostructure. On the left hand side the frame gate is depicted by the yellow rectangular frame, the Ohmic contacts by the black crossed squares, and the ﬁne gates by the thin black lines. The inset on the right hand side shows a false-colored SEM image the nanostructure of a similar sample. The red gates were used form the quantum point contact and are labeled 1 and 2, and are referred to as gates QPC1 and QPC2 in this chapter. The yellow gates are kept on ground except for the measurements in section 6.7. Adapted from Ref. [119].

56 6.2. Quantized conductance

with a base temperature of T = 60 mK using low-frequency AC lock-in techniques at a frequency of fAC = 30 Hz, with a time constant of 100 ms, 1 s wait time between data points and at 24 db ﬁlter slope.

6.2 Quantized conductance

We bias the frame gate to the voltage VFG = −0.85 V which is sufficient to deplete the electron gas underneath, thus isolating the inner part of the electron gas from the surrounding electron gas and the sample edges. A constriction can now be defined in the central region using the two fine gates colored red in the inset of Fig. 6.1 which create the QPC. We apply a fixed AC voltage bias of dVAC = 10 µV between the source (S) and drain (D) contact, measure the current dIAC and thereby determine the two-terminal differential conductance G = dIAC/dVAC.

Figure 6.2: Diﬀerential conductance G as a function of VQPC, the voltage applied to the two split gates, at a temperature of T = 60 mK and at zero external magnetic ﬁeld. Quantized conductance plateaus in steps of 2e2/h are clearly visible. Adapted from Ref. [119].

Fig. 6.2 shows G as a function of the voltage applied to the QPC gates VQPC, displaying three plateaus of conductance in steps of 2e2/h. The two QPC gates are biased slightly asymmetrically, such that VQPC1 = VQPC and VQPC2 = VQPC+300 mV. In order to assess whether the visibility of three steps is to be expected given the geometry of our sample, we estimate the number of modes observable. For this, we compare half the Fermi wavelength λF to the expected width of the QPC channel W according to n = 2W/λF [1]. For a two-dimensional system, we have p λF = 2π/kF = 2π/ns. (6.2) Inserting the density previously measured in a Hall bar of this wafer, we obtain λF = 35.1 nm. Considering a reduction of the lithographic width of the QPC due to the distance of the gates from the 2DEG, we assume an electronic width of the channel of W = 55 nm. This leads to an expected number of n = 3.13 modes

57 Chapter 6. Gate-deﬁned quantum point contacts

(where the real number of modes necessarily is an integer number) fitting into the channel, which coincides with the three steps observed in our measurement. The QPC pinches off the electron gas completely and no subtraction of a background conductance was needed, in contrast to previous experiments [98, 99]. The resistance at pinch-off is at least 108 Ω and the curves showed small hysteresis of the order of 10 mV between up and down sweep direction. We note that while the quantized conductance data in Fig. 6.2 shows good quantization on the first plateau at 2e2/h, the second and third plateaus are not exactly quantized at 4e2/h and 6e2/h anymore. In a non-interacting picture, the value of the quantized conductance on QPC plateaus can be reduced by backscattering. Weak backscattering of specific QPC modes may be caused by non-adiabatic coupling into the QPC or backscattering by disorder in the contact regions. Beyond that, time-dependent fluctuators near the QPC, or impurities leading to localized states are known to reduce the quality of plateaus and the transitions between them by producing resonance-like structure. All these effects can be mode-dependent, for example, because the different modes are subject to forward focusing with different strengths. This has been seen for instance in the branched flow through a QPC in scanning gate microscopy experiments [120, 121]. It can lead to mode-specific backscattering from impurities in the drain regions. To conclude, the reason for the difference in plateau quantization quality between the first, second, and third mode probably is related to mesoscopic details of the sample, which are hard to extract.

6.3 Finite bias spectroscopy

In order to map out the energy levels of the QPC, we perform finite bias spectroscopy by applying an additional DC bias voltage between source and drain leading to a bias voltage VDC across the QPC. This leads to an additional complication, since a part of the voltage drops across the Ohmic contacts, and therefore the voltage applied in the experimental setup will not exactly correspond to the voltage with which the QPC is biased. This effect is more pronounced when the resistance of the system under study is comparable to the contact resistance. This is the case for the QPC studied in this chapter, as its resistance can minimally be 25.813 kΩ in the regime where one spin-split mode is occupied, and the resistance of the contacts is around 10 kΩ. Therefore, the data have to be corrected to account for this voltage drop across the Ohmic contacts. This results in a new bias axis, onto which the data have to be rescaled. The procedure for doing so is outlined in the following. For this, an I-V characteristic of the contacts alone without operating any fine gates, i.e. keeping them at V = 0 has been recorded and can be seen in Fig. 6.3(a). When operating the device as intended, we assume that for the total current IDC flowing through the sample, we can write

I = Vapplied/(RC + RQPC), (6.3)

58 6.3. Finite bias spectroscopy

(a) (b)

Figure 6.3: (a) I-V characteristic of the Ohmic contacts obtained by measuring the current IDC as a function of the applied bias voltage VDC while the ﬁne gates are kept at zero voltage. (b) Contact resistance as a function of VDC. The blue curve shows the measured AC resistance and the green curve shows the numerical derivative of the DC component.

where Vapplied is the DC voltage bias applied to the sample, RC is the resistance of the Ohmic contacts, and RQPC is the resistance of the QPC. Multiplying both sides of equation 6.3 by its denominator and using Ohm’s law, we receive the simple equation

VDC, QPC = Vapplied − VDC, C, (6.4)

with the bias voltage dropping across the QPC, VDC, QPC, and across the contacts, VDC, C, respectively. Equation 6.4 now enables us to use the data of Fig. 6.3 (a) as a function to determine the bias drop across the Ohmic contacts as a function of IDC. To do so, the V -I-curve has to be interpolated for all values of IDC occurring in a given measurement, as it is a discrete data set. When recording a finite-bias measurement, the data are now treated as traces of Vapplied at fixed VQPC. For each measurement point in Vapplied, an actual VDC, QPC dropping across the QPC is then determined by using equation 6.4. Then, an equidistant axis between the smallest value of the maxima in VDC, QPC and the largest minima in VDC, QPC is created. Finally, using interpolation, appropriate current values are interpolated onto the grid given by the new, equidistant values of VDC, QPC. In the subsequent measurements in this section, whenever it is referred to an applied bias voltage VDC this value is already corrected for the bias voltage drop across the Ohmic contacts according to the procedure described above, unless stated otherwise. Fig. 6.3 (b) displays the contact resistance RC determined by measuring the AC component of the signal and by calculating the numerical derivative of the DC voltage signal. Both curves are in reasonable agreement and become indistinguishable when the DC signal is smoothed by a moving average filter before calculating its derivative, thereby suppressing the high-frequency noise component introduced by the derivative. It should be noted that the resistance is not constant as a function of

59 Chapter 6. Gate-deﬁned quantum point contacts

Level spacing Value

∆E12 2.73 meV ∆E23 2.31 meV ∆E34 2.1 meV

Table 6.1: Energy level spacings of the ﬁrst three modes of the QPC determined by ﬁnite bias spectroscopy. the bias voltage, and thus the bias compensation described here has to be conducted in order to obtain a correct gate voltage to energy conversion and one cannot simply subtract a constant contact resistance when varying the bias voltage.

2 4 6

Figure 6.4: Finite bias spectroscopy showing the transconductance dG/dVQPC as a function of VQPC and VDC. The diamond-shaped stripes of ﬁnite transconductance correspond to transitions between plateaus and are highlighted by white dashed lines, delimiting the dark plateau regions of zero transconductance which are labeled with the associated conductance values in units of e2/h. Adapted from Ref. [119].

Fig. 6.4 depicts the numerical derivative with respect to VQPC of the diﬀerential conductance dG/dVQPC, as a function of VDC and VQPC. The dark areas correspond to regions where the conductance does not change as a function of VQPC, i.e. conductance plateaus, and the bright regions mark transitions between them. The resulting diamond-shaped structure visualizes the magnitude of the mode spacings in energy, and the slope which has been highlighted by dashed white lines delivers a calibration from VQPC to energy. The extent of the diamonds in VDC direction decreases for increasing mode index n, corresponding to decreasing energy spacings of the modes. The magnitudes of the energy level spacings of the ﬁrst three modes can be found in Tab. 6.1. To assess the validity of these energies, we calculate the real space extent of the modes Ln assuming an harmonic oscillator potential of the frequency ω0, according

60 6.4. Behavior in a parallel magnetic ﬁeld

to m∗ 2 2 1 2 ω0 · Ln = ~ω0(n − 2 ), (6.5) ∗ with the electron effective mass m . Using ~ω0 = ∆Enm for the energy spacings of the respective modes determined from our previous finite bias measurements, we find the length scales as 2 ~ Ln = (2n − 1) · ∗ . (6.6) m ω0

For the ﬁrst three modes visible in this experiment, we determine L1 = 32 nm, L2 = 53 nm, and L3 = 72 nm. These length scales are within expectation considering the geometric dimensions of the constriction used in this experiment.

6.4 Behavior in a parallel magnetic ﬁeld

As a next step, we apply a magnetic ﬁeld Bk in parallel to the plane of the two- dimensional electron gas and along the transport direction of the QPC as depicted in the inset of Fig. 6.1 and investigate the resulting spin splitting of the modes. Fig. 6.5 shows the transconductance dG/dVQPC as a function of VQPC and Bk.

1 3 5

2 4 6

5 1 3

Figure 6.5: Transconductance dG/dVQPC as a function of VQPC and the magnetic ﬁeld Bk applied in plane of the two-dimensional electron gas along the direction of the QPC. The spin-degenerate conductance plateaus split according to the Zeeman eﬀect and are labeled with the associated conductance in units of e2/h. Adapted from Ref. [119].

The three diamond-shaped regions centered along a line through Bk = 0 T correspond to the spin-degenerate conductance plateaus of the ﬁrst three modes of the QPC. When increasing Bk these modes split and spin-polarized plateaus emerge as additional dark, diamond-shaped regions arranged in a regular pattern around the spin-degenerate plateaus. From the extent of the diamonds in Bk and the magnitude of their energy spacings extracted from the ﬁnite bias measurements shown in Fig. 6.4 we can determine the g-factor leading to the Zeeman splitting ∆E = gµB∆B. We

61 Chapter 6. Gate-deﬁned quantum point contacts

extract a value of |g| = 12.6, close to the bulk value g = −15 of InAs [122, 123]. Pre- vious measurements on InAs nanowire quantum dots found values of |g| = 8±1 [103] and measurements in etched QPCs in a narrow quantum well (thickness of 4 nm) determined |g| = 5.1 [100]. As this represents a substantial deviation from our results, we would like to ascertain whether the value determined in this experiment is realistic considering the much wider width of the quantum well of our sample. In order to do so, we utilize the expression 2 ∗ 2 2meP 1 1 g = 2 − 2 − 3 ~ E0 E0 + ∆0 02 (6.7) 2 2meP 1 1 + 2 0 − 0 0 3 ~ E0 − E0 E0 − E0 + ∆0 which yields a good approximation for the g-factor obtained from k · p theory [124]. In this equation, E0 is the energy gap between the Γ6C and Γ8V conduction and 0 valence bands, E0 is the energy gap between Γ7C and Γ8V bands, ∆0 is the spin- 0 orbit splitting between Γ8V and Γ7V valence bands, and ∆0 is the spin-orbit splitting 0 between Γ8C and Γ7C conduction bands [125]. P and P are momentum matrix elements corresponding to transitions between Γ6C and Γ8V + Γ7V and to transitions between Γ6C and Γ8C + Γ7C respectively [125]. To better visualize these various band edge parameters, Fig. 6.6 (a) shows a schematic band structure resulting from a 14×14 k·p calculation where the relevant bands, energy gaps, spin splittings, and matrix elements are labeled. This image results from a calculation for GaAs [125], but in the vicinity of the Γ-point, the dispersions of InAs and GaAs are fairly similar.

(a) E (b) E Γ8C EC Γ 7C Δ'0 } new P' E0 E0 Γ6C E'0

E EV 0 P z

Γ8V Δ0 }

Γ7V Figure 6.6: (a) Schematic band structure for the 14 × 14 k · p model for GaAs around the Γ point, adapted from [125]. (b) Schematic band edge diagram displaying the energies the conduction and valence bands as a function of the growth direction z for the InAs QW separated between two layers of higher band gap.

The numerical values of these band edge parameters are summarized in table 6.2.

62 6.4. Behavior in a parallel magnetic ﬁeld

Material GaAs AlAs InAs InSb InP m/m∗ 0.067 0.150 0.023 0.014 0.080 g∗ -0.44 1.52 -14.9 -51.6 1.26 E0 (eV) 1.52 3.13 0.42 0.24 1.42 ∆0 (eV) 0.34 0.29 0.38 0.82 0.11 P (eVA)˚ 10.49 8.97 9.20 9.64 8.85 0 E0 4.49 4.54 4.39 3.16 4.72 0 ∆0 0.17 0.15 0.24 0.33 0.07 P 0 (eVA)˚ 4.78i 4.78i 0.87i 6.32i 2.87i

Table 6.2: Band edge parameters of the semiconductors GaAs, AlAs, InAs, InSb, and InP. Taken from Refs. [1] and [2].

Inserting the values corresponding to InAs into equation 6.7 yields a bulk effective g-factor of g∗ = −14.75, in agreement with experimental results [122, 123]. For electrons trapped in a quantum well however, one needs to take into account that the band gaps are modified by the additional confinement energy. This energy can be readily calculated assuming a particle-in-a-box model for the quantum well of thickness d

2π2n2 E = ~ , (6.8) n 2m∗d2 where n is an integer number representing the occupied subbands. We know that n = 1, which means there is only one subband occupied given the density in the sample. For electrons, the confinement energy raises the energy level in comparison to the bulk conduction band, and for holes the quantized energy lies below the top new of the valence band, thus an effective band gap E0 is found, replacing the bulk band gap E0. This is visualized in Fig. 6.6 (b), where a band edge diagram of a quantum well sandwiched between two barrier materials of higher band gap can be seen. The first quantized levels for electrons and holes are portrayed by the blue and red dashed lines respectively. 0 We assume that the spin-orbit splittings ∆0 and ∆0 do not change when introducing the additional confining potential. Now we are able to calculate the effective g-factor g∗ for the electrons in our quantum well using equation 6.7 and taking into new account the modified energy gap E0 arising from quantization energies due to the finite quantum well width. This yields a value of g∗ = −13.58, which corroborates the experimental results. Therefore, we are able to attribute the high g-factor measured here to the fact that the quantum well investigated in this chapter is much wider than the ones used in previous experiments [100, 126].

63 Chapter 6. Gate-deﬁned quantum point contacts

6.5 Behavior in a perpendicular magnetic ﬁeld

Now we investigate the eﬀect of a magnetic ﬁeld perpendicular to the plane of the electron gas, B⊥, in order to demonstrate the magnetic depopulation of the quantized one-dimensional subbands. In Fig. 6.7 (a) the transconductance dG/dVQPC of the QPC is depicted as a function of VQPC and B⊥.

(a) (b)

Figure 6.7: (a) Transconductance dG/dVQPC as a function of VQPC and B⊥. Both magnetic depopulation and spin splitting are visible, and the white dashed lines show a fit to a model incorporating both phenomena for the first three modes of the QPC. The dark vertical stripes correspond to regions where the contacts are decoupled due to edge channels. The colored dashed lines correspond to the line cuts shown in (b). The white diamonds mark transitions between plateaus and are extracted from line cuts of the differential conductance. (b) Differential conductance G of the QPC as a function of VQPC along the dashed lines of equal color shown in (a). Spin split conductance plateaus in integer multiples of e2/h are emerging. Adapted from Ref. [119].

The three bright regions along a line through B⊥ = 0 T mark the transitions between the first three, spin-degenerate modes of the QPC. For increasing B⊥ they curve towards higher energy in a parabolic fashion and then transition into a linear slope as they merge with the Landau levels forming in high magnetic field. We can observe two parabolas emerging from a single step at B⊥ = 0 T, which are the magnetoelectric subbands [127]. They can be described by a model by Beenakker and van Houten [31] q 1 2 2 En = Eoff + (n − 2 ) · ~ ω0 + ωc + gµBB⊥, (6.9)

where ωc is the cyclotron frequency, and Eoff is an energy offset. A fit to this model is superimposed on the data as white dashed lines. The white diamonds mark transitions between plateaus and were extracted from line cuts of G as a function of VQPC. They serve as a guide to the eye in the regions where B⊥ is large and the magnitude of the transconductance is small. In order to better visualize the spin splitting occurring in perpendicular field, we show in Fig. 6.7 (b) line traces of

64 6.6. Shifting the quantum point contact in real space

the differential conductance as a function of VQPC at three different values of B⊥ depicted by the colored dashed lines in Fig. 6.7 (a). At B⊥ = 0.82 T, the plateau corresponding to the first magnetoelectric subband is not yet visible due to the broad transition to the first mode of the QPC, and the plateau at G = 3e2/h is a 2 narrow shoulder. For B⊥ = 1.2 T, clear steps in units of e /h are visible, and for B⊥ = 1.42 T and higher fields these plateaus become wider and more pronounced. The vertical black lines superimposed on the data in Fig. 6.7 (a) correspond to regions where no signal was measured due to a finite spacing between the Ohmic contacts and the edge of the frame gate. Therefore, at certain values of B⊥ where the bulk enters a fully developed quantum Hall state, the contacts become decoupled from the rest of the sample [128] and the signal vanishes periodically in 1/B⊥.

6.6 Shifting the quantum point contact in real space

In order to further characterize the QPC, we laterally shift the constriction in real space by individually changing the voltages applied to the two gates forming the QPC. The result can be seen in Fig. 6.8 where we plot the absolute value of the gradient of the diﬀerential conductance |∇G| with respect to the voltages applied to the two QPC gates, VQPC1 and VQPC2 as a function of these two gate voltages.

Figure 6.8: Absolute value of the gradient of the diﬀerential conductance |∇G| with respect to the voltages of the two gates forming the QPC, VQPC1 and VQPC2, as a function of these two voltages. Transitions between plateaus appear as arc-shaped bright lines. The vertical and horizontal lines correspond to the onset of depletion underneath gates QPC1 and QPC2 respectively. The white dashed line marks the path in gate voltage space along which all other measurements in this work were recorded. Adapted from Ref. [119].

