Electron spins in single and double quantum dots : transport, correlations and decoherence

This thesis is devoted to the study of electron spins in quantum dots. In recent
years, the quantum dots have evolved towards smaller sizes and a better control over
the number of electrons on a dot. Comparing to a decade ago, when the quantum
dots were containing hundreds of electrons and behaving unpredictably under small
changes of their electrostatic or magnetic environment, today, the state-of-the-art
quantum dots resemble rather atoms (with well-organized electronic shells and energy
gaps of several meV) than chaotic systems (with randomly distributed energy levels).
These man-made atoms display lucent quantum-mechanical properties, suitable for
use as a resource to extend the classical information processing to the ultimate quan-
tum one. The electron spin, being solely quantum by its nature, enriches the physics
of quantum dots and opens a new (quantum) dimension, which can be used as a
degree of freedom to store and process information. The need to coexist at the same
scale of both quantum degrees of freedom (to encode qubits) and classical degrees of
freedom (to use as local gates to control qubits) renders the scale of quantum information
processing to the meso-scale — the borderline between the worlds of classical
and quantum. The quantum dots are examples of refined mesoscopic systems, where
the quantum degrees of freedom can be deterministically controlled by classical gates
to implement the quantum Turing machine.
While this is the long-term goal and requires progressive development of appropriate
technologies, the present interest to quantum dots is focused on characterizing
the quantum dots (finding ways to learn their parameters), identifying the dominant
mechanisms of decoherence, and engineering interactions on demand, which are
needed for quantum computing. The flexibility of quantum dots to design and their
dynamical tunability promises an easy integration of quantum dots into (quantum)
circuits. Once a single building block with required physical properties is constructed,
it can be replicated to a large number of such blocks. Interestingly, this fundamental
building block has to consist of at least two quantum dots (two qubits), owing
to entanglement as a new resource in quantum information processing. This thesis
considers, thus, single and double quantum dots and studies their spin-related
phenomena in a variety of contexts.
Electron transport is a common method of studying quantum dots. A quantum dot
or a small quantum-dot circuit can be probed by leads at finite source-drain bias.
Based on transport measurements, specific regimes of interest can be identified, to
which the quantum dots can then be tuned by gates at any time. For quantum computing,
it is necessary to have access to specific coupling constants and to know their
dependence on gates. In this thesis, we study in detail two coupled quantum dots and
show that a number of parameters can be extracted from transport measurements.
In particular, the Heisenberg exchange interaction between spins of tunnel-coupled
quantum dots can be accessed in transport in the regimes of sequential tunneling,
cotunneling, and Kondo effect. The electron-electron correlations, such as entanglement,
are also of great interest in the physics of quantum dots and for applications
to quantum information processing. In Chaps. 2 and 3, we show that correlations
between electrons can be accessed in transport as well. Most vividly they show up
in the cotunneling conductance at the singlet-triplet transition and in some specific
strong heating regimes away from the singlet-triplet transition. Our results have recently
been used in experiment to access these correlations in a two-electron quantum
dot of elongated shape.
The Kondo effect is a correlated many-body phenomenon, distinct by peculiar features
in transport, such as unitary limit of conductance. It arises under specific conditions
that allow the many-body correlations to build up. In tunnel-coupled quantum dots,
the Kondo effect is sensitive to the exchange interaction between spins. The competition
between Kondo effect and exchange interaction brings the system close to a
quantum critical point (first studied for the two-impurity Kondo model in context of
magnetic impurities in metals). This quantum critical point can be studied experimentally
in the asymptotic limit of weakly coupled quantum dots. It gives rise to a
narrow peak in the linear conductance as function of the inter-dot tunnel coupling,
provided the latter is much smaller than the dot-lead coupling (t ≪ T). The competition between Kondo effect and exchange interaction is far more fragile than the
Kondo effect itself, and is thus characterized by a much smaller energy scale than the
Kondo temperature. Observation of the sharp peak in transport will therefore signify
that extremely fragile many-body correlations build up in the system, which alone is
fascinating. The long range nature of the Coulomb interaction between electrons also
reflects on the transport properties of coupled quantum dots. A wide peak emerges in
the linear conductance as function of inter-dot tunnel-coupling due to singlet-triplet
Kondo correlations. With applying an orbital magnetic field this peak turns into a
singlet-triplet Kondo effect, distinct by an enhanced Kondo temperature. Finally,
the Kondo correlations also enhance the signatures of Heisenberg exchange interaction
in the temperature dependence of linear conductance and further access to this
important for quantum computing parameter.
Coherence of electron spin is a fundamental question in solid state. In semiconductor
quantum dots, the spin coherence is limited by the dot intrinsic degrees of freedom,
such as phonons, spins of nuclei, particle-hole excitations in metallic gates, switching
impurities nearby the dot (1/f noise), electromagnetic fields, etc. The electron
spin interacts weakly with matter, which makes it a promising candidate for use in
quantum information processing. One important interaction in semiconductors is the
spin-orbit interaction. The spin-orbit interaction mediates coupling of the electron
spin to any degree of freedom that couples to the electron charge. In this thesis, we
study spin decoherence due to the spin-orbit interaction.
