Quantum dots
Quantum dot (QD) is a conducting island of a size
comparable to the Fermi wavelength in all spatial
directions.
Often called the artificial atoms, however the size is
much bigger (100 nm for QDs versus 0.1 nm for atoms).
In atoms the attractive forces are exerted by the
nuclei, while in QDs – by background charges.
The number of electrons in atoms can be tuned by
ionization, while in QGs – by changing the confinement
potential. This is similar by a replacement of nucleus by
its neighbor in the periodic table.
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Comparison between QDs and atoms
Parameter
Atoms
Quantum
dots
Level spacing
1 eV
0.1 meV
Ionization
energy
10 eV
0.1 meV
Typical magnetic
field
104 T
1-10 T
QDs are highly tunable. They provide possibilities to
place interacting particles into a small volume, allowing
to verify fundamental concepts and foster new
applications (quantum computing, etc).
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Phenomenology of quantum dots
F
AFM micrograph of the
gates structure to define
a QD in a Ga[Al]As
heterostructure.
The Au electrodes (bright)
have a height of 100 nm.
The two QPCs formed by
the gate pairs F-Q1 and F-Q2
can be tuned into the
tunneling regime, such that a
QD is formed between the
barriers.
Its electrostatic potential
can be varied by changing
the voltage applied to the
center gate
Lateral quantum dot
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Conductances of all QPCs
can be tuned by proper
gate voltages.
The F-Q1 and F-Q2 pairs
behave as perfect
quantized QPCs
The contact F-C cannot
be pinched off, but still
shows depletion
The central gate is designed to couple
well to the dot, but with a weak
influence on QPCs.
Blue arrow shows the working point.
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Gate voltage characteristics
Pronounced oscillations
The reason of the oscillations
was not clear in the beginning:
Coulomb blockade?
Resonant tunneling?
The usual way to find the
answer is to study magnetotransport
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The position of 22 consecutive
conductance resonances as
function of the gate voltage and
the magnetic field.
The QD has an approximately
triangular shape with a width
and height of about 450 nm.
The upper inset shows peak
spacings at B=0 as a function of
QD’s occupation. They are
consistent with theory (FockDarwin- model)
center gate
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Why the peaks are not equidistant?
• There is a smooth dependence on the gate voltage,
just because of change in the geometry (and
consequently, in capacitances);
• In addition to a smooth dependence there are
pronounced fluctuations – a rather rich fine
structure.
This fine structure is shown in the next slide,
where the smooth part is subtracted
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One can discriminate between
three main regimes:
1. Weak magnetic fields – the
spacings fluctuate, with a
certain tendency to bunch
together for small occupation
numbers;
2. Intermediate regime – quasiperiodic cusps;
3. High magnetic fields
Level fine structure for up
to 45 electrons on the dot
The observed structure needs
an interpretation!
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What one would expect for a QB device?
SET
Diamond stability diagram
V=10 μV
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Stability diagram for a Quantum Dot
Resembles diamond
structure for Coulomb
blockage (SET) system.
However, size of
diamonds fluctuates.
At low bias – resembles
usual CB oscillations;
At larger bias a fine
structure emerges, which
is absent in SETs
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Finally, the amplitude of
resonances can be tuned
by magnetic field:
Here we see amplitudes
of five consecutive
resonances versus
magnetic field.
The peak positions fluctuate by about 20% of their
spacing, while the amplitude varies by up to 100%.
Plenty of features are waiting for their explanation!
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What would follow from the picture of non-interacting
particles?
QD is a zero-dimensional system, its density of states
consists of a sequence of peaks, with positions
determined by size and shape of the confining potential,
as well as by effective mass of the host material.
To estimate the average spacing let us use the 2D model:
This energy should be compared with the typical
charging energy, since for an isolated dot the Coulomb
blockade must come into play. So we have to develop a
way to find the electron addition energy.
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The constant interaction (CI) model
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How much do we pay to add an electron to a quantum dot?
