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Interacting topological insulators out of equilibrium . Dimitrie Culcer. D. Culcer, PRB 84, 235411 (2011) D. Culcer, Physica E 44, 860 (2012) – review on TI transport . Outline. Introduction to topological insulators

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Interacting topological insulators out of equilibrium

Interacting topological insulators out of equilibrium

Dimitrie Culcer

D. Culcer, PRB 84, 235411 (2011)

D. Culcer, Physica E 44, 860 (2012) – review on TI transport


  • Introduction to topological insulators

  • Transport in non-interacting topological insulators

    • Liouville equation  kinetic equation

    • Current-induced spin polarization

  • Electron-electron interactions

    • Mean-field picture

    • Interactions in TI transport

    • Effect on conductivity and spin polarization

  • Bilayer graphene

  • Outlook

  • D. Culcer, Physica E 44, 860 (2012) – review on TI transport

    D. Culcer, PRB 84, 235411 (2011)

    D. Culcer, E. H. Hwang, T. D. Stanescu, S. Das Sarma, PRB 82, 155457 (2010)

    What is a topological insulator
    What is a topological insulator?

    • A fancy name for a schizophrenic material

      • Topological insulators ~ spin-orbit coupling and time reversal

  • 2D topological insulators

    • Insulating surface

    • Conducting edges – chiral edge states with definite spin orientation

    • Quantum spin-Hall effect – observed in HgTe quantum well (Koenig 2007)

  • 3D topological insulators

    • Insulating bulk

    • Conducting surfaces – chiral surface states with definite spin orientation

  • All the materials in this talk are 3D

  • The physics discussed is 2D surface physics

  • What is a topological insulator1
    What is a topological insulator?

    • Many kinds of insulators

      • Band insulator – energy gap >> room temperature

      • Anderson insulator – large disorder concentration

      • Mott insulator – strong electron-electron interactions

      • Kondo insulator – localized electrons hybridize with conduction electrons – gap

  • All of these can be topological insulators if spin-orbit strong enough

    • All of the insulators above have surface states which may be topological

    • When we say topological insulators ~ band insulators

    • Otherwise specify e.g. topological Kondo insulators

  • Also topological superconductors

    • Quasiparticles – Cooper pairs

  • All the materials in this talk are band insulators

  • What is a topological insulator2
    What is a topological insulator?

    • The first topological insulator was the quantum Hall effect (QHE)

      • QHE is a 2D topological insulator

      • No bulk conduction (except at special points), only edge states

      • Edge states travel in one direction only

      • They cannot back-scatter – have to go across the sample

  • Hall conductivity σxy= n (e2/h)

    • n is a topological invariant – Chern number (related to Berry curvature)

    • n counts the number of Landau levels ~ like the filling factor

  • QHE breaks time-reversal because of the magnetic field

  • The current generation of TIs is time-reversal invariant

  • C.L. Kane & E.J. Mele, Physical Review Letters 95 (2005) 226801.

    M.Z. Hasan& C.L. Kane, Reviews of Modern Physics 82 (2010) 3045.

    X.-L. Qi & S.-C. Zhang, Reviews of Modern Physics 83 (2011) 1057.

    X.-L. Qi, T.L. Hughes & S.-C. Zhang, Physical Review B 78 (2008) 195424.

    Why are some materials ti
    Why are some materials TI?

    Boring semiconductor

    • Surface states determined by the bulk Hamiltonian

  • Think of an ordinary band insulator

    • Conduction band, valence band separated by a gap

    • No spin-orbit – surface states are boring (for us)

  • Suppose spin-orbit is now strong

    • Think of tight-binding picture

    • Band inversion [see Zhang et al, NP5, 438 (2009)]

    • Mixes conduction, valence bands in bulk

    • Surface states now connect conduction, valence bands

  • Effective Hamiltonian on next slide

  • Bulk conduction


    Bulk valence

    Why are some materials ti1
    Why are some materials TI?

    • This is all k.p theory

      • Set kx = ky = 0

      • Solve for bound states in the z-direction: kz = -i d/dz

      • Next consider kx, ky near band edge

      • Surface state dispersion – Dirac cone (actually Rashba)

      • Chiral surface states, definite spin orientation

    • TI are a one-particle phenomenon

    Bulk conduction



    Bulk valence

    Zhang et al, Nature Physics 5, 438 (2009)

    How do we identify a ti
    How do we identify a TI?

    • In TI we cannot talk about the Chern number

    • Kane & Mele found another topological invariant – Z2 invariant

    • Z2 invariant related to the matrix elements of the time-reversal operator

      • Sandwich time reversal operator between all pairs of bands in the crystal

      • Need the whole band structure – difficult calculation

  • Z2 invariant counts the number of surface states

    • 0 or even is trivial

    • 1 or odd is non-trivial – odd number of Dirac cones

  • Theorem says fermions come in pairs – pair on other surface

  • In practice in a TI slab all surfaces have TI states

    • This can be a problem when looking at e.g. Hall transport

  • What is topological protection
    What is topological protection?