The spin decoherence time T2 — the lifetime of a coherent superposition of spin-up
and spin-down states — must be sufficiently long for quantum computing algorithms
and quantum error correction schemes to be implemented. The decoherence time T2
is limited by spin-flip processes occurring over the spin relaxation time T1. This gives
an upper bound for T2, T2 ≤ 2T1. The spin relaxation time T1 is extremely long in
quantum dots (measured values range from 100 μs to 20ms). Additional reduction of
T2 can occur only due to dephasing, i.e loss of phase of coherent Larmor precession,
which requires a (quantum) fluctuating magnetic field along the spin quantization
axis. Most solid state implementations of qubits suffer strongly from dephasing, which
makes T2 ≪ T1. In Chap. 4, we show that the spin-orbit interaction in quantum
dots is not responsible for a strong reduction of T2. We consider the Rashba and
Dresselhaus spin-orbit interactions in quantum dots and find that only spin relaxation
(i.e. no dephasing) is possible in the leading order in these interactions. We obtain
an effective spin Hamiltonian which contains only purely transverse fluctuations of
magnetic field in the leading order of the spin-orbit interaction. Our finding means
that the decoherence time T2 is close to its upper bound T2 = 2T1 for spin-decay
mechanisms based on the spin-orbit interaction (additional decoherence can occur
due to the hyperfine interaction in a quantum dot). We also calculate the relaxation
time T1 due to phonon emission and find an excellent agreement with experiment
if we use an independently measured spin-orbit length of (8 − 10) μm. Our results
indicate that phonon emission is a dominant mechanism of spin relaxation in GaAs
quantum dots.
For a two-electron quantum dot, we study spin relaxation between singlet and triplet
levels in Chap. 5. Since the orbital wave functions of singlet and triplet differ significantly
from each other, decoherence is expected to set in at a short time scale
(∼ 1 ns) due to charge noise. In contrast, the spin-flip transitions require spin-orbit
interaction, similarly to spin relaxation in a single-electron quantum dot, and therefore,
they have a much longer time scale (∼ 100 μs). The spin relaxation in a twoelectron
quantum dot can thus be used as an additional test to verify the dominant
mechanism of spin relaxation in quantum dots and to obtain an estimate for spin relaxation
between Zeeman sub-levels in single-electron quantum dots. In experiment,
the triplet-to-singlet spin relaxation is easier to study than the relaxation between
Zeeman sub-levels, since the latter requires application of a large magnetic field to
resolve in energy the spin-up and spin-down levels. The singlet and triplet levels are
well separated from each other already at zero magnetic field and and makes it possible
to study experimentally spin relaxation in the low B-field limit. With applying
an orbital magnetic field the two-electron quantum dot undergoes a singlet-triplet
transition, which allows one to probe the low energy part of the spectral function
of the environment at a finite magnetic field. Due to the spin-orbit interaction we
find avoided crossings of singlet and triplet levels at the singlet-triplet transition. We
calculate the relaxation rates due to phonon-emission and find a rich behavior of rates
as functions of magnetic field around the singlet-triplet transition. We also analyze
the effect of Coulomb interaction and show that it enhances the role of spin-orbit
interaction in the quantum dot, however it suppresses the interaction with phonons
for the singlet-triplet relaxation.
Motivated by recent spin read-out experiments, we study the effect of a QPC, functioning
close to a quantum dot, on the spin relaxation in the quantum dot. We derive
a microscopic model for the spin-charge interaction and calculate the spin relaxation
time T1 in Chap. 6. The interaction of spin with charge occurs due to the spin-orbit
interaction in the quantum dot at a finite magnetic field. We find a strong dependence
(1/r6) of the relaxation rate on the distance to the QPC.
In Chap. 7, we consider several spin read-out schemes based on spin-to-charge conversion
and subsequent charge detection by a QPC. We introduce the notion of n-shot
read-out, as opposite to the single-shot read-out, for the case when the “measuring apparatus”
has an systematic measurement inefficiency. This measurement inefficiency
is intrinsic to the measurement setup, i.e. it is not related to the signal-to-noise
ratio in a measurement output. In this case, for a qubit which is in either “spin-up”
or “spin-down” state, one has to repeat the preparation and measurement of qubit
some number of times n until the qubit state is known with a given infidelity α. We
characterize the spin read-out using the formalism of positive-operator-valued (POV)
measurements, and calculate n as a function of α and the POV probabilities (specific
to measurement setup). We introduce a measurement efficiency e (0 ≤ e ≤ 1) which
characterizes the measurement apparatus, and analyze examples for which e ≈ 0.5
and e ≈ 1. We show that e < 1 results in a reduced visibility in measurements of
coherent oscillations.
The shot noise of a double quantum dot is studied in Chap. 8 using a phenomenological
approach. We formulate a stochastic model for shot noise in a multilevel system.
We relate the parameters entering our model to the transmission/reflection amplitudes
of the scattering matrix. We study shot-noise close to the Kondo regime in the
double quantum dot. We find super-poissonian noise in the limit of weakly coupled
quantum dots.