Single-Layer and Bilayer Graphene Quantum Dot Devices Single-Layer and Bilayer Graphene

Christian Volk Christian Volk

Single-layer and bilayer graphene quantum dot devices Single-layer and bilayer graphene

This thesis reports low temperature transport experiments on sing- quantum dot devices le-layer and bilayer graphene quantum dot (QD) devices focusing on the analysis of excited state spectra and the investigation of the relaxation dynamics of excited states. Finite-bias spectroscopy measurements unveil the excited state spectrum of a graphene QD. RF pulsed gate spectroscopy is used to probe the relaxation dynamics of excited state to ground state transitions. Transient current spectroscopy yields an estimate of a lower bound for charge relaxation times on the order of 60 to 100 ns. The experimental results are compared to a basic model regarding electron-phonon coupling as the dominant relaxation mechanism. A bilayer graphene double quantum dot device is characterized. The capacitive interdot coupling can be tuned systematically by a local gate. The electronic excited state spectrum of a features a single-particle level spacing independent on the number of charge carriers on the QDs which is in contrast to single-layer graphene. Graphene nanoribbon based charge sensors are characterized as an important tool to detect individual charging events in close-by QDs. Back action effects of the detector on the probed QD are studied.

Dissertation 2015 · 2nd Institute of Physics · RWTH Aachen University Single-layer and bilayer graphene quantum dot devices | Christian Volk graphene quantum and bilayer Single-layer Single-layer and bilayer graphene quantum dot devices

Von der Fakult¨atf¨ur Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation

vorgelegt von

Diplom-Physiker Christian Volk aus K¨oln

Berichter: Universit¨atsprofessorDr. Christoph Stampfer apl. Professor Dr. Thomas Sch¨apers

Tag der m¨undlichen Pr¨ufung:26. Februar 2015

Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verf¨ugbar.

Abstract

Graphene quantum devices, such as single electron transistors and quantum dots, have been a vital field of research receiving increasing attention over the past six years. Quantum dots (QDs) made from graphene have been suggested to be an interesting system for implementing spin qubits. Compared to the well-established GaAs-based devices their advantages are the smaller hyperfine interaction and spin-orbit coupling promising more favorable spin coherence times. However, while the preparation, manipulation, and read-out of single spins have been demonstrated in GaAs QDs, research on graphene QDs is still at an early stage. So far, effects like Coulomb blockade, electron-hole crossover and spin-states have been studied in graphene QDs. This thesis reports low temperature transport experiments on single-layer and bilayer graphene quantum dot devices focusing on the analysis of excited state spectra and the investigation of the relaxation dynamics of excited states. Graphene nanoribbon based charge sensors are characterized as an important tool to detect individual charging events in close-by QDs. All devices are fabricated following the concept of width-modulated graphene nanostructures. Carving nanoribbons out of graphene sheets opens an effective band gap and offers the chance to circumvent the limitations originating from the gapless band structure in ”bulk” graphene. Shaping graphene into small islands (with typical diameters between 50 and 120 nm) connected to contacts via narrow ribbons (typical widths between 30 and 50 nm) allows the confinement of electrons and the formation of QDs with gate-tunable tunneling barriers. Graphene nanoribbon-based charge detectors coupled to QDs are characterized. They allow the resolution of charging events on the QD even in regimes where the direct current is below the noise floor and excited states of the QD can be

i resolved by this technique as well. The detector remains operating even in adverse conditions like under the influence of magnetic fields of several Teslas or while RF pulses in the MHz regime are applied to the QD. These findings are in particular relevant for experiments addressing spin states or relaxation processes in graphene QDs. Special emphasis lies on the influence of the detector on the transport properties of the probed QD. The effects of back action and counter flow are measured and discussed. The relaxation dynamics of excited states of graphene QDs is investigated. Finite-bias spectroscopy measurements unveil the excited state spectrum of the device. Long and narrow constrictions are used as tunneling barriers which allow driving the overall tunneling rate to a few MHz. In such a regime rectangular RF pulse schemes are applied to an in-plane gate to probe the relaxation dynamics of excited state to ground state transitions. Measurements of the current averaged over a large number of pulse cycles yield an estimate of a lower bound for charge relaxation times on the order of 60 to 100 ns which is roughly a factor of 5 to 10 larger than what has been reported in III/V quantum dots. The experimental results are compared to a basic model regarding electron-phonon coupling as the dominant relaxation mechanism. A bilayer graphene double quantum dot device is characterized. The capacitive interdot coupling can be tuned systematically by a local gate. The electronic excited state spectrum features a single-particle level spacing independent on the number of charge carriers on the QDs which is in contrast to single-layer graphene. In in-plane magnetic fields a level splitting of the order of Zeeman splittings is observed.

ii Zusammenfassung

Graphen-Quantenbauelemente wie Einzelelektronentransistoren und Quantenpunk- te sind seit sechs Jahren ein lebendiges Forschungsgebiet von steigendem Interesse. Graphen-Quantenpunkte werden als ein interessantes System zur Implementierung von Spin-Quantenbits angesehen. Im Vergleich zu den verbreiteten auf GaAs basie- renden Bauelementen besteht der Vorteil in einer kleineren Hyperfein- und Spin- Bahn-Wechselwirkung, was vorteilhafte Spin-Kohärenzzeiten verspricht. Während die Präparation, Manipulation und das Auslesen einzelner Spins in GaAs Quanten- punkten bereits gezeigt wurden, steht die Forschung an Graphen-Quantenpunkten noch am Anfang. Effekte wie Coulomb-Blockade, Elektron-Loch-Ubergang¨ und Spin-Zustände wurden bereits untersucht. Diese Arbeit zeigt Tieftemperatur-Transportexperimente an ein- und zweilagi- gen Graphen-Quantenpunkten mit dem Fokus auf der Analyse von Anregungsspek- tren und der Untersuchung der Relaxationsdynamik von angeregten Zuständen. Ladungssensoren basierend auf Graphen-Streifen werden charakterisiert und als wichtiges Instrument eingesetzt um Ladungsuberg¨ änge in einem benachbarten Quantenpunkt nachzuweisen. Alle Proben sind basierend auf dem Konzept der breiten-modulierten Nano- strukturen hergestellt. Wird Graphen in Streifen mit einer Breite auf Nanometers- kala geschnitten öffnet sich effektiv eine Bandlucke.¨ Das bietet eine Möglichkeit die Einschränkungen zu umgehen, die darauf beruhen, dass ausgedehntes Graphen keine Bandlucke¨ besitzt. Es werden kleine Inseln (typische Durchmesser zwischen 50 und 120 nm) hergestellt, die mit den Kontakten uber¨ schmale Streifen (Brei- te zwischen 30 und 50 nm) verbunden sind. Dadurch entstehen Quantenpunkte verbunden mit gate-steuerbaren Tunnelbarrieren. Ladungsdetektoren basierend auf Graphen-Nanostreifen gekoppelt an Quanten-

iii punkte werden charakterisiert. Sie erlauben Ladungsvorgänge des Quantenpunkts aufzulösen, selbst wenn der direkte Strom durch den Quantenpunkt unterhalb des Rauschniveaus liegt. Weiterhin können angeregte Zustände des Quantenpunkts mit dieser Technik nachgewiesen werden. Der Detektor funktioniert auch noch unter ungunstigen¨ Bedingungen, wie dem Einfluss eines mehrere Tesla starken Magnet- fels oder wenn Hochfrequenz-Pulse im MHz-Bereich an den Quantenpunkt angelegt werden. Diese Ergebnisse sind besonders fur¨ Experimente an Spin-Zuständen oder zur Untersuchung von Relaxationsprozessen interessant. Ein weiterer Schwerpunkt liegt auf dem Einfluss des Detektors auf die Transporteigenschaften des Quanten- punkts. Die Relaxationsdynamik von angeregten Zuständen in Graphen-Quantenpunkten wird untersucht. Bias-Spektroskopie-Messungen lösen das Spektrum der angeregten Zustände auf. Lange und schmale Zuleitungen zum Quantenpunkt ermöglichen, die Tunnelrate bis zu wenigen MHz zu reduzieren. In einem solchen Bereich werden Sequenzen aus Rechteckpulsen an ein Gate angelegt um damit die Relaxation von angeregten Zuständen zum Grundzustand zu untersuchen. Messungen des uber¨ mehrere Pulszyklen gemittelten Stroms erlauben eine untere Abschätzung der Ladungsrelaxationszeit in der Größenordnung von 60 bis 100 ns, was etwa 5 bis 10 mal größer ist als was fur¨ III/V-Quantenpunkte berichtet wurde. Die ex- perimentellen Ergebnisse werden mit einem vereinfachten Modell verglichen, das Elektron-Kopplung als den wesentlichen Relaxationsmechanismus annimmt. Ein Doppel-Quantenpunkt aus zweilagigem Graphen wird charakterisiert. Die kapazitive Kopplung zwischen beiden Quantenpunkten kann systematisch durch Gates gesteuert werden. Das elektronische Anregungsspektrum weist Einteilchen- Energieabstände auf, die (im Gegensatz zu einlagigem Graphen) unabhängig von der Anzahl der Ladungsträger auf dem Quantenpunkt sind. Ein paralleles Ma- gnetfeld fuhrt¨ zu einem Aufspalten der Energieniveaus in der Größenordnung der Zeeman-Aufspaltung.

iv Contents

Abstract i

Zusammenfassung iii

Contents v

1. Introduction 1

2. Graphene 5 2.1. Electronic band structure of single-layer graphene ...... 5 2.2. Scattering processes ...... 9 2.3. Electronic density of states in graphene ...... 9 2.4. Band structure of bilayer graphene ...... 10 2.5. Hexagonal boron nitride ...... 12

3. Transport through quantum dots 15 3.1. Constant interaction model ...... 16 3.2. Transport through quantum dots ...... 17 3.3. Transport through double quantum dots ...... 20 3.4. Energy scales ...... 21

4. Graphene quantum dot devices 25 4.1. Graphene nanoribbons and nanostructures ...... 26 4.2. Single electron transistors ...... 29 4.3. Quantum dots in width modulated nanostructures ...... 30 4.4. Charge detection ...... 31 4.5. Spin states in a graphene quantum dot ...... 34

v Contents

4.6. Gate-defined quantum dots in graphene nanoribbons ...... 36 4.7. Quantum dots defined by anodic oxidation ...... 38 4.8. Quantum dots on hexagonal boron nitride ...... 38 4.9. Soft-confinement of quantum dots in bilayer graphene ...... 41 4.10. Double quantum dots in graphene ...... 42 4.11. High frequency gate manipulation ...... 44

5. Sample preparation 47 5.1. Device fabrication ...... 47 5.2. Device design ...... 54 5.3. Graphene-hBN sandwich based devices ...... 56

6. Setup 61 6.1. 1K system ...... 61 6.2. Dilution refrigerator ...... 62 6.3. Measurement electronics ...... 64

7. Charge sensing 67 7.1. Introduction ...... 67 7.2. Device design and measurement setup ...... 69 7.3. Device characterization ...... 69 7.4. Charge detection ...... 70 7.5. Finite bias spectroscopy ...... 72 7.6. Sensitivity ...... 74 7.7. Pulsed gating ...... 75 7.8. Back action ...... 76 7.9. Counter ﬂow ...... 78 7.10. Temperature dependence ...... 83 7.11. Conclusion ...... 85

8. Relaxation times in graphene quantum dots 87 8.1. Introduction ...... 87 8.2. Device design and measurement setup ...... 88 8.3. DC low bias transport ...... 90

vi Contents

8.4. DC excited state spectroscopy ...... 92 8.5. Pulsed-gate spectroscopy ...... 95 8.6. Probing relaxation times ...... 101 8.7. Simulation ...... 104 8.8. Electron-phonon coupling ...... 109 8.9. Conclusion ...... 111

9. Bilayer graphene quantum dots 113 9.1. Introduction ...... 113 9.2. Fabrication and Characterization ...... 114 9.3. Tunneling barriers ...... 115 9.4. Finite bias spectroscopy, stability diagrams ...... 119 9.5. Capacitive interdot coupling ...... 122 9.6. Inter-dot tunnel coupling ...... 126 9.7. Excited state spectroscopy ...... 129 9.8. Magnetic ﬁeld dependence ...... 132 9.9. Spin blockade ...... 135 9.10. Conclusion ...... 135

10.Summary and outlook 137 10.1. Summary ...... 137 10.2. Further directions ...... 138 10.3. Outlook: QDs in graphene/hBN sandwiches ...... 139

A. Appendix 143

A.1. Process parameters: Graphene nanostructures on Si/SiO2 substrates 143 A.1.1. Substrate preparation ...... 143 A.1.2. Graphene deposition ...... 144 A.1.3. Device patterning ...... 145 A.1.4. Fabrication of ohmic contacts ...... 146 A.2. Process parameters: graphene/hBN sandwich based nanostructures 147 A.2.1. Graphene/hBN heterostructures ...... 147 A.2.2. Patterning sandwich based nanostructures ...... 148 A.2.3. Fabrication of ohmic contacts...... 149

vii Contents

A.3. List of samples ...... 150 A.4. Loss of mixture at the dilution refrigerator ...... 152 A.5. Optimization of the experimental setup for high frequency read-out 154

Publications 161

List of ﬁgures 167

Bibliography 169

Acknowledgements 190

Curriculum Vitae 191

viii 1 Introduction

The fast development in information technology over the past decades has been allowed by the steady improvement and miniaturization of transistors. The first germanium bipolar transistor was realized by Shockley, Bardeen and Brattain in 1947 who received the Nobel Price nine years later. 1960 the first field effect transistors (FETs) were fabricated using the material combination Si/SiO2. Other materials like GaAs were introduced in the late sixties. While the first transistor measured millimeters they have been shrinking further and further since that time. In late 2013 a transistor fabricated in the 14 nm technology has been demonstrated, further down-scaling to 10 nm is planned within the next two years. The steady miniaturization allowed increasing the integration density and thus in the end the computing power. Moore’s empirical law dating back to 1965 still seems to hold [1]. But the process of down-scaling CMOS devices will soon reach its limit. The feature size has already entered the regime where quantum mechanical effects have to be taken into account. Beside others, the influence of tunnel currents resulting in leakage through the gate dielectric becomes more relevant in even smaller devices. The implementation of new materials is one approach to cope with this issue. One material, which has been intensively studied over the past decade, is graphene. It consists of a monolayer of carbon atoms covalently bound in a hexagonal lattice and it is the first experimentally available two dimensional material. Theoreti- cal studies date back to 1947 when Wallace et al. calculated the band structure of graphite [2]. Only ten years ago, Novoselov and Geim first cleaved individual graphene layers from bulk graphite and deposited them onto a substrate. Their work triggered a cascade of experiments on graphene all over the world. In 2010 the

1 1. Introduction

Nobel Prize in Physics was awarded to Novoselov and Geim ”for groundbreaking experiments regarding the two-dimensional material graphene” [3]. Due to a number of unique properties graphene is interesting for fundamental studies and promising for future applications. The low energy dispersion relation takes the form commonly used to describe massless Dirac particles. This could be proved by the observation of the unconventional quantum Hall effect [4– 6]. Klein tunneling can be observed, another well-known effect from relativistic particles [7]. Graphene features an outstandingly high mobility. Recently, room temperature mobilities of up to 140000 cm2/Vs have been observed [8]. The low optical absorption of graphene together with the high sheet conductivity makes graphene an interesting candidate for display applications. Due to the strong in- plane covalent bonds graphene is one of the mechanically most stable materials with a Young’s modulus of 1 TPa [9]. It is thus also of great interest in the field of nano-electromechanical systems. An in-plane thermal conductivity exceeding 1000 Wm−1K−1 has been measured which is advantageous in electronic circuits as heat dissipation is a severe issue in electronics [10]. All these features make graphene a promising material complementing today’s semiconductor technology but it is also a an interesting candidate for alternative device concepts. Though quantum mechanical effects limit the further size reduction of state of the art devices, they path the way for a completely new class of devices. The principle is based on information processing using quantum bits (qubits) where entangled qubits allow parallel information processing [11, 12]. In 1998 Loss and DiVincenzo proposed quantum computation based on spin qubits in quantum dots (QDs) [13]. Their work triggered a lot of experiments over the past 15 years and QDs have been extensively investigated. Large progress has been made especially in GaAs/AlGaAs heterostructure double quantum dot systems. Coherent manipulation of coupled electron spins has successfully been demonstrated in systems of up to four coupled QDs [14–17]. However, III/V semiconductor based systems suffer from strong hyperfine interaction limiting the spin coherence times [18]. One way to cope with this issue is the polarization of the nuclear spin bath [19]. Another approach to avoid the influence of nuclear spins is the use of alternative materials, especially group IV

2 elements. Spin relaxation has been measured e.g. in Ge/Si nanowire qubits [20] and in silicon QDs [21]. Recently, electron spin resonance has been demonstrated in a Si/SiGe spin qubit with decay timescales signiﬁcantly larger compared to III/V QDs [22]. Among the group IV elements, carbon materials are of special interest. The hy- perﬁne interaction is small as 99% of natural carbon atoms are the isotope 12C[23] and the spin-orbit interaction is small due to the low mass of the nucleus [24, 25]. A valley-spin qubit has been studied in a carbon nanotube with Hahn echo measurements yielding a coherence time of ≈ 65 ns [26]. It is predicted that spin-qubits in graphene feature long coherence times [23]. However, this still has to be proven experimentally.

About this thesis. This thesis investigates quantum dot systems in single-layer and bilayer graphene. The focus lies on the analysis of excited state spectra and the investigation of the relaxation dynamics of excited states. Graphene nanoribbons based charge sensors are characterized as an important tool to detect individual charging events in close-by quantum dots. Chapter2 describes the band structure and the density of states of single-layer and bilayer graphene. The basic properties of hexagonal boron nitride are mentioned in this chapter as well. Chapter3 summarizes the transport phenomena in QD systems relevant for the experiments of this thesis. The constant interaction model, transport through quantum dots and double quantum dots as well as the relevant energy scales are discussed. Chapter4 reviews single-layer and bilayer graphene quantum dot devices. Single electron transistors, QDs and DQDs are taken into account. The chapter provides an overview of the experiments reported so far and compares diﬀerent fabrication techniques. The focus lies on width-modulated nanostructures as this technique has been used in this thesis. Chapter5 describes the device fabrication and typical device designs of etched graphene QD devices on SiO2. The fabrication of graphene/hBN heterostructure based devices is included. Details of the process ﬂow are given in appendices A.1 and A.2. The samples investigated in this thesis are listed in appendix A.3.

3 1. Introduction

Chapter6 describes and characterizes the experimental setup used for the presented experiments. The setup has been optimized for RF measurements as given in detail in appendix A.5. Chapter 7 presents charge sensing experiments with a graphene nanoribbon coupled to a close-by graphene QD. The device is characterized by DC low bias and finite bias spectroscopy. The sensitivity of the charge detector (CD) is studied as well as back action effects of the CD on the QD. The performance of the CD under the influence of a pulsed gate, a magnetic field and different temperatures is investigated. Chapter 8 addresses the relaxation dynamics of excited states of graphene QDs. DC measurements prove the tunability of the tunneling barriers. Finite bias spectroscopy allows the resolution the excited state spectrum of the device. Pulsed-gate schemes are used to probe the charge relaxation times of an excited state to ground state transition. A model of the electron-phonon coupling is used to understand the dominant relaxation mechanisms. Chapter 9 studies a bilayer graphene double quantum dot (DQD). The sample is characterized by DC transport measurements. The typical stability diagram allows the determination of e.g. coupling energies and the excited state spectrum. The capacitive interdot coupling can be controlled by an electrostatic gate. The evolution of excited states under the influence of an in-plane magnetic field is investigated. Chapter 10 summarizes the results of this work. First data obtained from a currently measured QD etched in a hBN/graphene/hBN sandwich structure are presented, giving an outlook on further developments in graphene QD technology.

4 2 Graphene

2.1. Electronic band structure of single-layer graphene

Long before the experimental discovery of graphene, as early as 1947, R. P. Wal- lace [2] performed calculations on its band structure in order to investigate the electronic properties of graphite. Carbon has six electrons in the configuration 1s2 2s2 2p2 with the 1s2 electrons tightly bound to the nucleus. To form a two-dimensional lattice, the s, px 2 and py-orbitals of the involved carbon atoms can hybridize in a sp configuration. Neighbouring atoms will form covalent σ-bonds with a binding angle of 120◦ resulting in a hexagonal crystal structure which can be regarded as a trigonal lattice with two atoms per unit cell. The bond length measures a0 = 1.42 A˚ (see Fig. 2.1 (a)). The strong overlap of the sp2 orbitals in the hexagonal lattice leads to a large energy separation of the bonding and antibonding σ-bands. The remaining pz-orbitals of the carbon atoms contribute to a conduction band. The electronic band structure can be derived following a tight-binding approach [2, 27, 28]. The Bravais lattice is defined by the lattice vectors a = √ √ 1 (a /2)(3, 3) and a = (a /2)(3, − 3). The vectors describing the reciprocal lat- 0 2 0 √ √ tice are thus given by b1 = (2π/3a0)(1, 3) and b2 = (2π/3a0)(1, − 3). The first Brillouin zone shows a hexagonal symmetry reminding of the graphene crystal structure in real space (see Fig. 2.1 (b)). The two inequivalent corners of the √ 0 √ Brillouin zone are K = (2π/3a0)(1, 3) and K = (2π/3a0)(1, − 3).

5 2. Graphene

a c y

4γ0 A

Unit cell a0

2γ0

a2 a1 x

0 Energy

b ky

-2γ0

b1 b2

G K kx M

K´

ky k reciprocal x 1. Brillouin zone lattice points

Figure 2.1.: Lattice structure and band structure of single-layer graphene. (a) Lattice structure of graphene in real space. The colors distinguish between the A and B sublattice. The unit cell is shaded in gray. (b) Reciprocal lattice of graphene. The ﬁrst Brillouin zone is highlighted together with the high symmetry points Γ, M, K and K’. (c) Band structure calculated in the limit of the nearest-neighbour tight-binding approximation (cf. equation 2.1). The π- and π∗-bands touch at the K- and K’-points.

6 2.1. Electronic band structure of single-layer graphene

Tight-binding calculations taking into account both nearest-neighbor (NN) and next-nearest-neighbor (NNN) hopping terms result in a band structure expressed by the relation [27–29],

p 0 E±(k) = ±γ0 3 + f(k) − γ0f(k), (2.1) with √ √ f(k) = 2 cos( 3kya0) + 4 cos( 3/2kya0) cos(3/2kxa0). (2.2)

The nearest-neighbour hopping energy measures γ0 ≈ 2.8 eV. According to ab- 0 initio calculations the next-nearest-neighbour hopping energy γ0 has been found to be on the order of −0.02γ0 to −0.2γ0 [30]. The positive and negative branch of the dispersion relation correspond to the π and π∗-band, respectively. The charge neutrality level lies in between them crossing the degeneracy points K and K’. 0 Finite γ0 breaks the electron hole symmetry. A detailed calculation can be found in literature [2, 27, 28]. Fig. 2.1 (c) shows the band structure for γ0 = 2.8 eV and 0 γ0 = −0.2γ0. At the center of the Brillouin zone, the Γ-point, the conduction and valence band are separated by 6t ≈ 17 eV, whereas the bands touch at the K-points of the Brillouin zone making graphene a semimetal. In transport measurements usually only low energies ( 1 eV) are accessible, thus only the dispersion relation close to the charge neutrality point is relevant. Wave vectors κ close to the K-point can be expressed by κ = K + k with |k| |K| being the wave vector relative to the K-point. Using this ansatz, the dispersion relation can be linearized around K [2, 31–33]:

3γ0a0 E±(k) = ±~ |k| = ±~vF |k|, (2.3) 2~

3γ0a0 6 where vF is the energy independent Fermi velocity given by 2 ≈ 10 m/s. 0 ~ Within this approximation the contribution of γ0 is neglected. This linear dispersion relation takes the form of the dispersion relation valid for massless relativistic particles (E = c · p, where c is the speed of light.). The K-points are therefore called Dirac points. Following this analogy, in the low energy regime the system can be described

7 2. Graphene by the Dirac equation in two dimensions

−i~vF σ · ∇Ψ(r) = EΨ(r), (2.4) with the vector of Pauli matrices σ = (σx, σy)[4, 27, 28, 33–35].

The Hamiltonian in momentum space reads as ! 0 kx − iky HK = ~vF = ~vF σ · k (2.5) kx + iky 0

∗ ∗ T for the K-point and HK0 = ~vF σ · k for K’, where σ = σ .

The corresponding wave functions take the form of two component spinors: ! ! 1 e−iθk/2 1 eiθk/2 Ψ±,K (k) = √ and Ψ±,K0 (k) = √ . (2.6) 2 ±eiθk/2 2 ±e−iθk/2

∗ Positive and negative sign correspond to the π and π band, θk = arctan(kx/ky) is the angle in the momentum plane. Rotating θk by 2π, a phase, the so-called Berry’s phase [36], of π is accumulated and thus the wave function changes sign.

In analogy to particle physics, σ describes a spin projection on the direction of motion, in graphene called pseudo-spin, which is completely independent of the electron spin [37]. A helicity (or chirality, what is the same for massless particles) can be attributed to the charge carriers which is deﬁned as the projection of the pseudo-spin on the direction of the momentum:

1 p hˆ = σ . (2.7) 2 |p|

Obviously, the helicity operator commutes with the Hamiltonian making it a good quantum number. The helicity can be either positive or negative. Consid- ering conduction band states at the K-point the wave vector and the pseudo spin are oriented in parallel (positive helicity) and anti-parallel for valence band states (negative helicity). The situation is vice versa at K’.

8 2.2. Scattering processes

2.2. Scattering processes

The scattering rate can be determined according to Fermi’s Golden Rule and thus 2 is proportional to the transition matrix element hΨf |U|Ψii with the scattering potential U, the initial and the final wave functions Ψi and Ψf . Let θi and θf describe the direction of propagation of the initial and final state in the momentum plane (cf. eq. 2.6), then θi − θf is the scattering angle. It can be shown that as a consequence of the helicity the matrix element shows a proportionality to the 2 term cos [(θi − θf )/2] which vanishes for θi − θf = π. Thus backscattering is suppressed as long as the pseudo-spin is conserved. In contrast to long range scattering potentials, short range scatterers can break the A-B lattice symmetry and introduce pseudo-spin flips [38]. Klein tunneling is another phenomenon observed in graphene due to the chiral nature of the electrons. A potential step is considered, which can be experimentally induced e.g. by a local gate voltage. In a conventional semiconductor an impinging electron will be transmitted through the barrier with a certain tunneling probability depending on the width and the height of the barrier. The situation is different for graphene where a band gap does not exist. At the barrier an electron can be scattered into a hole state with opposite momentum. This is a direct consequence of pseudo-spin conservation as electrons in region I and holes in region II share the same pseudo-spin orientation. Following Katsnelson et al. [39] and Castro Neto et al. [27] the transmission probability is given by

cos2 φ T (φ) ' . (2.8) 2 2 1 − cos (Dkx) sin φ

This yields a total transmission for perpendicular angle of impingement [40].

2.3. Electronic density of states in graphene

The electronic density of states (DOS) is deﬁned as the number of states per energy interval per unit volume [41]. It depends on the dimensionality of the system and the dispersion relation. In two dimension each states takes up a volume of (2π)2/V in k-space, where V is the volume of the crystal in real space. A two-dimensional

9 2. Graphene

2πk ring in the k-plane with radius k and width dk can be ﬁlled with dN = 4 (2π)2V dk states. Here, a factor of four has been included due to spin and valley degeneracy. 1 dN 2k The density of states in k space is thus given by D(k) = V dk = π . Applying the dispersion relation of graphene, E = ~vF |k|, the density of states is given by:

dk 2|E| D(E) = D(k(E)) = 2 2 (2.9) dE π~ vF

The density of states is linear in energy in contrast to conventional two-dimensional electron gases where it is constant [27, 28, 42].

2.4. Band structure of bilayer graphene

Bilayer graphene is formed by two graphene sheets Bernal stacked on top of each other and bound by van-der-Waals forces. The basis contains four atoms and the crystal can thus be described by four sublattices A1, B1, A2 and B2. The B2 atoms are directly positioned above A1 sites whereas A2 and B1 atoms do not have direct counterparts (see Fig. 2.2). In close analogy to single layer graphene the band structure can be obtained by a tight-binding approach taking into account the intralayer hopping energy

γ0 ≈ 2.8 eV and the interlayer hopping energy γ1 ≈ 0.4 eV [43] (cf. Fig. 2.2). Other interlayer hopping energies are neglected as they are signiﬁcantly smaller compared to γ1. The energy dispersion relation around the K-point is given by

q 1/2 2 2 2 2 4 2 2 2 2 E±(k) = ± ~ vF k + γ1 /2 ± γ1 /4 + γ1 ~ vF k . (2.10)

A minus sign in front of the inner square root describes the lowest energy subbands for electrons and holes, a plus sign the second electron and hole subbands which are split by 2γ1 ≈ 0.78 eV [27, 28, 44–46].

In the limit of low momenta (k γ1/(2~vF )) equation 2.10 simpliﬁes to

2 2 2 ~ vF k E±(k) = ± (2.11) γ1

10 2.4. Band structure of bilayer graphene

B 2 A2

c0 γ1

B 1 γ0 A1

Figure 2.2.: Lattice structure of bilayer graphene. The interlayer distance measures c0 = 3.35A˚, the intralayer hopping energy γ0 ≈ 2.8 eV and the interlayer hopping energy γ1 ≈ 0.4 eV [43]. The latter describes the coupling of B1 atoms in the lower layer and A2 atoms in the upper layer.

a 4 b 4 2 2

3 2 0 2γ1 0 Δ-Δ /2γ1 Energy Energy

Energy (eV) Energy (eV) -2 -2 Ö2Δ/hvF -4 -4 -6 -3 0 3 6 -6 -3 0 3 6 k (nm-1 ) k (nm-1 ) ky kx x ky kx x

Figure 2.3.: Band structure of bilayer graphene. (a) Band structure in the absence of an electric ﬁeld. (b) A perpendicular electric ﬁeld ∆ breaks the lattice symmetry and thus leads to a band gap opening.

11 2. Graphene describing a parabolic dispersion relation. As a consequence, the quasi-particles ∗ 2 have a ﬁnite eﬀective mass m = γ1/2vF originating from the transition energy γ1 between A1 and B2 sites [46]. The dispersion relation takes the typical form 2 2 ∗ of a two-dimensional electron gas: E±(k) = ±~ k /2m . As a consequence the density of states is constant:

γ1 D(E) = 2 . (2.12) π(~vF )

Please note that the parabolic approximation is only valid for small k, at higher momenta the effective mass has to be calculated by m∗ = ~2(∂2E/∂k2)−1. At large k (k γ1/(2~vF )) the dispersion relation converges towards the linear dispersion known from single-layer graphene [28], as shown in Fig. 2.3 (a). Bilayer graphene is of special interest as it allows the opening of a band gap by applying a perpendicular electric field. Taking into account this effect the energy dispersion relation reads as

q 1/2 2 2 2 2 2 4 2 2 2 2 2 E±(k) = ± ∆ /4 + ~ vF k + γ1 /2 ± γ1 /4 + (γ1 + ∆ )~ vF k , (2.13) where ∆ is a measure for the energy difference of the two graphene layers [27– 29, 45, 46]. Considering a finite perpendicular electric field such that ∆ 6= 0 but still ∆/2

γ0 the bands shift to larger energies |E| and a band gap will be opened close to the Dirac points. At k = 0 the gap of the ﬁrst subband measures ∆ and the one of the p 2 2 second subband 4γ1 + ∆ . As long as ∆ is small compared to γ0 the dispersion relation remains parabolic around the K-point while at larger ∆ the curve takes the typical ’mexican hat’-like shape [27–29, 45, 46]. The minimum band gap at √ 3 2 k = ∆/ 2~vF measures ∆ − ∆ /2γ1 . Fig 2.3 compares the band structure with and without an applied ﬁeld.

2.5. Hexagonal boron nitride

Hexagonal boron nitride (hBN) has a lattice structure similar to the one of graphite. An individual layer of hBN consists out of two sublattices, one consisting of boron

12 2.5. Hexagonal boron nitride

a y b

boron

nitride

c0 = 3.35 Å a0 = 1.44 Å

a2 a1 x

Figure 2.4.: Lattice structure of hexagonal boron nitride. (a) Lattice structure of single-layer hBN. The interatomic distance measures a0 = 1.44 A˚. (b) Stacking of multi-layer hBN. The interlayer distance measures c0 = 3.35A˚. and one of nitrogen atoms. The atoms form a hexagonal lattice structure (see Fig. 2.4 (a)) with an interatomic distance of 1.44 A˚ which corresponds to a relative lattice mismatch of only 1.7 % compared to graphene [47, 48]. In z-direction the layers are only weakly bound by van-der-Waals forces which allows an easy cleavage of the material. The stacking is diﬀerent compared to graphene since a boron atom is always positioned directly above a nitrogen atom and vice versa (see Fig. 2.4 (b)). Due to diﬀerent on-site energies the A-B lattice symmetry is broken which induces a wide band gap of 5.97 eV [49]. Thus hBN is insulating and transparent for optical light. Hexagonal boron nitride has an atomically smooth surface with no dangling bonds and a very low density of charge traps. This makes hBN an ideal material serving either as a substrate for graphene devices or for building encapsulated graphene devices [50, 51].

