Single spins in nanowire quantum dots with strong spin-orbit coupling

dots with strong spin-orbit coupling

-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 maandag 1 december 2014 om 10:00 uur

door

Johannes Wilhelmus Georg

VAN DEN

B

ERG

natuurkundig ingenieur geboren te ’s-Gravenhage, Nederland.

Dit proefschrift is goedgekeurd door de promotor: Prof. dr. ir. L. P. Kouwenhoven

Samenstelling promotiecommissie:

Rector Magnificus voorzitter

Prof. dr. ir. L. P. Kouwenhoven Technische Universiteit Delft, promotor Prof. dr. Yu. V. Nazarov Technische Universiteit Delft

Prof. dr. ir. R. Hanson Technische Universiteit Delft Prof. dr. E. P. A. M. Bakkers Technische Universiteit Eindhoven Prof. dr. J. M. van Ruitenbeek Universiteit Leiden

Dr. S. De Franceschi CEA Grenoble, Frankrijk

Prof. dr. Y. M. Blanter Technische Universiteit Delft, reservelid

Keywords: Nanowires, spin-orbit coupling, quantum dots, spins, qubits

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Front & Back: Electron microscope image and schematic illustration of a typical de-vice studied in this thesis.

Copyright © 2014 by J.W.G. van den Berg Casimir PhD Series, Delft-Leiden 2014-30 ISBN 978-90-8593-203-1

An electronic version of this dissertation is available at http://repository.tudelft.nl/.

1 Introduction 1

1.1 Quantum mechanics . . . 1

1.2 Quantum computation . . . 2

1.3 Spins as quantum bits. . . 3

1.4 Semiconductor nanowires . . . 4

1.5 Outline of this thesis . . . 4

References . . . 5

2 Theory 7 2.1 Quantum Dots . . . 7

2.1.1 Single quantum dots . . . 8

2.1.2 Double quantum dots . . . 9

2.1.3 Pauli spin blockade . . . 9

2.2 Spin-based qubits. . . 10

2.2.1 Spin rotations . . . 11

2.2.2 Relaxation and decoherence . . . 12

2.3 Spin-orbit coupling . . . 13

2.3.1 Spin-orbit coupling in crystalline solids . . . 14

2.3.2 Spin-orbit coupling due to structural inversion asymmetry . . . 16

2.3.3 Spin-orbit coupling in quantum dots . . . 17

2.4 Nuclear hyperfine interaction . . . 18

2.4.1 Effect of the nuclear field on the electron spin . . . 19

2.4.2 Nuclear spin dynamics. . . 20

References . . . 23

3 Device fabrication and measurement 29 3.1 Nanowire growth . . . 29

3.2 Device fabrication . . . 30

3.2.1 Electron beam lithography . . . 30

3.2.2 Gate fabrication . . . 31 3.2.3 Nanowire deposition . . . 32 3.2.4 Contact fabrication . . . 33 3.3 Measurement setup . . . 34 3.4 High-frequency experiments . . . 34 References . . . 36 vii

viii CONTENTS

4 Spectroscopy of spin-orbit qubits in indium antimonide nanowires 37

4.1 Introduction . . . 38

4.2 Device and readout . . . 38

4.3 Electric dipole spin resonance . . . 39

4.3.1 EDSR in the weak coupling regime . . . 39

4.3.2 EDSR in the strong coupling regime . . . 41

4.4 Anisotropy of the spin-orbit gap . . . 41

4.5 Supplementary Information . . . 45

4.5.1 Device fabrication and measurement details. . . 45

4.5.2 Widths of hyperfine and EDSR peaks. . . 45

4.5.3 Origins of S-T anticrossings in the EDSR spectrum. . . 46

4.5.4 The two-electron double dot energy level model . . . 48

4.5.5 Determining spin-orbit length lSO. . . 48

4.5.6 Additional anisotropy data . . . 50

4.5.7 Spectroscopy of strongly coupled spin-orbit qubits in indium ar-senide nanowires . . . 51

References . . . 52

5 Suppression of Zeeman gradients by nuclear polarization in double quan-tum dots 55 5.1 Introduction . . . 56

5.2 Bistable current for strongly coupled dots. . . 56

5.3 Electric dipole spin resonance spectroscopy . . . 58

5.4 A model for bistable behavior . . . 59

5.5 Supplementary Information . . . 63

5.6 Theoretical considerations . . . 68

5.6.1 Electron dynamics . . . 68

5.6.2 Dynamic nuclear polarization . . . 71

5.6.3 Maximum gradient achievable . . . 74

5.6.4 Fluctuations of the nuclear fields around the stable points. . . 74

References . . . 77

6 Fast spin-orbit qubit in an indium antimonide nanowire 81 6.1 Introduction . . . 82

6.2 Device . . . 82

6.3 Readout and EDSR . . . 82

6.4 Rabi oscillations . . . 84

6.5 Ramsey fringes and qubit coherence . . . 84

6.6 Supplementary Information . . . 88

6.6.1 Double dot occupation . . . 88

6.6.2 Rabi oscillations . . . 89

6.6.3 Manipulation fidelity . . . 89

6.6.4 Individual addressing of the qubits . . . 91

7 Electrical control of single hole spins in nanowire quantum dots 95

7.1 Introduction . . . 96

7.2 Holes in an InSb nanowire . . . 96

7.3 Few-hole double quantum dot . . . 97

7.4 EDSR in hole quantum dots . . . 99

7.5 Conclusion . . . 102

7.6 Supplementary information . . . 103

7.6.1 Bandgap of InSb nanowires . . . 103

7.6.2 Origin of the background triangles in figure 7.3 . . . 103

7.6.3 Zeeman splitting of QD lead resonances . . . 104

7.6.4 Spin blockade data in the weak coupling regime . . . 105

7.6.5 Comparison of hyperfine coupling strength between holes and elec-trons . . . 106

7.6.6 Additional measurements of the angular dependence of spin block-ade in the strong coupling regime . . . 106

7.6.7 Method for extracting the g -factors . . . 106

References . . . 108

8 Conclusions and outlook 113 8.1 Results . . . 113

8.2 Outlook . . . 114

8.2.1 Two-qubit gates . . . 114

8.2.2 Readout . . . 114

8.2.3 Improving coherence . . . 115

8.2.4 Coupling remote spins . . . 115

8.2.5 Topological quantum computing . . . 117

References . . . 117 Summary 121 Samenvatting 123 Acknowledgements 125 Curriculum Vitæ 127 List of Publications 129

1

I

NTRODUCTION

Our modern day society would not look the same without the (personal) computer. The invention of the field-effect transistor (FET) has made this development possible. Al-ready in 1926 the concept for such a transistor was patented by Julius Edgar Lilienfeld[1]. However, a practically working transistor was not realized until 1947 by John Bardeen, Walter Houser Brattain, and William Bradford Shockley in Bell labs. They made their device from the semiconductor germanium, work which earned them a Nobel prize in 1956. Nowadays, most computer chips are made out of metal-oxide semiconductor field-effect transistors (MOSFETs) in silicon. Continuing miniaturization has made tran-sistors ever smaller and allowed the number of trantran-sistors on a chip to grow exponen-tially over time (Moore’s law [2]). As the size of a transistor shrinks, at some point it be-comes so small that quantum mechanical effects start to become important (other than for explaining the properties of semiconductor materials which transistors are made of ). However, this behavior need not be a hindrance, but can actually be a valuable asset as we will see in the following.

