Electrical Control, Read-out and Initialization of Single Electron Spins

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Electrical Control, Read-out and

Initialization of Single Electron Spins

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Electrical Control, Read-out and

Initialization of Single Electron Spins

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 17 december 2013 om 15:00 uur door

Mohammad SHAFIEI

Master of Science, Chalmers University of Technology, Zweden geboren te Abadeh, Iran.

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Prof. dr. ir. L. M. K. Vandersypen Samenstelling van de promotiecommissie:

Rector Magnificus voorzitter

Prof. dr. ir. L. M. K. Vandersypen Technische Universiteit Delft, promotor Prof. dr. Yu. V. Nazarov Technische Universiteit Delft

Prof. dr. P. C. M. Planken Technische Universiteit Delft Prof. dr. ir. A. Brinkman Universiteit Twente

Prof. dr. M. A. Eriksson University of Wisconsin-Madison, Verenigde Staten Prof. dr. M. S. Rudner University of Copenhagen, Denemarken

Prof. dr. ir. R. Hanson Technische Universiteit Delft, reservelid

Published by: Mohammad Shafiei Printed by: Gildeprint, Enschede

Casimir PhD series Delft-Leiden 2013-35 ISBN: 978-90-8593-175-1

Copyright c 2013 by Mohammad Shafiei

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Chapter 1 Introduction

1.1 Quantum mechanics

In the realm of microscopic objects such as atoms, photons or electrons, particles behave nothing similar to what we see in everyday life. They behave to the best of our knowledge according to the rules of quantum mechanics. While in everyday life, objects can exist at only one place at a time, on atomic scales an object can be at different places at the very same time. This is called quantum superposition. For instance, a ball in a football field is at one specific position at a time, independent of whether you look at it or not. Instead, an electron (which many 19th century physi-cists imagined as a tiny ball) can exist in multiple places at a time and surprisingly its position depends on whether you look it or not. The act of looking (measuring) in fact forces the electron to be at one of its possible positions. This is called the collapse of an electron wavefunction; the wavefunction is an abstract mathematical tool to express the state of a quantum entity.

With no doubt quantum mechanics revolutionized the culture of scientific think-ing. While the philosophy of science was based on the assumption that if you know all details of a system you can precisely calculate its final state, quantum mechanics proclaims that on the microscopic scale we can only know the probability of finding the system in a specific final state. These bizarre characteristics and rules of quan-tum mechanics bothered many great minds of 20th century. As Einstein once asked one of his colleagues “Do you really believe that the moon exists only when you look at it?” [1]. Another inexplicable characteristic of quantum mechanics is quantum entanglement. Quantum entanglement implies that two objects even thousands of light years away from each other can be connected to each other so that if you look at one of them and for instance see its colour, the second one instantaneously knows which colour to be (infinitely faster than the speed of light). Einstein called this “spooky action at the distance”.

Despite these counter-intuitive concepts, quantum mechanics is extremely suc-cessful in the explanation of microscopic phenomena and resulted in the invention

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of lasers and transistors (logical devices in computers), both of which changed the course of human life.

1.2 Quantum computation

Over the last two decades, the size of the transistors in computers, a figure of merit for the computational speed, has become twice smaller every 18 months, following Moore’s law. But eventually this miniaturization is close to hit a limit, a limit where dimensions are so small that transistors behave quantum mechanically.

But is it really a problem to reach this limit? Can quantum mechanics help us in the computation? Can it be even used to simulate or compute difficult problems faster and more efficiently than classical computers. The answer to these questions is yes!

Nobel-prize winner Richard Feynman was one of the first scientists who specu-lated in 1982 that quantum computers can be used to simulate a quantum system efficiently, a so-called quantum simulator [2]. A decade later his speculation turns out to be correct [3]. On the other hand, David Deutsch and others were seeking computational problems in which a quantum computer works more efficiently than in a classical computer, and proposed the concept of quantum parallelism which I explain in simple terms in the following.

In our daily computer, the information is encoded into a bit, which can be 0 or 1. A processing unit is able to evaluate a function for one bit value at a time. For example, imagine a processor computing a function F (x) on an input bit x. To calculate F (0) and F (1) using the same processor, you need to run the function twice. It is still possible to speed up the computation by using two processors at the same time. This is called parallel computing. In a quantum computer, instead, a quantum bit (qubit |x i) can be both 0 and 1 at the same time, in a quantum superposition such as |x i=|0 i+|1 i. Therefore running function F (x) once, gives an output which contains both F (0) and F (1), speeding up the computation. Now imagine a register consists of n qubits, by running the same function on this register only once, we can compute a function on up to 2n_{input states. This is called}

quan-tum parallelism, a powerful tool to speed up the computation. However, a challenge arises when measuring the output state. As discussed, by looking at the result we only see one of the available outputs (either F (0) or F (1)) and more dramatically we even do not know which one. Therefore clever algorithms are required in order to decode the output of the computation. The search for interesting problems and efficient quantum algorithms is an active area of quantum information theory.

One of the interesting mathematical problems in number theory is the integer factorization of a number. Due to the difficulty of solving this problem, it is widely used for cryptography and secure communication. In 1994, Peter Shor presented a quantum algorithm where he showed that a quantum computer can solve this

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prob-1.3 Electrons- spin and charge

lem exponentially faster than even the fastest supercomputers [4]. This algorithm gave a new dynamic to the field of quantum information. Until now many other quantum algorithms are proposed where a quantum computer works more efficiently than the classical computers such as searching in an unstructured database.

Another question to address is what the requirements for physical implementation of a quantum computer are? The answer to this question gets more and more clear as the field of quantum information is developing. The most basic requirements are now formulated in 5+2 DiVincenzo criteria from which five are related to computation and two related to communication [5].

In principle, any quantum two-level system such as the discrete energies of an atom, the spin of an electron or the direction of current in the superconducting loop can be used as a qubit. Up to now, different platforms have been proposed and studied for the implementation of a quantum computer such as nitrogen vacancy centres in diamond, electron spins in quantum dots, trapped ions and superconduct-ing circuits. However, the pitfall of most quantum systems is decoherence, loss of information, due to the interaction of the qubits to the unknown and uncontrolled environment. It is still an open question which quantum system is the most suitable for the realization of the futuristic quantum computer.

To date, quantum algorithms have been executed on a handful of qubits, using e.g. nuclear magnetic resonance (NMR) [6, 7], superconducting circuits [8, 9], ion traps [10] and nitrogen vacancy centres [11].

1.3 Electrons- spin and charge

Current electronic devices like our televisions and computers make use of the charge of an electron. But beside the electric charge, an electron possesses a tiny magnetic moment, called spin. Exploiting the spin degree of freedom has opened up a new era in the field of semiconductor electronics providing a vast variety of applications which may revolutionize electronic devices.

For instance, the field of spin electronics (spintronics) makes use of the spin degree of freedom in order to create novel or more energy efficient electronic devices such as spin transistors, spin diodes and so forth. In fact, many of us have spintronic devices at home or in our offices, such as a computer hard disk.

With the recent advances in nanotechnology and using state-of-art fabrication techniques, it is now feasible to create tiny islands (called quantum dots) in a semi-conductor material to controllably trap single quanta of charge. In 1998, Loss and DiVincenzo proposed that the spin of an electron in a semiconductor quantum dot can be used for the realization of a quantum computer [12]. The electron spin, which can point parallel or antiparallel to a magnetic field, can be used as a qubit. In contrast to the charge of an electron, the spin of an electron is almost uncoupled from the electrostatic noise in the environment. This is the starting point of the

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research presented in this thesis.

