Mesoscopic superlattices

Mesoscopic superlattices Periodic arrays of mesoscopic elements. Lateral superlattices can be imposed onto a 2DEG by lithographic techniques. One-dimensional superlattices Patterned by holographic methods interference of laser beams, a > 200 nm. Shorter scales can be done using electron bean or scanning probe lithography, but accuracy is lower. The photo-resist is illuminated by the periodically modulated interference pattern of two laser beams. 1. Give examples of mesoscopic superlattices. Describe methods of their preparation. What could be dimension and spatial resolution of practically useful superlattices? What new physics and unusual features would one expect in superlattices? Superlattices 2

The resist can be used as Preparation a mask for lift-off process; for modulation of distance between the gate and 2DEG; as an etch mask Interesting manifestation Weiss oscillations, similar to Shubnikov-de Haas and also periodic in 1/B. They occur in weak magnetic fields and are anisotropic. Superlattices 3

Sh-dH Weiss oscillations The density modulation is only 2%, but the effect is very large! Why? The reason is the drift of the cyclotron orbits induces by periodic electric field of the superlattice. The local drift is proportional to ExB. Let us assume r c >> a. Then for most parts of the trajectory the drift averages to zero. The drift in x-direction is mostly accumulated near the turning points, and it is important that its signs are the same or opposite. 2. Describe Weiss oscillations. What is their periodicity in magnetic field? In what kind of superlattices do they take place? Are they anisotropic? Superlattices 4

Weiss oscillations If this drift has the same sign for the electron motion in the positive and negative x-direction, a net drift will remain for a complete cyclotron motion. The result from the calculation in the limit of r c >> a reads: It emerges from approximating the Bessel function It predicts that the oscillations are periodic in 1/B, that the oscillation amplitude increases as B is increased, and that the oscillations vanish for r c < a, in good agreement with the experiment. This result holds only for r c >> a, however. The theory assumes a sinusoidal superlattice potential. Other potential shapes will give phases that differ from π/4. 3. How can small electron density modulation lead to strong Weiss oscillations? Give semiclassical description of the oscillations by introducing cyclotron orbits and analysing local drift of electrons. Superlattices 5

Two-dimensional superlattices One can use the above procedure twice, second time after rotating by 90 o. The so-called antidot lattices are just 2D systems of holes in 2DEG. One can find the discrete values of magnetic field at which the orbits do not hit the scatterers so-called commensurate fields. For a square lattice they enclose 1,2,4,9,21, antidots 4. Give example of a two-dimensional superlattice. What are antidot lattices and their commensurate fields? How can commensurate fields result in increased resistivity? Superlattices 6

Classical and quantum mechanics of 2DEG Classical motion: Lorentz force: Perpendicular to the velocity! Newtonian equation of motion: m*v 2 /r c = evb; v=ebr c /m*; =v/2 r c ; c =2 =eb/m* Cyclotron orbit Cyclotron frequency, Cyclotron radius, In classical mechanics, any size of the orbit is allowed. Magnetotransport in 2DEG

m (emu) P (arb. u.) Antidots superlattice in YBa 2 Cu 3 O 7 film 0.0000000 B = 0.3 Oe T=72 K -0.0000002 2.00E-019-0.0000004 88 K 88.5 K 89 K 0.00E+000-0.0000006 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 B (T) 20000 40000 1/B (1/T) Superlattice is reflected in oscillations of magnetic moment of superconducting film. Superlattices 8

Antidots superlattice in YBa 2 Cu 3 O 7 film : MOI Optical MOI Superlattice guide magnetic flux motion in superconducting film. Planar sample size is 5 mm. Superlattices 9

2D self-assembly of nanoparticles Three-dimensional AFM image of Au nanoparticles on STO substrate deposited with 25 laser pulses at 780 C. The height of the pixels is scaled with the planar sizes in the plot.

Columnar structure of YBCO on 2D array of nanoparticles TEM image of a cross-sectional area inside YBCO film grown on substrate decorated with Ag nano-particles. A regular nanometerscale columnar structure is seen in the image.