Semicircular bright features correspond to transitions between conductance plateaus and show that they are inﬂuenced evenly by both gates. The bright horizontal and vertical lines, which only depend on VQPC1 or VQPC2 correspond to the pinch-oﬀ

65 Chapter 6. Gate-deﬁned quantum point contacts

underneath the respective gate. Impurities in the QPC can be seen as resonances in this type of plot [129], and their slopes correspond to their capacitances to the different QPC gates. When looking at imperfections which negatively influence the transport through a QPC in more detail, one has to differentiate between static disorder, which leads to a resonant structure in the energy-dependent transmission, and time-dependent fluctuations of the potential. The resonant structure and isolated peaks of the static disorder, as it has been seen for instance McEuen et al. [130], is not observed in our sample. Time-dependent fluctuators on the other hand can, despite the low temperature in our experiment, lead to smearing of the transition between two conductance plateaus, which would otherwise be sharper. In Fig. 6.8, one can see that the transition to the first plateau is strongly broadened compared to the transitions to the second and third plateaus. Additionally, there are regions in the gate voltage space spanned by the two QPC gates where the transitions are broader, and regions where they are narrower. A reason might be that there are regions in gate voltage space where time-dependent fluctuators are suppressed more strongly than in other regions. This is also why we chose to sweep the QPC gates asymmetrically along the path in gate voltage space marked by the white dashed line in Fig. 6.8, in order to curtail the effect of strong broadening of conductance steps. The onset to the first plateau, however, seems to be always broadened, no matter how one chooses to sweep the gates. This effect was already observed when sweeping the magnetic field, as discussed in section 6.5. We note that compared to GaAs QPCs with similar scattering times [8,9] the quantization of the plateaus is worse and noise is more pronounced, presumably caused by a background disorder potential. Similar conclusions have been drawn from measurements on quantum Hall states in InAs quantum wells [131]. The reason for this remains an open question and will need to be the subject of future investigation.

6.7 Inﬂuence of the coupling potential

In etched InAs QPCs, plateaus quantized at odd integer multiples of e2/h at zero external magnetic field have been reported [99, 100, 118]. We use the tunability of our structure by lateral gates to look for possible origins of such quantization, and to shine some light on the deviations of some of our plateaus from quantization at precise integer multiples of 2e2/h. By tuning the voltage on nearby gates surrounding the QPC we change the potential landscape around the QPC and thereby influence the coupling of electron waves from the contact regions into the channel. More specifically, we change the voltage Vsur applied simultaneously to the three neighboring gates of the QPC (colored yellow in the inset of Fig. 6.1) and VQPC and plot the resulting transconductance of the QPC at zero external magnetic field in Fig. 6.9 (a). For decreasing Vsur, the transitions between plateaus shift to more

66 6.7. Inﬂuence of the coupling potential

positive VQPC due to cross-capacitance, as expected. Furthermore, the onset to the ﬁrst plateau splits at Vsur = −0.15 V and a new conductance plateau arises. Other smooth changes of the conductance as a function of Vsur are evident in Fig. 6.9 (a).

(a) (b)

Figure 6.9: (a) Transconductance dG/dVQPC as a function of VQPC and Vsur the voltage applied to the ﬁne gates surrounding the QPC which are colored yellow in the inset of Fig. 6.1. As Vsur becomes more negative, additional plateaus (white regions) emerge. (b) Line cuts of the diﬀerential conductance G of the QPC as a function of VQPC along the dashed colored lines in (a) show the additional plateau-like features which are not quantized to any integer multiple of e2/h. Adapted from Ref. [119].

As examples, line cuts of the conductance at three values of Vsur corresponding to the colored dashed lines in Fig. 6.9 (a) are displayed in Fig. 6.9 (b). They show how the conductance traces are continuously deformed as Vsur changes. For example, the red trace (Vsur = −0.2 V) shows a new conductance plateau seemingly quantized at 2 1.6e /h, and the green trace (Vsur = −0.3 V) resembles a reentrant plateau feature at 2 VQPC = −1.45 V quantized at about e /h. Also the precise values of the conductance on plateaus quantized near integer multiples of 2e2/h changes slightly as a function of Vsur. These measurements exemplify how spurious plateau-like features at unconven- tional conductance values can be generated by suitably molding the potential landscape in the vicinity of the quantum point contact constriction. They show not only that adiabatic coupling of the QPC channel to the leads is of crucial importance, but also that this adiabaticity can be easily spoiled in our InAs samples. In section 6.6 we thoroughly discussed potential sources of imperfection, such as time-dependent fluctuations and static disorder, which are very local phenomena and take place on length scales of tens of nanometers. How adiabatic the coupling to the QPC is, or in other words, how smoothly the potential changes from ideally infinitely wide contact regions to a small constriction takes place on a much larger scale. The surrounding gates change the potential on both of these length scales, and it appears that both of these effects are at play. The structure which we create in our QPC conductance at Vsur = −0.3 V (green trace in Fig. 6.9 (b)) around

67 Chapter 6. Gate-deﬁned quantum point contacts

VQPC = −1.45 V resembles the resonant structure mentioned before and is an example for the influence of a local impurity. The general lowering of plateau values and the appearance of spurious new conductance plateaus for more negative Vsur might be related to a less adiabatic coupling, taking place on a much larger length scale. Therefore we may assume that changing the adiabaticity of the coupling to the QPC and changing the saddle point of the QPC potential by changing Vsur enables backscattering and resonant scattering in the vicinity of the constriction. The surrounding gates in this experiment let us turn these mechanisms on and off, and thereby show how novel plateaus, resonant and reentrant features can appear in a previously clean constriction. In comparison to fully etched constrictions, the degree of tunability provided by the additional gates around the QPC allow us to access these otherwise not attainable regimes. While adiabaticity of the coupling is related to the long-range variations of the potential, resonant structures in the energy-dependent transmission (see green curve in Fig. 6.9 (b)) can be caused by shorter range static disorder [130] in the vicinity of the constriction. Additionally, time-dependent fluctuators may lead to noise and broaden transitions between plateaus as observed for the first plateau in our sample and explained in section 6.6. Our results shown in Fig. 6.9 may be relevant for the interpretation of previously observed plateaus quantized at odd integer multiples of e2/h at zero external magnetic field [99, 100, 118].

6.8 Searching for signatures of a 0.7 anomaly

A few years after the experimental realization of the QPC it was observed that clean one-dimensional constrictions show an additional conductance plateau around 0.7(2e2/h)[132]. While this feature has been called the 0.7 anomaly it is not exactly quantized at its namesake conductance value and it displays conspicuous characteristics which are specified in the following. When applying a parallel magnetic field, the feature evolves into the first spin-split plateau at 0.5(2e2/h), suggestive of a spin-related origin of the phenomenon. Upon reducing the density of the 2DEG, the 0.7 anomaly becomes a more pronounced plateau [133]. Furthermore, it shows an unusual temperature dependence: whereas normal conductance plateaus become thermally smeared out with increasing temperature, the 0.7 structure becomes stronger [133]. The exact microscopic origin of the 0.7 anomaly has been a topic of debate for many years that is partly still going on [134, 135] with evidence gathered for a plethora of possible explanations, such as for instance spontaneous spin polarization [132, 133] or the formation of a Kondo-like correlated state [136]. When sweeping the voltage applied to the QPC gates and observing regular quantized conductance plateaus such as in Fig. 6.2 of section 6.2, no additional shoulder or plateau around 0.7(2e2/h) could be observed. A very distinctive signature of the 0.7 anomaly is its appearance as a zero bias anomaly when sweeping the source drain bias [136]. Similar measurements as performed by Cronenwett et al. [136] are

68 6.9. Conclusion

(a) (b)

Figure 6.10: (a) Differential conductance dI/dVDC of the QPC as a function of VDC while varying VQPC. (b) Zoom into area demarcated by red dashed rectangle in (a) showing the absence of a zero bias anomaly potentially corresponding to a 0.7 anomaly. In both figures the bias axis has not been rescaled. presented in Fig. 6.10, where we show the differential conductance of the QPC as a function of the bias voltage VDC. We plot multiple line traces while varying VQPC into the waterfall-type plot of Fig. 6.10. Dense regions, where many line traces are bunched together represent conductance plateaus, of which one can see the first and second modes at 2e2/h and 4e2/h respectively, as well as ”half-plateaus” at elevated bias forming at e2/h and 3e2/h in Fig. 6.10 (a). A zoom into the region below the first conductance plateau at small bias represented by the red dashed rectangle is shown in Fig. 6.10 (b). There, the traces are reasonably smooth as a function of VDC with only a slight shoulder visible around VDC = −0.5 mV on a few traces, which does not qualify as a zero bias anomaly that could be related to the presence of the 0.7 anomaly. The exact reason for the absence of this feature despite the high quality of the material at hand could not be ascertained thus far. It is important to note that all measurements presented in this chapter have been executed at a temperature of T = 60 mK. This represents a very unfavorable condition for observing the 0.7 anomaly, which usually becomes more pronounced at elevated temperatures.

6.9 Conclusion

In this chapter, we have presented measurements of quantized conductance in a fully gate-defined QPC in an InAs quantum well. In the absence of an external magnetic field, three steps quantized in 2e2/h could be observed. Upon the application of either a parallel or a perpendicular magnetic field, spin-resolved transport through the constriction could be observed. In finite bias measurements, the energy spacings of the modes could be determined. Combining this energy gauge and the magnetic field scale obtained in section 6.4, the g-factor in a parallel magnetic field could be

69 Chapter 6. Gate-deﬁned quantum point contacts

ascertained. Laterally shifting the QPC in real space and changing its coupling potential on both long and short ranges enabled the study of disorder and adiabaticity and their influence on plateau quantization. Signatures of the 0.7 anomaly could not be detected despite the high mobility of the heterostructure used, but could potentially be related to the low temperature of the measurement. A natural next step after the implementation of a QPC is the operation of more complex nanostructures such as single quantum dots, which will be presented in chapter7, or double quantum dots, which will be presented in chapter8. Further optimizing both sample design and growth with the goal of eliminating noise and fluctuations in these nanostructures is needed to pave the way towards achieving more ambitious goals such as realizing spin qubits in laterally defined InAs quantum dots. These could then enable performing qubit operations via the spin-orbit interaction similar to nanowires [137, 138] while benefiting from easier integration due to their two dimensional nature.

70 Chapter 7

Gate-deﬁned lateral quantum dot

There are thus far no reports on quantum dots in InAs two-dimensional electron gases, as their realization has been severely restricted by trivial edge conduction in structures based on etched mesas. By developing a method of defining a mesa entirely by electrostatic gating in chapter5 we were able to realize quantum point contacts which do not suffer from this issue in the previous chapter6. Building on this approach, we show that it is also feasible to realize quantum dots in InAs two- dimensional electron gases in this chapter. To begin with, we explain in section 7.3 how one tunes the structure into the regime of a few-electron quantum dot. This enables us to study Coulomb diamonds in finite bias in section 7.4 in order to obtain information on the energy scales of our quantum dot. In section 7.5 we study excited states and their behavior in magnetic fields applied in parallel and in perpendicular to the quantum well allowing us to gain insight into the energy level spectrum of the states in our quantum dot. Finally, we maintain few electron occupation on the dot but tune into a regime where it is strongly coupled to the leads, enabling the observation of the Kondo effect in section 7.6.

7.1 Introduction

A quantum dot is small island within a solid that can be populated with single electrons which are tightly confined in all three spatial directions and can thus only assume quantized energy levels [139]. In this sense, they are considered “artifical atoms” [140], where one can, just like for real atoms, investigate their energy spec- tra and ionization energies [11]. For our quantum dots, the surrounding solid is the semiconductor crystal of InAs, lending the electrons an effective mass instead of the free electron mass and modifying the dielectric permittivity. The attraction to the center of the artifical atom is not given by a positively charged nucleus as is the case for a real atom, but rather by a local potential minimum shaped by negatively charged gates surrounding the quantum dot structure [140]. Quantum dots are typically on the scale of 100 nm and thus much larger than real atoms, making it possible to explore regimes which are not accessible in atomic physics experiments.

71 Chapter 7. Gate-deﬁned lateral quantum dot

Electron-electron interactions for instance, are more important in large artificial atoms since with increasing size, the Coulomb energy decreases more slowly than the differences in orbital energies [140]. One can observe transitions between spin singlet and spin triplet ground state of a two-electron quantum dot at fields of a few Tesla [11], whereas for a real helium atom such a transition is only expected at fields that are five orders of magnitude larger [141], due to its much smaller size. Quantum dot technology is very mature for material systems like gallium arsenide and silicon, and many groundbreaking studies on the basic physics of quantum dots have been realized [12, 139]. Today, quantum dots in these two materials are so well understood [13, 142] that the challenges lie mostly in the engineering side of optimizing larger arrays [143, 144], increasing their quantum logic gate fidelities [145, 146], improving their readout [147, 148] and coupling schemes [149–151], all working towards achieving the goal of implementing one of the proposed schemes for realizing a quantum computer [6]. Due to trivial edge conduction present in InAs the realization of standard, mesa-etched quantum dots within a quantum well was not reported thus far. In this chapter, we show that it is possible to achieve realiable quantum dot behavior with a gate-defined mesa. We investigate in how far the behavior of the quantum dots is different from GaAs or Si, and whether effects due to spin-orbit coupling and the lower effective mass are present.

7.2 Characteristics of the sample used for deﬁn- ing a quantum dot

The sample used in the course of this chapter features an entirely electrostatically defined mesa with a frame gate and was fabricated according to the process described in chapter3. A key property of this sample is that 20 nm Al 2O3 layers were grown as gate dielectrics by ALD for both of the layers. These thinner gate dielectrics compared to the samples of chapters5 and6 bring along two main advantages. Firstly, the gates are effectively closer to the two-dimensional electron gas which allows the definition of smaller structures as the potential they shape in the quantum well will be sharper. A second advantage is that it allows the operation of all gates at lower voltages, reducing the risk of unwanted charging effects. The gate layout is similar to the one used for defining the quantum point contact in chapter6, shown there in Fig. 6.1. A scanning electron micrograph of the nanostructure of a similar sample is shown in Fig. 7.1. It consists of 10 finger gates shaped to form two separate quantum dots, which can be coupled to form a double quantum dot. The gates are labeled in Fig. 7.1 according to their function and this nomenclature will be used throughout this thesis. The left dot is defined by the left tunnel barriers LB1 and LB2, the left plunger gates composed of PGL1 and PGL2, and the middle barriers, MB1 and MB2. The right dot is then similarly formed in between the middle barriers, the right plunger gates PGR1 and PGR2, and the right barriers, RB1 and RB2. The gates were designed to be 40 nm wide and separated by

72 7.3. Tuning the quantum dot

PGL1 PGR1

LB1 MB1 RB1

100 nm

LB2 MB2 RB2

PGL2 PGR2

Figure 7.1: Scanning electron micrograph of a similar sample displaying the gate layout used for deﬁning a quantum dot in this chapter and a double quantum dot in chapter8. The nanostructure is composed of 10 ﬁnger gates which form left, middle, and right barriers as well as left and right plunger gates, and are labeled accordingly. a distance of 60 nm horizontally. The vertical separation was designed to be 120 nm between all of the three sets of barrier gates and 130 nm between the two sets of plunger gates.

7.3 Tuning the quantum dot

In order to characterize the device, we first verify that the frame gate is operating as intended and thereby also assess the effect of the thinner gate dielectric. The sample was cooled down in a dilution refrigerator with a base temperature of T = 60 mK. For all measurements in this chapter we were using low-frequency AC lock- in techniques at a frequency of fAC = 30 Hz and a time constant of 100 ms. We set the filter slope to 24 db and chose a wait time of 1 s between data points. To characterize the pinch-off of the electron gas underneath the frame gate, we proceed analogously to the measurements of section 5.2. A bias voltage is applied to the contacts outside of the frame gate, while the inner contacts are kept at ground and are therefore used as drain contacts. We record the conductance G between the inside and outside contacts as a function of the voltage VFG applied to the frame gate and plot the resulting data in Fig. 7.2. At VFG = −0.5 V, the conductance very sharply drops to zero indicating pinch-off underneath the frame gate. In the samples used in chapter5, the decrease in conductance was much broader and the pinch-off set in only past VFG = −0.8 V. We could therefore by employing a thinner gate dielectric successfully reach lower frame gate operation voltages and still achieve a stable, hysteresis-free, and leakage-free pinch-off of the frame gate, paving the way

73 Chapter 7. Gate-deﬁned lateral quantum dot

Figure 7.2: Conductance G between the outside and the inside of the frame gate as a function of the voltage VFG applied to the frame gate. Due to the thinner alumin oxide layer employed for this sample, pinch-oﬀ happens at a lower voltage than in previous samples.

for forming a quantum dot in this sample. After having characterized the frame gate, we set it to the voltage VFG = −550 mV which is suﬃcient to fully deplete the electron gas underneath, for all of the ensuing measurements in this chapter. We apply an AC source drain bias VAC between a source contact on the left of the nanostructure and a drain contact on the right of the nanostructure and record the current that is ﬂowing between these contacts.

Figure 7.3: Conductance G across the nanostructure while sweeping diﬀerent pairs of gates which are subsequently used to form a quantum dot. Due to the similar sizes of the separations, the depletion underneath the gates happens at the same voltage.

