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Landau levels • Simplest case: “free” 2d electrons in a magnetic field (applies to electrons in a semiconductor 2DEG) • Hamiltonian 1 H= 2m • Choose k x (p + eA) 2 eigenstate (x, y) = eikx x Y (y) A = Byx̂ Landau levels • One obtains 1 2m ✓ 2 d ~2 2 + (eB)2 y dy ~kx eB ◆2 ! Y = Y • This is a 1d simple harmonic oscillator with a frequency and center c eB = c cyclotron frequency ~kx y0 = = kx eB 2 r ~ = eB magnetic length Landau levels • Energy levels = Landau levels are n = ~⇥c (n + 12 ) • Each is highly degenerate due to n = 0, 1, 2, · · · independence of energy on kx • How many? Ly Lx 2 kx = i, i = 0, 1, 2, · · · Lx 0 < y0 = kx 2 < Ly A Lx Ly N= 0<i< 2 ⇥2 2 ⇥2 Landau levels • Degeneracy A e N= = AB = 2 2 ⇤ h ⇥ • Flux quantum = h/e ⇥ 4 10 15 T · m2 • This is basically the number of minimal quantized cyclotron orbits which fit into the sample area Dirac Landau Levels • We saw that Schrödinger electrons form Landau levels with even spacing. • It turns out Dirac electrons also form Landau levels but with different structure • We can just follow the treatment in the graphene RMP Dirac Landau levels ~ H = v~ · (~ p + eA) e~ ~ = i~v~ · (r + i A) ~ = ~v H ~v 0 i@x + @y + eB ~ y i@x @y + 0 eB ~ y ◆ (x, y) = eikx x (y) =E ✓ ✓ 0 kx + @ y + eB ~ y kx @y + 0 eB ~ y ◆ (y) = E (y) Dirac LLs `= ~v ` ✓ 0 kx ` + @y/` + eB ~ y` kx ` @y/` + 0 eB ~ y` ◆ r (y) = E (y) y/` (y) = (y/` + kx `) ~!c 0 p1 (@⇠ + ⇠) 2 p1 2 ( @⇠ + ⇠) 0 !c = ! p ~ eB 2v/` (⇠) = E (⇠) ~!c ✓ † 0 a a 0 ◆ Dirac LLs 1 a = p (@⇠ + ⇠) 2 1 † a = p ( @⇠ + ⇠) 2 (⇠) = E (⇠) [a, a† ] = 1 N = a† a N |ni = n|ni = ✓ |0i 0 ◆ ✓ 0 a a|0i = 0 † a 0 ◆ = ✓ Zero energy state: lives entirely on “A” sublattice 0 a|0i etc. ◆ =0 For the K’ point it lives on the B sublattice Dirac LLs ~!c ✓ 0 a † a 0 ◆ (⇠) = E (⇠) More general state ✓ † = ◆ ✓ |ni c|n 1i ✓ ◆ ◆ |ni a c|n 1i ~!c =E c|n 1i a|ni ✓ ◆ ✓ ◆ p c n|ni |ni ~!c p =E n|n 1i c|n 1i c = ±1 p E = ±~!c n Relativistic vs NR LLs A semiconductor 2DEG is formed by doping electrons into the conduction band. E Fermi in a semiconductor 2DEG is usually “high” 0 Fermi level is “in the middle” of 0th LL in undoped graphene !c /2 This is because there are a lot of electrons in graphene: 1 per C atom, filling the “negative” energy LLs Edge states • A simple way to understand the quantization of Hall effect, realized by Halperin • Consider Hall bar V(y) y x y Edge states V(y) y ~2 d 2 1 2 + m⇥ c y 2 2m dy 2 kx ⇤ 2 2 + V (y) Y = Y If V(y) is slowly varying, then we can approximate V (y) ⇡ V (kx 2 ) n ⇡ ~⇥c (n + 12 ) + V (kx ⇤2 ) Edge states n ⇡ ~⇥c (n + 12 ) + V (kx ⇤2 ) V(y) y kx Low energy states at the edges of the system Edge states n ⇡ ~⇥c (n + 12 ) + V (kx ⇤2 ) • Near the edge, we can linearize the energy kx = ±Kn + qx n F ± v n qx • This describes “right and left-moving chiral fermions” = edge states y x R L kx Edge states • Corresponds to semi-classical “skipping orbits”