1.4 Outline of the thesis

This thesis describes the experiments on electrical control, read-out and initialization of single electron spins in laterally defined quantum dots. The quantum dots are created by applying negative voltages on the metallic gates located on top of a GaAs/AlGaAs heterostructure. The outline of the thesis is as follows.

We start in chapter 2 with presenting the relevant theory background and methods used to perform the experiments in this thesis. We discuss the creation of lateral quantum dots, the spin states in such systems and the exchange coupling. We then introduce the interaction of an electron spin with its environment via the spin-orbit and hyperfine couplings.

Chapter 3 briefly presents the experimental setup. It additionally includes how we have improved the base temperature of our dilution fridge.

In chapter 4 we demonstrate independent single-shot read-out of two electron spins in a double quantum dot where we probe the anti-correlations between the two spins prepared in a singlet state. Using single-shot read-out, we also demonstrate the operation of the two-qubit exchange gate on a complete set of input states.

Chapter 5 presents experiments on electrical control of single electron spins where we use adiabatic rapid passage to invert an electron spin. We show that at high magnetic fields there is a clearly observable shift in the resonance condition of spin-orbit- and hyperfine- mediated electric dipole spin resonance (EDSR).

In chapter 6 we present a method for fast initialization of electron spin qubits making use of spin hot spots where the spin relaxation is enhanced by three orders of magnitude. Chapter 7 presents experiments on all-electrical independent address-ing of saddress-ingle electron spins in a double quantum dot where we observe well-separated Zeeman splittings between neighbouring dots.

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Chapter 2 Spins in few-electron GaAs

quantum dots

A quantum dot is a tiny artificial island in which the movement of electrons is constrained in all three spatial dimensions. Due to the quantum confinement, such a structure has a discrete energy spectrum. Quantum dots can be created in numerous ways and in different sizes. Some examples are lateral quantum dots [13–15], self-assembled quantum dots [16, 17] and quantum dots in semiconducting nanowires [18]. This thesis presents the experiments in electrostatically defined quantum dots referred to as lateral quantum dots. In this chapter, we present the relevant theory background as well as the methods used in performing the experiments.

2.1 Creation of lateral quantum dots

Laterally defined quantum dots are electrostatically defined islands in a quasi-two-dimensional electron gas, called 2DEG. All quantum dots reported in this thesis are created in a 2DEG formed in a silicon-doped AlGaAs/GaAs heterostructure. In the following we explain how the 2DEG is formed and can be used to create quantum dots.

When AlGaAs (typically with the composition of Al0.3Ga0.7As) is grown

di-rectly on top of a GaAs substrate a triangular shape quantum well is created at the heterointerface due to the mismatch between the bandgap of these two materials (AlGaAs has higher bandgap than GaAs1_{), see Fig. 2.1. One can now introduce free}

electrons in the heterostructure by n-type doping of AlGaAs layer with Si. These free electrons eventually accumulate at the interface between GaAs and AlGaAs. Due to the strong confinement of the electron in the growth direction, the elec-trons are confined in the quantum well and have discrete energy levels. At low temperatures (T.100 K) only the lowest subband of the quantum well is populated.

1_{Bandgap of GaAs: 1.42 eV and bandgap of Al}

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z n-AlGaAs GaAs AlGaAs GaAs 2DEG a b z + + + + + n-AlGaAs + AlGaAs GaAs + 100 nm

Figure 2.1: Two-dimensional electron gas. (a) A semiconductor heterostructure with a 2DEG formed approximately 100 nm below the surface, at the interface between GaAs and AlGaAs. The electrons in the 2DEG are coming from Si-donors in the n-AlGaAs doped layer. The thickness of the different layers is not to scale. (b) Conduction band close to the GaAs/AlGaAs interface. The electrons are accumulated in a triangular shape quantum well at the heterointerface. At low temperature, only the lowest energy level of the quantum well is populated. EF depicts the Fermi energy level. The gray solid curve presents the wavefunction of an electron confined in the quantum well. z-axis depicted by an arrow shows the growth direction.

Therefore, electrons can freely move only in a plane perpendicular to the growth direction. In this way, a 2DEG is formed. Since the electrons are separated from the Si-donors (scattering centres), the electron mobility can be very high (typically 105–106cm2/Vs at 4.2 K). Further improvement of the electron mobility is usually done by increasing the separation between the 2DEG and donors using an undoped AlGaAs layer.

Another property of a 2DEG is its low electron density (typically ∼ 3×1011cm −2) which results in a large Fermi wavelength2 _{(∼ 50 nm) and large screening length}3_.

Therefore, it is possible to locally deplete the electrons in a 2DEG by applying neg-ative voltages on metallic electrodes (gates) on the surface of heterostructure (∼ 100 nm above the 2DEG): applying negative voltage on the surface gates creates an electric field which repels the electrons in the 2DEG underneath via Coulomb interaction, see Fig. 2.2a. Therefore, using different surface gate geometries, we can electrostatically shape the 2DEG and confine the electrons in one- or zero-dimension. The first is used to create quantum channels and quantum point contacts and the latter is used to create quantum dots. Figures 2.2b,c show the schematic view and the gate pattern used to create a double quantum dot.

2.2 Charge stability diagram

In this section, we give a brief description of the electronic properties of electrons in a double quantum dot ignoring the spin of an electron.

2_{The Fermi wavelength of the 2DEG is given by λ} F =

q

2π

n where n is the electron density [19]. 3_{Thomas-Fermi screening length is on the order of Fermi wavelength.}

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2.2 Charge stability diagram c 2DEG Ohmic GaAs AlGaAs 30 channel

gate depleted_region

a b

300 nm

S D

Figure 2.2: Creation of a quantum dot in a 2DEG. (a) 2DEG beneath the gates is locally depleted by applying negative voltages on the metal surface gates. Therefore, depending on the gate pattern, electrons can be confined in one- or zero-dimension. (b) Schematic view of a laterally defined quantum dot device. The white region is where the 2DEG (depicted as light gray) is depleted using negative voltages on the surface gates. Ohmic contacts are used to make electrical contacts to the 2DEG. (c) Scanning electron microscope image of the metallic surface gates of an actual device. The two white circles present two quantum dots coupled to neighbouring reservoirs. Figure is taken from [20].

A laterally defined double quantum dot consists of two charge islands each tunnel coupled to a neighbouring reservoir and to one another. The equilibrium charge state of a double quantum dot for constant voltages on the surface gates is given by (N, M ) with N electrons in the left dot (dot 1) and M electron in the right dot (dot 2). The electrochemical potential of the left quantum dot, µ1(N, M ), is defined as

the energy required to add one electron to the (N − 1)-occupied left dot when there are M electrons in the right dot. Similarly, the electrochemical potential of the right quantum dot, µ2(N, M ), is defined as the energy needed to add one electron to the

(M − 1)-occupied right dot when the left dot is occupied by N electrons. Since there is a capacitive coupling between the surface gates and the quantum dots, the electrochemical potential can be controlled by gate voltages. Therefore, one can move the electrons between each quantum dot and reservoir or its neighbouring dot. For example when µ1(N, M ) lies above the Fermi energy of the reservoir, after

some time, an electron tunnels out of the quantum dot to the reservoir and the charge state of double dot is equilibrated to (N − 1, M ). A diagram presenting the equilibrium charge state of a double dot as a function of the voltage on surface gates is called “charge stability diagram”.

