in Small Quantum Dot Arrays
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus voorzitter van het College voor Promoties,
in het openbaar te verdedigen op vrijdag 21 juni 2013 om 12:30 uur door
Floris Rembrandt BRAAKMAN
Doctorandus in de natuurkunde geboren te Sas van Gent.
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 Univ. prof. dr. H. Bluhm Rheinisch-Westf¨alische Technische Hochschule
Aachen, Aken, Duitsland
Prof. dr. W. Wegscheider Eidgen¨ossische Technische Hochschule Z¨urich, Z¨urich, Zwitserland
Prof. dr. ir. R. Hanson Technische Universiteit Delft Prof. dr. Y. V. Nazarov Technische Universiteit Delft Prof. dr. ir. J. E. Mooij Technische Universiteit Delft
Published by: Floris Braakman
Coverdesign: Sia Braakman (www.siabraakman.nl) Printed by: Ipskamp Drukkers B.V., Enschede ISBN: 978-90-8593-161-4
Casimir PhD series, Delft-Leiden 2013-20 Copyright c 2013 by Floris Braakman
1 Introduction 1
1.1 Electrons . . . 1
1.2 The quantum world . . . 1
1.3 Quantum computation . . . 2
1.4 Electron qubits . . . 3
1.5 Electron spins in quantum dots . . . 4
1.6 Thesis outline . . . 5
References . . . 6
2 Device fabrication, measurement techniques and theory 7 2.1 Quantum dots . . . 8
2.1.1 Quantum dots in GaAs/AlGaAs heterostructures . . . 8
2.2 Tunnel coupled quantum dots . . . 12
2.3 Read-out of the charge on a quantum dot . . . 13
2.3.1 Transport and charge sensing . . . 13
2.3.2 Real-time charge sensing . . . 14
2.4 Spins in quantum dots . . . 15
2.4.1 Spin states in a single quantum dot . . . 15
2.4.2 Two-electron spin states in a tunnel coupled double quan-tum dot . . . 17
2.4.3 Spin blockade . . . 18
2.4.4 Spin-orbit interaction . . . 20
2.4.5 Hyperfine interaction . . . 21
2.4.6 Effects of spin-orbit and hyperfine interactions on electron spin coherence . . . 22
2.5 Photon-assisted tunneling . . . 23
2.5.1 Different regimes . . . 23
2.5.2 Single junction . . . 24
2.5.3 Single quantum dot . . . 24
2.5.4 Double quantum dots . . . 27 v
2.5.5 Generalizations . . . 31
2.6 Typical experimental setup . . . 31
2.6.1 Sample fabrication . . . 31
2.6.2 Dilution refrigerator . . . 32
2.6.3 Measurement electronics: DC . . . 32
2.6.4 Measurement electronics: RF . . . 35
2.6.5 Read-out reflectometry setup . . . 36
References . . . 36
3 Coupling artificial molecular spin states by photon-assisted tun-neling 43 3.1 Introduction . . . 44
3.2 Device and excitation protocol . . . 44
3.3 Interpretation of the photon-assisted tunneling spectra . . . 48
3.4 Extracting artificial molecule parameters . . . 51
3.5 Identification of the spin-flip mechanisms . . . 52
3.6 Methods . . . 55
3.6.1 Sample fabrication. . . 55
3.6.2 Measurement. . . 55
3.6.3 Simulation. . . 56
3.7 Additional Material . . . 56
3.7.1 Calibration of the detuning axis . . . 56
3.7.2 Triplet spin resonance . . . 58
3.7.3 The ∆m = 0 PAT transition . . . 60
3.7.4 Spin flip-tunneling mechanism . . . 62
3.7.5 Simulations of the PAT spectra - relaxation . . . 64
3.7.6 Full microwave quantum control of two spins within the (1, 1) manifold . . . 64
3.7.7 Measurement of spontaneous relaxation . . . 67
3.7.8 Power dependence . . . 69
3.7.9 Pure electric-dipole PAT transition . . . 70
References . . . 70
4 Mixing rates of spin-flip photon-assisted tunneling 75 4.1 Introduction . . . 76
4.2 Setup and spin-flip PAT . . . 76
4.3 Relaxation and mixing rates . . . 79
5 Long-distance coherent coupling in a quantum dot array 85
5.1 Introduction . . . 86
5.2 Long-distance coupling . . . 87
5.3 Detuning dependence long-range coupling . . . 89
5.4 Photon-assisted cotunneling: demonstrating coherence . . . 92
5.5 Methods . . . 96
5.6 Additional Material . . . 96
5.6.1 Additional charge stability diagrams . . . 96
5.6.2 Real-time traces for different detunings between outer dots 97 5.6.3 Cotunneling: effective tunnel coupling . . . 98
5.6.4 Lower bound estimation for charge T2 . . . 99
5.6.5 Calculation of the real-time transition rate . . . 100
5.6.6 Calibration of the detuning between middle and outer dot levels in Fig. 3b . . . 101
5.6.7 Frequency dependence of PAT and PACT . . . 102
References . . . 103
6 Photon- and phonon-assisted tunneling in the three-dimensional charge stability diagram of a triple quantum dot array 107 6.1 Introduction . . . 108
6.2 PAT in the triple quantum dot device . . . 109
6.3 Explanation of the PAT resonances near point C . . . 111
6.4 Phonon-assisted tunneling . . . 112
References . . . 113
7 Conclusions and future directions 117 7.1 Better qubit control and coherence . . . 118
7.2 Scaling up . . . 121
7.3 Follow-up experiments long-distance coupling . . . 124
References . . . 125
Appendix: Fabrication recipe 129
Acknowledgements 135
Summary 138
Samenvatting 141
Introduction
1.1
Electrons
Electromagnetism is by far the most dominant natural force in our life. Cer-tainly we appreciate that electricity allows us to power the countless electronic devices that surround us anywhere and at any time. But the influence of elec-tricity reaches much further than that. It is the electrically charged electrons, constituent particles of atoms and molecules, which to a very large degree de-termine the physical and chemical properties of substances. Electrons play an essential role in a plethora of physical phenomena, including electricity, thermal conductivity, superconductivity and magnetism [1]. Many of these effects are very well understood, but there remain some for which a good microscopic theory is lacking. It is clear then that electrons form a very interesting subject of inves-tigation. Through the use of nanotechnology, it is nowadays possible to make very small structures in which a small number of electrons can be isolated. Such few-electron systems, sometimes called artificial atoms for their similarity to real atoms, are well-suited for studying the physics of electrons at a microscopic level. In particular, few-electron systems are ideal for studying quantum physics. This is interesting from a fundamental point of view, but also very technologically relevant, as explained in section 3.
1.2
The quantum world
Events often appear to happen with a certain chance. We can be quite certain that the sun will rise again tomorrow, but we guess the outcome of a coin flip right only fifty percent of the time. Predicting the weather correctly often seems to follow the same odds. Events seem to occur with a certain probability. However, this seemingly probabilistic nature of events was long thought to be an illusion.
It was simply a lack of our knowledge which makes things uncertain. If we would know every little detail of the microscopic particles that make up the coin, the throwing hand, the air it flies in, and each of their interactions, we would be able to predict the outcome of every coin flip and be correct one hundred percent of the time. Probability would not exist in any fundamental way.
Over the last century the very succesful theory of quantum physics has been established. According to this theory, probability is in fact a much more funda-mental feature of nature than was thought before. It postulates that properties of objects, such as the position of an electron, do not always have well-defined values.
Instead, an electron can be in a state in which it is in different places at the same time. Such a state is called a superposition. However, when the position of the electron is measured, it will always be observed to be in only one place. At which location the electron will be measured to be is a matter of chance, with the measurement probability for each location calculable by quantum theory. It is as if prior to the measurement the electron is at several places at once, but by the act of measuring the observer has forced the electron to choose a location and stick with it. Quantum physics is succesful, because it can explain things that classical physics cannot, for instance why the negatively charged electrons surrounding the positively charged atomic nucleus do not simply collapse onto the nucleus, and the origin of magnetism.
