Graphene Quantum Dots

Diss. ETH No. 19582

Graphene Quantum Dots

A dissertation submitted to the ETH ZURICH for the degree of Doctor of Science

presented by Johannes G¨ uttinger Dipl. El.-Ing. ETH born June 24, 1982 citizen of Männedorf (ZH)

accepted on the recommendation of Prof. Dr. Klaus Ensslin, examiner Prof. Dr. Francisco Guinea, co-examiner Prof. Dr. Thomas Ihn, co-examiner 2011

Abstract In this thesis, transport experiments on graphene quantum dots and narrow graphene constrictions at cryogenic temperatures are presented. In a quantum dot, electrons are confined in all lateral dimensions, offering the possibility for detailed investigation and controlled manipulation of individual quantum systems. The recently isolated two-dimensional carbon allotrope graphene is an interesting host to study quantum phenomena, due to its novel electronic properties and the expected weak interaction of the electron spin with the material. Graphene quantum dots are fabricated by etching mono-layer flakes into small islands (diameter 60 - 350 nm) with narrow connections to contacts (width 2075 nm), serving as tunneling barriers for transport spectroscopy. Electron confinement in graphene quantum dots is observed by measuring Coulomb blockade and transport through excited states, a manifestation of quantum confinement. Measurements in a magnetic field perpendicular to the sample plane allowed to identify the regime with only few charge carriers in the dot (electron-hole transition), and the crossover to the formation of the graphene specific zero-energy Landau level at high fields. After rotation of the sample into parallel magnetic field orientation, Zeeman spin-splitting with a g-factor of g ≈ 2 is measured. The filling sequence of subsequent spin states is similar to what was found in GaAs and related to the non-negligible influence of exchange interactions among the electrons. Transport through individual graphene constrictions is characterized by sharp resonances and an extended region of suppressed conductance, larger than expected from a confinement induced band gap. The observed behavior is attributed to the formation of localized electron- and hole puddles around the charge-neutrality point, which are caused by imperfect edges and inhomogeneous doping. In further experiments, a narrow graphene constriction is placed in close proximity to the quantum dot and used as sensitive charge detector for electrons on the dot. In these experiments, carrier localization in graphene constrictions is directly observed. Due to the close proximity of detector and quantum dot in graphene, time-resolved detection of individual electrons hopping onto and out of the dot is achieved by using high tunneling barriers. By counting single hopping events, the tunneling rate of the barrier is extracted and its non-monotonic gate dependence is studied.

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Zusammenfassung In dieser Doktorarbeit wird der Ladungstransport in Quantenpunkten und schmalen Streifen aus Graphen bei Temperaturen unter 4.2 K untersucht. Ein Quantenpunkt ist eine Box, in der Elektronen in allen drei Raumrichtungen eingesperrt sind und ermöglicht detailierte Studien oder kontrollierte Manipulation von einzelnen Quantensystemen (zum Beispiel vom Elektronenspin). Graphen, eine zwei-dimensionale atomare Schicht aus Graphit, ist ein faszinierendes neues Material zum Studium von Quantenphenomenen. Das Interesse gr¨ undet sich vor allem in den spannenden elektronischen Eigenschaften und den erwarteten schwachen Wechselwirkungen von Elektronenspins mit der Umgebung. Um Quantenpunkte aus Graphen herzustellen, werden aus Graphenflocken kleine Inseln (Durchmesser 60-350 nm) mit schmalen Verbindungen zu den Kontakten (Breite 20-75 nm) geätzt. Die Verbindungen dienen als Tunnelbarrieren f¨ ur Spektroskopiemessungen u ¨ber Ladungstransport. Die Einsperrung von Elektronen in einem Graphenquantenpunkt kann u ¨ber die Messung von durch Coulomb-Wechselwirkung blockiertem Stromfluss und Transport u ¨ber angeregte Zustände verifiziert werden. Angeregte Zustände sind ein klares Indiz f¨ ur die Quantisierung von Energiezuständen in einem Quantenpunkt. Messungen in einem Magnetfeld senkrecht zur Probe erlauben den Energiebereich ¨ mit wenigen Ladungsträgern (Elektron-Loch-Ubergang), anhand des graphenspezifischen Landau-Niveaus beim Ladungsneutralitätspunkt zu identifizieren. Nach der Rotation der Probe in ein paralleles Magnetfeld, kann die Zeeman-Spin-Aufspaltung mit einem g-Faktor von g ≈ 2 gemessen werden. Die F¨ ullsequenz von nacheinanderfolgenden Spinzuständen ist vergleichbar mit jener, die in GaAs basierten Quantenpunkten gefunden wurde und ist ein Ausdruck von nicht vernachlässigbaren Einfl¨ ussen der Austauschwechselwirkung unter den Elektronen. Der Transport durch einzelne Graphenstreifen ist charakterisiert durch scharfe Resonanzen und einen ausgedehnten Energiebereich, in dem der Stromfluss unterdr¨ uckt ist. Jener Bereich ist grösser als von einer nur durch die Verengung verursachten Bandl¨ ucke zu erwarten wäre. Das beobachtete Verhalten wird der Bildung von lokalisierten Ladungspf¨ utzen um den Ladungsneutralitätspunkt aufgrund von nicht perfekten Rändern und ungleichmässiger Dotierung zugeschrieben. In weiteren Experimenten wird ein zusätzlicher schmaler Graphen-Streifen nahe

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beim Quantenpunkt hergestellt, welcher als sensibler Ladungsdetektor f¨ ur die Elektronen auf dem Quantenpunkt verwendet wird. In diesen Experimenten kann die Lokalisierung von Ladung in Graphen-Verengungen direkt beobachtet werden. Durch die schmale L¨ ucke und die daraus resultierende starke Kopplung zwischen Detektor und Quantenpunkt in Graphene, können bei genug hohen Tunnelbarrieren zeitaufgelöst einzelne Elektronen gemessen werden, die zwischen Quantenpunkt und Kontakt hin und her h¨ upfen. Durch Zählen einzelner Ereignisse kann die Tunnelrate der Barriere extrahiert und die nicht-monotone Gateabhängigkeit studiert werden.

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Contents 1 Introduction

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2 Graphene 2.1 Electronic structure of graphene . . . . . . . . . . . . . . 2.1.1 Band structure . . . . . . . . . . . . . . . . . . . 2.1.2 Linear expansion around K and Dirac equation . 2.1.3 Implications of the chirality in graphene . . . . . 2.1.4 Graphene in perpendicular magnetic fields . . . . 2.1.5 Spin-orbit interaction in graphene . . . . . . . . . 2.2 Transport properties of bulk graphene . . . . . . . . . . 2.2.1 Field effect in graphene . . . . . . . . . . . . . . . 2.2.2 Minimum conductivity and electron-hole puddles 2.2.3 Quantum Hall effect in graphene . . . . . . . . . 3 Sample fabrication 3.1 Graphene deposition and characterization . . . . . . . . 3.1.1 Characterization with a scanning force microscope 3.1.2 Raman imaging . . . . . . . . . . . . . . . . . . . 3.2 Fabrication of graphene nanostructures . . . . . . . . . . 4 Graphene quantum dots 4.1 Single-electron transport . . . . . . . . . . . . . . . . 4.1.1 Charge quantization and Coulomb blockade . 4.2 Graphene constrictions . . . . . . . . . . . . . . . . . 4.2.1 Opening a band gap in graphene constrictions 4.2.2 Transport gap in graphene constrictions . . . 4.2.3 Formation of charged islands in constrictions . 4.2.4 Localization due to edge disorder . . . . . . .

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6 Crossover from electrons to holes in a perpendicular magnetic field 6.1 ”Fock-Darwin” spectrum in graphene . . . . . . . . . . . . . . . . . . 6.2 Sample characterization at zero magnetic field . . . . . . . . . . . . . 6.3 Evolution of Coulomb resonances in perpendicular magnetic field . . . 6.3.1 Edge states and influence of constrictions . . . . . . . . . . . . 6.3.2 Comparison with calculations . . . . . . . . . . . . . . . . . . 6.4 Discussion and summary . . . . . . . . . . . . . . . . . . . . . . . . .

60 60 62 64 65 65 67

4.3

4.4

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4.2.5 Discussion and summary . . . . . . . . . . . . . . Coulomb blockade in graphene single electron transistors 4.3.1 Tunable Coulomb blockade . . . . . . . . . . . . . 4.3.2 Charging energies of the SETs . . . . . . . . . . . 4.3.3 Charging energy as a function of dot diameter . . Quantum confinement in graphene . . . . . . . . . . . . 4.4.1 Level statistics . . . . . . . . . . . . . . . . . . . 4.4.2 Excited-states and cotunneling onsets . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Charge detection in graphene quantum dots 5.1 Graphene constrictions as charge detectors . . . . . . . . . . . . . . 5.1.1 Detection of single electron charging . . . . . . . . . . . . . 5.1.2 Using the quantum dot as a charge detector . . . . . . . . . 5.2 Time-resolved charge detection in graphene . . . . . . . . . . . . . . 5.2.1 Time averaged charge detection . . . . . . . . . . . . . . . . 5.2.2 Time-resolved measurements . . . . . . . . . . . . . . . . . . 5.2.3 Analysis of charge detection time traces . . . . . . . . . . . 5.2.4 Tuning the tunneling barrier . . . . . . . . . . . . . . . . . . 5.2.5 Optimizing the signal-to-noise ratio by increasing the detector bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Measurement setup and performance of the detector . . . . . . . . . 5.3.1 Bandwidth of the detector . . . . . . . . . . . . . . . . . . . 5.3.2 Detector noise . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Discussion and summary . . . . . . . . . . . . . . . . . . . .

7 Measurement of spin states in graphene quantum dots 69 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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7.2

7.3

7.4

7.1.1 Experimental techniques . . . . . . . . . . . . . . . 7.1.2 Zeeman spin splitting in parallel magnetic fields . . 7.1.3 Effective g-factor . . . . . . . . . . . . . . . . . . . Measuring the Zeeman-effect . . . . . . . . . . . . . . . . . 7.2.1 Identification of peak pairs . . . . . . . . . . . . . . 7.2.2 Extraction of the g-factor . . . . . . . . . . . . . . Spin filling sequence . . . . . . . . . . . . . . . . . . . . . 7.3.1 Exchange interaction in graphene . . . . . . . . . . 7.3.2 Evolution of excited states in parallel magnetic field Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Summary and outlook 81 8.1 Further directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Appendices A Charge detection in double quantum dots . . . . . B List of samples . . . . . . . . . . . . . . . . . . . C Useful relations and constants in graphene . . . . C.1 Capacitance of discs and stripes to parallel D Processing of graphene on SiO2 samples . . . . .

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Publications

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Bibliography

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Acknowledgements

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Lists of symbols physical constants -e< 0 o h = 2π~ kB µB

Abbreviation 2DEG AC CB CNT DC DOS e-beam ES FIRST FWHM GS LL QD QPC PMMA SET SFM SNR SP

explanation electron charge dielectric permittivity vacuum dielectric constant Planck’s constant Boltzmann constant Bohr magneton

Explanation two dimensional electron gas alternating current Coulomb blockade carbon nanotubes direct current density of states electron beam lithography excitet state frontiers in research, space and time or simply our clean room full width at half maximum ground state Landau level quantum dot quantum point contact Poly (methyl methacrylate) single electron transistor scanning force microscopy signal to noise ratio single pixel e-beam lithography

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Symbol L,W,d A αg B CΣ γ Γ D ∆N EC EF en εN G g? I kF λF `B è m? µe µN ns ν Q vF rs re ρ SN T tox V ω0

Explanation system size (length, width, diameter) vector potential gate lever arm magnetic field self-capacitance of a quantum dot hopping parameter tunnel coupling density of states single-particle level spacing constant interaction energy Fermi energy voltage noise single particle energy of the Nth level conductance effective g-factor current Fermi wavenumber Fermi wavelength magnetic length elastic mean free path effective electron mass electron mobility electrochemical potential for the Nth electron electron sheet density filling factor charge Fermi velocity in graphene interaction parameter event rate resistance ground state spin quantum number temperature oxide thickness voltage confinement strength of a harmonic potential

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Chapter 1 Introduction An exciting new material Graphene is the first truly two-dimensional material which has become accessible experimentally [1, 2]. It consists of carbon atoms arranged in a hexagonal lattice which is only one atom thick. Graphene offers a number of unique properties making it very interesting for fundamental studies and future applications. The ultra-thin material fascinates a rapidly growing scientific community and its importance was very recently high-lighted by the 2010 Nobel-price for Kostya Novoselov and Andre Geim for their groundbreaking experiments on graphene. An early exciting result was the observation of an unusual quantum Hall effect by measuring electronic transport in graphene [3, 4]. Its specialty is linked to the peculiar low energy electronic properties of graphene which takes on the same mathematical form used to describe massless relativistic particles. Other exotic properties can be understood in this analogy, like Klein tunneling [5, 6] which describes the tunneling of particles through a classically forbidden region with a 100 % success rate. Further, the electrons are remarkably mobile in graphene [7, 8] despite the fact that the electron gas is exposed to the environment, which usually provides many scattering centers. On the other hand it is much easier to interact with a current at the surface, allowing for chemical sensing [9] or modification [10, 11] and many interesting experiments with microscopy tools (see e.g. [12–14]). Despite the fact that a graphene sheet is only one atom thick, it shows also remarkable mechanical and thermal properties due to the strong in-plane carbon bonds. The atomic sheet is able to sustain huge electrical currents [1, 15] and is an excellent heat conductor [16]. Experiments using graphene as a membrane showed its high flexibility, as it can be stretched like a balloon but still withstands pressure differences of several atmospheres while being impenetrable even for very small atoms such as Helium [17, 18]. These properties make graphene a very promising candidate for nano-electronic applications and open up an exciting and highly interdisciplinary field for fundamental research.

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Chapter 1. Introduction

Quantum physics in nanostructures The impressive miniaturization of electronic circuits in the past decades, vastly increased the possibilities for research on nanostructures and allowed to discover many interesting phenomenas such as Coulomb blockade, quantum tunneling or the quantum Hall effect, which are related to the non-intuitive laws of quantum mechanics. As todays computer chips are based on very small devices in the range of few tens of nanometers these phenomenas can no longer be neglected. A significant issue is for example caused by quantum tunneling which leads to considerable leakage currents and poses severe power and heat issues. On the other side, quantum physical phenomena could be used in new device concepts with superior properties. A very active research field is based on the idea to process information using quantum bits (qubits). The possibility to entangle individual qubits allows in principle for massive parallel processing of information with applications in the simulation of quantum systems, fast factorization or search algorithms [19, 20]. Among nanosystems, quantum dots (or artificial atoms) are the most intensively studied ones. This is because they allow to study fundamental quantum effects in confined geometries with the advantage that carrier density and confinement can be tuned externally. In particular, large progress has been made in the past decade to use quantum dots as hosts for qubits based on spins [21]. Using GaAs-based double quantum dot systems, both elementary qubit operations were shown namely coherent single spin rotation [22] and a two qubit entanglement operation [23]. The main drawback in GaAs-based systems are the limited spin-coherence times due to considerable hyperfine coupling and spin-orbit interactions. Part of this limitation can be overcome by polarizing the nuclear fields, as successfully shown in Ref. [24]. Graphene nanostructures Quantum dot systems based on carbon materials and in particular graphene are interesting as both limitations for the spin coherence times are expected to be significantly reduced. This is due to (i) weak hyperfine coupling as carbon materials consist predominantly of the nuclear spin free 12 C isotope (99%) and (ii) small spinorbit interaction as the carbon nuclei is light. An issue with nano-electronic graphene devices is the absence of a band gap in this material. Such an insulating energy region allows to control current flow by gating which is at the heart of every digital transistor, and predominantly used to electrostatically define quantum dots in semiconductor heterostructures. Several mechanisms to open a band gap in graphene have been suggested. These range from chemical modification, over symmetry breaking using an electric field in bilayer graphene or a special periodic substrate, to confinement in narrow constrictions. An overview of different theoretical proposals for graphene quantum dots can be found in Ref. [25].

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In 2007, shortly before the start of this PhD project, the first transport experiments on graphene nanostructures in the form of narrow ribbons and islands were reported [26–28]. In these ribbon measurements a sizable gap could be observed, however the extent of the insulating region in energy and the ribbon-width scaling of the gap was not expected and lead to an extended debate about the underlying mechanisms in the community. Structure of this thesis The focus of this thesis lies on electron transport through graphene quantum dots. In the second chapter we will have a closer look at the electronic properties of bulk graphene and their intriguing consequences for electronic transport. The fancy and original method to obtain graphene by peeling graphite with scotch tape is introduced in chapter three in addition to methods used for characterizing graphene and the fabrication of graphene nanostructures. In chapter four we will discuss narrow constrictions and show how to fabricate graphene quantum dots. Basic measurements of Coulomb blockade and quantum confinement are introduced and demonstrated. More insight about constrictions and quantum dots is gained in chapter five, where a constriction is used as a charge detector to sense the charge on the dot. Using a dedicated device, time-resolved charge detection is possible, allowing to monitor the tunneling of single electrons in real time. In chapter six and seven we study the properties of graphene quantum dots in magnetic fields. Orbital effects are studied in a magnetic field perpendicular to the sample plane in chapter six, focusing on the crossover from holes to electrons. For the measurements presented in chapter seven, the sample has been rotated into a parallel field orientation in order to investigate spin states in graphene quantum dots via the Zeeman spin-splitting. We conclude with a short outlook of the research on graphene quantum dots.

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Chapter 2 Graphene The experiments presented in this thesis were performed on graphene, a single atomic sheet of graphite. The isolation and electronic characterization of graphene on silicon-oxide (SiO2 ) in 2004 [1] and the measurements of the anomalous quantum Hall effect in 2005 [3, 4] triggered intensive research on this new material. In this chapter the electronic properties of graphene are briefly reviewed, with an emphasis on their intriguing consequences on transport phenomena like the quantum Hall effect. For a more detailed review see Ref. [29].

2.1

Electronic structure of graphene

The graphene lattice imaged by a transmission electron microscope is shown in Fig. 2.1 (a). The carbon atoms in graphene form strong covalent bonds made of two overlapping sp2 orbitals, leading to the planar hexagonal arrangement of atoms sketched in Fig. 2.1 (b). The Bravais lattice can be constructed from a unit √ cell 3/2) containing two basis √ √ atoms A and B using the basis vectors a1 = a(1/2, ˚ 3/2) with a = 2.46 A . The vectors t = a(0, 1/ 3) t and a2 = a(−1/2, 1 2 = √ √ a(−1/2, −1/2 3) t3 = a(1/2, −1/2 3) point from a B atom to the next neighbor A atoms.

2.1.1

Band structure

The first calculation of the electronic properties of graphene dates back to 1947 when P. R. Wallace calculated the electronic dispersion of graphene [31] in order to explore the usefulness of graphite as mediator in nuclear reactors (see also Ref. [32]). Here we follow the notation used by Ando and Ihn [33, 34]. The strong covalent in-plane bonds lead to a large energy separation of the bonding and anti-bonding state (σ-bands). On the other hand, the interaction between the pz orbitals is much weaker and leads to states around the Fermi level (called π-bands) which are mainly responsible for the electrical and low-energy optical behavior. The band structure around the Fermi level can be calculated by considering

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2.1. Electronic structure of graphene

(a)

(b) Unit cell

t1 t2

(c)

A

a2

B

a1

t3

y x Figure 2.1: (a) Transmission electron microscope image of a suspended graphene sheet (image courtesy of Zettl Group, Lawrence Berkeley National Laboratory and University of California at Berkeley [30]). (b) Graphene lattice with two carbon atoms per unit cell (shaded) denoted by A and B. only the π-bands. The periodicity of the lattice allows to use the Bloch wave function ansatz 1 X ikR e φtot (r − R) ψk (r) = √ N R

(2.1)

for the solution of Schrödinger’s equation. Here R = n1 a1 + n2 a2 (n1,2 integers) describes the location and N the number of lattice sites in the crystal whereas φtot (r) is the wave function for one unit cell. The latter is taken to be a linear combination of the pz wave functions φ(r) and φ(r − t1 ) of the two atoms in the unit cell [see Fig. 2.1 (b)], i.e., φtot (r) = A · φ(r − t1 ) + B · φ(r).

(2.2)

The Hamiltonian for the crystal lattice is given by the atomic potential of all the carbon atoms X p2 H= + (V0 (r − R − t1 ) + V0 (r − R)) , (2.3) 2m R where V0 (r) denotes the potential of a single atom. Considering only on-site and nearest-neighbor atoms and neglecting the small overlap integral between two pz orbital functions, the Hamiltonian is transformed into the following eigenvalue problem [34] γ

0 α? (k) α(k) 0

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A B

=E

A B

(2.4)

Chapter 2. Graphene

(a)

(b)

E ky

kx E qy

qx

ky

(c)

K

K‘ b2 K

b1

ī

kx

K‘

Figure 2.2: (a) Electronic dispersion of graphene. The π-band (red) and the π ∗ -band (blue) touch each other at singularity points (K-Points). (b) Zoom of the dispersion at the singularity points. The dispersion has a conical form with an approximately linear slope (see section 2.1.2). (c) Reciprocal lattice with first Brillouin zone (dashed). Only two of the six singularity points are independent (denoted by K√and K’). The reciprocal lattice is constructed by √ the vectors b1 = 2π/a(1, 1/ 3) and b2 = 2π/a(−1, 1/ 3). The coordinates of √ the K points are given by 2π/a(±1/3, ±1/ 3) and 2π/a(±2/3, 0). with the nearest neighbor hopping matrix element γ1 := hφA |H|φB i and α(k) = 1 + eik(t2 −t1 ) + eik(t3 −t1 ) . The energy eigenvalues are then given by q √ E(k) = ±γ1 |α(k)| = ±γ1 1 + 4 cos2 [kx a/2] + 4 cos[kx a/2] cos[ 3ky a/2]. (2.5) The dispersion relation is plotted in Fig. 2.2 (a) with the hopping parameter γ1 = −3.033 eV [35]. In this plot a non zero pz -orbital overlap integral σ = 0.129 eV is used, which leads to an electron-hole asymmetry [35]. Alternatively the slight mismatch to ab-initio calculations [36] is compensated by introducing second and third-nearest neighbor hopping terms γ2 ∼ 0.2γ1 and γ3 [29, 36, 37]. The lower red part of the curve is the (valence) π-band and the upper blue part forms the (conduction) π ∗ -band. The Fermi-surface cuts exactly through the degeneracy points because the two π-electrons (per unit cell) completely fill the π-band. As the degen-

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eracy points lie on the corner of the first Brillouin-zone, only two points denoted by K and K 0 are inequivalent as depicted in Fig. 2.2 (b).

2.1.2

Linear expansion around K and Dirac equation

Because the energy at the Γ point is around 10 eV above the Fermi-level, only the energy range around the K points is relevant for electronic transport [see Fig. 2.2 (b)]. In the vicinity of the K-points, a linear expansion of Eq. 2.4 with k = K + q leads to [33, 34] φA φA 0 qx − iqy = E(q) . (2.6) vF ~ φB φB qx + iqy 0 {z } | HK

√ with vF = 3γ1 a/2~ ≈ 106 m/s the constant Fermi velocity. The effective Hamiltonian can be written in the form of a Dirac-Weyl Hamiltonian HK = vF ~σ · q (2.7) using the 2D-Pauli-spin-matrices σ = (σx , σy ) [38, 39]. This Hamiltonian is used to describe massless Dirac Fermions in two dimensions and can be applied to describe transport in graphene [3, 4, 40, 41]. The spin of the relativistic particle is mimicked in graphene by the two component vector (φA , φB ) which is therefore also denoted as pseudo-spin vector. The massless character is found in the energy eigenvalues given by E(q) = ±vF ~|q| describing a linear dispersion relation with zero mass. Note that around K0 the Hamiltonian is given by HK0 = vF ~σ ∗ · q. The linear dispersion alone is not sufficient to understand the intriguing properties of graphene that are further linked to the pseudo-spin or the composition of the wave function from the two sublattices. Therefore it is worth to have a look at the eigenfunctions of the Dirac equation. The plane wave solution is given by [42] 1 ±eiϕ/2 φA (r) eik·r (2.8) hr|φi = =√ iϕ/2 φB (r) e 2 with the angle ϕ = arg(q). It is an interesting property that the direction of the pseudo-spin (φA , φB ) is coupled to the direction of the momentum q. For conduction band states at K the two vectors are parallel whereas their direction is anti-parallel for valence band states. This behavior is known as right-handed (positive) helicity or left-handed (negative) helicity [43]. Sometimes, also the more abstract term chirality is used which is identical with helicity only for massless particles. At K0 the helicity changes sign for the conduction and valence bands.

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Chapter 2. Graphene

2.1.3

Implications of the chirality in graphene

The chirality has important consequences for the scattering mechanisms in graphene. The interference amplitude of two plane waves with the direction determined by ϕ and ϕ0 is proportional to | hφϕ0 |φϕ i | ∝ cos

ϕ − ϕ0 . 2

(2.9)

Suppression of backscattering A consequence from the above equation is the suppression of direct backscattering (ϕ − ϕ0 = π). This can also be explained in terms of orbitals where a backscattering event implies a complete change of the wave function composition |φϕ+π i = (i, −i) |φϕ i as illustrated in Fig. 2.3 (a) by a bonding and an anti-bonding state. Such pseudo-spin-flip processes can only occur in the presence of a short range potential acting differently on the lattice sites A and B. Hence backscattering due to long range disorder is completely suppressed [44] giving rise to potentially high carrier mobilities (> 1000 000 cm2 /Vs) even at room temperature [45].

(a)

(b)

anti-bonding φA φB

K’

bonding

E k

φA φB

(c)

E

E

k

EF

k

EF

K EF

E

_

+

_ E x

x

Figure 2.3: (a) Momentum is related to the pseudo-spin i.e. the composition of the total wave function from the two sublattices. (b) Blockade of electron transport through a npn-barrier in a conventional gapped semiconductor. (c) In graphene transport through a npn-Barrier is possible due to Klein tunneling.