3 Transport through quantum dots

Quantum dots (QDs) are small objects where electrons are confined in all three dimensions. Small in this context means that the device dimensions are on the order of the Fermi wavelength λF = 2π/kF . In such a quasi zero-dimensional system the electronic excitation spectrum is discrete [52–55]. This is in analogy to atomic orbitals and thus QDs are sometimes called artificial atoms [56]. The energy to add (or remove) an additional electron to a QD required due to Coulomb repulsion is called charging energy. Following the analogy to atoms, this corresponds to the ionization energy. Since the 1980s, QDs have been an interesting field of research in solid state physics and nanoelectronics. QDs have been studied in a large number of material systems like single molecules [57], metal nanoparticles [58], planar two-dimensional electron gases [14, 15, 59–61], self-assembled semiconductor nanostructures [62], semiconductor nanowires [63–66], carbon nanotubes [26, 67–69] and more recently also in graphene [70–76]. Within this thesis, graphene QDs are studied via electronic transport experiments. For this purpose QDs are investigated in a configuration of a single electron transistor (SET) i.e. the QD is weakly coupled to lead electrodes and its electrochemical potential can be controlled by at least one gate electrode. A simple model of such a configuration is illustrated in Fig. 3.1. Tunneling transport between the QD and the leads is allowed, the gate is only coupled electrostatically [53, 77]. Transport through QDs is dominated by two major effects. Coulomb blockade is a consequence of the Coulomb interaction of electrons on the QD which leads to a repulsive force. Thus a certain amount of energy - the so-called charging energy - has to be supplied to add an additional electron to the QD. [78]. The confinement

15 3. Transport through quantum dots

S QD D

CS CD

VSD

VGL

Figure 3.1.: Model of a quantum dot coupled to leads and a gate electrode. Simple model of a quantum dot capacitively coupled to the source and drain leads and to one gate. Tunneling transport between the QD and the leads is allowed. These tunnel junction can be modelled by a capacitance and a resistance in parallel. in all dimensions leads to a discrete single-particle energy spectrum which is not aﬀected by interactions [53, 55].

3.1. Constant interaction model

The constant interaction model is based on the assumption that the Coulomb interaction of an electron on the QD and all other electrons inside and outside P the QD only depends on the total capacitance CΣ = CS + CD + i CGi where

CS and CD denote the capacitance to the source and the drain lead and CGi the capacitances to the diﬀerent gate electrodes. Additionally the single-particle energy is assumed to be independent of electron-electron interactions [55]. Within this model the total energy U(N) of the QD with N electrons can be expressed by [53]

!2 N 1 X X U(N) = · −e(N − N ) + C V + C V + C V + E . (3.1) 2C 0 S S D D Gi Gi n Σ i n=1

Here, VS, VD and VGi are the voltages applied to source, drain and gates, respec- PN tively. N0 is the number of electrons at VGi = 0 and n=1 En the sum over the energies of the occupied single-particle states. The electrochemical potential µ(N), i.e. the energy to add an additional electron

16 3.2. Transport through quantum dots to the QD already occupied by N − 1 electrons, measures

µ(N) = U(N) − U(N − 1) (3.2) !! e 1 X = e N − N − − C V + C V + C V + E . (3.3) C 0 2 S S D D Gi Gi N Σ i

For all N the electrochemical potential shows the same linear dependence on the gate voltages. The ratio αi = CGi /CΣ is called the lever arm of gate i. The diﬀerence between two subsequent electrochemical potentials

e2 Eadd(N) = µ(N + 1) − µ(N) = + ∆E = EC + ∆E, (3.4) CΣ

2 is called addition energy. The purely electrostatic contribution EC = e /CΣ is called charging energy and ∆E is the single-particle level spacing. The charging energy can be estimated by the dimensions of the dot. A lower bound for the total capacitance of a two-dimensional quantum dot CΣ can be given by approximating the QD with an isolated circular disc of diameter d. The capacitance of such a disc is given by CΣ ≈ 40d with the dielectric constant of the surrounding material. Thus an upper limit of the charging energy of EC ≈ 2 e /(40d) can be given, neglecting e.g. contributions to the total capacitances originating from contacts and gates.

3.2. Transport through quantum dots

In the following let us assume that the temperature is suﬃciently low, i.e. kBT <

∆,EC . Electron transport is only possible if the electrochemical potential of the

QD µ(N) is positioned between the electrochemical potentials of the source µS and the drain lead µD: µS ≤ µ(N) ≤ µD (or vice versa). As long as the bias transport window µS − µD = eVSD (with the bias voltage VSD applied to the leads) is small compared to the charging energy EC and the level spacing ∆, either one or no state can be in the transport window. If there is no state inside, no electrons can enter or leave the QD, the number of electrons occupying the QD is ﬁxed

17 3. Transport through quantum dots

(see Fig. 3.2 (a)). The system is in Coulomb blockade. By altering one of the gate voltages it is possible to shift an empty state into the transport window and thus lift the Coulomb blockade. An electron from the source lead can now enter the QD and subsequently leave the QD to the drain lead. The system is in the regime of sequential tunneling and a current can ﬂow. Measuring the current as a function of the gate voltage, regimes of conductance and of Coulomb blockade alternate, so-called Coulomb peaks appear (see Fig. 3.2 (c)). The proportionality between the peak spacing ∆VG and the addition energy is given by the lever arm

α = Eadd/e∆VG = CG/CΣ. It is a measure for the capacitive coupling of a gate to the QD. The height of the Coulomb peaks carries information on the overlap of wave functions of the QD states and states in the leads. Considering transport via an individual QD state, the peak conductance measures,

2 2e 1 ~ΓLΓR G0 = , (3.5) h 4kBT ΓL + ΓR where ΓL,R describe the tunneling rates between a QD state and the left (L) or right (R) lead, respectively. Please note that G0 can vary signiﬁcantly between neighbouring QD states. G0 scales inversely with temperature. The shape of a Coulomb peak is determined by the coupling of the QD to the leads and the temperature. If the thermal broadening is dominant (~Γ kBT ) the shape of a conductance peak is given by −2 αe∆VG G(∆VG) = G0 cosh . (3.6) 2kBT

The Coulomb peak shape allows the determination of the electron temperature. The tunnel coupling Γ = ΓLΓR , often called lifetime broadening, leads to a ~ ~ ΓL+ΓR Lorentzian broadening of the form

(~Γ)2 G(∆VG) ∝ 2 2 . (3.7) (~Γ) + (αe∆VG)

So far we have considered the case of low bias voltage eVSD ∆,EC , the so- called linear response regime. By increasing eVSD such that it exceeds the level

18 3.2. Transport through quantum dots

a 1 2 3 4

Eadd Δ

b V SD c IQD

3 ΔVG

N+1N N+2 4

Eadd = eΔVSD 1 2 VG VG

d IQD N-1 N N+1 N+2 ES GS

VSD

Eadd = αeΔVG

Figure 3.2.: Coulomb diamonds. (a) Schematics of four different configurations of a QD: (1) Zero bias and misalignment of the QD states with the lead potentials. (2) A QD state aligned with lead potentials. (3) The bias equals the addition energy. At least one state is within the transport window. (4) Ground state and first excited state are aligned within the transport window, two possible transport channels are open. (b) Illustration of finite bias spectroscopy measurements on a QD: The current IQD is plotted as a function of VSD and VG. In the white regions the device is in Coulomb blockade. The current and the number of electrons on the QD is fixed. In the blue shaded regime transport occurs. The lever arm α is the proportionality factor between the addition energy Eadd and the change in gate voltage ∆VG necessary to add the next electron to the QD. (c) Cut along the gate axis at a small bias (horizontal line in panel (b)). Coulomb peaks with a spacing of ∆VG appear. (d) Cut along the vertical green line in panel (b). The current increases each time another excited state enters the transport window.

19 3. Transport through quantum dots

spacing ∆ (but still eVSD < Eadd) it is possible that two quantum dot levels, a ground state (GS) and an excited state (ES), are positioned within the transport window. Electrons can now tunnel through the QD via GS and ES transitions. As two channels contribute to transport the current will increase (see Fig. 3.2 (d)). If the bias even exceeds the addition energy at least one ground state is always aligned within the transport window and thus the Coulomb blockade is lifted completely. If two ground states appear in the transport window, the number of electrons can vary between N-1, N and N+1. By bias spectroscopy measurements the current through a QD is recorded both as a function of the bias and the gate voltage. Diamond shaped regions (so- called Coulomb diamonds) of suppressed conductance occur when the system is in Coulomb blockade (see Fig. 3.2 (b)). Within such a region the number of charge carriers on the QD is constant. The extent of the diamonds in bias direction is a measure of the addition energy. Lines of enhanced conductance appearing in parallel to the diamond edges within the transport region originate from transport involving excited states. Employing the gate lever arm α the gate voltage axis can be converted into an energy scale (E = αeVG). Finite bias spectroscopy measurements allow the extraction of the charging energy and to study the excited state spectrum of a quantum dot. In large quantum dots the level spacing is small compared to the charging energy and the diamonds are almost equally sized.

3.3. Transport through double quantum dots

A double quantum dot can be modelled by two quantum dots connected with each other and to the leads via tunneling barriers. Capacitive coupling to the leads and gate electrodes as well as the interdot coupling have to be taken into account (see Fig. 3.3 (a)) [79]. Considering a double quantum dot which can be controlled by two gates, a so- called stability diagram (current as a function of the gate voltages) can be recorded (see Fig. 3.3 (b-d)). At low bias voltage and low temperature a current can ﬂow if both QD levels are aligned with the lead potentials (so-called triple points, see blue dots). If only one QD potential is aligned, co-tunneling can occur (dashed lines) at suﬃciently strong tunnel coupling. If both QDs are in Coulomb blockade

20 3.4. Energy scales the number of electrons occupying each dot are fixed. In the very simple case where G1 controls QD1 and G2 controls QD2 (i.e. crosstalk between G1 and QD2 and G2 and QD1 is neglected) and without capacitive interdot coupling the co-tunneling lines run in parallel to the gate axes (Fig. 3.3 (b)). Taking capacitive crosstalk into account the co-tunneling lines are tilted depending on the capacitances (Fig. 3.3 (c)). Experimentally, a finite capacitive interdot coupling has to be considered. An electron entering one QD increases the energy needed for another electron to enter the other QD. The triple points split into pairs with a separation depending on the coupling (see Fig. 3.3 (d)). The stability diagram shows the characteristic hexagonal pattern. At a finite bias voltage the pairs of triple points evolve into two overlapping triangular regions where a current can flow. Such a finite bias spectroscopy measurement allows the determination of the gate lever arms, addition energies and the mutual capacitive coupling. Experimental data are given in sections 9.4 and 9.5.

3.4. Energy scales

There are several energy scales that have to be considered performing transport experiments in graphene quantum dots.

2 Coulomb energy: The Coulomb energy EC = E /CΣ of a QD can be estimated by its self-capacitance. For quantum dots in planar two-dimensional electron gases (2DEGs) and in graphene the QD is commonly approximated by a circular disc of radius r embedded in a material with dielectric constant [35, 53]. Gates and contacts are neglected within this model. The self-capacitance then measures

C = 80r. Considering a graphene QD placed on a SiO2 substrate ≈ (SiO2 + 1)/2 ≈ 2.5 can be estimated as the average dielectric constant of the substrate material and air.

Level spacing in single-layer graphene: The level spacing of electronic excited states in graphene quantum dots can be derived from the density of states following Schnez et al. [80, 81] and G¨uttinger[82]. The DOS in single layer graphene is given

21 3. Transport through quantum dots

a e m Cm DV S QD QD D rg VLG CS CD DVlg C C GL,R CGR,L GL,L CGR,R DVrg dVlg

VSD m dVlg DVlg VGL VGR

VRG

b c d

N+1,M N+1,M+1 N+1,M N+1,M N+1,M+1 N+1,M+1

VLG VLG VLG

N,M N,M+1 N,M N,M N,M+1 N,M+1

VRG VRG VRG

Figure 3.3.: Electrostatics of a double quantum dot. (a) Model of a double quantum dot indicating the capacitances between the dots and the leads, the gates and in between each other. (b) Illustration of a low bias stability diagram of a DQD considering only capacitive coupling between the left dot and the left gate as well as the right dot and the right gate. (c) As in (b) including capacitive crosstalk. (d) The full model as depicted in (a) including interdot coupling. (e) Finite bias stability diagram with the relevant quantities to determine the L R lever arms αlg = VSD/δVlg and αrg = VSD/δVrg, the addition energies L L R R Eadd = αlg · ∆Vlg and Eadd = αrg · ∆Vrg and the mutual capacitive m m m m coupling Elg = αlg,L · δVlg , Erg = αrg,R · δVrg .

22 3.4. Energy scales by [27, 28] 2E D(E) = 2 . (3.8) π(~vF )

Approximating a quantum dot by a circular island with an area A = πr2 and number of charge carriers N, the total quantum mechanical energy is given by

Z EF 2A 3 EQM (N) = A dE E D(E) = 2 EF . (3.9) 0 3π(~vF )

Applying the dispersion relation for single layer graphene E = ~vF |kF | with √ p kF = πn = πN/A, equation 3.9 can be rewritten as

3 3 2A~ vF 3 2~vF p 3/2 4~vF 1 3/2 EQM (N) = 2 kF = π/AN = N . (3.10) 3π(~vF ) 3 3 d

The chemical potential µ can be expressed as following where the approximation is valid in the regime of large N. √ 4 v 1 N µ(N) = E (N +1)−E (N) = ~ F ((N +1)3/2−N 3/2) ≈ 2 v . (3.11) QM QM 3 d ~ F d

The excited state level spacing ∆ is thus given by the diﬀerence,

1 ∆(N) = µ(N + 1) − µ(N) ≈ ~vF √ . (3.12) d N

Level spacing in bilayer graphene: For bilayer graphene quantum dots the level spacing can be calculated following the same approach taking into account the diﬀerence in density of states and dispersion relation. In the vicinity of the Dirac points the dispersion relation of bilayer graphene is parabolic an thus the density of states equals the one of a two dimensional electron gas,

γ1 D(E) = 2 . (3.13) π(~vF )

23 3. Transport through quantum dots

This is in contrast to single-layer graphene and does not depend on the energy.

Thus, the total energy is quadratic in EF :

Z EF Aγ1 2 EQM (N) = A dE E D(E) = 2 EF . (3.14) 0 2π(~vF )

As derived in section 2.4 the low energy dispersion relation takes the form of a 2 2 ~ kF 2DEG, E = 2m∗ with the eﬀective mass deﬁned by the interlayer hopping energy: ∗ 2 m = γ1/2vF . Thus the energy reads as

2 2 2 2 2 Aγ1 π~ N π~ vF N EQM (N) = 2 ∗ = (3.15) 2π(~vF ) 2m A 2γ1 A and the chemical potential is given by

π 2v2 2N + 1 µ(N) = ~ F . (3.16) 2γ1 A

This results in an excited state level spacing ∆ which is independent of the number of electrons on the QD[83]:

π 2v2 1 ∆(N) = ~ F (3.17) γ1 A

Tunnel coupling: The tunnel resistance Rt to the leads has to be sufficiently low such that an electron is either located in one of the leads or on the QD. The minimum Rt to fulfill this condition can be estimated according to Heisenberg’s uncertainty relation ∆E · ∆t > h with the desired energy resolution ∆E and the time scale of a tunneling process ∆t = RtCΣ. This yields the relation Rt > h/CΣ∆E. Regarding the charging energy as the desired resolution the condition 2 Rt > h/e has to be satisfied [35, 53].

Thermal energy: As mentioned before, the Coulomb peaks are broadened proportional to cosh−2 αe∆VG . This implies that the temperature has to be suﬃ- 2kB T ciently low to resolve Coulomb charging eﬀects (kBT EC ) and the excited state spectrum (kBT ∆).

24 4 Graphene quantum dot devices

Quantum dots (QDs) allow controlled investigation and manipulation of individual quantum systems. They are in particular interesting as promising hosts for spin qubits [13]. These are reasons why QDs have been intensively investigated in different material systems over the past years. So far, most progress has been made in QDs in two-dimensional electron gases, especially in GaAs based heterostructures [14–16, 59–61] and elementary spin-qubit operations have been demonstrated. However, these systems suffer from limited spin decoherence times originating from spin-orbit and hyperfine interactions [18]. Ways to cope with this issue have been explored, e.g. the polarization of the nuclear spin bath [19]. In order to minimize the influence of nuclear spins alternative materials are of great interest, especially group IV elements. Spin relaxation has been measured e.g. in Ge/Si nanowire qubits [20] and in silicon QDs [21]. Recently, electron spin resonance has been demonstrated in a Si/SiGe spin qubit with decay timescales significantly larger compared to III/V QDs [22]. One way to further reduce the influence of nuclear spins is the use of isotopically purified silicon. Among the group IV elements carbon materials are an interesting alternative. The spin-orbit interaction is small due to the low mass of the nucleus [24, 25]. The hyperfine interaction is weak as 99% of natural carbon is the isotope C12 which has zero nuclear spin [23]. A valley-spin qubit in a carbon nanotube has been studied by Hahn echo measurements [26]. It is predicted that spin-qubits in graphene feature long coherence times [23]. Despite the advantages concerning hyperfine and spin-orbit interaction, graphene has a significant drawback. Due to the absence of a band gap and the pseudo-

25 4. Graphene quantum dot devices relativistic Klein tunneling effect it is challenging to confine electrons (see chapter2 or Refs. [27, 28, 39]). A lot of effort has been spent so far to overcome this limitation. Most approaches are based on carving nanostructures out of graphene sheets. It has been shown that an energy gap will be opened in graphene nanoribbons. Single electron transistors, QDs and DQDs have successfully been fabricated either by etching width- modulated nanostructures [70, 73, 74, 76, 83–88] or by electrostatic confinement of electrons in nanoribbons using gate electrodes [72, 89]. Another technique defines the nanostructures by local anodic oxidation [90, 91]. Recently, the confinement of electrons by magnetic fields has been demonstrated [92]. In addition, bilayer graphene offers the possibility to confine electrons by applying perpendicular electric fields [75, 93]. This chapter provides an overview of the experiments on graphene and bilayer graphene quantum dots reported so far and compares different fabrication techniques.

4.1. Graphene nanoribbons and nanostructures

As graphene nanoribbons are building blocks of most graphene quantum devices this paragraph summarizes the relevant transport properties of these structures. From carbon nanotubes (CNTs), which can be imagined as rolled-up graphene nanoribbons (GNRs), it is well known that a band gap opens depending on their orientation and their diameter [94]. The orientation of CNTs is named after their circumference while GNRs are classiﬁed according to their edges along the ribbon. N-aGNRs and N-zGNRs commonly denote armchair (a) and zigzag (z) GNRs with N dimers across the ribbon width. The zGNRs and aGNRs are well understood in theory. The band structure can be determined by applying vanishing boundary conditions to the Dirac Hamiltonian of graphene. In zGNRs one edge is made up by A atoms, the other one by B atoms and thus the boundary condition can be applied to the sublattices separately. In aGNRs the edges contain atoms of both sublattices and thus the boundary condition has to be fulﬁlled by both sublattices [95]. Alternatively the band structure can be determined following a tight-binding approach depending on the orientation and the width of GNRs. Armchair GNRs

26 4.1. Graphene nanoribbons and nanostructures have a metallic band structure if the condition N = 3m − 1 with integer m is fulﬁlled. Otherwise a semiconducting band structure occurs [96–98]. The band gap Eg scales inversely with the ribbon width W . An estimate is given by [82, 94],

Eg = 2~vF ∆kF = 2π~vF /W, (4.1) where ∆kF is the minimum allowed wave number across the ribbon width. DFT calculations taking into account next nearest neighbor hopping and a con- traction of the bond length at the edges have shown that even metallic aGNRs have at least a small band gap which depends on the ribbon width [99]. Zigzag GNRs have a gapless band structure independent on N. First-principle calculations have shown a high density of zero energy states at the edges which has been proved by scanning tunneling spectroscopy [100]. Magnetic ordering at the edges may lead to the opening of a small band gap in zGNRs [99, 101]. Theory assumes either pure armchair or pure zigzag GNRs where the edges are terminated by hydrogen atoms. Using the common experimental techniques it has not yet been possible to fulfill these conditions. Typically arbitrary edges occur when GNRs are etched out of graphene sheets. When using an O2-based plasma as the etchant the edges will probably be oxygen terminated. In the following, electronic transport through GNRs will be described on a more phenomenological basis. Field effect measurements have proven the presence of a transport gap (see e.g. Fig. 4.1(a,b) and Refs. [102, 104]) but still a number of sharp resonances can be observed within this gap. A common model to describe the electronic transport through etched GNRs (i.e. nanoribbons with rough edges) is based on stochastic Coulomb blockade [105, 106]. A disorder potential (e.g. substrate or edge disorder) can form electron and hole puddles close to the charge neutrality point. Due to the absence of a band gap in bulk graphene and due to the presence of the effect of Klein tunneling [39], transport between the electron and hole regions is possible. The situation is different in nanoribbons where a width dependent confinement gap separates electrons and hole puddles. Thus, effectively a large number of quantum dots or localized states formed and only tunnelling transport is possible [104, 107]. The presence of quantum dots has been demonstrated by finite bias spectroscopy

27 4. Graphene quantum dot devices

500 nm

b 0 c 100 20 20 w = 50 nm

e²/h) l = 500 nm -1 -3 G (10 50 0 10 0 VBG (V) 2 G (e²/h) -2 log

bias 0 0 -3 Current (nA) Bias V (mV) -10 -50 -4 Conductance

-20 -100 0 10 20 30 40 50 0 20 40 60

Back gate VBG (V) Back gate VBG (V) d e 50

30 Eg (w) = a/wexp(- b w ) 40 Eg(w) = a* /(w-w *)

20 30 BG

10 G 20 Δ V (V) E (meV)

10 0 0 20 40 60 80 100 20 40 60 80 100 Width w (nm) Width w (nm)

Figure 4.1.: Characteristics of etched graphene nanoribbons. (a) Scan- ning force micrographs of etched graphene nanoribbons with different lengths and widths. (b) Conductance through a graphene nanoribbon (50 nm wide, 500 nm long) as a function of the back gate voltage and thus the Fermi level. Regions of electron and hole transport are separated by a transport gap. The inset shows a measurement within the transport gap of a 200 nm wide nanoribbon. (c) Finite bias spectroscopy measurement allowing to determine the effective energy gap Eg and the transport gap correlated with ∆VBG. (d) Transport gap as a function of the width of different nanoribbons. (e) Effective energy gap as function of the width. Two models are fitted to the experimental data. Panels (a-c) adapted from Ref. [102], c 2011 American Institute of Physics. Panels (d,e) adapted from Ref. [103], c 2010 IOP Publishing.

28 4.2. Single electron transistors measurements on GNRs [86, 102, 104, 107]. Most importantly, two characteristic energy scales can be extracted, the eﬀective energy gap Eg and the transport gap

∆VG, see Fig. 4.1(b). The transport gap 4.1(c) is correlated with the maximum amplitude of the disorder potential. An eﬀective energy gap Eg can be deﬁned by the largest observed charging energy corresponding to the smallest quantum dot. It has been shown that this energy gap only weakly depends on the length of the nanoribbon [102]. The width dependence can be modelled by

−βW Eg(W ) = α/W e , (4.2) see Fig. 4.1(d) and Refs. [103, 106, 108].

4.2. Single electron transistors

Single electron transistors have been fabricated by carving the desired shape out of graphene sheets using electron beam lithography followed by reactive ion etching (see section 5.1).These devices consist of a graphene island connected to the source and drain electrodes via two GNRs. The devices make use of the fact that due to the narrow width of the GNRs an eﬀective transport gap is opened (see section 4.1) and thus they can be operated as tunable tunneling barriers. Ac- cording to equation 4.2 the gap scales approximately inversely with the ribbon width. Width modulation allows tailoring the transport gap along the ribbon axis. Close-by graphene gates and a global back gate tune the Fermi level in the nanostructure. A scanning force micrograph of a representative device is shown in Fig. 4.2 (a). Carefully tuning the voltages on the two outer gates (B1 and B2) it is possible to bring the device into a regime where the transport gaps of both constrictions cross the Fermi level. The central island is electrically isolated and only tunneling transport is possible between the island and the leads. The device can be operated as a SET. A series of distinct Coulomb peaks recorded as a function of the plunger gate voltage is shown in Fig. 4.2 (b). A charging energy of ≈ 3.4 meV has been determined by ﬁnite bias spectroscopy measurements (see Fig. 4.2 (c)) on a graphene SET with 50 nm wide tunneling barriers and an central

29 4. Graphene quantum dot devices

b c

Figure 4.2.: SET in a width-modulated graphene nanostructure. (a) False color scanning force micrograph of the device. Three lateral graphene gates are designed to locally tune the potential of the nanostructure. (b) Current through the SET as a function of the gate voltage. The inset shows a series of Coulomb peaks proving the operation as a SET. (c) Finite bias spectroscopy measurement in the same regime. Adapted from Ref. [84], c 2008 American Chemical Society. island measuring approximately 180 × 750 nm [84].

4.3. Quantum dots in width modulated nanostructures

The concept of width-modulated graphene nanoribbons described in the previous section can be employed to design a graphene quantum dot device. The smaller the central graphene island the more relevant quantum conﬁnement eﬀects become. Fig. 4.3 (a) shows a typical example of a graphene quantum dot with lateral graphene gates. Finite-bias spectroscopy is a method to investigate the excited level spectrum of a QD via direct transport experiments. If the bias exceeds the level spacing it is possible that two quantum dot levels, a ground state (GS) and an

30 4.4. Charge detection excited state (ES), are positioned within the transport window. Electrons can now tunnel through the QD via GS and ES transitions. As two channels contribute to transport the current will increase. Excited state resonances appear as lines parallel to the Coulomb diamond edge in ﬁnite-bias spectroscopy (see schematics in Fig. 3.2). A representative measurement is shown in Fig. 4.3 (b) where the diﬀerential conductance through a 140 nm wide QD is plotted [80]. From the size of the

Coulomb diamonds a charging energy of EC ≈ 10 meV can be determined. Excited state resonances are clearly visible in form of lines of increased conductance, most prominent at negative bias voltage (see white arrow). A level spacing of ∆ ≈ 1.6 meV has been extracted from the data set. QDs and double quantum dots based on width-modulated nanostructures have been fabricated and studied in transport experiments by a number of groups. Typical diameters of such devices measure between 50 and 150 nm. The addition energies follow roughly a trend inversely proportional to the diameter. The excited state level spacing is typically on the order of 1 to 2 meV. On a 40 nm wide QD a level splitting of up to 10 meV has been observed [70].

4.4. Charge detection

Charge sensors are commonly used in low-dimensional electronics to sensitively resolve individual localized charge states. Charge detectors (CDs) based on quantum point contacts (QPCs) [110] have extensively been used in two dimensional electron systems [111], especially in III/V heterostructures. In such devices e.g. coherent spin and charge manipulation [15, 112] and time resolved charge detection [113, 114] have been demonstrated. Moreover, QPC-based charge detectors are regularly used to read out spin qubits realized in double quantum dot systems in GaAs/AlGaAs heterostructures [19, 59]. This makes charge detection techniques interesting for read out of charge state of graphene QDs as well. An all-graphene device where a nanoribbon has been used as a charge detector for a close-by QD is shown in Fig. 4.3 (a) [109]. The nanoribbon measures a width of 45 nm and is separated from the QD by 60 nm. The close distance allows a capacitive coupling between the two objects. As shown in section 4.1

31 4. Graphene quantum dot devices

10 a B1 PG B2 c /h) 2 e

S D -3 5 300nm (10 qd

CD G 0

b 1.9 -2 d 14 /h) 2 e

1.8 2 -4 pg 12 V (V) (10 log(G(e /h)) 1.7 -4 cd 10 G

8 -10 -5 0 5 10 2.2 2.4 2.6 2.8

Vbias (mV) Plunger gate VPG (V) Figure 4.3.: QD in a width-modulated graphene nanostructure and charge detection. (a) Scanning electron micrograph of a graphene QD with a close-by nanoribbon. (b) Finite bias spectroscopy measurement on an etched graphene QD resolving excited states (see white arrow). (c) Conductance through the QD shown in (a) and (d) through the nanoribbon as a function of the plunger gate voltage. The dashed lines are a guide to the eye emphasising the coincidence of the Coulomb peaks with the conductance steps in the nanoribbon. Panels (a,c) adapted from Ref. [109], c 2008 American Institute of Physics. Panel (b) adapted from Ref. [80], c 2009 American Institute of Physics.

32 4.4. Charge detection transport through GNRs is characterized by resonances originating from localized sates. These resonances can be used to probe charging events on the coupled QD. Fig. 4.3 (c) shows a series of Coulomb peaks measured on the QD and the simultaneously measured conductance of the detector is shown in Fig. 4.3 (d). The overall peak shape is caused by a local resonance in the detector. Whenever an electron is added or removed from the QD the potential of the detector is changed resulting in a conductance step. The conductance steps are well aligned with the Coulomb peaks (see dashed lines as a guide to the eye). Using this method charging events of the QD can even be detected in regimes where the direct current through the QD is below the detection limit (see arrows in the regime between 2 and 2.2 V). The principle of charge detection can also be applied to finite bias spectroscopy measurements. Fig. 7.4 (b) shows the differential conductance through a 100 nm wide QD (for a scanning force micrograph see Fig. 7.4 (a)) [115]. A number of Coulomb diamonds and a rich spectrum of excited states (level spacing on the order of 1.5 to 2 meV) can be observed. The differential transconductance of the detector has been measured simultaneously (see Fig. 7.4 (c)). Prominent features well aligned with the Coulomb resonances in the QD conductance. The detector is most sensitive to the dominant tunneling barrier of the QD. For the two left Coulomb diamonds (marked by the black arrows) the dominant tunneling barrier changes with bias direction, indicating a rather strong capacitive coupling of the QD tunneling barriers to the source and the drain lead. Furthermore, transport via excited state transitions can be identified in the differential transconductance of the CD, highlighted by white arrows. Further studies have addressed the effect of back action of the CD on the QD. The peak current as well as the FWHM of the resonances increase with the applied voltage. It has even been shown that a further increase of the detector bias can fully lift the Coulomb blockade of the QD [115]. So far, the time-averaged current through the CD has been investigated. Real time detection measurements on graphene QDs give deeper insight into the tunneling processes of a graphene QD device (scanning force micrograph shown in Fig. 4.4 (a)) [116, 117]. A schematic of an individual conductance step of the charge sensor signal is shown in Fig. 4.4 (b). The step corresponds to one Coulomb peak

33 4. Graphene quantum dot devices in the QD conductance, when increasing the gate voltage the number of electrons occupying the QD changes by one. The regions labelled (1) and (3) correspond to a ﬁxed electron occupation of N and N+1, respectively. Region (2) - right on the conductance step - corresponds to a conﬁguration where electrons can enter and leave the QD. Experimental data of an individual conductance step is shown in Fig. 4.4 (c). The gate voltage is then parked at the position marked by the arrow and the time- resolved current through the CD is recorded (see Fig. 4.4 (d)). The two levels of current can be observed corresponding to an occupation of the QD with either N or N+1 electrons. Each jump corresponds to an electron entering or leaving the QD. The histogram in Fig. 4.4 (c) shows the current distribution recorded in a time span of 60 s. The occupation probabilities of N and N+1 can be altered by tuning the gate voltage.

4.5. Spin states in a graphene quantum dot

To address spin states in graphene QDs the magnetic field dependency of a 70 nm wide graphene QD device (see Fig. 4.5 (a)) has been studied [118]. Sequences of neighbouring Coulomb peaks have been recorded as a function of magnetic field at a bias voltage of 100µV which is significantly lower than the excited state level spacing. An out-of-plane magnetic field allows to determine the spin filling of orbital states. At an in-plane magnetic field the Zeeman splitting can be measured. Fig. 4.5 (b) shows the spacing of pairs of neighbouring Coulomb peaks as a function of B|| up to 12 T. The spacings depend linearly on the magnetic field with a slope of either zero or approximately ±2µB which is compatible with Zeeman spin splitting with a g factor of 2. Such slopes have been extracted from several pairs of neighbouring Coulomb peaks. The results are plotted as a function of the corresponding gate voltage in Fig. 4.5 (c). Orbital effects due to a misalignment of the magnetic field with respect to the sample orientation have been compensated. From the data a spin-filling sequence of ↓↑↑↓↓↑↑↓ can be determined in contrast to carbon nanotubes where a sequence of ↑↓↑↓ has been observed.

34 4.5. Spin states in a graphene quantum dot

a LG CG RG 1 N 2 N N+1 3 N+1

QD N+1 S D N N+1 N+1 CD N 100 nm b c d 1.6 2 N+1

(nA) 1 3 CD I out in N 0.4 1.58 1.60 0 0.5 1.0 1.5 40k 120k V (V) LG Time (s) # of points

Figure 4.4.: Time-resolved charge detection. (a) Scanning force micrograph of the investigated device. (b) Time-averaged current through the CD as a function of the gate voltage while scanning over Coulomb resonance. In regime (1) and (3) the QD is occupied by either N or N+1 electrons. Position (2) is a metastable state where the occupation oscillates between N and N+1. (c) Time-resolved current measured at position (2) marked in (b). (d) Histogram of the current values for a time span of 60 s. Adapted from Ref. [116], c 2011 American Physical Society.

