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Integer Quantum Hall effect • basics • theories for the quantization • disorder in QHS • Berry phase in QHS • topology in QHS • effect of lattice • effect of spin and electron interaction M.C. Chang Dept of Phys
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### Transcript of Integer Quantum Hall effect - Semantic Scholar...Integer Quantum Hall effect • basics • theories...

• Integer Quantum Hall effect• basics

• theories for the quantization

• disorder in QHS

• Berry phase in QHS

• topology in QHS

• effect of lattice

• effect of spin and electron interaction

M.C. Chang

Dept of Phys

• Hall effect (1879), a classical analysis

* *

ˆ; / 0 at steady state

dv v vm eE e B mdt c

B Bz dv dtτ

= − − × −

= =

*

*

/ // /

x x

y y

v Em eB ce

v EeB c mτ

τ⎛ ⎞ ⎛ ⎞⎛ ⎞

= −⎜ ⎟ ⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠

*

2

0*

2

11

x x xc

y y yc

m BE j jne necE j jB m

nec ne

ω ττ ρω τ

τ

⎛ ⎞⎜ ⎟⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟= =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟−⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠−⎜ ⎟⎝ ⎠

ρxy

B

j env= −*

0 *2 , cm

neBm ce

ρτ

ω ==

( )1 0

2

10

1

111

1

1

0 // 0

c

c

c

cc

c

c

nec Bnec B

ω τ

ω τ

ω τσω τω τ

ω τσ

ω τ

>

−⎛ ⎞= = ⎜ ⎟

+ ⎝ ⎠

−⎛ ⎞⎯⎯⎯→ ⎜ ⎟

⎝ ⎠−⎛ ⎞

⎯⎯⎯→ ⎜ ⎟⎝ ⎠

σ ρ2

0 *

nemτσ =

• Hall conductivity

• Hall resistivity

• Resistance and conductance

,x xx xy x x xx xy xy yx yy y y yx yy y

V R R I I VV R R I I V

Σ Σ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟Σ Σ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

Note:

detyy

xxRΣ

So it’s possible to have Rxx and Σxxsimultaneously be zero (provided Rxy and Σxyare nonzero).

0

3 :

2 :

y

y yxx xx yx yx

x xI

yxx xx yx yx

x

V E WL WD R RA I J A A

E WLD R RW J W

ρ ρ

ρ ρ

=

= ≡ = =

= = =

L

Wx

y

Quantum Hall effect 0, .

1 1det det

xx yx

yx yxyx yx

yx yx

const

RR R

ρ ρ

ρσ

ρ ρ

= =

→ Σ = = = = =

• Measurement of Hall resistance

2-dim electron gas (2DEG)

• GaAs/AlGaAs heterojunction

(broadened) Landau levels in a magnetic field

Ener

gy

μ

subband

Dynamics along z-direction is frozen in the ground state

• Ando, Matsumoto, and Uemura JPSJ 1975

Effect of disorder on σxy (theoretical prediction before 1980)

Kawaji et al, Supp PTP 1975

Si(100) MOS inversion layer

9.8 T, 1.6 K

• 1985

ρxy deviates from (h/e2)/n by less than 3 ppm on the very first report.• This result is independent of the shape/size of sample. • Different materials lead to the same effect (Si MOSFET, GaAsheterojunction…)

→ a very accurate way to measure α-1 = h/e2c = 137.036 (no unit)→ a very convenient resistance standard.

Quantum Hall effect (von Klitzing, 1980)

• An accurate and stable resistance standard (1990)

Kinoshita, Phys. Rev. Lett. 1995• theory

• experiment

• Condensed matter physics is physics of dirt - Pauli

dirty clean

• Flux quantization

0 2he

φ =

• Quantum Hall effect

• …

Often protected by topology, but not vice versa.

• The triangle of quantum metrology

QCP

e

(to be realized)

I

QHEV h / e 2

Josephsoneffect

f

e / h

• Quantum Hall effect requires • Two-dimensional electron gas• strong magnetic field • low temperature Note: Room Temp QHE in graphene (Novoselov et al, Science 2007)

Plateau and the importance of disorder

Why RH has to be exactly (h/e2)/n ?

