Quantum Hall Effect. Jessica Geisenhoff. December 6, 2017

Transcription

1 Quantum Hall Effect Jessica Geisenhoff December 6, 2017 Introduction In 1879 Edwin Hall discovered the classical Hall effect, and a hundred years after that came the quantum Hall effect. First, the integer Hall effect was discovered by Klaus von Klitzing in 1980 from measurements on a MOSFET (metal-oxide-semiconductor field-effect transistor) [4], then Tsui, Stormer, and Gossard discovered the fractional Hall effect in 1982 from measurements on GaAs-AlGaAs heterojunctions[8]. This field of study has since exploded and is still highly relevant today. In fact, the fractional Hall effect and the discovery of composite particles paved the way of anyonic symmetry and the concept of topological order which won a nobel prize in The quantized Hall effect is observed when a two dimensional electron gas (2DEG) is placed under the same conditions as the classic Hall bar. Except now the electrons are confined in the z direction but are free to move in the xy plane. A schematic of the Hall geometry is shown in figure (1). Figure 1: Photograph of a GaAs/AlGaAs sample. With indium paste and gold wires attached to measure the Hall resistance when a magnetic field is applied perpendicular to it. Hall geometry is shown in the schematic in the bottom right corner. Under the geometry shown in figure (1) one can measure the voltages V xx and V xy. Then the resistivity R xx = V xx /I and R xy = V xy /I can be found. From classical models, such as the Drude theory, one would predict these values to be R xx = 0 and R xy = B nec. However, measurements performed with a 2DEG deviates from this prediction. The Hall resistance, R xy, shows an abrupt stepwise instead of a smooth dependence and R xx is zero but becomes finite during transitions between plateaus of the Hall resistance. Values for the plateaus of the Hall resistance, shown in figure (2, a) take on values of 1

2 Figure 2: Experimental curves for the Hall resistance, R H = ρ xy, and the resistance R x = ρ xx as a function of the magnetic field. (a) The integer Hall effect Discovered by von Klitzing in 1980, and (b) the fractional Hall effect discovered by Tsui, Stormer, and Gossard in ρ xy = 2π 1 e 2 ν = 1, 2, 3,... (1) ν with extremely high accuracy, independent of cleanliness of sample or position contacts. Where ν in equation (1) takes on integer values for the integer Hall effect, figure (2, a) and fractional values for the fractional Hall effect, figure (2, b). It is the goal of paper is to elucidate the nature of the quantum Hall effect and the role the choice of material plays in this quantization process. The Classical Hall Effect First is of interest to review the Hall effect, and ultimately derive the conductivity and resistivity tensors of a 2-dimensional system. The Hall effect takes on the same symmetry shown in figure (1). A magnetic field is turned on that is perpendicular to the bar, in this case the z direction. A constant current (I) is made by a flow of electron is the x direction. The transverse voltage, V xy, is known as the Hall voltage.[7] The Hall effect arises from the behavior that is induced in the charged species by the applied magnetic field. Recall that the equation of motion for a particle of mass m and charge e in the presence of a magnetic field is m dv dt = e c v B (2) For the Hall effect we will take the magnetic field pointing in the z direction, B = (0, 0, B), and restrict the particles to the xy plane so the velocity is, v = (ẋ, ẏ, 0). Under this geometry, equation (2) yields to coupled differential equations. 2