35 4. Graphene quantum dot devices

2 a b B4-B3 c 1.5 g = ±2 0.2 B9-B8 A3-A2 B4-B3 1 A3-A2 0.1 B10-B9

Z 0.5 A4-A3 B5-B4 B8-B7 s d 0 0 B3-B2 /∆B (meV/T)

-0.5 Z A4-A3 B12-B11 pg -1 A5-A4 -0.1 A5-A4 B6-B5 -1.5 B7-B6 B11-B10

slope ∆E (4-9T) A2-A1 peak spacing ∆E (meV) -2 -0.2 B2-B1 -2.5 B2-B1 0 2 4 6 8 10 12 3.5 4 4.5 5 5.5

B|| (T) VPG (V)

Figure 4.5.: Spin states. (a) SFM image a 70 nm wide graphene QD. (b) Spacing of neighbouring Coulomb peaks as a function of B||. The dashed lines correspond to slopes of ±gµB for a g factor of 2. (c) Slopes of several Coulomb peak splittings. The obtained spin ﬁlling sequence is indicated on top of the plot. Adapted from Ref. [118], c 2010 American Physical Society.

4.6. Gate-deﬁned quantum dots in graphene nanoribbons

Graphene nanoribbons are also the basis for another approach to fabricate graphene SETs and QDs. In this technique graphene flakes are first patterned in GNRs (with a constant width along the GNR, typically below 100 nm) using electron beam lithography and reactive ion etching. In a subsequent lithography step a narrow gate finger is positioned on top of the GNR, separated by thin dielectric. Ohmic contacts are deposited at the ends of the nanoribbon. Due to the narrow width of the GNR a transport gap is opened which enables electrostatic confinement by top and back gate. A series of such devices with different geometries (GNR width ranging from 40 to 60 nm, length from 520 to 2000 nm and top gate width between 50 and 500 nm) has been studied [72]. Figures 4.6 (a,b) show a scanning electron micrograph and a schematic of a typical device. The local top gate and the global back gate allow tailoring the potential landscape along the nanoribbon. The gates locally dope the nanoribbon and thus can create tunable pn-junctions. Choosing appropri- ate gate voltages the device can be driven into a npn or pnp configuration (see Fig. 4.6 (c)). In such a regime either holes or electrons are confined in between the

36 4.6. Gate-deﬁned quantum dots in graphene nanoribbons

a b Graphene

D S TG 300 nm c d 0.4 pp’p 2

1 1.0 current (nA) 0 (nA) -550 -350 -510 0 0.5 top gate (mV) top gate (V) Current -1 0.0 0.1 npn -20 0 20 40 back gate (V)

E E current (nA) 0 I E I EII II -890 -870 -850 top gate (mV)

Figure 4.6.: Gate-defined quantum dot in a graphene nanoribbon. (a) Scanning electron micrograph of a top-gated graphene nanoribbon. The GNR is highlighted by dashed lines. (b) Schematic of the device. (c) Current as a function of back gate and top gate voltage. The insets illustrate the doping profile along the GNR in four different regimes. (d) Coulomb oscillations in the pp’p (VBG = 0 V) and npn (VBG = 81 V) configuration. Adapted from Ref. [72], c 2009 American Physical Society.

two pn-junctions effectively acting as tunneling barriers. Finite bias spectroscopy measurements prove the presence of Coulomb blockade that the device behaves as a SET with addition energies ranging from 0.5 to 3 meV. The addition energy scales inversely with the product of GNR width and top gate width. Even in nn’n and pp’p regimes (n’ and p’ denote strong n- and p-doping, respectively) Coulomb blockade effects can be observed (see Fig. 4.6 (d)). This is not expected due to the absence of tunneling barriers. Electron confinement is then attributed to disorder in a larger area than the gate geometry.

37 4. Graphene quantum dot devices

4.7. Quantum dots deﬁned by anodic oxidation

All devices discussed so far are based on plasma etched graphene nanostructures. The quality of such devices is expected to be influenced by the quality of the edges. Therefore the local anodic oxidation of graphene sheets has been introduced as an alternative technique [90]. In an controlled atmosphere (humidity typically ≈ 70 %) a biased conductive AFM tip carefully scans the a graphene flake along desired lines. The AFM is operated in contact mode, the tip current and the device resistance are monitored in situ. In the vicinity of the tip water molecules dissociate and the resulting radicals react with the carbon atoms thus creating locally nonconductive areas in the graphene sheet. Optimizing the humidity and the applied tip bias this technique is capable writing line widths down to 15 nm. Furthermore, the choice of another etchant in this process allows functionalizing the edges of the graphene structure. QDs with diameters as small as 20 nm - among the smallest graphene QDs reported so far - have been shaped. Fig. 4.7 (a) shows a scanning force micrograph of such a device. Addition energies of up to 50 meV have been observed as well as controlled tunability between electron and hole occupation of the QD. A finite bias spectroscopy measurement is shown in Fig. 4.7 (b). On another device the metal contacts have been evaporated using a shadow mask [91]. Thus it has been possible to avoid any electron beam lithography steps (and hence the use of any polymer resists) in the fabrication process. Excited state spectroscopy measurements have been performed on this device and applying an in-plane magnetic field a g-factor of ≈ 2 could be determined but no regular spin filling sequence.

4.8. Quantum dots on hexagonal boron nitride

Most graphene nanodevices including quantum dots are fabricated from graphene sheets resting on the host substrate SiO2. This material is believed to cause substrate induced disorder limiting the device performance, e.g. making it hard to tune QDs into the few carrier regime. Placing graphene on hexagonal boron nitride (hBN) is one approach to reduce the substrate induced disorder. Scanning tunneling microscopy studies have demonstrated that the size of charge puddles

38 4.8. Quantum dots on hexagonal boron nitride

a b 3.26 z (nm)

Figure 4.7.: Quantum dot deﬁned by anodic oxidation. (a) Scanning force micrograph of a graphene QD structure created by local anodic oxidation. The bright lines are the regions where the graphene has been oxidized. (b) Finite bias spectroscopy measurement on a QD measuring ≈ 20 nm in diameter. Adapted from Ref. [90], c 2010 Wiley-VCH Verlag GmbH & Co. KGaA. in graphene on hBN are one order of magnitude larger compared to graphene on

SiO2 where they measure a few tens of nm [51, 119]. In a competitive study graphene QDs of different diameters have been characterized on both SiO2 and hBN [120]. The graphene / hBN heterostructures have been fabricated by depositing exfoliated hBN flakes on a Si/SiO2 substrate. Graphene flakes are deposited on individual hBN flakes in a controlled way following a transfer process introduced in Ref. [47]. An optical image of a hBN / graphene stack is shown in Fig. 4.8 (a). The 20–30 nm thick hBN flake is visible in blue whereas the graphene is hardly visible (the dashed line is a guide to the eye). Scanning tunneling microscopy unveils a Moire pattern originating from the lattice mismatch of graphene and hBN (see Fig. 4.8 (b)) which indicates a high quality of the graphene. The unit cell vectors (a1, a2) measure approx. 3 nm in agreement with a mismatch of less than 5◦. Graphene nanostructures are then etched by reactive ion etching and contacted as described in section 5.1. The device design is adapted from typical QDs on SiO2. An example of a 300 nm wide QD device is shown in Fig. 4.8 (c). The devices are characterized by low and finite bias spectroscopy. Fig. 4.8 (d) shows a sequence of Coulomb peaks recorded on a graphene QD on hBN. For a quantitative comparison, the spacing between two subsequent Coulomb peaks is evaluated on several QDs on both substrates. Fig. 4.8 (e) shows a competitive analysis of the peak-spacing distribution of QDs of different sizes. The devices

39 4. Graphene quantum dot devices

a b c

d 0.6 e 0.2 0.4

SD 0.1 I0.2 (nA)

0 Standard deviation 0 5.2 5.3 100 200 300

Gate voltage VG (V) Island diameter (nm)

Figure 4.8.: Quantum dots on hexagonal boron nitride. (a) Optical micrograph of a graphene ﬂake on hBN. (b) Scanning tunneling micrograph of the stack (Fourier ﬁltered) showing a Moire pattern. (c) Scan- ning force micrograph of an etched graphene QD on hBN (diameter: 300 nm). (d) Coulomb peaks measured on a 180 nm wide QD on hBN. (e) Standard deviation of the distribution of Coulomb peak spacings as a function of the QD diameter for graphene on hBN and SiO2. Over 600 peaks have been evaluated on each device. Adapted from Ref. [120], c 2013 American Institute of Physics.

on hBN show a clear diameter dependence in contrast to the ones on SiO2. The constant standard deviation of the SiO2 devices hints that the substrate induces disorder is dominant as the contributions due to edge roughness are expected to scale with QD diameter. Vice versa, the decrease of the standard deviation with the diameter indicates the edge roughness being dominant. The data allow to estimate a reduction of the substrate induced disorder in graphene QDs on hBN by roughly one order of magnitude compared to SiO2.

40 4.9. Soft-conﬁnement of quantum dots in bilayer graphene

4.9. Soft-conﬁnement of quantum dots in bilayer graphene

The fact that a band gap can be opened by applying a perpendicular electric field to a bilayer graphene sheet allows for a completely different method to confine electrons. Bilayer graphene quantum dot devices with split-gates have been fabricated based on a suspended flake and on a hBN/bilayer graphene/hBN sandwich[75, 75]. Both devices have in common that a voltage difference between the top gates and the global back gate is applied. The electric field perpendicular to the graphene plane induces a local band gap under each of the gate fingers. The gap can be controlled by the voltage drop (cf. eq. 2.13). Adjusting the gate voltages while leaving the displacement field constant allows driving the Fermi level into the gap. Fig. 4.9 (d) shows a schematic of the band structure of a QD confined between the band gaps opened under the split gates. The soft confinement technique allows to electrostatically confine the electrons in the QD as it is commonly done in devices based on semiconductor heterostructures. The graphene flake does not need to be etched in the desired shape which avoids the influence of edge disorder. Furthermore, using hBN as the host material for the graphene sheet or suspending the flake reduces or even eliminates the influence of substrate induced disorder commonly observed in graphene devices resting on

SiO2. The sandwich device has been fabricated by successive transfer steps. The split gates are arranged on top of the heterostructure in the geometry of a 320 nm wide QD (see Fig. 4.9 (a)). The gate arrangement is reminiscent of layouts used to confine QDs in III/V heterostructures. A voltage drop of 60 V has been applied leading to a displacement field of ≈ 0.6 V/nm which should theoretically result in a band gap of ≈ 50 meV. The experimentally observed transport gap was significantly smaller and conductance could not be tuned below ≈ 1e2/h. Nevertheless Coulomb blockade has been achieved. An addition energy of 0.35 meV has been extracted from finite bias spectroscopy (see Fig. 4.9 (b)). The measurements did not show signatures of excited states which is in agreement with the relatively large diameter of the island. The suspended QD device has been fabricated by first contacting a bilayer

41 4. Graphene quantum dot devices

graphene flake resting on an Si/SiO2 chip and then covering it with a 150 nm thick SiO2 spacer layer. After depositing metal top gate electrodes the oxide substrate as well as the sacrificial layer have been etched away leaving the flake and the split gates suspended. Different lithographic device dimensions have been probed. Fig. 4.9 (c) shows a false colour SEM image of a representative device. Coulomb blockade measurements are in agreement with the geometric sizes (see Fig. 4.9 (e)). Conductance modulations only coupling to the back gate have been observed which hint parasitic conductance channels under the gates limiting the device performance. A suspended QD offers the chance to study the coupling of quantized electronic and vibrational degrees of freedom.

4.10. Double quantum dots in graphene

After successful realization of graphene QDs in width-modulated nanoribbons, the device concept has been extended to double quantum dots (DQDs): Two graphene islands separated by a GNR acting as the interdot tunneling barrier. DQDs are of interest as they can host solid-state spin qubits. Fig. 4.10 (a) shows an example of an etched graphene DQD device surrounded by 5 lateral gates. From a stability diagram, i.e. the current through the QD recorded as a function of two gate voltages, gate lever arms, charging energies, mutual capacitive coupling and excited sate spectra of the two QDs can be determined [73, 86, 122, 123]. The central gate voltage allows to tune the capacitive interdot coupling energy. As transport through GNRs is governed by a large number of sharp resonances, the coupling shows a highly non-monotonic behaviour. Fig. 4.10 (b) shows a detail of a finite bias stability diagram highlighting an individual pair of triple points. Triple points occur at gate configurations where both QDs are in resonance and a current can flow through the device. Fine-structure inside the triple points originates from excited states of the QDs but some of the observed resonances are also attributed to modulations of the tunneling barriers. Another example of a graphene DQD in is shown in Fig. 4.10 (c) [124]. Here a different device geometry has been chosen and metal gate electrodes have been placed in close vicinity of the QDs. The corresponding charge stability diagram is shown in Fig. 4.10 (d). The relative electron occupation of the two QDs is indicated in

42 4.10. Double quantum dots in graphene

a b SD SD V (mV) dI/dV (a.u.)

VTG2 (V)

c T4 d e +0.5 100 Energy plunger gate

T3 EC 2 T T1 EF

2Δ SD gate V (mV) top gates E dG/dV (e /h/V) V 2Δdot

T2 x –0.5 Cr/Au contact –100 9.2 VT12 (V) 9.3

Figure 4.9.: Soft-confinement in bilayer graphene. (a) Schematic of a hBN/graphene/hBN heterostructure with top gates to confine a quantum dot. (b) Finite bias spectroscopy on a device illustrated in (a). (c) False color scanning electron micrograph of a suspended bilayer QD device with suspended top gates. (d) Band structure along a cross section through a QD. Tunnel barriers are formed by inducing a bandgap tuned by an out-of-plane electric field. The back gate allows to induce charge accumulation in the dot and the leads. (e) Finite bias spectroscopy on the device shown in (c). Panels (a,b) adapted from Ref. [121], c 2012 American Physical Society. Panels (c-e) adapted from Ref. [75], c 2012 Nature Publishing Group.

43 4. Graphene quantum dot devices the plot. Not only the capacitive coupling can be influenced by a gate but also the quantum mechanical tunnel coupling. Gate-tunability of the tunnel coupling of a factor of 4 has been demonstrated [124]. Furthermore a significant dependency on the electron occupation has been observed. A DQD device can also be driven into a single QD regime in a controlled way. This effect is explained by a significant increase of the tunnel coupling as a response on the gate voltage [125, 126].

4.11. High frequency gate manipulation

Several experiments have addressed the mutual capacitive coupling, tunnel coupling and the excited state spectra in graphene DQDs. All experiments have been performed at DC. More recently, a graphene charge pump operating in the GHz regime has been demonstrated [76]. Such devices are of great interest among others for the realization of a current standard [127], single-photon generation [128] and read-out of spin-based graphene [23]. A DQD (diameter ≈ 200 nm) surrounded by ﬁve graphene gates tuning the coupling energies and the QD potentials has been fabricated. Two gates, designed as plunger gates to control the QDs, are connected to bias-tees. A RF voltage

VRF (t) is applied to these gates, a phase shifter controls the phase difference between them (see Fig. 4.11 (a)). A low bias charge stability diagram of the device is shown in Fig. 4.11 (b). Due to the RF signal the system describes a circle in the gate voltage plane which is illustrated in the stability diagram. Whenever the DC gate voltages are tuned such that the system encircles a triple point, one electron is shuttled through the system per pulse cycle. When crossing position (1) the potential of first QD is lowered such that an electron from the source is loaded. At (2), the electron tunnels to the second dot. Crossing position (3) the electron leaves the second dot to the drain. The frequency determines the rate of the transferred charges and thus the current through the device which is expected to be I = e · f. Fig. 4.11 (c) shows a measurement of the pumped current as a function of the DC gate voltages at fixed frequency (f = 1.5 GHz) and power. Depending on which triple point is enclosed in the cycle plateaus of different sign but same height are formed. The pumped current shows an almost linear frequency dependency with an error rate (i.e. a deviation of the measured current from the expected value

44 4.11. High frequency gate manipulation

a b I (pA) 100 nm 120 GC 3 GL

GR 100 2 L R GL V (mV) 1 CL 80 CR 0 S D -20 0 20

VRG (mV)

c d RP V (mV) QPC LP dI /dV (nS)

200 nm

VLP (mV)

Figure 4.10.: Double quantum dots in single-layer graphene nanostructures. (a) Scanning force micrograph of a DQD structure with all- graphene lateral gates etched out of a graphene ﬂake. (b) Stability diagram highlighting an individual pair of triple points (VSD = 6 mV) measured on the device in (a). (c) False color scanning electron micrograph of a DQD device with a capacitively coupled charge detector. The metal gate electrodes are placed in close vicinity to active structure. (d) Stability diagram recorded using the charge detector (dIQP C /dVLP plotted as a function of plunger gate voltages). Panels (a,b) adapted from Ref. [123], c 2010 IOP Publishing. Panels (c,d) adapted from Ref. [124], c 2013 Nature Publishing Group.

45 4. Graphene quantum dot devices

a V V b I (pA) c G1 G2 0.1 0.3 φ VRF (t) −0.44

(2) 240 120

(3) (V) 0

(2) G2

V (pA) (1) e I −0.46 −120 0.20

200 nm −240 0.18 (3) 0.18 (1) (V) 0.20 0.16 G2 V V 0.22 G1 (V) −0.32 −0.31 −0.30 −0.29 0.24 VSG VG1 (V)

Figure 4.11.: Charge pumping in a double quantum dot. (a) Scanning force micrograph of the device. A RF voltage VRF (t) is added to the DC gates controlling the QD potentials. A controllable phase diﬀerence φ is added to the signal on gate 2. (b) Stability diagram at bias voltage below 1µ V. The circle illustrates a possible pumping cycle around a triple point. Transitions (1)-(3) correspond to changes of the electron occupation of the QDs (see text). (c) 3D representation of the pumped current as a function of the DC gate voltages at ﬁxed frequency and power (f = 1.465 GHz, P = −15 dBm). Adapted from Ref. [76], c 2013 Nature Publishing Group. e · f) below 0.5% below 0.5 GHz. At higher frequencies the error rate increases up to a few percent due to an increase of failed pump cycles. It scales exponentially with f · RC. In metal charge pumps the same dependency is observed but due to the approximately ten times larger RC constant such devices the error rate of the graphene pump is smaller at the same frequency.

46 5 Sample preparation

5.1. Device fabrication

In this thesis single-layer and bilayer graphene quantum dot devices on Si/SiO2 substrates are investigated. The process technology can be summarized by these major steps:

• Preparation of substrates

• Micromechanical exfoliation

• Raman spectroscopy

• Device patterning

• Fabrication of ohmic contacts

Preparation of substrates: As the substrate material we use commercially available highly p-doped Si-wafers covered with a 290 nm thick layer of thermally grown

SiO2. Highly doped silicon is used in order to serve as a global back gate even at low temperatures. Standard electron beam lithography (EBL) and lift-oﬀ techniques have been used to pattern the wafers with metal markers for further EBL steps and optical alignment. The material combinations 5 nm Cr / 50 nm Au and 5 nm Cr / 50 nm Pt have been used for this purpose. Chromium has been chosen as an adhesion layer instead of titanium as it withstands hydroﬂuoric (HF) acid and thus allow HF dips in the following process steps. The wafers are then cut

47 5. Sample preparation into chips measuring 7 × 7 mm. Immediately before the deposition of graphene ﬂakes the substrates are cleaned in acetone (5-10 min) followed by isopropanol (5-10 min); sometimes an ultrasonic bath has been used. Subsequently they are exposed to an O2-plasma (300 W, 5 min) in a barrel reactor to remove organic residues and water.

Micromechanical exfoliation: Although a lot of progress has been made in the growth of graphene over the past years, graphene flakes exfoliated from natural bulk graphite still provide the best crystal quality. Thus research, especially transport experiments are mainly carried out on such flakes. The technique of micromechanical exfoliation has been introduced by Novoselov et al. in 2004 [129]. It makes use of the fact that in graphite the individual graphene layers are only weakly bond by van-der-Waals forces in contrast to the strong covalent intralayer bonds. This enables us to overcome the interlayer bonds using an adhesive tape and thus allows the cleavage of graphite crystals and the separation of individual graphene sheets. A tiny graphite crystal (lateral dimensions on the order of several mm, see Fig. 5.1 (a)) is placed on a strip of adhesive tape. The tape is then folded such that it encloses the graphite and opened again, all has done several times. In each step graphite crystals will be cleaved and thus thinned down. This process is repeated approximately 10-20 times such that the tape is covered by thin graphite (see Fig. 5.1 (b)). Cleaned substrates are pressed on the tape and peeled off (see Fig. 5.1 (c,d)). Again some graphite flakes will be cleaved and stick to the substrate due to van-der-Waals forces. We noticed that placing the tape with the substrates into an dry atmosphere box (humidity below 20%) for roughly one hour before peeling off the chips increases the amount of flakes transferred. We have chosen an adhesive tape with which it was possible to achieve a sufficiently large yield of single-layer and bilayer graphene flakes per chip while the amount of residual glue on the chip is comparably low. After the deposition of graphene the substrates are cleaned in acetone and isopropanol again. As far as graphene flakes have been deposited on the substrate no ultrasound is used anymore as we have noticed that it is likely to remove flakes by this treatment. The substrates are then searched under an optical microscope.

48 5.1. Device fabrication

a b

1 cm

c d

1 cm

Figure 5.1.: Micromechanical exfoliation. (a) The raw material, graphite crystals. (b) Graphitic ﬂakes distributed on a strip of adhesive tape after repeatedly folding and opening the tape. (c) A pre-patterned Si/SiO2 chip placed on a spot covered with graphitic ﬂakes on the tape. (d) The chip is pressed onto the tape.

49 5. Sample preparation

a b 10 µm 10 µm BL

FL SL

Figure 5.2.: Optical images of exfoliated flakes. (a) Optical microscopy image of a single layer graphene flake together with metal alignment markers. Residuals of the glue of the tape appear in light blue. (b) Flake with a region of bilayer and few-layer graphene. This flake has been selected to fabricate a double quantum dot device, see chapter 9.

Graphene is highly transparent (absorption ≈ 2.3%) but due to interference effects the visibility of graphene can be increased by tuning the oxide thickness and the wave length of the incident light. It has been shown that the contrast has maxima at oxide thicknesses of 90 and 300 nm for green light (λ ≈ 550 nm) [129–131]. With some experience it is possible to identify single-layer and bilayer graphene by eye. Figure 5.2 shows examples of different exfoliated graphene and few-layer graphene flakes on Si/SiO2-substrates.

Raman spectroscopy: Raman spectroscopy on graphene is an interesting ﬁeld in current research. For example, it can be used to investigate strain and doping in graphene [132]. Magnetoraman measurements shed light on electron phonon interaction processes [133]. In this work Raman spectroscopy is only used to reliably identify single-layer and bilayer graphene [134–137] among the ﬂakes that have been deposited onto the substrate. The description in this section is therefore limited to the basics of this technique. The G-peak, typically observed around 1580 cm−1 in graphite, graphene and carbon nanotubes, originates from the excitation of an in-plane optical phonon

50 5.1. Device fabrication at the Γ-point (i.e. with no momentum). Position and width of the peak can be used as a measure for doping and strain. Fig. 5.3 (a) illustrates the processes leading to the different Raman modes. The D-peak (≈ 1350 cm−1) can only be observed if intervalley scattering is allowed, which is the case in the presence of short range disorder. The absence of the D-peak indicates defect free graphene. The 2D-peak is a double resonance effect with a Raman shift around 2679 cm−1. The defect independent resonance involves two phonons (with momenta around K) 0 that interconnect the K and K cone in the electronic band structure. The number of available transitions depends on the number of subbands in the electronic band structure. This makes it an important tool to distinguish between graphene and multilayer flakes [134, 136]. As single layer graphene has one subband there is only one possible transition whereas bilayer graphene has two subbands resulting in four allowed transitions (see Fig. 5.3 (b)). Thus for single layer graphene a Lorentzian line shape of the 2D-peak is expected. In contrast, the 2D-peak of bilayer graphene will be a superposition of four Lorentzians. Fig. 5.3 (c,d) compare the Raman spectra measured on a single-layer and a bilayer graphene flake. A detailed study of Raman spectra of flakes of different thickness can be found in Ref. [136].

Device patterning: Although the fabrication of graphene quantum dots by atomic force microscope nanolithography has been demonstrated [138], plasma based dry etching is still the most common technique to pattern graphene nanostructures [4, 5, 139]. By the help of a CAD software an etching mask is designed individually for each graphene flake. The sample is spin coated with a poly- methylmethacrylate (PMMA) resist. The pattern is then transferred to the resist by electron beam lithography (EBL). After development the graphene is etched by reactive ion etching (RIE). The advantage of dry etching is its high anisotropy and selectivity. It has been proven that this technique does not introduce bulk defects in the graphene sheet [140]. The chosen argon / oxygen plasma (20% O2) combines the physical impact of the argon with the chemical reactivity of the oxygen ions. Short etching times of typically 5-8 s and low power of the radio frequency field (60 W) are sufficient to etch graphene. With increasing etching time or increasing power the PMMA is cross-linked by the plasma making it challenging to remove it with organic solvents. Common treatments to remove hardened resist like oxidiz-

51 5. Sample preparation

defect phonon phonon phonon Γ hω G hω D hω 2D

K K K‘ K K‘

K K‘ K K‘

c d G 2D-line 2D Intensity (a.u.) Intensity (a. u.)

1400 1800 2200 2600 2600 2650 2700 2750 Raman shift (cm-1 ) Raman shift (cm-1 )

Figure 5.3.: Raman spectroscopy. (a) Illustration of the processes leading to the different Raman modes. G-mode: First order process emitting an optical phonon at the Γ-point. D-mode: A defect is necessary to alow interband scattering. An optical phonon close to K is emitted. 2D-mode: Double resonance process emitting phonons close to K. (b) In bilayer graphene four double resonance processes contribute to the 2D-line. Only the first transition of the double resonance is sketched (see dashed lines). (c) Comparison of the Raman spectra of single- layer (blue) and bilayer (red) graphene obtained on the flakes used in chapters8 and 9. In single-layer graphene the 2D-line is at least twice as high as the G-line. (d) Close-up on the 2D-lines shown in panel (c). The bilayer peak is the sum of four Lorentzian line shapes (see dashed curves).

52 5.1. Device fabrication

a b e

c d

Figure 5.4.: Schematic process flow. (a) Graphene flake exfoliated on a highly p-doped Si substrate (blue) covered by 295 nm SiO2 (gray). (b) A layer of PMMA polymer resist (green) has been patterned by electron beam lithography. The graphene is etched by an Ar/O2-plasma (indicated by red arrows). (c) Etched graphene nanostructure. (d) Deposition of ohmic contacts by metal evaporation after a second EBL step. (e) Device after contacting which is now ready for bonding into a chip carrier or PCB. ing acids or plasma ashing cannot be used as these will destruct graphene. After the etching the samples are cleaned in acetone and dimethylsulfoxide (DMSO). The etched graphene structures are inspected by atomic force microscopy allowing to precisely measure the dimensions of the nanostructures. Contamination with residual resist can be detected as well as damaged structures, short circuits or flakes rolling up at the edges. Only structures looking promising will be pro- cessed further. Examples of scanning force microscopy (SFM) images of graphene nanostructures are shown in Fig. 5.6.

Fabrication of ohmic contacts: Bond pads and metal contacts connecting the graphene devices to the bond pads are defined within an additional EBL and lift-off step. 5 nm Cr are deposited as an adhesion layer followed by 50 nm Au. Transmis- sion line measurements have shown that graphene/metal contact resistances can be decreased by using other material combinations, especially palladium [141, 142]. In graphene nanostructures the contact resistance is not regarded as the limiting factor. The result of the lift-off process is monitored by optical and atomic force

53 5. Sample preparation

a b 200 µm

10 µm

Figure 5.5.: Contacted sample. (a) Flake shown in Fig. 5.2 (b) after contacting the etched nanostructure. The rectangular bond pads ar visible in panel (b). microscopy.

5.2. Device design

Single and double quantum dot devices have been fabricated where different device designs and geometries have been investigated. There are a number of parameters strongly influencing the device characteristics. The diameter of the graphene is- −1 lands directly influences the charging energy of the quantum dots EC ∝ d (cf. section 3.1) and the quantum mechanical level spacing ∆ ∝ d−1 and ∆ ∝ d−2 for single and bilayer graphene, respectively (cf. section 3.4). The effective transport gap opened within the nano ribbons connecting the quantum dots strongly depends on their width (cf. section 4.1). The capacitive coupling between the different elements of the devices (quantum dots, charge detectors, lateral gates) is governed by their distance. Quantum dots measuring between 50 and 120 nm in diameter have been fabricated where the charging energy is expected to be on the order of 10 to 20 meV and the level spacing on the order of a few meV. The constrictions have been designed to be significantly more narrow compared to the QDs, they typically measure 30 to 40 nm.

54 5.2. Device design

a b c 200 nm PG CD S D CD CD LG RG QD QD d CD CD S D QD QD CG 200 nm 200 nm S D 200 nm

Figure 5.6.: Atomic force microscope images of diﬀerent graphene quantum dot devices. (a) Bilayer graphene quantum dot with a close-by charge detector and three gates. (b) Quantum dot with two charge detectors on each side and two gates. (c) Double quantum dot device (d ≈ 50 nm) with ﬁve gates and two charge detectors. (d) Double quantum dot (d ≈ 100 nm) where gates and charge detectors have been positioned closer to the dots.

Fig. 5.6 (a) shows a bilayer graphene quantum dot device with a close-by charge detector. Two gates are positioned close to each constriction to control the tunneling rates. An additional gate is intended to act on the QD. Fig. 5.6 (b) presents a QD where the device design has been modiﬁed such that two gates are fabricated close to the QD but at the same time as far away from the constrictions and the source and drain leads as possible. This device layout has been chosen to enhance the coupling to the QD and to reduce the inﬂuence of the gates on the tunneling barriers. Two nanoribbons acting as charge detectors are positioned symmetrically on each side of the QD also being used as gates to locally control the tunneling barriers. A bilayer graphene double quantum dot device (Fig. 5.6 (c)) has been fabricated in a linear design. Five gates have been arranged on one side of the dots, two charge detectors on the other. The device architecture has been optimized such that the charge detectors and gates are positioned much closer to the QDs (see Fig. 5.6 (d)) to enhance the capacitive coupling and to allow a more selective control of each QD.

55 5. Sample preparation

5.3. Graphene-hBN sandwich based devices

Recently, Wang et al. have demonstrated µm size graphene/hexagonal boron nitride sandwich devices featuring room temperature mobilities of up to 150.000 cm2/Vs [8]. The reported technique allows the fabrication of hBN/graphene/hBN sandwiches without exposing the graphene sheet to any polymers or organic solvents [8, 143]. This is a signiﬁcant advantage to the commonly used transfer techniques where graphene and hBN are stacked step by step and a lot of eﬀort has to be done to remove residual polymers either by annealing [47, 144, 145] or even by mechanical cleaning [121, 146].

Fabrication of graphene-hBN heterostructures: Graphene flakes are exfoliated onto a Si/SiO2 chip as described in section 5.1. Flakes of the desired dimension are selected optically. Raman spectroscopy helps to identify single layer and bilayer graphene. A conventional object plate commonly used for optical microscopy is spin coated with a polymer resist. Both PMMA and PMMA/MAA have been used. Hexagonal boron nitride (hBN) is exfoliated onto this polymer the same way as graphene. The raw material, hBN crystals, has been obtained by Taniguchi and Watanabe [147]. By optical inspection and atomic force microscopy hBN-flakes with atomically smooth surface, lateral dimensions of at least 10 × 10 µm and a thickness of ≈ 10 nm are selected. The object plate is turned the hBN facing downwards and mounted to a standard mask aligner usually used for optical lithography. This tool allows the alignment of the hBN flake with respect to the graphene with a precision of a few µm. After proper alignment the chip with the graphene and the object plate are pressed together and subsequently separated again. As the adhesion due to van-der-Waals forces between the graphene and the flat hBN is stronger compared to the adhesion to the comparably rough SiO2, the graphene flake is lifted from its substrate and sticks to the hBN.

hBN flakes are exfoliated onto another Si/SiO2 chip and suitable flakes are selected optically. The graphene/hBN stack (still attached to the object plate) can be aligned on a suitable hBN flake. Repeating the same steps as described before it is possible to pick up the hBN and thus stacks of several graphene an hBN

56 5.3. Graphene-hBN sandwich based devices layers can be built. In the last cycle, chip and object plate are pressed together and heated up to temperatures between 90 and 120 ◦C degree. After separating them again the entire stack including the polymer sticks to the chip. The polymer is then dissolved in acetone. Please note that the graphene is neither in contact to the polymer nor to the organic solvent. Fig. 5.7 (a) shows an optical image of hBN/graphene/hBN sandwich structure. The encapsulated graphene is hardly visible in optical microscopy. Thus Raman spectroscopy is used as an imaging technique to determine the exact position of the graphene. For this purpose the full width half maximum (FWHM) of the characteristic 2D peak is plotted as a function of the position (see Fig. 5.7 (b)). The technique described above allows us to fabricate sandwiches with ﬂat areas up to 10 × 10 µm which is suﬃciently large to fabricate nanostructures. These dimensions are larger than what is usually achieved in graphene/hBN hybrid structures built by wet transfer techniques where it is challenging to design structures free of bubbles and wrinkles [47, 88, 148, 149].