We ﬁrst characterize the individual gate pairs which are going to be used to form the dot. For this we individually sweep the voltages applied to the left barriers, middle barriers, and left plunger gates, employing them as quantum point contacts

74 7.3. Tuning the quantum dot

and record their pinch-off characteristics while keeping all the other gates on ground. The result can be seen in Fig. 7.3, where we show a plot of the conductance for all three gate pairs. Pinch-off underneath all gates sets in at −0.7 V, as can be seen by the sharp drop in conductance. Past this point, a more negative voltage applied to the gate electrostatically narrows the constriction formed between the split gates. The slope of all three traces in this regime is similar, indicating that their lever on closing the constriction is similar. The trace corresponding to a voltage sweep of the left barriers, colored black in Fig. 7.3, and the left plunger gates, colored green in Fig. 7.3, lie almost exactly on top of each other. The transition from pinch-off of the electron gas underneath the gates to closing the electronic constriction happens at the same value of conductance, which implies that their electronic width is roughly equal. This is a slight surprise, as the separation between the plunger gates was designed to be 10 nm larger than between the barriers. However, this is less than a 10% difference in absolute separation, and additional fabrication imperfections might explain their similar behavior. In the sweep of the left barrier gates we recognize clear steps, indicative of a quantum point contact formation. As investigating a quantum point contact is not the focus of this chapter, the quantized values of the steps were not characterized further. There is no contact resistance subtracted from this two- terminal measurement, which is why the plateaus here are not at integer multiples of e2/h. Quantum point contacts in a similar samples are discussed in detail in chapter6. The sweep of the middle barrier gates shows the transition from pinch-off under the gates to a closing constriction at a slightly higher value of conductance, indicative of a slightly larger electronic width. The middle barriers are in the center of the structure and have the largest number of metallic gates surrounding them (4 on each side). These gates are kept on ground during the measurement, and are spatially closer to the middle barriers than the quantum well, which leads to field lines bending towards these gates rather than the quantum well, and thus screening of the applied voltage. Therefore, the middle barriers are less effective at the same gate voltage compared to the outside barriers. In the measurements in Fig. 7.3, we did not drive the gate pairs to the point of fully closing the constriction. The reasons are twofold: during the characterization we did not want to risk applying too high voltages that could potentially cause charging effects, and secondly, during quantum dot operation all six gate will be charged at the same time, thus their mutual cross capacitance and the negatively charged surroundings mean that we will require lower voltages to reach pinch-off than when just driving two individual gates. Once the individual gates are characterized, we start to tune the quantum dot. For this we simultaneously energize the left and middle barriers to a negative voltage and establish the point where both together pinch off the structure. Adjusting the voltage to the point where they form symmetric barriers, we start to charge the left plunger gates to a negative voltage, thus expelling electrons off the island formed in the middle. At some point, no charging events can be detected any more, since the plunger gates also act on the barriers due to cross capacitance and will close them.

75 Chapter 7. Gate-deﬁned lateral quantum dot

Therefore, at the current plunger gate voltage, one needs to readjust both tunnel barriers. This means that another 2D plot of left and middle barriers needs to be recorded, and the voltages set to the point where they form symmetric barriers, transparent enough to observe transport through the quantum dot. At this point, we charge the plunger gates to more negative voltages, depleting the dot further until the barriers are again closed too much. These two steps are then repeated until the desired regime is reached, ideally a dot that is populated by few electrons while remaining suﬃciently strongly tunnel coupled.

Figure 7.4: Color plot of the conductance G of the quantum dot as a function of the voltages applied to the two sets of tunnel barriers, VLB and VMB, which couple the dot to the leads. Both gates have a similar lever arm on the resonances, indicating that the dot is formed in the center of the lithographically deﬁned area with respect to the left and right side.

An exemplary color plot of the conductance G as a function of the voltages applied to the left and middle tunnel barriers, VLB and VMB, recorded at the end of the tuning procedure can be seen in Fig. 7.4. In the top right corner, where both gates are at the most positive voltages, the barriers are fully opened and current can flow through the system. When charging either set of barrier gates more negatively, one pinches this respective barrier off and thus stops the current, indicated by the conductance dropping to zero. This is true for the left barrier, as when one moves horizontally away from the top right corner, the conductance drops to zero, as well as for a more negative middle barrier, moving vertically downwards from the top right corner, where the conductance also drops to zero. When closing both barriers symmetrically, we surpass two sharp resonances between regions of zero conductance. These correspond to Coulomb resonances tantamount to removing a single electron from the dot and occur because also the barriers have a cross-capacitance and thus a finite lever arm on the energy levels of the dot. These resonances do have unit slope with respect to both gate voltages, indicating that both barriers have a similar lever arm onto the dot which in turn means that the dot is most likely located in the center between the left and middle barrier. One should not be misguided by

76 7.3. Tuning the quantum dot

the diﬀerent ranges of 200 mV for the middle barrier and 150 mV for the left barrier which, combined with the aspect ratio of the color plot make the slope look a bit shallower. The absolute value of voltage required to reach pinch-oﬀ is approximately 200 mV larger for the middle barriers than for the left barriers, which is consistent with the considerations that were made when discussing Fig. 7.3.

Figure 7.5: Color plot of the conductance G of the quantum dot as a function of the voltages applied to the upper plunger gate VPGL1 and to the lower plunger gate VPGL2, which control the occupation of the dot. The slope of the resonances indicate that the dot is formed in the middle of the area deﬁned by the gates.

Showcasing the effect of the plunger gates when tuning the dot, we turn to Fig. 7.5. There, we present a color plot of the conductance G as a function of the voltages applied to the upper and lower plunger gates, VPGL1 and VPGL2. We can see two sharp Coulomb resonances which correspond to an electron being loaded or unloaded off the quantum dot. The resonances are tuned both by the upper and the lower plunger gate, which means that the dot is located in the middle between these two gates. The slight bending of the resonances is an indication that the dot can be moved in space as a function of both plunger gates, entering regions where one of the two gates is more effective than the other. Together with the information gained from Fig. 7.4, we are able to state that the position of our dot is located in the center of the lithographically defined area and that it behaves very symmetrically with respect to variations of the voltages of the surrounding gates. When simultaneously going to more positive voltages on both plunger gates, the conductance of the system starts to increase, since the plunger gates also act on the tunnel barriers due to their cross-capacitance. During the tuning procedure of the dot, one would usually not record such a map for time efficiency, but rather sweep both plunger gates symmetrically, which is equivalent to a line cut from top right to bottom left corner of Fig. 7.5. The color plot shown here is helpful to spatially map out the dot and to assess how symmetric it is in space. We note that there are only two isolated Coulomb resonances visible, and the next resonance at more positive plunger gates is apparent only in line cuts, but

77 Chapter 7. Gate-deﬁned lateral quantum dot

already superimposed by an increasing background and strongly broadened caused by the tunnel barriers opening up. For more negative plunger gates, no additional resonances can be seen, even when expanding the gate range to multiples of the peak spacing. Even through multiple retunings of the barriers, it was not possible to observe additional resonances. The reason could be that either these are the last two electrons present in the dot, or that the plunger gates always close the tunnel barriers before the next resonance becomes visible. While we cannot unambiguously show it as there is no charge detector in this device, we have good reason to believe that these resonance indeed correspond to the last two electrons, and present a multitude of arguments for this throughout the rest of this chapter in sections 7.4, 7.5, and 7.6.

7.4 Coulomb blockade diamonds at ﬁnite bias

In the previous section we have outlined the procedure which was used to tune into the regime of a well-deﬁned quantum dot which is symmetric with respect to its surrounding gates. As a next step we apply a DC bias voltage VDC in order to determine the energy scales of the system.

(a) (b)

Figure 7.6: (a) Coulomb blockade diamonds visible in a color plot of the diﬀerential conductance dI/dVDC as a function of the plunger gate voltage VPGL and the bias voltage VDC. (b) Line cut through (a) at the white dashed line. Coulomb peaks indicative of charging an electron on the dot are visible in the conductance G as a function of VPGL.

In Fig. 7.6 we show Coulomb blockade diamonds visible in the diﬀerential conductance dI/dVDC of the dot as a function of the voltage applied to the plunger gate VPGL and the DC source drain bias voltage VDC. They can be seen as the regular, diamond shaped regions extending from zero bias. They are delimited by bright and sharp resonances corresponding to Coulomb peaks. At zero applied bias, the only points in VPGL where current can ﬂow are at the points where a level in the dot is resonant with the leads, and at these points Coulomb peaks are visible.

78 7.4. Coulomb blockade diamonds at ﬁnite bias

When applying a bias voltage one opens a bias window inbetween the levels of the source and drain leads, and while a level of the dot is in this bias window, current can flow. This region grows linearly with applied bias and is, in differential conductance, delimited by two peaks: one at the point where the level in the dot passes the upper Fermi level in one of the leads and current starts to flow, and one at the point where the level in the dot passes the lower Fermi level in the other lead and current flow ceases. This can be clearly seen in Fig. 7.6. A line cut at zero bias, denoted by the white dashed line in Fig. 7.6 (a) is shown in Fig. 7.6 (b). There, one can see the first two Coulomb resonances as well as for more positive VPGL an increase in the background and a broadened resonance once the barriers start to become more open. From the extent of the Coulomb blockade diamonds in Fig. 7.6 (a) we determine a charging energy of Ec = 3.8 meV. With a simple capacitance model using equation 5.1 we can get an estimate for the dot size of r = 40 nm which is within expectations given the lithographic dimensions of the structure.

Figure 7.7: Coulomb resonances persisting to a bias voltage of VDC = ±10 mV in Coulomb blockade diamond measurements of the diﬀerential conductance dI/dVDC as a function of plunger gate voltage VPGL and VDC. No hints of additional resonances become visible, supporting the hypothesis of the last electron.

We only see a singular Coulomb blockade diamond stemming from two Coulomb resonances which we assume to correspond to the last two electrons in the dot. In order to scrutinize this assumption we conduct an additional experiment shown in Fig. 7.7. We repeat the ﬁnite bias measurement from Fig. 7.6 (a) but expand the bias range to VDC = ±10 mV. As mentioned before, it is also conceivable that there are more than two electrons left in the dot, but at the required VPGL to bring their resonances into the bias window, the tunnel coupling becomes too small due to capacitive cross-coupling such that the additional resonances are never observed in transport. The Coulomb resonances remain clearly visible even for large bias but show no signatures of charging lines of additional electrons. This strongly supports the assumption that we are indeed dealing with the last two electrons. As can

79 Chapter 7. Gate-deﬁned lateral quantum dot

be seen in Fig. 7.7, that is exactly what happens in our case. The last Coulomb resonance that we can see extends from zero bias up to VDC = +10 mV, while remaining strongly coupled, with no signatures of additional Coulomb resonances crossing at high bias. For negative bias the resonance is coupled slightly weaker, but shows the same qualitative behavior. Therefore, we conclude that the ﬁnite bias data is consistent with our interpretation of observing the last electrons in this quantum dot.

7.5 Excited state spectroscopy

In the previous section we have been concerned with tuning our quantum dot in a regime where it contains few electrons and have applied ﬁnite bias spectroscopy to determine the charging energy. This corroborates the analogy of the artiﬁcial atom mentioned in the introduction, as it is the energy required to add an aditional electron to the atom, or the quantum dot in our case. Pursueing this analogy further, just like electrons in real atoms display a spectrum of excited states, this is also the case for electrons in quantum dots. Thus far, we have only been concerned with the ground states and have subsequently added electrons onto the dot in the sequential tunneling regime using the plunger gate.

E μT0 T0 μT+ T+ μS S

N 0 1 2 Figure 7.8: Schematic energy spectrum of the ground and excited states of a quantum dot containing up to two electrons in the presence of a small magnetic field. The energies required for transitions between the different states are depicted by dashed arrows and labelled according to the final state.

Before delving into measurements of these excited states, it is instructional to review the spectrum one would expect for a few-electron dot in order to be able to interpret the measurement results correctly. In Fig. 7.8 we can see a schematic spectrum of the ground and excited states of a quantum dot containing N = 0, 1, 2 electrons. For N = 0 the dot is empty and there are no states available. When one electron is loaded on the dot, it occupies the lowest orbital state and can either have a spin up or a spin down, which are degenerate at zero magnetic field barring an internal spin-orbit field large enough to have a preferential orientation. If we assume small magnetic field B, the degenerate ground state splits up into a spin oriented

80 7.5. Excited state spectroscopy

in parallel to the external magnetic field which is energetically favorable and which we shall call spin up and denote by an arrow pointing upwards in Fig. 7.8, and a spin down oriented antiparallel to the external magnetic field and denoted by an arrow pointing downwards. There are also higher orbital states for a single electron which are split into spin up and spin down states as well, which are not depicted in the schematic. When adding a second electron onto the dot, it can occupy the same orbital as the first electron in which case it must have an antisymmetric spin orientation to obey the Pauli exclusion principle, thus forming a spin singlet which is the lowest energy two-electron state and is labelled S in Fig. 7.8. It corresponds to 1 |Si = √ (|↑↓i − |↓↑i) (7.1) 2 as solution to the spin Hamiltonian HZ [1]. Beyond this singlet ground state, the electrons can occupy various excited states. When both spins are pointing in the same direction for instance, one of the electrons must occupy a higher orbital state in order to guarantee an antisymmetric orbital wave function to account for the now symmetric spin wave function. We receive the three triplet spin states

|T+i = |↑↑i (7.2) 1 |T0i = √ (|↑↓i + |↓↑i) (7.3) 2

|T−i = |↓↓i (7.4) which are degenerate at zero magnetic field but split up with a the Zeeman splitting ∗ EZ = g µBSzB where the Sz are the spin quantum numbers [1]. In the T+ state both spins are aligned in parallel to the external field, corresponding to Sz = +1, whereas in the T− state both spins are aligned antiparallel to the field and thus Sz = −1, and for the T0 state the spin projection vanishes Sz = 0. In Fig. 7.8 we show the singlet ground state and the two lowest energy triplet states. The electrochemical potential µelch or energy required for a transition between different one-electron and two-electron states is depicted by a dashed arrow for each of the transitions in Fig. 7.8. We always start from the one-electron ground state and can transition either to a ground or excited state upon addition of an electron. It should be noted that a transition from the spin up ground state to the T− state is forbidden by selection rules, as this would require a spin-flip. How an excited state is discernible from a transport measurement is schematically explained in Fig. 7.9. In (a) we depict a finite bias measurement where Coulomb blockade diamonds labeled with their corresponding electron number are sketched with black lines in the space spanned by the VDC and VPG. In (b), the DC current I along a line cut at fixed VDC corresponding to the red line in (a) between points 1 - 3 is shown. The differential conductance dI/dVDC can be seen in (c). The outlines of the Coulomb diamonds in (a) correspond to Coulomb resonances that are visible in the conductance. When following a line cut in plunger gate direction at fixed bias,

81 Chapter 7. Gate-deﬁned lateral quantum dot

(a) (b) I VDC

N=0 N=1 N=2 N=3 VPG 1 2 3 VPG dI (c) dVDC

1 2 3 VPG 1 2 3 Figure 7.9: (a) Schematic depiction of a finite bias measurement of a few-electron quantum dot. Coulomb blockade diamonds are labelled with their respective electron numbers and are delimited by Coulomb resonances in conductance. An excited state is visible as an additional resonance at point 2 . (b) Current as a function of VPG along the red line in (a). The resonances crossed at points 1 - 3 are marked on the VPG axis. (c) Conductance as a function of VPG along the same line cut showcasing resonances of different magnitudes at points 1 - 3 marked by the red line, one cuts through zones of different dot occupation. At first, N = 0 and the transport is blocked by Coulomb blockade. Once the energy level of the first electron enters the bias window, we see a resonance in the conductance and the dot occupation fluctuates between zero and one while transport is allowed as long as the level remains within the bias window. As soon as the level drops below the Fermi level of the drain contact µR, the dot is always occupied and we enter the Coulomb blockade in the N = 1 region, where no current is allowed to flow. At point 1 , the level of the two-electron ground state µS enters the bias window and the current flow starts again, while the electron number fluctuates between one and two. This is shown in a diagram of the energy levels in Fig. 7.10.

Following the red line and tuning VPG to more positive voltages at ﬁxed bias, we observe an additional resonance in conductance not corresponding to a Coulomb blockade diamond at point 2 . This can be rationalized by the two-electron excited state, the triplet state, entering the bias window at the Fermi level in the source contact µL, and the corresponding situation is depicted in the middle panel of Fig. 7.10. Now that both the singlet as well as the triplet state are within the bias window, transport can take place through both channels and the total current through the system increases, which is shown in Fig. 7.9 (b). This increase in the DC current in turn leads to a peak in its derivative, the conductance as can be seen in Fig. 7.9 (c),

82 7.5. Excited state spectroscopy

which explains the resonance visible in Fig. 7.9 (a).

μ 1 T 2 3 μ μ μ μ μ L S L T L μT μ μR S μR μS μR Figure 7.10: Energy level diagrams for situations 1 - 3 from Fig. 7.9 When the energy levels of the ground state µS or excited state µT enter the bias window between the energy levels in the left µL and right µR lead, transport is possible and a resonance in conductance occurs.

When charging VPG to an even more positive voltage, we enter the point where the energy level µS of the two-electron ground state passes below the level in the drain µR, leading to a resonance at point 3 as shown in Fig. 7.9 (a) and in the level diagram in Fig. 7.10. At this point, the dot is occupied by two electrons and any levels above the singlet shoot up by the charging energy and are thus outside of the bias window, and in turn current ceases as the dot enters the Coulomb blockade at N = 2. The decrease in current as a function of VPG shown in Fig. 7.9 (b) leads to another peak in the diﬀerential conductance visible in (c).

7.5.1 Behavior in a perpendicular magnetic field In the previous section we discussed the ground and excited states of a quantum dot occupied by up to two electrons, and have qualitatively derived their expected transport signatures. Based on this, we conduct finite bias measurements in order to elucidate whether these signatures are reproduced in our experiment. In the previous finite bias measurements in Figs. 7.6 and 7.7 only one Coulomb blockade diamond of the first electron was visible, as the dot was tuned to a regime where the barriers were quickly opening up with increasing VPGL. Therefore, the system was already not blockaded anymore when occupied with two electrons, making it impossible to observe excited states. As a consequence, we tune the dot into a regime where the tunnel barriers are more strongly closed, enabling the use of a more positive VPGL and in turn higher electron occupancy while keeping the system Coulomb blockaded. The resulting finite bias measurement in this regime can be seen in Fig. 7.11 (a). Coulomb blockade diamonds for the first three electrons are apparent. Due to the strongly closed barriers, the Coulomb resonances of the first electron are only very faintly visible. The Coulomb blockade diamonds for the first and second electrons are considerably larger than in Figs. 7.6 and 7.7, with the charging energy of the second electron exceeding 5 meV, indicating a smaller dot size in this configuration. At a bias voltage of VDC = −3.9 mV, an excited state of the two-electron dot is clearly visible. This situation corresponds to 2 in Fig. 7.9, which was explained in the previous section. The ground and excited states of a two-electron quantum

83 Chapter 7. Gate-deﬁned lateral quantum dot

(a) (b)

0 1 2 3 0 1 2

S T+

Figure 7.11: (a) Finite bias spectroscopy revealing an excited two-electron state in a color plot of the differential conductance dI/dVDC as a function of plunger gate voltage VPGL and DC bias VDC. Electron numbers are labelled within the respective Coulomb blockade diamonds. (b) VPGL sweeps while varying the perpendicular magnetic field B⊥ at a finite bias of VDC = −3.9 mV denoted by the dashed white line in (a). Electron numbers as well as one electron spin up and spin down, and two-electron singlet and triplet states are labelled. dot have different total spin, which can be verified from equations 7.1 to 7.4. This means that by applying a magnetic field and comparing their behavior to the theoretically expected one, it is possible to identify these states and validate whether the assumptions made thus far correspond to the experiment. Additionally, the spin degenerate one-electron state is expected to split up into spin up and spin down, and the degenerate triplet states are also expected to split in an applied magnetic field. This makes the magnetic field an important tool for obtaining information about the states in our dot.