1.1 Q

UANTUM MECHANICS

At the end of the 19th century, certain physicists—such as the (later) Nobel-prize winner Albert Michelson—held the view that the field of physics was almost finished. Only some minor problems remained and future work would only rest in finding applications for the established principles of nature and in making more precise measurements. When he started his studies in 1874, Max Planck was given a similar warning by his professor. However, this did not stop Planck from studying physics and subsequently becoming one of the key figures to lay the foundations of quantum mechanics.

In classical mechanics a particle follows a specific trajectory and can always be found at a specific place at a specific time. Quantum mechanics on the other hand, describes a particle by a wavefunctionΨ, which can be spread out over space. The probability to find the particle at position x at time t is given by |Ψ(x,t)|2_{. The evolution of this} wave-function is governed by the Schrödinger equation. Describing a particle as a wave has

1

2 1. INTRODUCTION

the peculiar consequence that it can be in more than one place at the same time. How-ever, trying to observe the particle forces it to be in a specific position and subsequent measurements will reveal the particle in that same position with certainty. This process is known as wavefunction collapse. Such behavior can be demonstrated using a double-slit experiment. A single particle like an electron fired at a screen containing two parallel slits, will travel through both slits at the same time, interfering with itself. When detected some distance behind the screen, the particle will still only appear at one point in space. When repeating this many times though, an interference pattern will emerge from the detected positions of the particles.

Although he played an important role in the development of quantum mechanics, Albert Einstein was never at ease with the theory and believed that in time a ‘more com-plete’ theory would be formulated. In letters to Max Born he expressed this concern by stating that “He [God] is not playing dice” (referring to the statistical nature of quan-tum mechanics) and that the theory implied “spooky action at a distance” (referring to entanglement)[3].

Einstein’s reluctance to accept quantum theory is not surprising considering it pre-dicts behavior that is very counterintuitive, because we do not observe it in the everyday world around us. Nonetheless, quantum mechanics has been very successful in explain-ing the world on a microscopic (atomic) scale and has played an important role in mod-ern technological inventions such as the laser and magnetic resonance imaging (MRI).

1.2 Q

UANTUM COMPUTATION

The power of quantum mechanics for computational purposes hinges on the very prin-ciples that make it so counterintuitive; namely quantum superposition and

entangle-ment. A classical computer takes an input, a number we can call x, and calculates the

corresponding output function f (x). If we want to know the output for a different input

x0_{, the computer needs to repeat the calculation to generate the output f (x}0_{). On a} quan-tum computer however, we can prepare a state that is a combination of every possible input value. Then by computing the function f (x) on this superposition state, we obtain the output for each possible input state simultaneously. For a system consisting of n

quantum bits (or qubits, the quantummechanical analogue to classical bits), that means

we can evaluate up to 2ninput states at the same time. The challenge remains though how to exploit this so-called quantum parallelism. After all, performing a measurement on the output will only give us one of the many possible answers. To this end, quantum algorithms are designed such that the final measurement will yield the desired answer. Already several quantum algorithms have been developed, the most famous being Shor’s

algorithm for factoring integers [4] and Grover’s algorithm for searching databases [5].

These algorithms are more efficient than any (known) classical algorithm.

One interesting application of quantum computers could be to (efficiently) simulate other quantum systems. This use of quantum computers as so-called quantum simula-tors was first envisioned by Richard Feynman in 1982 [6] and it was later shown that this idea was in fact possible [7].

Rudimentary implementations of quantum algorithms using just a few qubits have already been realized in several ways, such as by nuclear magnetic resonance on mole-cules [8, 9], superconducting circuits [10, 11], ion traps [12] and nitrogen vacancy

cen-1

ters [13]. Any useful quantum computer is generally expected to require on the order of 100 qubits though. It therefore remains an open question which type of system will be most suitable for realizing a quantum computer.

Although it may not be clear yet what physical implementation will best suited for a quantum computer, there is a set of requirements that any such system must ful-fill. These are the so-called DiVincenzo criteria, consisting of five requirements for the implementation of quantum computation, plus an additional two regarding quantum communication [14, 15].

1.3 S

PINS AS QUANTUM BITS

As we have seen, a quantum computer may be implemented in many different physical systems. In principle any quantummechanical two level system could encode a quan-tum bit. In this thesis, we are interested in the spin of a single charge carrier inside a semiconductor (i.e. an electron or hole).

In 1925 Samuel Goudsmit and George Uhlenbeck published a paper introducing the concept of a spinning electron in order to explain the fine structure present in atomic spectra [16]. The idea had in fact already occurred to Kronig before then, but after re-ceiving heavy criticism, he decided not to publish about it. After consulting with theo-rists such as Lorentz, Uhlenbeck and Goudsmit became worried that their idea was non-sensical and also wanted to refrain from publishing the paper. However, their supervisor Ehrenfest had already sent the paper off and told them it would appear soon, saying that they were “young enough to afford a stupidity” anyway [17]. Eventually it was Wolfgang Pauli who worked out the mathematical theory and incorporated electron spin in quan-tum mechanics [18]. Later, Paul Dirac formulated a complete relativistic wave equation that includes spin [19, 20].

Although a spinning electron carries only a tiny magnetic moment with it, it can nev-ertheless give rise to large scale effects, such as (ferro)magnetism; the alignment of a large number of spins leads to a macroscopic magnetic field. Remarkably, the reason behind this alignment of spins is not due to their dipolar interaction, but due to the exchange interaction, a result of electrons being fermions (and can thus only occupy a quantum state alone). Magnetism and thus spin has an important application in the storage on computer hard disks. Furthermore, the read-out of these disks nowadays relies on so-called giant magnetoresistance, an effect that was only recently (1988) dis-covered and for which the 2007 Nobel Prize in Physics was awarded.

A transistor based on the two possible spin orientations of an electron was proposed in 1990 by Datta and Das [21] (and hence often referred to as a spin-, or Datta-Das tran-sistor). Although this is another interesting example of the application of spin in elec-tronics (a field also known as spinelec-tronics), it still pertains to the use of spin only in a

clas-sical information processing context. It was proposed in 1998 by Loss and DiVincenzo

that electron spins could also be used for quantum computation [22]. They envisioned that a qubit could be encoded in the spin of an excess electron inside a quantum dot. Coupling such dots together would allow two-qubit gate operations and therefore quan-tum algorithms to be performed.

1

4 1. INTRODUCTION

1.4 S

EMICONDUCTOR NANOWIRES

Although we have established that we are interested in spin qubits, there are still var-ious ways in which to implement such a qubit. Examples include quantum dots in 2-dimensional electron gases (2DEGs), self-assembled quantum dots or nitrogen-vacancy (NV) centres in diamond. In this thesis though, we have chosen to investigate spins in semiconducting nanowires.

While silicon may be the most abundantly used material in the semiconductor in-dustry, there are a host of different semiconductor materials that receive attention for various applications. One can think of detectors, light-emitting diodes or solar cells, for example. When it comes to fabricating devices with different materials and differ-ent combinations of materials, nanowires are extremely versatile and can readily realize structures that may not be possible in bulk samples. Different crystal structures than found in bulk material can be obtained and heterostructures that would otherwise have prohibitively large strain can be grown in nanowires. This is due to the fact that strain originating from the combination of materials with different lattice constants can be re-laxed in the radial direction.

That many different applications have been found for nanowires should therefore be no surprise. Examples of these applications include solar cells [23], field-effect transis-tors [24], light-emitting diodes [25] and biochemical sensors [26].