Patterning sandwich based nanostructures. The process successfully used to etch graphene nanostructures is based on the resist PMMA as a mask and the plasma Ar/O2 as the etchant. Graphene as well as few-layer graphene is etched within seconds and thus the PMMA withstands this process. The etching rate of hBN in the same plasma is low (< 5 nm/min) and thus PMMA cannot be used. Fluorine based plasmas (e.g. SF6, CF4, CHF3) etch hBN significantly faster but these plasmas chemically modify the polymer resist and form teflon- alike compounds which are insoluble in organic solvents. Thus, a metal hard mask process is used to overcome these limitations. The etch mask is designed in a CAD software such that the devices are positioned in flat areas of the sandwich. The pattern is written by EBL into a layer of PMMA resist. Evaporation and lift-off creates an aluminum hard mask (10 nm Al). The pattern is then transferred into the sandwich by reactive ion etching using and

SF6/O2 plasma (20 sccm SF6, 5 sccm O2) with an RF power of 60 W. Etching time of 30 s are fully suﬃcient for typical stacks of thicknesses up to 40 nm. The hard mask is subsequently removed in tetramethylammonium hydroxide (TMAH). This patterning technique etches the desired shape and it exposes the edges

57 5. Sample preparation

a b -1 20 µm 2D FWHM (cm ) 24

10 µm 14 c d

1400 1800 2200 2600 2600 2650 2700 2750 Figure 5.7.: Optical micrograph and Raman spectroscopy on an hBN/graphene/hBN sandwich. (a) The two hBN sheets can clearly be seen whereas the graphene is invisible in this image. (b) Map of the FWHM of the 2D Raman peak of graphene. Data has been recorded in the area highlighted by the white dashed square in panel (a). (c) Typical Raman spectrum taken on a hBN/graphene/hBN sandwich. Additionally to the graphene G and 2D mode a characteristic peak originating from hBN can be detected centered around 1365 cm−1. (d) Close-up of the 2D line. The FWHM measures ≈ 17 cm−1.

58 5.3. Graphene-hBN sandwich based devices

a b c

200 nm 200 nm 5 µm

Figure 5.8.: Sandwich based graphene quantum dots. (a) Scanning electron micrograph of an aluminum hard mask on top of a hBN/graphene/hBN. (b) Etched nanostructure after removing the mask (same device as in (a)) The two nanoribbons have moved with respect to the QD, most probably during wet etching the metal mask. (c) Optical micrograph of two contacted quantum dot devices and three other structures which have not been contacted. All nanostructures have been etched out of the same hBN/graphene/hBN stack. of the graphene layer. The surface is still completely covered by hBN. Fig- ures 5.8 (a,b) shows scanning electron microscope images of a sandwich with the Al etch mask and an etched quantum dot device after stripping the mask. Fig- ure 5.8 (c) shows an optical image of two fully contacted devices.

Fabrication of ohmic contacts. The heterostructures are contacted via EBL, metal evaporation and lift-oﬀ. 5 nm of chromium have been used as an adhesion layer followed by 100 nm of gold. A one-dimensional contact to the graphene edges is established while the metal does not cover its surface. Wang et al. have characterized other material combinations using Ti, Pd, Ni and Al as the adhesion layer. The best results have been obtained for Cr/Pd/Au and Cr/Au [8] which is in contrast to conventionally contacted graphene sheets [141, 142].

6 Setup

6.1. 1K system

First measurements have been carried out in a home-built pumped 4He-cryostat. The system consists of a double walled stainless steel tube where the outer volume serves as an isolation vacuum, the inner one as the sample chamber. The entire system is placed into a liquid He dewar. The sample chamber is pumped continuously while the ﬂux of 4He from the dewar into the chamber is controlled via a needle valve. Making use of the energy consuming liquid to gas phase transition of the 4He, the temperature can be controlled by the 4He pressure in the sample chamber. A minimum stable base temperature of around 1.25 K could be achieved at approx. 0.9 mbar. The cooling power of the system can be estimated according to [150] p Q˙ = L V.˙ (6.1) RT The latent heat of 4He measures approximately 85 J/mol around base temperature [151] and the used rotary pump has a nominal pump power of V˙ = 34m3/h. This yields a cooling power of Q˙ ≈ 700µW . Please note that this value is an estimate as the nominal pump power is given for a pressure of 1 bar. Furthermore the pressure is measured outside the cryostat and thus it could diﬀer from the pressure in the sample chamber. The cooling power could be further using a pump with a higher pump power. The system can be operated continuously at a constant temperature for around 10 days, limited by the amount of Helium in the dewar. The sample is bonded in a ceramic chip carrier and mounted in a chip-socket at

61 6. Setup the bottom end of a cold finger. The sample is positioned close the bottom of the sample chamber. A temperature sensor (Cernox resistor) is attached at the back of the chip socket to measure the temperature as close to the sample as possible. The cold finger is equipped with a woven loom cable (24 lines) for DC measurements. The Cu/Ni alloy constantan has been chosen due its low thermal conductivity. With its length of approx. 1 m it has a series resistance of ≈ 100 Ω. Via a connector box at the top of the tube the measurement electronics is connected. In this box 1 kΩ resistances are mounted in series of each DC line. No additional filtering is used.

6.2. Dilution refrigerator

Most of the measurements presented in this thesis have been performed in a commercial Oxford Kelvinox MX400 dilution refrigerator. The operation principal of a dilution refrigerator can be found in detail e.g. in Ref. [152] and is not explained in this work. The cryostat is equipped with a superconducting magnet allowing magnetic fields of up to B = ±9 T. A base temperature of below 10 mK can be achieved at constant B-fields and even when sweeping the field repeatedly the temperature can be kept below 100 mK. A maximum cooling power of 400 µW can be achieved at T = 100 mK. The cryostat is wired with a 24 line constantan woven loom cable for DC measurements which is thermally anchored at each cooling stage. As the link to the measurement equipment an identical connector box as at the 1K setup is fixed directly on top of the cryostat. 1 kΩ resistances are inserted in series of each DC line. The entire DC wiring is very similar to the one of the 1K setup. Figure 6.1 illustrates the AC and DC wiring of the dilution refrigerator. Two semi-rigid stainless steel coaxial cables (UT-85-SS-SS) carry the AC signals. The advantages of stainless steel are its low thermal conductivity and the high mechanical stability (prevention of microphony effects). The RF signals are attenuated by 20 dB at the 4K stage to improve the thermal coupling of the inner conductor of the coaxial lines. Another attenuation stage of 10 dB is mounted at room temperature. The two attenuator stages allow to use higher pulse levels at the output of the wave form generator in order to enhance the signal-to-noise

62 6.2. Dilution refrigerator

DC lines (24x) RF lines (2x) dc connector box bias & gates pulse gates and SMA connectors at room temperature

1 kΩ 10 dB

4 K stage ~200 Ω

1 K stage 20 dB 700 mK stage

100 mK stage

mixing chamber

commercial Anritsu K251 cold-finger ~10 mK ~20 pF

constantan wire stainless steel coax copper coax Thermal anchoring

Figure 6.1.: Wiring of the dilution refrigerator. Schematics showing the AC and DC wiring of the cryostat including the cold ﬁnger.

63 6. Setup ratio. A home-built cold-finger is mounted below the mixing chamber. The sample is placed at its bottom end such that it is positioned in the center of the magnetic field. As in the 1K-system the sample is bonded in a ceramic chip carrier and mounted in a chip-socket. The sample can be placed either in parallel or perpendicular position with respect to the field. The cold-finger is made of oxygen- free copper to enhance the thermal coupling between the sample and the mixing chamber. The woven loom DC lines are wound around the cold-finger to ensure a good thermal anchoring and to create a continuous RC-filter: The wires measure an ohmic resistance of 100 Ω/m and the tightly wound cable establishes a capacitance to the grounded cold-finger of approximately 20 pF. Bias-tees of type Anritsu K251 (bandwidth 50 kHz - 40 GHz, rise time <7 ps), thermally anchored at the mixing chamber, mix AC and DC signals in order to apply them to the same gate. Between the bias-tee and the chip socket the RF signals are carried via two flexible coaxial cables (CuAg alloy). An impedance mismatch close to the sample - most likely occurring at the connection of the coaxial cable to the chip socket - doubles the pulse amplitude. Transmission measurements through the entire setup have shown that this effect together with the attenuators reduces the amplitude of the pulse at the sample by a factor 15.8 (24 dB) with respect to the one produced by the generator. The pulse shape remains almost unperturbed as can be see in Fig. 6.2. The pulse rise time detected close to the sample measures about 250 to 300 ps.

6.3. Measurement electronics

Fig. 6.3 (a) depicts the connection scheme of a typical measurement setup. The core of the DC measurement equipment is a home-built low-noise I-V-converter. This device allows to simultaneously apply bias voltages to the sample and to amplify currents up a factor of 1010. A circuit diagram of the I-V-converter is shown in Fig. 6.3 (b), the main operation principle is described in the caption. The converter mixes AC and DC voltages applied by external voltage sources. In order to minimize noise the DC voltage is divided by a factor of 100 and the AC voltage by a factor of 10.000. By this technique the DC bias applied to

64 6.3. Measurement electronics

1 a b

0.5

Pulse generator Amplitude (a.u.) Close to the sample 0 0 10 20 30 0 1000500 Time (ns) Time (ps)

Figure 6.2.: Typical pulse waveform. (a) Pulse shape of the arbitrary waveform generator Tektronix AWG 7082C. Comparison between the pulse measured at the end of a 2 m coaxial line connected directly to the output of the waveform generator (red, rise time 50 ps) and at the ﬁnal stage of the insert before RF optimization (blue, rise time 300 ps). In the latter case the signal has passed through the entire electronic set-up, including attenuators and bias-tee. The data sets are normalised. (b) Close-up of the data in (a). The vertical lines mark the rise time at the sample stage.

a DC-Source I-V Converter b RF Digital Mutlimeter Input A (Yokogawa 7651) (Agilent 34410A) Vref Out A - Output A Vbias (DC) - Out B Cw + VOUT DC-Source V (AC) In A In B Digital Mutlimeter + bias IS (Yokogawa 7651) (Agilent 34410A) VIN

-Vbias /2 R S Vref Source Drain DC-Source(s) (Yokogawa 7651) Gate(s) RF ID DC Connector Box Input B Pulse generator - Output B (Tektronix AWG Pulse gates - + VOUT 520 & 7082C) Sample + AC Connectors VIN

Magnet Vbias /2 power supply Cryostat Vref

Figure 6.3.: Measurement setup and I-V-converter.(a) Basic principle of a typical measurement setup. (b) The left operational amplifiers (blue) operated in feed-back mode drive a current through the feedback resistor RF and the sample (RS) such that the voltage differences at their input ports (+,-) equal zero. To achieve this the operational amplifiers set their output voltage to Vout = ±Vbias + IS/DRS. The right operational amplifiers subtract the bias voltage from Vout such that the voltages IS/DRS are measured at the outputs of the I-V- converter. The feedback resistor RF defines the amplification factor. Figure adopted with changes from Ref. [153].

65 6. Setup the sample can be modulated with a small AC voltage of a lock-in ampliﬁer.

The bias voltage is applied symmetrically (±Vb/2) to the sample with respect to ground. Additionally the voltages can be oﬀset by a reference potential Vref :

VS = Vref + Vb/2, VD = Vref − Vb/2. The voltage sources Yokogawa 7651 and the lock-in amplifiers Stanford Research SR830 are used. The output of the Yokogawa 7651 is low pass filtered (τ ≈ 1 ms) to reduce RF noise. The I-V-converter amplifies both the source and drain current by up to a factor of 1010 and converts them into a voltage which are then measured by multimeters (Agilent 34401A or 34410A). Amplification factors in powers of ten ranging from 105 to 1010 can be selected. According to the gain bandwidth product - which is approximately constant for an operational amplifier - the bandwidth scales inversely with chosen amplification. A characterization of our home-built amplifiers has determined a bandwidth below 70 kHz for a typical amplification of 108 [153]. The gate voltages are directly applied by Yokogawa 7651 voltage sources via a low pass (τ ≈ 1 ms). The pulse patterns are created by the arbitrary waveform generators Tektronix AWG520 and Tektronix AWG7082C. The AWGs allow pulse rise times as fast as 1 ns and 50 ps, respectively. Measurements are controlled by a LabView programme ”Step and Log” (programmed by Simon Gustavsson et al.) or by a MatLab based programme ”special measure” (programmed by Hendrik Bluhm et al.).

66 7 Charge sensing

This chapter has been published in parts in:

Graphene-based charge sensors C. Neumann, C. Volk, S. Engels, and C. Stampfer Biotechnology 24 44401 (2013) c 2013 IOP Publishing

Charge detection in a bilayer graphene quantum dot S. Fringes, C. Volk, C. Norda, B. Terr´es,J. Dauber, S. Engels, S. Trellenkamp, and C. Stampfer Physica Status Solidi B 248 2684–2687 (2011) c 2011 WILEY-VCH Verlag GmbH & Co. KGaA

Back action of graphene charge detectors on graphene and carbon nanotube quantum dots C. Volk, S. Engels, C. Neumann, and C. Stampfer Physica Status Solidi B, DOI:10.1002/pssb.201552445 (2015) c 2015 WILEY-VCH Verlag GmbH & Co. KGaA

7.1. Introduction

Charge sensors play an important role in low-dimensional electronic circuits, where detecting changes of localized charge states are crucial and challenging tasks. In

67 7. Charge sensing fact, nanoelectronic systems i.e. electronic systems with reduced dimensions show a variety of interesting physics including Coulomb blockade [53], Kondo effect [154] or Fano resonances [155], all closely related to the localization of electronic charge. Read out and manipulation of isolated electrons are key elements for studying these phenomena. In this context charge detectors (CDs) based on quantum point contacts (QPCs) [110] have extensively been used in two dimensional electron systems [111], especially in III/V heterostructures. In such devices coherent spin and charge manipulation [15, 112], full counting statistics [113], time resolved charge detection [113, 114] and controllable coupling to different quantum devices [156– 158] have been demonstrated. Moreover, QPC-based charge detectors are regularly used to read out spin qubits realized in double quantum dot systems in GaAs/AlGaAs heterostructures [16, 17, 19, 59]. In these experiments the charge detection fidelity is of great interest in order to maximize the read out speed. This figure of merit can be optimized by increasing the pinch-off slope and the capacitance between the QPC and the investigated device. While the capacitance can be tuned by reducing the distance between the charge detector and the system of interest (see illustration in Fig. 1(a)), the slope can be increased by replacing the QPC by a single electron transistor (SET). Recently, the use of charge detectors has been extended to hybrid systems where for example a nanowire quantum dot was probed by an underlying QPC detector [159, 160] or a metallic SET was used to detect charging events on a carbon nanotube quantum dot [161]. It has also been shown that narrow graphene ribbons can be used as well-working charge detectors [109]. This approach has been employed to perform charge sensing on individual graphene quantum dots [74, 87, 109, 117, 162], including time resolved detection of charging events on such systems [116]. Additionally, a carbon nanotube graphene hybrid device has recently been demonstrated where the charge state of a carbon nanotube quantum dot can be detected by the current through a nearby graphene nanoribbon-based charge sensor [88]. Here, graphene QD devices with integrated nanoribbon-based charge detectors are fabricated. QD and CD are characterized by low temperature transport measurements. The performance of the detector under the influence of an in-plane magnetic field, RF pulsed gating and temperature is studied. Special emphasis lies on the influence of the CD on the transport properties of the probed QD. The

68 7.2. Device design and measurement setup eﬀects of back action and counter ﬂow are addressed.

7.2. Device design and measurement setup

Graphene quantum dots devices with integrated graphene charge detectors are fabricated as described in chapter 5.1. The graphene QDs investigated in this chapter have diameters of around 100 nm and they are connected to source and drain leads via narrow graphene constrictions which act as tunable tunneling barriers [84]. Two nanoribbons with a width of about 70 nm, located at either side of the QD, act as charge detectors (see Fig. 7.1). By applying a reference potential to them, they can also be used as lateral gates, especially for tuning the transparency of the tunneling barriers. Additional plunger gates (PG) allow to electrostatically tune the QD potential as well as the tunneling rates. The highly p-doped Si substrate acts as a back gate and can be employed to tune the overall Fermi level in the graphene device. All measurements are performed in a dilution refrigerator, at a base temperature below 20 mK. Home-built low-noise DC ampliﬁers (ampliﬁcation factor of 108, bandwidth < 1 kHz, integration time 200 ms) allow us to measure currents with a precision better than 50 fA (cf. chapter 6).

7.3. Device characterization

Fig. 7.2 shows the transport characteristics of a graphene quantum dot and a graphene nanoribbon (cf. device shown in Fig. 7.1). At a constant bias voltage of

VSD = 1 mV and VCD = 0.5 mV, respectively, the current is measured on a large energy, i.e. large back gate voltage range (VBG = 10 V to 50 V). For small back gate voltages both devices show hole-dominated transport, while for large back gate voltages the transport is electron-dominated (see also insets in Fig. 7.2 (b)). These two regions are separated by the so-called transport gap, where the measured current is signiﬁcantly suppressed. The transport gap is located at positive gate voltages (around 20 to 37 V in Fig. 7.2 (a) and 27 to 37 V in Fig. 7.2 (b)) indicating a signiﬁcant p-doping of both structures, which is commonly observed in etched graphene nanostructures [102, 139, 163, 164] and which has also been

69 7. Charge sensing

a b

ICD

VBG I QD 200 nm

Figure 7.1.: Sample geometry. (a) Schematic and (b) scanning force micrograph of the measured device. The current through the quantum dot ISD and through the charge detector ICD can be measured simultaneously. The electrochemical potential of the QD is controlled by the plunger gate (PG). observed in the bilayer graphene double quantum dot presented in chapter 9 [83]. This p-doping arises most likely due to polymer resist residues and/or oxygen atoms bound to the graphene edges originating from the Ar/O2 plasma etching process [165]. In the transport gap regime the electronic transport is dominated by stochastic Coulomb blockade (cf. section 4.1). The observed large-scale current ﬂuctua- tions originate from local resonances in the graphene constrictions. The inset in Fig. 7.2 (a) shows a close up of the back gate characteristics of the graphene QD inside the transport gap (see highlighted region).

7.4. Charge detection

Coulomb resonances are observed if a state of the QD is aligned between the chemical potentials of the source and drain leads, otherwise transport is blocked due to Coulomb blockade, see Fig. 7.3 (a) and chapter 3.2. Individual conductance resonances in the transport characteristics of a graphene nanoribbon can be used to detect charging events on a capacitively coupled QD close by. Whenever the

70 7.4. Charge detection

a b 1.5 2 1 4 1 0 24.08 24.16 24.24 2 SD I (nA) 0.5 CD I (nA) 0 0 10 20 30 40 50 10 20 30 40 50

VBG (V) VBG (V)

Figure 7.2.: Back gate characteristics of a quantum dot and a charge detector. (a) Source-drain current ISD measured at a quantum dot as a function of back gate voltage VBG. A transport gap, where the current is strongly suppressed, is observed in the range VBG = 20 V to 37 V . Inset: Close up of the highlighted region inside the transport gap. distinct Coulomb resonances are observed. (b) Similar measurement as in panel (a) but measured on the charge detector. The inset highlights the hole and electron dominated transport regions.

overall charge of the QD changes, the electrostatic potential in the nanoribbon- based charge detector is altered due to the capacitive coupling of the two devices. A resonance in the ribbon will be shifted with respect to the gate voltage which is illustrated by the black and the gray curves in Fig. 7.3 (b). This results in sharp steps in the current measured through this ribbon (red curve). Thus, individual charging events in the QD can be probed by measuring the current passing through the nanoribbon, which consequently acts as a charge detector [74, 109, 162]. Ob- viously steep slopes within the conductance of the charge detector are favourable. In Fig. 7.3 (c) the simultaneously measured current through the quantum dot

(ISD) and the charge detector (ICD) are plotted as a function of the plunger gate voltage VPG. The back gate voltage VBG has been ﬁxed at 23 V such that the QD is operated in the transport gap and the CD in a regime with a number of sharp resonances. The current through the QD (ISD) shows almost equally spaced

Coulomb peaks which are perfectly aligned with the steps in ICD observed on top 2 of the resonance. Steep slopes of up to 0.08e /h per VPG can be observed in the

ICD-signal for VPG < −2 V while at VPG ≈ −1.8 V the slope of the resonance

71 7. Charge sensing

a b c 3 Dot Detector 0.4

1 2 0.2

SD 1 CD I (pA) 0 I (nA) CD I (arb. units) 0 1 2 -0.2 -2.5 -2 -1.5 -1 VPG (arb. units) VPG (V)

Figure 7.3.: Charge detection on a QD in the low-bias regime. (a) Schematic energy diagram of a QD in the Coulomb blockade regime and in the sequential tunneling regime. (b) Schematic illustration of the operation principle of the CD. Each individual charging event on the QD shifts the CD resonance (black and gray lines) due to the capacitive coupling of both devices. This mechanism results in steps in the gate dependent measurement (red line). (c) Simultaneous measurement of the QD and CD current as a function of the plunger gate voltage VPG (VSD = 1 mV, VCD = 0.5 mV). Please note that even at very low QD currents (e.g. at VPG = −1.22 V, VPG = −1.07 V) the CD can resolve individual charging events. is ﬂat. The CD can even resolve charging events that cannot be measured in the direct current (ISD) since the Coulomb peaks vanish in the noise ﬂoor (see regime

VPG > −1.5 V in Fig. 7.3 (c)). The noise ﬂoor measures roughly 100 fA at an integration time of 200 ms.

7.5. Finite bias spectroscopy

Fig. 7.4 (a) shows so-called Coulomb diamonds, i.e. the diﬀerential conductance dISD/dVSD measured on the QD depending on the gate voltage VPG and the bias

VSD. The addition energy measures roughly Eadd = 11.5 meV which is in good agreement with other quantum dots of a similar size (see e.g. chapter 8). The relative gate lever arm is approximately αPG = 0.08. A rich spectrum of excited states in the QD is observed as faint lines running parallel to the diamond edges. Their energies range from approximately 1.5 to 2.5 meV. The current through the detector ICD (Fig. 7.4 (b)) shows features well aligned with the Coulomb resonances visible in Fig. 7.4 (a). Those can be better resolved in the diﬀerential transconduc-

72 7.5. Finite bias spectroscopy

a 8 QD

4 0 SD V (mV) -4 dISD /dV SD (nS) -8 0 3 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 b 8 QDCD 4 0 SD

V-4 (mV) ICD (nA) -8 0.36 0.48 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 c 8 CD 4 0 SD

V-4 (mV) dICD /dV PG (nS) -8 -50 150 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5

VPG(V)

Figure 7.4.: Finite bias Spectroscopy. (a) Diﬀerential conductance dISD/dVSD as a function of VSD and VPG. Coulomb diamonds as well as numer- ous excited states are visible. (b) The simultaneously recorded charge detector current ICD shows an abrupt step for each charging event in the QD. (c) The diﬀerential transconductance dICD/dVPG exhibits features which can be associated with excited states (see white arrows).

tance dICD/dVPG in panel (c). Interestingly, for the two left Coulomb diamonds (marked by the black arrows) the dominant tunneling barrier changes with the sign of VSD, indicating a rather strong capacitive coupling of the QD tunneling barriers to the source and the drain lead. Apart from the ground state transport also transport via excited state transitions can be identiﬁed in the diﬀerential transconductance of the CD, highlighted by white arrows.

73 7. Charge sensing

7.6. Sensitivity

In the measurements discussed so far, the bias applied to the charge detector

(VCD) was optimized such that charging events in the QD could be easily detected.

The following section focuses on the influence of VCD on the detection sensitivity. Fig. 7.5 (a) shows a finite bias spectroscopy measurement of the detector, i.e. the transconductance of the detector dICD/dVPG as function of VCD and VPG. Remarkably, for small bias voltages well-resolved Coulomb diamonds with charging energies on the order of 1.5 meV occur. This is in agreement with the origin of the resonances in the charge detector nanoribbon. Within the diamonds, transport is blocked due to Coulomb blockade and thus the CD is completely insensitive to charging events in the nearby QD. If VCD exceeds the addition energy lines parallel to the VCD-axis appear in the differential transconductance which coincide with the Coulomb peaks of the QD.

In order to detect QD states over an extended VPG range the CD must be operated at a bias value outside the Coulomb blockade regime. But as a trade off the current through the CD increases with increasing VCD. This broadens the CD resonances resulting in a decreasing signal-to-noise ratio. This effect is demonstrated in Fig. 7.5 (b). The signal-to-noise ratio of three conductance steps of the CD (corresponding to three subsequent charging events of the QD, see arrows in panel (a)) is plotted as function of VCD. The signal-to-noise ratio is best for low VCD as long as the CD is not blocked due to Coulomb blockade and then decreases with V . From Fig. 7.5 (b) we extract an average charge sensitivity √CD −3 of 1.3 × 10 e/ Hz at VCD = 0.5 mV. Please note that the charge sensitivity strongly scales with VCD and the slope of the CD resonance where the charging event in the QD occurs. Magnetic field are necessary e.g. to detect individual spin states in graphene QDs. Thus the influence of high external magnetic field on the QD are investigated. Please note that it has been shown that magnetic fields may strongly alter the transport properties of graphene nanoribbons [166], rendering this a non-trivial task [149]. In Fig. 7.6 (a) two subsequent Coulomb resonances are measured for an in-plane magnetic field range from B =0 to 7.4 T. Both resonances can also be observed in the transconductance through the CD (Fig. 7.6 (b)) with a similar

74 7.7. Pulsed gating

1 2 3 a 3 b 140 3 4 0 2 100

0 -3 step noise CD 60 V (mV) Δ I /I -2 2 20 1 -4

CD LG -20 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 dI /dV(nS) 0 1 2 3 4 5 V (mV) VPG (V) CD

Figure 7.5.: Sensitivity of the charge detector. (a) Diﬀerential transconductance dICD/dVPG of the CD as a function of VCD and VPG. The charging events of the QD detected by the CD appear as white and dark vertical lines. Additionally, the CD shows clear Coulomb diamonds arising from the disorder induced isolated islands in the graphene nanoribbon. (b) Charge detector current step height divided by the average noise level of the current as function of VCD of three diﬀerent charging events in the QD (marked by the three arrows in panel (a)). The highest steps occur at the onset of the current at the edges of the Coulomb diamonds in the CD. resolution.

7.7. Pulsed gating

So far, the DC performance of the CD has been studied. RF pulsed-gate experiments are important to explore the relaxation dynamics of individual charge and spin states. Such techniques are commonly used on QDs based on III/V heterostructures [16, 167] and charge sensors often serve as sensitive read-out tools. RF gate manipulation of a graphene QD has been reported but a CD has not been used [168]. This paragraph focuses on the performance of the CD under the inﬂu- ence of pulsed-gating. Further details of RF pulsed-gate spectroscopy on graphene QDs are given in chapter 8. A rectangular pulse sequence is applied to the plunger gate (PG) (see Fig. 7.7 (b)), which shifts the QD potential by the electrostatic coupling. In Fig. 7.7 (b) the evolution of a Coulomb resonance for increasing pulse amplitude at a constant frequency of 20 MHz is plotted. The Coulomb resonance splits into two peaks with increasing pulse amplitude. The equivalent height of the

75 7. Charge sensing

a 0.3 b 30

0.2 6 6 0 0.1

4 0 4 -30 B (T) B (T)

2 2 SD I (pA) CD PG

0 0 dI /dV (nS) 2 2.1 2.2 2 2.1 2.2

VPG (V) VPG (V)

Figure 7.6.: Influence of the magnetic field. (a) Magnetic field dependence of two Coulomb resonances at VSD = −1.5 mV. (b) Both resonances can clearly be resolved in the CD differential transconductance over the entire magnetic field range. two peaks after splitting is in good agreement with the pulse duty cycle of 50%. This splitting is also observed in the charge detector transconductance as shown in Fig. 7.7 (c).

7.8. Back action

This section focuses on the inﬂuence of the charge detector on the quantum dot, the so-called back action. It can be observed that the Coulomb resonances of the QD broaden with increasing charge detector bias (see Fig. 7.8 (a). This effect is investigated systematically by ﬁtting several Coulomb peaks of the QD by

Imax ISD(VPG) = 2 , with the fit parameters peak position Vres, peak cosh [(VPG−Vres)/a] current Imax and line width a. An exemplary evolution of an individual Coulomb resonance with increasing VCD and the corresponding fits are displayed in Fig. 7.8 (a). Fig. 7.8 (b) shows the FWHM averaged over eight Coulomb peaks as a function of VCD. Varying VCD from 0 to 4.5 mV increases the FWHM by around 50%. At the same time the average peak height rises by about 400% (see Fig. 7.8 (c)). These measurements provide that the bias voltage applied to the CD has to be chosen carefully as it strongly influences the transport through the QD.

76 7.8. Back action

a b QD c CD 1 160 0.8 120 0.6 pulse

V 80 SD I0.4 (pA) CD PG I /V (pS) 40 0.2

0 0 2.3 2.35 2.4 2.3 2.35 2.4

Time VPG (V) VPG (V) Figure 7.7.: Influence of an RF pulse scheme. (a) Schematics of a symmetric rectangular pulse sequence with increasing amplitude. A bias-tee mixes AC and DC signals to the same gate. (b) Coulomb resonance under the influence of a 20 MHz pulse measured in the direct current ISD. It splits linearly with increasing amplitude (see dashed lines). (c) Differential transconductance through the CD measured simultaneously through the CD.

a b 1.6 c 6 VCD = 4.4 mV 3.6 mV 0 5 80 3.0 mV 1.4

2.7 mV 0 4 2.2 mV 1.2 mV 1.2 3

SD 40 I (pA) 0.3 mV QD Γ / 2 1

QD FWHM/FWHM 1 0 0.376 0.386 0 1 2 3 4 5 0 1 2 3 4 5

VPG (V) VCD (mV) VCD (mV)

Figure 7.8.: Back action eﬀects. (a) Evolution of a Coulomb resonance of the QD with increasing VCD. Red lines show the ﬁt to the experimental data (blue crosses) according to ISD(VPG) = 2 Imax/cosh [(VPG − Vres)/a]. (b) FWHM and (c) peak current averaged over 8 Coulomb resonances as a function of VCD (normalized to VCD = 0 mV). The FWHM increases by 50% at VCD = 4.5 mV while an average increase of the peak current by 400% can be observed.

77 7. Charge sensing

3 8 2

4 1 0 0 CD V (mV) -4

-8 SD

-0.46 -0.42 -0.38 -0.34 log|I | (log|pA|)

VPG (V)

Figure 7.9.: Back action. Logarithmic current through the QD (log(|ISD|)) as function of VPG and charge detector bias (VCD) for constant VSD = 0.5 mV. The Coulomb resonances of the Quantum dot broaden with increasing VCD such that a diamond-shaped pattern can be observed. The Coulomb blockade is completely lifted for VCD exceeding 9 mV.

The strong back action effect can also nicely be seen in Fig. 7.9 where the current through the QD is plotted as a function of VPG and VCD. With increasing VCD the Coulomb resonances in the QD broaden until the Coulomb blockade is lifted completely and a diamond-shaped pattern can be observed. This dependency on the current flowing through the CD indicates that the increase and broadening of the Coulomb resonances originate from noise and fluc- tuations in the CD nanoribbon [116]. Coupling of the nanoribbon to the QD via phonons seems less plausible as the phonons would have to couple via the SiO2 substrate due to the destroyed graphene lattice. Consequently, it is likely that photons play an important role and that processes related to photon assisted tunneling are responsible for lifting the Coulomb blockade in the transport through the graphene QD.

7.9. Counter ﬂow

In the previous section back-action eﬀects have been studied where the Coulomb resonances in the QD broaden as a function of the voltage VCD applied to the charge detector (cf. Figs. 7.8 and 7.9). Depending on the charge state of the QD the current through the QD can not only strongly increase but also completely reverse as a response to the applied voltage on the CD. This so-called counter ﬂow

78 7.9. Counter ﬂow

a 1.5 b 1.5 0 mV 20 mV SD 1 1 SD

0.5 0.5 Current I (pA) Current I (pA) SD 0 30 SD 0 120 15 60

Bias V (mV) -0.5 Bias V (mV) -0.5 0 0 -1 -1 -15 -60 -1.5 -30 -1.5 -120 24.06 24.08 24.1 24.12 24.14 24.16 24.18 24.06 24.08 24.1 24.12 24.14 24.16 24.18

Back Gate VBG (V) Back Gate VBG (V)

Figure 7.10.: Finite bias spectroscopy measurements of three neighbouring Coulomb resonances. Current through the QD measured as a function of the bias voltage VSD and the back gate voltage VBG The bias voltage applied to the charge detector measures VCD = 0 V in (a) and VCD = 20 mV in (b). The dashed lines are guides to the eye. effect is investigated in this section. Finite bias spectroscopy measurements have been performed for different voltages applied to the right charge detector. At VCD = 0 V typical Coulomb diamonds can be observed with - as expected - positive current at positive bias and vice versa (see Fig. 7.10 (a)). Fig. 7.10 (b) shows a finite bias spectroscopy measurement atVCD = 20 mV in the same range as shown in Fig. 7.10 (a). An overall increase of the transmission through the QD is observed which is in agreement with the back-action effect studied in section 7.8. Moreover, for some peaks the transition between positive and negative current moves away from the line of

VQD = 0 mV (see dashed lines in Fig. 7.10 (b)).