Klein tunneling Another consequence of the pseudo-spin is perfect transmission through a barrier. This is known as Klein tunneling in analogy to the well known relativistic effect [46]. In Fig. 2.3 (b,c) a rectangular potential barrier is shown for a conventional semiconductor and graphene. In the case of a gapped semiconductor, electron transport can only occur by tunneling through the barrier region. In graphene the behaviour is different. As there is no gap and due to pseudo-spin conservation, perfect transmission for angles at normal incidence occurs [46]. This behavior is illustrated in Fig. 2.3 (c). An electron coming from the left can be scattered into a hole moving

8


in opposite direction as they share the same pseudo-spin (purple). Experimental evidence of Klein tunneling in graphene has been observed by inducing a potential step with a narrow top gate (width comparable to the mean free path of the charge carriers) [5, 6]. Landau levels at half integer fillings The intuitive picture to explain the formation of Landau levels - a peaked density of states at high magnetic field - involves a semi-classical picture where electrons move on cyclotron orbits induced by the magnetic field. The density is peaked because only discrete particle energies are allowed at which the wavelength associated with the electron leads to positive interference while revolving. The phase accumulated by a wave propagating around a circle consists of a dynamic phase ∆φd and a Aharonov Bohm phase ∆φab [34]. In graphene there is an additional phase component due to the pseudo-spin. From Eq. (2.8) we see that by completing a circle the charge carrier acquires a phase shift of π, as φϕ+2π = −φϕ . This additional Berry phase [47] translates into a shift of the allowed frequencies and hence also shifts the Landau level energies by half an integer. The exact solution of the Dirac equation is presented in the next section 2.1.4.

2.1.4

Graphene in perpendicular magnetic fields

In this section the Landau level spectrum in graphene is derived, as first calculated in Ref. [48]. The influence of a magnetic field on the kinetics of a system can be incorporated by substituting q = −i∇ with −i∇ + (e/h)A [34]. Eq. (2.7) to e (2.10) HK = vF ~σ · −i∇ + A . ~ The energies for the Landau levels are q D Ei = sgn(n) 2e~vF2 B|n|,

n ∈ Z.

(2.11)

This solution includes the contributions from the different sublattices that are already combined. A more detailed calculation can be found in Ref. [34]. The Landau level degeneracy is, as in other systems, given by nL = eB/h. In graphene all Landau level states can be occupied by four electrons as we have a valley and a spin contribution. Two main differences p are observed when comparing the Landau level energies D in graphene En ∝ B(n + 1/2) with those in conventional (effective mass) 2D semiconductors EnS ∝ B(n + 1/2). First, the square root dependence of the Landau level energies on B and n in √ graphene is a manifestation of the linear energy dispersion (E D (ns ) ∝ ns ) In

9

Chapter 2. Graphene

(a)

B=5T

DOS (a.u.)

graphene

-200

-150

-100

(b)

50

0 E (meV)

50

100

150

200

20

30

40

DOS (a.u.)

m* = 0.067m0, no band gap

-40

-30

-20

10

0 E (meV)

10

Figure 2.4: (a) Landau levels in graphene at B = 5 T compared to Landau levels in a 2D-effective mass semiconductor (m∗ = 0.067m0 ). The broadening of the levels is caused by disorder and modeled with a Lorentzian with a full width at half maximum (FWHM) of 2meV. conventional 2D semiconductors, however, the density of states is independent of energy and hence E S (ns ) ∝ ns (see also Ref. [49]). Second, the half integer shift (n+1/2) arises from the pseudo-spin (two sublattice contributions) which induces an additional Berry phase as illustrated by the semiclassical argument in the foregoing section. The Landau levels can be probed by measuring the quantum Hall effect (see section 2.2.3).

2.1.5

Spin-orbit interaction in graphene

In general, the microscopic spin-orbit interaction is determined by the non-relativistic limit (momentum of particle small compared to mass) of the Dirac equation [50] HSO =

1 (∇V × p) · S . 2m2e c2

(2.12)

Usually the change of the potential ∇V is largest near the atomic nuclei and can be approximated by the intra-atomic contribution [51, 52] HSO = ∆SO L · S .

(2.13)

The strong dependence of the intra-atomic spin-orbit coupling constant ∆SO on the atomic number Z (∆SO ∝ Z 4 ) leads to a comparatively weak spin-orbit interaction for carbon materials where Z = 6 [53]. The strength of the intrinsic coupling is estimated as ∆SO = 6 − 12 meV [52, 54].

10

2.2. Transport properties of bulk graphene

Spin-orbit coupling in graphene is given by an intrinsic (Dresselhaus-like [55]) part ∆int determined from the symmetry properties of the lattice. In an ideal graphene sheet there is no first order coupling of pz orbitals [56]. Hence, only second order contributions account for the intrinsic part ∆int ∝ ∆2SO . Numerical estimates range from ∆int ∼ 1 µeV in Refs. [52, 54] to ∆int ∼ 24 µeV in a recent fully first principle calculation considering also the influence of d-orbitals [57]. The discrepancy to the larger intrinsic spin-orbit coupling found in graphite (order of 100 µeV) is attributed to coupling between the individual layers [52]. The inversion symmetry of the plane is broken by bending the sheet, applying an electric field perpendicular to the plane or by interaction with a substrate [58]. The broken symmetry is reflected by the introduction of a Rashba term [59] ∆R ∝ ∆SO . Numerical estimates give field induced contributions of ∆R,E ∼ 2 µeV for E = 50 V/300 nm [57] (∆R,E ∼ 22 µeV in Ref. [54]). The rippling of graphene produces inhomogeneous spin-orbit contributions which might compensate each other. Locally, contributions of ∆R,curv ∼ 15 µeV for a typical curvature radius rrip = 50 nm of a ripple are expected [52]. In general, the curvature contribution can be estimated by ∆R,curv = 0.8 meV/rrip (rrip in [nm]) [52, 53].

2.2

Transport properties of bulk graphene

The intruiging electronic properties around the Dirac point introduced in the previous part of this chapter can be experimentally probed by electronic transport measurements as described in the following.

2.2.1

Field effect in graphene

We begin this section by introducing the field effect which is the basic tool for transport measurements in graphene [1]. Measuring the conductance while varying the Fermi energy via the field effect, allows to obtain information about the band structure and the electronic quality of the material. A Hall-bar geometry as shown in Fig. 2.5 (a) is typically used to measure the basic material properties. By changing the potential applied to a global back gate (separated by the tox = 295 nm thick oxide), the charge carrier density is capacitively tuned by ns (Vbg ) = αVbg where α = 0 ox /(etox ) = 7.4 × 1010 cm−2 V−1 . Hence the density can be tuned via the back gate from holes to electrons manifesting itself in a decrease of the conductance while reducing the hole density until reaching a minimum at the charge neutrality point and increasing again with increasing electron density [see Fig. 2.5 (b)]. On both sides, the conductance varies linearly with the carrier density and can be modeled using Drude’s formula σ = ns eµe [60] for diffusive systems. The mobility µe,h ≈ 5000 cm2 /Vs for both carrier types independent of √ e πns ≈ 60 nm the charge carrier density translates to a mean free path of è = ~µ |e| 12 −2 at |ns | = 10 cm well within the diffusive regime of the µm-sized sample.

11

Chapter 2. Graphene

(b) ρxy d

s

SiO2

conductivity σxx (4e2/h)

14

ρxx

charge carrier density (1012 cm-2) -2 -1 0 1 2

13

D(E)

(a)

10 8 6

EF

E

4 2 0 -30 -20 -10 0 10 20 30 back gate voltage (V)

Si++ back gate

40

50

Figure 2.5: (a) Schematic of a contacted graphene Hall-bar device separated from the conductive silicon back gate by an insulating SiO2 layer. Longitudinal σxx and Hall conductivity σxy are measured in a four-probe geometry by driving a current from source (s) to drain (d). (b) Field effect i.e. conductivity as a function of back gate voltage Vbg at a fixed source-drain current Isd = 20 nA (T = 2 K). By varying the carrier density via the back gate charge transport is tuned from hole- to electron-like transport by passing the so-called chargeneutrality (or Dirac-) point at minimum conductivity. The charge-neutrality point is offset from Vbg = 0 due to residual doping of the graphene. In the inset the smearing of the carrier density around the charge neutrality point due to electron-hole puddles is sketched. A more involved model includes long and short range scatterers in a self-consistent Boltzmann equation for diffusive transport[61–64]: σ −1 = (ns eµc + σmin ) + ρs .

(2.14)

In addition to the Drude formula with the constant mobility µc determined by long range scatterers (charged-impurity Coulomb scatterers), σmin describes the residual conductivity at the charge neutrality point and ρs is a density independent contribution to the resistivity due to short range scattering. The resistivity contribution from short range scattering is relatively weak (ρs ∼ 100 Ω for graphene on SiO2 [64] and boron nitride [65]) but gets important at high densities giving rise to nonlinear dependence of the conductance on the density. Charged impurities on graphene can be greatly reduced in suspended devices [45, 66] or by using hexagonal boron-nitride as a substrate [65]. In combination with the removal of residues by annealing [15], mobilities up to 2000 000 cm2 /Vs have been achieved [45, 66].

12

2.2. Transport properties of bulk graphene

14 12 10 B=8T

6 4 2 -3 -2 -1 0 1 2 3 12 -2 charge carrier density (10 cm )

2.2.2

8

Uxx(k:)

ıxy (e2/h)

18 14 10 6 2 -2 -6 -10 -14 -18

0

Figure 2.6: Longitudinal resistivity ρxx and Hall conductivity σxy as a function of charge carrier density (∝ Vbg ) at B = 8 T (Isd = 100 nA, T = 2 K). Hall plateaus at half integer values are visible - characteristic for the quantum Hall effect in graphene. The plateaus are aligned with minima in the longitudinal resistance. The calculated filling factors are denoted by the vertical dashed lines.

Minimum conductivity and electron-hole puddles

At the charge neutrality point the conductivity stays finite with a typical value of σmin = 4e2 /h and a weak temperature dependence in disordered samples [28]. The weak temperature dependence is attributed to inhomogeneous impurity doping of the graphene giving rise to electron-hole puddles [12] around the charge neutrality point and increasing the average density of states as sketched in √ the inset of Fig. 2.5 (b). Hence, the temperature effect is weak up to kB Teh = ~vF πδns with Teh = 376/120 K for δns = 2.3 · 1011 cm−2 [12]. However, a significant temperature dependence of σmin is seen in suspended devices or for graphene on boron-nitride as a result of the reduced disorder density δns ≈ 1 · 109 cm−2 [45, 65]. The value of the minimum conductivity is theoretically expected to be 4e2 /πh for an ideal graphene sheet [67, 68]. This value differs by a factor of 1/π from the commonly observed 4e2 /h of graphene on SiO2 . This difference is due to several reasons: First, it has been noted by Tworzydlo and coworkers [68] that the minimum conductivity depends on the aspect ratio of the sample if W/L < 3. For a typical Hall bar geometry with W/L ≈ 1/2 σmin ≈ 4e2 /h is expected. Second, the quality of the material is important although an increase of charged impuritiers does not affect the minimum conductivity [8]. This observation can be explained by considering, on the one hand, electron-hole puddles around the charge neutrality point getting larger with higher impurity concentrations and therefore increasing the conductance. On the other hand the charged impurities act also as scattering centers giving rise to a reduction of the conductance. This ambiguous influence of disorder might also explain that high mobility samples show an even higher minimum conductivity such as 6 e2 /h in Ref. [65].

2.2.3

Quantum Hall effect in graphene

As mentioned above, a clear manifestation of the pseudo-spin is observed in the quantum Hall-effect (QHE) in graphene [3, 4, 69].

13

Chapter 2. Graphene

A typical measurement is shown in Fig. 2.6 where the Hall-conductivity σxy (black curve) and the longitudinal resistivity ρxx (red curve) are measured for varying charge carrier density at a magnetic field of B = 8 T. The period of the quantum Hall plateaus is related to the Landau levels. Whenever the Fermi energy is in between two Landau levels the current is carried via topologically protected edge states giving rise to vanishing longitudinal resistance and Hall-plateaus at quantized conductance values. The Hall resistance is determined by the number of allowed modes contributing each a conductance quantum e2 /h. The special Landau level energies in graphene (see section 2.1.4) lead to plateaus at σxy = ±4e2 /h(n + 1/2). The plateaus are shifted by half an integer with respect to the standard QHE sequence and have a four-fold degeneracy due to spin and valley degrees of freedom. The quantum Hall effect in graphene is particularly robust and it is possible to observe the effect even at room temperature in high magnetic fields [70]. This is mostly due to the large energy spacing between the zero energy and the first Landau level [compare Fig. 2.4 (a,b)]. This argument provides only part of the explanation. In conventional systems the quantum Hall effect could not be observed above 30 K, as localization is already completely destroyed at this temperature. Hence, the mechanism behind the broadening of Landau levels is different in graphene and goes beyond the understanding of traditional quantum Hall systems (see also Refs. [71, 72]).

14

Chapter 3 Sample fabrication Graphene is the basic material for other carbon allotropes such as the three-dimensionally stacked graphite, rolled up as one-dimensional carbon nanotubes (CNTs) [35, 73] or the zero-dimensional fullerenes including the famous C60 molecule [74]. Nonetheless, the two-dimensional form was the last to be controllably isolated. Starting in the 1970s, graphene has been grown epitaxially on top of other materials [75]. The interaction with the underlying material, however, significantly altered the electronic structure of these epitaxial graphenes. An alternative approach consisted of mechanical exfoliation from graphite by tailoring a graphite stack using an scanning force microscope (SFM) tip [76] or by writing with a so-called nanopencil consisting of a tiny piece of graphite at the end of an SFM cantilever [77]. Despite the sophistication of these methods, the breakthrough came by simply peeling graphite with adhesive tape to obtain individual sheets [1]. Although significant progress has been made in the chemical synthesis of graphene [78–80], the so-called ”scotch-tape technique” still provides the best electronic quality for transport measurements. This technique is used to obtain the graphene flakes investigated in this thesis.

3.1

Graphene deposition and characterization

As a substrate, highly p-doped silicon wafers with a thermally grown 295 nm thick oxide on top are used (supplier: Nova Electronics). Optical photolithography followed by metal evaporation and lift-off are used to define markers and bond pads on top of the silicon oxide. The markers are used to locate the graphene flakes and are necessary for the alignment during electron beam (e-beam) lithography. A detailed description of the process steps including the different parameters is attached as a processing sheet in Appendix D. This first step is made prior to the graphene deposition to reduce contaminations and to allow for wafer based processing, which is much more time efficient despite the lower graphene deposition yield observed when using prestructured samples. The processed wafers are diced into square chips of 7 mm length and are thor-

15

Chapter 3. Sample fabrication

ȝP

Figure 3.1: (a) Optical microscope picture of a graphene flake on a Si-chip covered with 295 nm SiO2 . The arrow points to the single-layer graphene part. This flake has been used for the fabrication of the quantum dot (labled sp2, see Appendix B) illustrated in this chapter and mainly studied in chapters 6,7.

oughly cleaned prior to the graphene deposition by ultrasonication in hot acetone and exposure to an oxygen plasma. Both sonication and ashing of the samples are no longer possible after the graphene deposition as the atomic sheets are destroyed. Therefore it is beneficial to minimize the amount of processing steps to reduce contaminations after the graphene deposition. Graphene flakes are obtained by peeling of natural graphite flakes (supplier: NGS Naturgraphit) with a blue sticky tape normally used as a suspension during wafer sawing. The advantage of the blue tape compared to the use of scotch tape is the reduced amount of glue residues on the chip and the flakes. However, more and larger flakes are obtained with the scotch tape, which is attributed to the superior adhesion. After several peeling steps with the tape, a preprocessed silicon chip is pressed onto the tape to transfer the flakes. Using this technique, many different graphitic flakes with a large variety in the number of layers are obtained. In order to find graphene flakes, it is crucial to have a fast detection technique at hand. Novoselov et al. [1] solved this issue by using an approximately 300 nm thick silicon oxide layer which maximizes the contrast between graphene and the background and enables to see single-layer graphene with a light microscope [1, 81]. An example of a graphene flake as observed with an optical microscope is shown in Fig. 3.1 (a). The single-layer part is marked with an arrow.

3.1.1

Characterization with a scanning force microscope

Quality and properties of the flakes are investigated using scanning force microscopy and Raman spectroscopy. From the SFM measurement shown in Fig. 3.2 (a) the exact shape and impurity concentration is analyzed by measuring the surface roughness. In this example the root mean square of the measured height variations on the graphene is Rq,gra = 0.22 nm in the marked region, comparable to the roughness of the initial SiO2 surface roughness (Rq,ox ≈ 0.2 nm) reported also elsewhere [65, 82]. From the cross-section along the dashed line it is possible to distinguish between different numbers of layers (see inset, averaged over 1 µm). However, the height of a single-layer varies between the interlayer distance of graphite (≈ 0.34 nm) and more than 1 nm (here ≈ 0.6 nm). The additional height might be related to wa-

16

3.1. Graphene deposition and characterization

1 0.63 nm

0 0

4 ȝP

2

Rq = 0.22 nm

Intensity (CCD counts)

2 nm

(a)

1200

2D

(b)

1000

ȝP

800

D (absent)

600

G

400 200 1200

1600 2000 2400 2800 Raman shift (cm-1)

Figure 3.2: (a) Atomic force microscope image of the flake presented in Fig. 3.1 after cleaning in hot acetone. Only few contaminations are visible on the flake manifesting in a low surface roughness of Rq,gra = 0.22 nm (measured in the 1 µm2 square). The surrounding SiO2 has Rq,ox ≥ 0.25 nm. (b) Raman spectrum taken in the center of the flake in (a). The sharp 2D-peak being higher than the G peak is characteristic for single-layer graphene. No D-peak located around 1350 cm−1 is visible, whereas the G-peak position is located around 1580 cm−1 and the position of the 2D peak around 2670 cm−1 .

ter molecules underneath the graphene flake [83] or other processes related to the surface chemistry of the substrate.

3.1.2

Raman imaging

A powerful tool to unambiguously determine the single or double layer nature of a graphene flake is Raman spectroscopy [84–86]. Fig. 3.2 (b) shows a Raman spectrum recorded at the center of the single-layer part of the flake. The two main signatures are the G peak at λ ≈ 1580 cm−1 and the 2D-peak at λ ≈ 2670 cm−1 . The G-peak is caused by scattering of the incoming laser light due to in-plane bond stretching phonons with zero momentum. The 2D peak is caused by scattering of the light with a breathing mode of the lattice. The process involves two phonons with momentum close to K. This process is highly sensitive to the phonon dispersion, resulting in one allowed phonon frequency (fit with a single Lorentzian) for monolayer graphene and four allowed transitions (fit with four Lorentians) for bilayer graphene. For three and more layers, the allowed transitions increase and it is difficult to identify the individual contributions with Raman spectroscopy. For multilayer graphene, measurements of the optical contrast are advantageous although the obtained values depend on many parameters which require an exact calibration. The D-line around 1350 cm −1 (with a 532 nm laser) is not visible due to the absence of intervalley scattering in defect free graphene [85].

17

Chapter 3. Sample fabrication

(a)

(b) 2.0 1.5

\ȝP

ȝP

z QP 10 etched (SiO2

1.0

5 0.5

graphene 0

0 0

0.5

1.0 1.5 [ȝP

ȝP

2.0

ȝP

Figure 3.3: (a) SFM image of an RIE etched graphene flake. (b) Optical image of the contacted device (sp2 in Appendix B) with an optical magnification of the square on the top left and a further zoom recorded with the SFM.

3.2

Fabrication of graphene nanostructures

Selected flakes are further processed into devices by etching the desired shape using reactive ion etching (RIE) and evaporation of electrical contacts. For both steps Polymethylmethacrylate (PMMA) masks are structured using e-beam lithography. An example for an etched graphene nanodevice is shown in the SFM-image in Fig. 3.3 (a). The mask for the smallest features is patterned by writing single-pixel (sp) lines with the e-beam. With this technique, 20 nm thin cuts in the graphene were achieved using a 45 nm thick layer of resist. For the etching, it is important to limit the amount of PMMA irradiation as it gets cross-linked under the ion bombardment. This makes the resist increasingly difficult to be removed afterwards. Therefore, the RIE-chamber is operated with as low electrical power as possible, just enough to ignite the plasma. Also the etch time is rather short with only 10 s when using the thin 45 nm resist. The flakes are contacted by deposition of 2 nm chromium (Cr) and 50 nm gold (Au) on the e-beam structured mask and subsequent lift-off of the metal covered PMMA mask in hot acetone. The Cr-layer is needed to provide good adhesion of the gold electrodes. The finished sample is shown in Fig. 3.3 (b) with two magnifications of the device on the left (optical- and SFM pictures). The order of the two process steps can also be interchanged and there is a trade-off between the yield of the etching step and the requirements on the contact resistances.

18

Chapter 4 Graphene quantum dots The term quantum dot stands for the confinement of electrons in all three spatial dimensions which leads to quantization of the energy spectrum. This is in analogy to atoms, where the positive charge of the nucleus traps the electrons. As a consequence, the underlying physics is very similar, allowing e.g. for the observation of shell filling in symmetric quantum dots [87]. Quantum dots are therefore also known as artificial atoms [88]. In contrast to real atoms, it is much easier to change system parameters and measure quantum dots as the size is typically 1000 times larger. This can for example be done by fabricating electrical gates and source/drain contacts to the dot in order to measure electronic transport, as it is done in the experiments reported here. However, the enlargement of the system comes with a reduction of the quantization energies (order of 1 meV), requiring cryogenic temperatures to experimentally resolve the energies. In this chapter, the basic properties of electron transport in quantum dots are discussed and illustrated with measurements. After a general introduction into single-electron transport (section 4.1), narrow graphene constrictions (or nanoribbons) are investigated in section 4.2. Graphene constrictions form the basic building block for carrier confinement in graphene nanostructures and are used as tunneling barriers for quantum dots. In section 4.3, basic measurements on large graphene island are presented, complemented by measurements on graphene quantum dots, where the atom-like energy quantization becomes visible (section 4.4). Further, the characteristic energy scales as a function of island diameter are summarized. The properties of graphene quantum dots in magnetic fields are discussed in chapter 6 and 7.

4.1

Single-electron transport

A quantum dot device for electronic transport consists of a small island that is weakly coupled to two conducting leads (source and drain) over tunneling barriers. A schematic is shown in Fig. 4.1 (a). The potential of the island can be tuned over a

19

Chapter 4. Graphene quantum dots

capacitively coupled gate. Over the past decades, quantum dot systems have been extensively studied in many different materials as reviewed in Refs. [87, 89, 90]. For the understanding of single-electron transport through such artificial atoms, it is helpful to keep in mind the analogy to natural atoms: These are usually investigated by optical spectroscopy (see also Ref. [88]). The minimum energy needed to remove an electron is the ionization potential and the maximum energy of photons emitted is the electron affinity. In quantum dots, electrons have to tunnel on and off the island for a current to flow. Similar to real atoms, for an electron to tunnel onto the dot an addition Energy Eadd has to be overcome. This addition energy arises due to quantization of charge and level spacing energy. The simplest model taking into account both quantizations is the constant interaction model, based on the assumptions of (i) constant capacitances and (ii) a single-particle spectrum N (B) which is unaffected by interactions [87, 91].1 From the total ground state energy of an island with N electrons E(N ), the electrochemical potential µN = E(N ) − E(N − 1) is given in this model as Cg 1 −e Vg − Vg(0) + N (B), (4.1) µN = Ec N − 2 CΣ with Cg the gate capacitance to the island, CΣ the total capacitance of the island and (0) Vg the gate voltage at which the island contains no electrons. The first two terms describe the electrostatic potential determined by the charging energy Ec = e2 /CΣ and a term depending on the gate voltage. The influence of the gate voltage is expressed as a lever arm αg = Cg /CΣ relating the gate voltage to the induced potential on the island. The last term in Eq. (4.1) is the single-particle energy defined by the confinement potential induced by geometry and magnetic field B. Within this model, the addition energy Eadd (N ) = µN+1 − µN is Eadd (N ) = Ec + ∆N .

(4.2)

The term ∆N = N +1 (B) − N (B) is the N-electron energy-level spacing, or short ∆ by assuming a constant level spacing.

4.1.1

Charge quantization and Coulomb blockade

For an electron to tunnel onto the island, the chemical potential of the leads (here µs ≈ µd ) has to be smaller or equal to the energy needed to load an additional electron onto the dot (µs,d < µN+1 ). If this is not the case the current is blocked as sketched in the schematic energy landscape in Fig. 4.1 (c). This condition is referred to as Coulomb blockade [92]. This blockade can be lifted by lowering the chemical potential of the dot with a positive voltage at the gate [Fig. 4.1 (d)] allowing a current to flow. 1

This model is also termed capacitance model whereas the constant interaction model describes the more general result obtained over the Hartree approximation which has the same form as Eq. (4.1) with the only difference of having (N − 1) instead of (N − 1/2) in the charging energy term and that Ec , αg are calculated self consistently [34].

20

4.1. Single-electron transport

source

dot

drain

(b) Isd (pA)

(a)

Vbias

gate Vg

Isd

25 20 15 10

(c) ȝs

(d)

ȝN+1 Eadd

ȝd ȝN+1

ȝN

5 0 -0.8 -0.6 -0.4

-0.2 0 Vg (V)

0.2

0.4 0.6

Figure 4.1: (a) Schematic picture of a quantum dot. The dot is tunnel-coupled to source and drain over two barriers. Applying a voltage to the gate shifts the chemical potential of the dot with respect to the leads. (b) Source-drain current as a function of gate voltage. Between two peaks the current is suppressed due to Coulomb blockade (c). (d) If the chemical potentials of dot and leads are aligned, the blockade is lifted and current can flow.

A typical measurement is shown in Fig. 4.1 (b) where the source-drain current Isd through the quantum dot is plotted as a function of the gate voltage Vg at low bias voltage. Sharp peaks appear whenever the chemical potential of the dot is aligned with the chemical potentials in the leads [Fig. 4.1 (d)]. The charging energy and the gate lever arm can be conveniently measured by varying the source-drain bias Vb while changing the chemical potential of the island. Fig. 4.2 (a) shows a schematic plot of the current as a function of bias and gate voltage. Starting from the current peaks at zero bias, the blocked regions shrink with higher bias voltage Vb giving rise to so-called Coulomb diamonds. This effect arises because a finite bias opens a window where transport is allowed [Fig. 4.2 (c-d)]. At the tip of the diamond the bias window matches the addition energy eVb = Eadd and the blockade is completely lifted. The gate lever arm is obtained from the spacing of two Coulomb peaks ∆Vg as αg = Eadd /(|e|∆Vg ). For the Coulomb blockade effect to be visible in transport, the dot has to be sufficiently small such that the charging energy is large compared to the thermal energy Ec kB T and the conductance through the tunneling barriers has to be low enough Gt < e2 /h in order to give rise to clear charge quantization. The condition is obtained from Heisenberg’s uncertainty relation ∆Ec ∆t ∼ ∆Ec CΣ /Gt > h with ∆t the time needed to charge the dot and Gt the conductance of the tunneling barrier [34]. For large artificial atoms the level spacing ∆ is small compared to the Coulomb repulsion of the electrons on the island (Eadd ≈ Ec ). In this case the Coulomb peak spacing is roughly constant and only determined over the charging energy. Because the current is switched on and off by charging the island with less than one electron, such artificial atoms are called single electron transistors.