Suppose that the highest level in the dot is
the next electron will occupy the level
the lowest energy.
. Then
having
To find the addition spectrum one has to add this
energy to the electrostatic gain, ΔE.
Correspondingly, if we want to remove an electron it is
necessary to subtract
,
According to the CI model, one assumes that the kinetic energies
independent of the number electrons on the dot, or ΔE and
are statistically-independent.
are
The CI model disregards electron correlations
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In general, it is not the case because of
• Screening
• Exchange & correlation effects
Essence of the CI model – adding the difference
between kinetic energies to the energy cost of
addition (removal) of an electron.
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The stability diagram consists of a semi-infinite set of
diamonds of similar shape.
However, their sizes (both along the V and VG axes)
fluctuate due to variation of the level spacing. Therefore,
the diamond structure is distorted.
Maximum extension
in V-direction:
Electrostatic
energy
Kinetic
energy
The peak spacing in gate voltage at small V:
Lever arm translating the addition energies to the gate voltages.
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So far so good, but what should we do with magnetic field?
Analytically solvable model (Fock, Darwin):
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The quantum numbers nx and ny can be expressed through
more natural quantum numbers – the radial, , n = 0,1,2 … ,
and orbital momentum,
Then the spectrum can be expressed as:
Spin
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Weak magnetic field:
At B = 0 the energy levels are just
At B=0 each level has orbital degeneracy of j,
in addition, there is a spin degeneracy 2.
Similar to the atomic spectra, we can speak
about jth Darwin-Fock shell.
Filled shells correspond to N=2, 6, 12, 20, ..
Magnetic field will remove both orbital and spin
degeneracy giving rise to rather complicated spectra.
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n, l, spin
The Darwin-Fock spectrum
for
Predicted evolution of conductance
resonance versus gate voltage and
magnetic field for
Note level crossings!
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The Darwin-Fock model is a good starting point
– it gives an idea about spectrum in magnetic field
- one can indicate filled shells at N=2, 6, 12, 20 …
- it predicts behavior of conductance resonances
However, the agreement with experiment is not
prefect
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Intermediate magnetic fields
We have explained the lowfield part of the curves by
the Fock-Darwin model.
Now we have to explain the
cusps.
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Intermediate magnetic fields
Since in a strong magnetic field confinement is not too
important it is reasonable to come back to Landau levels.
Let us define:
Then (spin is neglected)
In large magnetic field the confinement can be neglected
and m+1 is just the Landau level number.
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Transformation of the dot levels into LLs
What happens at the
levels’ crossings?
Different p
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Now let us assume that the filling factor is between 2 and
4, i. e., only two lowest Landau levels are occupied.
States with m=1 decrease in energy when magnetic field
increases, while the states with m=2 – increase.
Since the number of particles is conserved, the Fermi
level is switched between the Landau levels – the
electrochemical potential moves along the zigzag line.
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The period is approximately
,
the typical energy spacing being
Bright lines
correspond
to large
conductance
States belonging to
LL1 are closer to
the edge and better
coupled to the leads
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The qualitative results are similar for dot with
different shape. Below the results for hard wall
confinement are shown
However, the shape can be to some extent reconstructed
from the behavior of level spacings.
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Quantum rings – about 100 electrons
angular
moment
number of
flux quanta
Reconstruction of
energy spectrum
from resonances
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Quantum dots
Constant interaction model
Darwin-Fock model
“Magic” numbers
DF-model
Jumps of
the Fermi
level
Beyond the constant interaction model
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The CI model does not include exchange and correlation
effects, such as spin correlations, screening, etc.
Here we discuss some of such effects.
Hund’s rules in quantum dots
As known from atomic physics, Hund’s rules determine sequence of the levels’
filling:
1. The total spin gets maximized without violation the Pauli principle
(originates in exchange interaction keeping the electrons with parallel
spins apart)
2. The orbital angular moment must be maximal keeping restrictions of the
rule #1.