For a detailed analysis Fig. 7.11 shows corresponding line cuts at VQD = 0, ±0.1, ±0.2, ±0.3, ±0.4 and ±0.5 mV. Here, the red traces illustrate the data sets for

VQD = 0 mV. It can clearly be seen that already at VCD = 10 mV (see Fig. 7.11 (b)) all resonances tend to split into a peak of positive and one of negative current.

Elevated VSD can compensate for this eﬀect and thus suppress counter ﬂow. The

Coulomb peaks centered at VBG ≈ 24.11 V and 24.15 V show a negative shoulder on the left and a positive shoulder on the right for all VCD > 0. The peak centered at VBG ≈ 24.07 V shows the opposite behaviour at low VCD (see e.g.

Fig. 7.11 (b)). At VCD ≥ 40 mV the Coulomb blockade is lifted as can be seen from the gate voltage regimes between neighboring Coulomb resonances (see arrows in

79 7. Charge sensing

a b 20 4 0 mV 10 mV

10 2

0 0

Current (pA) -2 Current (pA) -10

-4 -20 c 60 d 150 20 mV 40 mV 40 100

20 50

0 0

Current (pA) -20 Current (pA) -50

-40 -100

-60 -150 24.06 24.08 24.1 24.12 24.14 24.16 24.18 24.06 24.08 24.1 24.12 24.14 24.16 24.18

Back Gate VBG (V) Back Gate VBG (V)

Figure 7.11.: Counter ﬂow. Current through the QD as a function of VBG recorded for diﬀerent bias voltages VSD = 0, ±0.1, ±0.2, ±0.3, ±0.4 and ±0.5 mV. The zero bias traces are highlighted in red. The charge detector is biased with VCD = 0, 10, 20 and 40 mV respectively, as indicated in each panel. The arrows in (d) emphasize the lifted Coulomb blockade at VCD = 40 mV.

Fig. 7.11 (d)). In order to discuss the observed behavior a simple model of transport through the QDs is employed which is derived from rate equations considering only one ground state in the QD:

Γ Γ I = Γ · (f (E) − f (E)) = − . (7.1) S D α(V0−VG)+VSD /2 α(V0−VG)−VSD /2 1 + e kB TS 1 + e kB TD

Here, V0 is the center of the Coulomb resonance, α the gate lever arm, Γ the overall tunnel tunneling rate which can be energy dependent and fS(E), fD(E) are the

Fermi distributions of the source and drain lead, respectively. The parameters TS and TD describe the broadening of the Fermi functions fS(E), fD(E) related to the two leads. A corresponding schematic of the model is shown in Fig. 7.12 (a). Here, the

80 7.9. Counter ﬂow

a b c 4 2 VCD = 30 mV 2 20 2 1 QD QD VSD 0 0

1 -2

Current I (pA) Current I-20 (pA)

-4 VCD = 10 mV 24.10 24.12 24.10 24.12 fS (E) TSD

Figure 7.12.: Comparison with the model of thermal broadening. (a) Il- lustration of the model assuming an asymmetric broadening of the Fermi distributions fS(E) and fD(E) in the leads and only considering a single ground state contributing to electronic transport. The blue areas indicate the broadening of the Fermi distribution in the source and drain reservoir. The two red lines indicate possible positions of QD levels and the corresponding direction of net current is indicated by the green arrows. (b) and (c) Comparison of experimental data (blue) and ﬁts (red) according to the model assuming single level transport (see eq. 7.1). Data for bias voltages VQD ranging from -0.4 mV to +0.4 mV are shown in each panel. The bias applied to the charge detector VCD measures 10 and 30 mV, respectively.

Fermi distributions fS(E) and fD(E) are assumed to exhibit a different amount of broadening with TS < TD and are offset by a positive bias VQD. This directly results in two scenarios where the gate voltage can either tune the QD state to a chemical potential E where fS(E)-fD(E) > 0 (scenario 1) or fS(E)-fD(E) < 0 (scenario 2). Scenario 1 thus results in an enhanced flow of electrons in bias direction, while in scenario 2 the electrons flow against the bias direction. Figs. 7.12 (b) and (c) show fits (red traces) of the described model to the experimental data (blue data points) at VCD = 10 mV (b) and VCD = 30 mV (c). Evidently, the model successfully reproduces the entire shape of the measured Coulomb resonances for different QD biases. In particular, the two scenarios 1 and 2, where the current is either enhanced (1) or flows against the direction of the applied bias (2) can be nicely explained (see e.g. arrows and labels in Fig. 7.12 (b)). To obtain more quantitative results, we fit the model to data sets measured for different VCD and extract the parameters TS and TD.

81 7. Charge sensing

a b 8 120

6 T S peak 80

TD 4 40

2 Peak current I (pA)

Effective temperature T (K) temperature T Effective 0 0 0 10 20 30 40 50 0 10 20 30 40 50

Charge detector bias VCD (mV) Charge detector bias VCD (mV)

Figure 7.13.: Effective temperature and peak currents. (a) Effective temperatures TS (blue) and TD (red) of the leads determined by the fits according to equation 7.1 (see e.g. Fig. 7.12). Measurements in a bias regime |VSD| ≤ 0.4 mV have been evaluated at each VCD. The solid lines represent linear fits to the experimental data. (b) Absolute peak Ipeak current of the negative shoulder of each resonance (see arrow labeled by (1) in (b)). The solid line represents a quadratic fit to the data.

Figure 7.13 (a) summarizes the results as a function of VCD. Roughly, a linear dependency of both eﬀective temperatures on VCD can be observed.

However, while TS only increases from ≈ 1.0 K at VCD = 0 mV to ≈ 2.5 K at VCD = 50 mV, TD shows a strong increase from ≈ 1.0 K to ≈ 8 K within the same range. This finding clearly shows an increasing asymmetry of the Fermi distribution broadening of both leads. An asymmetric influence of the charge detector on the two leads is in agreement with the device geometry (see Fig. 7.1 (b)) where one of the QD leads is closer to the CD than the other. As observed in Fig. 7.11 the current flowing against the bias direction increases with VCD. For a more quantitative analysis the absolute peak current of the negative shoulder of each Coulomb resonance centered around VG ≈ 24.11 V is plotted in Fig. 7.13 (b). A quadratic dependency of the peak current Ipeak on the detector bias is observed. This finding suggests that the current induced in the QD is approx. proportional to the electrical power dissipated in the CD. The model described above has a number of limitations. Neither does it not

82 7.10. Temperature dependence

provide the expected electron temperature for VCD = 0, nor does it adequately describe all observed transport phenomena. For example, Coulomb peaks ex- hibiting a total inversion of the current such as the one shown in Fig. 7.11 at

VBG ≈ 24.08 V cannot be explained within the limits of the single level transport model. A similar effect has been observed in a hybrid device of a carbon nanotube (CNT) QD coupled to a graphene nanoribbon CD [153]. For the discussion of that behavior a more advanced model has been employed which considers three levels, i.e. the ground state (GS) and two excited states (ES), contributing to transport through the QD [169]. The discussed transport model relies on an asymmetric broadening of the source and drain leads. The experimental results show that this broadening has to be induced by the interaction of the GNR based charge sensor with the QD. The quadratic dependency of the Coulomb peak current on VCD suggests a proportionality to the power dissipated in the CD. For typical VCD used in our experiments the dissipated power is on the order of hundreds of pW. However, the transfer of thermal energy from the graphene CD to the QD is hard to explain. While detector generated phonons are regarded as a relevant origin for back action in GaAs quantum dot devices [170], phonon mediated energy transfer via the substrate is unlikely in carbon based devices due to the mismatch of the phonon spectra of graphene and SiO2. It is more likely that energy is transferred via photons emitted by the current passing through the etched GNR based detectors [171]. Photon absorption in the leads of the QD devices could in turn lead to an effective broadening of its Fermi distribution. Moreover, it is possible that the occupation of the stochastic QDs formed in the GNR [107, 108, 171] fluctuates. This would alter the potential landscape of the capacitively coupled QD and allow charge pumping effects. Averaged over time this would have the same effect as an increase in temperature.

7.10. Temperature dependence

The inﬂuence of the temperature on the charge detector operation has been studied on a bilayer graphene quantum dot device (see Fig. 5.6)[162]. The back gate has

83 7. Charge sensing

a b 10 3.7 K 0.5 10 1 2.1 K 4 5 0 5 0.5 E 0 -0.5 0 0 3 10 0.5 10 1 3.1 K 1.7 K CD SD SD 5 0 5 0.5 CD 2 -0.5 0

0 Current I (nA) Current I (pA) 0 Current I (pA) 10 0.5 10 1 Current I (nA) 2.6 K 1.0 K Electron temperature TElectron temperature (K) 1 5 0 5 0.5

-0.5 0 0 0 0 0.8 0.9 1 1.1 1.2 1.3 1.4 1 1.1 1.2 1.3 1.4 1.5 0 1 2 3 4

Central Gate VCG (V) Central Gate VCG (V) Base temperature TMC (K)

Figure 7.14.: Temperature dependence. (a) The current through the quantum dot ISD (blue) and charge detector ICD (red) are plotted as function of the central gate voltage VCG for different base temperatures TMC as indicated in each panel. (b) Electron temperature TE extracted by fitting the data in panel (a) according to equation 7.2. The dashed line is a linear fit to the data points. been set such that the Fermi level is positioned in the transport gap. A series of Coulomb resonances has been recorded as a function of VPG and the base temperature TMC measured at the mixing chamber. The current through the QD shows a broadening of the Coulomb peaks developing towards Coulomb oscillations at elevated temperatures (see blue curves Fig. 7.14 (a)). The signal of the charge detector, in particular the current steps, exhibits also a significant temperature broadening (see red curves Fig. 7.14 (a)). Thus the sensitivity decreases with temperature but the characteristic steps are still visible for temperatures up to T = 4 K. Moreover, the transmission of the tunneling barriers is also affected by the increasing temperature. A fit of the Coulomb peaks by the formula

I0 I(VCG,TE) = 2 , (7.2) cosh [α(VCG − V0)/2kBTE] with the peak position V0 and the peak current I0 allows to determine the electron temperature of the system TE [172]. The gate lever arm α = 0.034 has been extracted from ﬁnite bias spectroscopy measurements. The values of the electron temperature TE averaged over ﬁve peaks are plotted as a function of TMC in Fig. 7.14 (b). As expected a linear dependency can be observed with a slope of

84 7.11. Conclusion

0.96 and an oﬀset of 0.18 K.

7.11. Conclusion

Graphene nanoribbon-based charge detectors coupled to graphene quantum dots have successfully been fabricated and characterized. Simultaneous measurements of the current through the QD and the CD prove that the detector allows the resolution of charging events on the QD even in regimes where the direct current is below the noise floor. Finite bias spectroscopy measurements show that excited states of the QD can be resolved with the CD. The CD keeps operating even if in-plane magnetic fields exceeding 7 T are applied or pulsed gates at 20 MHz are used to manipulate individual QD states. This is in particular relevant for experiments addressing spin states or relaxation processes where often the QD currents are too small to me measured directly. The characteristic steps in the CD current broaden with temperature whereas the detector retains its functionality. √ Average charge sensitivities on the order of 1.3 × 10−3e/ Hz can be achieved. The bias applied to the CD has been optimized to values slightly above the Coulomb blockade regime of the CD. In this case all QD resonances can be probed with a good signal-to-noise ratio while keeping the back action onto the QD at a minimum. At higher VCD back action becomes more and more relevant, completely lifting the Coulomb blockade in the QD. Beside increasing the overall transport through the QD, counter flow effects can be observed as a function of the bias applied to the CD. A simple model is employed which can explain the bias dependent behavior of isolated Coulomb peaks. It considers transport only involving a single ground state and an asymmetric broadening of the Fermi distribution in the leads of the QD. Further experiments have to be conducted to obtain a better understanding of the mechanism of energy transfer between the charge detector and the QD.

8 Relaxation times in graphene quantum dots

This chapter has been published in parts in:

Probing relaxation times in graphene quantum dots C. Volk, C. Neumann, S. Kazarski, S. Fringes, S. Engels, F. Haupt, A. M¨uller, and C. Stampfer Nature Communications 4 1753 (2013) c 2013 Nature Publishing Group

8.1. Introduction

A number of eﬀects have been studied in graphene quantum dots including Coulomb blockade [70], excitation spectra [70, 80, 84, 173], spin-ﬁlling sequence [118], and electron-hole crossover [85]. This chapter addresses the relaxation dynamics of excited states in graphene quantum dots via pulsed gate spectroscopy. QDs with highly tunable barriers formed by long and narrow constrictions have been fabricated. This design is motivated by studies on the aspect ratio of graphene nanoribbons [102] (see section 4.1). Earlier experiments have shown that width modulated nanostructures are suitable to realize tunnelling barriers that can be tuned down to the low MHz regime. Additionally, integrated charge detectors provide information on the asymmetry of the barriers [174]. This allows to reach suitable regimes for pulsed-gate transient current spectroscopy [167, 175].

87 8. Relaxation times in graphene quantum dots

a Quantum dot b Central gate Charge DC AC detector VCG VPP 200 nm

SiO2

ICD

Back gate Drain Source VLG VRG

IQD

Figure 8.1.: Sample geometry. (a) Schematic and (b) scanning force micrograph of the measured device. The central island is connected to source and drain electrodes by two 40 nm wide and 80 nm long constrictions, whose transparency can be tuned by the voltage applied to the nearby nanoribbons (VLG, VRG). The nanoribbon on the right hand side is also used as a charge detector (CD) for the dot. The central gate CG is connected to a bias-tee mixing AC and DC signals.

8.2. Device design and measurement setup

The samples are fabricated as described in detail in chapter 5.1. Graphene flakes ++ obtained by mechanical exfoliation are subsequently transferred to Si /SiO2 substrates where single-layer flakes are selected via Raman spectroscopy (see Fig. 8.2 (a)). The flakes are then patterned by electron beam lithography (EBL), followed by

Ar/O2 reactive ion etching. The devices consist of an etched graphene island (diameter d ≈ 110 nm) connected to source and drain leads by long and narrow (width approx. 40 nm) constrictions acting as tunnelling barriers [70] (see illustration in Fig. 8.1 (a)). Two graphene nanoribbons, located symmetrically on each side of the island, can be simultaneously used as gates to tune the transparency of the barriers and as detectors sensitive to individual charging events in the dot [74, 109, 115]. The electrostatic potential of the QD is controlled by a central gate (CG), on which

88 8.2. Device design and measurement setup

a b 100

60 G-Line 2D-Line QD I40 (nA) Intensity (a.u.)

0 1600 2000 2400 2800 0 10 20 30 40 50 60 -1 Raman Shift (cm ) Back gate VBG (V)

Figure 8.2.: Sample characterisation. (a) Raman spectrum of the measured graphene flake, recorded directly after exfoliation. The FWHM of the 2D line measures 34 cm−1 and the peak amplitude is almost twice as high as the G-line, proving that the flake is one monolayer thick [134]. (b) Back gate characteristics of the quantum dot measured at an electron temperature below 100 mK and a bias voltage VSD = 15 mV. For VBG < 31 V and VBG > 41 V, the device is in the hole- and in the electron-transport regime, respectively. In both cases, a significant current can flow through the dot. Vice versa, for 31 V < VBG < 41 V the device is in the transport gap and the current through the dot is strongly suppressed. The arrow indicates the back gate voltage VBG = 34.6 V at which all other measurements have been performed. a bias-tee mixes AC and DC signals (see Fig. 8.1 (b)). This allows performing pulsed-gate experiments with the same gate used for DC control. A back gate is used to control the overall Fermi level allowing to tune the charge-carrier density in a regime where transport is dominated by Coulomb blockade effects [70, 80, 84] (Fig. 8.2 (b)). All measurements are performed in a dilution refrigerator, at a base temperature below 20 mK. Home-built low-noise DC amplifiers (amplification factor of 108, bandwidth < 1 kHz, integration time 200 ms) allow measuring currents with a precision better than 50 fA. For the pulsed-gate experiments the arbitrary waveform generators Tektronix AWG520 and Tektronix AWG7082C are used. The bias-tee Anritsu K251 mixes AC and DC signals on the central gate. Two attenuation stages (10 dB at room temperature and 20 dB at 4 K) and impedance

89 8. Relaxation times in graphene quantum dots

(V) 1.0 LG

0.6 0

0.2 (pA) QD I Left barrier gate V 0.2 0.6 1.0 1.4 1.8 2.2

Right barrier gate VRG (V)

Figure 8.3.: Stability diagram. Current through the dot as a function of the left and right gate-voltages VLG, VRG, recorded at a source-drain voltage VSD = −1.5 mV and VCG = 0 mV. Resonances attributed to the dot (dotted line) or to localised states either in one of the constrictions (dashed lines) can be observed. mismatch reduce the amplitude of the pulse at the sample by a factor 15.8 (24 dB) with respect to the generator output. Further details on the setup are given in section 6.2

8.3. DC low bias transport

Here and in the following, the electron temperature measures Te < 100 mK, and the back-gate voltage is fixed at VBG = 34.6 V, corresponding to a Fermi level deep in the transport gap (see Fig. 8.2 (b)). The QD is first characterised by low-bias transport measurements (Fig. 8.3) as a function of the voltages applied to the two side gates controlling the transparency of the barriers [84]. Different families of resonances can be identified in this measurement, which can either be attributed to the dot (features with a relative lever arm 0.9; dotted line) or to localised states in one of the two constrictions (features with a relative lever arm of 5 and of 0.25, respectively; dashed lines). The device design (cf. section 5.2) has been optimised in order to allow individual tuning of the tunnelling rates of both barriers (ΓL,ΓR) over a wide range. The data indicate the possibility of controlling the current through the dot down to the sub-pA level by VLG and

VRG. By suppressing these rates the sequential tunnelling regime can be accessed. The

90 8.3. DC low bias transport

0.5 1

Detector 0.4 0.8

0.3 0.6 (pA) (nA) CD QD I I 0.2 0.4

Dot 0.1 0.2

0 0 -1 -0.5 0 0.4

Central gate VCG (V)

Figure 8.4.: Charge detection in the Coulomb blockade regime. Simul- taneous measurements of the current ﬂowing through the dot and through the right nanoribbon as a function of VCG. The bias voltage applied to the dot and to the nanoribbon are VSD = −1.5 mV and VCD = 0.2 mV, respectively. Barrier-gate voltages are VLG = 0.4 V and VRG = 0 V.

dot current exhibits approximately equally spaced Coulomb resonances as a function of the gate voltage VCG, with peak currents up to a few hundred fA (Fig. 8.4, blue curve). A current of 200 fA corresponds to an overall dot transmission rate of Γ < 1.25 MHz. The simultaneously measured current through the graphene nanoribbon on the right side of the QD shows characteristic sharp steps coinciding with the Coulomb peaks (Fig. 8.4, red curve). This indicates that the nanoribbon acts as a sensitive charge detector (CD) for the QD [74, 109]. Allowing to detect charging events even in regimes where the current through the QD is below the noise ﬂoor of approximately 50 fA. A detailed study of graphene charge detectors is presented in chapter 7.

91 8. Relaxation times in graphene quantum dots

8.4. DC excited state spectroscopy

Finite-bias spectroscopy is a common tool to probe the addition energy and the excited state spectrum of QDs [53]. Figures 8.5 (a,b) show the current and the differential conductance through the QD as a function of the bias voltage VSD and the voltage applied to the central gate VCD in a regime of weak tunnel coupling to the leads. A clear Coulomb diamond pattern can be observed. The measurements give an estimate of the addition energy Eadd ≈ 10.5 meV. In a simple disc-capacitor model, this corresponds to a QD diameter of 120 nm which is in good agreement with the geometric size of the device. Clear signatures of transport through excited states can be already observed in the current (see e.g. arrows in Fig. 8.5 (a)). Fig. 8.6 (a) shows a cut along constant VCG (see vertical dashed line in Fig. 8.5 (a). The two well-defined plateaus correspond to the ground state and the first excited state entering the bias-window. Transport through excited states becomes even more evident in the differential conductance (Fig. 8.5 (b)). A level spacing of about ∆ = 1.5 to 2.5 meV can be extracted which is in agreement with the electronic single-particle level spacing √ given by ∆ = (~vF )/(d N) ≈ 1.3 − 1.9 meV (see chapter 3.4). The number of carriers N on the dot, is assumed to be on the order of 10 to 20 [80]. Important information on the asymmetry of the tunneling barriers can be obtained from the differential transconductance dICD/dVCG of the CD [174], shown in Fig. 8.5 (c). Regimes where the coupling to the leads is comparable in magnitude are characterised by the appearance of regions of high dICD/dVCG in correspon- dence of both edges of a Coulomb diamond (cf. diamond centred around VCG ≈ 4.2 V). In contrast, regimes with one dominant tunneling barrier are indicated by a strongly one-sided dICD/dVCG (see diamond centred around VCG ≈ 4.45 V). This knowledge helps to identify regimes where the relaxation dynamics of the excited states can be investigated by pulsed-gate spectroscopy [167, 175]. This technique requires the pulse rise time (τrise) to be the fastest time scale in the system. Thus both tunnel rates (ΓL,ΓR) need to be significantly slower than −1 τrise. To fulfill this condition the system is tuned into a low current regime with rather symmetric tunneling barriers. Finite bias spectroscopy measurements on an individual Coulomb resonance in

92 8.4. DC excited state spectroscopy

a QD

0 (mV) SD V IQD (pA) -5 -5 0 5

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 b QD

5 0 (mV) SD V dIQD/dV SD (nS) -5 0 2.5

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 c CD G GG

5 L LR 0 (mV) SD V dICD/dV CG (nS) -5 0 5

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5

Central gate VCG (V)

Figure 8.5.: Finite bias spectroscopy. (a) Current through the quantum dot as function of VCG and VSD. The dashed lines are guides to the eye indicating the edges of the Coulomb diamonds. (b) Differential conductance of the QD, dIQD/dVSD. Resonances parallel to both edges of the Coulomb diamonds indicate transport through excited states. (c) Derivative of the current through the charge detector ICD with respect to VCG. Regions of high dICD/dVCG correspond to the onset of the transitions with the largest rate, thus providing information on the asymmetry of the tunnelling barriers. Note that this effect can be bias dependent (e.g. the diamond centred on VCG ≈ 4.45 V is more strongly coupled to the right lead for VSD > 0 and to the left one for VSD < 0) and it can be drastically influenced by the onset of transitions through excited states, as indicated by the appearance of kinks as those marked by the arrow.

93 8. Relaxation times in graphene quantum dots

dIQD/dVSD (nS) dICD/dV SD (nS) a b 0 2.5 c -5 0 5 4 8

2 4

1 0 (mV) (pA) QD SD I V 0 -4

-1 -8 -2-4 0 2 4 8 2 2.1 2 2.1

VSD (mV) VCG (V) VCG (V)

Figure 8.6.: Transport through excited states. (a) Line-cut at constant VCG = 4.86 V marked by the vertical line in Fig. 8.5 (a). The stepwise increase of the current is a signature of the discrete QD spectrum. (b) Simultaneous measurements of the diﬀerential conductance of the dot dIQD/dVSD and (c) of the transconductance of the charge detector dICD/dVSD in a regime of interest for pulsed-gate experiments.

94 8.5. Pulsed-gate spectroscopy

a b A B 1.6 B A

VPP

t T rise 1.2 (pA)

QD 0.8 c I 0.3

0.2 B 0.4 (pA)

QD I 0.1

A 0 0 0 0.5 1 2.02 2.06 2.1 VPP (V) VCG (V)

Figure 8.7.: Pulse scheme and low frequency pulsed excitation. (a) Sketch of a rectangular pulse scheme employed in the following measurements (duty cycle 50%). Low and high pulse level are labelled A and B, respectively. (b) Current through the dot whilst applying a 100 kHz pulse. Different lines correspond to VPP being varied from 0 to 1.4 V in steps of 50 mV (VSD = −1.5 mV, lines offset by 0.05 pA for clarity). (c) Height of the two resonances depending on the pulse amplitude. such a regime are shown in Fig. 8.6 (b). The charge detector acts as a highly sensitive electrometer such that even QD excited states can be clearly resolved in the differential transconductance dICD/dVSD (Fig. 8.6 (c)).

8.5. Pulsed-gate spectroscopy

To investigate the relaxation dynamics of excited states in a graphene QD, a pulsed-excitation scheme following the one introduced by Fujisawa et al. is used [167, 175, 176]. The basic idea is to probe transient phenomena in a quantum dot by measuring the average DC-current ﬂowing through the system in the presence of a small (ﬁxed) bias voltage VSD and of a square voltage pulse applied to one of the gates (see Fig. 8.7(a)).

95 8. Relaxation times in graphene quantum dots

As mentioned before, the crucial requirement is a rise time of the pulses much shorter than the inverse of the tunnel rates, such that the occupation of the quantum dot cannot follow the change of the potential adiabatically. Thus, by switching the pulse level from high to low (or vice versa), the system is brought into a state of non-equilibrium. A rectangular pulse scheme is applied to the central gate using a bias-tee (cf. Fig. 8.1 (b)). The current through the QD is recorded while the DC gate voltage scans over a Coulomb resonance. The frequency is kept constant while the amplitude is increased. If the pulse frequency is low (e.g. 100 kHz in Figs. 8.7 (b) and 8.8 (a)), the squared-wave modulation of the gate voltage results simply in the splitting of the Coulomb resonance into two peaks [168] which shift linearly with

VPP . Their height is approximately half the one of the original peak (see 8.7 (c)). These peaks originate from the QD ground state (GS) entering the bias-window at two different values of VCG, one for the lower pulse level (A), and one for the upper one (B). At larger frequencies (800 kHz in Fig. 8.8 (b)) an additional broadening of the peaks can be observed together with the splitting. The situation changes significantly at higher frequencies (from a few to tens of MHz), where a number of additional peaks appear together with the splitting of the ground state (compare Figs. 8.8 (a-c)). These resonances are caused by transient transport through quantum dot excited states (ES). The level spacing extracted from this measurement coincides with the one given by DC finite-bias measurements (compare Fig. 8.8 upper and lower panel). Each of the resonances corresponds to a situation in which the QD levels are pushed well outside the bias-window in the first half of the pulse, and then brought into a position where transport can occur only through the excited states in the second one, see Fig. 8.9. No current can flow in any of these two configurations in the stationary state. However, in the second half of the pulse electrons can tunnel from one lead to the other via the excited state, as long as the ground state remains unoccupied which contributes to a current through the QD. Fig. 8.10 illustrates the pulse excitation scheme in detail. Once the ground state gets filled, either because of direct tunnelling from the leads or by relaxation from the excited state, transport is blocked. The blockade is released in the next pulse cycle at the initial value of

96 8.5. Pulsed-gate spectroscopy

a b c 2 0.3 2 0.15 2 100kHz 800kHz 18MHz 0.6

0 0 0 (V) 1 (V) 1 (V) 1 PP PP PP PP V V V (pA) (pA) (pA) QD QD QD I I I 0 0 0 1 2.02 2.06 2.1 2.02 2.06 2.1 V (V) V (V) CG CG -2 0

(mV) -4 SD (nS)

V -6 QD SD

2.02 2.06 2.1 dI /dV

VCG (V)

Figure 8.8.: Pulsed excitation in the low and high frequency regime. (a) Colour-scale version of the data shown in Fig. 8.7. In (b) the pulse frequency is 800 kHz whereas in (c), upper panel the frequency measures 18 MHz. A number of additional resonances can be seen, corresponding to transient currents through excited states. The lower panel represents a DC ﬁnite-bias measurement at the same Coulomb resonance (same data as in Fig. 8.6). The dashed lines are guides to the eyes.

97 8. Relaxation times in graphene quantum dots

a ES GS x

b GS 8MHz

1.6

(pA) 0.8

QD I

0 0.32 0.36 0.4 0.44

VCG (V)

Figure 8.9.: Pulsed excitation in the high frequency regime. (a) Schematic of transport via ground state (GS), excited state (ES) and, on the left, of a possible initialisation stage. (b) Measurement similar to the ones shown in Fig. 8.7 (b), but for a diﬀerent Coulomb resonance. Here the pulse frequency is 8 MHz, VSD = −1 mV, and VPP is varied from 0 to 2 V in steps of 25 mV (traces oﬀset by 0.02 pA).

98 8.5. Pulsed-gate spectroscopy

a A B c d

VPP

trise T

A B

Figure 8.10.: Schematics of the pulsed excitation of the excited state (a) Schematic waveform of the pulse train. (b) Diagrams showing the position of the dot energy levels during the two pulse phases. (c) Schematic of the processes that can occur after the switch from low to high pulse-level: an electron can tunnel from the source either in to the excited state (ES) or in the ground state (GS) with a probability ratio ΓL,e/(ΓL,g + ΓR,g). (d) If the electron is injected into the ES, it can relax to the GS or tunnel out into the drain lead to give a net current. The direction of the current is set by the applied voltage bias, which should be large enough to saturate the current, but still suﬃciently small to allow energy resolution. the pulse-level. The schematic illustration shown in Fig. 8.10 considers the case where the excited state involved into transport is the one of the dot with N + 1 electrons, |N + 1, ei. In the following, the excited state of the dot with N electrons |N, ei plays a role. The two situations are conceptually equivalent and can be easily mapped one into each other. Under these conditions, a square-wave modulation of the gate voltage results in short current pulses, which can be resolved in DC-current measurements only if the frequency of the pulse is larger than the characteristic rate γ of the blocking processes. The lowest frequency at which signatures of transport through excited states emerge provides therefore an upper bound for γ. Figure 8.11 complements the pulsed-gate spectroscopy data in Figs. 8.8 and 8.9. The results are qualitatively independent of the frequency, indicating that it is faster than the characteristic blocking rate γ. Comparing the pulsed-gate to ﬁnite-bias spectroscopy data show a good agreement of the detected excited states.

99 8. Relaxation times in graphene quantum dots

a 8 MHz b 14 MHz c 22 MHz d 26 MHz 2 0.45 2 2 2 0.6 0.6 0.5

(V) 0 1 0 1 0 1 0 1 PP PP PP PP PP V (V) V (V) V (V) V (pA) (pA) (pA) (pA)

QD QD QD QD I I I I 0 0 0 0 40 40 40 40 0 0 0 0 -40 -40 -40 -40 SD SD SD SD V (mV) V (mV) V (mV) V (mV) (pS) (pS) (pS) (pS)

-6 -6 -6 -6 -6 -6 -6 -6 QD CG QD CG QD CG 0.34 0.44 0.34 0.44 0.34 0.44 0.34 0.44 QD CG dI /dV dI /dV dI /dV dI /dV VCG (V) VCG (V) VCG (V) VCG (V)

e 8 MHz f g 8 MHz 30 MHz 25 MHz h 25 MHz 2 2 0.7 2 0.3 2 0.6 2.5

0 0 0 0 1 PP PP PP PP V (V) V (V) V (V) V (V) (pA) (pA) (pA)

QD QD QD I I I QD I (pA) 0 1 0 1 0 0 1.2 100 6 0 0 0 -100 0

SD 0 SD SD V (mV) (nS) V (mV) (pS) (pS) SD V (mV)

V (mV) -6 -6 -6 -6 QD SD QD CG QD SD 2 2.1 2 2.1 QD SD dI /dV dI /dV dI /dV 4.56 4.62 4 4.15 dI /dV (nS) VCG (V) VCG (V) VCL (V) VCL(V)

Figure 8.11.: Pulsed-gate spectroscopy of excited states at different frequencies. The evolution of the Coulomb resonances is plotted as a function of the pulse amplitude at different pulse frequencies. (a-d) Data obtained at the same resonance as in Fig. 8.9 measured at frequencies of 8, 14, 22 and 30 MHz, respectively. (e,f) Same resonance as in Fig. 8.8, probed at frequencies of 8 and 30 MHz. (g,h) Pulsed- gate spectroscopy data of two additional resonances at a frequency of 25 MHz. In panel (h) two neighbouring states are shown. Upper and lower half of each panel correspond to pulsed-gate and to finite-bias spectroscopy, respectively.

100 8.6. Probing relaxation times

a b 12 c 1 (pA) QD

8 I

60 (V)

PP PP 0.8 V

4 QD SD dI /dV (nS) (nA) 0 QD 40 0 0 SD

V (mV) 20 -4 Current I QD SD

20 dI /V(nS) 10 SD

V (mV) 20 -8

0 -10 0 -12 0 5 10 15 20 25 -1.08 -1.04 -1 -1.02 -1 -0.98

Back Gate VBG (V) Right Gate VRG (V) VRG (V)

Figure 8.12.: Investigation of a second sample with the same geometry. (a) Back-gate characteristics recorded at a bias voltage of VSD = 5 mV. Ambipolar transport can be observed and between VBG = 10 V and VBG = 27 V, the device is in the transport gap. The arrow indicates the back gate voltage VBG = 21 V at which the other measurements have been performed. Inset: Scanning force micrograph of the device (scale bar: 200 nm). The sample has ex- actly the same geometry as the one presented in the main text. (b) Diﬀerential conductance dIQD/dVSD through the quantum dot as function of VCG and VSD. A number resonances originating from excited-states transitions can be clearly resolved. (c) Comparison of pulsed gate spectroscopy data (f = 20 MHz, upper panel) and DC ﬁ- nite bias measurements (lower panel) of a certain Coulomb diamond. The dashed lines are guides to the eyes.