21


Bias voltage Vb

(a)

(b)

(c)

(c) (b) N-1

(d) N

ȝN+1

Eadd/e

ȝN+1

Vg

ȝs

ȝN ȝN-1

ȝN ȝd

(d)

|e|Vb

ȝs

ȝd

ȝs

ȝN ȝd

ȝN

ȝd

ȝN ȝs

Figure 4.2: (a) Schematic Coulomb blockade diamonds. (b) Within the diamond-shaped region, the current is blocked due to Coulomb blockade. (c) Increasing the gate voltage brings the dot chemical potential into resonance with the source and a current can flow. (d) The current can flow as long as the dot level is within the window opened by the bias and ends when µN = µd . The addition energy is equal to the bias voltage at the tip of the diamond.

4.2

Graphene constrictions

The fabrication of quantum dots in graphene is not straightforward, as it is difficult to controllably confine carriers in the absence of a band gap (see also Klein tunneling in section 2.1.3). The formation of a band gap in graphene is of wide interest as the controlled switching between a conductive and an insulating state is at the heart of electronic applications such as digital transistors or sensors.

4.2.1

Opening a band gap in graphene constrictions

In order to open a band gap in graphene, it has been suggested to cut graphene into narrow ribbons in analogy to the rolled up carbon nanotubes (CNTs). In nanotubes, a band gap is formed depending on the orientation of the rolled up graphene stripe and its width (diameter) [35]. Note that contrary to carbon nanotubes, the nanoribbons are named as armchair or zigzag according to the edge along the direction of the ribbon [Fig. 4.3 (a)] whereas the nanortubes are denoted by the structure of their circumference. A first approximation of the band structure can be readily obtained from the energy dispersion of bulk graphene. The bulk dispersion is cut along the momenta allowed when applying periodic boundary conditions and the slices are projection onto the k-direction pointing along the direction of the tube or ribbon [35]. The dispersion is metallic if the slices of the dispersion include one K-point (see Fig. 4.3 (b) for a metallic zigzag ribbon) and semiconducting if not. A coarse estimation for the size of the band gap (if there is one) is then given by twice the sublattice spacing (λF = 2W ) ∆Econ = 2~vF ∆kF = 2π~vF /W.

(4.3)

For a W = 45 nm nanoribbon the gap is then estimated as ∆Econ ∼ 90 meV. A similar result is obtained from tight binding calulations [94–96] or analytically using the Dirac equation [97]. The subband energies for three distinct cases are

22

4.2. Graphene constrictions

(b)

(c)

A B

ī

(d) ac N=4

3

K

K‘

E/J1

armchair edge

(a)

(e) ac N=5

zz N=5

0

ky zigzag edge

-3

kx

0

k/a

S 0

k/a

S 0

k/a

S

Figure 4.3: (a) Lattice of a zigzag (armchair) graphene nanoribbon by extension in x- (y-)direction. (b) The confinement restricts the allowed k-states perpendicular to the ribbon axis (here for a zigzag ribbon). Tight binding calculations of nanoribbon subbands for (c) N = 3m − 2 armchair (ac) nanoribbon, (d) (b) metallic N = 3m − 1 armchair nanoribbon (m = 2) and (e) zigzag (zz) nanoribbon with N = 5 the number of zigzag chains/dimer lines in the armchair ribbon (Figs. (c-e) courtesy of K. Wakabayashi, see also Ref. [93]). displayed in Fig. 4.3 (c-e). The subbands in (c) originate from nanoribbons with armchair edges with N = 3m − 2 dimers and show a bandgap, whereas for an armchair nanoribbon with N = 3m − 1 dimers the K-points are intersected and the dispersion is metallic. Zigzag nanoribbons always show metallic behavior as the K points are intersected already for zero perpendicular momentum (Fig. 4.3 (d)). However, the flat band around the Fermi energy in (e) cannot be explained from cutting the graphene dispersion and a closer look at the boundary conditions is needed. For armchair ribbons, the boundary conditions at the edges require the wave functions to vanish at the edge on both sublattices in the armchair case. In the zigzag case, the nearest-neighbor boundary condition is satisfied even if the wave function vanishes only on one sublattice [sublattice B on lower border in Fig. 4.3 (a)] and is always opposite on both edges. This gives rise to the flat bands in the vicinity of the Fermi level for 2π/3 ≤ |k/a| ≤ π and this state is bound to one of the two edges occupying opposite sublattices [94, 95, 97]. More involved ab initio calculations [98–101] predict a band gap in all nanoribbons. In the N = 3M − 1 armchair case, an electronic gap opens due to contraction of bonds at the edges and by accounting for next nearest neighbor hopping. The gap is predicted to be of the order of 10 meV for a ribbon width of 20 nm assuming 3.5% reduction of the bondlength at the edges with a 1/W scaling [98]. For the other two configurations the gap can be approximated by ∆Econ = 2πvF ~/(3W ) [100]. In zigzag ribbons, the large density of states at the Fermi level results in a magnetic ordering already for small on-site repulsion [94, 96, 99]. The ground state is found as the antiferromagnetic configuration with opposite spins on the edges (and sublattices). Due to exchange potential differences on the two sublattices, a band gap is opened in analogy to electrically gated bilayer graphene or hexago-

23


nal boron nitride [58]. The minimum band splitting is estimated with ∆Econ = 9.33 eVnm/(W + 15 nm) [98] as ∆Econ (W = 30 nm) ≈ 30 meV. In summary a band gap of ∆Econ < 50 meV is expected in an armchair or zigzag constrictions of width W = 45 nm. Interesting for the experimental studies are also calculations with straight edges of arbitrary orientation [95, 102]. It is found that the zigzag edge is generic with edge states in all but perfect armchair ribbons. However, if the zigzag edge is imperfect the connection between individual edge states is reduced [95] and might give rise to completely localized states which change the transport properties significantly (more in section 4.2.4).

4.2.2

Transport gap in graphene constrictions

We now turn our attention to experimental conductance measurements of graphene constrictions pioneered by groups from the Colombia University and IBM in New York [26, 27]. SFM images of two typical graphene constrictions are shown in Fig. 4.4 (a,b). The constriction in (a) is elongated with a nonuniform width w ≥ 45 nm. This constriction design is intended to provide tunable tunneling barriers for quantum dots, as the change of the width should lead to a position dependent band gap which can in turn be used to tune the length and therefore also the strength of the barrier by gating (see Ref. [103] for more details). The second nanoribbon design provides a clearly defined length and has been used to study width and length dependence of transport through constrictions. As pointed out by several authors [104, 105], the length of the constriction is of minor importance and acts mainly on the distribution of the bias voltage along the ribbon (reduced electric field for longer ribbons). The transport properties are therefore mainly governed by the minimum width of the constriction, independent of the device design. In the following, the relevant properties are illustrated with measurements from the device in (a). A typical conductance measurement taken as a function of back gate voltage is shown in Fig. 4.4 (c) (T = 2 K). In contrast to the back gate characteristics of 2D-graphene shown in Fig. 2.5 (a), the measurement shows many reproducible conductance fluctuations. Further, transport between hole and electron conductance is suppressed within a voltage regime ∆Vbg , denoted as transport gap.p The suppression in back gate can be related to a gap in Fermi energy ∆EF ≈ ~vF 2πCg ∆Vbg / |e|, where Cg is the back gate capacitance per area [106]. This leads to an energy gap ∆EF ≈ 110 − 340 meV which is significantly larger than the band gaps ∆Econ < 50 meV in armchair graphene nanoribbons with W = 45 nm estimated from ab initio calculations. A direct energy relation is obtained when measuring the bias dependence of the conductance Fig. 2.5 (e). The size of the white suppressed region fluctuates strongly, but the gap extends roughly over Eg = e∆Vb ∼ 14 meV which is in agreement with the observations in Refs. [26, 27] for a 45 nm wide ribbon. This value is still

24


s

(b)

d

d

200nm s 200nm

(c) 50 hole

gap

electron

¨9bg

G (10-4e2 /h)

30 20

(d)

10

Bias (mV)

(e)

-5

0

5 10 15 20 Back gate Vbg (V)

0.1

20

8 6 4 2 0 7.82 7.83 7.84 Back gate Vbg (V)

25

(f)

-3

10

10 0

0 -10 -20 0

(f) 5 10 15 Back gate Vbg(V)

20

Isd(nA)

0

GVbg

(d) 10

-0.1

Bias (mV)

G (10-4e2 /h)

40

-4 0

-5

-10

-6 9

9.1 9.2 9.3 Back gate Vbg(V)

log[G(e2/h)]

(a)

-7

Figure 4.4: (a,b) Scanning force microscope image of two types of graphene constrictions with (a) nonuniform width (wmin = 45 nm) for quantum dot barriers in and (b) clearly defined geometry with W = 85 nm and L = 500 nm. (c) Conductance as a function of back gate voltage measured in constriction (a) at low source-drain bias (Vb = 300 µV). The hole and electron dominated conductance is separated by a region of suppressed conductance (transport gap) containing sharp resonances. (d) Magnification of a sharp resonance recorded in the transport gap [see arrow in (c)]. (e) Bias dependence of the conductance suppression in the transport gap (white). A close-up is shown in panel (f) where individual Coulomb diamonds are visible.

however much smaller than the gap in Fermi energy ∆EF ≈ 110 − 340 meV. Hence, in addition to the transport gap in Fermi energy ∆EF , there is a second energy scale involved denoted as source-drain gap Eg . Additional information can be gained by focusing onto a smaller back gate range within the transport gap where individual conductance resonances are resolved [see Fig. 4.4 (d,f)]. These resonances have to be distinguished from mesoscopic conductance fluctuations routinely observed in disordered nanometer-sized channels at el-

25


evated conductance values and low temperatures. In Fig. 4.4 (d) a particularly narrow resonance is magnified. The lineshape can be fitted with a thermally broadened cosh−2 [x(Vbg , Te )] function with x(Vbg , Te ) = (eαbg δVbg /2.5kB Te ) the gate and electron temperature dependence used to describe transport through strong localizations such as quantum dots [91, 107]. With the lever arm αbg = 0.2 (see below) an electron temperature of Te = 2.1 K is obtained in agreement with the T = 2 K base temperature of the measurements setup. Note that in other regimes the lineshape is much broader and characterized by the coupling of a localized state to the surrounding (see also section 5.2.4). Resonant suppression of the conductance is also observed arising from interference of localized with extended states; also known as Fano resonances [108]. Other similarities to quantum dot transport are observed in bias dependent measurements shown in Fig. 4.4 (f). Diamond shaped regions of suppressed conductance are observed as a signature of Coulomb blockaded transport. The individually extracted charging energies at the corresponding back gate voltage agree with the outline of the bias gap in Fermi energy shown in Fig. 4.4 (e) [106].

4.2.3

Formation of charged islands in constrictions

The Coulomb blockaded behavior described above and similar measurements by other groups [26, 109, 110] provide indications that the transport and the sourcedrain gap are related to the formation of charged islands in constrictions [see Fig. 4.5(a,b)]. This is in contrast to the expected band gap as predicted by theoretical calculations mentioned in section 4.2.1. Further indications for charged islands or quantum dots within constrictions are obtained from (i) variation of the relative lever arms of the localizations [106], (ii) individual charging events measured over capacitive coupling to the dot (see chapter 5.1.2) and (iii) from the width dependence of the energy gap [111, 112]. The width dependence can be best described with a model involving electron interactions [111] as Eg (W ) = α/W exp (−βW ). In order to provide a qualitative picture for the observed energy gaps, a model based on charged islands and disorder has been put forward [106, 109, 110]. Following Ref. [106], quantum dots along the constrictions can arise in the presence of a quantum confinement energy gap (∆Econ ) combined with a strong disorder potential ∆dis , as illustrated in Fig. 4.5 (c). A cross section at fixed energy then leads to the localized states visualized in Fig. 4.5 (b). As discussed above, the confinement energy ∆Econ (w) can neither explain the observed energy scale ∆EF , nor the formation of quantum dots in the constrictions. However, by superimposing a disorder potential giving rise to electron-hole puddles near the charge neutrality point [12], transport is blocked as the confinement gap separates different puddles and creates tunneling barriers (the confinement gap might be further supported by a Coulomb gap [113, 114]). Within this model, ∆EF depends on both the confinement energy gap and the disorder potential. An upper bound for the disorder potential is given over ∆dis ≤ ∆EF + ∆Econ allowing a comparison with disorder measure-

26


(a)

(c) w

z

E

¨(con ¨(F

x ¨dis

x

(b)

(d) E ¨(F

E x

localizations x

¨(con

Figure 4.5: (a) Schematic illustration of a graphene constriction. (b) Potential landscape at fixed energy with localized states (red) within the constriction. (c,d) Cartoon pictures explaining the formation of localized states due to disorder. In (c) charge impurities lead to the formation of electron-hole puddles characterized by the strength ∆dis of the charge neutrality point fluctuation. In bulk graphene transport across the puddles is still possible due to Kleintunneling. In constrictions, in combination with a confinement gap ∆Econ the puddles are separated and isolated states can form (red lines). (d) Edge disorder creates low energy states (red) which are laterally separated by local density of states minima across the constriction.

ments in bulk graphene. In Ref. [12], bulk carrier density fluctuations of the order of ∆n ≈ ±2 × 1011 cm−2 have been reported. This is in reasonable agreement with the extent of the observed transport gap as √ the corresponding variation of the local potential is given by ∆EF ≈ ∆Econ + ~vF 4π∆n ≈ 126 meV. The energy gap in bias direction Eg is not determined by the strength of the disorder potential (∆EF ) but by its spatial variations which are related to the size of the charge puddles. The minimum island size can then be related to the maximum charging energy Eg := Ec,max ∼ 14 meV. Note that this quantity is ill-defined for long nanoribbons as the bias potential drops across several puddles as pointed out by Han et al. [105]: There, this quantity is redefined via the electric field drop per unit ribbon length (see also next section). From comparison with addition energies from intentionally designed quantum dots [see Fig. 4.9 (a) below], we expect a corresponding island area of around 45×65 nm. The extent of the smallest localized puddle along the ribbon is therefore comparable to the width of the constriction. The size of the diamonds get generally smaller by tuning away from the charge neutrality point. This dependence can be attributed to the merging of individual puddles. The width scaling of Eg is proportional to that of EF [26, 105, 112] and hence approximately inversely proportional to the ribbon width. A 1/W -dependence is also used to decribe the addition energy of a quantum dot of size W × W (see Fig. 4.9 (a) with d the diameter of the island).

27


4.2.4

Localization due to edge disorder

Although the previously introduced simple model provides a good qualitative description of the observed properties, the introduction of a clean confinement band gap is unlikely in the presence of rough edges. As mentioned at the end of section 4.2.1, even by considering straight ribbons, a deviation from perfect armchair or zigzag orientations leads to disconnected edge states at low energies. Motivated by the first measurements [26, 27], the effect of rough edges was investigated in several calculations. Extremely rough edges with significant variation of the width were considered in Ref. [111] in order to explain the experimental data. However, already weak disorder with few removed atoms at the outermost edges resulted in localized states and significant modifications of the density of states [115–119]. The enhanced local density of states at edges with dislocations leads to a depletion in the ribbon center. If the localized edge-states are decoupled from transport channels, a barrier across the ribbon is likely to form [117] [see schematic in Fig. 4.5 (d)]. The localization length increases while moving away from the charge neutrality point [116]. Above an energy threshold (the effective mobility edge or ∆EF ), most states are extended across the whole ribbon in agreement with the measured resonances and the transport gap. The size of the localized states is estimated in Ref. [119] to be comparable with the ribbon width, in agreement with the size of the measured Coulomb diamonds. It is also worth mentioning that the dependence of the induced transport gap on the ribbon orientation is calculated to be rather weak in the presence of edge disorder [116] providing an explanation why no orientation dependence can be observed in experiments [26]. Transport in the presence of disorder induced localization (known as Anderson localization) has been studied in great detail in the literature (see Ref. [120] for a review). At finite temperature transport is possible by (i) activation to or above the mobility edge, (ii) activation to neighboring localized states known as activated transport or by (iii) variable range hopping to states located further away than the localization length. The larger hopping pays off if the state is close in energy [121, 122]. Temperature dependent measurements in long ribbons with L ≥ 500 nm revealed that the minimum conductance in the gap is characterized by a crossover at T ∗ from variable range hopping at low temperatures to activated transport at more elevated temperatures with an activation energy Ea [105, 119]. The transition temperature (T ∗ ∼ 17 K at W = 45 nm) shows a similar width dependence as Ea and ∆EF with each energy being an order of magnitude larger (kT ∗ < Ea < ∆EF ) [105]. In small constrictions, the activation energy Ea corresponds to the addition energy Eg of a single localized state. For the W = 45 nm wide constriction the obtained values ∆EF ≈ 150 meV Eg ≈ 14 meV are in good agreement with Ref. [105] and Eg kT ∗ ≈ 1.5 meV (from Ref. [105]).

28


4.2.5

Discussion and summary

Transport measurements in graphene constrictions can be understood both theoretically and experimentally by considering weakly disordered edges. However, the influence of charged impurities as an additional source of disorder cannot be neglected. First, the strength is in agreement with measurements in bulk graphene. Second, annealing of constrictions leads to a reduction of the transport gap and a shift of the center of the gap towards the charge neutrality point [123]. Further, Coulomb charging effects are found to be important, as the activation energy is comparable to the addition energies found in lithographically defined graphene quantum dots. This is also supported by measurements in which the dielectric substrate material was changed [105]. Although the localized states in graphene constrictions pose additional complications, they can readily be used as tunable tunneling barriers for transport measurements in graphene quantum dots. In short constrictions only few localized states play a role and can be controlled using dedicated gates. Nevertheless, the understanding of constrictions developed in this section and a careful analysis of the experiments is important to understand the observed behavior.

29


4.3

Coulomb blockade in graphene single electron transistors

Early measurements of Coulomb resonances in thin graphitic flakes have been performed using highly resistive contacts as tunneling barriers [124]. Such devices are limited in terms of island size and barrier controllability. By using narrow graphene constrictions as tunneling barriers, these limitations can be overcome [110, 125, 126]. In this section, graphene single electron transitors made of single and threelayer graphene with different island areas A1L = 0.06 µm2 and A3L = 0.1 µm2 are presented. A SFM-picture of the single-layer device is shown in Fig. 4.6 (a). The few-layer device has a similar geometry. The quantum dot structure is carved out of a graphene flake by etching as described in section 3.2. In addition to the (plunger) gate shown in Fig. 4.1 (a) there are two additional gates (lg ,rg) to tune the potential of the barriers.

4.3.1

Tunable Coulomb blockade

The current through the SET as a function of back gate is plotted in Fig. 4.6 (b) for low source-drain bias (Vb = 250 µV). The measurement is dominated by the transport gaps of the two constrictions, separating hole- from electron-transport (see section 4.2.2).

s

300 nm

(b) 40 1L

Ti/Au

pg

Current Isd (pA)

Graphene

(a)

d lg

rg

30

Vb =250μV T = 1.7K

20 10 0 -50

-40 -30 -20 -10 Back Gate Vbg (V)

0

Figure 4.6: (a) Graphene single electron transistor etched from a graphene sheet. The island is connected to source (s) and drain (d) over two narrow graphene constrictions tunable by the side gates lg and rg. The plunger gate (pg) is used to tune the dot and with the back gate (not shown) the global potential is adjusted. (b) Source-drain current as a function of back gate voltage. In between the hole- and the electron-transport regime, the barriers are closed due to the transport gaps in the constrictions. Using the side gates Vlg and Vrg , the two barriers can be tuned largely independently giving rise to a cross-shaped region of suppressed current in Fig. 4.7 (a). The vertical blue stripe is mainly tuned by Vlg and hence corresponds to the transport gap in the left constriction. The horizontal line is due to the pinch-off of the

30

4.3. Coulomb blockade in graphene single electron transistors

right constriction. A close-up of the gate dependence in the tunneling regime is shown in Fig. 4.7 (b). A similar dependence is observed in the few-layer device in Fig. 4.7 (c). In addition to the vertical and horizontal modulation of the current due to the resonances in the constrictions, finer diagonal oscillations are visible. This diagonal current modulations are equally affected by the two side gates and therefore attributed to Coulomb blockade in the graphene island.

-4.2

1L

10 0

-10 -20 -20 -10 0 10 20 Vlg (V)

Vrg (V)

Vrg (V)

20

(b)

Current log(I/A) -10 -11 -12

(c) 400

1L

-4.3 -4.4 -4.5 -5.9

Current (pA) 200 0

10.1 3L

Vrg (V)

Current (pA) 100 80 60 40 20 0

(a)

-5.8 -5.7 Vlg (V)

-5.6

10.0 9.9 9.8

-3.4 -3.3 -3.2 -3.1 Vlg (V)

Figure 4.7: (a) Source-drain current as a function of Vlg and Vrg in the singlelayer SET. The blue cross shaped region of low conductance is attributed to the transport gap in the two constrictions which are manly tuned by one of the two gates. (b) Close-up of (a) shows diagonal resonances equally tuned by both gates and attributed to Coulomb blockade of the island. (c) Simliar measurement as (b) for the 3-layer SET. The finer spacing of the diagonal oscillations in the 3-layer device is attributed to the larger island size (see text). Coulomb blockade is observed more clearly by changing the plunger gate which acts mainly on the island potential [Fig. 4.8 (a,b)]. In the smaller single-layer device pronounced Coulomb blockade peaks are visible and the larger three-layer device shows Coulomb blockade oscillations where the conductance is not completely suppressed in the blockaded regime. The difference is due to the larger capacitance CΣ and hence smaller charging energy Ec = e2 /CΣ in the latter device.

4.3.2

Charging energies of the SETs

The charging energy can be conveniently measured by variation of the bias voltage as explained before [Fig. 4.2]. In Fig. 4.8 (c,d), Coulomb diamond measurements are shown for both devices. The slight tilt of the diamonds compared to Fig. 4.2 (a) is due to asymmetric tunneling barriers. The more strongly coupled lead shows a steeper slope of the associated diamond edge (µN = µs,d ). In Fig. 4.8 (c) the drain is more strongly coupled and in Fig. 4.8 (d) it is the source. For the two devices, the single-particle level spacing is negligible and Eadd ≈ Ec with Ec ≈ 3.4 meV in the single layer and Ec ≈ 0.6 meV in the three-layer

31

(b) 5 Gsd(10-4e2/h)

0.6 0.4 0.2 0

(d) -3 -4 -5 -6 -7

log[Gdiff(e2/h)]

Vbias (mV)

(c)

0 -2 -4

4 3 2

40 ǻ9pp = 30.1±2 mV 30 20 0

Vpg (V) 4

1 0 2.7

-0.1 -0.06 -0.02 Plunger Gate Vpg (V)

4 1L 2

3L

0.8

2.8 2.9 3 3.1 Plunger Gate Vpg (V)

3.2

3L

0.4

-2.2 -2.6

0 -0.4

-3

-3.4 -0.8 2.9 2.95 3 3.05 3.1 3.15 Plunger Gate Vpg (V)

-0.1 -0.06 -0.02 Plunger Gate Vpg (V)

log[Gdiff(e2/h)]

0.8 1L

Vbias (mV)

Gsd(10-4e2/h)

(a)

ǻ9pp (mV)


Figure 4.8: (a,b) SET-conductance as a function of plunger gate voltage at low bias for the single-layer device (a) and the more open three-layer device (b). (c,d) Differential SET-conductance as a function of plunger gate and symmetrically applied bias voltage for the two devices. In the blue diamond shaped regions the conductance is suppressed due to Coulomb blockade. device. In the latter, the charging energy corresponds to 3.5 kB T at the measurement temperature of T = 2 K, comparable with the temperature broadening of the Fermi distribution in the leads. Hence, the current is never fully blocked as can bee seen in Fig. 4.8 (b). In the inset of Fig. 4.8 (b) the peak spacing of ∼ 150 subsequent Coulomb oscillations is plotted. The homogeneous peak spacing shows an average of Vpp = 30.1 ± 2 mV corresponding to a small variation of the charging energy of 7% or ∼ 40 µeV. The peak spacing in the single-layer device (not shown) fluctuates slightly more from 14%-19% (0.5 − 0.6 meV) in different regimes [103].

4.3.3

Charging energy as a function of dot diameter

In the two-dimensional case, a lower bound of the total capacitance of the island CΣ can be estimated by the self-capacitance of a metallic disc with diameter d: C = 40 d [34]. For the dielectric constant = (ox + 1)/2 ≈ 2.5 is used, an average between the underlying oxide and vacuum. An upper bound for the charging energy is then given as Ec = e2 /C = e2 /(40 d). In Fig. 4.9 (a), the addition energy of several graphene quantum dots is plotted as a function of the island diameter together with the charging energy estimated from the disk-model (red solid line).

32

4.3. Coulomb blockade in graphene single electron transistors

sp4

sp1

1

disc: e2/(4İ0İG 2 fit: ~ e /(9.4İ0İG

15 DDb

10

sp2 DDa

esd cd1

5 sp3

cd2

1L

3L 0 50 100 150 200 250 300 350 effective dot diameter d (nm)

25 20

Cbg/Cdp

20

(b) 30 BG capacitace (aF)

addition energy (meV)

(a) 25

0

15

esd

3L

sp3

10 5

1L

100 200 300 d (nm)

sp4 sp1 sp2

cd1 esd

Cdp Cdisc

sp3

0 50 100 150 200 250 300 350 effective dot diameter d (nm)

Figure 4.9: (a) Typical values for the addition energy extracted from Coulomb diamond measurements for varying dot size. The error bar for the energy includes the measured single-particle energies. An estimation of the charging energy based on the disc model (solid) gives roughly 2.5 times larger energies as a 1/d fit of the data (dashed). The data points labeled DDa and DDb originate from the same double dot measured in different cooldowns [127, 128]. The difference is attributed to lower source-drain coupling in the second cooldown. Sample cd2 is described in Ref. [14] and device esd in Ref. [129]. The other devices are listed in Appendix B. (b) Comparison of the measured back gate capacitance Cbg = αbg e2 /Ec to the capacitance of a disk on top of a plate in the two limits of a plate capacitor Cdp (green) for large diameter and an isolated disc Cdisc (red) for small diameter compared to the oxide thickness. In the inset, the ratio of the calculated and the measured capacitance Cbg /Cdp is shown, signifying the increased screening by the surrounding leads and gates for smaller structures.

