Quantum dots. Quantum computing. What is QD. Invention QD TV. Complex. Lego.

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1 Intel's New 49-qubit Quantum Chip & Neuromorphic Chip How To Make a Quantum Bit Quantum computing Quantum dots What is QD Invention QD TV Complex Lego

2 Electron orbitals and shells in atoms In atoms number of electrons per orbit is 2(2l+1): 2,6,10,14,18,22,26 In atoms number of electrons per shell is 2n 2 : 2, 8, 18, 32 In QDs number of electrons per shell is : 2, 6, 12, 20 Introduction to Nanophysics 2

3 Quantum dots: properties 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. 1. How can quantum dot be defined? What characteristic wavelength plays main role in its definition? Please explain why. Compare properties of quantum dots with properties of atoms. What holds electrons together in quantum dot? How can the number of electrons in quantum dot be changed? What are main applications of quantum dots? Quantum dots 3

4 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 to influence 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 4

5 QDs promise to make LCD screens more colourful and more energy efficient. Sony was the first to commercialize a quantum dot LCD TV in 2013 and there are now several companies (including Samsung) offering TVs with quantum dots. Quantum dots 5

6 Electron Coulomb blockade in a tunnel barrier Gate Dot Q = Ne Cost Repulsion at the dot E (Ne) = E ( (N+1)e) Q 0 = V g C At Attraction to the gate the energy cost vanishes! Single-electron transistor (SET) Introduction 6

7 QDs and atoms Schematic diagram, energy diagram and images of semiconductor quantum dots. L P Kouwenhoven et al. PRP, 2001 Quantum dots 7

8 QDs and atoms: periodic table There is two-dimensional shell structure with the magic numbers 2, 6, 12, 20,.... Current flowing through a two-dimensional circular quantum dot on varying the gate voltage (a), the addition of electrons to circular orbits (b) and periodic table for twodimensional elements (c). L P Kouwenhoven et al. PRP, 2001 Quantum dots 8

9 Electron orbitals and shells in atoms In atoms number of electrons per orbit is 2(2l+1): 2,6,10,14,18,22,26 In atoms number of electrons per shell is 2n 2 : 2, 8, 18, 32 In QDs number of electrons per shell is : 2, 6, 12, 20 Introduction to Nanophysics 9

10 Lateral quantum dot 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 Quantum dots 10

11 Lateral quantum dot: conductance 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 disconnected, but still shows drop in conductance. 2. Describe lateral quantum dot. What is the conductance as function of voltage between its different electrodes? How to disconnect central area of QD from the environment?. 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 11

12 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 3. Oscillations of what parameter are observed in quantum dots as a function of gate voltage? What could be the nature of these oscillations? What effects play major role in their appearance?. Quantum dots 12

13 Conductance oscillations: magnetic field dependence 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. center gate The upper inset shows peak spacing at B=0 as a function of QD s occupation. It is consistent with theory (Fock- Darwin- model) Quantum dots 13

14 Conductance oscillations: position Why the peaks are not equidistant? There is a smooth size dependence on the gate voltage, just because of change in the geometry (and consequently, in capacitances and distance between quantized levels); 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 14

15 (V G ) Conductance oscillations: fine structure 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 4. What are distinctive regimes in the voltage position of conductance oscillations as function of magnetic field that are typically observed in QDs? What could be the origin of their specific features in these regimes? Level fine structure for up to 45 electrons on the dot The observed structure needs an interpretation! Quantum dots 15

16 Coulomb blockade oscillations What one would expect for a QD device? Diamond stability diagram SET V=10 μv Quantum dots 16

17 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; 5. What is the shape of conductance oscillations in QD? Does it resemble Coulomb blockage oscillations? Could this effect be described by the diamond stability diagram for the Coulombblockade single electron transistor? How does amplitude of oscillations depend on magnetic field? At larger bias a fine structure emerges, which is absent in SETs Quantum dots 17

18 Conductance oscillations: amplitude behaviour 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 18

19 The constant interaction (CI) model What would follow from the picture of particles with constant interaction? 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 19

20 Density of electron states Number of states per volume per the region k,k+dk Density of states -Number of states per volume per the region E,E+dE. Since 3 Update of solid state physics 20

21 Adding an electron to a quantum dot Suppose that the highest level in the dot is the next electron will occupy the level the lowest possible 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 21