Quantum theory introduces some concepts that seemingly defy common sense (such as the superposition state), yet have been validified by a countless number of experiments. Another intriguing feature of quantum physics is the uncertainty principle. This principle states that certain properties of a system cannot be known simultaneously with arbitrary precision. For example, the velocity and position of an electron cannot both be exactly known at the same time. The uncertainty principle accounts for phenomena such as tunneling, by which par-ticles can travel through barriers energetically not accessible in a classical way. Such distinctly quantum mechanical effects cannot usually be observed in large objects; one needs to study small and relatively simple objects for those effects to become discernible. For electrons, being elementary particles, such quantum mechanical effects dominate their behaviour.
1.3
Quantum computation
There is a way to make powerful use of quantum physics. This comes in the form of information processing using quantum systems [2]. It is well know that the
fundamental logical unit is the bit. A classical bit can be in one of two states: it can be either 0 or 1. A quantum version of the bit, or qubit, is less limited. In addition to the states 0 or 1, a qubit can also be in a superposition of these two states. This superposition state is usually written as α| 0 i + β| 1 i, where α and β are arbitrary complex numbers satisfying the equality: |α|2 + |β|2 = 1. To illustrate the strength of quantum over ’normal’ classical computation, let us consider evaluating a function f (x), where the input value x can be 0 or 1. In a classical computer, to know the outcome of f (x) for all of its input states, f (x) has to be computed in separate, sequential steps for x = 0 and x = 1. However, in a quantum computer the input x can be in a superposition state: x = α| 0 i + β| 1 i. Computing f (x) now yields a superposition state of f (0) and f (1), hence f (x) was evaluated for both input values 0 and 1 simultaneously. The advantage a quantum computer has with respect to a classical computer becomes even more clear when more input values are allowed, by using more input (qu)bits. Evaluation of f (x) takes a classical computer one step per input value, while a quantum computer does the evaluation for all inputs in a single step. The fact that a measurement outcome collapses the superposition to a single-valued outcome at first sight complicates matters, since we only get one output value f (x). However, clever algorithms have been invented which deal with this problem. By choosing the right measurement, the single-valued outcome will give an answer equivalent to processing with all parts of the superposition state simultaneously. Quantum computation therefore can be exponentially more efficient in solving certain problems than classical computation, making it possible to solve problems that would take a classical computer (no matter how fast each step in this computer would be) as long as the lifetime of the universe.
So far we have been talking about qubits as abstract entities. Many possible physical realizations of qubits are being pursued in what has become a very active field of research. Here we focus on a single candidate for a real qubit: the electron.
1.4
Electron qubits
The spin of an electron forms in a natural way almost the perfect qubit [3, 4]. The spin of the electron is a tiny magnetic moment, which behaves as a two-level system. The two states point parallel or anti-parallel to an externally applied magnetic field, and can be assigned the labels | 0 i and | 1 i. The spin can also be in a superposition state of | 0 i and | 1 i, as required for a qubit. Importantly, due to the small magnitude of the magnetic moment the electron spin is only very weakly perturbed by influences from its environment. These weak interactions
make it robust against changes in its state which would result in errors in a computation. On the other hand, the weak interactions come at the price of not being able to very well control the qubit either. However, there is an elegant way out of this dilemma, for the charge of the same electron can be easily coupled to via small electric fields. The charge of the electron therefore provides an excellent handle to confine and, as we will see later (Chapter 2), initialize and read-out the electron spin qubit.
The electron can also be used as a qubit in a different way: by using its location in a double-well potential. The two states of the qubit then correspond to the electron being in the left or in the right well [5]. This type of qubit is called charge qubit, since only the charge of the electron is what is used here. This type of qubit is very easily manipulated via electric fields and also easy to read out using nanoscopic charge sensors. However, its large coupling to electric fields also makes the state of a charge qubit very responsive to electric noise. Because of this, a superposition state of a charge qubit can only be preserved on a nanosecond timescale [6].
The research reported in this thesis is mainly aimed at creating and controlling electron spin qubits, because of their superior lifetimes. However, inevitably the electron charge and how it is localized also plays a large role. This is clearly evidenced in Chapters 3 and 4, where we see that the state of the electron spin can be changed by moving the electron between two locations.
1.5
Electron spins in quantum dots
Electron densities in solid state materials can be very high: for instance
0.84x1023/cm3 for conduction electrons in copper [1] and 2x1011/cm2 for the
two-dimensional electron gases in the heterostructures used in the experiments de-scribed in this thesis (see Chapter 2). To study single electrons therefore seems like a daunting task. However, the rapid developement of lithographical tech-niques in the past twenty years now makes it possible to build structures on the nanometer scale, enabling us indeed to isolate few or even single electrons from the sea of conduction electrons. A prime example of such structures are quan-tum dots, and they provide a very controllable environment for studying single electrons. These single electrons can serve as the physical carriers of qubits, as discussed before. Many research groups around the world now work on making the building blocks of a quantum computer based on confined electron spins (and also charges). The direction this work has taken aims for creating long-lived superposition states, coupling several qubits and demonstrating simple quantum
algorithms. Another exciting opportunity given by the ability to confine single electrons, is that it makes for a very controllable playground to study quantum physics from the bottom-up: by starting from prototypical quantum few-level systems (such as the spin of a single electron), complexity can gradually be in-creased to study interactions, delocalization phenomena, quantum decoherence and many-body physics.
1.6
Thesis outline
Finally, we give a brief outline of this thesis:
• Chapter 2 gives a brief introduction to the main theoretical concepts as well as experimental techniques used in the later chapters.
• Chapter 3 describes an experiment with a double quantum dot, in which microwaves were used to induce electrons to tunnel between the dots and change their spin. These measurements show that it is possible to do spec-troscopy of two-electron spin states using microwaves, and possibly fully manipulate and read-out in the manifold of two-electron spin states. • Chapter 4 presents time-resolved measurements aimed at quantifying the
spin-flip rates of the experiment described in Chapter 3.
• Chapter 5 describes an experiment with a linear triple quantum dot which revealed that electrons can tunnel coherently between the outer two dots, without occupying the middle dot. A long-range coupling between distant dots, due to cotunneling through a virtual state of the middle dot, was found to be responsible. Such a coupling could potentially take the role of tunnel coupling at a distance, enabling a whole new range of experiments on quantum dot arrays.
• Chapter 6 presents measurements on photon- and phonon- assisted tunnel-ing resonances in a linear triple quantum dot, which can only be explained by considering the larger number of states available in this system. The results are discussed in the context of qubit leakage.
• Chapter 7 discusses outstanding issues in the field and possible new di-rections of research.
References
[1] Ashcroft, N. W. and Mermin, N. D. Solid State Physics. (Saunders, New York, US, 1974).
[2] Nielsen, M.A. and Chuang, I.L. Quantum computation and information. (Cam-bridge University Press, Cam(Cam-bridge, UK, 2000).
[3] Loss, D. & DiVincenzo, D. P. Quantum computation with quantum dots. Phys. Rev. A 47, 120126 (1998).
[4] Hanson, R., Kouwenhoven, L. P., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K. Spins in few-electron quantum dots. Rev. Mod. Phys. 79, 12171265 (2007).
[5] Hayashi, T., Fujisawa, T., Cheong, H. D., Jeong, Y. H. & Hirayama, Y. Coherent Manipulation of Electronic States in a Double Quantum Dot. Phys. Rev. Lett. 91, 226804 (2003).
[6] Petta, J. R., Johnson, A. C., Marcus, C. M., Hanson, M. P. & Gossard, A. C. Manipulation of a single charge in a double quantum dot. Phys. Rev. Lett. 93, 186802 (2004).