Most of the devices are much smaller than the two SETs (1L and 3L) discussed above and the addition energy contains contributions of the level spacing ∆. However, as the level spacing is comparable to or smaller than the indicated error bar (see also below), the charging energy provides the main contribution. The charging energy is overestimated by the disc model (∼ 2.5×) because of the capacitive coupling of the dot to the adjacent gates and leads which increases the total capacitance CΣ . The coupling is also reflected in the distribution of the charging energies of the devices. For dots coupled strongly to the leads, such as the device sp3, a smaller Eadd is observed, whereas for the extremely weakly coupled device sp4 the charging energy is comparably larger (this device is presented in more details in chapter 5.2). The comparably small charging energy of the few layer dot is not only caused by a larger coupling but also due to the increased volume compared to a single-layer dot. An alternative comparison of the dot size with the charging energy is given via the capacitance between the dot and the back gate Cbg = αbg e2 /Ec plotted in

33


Fig. 4.9 (b). The measured capacitance is compared to the capacitance of a metallic disk on top of a ground plate [130] with the limits given by an infinite plate capacitor with fringe capacitance Cdp = 0 ox [Adot /tox + 2d ln (2)] (green) for d tox and the isolated disc Cdisc = 0 (ox + 1)2d (red) for d tox . The difference to the measurement is attributed to screening of the electric field from the island by the leads and in-plane gates. The screening gets proportionally stronger for smaller devices as can be seen from the inset where the ratio Cbg /Cdp is plotted versus dot diameter (same trend is found for Cbg /Cdisc ). The screening in the device sp3 is particularly strong not only due to the strong coupling to the leads but also because of the small 20 nm trenches between the dot and the surrounding graphene gates compared to more than 60 nm gap in the device esd patterned without single-pixel (sp) e-beam lithography (see section 3.2).

4.4

Quantum confinement in graphene

So far we focused mainly on charge quantization. However, in devices with diameter d ≤ 200 nm, the energy quantization due to confinement becomes observable in transport at low temperatures. This manifests itself in fluctuations of the addition energy (level statistics, section 4.4.1) and in additional lines in Coulomb blockade diamonds (excited-states and co-tunneling onsets, section 4.4.2).

4.4.1

Level statistics

The distribution of eigenenergies in confined chaotic system has been studied in details in various different systems including quantum dot billiards [131]. For a nonsymmetric dot shape (classically chaotic billiard), the description of the statistics changes from Gaussian orthogonal ensemble (GOE) by applying a magnetic field to a Gaussian unitary ensemble (GUE) due to time reversal symmetry breaking [131]. For graphene quantum dots, there was a controversy about the expected statistics [126, 132, 133]. Interestingly, for Dirac-neutrino billiards, Berry and Mondragon predicted breaking of time reversal symmetry and a GUE distribution already at zero magnetic field due to chirality [42]. However, in graphene time reversal symmetry at B = 0 is restored by taking into account both valleys (valley degeneracy) [134, 135]. Numerical simulations of graphene billiards revealed GOE distributions in both, ideal classically chaotic shapes [133] and in disordered dots with edge roughness and defects [132]. Based on experimental data, it is generally very difficult to distinguish between a GUE and GOE distribution as the differences are rather small and require large statistics. Based on our data, we were not able to draw a conclusion because of significant changes in the charging energy due to variations of the coupling to the leads [see e.g. Fig. 4.7] which could not be isolated from changes of the single-particle spacing over a sufficiently large gate regime.

34

4.4. Quantum confinement in graphene

4.4.2

Excited-states and cotunneling onsets

In Coulomb diamond measurements, the single-particle level spacing can be measured independently of the charging energy. The energy quantization manifests itself in additional lines running parallel to the edges of the diamond and faint lines at constant bias within the blocked region [see arrows in Fig. 4.10 (a,b)]. Fig. 4.10 was recorded at Tel = 200 mK using sample sp2 with d ≈ 70 nm (see also chapters 6 and 7). The sketch in Fig. 4.10 (c) illustrates the first-order tunneling through excited states (black and blue dots) which leads to an increase of the dot current (line in differential conductance) as long as both ground and excited state(s) are within the bias window. Inside the diamond, the ground state stays occupied and the current is blocked even if the excited state level is still above the lead. However, second-order tunneling processes are possible within the diamond such as inelastic co-tunneling (red square) which sets in when eVb > ∆(N ) (dotted). If the bias is below the level spacing, only elastic co-tunneling (via the ground state) without energy dissipation is possible. Gqd (e2/h) 0 0.1

ȝN = ȝs

(b)

20 Bias (mV)

ȝN+1 = ȝd

(c) eVbias

(a)

N

N-1

N+1

¨1

Vgate

0 ¨1

-20 ȝs

4.5

Vpg (V)

5

20 mV

ȝs ȝN

ȝd

eVb ¨1 ȝd ȝN

ȝN+1 ȝs ȝd

Figure 4.10: (a) Coulomb blockade diamonds (i.e differential conductance as a function of bias and gate voltage) for a small d = 70 nm quantum dot. (b) Zoom around the diamond corner indicated by the arrow in (a). Co-tunneling onsets and a corresponding excited state lines are indicated. (Adapted from Ref. [136].) (c) First-order tunneling through excited states (black and blue dots) leads to an increase of the dot current (peak in differential conductance) outside the diamond (dashed). Within the diamond, the ground state stays occupied and first-order tunneling is blocked. However, higher-order processes are possible such as inelastic co-tunneling (red square) which sets in when eVb > ∆(N ) (dotted). (Schematics adapted from Ref. [137].)

Single-particle level spacing We proceed with a quantitative analysis of the level spacing. In Fig. 4.10 (b) the single-particle energies at the arrows can be read off as 2.3 meV in negative and

35


3.1 meV in positive bias direction. These energies are compared with an estimate of the single-particle level spacing ∆N in graphene dots, obtained from the graphene density of states D(E) [34, 129] which is given by D(E) =

f ·E 2πvF2 ~2

with f = 4 the valley and spin degeneracy of each state in graphene. The total confinement energy of the system is calculated as Z EF Econf (N ) = A D(E)dE 0

2 A = E3 3 πvF2 ~2 F 4 vF ~ 3/2 N , = 3 d with A = πd2 /4 the dot area and N the number of electrons on the dot at a given Fermi energy Z EF E2 D(E)dE = A 2F 2 . N =A πvF ~ 0 The change in confinement energy by the addition of one electron is (N + 1) = Econf (N + 1) − Econf (N ) 4 vF ~ = (N + 1)3/2 − N 3/2 3 d 2vF ~ √ N ≈ d

(4.4)

for N 1 . What we measure in Coulomb diamond measurements are changes of the chemical potential vF ~ ∆N = (N + 1) − (N ) ≈ √ d N

(4.5)

or in terms of energy in a 2D model ∆(EF ) ≈

1 1 . D(EF ) A

An alternative approach starting from the subband spacing δE = π~vF /W in a 1D graphene ribbon (width W ) with an additional confinement of length L in the direction of the ribbon, leads to a similar result [138] π~vF 1 π~vF W f · EF L π 1 1 ≈ . 2 D(EF ) W L

∆(EF ) ≈

36

(4.6) (4.7)

4.4. Quantum confinement in graphene

Hence in contrast to conventional 2D-semiconductors where the single-particle spacing is independent of energy at higher fillings [34], there is a 1/EF -dependence in graphene. Energy dependence of the level spacing In order to incorporate the energy dependence, the position of the charge neutrality point has to be known. A good estimation can be obtained by Landau level spectroscopy as described in chapter 6. Energy level spacings to the first excited state extracted from the Coulomb diamond measurements [such as Fig. 4.10 (b)] are plotted as a function of charge carriers on the dot N in Fig. 4.11 (a). The number of carriers is estimated from the distance to the charge neutrality point. For comparison, Eq. (4.5) is plotted as a dashed line with d = 70 nm.

(b) 7 P.

4 3

B=1.5T

2 0 -2 -4

9bg P9

2 1 sp2 0 0

10 30 20 40 FKDUJHFDUULHUV1

5 DYHUDJHGǻ1 PH9

9bP9

OHYHOVSDFLQJǻ1 PH9

(a) 5

4 3 2

esd

cd1

sp2 DDc

1

sp3

0

50

sp1

50

100 150 200 HIIHFWLYHGRWGLDPHWHUGQP

Figure 4.11: (a) Level spacing of the first excited state as a function of charges on the island estimated from Coulomb diamond measurements. The dashed line shows the energy dependence expected from the single-particle calculation in Eq. 4.5. The inset shows a Coulomb diamond measurements recorded around 500 meV below the charge neutrality point (measured in chapter 6) corresponding to N ≈ 40. The periodic spacing of the lines might result from the confinement potential for the electronic states on the dot but could also be attributed to phonon mediated transport, making dot transitions more probable if the energy difference between (in this case) drain and dot is a multiple of the confined phonon frequency. (b) Averaged level spacings observed in different samples and plotted as a function of effective dot diameter. The dashed line shows the diameter dependence expected from the model in Eq. 4.5 averaged over N = 1 . . . 30. Note that in larger devices d > 100 nm only few excited states have been observed and hence statistics is limited. A weak trend towards smaller level spacings for higher filling numbers can be observed. However, the dependence is non-monotonic and can’t be explained with the single-particle level spacing alone. Around the charge neutrality point where the

37


model is not applicable anymore, the spacing does not increase significantly. This is opposed to the energy spacing increase in the particle in a box picture especially for low fillings. Further investigations are needed to clarify this dependence.

Possibility of phonon mediated transport Interestingly, also the spacing between higher excited states does generally not decrease with increasing level index. An extreme example is shown in the inset of Fig. 4.11 (a) recorded at N ≈ 40 where up to 8 lines are visible (taken at B|| = 1.5 T, but qualitatively the same as at B|| = 0 ). The spectrum is roughly constant with an average spacing of δE = 0.5 meV for negative and δE = 0.65 meV for positive bias. The equidistant spacing could originate from the confinement potential on the dot, but could also be caused by phonon mediated transport. The wave length of confined phonon modes attributed with the observed energy spacings are given by λ/2 = hvs /2δE ∼ 60 and 80 nm with vs ≈ 200 000 m/s [139] the sound velocity in graphene. These dimensions agree very well with the 50 nm width and 80 nm length of the investigated quantum dot. Hence the change in differential conductance could arise from enhanced transport whenever a multiple of the phonon energy matches with the energy spacing between the chemical potential of the dot ground state and the drain (drain here because of the positive slope). As we have seen, the change of the differential conductance alone is not a sufficient criterion for an excited state. However, for most level spacings (all with N < 10) in Fig. 4.11 (a) a joining co-tunneling onset is observed, giving strong arguments for a process involving an excited state.

Dependence of the level spacing on the dot size In Fig. 4.11 (b), the averaged level spacing is plotted for different samples as a function of dot diameter d. For comparison, the dashed line shows the diameter dependence expected from the model in Eq. 4.5 averaged over N = 1 . . . 30. For the d = 200 nm device cd1, a comparatively large level spacing of ∆ ≈ 1.5 meV was observed in one gate region in several diamonds in a row. In the double dot device DDc (data from a third cooldown after additional processing, see Appendix A,B) two different energies have been observed in finite bias measurements. Step like changes in the current attributed to transport through excited states (∆ ∼ 1.7 meV), are superimposed onto periodic fluctuations with an average spacing of < 500µeV. The fluctuations have been attributed to an interference effect between two different dot hopping processes involving phonon mediated relaxation either in the first or the second dot [140].

38

4.5. Summary

4.5

Summary

Coulomb blockade effects arise in graphene not only in zero-dimensional geometries but already by cutting out one-dimensional narrow constrictions. Carrier localization in graphene constrictions is attributed to inhomogeneities in the density of states due to localization at zigzag edges and local variations of the Fermi level due to impurity doping. Coulomb blockade in the dot can be separated from parasitic effects by using side gates to geometrically locate the origin. The identification is increasingly difficult for smaller devices as the gates generally tune less selectively and the size and energy difference between the island and the localizations in the constrictions is reduced. Quantum confinement effects clearly manifest themselves in smaller devices as excited states and co-tunneling processes. While a general increase of the level spacing for smaller devices is observed, the energy dependence is not completely understood so far. Further, excited state lines without joining co-tunneling onset might have a different origin and be caused by density modulation in the leads or phonon mediated transport. Further studies of graphene quantum dots in perpendicular and parallel magnetic fields are presented in chapter 6 and 7.

39

Chapter 5 Charge detection in graphene quantum dots In the last chapter it was shown that graphene constrictions can be used as tunable tunneling barriers in graphene quantum dot devices. In this chapter it is demonstrated, that constrictions can also be used as sensitive detectors for charge changes on a nearby quantum dot. Charge detection techniques [141] have been shown to significantly extend the experimental possibilities with quantum dot devices: They can be used for detecting spin-qubit states [23, 142] and molecular states in coupled quantum dots [143]. Furthermore charge detectors have been successfully used to investigate shot noise on a single-electron level and full counting statistics for semiconductor-based devices [144]. This makes charge detection highly interesting for advanced investigations of graphene quantum dots and graphene nanosystems in general. In comparison with semiconductor quantum circuits, the negligible side depletion in graphene allows graphene-based nanostructures to be put closer together. The reduced spacing is expected to enhance the electrostatic coupling leading to larger signal-to-noise ratios and hence increased sensitivity in quantum measurements. In the first part of this chapter the detection principle is introduced and it is shown that charges down to a single electron are easily detected. The second part focuses on time-resolved charge detection which allows to study the transport properties of tunneling barriers in details.

5.1

Graphene constrictions as charge detectors

A scanning force microscope image of a graphene quantum dot device with integrated charge detector is shown in Fig. 5.1 (a). The dot (qd) with d ∼ 200 nm is tuned by the plunger gate (pg). The barrier constrictions to the leads have a minimum width of 35 nm and are tuned by the two other side gates (lg, rg). In addition, a 45 nm wide graphene nanoribbon is placed 60 nm next to the island and will be used as a charge detector (cd).

40

5.1. Graphene constrictions as charge detectors

(b) pg lg

300 nm rg

Iqd (nA)

(a)

qd

80

qd

0.15 0.1

60

50 mV

0.05

40 20

Vbcd Vbqd

Icd

Icd (pA)

cd

Iqd

0 40

cd

20 0 20

10

0 10 Back gate Vbg (V)

20

30

Figure 5.1: (a) Graphene quantum dot device (qd) with charge detector nanoribbon (cd) and lateral side gates (lg, pg, rg) to tune the barrier constrictions and the potential of the island. (b) Current as a function of back gate voltage through the quantum dot (upper panel) and charge detector (lower panel) measured at Vbqd = Vbcd = 500 µV and T = 1.7 K. The inset shows Coulomb blockade peaks of the dot as a function of back gate voltage measured in the transport gap. A characterization of the individual parts where the source-drain current is measured while varying the back gate voltage is shown in Fig. 5.1 (b). Both devices show an overlapping transport gap (-4. . . 15 V and 4. . . 14 V) at the measurement temperature of 1.7 K with 500 µV bias applied to both structures. The inset in Fig. 5.1 (b) shows Coulomb blockade peaks in the quantum dot recorded by focusing on a smaller back gate voltage range of 150 mV. From Coulomb diamond meaqd surements, an addition energy of Eadd ≈ 4.3 meV and a lever arm αbg = 0.34 are obtained (sample cd1 in Fig. 4.9). Detailed measurements of the charge detector constriction have already been presented before in Fig. 4.4 while introducing graphene constrictions in section 4.2. Due to localized states in the ribbon, transport in the gap is dominated by sharp resonances [Fig. 4.4 (d)] and the current through the constriction is highly sensitive to the surrounding electrostatic potentials. This high sensitivity can be used to detect single-electron charging on the quantum dot as demonstrated in Fig. 5.2 and Fig. 5.3 where the interplay of the two devices is studied.

5.1.1

Detection of single electron charging

For these measurements, the back gate voltage is set to Vbg = 6.5 V such that the quantum dot is close to the charge neutrality point as well as inside the transport gap of the charge detector [see arrow in Fig. 5.1 (b)]. We operate the charge detector at the border of the transport gap where strong resonances are accessible, in order

41

Chapter 5. Charge detection in graphene quantum dots

Gqd (10-3 e2/h)

(a)

Gcd (10-4 e2/h)

(b)

10

5

0 14 12 10 8

2.2 2.4 2.6 Plunger gate Vpg (V)

Figure 5.2: (a) Dot conductance Gqd and (b) charge detector conductance Gcd as a function of Vpg . The back gate voltage is fixed at Vbg = 6.5 V [see arrow in Fig. 5.1 (b)]. The charge detector signal shows an abrupt change whenever an additional electron is loaded to the dot. The arrows indicate Coulomb blockade resonances which can hardly be measured by conventional means because the current levels are too low. However, they can be detected by 2.8 the charge detector. Here Vbqd = 500 µV and Vbcd = 8.2 mV.

to make use of steep slopes of the conductance as a function of plunger gate voltage Vpg (here the slope is 4-6 10−4 (e2 /h)/100 mV in Vpg ). In Fig. 5.2 (a) almost equidistantly spaced (∆Vpg = 65±3.8 mV) Coulomb blockade resonances as function of Vpg at Vbqd = 500 µV are plotted. The strong modulation of the conductance peak amplitudes is due to a modulation of the conductance by the resonances in the barrier constrictions as discussed in the last chapter. Fig. 5.2 (a) shows the simultaneously measured conductance through the charge detector. On top of the peak-shaped detector resonance, we observe conductance steps which are well aligned (see dotted lines) with single charging events on the quantum dot. The conductance step is related to a shift of the charge detector resonance with respect to the plunger gate voltage. From an analysis of more than 300 charging cd events an average shift in plunger gate voltage of ∆Vpg = 34±6 mV is observed. Assuming a similar lever arm of the back gate to the dot and the constriction, the lever cd arm of the plunger gate on the constriction can be estimated as αpg = 0.014 by using the relative lever arms obtained in Fig. 5.3 below. The voltage shift then corresponds cd to a potential change on the constriction of ∆µcd = αpg |e|∆Vpg ∼ 0.5 ± 0.1 meV. The electrostatic coupling of the two devices is determined by the magnitude of the mutual capacitance Ccd,qd compared to the total capacitances CP ,cd and CP ,qd ∆µcd = e2

Ccd,qd . C ,cd · CP ,qd

(5.1)

P

Hence, the potential change can be increased by placing the detector closer to the quantum dot (larger Ccd,qd ) or reducing the screening of the two components (decreasing CP ,cd , CP ,qd ). The fact that in Fig. 5.2 (b) the shifts are slightly larger for lower plunger gate voltage might be related to the weaker coupling of the dot to the leads.

42

5.1. Graphene constrictions as charge detectors

The size of the step and hence the magnitude of the detector signal does not only depend on the electrostatic coupling, but also on the sensitivity of the detector conductance to potential changes (steepness of the detector slope). The detector performance is studied in more details during the discussion of time-resolved measurements in section 5.3.

¨9bgP9

(a) 15 dot

Gqd (10-3 e2/h) (b) 15

Gcd (10-4 e2/h) 8 detector 4

10

10

5

5

0

0

0

-2

2

2.2 2.4 2.6 9pg9

(c)

dGcdG9pg ȝ69 detector

5 0 -5

2

2.2 2.4 2.6 9pg9

2

2.2 2.4 2.6 9pg9

Figure 5.3: Conductance Gqd of the quantum dot (a) and the charge detector Gcd (b) as a function of plunger gate voltage Vpg and back gate voltage Vbg (Vbqd = Vbcd = 500 µV). The back gate voltage is converted to a relative scale starting at Vbg = 6.505 V with ∆Vbg = 0. In (a) the narrowly spaced periodic lines are Coulomb blockade resonances (long dashes), while the broad amplitude modulation which is less affected by the plunger gate is attributed to transmission resonances of the barriers (white dotted line). In (b) the broad line with increased conductance is even weaker affected the plunger gate voltage compared to the Coulomb blockade resonances in the dot and it is attributed to a local resonance in the charge detector (short dashes). In addition to this broad peak, faint lines with a slope corresponding to the Coulomb blockade resonances in the quantum dot are observed (arrows). (c) Numerical derivative of the charge detector conductance plotted in (b) with respect to plunger gate voltage (dGcd /dVpg ). The lines with short and long dashes indicate the two different relative lever arms on the dot and the detector. The origin of the steps in the detector are further investigated by measuring the conductance as a function of two gates (Vpg and Vbg ), plotted in Fig. 5.3 (a) for the dot and in Fig. 5.3 (b) for the detector (Vbqd = Vbcd = 500 µV). As the quantum dot is located closer to the plunger gate, the relative lever arm of the two gates on qd qd dot resonances αpg /αbg = 0.18 [long dashes, Fig. 5.3 (a)] is larger than the relative cd cd lever arm αpg /αbg = 0.04 [short dashes, Fig. 5.3 (b)] on the detector constriction. In the charge detector conductance, single charging events on the quantum dot are identified as conductance fringes [see arrows in Fig. 5.3 (b)]. This can be seen even better by numerical differentiation of Gcd vs. Vpg , as shown in Fig. 5.3 (c). Here the sharp conductance changes due to charging events in the dot are strongly pronounced, and both relative lever arms to the Coulomb blockade peaks and the constriction resonance in the charge detector are indicated by dashed lines.

43


The detection range can be improved by increasing the bias Vbcd , leading to a broadening of the constriction resonance as seen by comparing Fig. 5.3 (c) (Vbcd = 500 µV) with Fig. 5.2 (b) (Vbcd = 8.2 mV). However, the detector bias is limited by the extent of the transport gap in bias direction [see Fig. 4.4 (e)] outside which the sensitivity is greatly reduced. A side remark: The modulation of the Coulomb blockade resonances in Fig. 5.3 (a) is due to resonances in the tunneling constriction located around 300 nm away from the plunger gate [see Fig. 5.1 (a)]. This leads to an intermediate slope of 0.08 for these peak modulations (white dotted line). From the measurement shown in Fig. 5.2 (b) a nanoribbon conductance change of up to 10% can be exctracted for a single charging event. For lower bias voltages (e.g. Vbcd = 500 µV) the change in the conductance can be increased up to 60%. To summarize, a graphene constriction placed close to a quantum dot can be used as sensitive charge detector. Charge detection enhances the experimental readout of quantum dots to regimes where the current through the dot is too small to be measured conventionally. In contrast to state-of-the-art quantum point contact charge detectors, not the slopes to quantized conductance plateaus but local resonances in the graphene nanoribbon are used to detect charging.

5.1.2

Using the quantum dot as a charge detector

Because the quantum dot itself is also highly sensitive to potential changes in the surrounding, it can be used as a detector to investigate the charge flow in the constriction. dot

(b)

10-2

10-4

Vpg (V)

9.9

Gqd (e2/h)

Vpg (V)

10-3

9.9 9.8

9.8 9.39 9.4 9.41 9.42 9.43 Vbg (V)

cd 10

10

10

-5

9.39 9.4 9.41 9.42 9.43 Vbg (V)

10-4 Gcd (e2/h)

(a)

10-5

Figure 5.4: (a) Dot conductance as a function of plunger gate and back gate. The periodic Coulomb blockade peaks are shifted due to a charging event in the surrounding (dashed lines and white circles). This charging events coincides with the gate dependence of a resonance measured simultaneously in the constriction as shown in (b). Hence the dot can be used as charge sensor to show localization of charge in the constriction. In Fig. 5.4 (a,b) dot and constriction conductance are measured as a function of Vbg and Vpg similar to Fig. 5.3 but in a different gate regime. The steps in the

44

5.2. Time-resolved charge detection in graphene

constriction resonance [Fig. 5.4 (b)] stems from charging the dot with additional electrons and has been discussed above. Interestingly we can also observe small steps of the dot resonances whenever the resonance in the constriction is crossed [white circles in Fig. 5.4 (a)]. Hence, we observe complete charge localization in the constriction, providing additional indications that transport in constrictions is dominated by Coulomb blockade (as discussed in chapter 4.2.3). It is important to note that we observed such clear charging events in the constriction only rarely. The amount of localization is attributed to the strength of the coupling which is highly dependent on the particular regime as will be further elaborated in the next section.

5.2

Time-resolved charge detection in graphene

Additional information can be obtained by detecting the charge in a time-resolved way [145]. This allows to monitor single-electron charging events in real time and measure ultimately small currents by counting single-electron transitions [146]. The high resolution allows to investigate higher order correlations of the current and to extract for example the electron tunneling statistics or to probe electron-electron correlations [144, 147]. Time resolution further allows for single shot read out of spin-qubits (after spin to charge conversion), with potential applications in future quantum information processors [90].

5.2.1

Time averaged charge detection

The strength of charge detection lies in regimes where the source-drain current is too low to be detected by conventional current measurement methods (Iqd < few fA). The upper limit of the technique is determined by the detector bandwidth fbw . To count each electron contributing to a dot current of Iqd = 10 fA, a detector bandwidth of fbw > Iq d /e ≈ 60 kHz is required. Such a bandwidth was demonstrated in a hybrid system consisting of a nanowire quantum dot with strong coupling to a GaAs QPC charge detector underneath [148]. In lateral arrangements, roughly half of this bandwidth was achieved [142, 144]. All these measurements were performed in a conventional low temperature setup without high frequency optimization. In order to achieve low tunneling rates, narrow and slightly elongated barrier constrictions are used. In the investigated sample [Fig. 5.5 (a)], the barriers are 15 nm and 20 nm wide and 40 nm long. The narrower constriction turned out to be completely insulating and no tunneling between dot and source was observed. The quantum dot has a diameter of d ≈ 90 nm and is expected to show single-level transport at the measurement temperature of 1.7 K. The back gate characteristic of the 35 nm narrow charge detector in Fig. 5.5 (b) shows a transport gap extending over 12 V with sharp resonances. In Fig. 5.5 (c) a close up of a resonance recorded with the left gate shows the typical charge detection sawtooth pattern due to incremental charging of the dot as discussed in the previous

45

lg

pg

qd

s

(b)

rg

d

cd cdg

Gcd (e2/h)

(a)

0.15 0.10

Gcd (0.01e2/h)


4

(c)

2 0

í

0.05

100 nm

0 í

í

í Vbg (V)

0 1 Vlg (V)

0

Figure 5.5: (a) SFM micrograph of of the graphene quantum dot with source (s), drain (d) and lateral gates (pg, lg, rg) as well as the charge detector (cd) and its gate (cdg). (b) Measurement of the conductance through the charge detector as a function of back gate voltage Vbg at a bias of Vbcd = 10 mV. The cones indicate hole (left) and electron (right) transport, whereas the reddish area marks the transport gap. (c) Single resonance in the charge eff,cd detector conductance recorded as a function of Vlg at Vbg = −5.66 V with cd Vb = 7 mV. The effective Vbg contains the contribution from the back gate Vbg = −2.083 V and the influence of the charge detector gate Vcdg = −11.56 V while all other gates including Vbqd are at 0. The abrupt changes in Gcd arise from charging the dot with additional electrons.

section. The spatial locations of the charging events are identified by analyzing the influence of the gates on the jumps and resonances in Gcd (see Fig. 5.3). The influence (x) is characterized by a lever arm αg which relates the gate voltage Vg to the induced potential change in device x. The result is tabulated in Tab. 5.1 and shows a similar influence of the right and left gate on the abrupt changes of Gcd and superior coupling to the plunger gate. Such a behavior is expected if the abrupt changes result from charging the dot. Moreover, the absolute values of the lever arms can be estimated using dot bias spectroscopy by assuming symmetric electrostatic coupling to source and drain, revealing an addition energy of Ec = 19 meV. This is in good agreement with the charging energy for an isolated disc of diameter d = 90 nm Ecdisc = e2 /(40 d) ≈ 20 meV, expected (as an upper limit) from the dimensions of the island (see Fig. 4.9 device sp4 ). The spacing of the jumps in gate voltage are qd then related to an absolute plunger gate lever arm αpg = 0.10, based on which the other absolute lever arms in Tab. 5.1 are estimated. Compared to the previous sample the step in gate voltage ∆Vlg ≈ 0.13 V is roughly three times larger. However, the induced potential change in the detector cd estimated as αlg |e|∆Vlg ∼ 1 meV is not significantly larger because not only the cd detector is closer to the dot but also the screening side gates. The lever arm αlg is qd cd estimated from geometrical considerations to be αlg ∼ αcdg ≈ 0.02. 46


relative α αpg /αbg αrg /αbg αlg /αbg αcdg /αbg

abrupt change resonance 0.47 0.09 0.28 0.07 0.28 0.06 0.08 0.31

absolute α αpg αrg αlg αcdg αbg

αqd ≈ 0.10 ≈ 0.06 ≈ 0.06 ≈ 0.02 ≈ 0.21

Table 5.1: Lever arms of the different gates on abrupt quantum dot conductance changes and the resonances in the charge detector. The lever arms are extracted by comparing the influence of each gate relative to the back gate. The absolute plunger gate lever arm on the (dot-) charging events is extracted from dot-bias dependent measurements by assuming symmetric coupling to source and drain. The other absolute dot lever-arms are calculated based on this estimate.