In order to investigate this, we fix the source drain bias to VDC = −3.9 mV, which is the point where the excited state becomes clearly visible. We then sweep VPGL along this line, demarcated by the white dashed line in the finite bias measurement of Fig. 7.11 (a), while increasing the perpendicular magnetic field B⊥. The resulting differential conductance is shown in Fig. 7.11 (b). There, one can see a plethora of resonances which split and bend in B⊥. For example, the first Coulomb resonance corresponding to the charging of the first electron splits up into two branches moving away from each other, resembling the shifting of the spin states aligned in parallel and antiparallel to the applied magnetic field due to the Zeeman effect. When a magnetic field is applied in perpendicular there is not only this Zeeman shift due to the interaction with the spins of the electrons, but there is also an orbital contribution depending on the electrons spatial wave function. This explains the visble bending of all the resonances. In the two-electron regime, one can for instance see that the resonance corresponding to the singlet state labelled S is moving towards positive VPGL while the resonance of the lowest triplet state labelled T+ is moving

84 7.5. Excited state spectroscopy

towards more negative VPGL, an indication that these states have different total spins. Additionally, these resonances bend in a different way with increasing B⊥, suggesting that the electrons in these states are occupying different orbitals. However, similar as has been observed in section 6.5, horizontal stripes are superimposed on the data. These similarly correspond to regions where the ungated bulk of the sample enters a fully developed quantum Hall state, which is why these stripes only depend on B⊥, and thus the contacts become decoupled [128] and the signal vanishes. In the next section, we change the orientation of the sample with respect to the magnetic field and thus do not only remove this issue, but were also able to record higher quality data.

7.5.2 Behavior in a parallel magnetic field In this section we discuss measurements performed after rotating the sample in the cryostat, which changes the alignment of the magnetic field from perpendicular to parallel to the plane of the sample. First of all, this eliminates the primary issue that occurred in the perpendicular magnetic field: horizontal stripes of vanishing signal superimposed on the data caused by a quantum Hall state in the bulk of the sample which decoupled the contacts. A purely parallel magnetic field only influences the energy spectrum through the Zeeman effect related to the spin wave function, and does not cause any shifts in the orbital wave function of the electrons. Therefore, this removes the bending of the states seen in Fig. 7.11 (b) making it easier to identify the different spins of the ground and excited states that we are dealing with.

(a) (b)

T0 S T+

0 1 2

Figure 7.12: (a) Differential conductance dI/dVDC as a function of VPGL and parallel magnetic field Bk at finite bias VDC = −3.9 mV. Electron numbers and one- and two- electron ground and excited states are labelled. (b) Zoom into the yellow box in (a) showing a detailed view of the spin splitting of the first Coulomb resonance.

In Fig. 7.12 (a) we show a color plot of the diﬀerential conductance while varying VPGL at ﬁxed VDC = −3.9 mV, again along the white dashed line in Fig. 7.11 (a),

85 Chapter 7. Gate-deﬁned lateral quantum dot

and while increasing the parallel magnetic field Bk. We have labelled the regions containing zero, one, and two electrons, as well as the visible electronic transitions. Similar to the case in perpendicular magnetic field, the first Coulomb resonace splits into two resonances with Bk, indicating the splitting in a state with a spin oriented parallel to the magnetic field, here called spin up, and a state with a spin oriented antiparallel to the magnetic field, here called spin down. For better visibility, we show a zoom into the one electron region demarcated by the yellow dashed box in Fig. 7.12 (a) in the neighboring Fig. 7.12 (b). There, we can clearly see a singular resonance at Bk = 0 linearly splitting up with increasing field. From the ∗ energy splitting ∆E = gµB∆Bk we extract a effective g-factor of | g |= 16.4. This value is of appropriate magnitude for the material system at hand, but is larger than the expected bulk value g = −15 of InAs [122, 123] and also larger than the value we have determined in a quantum point contact of this material in section 6.5. The reason for this discrepancy is not immediately clear, however, there are experimental reports on quantum dots and similar nanostructures in InAs nanowires reavealing strongly enhanced g-factors [93, 152, 153], and recent theoretical work suggested that orbital contributions from higher subbands could lead to an enhancement in the g-factor [154]. Concluding the case of behavior of the first observed resonance in Bk, we state that it is consistent with the behavior expected from the first electron. Turning to the two-electron regime, we see that the first resonance of this regime which has been labelled S, as it is expected to be the transition for loading the spin singlet state, is shifting towards more positive VPGL with increasing Bk. The next resonance, which at Bk = 0 can be seen around VPGL = −455 mV, is strongly broadened. It actually encompasses two resonances, the first one stemming from the excited state, and the second one from the charging line corresponding to full occupation of the dot with two electrons and thus entering the second Coulomb blockade diamond, which is expected to happen when the two-electron ground state moves below the energy level in the source lead. This can be verified from Fig. 7.11 (a). The first of these two close resonances associated with the excited state splits up into multiple resonances with Bk, as is excpected from triplet states. The strongly coupled triplet resonance which moves towards more negative VPGL, which means decreasing energy, is recognized as the T+ state, and the faintly coupled line moving in parallel to the singlet is identified as the T0 state, as it has the same spin projection Sz = 0 as the singlet. The weak T0 state only splits off of the strong T+ state at finite magnetic field of around Bk = 1 T, why this is the case could at present not be ascertained. Due to the spin projection Sz = +1 of the T+ state it becomes energetically more and more favorable, the more the magnetic field is increased, up to the point where it crosses the singlet which has Sz = 0 and becomes the new ground state. Considering all these points, we state that also the behavior in the two-electron regime is consistent with a singlet two-electron ground state of spin projection Sz = 0 and a triplet excited state visible in finite bias. The triplet states split up in an external magnetic field in two states moving according to their respective spin projections Sz = +1 and Sz = 0. We note two more peculiar

86 7.5. Excited state spectroscopy

experimental observations. Firstly, when following the resonances to high Bk they do start to bend slightly. This is due to the fact that the rotation of the sample is done manually using a so-called Swedish rotator and the parallel and perpendicular positions are calibrated by eye outside of the dilution refrigerator and read off of an analog display, and the same value is then adjusted in the refrigerator when the sample is cold. This carries a certain degree of imprecision, and with increasing fields the small perpendicular component, that scales with the cosine of the angle of the sample times the total field, will at some point also have an orbital effect on the energy levels. The second observation is that there is a resonance of negative differential conductance that moves in parallel to the T+ state at a slightly higher energy. The origin of this resonance could not be determined in the current measurements, but its investigation might pose a topic of further measurements on this or on subsequent samples.

7.5.3 Singlet-triplet crossing While discussing the behavior of the two-electron ground and excited state in a magnetic field in the previous section, we noted that the triplet excited state T+ is moving down in energy due to its spin projection Sz = +1 aligned to the external magnetic field compared to the singlet excited state with vanishing spin projection Sz = 0. It is therefore expected that at a certain value of Bk, these two states cross and thus a transition of the ground state from a spin singlet to a spin triplet will take place. Additionally it is interesting to observe whether a crossing or an anticrossing occurs at the point where these two states meet, as in previous experiments in InAs nanowire quantum dots it was observed that singlet and triplet states are mixed by spin-orbit coupling and thus show an avoided crossing. The magnitude of the singlet-triplet anticrossing was used as a direct measure to determine the strength of the spin-orbit coupling [102, 103]. The region of interest lies within the cyan dashed box in Fig. 7.12 (a). A close-up view of this region is shown in Fig. 7.13 (a), where we can see the singlet and triplet transitions intersecting as resonances in the differential conductance as a function of VPGL and Bk. Within experimental resolution, there is no avoided crossing visible, the states intersect as completely straight lines apart from a very small bending occurring due to finite error in the magnetic field alignment, as described in the end of the previous section. This is in contrast to the experiments conducted in nanowire quantum dots, where the spin-orbit splitting δSO could be approximated as from the magnitude of the avoided crossing of singlet and triplet states [102, 103]. The direct implication of this experimental observation would be that the spin-orbit interaction in the quantum dot is vanishingly small. This is somewhat contradictory, as one would naively expect a large spin-orbit interaction in InAs and we have also measured a large g-factor from the Zeeman splitting of the Coulomb resonance of the first electron at finite bias. A key difference between this experiment and the ones conducted in nanowires, beyond the apparent difference of confinement

87 Chapter 7. Gate-deﬁned lateral quantum dot

(a) (b) I 1 2

VPG 1 2 dI dVDC

VPG 1 2

(c) μS (d) (e) μL μT+ μL μS μL μS μ μ μR T+ μR T+ μR Figure 7.13: (a) Singlet-triplet crossing in a color plot of the differential conductance dI/dVDC as a function of VPGL and Bk at finite bias VDC = −3.9 mV. Region corresponds to the cyan box in Fig. 7.12 (a). (b) Schematic of the current and differential conductance along the white dashed line in (a). (c)-(e) Schematics of the energy levels in the dot along the white dashed line in (a) illustrating the conversion of the singlet resonance from a peak to a dip in differential conductance. potential between a one-dimensional nanowire and a two-dimensional quantum well, is the crystal structure. Fasth et al. [103] used a nanowire with a wurtzite crystal structure, whereas our quantum well was grown in a zincblende structure. The relative orientation of the parallel magnetic field towards the crystal might lead to a direction in which the spin-orbit interaction vanishes, and this could be the case here. In our experimental setup it is not possible to rotate the magnetic field in the plane of the quantum well and therefore it could not be verified whether there is an orientation for which an avoided crossing is restored. Furthermore, it was observed in Fig. 7.13 (a) that the singlet state turns from a peak in differential conductance before the crossing with the T+ state into a dip in differential conductance once T+ becomes the ground state. This observation seems puzzling at first, but can be understood as competing transport and will be explained in the following. To illustrate the situation, we turn to Fig. 7.13 (b) where we show a schematic of the current and conductance along the dashed white line in Fig. 7.13 (a), and to Figs. 7.13 (c)-(e) where we show energy level schematics of the corresponding points. Starting from negative VPGL, we go towards more positive voltage and reach point 1 where the energy level of the T+ state becomes resonant with the Fermi energy in the source lead as shown in Fig. 7.13 (c) and we thus observe a steep increase in the current corresponding to a peak in the conductance both of

88 7.6. Kondo eﬀect at strong coupling to the leads

which are visualized in Fig. 7.13 (b). Further tuning VPGL more positive, we end up at 2 , where the singlet state, which is now higher in energy than T+, becomes resonant with the Fermi level in the source. This situation is depicted in Fig. 7.13 (d). Now both singlet and triplet states are within the bias window, and transport can in principle occur through either one of these states, shown by the two pairs of dashed arrows. However, while an electron is tunneling through the singlet state, the dot is occupied by two electrons for the time of the tunneling process, denoted by the blue dashed circle in Fig. 7.13 (e). Therefore, the T+ state shoots up by the charging energy, indicated by the red dashed arrow in Fig. 7.13 (e) and can not contribute to transport for this time. This means that during the tunneling process through either singlet or triplet, the respective other path is blocked due to Coulomb interaction. As is evident from the higher intensity and much broader linewidth in Fig. 7.13 (a), the T+ state is coupled much more strongly to the leads than the singlet state. This makes sense, as the triplet state requires one electron to be in a higher orbital state, which has a spatially larger wave function and thus has more overlap with the leads. Therefore, once the singlet state is an alternative option for tunneling through the dot at 2 , it competes with the more eﬃcient transport through the T+ state, thus lowering the total current through the system, which is shown in Fig. 7.13 (b). A decreasing total current corresponds to a dip in the conductance, and this explains the observation of resonances of negative diﬀerential conductance in our system.

7.6 Kondo eﬀect at strong coupling to the leads

The Kondo effect is a phenomenon of many-body physics which has been fascinating theorists and experimentalists alike. It first manifested itself as an increase in the resistance of metals upon decreasing the temperature, explained by a dilute amount of magnetic impurities [155]. One can understand the effect as a many-body state called the Kondo cloud forming between the unpaired spin of the magnetic impurity and the conduction band electrons trying to screen the localized spin. This many body state has a binding energy corresponding to kBTK where TK is the Kondo temperature. The increase in the resistance then comes about due to the fact that the scattering cross-section of the magnetic impurity is enlarged by the presence of the Kondo cloud screening it [156]. It may appear that this bulk effect is completely unrelated to semiconductor nanostructures, if not for the fact that it was theoretically proposed to occur between the Fermi sea in the leads coupled to a localized unpaired spin in a quantum dot [157, 158]. A few years later, the Kondo effect was indeed observed in GaAs quantum dot structures [159–163]. In contrast to the bulk Kondo effect where a peak in resistance is observed, the Kondo effect in quantum dots leads to a resonance in conductance. This can be understood by the formation of a spin singlet between an unpaired spin in the quantum dot and the spins in the Fermi sea in the lead. The spin in the quantum dot is effectively screened by successive virtual spin-flip tunneling

89 Chapter 7. Gate-deﬁned lateral quantum dot

events [160]. Therefore, the resonance in the density of states caused by the Kondo effect enables both spin species to partake in transport through the dot, which is why the unitary limit for the conductance of the Kondo resonance is 2e2/h whereas for a regular Coulomb resonance it is e2/h [164]. Thus far, the Kondo effect in quantum dots was observed in systems such as carbon nanotubes [165, 166], single-molecule transistors [167, 168], and InAs nanowires [152, 156, 169], but not in InAs quantum wells for the lack of quantum dots in these structures. During this section, we will show evidence that InAs two-dimensional electron gases can be added to the list of material systems exhibiting the Kondo effect.

7.6.1 Tuning into the Kondo regime by opening the dot The basic prerequisite necessary in order to observe the Kondo effect is the presence of a single unpaired spin in the quantum dot. This condition can be met in our system, as we have showed multiple pieces of evidence suggesting that we can reach the last electrons in our dot during the course of this chapter, and it again serves as a cross check to verify that the regime we identified as the last electron indeed shows an unpaired spin. The second basic condition is a temperature of the electronic system below TK, which for typical values of TK is reached in our dilution refrigerator. Another condition that must be met is very strong coupling to the leads, such that the electron in the dot is effectively shared with the leads and the Kondo many-body state can be formed. In the previous sections, we were interested in a dot with strongly closed tunnel barriers, where we applied low plunger gate voltages and high bias voltages in order to perform spectroscopy on excited states in the two-electron regime. In order to tune the dot to a regime where the Kondo effect can be observed we sequentially retune our dot while maintaining single electron occupation. To do so, we record a map of the dot conductance as a function of the two sets of tunnel barriers, and then adjust them to a value where they are more open and the conductance of the quantum dot starts to increase. Doing so, we usually passed two Coulomb resonances due to cross capacitance of the barriers. This was counteracted by adjusting a more negative plunger gate, fully depleting the quantum dot again and reaching a regime where it is weakly tunnel coupled. This process was iterated many times, until a regime was reached where the dot was strongly coupled to both reservoirs while we maintained the same low electron occupation.

7.6.2 Kondo eﬀect visible as a zero-bias anomaly

The resulting conductance of the quantum dot as a function of the plunger gate VPGL can be seen in Fig. 7.14 (a). There we can see the last two Coulomb resonances which are strongly broadened due to the large coupling to the leads. The strong coupling also manifests itself in the conductance of the resonances which is close to the unitary limit of e2/h, as can be seen for the ﬁrst resonance whereas one has to take into

90 7.6. Kondo eﬀect at strong coupling to the leads

(a) (b)

Figure 7.14: (a) Conductance G of the quantum dot as a function of the plunger gate voltage VPGL where a Kondo valley between the last two Coulomb resonances is visible. (b) Differential conductance dI/dVDC as a function of the bias voltage VDC through the middle of the Kondo valley at VPGL = −933 mV showing a narrow zero bias Kondo peak. account the strongly increasing backgrond for the second resonance. Inbetween these two resonances, the conductance does not drop down to zero, but stays above the values in the regimes left of the first and right of the second Coulomb resonances, where the dot is occupied with an even number of zero and two electrons respectively. This is the so called Kondo valley, where the conductance in the Coulomb blockade regime is enhanced due to the Kondo effect. The Kondo effect can be clearly seen in Fig. 7.14 (b) where we show the differential conductance as a function of the bias voltage VDC which exhibits a sharp resonance at zero bias. This is the Kondo peak, manifesting itself as a zero bias anomaly. The occurrence of the Kondo effect confirms that the region which we associated to occupancy of the dot by a one electron, shows a single, unpaired, spin and is thus consistent with our interpretation of the quantum dot data.

Displaying both the influence of VPGL and VDC into a single measurement we show a color plot of the differential conductance as a function of these two quantities in Fig. 7.15. There, we can see the Coulomb blockade diamond of the first electron which exhibits a strong Kondo peak. The zero bias anomaly is only visible within the Coulomb blockade diamond, where transport should be forbidden if not for the Kondo effect. This feature immediately vanishes when the dot is occupied by an even number of electrons, as becomes evident from the data. The yellow and cyan arrows mark the directions along which the line cuts in VPGL direction in Fig. 7.14 (a) and in VDC direction in Fig. 7.14 (b) have been recorded.