Semiconducting sensor device: structure

Shape-sensitive crystallization in colloidal superball fluids, Laura Rossia et al., 5286 5290 PNAS April 28, 2015 vol. 112 no. 17 Since antiquity it has been known that particle shape plays an essential role in the symmetry and structure of matter. A familiar example comes from dense packings, such as spheres arranged in a face-centered cubic lattice. For colloidal superballs, we observe the transition from hexagonal to rhombic crystals consistent with the densest packings. In addition, we see the existence of square structures promoted by the presence of depletion attractions in the colloidal system. (Scale bars: 1 μm.) SEM images of the silica superballs used for the experiments. Introduction 14

Electrostatic assembly of binary nanoparticle superlattices using protein cages, Mauri A. Kostiainen et al., NATURE NANOTECHNOLOGY VOL 8 JANUARY 2013 52 Protein-based nanocages provide a complex yet monodisperse and geometrically welldefined hollow cage that can be used to encapsulate different materials. Such protein cages have been used to program the self-assembly of encapsulated materials to form free-standing crystals and superlattices at interfaces or in solution. Here, we show that electrostatically patchy protein cages cowpea chlorotic mottle virus and ferritin cages can be used to direct the self-assembly of three-dimensional binary superlattices. Scale bar, 50 nm Introduction 15

Low field resistivity peaks These values correspond to increased resistivity. However: Resistivity maxima are not exactly at the expected positions; An unexpected negative Hall resistance has been observed How these features can be explained? Superlattices 17

Hall effect Consider results of modeling for a simple layout for measuring the Hall effect. An example is the antidot lattice with lattice constants a = 240 nm and b = 480 nm. The potential can be modeled as In the model, it is assumed = 2, while V 0 and β are chosen such that the antidot potential peaks out of the Fermi sea with a radius of 43 nm. 5. Introduce an approach to analyse Hall Effect and longitudinal resistance in antidot superlattices. What is realistic potential for electrons in isotropic and anisotropic antidot lattices and how it affects the resistance? Superlattices 18

Poincaré sections For square lattice: In the absence of electric fields the phase space consists of two separate regions, called the chaotic and regular regions. Electric field does not destroy pinning. 6. Describe technique of Poincaré sections. How can they shade light on the evolution of electron motion? Does bias voltage mix the chaotic and regular regions of the phase space? Is it correct that electrons in the regular regions are main carriers of current? Superlattices 19

Effect of driving electric field When a weak electric field applied, the electrons that started out inside a regular region remain therein, while all other electrons have escaped the monitored region (in the y-direction). This numerical result indicates that even a bias voltage does not mix the chaotic and regular regions of the phase space. Superlattices 20

Origin of conduction (a) Fraction of the phase space volume of the regular regions of the phase space for a square antidot lattice, (b) the longitudinal resistance of the chaotic electrons, and (c) the calculated resistance (full line) as compared with the measured one. The electrons in the regular regions cannot carry current, which is rather carried by electrons with chaotic dynamics. Therefore, the conductance is obtained from the contribution of the chaotic electrons Superlattices 21

Simulated trajectories The trajectories for r c = a/2 Arrows typical trajectories Insets positions of 1000 electrons after 135 ps in the area of 24x24 lattice periods Superlattices 22

Magnetic field dependency of conductance Main features: Peaks in σ xx and smooth σ yy is observed field increases; decrease of as magnetic At magnetic fields indicated by A and C the conductivity is approximately isotropic, while at the fields B and D they differ by a factor 25. These features can be understood from the shapes of typical trajectories. Superlattices 23

Conductivity tensor Magnetic field is applied in the z-direction, B = (0, 0, B) Important quantity is the product of the cyclotron frequency, by the relaxation time, S is a geometry factor Here v i are the components of the drift velocity vector. Solving this system of equations for j gives j = ^ σe with conductivity as a tensor, Resistivity tensor, Update of solid state physics 24

Electron trajectories At large magnetic field the electrons are trapped by antidots and conductance is very low. It is still easier to diffuse along x since a < b. At σ xx -peaks antidots channel electron along x-direction There is no preferred directions of diffusion Similar trends are seen from the picture of diffusing electron clouds (insets) Superlattices 25

