Integer Quantum Hall effect basics theories for the quantization - - PowerPoint PPT Presentation

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 v m eE e B m dt c B Bz dv dt τ = − − × − = =

• *

*

/ / / /

x x y y

v E m eB c e v E eB c m τ τ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠⎝ ⎠ ⎝ ⎠

* 2 * 2

1 1

x x x c y y y c

m B E j j ne nec E j j B m nec ne ω τ τ ρ ω τ τ ⎛ ⎞ ⎜ ⎟ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ = = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ − ⎜ ⎟ ⎝ ⎠

ρxy B

j env = −

• *

* 2 , c

m n eB m c e ρ τ ω = =

( )

1 2 1 1

1 1 1 1 1 / /

c c

c c c c c

nec B nec B

ω τ ω τ

ω τ σ ω τ ω τ ω τ σ ω τ

− << >>

− ⎛ ⎞ = = ⎜ ⎟ + ⎝ ⎠ − ⎛ ⎞ ⎯⎯⎯ → ⎜ ⎟ ⎝ ⎠ − ⎛ ⎞ ⎯⎯⎯ → ⎜ ⎟ ⎝ ⎠ σ ρ

2 *

ne m τ σ =

• Hall conductivity
• Hall resistivity

Resistance and conductance

,

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

V R R I I V V R R I I V Σ Σ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ = = ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ Σ Σ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

Note:

det

yy xx

R Σ = Σ

So it’s possible to have Rxx and Σxx simultaneously be zero (provided Rxy and Σxy are nonzero).

3 : 2 :

y

y y xx xx yx yx x x I y xx xx yx yx x

V E W L W D R R A I J A A E W L D R R W J W ρ ρ ρ ρ

=

= ≡ = = = = =

L

W

x y

Quantum Hall effect

0, . 1 1 det det

xx yx yx yx yx yx yx yx

const R R R ρ ρ ρ σ ρ ρ = = → Σ = = = = =

Measurement of Hall resistance

2-dim electron gas (2DEG)

GaAs/AlGaAs heterojunction

(broadened) Landau levels in a magnetic field Energy 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, GaAs

heterojunction…)

→ 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

2 h e φ =

• Quantum Hall effect
• …

Often protected by topology, but not vice versa.

The triangle of quantum metrology QCP e

(to be realized)

I QHE V h / e 2 Josephson effect 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

Broadened LL due to disorder

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

• see Laughlin’s argument below

( )

B c

k T ω <

Filling factor Aoki, CMST 2011 The importance

• f localized

states

Width of extended states?

256 states in the LLL. ε(Φ) periodic in Φ0

Aoki 1983

• Finite-Size

Scaling

~ , 1/ 2

x

E N x ν

−

Δ =

ΔE ΔE

Huo and Bhatt PRL 1992 Exponent for correlation length Li et al PRL 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)

2

1 ( ) ( ) 2

i i e i e i

e H p A r V V r m c ⎛ ⎞ = + + ⎜ ⎟ ⎝ + ⎠

∑

• 1

( )

x x i i x y x y x y

e e j A r m L L i x c c H c L L A L HΦ − ∂ ⎡ ⎤ = + ⎢ ⎥ ∂ ⎣ ⎦ ∂ ∂ = − = − ∂ ∂Φ

∑

• x

x

A L Φ =

x y solve

| | H E ψ ψ

Φ Φ Φ Φ

>= >

By the Hellman-Feynman theorem, one has

| | | | x

y

H E H E c j L ψ ψ ψ ψ

Φ Φ Φ 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.

2

( )

y x y y

V n e j c e E L n h − = − = 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 LLs

Gapless excitations at the edges

• Robust against disorder

(no back-scattering)