Figures 2.3b and c show charge stability diagrams of a double quantum dot without and with interdot capacitive coupling, respectively. When two quantum dots are totally uncoupled and neglecting the cross capacitances (Fig. 2.3a with Cm = C12 = C21 = 0), the electrochemical potential of each dot is independent of

the charge state of the neighbouring quantum dot, see Fig. 2.3b. The lines between charge states represent the gate voltages where the electrochemical potential of a dot is aligned with the Fermi energy of the reservoir. Once we introduce the tunnel coupling between the dots, charge transitions between dots are possible. If there is capacitive coupling between dots, the electrochemical potential of each dot also

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V₁ V₂ QD1 V_L CL Cm QD2 CR V_R C₁₁ C22 C₁₂ C21 gate gate a c d V2 V1 (0,0) (0,1) (0,2) (1,0) (1,1) (1,2) (2,0) (2,1) (2,2) V2 V1 (0,0) (0,1) (0,2) (1,0) (1,1) (1,2) (2,0) (2,1) b 1 1 2 3 µ₁(1,0) µ₂(0,1) 2 µ₁(0,1) µ₂(1,0) 3 µ₁(0,1) µ₂(1,0) 1 2 e V1 Reservoir Dot VSD QPC

(Charge meter) Time (ms)

out

IQPC

1.0 0

in

Figure 2.3: (a) Capacitive model of a lateral double quantum dot. Quantum dots (QD1 and QD2) are coupled to the left and right reservoirs and to each other. Gray line show the cross capacitances between gates and quantum dots. (b) Charge stability of a double quantum dot when the dots are uncoupled Cm = C12 = C21 = 0. (c) The same as (b) considering the tunnel coupling between the quantum dots and the cross capacitances. (d) Schematic diagram showing the electrochemical potentials of the dots at positioned depicted in (b) and (c). When the electrochemical potential is below the Fermi energy level of the reservoir, an electron will tunnel into the quantum dots. (e) QPC coupled to a quantum dot. QPC-current changes as an electron tunnels in or out of a quantum dot.

depends on the charge state of the other quantum dot. For a detailed description of charge stability diagrams, we refer to [21, 22].

A common method to probe the charge stability diagram is to use charge sen-sors in the vicinity of quantum dots. There have been different implementations of charge sensors coupled to quantum dots such as a single-electron transistor (SET) on the surface of the heterostructure [23], a quantum point contact (QPC) [24] and

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2.3 Compensation of capacitive coupling and virtual gates

a lateral quantum dot [25]. In the experiments presented in this thesis, we use two QPCs capacitively coupled to a double dot to measure the charge state of the quantum dots. The principle of charge detection via a QPC is as follows: when the QPC channel is biased in the tunneling regime (between two quantized conductance plateaus), the conductance is very sensitive to changes in the electrostatic environ-ment including any change in the number of electrons in nearby quantum dots. For example removing an electron from a quantum dot changes the electrostatic poten-tial of a neighbouring QPC and effectively increases its channel width, therefore the conductance increases. Usually the change in the conductance is on the order of a few percent of the conductance quantum (2e2_{/h). One can now probe the change in}

the conductance by measuring the current through the biased QPC channel [24] or using an RF reflectrometry technique [26]. Figure 2.3e depicts the change in a QPC current as an electron tunnels in and out of the quantum dot allowing the real-time detection of electron tunneling when the tunneling rate is within the bandwidth of the measurement.

2.3 Compensation of capacitive coupling and

vir-tual gates

In any complex quantum system, having orthogonal knobs to control the parameters of the system independently offers an important advantage and results in simplifying the control.

As discussed in the previous section, there are capacitive couplings between quantum dots (Cm) and capacitive coupling between gate 1 (2) and the right (left) dot. As a result, it is impossible to control the electrochemical potential of the left or right dot only by using gate 1 or 2, see Fig. 2.3. We therefore devise ”virtual gates” as a tool for independent control of the electrochemical potential of each dot, as explained in the following.

Consider a charge stability diagram of a double dot as shown in Fig. 2.4a. The charge transition between (1,1) and (0,1) shows the gate voltages at which the electrochemical potential of the left dot aligns with the Fermi energy level of the left reservoir. In other words, moving parallel to the left dot charge transition line results in a change in the electrochemical potential of the right dot while keeping the electrochemical potential of the left dot unchanged (independent control of the right dot’s electrochemical potential). An analogous argument holds for independent control of the left dot’s electrochemical potential.

Therefore one can define new voltage axes along the charge transitions, see Fig. 2.4a. We refer to these voltage axes, vL and vR, as virtual gate voltages, as they do not present a voltage on a single gate but rather a combination of voltages on different gates. Transformation of the vLP and vRP axes to the new coordinates of

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vL and vR can be done simply via the following relation: vL vR = cos γLP − sin γRP − sin γLP cos γRP −1 vLP vRP (2.1)

where γLPand γRP are the angles between the different coordinates, as shown in Fig.

2.4a. Figure 2.4b shows the schematics of a double dot charge stability diagram as a function of virtual gate voltages. We point out that the above transformation is valid in the range of gate-voltages where constant interaction model is valid: (1) Coulomb interactions between the electrons in the dot and all other charges are constant in the gate voltage space and independent of the number of electrons in the dot. (2) The single particle energy spectrum is independent of these interactions.

VR VL b (0,0) (1,1) (1,0) (0,1) VRP VLP a γ_RP γ LP (0,0) (1,1) (1,0) (0,1)

Figure 2.4: Double dot charge stability diagram as a function of (a) top gates vLP and vRP and (b) virtual gates. We transform axes presenting the voltage on the metallic plunger gates (green arrows) to the axes of the virtual gates (red arrows). γLP and γRP are the angles between the axes. (n, m) indicate the number of electrons in the left and right dot, respectively.

2.4 Coupled quantum dots

So far we neglected the effect of tunnel coupling on the energy states of a quantum dot. Here we would like to discuss how tunnel coupling modifies the eigenenergies of a double dot.

Consider two ground states of an uncoupled double dot |(0, 1) i and |(1, 0) i. Without tunnel coupling, the electron is completely localized in either of the dots. When we introduce tunnel coupling between the dots, an electron can coherently tunnel between the dots. Consequently, the orbital wavefunction of an electron is de-localized over the coupled dots and the eigenenergies are modified. The Hamiltonian

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2.5 Spin states in a quantum dot

of a tunnel coupled two level system is given by H = 1 2 tc t∗_c − (2.2) where is the energy detuning between the uncoupled energy levels and tc is the

tunnel coupling strength. Figure 2.5b shows the eigenenergies of the Hamiltonian H as a function of detuning. The energy difference between the new eigenstates (bonding and antibonding states) is given byp2_{+ (2t}

c)2. For detailed discussion

see [22]. Note that so far we did not include the spin of an electron.

VR VL a ε ε Detuning ε (µeV) Energy (µ eV ) b 0 2t_c (0,0) (1,0) (1,1) (0,1) (1,0) (0,1)

Figure 2.5: (a) The charge stability diagram of a tunnel coupled quantum dots. shows the detuning axis between (1,0) and (0,1) electrochemical potential. (b) Introducing the tunnel coupling modifies the eigenenergies. Dashed lines are the eigenenergies of uncoupled double dot. Sold lines are the eigenenergies in the presence of tunnel coupling. Due to the constant tunnel coupling, there is an avoid crossing at zero detuning.

2.5 Spin states in a quantum dot

When an electron confined in a quantum dot is placed in a magnetic field, it feels a Zeeman interaction given by H = gµBS.B. Here g is the electron g-factor, S =

(Sx, Sy, Sz) is the spin operator, µB is the Bohr magneton and B is the external

magnetic field. To minimize the perturbation of the orbital states in a quantum dot, we typically apply the external magnetic field parallel to the 2DEG plane. The Zeeman interaction results in splitting between two electron spin states, | ↑ i (parallel to the magnetic field) and | ↓ i (antiparallel to the magnetic field) with the energy difference given by the Zeeman energy gµBBz. For convenience, we consider the

z-direction as the direction of magnetic field. In GaAs quantum dots, the electron g-factor has a negative sign and therefore spin | ↑ i is the ground state.