3 mẍ = ebẏ (3) mÿ = ebẋ (4) with the general solutions x(t) = X Rsin( eb t + φ) mc (5) y(t) = Y Rcos( eb t + φ) mc (6) This indicates the particles move around a circle in an anticlockwise direction. The radius R, position (X, Y), and phase are all arbitrary. However, the frequency with which the particle moves along this circle is fixed ω B = eb (7) mc Which is known as the cyclotron frequency. The classical Drude model considers a damping term in which the collisions of electrons are considered. Therefore, the equations of motion of a particle should include a relaxation time, /tau that is the average time between collisions for electrons. m dv dt = ee ev c B mv τ We are interested in the steady state solutions of the equation, dv dt = 0. As electrons move through the Hall bar the negatively charged species get pushed to one side of the bar creating this Hall resistance. However, as more current is pushed through this bar these electrons move through the bar unhindered since they are balanced by an electric and magnetic force acting on the electrons, which balance one another, which is the steady state. (8) Recall that the current density is 0 = ee x ω B p y p x τ 0 = ee y + ω B p x p y τ (9) (10) By multiplying equations (10) by neτ m ( 1 ωb τ ω B τ 1 J = nev (11) we get an equation of the form ) J = e2 nτ m E (12) By inverting this matrix, we can derive Ohm s law, J = σe. Where σ becomes the conductivity tensor and the inverse of the conductivity tensor is the resistivity. ( ) ρ = σ 1 ρxx ρ = xy ρ xy ρ xx = m e 2 nτ ( 1 ) ωb τ ω B τ 1 (13) 3

4 Now we can construct equation for observable values, such as resistance. Assuming the Hall bar is a rectangle of area L x L y R xy = V y = L ye y = ρ xy = B I x L y J x nec R xx = V y = L xe x = L x ρ xx = L x m I x L y J x L y L y ne 2 τ (14) (15) Figure 3: Predicted classical resistivity. Now we can see that the Hall resistance or transverse resistance is independent of both the geometry of the sample as well as the scattering time, τ. Uniquely, this resistance does not depend on the scattering time τ. Using this information, we can make experimental predictions. The Hall resistance or resistivity should have a linear relation with the applied magnetic field magnitude, and the longitudinal resistance should be a constant and in the limit τ (like for a perfect conductor) this becomes zero. These values were already discussed in the introduction and the predictions for resistivity are shown qualitatively in figure (3). Two-dimensional Electron Systems Before we start discussing the quantum Hall effect it is important to understand some of the materials which are classified as two-dimension electron gases. These materials confine an electron in only one direction and allow the physics of the quantum Hall effect to be possible. MOSFETs and Aluminum Gallium Arsenide/Gallium Arsenide One of the most well-known methods for confining electrons to a two-dimensional surface is by trapping them at a surface between the surface of a semiconductor (silicon, GaAs) and an insulator (SiO 2, AlGaAs). Although, the integer Hall effect was originally observed in a MOSFET, GaAs/AlGaAs gives more control over impurities so from here on out we will focus on this material. First the semiconducting layer of GaAs is created via molecular beam epitaxy process yielding a highly pure species. Then a layer of AlGaAs is layered on top of the GaAs. These materials have very similar crystal structures, so lattice mismatch is minimized. As this layer is deposited modulation doping is performed and silicon impurities are added the solid [1]. These silicon atoms sit in the lattice position of the gallium. The silicon ions subsequently lose an electron which wants to fill lower energy states in the GaAs material. However, these now positively charged silicon 4

5 ions begin attracting these electrons. These two opposing forces bound to this interface creating a two-dimensional electron system. Figure 4: Calculations of the electric sub bands of the GaAs/AlGaAs heterostructure. As well as a schematic for a cross section of a typical Hall bar. For a less qualitative description of these 2-dimensional layers one can estimate the electric sub bands by modeling an electron in a triangular potential where at z = 0 there is an infinite barrier and for z 0 the potential is a constant electric field, F These energy layers become E j = ( 2 2m ) ( 2 πef ) (j + 4 ) 2 3 j = 1, 2, 3,... (16) By lowering the temperature and controlling the carrier density through silicon doping one can fill only the lowest electric sub band and create a strictly 2-dimensional system with an energy spectrum of that of a free electron. Graphene Briefly I would like to mention the unique class of layered materials where the layers are held together by van der Waals forces. When these layers are cleaved one creates a crystal lattice that only consists of one layer that is the thickness of a single atom. This material then directly confines an electron to a two-dimensional lattice. The most prominent material of this nature would be graphene which is a 2-dimensional layer of carbon atoms in a hexagonal geometry. However, these materials have more complications, for instance graphene is a semimetal. Integer Hall Effect As shown in figure 2 ρ xy starts exhibiting plateaus when measured with increasing magnetic fields. Comparing this to figure 3 we can see that this straight line starts exhibiting quantum effects via plateaus at integer values. Therefore, we wish to derive the electron energy spectrum in the presence of a quantizing magnetic field. First, we must construct the Hamiltonian under the landau gauge. 5

