Download Landau levels - UCSB Physics

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”