In this thesis we are specifically interested in nanowires made of indium arsenide or indium antimonide. Both are materials in which the orbital motion of the electron is strongly coupled to its spin. This has the advantage that electric fields can easily and quickly manipulate the electron spin through electric dipole spin resonance (EDSR) [27].

1.5 O

UTLINE OF THIS THESIS

The remainder of this thesis is built up in the following way:

Chapter 2 introduces the theoretical background relevant to the experiments

dis-cussed in this thesis, starting with the confinement of single charge carriers in a quan-tum dot and electrical transport through such a structure. Furthermore the phenomena of spin-orbit coupling and hyperfine coupling are discussed.

The growth of semiconducting nanowires and fabrication of devices with these nano-wires is discussed in chapter 3, followed by an outline of the experimental setup used to measure these devices.

In chapter 4 electric dipole spin resonance is used to probe the spectrum of two-electron eigenstates in a few-two-electron double quantum dot.

Dynamic nuclear polarization, suspected to be behind bistable behavior of current through a nanowire quantum dot, has been investigated in chapter 5.

The realization of a qubit utilizing the spin-orbit coupling inside an indium anti-monide nanowire is the subject of chapter 6.

Chapter 7 demonstrates the possibility of confining single holes inside an indium

antimonide nanowire and the ability to control them using microwave electric fields, similar to electrons.

Finally, a conclusion is given in chapter 8 together with an outlook on possible future research.

1

R

EFERENCES

[1] J. E. Lilienfeld, Method and apparatus for controlling electric currents, (1930). [2] G. Moore, Cramming More Components Onto Integrated Circuits, Proceedings of the

IEEE 86, 82 (1998).

[3] R. Andrews, The New Penguin Dictionary of Modern Quotations (Penguin Books Limited, 2003).

[4] P. Shor, Algorithms for quantum computation: discrete logarithms and factoring, in Proceedings 35th Annual Symposium on Foundations of Computer Science (IEEE Comput. Soc. Press, 1994) pp. 124–134.

[5] L. Grover, Quantum Mechanics Helps in Searching for a Needle in a Haystack, Phys-ical Review Letters 79, 325 (1997).

[6] R. P. Feynman, Simulating physics with computers, International Journal of Theo-retical Physics 21, 467 (1982).

[7] S. Lloyd, Universal Quantum Simulators, Science 273, 1073 (1996).

[8] L. M. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, Experimental realization of Shor’s quantum factoring algorithm using

nu-clear magnetic resonance. Nature 414, 883 (2001).

[9] N. Xu, J. Zhu, D. Lu, X. Zhou, X. Peng, and J. Du, Quantum Factorization of 143 on a

Dipolar-Coupling Nuclear Magnetic Resonance System, Physical Review Letters 108,

130501 (2012).

[10] L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Demonstration of

two-qubit algorithms with a superconducting quantum processor. Nature 460, 240

(2009).

[11] E. Lucero, R. Barends, Y. Chen, J. Kelly, M. Mariantoni, A. Megrant, P. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, Y. Yin, A. N. Cleland, and J. M. Martinis,

Computing prime factors with a Josephson phase qubit quantum processor, Nature

Physics 8, 719 (2012).

[12] S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Häffner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Implementation of the Deutsch-Jozsa algorithm on

an ion-trap quantum computer. Nature 421, 48 (2003).

[13] T. van der Sar, Z. H. Wang, M. S. Blok, H. Bernien, T. H. Taminiau, D. M. Toyli, D. A. Lidar, D. D. Awschalom, R. Hanson, and V. V. Dobrovitski, Decoherence-protected

quantum gates for a hybrid solid-state spin register. Nature 484, 82 (2012).

[14] D. P. DiVincenzo, Topics in Quantum Computers, in Mesoscopic Electron Transport, edited by L. L. Sohn, L. P. Kouwenhoven, and G. Schön (Springer Netherlands, Dor-drecht, 1997) pp. 657–677.

1

6 REFERENCES

[15] D. P. DiVincenzo and D. Loss, Quantum computers and quantum coherence, Journal of Magnetism and Magnetic Materials 200, 202 (1999), arXiv:9901137 [cond-mat] . [16] G. E. Uhlenbeck and S. Goudsmit, Ersetzung der Hypothese vom unmechanischen

Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elek-trons, Die Naturwissenschaften 13, 953 (1925).

[17] G. E. Uhlenbeck, FIFTY YEARS OF SPIN: Personal reminiscences, Physics Today 29, 43 (1976).

[18] W. Pauli, Zur Quantenmechanik des magnetischen Elektrons, Zeitschrift für Physik

43, 601 (1927).

[19] P. A. M. Dirac, The Quantum Theory of the Electron, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 117, 610 (1928).

[20] P. A. M. Dirac, The Quantum Theory of the Electron. Part II, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 118, 351 (1928). [21] S. Datta and B. Das, Electronic analog of the electro-optic modulator, Applied Physics

Letters 56, 665 (1990).

[22] D. Loss and D. P. DiVincenzo, Quantum computation with quantum dots, Physical Review A 57, 120 (1998).

[23] B. Tian, X. Zheng, T. J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, and C. M. Lieber,

Coaxial silicon nanowires as solar cells and nanoelectronic power sources. Nature

449, 885 (2007).

[24] Y. Li, F. Qian, J. Xiang, and C. M. Lieber, Nanowire electronic and optoelectronic

devices, Materials Today 9, 18 (2006).

[25] E. D. Minot, F. Kelkensberg, M. van Kouwen, J. A. van Dam, L. P. Kouwenhoven, V. Zwiller, M. T. Borgström, O. Wunnicke, M. A. Verheijen, and E. P. A. M. Bakkers,

Single quantum dot nanowire LEDs. Nano Letters 7, 367 (2007).

[26] Y. Cui, Q. Wei, H. Park, and C. M. Lieber, Nanowire nanosensors for highly sensitive

and selective detection of biological and chemical species. Science 293, 1289 (2001).

[27] K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, and L. M. K. Vandersypen, Coherent

2

T

HEORY

This chapter will introduce the theoretical background relevant to the experiments dis-cussed in this thesis. It aims at a concise introduction of all important concepts, while a more extensive discussion of these subjects may be found in various books and reviews [1–3]. We will start by discussing quantum dots and a simple model (the constant in-teraction model) to understand transport through a quantum dot when it is coupled to source, drain and gate electrodes. Both single and double quantum dots will be consid-ered. Next, the concept of spin blockade will be covered, which is an important read-out mechanism in our experiments. Then we will continue with a discussion about the use of spins as qubits and how these may be controlled. Any physical qubit suffers from the loss of quantum information, which will therefore be included in the discussion. We will follow with the two most important interactions that couple a spin to the environment; spin-orbit coupling and hyperfine coupling. Since the hyperfine coupling depends on the nuclear spin bath, we will also consider several mechanisms that influence its evolu-tion.

2.1 Q

UANTUM

D

OTS

Quantum dots are small semiconducting islands in which single charge carriers are con-fined in all dimensions [4]. There are many different structures in which quantum dots can be realised, such as a two-dimensional electrons gas (2DEG), nanocrystals or nano-wires. The latter of course are the focus of this thesis. In the next chapter the growth of such nanowires and fabrication of quantum dot devices will be discussed, while here we will introduce some theoretical concepts for understanding the measurements on quantum dots.