• Classical picture

Chiral 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

2

Streda 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 edge

ˆ c M z = ∇ ×

Degeneracy of a LL: D=BA/Φ0

Current response: conductivity

• Vector potential of

an uniform electric field

2 2

1 ( ) 2 '

latt i t

e e H p A V H A p O A m c m c e H p A e m c

ω ω −

⎛ ⎞ = + + = + ⋅ + ⎜ ⎟ ⎝ ⎠ = ⋅

• 1

( ) ( ) ( ) , then ( ) ;

i t i t

A t E t c t i E t E e A t A e E A c

ω ω ω ω ω ω

ω

− −

∂ = − ∂ = = =

• 2

, ( )

m m m m m m m m m m

f f v v e iV v v

α α β α β α

σ ω ω ω ω ψ ω ω ω ψ − = + ≡ − ≡

∑

• 1st order perturbation in E →

Kubo-Greenwood formula

j E

α αβ β

σ =

1 = =

m m m m m

p m u u k u m u k k i

α α α α α

ε δ ω ∂ ∂ + ∂ ∂ ∂ +

• Quantization of Hall conductance

Thouless et al’s argument (1982)

3

2 2 2 2

1 2

nk nk nk nk DC m m m m m m nk nk

u u u u i p p p p e f im V e f V k k k k

α β α β α α β β β α

σ ω

≠

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

∑ ∑

• ℓ, m = (n, k)
• Berry curvature

( ) ( )

2 2

( , , are ( ( ) ) cyclic) 1 2

nk nk nk nk n z H n BZ n

e u u u u k k k i k d k k k h

γ α β β α

α β γ σ π ⎛ ⎞ ∂ ∂ ∂ ∂ Ω ≡ − ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ∂ ⎝ Ω ⎦ ⎠ ⎡ ⎤ = ⎢ ⎥ ⎣

∫

• Hall conductivity for the n-th band

cell-periodic function um an integer for a filled band ( ) ( )

n n n k k n k

k i u u A k Ω = ∇ × ∇ = ∇ ×

• ( )

n n n k

A k i u u ≡ ∇

• Berry curvature (for n-th band)
• Berry connection

2

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

BZ b 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 k k 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 n θ θ θ θ π = − ∇× + − =

∫

• Pf:

a b c d

Brillouin zone

• 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

2

1 intege ( r ) 2

B z Z

k k d n π Ω =

∫

• [

]

1 2 1 2 1 2 1 2

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) i a a b i b b c i d c d i a d a i a b d a a 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

2

( ) , ( ) 2

d

g L L g d σ β

−

= = −

• For large g (good conductor)
• For small g (insulator)

/

( ) , ( ) ln

L c c

g g L g e g g

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=<Ei+1-Ei> (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<<1) ～ sβ

β=1

From many nuclei

Fig from Altshuler’s ppt

Wigner-Dyson classes

GOE GUE GSE AI A AII

Altland- Zirnbauer classes

Quantization of magnetic 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 Shnir, Magnetic monopoles

Note:

2 2 2 / N S ig ig N S ieg c

A A ie e e

ϕ ϕ ϕ

ψ ψ

− ⋅

− = − ∇ =

• 2eg

n c =

Analogy in QH system

• Gauge transformation

Kohmoto, Ann. Phys, 1985

• Two atlases

, , ,

( , ; ) ( ) ( )

n n n

H r p x E x

λ λ λ

λ 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 i

H 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

, ( ) , ( ' ( ) ' )

T n

i dt E n t n T

e

λ λ

ψ ψ

− ∫

=

• Dynamical phase

Adiabatic evolution of a quantum system λ(t) E(λ(t))

( ) (0) T λ λ =

• x

x

n n+1 n-1

• Phases of the snapshot states at different λ’s

are independent and can be arbitrarily assigned

( , ( , ( ) ) )

n

n t n t i

e

λ λ γ λ

ψ ψ →

• Do we need to worry about this phase?
• Energy spectrum:

( , ; ) H r p λ

, , n n n

i

λ λ

γ ψ ψ λ λ ∂ = ⋅ ≠ ∂

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

No! Pf :

( ) ( ) H t i t t

λ λ

∂ Ψ = Ψ ∂

• '

( ') ( ) ,

( )

t n n

i dt E t i n

t e e

γ λ λ λ

ψ

−

Ψ = ∫

• Consider the n-th level,

Stationary, snapshot state

, , n n n

H E

λ λ

ψ ψ =

• ( )

, ,

'

n

n n i

e φ

λ λ λ

ψ ψ =

• n

φ λ ∂ = − ∂

• Choose a φ (λ) such that,

Redefine the phase,

Thus removing the extra phase

An’(λ) An(λ) An’(λ)=0

• One problem:

( ) A

λφ

λ ∇ =

• does not always have a

well-defined (global) solution.