• see Laughlin’s argument below

( )B ck T ω<

Filling factor

Aoki, CMST 2011

The importance of localized states

• Width of extended states?

256 states in the LLL. ε(Φ) periodic in Φ0Aoki 1983

• • Finite-Size Scaling

~ , 1/ 2xE N x ν−Δ =

ΔE ΔE

Huo

and Bhatt P

RL 1992

Exponent for correlation length

Li et al PR

L 2005

• experiment Ensemble average over 100-2000 disorder configurations

States that can carry Hall current (with non-zero Chern number)

• Quantization of Hall conductance, Laughlin’s gauge argument (1981)21 ( ) ( )

2 i i eiei

eH p A r V Vrm c⎛ ⎞= + +⎜ ⎟⎝

+⎠

1 ( )

x x iix y

x y x y

e ej A rm L L i x c

c H cL L A L

− ∂⎡ ⎤= +⎢ ⎥∂⎣ ⎦

∂∂= − = −

∂ ∂Φ

x xA LΦ =

x

y

solve | |H Eψ ψΦ Φ Φ Φ>= >By the Hellman-Feynman theorem, one has

| | | |

xy

H EH

EcjL

ψ ψ ψ ψΦ ΦΦ Φ Φ Φ Φ

Φ

∂ ∂∂< >= < >=

∂Φ ∂Φ ∂Φ∂

∴ = −∂Φ

• EF at localized states, no charge transfer whatever Φ is.

• EF at extended states, only integer charges may transfer along y when Φ is changed by one Φ0. 0

2( ) yx y

y

Vn ej c e EL

nh

−= − =

Φ

• Due to gauge symmetry, the system needs to be invariant under Φ→ Φ+ Φ0,

• Simulate a longitudinal EMF by a fictitious time-dependent flux Φ

1

• Edge state in quantum Hall system

• Bending of LLsGapless excitations at the edges

• Robust against disorder (no back-scattering)

• Classical pictureChiral edge state (skipping orbit)

• number of edge modes = n

• Inclusion of lattice (more details later)

• Bulk states: En(kx,ky) (projected to ky); Edge states: En(ky)

• when the flux is changed by 1 Φ0, the states should come back.

→ Only integer charges can be transported.

Figs from Hatsugai’s ppt

• 2Streda formula (1982)

Giuliani and Vignale, Sec 10.3.3

• If ν bands are filled, then the number of electrons per unit area is n=νeB/hc

∴ σH=νe2/h

L

R

Nonzero along edgeˆc M z= ∇ ×

Degeneracy of a LL: D=BA/Φ0

• Current response: conductivity

• Vector potential of an uniform electric field

22

00 0

0

1 ( )2

'

latt

i t

e eH p A V H A p O Am c m ceH p A e

m cω

ω−

⎛ ⎞= + + = + ⋅ +⎜ ⎟⎝ ⎠

= ⋅

1 ( )( )

( ) , then ( ) ;i t i t

A tE tc t

iE t E e A t A e E Ac

ω ωω ω ω ω

ω− −

∂= −

= = =

2

,

( ) m m m

m m m

m m m

m

f f v veiV

v v

α

α

β

α

β

ασ ω ω ω

ω ψ

ω

ω ω ψ

−=

+

≡ − ≡

• 1st order perturbation in E →

Kubo-Greenwood formula

j Eα αβ βσ=

• 00

1=

=

m

m m

m

m

pm

u

u k um

uk k

i

α

α

α

α

α

ε δ ω∂ ∂+∂ ∂

∂ +

Quantization of Hall conductanceThouless et al’s argument (1982)

3

2

2 20

2

1

2 nk nk nk nk

DC m m m m

m m

nknk

u u u ui

p p p pe fim V

e fV k k k k

α β

α β

α

α β

β

β

α

σω≠−

=

⎛ ⎞= ⎜ ⎟⎜ ⎟

∂ ∂ ∂ ∂−

∂ ∂ ∂ ∂⎝ ⎠

ℓ, m = (n, k)

• Berry curvature

( ) ( )2

2

( , , are

(

( )

)

cyclic)