Pulsed gate spectroscopy measurements have been performed on other devices with the same lithographic design (see inset in Fig. 8.12 (a)), as well. Fig. 8.12 (a) shows the back gate characteristics and Fig. 8.12 (b) a typical ﬁnite-bias spectroscopy measurement of the second sample. Both devices show a similar behaviour. Figure 8.12 (c) compares pulsed gate spectroscopy data and DC ﬁnite bias measurements. Excited states energies extracted from the two measurements are in good agreement.

8.6. Probing relaxation times

A more accurate estimate of the relaxation time of the excited states can be obtained by a diﬀerent pulse scheme (see Fig. 8.13 (a)) following Fujisawa et al.,

101 8. Relaxation times in graphene quantum dots

a b 0.60 IQD (pA) 100

TA TB (ns)

c A |N, e> T Current t |N, g> |N+1, g> 2 x 2.02 2.1 VCG (V)

Figure 8.13.: Transient current spectroscopy. (a) Pulse scheme used to study transient currents through excited states: time TB is ﬁxed while time TA is varied. (b) Current through the QD as a function of VCG and TA for VSD = −1 mV, VPP = 0.9 V, TB = 30 ns. Dashed lines are a guide to the eye indicating the peaks due to transport via ground (blue) and excited (red) states. (c) Schematic of the transitions occurring during TA for the central peak in (b). |N, gi and |N, ei indicate the ground and excited state of the dot with N excess electrons.

where TA is varied while keeping TB and VPP fixed [167, 175, 177]. Such a measurement is shown in Fig. 8.13 (b) for the same Coulomb resonance investigated in Figs. 8.7 and 8.8. All lines are bent due to a parasitic effect of the bias-tee adding a duty-cycle dependent DC-offset to VCG (VPP (TA − TB)/2(TA + TB)). The two outermost peaks correspond to transport via the dot ground state when this is in resonance with the bias window during TA (right peak) and TB (left peak). The inner peak results from transport via an excited state, as indicated in the scheme sketched in Fig. 8.13 (c). The transition |N, gi → |N + 1, gi is not energetically allowed in the range of values used for VSD and VCG during TA making |N, gi an ’absorbing’ state. At the beginning of each cycle, the dot is initialised in the state with N + 1 electrons during TB (not shown). If the system is in the excited state |N, ei the competing processes |N, ei → |N, gi and |N, ei → |N +1, gi are possible. For each of the peaks in Fig. 8.13 (b) the average number of electrons tunnelling through the device per cycle can be estimated by hni = I(TA + TB)/e. Fig. 8.14

102 8.6. Probing relaxation times

(a) shows the number of electrons transmitted via the ground state (blue) and via the excited state (red) as a function of the pulse length TA. While the first one increases linearly with the duration of the lower pulse level TA, the number of electrons transmitted via the excited state tends to saturate, indicating a transient effect. Figs. 8.14 (b-d) shows data obtained by transient spectroscopy measurements on the same state but for different combinations of voltages at the barrier-gates

VLG, VRG. The characteristic time constant 1/γ of this saturation process as well as the asymptotic number of transmitted electrons nsat varies when changing these voltages, as these aﬀects the tunnelling rates. Solving the rate equations describing the tunneling and relaxation processes the average number of electrons transported through the QD per pulse cycle can be written as:

−γ+TA −γ−TA n(TA) = nsat − n+e − n−e (8.1) with the saturation value

2Γ2(Γ1 + Γτ ) Γ2 nsat = − , (8.2) Γ1(Γ1 + Γ2 + Γτ ) + 2Γ2Γτ Γ1 + Γ2 and the rates

1 r 1 γ = Γ + Γ + Γ ± δ with δ = Γ Γ + (Γ − Γ )2 (8.3) ± 1 2 2 τ 1 2 2 2 τ and 1 Γτ (2Γ2 − Γ1 − Γτ ) n± = nsat ± . (8.4) 2 4δ(Γ1 + Γτ )

Γτ is the intrinsic relaxation rate of the excited state and Γ1,Γ2 the tunneling rates to the leads.

−γ+TA Typically the condition δT 1 is fulﬁlled such that γ+ γ− and thus e ≈

0 for relevant TA. Equation 8.1 simpliﬁes to a single exponential function

n(TA) = nsat[1 − exp(−γTA)], (8.5) where nsat is the saturation value for long TA [167] and γ the characteristic rate of

103 8. Relaxation times in graphene quantum dots

the blocking processes. In the limit of Γ2 Γ1, Γτ the observable rate γ is related 1 to the relaxation rate Γτ and the tunneling rate Γ1 via γ = Γτ + 2 Γ1. In regimes Γ2 > Γ1, Γτ equation 8.3 at least yields the relation γ > Γτ and thus γ gives an upper limit for Γτ . Fitting the data-set shown in Fig. 8.13 (b) with equation 8.5, the characteristic rate of the blocking processes γ = 12.8 MHz can be extracted. This gives a −1 lower bound τ = Γτ > 78 ns for the life-time of the quantum dot excited state. Studying further electronic excited states with energies in the range of 1.7-2.5 meV (see Figs. 8.11 and 8.14), a lower bound for the relaxation time in the range 60-100 ns can be estimated.

8.7. Simulation

Transient currents through a quantum dot can be calculated by solving the rate equations. The time evolution of the system can be described by the following master equation [78, 178–180]

X dtpn = Wnmpm or short ˙p = Q · p, (8.6) m with the occupation probability p of the considered quantum states and the transition matrix Q:   W11 ...W1M  . . .  Q =  . .. .  (8.7)   WN1 ··· WNM

The elements Wnm of Q describe the transition rate between two quantum state m and n. They include contributions from relaxation and tunneling processes. The transition rates of tunneling in and out event are given by ΓL,Rf L,R and ΓL,R(1 − f L,R), respectively. ΓL,R are tunneling rates of the left and the right barrier and f L,R the corresponding Fermi distributions.

−1 f L,R = 1 + e(En−µL,R)/kB T (8.8)

104 8.7. Simulation

a b 0.2 0.4 1/γ = 78ns GS 1/γ = 81ns

0.3 0.15 ES

0.2 0.1

0.1 0.05 Electrons per pulse cycle Electrons per pulse cycle

0 0 0 50 100 0 50 100 c d 0.4 1/γ = 92ns 1/γ = 100ns 0.3

0.3

0.2 0.2

0.1 0.1 Electrons per pulse cycle Electrons per pulse cycle

0 0 0 50 100 0 50 100

TA (ns) TA (ns)

Figure 8.14.: Transient current through excited states. Average number of electrons transmitted per cycle via GS (blue circles) and ES (red circles) transitions as a function of the pulse length TA recorded at four different configurations of the tunneling barriers. Solid lines represent linear and exponential fits to the experimental data. (a) Data extracted from the peaks shown in Fig. 8.13 (b). Voltage applied to the barrier-gates: VRG = 0 mV and VLG = 420 mV. Parame- ters obtained by the fit to the transport data via the excited state: 1/γ = 78 ns and nsat = 0.313. (b) VRG = 50 mV, VLG = 420 mV, nsat = 0.176, 1/γ = 81 ns. (c) VRG = 0 mV, VLG = 410 mV, nsat = 0.376, 1/γ = 92 ns. (d) VRG = 0 mV, VLG = 400 mV, nsat = 0.314, 1/γ = 100 ns.

105 8. Relaxation times in graphene quantum dots

The Fermi distribution depends on the diﬀerence of the energy En of the dot state n and the electrochemical potentials µL,R of the leads and it is broadened by the temperature T .

Using the ansatz p = eκt · v, the master equation ˙p = Q · p simpliﬁes to the eigenvalue problem Q · v = κ · v with eigenvalues κ and eigenvectors v [178, 179]. This yields the time dependent occupation probability p(t). The current evaluated at the drain lead can be written as

X I(t) = −e Wn,mpn(t). (8.9) n=m

Considering a pulsed gate experiment the transition matrix Q is no longer constant as the state energies En are altered periodically with respect to the chemical potentials of the leads µL,R. Thus the current has to be evaluated separately for both levels A and B of the pulse cycle. The current averaged over one pulse cycle is then given by ! Z TA Z TA+TB hIi = IA(t)dt + IB(t)dt / (TA + TB) . (8.10) 0 TA

To simulate the pulsed gate experiments a system with the 0-electron ground state |0gi, the 1-electron ground state |1gi and the 1-electron excited state |1ei is considered. p can be deﬁned as

p(|1ei)   p = p(|1gi) . (8.11) p(|0gi)

In this notation the tunneling matrix reads as

L L R R L L R R ! −Γ 1 − fe − Γ 1 − fe − Γτ 0 Γ fe + Γ f − e L L R R L L R R Q = Γτ −Γ 1 − fg − Γ 1 − fg Γ fe + Γ fg . L L R R L L R R L L R R L L R R Γ 1 − fe + Γ 1 − fe Γ 1 − fg + Γ 1 − fg −Γ fe − Γ f − e − Γ fe − Γ fg (8.12) The excited state to ground state transition |1ei → |1gi is the only relaxation process while the others are tunneling in and out transitions to the leads. The

106 8.7. Simulation

-1 -1 -1 -1 -1 -1 a ΓRL/Γ =20ns/20ns=1 b ΓRL/Γ =200ns/20ns=10 c ΓRL/Γ =20ns/200ns=0.1 1 0.8 0.6

I1 0.4

P (P g>) 0.2 0 0.3 Th =10ns |1, e> T =70ns Current 0.2 h t I1

P (P e>) 0.1 |1, g> x |0, g> 0 1 0.8 0.6 I0g 0.4 P (P 0.2 >) 0 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 T(ns) T(ns) T(ns)

Figure 8.15.: Time evolution of the occupation probability of the relevant quantum dot states |1ei), |1gi) and |0gi). The ratios of −1 −1 the tunneling rates measure (a) ΓL /ΓR = 20ns/20ns = 1, (b) −1 −1 −1 −1 ΓL /ΓR = 200ns/20ns = 10 and (c) ΓL /ΓR = 20ns/200ns = 0.1. The initialization times measure TB = 10 ns (blue) and TB = 70 ns −1 (red). The relaxation time has been set to Γτ = 50 ns. The inset illustrates the allowed relaxation and tunneling transitions. transition |1gi → |1ei is energetically forbidden. A solution for the time-dependent occupation probability p(t) can be obtained numerically. Important parameters are the two tunneling rates ΓL and ΓR, the relaxation rate Γτ , the electro-chemical potentials µL and µR, the excited state level spacing ∆ and the pulse lengths TA and TB. For simplicity reasons the shift of the dot potential (µgate) with respect to the leads as a response to the (pulsed) gate is included in the chemical potentials as µL,R = µgate ∓ eVSD/2. Figure 8.15 shows the time evolution of the occupation probability p(t) for three diﬀerent ratios of ΓL and ΓR and two initialization times. Experimentally reasonable values have been selected for the other parameters: VSD = 1 mV, ∆ = −1 1.7 meV, Γτ = 50 ns, TA = 200 ns. During the initialization pulse the quantum dot is depopulated, thus p(|1ei) and p(|1gi) decrease while p(|0gi) increases. Short pulse lengths result in incomplete initialization which can be easily seen by the blue curve (TB = 10 ns). The initialization time of 30 ns typically used in the experiment leads to an eﬃciency of approximately 95%. At the onset of the read out pulse the quantum dots gets populated, both p(|1gi)

107 8. Relaxation times in graphene quantum dots

a A B b c CG CG A B V (t) V (t) I (t) I (t)

Time t Time t

d A B e f CG CG V (t) V (t) I (t) I (t)

Time t Time t

Figure 8.16.: Simulation of the time-resolved current through the QD. (a- c) Transport through the excited state. (a) Schematics illustrating the possible transition during the two pulse phases. (b) Applied pulse scheme (red). Current through the QD via ES transitions (blue). Please note that the current is negative during the initialization step. (c) Same as in (b) but for a long TB. The current vanishes if the system remains in step B for a sufficiently long time. (d-f) Transport through the ground state. (d) Schematics similar to the one in (a). No relaxation processes are involved. (e,f) Current through the QD via GS transitions for a short and a long read out time TB. In contrast to (c) the current saturates for long TB. and p(|1ei) increase. In order to obtain a significant occupation of the exited state |1ei during the read out pulse the tunneling rates should be either symmetric (see panel a), or the source barrier should be the faster one (panel b). At an inverse ratio the occupation probability of the excited state is reduced drastically −1 −1 (panel c). In the experiment a ΓL /ΓR -ratio between 1 and 10 is assumed. After switching to the read-out pulse, the system starts to converge asymptotically to a complete population of the state |1gi. The time-dependence of the occupation probabilities allows to simulate the time- resolved current. Figure 8.16 shows the results for transport through the excited state (panel a-c) and the ground state (d-f) for two different pulse lengths TA. Considering ES transport, the current vanishes if the system remains in configuration A for a sufficiently long time which is in contrast to GS transport where the current saturates at a finite value. Taking these results, the current and thus the time-resolved number of electron tunneling through the QD per pulse cycle as a function of the read out time TA

108 8.8. Electron-phonon coupling

0.4

0.3

0.2

0.1

Electrons per pulse cycle 0 0 50 100 150 200

TA (ns)

Figure 8.17.: Simulation of the number of electrons tunneling per cycle. The average number hni of electrons tunneling through the QD per pulse cycle simulated for a relaxation time τ = 100 ns and three −1 −1 diﬀerent ratios of the tunneling rates: ΓL /ΓR = 20ns/20ns = 1, −1 −1 −1 −1 ΓL /ΓR = 100ns/20ns = 5 and ΓL /ΓR = 200ns/20ns = 10.

can be estimated. Figure 8.17 shows an example for three diﬀerent ratios of the tunneling rates. The results are qualitatively in reasonable agreement with the experimental data (cf. Fig.8.14).

8.8. Electron-phonon coupling

A reasonable mechanism leading to charge relaxation in a graphene QD is electron- phonon interaction. An estimate of the charge relaxation time τ can be obtained 1 2π P 2 according to Fermi’s golden rule = |hΨf |Hel−ph|Ψii| δ(Ef −Ei +Eph(q)), τ ~ q by making some simplified assumptions on the electron-phonon coupling Hamil- tonian Hel−ph and the shape of the electronic wave functions Ψi and Ψf . The graphene QD is modelled as a circular disc of radius R = 55 nm confined by an m m0 infinite mass term [36]. The dot states Ψi = Ψn (Ψf = Ψn0 ) are then eigenfunc- tions of the following Hamiltonian: HQD = vF ~σ · ~p+κV (r)σz, where the first term on the right hand side is the Dirac-like graphene Hamiltonian and the second one includes a mass-related potential energy term V (r)(κ is here the valley index), which is zero inside the quantum dot but tends to infinity towards the edges, i.e. V (r) = 0 for r < R and V (r) → inf for r ≥ R [36, 181]. The wave functions of

109 8. Relaxation times in graphene quantum dots such a Hamiltonian are given by: ! J (k r)eimφ Ψ(r, φ)m = Ψm = Cm m n , (8.13) n n n im+1φ iJm+1(knr)e

m where Jm(x) are Bessel functions of the first kind, Cn normalization constants and kn the solutions of the equation Jm(kR) = κJm+1(kR)[181]. The main electron-phonon coupling mechanisms in graphene are the deformation potential, which is due to an area change of the unit cell, and the bond-length change mechanism, which corresponds to a modified hopping probability [182–184]. At low energies (meV regime) only acoustic phonons have to be considered resulting in six possible relaxation channels: longitudinal-acoustic (LA), transversal- acoustic (TA), and transversal out-of plane (ZA) phonons, each of which can be coupled to electrons either via deformation potential (D = g1 ≈ 18−29 eV) or the bond-length change (g2 ≈ 1.5 eV) mechanism [183, 184]. The latter mechanism only gives a marginal contribution compared to the first one due to the smaller coupling constant. Moreover, since only supported graphene is considered, flexural (ZA) phonons are quenched and can therefore be neglected. The coupling to TA phonons via deformation potential can be also disregarded, as it turns out to be a two-phonon process [182]. Thus the coupling to the LA phonon mode via deformation potential can be regarded as the dominant cause of relaxation. Furthermore assuming that at the low temperature of the experiment the phonons are in their ground-state, the electron-phonon coupling Hamiltonian takes the form [182]

s ! q~ 1 0 + i~q~r −i~q~r Hel−ph(~r) = iD (bq e − bqe ) (8.14) 2Aρvs 0 1

where D is the deformation potential, A the area of the graphene quantum dot, −7 1 4 ρ = 7.5 × 10 kg/m the mass density and vs = 2 × 10 m/s the sound velocity + of the LA mode. The creation operator bq describes the emission of one phonon. With these ingredients, the inverse relaxation time can be written as

2 2 1 D q0 m0 m 2 m0−m m0−m 2 = 2 |Cn0 Cn | [Mmm0,nn0(q0) + Mm+1m0+1,nn0(q0)] (8.15) τ 2ρ~vs

110 8.9. Conclusion

500

400

300

200

Relaxation Time t (ns) Time Relaxation 100

0 0 2 4 6 8 10 Level Spacing (meV)

Figure 8.18.: Calculated relaxation times due to electron-phonon coupling. Relaxation times due to coupling to longitudinal acoustic (LA) phonons via the deformation potential (here D = 20 eV is assumed). An ideal circular graphene quantum dot with a diameter of d = 110 nm is considered, (n,m) are limited to nmax = 30 and mmax = 15. τ is plotted as a function of the level spacing, i.e. as a function of the energy of the phonon causing the relaxation Eph = ~ωph = ∆. The regime of level spacings relevant for the measurements is highlighted in red.

l R R where (mm0, nn0) (q0) = 0 drrJl(q0r)Jm0(kn0r)Jm(knr) and q0 = ∆/~vs, is the wave vector of a phonon whose energy matches the quantum dot level spacing ∆. A numerical solution to this equation is shown in Fig. 8.18 where the relaxation times τ as function of the level spacing ∆ is plotted. For a level spacing in the range of 1.5-2.5 meV, coupling to LA phonons results in relaxation times on the order of 40-400 ns which is in reasonable agreement with the experimental ﬁndings. These values should be regarded as an estimate as e.g. the exact shape of the QD and the inﬂuence of the leads are neglected.

8.9. Conclusion

Graphene quantum dots with highly tunable tunneling barriers have been fabricated and characterized. A device design with long and narrow constriction has been chosen to allow tuning the tunneling rates into the low MHz regime. A capacitively coupled nanoribbon acting as a sensitive charge detector is a tool providing information on the asymmetry of the coupling between the QD and the leads. The

111 8. Relaxation times in graphene quantum dots excited state spectra are studied by DC ﬁnite bias spectroscopy. The relaxation dynamics of excited states is addressed by transient current spectroscopy. Rectangular RF pulse sequences are applied to a side gate while the average DC current through the QD is recorded. By this technique a lower bound for the relaxation time of 60 to 100 ns can be obtained which is roughly a factor of 5 to 10 larger than what has been reported in III/V quantum dots [167, 175–177]. The lifetime of electronic excitations is most likely limited by electron-phonon interaction. The comparably long relaxation times can be a hint for reduced electron- phonon interaction in graphene due to the absence of piezoelectric phonons. A model assuming the coupling to longitudinal acoustic (LA) phonons via the deformation potential to be the most dominant relaxation mechanism yields relaxation times in the range of 40 to 400 ns for an circular graphene QD. This is in agreement with the experimental results giving a lower bound of 60 to 100 ns. Future experiments shall address spin relaxation processes. For this purpose three level pulse schemes can be used [175]. The sample design can be further optimized such that the RF gate couples strongly to the QD while only showing weak inﬂuence on the tunneling barriers.

112 9 Bilayer graphene quantum dots

This chapter has been published in parts in:

Electronic excited states in bilayer graphene double quantum dots C. Volk, S. Fringes, B. Terr´es,J. Dauber, S. Engels, S. Trellenkamp, and C. Stampfer Nano Lett. 11 3581-3586 (2011) c 2011 American Chemical Society

Tunable capacitive inter-dot coupling in a bilayer graphene double quantum dot S. Fringes, C. Volk, B. Terr´es,J. Dauber, S. Engels, S. Trellenkamp, and C. Stampfer Physica Status Solidi C 9 169–174 (2012) c 2011 WILEY-VCH Verlag GmbH & Co. KGaA

9.1. Introduction

Most of the studies on graphene quantum dots reported so far were based on single- layer graphene and showed a number of device limitations related to the presence of disorder, vibrational excitations and to the fact that the missing band gap makes it diﬃcult to realize soft conﬁnement potentials. In particular, it has been shown that intrinsic ripples and corrugations in single-layer graphene can lead to unintended vibrational degrees of freedom [185] and to a coherent electron-vibron coupling

113 9. Bilayer graphene quantum dots in graphene QDs [186]. Bilayer graphene is a promising candidate to overcome some of these limitations. In particular it allows opening a band gap by an out- of-plane electric field [187–189] and thus definition of QDs via soft confinement potentials. Furthermore, it has been shown that ripples and substrate-induced disorder are reduced in bilayer graphene [190], increasing the mechanical stability and suppressing unwanted vibrational modes. In this chapter, a bilayer graphene double quantum dot (DQD) device with a number of lateral gates is studied. The local gates are designed to tune the tunneling barriers as well as the electrochemical potentials of the two dots. The sample is characterized by DC transport measurements. Stability diagram allow the determination of e.g. coupling energies, charging energies and the excited state level spacing. The excited state spectrum is studied and compared to the theory for single-layer and bilayer graphene. The influence of an in-plane magnetic field is investigated.

9.2. Fabrication and Characterization

The investigated graphene double quantum dot device is fabricated following the ++ technique described in chapter5. Graphene flakes deposited on a Si /SiO2 substrate are inspected by Raman spectroscopy which allows to reliably identify bilayer graphene flakes. Detailed studies of the Raman spectra of graphene in different thicknesses can be found here [134, 136]. Figure 9.1 (a) shows the Raman spectrum the present device, Figure 9.1 (b) highlights the 2D peak of the spectrum. The characteristic 2d line of bilayer graphene is the sum of four Lorentzians which is in contrast to single layer graphene. Following Graf et al. [136], the spacing between the two peaks (emphasized by the dashed lines) is an important criterion to distinguish between different graphene thicknesses. Here, this spacing measures 18.8 cm−1 which is a strong indicator for bilayer graphene. The device etch mask is designed such that the two quantum dots measure 50 nm in diameter whereas the constrictions leading to the dots are 120 nm long and 35 nm wide. The interconnection between the dots should measure 50 nm in length and 35 nm in width. Five lateral gates are designed in a distance of 40 nm from the active device to control the electrostatic potential on each QD as well as

114 9.3. Tunneling barriers

a b G 2D-line

2D Intensity (a.u.) Intensity (a. u.)

1400 1800 2200 2600 3000 2600 2650 2700 2750 Raman shift (cm-1 ) Raman shift (cm-1 )

Figure 9.1.: Raman spectrum of a bilayer graphene flake. (a) Raman spectrum of the investigated graphene flake. (b) 2D-line of the spectrum fitted with four Lorentzians. The separation of the inner peaks measures 18.8 cm−1. the tunneling rates through the three barriers individually. Thin nanoribbons have been designed on the opposite side of the QDs which are supposed to act as charge sensors. The device design has been chosen completely symmetric. Electron beam lithography followed by reactive ion etching (Ar/O2) is used to transfer the pattern to the graphene flake. Figure 9.2 (a) shows a scanning force micrograph of the etched DQD device. The gates further on referred as left gate (LG) and right gate (RG) are used to control the electrochemical potential and thus the occupation of electrons of the QDs. The central gate (CG) is used to tune the inter-dot coupling while the outer left gate (LG*) controls the left tunneling barrier. The fifth gate as well as the charge detectors unfortunately did not show any effect and have not been used throughout the measurements. The back gate (BG) is used to tune the overall Fermi level.

9.3. Tunneling barriers

First transport measurements have been performed in a pumped 4He system with a base temperature of 1.3 K (see section 6.1. A symmetric dc bias voltage VSD is applied to the source and drain contacts. Figure 9.3 shows the current through

DQD as a function of the back gate voltage VBG.

115 9. Bilayer graphene quantum dots

a 200 nm

Source Drain

BG LG* LG CG RG

b Electrons

LQD RQD Energy

Tunneling Barriers Holes

Figure 9.2.: Scanning force micrograph and eﬀective band structure. (a) Scanning force microscope image of the investigated double quantum dot device. The diameters of the etched quantum dots (QDs) measures roughly 50 nm while the width of the 100 nm long constrictions leading to the dots measure 30 nm. (b) Schematic illustration of the eﬀective band structure of the device highlighting the three tunneling barriers (hatched areas) induced by the local constrictions (see panel a).

a b 20 20

15 15

10 10

Current I (nA) 5 Current I (nA) 5

0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60

Back Gate Vbg (V) Back Gate Vbg (V)

Figure 9.3.: Back gate characteristics. (a) Current as a function of the back gate voltage recorded at VSD = 20 mV and T = 1.3 K. The arrow marks the BG regime where the following measurements have been taken. (b) Back gate characteristics of the same device recorded within a diﬀerent cool-down cycle.

116 9.3. Tunneling barriers

In a region roughly ranging from VBG = 32 V to VBG = 46 V (see Fig. 9.3 (a)) the current is suppressed, the device is in the so-called transport gap [104]. The back gate tunes the Fermi level such that it crosses the gaps opened up locally within the narrow constrictions (see schematics in Fig. 9.2 (b)). Electronic transport is dominated by stochastic Coulomb blockade originating mainly from lateral confinement in combination with a significant disorder potential, arising from bulk disorder and edge roughness. Details on the transport through graphene nanostructures are given in section 4.1 and Refs. [102, 104, 171]. Reproducible peaks can be observed occurring due to transport through localized states in the constrictions. At gate voltages below 32 V the Fermi level is pushed out of the gaps into the valence band (the device is in the hole transport regime) whereas at voltages exceeding 46 V it is pushed into the conduction band (the device is in the electron transport regime) [71, 84, 85]. In these regimes resonances in the constrictions lead to peaks of elevated conductance. The transport gap appears at significantly positive back gate voltages indicating a strong overall hole doping of the device. This effect which is commonly observed in graphene nanostructures can possibly be attributed to atmospheric

O2 binding [165], residues of polymer resist and influences of substrate disorder [102, 139, 163, 164]. In the following measurements the back gate voltage is fixed in the transport gap as denoted by the arrow in Figure 9.3 (a). The Fermi level crosses the three energy gaps defined by the constrictions (cf. Fig. 9.2). The sample has shown a high stability and reproducibility over a long period of time. After a second cool- down charge rearrangements shifted the charge neutrality point to significantly higher back gate voltages. The transport gap could still be accessed within a reasonable VBG-range (see Fig. 9.3 (b)). At constant bias and back gate voltages the current is measured as a function of the central gate (VCG)) and the two outermost gates (Vlg∗ = Vrg), see Figure 9.4 (a). Three regimes with different levels of conductance can be observed. Schematic energy diagrams for each of the regimes are shown in Fig. 9.4 (b). In regime (2) all side gates are tuned to very negative voltages, such that all three effective tunneling barriers are pushed in the conduction band. Hole transport takes place throughout the structure and currents on the order of 100 pA can be measured. Increasing

117 9. Bilayer graphene quantum dots

a b 200 1 LQD RQD 20 a 1 E 160 F 10

cg 2 120 0

80 Current (pA) EF Central Gate V-10 (V) b 3 40 -20 2 3 0 EF -20 -10 0 10 20

Side Gates Vlg* = Vrg (V) Tunneling Barriers

Figure 9.4.: Characterization of the tunneling barriers. (a) Source-drain current as a function of the central gate voltage (Vcg) and the voltages applied to the outer side gates (Vlg∗ =Vrg) at VSD = 20 mV and T = 1.3 K. Two characteristic slopes separate three areas of diﬀerent levels of conductance. (b) Schematic band structure illustrating the regimes marked in (a): DQD regime (1), hole transport regime (2) and single quantum dot regime (3).

voltages on the left and right gate push the two outer tunneling barriers downwards such that they cross the Fermi level. A quantum dot regime (regime (3)) confined by the outer barriers is formed and thus the current is decreased. An increase of the central gate voltage pushes even the central tunneling barrier downwards such that the device enters a DQD configuration. The current is significantly suppressed, sharp resonances with peak currents on the order of 10 pA appear.

Two dominant slopes occur throughout the data which coincide with the slopes separating the three regimes. They are determined by the relative lever arms of the lateral gates (CG, LG* and RG) on the three tunneling barriers. While the two outer tunneling barriers can be controlled well by the left, right and the central gate (α = ∆Vlg∗,rg/∆Vcg ≈ 0.55), the dominant central barrier can only be weakly tuned by Vlg∗ and Vrg (β = ∆Vcg/∆Vlg∗,rg ≈ 0.13), which is in good agreement with the geometry of the device.

118 9.4. Finite bias spectroscopy, stability diagrams

a -2 -9 1.5 -10 1 -11

lg 0.5 -12 Current (log[A]) 0 -13 Left Gate V (V) -0.5

-1 b -2 -1.5 lg V (V) -2 -4 -3 -2 -1 0 1 2 3 20 25

Right Gate Vrg (V) Vrg (V)

Figure 9.5.: Charge stability diagram. (a) Finite bias (VSD = 10 mV) charge stability diagram recorded in the double quantum dot regime (VCG = 12 V, T = 1.3 K), see black horizontal bar in Fig. 9.4. The dashed lines emphasize the co-tunneling lines. (b) Local resonances in the right constriction as function of the left and right gate voltages measured in regime (3) with additionally lifted left tunneling barrier (Vlg∗ = 0 and Vlg < 0), such that only the right tunneling barrier crosses the Fermi level. Both panels have the same aspect ratio.

9.4. Finite bias spectroscopy, stability diagrams

In the following, the central gate voltage is set such that the device is operated in the DQD regime (cf. region (1) in Fig. 9.4). Figure 9.5 (a) shows a so-called stability diagram where the current is recorded in dependence of the left and the right gate voltage. A regular hexagonal pattern which is characteristic for the charge stability diagram of a DQD is visible over a large range of gate voltage [79]. Electronic transport through the DQD is only possible in the case where the energy levels in both dots are aligned within the source-drain bias window (see triangular regions of elevated conductance). In case only one dot is aligned with the bias window, co-

119 9. Bilayer graphene quantum dots tunneling can occur which can be observed as lines connecting neighbouring triple points. This effect is clearly visible in the regime of elevated conductance (see dashed lines and arrows in Fig 9.5 (a)). The co-tunneling lines separate hexagonal shaped areas in which both QD levels are outside the transport window. There, the number of charge carriers of both QDs is fixed. The relative lever arms between both gates acting on the two dots (LQD and RQD) are given by the slopes R L of these co-tunneling lines, αrg,lg = 0.85 and αlg,rg = 0.12. Interestingly, these results indicate that the RQD is almost equally tuned by both gates which is consistent with the device geometry. The significant variation of the current between different triple points originates from a modulation of the transparency of the tunneling barriers due to local resonances. Already between two neighbouring dots the current can vary by more than one order of magnitude (see Figure 9.5 (a)). Figure 9.5 (b) shows a charge stability diagram recorded in regime (3) but with lifted left barrier (Vlg∗ = 0 and Vlg < 0). Thus the Fermi level only probes localized states in the right tunneling barrier. Highly non monotonous variations of the conductance with the gate voltages appear. Only one slope (∆Vlg/∆Vrg ≈ 0.2, dashed line) is visible corresponding to the slope of the transmission modulation in Figure 9.5 (a). Figure 9.6 (a) shows a high-resolution close-up of Figure 9.5 (a) but with reversed bias (VSD = -10 mV). The expected shape of the triple points, two overlapping triangles, can clearly be identified. The relation between their size which depends on VSD and the distance of two neighbouring triple points gives the lever arms of the side gates to the QDs, see illustration in Fig. 9.6 (b) [79]. L The lever arm between the left gate and the left dot is αlg = VSD/δVlg = 0.056, whereas the lever arm between the right gate and the right dot measures R αrg = 0.019 [79, 86]. This allows to extract the addition energies of each QD L L R R Eadd = αlg · ∆Vlg = 23 meV and Eadd = αrg · ∆Vrg = 13 meV, reflecting an asymmetry in the QD island sizes. As described before, the QDs are delimited by the tunneling barriers defined within the thin constrictions due to the local disorder potential and the confinement. Thus the disorder potential landscape will strongly influence the position and the extent of the barriers [104]. In a first approximating the quantum dot can be modelled as a metallic circular

120 9.4. Finite bias spectroscopy, stability diagrams

a b 0.6 -10 m DVrg 0.4 DVlg -11 0.2 DVrg lg dVlg 0 m

-12 Current (log[A]) dV DVlg Left Gate V (V) V lg -0.2 lg

-0.4 -13 Vrg -3 -2 -1

Right Gate Vrg (V)

Figure 9.6.: Honeycomb pattern. (a) High-resolution close-up of Figure 9.5 (a) but with reversed bias (VSD=-10 mV) highlighting the double dot characteristic honeycomb pattern. (b) Schematic charge stability diagram denoting all quantities necessary to deduce the gate lever arms, addition energies and the mutual capacitive coupling energy.

disc surrounded by SiO2 and vacuum. The self-capacitance C = 40d gives a lower bound for the total capacitance CΣ [35]. Here, an average dielectric constant

= (ox + vacuum)/2 ≈ 2.5 is used [82]. Assuming a dot diameter of d = 50 nm, 2 2 EC = e /C = e /(e0d) ≈ 36 meV is an upper limit for the charging energy. This model overestimates the charging energy by roughly a factor of 2 which is in agreement with earlier studies [82]. So far, all measurements have been carried out at a temperature of around 1.3 K where already a fine structure within the triple points can be observed. To achieve a better energy resolution the sample has been transferred to a dilution refrigerator with a base temperature of below 10 mK. All electron temperature is estimated with 100 mK. The following measurement have been carried out at this temperature. The back gate characteristics of the device differs significantly from the one measured in the first cool-down cycle (compare Figs. 9.3 (a) and (b)). It is likely that trapped charges within the oxide have rearranged such that the potential landscape has changed drastically. Again the Fermi level has been fixed within

121 9. Bilayer graphene quantum dots

3.3

3.2 -10

3.1

3.0

lg -11 2.9

2.8 Left Gate V (V) Current (log[A])

2.7 -12

2.6

2.5 -13 -2.5 -2 -1.5 -1 -0.5

Right Gate Vrg (V)

Figure 9.7.: Charge stability diagram at Te = 100 mK. Similar measurement as in Fig. 9.6 but in a diﬀerent cool down cycle. The well deﬁned triple points allow to extract the inter-dot coupling energy (see arrows). A rich spectrum of excited states can be observed.

the transport gap (VBG = 52 V). Figure 9.7 shows a stability diagram recorded at a base temperature of 10 mK. L R The addition energies have changed to Ea = 18 meV and Ea = 21 meV, respectively, indicating a change of the dot sizes. Interestingly, the extracted addition energies are in reasonable agreement with values from single-layer graphene QDs of a similar size [118]. The system is now in a more symmetric regime.