All of our measurements in this thesis involve electrical transport through quantum dots, so they are connected to source and drain leads. In addition, electrostatic gates below the nanowire help define the quantum dots and control the number of charge carriers in the dots. The constant interaction model [5] explains well the transport char-acteristics of a quantum dot coupled to gate electrodes and connected to source/drain leads. Therefore we will use it here to describe the transport through our devices.

2

8 2. THEORY

2.1.1. S

We will start by considering a single quantum dot coupled to a source and drain elec-trode, as well as to a plunger gate that can control the number of electrons in the dot. Figure 2.1(a) schematically depicts such a configuration. In the constant interaction model the Coulomb interactions between electrons1are captured by a single, constant capacitance C , which is the sum of the capacitances to source CS, drain CD and gate

CG. Secondly, in the constant interaction model the single-particle energy levels are ass-sumed to be independent of the Coulomb interactions. Using this the total energy U (N ) on the quantum dot can be calculated.

To understand the transport through the quantum dot, it is useful to consider the electrochemical potential on the dot, which is the amount of energy needed to add the

N th electron to the dot;µ(N) ≡ U(N)−U(N −1). The electrochemical potential depends

linearly on the gate voltage and thus the amount of electrons on the dot can be changed by changing the gate voltage. When the electrochemical potential on the dot aligns with those of the leads, electrons can move on and off the dot, allowing current through the device. However, when the chemical potential of the dot lies outside of the bias window set by the leads, only one charge state is energetically allowed and current is blocked. The dot is then said to be in Coulomb blockade. Thus transport is only possible when

µS≥ µ ≥ µD. This principle is illustrated in figure 2.1(b).

The spacing between consecutive potential energies is the addition energy Eadd≡

µ(N+1)−µ(N) = EC+∆E. This addition energy consists of the charging energy EC= e2/C , accounting for the Coulomb repulsions between electrons, and the energy spacing_∆E between two discrete energy levels of the dot. Note that∆E can be 0 when two electrons are added to the same energy level, such as is the case for spin degenerate orbitals.

a

QD

b

=

Figure 2.1: (a) Schematic circuit diagram of a single quantum dot coupled to source and drain electrodes and a gate to control the charge on the dot. (b) Current through the quantum dot for small bias. As illustrated above the graph, when the electrochemical potential aligns with that of the leads, transport is enabled; when it lies outside the bias window, no current flows and the dot is said to be in Coulom blockade.

1_{Note that while we consider electrons here, the model describes just as well the basic electronic properties} for holes. So wherever is written ‘electrons’, one may also read ‘holes’, with the understanding that they have opposite charge.

2

2.1.2. D

Now that we have introduced a single quantum dot connected to source and drain leads, we will consider two quantum dots coupled to each other and connected (in series) to source and drain electrodes, as shown in figure 2.2(a). By measuring the current through the dots as a function of the gate voltages VLand VR, controlling the number of electrons on the left and right dot respectively, a so-called stability diagram can be mapped out. This gives rise to a honeycomb pattern indicating regions of constant charge (NL, NR) on the double quantum dot (figure 2.2(b)). At the triple-points where three different charge states meet, the chemical potentials of the dots and leads align and transport through the double dot is enabled. These points expand into triangular regions where transport can occur for high applied bias.

a

b

Figure 2.2: (a) Schematic picture of a double quantum dot connected to source and drain leads. (b) So-called stability diagram in which current through the double dot is plotted as a function of the plunger gate voltages. The diagram maps out regions of constant charge in the double dot, giving rise to a honeycomb pattern. At the points where these regions meet the electrochemical potentials of the dots align with the leads and current through the device is possible. For larger bias these so-called triple-points expand into triangular regions.

2.1.3. P

So far we have only considered the influence of an electron’s charge and a discrete en-ergy spectrum on the transport through a quantum dot. An electron also possesses spin however, and this can complicate the transport through a double quantum dot. In par-ticular, current through the double quantum dot can become suppressed for certain spin configurations in the dots. The simplest case that can be considered is the transition be-tween the (1,1) and (0,2) charge states in the double dot. The two electrons in these charge state can occupy either a spin singlet state S =↑↓ − ↓↑ (excluding normalization), or one of three triplet state: T_{+=↑↑, T}0=↑↓ + ↓↑ and T−=↓↓, where ↑ and ↓ respectively indicate up and down spin states of a single electron. Because tunneling between the two dots conserves spin, this can lead to so-called Pauli spin blockade. Pauli spin block-ade occurs when the double dot is loblock-aded with a (1,1) triplet state. From such a state, the double dot cannot transition to the (0,2) charge state, because spin selection rules forbid a transition to the (0,2) singlet state and a transition to a (0,2) triplet state is energetically not accessible (except for large applied bias that exceeds the orbital energy of the dot). The electrons will thus become trapped and current through the dots will be suppressed.

2

10 2. THEORY

The current carrying cycle can only be restored when one of the electron spins is flipped. Note that this spin blockade only occurs in the forward bias direction. For reverse bias, electrons will initially be loaded into the S(0,2) singlet ground state, which can always transition to the S(1,1) state and thus no blockade will occur.

0 I (pA) 150 -165 -185 VSD = 1.3mV VSD = -1.3mV VR (mV) -165 -185 VR (mV) VL (mV ) -245 -270 S02 T02 T11 S11 S02 T02 T11 S11

a

b

c

d

Figure 2.3: Illustration of the Pauli spin blockade principle. (a) Energy diagram of a double dot under forward bias around the (1, 1) → (0,2) → (0,1) → (1,1) transport cycle. When a (1,1) triplet state is loaded, the S(0,2) state will be inaccessible and transport will be suppressed. (b) The same diagram under reverse bias. Now the S(0,2) charge state will be loaded first and no blockade occurs. (c) Stability diagram corresponding to (a), the triangles indicate regions where transport is energetically allowed, but blocked by spin blockade. Note that at the edges of the triangles current still occurs due to spin exchange with the leads. (d) For reverse bias, current flows normally.

2.2 S

PIN

-

BASED QUBITS

A quantum bit, or qubit, can be implemented in many physical systems. Daniel Loss and David DiVincenzo first suggested in 1998 that the spin of an electron trapped inside a quantum dot would be a suitable candidate for a qubit[6]. The up | ↑〉 and down | ↓〉 spin states are natural candidates for the |0〉 and |1〉 states of the qubit. As will become clear in a moment, it is useful to write the state of a qubit in the following way:

|Ψ〉 = cos(θ/2)| ↓〉 + sin(θ/2)eiφ| ↑〉, (2.1) where we have omitted a global phase factor. The state of a qubit can then conveniently be represented visually on a so-called Bloch sphere, see figure 2.4, whereθ and φ are the polar and azimuthal angles respectively. On the sphere, the poles represent the up | ↑〉 and down | ↓〉 state of the spin, while intermediate points on the surface of the sphere

2

are superpositions of these states. Points inside the sphere describe mixed states (states about which we have only partial information) [7].

x y z

a

b

c

Figure 2.4: (a) Bloch sphere visualizing a qubit state as described by equation 2.1. (b) Relaxation is the decay of the qubit to its ground state, and thus represents a loss of information about the angleθ. In addition, relaxation contributes to a decay of the transverse polarization. (c) Dephasing corresponds to a loss of information stored in the angleφ.