In the following sections, we briefly explain the two most important interactions of an electron spin in a GaAs quantum dot with its environment: spin-orbit coupling and hyperfine interaction.

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2.6 Spin-orbit interaction

Special relativity suggests that motion of an electron in an electric field, such as from a lattice potential, creates a kinetic term in the Hamiltonian. Once we are in the rest frame of the electron, this kinetic term translates into an effective mo-mentum dependent magnetic field through which the electron spin is coupled to the electric field (see for example [27]). This mechanism through which spin and orbital degrees of freedom are coupled is called “spin-orbit interaction”. Since the orbital degree of freedom is affected by the lattice fields, the spin of an electron feels the crystal symmetry through spin-obit interaction. Here we briefly discuss spin-orbit interaction from an experimentalist point of view. A detailed discussion requires band structure calculation, see [28] for the theory of spin-orbit in two dimensional systems.

The motion of an electron in the periodic potential of a crystalline solid is char-acterized by a band structure energy E(k) where k is the wave vector. For the electrons in a diamond lattice (like Si and Ge), the inversion symmetry implies that the energy does not change with the sign of the wave vector: E(k) = E(−k). Together with the Kramers degeneracy, E↑(k) = E↓(−k)4, one can conclude that

inversion symmetry results in at least a double degeneracy of the Bloch waves. Zincblende structures such as GaAs, instead, lack inversion symmetry and spin splitting can occur even at zero magnetic field5_{. For a GaAs two dimensional electron}

gas grown along [001]6_{, the spin-orbit Hamiltonian is given by [21]}

H_D2D,(001) = β[−pxσx+ pyσy] (2.3)

where x and y are the crystallographic directions of [100] and [010], respectively. σx

and σy are the Pauli X and Y matrices. β is the Dresselhaus spin orbit coupling

strength (coefficient) and it is material dependent. The above Hamiltonian can be derived from the bulk Dresselhaus Hamiltonian with two considerations: (1) the electrons are confined in the growth direction (2) the confinement is very strong so that the linear term is the dominant term7 _{(see [21]).}

Additionally, in a GaAs 2DEG there is an asymmetric confinement from the potential along the growth axis. This lack of inversion symmetry (so-called struc-tural inversion asymmetry or SIA) results in what is known as Rashba spin-orbit

4_{Under time reversal not only the motion sign but also the spin sign is reversed.}

5_{In addition to the spin-splitting, spin-orbit interaction results in a deviation of the electron}

g-factor from the one of free electrons, g0∼2. For bulk GaAs the effective g-factor is g∗ = −0.44.

For a GaAs 2DEG, the reduced symmetry along the growth axis results in an anisotropic g-factor which among others depends on the width of the quantum well.

6_{For GaAs, [100], [010] and [001] directions are equivalent.}

7_{Theoretically, Dresselhaus Hamiltonian includes terms which are cubic in momentum}

com-ponents. When the confinement is very strong so that hp2

zi p2x, p2y, the cubic terms can be

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2.6 Spin-orbit interaction p x||[100] p y||[010] p x||[100] p y||[010] ||[110] ||[110] ||[110] ||[110] a b

Figure 2.6: An electron moving with momentum p feels an effective magnetic field via (a) Dresselhaus (b) Rashba interaction. The arrows show the orientation of this magnetic field for different movement direction respect to the crystallographic axes. Here the assumption is α and β are positive. In this case, an electron moving along [110] feels maximum effective magnetic field. When α and β have different signs as the case for our heterostructure, an electron moving along [110] feels minimum effective magnetic field.

coupling. In this case the spin-orbit Hamiltonian is given by

H_R2D = α(−pyσx+ pxσy) (2.4)

where α is the Rashba spin-orbit coupling coefficient, which depends both on the material and on the exact shape of the confinement potential.

Figure 2.6 shows the direction of the effective magnetic field for Dresselhaus and Rashba spin-orbit interactions. For the Rashba spin-orbit term, an electron feels an effective magnetic field perpendicular to its direction of movement. However for the Dresselhaus term, the direction of the internal magnetic field is anisotropic and it points perpendicular to the momentum only when an electron moves along [110] or [1¯10], see Fig. 2.6b. An electron moving in the 2DEG feels both spin-orbit terms and therefore depending on the relative sign and amplitude of α and β, the direction of the net effective magnetic field varies. For example, the effective magnetic field that an electron moving along [1¯10] feels is minimum for αβ > 0 and maximum for αβ < 0, see Fig. 2.6a.

The spin of an electron traveling in the 2DEG rotates due to the magnetic field. The angle by which it rotates depends on the distance traveled and the strength of the spin-orbit coupling. Therefore, it is common to characterize the spin-orbit coupling strength by the distance over which the electron spin rotates by π, known as spin-orbit length. The spin-orbit length for bulk GaAs (α = 0) is defined as lSO =

~/βm∗ with m∗ the effective mass of an electron (m∗ = 0.067me). β is estimated to

be between 1 − 3 × 103m/s [28] which leads to a spin-orbit length of lSO= 1 − 10 µm

consistent with the measured data [29]. For a GaAs 2DEG and considering both Rashba and Dresselhaus spin-orbit terms, the spin orbit length is modified. It is convenient to introduce two distinct spin-orbit lengths l_SO± _{= ~/(α ± β)m}∗ along

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the [110] and [1¯10] crystallographic axes. One noteworthy regime is when α and β are equal. In this case Rashba and Dresselhaus spin-orbit interactions cancel out along [1¯10] direction and spin-orbit interaction cannot result in spin rotation of an electron moving along this direction.

2.6.1 Spin-orbit interaction in quantum dots and spin

re-laxation

In a GaAs quantum dot, the electron is bound to the 2DEG plane and the size of a quantum dot (∼ 100 nm) is much smaller than the orbit length. Therefore, spin-orbit coupling can be seen as a small perturbation to the spin-orbital energy spectrum.

When an electron is trapped in a quantum dot (thus has no linear momentum hpx,yi = 0), there is no direct coupling between the Zeeman sublevels via spin-orbit

interaction, as hn ↓| HSO|n ↑ i ∝ hn| px,y|n i h↓| σx,y| ↑ i = 0 with n representing the

orbital and HSO = HD2D + HR2D. Instead, spin-orbit interaction couples different

spin states of different orbitals [30]. As a result, pure spin states are not any more eigenstates of the Hamiltonian. Instead, the eigenstates are pseudo-spin states8_{. For}

example, pseudo spin-down state in the nth orbital is the spin down state of the nth orbital mixed with the spin up states of other orbitals: |n ↓ i = |n ↓ i +P

i6=nκi|i ↑ i

where κi is proportional to the spin-orbit coupling strength between |n ↓ i and |i ↑ i

divided by their energy difference (normalization factor is omitted for brevity). In this thesis, we use the term spin states instead of pseudo-spin states for brevity.

An important consequence of spin-orbit coupling is that it couples an electron spin to electric fields through its charge (orbital state). This coupling has both benefits and drawbacks. The advantage of such coupling is that it is possible to control spin transitions by using electric fields only, either coherently as proposed by [33–36] or via adiabatic rapid passage as will be discussed in chapter 5. On the other hand, unwanted electric field fluctuations can also couple to the electron spin which results in spin relaxation [30, 37–40].