Device fabrication, measurement
techniques and theory
In this chapter, background theory is discussed which will be useful for under-standing the later chapters. First our system of choice for studying confined electrons in the solid state is introduced: laterally defined quantum dots in semi-conductor heterostructures. Basic properties and measurement techniques of this system are explained before we move on to discuss the physics of few-electron spins in quantum dots, in the context of quantum information processing. Then photon-assisted tunneling of electrons is discussed in some detail. Different as-pects of this phenomenon were studied and used in the research reported in the later chapters. Finally, details of the experimental setups that were used to obtain the experimental results are given at the end of this chapter.
2.1
Quantum dots
Quantum dots are isolated islands of charge. The number of charges confined on these islands can be quite small, it can be reduced to a single electron or hole. Quantum dots are often referred to as artificial atoms, because their properties are quite similar to those of atoms. Although typically larger than atoms, quantum dots are still small enough structures (∼1-100 nm) to exhibit clearly discernible discrete energy levels at low temperatures. Much of the quantum physics of atoms therefore also applies to quantum dots. One important advantage of quantum dots is that many properties, such as charge occupation number, tunnel barriers to other dots and to electron reservoirs can be tuned over a large range by design and even during an experiment. This makes quantum dots ideal for studying the quantum physics of few electrons in the solid state. This is not only an interesting topic in itself, but is also useful for instance for quantum information and quantum simulation purposes.
Quantum dots come in many different forms (self-assembled dots [1], quantum dots defined in nanowires [2, 3] or nanotubes [4, 5]). In this thesis we focus exclusively on dots defined electrostatically by metallic gate electrodes in the two-dimensional electron gas (2DEG) of doped GaAs/AlGaAs heterostructures [6].
2.1.1
Quantum dots in GaAs/AlGaAs heterostructures
GaAs and AlxGa1−xAs (with x typically ∼ 0.3) are semiconductor alloys with the
same crystal structure (Zinc-blende) and very similar lattice constant (∼ 0.565 nm). This makes it possible to stack these materials on top of each other nearly seamlessly using growth processes such as metal-organic chemical vapour depo-sition (MOCVD) or molecular beam epitaxy (MBE). Due to the difference in bandgap, a discontinuity of the conduction band (typically ∼ 0.3 eV) occurs at an interface of the two alloys. For electrons, this discontinuity acts as one side of a trapping potential. In the heterostructures used here, a plane of the AlGaAs (located ∼ 50 nm above the GaAs/AlGaAs interface) is n-doped with Si. Elec-trons originating from these dopants can travel through the heterostructure and end up in the GaAs region below the AlGaAs layer, leaving behind positively charged dopants. At this point, the discontinuity in bandgap prevents the elec-trons from returning to the dopants. The now positively charged Si ions produce an electric field, causing the electrons to accumulate at the interface and form a thin (∼ 5-10nm) sheet of highly mobile electron gas, the 2DEG. Although very restricted in the growth direction, the electrons are highly mobile in the plane perpendicular to this direction (mobilities of a few million cm2/Vs are typical in
GaAs AlGaAs Si-dop ed A lGaAs GaAs ~100nm Gate Ohmic contact Electron G as
Figure 2.1: Schematic view of a GaAs/AlGaAs heterostructure with Au/Ti gates patterned on the surface and Ohmic contacts to connect to the two-dimensional electron gas. The gate pattern displayed here can be used to create three quantum dots in series and an additional quantum dot for use as a charge sensor (dashed circles).
the experiments presented in this thesis). The seamlessness of the interfaces of the heterostructure is critical to attain these high mobilities, since imperfections lead to electron scattering. Because of the extremely high quality of GaAs/AlGaAs heterostructures that can be attained nowadays, they form an ideal prototypical system to study the physics and applications of quantum dots.
Bandgap engineering therefore allows one to confine electrons in a two-dimensional plane. The electron gas can be confined laterally by the application of bias volt-ages on nearby gate electrodes. For this purpose, nanoscale metallic gates can be patterned on the top surface of the heterostructure, using electron beam lithog-raphy. Due to the relatively low density of the 2DEG (∼ 2 × 1011 cm−2), a
negative gate voltage of a few Volt is sufficient to selectively deplete parts of the 2DEG. This allows the formation of quantum dots in the 2DEG, separated from each other and other parts of the 2DEG by depleted regions. When small enough, these depleted separations can act as tunnel barriers. Electrons can tunnel through, with a rate set by the size of the barrier. By applying a more negative gate voltage, the number of charges occupying a quantum dot can be reduced one electron at a time. Tuning a quantum dot to hold a single electron is now a routine operation. The 2DEG can be electrically contacted by annealing Ni-AuGe surface pads into the heterostructure, forming ohmic contacts [7].
500 nm 100 nm IQPC IQPC IQPC ISQD
a
b
Figure 2.2: Scanning Electron Micrographs of double quantum dot a, and triple quantum dot samples b.
A large part of the behaviour of the system discussed so far can be captured by associating with each element (dots, gates, source and drain reservoirs) a ca-pacitance, voltage and a charge. These quantities are then used in the so-called Constant Interaction (CI) model to calculate the energy spectrum of the sys-tem [8, 9, 10]. The CI model makes two basic assumptions. First, all Coulomb in-teractions of an electron with all other charges are parametrized by a capacitance C. Second, the interactions between electrons do not modify the single-electron energy spectrum. In the CI model, each element i of a sample is associated with a capacitance Ci, a voltage Vi and a charge Qi. Charges accumulate on an element
according to Qi = CiVi. Here Ci can be written as the sum of all capacitances
with each of the other elements, Ci = P Cj. This is illustrated in Figure 2.3
for a linear triple quantum dot array, with electron reservoirs on each side of the array, a charge sensing dot (explained below) and a gate for each of the dots.
The total ground-state energy of a dot containing N electrons can be approx-imated with the CI model to be:
E(N ) = [e(N − N0) − X i CgiVgi ]2/2C +X N En,l(B) (2.1)
where N = N0 for Vg = 0, the total capacitance of the dot C = Cs+ Cd+PiCgi.
The last term of the equation is a sum of the energies of the occupied single-particle states, which in general depend on magnetic field. Note that only this last term makes the model non-classical. We can now define the electrochemical potential of a dot to be:
V
g2V
g1V
g3V
g3V
g2V
g1Q
1V
1V
LQ
2V
2Q
3V
3V
RV
g1V
g3V
g2Q
sV
sV
sLV
sRFigure 2.3: Electrostatic representation of a triple quantum dot and additional charge sensing dot. Here Ci, Viand Qi are the total capacitance, voltage and charge associated
with element i. Lines connecting elements through a rectangular box indicate tunnel barriers.
This is a useful quantity for studying electron transport, since it can give the energy needed to remove an electron from one location in order to add it to another location. The electrochemical potential depends linearly on each of the gate voltages Vg. The gates can therefore be used to straightforwardly change the
electrochemical potential of a dot. This is illustrated in Fig. 2.4, where a ladder of electrochemical potentials of a dot is drawn next to the Fermi levels of the source and drain reservoirs, which are separated from the dot by tunnel barriers. Electrons can be exchanged elastically between the dot and a reservoir if an electrochemical potential of the dot is aligned with the Fermi level of the reservoir (Fig. 2.4b). When a voltage bias is applied over the source and drain, their Fermi levels differ and a so-called bias window is opened (see Fig. 2.4c). In this case, a current can run through the quantum dot whenever an electrochemical potential of the dot is positioned within the bias window. This makes the conductance of the dot highly dependent on the gate voltage. In Figure 2.5a the conductance through the dot as a function of the gate voltage is shown, with a source-drain bias voltage of 100 µV applied. The peaks in conductance are called Coulomb peaks, with the Nth peak corresponding to µdot(N ) being aligned in the bias
window. When the temperature is small compared to the charging energy of the dot and the level separation, the normalized expression for the Nth Coulomb
µ(N-1) µ(N+1) µ(N) µ(N-1) µ(N+1) µ(N) µ(S) µ(S) µ(D) µ(N-1) µ(N+1) µ(N) µ(S) µ(D) µ(D)
a
b
c
Figure 2.4: Different alignments of electrochemical potentials of a dot relative to source and drain Fermi levels. In a,, the dot is in Coulomb blockade and no current can run. b, Elastic transport through the dot can take place. c, Inelastic transport through the dot can take place
peak in conductance is given by Gnorm = cosh−2((µdot(N ) + C)/2kBT , with C a
constant [11]. This expression can be used to determine the electron temperature.
