THE INTEGER QUANTUM HALL EFFECT

Justin H. Wilson

Wednesday, September 18, 13

QUANTUM HALL EXPERIMENTK. V. Klitzing, G. Dorda, and M. Pepper PRL 45, 494 (1980)

Wednesday, September 18, 13

THE BASICS

•Magnetic Field

• Charge

• Effective Mass

• g-factor

Two scales are important:

Cyclotron frequency

Magnetic length

B

e

m

g⇤

!c = eB/m

`B =p

~/(eB)

Wednesday, September 18, 13

THE HALL EFFECT IN A PURE SYSTEM

E = (n+ 1/2)~!c ± 12g

⇤µBB

Ne = nA, N� = BA/(h/e)

⌫ =Ne

N�=

nh

eB

vdrift = E/B, I = envdriftw, VH = Bvdriftw

Landau Levels

Filling fraction

Hall Resistance

RH =VH

I=

B

ne=

1

⌫

h

e2

Wednesday, September 18, 13

LANDAU LEVELS AND DISORDER

⇢(E)

ELocalized states

Extended states

Plot (right):Advanced Quantum Mechanics II PHYS 40202 : taken from

- http://oer.physics.manchester.ac.uk/AQM2/Notes/Notes-4.4.htmlhttp://creativecommons.org/licenses/by-nc-

sa/3.0/

Wednesday, September 18, 13

PERCOLATION

V (x, y)

Localized and extended states

Width of states`B =

p~/(eB)

Correlation length of disorder� � `B

Wednesday, September 18, 13

NEW PUZZLE

• If we have reduced the number of current carrying states, how is the Hall resistance not affected?

Wednesday, September 18, 13

EDGE STATES

S D

y

x

Empty Landau level

µD

y

V(y)

VSDµS � µD = eVSD

I =ev

L⇥ L(µS � µD)

hv=

e2

hVSD

Current per state Difference in states on top and bottom

Wednesday, September 18, 13

FLUX INSERTION

IV

�+ �̃(t) V = �d�̃

dt

I = �xy

V

t

�0 = h/e

�tot

(t)

Change of one flux quantum returns us to initial state

integer⇥ e = Q = �xy

�� = �xy

h

e

�xy

= integer⇥ e2

h

Wednesday, September 18, 13

CHERN NUMBER

⇥

�(t)H =

1

2

"✓�i~@

x

+e⇥

Lx

◆2

+

✓�i~@

y

+ eBx+e�

Ly

◆#+ V (x, y)

(x, y + Ly) = (x, y)

(x+ L

x

, y) = e

iyL

x

/`

2B (x, y)

jx

Lx

=@H

@⇥

hIx

i = @⇥E � i~[h@⇥ |@� i � h@� |@⇥ i]@t�

�xy

�̄xy

= � i~(h/e)2

ZZh/e

0d� d⇥r⇥ v =

e2

h

1

2⇡i

Iv · dl

Integer

TKNN, PRL 49, 405 (1982)

Wednesday, September 18, 13

SCALING FLOW DIAGRAMg(L) =

h

e2�xx

(L)Ld�2

�(g) =@ ln g

@ lnL

Anderson insulator (good) Metal

g(L) / Ld�2, �(g) = d� 2g(L) / e�L/⇠, �(g) < 0

�(g)

ln(g)

d = 3

d = 2

d = 1

g⇤d > 2

Insulator Metal

Wednesday, September 18, 13

QUANTUM HALL SCALING FLOW

Hall conductance adds another parameter that can “flow”

H. Levine, S. B. Libby, and A. M. M. Pruisken, PRL 51, 1915 (1983)

D. E. Khmel’nitzkii, JETP Lett. 38, 552 (1983)

Plateau transition

Quantum Hall Plateau

(e2/h)

Wednesday, September 18, 13

CONCLUSION

• The Quantum Hall effect, due to a magnetic field, crucially depends on disorder in a system.

• The localized states don’t contribute to conductance, and the quantization can be found with either :

• Edge states

• Flux insertion

• Chern numbers

• Scaling flow diagrams due of conductance provide a description of the Quantum Hall transition with disorder.

Wednesday, September 18, 13