22 The constant interaction model: deficiencies In general suggesting that kinetic energies are independent of the number electrons on the dot is not correct because of: electron electron interactions Screening Exchange & correlation effects Essence of the CI model adding the difference between kinetic energies to the electrostatic energy cost of addition (removal) of an electron. 6. Explain main assumptions of constant interaction (CI) model. Define the distance between the energy levels in 2D quantum dot. How does it depend on the area of QD? What is other energy scale that is important for CI model? How to calculate the addition energy of one electron to QD? How to modify single-electron tunnelling model to get CI model? What does constant interaction model not take into account? Quantum dots 22

23 Fulton & Dolan, 1987 The SET transistor An extra electrode (gate) defined in a way to have very large resistance between it and the island. That allows to tune induced charges by the gate voltage The Coulomb gap is given by the onset of the same tunnelling events as for the single island studied above. Now, however, the Coulomb gap depends upon the gate voltage. The energy differences at electron tunnelling are: Coulomb blockade is established if all four energy differences are positive. Single electron tunneling 23

24 Stability diagram of Single Electron Transistor (SET) Coulomb diamonds: all transfer energies inside are positive Conductance oscillates as a function of gate voltage Coulomb blockade oscillations Stability diagram of a single-electron transistor. Within the diamonds, Coulomb blockade is established, while outside, a current flows between source and drain. The slopes of the boundaries are given by C 1G /(C 11 C 1S ), and by C 1G /C 1S. Single electron tunneling 24

25 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: Quantum dot: stability diagram 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 25

26 Fock-Darwin model So far so good, but what should we do with magnetic field? Analytically solvable model (Fock, Darwin): Electron motion is no longer free in the third direction, and the parabolic potential is orientated in the (x, y) plane. The quantum dot is circular in shape and has a parabolic confinement potential. Quantum dots 26

27 Fock-Darwin model: natural quantum numbers 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: 7. Describe Fock-Darwin model. What are its assumptions about the shape of QD and confinement of its electrical potential? What is the result of Schrödinger equation solution for this model and what are natural quantum numbers that are used to express this solution? Spin Quantum dots 27

28 Fock-Darwin model: 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. 8. Introduce the jth Fock Darwin shell. Illustrate splitting of Fock Darwin levels by magnetic field. How is it reflected in experimentally observed behaviour of conductance peaks in low magnetic fields? Quantum dots 28

29 Fock-Darwin model: splitting of levels n, l, spin The Darwin-Fock spectrum for Note level crossings! Predicted evolution of conductance resonance versus gate voltage and magnetic field for Quantum dots 29

30 Fock-Darwin model: summary At B = 0, the energy levels are located at (j + 1)ħω 0, with j = 2n + l, and with an orbital degeneracy of j. The Darwin-Fock model is a good starting point it gives an idea about spectrum in magnetic field - one can construct filled shells at N=2, 6, 12, 20 - it predicts behavior of conductance resonances However, the agreement with experiment is not prefect Quantum dots 30

31 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 31

32 Since in a strong magnetic field confinement is not too important it is reasonable to come back to Landau levels. Let us define: Introducing Landau level number Then (spin is neglected) In large magnetic field the confinement can be neglected and m+1 is just the Landau level number. Intermediate magnetic fields Quantum dots 32

33 Transformation of the dot levels into LLs What happens at the levels crossings? Different p 9. Explain the behaviour of conductance peaks in intermediate magnetic field. Introduce Landau level quantum numbers. How do parabolic potential dot levels transform into Landau levels? Explain switching between Landau levels at occupation numbers between 4 and 2. Quantum dots 33

34 Transitions between Landau levels 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 34

35 Filling factor in Landau quantization Usually the so-called filling factor is introduced as For electrons, the spin degeneracy Magnetic field splits energy levels for different spins, the splitting being described by the effective g-factor - Bohr magneton For bulk GaAs, Magnetotransport in 2DEG

36 Conductance peaks in intermediate fields The period is approximately The typical energy spacing States belonging to LL1 are closer to the edge and better coupled to the leads Bright lines correspond to large conductance 10. What is magnetic field period and typical energy spacing in the spectrum of conductance peaks in QD in intermediate fields? How does the picture change in a hard wall-confinement? What is the behaviour that is expected in a quantum ring? Quantum dots 36