9.5. Capacitive interdot coupling

Due to the improved energy resolution the edges of the triangles are much more deﬁned allowing the quantitative extraction of the mutual capacitive coupling energy (Em) between the two QDs. Figure 9.8 (a) shows a schematic of a pair of triple points highlighting the voltage diﬀerences, which allow the determination m m of the mutual capacitive coupling energies, Erg and Elg . These energies can be

122 9.5. Capacitive interdot coupling

a b 0.7 c x0.5 12

δV lg 0.6 lg 8

m 0.6 δV lg 4 Current (pA) Left Gate V (V) m δVrg δVrg 0.5 0 8.8 9.0 9.2 8.5 8.7 8.9

Right Gate Vrg (V) Right Gate Vrg (V)

Figure 9.8.: Triple points with diﬀerent mutual capacitive coupling. (a) Illustration of a pair of triple points labelled with the quantities necessary to deduce the coupling energy. (b) and (c) Very same triple points in a stronger (Vcg = 7.42 V) and weaker (Vcg = 7.54 V, VSD = 6 mV) inter-dot coupling regime respectively.

m directly extracted from the gate lever arms and the gate voltage diﬀerences δVrg m and δVrg , respectively:

m m m m Erg = αrg,R · δVrg and Elg = αlg,L · δVlg . (9.1)

For the pair of triple points highlighted by the arrows in Figure 9.7 the capacitive m L R m coupling measures E = αlg · ∆V mlg = αrg · ∆Vrg =2.4 meV. By changing the central gate voltage the coupling energy between the left and right QD can be tuned. Figures 9.8 (b) and (c) compare close-ups of the very same m triple point at two different inter-dot coupling energies E = 4.8 meV (Vcg=7.42 V) m and E = 3.0 meV (Vcg=7.54 V), where the different separation of the overlapping triple points are obvious. m m Figure 9.9 compares the coupling energies Elg and Elg determined by left and right gate voltage values according to equation 9.1. In principle, both energies should be equal, but limited resolution and partial charge rearrangements lead to smeared out edges of the triangles and thus a scattering of both results. No systematic dependency is observed. The insets show the investigated pair of triple points in three different coupling regimes. Figure 9.10 presents a detailed analysis of the coupling energy as a function of

Vcg for three individual nearby triple points. Only a relative numbers of charge

123 9. Bilayer graphene quantum dots

5.0 m E R

4.6 m E L m R,L

E 4.2

I (pA) 3.8 3

2 3.4 1 Coupling Energy (meV)

3.0 0 7.4 7.41 7.42 7.43 7.44 7.45 7.46 7.47

Central Gate Vcg (V)

Figure 9.9.: Capacitive interdot coupling. Capacitive inter-dot coupling energy as a function of the central gate voltage Vcg for left (open circle) and right (full circle) gate voltage values. The insets show pair of triple point in the regime of weak (Vcg = 7.4 V), medium (Vcg = 7.43 V) and strong (Vcg = 7.46 V) interdot coupling. carriers can be given for each pair of triple points. Data for occupation numbers (M,N), (M+1,N) and (M+5,N-1) are shown with M and N unknown. The examples of in Figure 9.8 (b,c) are highlighted by blue arrows. The coupling energies of the neighboring triple points (M,N) and (M+1,N) increase with Vcg whereas it decreases by more than a factor of two for the triple point (M+5,N-1). This observation is in agreement with earlier studies on single layer graphene QDs [89, 191] showing a strongly non-monotonic dependence of the conductance on gate voltages due to the sharp resonances in the constrictions. The extracted coupling energies are roughly by a factor of two larger, as compared to these studies [89, 191], which might be related to the considerably smaller size of the investigated device. Figure 9.11 (a) shows the triple points corresponding to the data of the black curve in Figure 9.10. With increasing central gate voltage the spacing between the triangle tips and thus the capacitive coupling energy decreases. Interestingly, the current within the triangles modulates with the coupling energy. Taking a closer look at the excited state with an energy of ≈ 2 meV a change of its coupling can ce observed (see black arrows). At Vcg = 7.44 V this state is almost suppressed

124 9.5. Capacitive interdot coupling

(M+1,N) Vb = 8 mV

6 (M,N) Vb = 6 mV

(M+5,N-1) Vb = 6 mV I (pA)

(M+5,N-1) Vb = -6 mV (V) 1.6

lg 3 5 (M+1,N) 1.4 1.5 (M,N)

m 1.2 Left Gate V

E 0 4 5.5 6 6.5 7 7.5 8 Right Gate Vrg (V)

3 Coupling Energy (meV) 2

7.4 7.42 7.44 7.46 7.48 7.5 7.52 7.54 7.56 7.58 7.6

Central Gate Vcg (V)

Figure 9.10.: Capacitive interdot coupling at diﬀerent occupations. Mu- tual capacitive inter-dot coupling energy Em plotted as a function of central gate voltage. The relative numbers of charge carriers measure (M,N), (M+1,N) and (M+5,N-1), respectively. Data has been recorded for negative and positive bias voltages. The inset shows the position of the triple points in the charge stability diagrams revealing (M, N).

125 9. Bilayer graphene quantum dots

while it is well pronounced at Vcg = 7.54 V. The central gate voltage influences the overall position of the triple point pair within the gate voltage plane as a function of VCG via capacitive cross-talk. Regarding the device geometry, the influence of cross-talk is not surprising. The behaviour of the coupling energy remains the same at reversed bias voltage (see dashed line), though an offset in energy is observed. Figure 9.11 (b) shows the corresponding triple points. The dotted line emphasizes a charge rearrangement which is obviously influenced by the gate voltage. Increasing VCG moves the pair of triple points to lower plunger gate voltage values. The rearrangement is located very close to the right barrier and couples only weakly to the central gate.

Therefore it moves through the triangles with varying gate voltage Vcg. The investigated device showed coupling energies between 2 and 6 meV (see Fig. 9.10) which is roughly twice the value reported for a larger (d ≈ 90 nm) single-layer double quantum dot [191]. The capacitive interdot coupling shows a non-monotonous dependence on the gate voltage and the charge carrier occupation of the QDs. This eﬀect can most likely be attributed to the disorder dominated electrostatic landscape in the narrow constriction connecting the two quantum dots, which is in agreement with other experiments on graphene tunneling barriers [116].

9.6. Inter-dot tunnel coupling

Transport through the quantum dot states can be analyzed more quantitatively using the result from Stoof and Nazarov for resonant tunneling [79, 89, 192]. In the limit of weak inter-dot tunnel coupling tm Γin,out, where tm is the inter-dot tunnel coupling, and Γin,out the incoming and outgoing tunnel rates, the current I follows a Lorentzian line shape as a function of the detuning energy ε,

2 4etm/Γout I(ε) = 2 . (9.2) 1 + (2ε/hΓout)

Figure 9.12 (a) shows the absolute value of the current along the detuning axis of two diﬀerent triple points recorded at positive and negative bias. Lorentzian line ﬁts to the ground state resonance according to equation 9.2 are plotted in red.

126 9.6. Inter-dot tunnel coupling

a 0.70 7.40V 7.42V 7.44V 7.46V 7.48V I (pA) 15 0.65 (V) lg 0.60 7.5 V

0.55 0 8.8 9.0 9.2 8.8 9.0 9.2 8.8 9.0 9.2 8.7 8.9 9.1 8.7 8.9 9.1 0.65 7.50V 7.52V 7.54V 7.56V 7.58V I (pA) 8 0.60 (V) lg 0.55 4 V

0.50 0 8.5 8.7 8.9 8.5 8.7 8.9 8.5 8.7 8.9 8.4 8.6 8.8 8.4 8.6 8.8

Vrg (V) Vrg (V) Vrg (V) Vrg (V) Vrg (V) I (pA) b 7.40V 7.44V 7.48V 7.50V 7.54V 7.58V 0

-3 (a.u.) lg V x2 x2 x2 x1 x1 x1 -6 8.8 9.1 8.7 9.0 8.6 8.9 8.5 8.8 8.4 8.7 8.3 8.6 Vrg (V) Vrg (V) Vrg (V) Vrg (V) Vrg (V) Vrg (V)

Figure 9.11.: Series of triple points with different mutual capacitive coupling. Current through the DQD as a function of the side gate voltages for different central gate voltages. The very same triple point is measured at (a) Vb=6 mV and (b) Vb = −6 mV, for increasing Vcg, ranging from 7.40 V to 7.58 V. The capacitive inter-dot coupling energies for (a) and (b) are represented by the black and gray line in Fig 9.10. The black crosses in panel (a) are fixed in the gate voltage plane and highlight the adjustment of the triple point position due to capacitive cross-talk. The dotted line in panel (b) highlights the position of a charge rearrangement.

127 9. Bilayer graphene quantum dots

a b 1.48 I (pA) 2 3 2 1 1.43 ε 1

0 Left Gate (V) 1.38 0 Current I (pA) 6.6 6.75 6.9 4 I (pA) 3 3.45 0 2 3.4 -2 1 ε 0 3.35 Left Gate (V) -4 Current | I (pA) -1 01 2 3 4 5 6 2.2 2.4 2.6 Detuning Energy ε (meV) Right Gate (V)

Figure 9.12.: Interdot tunnel coupling. (a) Current through the DQD along the detuning energy axis of two diﬀerent triple points, as shown by the arrows in panel (b). Lorentzian functions (cf. eq. 9.2) are ﬁtted to the ground state (red lines). (b) Two pairs of triple points from which the data in (a) has been extracted (Vcg = 7.465 V, VSD = 6 mV and Vcg = 2 V, VSD = −6 mV, respectively).

In both traces in Fig. 9.12 (a) excited states are clearly visible (see arrows). The corresponding triple points are shown in Figure 9.12 (b). Following Ref. [89] the ﬁt is restricted to data points outside the bias triangle in order to minimize the contribution from inelastic transport.

The ﬁts yield a tunnel rate between the right dot and the drain of hΓR→d ≈

210 µeV and an inter-dot tunnel rate of htm ≈ 1.5 µeV. Measurements at negative bias allow to obtain the tunnel rate from the left dot to the source of hΓL→s ≈

420 µeV and an inter-dot tunnel rate of htm ≈ 1.7 µeV. The tunnel rates to the leads are on the same order of magnitude compared to earlier results by Liu et al. [89] whereas the measured inter-dot tunnel rate is by a factor of six smaller. This might be explained by the diﬀerent device design where a DQD is deﬁned by two gates on top of a 20 nm wide graphene nanoribbon. Their tunneling barriers seem to be more transparent. Please note that the extracted inter-dot tunnel rate is also consistent with the upper bound of t ≤ 20 µeV estimated by Molitor et al. [123] for a single-layer graphene double quantum dot.

128 9.7. Excited state spectroscopy

a 12 b 16 c 3.1 4 1.94 e=D 2.50 12 3 lg 8 lg lg NM 8 2 4 3.0 1.86

Current (pA) 2.40 4 Current (pA) 1 Current (pA)

Left Gate V (V) V = -10 mV Left Gate V (V) V = -10 mV Left Gate V (V) V = -6 mV b 0 b 0 b 0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 3.5 3.7 3.9 4.1 4.9 5.1 5.3

Right Gate Vrg (V) Right Gate Vrg (V) Right Gate Vrg (V)

Figure 9.13.: Pairs of triple points featuring excited states. (a) High- resolution close-up of a pair of triple points presented in Fig. 9.7 showing a dense set of excited states. Inset: Schematic energy diagram showing the ground states of the DQD system and the ﬁrst excited state of the left dot. (b,c) Diﬀerent examples of triple points featuring similar excited state energies (VSD = −10 and −6 mV, respectively). The current is plotted in absolute values.

9.7. Excited state spectroscopy

Already at a temperature around 1.3 K signatures of excited states can observed (cf. Figure 9.6), but the increased energy resolution in the dilution refrigerator allows a more systematic analysis of the excited sate spectrum of the device. The data presented in Figure 9.7 exhibit distinct lines of increased conductance parallel to the base line of the triple points. Figure 9.13 (a) shows a close-up of an individual pair of triple points highlighting a set of electronic excited state resonances (see black arrows). Along such a resonance the inter-dot detuning energy ε is constant and the current is increased due to excited state transport. The configuration for transport via the 1st excited state is illustrated in the inset in Fig. 9.13 (a)). The electronic nature of these excited states can be shown by the magnetic field dependence as discussed in the following section. The detuning energies of the five visible excited state resonances measure ε = 1.7, 3.3, 4.9, 6.5 and 8.1 meV, respectively. Figure 9.13 (b) shows a pair of triple points recorded in a different gate voltage regime. Excited states parallel to the base line (see black arrows) as well as a resonance parallel to its right edge can be observed. The first two excited states are again found at detuning energies of ε = 1.7 and 3.8 meV. The suppressed conduction near the right edge can be attributed to variations of the coupling between energy levels of the left dot and the source lead [86]. Figure 9.13 (c) shows another pair of triple points featuring very similar

129 9. Bilayer graphene quantum dots excited state energies (ε = 1.8 and 3.5 meV). In total more than 50 excited states covering a wide energy range have been analysed. The energy spacing of two subsequent electronic excited states has been found to be constant over the entire range with a value of ∆ = 1.75 ± 0.27 meV. Interestingly, this value is in good agreement with the single-particle conﬁnement energy in disk-like bilayer graphene QDs, which can be estimated by using the 2 density of states for bilayer graphene DBL(E) = γ1/π(~vF ) , with the Fermi velocity vF and the inter-layer hopping energy γ1 = 0.39 meV. Consequently, the single-particle level spacing is given by

2 2 4~ vF 1 ∆BL = 2 , (9.3) γ1 d where d is the diameter of the bilayer QD. Please note that this result only depends on the QD diameter, especially it is independent on the number of charge carriers. Assuming a reasonable QD diameter of d = 50 nm, this leads to a level spacing of ∆ = 1.71 meV, which is in good agreement with the experimental data. The charge carrier number independent level spacing is in contrast to single- 2 layer graphene QDs. The linear density of states DSL(E) = 2E/π(~vF ) yields a single-particle level spacing of √ ∆SL(N) = ~vF /d N, (9.4) with the number of carriers N [80]. The excited state level spacing of single layer and bilayer graphene QDs is derived in section 3.4. Figure 9.14 compares the experimental data with the calculated single-level spacing ∆ for 50 nm diameter single-layer and bilayer graphene QDs. As the total number of carriers (N,M) is unknown the excess number of carriers on the left dot, ∆N = N − N is plotted. The horizontal solid line marks the constant level 0 √ spacing calculated for the bilayer QD, whereas the three 1/ N-dependent curves belong to a single-layer QD with diﬀerent absolute numbers of charge carriers on the dot (N0 = 5, 25 and 50). The experimental data (circles) reveal the change of the number of carriers on the left QD (∆N). The presented data have been extracted from charge stability

130 9.7. Excited state spectroscopy

4.5

4.0 N0 = 5

3.5 d = 50 nm

N0 = 25 3.0 D 2.5

N0 = 50

2.0 Level spacing (meV)

1.5 bilayer

1.0 single-layer

N = N0 + DN 0.5 -20 0 20 40 60 80 Excess number of carriers D N Figure 9.14.: Excited state statistics. Single-level spacing as a function of the excess number of charge carriers on the left dot. The circles represent the experimental data and the dashed horizontal line marks their mean value (∆ = 1.82 meV). The three curves represent the model of a single-layer QD for diﬀerent absolute occupation numbers N0. The solid horizontal line marks the constant level spacing estimated for a bilayer dot with a diameter of 50 nm.

131 9. Bilayer graphene quantum dots

diagrams (see e.g. Fig. 9.7) measured at VSD < 0. For large numbers of excess carriers (∆N > 40) relative gate lever arms have been used to estimate ∆N. Since charge rearrangements cannot be fully excluded these data have to be taken with care. The average value of excited sate level spacings yields a constant value of ∆ = 1.82 ± 0.14 meV (dashed line). A constant level spacing might in principle also result from an effective constant density of states induced by disorder. However, following Güttinger et al. [85], the effective dot disorder potential can be estimated to be on the order of the addition energy of 5-10 charge carriers. Since the experiments cover a significantly larger energy range the disorder can be ruled out as the reason of the observed constant level spacing. Consequently, the model suitable for describing single-layer √ QDs (∆(N) ∝ 1/ N) does not fit the experimental data for any reasonable N0 whereas the model for a bilayer QD (∆(N) = const.) is in good agreement. This yields the conclusion that the investigated QDs are indeed extending over both graphene sheets forming the bilayer system.

9.8. Magnetic ﬁeld dependence

In this section the evolution of excited states as function of the magnetic ﬁeld oriented in parallel to the bilayer graphene plane is studied proving the electronic nature of these states. Figure 9.15 compares a pair of triple points recorded at

Bk = 0 T and at Bk = 2 T, respectively. Excited states parallel to the base line as well as one parallel to the left edge of the triangle are visible (dotted lines). At zero magnetic ﬁeld again a level spacing of 1.8 meV (see e.g. arrow in panel (a)) can be observed. When applying a parallel magnetic ﬁeld (panel (b)) the position of the triangles with respect to the gate voltage plane almost does not change at all (see crosses and solid lines in Fig. 9.15).

This holds even for Bk-fields up to 9 T proving the orbital contributions of the wave functions are unaffected. The parallel magnetic field has two effects on the triple points. First, the conductance at large detuning values slightly increases with increasing Bk. This is especially true for the left triangle, where even parts are strongly suppressed at low Bk-fields (see left edge in Fig. 9.15). Since the associated slope differs from the

132 9.8. Magnetic ﬁeld dependence

a e B|| = 0 T b B|| = 2 T 3 3.45

lg 2 3.40 x x 1 Current (pA) Left Gate V (V) 3.35 x x 0 2.3 2.4 2.5 2.6 2.3 2.4 2.5 2.6

Right Gate Vrg (V) Right Gate Vrg (V)

Figure 9.15.: Pair of triple points under the influence of a magnetic field. The very same individual triple points measured at (a) Bk = 0 T and (b) Bk = 2 T, respectively (VSD = −6 mV). actual triple point edge this effects can be attributed to a modulated transmission from the drain to the right dot (VSD < 0). Second, the 1st excited state parallel to the base line splits into two separate peaks (see dotted lines in Fig. 9.15 (b)). This splitting is linear in magnetic field up to roughly Bk = 5 T as shown in Figure 9.16 (a), where current line cuts along the detuning energy axis ε (see dashed line in Fig. 9.15 (b)) are plotted for different B-fields. The characteristic linear peak splitting measures 0.2 meV/T (see diverging dashed lines). In contrast, the position of the base line (ε=0) and the 2nd excited state (ε = 3.6 meV) are constant up to 9 T as function of the

Bk-field (see vertical dashed lines). Different triple points show a similar peak splitting. Figure 9.16 (b) shows an example of a measurement, where the 3rd excited state (ε = 5.3 meV) splits. The current is plotted as a function of detuning energy and parallel magnetic field. The inset shows the corresponding triple points (detuning axis indicated by dashed line). While the position of the baseline and the 1st excited state are constant in nd rd Bk, the 2 excited state shows a down-shift in energy and the 3 excited state clearly splits linearly into two peaks (see dashed lines). The base line as well as the 1st and 2nd excited state completely vanish at increased parallel magnetic fields.

The observed Bk dependent peak shifts have a slope of either 0 or ± 0.1 meV/T.

133 9. Bilayer graphene quantum dots

a b 5 20 6 9T 18 5 4 16 7T 4 14 3 12 3 5T 2 10

8 1 Current (pA) 3T 2 6 Magnetic field B (T) 0

4 1T 1 2

0 Current (pA)

0 -1 0 1 23 4 5 6 0 2 4 6 8 10 Detuning Energy (meV) Detuning Energy (meV)

Figure 9.16.: Excited states under the influence of a parallel B-field. (a) Absolute current as a function of the detuning energy measured along the dashed line indicated in Fig. 9.15. Each curve corresponds to a different value of Bk covering a range from 0 to 9 T. The traces are offset by 2 pA for clarity, VSD = -6 mV. (b) Current as a function of the detuning energy and Bk (VSD = -8 mV) recorded at a different triple point shown in the inset (the scale bars are 50 mV in gate voltages.

134 9.9. Spin blockade

Most likely, the observed peak splittings originate from Zeeman spin splitting.

This is motivated by the linear Bk-ﬁeld dependence of the peak splittings (up to high ﬁelds) and their characteristic energy scale, which is in reasonable agreement with gµB = 0.116 meV/T, assuming a g-factor of 2. This g-factor would be in agreement with previously published spin-resolved quantum interference measurements in graphene [193] and observations of spin states in a single-layer graphene quantum dot [118].

9.9. Spin blockade

The Pauli spin blockade is a common tool to read out the state spin qubits. The so-called spin to charge conversion uses that individual transitions between the two QDs are forbidden due to Pauli blockade. This technique is widely used in III/V-heterostructure based DQDs [14, 194] and has been demonstrated for in a carbon nanotube device [195] but has not yet been observed in graphene. Figure 9.17 hints the presence of spin blockade. At a positive bias voltage the measurement shows a typical pair of triple points featuring excited states. At reversed bias the base line is almost completely suppressed (compare the experimental data and the dashed triangles in panel (b)). This observation is in agreement with Pauli spin blockade which prevents e.g. the triplet (1,1) to singlet (2,0) transition whereas a transition from the singlet (2,0) to the singlet (1,1) is possible. Additionally, the overall current measured at VSD < 0 is significantly smaller than at VSD > 0 A suppression of the base line can originate from spin blockade but it is not a proof. Other triple points did not show the effect as it would be expected in case of spin blockade [196]. Thus it is more likely that state and energy dependent tunneling barriers cause the observed an effect.

9.10. Conclusion

A bilayer graphene double quantum dot based on a width-modulated graphene nanostructure has been characterized. In-plane gates allow tuning the transport over a wide energy range and the device can be driven into a single or double

135 9. Bilayer graphene quantum dots

a b 3 V = -6 mV VSD = 6 mV 0.60 SD 0.60 LG LG 0.56 2 0.56

0.52 0.52 1 Left gate V (V) Left gate V (V) 0.48 0.48 0 Absolute current |I| (pA) 8.4 8.5 8.6 8.7 8.3 8.4 8.5 8.6

Right gate VRG (V) Right gate VRG (V)

Figure 9.17.: Indications of spin blockade. (a) Pair of triple points recorded at VSD = 6 mV. The dashed lines are a guide to the eye highlighting the edges of the triple points. (b) Same measurement at VSD = −6 mV. The dashed triangles have the same shape as in panel (a) emphasizing that the base line of the triple points is suppressed. The current scale is the same in both panels. dot configuration. In the DQD regime the characteristic honeycomb-like charge stability has been recorded. The coupling energies of the gates to the tunneling barriers and to the QDs have been determined. By means of a local gate the mutual capacitive coupling energy can be tuned systematically. This tunability is relevant for further graphene DQD based experiments. The capacitive coupling energy scale as well as the inter-dot tunnel coupling are in agreement with earlier results on single-layer graphene DQDs. The electronic excited state spectrum has been investigated over a wide energy range. In contrast to single-layer graphene the measured single-particle level spacing is independent of the number of charge carriers on the QDs which is in agreement with theory taking into account the QD diameter. The device remains stable under the influence of an in-plane magnetic field. A splitting of the energy levels on the order of Zeeman splittings is observed. These results show that bilayer graphene DQDs can well controlled by lateral gates, that the energy spectrum is in agreement with theory and that spin degeneracy can be lifted by an in-plane magnetic field. Thus, bilayer graphene DQD devices are promising for future experiments addressing spin related phenomena.

136 10 Summary and outlook

10.1. Summary

In this thesis single-layer and bilayer graphene quantum dot devices were studied. All devices were fabricated as width-modulated nanostructures carved out of graphene sheets. The device design was optimized such that both the QDs as well as the tunneling barriers could be well controlled by lateral gates with as little cross talk as possible. The QDs were connected to the leads via long and narrow constrictions allowing for driving the tunneling rates into the low MHz regime which is essentially necessary for the pulsed experiments. Most of the devices were equipped with capacitively coupled graphene nanoribbons acting as charge sensors. Charge detectors commonly used in III/V semiconductor experiments, e.g. to read out spin qubits. Following the idea of realizing graphene based qubits, high quality charge detectors are of great interest. The investigated charge detectors proved to be suitable as sensitive electrometers resolving charging events on the QD where the signal measured through the QD was below the detection limit. It was even possible to resolve individual excited state transitions of the QD. Beside this, the charge detectors provided information on the asymmetry of the coupling between the QD and its leads which is important for relaxation time measurements. The charge detectors kept functional under adverse conditions. They were successfully operated under the influence of high magnetic fields and RF pulses, valuable attributes for application in qubit experiments. The functionality remained even under varying temperatures. The influence of the charge detectors on the QDs was studied systematically. Choos-

137 10. Summary and outlook ing a too large bias applied to the detector lead to significant back action which could even completely lift the Coulomb blockade of the probed QD. Counter flow occurred in the QD at elevated voltages applied to the CD. A simplified model explains this by an effective temperature broadening the Fermi distribution in the lead closer to the CD. These findings are important to operate the CD in a regime with a good signal-to-noise ratio and as little back action as possible. Graphene QDs were intensively studied over the past five years and among others, Coulomb blockade, excitation spectra, spin-filling sequence, and electron-hole crossover have been demonstrated. This thesis addresses the relaxation dynamics of excited states in graphene QDs for the first time. The excited state spectra were studied by DC finite bias spectroscopy. Different RF pulse sequences were employed to perform transient current spectroscopy. These measurements yielded a lower bound for the charge relaxation time of 60 to 100 ns. The results are in reasonable agreement with a model mostly considering longitudinal acoustic (LA) phonons accounting for the relaxation. A bilayer graphene double quantum dot was characterized. The work focused on excited state spectroscopy and on the tunability of the interdot coupling. A systematic control of the mutual capacitive coupling was demonstrated. The electronic excited state spectrum was investigated by finite bias spectroscopy covering a wide energy range. In agreement with theory the measured single-particle level spacing turned out to be independent of the number of charge carriers. A splitting of the energy levels on the order of Zeeman splittings was observed in an in-plane magnetic field. Bilayer graphene DQD devices are promising for future experiments addressing spin related phenomena. However, spin blockade has not been demonstrated yet.

10.2. Further directions

The results of this thesis point a possible direction of future research on graphene QD devices. Further experiments and simulations are necessary to gain a better understanding of the relevant relaxation mechanisms and of the coupling of a graphene QD to its environment. In this context QDs on diﬀerent substrates are of interest. In suspended graphene QDs out-of-plane phonon modes may become

138 10.3. Outlook: QDs in graphene/hBN sandwiches relevant. Charge relaxation can be regarded as an interstation on the way to spin relaxation. To address this issue three level pulse schemes can be used. Long spin coherence times as promised by weak hyperﬁne and spin orbit interaction still have to be demonstrated. The sample design can be further optimized in a way that the QD and the tunneling barriers can be tuned more individually which is especially important for pulse gated experiments. One way could be the use of top gates. Spin blockade still needs to be searched for. This eﬀect may allow for realization of spin to charge conversion in graphene DQDs and thus would be an important step towards spin qubits in graphene. Further studies on charge sensing may provide deeper insights into the details of the coupling and the energy transfer between quantum dots and charge detectors.

10.3. Outlook: QDs in graphene/hBN sandwiches

Etched graphene nanostructures, including QDs, resting on SiO2 suffer from disorder. This limits e.g. the performance of tunneling barriers and makes it challenging to tune devices into the few electron regime or to operate it at high magnetic fields. The substrate and the edges are expected to be the main origins of disorder in such devices. Hexagonal boron nitride is regarded as a promising substrate material for graphene devices as it provides an atomically smooth surface with no dangling bonds and a very low density of charge traps [50, 51]. It has been shown that the size of charge puddles in graphene on hBN are about a factor of ten larger compared to graphene on SiO2 where they measure a few tens of nm [51]. Engels et al. [120] compared the statistics of Coulomb peak spacings of graphene QDs on both substrates. It has been found that for QDs larger than 100 nm in diameter the standard deviation of the normalized peak spacing distribution is smaller on hBN than on SiO2. Epping et al. [149] reported SETs which are stable in perpendicular magnetic fields up to

9 T which is in contrast to measurements reported on SiO2 so far [166]. These results indicate that hBN is an advantageous substrate for graphene QDs. Wang et al. have demonstrated a µm size graphene/hBN sandwich devices featuring room temperature mobilities of up to 150.000 cm2/Vs [8]. This ﬁnding raises the question if the stability of QDs can be further increased following this

139 10. Summary and outlook

a 120 b 1 -8

100 -9 80 0.5 200 nm RG 60 -10 0 40 Current I (nA) Current (log[A]) Right gate V (V) -11 20 -0.5 0 -12 -15 -10 -5 0 5 10 15 -0.5 -0.25 0 0.25 0.5

Back gate VBG (V) Left gate VLG (V)

Figure 10.1.: Characterization of sandwich based quantum dot. (a) Cur- rent through the QD as a function of the back gate voltage recorded at VSD = 1 mV and TMC ≈ 20mK. The ambipolar behavior is clearly visible. The transport gap extends from ≈ -2 to +2 V. Inset: Scan- ning electron micrograph of the device. The central island measures 120 nm in diameter, the constriction are 35 and 50 nm wide. The left nanoribbon has moved after etching the structure. (b) Stability diagram recorded at VBG = −1.2 V. The dashed lines are a guide to the eye emphasizing resonances attributed to the constrictions (slopes of 14 and 0.6). The solid line highlights the resonances attributed to the QD itself. Features characteristic for a DQD can be observed in some regions (see e.g. dashed circle). technique. So far, no sandwich-based nanostructures have been reported. This section shows preliminary results of a QD carved out of a graphene/hBN sandwich. The fabrication follows Wang et al. as described in section 5.3. A quantum dot measuring 120 nm in diameter has been etched out of a hBN/single-layer graphene/hBN heterostructure. Two nanoribbons and a gate a deﬁned close-by (see inset in Fig. 10.1 (a)). The back gate characteristics shows a transport gap centered around zero gate voltage and only covers a range of 4 V (see Fig. 10.1 (a)).

This is in contrast to similar-sized graphene QDs on SiO2 where the gap can ex- tend up to 20 V and usually a strong p-doping is observed (cf. Fig. 7.2). A stability diagram has been recorded operating the two nanoribbons as gates showing resonances with relative lever arms of 14 and 0.6 (see Fig. 10.1 (b)). This is in reasonable agreement with the device geometry where the left ribbon has moved during fabrication after the etching step. The narrow-spaced lines with a relative lever arm of 2.1 are attributed to the QD. Regions with triple points indicating

140 10.3. Outlook: QDs in graphene/hBN sandwiches

a 5 10 b 5 4

8 3 SD

6 2

SD 0 SD 0 4 1 Bias V (mV) 2 Bias V (mV) 0

0 conductance dI/dVDiff. (µS) -5 -5 -0.8 -0.7 -0.6 -220 -170 -120

Left gate VLG (V) Left gate VLG (V)

Figure 10.2.: Finite-bias spectroscopy. (a) Diﬀerential conductance as a function of VSD and VLG. VBG = −1.2 V, VRG = −0.34 V. (b) Close-up on a Coulomb diamond where signatures of excited states can be observed (see arrows). the formation of a double quantum dot can be identiﬁed. Finite-bias spectroscopy measurements (see Fig. 10.2 (a)) yield addition energies between 3 and 7 meV. Together with the observed charge rearrangements this indicates that the device is still in an unstable regime. Nevertheless, excited states could be observed (level spacing between 1.6 and 1.9 meV, see Fig. 10.2 (b)).