5.2.2

Time-resolved measurements

After this initial characterization of the device we will investigate the dot occupancy in real time with the charge detector. Fig. 5.6 (a) shows a zoom of the time averaged current across a step measured using a typical integration time of 0.2 s. Here a lower bias of Vbcd = 0.5 mV is applied. We don’t observe a single step (see gray line) but the signal is noisy and the transition is smeared out. These are indications that the timescale of the transition is comparable to the measurement integration time. Indeed, by measuring a time trace at the position indicated by (b) and shown in Fig. 5.6 (b), a two-level random signal switching in intervals of around 0.5 s is observed. The two-level signal shows a large signal-to-noise ratio (SNR) of ∆Istep / hInoise i ≈ 30 at a measurement p system bandwidth of around 400 Hz. The noise hInoise i is defined as the variance Var(I) of the detector current on each of the two current levels. The two levels indicate whether there are N electrons (lower) or N + 1 electrons (higher level) on the dot. Hence, real time detection of single charge carriers tunneling into and out of the dot is possible. The corresponding dwell times τin and τout are indicated in the figure. The histogram in Fig. 5.6 (c) shows the distribution of the current for a 60 s time trace with a clear separation of the two states. The slight asymmetry in the distribution of the upper level is attributed to an additional weakly coupled charge fluctuator sometimes appearing in the N + 1 state. Note that the state with larger detector current can in principle also denote the lower electron occupancy if the current has a positive slope in gate voltage. By tuning the dot potential away from the resonance condition, the occupation probability of the lower N [upper N + 1] level is reduced as seen in Fig. 5.6 (e,d)[f,g]. In Fig. 5.6 (g) for example the chemical potential for the N + 1 transition lies below the chemical potential in the lead (see schematic). Therefore the dwell time for the empty state τin is much smaller than τout .

47

1.6 1.2 0.8 0.4

(b)

(a) (b) (e)

(c)

(g)

N+1 (f)

Ĳout

Ĳin

0.5

1.0 1.5 Time (s)

N

(d)

1.581.60 0 Vlg (V)

Icd (nA)

Icd (nA)


1.6 1.0 1.6 1.0 1.6 1.0 1.6 1.0

40 120 103 points

(d)

N+1

(e) N+1

(f) N+1

(g) N+1

0

1

2 3 Time (s)

4

5

Figure 5.6: (a) Time averaged current through the charge detector as a function of Vlg while scanning over a similar quantum dot resonance as shown in Fig. 5.5 (c). The step indicates a single change from N to N+1 electrons on the dot at Vbcd = 0.5 mV. (b) Time-resolved current through the charge detector taken at the point labeled with (b) in (a). The time an electron needs to tunnel into or out of the dot is marked with τin and τout respectively. (c) Histogram of the Icd values for the whole time trace of 60 s. (d-g) Time traces taken at different Vlg marked in (a), show a gradual change of the dot electron occupancy from N (lower level) to N+1 electrons (see schematics).

5.2.3

Analysis of charge detection time traces

From a quantitative analysis of time traces, detailed information about the transport process is obtained. This includes (i) the number of dot-levels participating in transport, (ii) the tunneling coupling, (iii) the carrier temperature and distribution in the leads and (iv) the individual tunneling rates [149]. For this analysis the right barrier is slightly opened in order to improve the statistics. In Fig. 5.7 (a), the distribution of the dwell times is plotted for a situation where the chemical potential of the dot state µN +1 is slightly above the chemical potential of the drain µd (see schematic). The dwell times are exponentially distributed indicating that only a single dot level is involved in the transport process, as expected at this temperature by considering the small size of the dot. Hence the probability density pin/out (t) which is the number of counts with dwell time τin/out = t normalized by the total number of counts - is given by ! t 1 exp −

dt. (5.2) pin/out (t)dt =

τin/out τin/out

48


1 hτin i

10

2

Ƚin = 47 Hz Ƚout = 102 Hz

*in N+1

*out

101 10

0

0

0.05 0.10 time (ms)

0.15

1 0.8 0.6 0.4 0.2 0 30 20 10 0

(b)

= 47 Hz

N+1

N ȝN+1

ȝN+1 ȝN

ȝN

(c)

10

(a) kT 3.55 Dlg

10

events

# of counts

10

3

events/second

(a)

PN+1

Fitting this equation to the data in Fig. 5.7 (a), tunneling rates Γin = 1 and Γout = hτout = 102 Hz are extracted. i

10 10

* 4

4

3

(d)

2 1 0

10 í

0

0.03

įV (V) lg

2.08 2.09 2.10 2.11 2.12 2.13 Vlg (V)

Figure 5.7: (a) Distribution of the dwell times recorded at a position where the dot chemical potential is slightly above the drain chemical potential [see schematic and indicated by the arrow in (c)]. The distribution shows an exponential behavior described by Eq. 5.2 with Γin = 47 Hz and Γout = 102 Hz. (b) Occupation probability for the N+1 electrons dot state P (N + 1) as a function of Vlg . The solid line is a fit to 1 − f (E) with f the Fermi function. Assuming T = 1.7 K a Vlg lever arm of αlg = 0.077 is obtained. For comparison the dashed line shows the result obtained in (c). Here the events per second are plotted for varying Vlg . Comparing these event rates with Eq. (5.3) the tunneling coupling and the lever arm are given as Γ = 143 Hz and αlg = 0.067 (solid line). The difference of the lever arms extracted from (b) and (c) (dashed line) is attributed to the limited statistics. (d) The data in (c) is fitted in addition with a tunnel-coupling broadened Lorentzian line shape ∝ Γ/ (αlg (δVlg )2 + Γ2 ) (dashed line) and plotted in logarithmic scale, confirming the thermal broadening of the resonance. This measurement has been conducted at a dot resonance close to the one analyzed in Fig. 5.6 with faster tunneling rates [see Fig. 5.8 (a)]. As shown in Fig. 5.6 (d-g), these rates change by tuning the potential of the dot. Fig. 5.7 (b) shows the occupation probability of the N+1 state changing from 0 to 1 while lowering the chemical potential of the dot for increasing Vlg (see schematics). If a gate voltage-independent tunneling coupling to the lead is assumed, the occupation probability is determined by the distribution of the charge carriers in the lead. In the lead a Fermi distribution f = [1+exp (∆µ/kTe )]−1 of the electron energies is assumed, with ∆µ the difference between the chemical potential of the dot changed by the left gate over the lever arm αlg according to ∆µ = αlg |e|(Vlg − Vres ). The occupation probability is then given as PN +1 (∆µ) = 1 − f (∆µ) and we can extract αlg ≈ 0.077 under the assumption that the electron temperature Te is equal to the bath temperature of 1.7 K. Combining single-level transport and constant tunnel coupling, it is possible to

49


extract the tunnel coupling Γ by counting the number of tunneling events. The event rate re for tunneling in is given by re =

1 = Γ · f (1 − f ). hτin i + hτout i

(5.3)

The best fit yields Γ = 143 Hz and αlg ≈ 0.067. The fit obtained in Fig. 5.7 (b) is plotted for comparison as a dashed line in Fig. 5.7 (c) and vice versa in Fig. 5.7 (b). The discrepancy between the two fits is explained by the larger influence of the data points around the resonance in Fig. 5.7 (b) compared to Fig. 5.7 (c). In Fig. 5.7 (d), a Lorentzian broadened line shape (dashed line) and a thermally broadened fit (solid line) are plotted on a logarithmic scale for comparison. As expected for low tunneling coupling, the thermally broadened line shape describes the data much more accurately. A side remark: Knowing the event rate re and the tunnel coupling, full information about the charge noise is available. It can be calculated as Var(Q) =

e2 re Γ

(5.4)

with the power spectral density given by [150] SQ =

5.2.4

e2 re2 . πΓ ω 2 + re2

(5.5)

Tuning the tunneling barrier

The barrier and the corresponding tunneling coupling can be tuned by changing the associated (side) gate [125, 127]. Fig. 5.8 (a) shows the number of events per second as a function of Vlg and Vrg . The measurement analyzed in Fig. 5.7 is a cut at Vrg = −0.37 V. The diagonal line corresponds to the resonance condition where an additional electron is loaded onto the dot (N → N+1). The potential of the dot is affected equally by both gates as expected from the geometry [Fig. 5.5 (a)]. In addition, a change in the number of counts is observed especially at lower Vrg . This change in the barrier transmission can be seen more easily in Fig. 5.8 (b) where the corresponding tunneling coupling Γ [obtained by fitting to Eq. (5.3)] is plotted as a function of Vrg (A, red circles). The same behavior is also observed in a second measurement (B, blue squares) taken in the same gate regime after a charge rearrangement and scaled by a factor of 10 for clarity. While the tunneling coupling varies strongly by changing Vrg , the full width at half maximum (FWHM) is approximately constant. Under the stated assumptions the FWHM is inversely proportional to the lever arm which shows an average of αlg = 0.062 at T = 1.7 K [see Fig. 5.8 (c)]. Unlike GaAs-based quantum dots where the tunneling rate increases monotonically while gating due to depletion of the electron gas, the non-monotonic exponential changes of the tunneling rate in a graphene sample are an indication for the

50


80

events/second 60 40 20

0

ī+]

-0.4 -0.3 -0.2

103 10

A

B

x10

2

101

(c) Įrg H

9rg9

(b)

-0.1 0 1.7 1.8 1.9 2 9lg9

2.1

1 0.8 0.6 0.4 0.2 0

30 20 10 0

-0.4 -0.3 -0.2 -0.1 0 9rg9

):+0PH9

(a)

0.1

Figure 5.8: (a) Number of events per second measured as a function of Vlg and Vrg while crossing a dot resonance. A single trace taken at Vrg = −0.37 V (arrow) is analyzed in Fig. 5.7. (b) Tunneling coupling as a function of Vrg (and ∼ −Vlg ) plotted in logarithmic scale. The red circles are obtained from the data shown in (a) (regime A), whereas the blue squares are deduced from a similar measurement in a slightly shifted gate regime B around Vrg = 0 and Vlg = 1.75 V after a charge rearrangement (10x magnified for clarity). The non-monotonic behavior can be explained by modeling the constriction as asymmetric double barrier with Lorentzian shaped resonances from the weak barrier (~ΓA = 6.2 meV and ~ΓB = 6.4 meV) and an exponential suppression due to the strong barrier. (c) Peak width plotted as inverse lever arm 1/αrg and FWHM (right scale) as a function of Vrg in both regimes with an average αrg = 0.062 . Here Vbcd = 0.5 mV and the counting time is 60 s per point.

presence of resonances in the constrictions [14, 125, 127]. It is important to note that no charging effect of the constriction is seen in this measurement. However, it is possible to tune the device into a regime where the typical hexagon pattern of a double dot [151] (see also Appendix A) is measured while changing Vrg versus Vlg in Fig. 5.9 (a). In other regimes, parasitic localized states are visible as additional charge fluctuations contributing to the noise as shown in Fig. 5.9 (b,c). From the difference in the step height, the electrostatic coupling of the additional charge fluctuator is estimated to be ten times weaker than that of the dot to the detector. In the gate regime investigated in Fig. 5.8, the influence of localized states on the dot energy is negligible but still the barrier transmission is modulated. Such a behavior might occur if the additional localized state is strongly coupled to the lead but only weakly to the dot. This situation can be modeled with a one-dimensional asymmetric double barrier with ΓR ΓL . For noninteracting electrons in the case of kT hΓ ∆ (with ∆ the level spacing), the total transmission is given by [152–

51


0.1

(i)

0

-2

-0.1 -3

0

1 2 Vrg (V)

Icd (nA)

-1

12 8

(b)

4

0 12

Icd (nA)

0.2

(iii) (ii)

dIcd /dVrg (a.u.)

Vlg (V)

(a) 0

8

(c)

4 0 0

3

50

100 Time (ms)

150

200

Figure 5.9: (a) Derivative of the charge detector current as a function of left and right gate. Apart from the slow modulation of the charge detector current, three different resonances can be identified. Resonance (i) is equally affected i by both side gates (αlg/rg = 1) and attributed to the main dot. Resonance (ii) couples to the dot resonance (anti-crossings) and is attributed to a localized state in the right constriction as it is stronger affected by the right side gate ii (αlg/rg = 0.4). The third resonance (iii) does not couple significantly to the iii dot as it is located even further on the right side (αlg/rg = 0.17). (b,c) Charge induced fluctuations on the two detector current levels in a different gate regime (Vbg = −7.91 to −7.89 V, Vrg = 2.44 V and Vlg = Vcdg = 0). If the charging of localized states is comparably slow (τ ∼ 10 ms), additional steps are visible in the counting trace (red circle). Note that all measurements shown in this figure are recorded in the high bias regime at Vcdb = 10 mV. 154] Ttot =

X p

ΓL

(ΓR

/2)2

ΓR . + (EF − Ep )2

(5.6)

Here, ΓL,R are assumed to be independent of the resonance p. In this approximation, the Lorentzian shape is caused by the weak barrier while the overall amplitude is determined by the tunneling coupling of the high barrier. For the high barrier an exponential dependence of ΓL on the gate voltage is assumed while the gate dependence of the lower barrier is neglected. The dashed lines in Fig. 5.8 (b) are the corresponding fits with two resonances (p = 1, 2). The extracted tunneling coupling of the weak barrier is similar in both measurement with ΓR,A = 6.2 meV and ΓR,B = 6.4 meV assuming a typical lever arm of αrg,loc = 0.1 on the localized state. For the left barrier the WKB result with a linearized exponential ΓL = Γ0 exp [−κ(δU − δE)] is used [155]. The details of the barrier are described by Γ0 and κ while δE (δU ) describes small perturbations of the dot energy (barrier potential). By keeping the dot energy constant and assuming a linear gate dependence of the barrier potential barrier we get ΓL = Γ0 exp [−βVrg ] where β = καrg . For the two measurements we −1 obtain Γ0,A = 0.1 Hz, βA = 6 V and Γ0,B = 0.2 Hz, βB = 11 V−1 . The different

52


parameters in the two regimes are attributed to a change of the left barrier potential between the two measurements.

15 10 5 0 0

1

5 4 (b) 200 3 150 2 100 1 50 0 0 0 1 2 3 4 5 6 7 cd Vb (mV)

2

3 4 cd Vb (mV)

5

FWHM (meV)

20

noise (pA)

25

*/*0

(a)

signal (nA)

¨,step½,noise›

30

6

7

1.6 1.4 1.2 1 1

(c)

(d)

0.8 0.6 0.4

0

1

2

3 4 5 cd Vb (mV)

6

7

Figure 5.10: (a) Signal-to-noise ratio for increasing charge detector bias Vbcd . Every data point is obtained from the average step height (signal) and the average noise on the two levels from 60 time traces (Vlg = 1.735 − 1.79 V) each 25 seconds long at Vrg = 0. (b) Signal and noise versus Vbcd plotted independently. The bold arrow marks the onset of the stronger increase in noise giving rise for the saturation of the signal-to-noise ratio in (a). (c,d) Dependence of the dot events on the detector bias. Here, measurement B shown in Fig. 5.8 (b,c) for Vbcd = 0.5 mV is repeated for different values of Vbcd with −150 mV< Vrg < 25 mV. In (c), the relative change of the tunneling rate Γ is plotted. In order to compensate for variations in the tunneling rate by changing Vrg , Γ0 (Vrg ) is defined as the average tunneling rate for Vbcd < 2 mV for each Vrg . In (d), the FWHM of the peaks [as shown in Fig. 5.7 (c)] averaged over the different Vrg is plotted for increasing Vbcd .

5.2.5

Optimizing the signal-to-noise ratio by increasing the detector bias

In the measurements shown so far, the bias in the charge detector was kept low (Vbcd = 500 µV) in order to prevent back-action of the detector on the dot [156, 157]. On the other hand, a higher charge detector bias leads to an increase of the signal (∆I). In order to maximize the performance, the signal-to-noise ratio (SNR) is investigated as a function of charge detector bias in Fig. 5.10 (a,b). The value of the noise is obtained from the standard deviation of the current on the two levels hInoise i. For a discussion of the frequency dependence we refer to section 5.3.2. The SNR is maximized for Vbcd = 2 mV and gets smaller for Vbcd > 2 mV due to a stronger increase of the noise in Fig. 5.10 (b)(bold arrow) where the signal and the noise are plotted independently. This higher noise is correlated with an increase in tunneling events and a broadening of the dot event peak as can be seen in Fig. 5.10 (c,d). The

53


data is obtained by recording a left-gate right-gate map of the charging events [such as Fig. 5.8 (a)] for different Vbcd . The extracted FWHM of the peak and tunneling rates are averaged over 15 right gate values (Vrg = [−150 − 25 mV]) in regime B (see blue squares in Fig. 5.8 (b,c) where the tunneling coupling and the FWHM are shown as a function of Vrg at Vbcd = 0.5 mV). In order to account for variations of the tunneling rate with Vrg , an average tunneling rate Γ0 is defined as the average tunneling rate for Vbcd < 2 mV for each Vrg value. In Fig. 5.10 (c), the average rate Γ/Γ0 increases up to 40% from Vbcd ≤ 2 mV to Vbcd = 7 mV. Note also the increase of the standard deviation of the average, reflected in the errorbar. In Fig. 5.10 (d), the FWHM is calculated from the peak width using the leverarm αlg = 0.6 obtained from Fig. 5.8 (c). Similar to the tunneling rate, the peak width depends approximately linear on the detector bias above Vbcd = 2 mV. Due to the correlation with the noise in the charge detector, the back-action from the detector on the dot is attributed to mainly arise from shot noise generated in the detector constriction [156, 158]. Photon emission and absorption is possible in a wide range of energies in graphene due to the linear, zero-bandgap electronic dispersion. Heating due to acoustic phonons [159, 160] is less plausible because the phonons have to couple via the SiO2 substrate across a different material. In Ref. [161], a GaAs quantum point contact heated the reservoirs of a double dot by 1 K with a current of ≈ 160 nA. Here, the maximum detector current of Icd = 22 nA at Vbcd = 7 mV (Icd = 4 nA at Vbcd = 2 mV) is small compared to the change in FWHM which would correspond to a change in Tel from 1.7 K to 3.1 K. In addition, graphene has a high thermal conductivity (≈ 5000 W/m/K [16]) compared to SiO2 (1.3 W/K/m, both at RT) and therefore the generated heat is expected to thermalize in the leads of the graphene constriction (rather than to heat the dot leads via the oxide). However, in order to clarify this mechanism, further experiments including energy dependent absorption mechanisms in single [162] and double dots [158, 159, 163, 164] are required.

5.2.6

Summary

Using a graphene constriction as a detector, it is possible to detect individual charging events in a quantum dot with high signal-to-noise ratio but limited bandwidth (see also section 5.3) in a time-resolved fashion. For the measurements recorded at low detector bias, the tunneling rate can be modeled conventionally by a single time-dependent process with a temperature broadened energy distribution of carriers in the lead. Gating of the tunneling barrier reveals a non-monotonic gate dependence of the tunneling coupling by counting individual charging events. This behavior is attributed to resonant tunneling through localized states in the barrier strongly coupled to the lead with ~Γ ∼ 6 meV. For a detector bias Vbcd > 2 mV, we see a symmetric broadening of the energy distribution of the tunneling events accompanied by an increase in the detector noise. The back-action is suspected to originate from shot noise in the charge detector.

54

5.3. Measurement setup and performance of the detector

5.3

Measurement setup and performance of the detector

As mentioned in the introduction of this chapter the signal-to-noise ratio is expected to be larger in graphene compared to conventional semiconductor circuits due to the enhanced electrostatic coupling coming along with the reduced spacing of the charge detector and the quantum dot. Nevertheless the maximum bandwidth in the presented device is lower than what has been achieved in semiconductor quantum circuits. In this section we will see that among other issues an enhanced noise level is responsible for this limitation. To understand the origin of the noise we have to take a closer look at our measurement setup.

Bandwidth of the detector

Rf efn

cold Cc

Rcd n ecd

(b) 104

IV-converter

(a)

~

Rcd = 100kȍ

~ ~ -

n eamp

CF Vout

+ Filter

Signal amplification

5.3.1

10

3

102 101 1.0

b Vcd

0.1 101

ADC

Rf 0ȍ Rf 0ȍ Rf 0ȍ 102 103 104 Frequency (Hz)

105

Figure 5.11: (a) Measurements setup used for time-resolved experiments. The current from the charge detector is amplified by a room temperature I-V converter circuit with the amplification depending on the feedback resistor Rf . The shunt capacitance Cf is used to reduce the capacitive noise gain at higher frequencies. The dominating noise contributions are indicated in red (see text). This figure is adapted from Ref. [165]. (b) Frequency response of the I-V converter for different values of Rf . The amplification is determined by the ratio Rf /Rcd . The bandwidth is limited by the low-pass formed between the feedback elements [fbw ∼ 1/(2πRf Cf )]. A shunt capacitance Cf = 2 pF is used in the calculation. The measurement setup used for time-resolved charge detection is shown in Fig. 5.11 (a). The signal amplification of the operational amplifier (opamp) circuit

55


is given by Vout =

Zf v(f ) Vin Rcd v(f ) + 1

(5.7)

with Zf = Rf /(1 + 2πiRf Cf ) the feedback impedence, Rcd the resistance of the detector and v(f ) = v0 /(1+if /fg ) the frequency dependent gain of the op-amp (with v0 = 106 and fg = 16 Hz). In Fig. 5.11 (b), the amplification is plotted for different feedback resistors Rf using Rcd = 100 kΩ (solid). Up to 10 kHz, the frequency dependence of the gain can be neglected and the amplification is determined by RZcdf (dashed). The bandwidth of the I-V converter is limited by the feedback resistor and the shunt capacitance Cf to fbw ∼ 400 Hz (5 kHz) for the feedback resistors used with RF = 100 MΩ (10 MΩ).

5.3.2

Detector noise

In Fig. 5.12, the frequency spectrum of the detector current noise is plotted for Rf = 10 MΩ and 100 MΩ in (a) and a roughly 50 times higher detector resistance Rcd in (b). The spectra are obtained by averaging the power spectral density of ten 0.5 s long time traces without charging events.1 Note that the plotted current noise is obtained from the output voltage noise by dividing with the sample resistance to allow comparison with the detector signal. The total noise spectrum is dominated by three noise contributions indicated in red in Fig. 5.11. At high frequencies √ the most important contribution is the amplifier input voltage n noise eamp ∼ 5 nV/ Hz [167], responsible for an output voltage noise of Rf 1 + i2πCc Rcd f out eamp = 1 + · enamp . (5.8) 1 + i2πCf Rf f Rcd The increase of the amplifier noise above fcng ∼ 1/(2πRcd Cc ) (here ≈ 160 Hz) is called capacitive noise gain and is limited at higher frequencies due to a low pass filter formed by the feedback resistor and shunt capacitance 1/(2πRf Cf ) (see short dashed lines). At even higher frequencies, the finite bandwidth of the gain v(f ) = v0 /(1 + if /fg ) with v0 = 106 and fg = 16 Hz, leads to a reduction of the noise (see kink around 2 kHz in the spectrum plotted in Fig. 5.12 (b), not modeled in calculation). The capacitive noise gain can be greatly reduced by using a cold amplifier which can be placed closer to the sample minimizing the cable capacitance [168, 169]. An alternative approach is to build a matched radio frequency circuit to read out the detector [145, 170, 171]. A cold amplifier would also reduce the thermal noise generated by the feedback resistor p (5.9) enf = 4kB T Rf . However, the thermal noise poses no limitation in the current measurement as the low-frequency spectrum is dominated by sample noise. 1

For the noise spectrum of the setup without sample see Ref. [165].

