Supersymmetry and Quantum Hall effect in graphene

Transcription

1 Supersymmetry and Quantum Hall effect in graphene Ervand Kandelaki Lehrstuhl für Theoretische Festköperphysik Institut für Theoretische Physik IV Universität Erlangen-Nürnberg March 14, Introduction 1.1 Topic and message of the report Graphene as a new material Apparent importance of materials with specific properties for the development of new or better electronic devices leads to a huge interest in those materials. One of them, thoroughly investigated in recent times, is graphene. Graphene is a single planar sheet of carbon atoms. It can be regarded as a layer of graphite, which is a composite of many parallel carbon sheets. An overview of its properties was given by Anton Lopatin in his talk at MB-JASS 007. Figure 1: Structure of graphene Central topic of this work consists of the specific Hamiltonian in graphene and its impact on the Quantum Hall effect Graphene Lattice Graphene has a hexogonal two-dimensional lattice with a diatomic basis. The reciprocal lattice obtained for this type of real space lattice is also two-dimensional and hexagonal. Figure : Graphene lattice in real l. and reciprocal r. space If R is a lattice vector in real space than we can calculate the coordinate vector of atom types contained in the lattice by addition of corresponding displacement vectors in the Wigner- Seitz cell: v 1 + R A, v + R B. 1

2 1. Goal Similar to the real space lattice the reciprocal lattice of graphene consists of two equal sublattices which are shifted with respect to each other. We call the representatives of these two sublattices K and K. These two points are independent and symmetric to each other with respect to time-inversion t t. Figure 3: Brillouin zone of graphene Our first important goal is the effective Hamiltonian around K and K. 1.3 Motivation of the Hamiltonian Tight Binding We write down the wave function of the system corresponding to the tight binding approach: ψ k r = R e i k R χ r R 1 ψ k r = α R = R =: R e i k R+ v 1 f r R v 1 + β e i k R+ v f r R v R [ ] αe i k v 1 f r R v 1 + βe i k v f r R v 3 e i k R e i k R αϕ α k r + βϕ β k r = αψ α k r + βψ β k r Obtaining Hamiltonian Ĥψ k = ε k ψ k, ψ k = αψ α k + βψ β k 5 Ĥ acts on vectors in -dimensional space spanned by 1 0 ψ α k A sublattice and 0 1 ψ β k B sublattice. We now calculate the matrix elements, e.g. Ĥ αβ = ψ α ψ k Ĥ β k = e i k R+ v 1 R v f r R v 1 Ĥ f r R v = 6 R, R = e i k R+ v 1 R v f r R + v 1 R v Ĥ f r 7 R, R j te i k u j, u j = R + v 1 R v, u j = v 1 v, 8 where we only consider contributions from neighbouring atoms.

3 These matrix elements form a -matrix, which represents the Hamiltonian Ĥ: t j 0 t j e i k u j 0 e i k u j 9 Taylor-expansion around k K k = kk + k yields 0 k x ik yv k x + ik yv 0 More general, using effective mass method: 0 ˆp x iˆp y v ˆp x + iˆp y v 0 We obtained the Hamiltonian we will discuss thoroughly from now on. Figure 4: Brillouin zone Properties of Hamiltonian.1 Without magnetic field.1.1 Dirac-Hamiltonian First recall the effective Hamiltonian around K derived above: 0 ˆp x iˆp y v ˆp x + iˆp y v 0 This is the Dirac Hamiltonian of a relativistic particle! The only important difference to the usual Dirac-Hamiltonian is that no spin is involved in the problem. Besides, we do not have the mass termmc in the Hamiltonian, so we are dealing with extremely relativistic quasiparticlesweyl-hamiltonian. Systems of massless particles and quasiparticlese.g. neutrinos and antineutrinos can be described in this way. One should pay attention to the fact, that this Hamiltonian acts around K, but a similar one acts around K..1. Eigenvectors without magnetic field 0 ˆp x iˆp y v, 13 ˆp x + iˆp y v 0 which was derived earlier must have eigenvectors α ψ k = e i k r, p = β k Using Schroedinger-equation the values α and β can be determined: ψ ± k = 1 e i k r ±ie i θ k, ε ± = ± v k, 14 e i θ k where k x + ik y = k e i π θ k Figure 5: Dispersion relation