7.6.3 Splitting of the zero-bias anomaly in a magnetic ﬁeld In the previous section, we have observed the Kondo eﬀect as a peak in the differential conductance at zero bias. This is explained by a resonance in the density

91 Chapter 7. Gate-deﬁned lateral quantum dot

Figure 7.15: Coulomb blockade diamond exhibiting a Kondo eﬀect manifesting itself as a zero bias peak in the diﬀerential conductance dI/dVDC as a function of the plunger gate voltage VPGL and the bias voltage VDC. The yellow and cyan arrows correspond to the directions in which the VPGL and VDC cuts in Fig. 7.14 were taken.

of states at the Fermi level. When applying a magnetic field, the spin degenerate level in the dot splits and the peaks in the density of states move down in energy for the spin aligned in parallel and up in energy for the spin aligned antiparallel to the external magnetic field. This suppresses the conductance peak at zero bias, but the resonance can be restored when varying the chemical potential such that it becomes resonant with the energy of the upper or lower spin respectively [170]. Therefore, we expect two peaks split by ∆E = 2gµBBk [156]. The splitting of the Kondo resonance therefore enables us to measure the g-factor in a different regime, and can be compared to the result obtained from the excited state spectroscopy in section 7.5.

In Fig. 7.16 (a) we apply a magnetic field Bk in parallel to the quantum well and measure the differential conductance as a function of VPGL and VDC, recovering the familiar Coulomb blockade diamond. However, one can clearly see that the Kondo peak that was visible as a zero bias anomaly in Fig. 7.15 has now been split up due to the external magnetic field. Otherwise, the data is not influenced by Bk. In order to quantify the effect of the magnetic field, we fix the plunger gate to the middle of the Kondo valley at VPGL = −933 mV and analyze the splitting of the Kondo peak as a function of Bk and VDC. The resulting differential conductance is shown in a color plot in Fig. 7.16 (b). There we see a linear Zeeman splitting of the zero bias peak as Bk increases. Using the equation ∆E = 2gµBBk, we determine a g-factor of | g |= 16.3. This value is in excellent agreement with the result obtained in section 7.5, thus corroborating the Kondo physics detected in our system.

92 7.6. Kondo eﬀect at strong coupling to the leads

(a) (b)

Figure 7.16: (a) Coulomb blockade diamond at a magnetic field of Bk = 150 mT. The Kondo peak visible in the differential conductance dI/dVDC as a function of the plunger gate voltage VPGL and the bias voltage VDC exhibits a Zeeman splitting. (b) Linear Zeeman splitting of the zero bias anomaly visible in the differential conductance as a function of VDC and Bk.

7.6.4 Temperature dependence of the zero-bias anomaly Another hallmark of the Kondo eﬀect is its strong temperature dependence. The Kondo peak broadens and decreases in height with increasing temperature and is reduced to half of the zero-temperature conductance at the Kondo temperature. We can associate the Kondo temperature with the point where the thermal energy equal to the binding energy kBTK of the many body state.

(a) (b)

Figure 7.17: (a) Conductance G of the Kondo peak as a function of the bias voltage VDC for diﬀerent temperatures T showing a strong decrease in peak height with increasing temperature. (b) Fit of the peak conductance values at zero bias for diﬀerent temperatures (black points) to a phenomenological model (blue line) used to extract TK.

Multiple traces of the conductance of the Kondo resonance as a function of VDC recorded at diﬀerent temperatures can be seen in Fig. 7.17 (a). Using a heater

93 Chapter 7. Gate-deﬁned lateral quantum dot

located at the mixing chamber, the temperature of the sample was adjusted ranging from a base temperature of T = 57 mK up to T = 952 mK. Between T = 527 mK and T = 900 mK, the temperature was too unstable due to the circulation breaking down, and no measurement points could be recorded. One can see that the peak height is very sensitive to temperature and decreases strongly when increasing T , accompanied by a signiﬁcant broadening of the peak. In order to extract the Kondo temperature, we use the phenomenological expression

0 2 −s G(T ) = G0[1 + (T/TK) ] , (7.5)

with the conductance at zero temperature G0 [156]. The Kondo temperature can 0 then be extracted from TK using

0 1/s 1/2 TK = TK/(2 − 1) , (7.6) where the parameter s = 0.22 [156]. In previous work, this value of the parameter s was established to best reproduce numerical renormalization group calculations of a spin 1/2 impurity [156, 171]. Applying this formalism to our data we show in Fig. 7.17 (b) the peak conductance values Gpeak at zero bias on a semilogarithmic plot as a function of temperature. The black diamonds indicate the data points, and the blue line is a fit to equation 7.5. We have found a best fit for s = 0.15, which results in a Kondo temperature of TK = 801 mK and is applied in Fig. 7.17 (b). Using s = 0.22, the fit becomes slightly worse but we receive an almost unchanged TK = 797 mK.

7.7 Conclusion

During the course of this chapter we have shown that is is possible to use the entirely gate-defined mesa technique introduced in chapter5 to define high quality quantum dots in an InAs quantum well, which up to date was not possible. We have elucidated the process of tuning a quantum dot based on three sets of opposing fine gates in section 7.3. Reaching a stable few-electron regime, we measured Coulomb blockade diamonds, determined the charging energy of the dot and presented evidence for reaching single electron occupation in section 7.4. During the subsequent section 7.5 we performed spectroscopy on the ground and excited states in the one- and two- electron regimes of the quantum dot. We determined the g-factor from the splitting of spin up and down states and identified spin singlet and spin triplet states in a parallel magnetic field. When increasing the field, these two states surprisingly showed a direct crossing where one would expect an avoided crossing due to spin- orbit interaction. Past this crossing, the ground state changed from a spin singlet to a spin triplet. Tuning the dot to a regime where it is strongly coupled to the leads we were able to demonstrate the Kondo effect in our dot in section 7.6. We exctracted the g-factor from a Zeeman splitting of the zero bias peak which corresponded to the value determined with the excited state spectroscopy. The peak was found to be

94 7.7. Conclusion

strongly temperature-dependent in accordance with past experiments demonstrating Kondo physics. In the next chapter we will build on the results achieved in this chapter and show that two of these quantum dots can be coupled together to realize a double quantum dot structure.

95 Chapter 8

Double quantum dots

During the course of this thesis, we have outlined the way towards realizing quantum devices in InAs two-dimensional electron gases step by step. Through the elimination of trivial edge conductance, first unsuccessfully attempted by chemical passivation in chapter4, and ultimately achieved by electrostatic definition of a mesa in chapter5, the main obstacle for nanostructures was overcome. Based on the definition of tunnel barriers resulting in the quantum point contacts of chapter6 it became feasible to define lateral quantum dots that showed a rich spectrum of physics explored throughout the previous chapter. All of the prior work culminates in this chapter, where the most advanced quantum device of this thesis is presented: a gate tunable double quantum dot achieved by tunnel coupling of two single quantum dots. We begin by outlining in section 8.2 how exactly this coupling process was executed. The emerging charge stability diagram of two dots is characterized in section 8.3. The triple points in such a diagram extend to finite bias triangles observed in section 8.4 when a voltage between source and drain contacts is applied. Finally, we were able to realize singlet-triplet spin blockade and investigated it at strong interdot coupling in section 8.5 and at weak interdot coupling in section 8.6.

8.1 Introduction

When introducing the concept of a semiconductor quantum dot in the previous chapter, we have evoked the image of an artificial atom [140] due to the possibility of studying atomic physics in a solid state system. Extending this analogy, we can create an artifical molecule by coupling two quantum dots together [13]. Due to the high tunability provided by the plethora of gates that the structure is composed of it is possible to tune the occupation of each of the individual dots, their coupling to the leads, and even the coupling between them, simulating ionic-like bonding of our artificial molecule at weak interdot coupling and covalent-like bonding at strong interdot coupling [13]. These double quantum dots lie at the heart of the technological promise that is quantum computation. A logical quantum bit or qubit can for instance be formed by the single spin of an electron in a quantum dot, where the

96 8.2. Coupling two quantum dots

projection of its spin encodes the logical states 0 and 1 [6]. Having two qubits, one can also encode information in the charge state of the quantum dot [172–174], which could for instance be imagined as being encoded with a single electron occupying either the left or the right dot. Within the last years, many more possible implementations of qubits within double or even triple quantum dots have been suggested and demonstrated. Among these are the singlet-triplet qubit [175–177] making use of the two-electron ground and excited states, utilizing the exchange interaction [178, 179] in a resonant exchange qubit [180] or in a quadrupole qubit [181]. Thus far, these proposals have been realized in the technologically very mature materials GaAs and Si. During this chapter, we will show that it is possible to deﬁne stable, few-electron double quantum dots in InAs. This in principle paves the way towards realizing such proposals in a material featuring a strong spin-orbit interaction, and to arouse interest for new proposals that could potentially harness this interaction in a double quantum dot.

8.2 Coupling two quantum dots

During the course of this section, we will outline the procedure that was used to tune our sample with the gate layout shown in Fig. 7.1 into a double quantum dot. There are two general approaches that could be taken. Firstly, having already defined the left quantum dot of our structure, one can successively charge the right plunger and right barrier gates negatively, while increasing the voltage on the gates of the left dot to maintain the same charge occupation and not pinch off completely. This was attempted, but did not turn out to be successful, as the middle barriers start out at pinch-off and right barriers need to be driven to pinch-off while one constantly has to counteract for the cross-capacitance closing the middle barrier even more. When transport vanishes, it is not possible to tell whether the middle or left barriers are closed too much, complicating this tuning procedure. A second method of tuning the double dot that was found to be superior in this particular case is starting to define one large and elongated dot between the left and right barriers while the left plunger, middle barrier, and right plunger gates are collectively operated as plunger gates. Once such a dot has been formed, it can be cut in half by successively decreasing the voltage on the middle barrier. This has the added benefit that once the double dot regime is reached it is clear that this dot is in the strong interdot coupling regime, as opening the middle tunnel barrier more causes the two dots to merge. Beginning this tuning procedure is similar as for a single dot. We start with both pairs of plunger gates and the middle barrier energized to a voltage where the electrons underneath the gates are depleted but where they do not start to pinch off the electron gas. We chose V = −450 mV in this case. Then, pinch-off traces of both the left and right barriers are taken separately to find the point where they start to pinch off the electron gas thus tunnel coupling the dot to the reservoirs.

97 Chapter 8. Double quantum dots

Figure 8.1: Conductance G of the double quantum dot as function of the voltage applied to the middle barrier gates VMB during tuning of the double quantum dot. Coulomb resonances are visible as the middle barrier acts as plunger gate of a large quantum dot between all gates.

Once both barriers are set to the point of steepest slope on their respective pinch-oﬀ curves, a sweep of the middle barrier gates is recorded and can be seen in Fig. 8.1. A plethora of Coulomb resonances is visible over a large range in the voltage VMB applied symmetrically to the middle barrier gates. This indicates that the middle barrier acts as a plunger gate for the large single quantum dot that we have formed in our nanostructure, expelling individual charge carriers one after another without stopping current ﬂow.

(a) (b)

Figure 8.2: (a) Conductance G of the double quantum dot as a function fo the voltage applied to the left plunger gates VPGL during tuning of the double quantum dot. Coulomb resonances indicate expelling of electrons oﬀ of the left dot until the current ceases due to cross capacitance to the barriers. (b) G as a function of the voltage applied to the right plunger gates VPGR, showing similar behavior.

We thus set the middle barrier to a voltage inbetween the last few resonances

98 8.2. Coupling two quantum dots

around VMB = −800 mV, such that the large dot is depleted of more carriers and in order to move closer to partitioning the dot. As a next step, we take individual gate sweeps of both sets of plunger gates. The dot conductance for a sweep of the voltage applied to the left plunger gate VPGL is shown in Fig. 8.2 (a), and for a sweep of the voltage applied to the right plunger gate VPGR in Fig. 8.2 (b). Both show a qualitatively similar behavior of a few Coulomb resonances visible after which the conductance drops to zero. Cross capacitance to the middle barriers and the tunnel barriers of the respective side of the plunger gate is the reason for the ceasing current through the dot. For the next step of tuning to a double dot, we use the knowledge gained in the single sweeps of each plunger gate about the range in which we can expect to see Coulomb oscillations and combine them to measure a map of the conductance as a function of both plunger gate voltages. A color plot of such a measurement is shown in Fig. 8.3 (a). There we can see Coulomb resonances as lines of unit slope, inﬂuenced evenly by both pairs of plunger gates. This means that we still have a single large quantum dot that is tuned to a similar extent by changing the voltage on either the left or right plunger gate. For a double quantum dot, one would expect the characteristic honeycomb pattern, where both plunger gates mostly eﬀect the dot on their respective side with only a weak capacitive cross coupling.

(a) (b)

Figure 8.3: (a) Conductance G of the quantum dot in a color plot as a function of the voltage applied to the left plunger gates VPGL and to the right plunger gates VPGR. Reso- nances of equal slope with respect to both sets of plunger gates and no honeycomb pattern is visible, indicating tuning of one large quantum dot. (b) G as a function of the voltage applied to the left barrier gates VLB and to the right barrier gates VRB necessary in order to readjust the tunnel barriers.

Having gained information about the state of the large dot, we keep the plunger gate voltages at the center of the scan frame of Fig. 8.3 (a) as we do want them to be more positive than the barriers in order to have well deﬁned circular quantum dots inbetween the barriers. During the tuning procedure we have signiﬁcantly reduced VMB, but have still not partitioned the large dot. In order allow for an

99 Chapter 8. Double quantum dots

even more negative VMB without completely pinching off the current through the dot, we need to readjust the barriers. In Fig. 8.3 (b) we show a color map of the dot conductance as a function of the voltages of the left and right barriers, VLB and VRB. The characteristic pinch-off of both gates can be seen when the finite conductance in the top right corner ceases once either or both barriers are charged to a more negative voltage. One can faintly see resonances crossing this plot due to the finite cross capacitance to the energy levels in the dot. From this measurement, we choose a point where both barriers are opened symmetrically to the point where they just pinch off. This enables us to reiterate all measurements described previously in this section, moving to a more and more negative VMB until the large single dot gets partitioned into two single dots which are tunnel coupled. The resulting charge stability diagram of a double quantum dot is shown and explained in the next section.

8.3 Charge stability diagram

After successful tuning of the double quantum dot according to the procedure described in the previous section, we have reached a situation where two quantum dots in our sample are connected in series between source and drain leads. This situation is schematically depicted in Fig. 8.4.

RLB,CLB RMB,CMB RRB,CRB

S NL NR D

CPGL CPGR

VPGL VPGR Figure 8.4: Schematic network model of a serially coupled double quantum dot containing NL electrons on the left and NR electrons on the right dot. The two dots are tunnel coupled, characterized by a tunneling capacitance and a tunneling resistance, to each other and on the left side to the source and on the right side to the drain leads. The left and right dot are capacitively coupled to the left and right plunger gates.

The two dots with occupation NL and NR on the left and right dot are tunnel coupled to each other and on the left side to the source electrode and on the right side to the drain electrode. The tunnel coupling can be descriped by a tunneling resistor and a tunneling capacitor that are connected in parallel. The tunnel couplings are achieved via the left barriers, middle barriers, and right barrers, and the respective capacitances and resistances are labelled accordingly in Fig. 8.4. The left and right quantum dot are furthermore capacitively coupled to the left and right plunger gates.

100 8.3. Charge stability diagram

The charge stability diagram [182] is usually shown as the conductance G of the double dot system within the two-dimensional parameter space spanned by the voltages on the left andd right plunger gates VPGL and VPGR. There, regions of ground state electron occupation in each dot can be assigned which are, at nonzero bias voltage, separated by lines of finite conductance where transitions can take place. One can calculate the electrostatic energy of the system within a capacitance model [1, 13] in order to derive the shape of such a charge stability diagram. For two non-interacting dots, one would expect a checkerboard pattern, where the occupation of the left dot only depends on VPGL and the occupation of the right dot only depends on VPGR. Due to finite capacitive coupling, the charges accumulated on one dot affect the energy levels in the neighboring dot and can cause transitions there. This leads to a deviation from the checkerboard pattern, where the transition lines acquire a finite slope and the crossing points split up into two triple points, leading to a hexagonal shape of the charge stability regions. At zero bias, the triple points are the only points in the charge stability diagram where elastic tunnel current flow is allowed, and electrons can be shuttled sequentially across the double quantum dot since its energy levels are resonant with the Fermi levels in the source and drain leads.

(0,4) (1,4) (2,4) (0,3) (1,3) (2,3) (0,2) (1,2) (2,2) (0,1) (1,1) (2,1) (0,0) (1,0) (2,0)

Figure 8.5: Charge stability diagram of the double quantum dot visible in a color plot of the logarithm of the conductance as a function of the voltages applied to the plunger gates of the left and right dot, VPGL and VPGR. Labels indicate the charge state in each section where the left and right numbers denote the number of electrons in the left and right dot respectively.

We measure this charge stability diagram in Fig. 8.5 and recover its expected characteristic shape, due to which it is also known as honeycomb diagram. The

101 Chapter 8. Double quantum dots

logarithm of G is plotted as a function of VPGL and VPGR and the labels (NL,NR) within regions of stable charge denote the occupation in the left and right dot. For low electron occupation, only the triple points are visible and the cotunneling lines demarcating transitions between charge states have completely vanished as the barriers are strongly closed. In the top right region, where all tunnel barriers become more open due to cross capacitance from the plunger gates, we enter the regime of strong interdot coupling. The slope of the resonances changes and the charge stability diagram starts to resemble that of a single quantum dot with almost straight lines of unit slope with respect to left and right plunger gates. In this region, the electrons become delocalized and are shared between both dots.