In a GaAs quantum dot, there exist various sources of random electric field fluc-tuations. Some examples are electric noise on the gate potentials (such as Nyquist noise [41]), background charge fluctuations [42], charge fluctuations caused by elec-trons passing through the QPC [43] or the electrical noise created by the lattice phonons [37, 39]. Among them, spin coupling to the electric-field fluctuations cre-ated by the lattice phonons is the major cause of spin relaxation. The physics of spin relaxation with phonons is very involved and has been studied extensively in [30, 37–40, 44–46]. The pseudo-spin states are coupled to the phonon bath through spin-orbit coupling. The relaxation rate depends on (i) the density of states of phonons at the energy equal to the Zeeman energy (once a spin flips, its energy is

8_{In a quantum dot, the lateral confinement via gates influences the electron g-factor through}

spin-orbit interaction. It has been proposed that using the gate potential one can control the g-factor of an electron [31, 32]

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2.6 Spin-orbit interaction

absorbed by a single phonon) and (ii) the strength of the pseudo-spin state cou-pling to the phonons. For the energy range applicable in a GaAs quantum dot, only acoustic phonons (via piezoelectric coupling and neglecting the deformation potential coupling) are relevant, whose density of state scales as E_Z2. Moreover, the strengths of the coupling to the phonons scales as E_Z3 for the piezoelectric phonons. Therefore, a relaxation rate dependency of ∝ E5

Z is expected [37] and it has been

verified experimentally in [44]. Due to spin-orbit coupling, the spin relaxation rate also depends on the orbital level spacing, ∆, (or dot size) and scales as ∆−4 at low temperatures. See [21] for a review on the spin relaxation in GaAs quantum dots. Note that the above argument is not valid at low magnetic fields (B< few 100 mT) where electron-nuclear flip flops are the dominant source of relaxation.

As we discussed above, the spin relaxation is directly related to the degree of admixing of orbital and spin states. In chapter 6 of thesis, we show that in a coupled double dot close to the avoided crossing the degree of admixing is complete and the spin relaxation is enhanced by several orders of magnitude. More specifically, when the Zeeman energy exceeds the tunnel coupling of a double dot, ”hot spots” occur where the spin splitting matches the quantized orbital level spacing. In this case, spin-orbit (and hyperfine interactions) effectively mix the spin and orbital excited states.

2.6.2 Electric dipole spin resonance (EDSR) mediated by

spin-orbit interaction

Now we briefly discuss how spin-orbit coupling can be exploited for electrical manip-ulation of an electron spin. As discussed theoretically in [36], when a homogeneous time dependent electric field, E(t), at the frequency equal to the Larmor frequency of an electron (gµB|Bext|/h with Bext the external magnetic field and h Planck’s

constant) is applied to an electron in a harmonic potential (Fig. 2.7), the centre of the potential is displaced by an amount equal to r(t) = −_meE(t)∗_ω2

0. Here m

∗ _{is the}

elec-tron effective mass. ~ω0 is the orbital energy splitting with ~ the reduced Planck’s

constant and ω0 the oscillator frequency of the harmonic potential. Through

spin-orbit interaction (HSO), the oscillating electron feels an alternating magnetic field

orthogonal to the direction of the external field, given by [36, 47]: Beff(x, y) = n ⊗ Bext nx= 2m∗ ~ (−αy − βx) ; ny = 2m∗ ~ (αx + βy) ; nz = 0 (2.5)

where n points toward the direction of electron’s displacement with respect to the crystallographic axes x and y defined as [100] and [010], respectively.

As expected, equation 2.5 is very anisotropic. One can now extract the effective time varying magnetic field that an electron feels when it is moving along the [110]

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0.3 mµ Position E nergy B_ext

E

DC+ r(t) a b

Figure 2.7: Spin-orbit mediated EDSR. An oscillating electric field coupled to a quantum dot results in a movement of the electron wavefunction along the direction of electric field. (a) The scanning electron microscope image of a device where an oscillating electric field is applied between two surface gates. (b) The solid black and gray lines are the potential wells defining the quantum dot. The red solid line is the electron wavefunction. Arrows show the spin orientation as it is rotated by moving the wavefunction. Adapted from [47]

or [1¯10] directions [36, 47]: |Beff(t)| = 2|Bext|

|r(t)| lSO

, (2.6)

with lSO = ~/m∗(α ∓ β) the spin-orbit lengths. This oscillating effective

mag-netic field results in a spin rotation with a frequency (Rabi frequency) fRabi =

(gµB|Beff|)/2h.

2.7 Hyperfine Interaction

The interaction between magnetic moment of an electron with the magnetic mo-ments of nearby nuclei is called hyperfine interaction. This interaction is very well known in the context of molecular spectroscopy and atomic physics and it is responsi-ble for the atomic-fine structure, see for example [48]. Usually hyperfine interaction can be decomposed into two parts: isotropic and anisotropic. The first part ex-ists when there is an overlap between electron and nuclear wavefunctions (like an electron in an s-orbital) and it is known as Fermi contact interaction. The second part results from the electron nuclear dipole-dipole interaction (like an electron in a p-orbital) and it is negligible for s-orbital electrons.

The Fermi contact interaction between an electron spin S and a nuclear spin I is given by [49]

HHF =

2µ0

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2.7 Hyperfine Interaction

where µ0 = 4π · 10−7Vs/Am, g0 is the free-electron g-factor and γN is the nuclear

gyromagnetic ratio. µBis the Bohr magneton and ~ is the reduced Planck’s constant.

One can now integrate the above Hamiltonian over the orbital state of the electron to derive the hyperfine interaction in the subspace of the orbital state given by[49]

HHF =

2µ0

3 g0µBγN~Ii· S|ψ(ri)|

2

(2.8) where |ψ(ri)|2 is the probability amplitude of the electron wavefunction at the

po-sition of the nucleus.

Contrary to an atom, an electron in a quantum dot couples to many lattice nuclei of the host material. In GaAs, there are three isotopes, α, of 69_Ga, 71_{Ga and} 75_As

with natural abundances, ηα, of 1, 0.6 and 0.4, respectively. All these three isotopes

have nuclear spin I=3/2. Due to the s-type nature of the GaAs conduction band, the Fermi contact hyperfine Hamiltonian is the relevant Hamiltonian, given by

HHF =

X

i,α

Ai,αIi,α· S. (2.9)

Here, the subscript i labels the nuclei. Ai,α = v0Aα|f (ri)|2, where v0 is the volume

of the unit cell and f (r) is the electronic envelope wavefunction. The hyperfine coupling strength, Aα, is given by

Aα =

2µ0

3 g0µBγN,αηαdα. (2.10)

The electronic density, dα, is estimated to be ∼98 ˚A−3for75As nuclei and ∼58 ˚A−3for

both Ga isotopes [49]. Aα is estimated to be 46.8 µeV, 23.17 µeV and 19.86 µeV for 75_As, 69_{Ga and}71_{Ga, respectively. Therefore, the total hyperfine coupling strength}

is A ∼ 90µeV .

An alternative way to describe the effect of nuclei on the electron spin is through an effective magnetic field BN = PN_i=1Ai,αIi,α/gµB, known as Overhauser field

which acts on the electron spin as gµBBN [50, 51]. When all nuclear spins are

polarized, the maximum nuclear field of about BN,max = 5.3 T 9 is obtained. In

thermal equilibrium (typically T≥ 10 mK and |Bext| ≤ 12 T) and in the absence of

dynamic nuclear polarization, the thermal energy dominates both nuclear Zeeman energy and hyperfine coupling (HHF). As a result, a thermal equilibrium average

polarization according to the Boltzmann distribution occurs. In addition, there is a statistical fluctuations in the nuclear field around this average polarization which follows a Gaussian distribution for large N with the standard deviation given by σN ∼ BN,max/

√

N [50, 52, 53]. For a GaAs quantum dot, the electron wavefunction overlaps with about N ∼ 106 _{nuclear wavefunctions giving σ}

N in the range of a few

mT, consistent with the measurements [54–57].

9_{For fully polarized nuclear spins, hyperfine interaction, equation 2.9, becomes H}

HF = AI.