4 .1.3 Introducing Pauli-matrices Although no normal spin appears in this probem, we use quasi-spin since the Hamiltonian can be described by Pauli-matrices: i 1 0 σ x =, σ 1 0 y =, σ i 0 z = and their combinations σ + =, σ 0 0 =, σ 1 0 ± = 1 σ x ± iσ y 18 Now we can rewrite Ĥ: 0 k x ik y v = vσ k x + ik y v 0 x k x + vσ y k y = 19 = v σ k = v k k σ =: v k χ k k Chirality operator σ = σ x, σ y, k = k x, k y, k ± = k x ± ik y 1 Last expression can be written as using the chirality operator v k χ k χ k = σ k k 3 Since χ k commutes with Hamiltonianwhich means conservation of chirality chirality is as the projection of quasi-spin in the k direction a good quantum number, quasi-spin itself is a bad one. Solutions can be described completely by k and χ k.. With magnetic field..1 Hamiltonian in magnetic field We now intoruce magnetic field B along the z axis and corresponding vector potential A: A = 0, Bx, 0, B = A Landau gauge 4 This introduction leads to a change of the Hamiltonian according to replacements ˆ p = i ˆ π = ˆ p e c A, v σ p v σ π = v σ ˆ π 5 Here we denote as ˆ π = ˆπ x, ˆπ y the two-dimensional momentum. We get: σ p = σ + p + σ p + σ ˆ π = σ +ˆπ + σ ˆπ + 6 ˆπ ± = ˆp ± i e c Bx [ˆπ +, ˆπ ] = e c B 7 4

5 .. Analogy with ladder operators We rewrite this important result in a few different ways [ˆπ +, ˆπ ] = e c l H = c eb B = eb c sgneb = sgneb =: γ sgneb 8 lh is called magnetic length 9 We see that the commutator of ˆπ + and ˆπ is a number, whose sign is sgneb. Recall: raising and lowering operators of HO are defined as a n = n n 1, a n = n + 1 n [ ] a, a = 1, a a = ˆN 31 ˆ π ± can therefore be understood as raising/lowering operators depending on sgneb. 3 Supersymmetry methods 3.1 Basics of Supersymmetry What is supersymmetry? In the section above we found an analogy of operators in our problem with ladder operators of the harmonic oscillator. More general concept with the same properties is that of supersymmetric operators. We consider a system with bosons and fermions with variable number of both particle types. Supersymmetry refers to the symmetry between bosons and fermions, i.e. to specific transformations from one type to the other. Supersymmetric states are denoted by n B, n F, where n B = 0, 1,,..., n F = 0, Suppersymmetric operators Operators of crucial importance are the annihilationb, f and creationb, f operators: b n B, n F = n B n B 1, n F, b n B, n F = n B + 1 n B + 1, n F 3 [ ] b, b = 1, b b = ˆN B 33 for bosons and f n B, n F = n F n B, n F 1, f n B, n F = n F + 1 n B, n F { } f, f = ff + f f = 1, f f = ˆN F 35 for fermions. Now we define operators which transform bosons to fermions and vice versa: Q + = bf Q + n B, n F n B 1, n F Q = b f Q n B, n F n B + 1, n F 1 37 Their properties can be derived from relations applying for f and b: f = f = 0 Q + = Q =

6 Besides, let us define their hermitian linear combinations: Q 1 = Q + + Q, Q = iq + Q {Q 1, Q } = 0 39 First, one should consider the simplest Hamiltonian consisting of operators defined above: {Q +, Q } = Q 1 = Q 40 [H, Q] = 0, where Q = Q +, Q, Q 1 or Q 41 Therefore the values of these operators are conserved in such systems. 3. General Hamiltonian 3..1 Generalizing Hamiltonian Back to our Hamiltonian 0 vπ vπ We introduce the mass term : vπ vπ + 43 Recall: there are two independent points in momentum space: K and K. They act in different subspaces, so the total Hamiltonian has the form ĤK Ĥ K 3.. Suppersymmetric operators Figure 6: Dispersion relation withoutl. and withr. mass Figure 7: Brillouin zone Summarizing Hamiltonian around K and K we get vπ vπ + vπ + 45 vπ σz + σ = + π + σ π + v σ z + σ + π + + σ π v If we interchange the last two basis vectors we get σz + σ + π + σ π + v 0 0 σ z + σ + π + σ π + v 47 We have obtained a four-dimensional Hamiltonian consisting of two two-dimensional ones which differ only by the sign of. To unify these two, we try to calculate the square of one of these two two-dimensional Hamiltonians. We will then see that this square is at the same time the square of the second two-dimensional part. 6