8.4 Finite bias triangles emerging from triple points

In this section, we study the effect of a finite source drain bias VDC on the transport behavior of our double quantum dot. During chapter7 we saw that an applied bias voltage opens a bias window between the Fermi level in the source and drain contacts. For as along as a specific state of a single quantum dot is within this bias window, transport through the dot is possible while the electron number fluctuates by one as charge carriers continuously hop on and then off the dot. The one-dimensional charge stability of a single dot recorded in a plunger gate sweep is extended into Coulomb blockade diamonds when recording these sweeps while varying the applied bias. The charge stability diagram of a double quantum dot is already recorded within a two-dimensional parameter space, and therefore we have to investigate how these diagrams change depending on the applied bias voltage. Three such charge stability diagrams can be seen in Fig. 8.6. They were all recorded in the vicinity of the triple points at (0,2)/(1,2)/(0,3) and at (1,2)/(0,3)/(1,3) occupations. The first diagram shown in (a) has been recorded at bias voltage of VDC = 100 µV and show the current IDC as a function of voltages of the left and right plunger gates, VPGL and VPGR. This bias voltage is very small, and therefore no current flows in most of the parameter space, except at the two triple points, where the levels in the dot are aligned the source and drain leads and a resonant tunneling process can occur. In the next step in Fig. 8.6 (b) we show the same region of the charge stability diagram at VDC = 250 µV. One can clearly see that the triple points have grown both in size and in intensity and assume the shape of small triangles. In Fig. 8.6 (c) we increase the bias voltage again to VDC = 500 µV and measure the same region in VPGL and VPGR. There, two clear triangular regions in which current is allowed to flow have emerged from the two triple points. In order to understand how these so called finite bias triangles come about, we turn to the energy levels within the dot at the specific points within these triangles. Fig. 8.7 (a) shows a zoom into the plot of Fig. 8.6 (c), the charge stability diagram around (0,2) and (1,3) occupation at VDC = 500 µV. The three corners of the left

102 8.4. Finite bias triangles emerging from triple points

(a) (b)

(1,3) (0,3)

(0,2) (1,2)

VDC = 100 μV VDC = 250 μV

(c)

VDC = 500 μV

Figure 8.6: (a) Current IDC at a bias voltage of VDC = 100 µV as a function of voltages of the left and right plunger gates, VPGL and VPGR, between (0,2) and (1,3) charge occupation. Only the triplet points are visible. (b) IDC at VDC = 250 µV shows the emergence of ﬁnite bias triangles. (c) At VDC = 500 µV, large and regular ﬁnite bias triangles in IDC have developed.

finite bias triangles are marked with green, blue and red points and schematics of their corresponding energy levels are depicted in Fig. 8.7 (b)-(d). Within the finite bias triangle we show two new axis corresponding to a rotated coordinate frame. They are the ε-axis along the baseline of the triangle, and the δ-axis perpendicular to that along the altitude of the triangle and their meaning is explained in the following. At the green point in the finite bias triangle the energy levels are aligned as shown in Fig. 8.7 (b). The energy levels in the double dot system become resonant with the level in the source lead and transport becomes possible. When now simultaneously decreasing the voltages on both plunger gates, the energy levels in both the left and the right dot move to lower energies, until we reach the other end of the baseline of the finite bias triangle, denoted by the blue point. There, as shown in Fig. 8.7 (c), the energy levels of the double dot have moved through the whole bias window and are now resonant with the drain lead. If they were to move below the Fermi level

103 Chapter 8. Double quantum dots

(a) (b)

(d) V = −500 μV VDC = +500 μV DC ε δ

Figure 8.7: (a) Current IDC at a bias voltage of VDC = 500 µV as a function of VPGL, the voltage on the left, and VPGR, the voltage on the right plunger gate, recorded around (0,2) and (1,3) charge occupation. δ and ε define a rotated coordinate system along the axis of the finite bias triangle. (b)-(d) Energy level schematics at the green, blue, and red points within the finite bias triangle of (a). The detuning axis δ and the energy axis ε are defined.

in the drain, the dot would become occupied by one more electron and transport ceases as we leave the ﬁnite bias triangle. Therefore, moving along the ε-axis we simultaneously change both energy levels of the double dot system. If one adjusts the energy levels in a way that they are in the middle of the bias window, which corresponds to the point halfway between the green and blue points one can reach the red point at the tip of the triangle by charging VPGL to a more negative and VPGR to a more positive voltage. This corresponds to raising the energy level in the left dot and decreasing the energy level in the right dot and the level alignment is shown in Fig. 8.7 (d). Having moved along the δ-axis, we have changed the diﬀerence between levels in the left and right dot, therefore it is referred to as the detuning axis.

As we have seen in Fig. 8.6, the ﬁnite bias triangles grow linearly with VDC, as their size directly corresponds to the size of the bias window. Similar to the case of a single quantum dot, this enables us to get an energy calibration and determine the lever arm of our gates on the energy levels in the double dot system. This can be done for both of the dots separately. For instance, moving along a line from the green to the red point, we shift the level in the right dot across the full bias window, while red green the level in the left dot stays unchanged. This means that ∆VPGR = VPGR − VPGR divided by VDC lets us determine the lever arm of the right plunger gate on the right dot. As the right plunger gate also has a ﬁnite lever arm on the level of the left dot, the left plunger gate needs to be adjusted slightly such that the left dot stays on resonance with the Fermi level in the source lead. The voltage required for this green red is ∆VPGL = VPGL − VPGL, and the ratio of ∆VPGR and ∆VPGL gives us information about the ratio of the lever arms of the right plunger gate on the levels in the right

104 8.5. Singlet-triplet spin blockade by Pauli exclusion at strong interdot coupling

and left dot. The same analysis can be done for the line connecting the red and blue points to determine the lever arms of both gates on the levels in the left dot.

8.5 Singlet-triplet spin blockade by Pauli exclusion at strong interdot coupling

In the previous section we outlined how at finite bias in a double dot triangular regions of allowed current flow are formed. At low electron numbers and above closed shells however, a vivid manifestation of quantum mechanics and spin physics can alter this picture. Current rectification, or allowed flow in one bias direction but suppressed flow in the other bias direction, can occur due spin selection rules at specific transitions. This phenomenon called spin blockade has been observed in single quantum dots [183] as well as double quantum dots [184, 185]. We focus on the singlet-triplet spin blockade of a double quantum dot in this section and begin by detecting its presence and explaining its origin. All measurements shown in this section are carried out at strong interdot coupling, where the electrons are strongly delocalized between the two quantum dots. In section 8.6 we revisit the spin blockaded transition again at weak interdot coupling in order to investigate how the coupling between the dots affects the transport behavior.

(a) μ (b) T μ S μS

μT (c) μ S μS

Figure 8.8: (a) Measurement of ﬁnite bias triangles across the transition from (1,3) to (0,4) charge occupation at a bias voltage of VDC = 500 µV. The current IDC as a function of VPGL, the voltage on the left, and VPGR, the voltage on the right plunger gate, vanishes within the triangles due to spin blockade and only faint lines at the left and right boundary lines of the left and right triangles respectively are visible. (b) Energy level schematic at the yellow point within the ﬁnite bias triangle illustrating spin blockade due to a forbidden transition from (1,1) triplet to (0,2) singlet. (c) Energy levels at the cyan point where an electron in the (1,1) triplet state can be exchanged with the leads and tunnel back in the (1,1) singlet state, allowing transport.

105 Chapter 8. Double quantum dots

Fig. 8.8 (a) shows a color plot of the current IDC as a function of left and right plunger gates voltages, VPGL and VPGR around the region of total occupation of the dot with four electrons. The applied bias voltage of VDC = +500 µV has been chosen in such a way that we probe the transition from the charge state (1,3) to the charge state (0,4). However, instead of observing finite bias triangles introduced in section 8.4 we see a vanishing current except for two very faint lines of suppressed current. In the white dashed lines we plot the region of the finite bias triangles expected for this bias, and we see that the two faint lines align with the left side of the left triangle and the right side of the right triangle. In order to understand this current suppression, we turn to the energy level diagrams of the double dot system both within the suppressed region at the yellow point in Fig. 8.8 (a) and at the outer boundary of the triangles at the cyan point, where a small current is flowing. The two extra electrons in the right dot form a closed first shell and lie energetically far below the levels contributing to transport, which is why they are not relevant for the spin blockade and we consider a situation of transport from (1,1) to (0,2) which is identical to (1,3) to (0,4). In Fig. 8.8 (b) we see a schematic energy level diagram at the yellow point within the expected region of the finite bias triangle. There, both the (1,1) singlet as well as the (1,1) triplet are within the bias window on the left dot, whereas the right already contains an electron and the (0,2) singlet is the only state that energetically can allow an electron to sequentially tunnel through the dot. If an electron from the left lead enters the double dot in the (1,1) singlet, it is free to tunnel into the (0,2) singlet and then enter the right lead. However, if the electron tunneling into the dot has the same spin as the electron of the right dot, it cannot occupy the (1,1) singlet and has to tunnel into the (1,1) triplet. The electron now can not tunnel into the (0,2) singlet due to the Pauli exclusion principle, as it has the same spin as the electron already occupying the right dot. Therefore, barring a spin flip, the electron is stuck in the (1,1) triplet configuration and the current comes to a stop due to spin blockade. Fig. 8.8 (c) depicts the situation which allows a finite current to flow on the outer edges of the finite bias triangles, such as for instance at the cyan point in Fig. 8.8 (a). As can be seen in the schematic, the energy of the (1,1) triplet at this point lies within the thermal broadening of the Fermi level in the source lead. Therefore an electron stuck in the (1,1) triplet state can with a finite probability tunnel back into the lead and another electron of a different spin can tunnel back into the dot, then occupying the (1,1) singlet state which does not lead to spin blockade. This effective exchange of spin with the lead allows a finite current to flow along this edge. A similar process for holes allows current to flow on the outer edge of the right triangle [185]. Understanding the origin of this singlet-triplet spin blockade, we acknowledge that it is bound to occur whenever we have a situation of current flow from a (1,1) triplet to a (2,0) or (0,2) singlet, and also at higher electron occupations where we have closed shells beneath the singlet or triplet states in either dot of the double dot system. We show a schematic charge stability diagram in Fig. 8.9 where the charge

106 8.5. Singlet-triplet spin blockade by Pauli exclusion at strong interdot coupling

(0,4)

(0,3) (1,3)

(0,2) (1,2) (2,2)

(0,1) (1,1) (2,1)

(0,0) (1,0) (2,0)

Figure 8.9: Schematic charge stability diagram with arrows indicating whether a transition is expected to be allowed (green) or spin blockaded (red). Labels within charge stability regions denote electron occupation of left and right dot, respectively. stability regions are labelled with the respective electron occupations of the left and right dots. The transitions between regions of the same total charge are marked with green arrows where current is expected to ﬂow in this direction, and with red arrows where current ﬂow is expected to be spin blockaded in the respective direction. In the following, we are measuring all of the transitions expected to be blockaded in this diagram in order to characterize the singlet-triplet spin blockade in our system and to see whether it matches this pattern.

8.5.1 Investigating the transition between (0,3) and (1,2) charge occupation We begin by investigating the transition at total charge occupation of three at the transition between (0,3) and (1,2). As can be seen from the schematic charge stability diagram in Fig. 8.9, we expect this transition to be allowed in both forward and reverse bias. This is due to the fact that the closed shell occupied with two electrons in the right dot does not affect the transport, which means that we effectively shuffle a single spin between two dots of zero total spin, comparable to a (0,1) to (1,0) transition. In Fig. 8.10 we can see the resulting transport behavior. At a positive bias of VDC = +500 µV shown in Fig. 8.10 (a), we drive the (1,2) to (0,3) transition and two regular finite bias triangles appear in the magnitude of IDC as a function of VPGL and VPGR. The same situation occurs in Fig. 8.10 (b) for reverse bias of VDC = −500 µV which corresponds to the (0,3) to (1,2) transition. The scan frames in this section

107 Chapter 8. Double quantum dots

(a) (b)

VDC = +500 μV VDC = −500 μV

Figure 8.10: (a) Finite bias triangles visible in the magnitude of IDC as a function of VPGL and VPGR at VDC = +500 µV, corresponding to the allowed transition (1,2) to (0,3). (b) Finite bias triangles in the magnitude of IDC in the reverse direction at VDC = −500 µV. The (0,3) to (1,2) transition is allowed as well.

are chosen in such a way that the range in both VPGL and VPGR are always 10 mV so that the ﬁnite bias triangles can be compared directly. Due to a mistake during the measurement, data was not collected for all points in Fig. 8.10 (b), which is why these points are left blank. These measurements conﬁrm the expectation that both directions of this transition should be allowed. (a) (b) B = 100 mT B = 100 mT

VDC = −500 μV VDC = +500 μV

Figure 8.11: (a) Allowed transition from (1,2) to (0,3) occupation at VDC = +500 µV and at Bk = 100 mT. The Zeeman splitting of the resonances can be seen within the ﬁnite bias triangles. (b) Reverse direction at VDC = −500 µV from (0,3) to (1,2) at Bk = 100 mT. Similar to (a), a Zeeman splitting within the ﬁnite bias triangles occurs.

We repeat the measurements shown in Fig. 8.10 at an in-plane magnetic ﬁeld of Bk = 100 mT. The corresponding ﬁnite bias triangles for the (1,2) to (0,3) direction

108 8.5. Singlet-triplet spin blockade by Pauli exclusion at strong interdot coupling

at forward bias are shown in Fig. 8.11 (a) and the finite bias triangles for the (0,3) to (1,2) transition are shown in Fig. 8.11 (b). The magnetic field leads to a small Zeeman splitting of the resonances of the dot as now a single electron in the left or right dot does not have a single, spin-degenerate state but rather two energetically split states for the two spin species. This is witnessed by a double step of the edges of the finite bias triangles visible for both transport directions in Fig. 8.10 (a) and (b).

8.5.2 Investigating the transition between (1,3) and (2,2) charge occupation In this section we add one additional electron to the double dot system to reach a total charge of four and probe the transition between (1,3) and (2,2) occupation. Comparing with Fig. 8.9, one expects spin blockade to inhibit transport from the (1,3) to the (2,2) state whereas from (2,2) to (1,3) current ﬂow should proceed unimpeded.

(a) (b)

VDC = +500 μV

VDC = −500 μV

Figure 8.12: (a) Magnitude of IDC as a function of VPGL and VPGR at VDC = +500 µV, corresponding to the transition from (2,2) to (1,3). Finite bias triangles indicate transport in this direction. (b) Same region at reverse bias VDC = −500 µV. The (1,3) to (2,2) transition is strongly suppressed by spin blockade.

The measurement of the absolute value of IDC in the two-dimensional parameter space spanned by VPGL and VPGR encompassing the transition between (1,3) and (2,2) occupation can be seen in Fig. 8.12. In (a) we apply a forward bias of VDC = +500 µV, driving the transition from (2,2) to (1,3). We expect this process to be possible, as either of the two electrons in the left dot can tunnel to the right dot, which exhibits a closed shell of two electrons. As can be seen in the data, two ﬁnite bias triangles are visible indicating the expected current ﬂow. Fig. 8.12 (b) shows the same transition at reverse bias of VDC = −500 µV. There, transport should occur from the (1,3) to the (2,2) state and is expected to be spin blockaded, as an electron entering the right dot in a triplet state can not pass into the (2,2) singlet.

109 Chapter 8. Double quantum dots

This can also be seen from the data, where the current within the bias triangle is strongly suppressed such that it can not be differentiated from the background. A finite amount of current is permitted to flow at the outside egdes of the triangles as there the triplet state is within the thermal broadening of the Fermi level in the source lead and can exchange an electron with the leads which can tunnel back into the dot in a singlet state, as discussed in the context of Fig. 8.8.

8.5.3 Investigating the transition between (0,4) and (1,3) charge occupation Having investigated the transition from (1,3) to (2,2) occupation in the previous section, we now turn to the other transition from the (1,3) charge stability region at a total occupation of four and probe the transition between the (0,4) and (1,3) stability regions. At this transition, we expect the direction from (1,3) to (0,4) which occurs at positive bias to be spin blockaded as the electron in the (1,3) triplet can not tunnel into the (0,4) singlet due to the Pauli exclusion principle. Consequently, transport from (0,4) to (1,3) at negative bias should proceed uninhibited, as an electron in the (0,4) singlet can always tunnel into the left dot to occupy a (1,3) state.

(a) (b)

VDC = +500 μV

VDC = −500 μV

Figure 8.13: (a) Magnitude of IDC as a function of VPGL and VPGR at VDC = +500 µV, corresponding to the transition from (1,3) to (0,4). The current flow is suppressed due to spin blockade. (b) Same region at reverse bias VDC = −500 µV. Current flows in the direction of (0,4) to (1,3) as finite bias triangles become visible in IDC.

The corresponding measurement can be seen in Fig. 8.13 where we plot the magnitude of IDC in the region of VPGL and VPGR corresponding to the transition between the occupations (0,4) and (1,3). A bias of VDC = +500 µV is applied in Fig. 8.13 (a) which means that the transition is driven from (1,3) to (0,4) and as is expected, spin blockade is observed. Current within the triangle is completely suppressed and only the outside edges of the ﬁnite bias triangles carry a current. Applying the opposite bias voltage of VDC = −500 µV and therefore operating the transition from (0,4) to

110 8.5. Singlet-triplet spin blockade by Pauli exclusion at strong interdot coupling

(1,3) should lead to an observable current. Fig. 8.13 (b) shows this situation and the corresponding current flow within the finite bias triangles, and thus proves that the rectifying behavior expected from the spin blockade between forward and reverse bias holds true also for this transition. At this point we would like to note that all the experiments thus far have been performed at zero external magnetic field and spin blockade was clearly observed. This is quite surprising, as in early experiments in GaAs it has been shown that spin blockade is not pronounced at zero magnetic field due to hyperfine coupling between electron and nuclear spins [186–188]. An electron in a semiconductor quantum dot is exposed to a large number of nuclear spins from the atoms of the crystal surrounding it, which produce a fluctuating magnetic field. This so-called Overhauser field is generally going to be different for the electrons in the left and right dot of a double quantum dot. Due to the randomly fluctuating fields in both dots, the spins in these dots are not in the same singlet-triplet basis anymore, but have eigenstates that are admixtures of these [187]. Therefore, a finite leakage current can at zero magnetic field flow in the region which is expected to be spin blockaded. The blockade can be restored by the application of a small magnetic field, which in previous experiments was of the order of 20 mT[187]. The small magnetic field polarizes the nuclear spins in the crystal and thus guarantees that the spins in both dots obey the same quantization axis. Thereby, the singlet-triplet mixing is suppressed and an electron can not tunnel from a (1,1) triplet to an (0,2) singlet state. At all transitions investigated in our sample at strong interdot coupling, a strong spin blockade was found already at zero magnetic field, in contrast to what has been observed previously in GaAs. Comparing nuclear spins of the crystal, the two most abundant isotopes of gallium, 69Ga and 71Ga each have a nuclear spin of −3/2, whereas the two most abundant isotopes of indium, 113In 115In both feature a nuclear spin of +9/2. The arsenic atoms, purely 75As with spin −3/2, are common to both InAs and GaAs. Therefore, we do not expect a smaller contribution from hyperfine coupling which could prevent singlet and triplet states from mixing. The spin-orbit interaction however, which can also influence mixing between singlet and triplet states, is expected to be much stronger in InAs. Theoretical work suggested that, for small magnetic fields where the (1,1) states are not strongly Zeeman split they are rearranged by the spin-orbit coupling to four new states [189]. Of these, three states are blocked and one can decay via the (0,2) state, which is why spin blockade is conserved at these small external fields [189]. When the magnetic field is increased, it couples the blocked states to the decaying one, which eventually leads to a lifting of the spin blockade [189]. Therefore, a dip in the current as a function of magnetic field centered around zero is expected within this model [189] and similar features have been seen in experiments on InAs nanowire double quantum dots [190, 191]. We investigate the dependence of the current within the spin blockaded region on the parallel magnetic field Bk in Fig. 8.14 (a) while also varying the detuning δ. Around Bk = 0 a region of clear current suppression, or pronounced spin blockade, is visible. The regions of current flow which form with increasing Bk take the shape of

111 Chapter 8. Double quantum dots

(a) (b)

Figure 8.14: (a) Dependence of the leakage current IDC in the spin blockaded transition from (1,3) to (0,4) on the parallel magnetic field Bk and the detuning δ. Surprisingly, the current is strongly suppressed at zero applied field. (b) Line cuts of IDC as a function of Bk at different values of δ. resonances and their slope corresponds to the g-factor measured from excited state spectroscopy, when converting the δ-axis to energy by taking into account the lever arm determined from the finite bias measurements. In Fig. 8.14 (b) we show line cuts at small values of detuning as a function of Bk. There, the dip in the current around zero magnetic field becomes evident again. What is more however, one can notice a small additional but sharper dip that is superimposed on the large dip around zero magnetic field. Especially for the green and red data points corresponding to δ = 2 mV and δ = 3 mV this narrow dip becomes very apparent in Fig. 8.14 (b). The origin of this additional feature is related to spin-orbit coupling modifying the inter-dot tunnel coupling, causing it to acquire non-spin conserving elements.