With A=90 µeV, we get HHF=135 µeV which is about 5.3 T. Here, we used the electron g-factor

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a b BN S _B N

B

_N BN z y x

Figure 2.8: (a) Electron in a quantum dot interacts with many nuclear spins of GaAs. (b) Electron spin interacts with the effective nuclear field. The nuclear field can be decom-posed into longitudinal and transverse components. The fluctuations of the longitudinal component (δBz_N) result in pure dephasing.

Now we briefly discuss the effect of hyperfine interaction on the time evolution of electron spin (for more detailed review see [21]). In the presence of nuclear field, an electron spin (S) precesses about the vector sum of the nuclear Overhauser field and the external magnetic field. BN can be decomposed into a longitudinal

com-ponent (parallel to the external field), Bz

N, and the transversal components, B x,y

N ,

see Fig. 2.8. The longitudinal component adds to the external field and changes the electron Larmor frequency. Any random fluctuations of this component lead to an uncertain precession frequency and thereby reduction of the phase coherence (dephasing). Considering that Bz

N is sampled from a Gaussian distribution with

a standard deviation σN, the time-ensemble-averaged dephasing time can be

ex-pressed as T₂∗ = √_2~/gµBσN [50]. Using typical values for a GaAs quantum dot

(σ =2.2 mT and g=0.35 [54]), we get a dephasing time of about 20 ns. The transver-sal components also can change the precession frequency. However, this effect can be neglected at high magnetic fields (|Bext| BNx,y)): transverse components only

tilt the rotation axis by an small amount of θ ∼ BN/B. Therefore, the

dephas-ing rate is θ2 smaller than the one due to the contribution of fluctuations in the longitudinal component [50]. At low magnetic fields by contrast (|Bext| B_Nx,y),

transverse components cannot be neglected. In this case, the electron spin rotates around vector sum of these components and a spin flip can occur.

2.8 EDSR mediated by Hyperfine Interaction

An oscillating electric field ˆE(t) generated by an oscillating voltage applied to a gate coupled to a quantum dot results in the movement of the dot center (see section 2.6.2). In the presence of nuclei, a moving electron feels a spatially inhomogeneous Overhauser nuclear field (BN(r(t), t)). In other words, the moving electron feels a

position dependent Zeeman interaction through which spin and orbital degrees of freedom are hybridized, analogous to an external gradient field [58, 59]. Therefore,

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2.9 Spin states and exchange interaction in coupled quantum dots the spin of an electron feels an oscillating effective magnetic field about which it rotates. This electric-dipole spin resonance (EDSR) happens when the frequency of the oscillating electric field matches the electron Larmor frequency. In, ref. [60], Laird et. al. gave a theoretical description for this mechanism where an electron confined in a parabolic quantum dot is coupled to the oscillating electric field and the hyperfine field of ensemble of nuclei. As a result of this coupling, the moving electron experiences the spatial inhomogeneity of the nuclear field. Therefore spin transition occurs with a characteristic Rabi frequency given by: ΩR = e|E(t)|A_~Λ

q

I(I+1)

8πdn [60]

where n is the nuclear spin concentration, e is the electron charge, A is the hyperfine coupling constant (an average hyperfine coupling constant weighted by the natural abundances of all three isotopes). d is the vertical confinement and Λ is the orbital level splitting. Note that the x and y components of the nuclear spin define an instantaneous Rabi frequency with a Gaussian distribution with standard deviation ΩR. Furthermore, the z-component of the random Overhauser field displaces the

Larmor frequency by an amount characterized by a Gaussian distribution.

Here we give three interesting characteristics of hyperfine mediated EDSR: (1) Hyperfine mediated EDSR is possible even without the existence of spin-orbit cou-pling. (2) The transition is governed by random nuclear fields. As a consequence of averaging over these fluctuations, the Rabi oscillations vanish (incoherent transi-tion). (3) The Rabi frequency is independent of the external magnetic field, contrary to the magnetic field dependence of the spin-orbit mediated EDSR Rabi frequency. In chapter 5 of this thesis, we show experiments where these two mechanisms are resolved.

2.9 Spin states and exchange interaction in

cou-pled quantum dots

So far we have considered a single electron only. Here we briefly describe the spin states of two electrons in a double quantum dot.

When two electrons are confined in a single quantum dot in the absence of a magnetic field, the energy spectrum depends on the total spin of two electrons. The ground state of such a system is a spin singlet state with a total spin number S = 0, |S i = √1

2(| ↑↓ i − | ↓↑ i) where two electrons occupy the lowest orbital state

and the first excited states are spin-triplet states with a total spin number S = 1, |T0_{i =} _√1

2(| ↑↓ i + | ↓↑) i, |T

+_{i = | ↑↑ i, |T}−_{i = | ↓↓ i where electrons occupy the}

first two orbital states. The energy difference between singlet and triplet states is a result of Coulomb interaction between two electrons and the Pauli exclusion principle (exchange interaction) [61].

Now let’s consider two electrons in a double quantum dot in the absence of a magnetic field where each electron occupies a single quantum dot. Analogous to a

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Hydrogen molecule, the Coulomb interaction between these two electrons leads to a Heisenberg exchange Hamiltonian acting on the spins [62, 63]: H = J S1.S2. One

can rewrite the Heisenberg Hamiltonian using flip-flop terms: Hss = J S1zS z 2 + J 2 S + 1 S − 2 + S − 1S + 2 (2.11) where J is the exchange coupling. S_i+and S_i−are the raising and lowering operators: S_i+ = Sx i + iS y i = | ↑ iih↓|i and S − i = Six− iS y i = | ↓ iih↑|i. Here we use ~ = 1.

The eigenstates of the above Hamiltonian are one singlet (|S(1, 1) i) and three triplet states(|T0(1, 1) i, |T+(1, 1) i, |T−(1, 1) i). Note that, for all these states,

elec-trons are positioned in different dots. The convention in this thesis is as follows: ↑↑ in the (1,1) charge state denotes ↑L↑R where ↑L (↑R) refers to the spin state in the

left (right) dot.

At zero magnetic field, all triplet states are degenerate and the energy differ-ence between the triplet and singlet states (ET0 − ES) is equal to J . Note that the

exchange coupling strength is a function of the overlap between the wavefunctions of two electrons. Therefore one can control the exchange coupling by controlling the separation between the electrons’ wavefunctions. This can be done in a double quantum dot by pushing the wavefunction of an electron in one dot to the neigh-bouring dot. At constant magnetic field, the total spin Hamiltonian besides the spin-spin interaction Hamiltonian includes the Zeeman terms: H = Hz + Hss. The

Zeeman term is given by Hz =

P

i=1,2gµBBzSz where z points along the external

magnetic field. In this case, the energy degeneracy between triplet states is lifted and the |T+_{i and |T}−_{i states are separated from |T}0_{i state by an amount equal to}

the Zeeman energy of an electron.

