THE INTEGER QUANTUM HALL EFFECT · CONCLUSION • The Quantum Hall effect, due to a magnetic ﬁeld, crucially depends on disorder in a system. • The localized states don’t contribute

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)


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)


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


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/


PERCOLATION

V (x, y)

Localized and extended states

Width of states`B =

p~/(eB)

Correlation length of disorder� � `B


NEW PUZZLE

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


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


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


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)


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


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)


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.


Documents

THE INTEGER QUANTUM HALL EFFECT · CONCLUSION • The Quantum Hall effect, due to a magnetic ﬁeld, crucially depends on disorder in a system. • The localized states don’t contribute