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

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

OutlookD. 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? 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 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 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 invariantC.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? 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

Eg

Boring semiconductor

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

Surface states

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

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? 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 - Bi2Se3

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

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 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 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 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, Physica E 44, 860 (2011)

Interactions + chirality - nontrivial

Exotic phases with correlations cf. talk by Kou Su-Peng this morning 流光溢彩 See also Greg Fiete, Physica E 44, 844 (2012) review on spin liquid in TI + ee

TI Hamiltonian – no interactions H = H0 + HE + U

H0 = band

HE = Electric field

U = Scattering potential

Impurity average

εF τp >> 1

τp = momentum relaxation time

εF in bulk gap – electrons

T=0 no phonons, no ee-scattering

Bulk conduction

Bulk valence

Surface states

εF

TI vs. Familiar Materials

Unlike graphene

σ is pseudospin

No valleys

Unlike semiconductors

SO is weak in semiconductors

No spin precession in TI Semiconductor with SO

Effective magnetic field

kx

ky

Spin-momentum locking Equilibrium picture

General picture at each k

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

Effective magnetic field

Spin

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 Reduce to equation for f – like Boltzmann equation

Scattering term

This is 1st Born approximation – Fermi Golden Rule

Spin precession ScatteringDriving term due to the electric field

Scattering in Scattering out

Scattering term Density matrix = Scalar + Spin

Spin

Scattering term – in equilibrium only conserved spin

Suppression of backscattering

Conserved spin

Non-conserved spin

Effective magnetic field Spin

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)

Conductivity Conserved spin ~ like Drude conductivity

Precessing spin ~ extra contribution

Needs some care

Produces a singular contribution to the conductivity

Cf. graphene Zitterbewegung and minimum conductivity

Momentum relaxation time

ζ contains the angular dependence of the scattering potential. W is the strength of the scattering potential.

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 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

Insulating bulk

Conducting edge

Spin-polarized current

E // x No E

kx

kx

ky

ky

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 (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)

Majority

Minority

EF

DOS

Interacting TI The Hamiltonian has a single-particle part and an

interaction part

Matrix elements

Matrix elements in the basis of plane waves D. Culcer, PRB 84, 235411 (2011)

This is just the band Hamiltonian – Dirac

This is the Coulomb interaction term

This is just the electron-electron Coulomb potential

Plane wave states

Screening 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 interactions Screening – RPA

ee-Coulomb potential also screened

Mean-field Hartree-Fock calculation

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 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 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? 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

D. Culcer, PRB 84, 235411 (2011)

This is why the net effect is a rotation

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

Enhancement and precession

kx

ky

kx

ky

Non-interacting Interacting

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

D. Culcer, PRB 84, 235411 (2011)

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

D. Culcer, PRB 84, 235411 (2011)

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.

Majority spin subband, spins save energy. Polarization enhanced.

TI Rashba

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)

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

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)

εF

Bilayer graphene Screening – RPA

Conductivity renormalization

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

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 graphene Fractional change

Outlook 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)