2.2
Tunnel coupled quantum dots
In Fig. 2.3 there is an electrostatic, capacitive coupling between quantum dots. Tunnel coupling is another type of coupling, in which electrons can tunnel co-herently from one dot to another. The new charge eigenstates in the presence of tunnel coupling are delocalized states extending over the two dots. Consider a two-level system, in which intially the two states correspond to the localized ground states of an electron on dot 1 or dot 2, |φ1i and |φ2i. These are the
eigenstates of the system without tunnel coupling. With a finite tunnel coupling, the Hamiltonian becomes [12]:
H = ε/2 tc tc −ε/2 (2.3)
with ε the energy detuning between |φ1i and |φ2i, and tc the strength of the
tunnel coupling . The new eigenstates of the Hamiltonian are the symmetric and anti-symmetric delocalized states:
|ΨSi = − sin θ 2e −iϕ/2|φ 1i + cos θ 2e iϕ/2|φ 1i (2.4) |ΨAi = cos θ 2e −iϕ/2|φ 1i + sin θ 2e iϕ/2|φ 1i
where tan θ = 2|tc|/ε. with eigenvalues: ES = 1 2(Eφ1 + Eφ2) − r 1 4(ε) 2+ |t c|2 (2.5) EA = 1 2(Eφ1 + Eφ2) + r 1 4(ε) 2+ |t c|2
The new level splitting is therefore given by εnew = p(ε)2+ (2|tc|)2, which
has an avoided crossing at zero detuning of size 2|tc|. As we will see later,
inter-dot tunnel coupling provides the basis for many exciting experiments, studying and using for instance the exchange interaction and Landau-Zener-St¨uckelberg interferometry.
2.3
Read-out of the charge on a quantum dot
2.3.1
Transport and charge sensing
Transport through a quantum dot as described above can be used to determine the number of charges occupying the dot. In between the Nth and (N + 1)th
Coulomb peaks, the charge on the dot is in principle fixed to N electrons. By counting back to the first peak, one can therefore establish the occupancy of the dot. Quite often however, the tunnel barriers between dot and reservoirs increase as the dot is emptied, making it difficult to measure transport through the dot for low occupations. In this case the first discernible Coulomb peak does not correspond to the addition of the first electron to the quantum dot. Other than determining the occupation of the quantum dots, transport measurements can be used for instance for excited state spectroscopy of the dots. Since transport mea-surements were not used extensively in the experiments reported in this thesis, we will refer to other work
A different type of measuring the charge on the quantum dot makes use of a capacitively coupled sensor. A nearby quantum point contact (QPC) in the 2DEG, can be used for this purpose. When biased at the flank between conductance plateaus the conductance through the QPC is extremely sensitive to fluctuations in the local electrostatic potential and can thereby easily detect a charge change of a single electron on the dot [13]. Similarly, instead of a QPC, a nearby quantum dot can be used as a charge sensor [14]. In this case the sensor is biased to be at the flank of a Coulomb peak (Fig. 2.5a). A quantum dot is
V
gate(mV)
I
transport (pA)I
sensor (pA)0
20
40
60
80
20
40
60
80
40
80
120
160
Figure 2.5: Current through a a quantum dot and b a charge sensor capacitively coupled to a quantum dot as a function of the voltage on a gate closeby the dot. The slow background modulation of the charge sensor signal results from cross-capacitance of the gate with the charge sensor.
often a more sensitive charge sensor than a QPC. Figure 2.5b shows the current through a charge sensor as a function of dot gate voltage.
A charge sensor is sensitive to the dot occupation, irrespective of the dot’s tunnel barriers. In practice, this means that with a charge sensor one can measure a change in the occupation of the dot if the tunneling rate between dot and reservoir is higher than the rate at which measurement points are acquired. This also makes measuring the charge on a few-electron dot more easy than using transport measurements. The passing of a current through a charge sensor can occur via inelastic processes, in which the charge carriers can lose energy by the emission of acoustical phonons [15]. It was shown that these phonons can be reabsorbed by the quantum dots under study and lead to resonant tunneling of the electrons off and on the dots [16, 17].
2.3.2
Real-time charge sensing
When the tunneling rate in and out of a quantum dot falls within the bandwidth of the charge sensor, the tunneling events can be monitored in real-time, see Fig. 2.6. This enables a variety of new measurements, for instance on acquiring counting statistics of transport [18], tunneling spectroscopy [19] and single-shot read-out of spins [20]. See section 2.6.5 for a description of an RF reflectometry setup, which enables an increased read-out bandwidth of the charge sensor.
0
1
2
3
4
5
6
7
Time (ms)
I
SQD(a.u.)
Figure 2.6: Charge sensor signal as a function of time. The steps correspond to electrons tunneling in (down) and out (up) of the quantum dot.
In case of tunneling between a dot and a reservoir, the tunnel rates in and out of the dot depend on the alignment of the dot electrochemical potentials with the Fermi level of the reservoir, with the tunnel coupling setting the maximum rate. The total tunnel rate Γ can be extracted from the real-time measurements in the following way. Γ can be expressed in terms of the rate of tunneling in, or out of the dot as follows: Γ = Γin/F = Γout/(1 − F ), where F is the Fermi distribution
of the reservoir. The rates Γin and Γout are inversely proportional to the average
times between tunnel events that an electron spends on or off the dot, < τin >
and < τin > respectively. We therefore get Γ−1 = F (1 − F ) (hτini + hτouti). The
value of F can be tuned to be 0.5 by aligning µdot exactly with the Fermi level of
the reservoir, or it can be established using the relation F = hτini/ (hτini + hτouti).
2.4
Spins in quantum dots
When studying the physics of single electrons in quantum dots, the spin of the electrons cannot be ignored. Moreover, quantum bits can be defined by two spin states and much of the research in this field is directed towards the use of these spin qubits in quantum information processes. In this section we discuss the physics of electron spins confined in quantum dots. The relevant spin states are described, their transport properties in and through quantum dot structures and their interactions with the environment. We end with a discussion of coherence times of spin-based quantum bits.
2.4.1
Spin states in a single quantum dot
The spin of a single electron is a two-level system, with the two states denoted as | ↑ i and | ↓ i. When a magnetic field ~B is applied, the | ↑ i state (parallel to ~B) is split off in energy from the | ↓ i state (anti-parallel to ~B) by the Zeeman energy
∆EZ = gµBB. Here g is the g-factor (∼ −0.4 in GaAs/AlGaAs heterostructures,
making | ↓ i the state with lowest energy) and µB the Bohr magneton [21].
