PHYS-E0551 Homework assignment 7: Graphene (due Nov 5) Homework assignment 7: Graphene (due Nov 5) ˆ = P 0 Uk0 −k [ˆ 1. Scattering in graphene. Consider the potential U c†A,k0 cˆA,k + cˆ†B,k0 cˆB,k ]. This corresponds kk R to a U (r) = dk e−ik·r Uk potential in position representation, such that the potential is the same on the A and B sublattice sites. ˆ |+, qi for backward scattering from momentum q to (a) Calculate the matrix element M = h+, −q|U −q in graphene. The scattering rate would be ∝ |M |2 according to Fermi golden rule. The eigenstates are as derived in the lectures: |s, qi ≈ (K) where ψˆs s = ±1. cˆA,K+q cˆB,K+q † ψˆs(K) (q)|0i , (1) √ (q) = (eiθq /2 , se−iθq /2 )/ 2, is the eigenstate vector for energy s,q = s~vF |q|, and (b) Compare the result to Klein tunneling. (c) What happens if the potential is not the same on A and B sublattices? 2. Graphene in magnetic field. Calculate the Landau levels, i.e., eigenenergies in a magnetic field: 0 −i∂x − ∂y + eAx /~ − ieAy /~ a a ˆ ˆ H = ~vF , H = . −i∂x + ∂y + eAx /~ + ieAy /~ 0 b b (2) The magnetic field is included via the minimal substitution ~k 7→ ~k−qA, k = −i∇. Consider the vector potential A = (−By, 0, 0) corresponding to a magnetic field B = ∇ × A perpendicular to graphene. (a) Find the eigenenergies n . Make the Ansatz a(x, y) = eikx x a(y), b(x, y) = eikx x b(y), eliminate a to obtain a second-order equation for b. Note that the harmonic oscillator equation [−∂y2 /(2m) + pc c 1 2 2 (y − y0 ) ]f (y) = Ef (y) has eigenenergies En = m [n + 2 ], n = 0, 1, 2, . . .. 2 2 ~ ~ 2 2 (b) Compute the LL energies in 2D electron gas, [ 2m ∗ (−i∂x − eBy/~) + 2m∗ (−i∂y ) + U0 ]ψ(x, y) = ψ(x, y), using the same approach as above. How do the results differ from graphene? (c) Calculate the degeneracy gn of graphene Landau levels in a sample of size Lx ×Ly . Do this by noting that: (i) the LL energy is independent of kx , (ii) kx is box-quantized as usual, kx = (2π/Lx )nx , nx ∈ Z, (iii) the oscillator location y0 , which gives the y coordinate around which the wave function is localized, must be inside the sample 0 < y0 < Ly . The degeneracy gn of LL n is then the number of allowed nx -values corresponding to the energy n , times 4 (spin and valley). (d) Calculate the coarse-grained density of states per volume, assuming the interval d contains a large number of n levels (= assume n is continuous variable). That is, N () = LgxnLy dn d . How does the result compare to the situation without the magnetic field? How does the density of states look like without the coarse-graining? The unusual LL structure of the Dirac Hamiltonian can be observed using a scanning tunneling miR croscope. The tunneling current is I(V ) = g dE N (E)[f (E − V ) − f (E)], so at zero temperature RV dI I(V ) = g 0 dE N (E) so that G(V ) = dV ∝ N (E = V ). You can check if you understand the results from the following article: http://www.nature.com/nphys/journal/v3/n9/full/nphys653.html (updated 2014-10-28 16:30:05)