    • Topological protection really comes from time reversal.

      • So it really is a schizophrenic insulator

  • Disorder

    • Like a deformation of the Hilbert space

    • Non-magnetic disorder – TI surface states survive

  • Electron-electron interactions

    • Coulomb interaction does not break time reversal, so TI surface states survive

  • Protection against weak localization and Anderson localization

    • No backscattering (we will see later what this means)

  • The states can be in the gap or buried in conduction/valence band

    • The exact location of the states is not topologically protected

  • Most common ti bi 2 se 3
    Most common TI - Bi2Se3

    Zhang et al, Nature Physics 5, 438 (2009)

    More on bi 2 se 3
    More on Bi2Se3

    • Quintuple layers

      • 5 atoms per unit cell – ever so slightly non-Bravais

  • Energy gap ~ 0.3 eV

  • TI states along (111) direction

  • High bulk dielectric constant ~ 100

  • Similar material Bi2Te3

    • Has warping term in dispersion – Fermi surface not circle but hexagon

    • Bulk dielectric constant ~ 200

    • Surface states close to valence band, may be obscured

    • The exact location of the surface states is not topologically protected

  • Surface states exist – demonstrated using STM and ARPES

  • Current experimental status
    Current experimental status

    • STM enables studies of quasiparticle scattering

      • Scattering off surface defects – initial state interferes with final state

      • Standing-wave interference pattern

      • Spatial modulation determined by momentum transfer during scattering

      • Oscillations of the local DOS in real space

    Zhang et al, PRL 103, 266803 (2009)

    Current experimental status1
    Current experimental status

    • ARPES

      • Also measures local DOS

      • Map Fermi surface

      • Map dispersion relation

    • Fermi surface maps measured using ARPES and STM agree

    • Spin-resolved ARPES

      • Measures the spin polarization of emitted electrons – Hsieh et al, Science 323, 919 (2009).

    Alpichshev et al, PRL 104, 016401 (2010)

    Current experimental status2
    Current experimental status

    • Unintentional Se vacancies – residual doping

    • Fermi level in conduction band – most TI’s are bad metals

    • Surface states not clearly seen in transport – obscured by bulk conduction

      • Seen Landau levels but no quantum Hall effect

  • Experimental problems

    • Ca compensates n-doping but introduces disorder – impurity band

    • Low mobilities, typically < 1000 cm2/Vs

    • Atmosphere provides n-doping

    • TI surfaces remain poorly understood experimentally

  • All of these aspects discussed in review

    • D. Culcer, PhysicaE 44, 860 (2011)

  • Interactions chirality nontrivial
    Interactions + chirality - nontrivial

    Exotic phases with correlations cf. talk by Kou Su-Peng this morning


    See also Greg Fiete, Physica E 44, 844 (2012) review on spin liquid in TI + ee

    Ti hamiltonian no interactions
    TI Hamiltonian – no interactions

    • H = H0 + HE + U

      • H0 = band

      • HE = Electric field

      • U = Scattering potential

        • Impurity average

      • εFτp >> 1

      • τp= momentum relaxation time

      • εF in bulk gap – electrons

      • T=0  no phonons, no ee-scattering

    Bulk conduction




    Bulk valence

    Ti vs familiar materials
    TI vs. Familiar Materials

    • Unlike graphene

      • σis pseudospin

      • No valleys

    • Unlike semiconductors

      • SO is weak in semiconductors

      • No spin precession in TI

    Semiconductor with SO

    Effective magnetic field
    Effective magnetic field

    Effective magnetic field




    General picture at each k

    Out of equilibrium the spin may deviate slightly from the direction of the effective magnetic field

    Spin-momentum locking

    Equilibrium picture

    Liouville equation
    Liouville equation

    • Apply electric field ~ study density matrix

      • Starting point: Liouville equation

      • Method of solution – Nakajima-Zwanzig projection (中岛二十)

  • Project onto k and s kinetic equation

  • Divide into equations for diagonal and off-diagonal parts

  • Kinetic equation
    Kinetic equation

    • Reduce to equation for f – like Boltzmann equation

    • Scattering term

      • This is 1st Born approximation – Fermi Golden Rule

    Driving term due to the electric field

    Spin precession


    Scattering in

    Scattering out

    Scattering term
    Scattering term

    Effective magnetic field


    • Density matrix = Scalar + Spin

    • Spin

    • Scattering term – in equilibrium only conserved spin

    • Suppression of backscattering

    Conserved spin

    Non-conserved spin

    Kinetic equation1
    Kinetic equation

    • Conserved spin density

    • Precessing spin density

    • Solution – expansion in 1/(AkFτ)

