Landau levels

• Simplest case: “free” 2d electrons in a magnetic field (applies to electrons in a semiconductor 2DEG)

• Hamiltonian

• Choose kx eigenstate

H =1

2m(p+ eA)2 A = Byx̂

�(x, y) = eikx

xY (y)

Landau levels

• One obtains

• This is a 1d simple harmonic oscillator with a frequency and center

�c =eB

ccyclotron frequency magnetic length

� =

r~eB

1

2m

�~2 d2

dy2+ (eB)2

✓y � ~k

x

eB

◆2!Y = �Y

y0 =~k

x

eB= k

x

�2

Landau levels

• Energy levels = Landau levels are

• Each is highly degenerate due to independence of energy on kx

• How many?

�n = ~⇥c(n+ 12 )

Lx

Ly

i = 0, 1, 2, · · ·

0 < i <Lx

Ly

2�⇥2N =

A

2�⇥2

kx

=2�

Lx

i,

0 < y0 = kx

�2 < Ly

n = 0, 1, 2, · · ·

Landau levels

• Degeneracy

• Flux quantum

• This is basically the number of minimal quantized cyclotron orbits which fit into the sample area

N =A

2�⇤2= AB

e

h=

�

⇥

� = h/e ⇥ 4� 10�15T ·m2

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 levelsH = v~� · (~p+ e ~A)

= �i~v~� · (~r+ ie

~~A)

= ~v✓

0 �i@x

� @y

+ eB

~ y�i@

x

+ @y

+ eB

~ y 0

◆

H = E (x, y) = e

ik

x

x

�(y)

~v✓

0 kx

� @y

+ eB

~ ykx

+ @y

+ eB

~ y 0

◆�(y) = E�(y)

Dirac LLs

y/`

~v`

✓0 k

x

`� @y/`

+ eB

~ y`kx

`+ @y/`

+ eB

~ y` 0

◆�(y) = E�(y)

�(y) = �(y/`+ kx

`)

` =

r~eB

!c =p2v/`

~!c

0 1p

2(�@⇠ + ⇠)

1p2(@⇠ + ⇠) 0

!�(⇠) = E�(⇠)

Dirac LLs~!c

✓0 a†

a 0

◆�(⇠) = E�(⇠)

a =1p2(@⇠ + ⇠)

a† =1p2(�@⇠ + ⇠)

[a, a†] = 1 N = a†a

N |ni = n|ni a|0i = 0 etc.

� =

✓|0i0

◆ ✓0 a†

a 0

◆� =

✓0

a|0i

◆= 0

Zero energy state: lives entirely on “A” sublatticeFor the K’ point it lives on the B sublattice

Dirac LLs~!c

✓0 a†

a 0

◆�(⇠) = E�(⇠)

More general state � =

✓|ni

c|n� 1i

◆

~!c

✓a†c|n� 1i

a|ni

◆= E

✓|ni

c|n� 1i

◆

~!c

✓cpn|nip

n|n� 1i

◆= E

✓|ni

c|n� 1i

◆

c = ±1 E = ±~!cpn

Relativistic vs NR LLsE

0

!c/2

Fermi level is “in the middle” of 0th

LL in undoped graphene

This is because there are a lot of electrons in graphene: 1 per C atom, filling the “negative” energy LLs

Fermi in a semiconductor 2DEG is usually

“high”

A semiconductor 2DEG is formed by doping electrons into the conduction band.

Edge states• A simple way to understand the

quantization of Hall effect, realized by Halperin

• Consider Hall bar

y

V(y)

x

y

Edge states

y

V(y)

If V(y) is slowly varying, then we can approximate

� ~22m

d2

dy2+

1

2m⇥2

c

�y � k

x

⇤2�2

+ V (y)

�Y = �Y

V (y) ⇡ V (kx

�2)

�n

⇡ ~⇥c

(n+ 12 ) + V (k

x

⇤2)

Edge states

y

V(y)

kx

Low energy states at the edges of the system

�n

⇡ ~⇥c

(n+ 12 ) + V (k

x

⇤2)

Edge states

kx• Near the edge, we can linearize the energy

• This describes “right and left-moving chiral fermions” = edge states

R

L

y

x

�n

⇡ ~⇥c

(n+ 12 ) + V (k

x

⇤2)

kx

= ±Kn

+ qx

�n

� �F

± vn

qx

Edge states

• Corresponds to semi-classical “skipping orbits”