Conductance description One can model diffusion instead of conductance using the Einstein relation. On the other hand, the current must be proportional to the gradient of the electrochemical potential, Thus Now, how one can model diffusion? 7. Can description of conductance be given using a diffusion model? Is Kubo formula applicable in this case? Can you comment on classical and quantum Kubo formulae? What is the diffusion coefficient in the Kubo formula in an absence of magnetic field and driving force? Superlattices 26

Brownian motion An explanation of electrons behaviour is in the theory of Brownian motion, see courses in statistical physics. In this theory a particle is assumed to be subjected to random forces, which make velocity a random quantity. Then: classical Kubo formula Here: Superlattices 27

Kubo formula The expression on previous slide is sometimes called the classical Kubo formula. According to quantum mechanics, velocities are operators which do not commute at different times. Then calculation is much more difficult. In the absence of magnetic field and driving force, Thus where d is the dimensionality of the system Superlattices 28

Quantum effects In hexagonal antidot lattices, additional oscillations can be observed, which are periodic in B. It is the result of enhanced backscattering due to phase coherent Altshuler Aronov Spivak (AAS) oscillations (a) Longitudinal magneto-resistivities in square and hexagonal antidot lattices (the lattice geometries are shown in the inset). (b) Enlargement of xx for the hexagonal lattice around B = 0, which oscillates with a period of B = h/2e A, where A is the average area of one antidot. The strong temperature dependence indicates a phase coherent origin. 8. Can quantum effects be seen in the behaviour of superlattices in strong magnetic field? Does anything unusual happen in hexagonal antidot lattice? If yes, can you give possible quantum interpretation of the effect? Superlattices 29

Bohr-Sommerfeld quantization rule The number of wavelength along the trajectory must be integer. Only discrete values of the trajectory radius are allowed Energy spectrum: ω c τ 1 Landau levels Wave functions are smeared around classical orbits with r n = l B (n+1) 1/2 ; l B = (ħ/ c m) 1/2 l B is called the magnetic length. It is radius of classical electron orbit for n = 0. v/r; r v/ ; mv 2 /2= ħ c (n+1/2); v n 0 = (ħ c /m) 1/2 ; l B = r n 0 = (ħ/ c m) 1/2 Magnetotransport in 2DEG

Density of states Two dimensional system, periodic boundary conditions Momentum is quantized in units of A quadratic lattice in k-space, each of them is g-fold degenerate (spin, valleys). Assume that, the limit of continuous spectrum. Number of states between k and k+dk: Update of solid state physics 31

Density of states in 2D 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 32

Modification of density of states The levels are degenerate since the energy of 2DEG depends only on one variable, n. Number of states per unit area per level is m* Realistic picture ω c τ 1 Finite width of the levels is due to disorder Magnetotransport in 2DEG

Varying occupation of Landau levels Metal Insulator A series of metal-to-insulator transitions Magnetotransport in 2DEG

Weak periodic potential in strong magnetic field Weak periodic potential lifts the degeneracy (proportional to B) of the Landau levels and induces minibands separated by bandgaps. a) b) which means that q flux quanta penetrate p unit cells. (a) Calculated energy spectrum for one Landau band in a square superlattice. The ratio p/q 1/B measures the number of flux quanta h/e in units of BA, where A denotes the area of the unit cell. The numbers inside the figure indicate the value of the Hall resistance in the corresponding minigaps, in units of e 2 /h. (b) Effect of the superlattice on the Landau fan. 9. Describe the Hofstadter butterfly effect and the lift of degeneracy in the Landau fan by weak periodic potential. How is Hofstadter butterfly reflected in transport properties of superlattices? http://en.wikipedia.org/wiki/landau_quantization Superlattices 35

Hofstadter butterfly Predicted by Hofstadter for natural crystals, where its observation requires extremely strong magnetic fields of thousands of teslas. a) b) (a) Calculated energy spectrum for a Landau band in a superlattice. (b) Transport signatures of the Hofstadter butterfly. The Shubnikov de Haas resonances show a fine structure, the Hall resistance jumps between different steps within one quantum Hall plateau (the integer numbers indicate the Landau level index). This fine structure reflects the motion of the Fermi level across a butterfly within a Landau level. Superlattices 36