56


-11

10-2

10-3 10

-12

10

-13

10-14 101

total noise Rcd 1/f noise amp noise Rf thermal noise

10-4

10-5 2

3

10 10 Frequency (Hz)

10-10 Rcd 0ȍ615

10

sample 1/f noise

amp noise

lim. gain

10-1

10-12

10-2

10-13

10-3

4

10

10-11

1

2

3

10 10 Frequency (Hz)

current noise (A/Hz1/2)

10

(b) Rf 0ȍ615 Rf 0ȍ615

charge noise (e/Hz1/2)

current noise (A/Hz1/2)

(a) 10-10

10-14 104

Figure 5.12: (a) Current- and charge-noise versus frequency for different feedback resistors Rf = 10 MΩ (blue) and 100 MΩ (red) measured at a bias Vbcd = 1.7 mV with optimal signal-to-noise ratio (Rcd = 570 kΩ). The noise is obtained by averaging the noise spectra of 10 time-traces each 0.5 s long. The total modeled noise (black) is mainly determined by 1/f-noise of the sample (long dashes) and the amplifier noise (short dashes). The thermal noise of the feedback resistor is lower in magnitude (dotted-dashed). On the right scale the charge noise is plotted. It is obtained by dividing the noise by the step height of the signal Isignal = 1.9 nA induced √ by charging the dot with an additional −3 electron. The value below 10 e/ Hz above f = 1 kHz is comparable with the resolution shown in Ref. [166]. A frequency independent indicator is the signalto-noise ratio (SNR) extracted from the time traces. For Rf = 10 MΩ we have SNR = 19 with a time resolution (half step) of 2.5 ms and for Rf = 100 MΩ SNR = 31 with a time resolution of 250 µs. (b) Current- and charge-noise versus frequency measured in a regime with larger sample resistance Rs = 25 MΩ at a bias of Vbcd = 0.5 mV and Rf = 100 MΩ. Despite a reduction in current noise, the overall charge noise is larger due to the reduction of the signal amplitude. At low frequencies, the noise is mainly limited by 1/f-noise of the sample. At intermediate frequencies, the noise spectrum is dominated by amplifier noise which is cut off at higher frequencies by the reduced gain of the opamp. The sample noise can be described by a 1/f-dependence adding to the output noise according to the signal transfer function given in Eq. (5.7) (long dashed lines). As independent noise sources add up incoherently, the total noise at the output is given by X e2tot = e2x . (5.10) x

57


with eout x the individual noise contributions at the output. The amplitude of the sample noise is obtained by comparing etot with the measured spectrum. In Fig. 5.12 (a) we obtain encd (1 Hz) = 0.3 . . . 0.4 µV and encd (1 Hz) = 20µV in Fig. 5.12 (b). Compared with the frequency dependence of GaAs QPC detectors in Refs. [166, 171], the sample noise in our detector is roughly three times larger. Note that additional low frequency noise due to charging events is present in the referenced papers leading to a steeper low frequency dependence. The 1/f-noise is characteristic for charge fluctuators in imperfect contacts or traps in the substrate. Here it might also be due to parasitic resonances in the constrictions [see Fig. 5.9 (c)]. To clarify the origin, further investigations of the contact noise and different detector geometries are desirable. Additional noise sources √ not described above include thermal charge noise of Rcd the detector (SI ∼ 15 fA/ Hz for√Rcd = 570 kΩ) and the current noise of the operational amplifier SIamp = 1.6 fA/ Hz. Both sources are comparatively small in magnitude and can be neglected.

5.3.3


In the presented noise spectrum √ device, a current step of up to 2 nA with a relevant cd below 2 pA/ Hz is measured at a detector bandwidth of 4 kHz (V√ b = 1.7 mV). This corresponds to a charge sensitivity of the detector below 10−3 e/ Hz comparable to what has been reported in Ref. [166]. Although in principle a bandwidth of up to 30 kHz can be achieved with a conventional room temperature amplifier setup, the bandwidth is limited in the presented measurements to below 1 kHz to ensure sufficient SNR for counting. The first limitation is posed by the larger sample noise of our detector. The negative influence on the SNR is limited by reducing the bandwidth with a larger feedback resistance. Concerning the amplifier noise at higher frequencies, the comqpc paratively high resistance of our detector (Rcd = 500 kΩ compared to Rcd = 35 kΩ in QPC-based detectors [166, 172]) brings a weaker signal amplification and hence the amplifier noise gets proportionally more important. Here as well an increase of the feedback resistance is beneficial as the amplification is restored. A second limitation for the SNR is related to the general difficulty using SETbased charge detectors to maintain the working point with optimal sensitivity. In principle it is possible to compensate the influence of any gate on the charge detector with a charge-detector gate. However, in the presented sample the change of Vcdg induced additional charge fluctuations in the detector and was therefore kept at a constant value during measurements. A more general issue for time-resolved charge detection in graphene is the limited tunability of the dot barriers. Although the current through etched graphene quantum dots can be tuned over several orders of magnitude, the height of the tunneling barrier is limited by the width of the constriction. Despite the mentioned issues with the current implementation, graphene charge

58


detectors show high sensitivity with signal changes of more than 50% at moderate cryogenic temperatures of T = 1.7 K. This is because graphene offers the possibility for very strong electrostatic coupling between charge detector and quantum dot as the spacing between them can in principle be made even smaller than the 35 nm in this device. The coupling could be even further increased by making use of the monoatomic thickness of graphene and vertically stacking dot and detector [148]. In addition, the bandwidth can be improved using a radio frequency setup and a low temperature amplifier where √ a bandwidth of up to 1 MHz was shown with a charge −4 sensitivity of ∼ 2 · 10 e/ Hz in Ref. [145]. This would allow to measure at higher frequencies where the 1/f - sample noise is much smaller.

59

Chapter 6 Crossover from electrons to holes in a perpendicular magnetic field The spectrum of a quantum dot containing many electrons is usually difficult to understand in detail. It is therefore a general task in quantum dot physics to reach the regime where the dot is only filled by one or two electrons. In small bandgap semiconducting systems, the few electron regime is characterized by a large spacing between Coulomb peaks (or large Coulomb diamond) separating the first electron and hole [173, 174]. In graphene quantum dots it is difficult to observe the increased spacing, as not only the confinement gap Econ but also the charging energy Ec scales inversely with the diameter (width) and both are comparable in size (see chapter 4). An alternative way is to analyze the magnetic field dependence of Coulomb peaks to distinguish between electrons and holes. On a first glance, the diamagnetic shift of Coulomb peaks could be investigated for this purpose. However, the diamagnetic shift is negligible in graphene due to the atomic thickness of the 2D electron gas. In the following it is shown that the electron-hole crossover can be pinned down to a few states via the formation of the graphene specific zero energy Landau level at high magnetic fields.

6.1

”Fock-Darwin” spectrum in graphene

The energy spectrum of a confined system in the presence of a magnetic field is typically described with the Fock-Darwin energy diagram using a single-particle approximation and assuming a parabolic confinement potential with confinement energy ~ω0 [175, 176]. The corresponding Hamiltonian is given by H=

(p + |e|A)2 1 ∗ 2 2 + m ω0 r . 2m∗ 2

(6.1)

An example of the resulting energy spectrum is plotted in Fig. 6.1 (a) for m∗ = 0.067m0 (GaAs) and ω0 = 1 meV. The Fock-Darwin spectrum provides a good

60

6.1. ”Fock-Darwin” spectrum in graphene

qualitative description for the measured magnetic field dependence of radially symmetric quantum dots in semiconductors [87] (see also Ref. [34]).

(a)

(b)

n = -1

h

n=0

e

n=1

30 50 nm

B (T)

B (T)

3 2 1 0 0

2 4 6 8 Energy (meV)

10

20 10 0 -0.2

-0.1

0

0.1

0.2

Energy eV

Figure 6.1: (a) Fock-Darwin spectrum of a conventional semiconductor quantum dot with effective mass (m∗ = 0.067m0 for n-type GaAs and ω0 = 1 meV). (b) Numerical calculation of the perpendicular magnetic field dependence of a 50×80 nm graphene quantum dot (see inset) comparable to the island investigated in the experiment in Fig. 6.2 (a). The simulation is performed using a third-nearest neighbor tight-binding approximation where the contribution from edge states at the zigzag boundaries is removed by employing a strong on-site potential of ±2eV on the carbon atoms of the two sublattices with dangling bonds [102] (see also edge-states section below). In graphene, the overall behavior is markedly different as can be seen by comparing Fig. 6.1 (b) with Fig. 6.1 (b), where the result of a numerical tight-binding calculation is shown [for more details see Ref. [132, 177]]. Apart from the fact that electrons and holes contribute to the spectrum, there are several other important differences in graphene: At low magnetic field, the density of states is low near E = 0 and increases linearly with E reflecting the linear dispersion of the Dirac cone (see p section 2.1.2). Similar as in semiconductor dots where the magnetic length `B = ~/eB becomes comparable or even smaller than the dot size, the spectra are increasingly dominated by the magnetic field and the density of states converges towards Landau levels (see top of Fig. 6.1 (a)nd section 2.1.4). In bulk graphene, the peculiar n = 0 Landau level with a field-independent energy E0 = 0 marks the electron-hole crossover point. Single-particle levels around the Dirac point converge from negative and positive energies towards E = 0. These qualitative results are confirmed by analytical calculations for circular dots where an infinite mass boundary is introduced [178, 179]. The transition region between the two limits is characterized by various avoided crossings with states diverging to different Landau levels below filling factor ν = 2 [see also magnification of the dashed square in Fig. 6.5 (a)]. The avoided crossings are a consequence of the finite coupling between the K and K’ valley. The coupling

61

Chapter 6. Crossover from electrons to holes in a perpendicular magnetic field

is possible if the sublattice symmetry is broken which is the case for atomically sharp edges or short range impurities [177]. At ν = 2, this leads to a pronounced kink due to the abrupt disappearance of anti-crossings for higher fields as discussed below [Fig. 6.5]. For small magnetic fields, a linear B-dependence is predicted for massless Dirac Fermions without valley scattering [180]. The dependence gets quadratic in the presence of atomically sharp edges or other short range scatterers and can be used to measure the amount of inter-valley scatterers in the experiment [177].

6.2

Sample characterization at zero magnetic field

In the following, transport spectroscopy is used to investigate the magnetic field dependence of a graphene quantum dot and the results are compared to the theoretical calculations presented above.

(a) Bŏ B||

s

0.2 0.1

h

0

d

e

10 Vpg (V)

pg bg

(b)

(c)

B A

10-1 10-2

5

10-3

C

0 -20

-10

0 10 20 Back gate Vbg (V)

30

G (e2/h)

G (e2/h)

0.3

10-4

Figure 6.2: (a) Scanning force microscope picture of the investigated quantum dot sample. The perpendicular and parallel magnetic field orientation are indicated by the arrows. The potential energy of the dot is mainly tuned by the back gate (bg) and the plunger gate (pg). (b) Source-drain conductance as a function of back gate (Vb = 4 mV). (c) Source-drain conductance as a function of back gate and plunger gate. Three main measurement regimes across the transport gap (white regime) are indicated. Regimes A and B are measured at fixed back gate voltage Vbg = −0.9 V and -1.8 V respectively whereas regime C is measured at constant plunger gate voltage (Vpg = 2 V). An image of the sample is shown in Fig. 6.2 (a). The ≈ 50 nm wide and ≈ 80 nm long quantum dot is connected to source (s) and drain (d) via two 25 nm wide and 10 nm long tunneling constrictions. The device is tunable by the back gate (bg) and several in-plane gate electrodes, including the plunger gate (pg). All measurements reported in this and the next chapter were performed in a dilution refrigerator with a base temperature of 90 mK.

62

6.2. Sample characterization at zero magnetic field

D2

-2

, A1

A4

-3

,,

-4

-1.5 B

B1

-2 3.0

(b) Bias (mV)

20

3.5

4.0 4.5 5.0 5.5 Plunger gate Vpg (V)

(c) 0.1 G (e2/h)

Vbg (V)

-0.5 A -1

-1

D1

10 mV

0

(a)

log (G) (e2/h)

In the back-gate characteristic of the device (Fig. 6.2 (b), taken at Vb = 4 mV) the crossover from hole- to electron conductance is visible with the transport gap region extending from 0 < Vbg < 10 V (see also section 4.2.2). The quantum dot charge can be selectively tuned by the nearby plunger gate, allowing tunneling spectroscopy of dot states by reducing the influence of parasitic resonances in the constrictions (see section 4.2) Low-bias conductance measurements as function of back- and plunger gate are shown in Fig. 6.2 (c). The transport gap (bright area) measured as a function of the back gate shifts with a slope given by the relative gap lever arm αpg,bg ≈ 1.3. Three different measurement regimes are indicated by the labels A-C for which the magnetic field dependence is studied in more details in this and the next chapter.

6.0

0

20 mV

A

,

0

A1

A2

A4

A5

,,

A10

-20 2.5

3.0

3.5 4.0 Plunger gate Vpg (V)

4.5

5.0

5.5

Figure 6.3: (a) Dot conductance as a function of plunger gate and back gate in the gate regime indicated by the box in Fig. 6.2 (c) (Vb = 100 µV). Coulomb blockade peaks of the quantum dot are visible as parallel lines. At higher plunger gate voltage (II) additional resonances with a weaker plunger gate dependence (α2 ) are observed. (b) Coulomb blockade diamond measurement in regime A. (c) Close-up of (b) showing excited states and co-tunneling lines (arrows). Much more fine structure in the Vpg - Vbg parameter plane appears on finer voltage scales [Fig. 6.3 (a), close-up of box in Fig. 6.2 (c)]. We observe a sequence of essentially straight tilted lines signifying Coulomb blockade resonances of the quantum dot. The corresponding relative plunger/back gate lever arm is given by

63


qd α1 = αpg,bg = 1.27 [left dashed line in Fig. 6.3 (a)], comparable to the influence of gap the plunger gate on the gap (αpg,bg ). A Coulomb blockade diamond measurement in regime A [solid line A in Fig. 6.3 (a)] is shown in Fig. 6.3 (b). We observe strong fluctuations of the addition energy ranging from 13−32 meV. Individual Coulomb peaks are labeled (A1,A2,. . . ) and can be identified in both measurements by comparison of the peak spacings. A close-up of the boxed region in Fig. 6.3 (b) [see Fig. 6.3 (c)] shows a well resolved and rich excited state spectrum with corresponding co-tunneling onsets (see arrows) and analyzed in more details in section 4.4.

6.3

Evolution of Coulomb resonances in perpendicular magnetic field

The evolution of Coulomb resonances as a function of a magnetic field B⊥ perpendicular to the sample plane is shown in Fig. 6.4 for two different regimes A and C. In Fig. 6.4 (a) we tune over 50 Coulomb resonances from hole to electron transport since we cover the whole transport gap [vertical line A in Fig. 6.2 (c)]. The measurement also includes those resonances discussed in Fig. 6.3 (a,b) indicated by the labeled Coulomb resonances. The measurement has been taken in two steps with an overlapping part (below Vpg = 7 V) to reduce the number of charge rearrangements coming along with larger gate sweeps. In the overlapping part all features are reproduced except for a small, compensated shift of ∆Vpg = 50 mV. 3

4

Plunger gate Vpg (V) 5 6

8

9

5 0

(b)

0

1

A10

2

3

4

* (V) Effective plunger gate Vpg 5 6

5 -3

-2

-1 -2 -3 -4

7

10 C 0

-1 -2 -3 -4

,,

, A1

B (T)

7

10 A

log (G) (e2/h)

2

log (G) (e2/h)

B (T)

(a) 1

-1

0

1 2 Back gate Vbg (V)

3

4

5

6

Figure 6.4: Coulomb blockade peaks as a function of perpendicular magnetic field in gate regime A (a) and regime C (b). For comparison the back gate voltage in (b) is translated into an effective plunger gate scale (for regime A) given at the top of the graph by using the relative lever arm αpg/bg of the dot. The evolution of Coulomb resonances with (perpendicular) magnetic field shows a common trend at lower gate voltages (see e.g., the box at Vpg ≈ 2.6 V) to bend

64

6.3. Evolution of Coulomb resonances in perpendicular magnetic field

for increasing B-field towards higher energies (higher Vpg ). In contrast, we find at higher gate voltages the opposite trend (see e.g., the box at Vpg ≈ 6 V), where resonances tend to shift to lower energies for increasing B-field.

6.3.1

Edge states and influence of constrictions

This overall pattern is disturbed by additional features such as localized states, regions of multi-dot behavior and strong amplitude modulations attributed to constriction resonances and edge states in the dot. For example, we observe around Vpg = 3 V (marked with I) a weakly coupled state crossing the resonances to the left at B = 5 and 9 T. At B = 0 this state is not visible in transport but leads to a large diamond and an extended peak-to-peak spacing [I in Fig. 6.3 (a,b)]. The weak magnetic field dependence of this state and its low visibility in transport is compatible with a localized edge state. The edge state density is estimated by Wimmer and coworkers in Ref. [181] and their result is briefly summarized here. As already discussed in section 4.2 localized edge states are expected to form in the presence of random edges. The zero energy edge states in a zigzag nanoribbon with cristallographically clean edges [see Fig. 4.3 (e)] disperse if the sublattice symmetry is broken (by next nearest neighbor hopping, passivating atoms at edges) and in a confined geometry also due to wave function overlap (finite size effect). For a typical dot size the dispersion is dominated by the broken sublattice symmetry (order of 100 meV) giving rise to a constant density of states ρedge ≤ 2d[nm]/|E0 | within the energy interval 0 < E < E0 with E0 = ∆−γ2 where ∆ denotes the edge potential and γ2 the next nearest neighbor hopping term. This is an upper limit as the number of edge states is significantly reduced (factor 0.5) in the presence of additional edge disorder [181]. Typical energies (γ2 = 0.1γ1 ) give an edge state density of ρedge = 0.5 meV−1 over 300 meV for a dot with d = 70 nm, comparable to the single-particle spacing. Based on these calculations edge states represent a significant contribution to the low energy spectrum in graphene quantum dots. Although we see individual signatures of edge states in the investigated regimes their occurrence is rare, which might be attributed to an enhanced edge roughness. In addition to strongly localized states, we observe several level crossings and splittings in the region of Vpg = 5 − 6 V (starting with II). These crossings are attributed the presence of additional resonances following a different relative lever arm [αpg,pg = α2 ≈ 0.53 see upper right of Fig. 6.3 (a)]. Mixing of these states with dot states also leads to deformed Coulomb-blockade diamonds [see II in Fig. 6.3 (b)].

6.3.2

Comparison with calculations

In order to compare the measurements with theoretical calculations peak positions are extracted from Fig. 6.4 and plotted in Fig. 6.5 (b) for regime A. Reproducible peak shifts for all Coulomb blockade peaks in this gate range were compensated

65


based on a linear fit to the resonance at 4.25 V. In addition the weakly coupled state around Vpg = 3 V is removed (no edge states are considered in the calculation). h

(c) A

e

12 Bŏ (T)

B (T)

12 8 4 0

-400

-200

(b)

0 200 Energy (meV)

B (T)

p2p (meV) 3

5 7 4 6 Plunger gate Vpg (V)

8

12

8

8

4

4

(d)

4 2

A3

0

0 3.6

400

12 A 8 0 1

A2

9

Gqd (10-2e2/h)

(a)

3.9 Vpg (V)

22 21 20 19

A C Theory

0 2 4 6 8 10 12 B (T)

Figure 6.5: (a) Coulomb peak spectrum derived from the calculated eigenenergies by assuming a constant charging energy and spin degenerate states. The dashed line indicates filling factor ν = 2 above which all eigenstates continuously evolve into the zero energy Landau level. (b) Coulomb peak position extracted from the measurement shown in Fig. 6.4 (a). (c) Perpendicular magnetic field dependence of a finite bias Coulomb diamond cut [Vb = 11 mV, see dashed line A2-A3 in Fig. 6.3 (b)]. The kink is attributed to the crossing with filling factor ν = 2 [see (a)]. (d) Averaged peak-to-peak spacing as a function of magnetic field for peaks of two regimes. The peaks of regime B are lower in energy in agreement with the kink at higher B field also observed in the calculated spacing from (b) (solid for peaks of regime A, dashed for B). The measurement can be compared with the calculation of a comparable dot geometry shown before in Fig. 6.1 (b). As the measured charging energy (Ec ≈ 18 meV) is much larger than the single-particle level spacing (δE ≈ 4 meV) a constant Ec is included [see Fig. 6.5 (a)]. In addition, the states are assumed to be spin degenerate at B = 0 with a weak (Zeeman) splitting at finite fields (see also chapter 7). In addition to the convergence of the levels towards E = 0 around the Dirac point a clear change of the B-dependence at lower magnetic field is observed (dashed line). In this regime the magnetic field induces level crossings of the single-particle levels, being visible as small fluctuations of the peak positions [barely visible in Fig. 6.5 (a)]. In analogy to Ref. [182] no crossings appear beyond the filling factor ν = 2 boundary (ridge of the first Landau level). In contrast to GaAs quantum dots this leads to pronounced “kinks” in the peak positions [dashed line in Fig. 6.5 (a)] because in graphene the n = 0 Landau level does not shift in energy with increasing magnetic

66

6.4. Discussion and summary

field. This behavior is a unique consequence of the interplay between the linear two dimensional (2D)-dispersion of graphene, and the finite-size dot. Comparing the numerical data and the measurement [Fig. 6.5 (a) and Fig. 6.5 (b)] we find the same qualitative trend of states running towards the center (E0 ). Some kinks in the experimental data may be attributed to level crossings with the filling factor ν = 2 boundary [see also finite bias cut in Fig. 6.5 (c) and arrows in Fig. 6.5 (b) and Fig. 6.4 (b)]. Beyond the data presented here, these kinks in the magnetic field dependence of Coulomb resonances may be used in future experiments to identify the few-electron regime in graphene. The convergence of levels towards E = 0 can be quantified by the nonmonotonic evolution of the mean level spacing plotted in Fig. 6.5 (d). Measured peak-to-peak spacing within Vpg = 2.4 − 4.9 V [at B = 0 in Fig. 6.5 (b)], are compared with calculated level spacings for −315 meV < E < −15 meV [at B = 0 T in Fig. 6.5 (a)]. The energy range for the comparison is obtained by converting the plunger gate and back gate voltage into energy using the lever arms αpg = 0.125, αbg = 0.1, and allowing for an energy offset of 530 meV obtained by comparison of Fig. 6.5 (b) and Fig. 6.5 (c). In the experiment we find a maximum at B ≈ 4 T followed by a decrease for increasing B, consistent with the theoretical expectation (solid line), reflecting the convergence of single-particle levels towards E = 0. The slower decrease in the experiment may be attributed to the presence of additional localized states as described above. These findings are confirmed by additional measurements taken in a different regime sweeping the BG across the charge neutrality point [Fig. 6.4 (b)] with the same qualitative behavior. In Fig. 6.5 (d) (triangles) the mean peak-topeak spacing of resonances −1.3 < Vbg < 1.7 V is plotted. The data points have been shifted down by -2.5 meV for better comparison with theory (correspondong to Ec = 20.5 meV). By using the same energy offset (530 meV) and lever arms as before the corresponding energy interval −415 meV < E < −115 meV is obtained. Also for this regime the shape of the measured mean energy spacing agrees well with that obtained by theory. The maximum is now shifted to higher magnetic field (B ≈ 7 T) as expected for an energy window shifted to larger |E|. By comparison with theory, we assign the location of the transition region between hole and electron transport to the states ranging from Vpg = 4.3 V to 5.5 V qd (Vbg = −0.9 V). Using the relative lever arm αpg,bg ≈ 1.27 [see dashed line in Fig. 6.2 (c)] this region is converted to an effective back gate voltage scale and plotted as a gray shaded box in Fig. 6.2 (b,c). The extracted crossover region is in good agreement with the center of the transport gap as expected if the quantum dot and the constriction are similarly doped.

6.4


The perpendicular magnetic field dependence of Coulomb blockade peaks reveals unique graphene specific properties over the formation of the magnetic field inde-

67


pendent n = 0 Landau level at high magnetic fields. The evolution of Coulomb peaks in the vicinity of the electron-hole crossover is characterized by sharp kinks at filling factor ν = 2, above which ideally no anti-crossings with eigenstates running towards different Landau levels are observed anymore. In principle it is easier to observe the Landau level spectrum in larger graphene quantum dots as lower magnetic field are required. So far this transition could not be observed in larger systems either due to limited conductivity of the barriers [178] or due to double dot formation at high magnetic fields in a quantum dot with strong coupling to the leads [183]. Although the charge neutrality point can be pinned down to only a few Coulomb peaks the single-electron/hole regime could not be precisely identified. A first difficulty arises from localized edge states which are expected to occur in the vicinity of the crossover and exhibit the same weak magnetic field dependence as the other states around the crossover. Second, no monotonous increase of the excited state energies (or addition energy) while approaching the charge neutrality point is observed [Fig. 4.11 (a)]. However, in the few electron regime, the theoretical prediction of the excited states energies from the single-particle picture has only limited validity due to the divergence for N = 0. In addition many-particle effects may play a role even in the vicinity of the crossover if particle and hole excitations are not clearly separated in energy (see also next chapter). A third issue are the influences of parasitic resonances in the constrictions. By coupling to the dot the charging energy is significantly modulated and the Coulomb peak spacing gets hard to interpret [region II in Fig. 6.3 and Fig. 6.4 (a)]. We will have a look at possible solutions to these issues after the discussion of the parallel magnetic field behavior in the last chapter.

68

Chapter 7 Measurement of spin states in graphene quantum dots In the last three chapters, we focused on the charge and the orbital wave functions in graphene quantum dots. However, as motivated in the introduction, graphene provides also an interesting host to process information using spins. First experiments in the direction of spin transistors explored the spin coherence length by spin injection [184, 185]. This spin transport experiments could not confirm long spin life times in graphene so far because the spin coherence length is most probably limited by impurity scattering [186]. Further experiments explored the Zeemansplitting using universal conductance oscillations in graphene and nicely confirmed the expected value for the g-factor in bulk graphene to be |g| = 2 [187]. Quantum dots offer an other powerful tool to investigate spin properties, as they allow to address individual spins and can serve as a host to process quantum information using spin-qubits. So far the largest progress in this direction has been made in GaAs-based quantum dots, where single spin preparation, manipulation and read-out has been demonstrated [22, 23, 142, 188]. In this chapter we want to explore spin states in graphene quantum dots by measuring the Zemann effect.

7.1 7.1.1

Introduction Experimental techniques

The small energy scale on which Zeeman splitting occurs poses difficulties in taking well-resolved measurements. Several techniques have been employed to measure the Zeeman-spin-splitting in quantum dots. The splitting can be measured by analyzing the differences in the addition energy of subsequent Coulomb peaks for increasing in-plane field (B|| ) [189–192]. Energy fluctuations induced by potential changes are minimized by investigating the Zeeman-splitting of ground and excited states in Coulomb blockade measurements [191, 193, 194]. Further accuracy can be gained

69

Chapter 7. Measurement of spin states in graphene quantum dots

by analyzing the Kondo-peak splitting [194, 195] or by investigating spin dependent co-tunneling onsets in B|| [194, 196]. Due to the lack of the two latter features in the investigated device we will focus on the evolution of Coulomb peaks and excited states in this chapter. The Kondo-effect [197] has not been observed in etched graphene quantum dots so far. This might be attributed to the coupling of excess spins on the quantum dot with parasitic states in the constrictions (discussed in chapter 4), rather than the coupling to a collective cloud of electrons in the Fermi seas of the leads.