2.2.1. S

To manipulate a qubit, we must be able to drive transitions between its basis states; for a spin these are the two Zeeman-split levels. To drive transitions between these states the most basic technique that can be used is electron spin resonance (ESR) [8]. With this technique a rotating magnetic field B1is applied perpendicular to the external field Bext and resonant with the spin’s Larmor frequency fac= g µBBext. The spin will then revolve around B1, making a simple circular motion in the frame of reference rotating with the spin’s Larmor precession, while following a more complex, spiraling path when viewed in the lab frame (figure 2.5). In practice, the magnetic field applied resonantly with the spin will not be a rotating magnetic field, but simply an oscillating magnetic field Bac, as this is easier to generate. This field can be decomposed into two components (each of amplitude B1= Bac/2), one rotating in the same direction as the spin, while the other rotates in the exact opposite direction. According to the rotating wave approximation the field rotating with the spin will then drive ESR, while the other field can be neglected as it is far off-resonant (as long as BextÀ B1).

Although ESR can be used to manipulate a spin qubit [9], alternating magnetic fields can be difficult to generate locally on a chip. It is therefore desirable to perform spin rotations using electric fields, which can be more easily applied locally. Electric dipole spin resonance (EDSR) allows such electric field induced spin rotations [10–13]. How-ever, since a spin does not couple directly to electric fields, an additional mechanism is required to achieve this coupling. This can be provided by e.g. a magnetic field gradient [12, 14], g -tensor modulation [13] or by spin-orbit coupling [10, 15, 16]. The latter will be discussed further in section 2.3 and is the relevant mechanism for the work in this thesis. Applying an alternating electric field to an electron causes it to move back and forth. When a magnetic field gradient is present, the electron spin will experience a varying magnetic field as a function of time. A magnetic field gradient can be generated

ex-2

Figure 2.5: Electron spin resonance (a) in the lab frame, where the spin spirals down over the Bloch sphere, and (b) in a frame rotating with the electron Larmor frequency around the z-axis. The spin then simply rotates around the magnetic field B1.

ternally, e.g. by a micromagnet [12, 14], but can also originate from the magnetic field generated by the nuclei [11]. From the electrons’ frame of reference it will be as if an oscillating magnetic field is applied to it. Therefore, as with ESR, the oscillating electric field will induce spin rotations. Similarly, in the case of spin-orbit mediated EDSR, an oscillating electric field generates an effective oscillating magnetic field through spin-orbit coupling, thus driving spin rotations. Chapter 6 demonstrates this mechanism in an indium antimonide nanowire.

2.2.2. R

Any qubit interacts with its environment. Such interactions disturb the quantum state of the qubit in an uncontrolled manner and therefore lead to a loss of information stored in the qubit. For a qubit encoded in a single spin two important interactions with the environment are the spin-orbit interaction and the hyperfine interaction. These inter-actions will be discussed in subsequent sections. We will first however consider here two general processes that lead to the loss of information stored in a qubit: relaxation and dephasing.

When a qubit interacts with its environment in such a way that it dissipates energy, the qubit will relax to its ground state. The timescale on which this happens is typically called T1. As illustrated in figure 2.4, this relaxation process can be viewed as a loss of information stored in the angleθ in the qubit state as described by equation 2.1.

Dephasing on the other hand is the loss of information stored in the angleφ and

leads to a decay of an initial transverse polarization 〈σ⊥〉 (whereσ⊥can include both the Pauli spin matricesσxandσy). However, the dissipative process of relaxation also gives a contribution to such loss of transverse polarization, or decoherence. Pure dephasing is characterized by a timescale T_φand decoherence by a timescale T2. They are related through 1/T2= 1/2T1+ 1/Tφ[17].

Using a so-called Ramsey sequence, the dephasing of a spin can be measured. This is illustrated in figure 2.6 in the rotating frame. After initializing the spin into one of its eigenstate (i.e. | ↑〉 or | ↓〉), a π/2-pulse is applied to rotate the spin to the x y-plane. Sub-sequently the spin is allowed to evolve freely for a timeτ. Finally, another π/2 rotation is performed (around the same axis as the initial pulse). If the spin is undisturbed during

2

a

b

c

Figure 2.6: Ramsey sequence (in the rotating frame). (a) The spin is rotated into the x y-plane by aπ/2 rotation. (b) During a timeτ the spin evolves freely. (c) At the end of the sequence another π/2 pulse is applied. The final state depends on the free evolution of the spin, as indicated by differently shaded arrows.

the free evolution, a spin initially in the up state | ↑〉 will end up in the state | ↓〉. How-ever, if during this time the spin rotates over an angleπ in the x y-plane, it will end up in the state | ↑〉. Such a sequence thus allows a decay of the transverse polarization to be mapped onto the longitudinal polarization, which can be easily measured. The decay characteristics are determined by the noise spectrum of the fluctuations causing the de-phasing. For Gaussian distributed noise and when low frequency fluctuations dominate, this leads to a decay 〈σx〉(τ) ∝ exp

³ −¡ τ/T∗ 2 ¢2´ .

To further characterize qubit coherence a Hahn echo sequence can be used, which is illustrated in figure 2.7. It is similar to the Ramsey sequence, but it includes an extraπ rotation in the middle of the free evolution period. Thisπ-pulse effectively interchanges the | ↑〉 and | ↓〉 states. If the fluctuations causing the dephasing are constant during the duration of the pulse sequence, the random phase picked up during the first free evolu-tion period, is canceled in the second. The transverse spin polarizaevolu-tion is then refocused at the end of the sequence. The decay time obtained from this method can be referred to as Techo, which can clearly be much longer than the inhomogeneous dephasing time

T₂∗.

This spin echo sequence can be generalized to include moreπ-pulses (Carr-Purcel pulses [18]) in the free evolution time_{τ. The more pulses are included, the less sensitive} the decay will be to low frequency fluctuations. Various pulse sequences can be em-ployed to counter noise sources with specific spectral densities and relations between the corresponding coherence times can be derived [19].

2.3 S

PIN

-

ORBIT COUPLING

With the introduction of the concept of a spinning electron, Uhlenbeck and Goudsmit proposed that the property of spin can provide an explanation for the fine structure present in atomic spectra [20]. This essentially introduced the phenomenon of spin-orbit coupling (or interaction), which indeed plays an important role in atomic physics in the description of fine structure as is now known. The spin-orbit coupling for an elec-tron moving through a potential V is given by the following Hamiltonian [3]:

2

d

c

b

a

Figure 2.7: Hahn echo sequence. (a) As for a Ramsey sequence an initialπ/2 pulse rotates the spin into the x y-plane. (b) (top view of the x y-plane of the Bloch sphere) For a time t = τ/2 the spin evolves freely, after which aπ pulse flips the spin. (c) During the second half of the free evolution the spin is refocused, because it feels a nuclear field that is effectively opposite. (d) A finalπ/2 rotation prepares the spin for read-out.

HSO= − ħ

4m2₀c2σ · ¡p × ∇V ¢ (2.2)

Here m0is the free electron mass, c the speed of light, p the momentum operator and

σ = (σx,σy,σz) a vector of the Pauli spin matrices. This equation can be derived from

a non-relativistic approximation to the Dirac equation, though that will not be covered here. Instead we will consider a more classical derivation leading to the same functional form. An electron electron moving through an electric field E with velocity v = p/m experiences in its rest frame an effective magnetic field Beff= _mc−12p × E. Because the

electron spin S possesses a magnetic momentµ =_meS

0, it interacts with this field, leading

to a spin-orbit Hamiltonian HSO= −µ · Beff. Although this leads to the correct depen-dence, the predicted magnitude of the spin-orbit coupling will be off by a factor of two. The reason for this is that the electron orbits around the nucleus and is thus being accel-erated. A correction term therefore has to be added due to the Lorentz transformation to the frame of the electron, as first shown by Thomas [21]. This term is exactly half of the expression derived from the naive picture and opposite in sign, thus solving the dis-crepancy. From the equation 2.2 it is clear that spin-orbit coupling is strongest near the atomic core, since there the momentum of the electron and the Coulomb potential are the largest. For the same reason, heavier elements generally have stronger spin-orbit coupling.