      • AkFτ ~ (Fermi energy) x (momentum scattering time)

      • Assumes (AkFτ) >> 1 – in this sense it is semiclassical

      • Conserved spin gives leading order term linear inτ

      • Precessing spin gives next-to-leading term independent ofτ

    Culcer, Hwang, Stanescu, Das Sarma, PRB 82, 155457 (2010)


    • Conserved spin ~ like Drude conductivity

    • Precessing spin ~ extra contribution

      • Needs some care

      • Produces a singular contribution to the conductivity

      • Cf. grapheneZitterbewegung and minimum conductivity

    ζ contains the angular dependence of the scattering potential.

    W is the strength of the scattering potential.

    Momentum relaxation time

    Topological protection
    Topological protection

    • Protection exists only against backscattering – π

      • Can scatter through any other angle – π/2 dominates transport

      • Transport theory results similar to graphene

      • Conventional picture of transport applies

        • Electric field drives carriers, impurities balance driving force

  • There is nothing in TI transport that makes it special

    • States robust against non-magnetic disorder

    • Disorder will not destroy TI behavior

    • But transport still involves scattering, dissipation

    • Remember transport is irreversible

  • Careful with metallic contacts – not localized

    • May destroy TI behavior if too big

  • Spin polarized current
    Spin-polarized current

    • Current operator proportional to spin

    • No equivalent in graphene

    • Charge current = spin polarization

      • 10-4 spins/unit cell area

      • Spin polarization exists throughout surface

      • Not in bulk because Bi2Se3 has inversion symmetry

      • This is a signature of surface transport

      • Smoking gun for TI behavior?

      • Detection – Faraday/Kerr effects

    Conducting edge


    Insulating bulk

    Spin polarized current1
    Spin-polarized current





    E // x

    No E

    Electron electron interactions
    Electron-electron interactions

    • TI is a single-particle phenomenon

      • Recall topological protection – transport irreversible

      • TI phenomenology – robust against disorder and ee-interactions

      • But this applies to the equilibrium situation

  • Out-of-plane magnetic field – out-of-plane spin polarization (Zeeman)

    • In-plane magnetic field does NOTHING

  • In-plane electric field – in-plane spin polarization (similar to Zeeman)

    • Because of spin-orbit

  • How do electron-electron interactions affect the spin polarization?

  • Can interactions destroy the TI phase out of equilibrium?

  • D. Culcer, PRB 84, 235411 (2011)

    Exchange enhancement
    Exchange enhancement

    • Exchange enhancement (standard Fermi liquid theory)

      • Take a metal and apply a magnetic field – Zeeman interaction

      • ee-interactions enhance the response to the magnetic field

      • Enhancement depends on EXCHANGE and DENSITY OF STATES

    • Stoner criterion

      • If Exchange x Density of States large enough …

      • This favors magnetic order

      • Electric field + SO = magnetic field

      • Can interactions destroy TI according to some Stoner criterion?





    D. Culcer, PRB 84, 235411 (2011)

    Interacting ti
    Interacting TI

    • The Hamiltonian has a single-particle part and an interaction part

    • Matrix elements

    • Matrix elements in the basis of plane waves

    This is just the band Hamiltonian – Dirac

    This is the Coulomb interaction term

    Plane wave states

    This is just the electron-electron Coulomb potential

    D. Culcer, PRB 84, 235411 (2011)


    • Quasi-2D screening, up to 2kF the dielectric function is (RPA)

    • Effective scattering potential

      • All potentials renormalized – ee, impurities (below)

      • Quasi-2D, screened Coulomb potentials remain long-range

  • rs measures ratio of Coulomb interaction to kinetic energy

    • In TI it is a constant (same as fine structure constant)

  • Culcer, Hwang, Stanescu, Das Sarma, PRB 82, 155457 (2010)

    Electron electron interactions1
    Electron-electron interactions

    • Screening – RPA

      • ee-Coulomb potential also screened

  • Mean-field Hartree-Fockcalculation

  • Analogous to Keldysh – real part of ee self energy (reactive)

  • Interactions appear in two places: screening and Hartree-Fock mean field

  • No ee collisions (i.e. no extra scattering term = no ee dissipative term)

    • This is NOT Coulomb drag

  • D. Culcer, PRB 84, 235411 (2011)

    Mean field
    Mean field

    • Kinetic equation – reduce to one-particle using Wick’s theorem

      • Interactions give a mean-field correction BMF

      • Think of it as an exchange term

  • BMF – effective k-dependent ee-Hamiltonian

  • Spin polarization generates new spin polarization – self-consistent

    • Renormalization (BMF goes into driving term)

  • D. Culcer, PRB 84, 235411 (2011)

    Electron electron interactions2
    Electron-electron interactions

    • Renormalization of spin density due to interactions

      • Correction to density matrix called See

  • Comes from precessing term – i.e. rotation

    • This is the bare correction

    • How can spin rotation give a renormalization of the spin density?