1

2

nk nk nk nk

n zH nBZ

n

e

u u u ukk k

ik

d k k

k

h

γα β β α

α β γ

σπ

⎛ ⎞∂ ∂ ∂ ∂Ω ≡ −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝

Ω⎦

⎡ ⎤= ⎢ ⎥

⎣∫

• Hall conductivity for the n-th band

cell-periodic function um

an integer for a filled band

( )

( )

n n nk k

nk

k i u u

A k

Ω = ∇ × ∇

= ∇ ×

( )n n nkA k i u u≡ ∇

• Berry curvature (for n-th band)

• Berry connection

• 2

( ,0) ( , ) ( , ) (0, )

BZb c d a

a b c d

x x x x x y y y x y y y

d k

dk A dk A dk A dk A

dk A k A k g dk A g k A

A

k→ ↑

= ⋅ + ⋅ + ⋅ + ⋅

⎡ ⎤ ⎡ ⎤= − + −⎣ ⎦

×

∫ ∫ ∫ ∫∫ ∫

1 2( ) ( )ˆ ˆ

2 2

,

( ,0) ( , ) ( ) ( )

etc

y x

x y

i k i kk k g x k k g y

x x x x x y

u e u u e u

dk A k A k g a b

θ θ

θ θ

+ +

= =

⎡ ⎤− = −⎣ ⎦∫

2

2 2 1 1( ) ( ) ( ) ( )2

BZ

d k

a b d a

A

nθ θ θ θπ

= −

∇×

+ −=

Pf:a b

cd

Brillouinzone

• Niu-Thouless-Wu generalization to system with disorder and electron interaction (PRB 1985).

BZ

Zeros and vortices

total vorticity in the BZ

Czerwinski and Brown, PRS (London) 1991

gx

gy

21 intege( r )2 B

zZ

k kd nπ

Ω =∫

[ ]

1

2

1

2

1 2 1 2

( )

( )

( )

( )

( ) ( ) ( ) ( )

i aa b

i bb c

i dc d

i a b d aa a

u e u

u e u

u e u

u e u

u e u

θ

θ

θ

θ

θ θ θ θ

+ − −

=

=

=

=

∴ =

• Connection with localization in disordered system (Anderson, 1958)

• one-parameter scaling hypothesis(Abrahams et al, 1979 < Thouless, Landauer…): assumeβ(g) depends only on g

Localized

extended

Quasi-extended

20( ) , ( ) 2

dg L L g dσ β−= = −

• For large g (good conductor)

• For small g (insulator)/( ) , ( ) lnLc

c

gg L g e gg

ξ β−= =

Lagendijk et al, Phys Today 2009

• All wave functions of disordered systems in 1D and 2D are localized.

• QHE belongs to a new class of disordered systems.

This analysis does not apply to the QHS, since the extended states are crucial there.

Flow follows the increase of L

MIT

conductance

• Fig from Altshuler’s ppt

Spectral distribution of random matrix (rank N>>1)• eigenvalues Ei• mean level spacing d1= (taking ensemble average)

• spacing between NN s=(Ei+1-Ei)/d1• P(s): distribution function of s

• spectral rigidity: P(0)=0

• level repulsion: P(s

• Fig from Altshuler’s ppt

Wigner-Dyson classes

GOE

GUE

GSE

AI

A

AII

Altland-Zirnbauerclasses

• Quantization ofmagnetic monopole (see Sakurai Sec 2.6)

• Vector potential (use 2 “atlas” to avoid Dirac string)

• gauge transformation between 2 atlas

→ monopole charge is quantized

YM

Shn

ir, M

agne

tic m

onop

oles

Note:

2 2

2 /

N S ig ig

N S ieg c

A A ie ee

ϕ ϕ

ϕψ ψ

− = − ∇

=

2eg nc=

• Analogy in QH system

• Gauge transformation

Kohmoto, Ann. Phys, 1985

• Two atlases

• , , ,( , ; ) ( ) ( )n n nH r p x E xλ λ λλ ψ ψ=

• Fast variable and slow variable

• “Slow variables Ri” are treated as parameters λ(t)

(Kinetic energies from Pi are neglected)

• solve time-independent Schroedinger eq.