2.3.1. S

-

The motion of electrons in a crystalline solid like a semiconductor is dictated by energy bands En(k), where n indicates band number and k is the wave vector. Kramers the-orem tells us that due to time reversal symmetry, we should have that E_↑(k) = E↓(−k), with ↑,↓ indicating the (pseudo-)spin direction of the Kramers doublet. When the crys-tal lattice through which the charge carriers move possesses inversion symmetry, such as is the case for diamond, we furthermore have that E_↑(k) = E↑(−k). Together this leads to E_↑(k) = E↓(k), i.e. spin degeneracy. However, certain crystal structures lack this

inver-2

sion symmetry, or, as it is also commonly referred to, exhibit bulk inversion asymmetry (BIA). Zinc blende and wurtzite crystals (see figure 2.8), which are relevant to this thesis, are examples of crystals with BIA. The lack of inversion symmetry means that in these crystals spin splitting can occur at zero external magnetic field.

a

b

c

d

Figure 2.8: (a),(b) Zinc blende crystal structure. Different colors represent different atomic species. In (b) the crystal structure is shown from the [001] direction with the numbers indicating the height of the atoms (as fractions of the lattice constant a). (c),(d) Wurtzite crystal structure. The crystal is shown in the [0001] direction in (d), with other crystal directions indicated and heights shown in units of the lattice constant c.

In a crystal lacking inversion symmetry, spin-orbit coupling arises from the same microscopic origin as described by equation 2.2. The spin-orbit coupling due to bulk inversion asymmetry in a zinc blende crystal is given by [22]:

H_BIAZB _{= γ}hpx ³ p2_{y− p}2_z´σx+ py¡p2z− p2x ¢ σy+ pz ³ p2_{x− p}2_y´σz i , (2.3)

with_{γ the coupling constant and x,y,z corresponding to the main crystallographic} di-rections [100],[010] and [001]. The different symmetry in a wurtzite crystal leads to a Hamiltonian of the form [23]:

H_BIAW =hλ + λlp2z+ λt ³

k2_x+ k2y ´i

(p × z) · σ, (2.4)

where z is a unit vector pointing in the [0001] direction. Although these symmetry con-siderations allow us to derive the functional form of the spin-orbit coupling, knowledge

2

16 2. THEORY

of the precise band structure is needed for the strength of the coupling. The band struc-ture near a specific k-point can be calculated using the so-called k · p method [24]. This method starts from a Bloch wave description of the electronic states:Ψ(r ) = ei krunk(r ), where unk(r ) is the Bloch function for a certain k-vector and energy band n. The k · p method then allows easy evaluation of the Bloch functions. Of particular interest to us is the region near the fundamental bandgap, around theΓ-point at k = 0. In the extended Kane model interactions between the topmost p-like valence band states (X,Y,Z), as well as the s-like (S) and p-like (X’,Y’,Z’) states in the lowest conduction bands are taken into account exactly for a band structure calculation. Interactions with other bands can be in-cluded perturbatively into the model. A convenient characteristic of the extended Kane model is that it takes bandstructure parameters (such as bandgap, spin-orbit gap and transition matrix elements, which can be measured (e.g. by optical experiments) and connects them to quantities such as the effective mass m∗and the effective Landé g -factor g∗[3]. As a side note, the band structure parameters are not always be known, as is the case for wurtzite InAs for example. This is due to the fact that the wurtzite crystal structure can not be realized in bulk InAs, but only in nanowires. Band structure pa-rameters are expected to differ by roughly 10 % due to a difference in crystal structure [25].

2.3.2. S

-

Apart from spin-orbit coupling due to bulk inversion asymmetry, spin-orbit coupling can also arise due to asymmetric confinement potentials. This is also referred to as struc-ture inversion asymmetry (SIA). In a 2DEG this leads to the well known Rashba spin-orbit Hamiltonian: H_R2D= α(−pyσx+ pxσy). For a nanowire the form of spin-orbit coupling depends on the details of the confinement. We can distinguish between two extreme cases, after references [26, 27]. The first case we consider is that of a quantum dot much more strongly confined in the longitudinal direction than in the transverse direction. For such a nanowire quantum dot the spin-orbit coupling is of Rashba type:

H_SOt = α¡p × c¢ · σ, (2.5)

with c being the unit vector along the growth direction of the nanowire, andα a coupling constant depending on the details of the material and confinement.

In the other extreme, we consider a very elongated dot, where the strongest confine-ment is provided by the nanowire surface. In this case the the spin-orbit coupling can generally be written as [26, 27]:

H_SOl = (p · c)(η · σ). (2.6)

Hereη is a vector of coupling constants, capturing the effects of the confinement. This

Hamiltonian can describe structural as well as bulk inversion asymmetries [28, 29]. Prac-tically, the confinement of an experimentally realized nanowire quantum dot will lie in between these extremes, making the analysis less straightforward.

To characterize the spin-orbit strength in a nanowire, the spin-orbit length lSOcan be used. This is the length over which an electron will flip its spin due to spin-orbit cou-pling, as it moves through a semiconductor [2]. Typical values for the spin-orbit length

2

in InAs and InSb are a few hundred nanometers. As this is about two orders of magni-tude shorter than in a GaAs 2DEG, this illustrates that these are semiconductors with strong spin-orbit coupling. In chapter 4 we will demonstrate that EDSR can be used to probe the spin-orbit strength in a double quantum dot. The g -factor in a semiconductor is also strongly influenced by confinement [30], and can furthermore be anisotropic [31] (see also chapter 4). This is not surprising if we consider that the effective g -factor is essentially also a result of spin-orbit interaction. Confinement effectively increases the energy gap, thus changing the band-structure and consequently the g -factor.

2.3.3. S

-

Quantum dots are quasi-zero dimensional objects, meaning that particles are confined in all directions. Therefore, the average momentum of an electron in a dot is zero in all direction; 〈px,y,z〉 = 0. As a result, spin-orbit interaction does not directly couple Zeeman split sublevels, because 〈l ↓| HSO|l ↑〉 ∝ 〈l | px,y,z|l 〉 〈↓| σ| ↑〉 = 0, where l labels the quan-tum dot orbitals. Instead, only states with different orbital as well as different spin parts couple [32]. It is thus also due to spin-orbit coupling that what we refer to as spin-up and spin-down states are actually admixtures of both spin and orbital states. When the or-bital level spacing is much larger than the Zeeman and spin-orbit splitting, these effects can be included in the eigenstates perturbatively [2]:

|l ↑〉SO= |l ↑〉 +X l0_6=l ­l0_↓¯ ¯H_SO|l ↑〉 El− El0− ∆EZ|l 0_↓〉, |l ↓〉SO= |l ↓〉 +X l0_6=l ­l0_↑¯ ¯H_SO|l ↓〉 El− El0− ∆EZ|l 0_↑〉. (2.7)

Here l and l0again refer to orbital quantum numbers and∆EZis the unperturbed spin splitting. From these equations it is clear that the largest admixture comes from the first orbital excited state and the degree of admixture increases with spin-orbit strength.