We now turn to the case of two electrons occupying a single quantum dot [21]. For an unperturbed Hamiltonian (i.e. without spin-orbit or hyperfine coupling terms, which is to first order a good approximation for our system), the two-electron state is the product of an orbital and spin part. Since two-electrons are fermions, the total two-electron state is anti-symmetric under exchange of the two electrons. Therefore, if the orbital part is symmetric, the spin state must be anti-symmetric, and vice versa. The total spin S is 0 or 1. The state corresponding to S = 0 is the anti-symmetric spin singlet |S i:
|S i = | ↑↓ i − | ↓↑ i√
2 (2.6)
For S = 1, there are three possible states, corresponding to ms = −1, 0, 1 (ms is
the quantum number for the z-component of the spin). These are the so-called triplets (|T+i, |T0i and |T−i):
|T−i = | ↓↓ i |T0i =
| ↑↓ i + | ↓↑ i √
2 |T+i = | ↑↑ i (2.7)
In a finite magnetic field, the three triplet states are split by the Zeeman splitting, ∆EZ. Together with the Coulomb interaction between the two electrons, the
anti-symmetrization requirement of the total two-electron state causes the triplets to differ in energy from the singlet even at zero magnetic field [22, 23]. When in a singlet state, the two electrons are allowed to both occupy the lowest available orbital. This is not possible for the triplets, which need to have one electron occupying a higher orbital, raising the triplet energy by the orbital splitting Eorb.
However, the triplet energy gets reduced by an amount EK, both because of
the anti-symmetry of the orbital part of the two-electron wavefunction (resulting in the exchange energy) and because of the reduced direct Coulomb interaction between the two electrons occupying different orbitals. At zero magnetic field, the singlet is the ground state and we can write the energies of the four spin states as:
E(S) = 0, E(T−) = E(S) + EST + ∆EZ (2.8)
E(T0) = E(S) + EST, E(T+) = E(S) + EST − ∆EZ
Note that the exchange originates in the Coulomb interaction but leads to an energy separation between states of different total spin.
Energy
Detuning ε
S(0,2) S(0,2) T-(1,1) T+(1,1) S(1,1)/T0(1,1) T0(1,1)/S(1,1) 0 EZ=gµBB 2√2tcFigure 2.7: Energy levels of two-electron spin states in a tunnel coupled double quantum dot as a function of detuning ε, at a finite value of the magnetic field B.
2.4.2
Two-electron spin states in a tunnel coupled double
quantum dot
In tunnel coupled double quantum dots, electrons can be moved between the dots by changing the detuning of the electrochemical potentials of the two dots relative to each other, using gate voltages. For two electrons occupying the double dot, the relevant detuning is ε = µl(1, 1) − µr(0, 2) (or analogously µr(1, 1) −
µl(2, 0)), with the electrochemical potentials those of the ground states with
charge configurations (1, 1) and (0, 2). The subscripts l and r refer to the addition of an electron to the left or right dot respectively. The strength of the coupling between the charge states (1, 1) and (0, 2) is given by the matrix element tc.
Taking the spin of the electrons into account, we now have eight spin states, one singlet and three triplets for both the charge states (1, 1) and (0, 2). Since tunnel coupling (in the absence of spin-orbit coupling or a magnetic field gradient across the dots) is spin-conserving, S(1, 1) is only coupled to S(0, 2) and the same goes for each of the triplets. As illustrated in Fig. 2.7, this creates an anticrossing of size 2√2tc between S(1, 1) and S(0, 2) at ε = 0. Near the anticrossing, the
S(1, 1) and S(0, 2) states are no longer the eigenstates of the system. Instead, the eigenstates are the hybridized bonding and anti-bonding states [10] (as in section 2.2):
and
ΨA= cos(Θ/2)e−iφ/2 S(1, 1) + sin(Θ/2)eiφ/2 S(0, 2) (2.10)
with Θ = arctan(2|tc|/ε. The energy of the (0, 2) triplets is typically ∼ 500µeV
higher than the other states, due to one of their electrons occupying a higher orbital, as just discussed. This shifts the anticrossing for each of the triplets to a higher value of the detuning. Therefore, for ε ≈ 0 we can neglect hybridization and charge transitions between the (1,1) and (0,2) triplet states, and we can therefore focus on the other five states, S(0, 2) and S(1, 1) (or ΨB and ΨA),
T+(1, 1), T0(1, 1) and T−(1, 1). The Hamiltonian in this basis can be written as:
H0 = − ε|S(0, 2) i hS(0, 2)| + √ 2t|S(1, 1) i hS(0, 2)| + |S(0, 2) i hS(1, 1)| − gµBBext |T+(1, 1) i hT+(1, 1)| − |T−(1, 1) i hT−(1, 1)| , (2.11)
The singlet-triplet energy splitting in the two-electron double quantum dot can be written as the energy difference between T0(1, 1) and the hybridized singlet
S(1, 1)/S(0, 2), yielding an ’effective’ exchange energy J that depends on the detuning ε. This energy enters the Hamiltonian the same way as the exchange energy: HJ = J ~S1· ~S2, with ~S1and ~S2 the spin operators of the electrons [24]. An
extremely useful feature of this exchange energy is that its value can be tuned by changing the tunnel coupling tc or the detuning ε. Control over these parameters
thereby enables entangling operations between the two spins, such as swapping of the spins [25, 26] or controlled-phase gates [27].
2.4.3
Spin blockade
The combination of the Pauli exclusion principle and Coulomb repulsion not only results in the exchange energy splitting between singlets and triplets on a double dot, but also leads to effects in the dynamics of electrons that tunnel between the two dots. As we saw, for small enough detuning ε, only the singlet state can tunnel from (1, 1) to (0, 2), while for the triplets this tunneling transition is forbidden. This effect is termed Pauli spin blockade and can be observed in transport measurements as a strong dependence of the size of the current on the direction of the voltage bias across the double dot [28, 29, 21].
We focus on transport near the (1, 1)-(0, 2) interdot charge transition. We consider transport through the electron cycle: (1, 1) → (0, 2) → (0, 1) → (1, 1), and look first at the case of forward bias. For ε ≥ 0, two possible situations can occur. First, an electron entering the left dot from the left reservoir can form the singlet state S(1, 1) with the electron in the right dot. In this case, it is
EST T(1,1) a b c T(1,1) S(1,1) S(1,1)S (0,2) S(0,2) T(0,2) T(0,2) T(0,2) S(0,2) S(1,1) T(1,1)
Figure 2.8: Schematic of Pauli spin blockade. a, At small forward bias, current through the double dot can be blocked when the system gets loaded with a spin triplet. b When the bias becomes larger than the singlet-triplet splitting, current can flow again. c For small reverse bias, current can always flow.
possible for the left electron to move on to the right dot and form the singlet state S(0, 2), allowing the next steps in the transport cycle and contributing to current. The other possibility is that the electron entering the left dot forms a triplet with the electron in the right dot. As shown in Fig. 2.8a, it is then not possible for the electron on the left dot to move to the right dot, since the triplet (0, 2) states have much higher energy in a single dot, EST. The double dot is now
in the (1, 1) configuration and cannot proceed to the next step in the cycle, (0, 2). Transport is therefore halted until the triplet decays to a singlet or an electron is exchanged with a reservoir. However, when increasing the voltage bias beyond EST, transport can also proceed via the T (0, 2) state, lifting the spin blockade
(Fig. 2.8b). For a small reverse bias (Fig. 2.8c), only singlet states can be loaded and a current can now always flow within the limits placed by Coulomb blockade. Similarly, measurements using a charge sensor can be used to probe spin blockade. In the case of a DC charge sensor, one can use the following pulse sequence: 1) empty by pulsing to for instance (0, 1), 2)initialize to a random two-electron spin state by pulsing to (1, 1), 3)pulse to positive detuning where only the singlet can go to charge state (0, 2) and wait there for some time Tw.
The last stage is where the charge sensor can give a contrast between spin states, since it gives a different signal for (1, 1) than (0, 2). For random spin injection, the measured charge signal will be between that of (0, 2) and (1, 1). One then repeats this pulse sequence many times to obtain good contrast in the DC read-out of the charge sensor current. Using a fast charge sensing setup, for instance using RF reflectometry, this procedure can also be done in a single-shot way.
Spin blockade is an invaluable tool for measuring the spin state of two electrons occupying the double dot. Furthermore, it can be used to study mechanisms which lead to relaxation and dephasing of these spin states, such as the
spin-orbit and hyperfine interactions (treated in the next subsections).