6 Landau Gauge Remember that a The Lagrangian for a particle of charge e and mass m moving in a magnetic field (B = A) is where A is the vector potential. The canonical momentum then becomes L = 1 2 mẋ2 eẋa (17) And the Hamiltonian becomes p = L ẋ = mẋ ea (18) H = ẋp L = 1 2 (p + e c A)2 (19) The choice of gauge is arbitrary if it satisfies that the magnetic field is only points in the z direction in the same geometry we discussed in the classical Hall effect discussed above. Here the landau gauge is chosen, therefore This means the Hamiltonian for this system is A = Bẑ (20) A = xbŷ (21) H = 1 2m (p2 x + (p y + eb c x)2 ) (22) Therefore, the Schrodinger equation becomes Potential V (x) = 0 ( 1 2m (p2 x + (p y + eb c x)2 ) + V (x))ψ(x, y) = ɛψ(x, y) (23) By taking the potential term to be zero we can solve the Schrodinger equation in equation (23) with ease. We start by a separation of variables. H = 1 2m p2 x + m 2 (ω2 B(x + k eb )2 (24) ( 1 2m p2 x + m 2 (ω2 B(x + k eb )2 ))ψ(x, y) = ɛψ(x, y) (25) Since p y commutes with the Hamiltonian we can replace it with k and the y dependence of the wavefunction becomes a plane wave. ψ k (x, y) = e iky f k (x) (26) We can separate the x and y components since its y dependence is a plane wave. We have also been able to rearrange the Hamiltonian in a way that it resembles the Hamiltonian for a harmonic 6

7 oscillator with frequency ω B centered at k eb. This means that we know the eigen energies and wave function for this Hamiltonian. ɛ n = ω B (n ) (27) ψ n,k = e iky H n (x + kl 2 B)e (x+kl2 B )2 /2l 2 B (28) Where H n is a Hermite polynomial and l B = eb is known as the magnetic length, (which is the smallest size of a circular orbit in a magnetic field which is allowed by the uncertainty principle) and x 0 = klb 2 is the guiding center of the wavefunction (in the x direction the wavefunction is exponentially localized around this point. If we pick out a rectangle with sides L x and L y we can find the number of states in the system per value of k. If x 0 falls between 0 x 0 < L x to satisfy periodic boundary conditions Lx k < 0 lb 2 must be true. A similar argument can be made for the y direction, but a plane wave satisfying periodic boundary conditions is more familiar yielding k i = 2πi L y where i is any integer. Therefore, the density of states is N = AB φ 0 (29) φ 0 = 2π e (30) Where A is the area L x L y and φ 0 is known as the quantum of flux. The degeneracies per k value is very large and the landau levels of the system begin to resemble those shown in figure (5). Figure 5: Representation of landau levels with no potential bounding the wavefunction. Conductivity revisited, V (x) = ee x Now we have the tools to start to construct equations that describe the Hall resistance from a quantum perspective. Recall that the velocity of the particle is described by And the total current is given by mẋ = p + ea (31) 7