C A d λ

⋅ =

∫

• C A d λ

⋅ ≠

∫

• Vector flow A
• Contour of φ

C Vector flow Contour of φ

A

• φ is not

defined here C

' ( ') ( ) (0)

C T

i dt t T i E

e e

γ λ λ

ψ ψ

− ∫

=

• C

C

i d

λ λ

γ ψ ψ λ λ ∂ = ⋅ ≠ ∂

∫

• Berry phase (path dependent)
• M. Berry, 1984 :
• Parameter-dependent phase

NOT always removable! Index n neglected

• Berry connection (or Berry potential)
• Berry curvature (or Berry field)

( ) A i

λ λ λ

λ ψ ψ ≡ ∇

• ( )

( ) F A i

λ λ λ λ λ

λ λ ψ ψ ≡ ∇ × = ∇ × ∇

• C

C S

A d A da

λ

γ λ = ⋅ = ∇ × ⋅

∫ ∫

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

( )

( ) ( ) ( ) ( )

i C C

e A A F F

φ λ λ λ λ

ψ ψ λ λ φ λ λ γ γ → → −∇ → →

• i

i

• i

i

Redefine the phases of the snapshot states Berry curvature and Berry phase are gauge invariant

2

λ

3

λ

1

λ

( ) t λ

• C

S Some terminology

λ→ k in QHS

spin × solid angle

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

B B

H B

λ

μ σ

= =

⋅

• y

z x

( ) B t

• C

S

• Real space
• Parameter space

Berry curvature

a monopole at the origin

Berry phase

S

1 = ( ) 2 F da C γ ±

± ⋅

= Ω

∫

• ∓

y

B

z

B

x

B

( ) B t

• C

E(B) B

Level crossing at B=0 − +

2 , ,

ˆ 1 ( ) 2

B B B B

B F B i B ψ ψ

± ± ±

= ∇ × ∇ =

• ∓

Examples of the Berry phase:

Magnetic monopole / Berry phase / fiber bundle

( ) A k i

λ λ λ

ψ ψ ≡ ∇

• in parameter space

Berry connection Berry curvature (in 3D)

( ) ( ) F A

λ

λ λ ≡ ∇ ×

• S

( ) =

C C A

d F da γ λ λ = ⋅ ⋅

∫ ∫

• Berry phase

Total curvature 1 ( ) integer 2 F da λ π ⋅ =

∫

• 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 ( ) integer 4 B r da π ⋅ =

∫

• 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≠0 K≠0 G=0

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

G>0 G=0 G<0

• Positive and negative

Gaussian curvature

外在 內在

• Berry phase ≒ anholonomy angle in differential geometry
• Berry curvature ≒Gaussian curvature

2

1 lim

A

A R α

→

=

• Gaussian curvature G ≣
• anholonomy angle α= 內角和-180

Euler characteristic

歐拉特徵數

2 ( ), 2(1 )

M da G

M g πχ χ = = −

∫

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

g = 1 g = 2 g =

The most beautiful theorem in differential topology

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

( )

,

2

g M M

M M

da G ds k π χ

∂

∂

+ =

∫ ∫

Marder, Phys Today, Feb 2007

• Can be generalized to higher dimension.

• Nontrivial fiber bundle

Möbius band

• Trivial fiber bundle

(a product space R1 x R1)

Simplest examples: R1 R1

base 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

Hofstadter spectrum

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 1 10 10 29 1 3 1 87 − = = +

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).

Hofstadter’s butterfly (Hofstadter, PRB 1976)

• 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).

Width of a Bloch band when B=0 Landau subband

集異璧 著作：Douglas R. Hofstadter 翻譯：郭維德

Pulitzer 1980

MIT: http://ocw.mit.edu/high-school/courses/godel-escher-bach/

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 r

r 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 / 5 2 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) 2(0) 5(0)

r r

p q r t s = = + = + = + = − + = + = − + = +

See Xiao et al RMP 2010 for another derivation

1 /

H tot tot plaq plaq H r

n ec B N r r n A q A q BA p q 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-crossing

Lee, Chang, and Hong, PRB 1998

Φ=2/5 Lattice with edges

• Energy dispersion of edge states

Hatusgai, J Phys 1997