3. For a given term, in an atom with outermost subshell half-filled or less, the
level with the lowest value of the total angular momentum quantum
number J lies lowest in energy. If the outermost shell is more than halffilled, the level with the highest value of J is lowest in energy.
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Filling of the Fock-Darwin
potential by first 6 electrons
at B = 0.
Configurations are labeled as
in atomic physics, 2S+1LJ.
Here S is the spin, J is the
total moment, L is the orbital
moment.
What happens in strong magnetic field, above the
threshold for cusps, i. e. for filling factor below 2?
The CI model even with account of Hund’s rules fails
and correlation effects become extremely important.
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Magnetic field dispersion of
the levels from 30 to 50
Energy of 39th level
The crossings are only due
to spin – no orbital crossings
Relatively rare crossing
are expected, however,
rapid oscillations in the
peak positions were
found.
Experimental summary
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What is the reason?
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To explain the observed features let us revisit the picture
of edge channels.
Guiding center
lines for ν=2
(metallic states)
Screening in the
metallic regions
Corresponding
LLs
Potential drops
concentrate in
the insulating
regions
Density profile
Edge states
evolve into
metallic stripes
(Chklovskii et
al.)
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Let us adapt this picture to circular dots.
We arrive at a metallic ring and a disk
separated by an insulating ring. The
concrete structure depends on effective
g-factor.
Now we have to discuss the Coulomb
blockade in such system.
The electrostatic cost of the
electron transfer between the
spin-down and spin-up sublevels
should be taken into account. It
can be done using an equivalent
circuit.
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The electrostatic is not simple since capacitances depend
on magnetic field, coupling of the gate voltage to
different island is different, etc.
The theory turned out to be rather successful.
Theory
Experiment
P. McEuen et al., PRB (1992)
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Thus we arrive at the following summary:
• In weak magnetic field Fock-Darwin model
allows for the conductance resonances;
• In intermediate magnetic fields the fieldinduced repopulation of LLs becomes
crucially important;
• In strong magnetic field the CI model
fails, and correlation effects become
important. The most important are spin
correlations in combination with screening
effects (stripes).
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Quasi-chaotic
The upper levels depend on
magnetic field in a quasi-chaotic
fashion.
Sometimes people call such a
behavior the quantized chaos.
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The distribution of nearest neighbor spacings
At large occupation numbers and in weak magnetic fields
many levels are involved, and the energy spectrum
becomes very complicated.
Is it any way to find universal properties avoiding
concrete energy spectrum?
The proper theory is referred to as quantized chaos.
The classical system is called chaotic if its evolution in
time depends exponentially on changes of the initial
conditions.
Example - a particle in the box, classical
dynamics with specular reflection.
The trajectory depends on the initial
condition, p(0) and r(0).
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Let us discuss how the difference between 2 trajectories,
evolves in time for the time much larger then the
elementary traversal time provided the position difference
at the initial time is infinitesimally small.
The answer strongly depends on the shape of the cavity.
If it diverges exponentially, then the cavity is called
chaotic. Otherwise it is called regular.
Most shapes – like the Sinai billiard –
show chaotic dynamics.
Quantization of chaotic dynamics is a
tricky business
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Universal distribution of the nearest-neighbor peak
separations (NNS) and distribution of conductance
resonances are the main topics discussed in context of
QDs.
The separations
are plotted as a
histogram, and then fitted by some distribution function.
For the Fock-Darwin system at B=0 we obtain
For a regular system the distribution is non-universal.
In chaotic systems the distributions are universal, but
not random (Poissonian)! The concrete form of the
distribution depends on the symmetry of the Hamiltonian.
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Random matrix theory
Hamiltonian is presented as a matrix in some basis, the
matrix elements being assumed random, but satisfying
symmetry requirements.
If the Hamiltonian is invariant with respect to time
inversion, then the matrix should be orthogonal.
If time reversal symmetry is broken, then the matrix is
unitary.
These two cases are called the Gaussian orthogonal
ensemble (GOE) and Gaussian unitary ensemble (GUE),
respectively.