2.4.4
Spin-orbit interaction
An electron moving in a static but non-homogeneous electric field ~E (such as the electric field exerted by the proton on the orbiting electron in a hydrogen atom) experiences in its rest frame a time-varying electric field, which induces a magnetic field. In turn, this magnetic field couples to the spin of the electron. This spin-orbit interaction can be written as a relativistic correction to the (non-relativistic) Schr¨odinger equation: HSO = 4m~2
0c2
( ~E)×p·~σ, where m0 is the free electron mass,
c is the speed of light, p the electron momentum, and σ the Pauli matrices. In the solid state, electrons can also be subject to a spin-orbit interaction, due to electric fields originating from the crystal potential landscape [22].
In the GaAs/AlGaAs 2DEGs under investigation here, two main mechanisms can generate inhomogeneous electric fields resulting in spin-orbit coupling. First, since the crystal structure of GaAs is Zinc-Blende, there is a spatial inversion asymmetry of the lattice atoms. The resulting electric field contributes to spin-orbit coupling in the form of the so-called Dresselhaus [30, 21] term in the Hamil-tonian, which in two dimensions reduces to: HD = β(−pxσx + pyσy) + O(|p3|),
with the |p3| term much smaller than the linear-momentum terms due to strong
confinement in the z-direction. Here, β depends on crystal composition and the confinement in the z-direction, and x, y correspond to crystallographic directions [100],[010].
A second mechanism contributing to spin-orbit interaction is the asymmetry of the potential confining the 2DEG in the z-direction. This enters the spin-orbit interaction Hamiltonian as the Rashba term [31, 21], HR = α(−pyσx + pxσy).
Here α does not only depend on the strength of the confining potential, but also on the crystal composition at the location of the 2DEG.
The strength of the orbit coupling can be expressed in terms of the spin-orbit length: the length over which a spin has to travel to undergo a π-rotation due to the spin-orbit interaction. This length is independent of the momentum, because when the electron moves faster, its spin rotates faster, but the electron also traverses a given distance in a shorter time. The spin-orbit length has been measured to be 1 − 10µm in GaAs [32].
The spin-orbit interaction offers a handle to manipulate the spin of confined electrons using AC electric fields [33]. These are preferred over AC magnetic fields, since they are usually easier to generate and cause less side-effects such as heating of the system. However, as will be discussed in a later subsection, together with the hyperfine interaction, the spin-orbit interaction also leads to
decoherence of the electron spin states.
2.4.5
Hyperfine interaction
In III-V semiconductors, the nuclei have non-zero spin. Through magnetic dipole-dipole interactions these spins couple to the electron spin, an effect called hyper-fine interaction. For electrons inhabiting s-orbitals (which is the case for the conduction band electrons confined in the quantum dots investigated in this the-sis), the electronic wavefunction is finite at the position of the nuclei and the Hamiltonian assumes the form of a Fermi contact hyperfine interaction [34].
In general, the electron spin is in contact with N nuclear spins, in which case the contributions to the hyperfine Hamiltonian from the N nuclei can be summed, yielding: HHF = N X k AkIk· S, (2.12)
with Ak the hyperfine coupling constants for the different nuclei. In the quantum
dots studied here, N is of the order of 106. The coupling constants can be written as Ak = νA|ψ(rk)|2, with ν the volume of a crystal unit cell containing
one nuclear spin. The average hyperfine coupling constant A is independent of N and in GaAs A ∼ 90µeV.
Since the typical timescale of nuclear spin dynamics is much larger than that of the electron spin dynamics (seconds vs. nano- to milliseconds), a semi-classical approximation can be made which replaces the hyperfine interaction of each in-dividual nuclear spin with the electron spin by an effective magnetic field of the whole nuclear spin bath acting on the electron spin. This effective field is called the Overhauser field BN and the new hyperfine Hamiltonian can be written as:
HHF = gµBBN · S. The Overhauser field is maximal when all nuclear spins are
aligned, yielding in GaAs |BN| ∼5 T [35], independent of N . Because under
typical experimental conditions the thermal energy kBT is much larger than the
nuclear Zeeman energy associated with the Overhauser field, the nuclear bath will not be polarized. Instead, the Overhauser field will be given by a Gaussian distribution around zero of width σN = IA/gµB
√
N in all three directions [36]. The quasi-static Overhauser field is sampled from this distribution. For different measurements, the Overhauser field contribution typically differs by σN. Note
that the 1/√N behaviour of σN means that an electron confined in a quantum
dot will experience larger fluctuations in the Overhauser field than an electron in the bulk or unconfined 2DEG. In GaAs quantum dots, σN has been measured to
be a few mT [37, 38]. The Overhauser fields in two adjacent quantum dots can be different, which can lead to the coupling of two-electron spin states [39].
2.4.6
Effects of spin-orbit and hyperfine interactions on
electron spin coherence
Since the electron spin has only a very small magnetic moment it is very much decoupled from its magnetic surroundings, leading to excellent coherence times. The most prominent interactions of electron spins confined in a quantum dot with its environment occur via the spin-orbit coupling, the hyperfine coupling with the lattice nuclear spins, tunneling processes with nearby Fermi reservoirs, and, for two-electron spin states, exchange noise [21]. For a single electron spin, the decay time from excited to ground state T1 is mainly limited by spin-orbit
mediated relaxation via the electric field fluctuations in the form of phonons. One can theoretically derive the spin relaxation time due to this mechanism through a calculation involving the phonon density of states (at the Zeeman energy ∆EZ)
and the electric field amplitude of the phonons. As the phonon density of states depends on the Zeeman energy, the spin relaxation time in general depends on magnetic field. As a final factor, one has to take into account the dot size as compared to the phonon wavelength. Taken altogether, T1 is theoretically
pre-dicted to be proportional to ∆EZ−5 when proceeding via piezolelectric phonons and ∆EZ−7 with deformation potential phonons [40]. For typical fields of a few Tesla, relaxation occurs mainly via piezoelectric phonons. Measured values of T1
range from the millisecond to second range, for fields from 14 to 1 T [20, 41, 42]. The dephasing time T2 is to first-order not limited by spin-orbit coupling [43],
but by the hyperfine coupling with the fluctuating bath of nuclear spins. The fact that these fluctuations cause dephasing can be realized intuitively, by con-sidering the Overhauser field experienced by the electron spin due to the nuclear spins. At any given point in time, the Overhauser field points in a random di-rection, with a random amplitude. Relevant for dephasing, the component of the Overhauser field along the quantization axis of the electron spin (which can be set by an externally applied magnetic field) slightly changes the Larmor pre-cession frequency of the electron spin, leading to dephasing. An often measured quantity is T2∗, which represents the time-averaged dephasing time of the electron spin (corresponding to averaging over many values of the Overhauser field). This value is on the order of 10 ns [25, 44]. The nuclear bath changes only on a second timescale, making it possible to partially revert dephasing. These so-called echo or dynamical decoupling schemes first let the electron spin evolve under influence of the Overhauser field, accumulating extra phase. Next, the electron spin is
flipped, and the Overhauser field now rewinds the extra phase that was built up. This can be repeated for better performance. Using these schemes, T2 times in
excess of 200 µs were measured [45] in GaAs/AlGaAs-based quantum dots.
2.5
Photon-assisted tunneling
Thus far we have only considered resonant transfer of electrons between quantum dots which occur when aligning various electrochemical potentials, either of dots or leads. However, interaction with a high-frequency electric field can also lead to electronic transitions between levels that are detuned in energy, yielding inelastic tunneling events. Since this is a recurring theme in the experiments described in this thesis, we will discuss these so-called photon-assisted tunneling (PAT) effects in more detail. In this discussion, the focus will lie on the phenomenology of microwave-induced tunneling transitions in quantum dot systems. Relevant reviews are: Van der Wiel et al. [10], Platero et al. [46], Shevchenko et al. [55] and Van der Wiel et al. [47].