141

A Appendix

A.1. Process parameters: Graphene nanostructures

on Si/SiO2 substrates

A.1.1. Substrate preparation

• Substrate material: p-doped Si-wafers with 285 nm thermally grown SiO2. Supplier: Nova Electronics.

• Cleaning and coating with EBL resist

– Chemical cleaning: 5 min acetone (ultrasonic bath), 2 min cleaning in isopropanol, blowing dry with nitrogen.

– Plasma ashing: 5 min oxygen plasma (barrel reactor, 0.2 mbar) at 300 W.

– Spin coating PMMA 200k 4% (Allresist ARP 649.04) at 6000 rpm, 30 s (yields a thickness of ≈ 120 nm, checked by ellipsometry). Prebake for 10 min at 180 ◦C.

– Spin coating PMMA 600k 4% (Allresist ARP 669.04) at 6000 rpm, 30 s (yields a thickness of ≈ 180 nm). Prebake for 10 min at 180 ◦C.

• Electron beam lithography of alignment markers for further EBL steps and optical alignment.

143 A. Appendix

– Beam voltage 50 kV, dose 450 µC/cm2, beam current 5 nA, beam step size 50 nm. No proximity correction. – Development in MIBK based developer AR 600-55 (Allresist) for 70 s. Stopping in isopropanol, 30 s. – Removal of residual resist: 5 s oxygen plasma (0.2 mbar) at 300 W.

• Metal deposition – Deposition of 5 nm Cr at a rate of 0.2 nm/s (Balzers 500 electron beam evaporator). – Deposition of 50 nm Au at a rate of 0.5 nm/s

• Lift-oﬀ – Lift-oﬀ in acetone (ultrasonic bath). – Cleaning in fresh acetone and isopropanol.

• Dicing of chips – Spin coating of a protection layer for dicing: Photoresist AZ-5214, 4000 rpm. – Prebake for 5 min at 90 ◦C. – Dicing with a wafer saw into chips of 7 × 7 mm.

A.1.2. Graphene deposition

• Substrate preparation. – 5 min acetone, ultrasonic bath, 2 min cleaning in isopropanol, blowing dry with nitrogen. – 5 min oxygen plasma (0.2 mbar) at 300 W.

• Micromechanical exfoliation. – A tiny graphite ﬂake is placed on a strip of adhesive tape. The tape is folded and separated again about 10 to 20 times. Prepared chips are pressed onto the tape.

144 A.1. Process parameters: Graphene nanostructures on Si/SiO2 substrates

– Leaving the tape with chips in a dry atmosphere box (humidity below 20%) for 1 hour.

• 5 min cleaning in acetone, no ultrasonic bath, 2 min cleaning in isopropanol, blowing dry with nitrogen.

A.1.3. Device patterning

• Coating with EBL resist.

– Chemical cleaning: 5 min acetone (no ultrasound), 2 min isopropanol, blowing dry with nitrogen.

– Dehydration: 5 min at 180 ◦C.

– Spin coating PMMA 50k 4% (Allresist ARP 639.04) at 6000 rpm, 30 s (yields a thickness of ≈ 75 nm. Prebake for 10 min at 180 ◦C.

• Electron beam lithography.

– Beam voltage 50 kV, dose 200 µC/cm2, beam current 100 pA, beam step size 1 nm. No proximity correction.

– Development in AR 600-55 (Allresist), 50 s. Stopping in isopropanol, 30 s.

• Reactive ion etching.

– Cleaning of the RIE chamber immediately before etching the sample:

Ar/O2 plasma, ﬂow rates 20/20 sccm, pressure 0.025 mbar, RF power 300 W, 15 min.

– RIE etching of the sample: Ar/O2 plasma, ﬂow rates 32/8 sccm, pressure 0.025 mbar, RF power 60 W, 5 s.

• Chemical cleaning: 15 min cleaning in acetone, 15 min cleaning in dimethyl- sulfoxid (DMSO), 2 min cleaning in isopropanol, blowing dry with nitrogen. Using a pipette facilitates the removal of residual PMMA.

145 A. Appendix

A.1.4. Fabrication of ohmic contacts

• Coating with EBL resist. – Chemical cleaning: 5 min acetone (no ultrasound), 2 min isopropanol, blowing dry with nitrogen. – Dehydration: 5 min at 180 ◦C. – Spin coating PMMA 200k 4% (Allresist ARP 649.04) at 6000 rpm, 30 s (yields a thickness of ≈ 120 nm). Prebake for 10 min at 180 ◦C. – Spin coating PMMA 600k 4% (Allresist ARP 669.04) at 6000 rpm, 30 s (yields a thickness of ≈ 180 nm). Prebake for 10 min at 180 ◦C.

• Electron beam lithography. – Beam voltage 50 kV, no proximity correction. – Fine contacts (up to 100 nm wide): dose 510 µC/cm2, beam current 100 pA, beam step size 1 nm. – Medium contacts (up to 2 µm wide): dose 510 µC/cm2, beam current 100 pA, beam step size 5 nm. – Coarse contacts (up to 10 µm wide): dose 510 µC/cm2, beam current 150 nA, beam step size 50 nm. – Bond pads (130×130 µm): dose 400 µC/cm2, beam current 150 nA, beam step size 50 nm. – Development in AR 600-55 (Allresist), 70 s. Stopping in isopropanol, 30 s.

• Metal deposition. – Deposition of 5 nm Cr at a rate of 0.2 nm/s (Balzers 500 electron beam evaporator). – Deposition of 50 nm Au at a rate of 0.5 nm/s

• Lift-oﬀ. – Lift-oﬀ in acetone (without ultrasonic bath). A pipette is used to remove the metal.

146 A.2. Process parameters: graphene/hBN sandwich based nanostructures

– Cleaning in fresh acetone and isopropanol.

A.2. Process parameters: graphene/hBN sandwich based nanostructures

A.2.1. Graphene/hBN heterostructures

• Preparation of chips with graphene and hBN ﬂakes.

++ – Si /SiO2 chips patterned with alignment markers are prepared as in A.1.1. – Micromechanical exfoliation of graphene and hBN ﬂakes onto diﬀerent

Si/SiO2 chips as in A.1.2. – hBN raw material: hBN crystals by Taniguchi and Watanabe [147].

• Preparation of hBN ﬂakes on object plate. – Spin coating of PMMA/MAA copolymer resist on a conventional object plate for microscopy at 4000 rpm. Prebake for 10 min at 120 ◦C. – Micromechanical exfoliation of hBN ﬂakes onto the coated object plates similar to A.1.2.

• Pick-up technique. – The object plate is mounted in a mask aligner (conventionally used for optical lithography) at the position where usually the mask is placed. The hBN faces downwards. – Chip with graphene flakes placed on the chuck. – Positioning of a desired hBN flake above a graphene flake using the micromanipulator stage of the mask aligner. – Chip and object plate are brought into contact and are subsequently separated again. The graphene sticks to the hBN and is picked off the chip.

147 A. Appendix

– Repeating these steps it is possible to build graphene/hBN stacks of multiple layers.

• Placing the stack on the desired substrate. – Chip with hBN ﬂakes placed on the chuck. – Alignment of the graphene/hBN stack above a hBN ﬂake. – Chip and object plate are brought into contact. – Heating up to 90 to 120 ◦C to soften the polymer. – Separating the chip from the object plate leaves the entire stack on the chip. – Dissolving the polymer in acetone (15 min, no ultrasound), cleaning isopropanol.

A.2.2. Patterning sandwich based nanostructures

• Coating with EBL resist. – Chemical cleaning: 5 min acetone (no ultrasound), 2 min isopropanol, blowing dry with nitrogen. – Dehydration: 5 min at 180 ◦C. – Spin coating PMMA 950k 2% (Allresist ARP 679.02) at 3000 rpm, 30 s (yields a thickness of ≈ 80 nm). Prebake for 10 min at 180 ◦C.

• Electron beam lithography. – Beam voltage 50 kV, beam current 100 pA, beam step size 2.5 nm. Base dose 170 µC/cm2. Proximity correction yields a factor of 2.2. – Development in AR 600-55 (Allresist), 70 s. Stopping in isopropanol, 30 s.

• Fabrication of metal hard mask. – Deposition of 10 nm Al at a rate of 0.1 nm/s (Pfeiﬀer 570 electron beam evaporator).

148 A.2. Process parameters: graphene/hBN sandwich based nanostructures

– Lift-oﬀ in acetone (without ultrasonic bath). A pipette is used to remove the metal. – Cleaning in fresh acetone and isopropanol.

• Reactive ion etching. – Cleaning of the RIE chamber immediately before etching the sample:

SF6/Ar plasma, ﬂow rates 20/20 sccm, pressure 0.025 mbar, RF power 300 W, 10 min.

– RIE etching of the sample: SF6/Ar plasma, ﬂow rates 20/5 sccm, pressure 0.025 mbar, RF power 60 W, 30 s.

• Mask removal. – Etching of aluminum and alumina in tetramethylammonium hydroxide (TMAH) for 2 min. – Rinsing in DI water for 10 min.

A.2.3. Fabrication of ohmic contacts.

• The procedure follows strongly A.1.4.

• 5 nm of Cr and 100 nm of Au are evaporated.

149 A. Appendix

A.3. List of samples

• Characterization of tunneling barriers CG CG • Charge detection CD CD Sample • Finite bias WAC 87 A4 QD spectroscopy single-layer ﬂake (see chapters7, 8) • Pulsed gate S D spectroscopy 200 nm • Relaxation dynamics

Sample • Charge detection WAK 11 C1 left • Back action single-layer ﬂake • Counter ﬂow (see chapter7)

200 nm

150 A.3. List of samples

Sample WAK 11 C1 • Pulsed gate right spectroscopy single-layer ﬂake (see chapter8)

200 nm

• Characterization of tunneling barriers 200 nm Sample • Stability diagram S D NT4 14 B2 left bilayer ﬂake Interdot coupling (see chapter9) • Excited state spectroscopy

• Coulomb blockade CD Sample • Charge detection LG QD RG WAB 32 D4 bilayer ﬂake • Temperature S D (see chapter9) dependence CG 200 nm

Sample • Basic Alex 122 C1 characterization hBN/graphene/hBN sandwich • Coulomb blockade (see section 10.3) 200 nm

151 A. Appendix

A.4. Loss of mixture at the dilution refrigerator

Usually the automatic warm up routine has been used to warm the Oxford Kelvi- nox 400 dilution refrigerator. The programme stops the circulation, closes the 1K pot, heats mixing chamber, still and sorb and recovers the mixture to a dump vessel.

During one warm up procedure the N2 cold traps had accidently been warmed up before starting the routine. Finishing the automatic sequence the dump pressure reached a value about 100 mbar too high compared to the common value. This corresponds to approx. 10 l of gas at ambient pressure. The pressure difference was attributed to contaminations pumped into the dump. The system was opened for sample exchange and cooled down again. While condensing the mixture it was purified by the cold traps. After warming the system the next time the dump pressure showed 110 mbar too little compared to the standard value. This indicates a loss of approx. 11 l of mixture. Most probably a certain amount of mixture remained in the system after finishing the first warm up sequence and was lost when opening the system. A leak at the insert or the pumping lines was excluded using a leak detector. Even with the reduced amount of mixture it was possible to run the system and a stable base temperature of ≈ 30 mK could be achieved. To determine the concentration of 3He in the mixture, the so-called single-shot procedure was performed. First the mixture is fully condensed in and the circulation started. The circulation is then interrupted by closing the valve returning the mixture to the cold traps and opening the valve leading to the dump. All gas is pumped back to the dump while the system pressures and temperatures are monitored. The power applied to the still heater is adjusted to keep the still pressure and thus the evaporation rate constant. For further details of the procedure refer to the manual [197]. Due to the higher vapor pressure first mainly 3He is pumped. The temperature of the mixing chamber is expected to stay constant or even to decrease as no warm gas is returned. When all 3He is extracted from the mixture, the cooling process stops and the temperature increases significantly. The pressure measured at the dump at this time is an upper limit of the 3He volume. The results of the single-shot measurement are shown in Fig A.1. The temper-

152 A.4. Loss of mixture at the dilution refrigerator

1000 160 800 120 600 MC

80 400

40 200 Temperature T Temperature (mK) Dump Pressure G2 (mbar) 0 0 0 100 200 300 400 500 600 700 Time (min)

Figure A.1.: Single-shot measurement. Pressure at the dump (pressure gauge G2 of the gas handling system; red curve) and temperature of the mixing chamber TMC (blue curve) as a function of time. The start of the single shot procedure is indicated by the black arrow.

ature of the mixing chamber did not fall in the beginning as expected. It first increased and then remained in a regime below 500 mK for several hours. After approximately 480 min (see dashed line) the slope of the curve changed significantly and the temperature started to increase linearly with time. The pressure at the dump measured ≈ 145 mbar at this time. It was not possible to keep the still pressure constant and the entire procedure took significantly longer than expected according to the manual (approx. 1.5 h to extract all 3He). This was possibly caused by a too little amount of mixture leading quickly to an empty still. The single-shot was repeated twice yielding a similar result.

We assume that the mixture contained approx. 145 mbar of 3He (see dashed line in Fig. A.1) corresponding to a loss of approx. 0.5 l 3He and 10.5 l 4He. During the next circulation 9 l of 4He gas were added in steps of 1.5 l with at least 2 hours between each step. The system parameters remained almost unaﬀected, the base temperature decreased to ≈ 28 mK.

153 A. Appendix

A.5. Optimization of the experimental setup for high frequency read-out

Measurements have shown that the system can be further optimized for high frequency applications: E.g. during the pulsed gate experiments we have encoun- tered resonances in the system at frequencies between approx. 1 and 8 MHz. The sample has been bonded into a ceramic chip carrier and all lines (including the coaxial cables carrying the AC signals) have been soldered to the back of the chip socket. This induces an impedance mismatch being a source for resonances in the system. Furthermore the functionality of the system shall be upgraded to allow high frequency read-out measurements. We redesign our setup motivated by the one used by Foletti et al. [198]. The wiring of the insert is modified, a new cold-finger is mounted and the chip socket is replaced by a PCB. A wiring scheme including the low-temperature amplifier for RF read-out is shown in Figure A.2 (b). In this concept two RC filters in series are used for the DC lines. The resistances of 1 kΩ in the connector box at room temperature, the constantan woven loom cables (approx. 200 Ω) and 470 pF feed through capacitances at the head of the cold finger form the first RC filter (cut-off frequency ≈ 280 kHz). The second one is formed by two 1 kΩ resistances in series at the cold finger and 10 nF capacitances on the PCB close to the sample (cut-off frequency ≈ 8 kHz). The resistances are thermally anchored in two copper plates at the cold finger. The AC lines are thermally anchored via attenuators at three stages (1 K, 100 mK, 10 mK). At the cold-finger Cu coaxial cables are used as a high thermal conductivity is desired between the mixing chamber and the sample. A specially designed PCB is used to allow impedance matching as close to the sample as possible. A 50 Ω stripline is patterned for the RF read-out circuit. On-chip bias tees replace the commercial ones. The wiring of the insert with the different attenuator stages and the RF read-out equipment is displayed in Fig. A.3. Figure A.4 shows the new cold-finger with all installed components and wires together with a technical drawing. The entire cold-finger can be shielded by a copper cylinder.

154 A.5. Optimization of the experimental setup for high frequency read-out

a DC lines (24x) RF lines (2x) b DC lines (24x) RF lines (4x) RF read-out c bias & gates pulse gates bias & gates e.g. pulse gates in out

dc connector box 1 kΩ and SMA connectors 1 kΩ

10 dB at room temperature

~200 Ω 4 K stage ~200 Ω

1 K stage 20 dB 700 mK stage

100 mK stage

directional coupler mixing chamber feed through 470 pF capacitances

commercial 2 kΩ Anritsu K251 cold-finger ~10 mK on-chip filter capacitances 10 nF

~20 pF on-chip bias tees

constantan wire stainless steel coax copper coax Thermal anchoring

Figure A.2.: Wiring of the dilution refrigerator. (a) Schematics showing the AC and DC wiring of the cryostat including the cold ﬁnger. (b) Wiring scheme of the setup prepared for RF read-out. (c) Photogra- phy of the Insert indicating the positions of the cooling stages. For detailed images refer to Fig. A.3.

155 A. Appendix

a b c constantan woven-loom 4K stage cables additional connectors (not in use) micro D-sub connector cryogenic semi-rigid amplifier stainless steel 3 dB coaxial cables attenuator base plate circulator

10 dB filter box with attenuator feed-through 20 dB capacitances attenuator 1K stage 100 mK stage semi-rigid Cu coaxial cables 20 dB attenuator

Figure A.3.: Wiring of the insert of the dilution refrigerator. (a) Top part of the insert showing the space between 4K and 1K stage. A cryogenic amplifier and a circulator are used for RF read-out (c.f. Fig. A.2 (b)). The 20 dB attenuators of the gates are thermally anchored at the 1K stage (due to limited space positioned above and below). (b) 10 dB attenuators at the 100 mK stage. (c) 3 dB attenuators anchored at the base plate. The mixing chamber is positioned at the back side. The cold-finger (c.f. Fig. A.4) is attached to the base plate. The filter box for the DC lines and the semi-rigid Cu coaxial lines can be seen.

Figure A.5 shows the PCB prepared for high frequency measurements. On each DC line 10 nF capacitors to ground are installed as RF ﬁlters. On-chip bias-tees (10 nF / 4.7kΩ) allow to couple AC and DC signals. NPO capacitors are used due to their thermal stability over a large range. The setup has been characterized using a vector network analyzer (VNA) and a sampling oscilloscope. Fig. A.6 (a) shows the forward gain (transmission S21) measured with the VNA. RF signals ranging from 300 kHz to 20 GHz have been sent through the stainless steel coaxial cables of the insert including the three attenuator stages of 20, 10 and 3 dB. Obviously the signal is already damped by 33 dB at low frequencies. An additional damping of 3 dB occurs at approximately 240 MHz. The transmission of signals exceeding a few GHz can be completely neglected. A square pulse wave form generated by an arbitrary wave form generator has been sent through the setup to determine the pulse rise time. Fig. A.6 (b) shows an 8 MHz pulse train measured at the mixing chamber. The inset is a zoom on

156 A.5. Optimization of the experimental setup for high frequency read-out

a Shielded filter box with Semi-rigid Cu DC lines with 2x 1kΩ SMA connectors Micro D-sub feed through capacitors coaxial cables resistances in series connector

b Micro D-sub c connector

PCB mounted in cold-finger

d PCB SMA connectors feed through 470 pF Resistances capacitors Resistances to ground Filter box

Figure A.4.: Coldfinger for high frequency measurements. (a) Coldfinger without shielding cylinder. A shielded filter box is positioned at the head of the coldfinger. Two 1kΩ resistances are inserted in series of the DC lines. Semi-rigid Cu coaxial lines carry the AC signals. (b) Close-up of the open filter box. 470 pF feed through capacitors are installed for each DC line. (c) PCB mounted at the tail of the coldfinger. The DC lines are connected via a micro D-sub plug, the RF lines via SMA connectors. (d) Technical drawing of the coldfinger.

157 A. Appendix

a RF filters on dc lines: Bias-tees Strip line 10 nF capacitors to GND 10 nF / 4.7 kΩ 50 Ω

c Bias-tees

4.7 kΩ DC input sample AC input 10 nF

b SMA connectors Semi-rigid Micro D-Sub enamel-insulated MMCX connector coaxial cables (Cu) connector (DC) Cu wires RF read-out line

Figure A.5.: PCB for high frequency measurements. (a) Front and (b) back view of the PCB. The sample space is positioned in the center surrounded by bond pads. 10 nF capacitors are used on the DC lines as RF filters. 4 bias-tees (10 nF / 4.7kΩ) have been installed e.g. for AC and DC control of the same gate. One AC line is designated as RF read-out line. the rising flank highlighting the 20%/80% rise time of approximately 1 ns. The losses of the semi-rigid Cu coaxial cables at the cold-finger are negligible compared to the stainless steel cables (see Fig. A.6 (c)). The RF wiring of the cold-finger has been checked with a sampling oscilloscope. In Fig. A.6 (d) the impedance is plotted as function of time which is correlated with the length of the cable. Small impedance mismatches at the SMA connectors can clearly be seen. Kinks in the cables at the head of the cold finger can be resolved as well as small wiggles in the regime between 0 and 0.5 ns. The coaxial cable ends on the PCB with an open end.

158 A.5. Optimization of the experimental setup for high frequency read-out

a b -30 1 -40 3 dB 240 MHz 0.8 0.6 -50 1 0.4 -60 0.2 Amplitude (a.u.) 0 -70 0

Transmission S21 (dB) Transmission 0 1 2 10-4 10-3 10-2 10-1 1 10 102 0 20 40 60 80 c Frequency (GHZ) d Time (ns) 1 100

75 SMA connector SMA connector at mixing chamber at PCB 0 50

open wire on PCB 25 Impedance (Ohm) Transmission S21 (dB) Transmission -1 0 10-2 10-1 1 10 102 0 1 2 3 4 Frequency (GHZ) Time (ns)

Figure A.6.: Characterization of the optimized setup. (a) Forward gain (transmission S21) of the stainless steel coaxial cables including three attenuator stages of 20, 10 and 3 dB. The dashed lines mark the in- tentional damping of 33 dB and an additional damping of 3 dB at a frequency of 240 MHz. (b) 8 MHz square pulse generated by the Tektronix AWG7082C, measured at the mixing chamber. The inset shows a zoom of the rising flank of the pulse. The dashed vertical lines mark the measured 20%/80% rise time of 1 ns. (c) Forward gain (S21) of the semi-rigid Cu coaxial cables of the cold-finger. (d) Impedance measurement of the RF lines of the cold-finger including the coaxial cables at the PCB. Reflections at each connector and at the open end can be seen.

159

Publications

Disorder induced Coulomb gaps in graphene constrictions with diﬀer- ent aspect ratios B. Terr´es,J. Dauber, C. Volk, S. Trellenkamp, U. Wichmann, and C. Stampfer Appl. Phys. Lett. 98 032109 (2011)

Transport in kinked bi-layer graphene interconnects B. Terr´es,N. Borgwardt, J. Dauber, C. Volk, S. Engels, S. Fringes, P. Weber, S. Trellenkamp, U. Wichmann and C. Stampfer Proceedings of the 6th IEEE International Conference on Nano/Micro Engineered and Molecular Systems, Kaohsiung, Taiwan, February 20-23 2011, 1037–1040

Transport in Graphene Nanostructures C. Stampfer, S. Fringes, J. Güttinger,F. Molitor, C. Volk, B. Terrés,J. Dauber, S. Engels, S. Schnez, A. Jacobsen, S. Dröscher,T. Ihn, and K. Ensslin Frontiers of Physics 6 271–293 (2011)

Electronic excited states in bilayer graphene double quantum dots C. Volk, S. Fringes, B. Terr´es,J. Dauber, S. Engels, S. Trellenkamp, and C. Stampfer Nano Lett. 11 3581-3586 (2011)

Charge detection in a bilayer graphene quantum dot S. Fringes, C. Volk, C. Norda, B. Terr´es,J. Dauber, S. Engels, S. Trellenkamp, and C. Stampfer Physica Status Solidi B 248 2684–2687 (2011)

161 A. Appendix

Fabrication of coupled graphenenanotube quantum devices S. Engels, P. Weber, B. Terr´es,J. Dauber, C. Meyer, C. Volk, S. Trellenkamp, U. Wichmann, and C. Stampfer Nanotechnology 24 035204 (2013)

Probing relaxation times in graphene quantum dots C. Volk, C. Neumann, S. Kazarski, S. Fringes, S. Engels, F. Haupt, A. M¨uller, and C. Stampfer Nature Communications 4 1753 (2013)

Etched graphene quantum dots on hexagonal boron nitride S. Engels, A. Epping, C. Volk, S. Korte, B. Voigtl¨ander,K. Watanabe, T. Taniguchi, S. Trellenkamp, and C. Stampfer Appl. Phys. Lett. 103 073113 (2013)

Graphene-based charge sensors C. Neumann, C. Volk, S. Engels, and C. Stampfer Nanotechnology 24 44401 (2013)

Etched graphene single electron transistors on hexagonal boron nitride in high magnetic ﬁelds A. Epping, S. Engels, C. Volk, K.Watanabe, T. Taniguchi, S. Trellenkamp, and C. Stampfer Phys. Status Solidi B 25 2682 (2013)

162 A.5. Optimization of the experimental setup for high frequency read-out

Reducing disorder in graphene nanoribbons by chemical edge modiﬁ- cation J. Dauber, B. Terr´es, C. Volk, S. Trellenkamp, and C. Stampfer Appl. Phys. Lett. 104 083105 (2014)

Back action of graphene charge detectors on graphene and carbon nanotube quantum dots C. Volk, S. Engels, C. Neumann, and C. Stampfer Physica Status Solidi B, DOI:10.1002/pssb.201552445 (2015)

Graphene quantum dots C. Volk, C. Neumann, S. Engels, A. Epping, and C. Stampfer Chapter in K. Sattler: Carbon Nanomaterials Sourcebook. In press.

163

List of Figures

2.1. Lattice structure and band structure of single-layer graphene . . . 6 2.2. Lattice structure of bilayer graphene ...... 11 2.3. Band structure of bilayer graphene ...... 11 2.4. Lattice structure of hexagonal boron nitride ...... 13

3.1. Model of a quantum dot coupled to leads and a gate electrode . . . 16 3.2. Coulomb diamonds ...... 19 3.3. Electrostatics of a double quantum dot ...... 22

4.2. SET in a width-modulated graphene nanostructure ...... 30 4.3. QD in a width-modulated graphene nanostructure and charge detection ...... 32 4.6. Gate-defined quantum dot in a graphene nanoribbon ...... 37 4.7. Quantum dot defined by anodic oxidation ...... 39 4.9. Soft-confinement in bilayer graphene ...... 43

5.1. Micromechanical exfoliation ...... 49 5.2. Optical images of exfoliated flakes ...... 50 5.3. Raman spectroscopy ...... 52 5.4. Schematic process flow ...... 53 5.5. Contacted sample ...... 54 5.6. Different device designs ...... 55 5.7. Optical micrograph and Raman spectroscopy on an hBN/graphene/hBN sandwich ...... 58 5.8. Sandwich based graphene quantum dots ...... 59

165 List of Figures

6.1. Wiring of the dilution refrigerator ...... 63 6.2. Typical pulse waveform ...... 65 6.3. Measurement setup ...... 65

7.1. Sample geometry ...... 70 7.2. Back gate characteristics of a quantum dot and a charge detector . 71 7.3. Charge detection on a QD in the low-bias regime ...... 72 7.4. Finite bias Spectroscopy ...... 73 7.5. Sensitivity of the charge detector ...... 75 7.6. Influence of the magnetic field ...... 76 7.7. Influence of an RF pulse scheme ...... 77 7.8. Back action effects ...... 77 7.9. Back action ...... 78 7.10. Finite bias spectroscopy ...... 79 7.11. Counter flow ...... 80 7.12. Comparison with the model of thermal broadening ...... 81 7.13. Effective temperature and peak currents ...... 82 7.14. Temperature dependence ...... 84

8.1. Sample geometry ...... 88 8.2. Sample characterisation ...... 89 8.3. Stability Diagram ...... 90 8.4. Charge Detection ...... 91 8.5. Finite bias spectroscopy ...... 93 8.6. Transport through excited states ...... 94 8.7. Pulse scheme and low frequency pulsed excitation ...... 95 8.8. Pulsed excitation in the low and high frequency regime ...... 97 8.9. Pulsed excitation in the high frequency regime ...... 98 8.10. Schematics of the pulsed excitation of the excited state ...... 99 8.11. Pulsed-gate spectroscopy of excited states at diﬀerent frequencies . 100 8.12. Investigation of a second sample with the same geometry . . . . . 101 8.13. Transient current spectroscopy ...... 102 8.14. Transient current through excited states ...... 105

166 List of Figures

8.15. Time evolution of the occupation probability of the relevant quantum dot states ...... 107 8.16. Simulation of the time-resolved current ...... 108 8.17. Simulation of the number of electrons tunneling per cycle . . . . . 109 8.18. Calculated relaxation time due to electron-phonon coupling . . . . 111

9.1. Raman spectrum of a bilayer graphene ﬂake ...... 115 9.2. Scanning force micrograph and eﬀective band structure ...... 116 9.3. Back gate characteristics ...... 116 9.4. Characterization of the tunneling barriers ...... 118 9.5. Charge stability diagram ...... 119 9.6. Honeycomb pattern ...... 121

9.7. Charge stability diagram at Te = 100 mK ...... 122 9.8. Triple points with different mutual capacitive coupling ...... 123 9.9. Capacitive interdot coupling ...... 124 9.10. Capacitive interdot coupling at different occupations ...... 125 9.11. Series of triple points with different mutual capacitive coupling . . 127 9.12. Interdot tunnel coupling ...... 128 9.13. Pairs of triple points featuring excited states ...... 129 9.14. Excited state statistics ...... 131 9.15. Pair of triple points under the influence of a magnetic field . . . . 133 9.16. Excited states under the influence of a parallel B-field ...... 134 9.17. Indications of spin blockade ...... 136

10.1. Characterization of sandwich based quantum dot ...... 140 10.2. Finite-bias spectroscopy ...... 141

A.1. Single-shot measurement ...... 153 A.2. Wiring of the dilution refrigerator ...... 155 A.3. Wiring of the insert of the dilution refrigerator ...... 156 A.4. Coldﬁnger for high frequency measurements ...... 157 A.5. PCB for high frequency measurements ...... 158 A.6. Characterization of the optimized setup ...... 159

167

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188 Acknowledgements

First of all, I would like to thank Prof. Christoph Stampfer for offering me the chance to join his newly established research group as a PhD student. This gave me the opportunity to dive into the fascinating field of graphene research. I appreciated the inspiring and instructive discussions and his enthusiastic view on new experiments. I would like to thank Prof. Thomas Schäpers for being the second advisor of my thesis. I am grateful for the help and support by the members of the group on which I could always rely on. With Christoph Neumann I performed a lot of experiments together during his master’s thesis. We were a good team in measurements and data analysis and we wrote two publications together. He was a nice office mate and I enjoyed the scientific and non-scientific conversations we had. Stefan Fringes joined the bilayer graphene project as a master’s student. He was a great help in sample fabrication, measurements and data analysis and we wrote three publications together. With Stephan Engels I had many helpful discussions, he always kept a critical eye on manuscripts and he spent a lot of time with me setting up the dilution refrigerator. With Alexander Epping I measured my last device in the group and he will be my successor in the field of graphene quantum dots. Hopefully the inherited devices will be useful for him. Sebastian Kazarski and Sven Reichardt contributed to the graphene QD project by numerical simulations on relaxation processes and electron phonon interaction. We had a lot of helpful discussions. Bernat Terres and Jan Dauber were my first colleagues when I joined the group and they introduced me to fabrication of graphene devices. Matthias Goldsche was

189 Bibliography always willing to share his fabrication experience and I thank for many interesting discussions and being office mate for a long time. Frederic Buckstegge helped with the fabrication of graphene/hBN heterostructures. Thanks to Regine Ockelmann and Andre Müllerfor always being willing to help, for critical proofreading and for many nice conversations. I would like to thank Uwe Wichmann for keeping our electronic equipment running, troubleshooting problems and for sharing his knowledge in electronics with me. He always had an open ear for any kind of question. Many thanks to all members and former members of the group for creating this pleasant and welcoming atmosphere. I appreciated being part of this team and I enjoyed our common group activities. With Federica Haupt I had a number of theory related discussions and I am especially thankful for the valuable input for our manuscript. I would like to thank all members of the HNF and the former PGI-PT at the Forschungszentrum Jülich who kept the clean room running and supported me with fabrication related issues. Special thanks to Dr. Stefan Trellenkamp for the electron beam lithographies and for his expertise in all lithography related questions. Furthermore, I would like to thank the members and former members of the PGI-9 with whom I spent time in the clean room, during lunch or in the coffee breaks. Thanks to JörgSchirra and Sascha Mohr for the almost non-intermitted supply with cryogenics and thus allowing my long term measurements. The mechanical work shop always helped with technical problems even on short notice. I would like to thank all of my friends who were an important part of my life outside university. Finally, I would like to thank my family who supported me all over the past years.

190 Curriculum Vitae

Christian Volk born May 15, 1984 in K¨oln

2/2010- PhD student in the group of Prof. Christoph Stampfer, 6/2014 2nd Institute of Physics, RWTH Aachen University, Germany Single-layer and bilayer graphene quantum dot devices

12/2008- Diplom thesis in the group of Prof. Thomas Sch¨apers, 12/2009 Institute for Bio- and Nanosystems, Forschungszentrum J¨ulich, Germany Single electron tunneling in semiconductor nanowires

8/2007- Studies as an exchange student 3/2008 at Lund University, Sweden

4/2004- Studies of physics at Cologne University, Germany 12/2009 with the focus in experimental solid state physics

6/2003 Abitur at Gymnasium Lechenich, Erftstadt, Germany

191