“snapshot” solution

{ }( , ; , )i iH r p R P

electron; {nuclei}

Born-Oppenheimer approximation

e－

H+2 molecule

nuclei move thousands of times slower than the electron

Instead of solving time-dependent Schroedinger eq., one uses

Connection with Berry phase First, a brief review of Berry phase:

• • After a cyclic evolution

0

, ( ) , (

'

0

(

)

' )T

ni dt E

n

t

n T eλ λψ ψ− ∫=

Dynamical phase

Adiabatic evolution of a quantum system

0 λ(t)

E(λ(t)) ( ) (0)Tλ λ=

xx

n

n+1

n-1

• Phases of the snapshot states at different λ’sare independent and can be arbitrarily assigned

(, ( , ( )

))

nn t n t

ieλ λγ λψ ψ→

• Energy spectrum:

( , ; )H r p λ

• , , 0n n ni λ λγ ψ ψ λλ∂

= ⋅ ≠∂

≣An(λ)

• Fock, Z. Phys 1928• Schiff, Quantum Mechanics (3rd ed.) p.290

No!

Pf :

( ) ( )H t i ttλ λ∂

Ψ = Ψ∂

0' ( ')( )

,( )t

nni dt E ti

nt e eγ λ

λ λψ−

Ψ = ∫

Consider the n-th level,

Stationary, snapshot state

, ,nn nH Eλ λψ ψ=

( ), ,'

nn n

ie φ λλ λψ ψ=

nφλ

∂= −

∂Choose a φ (λ) such that,

Redefine the phase,

Thus removing the extra phase

An’(λ) An(λ)

An’(λ)=0

• • One problem: ( )Aλφ λ∇ =does not always have a well-defined (global) solution.

0C

A d λ⋅ =∫ 0C A d λ⋅ ≠∫

Vector flow A

Contour of φ

C

Vector flow

Contour of φ

A φ is not defined here

C

0' ( ')

( ) (0)C

Ti dt t

Ti Ee eγλ λψ ψ

− ∫=

0C C i dλ λγ ψ ψ λλ∂

= ⋅ ≠∂∫

• Berry phase (path dependent)

M. Berry, 1984 : • Parameter-dependent phase NOT always removable!

Index nneglected

• • Berry connection (or Berry potential)

• Berry curvature (or Berry field)

( )A i λλ λλ ψ ψ≡ ∇

( ) ( )F A iλ λ λλ λλ λ ψ ψ≡ ∇ × = ∇ × ∇

C C SA d A daλγ λ= ⋅ = ∇ × ⋅∫ ∫

• Stokes theorem (3-dim here, can be higher)

• Gauge transformation

( )

( ) ( )

( ) ( )

i

C C

e

A A

F F

φ λλ λ

λ

ψ ψ

λ λ φ

λ λγ γ

→ −∇

→→

i

i

ii

Redefine the phases of the snapshot states

Berry curvature and Berry phase are gauge invariant

( )tλ

C

S

Some terminology

λ→ k in QHS

• spin × solid angle

Example: spin-1/2 particle in slowly changing B field

BBH Bλ μ σ= = ⋅

y

z

x( )B t

CS

• Real space • Parameter space

Berry curvature

a monopole at the origin

Berry phase

S

1= ( )2

F da Cγ ± ± ⋅ = Ω∫ ∓

yB

zB

xB( )B t

C

E(B)

B

Level crossing at B=0

+

2, ,

ˆ1( )2B BB B

BF B iB

ψ ψ± ± ±= ∇ × ∇ = ∓

• Examples of the Berry phase:

• Magnetic monopole / Berry phase / fiber bundle

( )A k i λ λ λψ ψ≡ ∇

in parameter space

Berry connection

Berry curvature (in 3D)

( ) ( )F Aλλ λ≡ ∇ ×

S

( )

=

C CA d

F da

γ λ λ= ⋅

∫∫

Berry phase

Total curvature

1 ( ) integer2

F daλπ

⋅ =∫

in real space

Vector potential

Magnetic field

( ) ( )B r A r≡ ∇×

( )A r

( )