Because it mixes the spin and orbital degrees of freedom, spin-orbit coupling cre-ates a coupling of a spin to electric fields. Therefore, carefully controlled electric fields can be used to manipulate electron spins, see references [10, 16], as well as chapter 6 of this thesis. However, uncontrolled electric field fluctuations can be an important source of spin relaxation [33]. Such fluctuations can originate from electrical noise in the setup, but also from lattice phonons [34, 35]. These phonons can either generate electrical field fluctuations through inhomogeneous deformation of the crystal lattice, thereby locally changing the bandgap; or through the piezoelectric effect. In the relax-ation process a phonon is created that carries of the energy EZ. The relaxation rate there-fore depends strongly on this Zeeman splitting EZ, through the following dependencies [2]: (i) the phonon density of states, which scales as E_Z2, (ii) coupling of the phonon to the dot (∝ EZ), (iii) the strength of the electric field of a single phonon, leading to a de-pendence ∝ E±_Z for piezoelectric or deformation potential phonons respectively, and (iv) the strength of spin-orbit interaction, proportional to EZ. This leads to a relaxation rate dependence on the magnetic field as E_Z5for piezoelectric phonons and E7_Zfor defor-mation phonons (all but (i) contribute quadratically to the relaxation rate). It has been demonstrated experimentally that the expected 1/T1∝ Bext5 dependence for

piezoelec-2

18 2. THEORY

tric phonons holds in GaAs quantum dots[2]. Furthermore, it has been shown that in such a system the relaxation time of a single electron spin can exceed 1 s at low magnetic fields [36].

2.4 N

UCLEAR HYPERFINE INTERACTION

It is not just electrons that carry spin inside a semiconductor, also the nuclei of the host material can posses spin. In particular in III-V semiconductors, all atomic nuclei have non-zero spin. This gives rise to the hyperfine interaction, a phenomenon which lends its name from atomic physics, where it explains the hyperfine structure in atomic spec-tra. The coupling between the magnetic moment of an electron spinµeand that of a nucleus_µNis given by the following Hamiltonian:

H =µeµN r3 −

3(µer )(µNr )

r5 (2.8)

where r is the position vector pointing from the nucleus to the electron. This expression only holds for electrons whose wavefunctions do not overlap with the nucleus, i.e. for electrons with non-zero orbital angular momentum. For electrons with s-type symme-try however (which is the case for the conduction band of III-V semiconductors [37]), a relativistic correction is necessary, since these do overlap with the nucleus. This leads to the Fermi contact hyperfine interaction between an electron spin S and nuclear spin I [38, 39]: HHF= 2µ0 3 g0µBγnħ ¯ ¯ψ(0)¯¯ 2 I ··· S (2.9)

Here_µ0= 4π · 10−7Vs/Am is the permeability of free space, g0≈ 2 the freeelectron g -factor andµBthe Bohr magneton. γnis the nuclear gyromagnetic ratio and |ψ(0)|2the magnitude of the electron’s wavefunction at the position of the nucleus.

An electron in a quantum dot interacts with with many nuclei as opposed to an elec-tron in an atom. The wavefunction for the elecelec-tron can be written as a product of an envelope wavefunction Ψ(r ), determined by the macroscopic confinement potential, and a Bloch function u(r ) resulting from the periodic lattice of the host material. The contact hyperfine interaction for an electron in a quantum dot can then be written as the sum of the contributions of all N nuclei:

HHF= 2µ0 3 g0µBγNħ N X i =1 ¯ ¯ψ(r_i) ¯ ¯ 2 Ii· S (2.10)

where ri is the position of nuclear spin Ii. Defining a hyperfine coupling constant A = 2µ0

3 g0µBγNħ |u(0)|2, we can rewrite the above equation to:

HHF= N X i =1

AiIi· S (2.11)

where Ai= v0A|Ψ(ri)|2and v0is the volume of the unit cell. An r.m.s. average hyperfine coupling constant can be used for materials consisting of different nuclear isotopes j

2

(with corresponding relative abundance_νj):

A =

s X

j

νj(Aj)2 (2.12)

Table 2.1 lists the nuclear spins, gyromagnetic ratios and hyperfine coupling con-stants for various isotopes relevant to the work presented in this thesis.

Table 2.1: Nuclear spin, gyromagnetic ratio, contact hyperfine coupling strengths in InAs and InSb, quadrupole moments and relative abundances of several isotopes. Table based on [37, 40]. Note that values for Ajdepend on the normalization used. 1 mb (millibarn) = 10−31m2.

I γj (rad T−1s−1) Aj(µeV) Qj(mb) νj 75_{As 3/2 4.60 · 10}7 _{86 314 1} 113_{In 9/2 5.88 · 10}7 _{110 759 0.04} 115_{In 9/2 5.90 · 10}7 _{110 770 0.96} 121_{Sb 5/2 6.40 · 10}7 _{208 -360 0.57} 123_{Sb 7/2 3.47 · 10}7 _{113 -490 0.43} 29_{Si 1/2} −5.32 · 107 0 0.05

2.4.1. E

The hyperfine interaction between a single electron spin and many nuclear spins can lead to complex many-body behavior of the system. In principle entanglement can even be created between the nuclear spin bath and the electron spin, provided their initial states are sufficiently pure [41]. It can also lead to complex effects such as dynamic nu-clear polarization (DNP) [42–44]. In chapter 5 we will discuss effects observed on trans-port and EDSR in an InAs quantum dot as a result of DNP. For now, however, we will ignore the effect an electron can have on the nuclear spin and only consider the effect the nuclei have on the electron spin. Generally the nuclear spin system evolves slowly compared to the evolution of the electron spin. Using this, the effect of the nuclear spin bath can be approximated as an effective magnetic field acting on the electron spin [45]:

HHF= N X i =1

AiIi· S = g µBBNS, (2.13)

where BNis the nuclear Overhauser field originating from the hyperfine interaction of the N nuclear spins combined. With this approximation, we have replaced the nu-clear spin operators with a single classical magnetic field. In InAs, this field can be |BN,max| ∼ 1 T when the nuclear spins are fully polarized. For InSb |BN,max| is a few hun-dred mT. This field is independent of the number of nuclei N ; in larger dots the elec-tron will interact with more nuclei, but each nucleus will also contribute proportionally less to the Overhauser field. Under the present experimental conditions however (tem-perature_?100 mK and magnetic field_>10 T), the thermal energy kBT dominates the nuclear Zeeman energy and thus the nuclear spin bath is mostly unpolarized. The nu-clear field furthermore undergoes statistical fluctuations following a Gaussian distribu-tion with standard deviadistribu-tionσN∼ BN,max/

p

2

20 2. THEORY

N ∼ 106, leading to fluctuations on the order of_σN∼ 1 mT. This estimate for the for the nuclear field fluctuations is consistent with optical [46, 47] and electrical measurements [48, 49] performed on quantum dots.

Having such a random field add to the externally applied magnetic field, changes the Larmor precession frequency and thus leads to dephasing, as previously discussed in section 2.2.2. An extra phase ofπ is picked up within ∼ 5 ns for a typical nuclear field of

B_Nz∼ 1 mT in an InAs quantum dot (taking g = 9 and Bextin the z-direction). A Gaussian nuclear field distribution leads to dephasing also following a Gaussian shape: e−(t /T2∗)2_,

where T₂∗=p2ħ/g µBB_Nz. The transverse components B x,y

N of the nuclear field do not have a significant effect on dephasing if the external magnetic field is large (BextÀ BN). Their effect is a tilt in the precession axis of ∼ BN/Bextand a change in the precession frequency of gµBB_N2/Bext.