2.5.1
Different regimes
The response of electrons in quantum dots to an AC electric field strongly depends on the frequency f of the excitation, as compared to other timescales and energies of the system. Before going on to discuss PAT in more detail, we can already make the following broad divisions:
1. When f << Γ, with Γ the tunneling rate across a potential barrier, the response to the AC excitation can be completely understood in terms of the DC characteristics of the system. In case of a single quantum dot embedded between reservoirs, the AC excitation simply modulates the electrochemical potential of the dot sinusoidally. As a result, the levels of the dot get detuned from the Fermi level of the connected reservoirs by an amount proportional to the amplitude of the AC signal. In transport measurements, the current through the dot as a function of Vg (where Vg is a DC gate
voltage used to set the electrochemical potential of the dot) is given by a convolution of the DC current (Coulomb peaks) with the AC signal. This yields a DC current with side shoulders to the main Coulomb peak. The separation of the shoulders is set by the amplitude, not the frequency, of the AC excitation. The quantized nature of the electric field therefore does not enter the stage in this regime.
2. When f >> Γ, each electron experiences many cycles of the AC excitation before tunneling. This is the non-adiabatic regime and will be treated in the following subsections.
3. When hf << 4kBT , with T the electron temperature, single-photon
reso-nances can no longer be distinguished, making this the classical regime. 4. Lastly, when hf > ∆E, with ∆E the level splitting of a quantum dot, the
excited orbital states of the dot can also play a role and PAT can be used to do excited state spectroscopy.
In the following, we will focus on regime 2 and the reverse of regime 3 (hf > 4kBT ). This is the regime where absorption and emission occurs in quanta of
the AC electric field. The microwave frequencies we will consider will mostly be smaller than ∆E, with the exception of section 2.5.3.
2.5.2
Single junction
A lot of the theory on photon-assisted transport in quantum dots is based on the description given by Tien and Gordon for the case of a single junction between two reservoirs (two superconducting films separated by an insulating film in the case of Tien and Gordon [48]). In their model, under the influence of microwave excitation, the density of states of the reservoirs is changed, allowing electrons to tunnel into states with energies E, E ± hf, E ± 2hf, ....
In the case that f >> Γ, the DC current through the junction is modified by the presence of microwave excitation as follows [48, 47]:
Inew(VSD) = ∞
X
n=∞
Jn2(eVAC/hf )I(VSD+ nhf /e) (2.13)
Here Jn is the Bessel function of the first kind with n = 0, 1, 2, ... and gives the
probability that tunneling electrons absorb or emit n photons with energy nhf . Furthermore, VAC is the microwave amplitude, I the current without microwaves
applied and VSD the DC source-drain bias. The end result is that the current
Inew shows steps or kinks for VSD= nhf /e.
2.5.3
Single quantum dot
Since in quantum dots, the electrons are confined in all directions, new energy scales come into play. First of all the Coulomb interaction plays a large role, setting the charging energy EC. Next, the confinement leads to discretized states
in the dot with energy difference ∆E. The discrete nature of the dot spectrum only becomes apparent when ∆E > kBT . Analogous to the Tien-Gordon
for-mula Eq. 2.13, the tunnel rate Γnew(E) through each barrier in the presence of
microwave excitation can be written as: Γnew(E) =
∞
X
n=−∞
Jn2(eVAC/hf )Γ(E + nhf ) (2.14)
where Γ is the tunnel rate without microwave excitation. The tunnel rates in and out of the dot for the left and right barrier as a function of gate voltage Vg
will vary as the alignment of the dot levels with the Fermi levels of the reservoirs shifts. Explicitly, the tunnel rates can be written as [47]:
Γinl/r,j = Γl/r,j X n Jn2(eVACl/r/hf )F (µj− nhf + ηl/reVSD; Tl/r) Γoutl/r,j = Γl/r,jPnJn2(eV l/r AC/hf )[1 − F (µj − nhf + ηl/reVSD; Tl/r)]
with Γl/r,j the tunnel rate through the left or right barrier into or out of energy
level j, eVACl/r the amplitude of the microwave field at the left or right barrier, F the Fermi distribution, µj the electrochemical potential of level j, Tl/r the
temperature of the left or right reservoirs and ηl/r describes the asymmetry of
the DC voltage drop across the two barriers.
The current through the dot under microwave excitation can be found using a master equation approach, in which one keeps track of the available levels an electron can tunnel to and with which rates the electron can tunnel in and out through both barriers [49, 47].
Pumping effects
So far, the two barriers separating the quantum dot from the source and drain reservoirs have been treated in a symmetric way. However, the microwave-induced AC voltage drop over each barrier need not be the same. In this situation, pump-ing of electrons across the dot can take place and a current can run even when VSD = 0. When for instance the AC voltage drop over the left barrier is much
larger than that over the right barrier, photon-assisted tunneling occurs prefer-entially on the left side. This means that for a dot level lying below the Fermi levels of the reservoirs, photon-assisted tunneling can empty it through the left reservoir, while it can be filled (without the help of the microwaves) from both reservoirs, resulting in a net current. This pumping process is very similar to the
e
g
Figure 2.9: Alignment of electrochemical potentials which can result in current through an excited state, induced by microwaves.
case of asymmetric reservoir temperatures. Additionaly, the height of the two tunnel barriers may be modulated by the microwaves asymmetrically [50, 51]. In these situations, pumping of electrons across the dots can take place. It is im-portant to be aware of these processes, since pumping can make PAT resonances less clear and thus harder to observe. Asymmetric AC voltage drops can occur because of standing waves in the cavity formed for instance by the sample holder. Small frequency shifts can change the asymmetry, thereby allowing to tune away pumping effects.
PAT involving excited states
Using frequencies higher than the discrete level separation, hf > ∆E, PAT can be used to study the spectrum of a single quantum dot [52]. PAT involving an excited state of the quantum dot can proceed as illustrated in Fig. 2.9. The excited state is lined up inside the bias window. However, since an electron is occupying the ground state, the excited state cannot be populated due to Coulomb blockade. When the difference between the ground state energy and one of the Fermi levels of the reservoirs is lower than nhf , PAT can bring the electron occupying the ground state to the lead (this does not need to be resonant and also does not contribute to net current, the ground state only needs to be emptied). When that happens, the excited state can become occupied, and a current can run through the excited state as long as the ground state stays empty. Additional PAT processes via an excited state can occur when the excited state energy is offset by ±nhf . Note that relaxation processes that bring electrons from the excited to the ground state decrease the height of the PAT peak.
a b c hf ε hf ε (a.u.) Isensor (a.u .) 0
Figure 2.10: Illustration of PAT in a double quantum dot, with zero source-drain bias. a, At negative detuning, absorption of a photon can cause tunneling from the right to the left dot. b, At positive detuning, photon absorption leads to tunneling from left to right. c, Current through charge sensor as a function of detuning ε, with microwaves applied to one of the gates defining the double dot. Multiple PAT peaks are visible on both sides of the charge transition (around ε = 0). The peaks are positive on one side and negative on the other side.
2.5.4
Double quantum dots
The PAT discussed above involved always at least one Fermi reservoir, that is a continuum of states. In a tunnel coupled double quantum dot, interdot PAT can occur between two discrete states. We can therefore treat this case as a driven two-level system. As we will see, this makes it possible to model the system as a cascade of Mach-Zehnder interferometers, using the Landau-Zener-St¨uckelberg (LZS) formalism. First we will, however, follow the approach of Van der Wiel et al. [10] and make a distinction into two different regimes of tunnel coupling, the weak (tc < hf ) and strong (tc < hf ) coupling regimes. In the case of weak
coupling (sometimes called ionic bonding), the electrons are strongly localized on the individual dots, whereas for strong coupling (covalent bonding), the electrons can become delocalized over the two dots.