37 Hard wall confinement (a) Landau levels in a quantum dot with an approximate hard-wall confinement. (b) A section of the calculated energy level diagram for 2 ν 4. (c) A corresponding set of experimental data. The shape can be to some extent reconstructed from the behavior of level spacing. Quantum dots 37

38 Quantum ring Quantum ring about 100 electrons angular moment number of flux quanta Reconstruction of energy spectrum from resonances Quantum dots 38

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

40 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. Quantum dots 40

41 Hund s rules in QD 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. J = L + S. 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: correlation effects become extremely important. 11. Explain Hund s rules for a quantum dot. How would the filling of the Fock-Darwin potential by first 6 electrons at B = 0 take place? Are Hund s rules applicable for filling factor below 2? Quantum dots 41

42 Conductance peaks in high fields 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. What is the reason? Quantum dots 42

43 Conductance in high fields: role 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.) 12. What happens with conductance peaks at filling factor below 2? Has it anything to do with edge channels? Explain spin separation model for circular dot and the role of intra-dot capacitance. Quantum dots 43

44 Phenomenology of circular dot 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 44

45 Role of intra-dot capacitance 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) The period of the zigzag lines of the transmission peaks as a function of B are now a measure for the intra-dot capacitance. Quantitative calculations within such a model agree well with the experimental data. Quantum dots 45

46 Quantum dot behaviour 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. Quantum dots 46

47 Quasi-chaotic behaviour Quasi-chaotic The upper levels depend on magnetic field in a quasi-chaotic fashion. Sometimes people call such a behavior the quantized chaos. 13. Does quasi-chaotic behaviour take place in quantum dots? If yes, how is it expressed? Is it classical or quantum chaos? What is the difference? Quantum dots 47

48 The distribution of nearest neighbor spacing At large occupation numbers and in weak magnetic fields many levels are involved, and the energy spectrum becomes very complicated. Is 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 48

49 Chaos in classical and quantum systems 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 49

50 Universal distributions 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. 14. Explain main features of the nearest-neighbour conductance peak separation in QD. What are statistical distributions and techniques that could be used to describe it? Are there universal distributions? Is random matrix theory applicable in this case? Can you comment on universal Wigner-Dyson distributions and their applicability to QDs? Does experiment show behaviour different from these distributions? Quantum dots 50

51 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 51

52 Wigner surmises: two level system distributions The results of rather complicated analysis shows that these cases are covered by universal Wigner-Dyson distributions. With spin-degeneracy Quantum dots 52

53 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 53

54 Eigenvalues (in general): Eigenvalues and splitting Splitting: Quantum dots 54

55 Derivation 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 55

56 Derivation: result 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 56

57 Universal distributions 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 57

58 Nearest-neighbour peak separation distribution Experiment 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 58

59 NNS distribution summary Studies of nearest-neighbor peak separation distributions is a powerful tool for optimizing various model for residual interactions in small systems. This area is still under development. Quantum dots 59

60 Amplitude and shape of conductance resonances 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. 15. What are approaches that could be taken to explain amplitude and shape of conductance resonances? Can you explain amplitude variation in a simple diagram model of quantised levels for a QD separated by two tunnelling barriers? What is the role of electrostatic charging energy there? How can single-particle levels be determined by high-bias transport measurements? Can shape of resonances be easily analysed? Quantum dots 60

61 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 61

62 Conductance resonances: amplitude change Free energy diagrams of a quantum dot with a bias voltage comparable to the level spacing Δ applied. Full circles denote occupied dot states, while open circles indicate empty states. To the lower left, the resulting conductance resonance and the corresponding energies for each scenario (a) (g) are sketched. It is a powerful spectroscopy of single-electron levels in small structures. Quantum dots 62

63 High-bias transport measurements For spectroscopy of single-particle levels, in fact, gates are not necessary. Single-electron levels manifest themselves as peaks in the differential conductance. Quantum dots 63

64 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 64

65 Other types of quantum dots 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 65

66 Other QDs and molecular electronics Selfassembled arrays Ge-in-Si Components of molecular electronics 16. Give examples of other than semiconducting quantum dots. Could they be used as components of molecular electronics? Can nano-crystals be used as QD? Can you give examples of these? Hybrid structure for CNOT quantum gate Quantum dots 66

67 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 67

68 Summary 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 68