This dephasing caused by the nuclear field can be suppressed in different ways. First of all, by polarizing the nuclear spin system, the fluctuations in the nuclear field can be reduced. If a proportion p of the nuclear spin bath is polarized, the nuclear field distribution will be narrowed by a factor of 1/_{pN(1 − p}2_{) [50, 51]. It is however difficult} to experimentally achieve a polarization large enough to see a significant improvement; e.g. a factor of 100 increase in T∗

2 requires a polarization of over 99.99%.

2.4.2. N

Although a Hahn echo or more complex pulse sequences can extend the qubit coherence time as we have seen in section 2.2.2, this time cannot be extended indefinitely. This is due to the fact that the nuclear field is not entirely static and is thus limiting the co-herence. To understand the nuclear field dynamics, we consider here three interactions contributing to its evolution: (i) hyperfine interaction, (ii) dipole interaction between adjacent nuclear spins and (iii) quadrupole coupling of individual nuclei with an elec-tric field gradient. In particular for indium atoms, the latter may be important, as indium nuclei possess a relatively large quadrupole moment.

HYPERFINE INTERACTION

Just as the nuclei act on the electron through the hyperfine interaction, so does the elec-tron act back on the nuclei through the same interaction. Each nucleus is subject to the so-called Knight shift, a small magnetic field ∼ A/N . This results in a precession of the nuclear spins when the externally applied magnetic field is low (Bext¿ A/N ), and thus in a change of nuclear field. For large external field, both the electron and nuclear spin precess around this same field and precession due to the Knight field (and therefore a change in the nuclear field) is strongly suppressed. By rewriting the hyperfine interac-tion Hamiltonian from equainterac-tion 2.11, we can see how it affects the nuclear field:

HHF= N X i Ai¡IixS x + IiyS y + IizS z¢ = N X i Ai¡Ii+S++ Ii−S−+ 2I z iS z_{¢ ±2.} (2.14)

Here I±and S±are raising and lowering operators for the nuclear and electron spin re-spectively. The first two terms in this expression describe electron-nuclear flip-flops,

2

which cause fluctuations in the longitudinal component B_Nz of the nuclear field. At finite external field Bext, this process becomes suppressed due to the Zeeman energy differ-ence between the electron and nuclear spin. Virtual processes in which a flip-flop with one nucleus i is followed by another with nucleus j , can be efficient to higher magnetic fields, since they conserve the electron spin (and associated energy cost). The nuclear field then changes if Ai6= Aj. The complex interplay between the electron spin and nu-clear dynamics can lead to so-called non-Markovian dynamics, i.e. a decay characteristic different than the usual exponential decay [50].

DIPOLE INTERACTIONS

Dipole-dipole interactions between neighboring nuclear spins are described by the fol-lowing Hamiltonian [18]: HDD= X i <j µ0γiγjħ2 4π¯¯ri j ¯ ¯ 3 à Ii· Ij− 3 ¯ ¯r_{i j} ¯ ¯ 2¡Ii· ri j¢ ¡Ij· ri j ¢ ! , (2.15)

whereγiis the gyromagnetic ratio of nucleus i . In the secular approximation this Hamil-tonian simplifies to:

H_{i , j}= D³I_i+I−_j + Ii−I+j − 4I z iI z j ´ ±2 (2.16)

for strong magnetic fields (Zeeman energies larger than the interaction strength D). For InAs we have D ∼ 1/50 µs [52]. The first two terms in this expression represent flip-flips of nuclear spin pairs, changing the nuclear field B_Nz and thus affecting the coherence of the electron spin. The final term in equation 2.16 is responsible for changing B_Nx,y, which for GaAs is expected to occur on a 100µs timescale [2]. The drift in B_Nz however may be on a much longer timescale. In particular, the flip-flop rate will be suppressed for |Ai− Ai +1| > D. Theoretically, the contribution of dipole-dipole interactions to the elec-tron spin coherence are estimated as ∼10–100µs [53–56]. An important consideration is that the spin coherence time not only depends on the typical correlation time of the nuclear spin bath, but also on the amplitude of nuclear field fluctuations. Interestingly, in GaAs quantum dots at low magnetic fields (< 200 mT) revivals of the coherence were observed within some time after initially collapsing [57]. This behavior was theoretically predicted and is attributed to precession of the gallium and arsenic nuclear spins in the external magnetic field [19, 58].

QUADRUPOLE COUPLING

Another interaction that may be relevant to the nuclear dynamics, is the quadrupole coupling of individual nuclear spins to electric field gradients. In particular this could be important for InAs and InSb nanowires, since indium possesses a large nuclear quadru-pole moment (see table 2.1). Electric field gradients can originate for example from im-perfections in the crystal lattice structure. Quadrupole coupling can cause shifts in the Zeeman energy [17] and can cause faster nuclear relaxation [59].

Quadrupolar effects arise because the nucleus is generally not spherical. For exam-ple, if we consider an elongated nucleus in a magnetic field gradient, as illustrated in figure 2.9, it is obvious that the electrostatic energy depends on the orientation of the nucleus. The Hamiltonian describing the quadrupole coupling [17] is given by:

2

a

b

c

Figure 2.9: (a),(b) An elongated nucleus in the field field of four charges. The system favors configuration (b) as that will put the nucleus closest to the negative charges. (c) Zeeman split energy levels of an I = 3/2 nucleus with (right) and without (left) taking quadrupole coupling into account (to first order). Figure after [17].

HQ= eQ 6I (2I − 1) X α,β V_α,β· 3 2 ³ IαIβ+ IβIα´− δα,β ¸ (2.17)

where eQ is the nuclear quadrupole moment and Vα,β=_∂x∂_α2_∂xV_β, with x_α,βdenoting the

x,y or z-direction. As we did for the hyperfine interaction, we can rewrite this equation

in terms of the nuclear raising and lowering operators I±to make clear the effect on the nuclear field: HQ= eQ 4I (2I − 1) h V0¡3Iz2− I2¢ +V+1(I−Iz+ IzI−) +V−1¡I+Iz+ IzI+¢ +V+2(I−)2+ V−2¡I+ ¢2i . (2.18)

Here V0, V±1and V±2are linear combinations of the gradients, defined by:

V0= Vzz

V_±1= Vzx± iVz y

V_±2=1

2(Vxx− Vy y) ± iVx y

(2.19)

In general, for a magnetic field axis not aligned with one of the principal axes of the elec-tric field gradient tensor, each of these elements is non-zero. As a result, quadrupole coupling can cause a change in the longitudinal polarization of the nuclear states. Elec-tric field gradients giving rise to quadrupole coupling may originate from wurtzite crystal symmetry (for cubic crystal symmetry it vanishes), strain, dopants or defects (either in-side or on the surface of the nanowire). External sources such as gates or impurities in an underlying dielectric layer may also produce field gradients. Interestingly, quadru-pole coupling leads to a shift in Zeeman levels, as illustrated in figure 2.9(c). Such a shift has been observed in GaAs 2DEGs, giving rise to multiple-quantum∆m = ±2 transitions [60, 61]. Quadrupole coupling can be an important mechanism for nuclear spin relax-ation and thus lead to fast nuclear dynamics. In the presence of strong electric field