Weak coupling
In the case of weak tunnel coupling, PAT occurs between states with charge localized on each individual dot, |φ1i and |φ2i. In the following, we will assume
that |φ1i and |φ2i are the ground states of the electrons on the two dots. The
coupling tcallows tunneling between these states, yet is sufficiently small that at
most values of detuning ε = µ1(N ) − µ2(M ) the eigenstates of the Hamiltonian
including tc are still very close to |φ1i and |φ2i (see section 2.2). Photon-assisted
tunneling events between the dots can occur when an integer multiple of the microwave energy equals the detuning of the electrochemical potentials of these
a
b
hf
hf
V
SDFigure 2.11: Illustration of PAT in a double quantum dot, with finite source-drain bias. Now PAT at positive as well as negative detunings lead to current of the same sign.
two states, ie. ε = nhf .
The photon-assisted interdot tunneling events can give rise to a finite current, from source to drain reservoir, through the double dot. In the configurations of electrochemical potentials shown in Fig. 2.10, absorption of a photon leads to a pumping current at zero source-drain voltage VSD. Note that in this
con-figuration, spontaneous photon emission does not produce a net current. In a measurement performed with a charge sensor (Fig. 2.10c), PAT peaks of op-posite sign flanking an interdot charge transition can be observed. They both correspond to PAT under absorption of photons, as indicated in Fig. 2.10
In the presence of a finite value of VSD, both photon absorption and emission
can lead to a net current of the same sign through the double dot (see Fig. 2.11). A theoretical study is given in Stoof et al. [53]. There, and also in the Landau-Zener approach discussed later, the microwaves are assumed to sinusoidally modulate the detuning, therefore: ε(t) = ε0 + eVACcos(2πf t). The Hamiltonian of the
system then becomes: H(t) = tcσx+
ε(t)
2 σz (2.15)
with σx and σz Pauli matrices in the basis formed by |φ1i and |φ2i. In the
case where the tunnel coupling amplitude is small compared to all other relevant energies in the system, tc < eVAC, hf, hΓl,r, Stoof et al. find the following
ex-pression for the DC current through the double dot in the presence of microwave excitation: Inew = et2c ∞ X n=−∞ Jn2(eVAC) Γr Γ2 r 4 + (n2πf − ∆E/h) 2 (2.16)
The result is similar to the Tien-Gordon formula for PAT through a single junc-tion. This makes sense, since the nth order Bessel function of the first kind,
evaluated at eVac/hf gives the probability that an electron absorbs (n > 0) or
emits (n < 0) n photons of energy hf . In Eq. 2.16 Γr enters and not Γl, since in
the transport setting with forward bias used here Γr represents the decay of the
resonant state formed by PAT. Therefore, when the tunnel rate from reservoir to dot is much larger than the tunnel rate between the dots, the width of the PAT peaks is ∼ Γr. Finally, for weak coupling the energy separation between the PAT
peaks and the DC charge transition depends linearly on the frequency. Strong coupling
We can speak of strong coupling when the tunnel coupling strength is of the same order of magnitude as the microwave energy ( tc ∼ hf ). Then, the separation
between PAT peaks and DC charge transition can depend non-linearly on the frequency. The reason for this is that now for a large range of the detuning be-tween µ1(N ) and µ2(M ), the eigenstates of the double dot are the delocalized
symmetric and anti-symmetric states ΨS and ΨA. As shown in section 2.2 these
states have an energy splitting of εnew = 2pε2+ (2tc)2) (for eVAC < hf ).
Reso-nant absorption of microwaves thus requires the frequency to match this energy. Note that the minimum value of εnew is 2tc, making PAT for frequencies below
2tc impossible. The frequency-dependence of the PAT peak separation makes it
possible to easily establish a value for tc [54] (see Additional Material Ch. 5).
Landau-Zener-St¨uckelberg interferometry
The Landau-Zener-St¨uckelberg (LZS) formalism is a very useful framework to illustrate the coherent quantum dynamics of photon-assisted tunneling [55, 56]. When a two-level system is swept through an anti-crossing at a rate v = dE/dt (with E is the energy difference of the uncoupled levels), it can reach a superpo-sition of its ground and excited states. Starting out from the ground state, the probability that the system will make a transition to the excited state during one traversal of the anti-crossing is given by the Landau-Zener formula [55],
PLZ= exp −2π(2tc) 2 ~v . (2.17)
with 2tcthe size of the anti-crossing. In general, the system will end up in a
super-position of ground and excited states. Away from the anti-crossing, the two parts of the superposition evolve independently and acquire a relative (St¨uckelberg) phase, since they differ in energy. When the system is now swept back through the anti-crossing, the Landau-Zener formula applies again, this time starting out from the evolved superposition. The end result will be that the system can end
Detuning ε (µeV) 10 15 0 5 -5 0 200 400 -200 -400 Power (dBm) dISQD dV (a.u.) lock-in
Figure 2.12: Derivative of charge sensor current (measured through a lock-in ampli-fier), as a function of detuning and applied microwave power. The microwaves were chopped at the reference frequency of the lock-in amplifier.
up in the ground or excited state, as a function of the acquired phase. This is equivalent to the interference pattern from a Mach-Zehnder interferometer [57]. The two traversals of the anticrossing form the analog of the beamsplitters of the interometer, while the energy difference leading to the relative phase between ground and excited state forms the effective difference in optical path length.
In the double quantum dot, the applied microwaves are taken to sinusoidally modulate the detuning between the two tunnel coupled states, thus taking the system repeatedly through an anti-crossing. This can yield the interference pat-tern just described (in this case fringes in probability amplitudes for the electron to end up on the right or on the left dot). The amplitude of the microwaves can be used to change the acquired St¨uckelberg phase, leading to interference fringes along the y-axis of Fig. 2.12 Moreover, since the microwaves are applied quasi-continuously, the anti-crossing is traversed periodically for many times. The phase built up during one period (two traversals plus times away from the anti-crossing) will determine the initial superposition the system is in for the next period. Therefore, there will be not just interference within one period, but also between consecutive cycles of the microwaves. This gives an extra condition for constructive interference due to the St¨uckelberg phase. Constructive interference of the excited state can now only occur if the detuning ε = nhf , with n an inte-ger. The fringes in the the total interference pattern resulting from this condition
can be identified as being multi-photon processes.
It is important to note that the oscillations along the y-axis of Fig. 2.12 mean that a coherent superposition of ground and excited state is created and maintained between two passings through the anti-crossing. This can even be the case when the charge dephasing time is much shorter than the microwave period (TCh
2 << 1/hf ). This can be understood by noting that at a finite detuning, only
the ’tip’ of the microwave sine wave brings the system through the anti-crossing and back [58]. The interval between two passages through the anti-crossing can therefore be a fraction of the whole microwave period. The width of the fringes along the detuning axis however is proportional to the inverse of T2Ch (when other broadening effects such as heating, power broadening etc. are relatively small). Hence, a small TCh
2 will lead to a merging of multiphoton resonances.
2.5.5
Generalizations
The photon-assisted phenomena described in the previous subsections involved electron charges tunneling between neighbouring elements (dots, reservoirs). This can be made more general by considering not only the electron charge but also the electron spin. In Chapters 3 and 4, we show that in the presence of spin-orbit coupling between two dots, photon-assisted tunneling can occur between states of different spin.
Another generalization we can make is to also consider higher order tunneling processes. In Chapter 5, photon-assisted transfer is demonstrated between two dots with another dot in between. There, the coupling term is shown to be provided by cotunneling, rather than direct tunneling. Except for substituting the coupling element in the Hamiltonian, the formalism for photon-assisted transfer remains the same.
2.6
Typical experimental setup
2.6.1
Sample fabrication
Turning a wafer containing a GaAs/AlGaAs heterostructure into samples with laterally gated quantum dots requires some fabrication steps in a cleanroom. The following steps outline such a fabrication recipe:
1. Clean sample.