7.1.2

Zeeman spin splitting in parallel magnetic fields

In a parallel magnetic field the spin-dependent energy splitting due to the Zeeman Effect [198] can be measured, as it is not completely masked by the strong orbital shifts induced in perpendicular orientation. In conventional semiconducting 2DEGs the parallel magnetic field dependence is defined via the Schrödinger Hamiltonian HS =

(p + |e|A)2 + g ∗ µB Bk S, 2m∗

(7.1)

consisting of a kinetic and a Zeeman term with the effective g-factor g ∗ and S the spin quantum number along Bk . The gauge can be chosen to be A = Bk /2(0, −z, y) and we get HS ≈

|e|2 B||2 2 p2 ∗ 2 + Sg µ B + y + z . B k 2m∗ 8m∗

(7.2)

Here the small (Lx ) orbital part proportional to B|| has been neglected. The parallel magnetic field dependence is then given via a linear Zeeman term and a quadratic diamagnetic shift which depends on the roughly level-independent cross section of the wave function in the direction of the magnetic field. As the diamagnetic shift is also affected by lateral confinement present in quantum dots, the effect on the leads and the island is not the same and leads to a parabolic shift of Coulomb peaks towards higher gate voltage for electron-like states. The corresponding measurement for GaAs quantum dots together with a good description of the effect can be found in section IV of Ref. [190]. In graphene the situation is different as we have a linear instead of a quadratic dependence on the vector potential A. With the in-plane magnetic field pointing in x-direction there is one possible gauge where A has no z-component given by A = 0, −zB|| . Without spin the Dirac Hamiltonian is given by HD = vF σp + vF |e|σy zB|| | {z } | {z } H0

(7.3)

λH1

The strength of the magnetic field dependence can be estimated perturbatively by defining HD = H0 + λH1 . The first order approximation for the energy is obtained

70

7.2. Measuring the Zeeman-effect

as

En1 = hφn |H1 |φn i = φn0 |vF |e|σy zBk |φn0 . In contrast to 2DEGs formed in semiconductor heterostructures the z-extent of the electron gas is negligible in graphene and En1 ≈ 0. In second order perturbation the energy is P 2 n6=m | hφn0 |H1 |φm0 i | 2 En = . 0 En0 − Em The energies of the sp2 -bonds (σ-bands) are far away from the Fermi-level. This leads to a large energy difference in the denominator and therefore the second order contribution is also negligible. Hence, in a flat graphene sheet we expect the energy dependence of an in-plane magnetic field to be solely given by the Zeeman splitting. (B|| ) = Sg ∗ µB B|| .

7.1.3

(7.4)

Effective g-factor

The effective g-factor is a material parameter and can be estimated using k · ptheory [34, p. 45]. The deviation from g = 2, for a free electron in vacuum, is determined by the strength of the spin-orbit interaction in the material. In a material with spin-orbit interaction the exact value of g ∗ depends also on the energy separation of different levels or bands and hence might differ significantly from the bulk value in confined systems. This size dependence of the effective g-factor has been nicely shown in Ref. [192]. In graphene the spin-orbit interaction is expected to be weak and hence the gfactor in a graphene quantum dot is expected to be close to g = 2 as found in bulk graphene [187].

7.2 7.2.1

Measuring the Zeeman-effect Identification of peak pairs

In general, the extraction of a small Zeeman splitting from the magnetic field dispersion of individual conductance resonances is difficult because any small orientational misalignment of the sample causes a significant perpendicular magnetic field component. Typically, the dispersion of states in the perpendicular field dominated by orbital effects is much stronger than the faint Zeeman splitting. Hence, a small misalignment severely hampers the extraction of precise g-factor values from the dispersion of individual peaks. The problem can be circumvented by analyzing the peak-to-peak spacing of spin-pairs, i.e., two subsequently filled electrons occupying the same orbital state with opposite spin orientation. This is because orbital contributions can be significantly reduced by subtracting the positions of individual peaks sharing the same orbital shift [190, 199].

71


Peak pairs sharing a similar perpendicular field dependence can be found in Fig. 6.4 (a) highlighted by the box at Vpg ≈ 2.6 V or in Fig. 6.5 (c), both in gate regime A [Fig. 6.2 (c)]. In the latter, the comparable evolution of the two diamond edges (white arrows) could also arise from slightly different orbitals as the Coulomb peaks around the electron-hole crossover exhibit similar magnetic field dependencies [see Fig. 6.5 (a)]. 5

5.2 5.4 5.6

B

5

4

10

2

15 20

(b) 0 B|| (T)

0

10-4 2 B1 B2 B3 B4 B5 B6 B7 B8 B9 10 11 4 10-3 6 8 10-2 10 12 10-1 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 Vpg (V)

(c)

Vpg,B (V)

5.7 5.8 5.9 6

6.1 6.2

perp. parallel B=0

5

100 pA

6

4.2 4.4 4.6 4.8

Gqd (e2/h)

Bŏ (T)

8

4

Gqd (10-2e2/h)

Vpg (V)

(a)

5.1 5.2 5.3 5.4 5.5 Vpg [= Vpg,A] (V)

Figure 7.1: Rotation of the sample orientation with respect to the magnetic field. (a) Measurement of Coulomb blockade peaks as a function of perpendicular magnetic field B⊥ in gate regime B [with Vbg = −1.8 V, see Fig. 6.3 (a)] at Vb = 100 µV. To ease comparison with regime A an effective plunger gate voltage Vpg := Vpg,A = Vpg,B − 0.7 V is plotted, incorporating the difference in back gate voltage [see also (c)]. The magnetic field dependence is similar to that observed in regimes A and C (Fig. 6.4). The conductance is plotted in a linear scale to enhance the visibility of amplitude changes. (b) After rotating the sample by 90◦ the same Coulomb peaks are measured in parallel magnetic field. To enhance the visibility of the peaks the conductance is plotted in logarithmic scale. (c) Comparison of Coulomb peaks measured at B = 0 in parallel orientation (red) and in perpendicular orientation (black). Potential spin pairs are also identified in the gate regime B indicated in Fig. 6.3 (a). In Fig. 7.1 (a) the evolution of Coulomb peaks with increasing perpendicular field is displayed. The lowest two peaks (B1, B2) and the following two (B3, B4) are identified as potential spin pairs due to their similar peak evolution. In Fig. 7.1 (b) a measurement of the same peaks is shown after careful in-situ rotation of the sample into an orientation parallel to the magnetic field. Due to friction the base temperature of 90 mK rose while tilting, but did not exceed 500 mK and the sample regime could be maintained. This can be seen best in Fig. 7.1 (c) with Coulomb peaks measured at B = 0 before and after the rotation showing similar Coulomb peak spacing and amplitudes. The step size in between Coulomb peaks is reduced

72

7.2. Measuring the Zeeman-effect

(see crosses) to save measurement time while maintaining a high resolution for the determination of the exact Coulomb peak position. Note that in order to ease comparison between different regimes the plunger gate voltage applied in regime B [Vpg,B top axis in Fig. 7.1 (c)] is converted to an effective plunger gate voltage in regime A: Vpg,A = Vpg,B − 0.7 V =: Vpg [bottom axis in Fig. 7.1 (c)], used throughout the chapter.

7.2.2

Extraction of the g-factor

To analyze the movement of the peaks shown in Fig. 7.1, the peak positions are extracted by fitting. The results for the two pairs (B1-B4) in Fig. 7.1 and the pair (A2-A3) in Fig. 6.5 (c) are shown in Fig. 7.2 (a). Peak positions are plotted with suitable offsets in Vpg such that pairs coincide at B = 0 T. We attribute the kink in peak B4 around 6 T to a crossing of two eigenstates with magnetic field dependencies differing in sign. A corresponding kink of peak B3 is observed around 10 T (see also Ref. [129]). For low perpendicular magnetic field (B⊥ < 3 T for A2/A3, B1/B2; B⊥ < 6 T for B3/B4) the orbital states of each pair have approximately the same magnetic field dependence. Hence spurious orbital contributions to the peak spacing resulting from slight misalignment are limited. The data in Fig. 7.2 (a) for parallel orientation of the magnetic field shows roughly linear splitting of pairs with increasing Bk . We interpret this splitting as the Zeeman splitting of two spin states with the same orbital wave functions. The magnitudes of the Zeeman splitting of the three pairs discussed above and for two additional peak-spacings from region A are plotted in Fig. 7.2 (b). The data is obtained in two steps. First, the peak-to-peak spacing from Fig. 7.2 (b) and corresponding measurement in regime A is extracted. Second, the peak spacing is linearly fit from 4.2 T to 13 T and the offset obtained from the fit is subtracted. The Zeeman spin splitting ∆E Z is given by [189, 190] Z ∆EN +1 = (SN +1 − 2SN + SN −1 ) gµB Bk + const.

Here SN is the spin quantum number along the B|| -direction of the dot with N (1) electrons. Spin differences of successive ground states ∆SN +1 = SN +1 − SN are half-integer values (e.g. ∆S (1) = 1/2 for adding a spin up electron or ∆S (1) = 3/2 for adding a spin up electron while ”flipping” another spin from down to up). Hence differences between three successive spin ground states take on integer values (2) ∆SN +1 = SN +1 − 2SN + SN −1 = 0, ±1, ±2, . . .. Apart from the splitting of the pair B2-B1 the splitting of all pairs in Fig. 7.2 (b) show slopes ∆E Z = ∆S (2) gµB B|| with integer values ∆S (2) = 0, ±1 and a g-factor value of approximately 2 [black dashed lines in Fig. 7.2 (b)].

73


(b)

100 mV

B4 B1

Vpg

B3

B2

peak spacing 'EZ (meV)

(a) B4 B3 B1 A3 A2

B2 A3 A2

12 10 8 6 4 2 0 2 4 6 8 10 12 Bŏ (T) B|| (T)

2 g = ±2

1.5 1

B4-B3 A3-A2

0.5

A4-A3

0

-0.5 -1

A5-A4

-1.5 -2 -2.5 0

B2-B1

2

4

6 8 B|| (T)

10 12

Figure 7.2: (a) Comparison of the evolution of three Coulomb peak pairs in perpendicular (left) and in-plane (right) magnetic field. The peaks are extracted by fitting from measurements such as Fig. 7.1 and offset in plunger gate voltage such that the pairs coincide at B = 0. (b) Peak spacing as a function of parallel magnetic field for the three pairs in (a) and two subsequent peak spacings (A4-A3 and A5-A4) in regime A. The expected Zeemann splitting ∆E Z = ±|g|µB B for a g factor |g| = 2 is plotted as dashed line. For the energy conversion the lever arm αpg = 0.13 ± 0.01 is used.

7.3

Spin filling sequence

In addition to the value of the g-factor in graphene quantum dots, it is instructive to know more about the sequence how spins are filled into the dot. However, to obtain the filling sequence we have to be able to extract spin-related shifts from individual unpaired conductance resonances despite the inherent difficulty mentioned in the beginning of the chapter. The final goal is to reconstruct a plausible spin-filling sequence of the quantum dot in regimes A and B. The following strategy is applied: first, the misalignment angle β between the sample plane and the magnetic field is estimated, such that second, the peak shifts in parallel field can be corrected for the orbital effects arising from the misalignment. The misalignment angle is estimated here, by assuming Zeeman spin splitting with |g| = 2 and linear peak shifts µ⊥ (i) in B⊥ in the range 0 ≤ B⊥ ≤ 2 T and µk (i) in Bk in the range 4.2 T ≤ Bk ≤ 13 T. The misalignment can then be compenstated (2) with µk (i)0 = µk (i) − βµ⊥ (i). If we consider the three options ∆SN +1 = 0, ±1 for the slope of the peak-to-peak spacing (= µk (i + 1)0 − µk (i)0 ), we can calculate a corresponding ”misalignment angle” for each case. The result is shown in Fig. 7.3, where the three misalignment angles for each peak-to-peak spacing are plotted. The best agreement is obtained for the sequence connected by the dotted line with a misalignment angle of βB = 3◦ ±1◦ (dashed). Note that the peak-spacings of the two pairs B2-B1 and B4-B3 are not considered in the fit because this procedure is very

74

7.3. Spin filling sequence

sensitive to small differences in the evolution of individual peaks (µ⊥ (i + 1) − µ⊥ (i)) as a function of perpendicular field. This is also the reason why some values in Fig. 7.3 are scaled to fit into the figure (1/10× for peak pair B2-B1 and B4-B3). 20 15

x 0.1 x 0.1

ȕB (degree)

10 5 0 -5 -10 -15 -20 0

¨6(2) = 0 ¨6(2) = -1

x 0.5

(2)

¨6 = 1 PLQȕB= 3°)

2

4 6 8 10 peak spacing (Bx+1-Bx)

Figure 7.3: Misalignment angle β for the three spin filling combinations ∆S (2) = 0, ±1 of the peak-spacings in regime B. The angle βB is obtained by compensating the peak shifts in Bk with a contribution from the individual perpendicular field dependence i (B⊥ ) ∼ ci B⊥ (where B⊥ = βB · Bk ), such that the resulting peak spacing (2) Z 12 corresponds to ∆E = ∆SN +1 gµB B|| (2) for |g| = 2 and ∆SN +1 = 0, ±1.

In regime A a different misalignment angle βA = 0 ± 0.5◦ is found because the sample had been rotated into perpendicular field orientation between measurements B and A. Now we will construct the filling sequence. The misalignment-compensatedslopes of the peak-spacings are shown in Fig. 7.4 as a function of the plunger gate voltage Vpg . The gray arrows indicate the change of the slope by the compensation of the angular misalignment. The error bar includes the error from both the lever arm estimation (10% of the slope) and the variance from the linear fit of the peakto-peak spacings. Three dashed black lines indicate the slopes corresponding to ∆E Z /∆B = {0, ±1} × 2µB and the data points are colored according to their closest ∆S (2) value (blue for ∆S (2) = −1, green for ∆S (2) = 0 and red for ∆S (2) = +1). Note that the two regimes A and B overlap: pairs B2-B1 and A5-A4 are in fact the same Coulomb peaks as it can be seen from a conductance measurement in dependence of Vpg and Vbg [see Fig. 6.3 (a)]. We can extract the spin-filling-sequence ↓↑↑↓↓↑↑↓ from peak A2 to B6 (dotted gray line). We emphasize that we see a clear deviation from ↑↓↑↓ as it has been seen in carbon nanotube quantum dots [173, 200– 202]. However, as we will see in the next section, this result can be made plausible by considering the exchange interaction between electrons in graphene. In addition to the peak spacing it is instructive to have a look at the individual Coulomb peak evolution in B|| . In Fig. 7.5 (a-c) the evolution of four peaks A2-A5 in regime A are plotted in pairs together with their mean values (black dots) as a function of perpendicular magnetic field for B⊥ = 4 to 13 T. Note that no correction for orbital effects is needed in regime A. A linear fit to the average energies reveals slopes of 23 µeV/T, 75 µeV/T and 21 µeV/T. Because we have a negligible diamagnetic shift in graphene, two slopes ±g/2µB = 58 µeV/T are expected for the individual peaks if SN = ±1/2. This picture is in agreement with the measurement and the spin sequence (A2:↓, A3:↑, A4:↑, A5:↓),

75


slope'EZ/'B (meV/T)

0.2

B9-B8 B4-B3

A3-A2

0.1

+1

B10-B9 B5-B4 B8-B7 B3-B2

0

B12-B11

A4-A3

-0.1

A5-A4

B6-B5 B7-B6

-0.2

'EZ JȝB'B

A2-A1

3.5

B11-B10

0

-1

(4-9T)

B2-B1

4

4.5 Vpg(V)

5

5.5

Figure 7.4: Slopes of peak-to-peak spacings [Fig. 7.2 (b)] obtained from a linear fit for B|| = 4.2 − 13 T for subsequent peaks in regime A and B. The slopes of measurement B are compensated by a parallel magnetic field misalignment of βB = 3◦ with the help of the perpendicular magnetic field dependence. The shift induced by this compensation is depicted by the gray arrows. The peak B7-B6 has a strong kink at B|| ≈ 9 T and is fitted only up to 9 T. The different slopes group in three categories depending on their spin filling (right side) and are colored accordingly. The dashed lines indicate the expected peak spacings for g = 2 which is ∆E Z /∆B = ±116 µeV/T for subsequently filling opposite spins and ∆E Z /∆B = 0 for filling twice the same spin. by taking into account an additional systematic slope of ∼ 20 µeV/T for the whole measurement (e.g. due to a slightly varying background potential). The difference in B⊥ together with misalignment can not explain this systematic change of the slope as a compensation would shift the peaks in opposite directions (for A3-A2: ≈ 0.06 µeV/T/◦ , A4-A3: ≈ −37 µeV/T/◦ and A5-A4: ≈ 24 µeV/T/◦ ). Unfortunately peaks A6 and higher are very noisy and the slope of peak A1 changes at 7T (−150 µeV/T from 0-7 T and 170 µeV/T from 7-13 T). The comparatively large change with B|| for peak A1 above 7 T could be due to the flipping of an excess spin on the dot (↓→↑) while loading an additional electron (↑). Such a process leads to a total spin change of +3/2 or ∆E Z /∆Bk = 3g/2µB = 174 µeV/T comparable with the observed slope. In regime B this analysis gives not as clear results, probably because of the error induced by the compensation of the misalignment angle.

7.3.1

Exchange interaction in graphene

In order to understand the observed filling sequence we compare the exchange interaction in graphene with other materials. The strength of the Coulomb interaction in a material can be characterized by the ratio rs between interaction energy and

76

7.3. Spin filling sequence

0.8

Energy (meV)

0.6 0.4

0.8

(a)

0.6

A2 A3

ȝH97

-0.2

4

6

0.2 ȝH97

0

ȝH97

0 -0.2

-0.4

8 10 12 B|| (T)

A4 A5

0.4

-0.2 (A2+A3)/2

(c)

0.6

0.2

0

-0.4

ȝH97

A3 A4

0.4

0.2

0.8

(b)

(A3+A4)/2 4

6

-0.4

8 10 12 B|| (T)

(A4+A5)/2 4

6

8 10 12 B|| (T)

Figure 7.5: Fitted individual peak positions of A3-A5 together with the mean value of of the two slopes. (a) ”Spin pair” A3-A2 shows an average slope of 23 µeV/T. (b) Peak A3 and A4 have similar slopes of 75 µeV/T ≈ 17 µeV/T +g/2µB . (c) Peaks A4 and A5 show similar behavior as the peaks in (a) but with opposite order of the slopes. The observed slopes are in agreement with the spin filling sequence by assuming a systematic shift of ≈ 21 µeV/T. kinetic energy [203]. First we consider the example of semiconducting carbon nanotubes. In one dimension the interaction parameter is given over rs =

1 Eint ≈ ∗ , EF aB ns

(7.5)

with a∗B = ~2 /(e2 m∗ ) the effective Bohr radius and ns the charge carrier density. In the case of a large band gap (large effective mass m∗ ) and low fillings interaction are dominating (rs 1). In the extreme case the mutual repulsion of the electrons can lead to the formation of an ordered Wigner crystal [204]. On the other hand, in small bandgap or metallic nanotubes the interaction parameter is small (rs 1) except for low densities. In this case fourfold shell filling is routinely observed and the spectrum can be well described using the constant interaction model [200–202]. In contrast to conventional 2D semiconductors with parabolic bands (and also 1/2 bilayer graphene) where rs ∼ ns , in graphene this ratio is rs =

e2 /(4π0 ree ) Eint = p = αg ≈ 0.9, EF π~2 vF2 ns

(7.6)

independent of the charge carrier density ns (see also a recent review by Das Sarma et al. [205]). Here ree = (πns )−1/2 is the mean electron separation and αg = α·vF /r c the graphene fine-structure constant. For the dielectric constant a mixture of vacuum and SiO2 = 2.5 is used. Note that the same expression for rs is also obtained by compar-

77


ing the interaction √ energy with the confinement energy in a graphene quantum dot [(N + 1) ≈ 2vF ~ N /d, see Eq. (4.4)]. The splitting of the eigenstate energies due to the exchange interaction can be estimated using ξ(rs = 1) ≈ 0.6∆ [206–208]. The exchange energy splitting is comparable to the single-particle-level-spacing ∆ and can therefore lead to ground-state spin polarization S > 1/2 in agreement with our interpretation and observations in GaAs quantum dots [189, 190]. An alternative reason for spin-polarization in graphene could arise from the valley degeneracy. However, the presence of atomically sharp edges leads to a coupling of the two valleys and the valley degeneracy is expected to be lifted [25, 209] (see also section 4.2.1). We end this section with a comment regarding the opposite spin filling of the two spin pairs B2-B1 and B4-B3 in regime B [see Fig. 7.4]. This sign change of the ground state spin can be understood as a consequence of the interaction. Due to the spin-dependent interaction of a state with other electrons in the dot, the eigenenergies of opposite spins are no longer the same and the degeneracy is lifted. Depending on the location of the wavefunction and the number of electrons in the dot, the eigenenergy of the ”spin up” state might be lower or higher than that of the ”spin down” state from the same orbital.

7.3.2

Evolution of excited states in parallel magnetic field

Next we describe an attempt to observe the Zeeman splitting of excited states in parallel magnetic field. Fig. 7.6 (a) shows part of a Coulomb blockade diamond measurement at B = 0 with a zoom to the positive bias part of the Coulomb peak A11 [see dashed box in Fig. 6.3 (b)]. Along the white bar at Vb = 12 mV the evolution of the excited-state peaks is measured in parallel magnetic field and shown in Fig. 7.6 (b). The six differential conductance peaks labeled in Fig. 7.6 (b) are linked to the stronger resonances in Fig. 7.6 (a). The magnetic field dependence of the two faint lines between resonance 2 and 5 in Fig. 7.6 (a) could not be traced in Fig. 7.6 (b). The peaks 1, 3 and 4 are caused by ground and excited states aligned with the source Fermi-energy, while peaks 2, 5 and 6 are linked to the drain. The evolution of the peaks can be attributed to two different slopes. For comparison black dashed lines with slopes differing by the Zeeman energy ∆E Z = |g|µB B with g = 2 are plotted on top. Despite a slight underestimation of the splitting, the lines agree reasonably well with the measurement. Hence we attribute the two slopes to Zeeman spin splitting. Peak 2 in Fig. 7.6 (b) could also be split-off from peak 3, but the amplitude difference indicates that the peak is rather caused by an excited state aligned with drain, giving rise for the faint resonance in the diamond at ≈ 9 meV below the ground state energy in Fig. 7.6 (a). Peak six has a kink at around 9 T which might arise from an avoided crossing with peak five. The absence of clear peak splittings that would be induced by lifting the spin degeneracy with B can be seen as another indication of significant charge carrier interactions.

78

7.4. Summary

(b)

Gqd (10-3e2/h) -20 0 20 40 60 80

Gqd (10-3e2/h) -20 0 20 40 60 80

20

12.5

10

10

0

1

B|| = 0T

-10

-20

4.9

6

2 B|| = 0T

B|| (T)

Vb (mV)

(a)

5

5 4.95 5 Plunger gate Vpg (V)

7.5 5

4 3

5.05

Vb=12mV

1

2.5 4.95

2

3

4

5

4.975 5 5.025 Plunger gate Vpg (V)

6 5.05

Figure 7.6: (a) Coulomb diamond measurement close to the electron-hole crossover and close-up of excited states at positive bias of the right Coulomb peak [A12, see also dashed box in Fig. 6.3 (b)]. Excited states are indicated by guides to the eye to ease comparison with the magnetic field dependent data. (b) Fixed bias cut at Vb = 12 mV [white line in (a)] as a function of in-plane magnetic field. The slopes of the dashed lines correspond to the Zeeman spin splitting with g = 2 and are plotted over the raw data for comparison.

7.4

Summary

In this chapter we have seen how spin states in graphene quantum dots can be accessed in a parallel magnetic field. The evolution of Coulomb peaks and excited states show a linear and g = |2| compatible Zemann splitting with magnetic field. The agreement with the bulk g-factor, is expected in a material with weak spin-orbit coupling such as graphene. Note that not all conductance peaks were stable enough for the required high resolution measurement of the peak position and some followed a different behavior as expected from the Zeeman splitting similar to observations in GaAs quantum dots [189, 190]. The observed spin sequence with spin filling polarization and the absence of clear degenerate ground and excited states are in agreement with the strength of Coulomb interactions expected from a comparison of interaction and kinetic energy of electrons in graphene. In contrast to GaAs or nanotube quantum dots we do not expect a significant dependence of the exchange interaction on the carrier density (see Ref. [190, 208]). The dependence of the state energies on the microscopic details of the wave functions and possibly also on spin polarized edge states makes it difficult to predict or interpret the exact spin filling sequence even if only few electrons are on the etched graphene quantum dot. One posibility to reduce the influence of interaction is to embed graphene in a high- dielectric. An other specialty of graphene is the absence of a diamagnetic shift due to the real two-dimensional nature of the electron gas. However, as the diamagnetic shift is assumed to be independent of the state it is visible only in the evolution of individual Coulomb peaks. For the most stable peaks a small and roughly linear systematic shift (∼ 20 µeV/T) is found in addition to the spin contribution. This shift might

79


be attributed to a small drift in the electrostatic potential but has not been checked in details. To further clarify this issue additional measurements, hopefully with even better energy resolution are desirable. In general, apart from the electrostatic stability also the corrugations present in graphene can alter the observed behavior in Bk [210]. From scanning tunneling microscopy measurements the ripples are estimated to be roughly 0.5 nm high (rms) with corrugation lengths of 5−10 nm [13, 211] corresponding to misalignment angles of ∼ 2 − 4 degrees. Hence, local perpendicular magnetic field components might be present even with perfect alignment of the sample plane. In the following outlook will discuss future directions to enhance control over spin states in graphene quantum dots.

80

Chapter 8 Summary and outlook The intriguing electrical properties combined with its unique geometry make the ’wonder’ material graphene a fascinating nano-electronics research object. Despite the short time since graphene is available for experiments a wealth of experiments have been performed and many interesting properties were found.

Towards single electrons and spins in graphene quantum dots In this thesis transport through graphene nanostructures with emphasis on the energy and spin states in graphene quantum dots has been presented. The fact that graphene does not exhibit a band gap poses a significant challenge for tunable carrier confinement. Part of the problem can be overcome by cutting (etching) graphene into narrow constrictions allowing to control the transmission via gating. Such constrictions can then be used as tunable tunneling-barriers for single electron transistors and quantum dots. By the observation of Coulomb blockade and the match of the related energy to the island size of the device, the expected functionality could be confirmed (chapter 4.3). Here the use of in-plane graphene gates has several advantages as it allows to tune the individual parts of the device independently without the necessity of additional processing steps needed e.g. for top gates. By shrinking the diameter of the island below 200 nm quantum-confinement effects become visible in transport (see chapter 4.4). The observed single-particle energy scales with the inverse device size as expected. However, the measured spectrum in an individual device is hard to understand as it shows no clear dependence on the number of electrons on the island. This is similar to observations in semiconductor quantum dots with many electrons where their mutual interaction mixes up the electron filling due to the fluctuating Hartree and exchange energy which depend on the microscopic details of all dot states filled. This observation is supported (i) by the absence of an even-odd spin filling behavior when measuring the Zeeman effect in a magnetic field parallel to the sample plane and (ii) by the comparable magnitude of kinetic and Coulomb energy in graphene which is curiously independent of the electron density (chapter 7).