For isolated quantum dots, the overlap between the wavefunctions is negligible and so is the exchange coupling (singlet and triplet states are degenerate). Once we introduce tunnel coupling between two dots (tc), transitions from (1,1) to(0,2) charge

states are possible. This hybridizes the charge states of similar spin states (tunneling preserve spins, to a good approximation), see section 2.4 for charge hybridization in coupled quantum dots. Since there is a large energy difference between single-dot singlet and triplet states, the charge hybridizations between singlet and triplets of (1,1) and (0,2) charge states happen at different energy detunings, see Fig. 2.9a. Us-ing the Hamiltonian H in equation 2.2 between |S(1, 1) i and |S(0, 2) i charge states, one can derive the spin exchange coupling as a function of detuning and tunnel cou-pling. Exchange coupling can be approximated as J () =

2 −p(

2)

2_{+ |t}

c|2 where

is the detuning between S(1, 1) and S(0, 2) charge states. The approximation is valid in the range of detuning where the hybridization between the triplet states can be ignored [53] One can understand the detuning dependence of the exchange coupling as follows: exchange coupling depends on the overlap between two elec-trons’ wavefunctions. The detuning changes the ratio between the (1,1) and(0,2) occupation and thereby the amount of overlap between the wavefunctions. As a

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2.9 Spin states and exchange interaction in coupled quantum dots ε [µeV] Energy [ µeV] -600 0 600 -200 0 200 2tc2 Ec ~ 2 3 1 1 S11 T11 ε S11 S02 T11 2 S11 T11 S02 3 S11 T11 S11 T02 T02 ε [µeV] Energy [ µeV] -200 0 200 -50 0 50 a b ∼∆BN,z ∆ΕZ J(ε) S (1,1) S (0,2) T (0,2) T (1,1) 2tc T₊ T -T₀

Figure 2.9: Energy diagram of spin states in coupled quantum dots. Charge states of similar spin states hybridize at the (1,1)-(0,2) charge transition. (a) Energy diagram together with the position of electrochemical potentials at different detuning values (). Here, external magnetic field is zero, tunnel coupling is tc =20 µeV and singlet-triplet splitting is 400 µeV. (b) Applying an external magnetic field results in energy splitting of triplet states. Here the external magnetic field is 1 Tesla and tunnel coupling is 10 µeV.

result, the exchange coupling is dependent on the detuning.

At zero detuning where singlet states have maximum charge hybridization, the exchange energy is equal to J = tc. In the middle of the (1,1) charge state where

= −Ec₂ 10_{, the exchange coupling is J () ∼} 2t2 c Ec

11_{. In coupled lateral quantum dots,}

typically Ecis in the order of couple of meV and tcis about tens of µeV. Therefore,

the exchange coupling becomes very small in the middle of the (1,1) charge state. In other words, it is possible to turn off the exchange interaction by pulsing of the detuning into the middle of (1,1) charge state.

To simplify the discussion, we define a Bloch sphere where the | ↑↓ i and | ↓↑ i states are positioned at the north and south poles. Starting from the initial state | ↑↓ i, we turn on the exchange interaction (J). In this case, the Bloch vector rotates with an angular frequency of J/~ about a vector pointing along the x axis of the Bloch sphere. After a time τp = ~π/J, the spin states are exchanged (swapped).

Therefore, the final spin state is | ↓↑ i. Loss and DiVincenzo proposed that using exchange interaction for a time τp/2, known as

√

SWAP operation, and single qubit gates, one can construct a controlled-NOT (CNOT) gate [12]. CNOT and single qubit gates are a universal set of gates for quantum computation: an arbitrary unitary operation on n-qubits can be done using combinations of these two gates [64].

Now we briefly discuss the effect of the hyperfine interaction. When two elec-trons are separated in a double dot, the electron in each dot experiences a different

10_{where electrochemical potential of two dots lines up (µ}

L(1, 1) = µR(1, 1)). 11_{Here we use a Taylor expansion of J (t}

c/) which in the first approximation gives J () ∼ 2t

2 c

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nuclear Overhauser field through hyperfine interaction with different nuclear baths. These two nuclear fields affect the spin-spin interaction. The nuclear field difference along the external magnetic field (∆Bz

N) mixes |S(1, 1) i and |T0(1, 1) i spin states.

Moreover, the x and y components (∆B_Nx,y) couple |T+(1, 1) i and |T−(1, 1) i states

to |S(1, 1) i. The degree of admixing depends on the energy difference between these states. For large negative detuning deep into the (1,1) charge configuration where J () → 0, the difference between nuclear fields (∆BN) dominates the exchange

in-teraction. At high magnetic fields, |T+(1, 1) i and |T−(1, 1) i are well separated from

|S(1, 1) i state and therefore they do not mix with the singlet any more. On the other hand, the nuclear field difference along the external magnetic field (∆Bz

N)

mixes the |S(1, 1) i and |T0(1, 1) i. Therefore, the eigenstates of the spin

Hamilto-nian are no longer singlet and triplet states but | ↑↓ i, | ↑↑ i, | ↓↑ i and | ↓↓ i. Note that nuclear fields have a slow dynamics and the energy between these eigenstates varies slowly over the time.

When the exchange interaction is much higher than the nuclear field difference, the eigenstates can be well approximated by singlet and triplet states. In the in-termediate regime, the Bloch vector precesses about the vector sum of the nuclear field ∆Bz

N and the exchange interaction.

2.10 Conclusion

In conclusion, we have presented the theory background and the methods relevant to the measurements performed in this thesis. We explained the spin states in coupled quantum dots and discussed the interaction of an electron spin with its environment through spin-orbit coupling and hyperfine interaction.

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Chapter 3 Experimental setup

In this chapter we briefly describe the experimental setup used for performing the measurements presented in this thesis.

3.1 Device cooling and dilution refrigerator

To bound the electrons in a 2DEG and create lateral quantum dots, it is crucial to cool down the heterostructure so that thermal excitations are strongly suppressed. Therefore, the quantum-dot device needs to be cooled down below 1 K. As described in the next chapter, for single shot read-out of the electron spins, the thermal energy corresponding to the electrons in the reservoirs needs to be well below the Zeeman energy (gµBBext > 4kBT ). Therefore, we cool down the device using a Kelvinox

400HA dilution refrigerator to a temperature of about 80-100 mK (with a cooling power of 400 µW at 100 mK). The lattice phonons of the semiconductor are thermal-ized to the base temperature of the dilution refrigerator1_{. However, the electrons in}

the 2DEG have a higher temperature (150-250 mK) because of the weak electron-phonon coupling [66, 67]. Instead, the electrons in the 2DEG are mostly cooled via DC/RF wires connected to the ohmic contacts of the device. In our devices, ohmic contacts are connected to DC wires which are thermally anchored at different stages of the dilution fridge. Note that radiation and noise in the DC/RF wires can also heat up the electrons. Therefore, we use different filtering stages at different tem-peratures to damp the noise (see section 3.3.1). In our dilution fridge, the radiation shield has been removed to increase the available space inside the fridge. There-fore, to suppress heating of the sample by the 4 K radiation of the inner vacuum chamber (IVC), we mount our device in a copper can which is attached to mixing chamber via cold finger2. Additionally, we cover the PCB with a copper housing

1_{Note that usually the device is glued to the PCB with FR4 as dielectric material. Therefore}

the cooling efficiency is limited by FR4 thermal conductivity [65] and the glue.

2_{4 K radiations from the IVC adds a heat load on the mixing chamber plate, thus increasing}

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for extra damping of radiation. In the experiments presented in this thesis, electron temperature is measured to be about 250 mK. After performing the measurements of chapters 4–7, we reduced the base temperature of the fridge and thereby the elec-tron temperature (see section 3.6). The dilution fridge used for the cooling down of our sample is equipped with a superconducting magnet which can be used to apply magnetic fields up to 14 T. 56pF 4.7nF 10

a

1.7 K stage mixing chamber stage (m/c) Super-conducting coax lines

b

DC wires copper can

Cu-powder filters RC filters PCB m/c 1 2 1 56pF 47nF 10 DC DC 2 sample

c

100pF DC DC 3 3 SMA SMA connector connector

Cold finger (see b)

Figure 3.1: (a) Insert of Kelvinox 400HA dilution refrigerator from 1 K stage to mixing chamber stage. (b) Cold finger connected to the mixing chamber with (left picture) and without (right picture) copper can. PCB is covered with a copper housing. RC filters, Cu-powder filters and PCB are positioned inside the copper can. (c) PCB with a sample glued on it. Sample is connected via wired bonds to the PCB. The values of bias-tees are shown. All DC wires are connected with a 100 pF to ground.