7 3..3 Squared Hamiltonian In the last section we derived the total effective Hamiltonian: σz + σ + π + σ π + v 0 0 σ z + σ + π + σ π + v 48 Let us look at the square of the Hamiltonian around K: σ z + σ + π + σ π + v = σ z + σ + π + σ π + v + v {σ z, σ } ˆ π = 49 = σ z + σ + π + σ π + v = 50 = σ z + σ + π + σ π + v 51 Since {σ z, σ } = 0 5 Ĥ = σ z + σ + π + σ π + v contains therefore both solutions for K and K. Squared Schroedinger-equation has the form Ĥ ψ = ε ψ = σ z + vσ π ψ = + v σ π ψ 53 where we have to tackle the right side yet: σ ˆ π = 1 [ ] { π+ ˆ, πˆ } γ sgnebσ z with 54 { π ˆ +, πˆ } n = γ n + 1 n 55 which applies always since one of the operators π ˆ ± is always the lowering operator, the other is always the raising operator. Only which one is which depends on sgneb. Hence, σ π n α = 1 γ [n + 1 sgnebσ z ] n α 56 Landau levels correspond to bosons n, quasi-spin to fermions α. ε = + v γ n + 1 sgnebσ z 57 ε n, = ε n 1, = + v γ n provided eb < 0 58 Here we denoted by and the fermion states: σ z = 1, σ z = 1. This equality of the energy values implies degeneracy of energy levels. However, there is no such degeneracy for the 0 state because there is no corresponding state 1. All levels except the ground level are therefore double-degenerate. This has a great impact on the Quantum Hall effect in graphene. 7

8 4 Quantum Hall effect 4.1 Quantum Hall effect Basics of Quantum Hall effect Quantum Hall effect is the Quantum-mechanical version of the classical Hall effect observed in two-dimensional electron systems. Namely, at low temperatures and strong magnetic fields the Hall conductance takes on quantized values: σ = ν e. h An Illustration of the Hall effect is given in Fig. 8. Circular currents induced by magnetic field add up to edge currents flowing in different directions. Voltage applied across the probe induces difference of these edge currents which leads to a total non-zero current. In practice we first produce this current and then apply voltage across the probe in order to reach equilibrium. V AB = 0 V AB 0 Figure 8: Edge currents in two-dimensional systems leading to Hall effect We use the standard formulae to describe the Quantum Hall effect in our systemgraphene: filling factor: ν B = ρ N B, σ xy = ecρ B, ρ = n e n h 59 Recall: N B = 1 πl H From that we get the following expression for the conductance: ν B = πc ρ 60 eb 61 σ xy = e c e ρ sgnebsgnµ = eb π ν BsgneBsgnµ Unusual Quantum Hall effect in graphene Our main goal is to investigate the specific properties of the Quantum Hall effect in graphene. First, let us calculate the conductance using the Bose-Einstein-distribution: 1 n e = e ε µ T + 1, n h = 1 n e 63 n e n h = tanh µ ε 64 T 8

9 We have to sum up contributions from all energy levels and take care of the right degeneracy discussed above: ν B sgnµ = 1 tanh µ + + tanh µ 65 T T + tanh µ + ε n + tanh µ ε n 66 T T n=1 Now we recall the expression for the energy levels and the limit behaviour of tanh: 67 ε n = + γ v n 68 ω tanh sgnω for T 0 69 T So we count conducting electrons with the right degeneracy. This is shown in Fig. 9 & 10. Figure 9: Intersections of the chemical potential with Landau-levels deformed near the edges correspond to the conducting electrons. Be aware of degeneracies! Figure 10: Unusual first half-step of σ xy confirmed by the number of intersections in Fig. 9. In spite of typical integer steps we observe a half-step in the middle and integer steps from their on. This interesting property of graphene emerges as shown above from the form of its Hamiltonian. 9

10 Contents 1 Introduction Topic and message of the report Graphene as a new material Graphene Lattice Goal Motivation of the Hamiltonian Tight Binding Obtaining Hamiltonian Properties of Hamiltonian 3.1 Without magnetic field Dirac-Hamiltonian Eigenvectors without magnetic field Introducing Pauli-matrices Chirality operator With magnetic field Hamiltonian in magnetic field Analogy with ladder operators Supersymmetry methods Basics of Supersymmetry What is supersymmetry? Suppersymmetric operators General Hamiltonian Generalizing Hamiltonian Suppersymmetric operators Squared Hamiltonian Quantum Hall effect Quantum Hall effect Basics of Quantum Hall effect Unusual Quantum Hall effect in graphene