=C

S

A r dr

B da

Φ = ⋅

∫∫

Magnetic flux

Monopole charge

1 ( ) integer4

B r daπ

⋅ =∫

connection

curvature

1st Chern number

horizontal lift (along a U(1) fiber)

U(1) fiber bundle

A

F

γ

C1

(QHE: λ→ k in BZ)

• Connection with geometry First, a brief review of topology:

K≠0

G≠0K≠0

G=0

• extrinsic curvature K vs• intrinsic (Gaussian) curvature G

G>0

G=0

G

• Euler characteristic 歐拉特徵數

2 ( ), 2(1 )M

da G M gπχ χ= = −∫

• Gauss-Bonnet theorem (for a 2-dim closed surface)

0g = 1g = 2g =

The most beautiful theorem in differential topology

• Gauss-Bonnet theorem (for a surface with boundary)

( ),2gM M M Mda G ds k π χ∂ ∂+ =∫ ∫

Marder, Phys Today, Feb 2007

• Can be generalized to higher dimension.

• • Nontrivial fiber bundleMöbius band

• Trivial fiber bundle(a product space R1 x R1)

Simplest examples:

R1

R1base

fiber

• Fiber bundle ~ base space × fiber space

Fiber bundle: a generalization of product space

• In physics, a fiber bundle ~ Physical space × Inner space

• In QHS, we have T2 x U(1)

• The topology of a fiber bundle is classified by Chern numbers ～the topology of a closed surface is classified by Euler characteristics

(spin, gauge field…)

base space

fiber space

• Lattice electron in a magnetic field: magnetic translation symmetry

consider a uniform B field

Indep of r• Magnetic translation operator

Commute if this is 1

Xiao et al, RMP 2010

B

a1

a2

• Simultaneous eigenstates: magnetic Bloch states

• If Φ=(p/q)Φ0 per plaquette, then Magnetic Brillouin zone = BZ/q.

e.g., p/q=1/3

Band structure of a 2DEG subjects to both a periodic potential V(x,y) and a magnetic field B.

Can be studied using the tight-binding model (TBM).

B The tricky part:

q=3 → q=29 upon a small change of B!Also, when B → 0, q can be very large.

1

3 110

1029

13

187−

= = +

Surprisingly complex spectrum!Split of energy band depends on flux/plaquette.If Φplaq/Φ0= p/q, where p, q are co-prime integers, then a Bloch band splits to q subbands (for TBM).

• A fractal spectrum with self-similarity structure

Self-similarity (heirarchy)

B → 0 near band button, evenly-spaced LLs

• The total band width for an irrational q is of measure zero (as in a Cantor set).

Wid

th o

f a B

loch

ban

d w

hen

B=0

Landau subband

• 集異璧

Pulitzer 1980M

IT: h

ttp://

ocw

.mit.

edu/

high

-sch

ool/c

ours

es/g

odel

-esc

her-b

ach/

• C1 = 1

C2 = −2

C3 = 1

Bloch energy E(k) Berry curvature Ω(k) p/q=1/3

• Distribution of Hall conductance among subbands(Thouless et al PRL 1982)

r rr pt qs= +• Diophantine equation

• for rectangular lattice: sr should be as small as possible

• for triangular lattice: sr and tr cannot both be odd(Thouless, Surf Sci 1984)

e.g., / 2 / 52 5

5 2(0) 5(1)4 2(2) 5(0)3 2( 1) 5(1)2 2(1) 5(0)1 2( 2) 5(1)0 2(0) 5(0)

r r

p qr t s

== += += += − += += − += +

See Xiao et al RMP 2010 for another derivation

1

/

H

tot

tot plaq

plaq

H r

necB

N r rnA q A q

BApq hc e

t

σ

σ

∂=

= =

=

∴ =

• Streda formula

• for weak magnetic field: (σH)r = tr - tr-1• for strong magnetic field: (σH)r = sr - sr-1

• Jump of Hall conductance induced by band-crossingLee, Chang, and Hong, PRB 1998

• Φ=2/5

Lattice with edges

• Energy dispersion of edge states

Hatusgai, J Phys 1997