81

Chapter 8. Summary and outlook

In general, the full understanding of the energy spectrum in quantum dots is therefore often only possible in the few electron regime where only one or two electrons are on the dot. In contrast to other materials it is not straight forward to find this regime in graphene as the confinement band gap, the charging energy and the single-particle level spacing all scale inversely with the device size. Therefore is very hard to distinguish the first hole from the first electron just by investigating the Coulomb-blockade peak spacing. Additional information can be gained by analyzing the evolution of orbital states under the influence of a perpendicular magnetic field (chapter 6). With increasing field the states around the crossover evolve towards the graphene specific zero energy Landau level marking the electron-hole crossover. Using this technique the crossover could be narrowed down to a few Coulomb peaks, but no signatures for an empty dot was observed in this region. This might have several reasons such as the possibility of excitations between hole and electron like states but is difficult to nail down due to the presence of localized states in the constrictions and edge states. Much more Coulomb blockade Detailed transport measurements on individual (short) constrictions revealed sharp conductance resonances and other features typical for Coulomb blockaded transport (chapter 4.2). The high sensitivity of the constriction conductance on the surrounding electrostatics can be harvested by using it as a charge detector to detect charge changes on a nearby quantum dot, independently of the dot source-drain current. On the other hand the interplay between the two devices can also be employed to study the properties of constrictions. Using the dot as a detector, charging of localized charge puddles in the constriction similar to quantum dots is observed. The formation of disconnected puddles in graphene constrictions is attributed to strong carrier localization at zigzag edges and inhomogeneous doping giving rise to disconnected electron-hole puddles around the charge neutrality point. Despite of these additional complications it is possible to detect individual electron transitions in real time using such a graphene detector. The high sensitivity is attributed to the large electrostatic coupling between the two graphene devices which can be placed very close to each other due to the ultimate flatness and negligible carrier depletion in graphene. Time resolution allows to analyze the tunneling processes in more details. In a particular gate regime the tunneling rate shows a non-monotonic energy dependence with a Lorentzian shape attributed to a resonant state in the constriction which is strongly coupled to the lead. In other gate regimes resonant states in the constriction are completely localized and charging events can be observed as described before. The independent control of the individual barriers and localized states is even more difficult in graphene double dots which have been studied recently in several experiments [127, 128, 212–214]. The measurement of Pauli spin-blockade in a graphene double dot would be highly interesting as it provides an excellent tool to

82

8.1. Further directions

investigate spin properties, such as the spin coherence time [90]. However, the effect has not been observed so far in graphene. In a single graphene quantum dot a g-factor close to 2 has been measured (chapter 7) in agreement with the expected weak spin-orbit interaction. Nevertheless, many open questions regarding the spin in graphene quantum dots remain, including the interplay of dot spins with spins of edge states and localized states in the constrictions.

8.1

Further directions

It always facilitates the understanding and controllability of a complex system if the properties of interest can be well isolated from other effects. Hence, it is highly desirable to reduce the amount of disorder and the resulting parasitic localized states and edge states in graphene nanostructures. This highly active research area can be grouped in three directions: (i) reduction of bulk disorder (ii) reduction of edge disorder and (iii) alternative ways to create a band gap or confine carriers in graphene: (i) It has been shown, that a significant part of the bulk disorder results from the interaction of the graphene with the conventional silicon-oxide substrate. Record carrier mobilities and the fractional quantum Hall effect could be observed by etching the oxide and suspending graphene [66, 215–217]. The suspension of graphene nanostructures allows to study the interplay of electrical with the unique mechanical properties of graphene. However, free standing structures are harder to gate and not only electrostatically but also mechanically delicate. An interesting alternative is the use of a different substrates such as hexagonal boron-nitride where the negative influence of the substrate is greatly reduced while containing the mechanical stability [65]. With both methods it remains important to limit the amount of contaminations and to use annealing procedures to remove as many residues as possible. The potential of clean devices is exemplary in carbon nanotubes, where a wealth of new phenomena could be observed in ultra-clean devices by growing the tubes as a last fabrication step [174, 204, 218]. (ii) The reduction of bulk disorder alone might not be sufficient to significantly simplify the observations in graphene nanostructures. This is because all graphene nanostructures with sharp edges feature edge states around the charge neutrality point, except those with perfect armchair orientation. In order to construct atomically sharp edges bottom-up fabrication methods based on molecular precursors [219], unzipping of carbon nanotubes [220] or edge sensitive chemical cutting methods are proposed [221]. (iii) An other alternative is to find a different way to control charge carriers in graphene. It has been shown that a band gap can be opened in bilayer graphene due to breaking of the sublattice symmetry in an electric field [222, 223]. A remaining issue with this approach is the disorder which is comparable to the band gap and

83

Chapter 8. Summary and outlook

so far hinders complete electrical isolation. Chemical functionalization is an other promising route for the creation of a soft confinement [224, 225]. The big and still growing interdisciplinary community gives confidence that the current issues can be overcome and that measurements on devices of improved quality allow to extract many more interesting details of confined quasi-particles in graphene quantum dots and allow to make use of the special physical properties in future graphene nanodevices.

84

Appendices A

Charge detection in double quantum dots

Transport trough a double dot at finite bias is given by pairs of tripple points by tuning the dot potential over adjacent side gates (Vgr and Vgl ) as shown in Fig. A.1 (a). In this triangles the chemical potentials of source, left dot, right dot and drain are arranged in staircase-like fashion allowing for a current to flow (more details can be found e.g. Ref. [151]). From such a measurement the main energy scales including the addition energies and the corresponding lever arms are obtained. Here the lever arms are estimated as αgl,l = 0.16e, αgl,l = 0.12e, αgl,r = 0.06e and αgr,l = 0.05e. R L The addition energies are given by Eadd = 9.4 meV and Eadd = 18.1 meV.

(a)

3

Idot (pA)

2

1

0

(b)

dIlcd/dVgr (a.u.)

1

0

í

-750 113 mV

180 85 mV

160 Vgl (mV)

Vgl (mV)

-800 78 mV

-850

140

-900

100

-950

80 0

50 100 Vgr (mV)

75 mV

120

60

150

80 100 120 140 160 Vgr (mV)

Figure A.1: (a) Current trough a double dot as a function of the two side gates voltages Vgr and Vgl at finite source-drain bias of Vbdot = 5 mV (Vbg = 0 V). (b) Derivative of the left charge detector current as a function of Vgr and Vgl (Vbg = 28.65 V). This rather large asymmetry of the energies has not been observed in previous cool-downs of the sample [127, 128] (before the fabrication of charge detectors) and R is reduced in a different back gate regime [see Fig. A.1 (b)] where Eadd = 8.4 meV L and Eadd = 13.1 meV. In this regime no measurable current flows through the double 85

dot but the change in dot charges can still be detected over a jump in the charge detector current [see Fig. A.1 (b)]. The change of the dot charges is reflected in a hexagon pattern. Charging of the right dot leads to a weaker signal in the left charge detector as the electrostatic coupling is smaller (see dashed line).

(a)

(b)

9 135 mK

7.2

8.6

Gcd(10-1e2/h)

Gcd(10-2e2/h)

8.8 8.4 8.2 8 7.8

7.6 -1.5 -1

-0.5 0 0.5 dVgr (mV)

1

7.1 7 6.9 6.8 -1

1.5

158 mK

-0.5

0 0.5 dVgr (mV)

1

(c) Gcd(10-1e2/h)

1.21

W ȝH9

1.2 1.19 1.18 1.17 1.16

-0.4

-0.2 0 0.2 0.4 dot detuning (meV)

Figure A.2: (a,b) Change of the left charge detector conductance as a function of Vgr by loading an additional electron to the left dot (a) and the right dot (b). The dashed line is a fit with a temperature broadened Fermi function. (c) Change of the detector conductance as a function of an interdot tunneling event. From the fit (see text) the interdot tunneling coupling is obtained as t = 52 µeV for an average electron temperature of 144 mK. From the exact shape of the detector conductance at a charging line information about the broading of the density distribution in the leads and the dot levels can be obtained. In Fig. A.2 the change of the charge detector conductance as a function of Vgr (a,b) and level detuning (c) is shown for the three different charging events (see insets). The addition of an electron from a lead to the nearby dot is governed by the Fermi broadened density of states in the corresponding lead. From a corresponding fit (see also Fig. 5.2) the electron temperatures in the leads are obtained as Tel = 135 mK and Tel = 158 mK in source and drain respectively. As expected the electron temperature is slightly above the base temperature of the measurement system of

86

Tb = 120 mK. From five different fits an average electron temperature of 144 mK is obtained. In Fig. A.2 (c) the charge detector conductance as a function of energy detuning δ between the two dot levels (see inset) is shown. The conductance has been smoothened over a five point window. Following to DiCarlo and coworkers [143] the change in charge detector conductance is described by ! √ δ 2 + 4t2 δ . tanh Gcd (δ) = Gbgr (δ) + ∆Grl √ 2kB Te δ 2 + 4t2 Here Gbgr (δ) is the background charge detector conductance affected by the capacitive crosstalk to the swept gates, ∆Grl the change in conductance due to the excess electron moving from the right to the left dot (0, 1) → (1, 0) and t the tunneling coupling as main fit parameter. From the previous analysis an average electronic temperature of Te = 144 mK has been estimated. Using the previously obtained value Te = 144 mK for the electron temperature, a fit to the data yields t = 52 µeV comparable to t ∼ 14 µeV obtained previously by analyzing the low bias dot current at triplepoints [128].

87

B name 1L

List of samples picture Ti/Au

pg

300 nm s

d lg

rg

3L 300 300 nm nm

pg

Ti/Au

ss

d d lg lg

rg rg

cd1 pg lg

300nm rg

s

d cd

sp2 100nm s

d

pg

device info single-layer flake NoC06 etching: no sp used, contacts: Ti/Au, processed: 07/2007 ddot,eff ∼ 250 nm wbarr ∼ 50 nm

measurement Single electron transistor Coulomb blockade Tunable barriers see chapter 4

three-layer flake NoC06 etching: no sp used, contacts: Ti/Au, processed: 07/2007 ddot,eff ∼ 355 nm wbarr ∼ 60 nm

Single electron transistor Coulomb blockade Tunable barriers see chapter 4

single-layer flake NoH07 etching: no sp, contacts: Ti/Au, processed: 02/2008 ddot,eff ∼ 200 nm wbarr ∼ 35 nm wcd ∼ 45 nm `gap ∼ 60 nm

Constrictions Transport gap Charge detection see chapters 4, 5

single-layer flake JCJ1 E2 etching: sp EBL, contacts: Cr/Au, processed: 09/2008 ddot,eff ∼ 70 nm wbarr ∼ 25 nm

Excited states Cotunneling Electron-hole crossover Spin-states see chapters 4, 6, 7

88

name sp3

picture 150 nm

s

d

lg

pg

rg

lg

pg

rg

sp4

qd

s

d

cd cdg

100 nm

dd lcd

cg

L

100 nm

s

rcd R

d

device info single-layer flake JCJ14 etching: sp EBL, contacts: Cr/Au, processed: 10/2008 ddot,eff ∼ 140 nm wbarr ∼ 75 nm wsp ∼ 20 nm

measurement Open quantum dot see chapter 4

single-layer flake J04 B5 etching: sp EBL, contacts: Cr/Au, processed: 01/2009 ddot,eff ∼ 90 nm wbarr ∼ 20 nm wcd ∼ 35 nm `gap ∼ 35 nm

Closed quantum dot Time resolved charge detection see chapter 5

single-layer flake FD5 etching: sp EBL, contacts: Cr/Au, processed: 02/2009, 03/2010 ddot,eff ∼ 90 nm wbarr ∼ 30 nm

Double quantum dot: - couldowns: DDa, DDb Additional fabrication of charge detectors: - couldown: DDc see chapter 4, App. A

89

C

Useful relations and constants in graphene

Carbon-carbon bond distance a0 in the hexagonal lattice [35, 36] Graphene lattice constant a= Nearest neighbor hopping or interaction parameter γ1 [35] (also denoted as t or V ) Fermi velocity around K points (vF = 106 m/s ⇒ γ1 = 3.09 eV)

a0 = 0.142 nm.

(9.1)

√ 3a0 = 0.246 nm.

(9.2)

γ1 = −3.033 eV.

(9.3)

vF = 3γ1 a0 /(2~) ≈ 106 m/s.

Fermi wave vector kF =

√

πns =

EF . ~vF

(9.4)

(9.5)

Fermi wavelength 2π 2π~vF λF = = = kF EF Energy as a function of charge charrier density

r

4π . ns

√ E = ~vF πns .

(9.6)

(9.7)

Density of states √ ns E D(E) = 4 =4 √ . 2 2 2π~ vF 2 π

90

(9.8)

C.1

Capacitance of discs and stripes to parallel plates

Disc The capacitance of a metal disc with diameter d separated by a dielectric (thickness tox ) from a parallel ground plate can be approximated by [130] Cdp =

π0 (ox + 1)d . −1 4tox (ox +1) tan dox

If tox d the tan−1 -term can be approximated with π/2 leading to Cdp = 40 (ox + 1)/2 · d = 40 eff d, equal to the self-capacitance of an isolated disk. In the opposite limit with d tox the capacitance can be approximated by a plate capacitor with an additional fringe capacitance depending on the perimeter π(d/2)2 πd ln (2) Cdp = ox 0 + . tox π Stripe The capacitance of a metal stripe with width w and length L separated by a dielectric from a parallel ground plate can be approximated by [130] ! √ 2( cosh α + 1) √ Csp = 2π0 (ox + 1)L/ ln cosh α − 1 with α=

πox W 4tox (ox + 1)

√ and valid if 1/ 2 > tanh α which is for our purpose equivalent with tox W . In the opposite case with tox W the plate capacitor model with perimeter fringe capacitance is valid π(d/2)2 2L ln (2) Csp = ox 0 + . tox π In graphene separated from the back gate by 300 nm SiO2 the bulk capacitance is given by Cg,2D = 7.2 × 1010 cm−2 V−1 |e|.

91

D

Processing of graphene on SiO2 samples

Process

Description of process steps

Photolitho markers (wafer):

1. HMDS coating

a) 2-5 Droplets on paper in glassware (not on glass) b) put wafer (oxide on top) on a glass grid in the glassware

2. Spinning

a) resist: ma-N 1405 (cover at least 1/3 of wafer) b) use new spinner with 3000rpm/2/40s c) softbacke: 100◦ C for 60s

3. Exposure

a) MA6 maskaligner, channel 1 (365 nm) if problem use CP b) lamp test with correct sensor (365 nm) c) intensity 350 mJ/cm2 (around 50-70 s)

4. Development

a) use ma-D 533s, 60-70s gentle agilitating b) rinse in water (2min, frequently tilting sidewards to exchange water)

Marker evaporation

1. Plasma asher:

30s at 200W

2. Metal evaportion

2 nm Cr (or Ti), 40-50 nm Au. Trade off: thick (200 nm): better e-Beam contrast (alignment, focus), thin (50 nm) tendency of more flakes

3. Lift off

around 10 minutes in 50◦ C acetone until gold starts to lift, blow acetone with pipette and use US (up to power 9) to remove gold.

4. Wetbench: clean sample Dicing wafer

Deposit graphene

Contacting graphene

a) 2min in acetone at power 9 (US bath) b) 2min in isopropanol at power 9 (US bath)

1. Protection resist:

Spin Shipley 1805 60 s with 5000 rpm. Hard bake 2 min at 115◦ .

2. Dicing

dice into 7.1x7.1mm2 chips (Hansjakob Rusterholz)

1. Clean chips (2x)

a) 2 min in aceton at power 9 (US bath) b) 2 min in isopropanol at power 9 (US bath)

2. Plasma asher

2 min at 200 W (0.7 Torr)

3. Gray room

cut tape pieces from blue tape next to wafer saw

4. Exfoliation

disperse graphite flakes over piece of blue tape, fold 6-10 times (stop if it does not stick well anymore!), put chip on tape and press gently from top (not sidewards)

5. US test

short US pulse in acetone, power 1, to make sure the flakes stick well

6. Cleaning

leave sample in 50◦ C acetone for at least 30 min in order to remove glue remainings (no additional US because this may rip the flakes!)

1. Clean samples

rinse in aceton, isopropanol, blow dry N2

2. Soft bake

2 min at 120◦ C (remove moist, optional)

3. Spinner

a) P(MAA/MMA) 1:1 CB, 1000 rpm for 1 s, 5000 rpm for 45 s; bake 5 min at 180◦ C b) PMMA 950K 1:1 CB, 1000rpm for 1 s, 5000 rpm for 45 s; bake 5 min at 180◦ C

4. eBeam 5. Development

MIBK:IPA 1:3 60 s (no stirring), isopropanol 60 s, N2 blow dry

6. Metal evaportion

2 nm Cr (or Ti), 40-50 nm Au

7. Lift off

some minutes (∼ 10) in warm acetone until gold starts to lift, stir in acetone or blow acetone with pipette. If some parts cannot be removed: short US pulse, start with power 1

92

Etching graphene

1. Clean samples

rinse in aceton, isopropanol, blow dry N2

2. Soft bake

2 min at 120◦ C (remove moist, optional)

3. Spinner

(i) multilayer etch: PMMA 950K 1:1 CB, 2000rpm for 2 s, 5000 rpm for 45 s. Bake 5 min at 180◦ C (thickness: ∼ 135 nm) (ii) for sp-line: PMMA 950K 2:5 CB 1000 rpm/2 s, 6000 rpm/45 s. Bake 5 min at 180◦ C (thickness ∼ 45 nm thickness)

4. eBeam 5. Development 6. Etching RIE80

Packaging

MIBK:IPA 1:3 60 s (no stirring), isopropanol 60 s, N2 blow dry a) plasma cleaning, 30 s with O2 and SF6 , 15 min O2 only. Parameters: O2 flow: 100 sccm, SF6 flow: 30 sccm, forward power 300 W, pressure 200 mTorr, (recipe: Christoph O2 clean) b) load sample c) etch multilayer graphene: etch 20 s with parameters: O2 flow: 5 sccm, Ar flow: 40 sccm, forward power 60 W, pressure 30 mTorr (recipe: Johannes C-etch thick). etch sp-line: etch 10 s with parameters: O2 flow: 20 sccm, Ar flow: 40 sccm, forward power 65 W, pressure 35 mTorr (recipe: Johannes C-etch thin).

1. Prepare conductive glue

H20E A,B (1:1), approx 15 g of each, mix with toothpick

2. Glue sample

contact the back gate: put conductive glue from the side of the wafer to the ground plate of the chip. Optionally use dummy chip as spacer for better access with the SFM.

3. Bake

in annealing oven, with Formier gas approx 10 m3 /h, 400 mBar, 20 min at 120◦ C

93

Publications Time-resolved charge detection in graphene quantum dots J. G¨ uttinger, J. Seif, C. Stampfer, K. Ensslin, and T. Ihn Phys. Rev. B 83 , 165445 (2011). Raman spectroscopy on etched graphene nanoribbons D. Bischoff, J. G¨ uttinger, S. Dröscher, T. Ihn, K. Ensslin, and C. Stampfer J. Appl. Phys. 109, 073710 (2011). Transport through a strongly coupled graphene quantum dot in perpendicular magnetic field J. G¨ uttinger, T. Frey, C. Stampfer, T. Ihn, and K. Ensslin Nanoscale Research Letter 6 , 253 (2011). Coherent Electron-Phonon Coupling in Tailored Quantum Systems uller, S. Dröscher, P. Roulleau, S. Baer, T. Choi, F. Molitor, J. G¨ uttinger and T. M¨ K. Ensslin, and T. Ihn Nature Communications 2 , 239 (2011). Imaging Localized States in Graphene Nanostructures S. Schnez, J. G¨ uttinger, M. Huefner, C. Stampfer, K. Ensslin, and T. Ihn Phys. Rev. B 82, 165445 (2010). Spin States in Graphene Quantum Dots J. G¨ uttinger, T. Frey, C. Stampfer, T. Ihn, and K. Ensslin Phys. Rev. Lett. 105 , 116801 (2010). Transition to Landau Levels in Graphene Quantum Dots F. Libisch, S. Rotter, J. G¨ uttinger, C. Stampfer, and J. Burgdörfer Phys. Rev. B 81 , 245411 (2010). Observation of excited states in a graphene double quantum dot F. Molitor, H. Knowles, S. Dröscher, U. Gasser, T. Choi, P. Roulleau, J. G¨ uttinger, A. Jacobsen, C. Stampfer, K. Ensslin, and T. Ihn Euro Phys. Lett. 89 , 67005 (2010).

94

Graphene single-electron transistors T. Ihn, J. G¨ uttinger, F. Molitor, S. Schnez, E. Schurtenberger, A. Jacobsen, S. Hellm¨ uller, T. Frey, S. Dröscher, C. Stampfer and K. Ensslin Materials Today 13 , 44 (2010) . Energy and transport gaps in etched graphene nanoribbons F. Molitor, C. Stampfer, J. G¨ uttinger, A. Jacobsen, T. Ihn, and K. Ensslin Semicond. Sci. Technol. 25 , 034002 (2010). Graphene quantum dots in perpendicular magnetic fields J. G¨ uttinger, C. Stampfer, T. Frey, T. Ihn, and K. Ensslin Physica Status Solidi B 246 , No. 11, 2553 (2009). Electron-Hole Crossover in Graphene Quantum Dots J. G¨ uttinger, C. Stampfer, F. Libisch, T. Frey, J. Burgdörfer, T. Ihn, and K. Ensslin Phys. Rev. Lett. 103 , 046810 (2009). Transparency of narrow constrictions in a graphene single electron transistor C. Stampfer, E. Schurtenberger, F. Molitor, J. G¨ uttinger, T. Ihn, and K. Ensslin Int. J. Mod. Phys. B 23 , 2647 (2009). Transport through graphene double dots F. Molitor, S. Dröscher, J. G¨ uttinger, A. Jacobsen, C. Stampfer, T. Ihn, and K. Ensslin Appl. Phys. Lett. 94 , 222107 (2009). Energy gaps in etched graphene nanoribbons C. Stampfer, J. G¨ uttinger, S. Hellm¨ uller, F. Molitor, K. Ensslin, and T. Ihn Phys. Rev. Lett. 102 , 075426 (2009). Transport gap in side-gated graphene constrictions F. Molitor, A. Jacobsen, C. Stampfer, J. G¨ uttinger, T. Ihn, and K. Ensslin Phys. Rev. B 79 , 075426 (2009). Observation of excited states in a graphene quantum dot S. Schnez, F. Molitor, C. Stampfer, J. G¨ uttinger, I. Shorubalko, T. Ihn, and K. Ensslin Appl. Phys. Lett. 94 , 012107 (2009).

95

Charge Detection in Graphene Quantum Dots uller, F. Molitor, T. Ihn, and K. Ensslin J. G¨ uttinger, C. Stampfer, S. Hellm¨ Appl. Phys. Lett. 93 , 212102 (2008). Coulomb oscillations in three-layer graphene nanostructures J. G¨ uttinger, C. Stampfer, F. Molitor, D. Graf, T. Ihn, and K. Ensslin, New Journal of Physics 10 , 125029 (2008). Tunable graphene single electron transistor C. Stampfer, E. Schurtenberger, F. Molitor, J. G¨ uttinger, T. Ihn, and K. Ensslin, Nano Lett. 8 , 2378-2383 (2008). Tunable Coulomb blockade in nanostructured graphene C. Stampfer, J. G¨ uttinger, F. Molitor, D. Graf, T. Ihn, and K. Ensslin, Appl. Phys. Lett. 92 , 012102 (2008). Local gating of a graphene Hall bar by graphene side gates F. Molitor, J. G¨ uttinger, C. Stampfer, D. Graf, T. Ihn, and K. Ensslin, Phys. Rev. B 76 , 245426 (2007). Electron Shuttle Instability for Nano Electromechanical Mass Sensing C. Stampfer, J. G¨ uttinger, C. Roman, A. Jungen, T. Helbling, and C. Hierold, Nano Lett. 7 , 2747 (2007).

96

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Acknowledgements During the past years in the Nanophysics Group I had the chance to work together with many talented people, without whom this work would not have been possible. First of all I would like to thank Klaus Ensslin for giving me the opportunity to do my Ph.D. in his group. It was a pleasure to profit from your large physics knowledge and your managing skills. Your trust and understanding that decent scientific pages can also be written in Vienna means a lot to me. I’m very grateful to Thomas Ihn for being an excellent teacher and very inspiring co-advisor. Thanks to your patience and time I got access to many hidden physical beauties. I always had a bunch of additional work but also a good feeling when I walked out of your office. In addition I like to thank my second co-examiner Paco Guinea for taking his time to read this thesis and to come to Zurich. A special thanks to Christoph Stampfer, who introduced to me the fascinating world of science and motivated me to work on graphene. It was a big pleasure to push projects and celebrate the results with you! Additionally, I like to thank the other members of the graphene team: Davy Graf and Fran¸coise Molitor for pioneering the graphene work in the group, together with Susanne Dröscher, Arnhild Jacobsen and Dominik Bischoff for the teamwork and nice atmosphere. Our team was frequently extended by excellent students. A special thanks to Sarah Hellm¨ uller, Tobias Frey, Johannes Seif and Achille Capelli for their contributions to this work. Big thanks also to Florian Libisch and Stephan Schnez for fruitful collaborations and stimulating discussions. Dealing with experiments I could always count on the help of Paul Studerus concerning electronics and Cecile Barengo for cryogenics and alternative approaches. Claudia Vincenz does an excellent job in the administration, and in addition creating a warm and lively atmosphere on our floor. I thank also all the other members of the group for many interesting discussions and helpful hands. Here, I further acknowledge the support from the ETH FIRST lab and the inspiring exchange with many researchers within the lab. My position was founded by the Swiss National Science Foundation. Thanks a lot! Apart from the scientific world I thank my friends for very enjoyable times e.g. by playing squash and Siedler, riding snowboard or partying to good music. A special thanks to my mother for always having an open ear and sending delicious energy in busy times. Finally I would like to thank Niki Bauer for all her support during the past years. Without you my world would not have half of its color.

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