Integer Quantum Hall effect - Semantic Scholar...Integer Quantum Hall effect • basics • theories...

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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

    Broadened LL due to 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 ad a

    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→

    • Do we need to worry about this phase?

    • 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

  • 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

    3 110

    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).

  • 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).

    Wid

    th o

    f a B

    loch

    ban

    d w

    hen

    B=0

    Landau subband

  • 集異璧

    著作:Douglas R. Hofstadter翻譯:郭維德

    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