Cloning of Dirac fermions in graphene superlattices, L. A. Ponomarenko et al., 594 NATURE VOL 497 30 MAY 2013 Superlattices have attracted great interest because their use may make it possible to modify the spectra of two-dimensional electron systems and, ultimately, create materials with tailored electronic properties. In previous studies, itproved difficult to realise superlattices with short periodicities and weak disorder. Evidence for the formation of superlattice minibands (forming a fractal spectrum known as Hofstadter s butterfly) has been limited to the observation of new low-field oscillations and an internal structure within Landau levels6. Here transport properties of graphene placed on a boron nitride substrate and accurately aligned along its crystallographic directions is reported. The substrate s moire potential acts as a superlattice and leads to profound changes in the graphene s electronic spectrum. Second generation Dirac points appear as pronounced peaks in resistivity, accompanied by reversal of the Hall effect. Introduction 37

Cloning of Dirac fermions in graphene superlattices, L. A. Ponomarenko et al., 594 NATURE VOL 497 30 MAY 2013 a) Longitudinal conductivity as a function of n and B. c) Hofstadter-like butterfly for the graphene on-hbn superlattice. The electronic states are shown by black dots. For simple fractions p/q, we plot energies of the states in red. The regions around = 0 p/q are empty because the corresponding supercells are too large to do the necessary calculations. The blue curves show several low Landau levels for small B, which were calculated analytically for main and secondary Dirac fermions with parameters of the zero-b spectrum. Introduction 38

Hofstadter s butterfly and the fractal quantum Hall effect in moire superlattices, C. R. Dean et al., 598 NATURE VOL 497 30 MAY 2013 Here we demonstrate that moire superlattices arising in bilayer graphene coupled to hexagonal boron nitride provide a periodic modulation with ideal length scales of the order of ten nanometres, enabling unprecedented experimental access to the fractal spectrum. We confirm that quantum Hall features associated with the fractal gaps are described by two integer topological quantum numbers, and report evidence of their recursive structure. Observation of a Hofstadter spectrum in bilayer graphene means that it is possible to investigate emergent behaviour within a fractal energy landscape in a system with tunable internal degrees of freedom. Introduction 39

Quantum cascade laser Adapted from the Bell Labs web-site A quantum-cascade laser is a sliver of semiconductor material about the size of a tick. Inside, electrons are constrained within layers of gallium and aluminum compounds, called quantum wells are nanometers thick -- much smaller than the thickness of a hair. 10. Explain the action of superstructure in quantum cascade structures. How is it utilised in quantum-cascade lasers? Nanophotonics 40

Distributed quantum cascade laser Superlattices 41

Photonic crystals Optical semiconductors Photonic crystals are periodic optical nanostructures that are designed to affect the motion of photons in a similar way that periodicity of a semiconductor crystal affects the motion of electrons. SEM micrographs of a photonic -crystal fiber produced at US Naval Research Laboratory. The diameter of the solid core at the center of the fiber is 5 µm, while (right) the diameter of the holes is 4 µm To create a biosensor, a Photonic Crystal may be optimized to provide an extremely narrow resonant mode whose wavelength is particularly sensitive to modulations induced by the deposition of biochemical material on its surface. 11. Give examples of periodic optical nanostructures. Describe the structure and possible applications of photonic crystals. Nanophotonics 42

Summary The motion in the antidot lattice in magnetic field is partly chaotic it contains stability islands and chaotic seas. Electron dynamics in superlattices and other periodic arrays is a laboratory for studying both coherent and incoherent electron transport in nanodevices. They allow checking fundamental concepts, testing new numerical methods, and characterizing novel nanostructures. Quantum behaviour is observed in superlattices. Superlattices are very promising for various applications. Superlattices 43

Home activity a) Read: TH, Chapter 11 Mesoscopic Superlattices (pp 328-340) again. Try to summarize what we learned about properties of nanoscale superlattices. Identify issues that you would like to discuss on practical. b) Refresh: TH, Chapter 6 Magneto-transport Properties of Quantum Films (pp 166-195). c) Read: TH, Chapter 12 Spintronics (pp 342-360). Try to prepare for student s active learning day on May 9. d) Try to address questions in TH, p 340. e) Work with questions to Mesoscopic Superlattices: and other questions. Introduction 44