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The results of rather complicated analysis shows that
these cases are covered by universal Wigner-Dyson
distributions.
With spin-degeneracy
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Derivation based on Wigner surmise:
Hamiltonian:
Orthogonal transformation:
with
It follows from the equation
that
Now we have to calculate the level splitting.
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Eigenvalues (in general):
Splitting:
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Now,
s
Δ
h
It is assumed here that p1 and p2 are smooth functions.
From the invariance with respect to unitary
transformation one would get
Then there are three independent variables, Δ and
h1=Re h, h2=Im h.
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Then we have
Thus, now we have a sphere in 3D space, and
Though not strictly proved,
RMT agrees with experiments
in many systems (excitation
spectra of nuclei, hydrogen
atom in magnetic field, etc.),
as well as with numerical
simulations
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An example of numerical
calculations for Sinai billiard
(about 1000 eigenvalues) –
histogram.
Comparison with GOE Wigner
surmise and Poisson
distribution is also shown
NNS distributions “know” whether the states are
extended or localized.
This property is extensively used in numerical studies of
localization.
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In quantum dots, one subtracts single-electron charging
energy from the measured addition spectrum.
Experiment
There is no signature of
bimodal distribution!
Why?
This is probably due to
spin-orbit interaction,
which is beyond the CI
model
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Studies of NNS distributions is a powerful tool for
optimizing various model for residual interactions in
small systems.
This area is still under development.
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The shape of conductance resonances
and current-voltage characteristics
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Earlier we discussed only the information emerging
from peak positions.
What can be found form the amplitude and shape of
conductance resonances?
Clearly, the peak shape and amplitude depend on the
coupling to the leads. This fact can be used to find
the properties of the wave functions.
This is in contrast to SET where many states are
coupled to the leads and the peak amplitudes are
almost constant.
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To illustrate new possibilities let us revisit the earlier
discussed conductance resonances
No current flow,
electrochemical
potentials are
inside Coulomb gap
As VG increased, a
new process
emerges. That leads
to the scenario
shown below
As VG is increased further
Now 2 levels
contribute to
transport
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Evolution of conductance with
increase of the gate voltage
Thus, we have a powerful
spectroscopy of singleelectron levels in small
structures
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Single-particle levels can be determined by high-bias
transport measurements.
In fact, gates are not necessary.
Single-electron levels manifest themselves as peaks in
the differential conductance.
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What about shape of resonances?
This is a complicated problem since all important
parameters – kBΘ, Δ, and hΓ - are usually of the same
order of magnitude.
No analytical expression for the shape in general case
since
•Coulomb correlation of tunneling through different
barriers;
•Electron distribution function inside the dot is nonequilibrium
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Other types of quantum dots
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Granular materials
MO composites:
Encapsulated 4 nm Au
particles self-assembled
into a 2D array
Vertical dots
Surface clusters
Individual grains
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Selfassembled
arrays
Ge-in-Si
Components of molecular
electronics
Hybrid structure for CNOT
quantum gate
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Nanocrystals
Nanocrystals are aggregates of anywhere from a few
hundreds to tens of thousands of atoms that combine into
a crystalline form of matter known as a "cluster."
Typically around ten nanometers in diameter, nanocrystals
are larger than molecules but smaller than bulk solids and
therefore frequently exhibit physical and chemical
properties somewhere in between.
Given that a nanocrystal is virtually all surface and no
interior, its properties can vary considerably as the crystal
grows in size.
Promising for applications in
electronics, medicine, cosmetology,
etc.
The rod-shaped nanocrystals
to the far left can be stacked
for possible use in LEDs,
while the tetrapod to the far
right should be handy for
wiring nano-sized devices.
Adapted from the web-page of the P. Alivisatos group
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Quantum dots are main ingredients of modern and
future nanoscience and nanotechnology.
There was a substantial progress in their studies, many
properties are already understood.
However, many issues, in particular, role of electronelectron orbital and spin correlations, remain to be
fully understood.
This is a very exciting research area.
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