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.
Comparison between QDs and atoms Parameter
Typical magnetic field
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). Quantum dots
Phenomenology of quantum dots
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 Quantum dots
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. Quantum dots
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
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 Quantum dots
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
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! Quantum dots
What one would expect for a QB device? SET
Diamond stability diagram
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
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!
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. Quantum dots
The constant interaction (CI) model
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.
The CI model disregards electron correlations Quantum dots
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.
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
The peak spacing in gate voltage at small V:
Lever arm translating the addition energies to the gate voltages. Quantum dots
So far so good, but what should we do with magnetic field? Analytically solvable model (Fock, Darwin):
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:
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. Quantum dots
n, l, spin
The Darwin-Fock spectrum for
Predicted evolution of conductance resonance versus gate voltage and magnetic field for
Note level crossings! Quantum dots
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
Intermediate magnetic fields
We have explained the lowfield part of the curves by the Fock-Darwin model. Now we have to explain the cusps.
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. Quantum dots
Transformation of the dot levels into LLs
What happens at the levels’ crossings?
Different p Quantum dots
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.
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
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. Quantum dots
Quantum rings – about 100 electrons
number of flux quanta
Reconstruction of energy spectrum from resonances Quantum dots
Constant interaction model Darwin-Fock model “Magic” numbers
Jumps of the Fermi level
Beyond the constant interaction model
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.
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. Quantum dots
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 Quantum dots
What is the reason? 35
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
Potential drops concentrate in the insulating regions
Edge states evolve into metallic stripes (Chklovskii et al.) Quantum dots
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. Quantum dots
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) Quantum dots
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).
The upper levels depend on magnetic field in a quasi-chaotic fashion. Sometimes people call such a behavior the quantized chaos.
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). Quantum dots
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 Quantum dots
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. Quantum dots
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.
The results of rather complicated analysis shows that these cases are covered by universal Wigner-Dyson distributions. With spin-degeneracy
Derivation based on Wigner surmise: Hamiltonian:
Orthogonal transformation: with
It follows from the equation
Now we have to calculate the level splitting.
Eigenvalues (in general):
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. Quantum dots
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 Quantum dots
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. Quantum dots
In quantum dots, one subtracts single-electron charging energy from the measured addition spectrum.
There is no signature of bimodal distribution! Why?
This is probably due to spin-orbit interaction, which is beyond the CI model Quantum dots
Studies of NNS distributions is a powerful tool for optimizing various model for residual interactions in small systems. This area is still under development.
The shape of conductance resonances and current-voltage characteristics
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.
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
Evolution of conductance with increase of the gate voltage
Thus, we have a powerful spectroscopy of singleelectron levels in small structures Quantum dots
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. Quantum dots
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
Other types of quantum dots
Granular materials MO composites:
Encapsulated 4 nm Au particles self-assembled into a 2D array
Individual grains Quantum dots
Selfassembled arrays Ge-in-Si
Components of molecular electronics
Hybrid structure for CNOT quantum gate Quantum dots
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
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.