8 I = eẋ (32) I = e ψ i + ea ψ (33) m filledstates Here, we are still working in the landau gauge. Not only do we have a magnetic field in the z direction, but we also turn on an electric field in the x direction. This is constructed by making the potential V (x) = ee x. Now we can find the current in both the x and y direction. We can separate the current in the x and y direction I x = e m v n=1 k ψ n,k i x ψ n,k (34) = 0 (35) I y = e m v ψ n,k i y + exb ψ n,k (36) n=1 = ev k k E B This sum over k is over the density of state in equation (30) We then divide by the area to get the current density. 13 ( ) E E = (38) 0 ( ) 0 J = eve (39) φ 0 This yields the resistivity (37) ρ xx = 0 (40) ρ xy = φ 0 ev = 2π e 2 (41) v These resistivitys accurately predict figure 2 and are the same resistivity that are found along the plateau states. Sample bounded by infinite potential Let s now consider the case in which our system is bounded at x = 0 by an infinite potential wall. This is depicted in figure (6). As the electron moves towards this barrier it becomes confined by this infinite potential well causing an increase in the energy states of the system shown as a bending of the bands in figure (6). In this bounded system the density of states in the energy gap between landau levels becomes finite. This means that now the chemical potential can be fixed at any level between these two energy levels. This dependence on energy depends on the guiding center x 0 = klb 2.[2] 8

9 Figure 6: Inset shows the effective potential well formed from the infinite potential wall at x = 0 and the rest of the schematic depicts the energy dependence the first two landau levels have on this infinite well potential. The electric current now has a dependence along the edge states, now we can calculate the current again for a cross section of our set up. This gives us I = e h (µ right µ left ) (42) Where µ is the chemical potential at the left and right edges of the cross-sectional area.[3] From this equation we can see that when the system is in equilibrium these chemical potentials compensate one another. It is only when these two chemical potentials differ that the quantum Hall effect is observed. In this model we can also say that the material behaves as an insulator in the bulk and a metal at the edge states. When an electric field is turned on we can consider that some electrons are depleted on one side of the band and more are filled on the other. When µ = ev H we derive the Hall conduction. Figure 7: How the edge states change in occupation after an electric field is applied. Fractional Hall Effect Very briefly I will discuss the fractional quantum Hall effect, which arises when ρ xy = 2π 1 e 2 ν ν = 1/3, 2/5,... (43) the ν takes on fractional values.[5][6] This result was very surprising since it meant that particles themselves must be carrying fractional charge as opposed to integer charge. These are called composite particles. 9

10 Figure 8: Schematic showing vortex - electron attachment for the ν = 1 3 case. The impinging magnetic field creates little vortices that span the entire material. These vortices are equal to the number of flux quantum generated. However, there can be more flux quantum available than there are electrons in these systems. It then lowers the coulomb interaction for the electrons to attach to these vortices. When this happens, you get composite particles of fractional charge. Figure (8) shows this happening for the case of ν = 1 3. Conclusion When two dimensional systems are subjected to high magnetic field reveals a new subset of physics. This landau quantization of electron motion is only observed in electrons confined to two dimensions. The integer Hall effect eventually led to the discovery of the fractional Hall effect in which states of fractional charge were discovered. Both fields have blown up and state of the art experiments are being conducting to further our knowledge with the use of both phenomena. References [1] A Y Cho. Morphology of Epitaxial Growth of Gaas by a Molecular Beam Method - Observation of Surface Structures. J. Appl. Phys., 41(7): , [2] V T Dolgopolov. Integer quantum Hall effect and related phenomena. Physics-Uspekhi, 57(2): , [3] Quantized Hall. E,. 25(4), [4] K. V. Klitzing, G. Dorda, and M. Pepper. New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Phys. Rev. Lett., 45(6): , [5] R. B. Laughlin. Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett., 50(18): , [6] Horst Stormer. Nobel Lecture: The fractional quantum Hall effect. Rev. Mod. Phys., 71(4): ,

11 [7] David Tong. Lectures on the Quantum Hall Effect. (January), [8] D. C. Tsui, H. L. Stormer, and A. C. Gossard. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett., 48(22): ,