8.6 Singlet-triplet spin blockade by Pauli exclusion at weak interdot coupling

In the previous section we have investigated spin blockade in the regime of strong interdot coupling, which in the analogon of the artificial molecule corresponds to a covalent bonding, with the electrons being shared between the two atoms. There, we observed a pronounced spin blockade at zero magnetic field which can be explained by spin-orbit mediated mixing of singlet and triplet states [189]. Now we tune the dot to the regime of weak interdot coupling, where the electrons are strongly localized on their respective quantum dot emulating ionic binding of the artificial molecule. In the following, we measure spin blockade at the same transitions that we have investigated previously in order to compare the effect of the interdot coupling, and additionally also review the transitions at (1,1) and (0,2), where no excess electrons are present in either of the quantum dots. We then compare our results qualitatively

112 8.6. Singlet-triplet spin blockade by Pauli exclusion at weak interdot coupling

and quantitatively to the behavior that has been observed both in double dots in InAs nanowires as well as in GaAs two-dimensional electron gases.

8.6.1 Investigating the transition between (1,3) and (2,2) charge occupation The first transition that we revisit at weak interdot coupling is between the charge stability regions of (1,3) and (2,2) occupation. For strong interdot coupling we have seen in Fig. 8.12 of section 8.5.2 that current flow is allowed for positive bias in the direction from (2,2) to (1,3) which corresponds to excess electrons transitioning from the (2,0) singlet via the (1,1) state into the drain lead. For negative bias, we observed that the transition from the (1,1) triplet to the (2,0) singlet was forbidden due to spin blockade. Contrary to GaAs, we found that this spin blockade was pronounced at zero field and exhibited a pronounced dip as a function of the magnetic field.

(a) (b)

VDC = +500 μV VDC = −500 μV

Figure 8.15: (a) Magnitude of IDC as a function of VPGL and VPGR at VDC = +500 µV and at weak interdot coupling. Transport occurs as the transition is driven from (2,2) to (1,3). (b) Same region at reverse bias VDC = −500 µV. The (1,3) to (2,2) transition is strongly suppressed by spin blockade, but in contrast to strong coupling current ﬂow is not suppressed at a line of δ = 0.

In Fig. 8.15 (a) we show the magnitude of IDC as a function of the voltage on the left plunger gate VPGL and the voltage on the right plunger gate VPGR at a bias voltage of VDC = +500 µV. Just like for strong interdot coupling, transport occurs and finite bias triangles are visible. What is different here however, is a very pronounced baseline of the finite bias triangles at zero detuning, connecting the two triple points. The color scale has been chosen to be the same for all figures of the spin blockade in order to be able to better compare the degree of current suppression in the blockaded regime, but on the baseline IDC reaches values of the order of 50 pA, a factor of five larger than the current in the strong coupling regime in this region. The current strongly decreases when going from the baseline to the tip

113 Chapter 8. Double quantum dots

of the triangles, which is also a qualitative difference to the strong interdot coupling, where the current level was mostly uniform within the finite bias triangle. At the reverse bias of VDC = −500 µV we encounter spin blockade of the current in the transition from (1,3) to (2,2), where the (1,1) triplet of excess electron can not tunnel into the (2,0) singlet due to conservation of spin. We immediately recognize three qualitative differences to the strong coupling regime of Fig. 8.12 (b). The current at the triple points is stronger for weak interdot coupling. The edges of the triangles corresponding to current flow enabled by spin exchange with the leads while the (1,1) triplet is within thermal broadening of the Fermi level in the source lead, is not very pronounced as compared to strong interdot coupling. Thirdly, and most interestingly, the spin blockade is lifted for zero detuning, along a line connecting the two triple points corresponding to the levels in the two dots being on resonance, where finite current flows at weak interdot coupling, which was not the case for strong interdot coupling. In order to better understand why this is the case, we study the Bk dependence of the current in the spin blockaded region.

(a) (b)

Figure 8.16: (a) Magnitude of IDC at the spin blockaded (1,3) to (2,2) transition as a function of the detuning δ and the magnetic ﬁeld Bk. The peak at zero detuning splits up into two Zeeman splitting branches and one sharp resonance at Bk = 0. (b) Zoom into the yellow dashed rectangle of (a) showin the resonance at zero external magnetic ﬁeld. The resonance persists over a small range in δ and is very in Bk.

Fig. 8.16 (a) shows how the current IDC in the spin blockaded region depends on the external magnetic field Bk and the detuning δ. The pronounced baseline of the spin blockaded finite bias triangle that we saw in Fig. 8.12 (b) corresponds to the finite current at zero field and zero detuning. This point splits up into three branches. As also observed for strong interdot coupling, we see a Zeeman splitting of the resonances corresponding to the two triplet states where spin blockade is lifted. More interestingly however, we see a straight line that is extremely narrow in Bk and centered around zero field. With increasing detuning, this resonance becomes weaker while the two Zeeman split resonances increase in current. In order to investigate this feature in more detail we zoom into the region around zero field demarcated by the

114 8.6. Singlet-triplet spin blockade by Pauli exclusion at weak interdot coupling

Isotope Abundance (%) Nuclear spin I magnetic moment µ 75As 100 3/2 1.4395 113In 4.29 9/2 5.5289 115In 95.71 9/2 5.5408

Table 8.1: Nuclear magnetic resonance data for the relevant isotopes of the elements composing InAs. Taken from Ref. [3]. yellow dashed rectangle in (a). The corresponding data can be seen in Fig. 8.16 (b) where the narrow resonance with a full width at half maximum (FWHM) of the order of 500 µT becomes evident. The resonance decays quickly with increasing detuning. It is not centered exactly at Bk = 0 due to residual magnetization of the magnet in our experimental setup. A peak in the current at zero magnetic field at weak interdot coupling has been observed previously in other experiments on double dots in GaAs [187] and InAs nanowires [190, 191]. This peak corresponds to a lifting of the spin blockade due to a mixing of singlet and triplet states mediated via the hyperfine interaction as described previously. By applying an external magnetic field, spin blockade is restored once the Zeeman energy is larger than the energy EN = gµB|∆BN| that corresponds to the difference in effective nuclear magnetic fields in the two quantum dots [187]. ∆BN = BNL − BNR where BNL and BNR are the effective nuclear fields in the left and right quantum dots√ that are fluctuating in an uncorrelated fashion [187]. These fields scale as BN,max/ n with a maximal magnitude of the nuclear field BN,max in case that all nuclei are polarized and n the number of nuclei in a particular quantum dot [192, 193]. A fit to the shape of the current peak as a function of magnetic field can be used to extract the magnitude of the fluctuating Overhauser field [187]. In GaAs, fully polarized nuclear spins contribute to a maximum nuclear field of 6 BN,max ≈ 5 T [194] and one can for typical dot sizes assume n ≈ 10 nuclei resulting in a fluctuating nuclear field of roughly 5 mT in each of the dots, which corresponded also to the width of the resonance, or the field that had to be applied in order to restore spin blockade [187]. Here, we have a resonance that is about a factor of 10 sharper, which means that one has to assume that the magnitude of the fluctuating magnetic field is also a factor of 10 smaller in the vicinity of our InAs quantum dots than in the GaAs quantum dots. In order to qualitatively understand this, we need to consider the two main differences to GaAs quantum dots, which are the different nuclear spins of In compared to Ga atoms, as well as the larger electronic g-factor. The nuclear spins of the relevant isotopes of In and As are given in table 8.1. The resulting Zeeman energy of a nuclear spin is given as µ E = −g µ I B = − µ I B, (8.1) Z N N z I N z for a nuclear spin projection Iz and in an applied magnetic field B. The maximum spin splitting, or Zeeman energy difference between Iz = +I and Iz = −I, is there-

115 Chapter 8. Double quantum dots

fore 1.05 mK T−1 for 75As and 4.05 mK T−1 for 113In and 115In. This means that at the temperature of our systems huge magnetic fields would be required to polarize the nuclear spins and we can thus consider them randomly distributed. The probability distributions of Iz of both In and As atoms have zero mean and variances of 2(81/4 + 49/4 + 25/4 + 9/4 + 1/4) 33 Var(I(In)) = = , (8.2) z 4 4 and 9/4 + 1/4 + 1/4 + 9/4 5 Var(I(As)) = = . (8.3) z 4 4 For a large number N of nuclei the probability distribution of the total spin projection approaches a normal distribution according to the central limit theorem. Its mean is again zero and its variance is N ·Var(Iz). Therefore, we denote the standard deviation of the distribution as q r (In) (In) 33N σ = N · Var(Iz ) = , (8.4) Iz 4 q r (As) (As) 5N σ = N · Var(Iz ) = . (8.5) Iz 4 The polarizations can now be obtained by dividing the standard deviation by the product of the number of nuclei N and their respective nuclear spin I. These are of similar size, √ σ(In) 33 0.64 Iz = √ ≈ √ , (8.6) NI(In) 9 N N √ σ(As) 5 0.75 Iz = √ ≈ √ . (8.7) NI(As) 3 N N We can use these results to estimate the typical magnitude of the magnetic field stemming from the nuclear spin bath of the lattice that we would expect in the quantum dot. The Hamiltonian of the hyperfine interaction between the nuclear spins and an s-type conduction band electron is [194]

X 2µ0 X H = g µ µ d I(α,i)S |Ψ(r )|2, (8.8) HF 3 N N B α z z i α i∈α with the envelope wave function Ψ(r) of the electron localized in the quantum dot, the parameter α ∈ In, As specifying the type of nucleus, and dα the electronic density at the respective nucleus. This equation can be transformed to

X X (α) (α,i) HHF = Ai Iz Sz, (8.9) α i∈α

(α) (α) 2 (α) with Ai = A (v0/2)|Ψ(ri)| . Here, A are the hyperﬁne coupling energies of In and As under the assumption of two electrons in a unit cell of volume v0. These

116 8.6. Singlet-triplet spin blockade by Pauli exclusion at weak interdot coupling

coupling energies are material constants with the values of A(In) ≈ 170 meV [195] and A(As) ≈ 86 meV [194]. Assuming that the electron in a quantum dot has a constant (α) wave function inside of the dot that vanishes outside of the dot, we simplify Ai = A(α)/2N when to dot contains N primitive unit cells. Therefore the Hamiltonian of the hyperﬁne interaction now becomes " # 1 X X H = A(In) I(In),i + A(As) I(As),i S . (8.10) HF 2N z z z i∈In i∈As

Comparing this to a Zeeman Hamiltonian of the form H = gµBBN Sz we can read off the effective nuclear field BN " # 1 (In) X (In),i (As) X (As),i BN = A Iz + A Iz . (8.11) 2gµBN i∈In i∈As The standard deviation of this nuclear field can be determined by making use of the standard deviations of the nuclear spin projections and amounts to 1 h i σ = A(In)σ(In,N) + A(As)σ(As,N) . (8.12) BN Iz Iz 2gµBN With the values of these standard deviations calculated earlier, we receive √ √ 1 h (In) (As) i σBN = √ A 33 + A 5 (8.13) 4gµB N 316 mT ≈ √ , (8.14) N when using a g-factor of 16 as determined from the splitting of the first electron resonance and the splitting of the Kondo resonance in chapter7. Assuming a quantum dot size similar to typical GaAs devices, we can assume N ≈ 106 nuclei in the µ vicinity of the quantum dot, resulting in a fluctuating field of σBN ≈ 320 T, very similar to what has been observed in Fig. 8.16. We can thus see that the effect of the larger nuclear spin of 9/2 of the In isotopes compared to the nuclear spin of 3/2 of the Ga isotopes√ does not√ have a significant influence on the√ fluctuating nuclear 1 (In) (As) (Ga,As) field as 2 [A 33 + A 5] for InAs is replaced by A 5 for GaAs which is a factor 0.33 smaller. The main difference stems from the g-factor of InAs which is in our case a factor of 40 larger than the g-factor of GaAs. Taking both effects into account, this leads to a fluctuating nuclear field that is expected to be 40 · 0.33 ≈ 13 times smaller for an InAs quantum dot compared to a GaAs quantum dot of the same geometry.

8.6.2 Investigating the transition between (0,4) and (1,3) charge occupation To further analyze the behavior of the spin blockaded transitions at weak interdot coupling and to see whether the behavior we observed in the previous section is

117 Chapter 8. Double quantum dots

reproduced we turn to the second spin blockaded transition that is present at the opposite boundary of the (1,3) charge stability region. We apply a bias of VDC = +500 µV and drive a transition from (1,3) to (2,2) occupation. The resulting current as a function of VPGL and VPGR can be seen in Fig. 8.17 (a). The current is clearly spin blockaded, and we see qualitatively similar features as in the (1,3) to (2,2) transition. At resonant tunneling between the dots, the spin blockade is lifted and thus a baseline of the ﬁnite bias triangles at δ = 0 is visible. Additionally, at both triplet points a strong current is permitted to ﬂow. The baseline of this transition shows a small jump which is due to a charge rearrangement that proved to be reliable over subsequent measurements of this area.

(a) (b)

VDC = +500 μV

Figure 8.17: (a) Magnitude of IDC as a function of VPGL and VPGR for the spin blockaded transition from (1,3) to (0,4) occupation at VDC = +500 µV at weak interdot coupling. Current is strongly suppressed except for resonant tunneling at δ = 0. (b) Dependence of the current in the spin blockade on the detuning δ and the magnetic ﬁeld Bk. The peak at zero detuning shows Zeeman splitting and a narrow resonance at Bk = 0 which vanishes with δ.

In Fig. 8.17 (b) we study the resonant current through the dot at the baseline of the finite bias triangles connecting the two triple points in more detail. Plotting the magnitude of IDC we vary Bk and the detuning δ. We see a similar behavior as has been found in the previous section 8.6.1. At Bk = 0, a sharp resonance is visible that decays as δ is increased. This corresponds to the region where the spin blockade is lifted due to the randomly fluctuating magnetic fields from the spins of the nuclei which are different in both quantum dots. At larger magnetic fields, we see two resonances that imply a Zeeman split state contributing to transport.

8.6.3 Investigating the transition between (0,2) and (1,1) charge occupation At strong interdot coupling in section 8.5 only transitions that had a closed shell in the right dot were accessible for spin blockade experiments. There, we made use of

118 8.6. Singlet-triplet spin blockade by Pauli exclusion at weak interdot coupling

the fact that the closed shell lies energetically far below the Fermi level in the source and drain leads and could demonstrate blockade between the (1,1) triplet and the (0,2) singlet of the excess electrons. For weak coupling, where a much more negative voltage is applied to the middle barrier gates, we can excess the region without a closed shell below the singlet and triplet states in the dot. During this section and in the following one, we investigate spin blockade in a regime of total occupation of the double dot with two electrons.

(a) (b)

VDC = +500 μV

Figure 8.18: (a) Magnitude of IDC as a function of VPGL and VPGR for the spin blockaded transition from (1,1) to (0,2) occupation at VDC = +500 µV at weak interdot coupling. Current flows only when the two dots are on resonance at δ = 0 and on the outer edges of the finite bias triangles due to exchange of electrons of the left dot with the source lead within thermal broadening. (b) Dependence of the current in the spin blockade on the detuning δ and the magnetic field Bk. The peak at zero detuning shows Zeeman splitting and a narrow resonance at Bk = 0 which vanishes with δ.

A color plot of the magnitude of IDC in the vicinity of the (1,1) to (0,2) transition can be seen in Fig. 8.18 (a). We drive the transition from (1,1) to (0,2) with a bias voltage of VDC = +500 µV and a pronounced spin blockade is visible. Similar to the transitions with a closed shell below in the right dot investigated in the previous two chapters, strong current flow is possible at the triple points and at a straight line connection these, corresponding to δ = 0 and the dots being on resonance with each other. In this regime, we can see the finite current at the outer edges of the triangles that is possible because the electron in the (1,1) triplet can tunnel back into the source lead and then back into the dot with a flipped spin while it is within thermal broadening of the lead. Characterizing the leakage current in the spin blockade that occurs at the baseline of the finite bias triangles in an analogous fashion to the previous transitions, we tune the sample to a point on the baseline connecting the two triple points in the middle of the left triangle and observe the response to a magnetic field Bk while varying the detuning. A color plot of the resulting current is shown in Fig. 8.18 (b). The behavior is identical to the previous two transitions with an additional closed

119 Chapter 8. Double quantum dots

shell. The very narrow resonance at Bk = 0 wherein the spin blockade is lifted is visible and is attenuated with increasing detuning. Starting from δ = 0 we see two resonances that show a Zeeman splitting and that get more pronounced with increasing Bk, similar to the previously investigated transitions.