  • Remember the current operator is proportional to the spin

    • Whenever we say charge current we also mean spin polarization

    • Whenever we say spin polarization we also mean charge current

  • D. Culcer, PRB 84, 235411 (2011)

    What happens
    What happens?

    • Spin-momentum locking

      • Effective SO field wants to align the spin with itself

    • Many-body correlations – think of it as EXCHANGE

      • Exchange wants to align the spin against existing polarization

      • Exchange tilts the electron spin away from the effective SO field

      • If no spin polarization exchange does nothing

    • This is why the net effect is a rotation

    • It shows up in the perpendicular part of density matrix because it is a rotation

    D. Culcer, PRB 84, 235411 (2011)

    Enhancement and precession
    Enhancement and precession







    Electron electron interactions3
    Electron-electron interactions

    • First-order correction

    • Same form as the non-interacting case, same density dependence

      • Because of linear screening – kTFkF

      • Not observable by itself

      • Embedded as it were in original result

  • Kinetic equation solved analytically to all orders in rs

  • D. Culcer, PRB 84, 235411 (2011)

    Reduction of the conductivity
    Reduction of the conductivity

    D. Culcer, PRB 84, 235411 (2011)

    Why reduction
    Why reduction?

    • Interactions lower Fermi velocity

      • They enhance the density of states

  • Another way of looking at the problem

    • TI have only one Fermi surface

    • Rashba SOC, interactions enhance current-induced spin polarization

  • Majority spin subband, spins save energy. Polarization enhanced.

    Polarization reduced.

    TI is like minority spin subband.

    Spins gain energy by lining up with the field.



    Minority spin subband, spins gain energy. Polarization reduced.

    D. Culcer, PRB 84, 235411 (2011)

    Current tis
    Current TIs

    • Current TIs have a large permittivity ~ hundreds

      • Large screening

      • rs is small (but result holds even if rs made artificially large)

      • Coulomb potential strongly screened

      • Interaction effects expected to be weak

    • For example Bi2Se3

      • Relative permittivity ~ 100

      • Interactions account for up to 15% of conductivity

    • Bi2Te3 has relative permittivity ~ 200

    • This is only the beginning – first generation TI

    D. Culcer, PRB 84, 235411 (2011)

    I nteractions out of equilibrium
    Interactions out of equilibrium

    • T = 0 conductivity of interacting system

      • Same form as non-interacting TI

      • But renormalized – reduction factor

  • Reduction is density independent

    • Peculiar feature of linear dispersion – linear screening

    • The only thing that can be `varied’ is the permittivity

  • No Stoner-like divergence

  • Is TI phenomenology robust against interactions out of equilibrium?

    • YES

  • This is an exact result (within HF/RPA)

  • D. Culcer, PRB 84, 235411 (2011)

    Bilayer graphene
    Bilayer graphene


    • Quadratic spectrum

      • Perhaps renormalization is observable

  • Chirality

    • But pseudospin winds twice around FS

    • Gapless

    • Gap can be induced by out-of-plane electric field

  • As Dirac point is approached

    • Competing ground states

    • See work by A. H. MacDonald, V. Fal’ko, L. Levitov

  • Wei-Zhe Liu, A. H. MacDonald, and D. Culcer (2012)

    Bilayer graphene1
    Bilayer graphene

    • Screening – RPA

    • Conductivity renormalization

    Wei-Zhe Liu, A. H. MacDonald, and D. Culcer (2012)

    Bilayer graphene2
    Bilayer graphene

    • BLG and TI interactions in transport

      • Interestingly: 大同小异

    • WHY?

    • Gain a factor of k in the pseudospin density

    • Lose a factor of k in screening

    • Overall result

      • Small renormalization of conductivity

      • Weak density dependence

    Wei-Zhe Liu, A. H. MacDonald, and D. Culcer (2012)

    Bilayer graphene3
    Bilayer graphene

    • Fractional change


    • TI thin films with tunneling between layers

      • Mass term but does not break time reversal – see work by S. Q. Shen

      • Exotic phases – e.g. QAH state at Dirac point

      • What do Friedel oscillations look like?

  • Interactions in non-equilibrium TI – other aspects

    • Kondo resistance minimum

    • So far few theories of the Kondo effect in TI

    • Expect difference between small SO and large SO

  • D. Culcer, PRB 84, 235411 (2011)

    D. Culcer, Physica E 44, 860 (2012) – review on TI transport

    Wei-Zhe Liu, A. H. MacDonald, and D. Culcer (2012)