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3.2 Printed circuit board (PCB)

Figure 3.1c shows a PCB similar to the one used for the experiments presented in this thesis. The sample is glued on the FR4 material of a PCB (printed by Eurocircuits). Then, the gates and the ohmic contacts are connected to the PCB via Al-wire bonds. The PCB is equipped with four SMA connectors to apply voltage or microwave (MW) pulses on the gates. In the measurements in this thesis, we use voltage pulses on the plunger gates to control the electrochemical potential of the dots and MW pulses on the side gates to manipulate electrons spins (see Fig. 3.2). To combine the voltage and MW pulses with the DC voltages, we use RC type bias-tees mounted on the PCB. For the bias-bias-tees connected to the plunger gates, we used R=10 MΩ and C=47 nF, giving a settling time of ∼470 ms. This allows applying voltage pulses as long as tens of ms. For the lines used for the MW pulses we use bias-tees with R=10 MΩ and C=4.7 nF giving a time constant of ∼ 47 ms (we use a higher cut-off frequency for the these bias-tees to damp the low frequency noise coming from the MW source).

3.3 Measurement electronics

Figure 3.2 shows the schematic of the measurement electronics connected to the sam-ple. The measurement electronics mainly consist of (i) digital-to-analog converters (DACs) and optically isolated voltage sources (ii) current-to-voltage (IV) converters. They are all designed by Raymond Schouten at Delft University of Technology. The gates of our device are voltage biased using 16-bit DACs providing voltages from -2 to 2 V (resolution of about 0.06 mV). The voltage fluctuation of the DACs is very low and is measured to be about 4 µV during an hour. For the plunger gates, which are used to control the electrochemical potential of the dots, we use a combination of two 16-bit DACs of which one is down-scaled with a factor of 100. Therefore, the voltage resolution (thus control over the electrochemical potentials) is improved to 600 nV. DACs are controlled by computer via an optical fiber. In this way, the measurement computer is decoupled from the measurement setup. All measurement electronics connected to the sample are battery-powered and optically isolated via analog opto-couplers from commercial equipment connected to the mains power line.

A DC transport measurement of quantum dots usually includes voltage biasing ohmic contacts across the quantum dots (QDs) or quantum point contacts (QPCs) and measuring the current flow. To voltage bias the ohmic contacts (source and drain), we use optically isolated voltage sources. An optically isolated voltage source converts the voltage of a DAC into an optical signal. Then, a photodiode is used to convert the optical signal into an output voltage (photocurrent of the photodiode passed through a resistance of 10 Ω so that a voltage between -2 and 2 mV with

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cold V_s DAC 1 DAC 2 117 Ω 1.5 nF DAC 3

...

DAC 16 fiber computer 470 Ω 1.5M Ω 10 nF 1.5 nF 0.27 nF 100 pF 5 Ω 5 Ω 10 MΩ x104 Digital Oscillo-scope R_f LAN IV-converter Isolation amplifier computer feedthrough capacitor RC filters

Room temperature

Base

temperature

100 mK-300 K

IsoAmp x10 5 Cu-Powder filters Pi-filters Wire resistance Twisted pairs Thermal anchoring 0.4 nF fast line gate connection Ω 5 Ω 1 mV/V Cold finger 0.4 nF 470 Ω 0.2 µ H plunger gate plunger gate side gate side gate

Figure 3.2: Electrical circuit schematics of the measurement electronics connected to the device. DACs are used to apply voltages on the gates. An optically isolated DAC together with an IV-converter is used to perform voltage-biased current measurement.

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JFET-3.3 Measurement electronics

based IV-converter. We used the same IV-converter as the one used in [68] and one copy of it, to measure the current of two QPCs adjacent to a double quantum dot. The output voltage of a IV-converter, which has been optically isolated using an isolation-amplifiers (ISO-amp)3, can be read out using any commercial voltmeter. We use a Keithley 2700 for the low bandwidth voltage measurements and a LeCroy WaveRunner 44Xi oscilloscope for high bandwidth measurements. In the design of the IV-converter special attention is made (i) to minimize the input noise of the IV-converter and (ii) to have sufficient bandwidth for the single-shot measurements presented in this thesis. The input noise of the IV-converter includes the input noise of the amplifier and the thermal noise from the feedback resistor. The first stage of the amplification is based on a low-noise JFET (Interfet 3602) resulting in an input voltage noise of 0.8 nV/√Hz. The gain and resistance of the IV-converter are 104 and 10 MΩ giving an input resistance of 1 kΩ.

The input of each IV-converter is connected to an ohmic contact of the device using Cu-wires. The input resistance (Ri) of the IV-converter together with the

capacitance of the line from the IV-converter to the ohmic contact creates a low pass filter which limits the bandwidth of the measurement. The line capacitance originates from the capacitance of the wires and capacitance of filtering stages to ground. Note that as a result of line capacitance, the output voltage-noise of the IV-converter increases as a function of frequency. Therefore, it is very important to minimize the line capacitance to ground. This is definitely a compromise, since it means decreasing the filtering in the line. To decrease the line capacitance (i) we used a short wire (1 m) between the fridge feedthrough and the IV converter (∼ 100 pF/m). (ii) The filtering is drastically decreased to minimize the capacitance to the ground: we use a small capacitance (100 pF) instead of pi-filters and we use a low-capacitance-first order RC filter (270 pF), for details of the filtering stages see section 3.3.1). (iii) The drain ohmic contact is grounded at the mixing chamber plate (“cold ground”) rather than room temperature. In other words, we avoid us-ing extra wires (which are normally connected to the filter stages) to minimize the line capacitance. Therefore, a single wire connected to the source ohmic contact is used to voltage bias and measure the current. In order to voltage-bias the ohmic contacts, we bias the IV-converter (see figure 3.2). Note that there is a voltage difference between the cold ground and room temperature ground (thermo-voltage). Therefore, we connect the ground of the optically isolated voltage source to the cold ground as well. The total capacitance of the line from room temperature to the sample is then measured to be ∼1 nF. Therefore the bandwidth of the IV converter is ∼160 kHz. In reality, in order to obtain a sufficient signal-to-noise ratio, we need to use additional low-pass filters to the attenuate the noise (∼ 40 kHz cut-off fre-quency). We use a tunable Butterworth low-pass filter after the ISO-amp for this purpose (double-channel Kron-Hite model: 3202).

Electrical Control, Read-out and Initialization of Single Electron Spins

Electrical Control, Read-out and

Initialization of Single Electron Spins

Electrical Control, Read-out and

Initialization of Single Electron Spins

Proefschrift

Mohammad SHAFIEI

Contents

Chapter 1

Introduction

1.1

Quantum mechanics

1.2

Quantum computation

1.3

Electrons- spin and charge

1.4

Outline of the thesis

Chapter 2

Spins in few-electron GaAs

quantum dots

2.1

Creation of lateral quantum dots

2.2

Charge stability diagram

2.3

Compensation of capacitive coupling and

vir-tual gates

2.4

Coupled quantum dots

2.5

Spin states in a quantum dot

2.6

Spin-orbit interaction

2.6.1

Spin-orbit interaction in quantum dots and spin

re-laxation

2.6.2

Electric dipole spin resonance (EDSR) mediated by

spin-orbit interaction

E

2.7

Hyperfine Interaction

B

2.8

EDSR mediated by Hyperfine Interaction

2.9

Spin states and exchange interaction in

cou-pled quantum dots

2.10

Conclusion

Chapter 3

Experimental setup

3.1

Device cooling and dilution refrigerator

a

b

c

3.2

Printed circuit board (PCB)

3.3

Measurement electronics

...

Room temperature

Base

temperature

100 mK-300 K