8.6.4 Investigating the transition between (2,0) and (1,1) charge occupation In order to fully characterize our double dot system we now investigate the other remaining transition at two electron occupation. We tune the dot into the (1,1) charge stability region and drive the transition towards the (2,0) state applying a negative bias of VDC = −500 µV. We show the resulting current IDC in a plane of VPGL and VPGR in Fig. 8.19 (a). As expected spin blockade between the (1,1) triplet and the (2,0) singlet suppresses the current in the region of the ﬁnite bias triangles. Similar to all previous spin blockaded transitions investigated at weak interdot coupling, a leakage current is observed for δ = 0 when the two dots are resonant. We note an irregularity between the two triple points, where the upper one is untypically broadened. Additionally, the outside edges of the ﬁnite bias triangles carry current enabled by thermal broadening in the vicinity of the Fermi level of the source lead.

(a) (b)

VDC = −500 μV

Figure 8.19: (a) Magnitude of IDC as a function of VPGL and VPGR at weak interdot coupling and at VDC = −500 µV. This drives the transition from (1,1) to (2,0) occupation, where spin blockade occurs. The blockade is lifted when the two dots are on resonance at δ = 0 and on the outer edges of the ﬁnite bias triangles due to exchange of electrons of the left dot with the source lead within thermal broadening. (b) Magnetic ﬁeld dependence of the leakage current in the spin blockade while increasing detuning δ. The peak at zero detuning splits into two Zeeman splitting branches and a narrow resonance at Bk = 0.

The effect of the magnetic field Bk on the spin blockaded finite bias triangle is shown in Fig. 8.19 (b). There, starting from a point at δ = 0 we vary the detuning towards the tip of the finite bias triangle in addition to Bk. The leakage current on

120 8.7. Conclusion

the baseline shows two Zeeman splitting branches and one much narrower resonance at Bk = 0, in accordance with what has been observed for the previous transitions at weak interdot coupling.

8.7 Conclusion

In this chapter we have built on the results of chapter7 and have shown that two single quantum dots in an InAs quantum well enabled by an entirely gate-defined structure can be coupled together to achieve few-electron double quantum dots. The tuning and coupling procedure of this system was described in detail in section 8.2. The resulting charge stability diagram is explained and presented during the course of section 8.3. In the next section 8.4 we apply a source drain bias and observe the resulting finite bias triangles arising at the triple points between charge stability regions. As our double quantum dot is in the few-electron regime, an interesting irregularity in the finite bias triangles between positive and negative bias at certain transitions due to a quantum effect has been observed. This so-called singlet-triplet spin blockade by Pauli exclusion between the (1,1) triplet and the (0,2) or (2,0) singlet states was characterized at strong interdot coupling in section 8.5 and at weak interdot coupling in section 8.6. At strong interdot coupling, we consistently observe a pronounced spin blockade even at zero external magnetic field. This could be explained by rearrangement of the (1,1) states into four new states while their Zeeman splitting is small. Of these states, one allows transport and the other three are blockaded until a magnetic field couples them and thereby destroys spin blockade. For weak interdot coupling, spin blockade is lifted for the resonant tunneling process at zero detuning. This can be understood as an effect of hyperfine coupling to the nuclear spin bath in the environment of the dot, which mixes singlet and triplet states. In a magnetic field, this leakage current reveals a narrow resonance around zero field which we believe to be associated to the magnitude fluctuating Overhauser field caused by the nuclear spins.

121 Chapter 9

Conclusion and outlook

9.1 Conclusion

The main objective of this thesis was the realization of nanostructures which allow the observation of quantum transport phenomena in InAs two-dimensional electron gases. We have identified and analyzed one crucial issue that prevented operating quantum devices in the way in which it was executed on GaAs: Fermi level pinning at the surface of InAs leads to electron accumulation at the sidewalls of an etched mesa structure. These accumulated charge carriers in turn lead to trivial edge conduction, which introduces a parasitic conductive channel inhibitive to quantum devices which rely on achieving full pinch-off of the electron gas. The edge conduction was studied in chapter4, where methods of chemical passivation intended to reverse this Fermi level pinning during the fabrication process were investigated. Treatments with ammonium sulfide, TAM, magnesium borohydride, and PCBM showed no or adverse effects on edge conduction, when compared to a reference sample passivated with only a silicon nitride dielectric layer. Edge conduction was also shown to exist and characterized in samples with purposely oxidized edges. Direct passivation with an aluminum oxide dielectric layer in an ALD process based on a trimethylaluminum precursor proved to be most promising, resulting in a linear edge resistivity of up to 72.8 kΩ µm−1. Entirely overcoming the technological challenge of trivial edge conduction de- manded a different solution, which was found in chapter5. We proposed and realized a device geometry based on a fabrication process without physical etching, where the mesa is defined electrostatically utilizing two layers of gates that are insulated by an aluminum oxide dielectric layer from the wafer surface and from each other. Experiments demonstrating the validity of this approach were conducted which in- cluded the realization of full pinch-off of a quantum point contact and the proof of principle operation of a quantum dot. A switch to heterostructures grown on a GaSb substrate enabled moving beyond proof of concept operation of quantum devices. In chapter6 quantum point contacts, which in contrast to previous works on etched constrictions facilitated full pinch-

122 9.2. Outlook

off and did not require subtraction of a parasitic conductance, were successfully realized. Three steps of quantized conductance were observed which showed spin splitting in a parallel magnetic field. The g-factor was determined to be |g| = 12.6, in accordance with estimations based on k · p calculations. Upon application of a perpendicular magnetic field, magnetoelectric subbands could be characterized and agreed with a theoretical model. We could show that the shape of the conductance trace is very sensitive to the coupling potential both on small and large length scales and that reentrant features and plateau-like regions at non-integer multiples of e2/h can be introduced by changing this potential. Building on these results we realized a gate-defined few electron quantum dot in chapter7. Measurements of Coulomb blockade diamonds allowed us to determine the energy scales of the system. At finite bias, excited state spectroscopy of the one- and two-electron states was performed, extracting a g-factor of |g| = 16.4 from the Zeeman splitting of the first Coulomb resonance. Singlet and triplet states were identified as ground and excited states for occupation with two electrons and intersected in a parallel magnetic field. The lack of an avoided crossing hints at a vanishing spin-orbit splitting between these states, likely resulting from an interplay of the direction of the in-plane field and the particular crystallographic orientation of transport. At strong coupling to the leads, the Kondo effect was observed in excellent agreement to theory. The magnitude of the splitting of the Kondo resonance in a magnetic field confirmed the value of the g-factor extracted from the excited state spectroscopy. Coupling two of these dots enabled the formation of double quantum dots in chapter8. At finite bias, the triple points in the charge stability diagram gave rise to finite bias triangles. Singlet-triplet spin blockade due to the Pauli exclusion principle was observed and characterized at multiple transitions. For strong interdot coupling, pronounced current suppression in the blockaded regions was found already at zero external magnetic field. In the region of weak interdot coupling, the spin blockade was lifted for the resonant tunneling process giving rise to a narrow peak in a magnetic field measurement, likely related to hyperfine interactions with nuclear spins of the surrounding crystal.

9.2 Outlook

The technology of utilizing multiple overlapping layers of gates to electrostatically deﬁne a mesa structure and a nanostructure within it was successfully developed in this thesis. Quantum devices which thus far in InAs two-dimensional electron gases have remained outside of the scope of etch-based means of fabrication have been realized using this technique. For the single and double quantum dots investigated in chapters7 and8, strong indications for reaching the regime of the last charge carriers are presented, yet without a charge detector this can not be unanimously proven. A natural next step

123 Chapter 9. Conclusion and outlook

would therefore be the fabrication of a double quantum dot device with a nearby quantum point contact or quantum dot that enables charge sensing [196, 197]. Follow-up experiments in such a double dot with a charge sensor can, due to the good tunability and stability of the material, already be geared towards basic qubit experiments such as the realization of a spin-orbit qubit [137, 138], or a singlet-triplet qubit [175, 176]. Extending the system to a triple quantum dot is straightforward and broadens the range of possible qubit implementations that can be explored. Additionally, one can expand the device by adding a superconducting microwave resonator and connecting it to one of the plunger gates of the quantum dot [198– 200]. This could enable the investigation of coupling between photons and and the spin of the qubit in an environment where strong spin-orbit coupling is present. Ultimately, going beyond the limitations of the material system InAs it is possible to expand the gate-deﬁned mesa technique introduced here to other narrow bandgap materials that suﬀer from similar Fermi level pinning issues. As most developments in modern physics are driven by ever increasing material quality or the discovery of new materials, it is conceivable that a new material will become of interest which is not processable by etching techniques. In that case and if the material is gateable, the multi-layer gating geometry can be readily applied to enable future quantum devices.

124 Publications

Experimental Signatures of the Inverted Phase in InAs/GaSb Coupled Quan- tum Wells M. Karalic, S. Mueller, C. Mittag, K. Pakrouski, Q. S. Wu, A. A. Soluyanov, M. Troyer, T. Tschirky, W. Wegscheider, K. Ensslin, T. Ihn Phys. Rev. B 94, 241402(R) (2016)

Lateral p-n Junction in an Inverted InAs/GaSb Double Quantum Well M. Karalic, C. Mittag, T. Tschirky, W. Wegscheider, K. Ensslin, T. Ihn Phys. Rev. Lett. 118, 206801 (2017)

Edge Transport in InAs and InAs/GaSb Quantum Wells S. Mueller, C. Mittag, T. Tschirky, C. Charpentier, W. Wegscheider, K. Ensslin, T. Ihn Phys. Rev. B 96, 075406 (2017)

Edgeless and Purely Gate-Deﬁned Nanostructures in InAs Quantum Wells C. Mittag, M. Karalic, Z. Lei, T. Tschirky, W. Wegscheider, T. Ihn, K. Ensslin Appl. Phys. Lett. 113, 262103 (2018)

Gate-Tunable Electronic Transport in p-Type GaSb Quantum Wells M. Karalic, C. Mittag, M. Hug, K. Shibata, T. Tschirky, W. Wegscheider, R. Winkler, K. Ensslin, T. Ihn Phys. Rev. B 99, 115435 (2019)

125 Phase Slips and Parity Jumps in Quantum Oscillations of Inverted InAs/GaSb Quantum Wells M. Karalic, C. Mittag, S. Mueller, T. Tschirky, W. Wegscheider, L. Glazman, K. Ensslin, T. Ihn Phys. Rev. B 99, 201402(R) (2019)

Electric-Field-Induced Two-Dimensional Hole Gas in Undoped GaSb Quan- tum Wells K. Shibata, M. Karalic, C. Mittag, T. Tschirky, C. Reichl, H. Ito, K. Hashimoto, T. Tomimatsu, Y. Hirayama, W. Wegscheider, T. Ihn, K. Ensslin Appl. Phys. Lett. 114, 232102 (2019)

Quantum Transport in High-quality Shallow InSb Quantum Wells Z. Lei, C. Lehner, E. Cheah, M. Karalic, C. Mittag, L. Alt, J. Scharnetzky, W. Wegscheider, T. Ihn, K. Ensslin Appl. Phys. Lett. 115, 012101 (2019)

Single and Double Quantum Dots in an InAs Two-Dimensional Electron Gas in preparation

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140 Acknowledgements

“Nada en esta vida vale dos duros si no tienes alguien con quien compartirlo.” – Carlos Ru´ızZaf´on

A PhD project is always more than just the work of one person and hence this thesis has only been made possible by the continuous support from and collaboration with a great number of truly wonderful people. Being in Switzerland, it appears natural to me to draw the analogy to a train ride. As you get on, you have certain people around you. During the journey your company constantly changes as the old knowns gradually get oﬀ while new faces join. Similar to a train breezing through the mountaineous landscape of the Alps, also the motivation of a PhD student everchangingly seesaws between deep valleys and vertiginous heights, depending on whether a measurement has produced an interesting result or a sample has just died. First and foremost, I would like to thank Klaus Ensslin who made this work possible by giving me the opportunity to embark on the journey that is pursueing a PhD. The way you manage the group is exemplary, knowing that happy PhD students produce the best research. You always have an open door for us to walk in and discuss any issue that occurred, no matter how busy your schedule is. I am thankful that I could learn from your intuition and experience on electronic transport, as well as from your pragmatic approach to solving problems. The atmosphere you managed to create in our group is unmatched, full of positivity, helpfulness, and mutual encouragement to strive to unravel more and more of the physics in our samples. I am not sure how to thank Thomas Ihn enough. Your immense knowledge and intuition on physics never cease to amaze me, and discussing new measurements with you made my project advance at lightning speed. We dug through every detail of the measurements until we knew exactly what was going on. The way you can ad-hoc derive important equations and make good estimates for just about any quantity based on a few reasonable assumptions is impressive. I still vividly remember the times when we went to the basement lab together and dusted oﬀ one of your favorite instruments, the spectrum analyzer, to examine the noise or the signal of my measurement in more detail. Having Daniel Loss as a co-referee on the committee of my PhD thesis is a great honor to me. I am keeping in good memory our encounter at a conference in Madrid where I was impressed by your presentation and numerous questions and comments

141 during the sessions. The friendly and informative discussion we had in the coffee break was very helpful to me along the way and I am thankful for your insightful comments and evaluation of my thesis. I would like to thank Werner Wegscheider and Thomas Tschirky for providing us with wafer material to perform our experiments on, and for all the meetings in which you shared you expertise on crystal growth and heterostructures. Reading and evaluating my thesis as a co-referee is something for which I am thanking Werner especially. The collaboration with Mike Manfra and Candice Thomas was an example of pro- fessional and efficient teamwork. Within short time after contacting you we were supplied with material, and once we had results we could always count on your fast feedback. In yet another efficient collaboration, Maksym Kovalenko and Olga Nazarenko helped us with the passivation project. Despite not managing to remove the edge states by passivation, we could learn a lot from you in this collaboration for which I am very grateful. Peter Märkiis a wizard of electronics and to me he resembles the human incar- nation of Gyro Gearloose. His numerous ingenious devices have made our lives in the lab easier and more productive, reduced measurement noise, and reinvigorated experimental setups, for which I am very thankful. Your contributions to the men- tal wellbeing of our group by providing apples, experiments, and interesting riddles should also not be forgotten. In a similar vein I would like to thank Erwin Studer and Thomas Bählerfor providing technical assistance and maintaining our pumps and vacuum equipment in the best of shape. Claudia Vinzens managed all organizational issues in a heartbeat and with great care, and on top of that always spread a good mood with a smile and a refreshing conversation, thank you very much for that. I would like to thank the whole FIRST cleanroom team, who maintain the facility very well and are always willing to help and have an open ear for improvements and requests. The other two members of our original InAs/GaSb subgroup are Susanne Müller and Matija Karalic. Susanne took great care of getting me started with my PhD, provided me with a place to stay while I was visiting apartments, introduced me to our cleanroom processing, and guided me along my first transport measurements, for all of which I am very thankful. Matija is the person with whom I worked together most closely, and we went through all the highs and lows of this project, always somehow coming up with an idea for a future experiment. We shared egregious fabrication runs, adventurous cool-downs and warm-ups in B17, complicated data analysis, but most importantly, many laughs and interesting conversations along the way. We cleared our mind from whatever was not working according to plan in the lab on some Tuesday or Friday nights over a game of cards. You are one of the most hard-working, dedicated, creative, and

142 smart people that I know and I do not think that my project would have been half as successful if it were not for you. For the single and double quantum dot I got a lot of help in tuning the device and interpreting data from Jonne Koski. Your expertise in quantum mechanics and the tuning of dots is unrivaled, and I am very grateful for your help. Andrea Hofmann, Beat Bräm,Tobias Krähenmann,Szymon Hennel, Pauline Si- monet, and Richard Steinacher were already on the train when I boarded and are the core members in the image of the Ensslin group that I carry in my mind. You made me feel part of the team even in the very moment in which I had just joined it. I am very thankful to all of you for the help I received, for the Friday beers we shared, for the Swiss German lessons, the soccer games, and for generally making my time here so enjoyable. Our new group members cherish this tradition and positive attitude, which makes me certain that the Ensslin group is headed to a great future personally and professionally. In Marius Eich, Giorgio Nicol`ıand Marc RöösliI have not only found exemplary lab members but also some of the best friends that I could have wished for. Together we shared so many laughs, cooked delicious dinners, worked out in the gym, ran in the forest, organized board game nights or parties, and even went on holidays and weekend trips. I am very thankful for all the great moments we had both in- and outside of the lab. Working on optics in diamond vacancies, Jonathan Zopes always provided me with a lot of invaluable advice from an outside perspective. We met within the first months of my PhD and I am very thankful for the great friendship that has developed. Going on a coffee break or a mountain bike tour together with you was always a great way to take a breather from what was going on in the lab. I am incredibly happy that Philipp Gadow, Andreas Rauscher and I could maintain our friendships even with our ways parting after the end of our studies together. Catching up with our lives when we were meeting in Munich or when you found the time to visit me always filled me with joy. Mama und Papa, I am so thankful to you for everything you have done for me, that I do not think that I can find the right words to express me. The amount of love and support you have for me brings tears to my eyes as I type. While your understanding for my demanding work and the distance between us remained boundless, I wish that I would have found the time to visit you more often. Now it is time for me to get off this train, but I am not doing so alone. Throughout endless journeys in buses, trains, and airplanes, you continuously supported me and provided a haven of bliss. Here we are, Julia - hand in hand - ready to begin a new chapter, eagerly looking forward to the start our common future.

143 Curriculum Vitae

Name: Christopher Mittag born: May 27, 1990 in Borna (Germany) citizen of Germany

01/2016 Start of Ph.D. ETH Zurich, Switzerland Laboratory for Solid State Physics, Nanophysics group Prof. Dr. K. Ensslin and Prof. Dr. T. Ihn

10/2013–12/2015 Master of Science in Physics Technical University of Munich, Germany Master thesis: “Mid-Infrared Frequency Comb Generation in Silicon Micro-Ring Resonators”

11/2014–11/2015 Research project (external Master thesis) Harvard University, Cambridge, USA Laboratory for Nanoscale Optics Prof. Dr. M. Lonˇcar

10/2010–08/2013 Bachelor of Science in Physics Technical University of Munich, Germany

08/2005–06/2009 Gymnasium “Am Breiten Teich”, Borna, Germany

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