Level spacing 1 ev 0.1 mev Ionization energy Typical magnetic field 10 4 T 1-10 T

Transcription

1 Quantum dots

2 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. Quantum dots 2

3 Comparison between QDs and atoms Parameter Atoms Quantum dots Level spacing 1 ev 0.1 mev Ionization energy Typical magnetic field 10 ev 0.1 mev 10 4 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). Quantum dots 3

4 Phenomenology of quantum dots

5 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-Q 1 and F-Q 2 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 5

6 Conductances of all QPCs can be tuned by proper gate voltages. The F-Q 1 and F-Q 2 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 6

7 Gate voltage characteristics The reason of the oscillations was not clear in the beginning: Coulomb blockade? Resonant tunneling? Pronounced oscillations The usual way to find the answer is to study magnetotransport Quantum dots 7

8 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 (Fock- Darwin- model) center gate Quantum dots 8

9 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 Quantum dots 9

10 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 10

11 What one would expect for a QB device? SET Diamond stability diagram V=10 μv Quantum dots 11

12 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 Quantum dots 12

13 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! Quantum dots 13

14 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 14

15 The constant interaction (CI) model Quantum dots 15

16 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 Quantum dots 16

17 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. Quantum dots 17

18 The stability diagram consists of a semi-infinite set of diamonds of similar shape. However, their sizes (both along the V and V G 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. Quantum dots 18

19 So far so good, but what should we do with magnetic field? Analytically solvable model (Fock, Darwin): Quantum dots 19

20 The quantum numbers n x and n y 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 Quantum dots 20

21 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 j th 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 21

22 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 22

23 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 Quantum dots 23

24 Intermediate magnetic fields We have explained the lowfield part of the curves by the Fock-Darwin model. Now we have to explain the cusps. Quantum dots 24

25 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 25

26 Transformation of the dot levels into LLs What happens at the levels crossings? Different p Quantum dots 26

27 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. Quantum dots 27

28 The period is approximately, the typical energy spacing being States belonging to LL1 are closer to the edge and better coupled to the leads Bright lines correspond to large conductance Quantum dots 28

29 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 29

30 Quantum rings about 100 electrons angular moment number of flux quanta Reconstruction of energy spectrum from resonances Quantum dots 30

31 Quantum dots Constant interaction model Darwin-Fock model Magic numbers DF-model Jumps of the Fermi level

32 Beyond the constant interaction model Quantum dots 32

33 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. Quantum dots 33

34 Filling of the Fock-Darwin potential by first 6 electrons at B = 0. Configurations are labeled as in atomic physics, 2S+1 L J. 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 34

35 Magnetic field dispersion of the levels from 30 to 50 Energy of 39 th 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 What is the reason? Quantum dots 35

36 To explain the observed features let us revisit the picture of edge channels. Guiding center lines for ν=2 (metallic states) Corresponding LLs Density profile Screening in the metallic regions Potential drops concentrate in the insulating regions Edge states evolve into metallic stripes (Chklovskii et al.) Quantum dots 36

37 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 37

38 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. Experiment Theory P. McEuen et al., PRB (1992) Quantum dots 38

39 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). Quantum dots 39

40 Quasi-chaotic The upper levels depend on magnetic field in a quasi-chaotic fashion. Sometimes people call such a behavior the quantized chaos. Quantum dots 40

41 The distribution of nearest neighbor spacings

42 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 42

43 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 43

44 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 44

45 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. Quantum dots 45

46 The results of rather complicated analysis shows that these cases are covered by universal Wigner-Dyson distributions. With spin-degeneracy Quantum dots 46

47 Derivation based on Wigner surmise: Hamiltonian: Orthogonal transformation: with It follows from the equation that Now we have to calculate the level splitting. Quantum dots 47

48 Eigenvalues (in general): Splitting: Quantum dots 48

49 Now, s Δ h It is assumed here that p 1 and p 2 are smooth functions. From the invariance with respect to unitary transformation one would get Then there are three independent variables, Δ and h 1 =Re h, h 2 =Im h. Quantum dots 49

50 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 50

51 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 51

52 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 Quantum dots 52

53 Studies of NNS distributions is a powerful tool for optimizing various model for residual interactions in small systems. This area is still under development. Quantum dots 53

54 The shape of conductance resonances and current-voltage characteristics Quantum dots 54

55 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. Quantum dots 55

56 To illustrate new possibilities let us revisit the earlier discussed conductance resonances No current flow, electrochemical potentials are inside Coulomb gap As V G increased, a new process emerges. That leads to the scenario shown below As V G is increased further Now 2 levels contribute to transport Quantum dots 56

57 Evolution of conductance with increase of the gate voltage Thus, we have a powerful spectroscopy of singleelectron levels in small structures Quantum dots 57

58 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 58

59 What about shape of resonances? This is a complicated problem since all important parameters k B Θ, 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 Quantum dots 59

60 Other types of quantum dots Quantum dots 60

61 Vertical dots Granular materials MO composites: Encapsulated 4 nm Au particles self-assembled into a 2D array Surface clusters Individual grains Nano-pendulum Quantum dots 61

62 Selfassembled arrays Ge-in-Si Components of molecular electronics Hybrid structure for CNOT quantum gate Quantum dots 62

63 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. Adapted from the web-page of the P. Alivisatos group 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. Quantum dots 63

64 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. Quantum dots 64