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Interacting topological insulators out of equilibrium

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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 Electron-electron interactions Bilayer graphene Outlook

- Introduction to topological insulators
- Transport in non-interacting topological insulators
- Liouville equation kinetic equation
- Current-induced spin polarization

- Mean-field picture
- Interactions in TI transport
- Effect on conductivity and spin polarization

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? 2D topological insulators 3D topological insulators All the materials in this talk are 3D The physics discussed is 2D surface physics

- A fancy name for a schizophrenic material
- Topological insulators ~ spin-orbit coupling and time reversal

- Insulating surface
- Conducting edges – chiral edge states with definite spin orientation
- Quantum spin-Hall effect – observed in HgTe quantum well (Koenig 2007)

- Insulating bulk
- Conducting surfaces – chiral surface states with definite spin orientation

What is a topological insulator? All of these can be topological insulators if spin-orbit strong enough Also topological superconductors All the materials in this talk are band insulators

- 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 the insulators above have surface states which may be topological
- When we say topological insulators ~ band insulators
- Otherwise specify e.g. topological Kondo insulators

- Quasiparticles – Cooper pairs

What is a topological insulator? Hall conductivity σxy= n (e2/h) QHE breaks time-reversal because of the magnetic field The current generation of TIs is time-reversal invariant

- 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

- n is a topological invariant – Chern number (related to Berry curvature)
- n counts the number of Landau levels ~ like the filling factor

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? Think of an ordinary band insulator Suppose spin-orbit is now strong Effective Hamiltonian on next slide

Boring semiconductor

- Surface states determined by the bulk Hamiltonian

- Conduction band, valence band separated by a gap
- No spin-orbit – surface states are boring (for us)

- 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

Bulk conduction

Eg

Bulk valence

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

Surface

states

Bulk valence

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

How do we identify a TI? Z2 invariant counts the number of surface states Theorem says fermions come in pairs – pair on other surface In practice in a TI slab all surfaces have TI states

- 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

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

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

What is topological protection? Disorder Electron-electron interactions Protection against weak localization and Anderson localization The states can be in the gap or buried in conduction/valence band

- Topological protection really comes from time reversal.
- So it really is a schizophrenic insulator

- Like a deformation of the Hilbert space
- Non-magnetic disorder – TI surface states survive

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

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

- 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 Energy gap ~ 0.3 eV TI states along (111) direction High bulk dielectric constant ~ 100 Similar material Bi2Te3 Surface states exist – demonstrated using STM and ARPES

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

- 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

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 Experimental problems All of these aspects discussed in review

- 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

- 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

- D. Culcer, PhysicaE 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

- εFτp >> 1
- τp= momentum relaxation time
- εF in bulk gap – electrons
- T=0 no phonons, no ee-scattering

Bulk conduction

εF

Surface

states

Bulk valence

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

ky

Spin

kx

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 Project onto k and s kinetic equation Divide into equations for diagonal and off-diagonal parts

- Apply electric field ~ study density matrix
- Starting point: Liouville equation
- Method of solution – Nakajima-Zwanzig projection (中岛二十)

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

Scattering in

Scattering out

Scattering term

Effective magnetic field

Spin

- Density matrix = Scalar + Spin
- Spin
- Scattering term – in equilibrium only conserved spin
- Suppression of backscattering

Conserved spin

Non-conserved 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. 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 There is nothing in TI transport that makes it special Careful with metallic contacts – not localized

- 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

- States robust against non-magnetic disorder
- Disorder will not destroy TI behavior
- But transport still involves scattering, dissipation
- Remember transport is irreversible

- 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

Conducting edge

Insulating bulk

Electron-electron interactions Out-of-plane magnetic field – out-of-plane spin polarization (Zeeman) In-plane electric field – in-plane spin polarization (similar to Zeeman) How do electron-electron interactions affect the spin polarization? Can interactions destroy the TI phase out of equilibrium?

- 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

- In-plane magnetic field does NOTHING

- Because of spin-orbit

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?

DOS

EF

Minority

Majority

D. Culcer, PRB 84, 235411 (2011)

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)

Screening rs measures ratio of Coulomb interaction to kinetic energy

- 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

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

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

Electron-electron interactions 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)

- Screening – RPA
- ee-Coulomb potential also screened

- This is NOT Coulomb drag

D. Culcer, PRB 84, 235411 (2011)

Mean field BMF – effective k-dependent ee-Hamiltonian Spin polarization generates new spin polarization – self-consistent

- Kinetic equation – reduce to one-particle using Wick’s theorem
- Interactions give a mean-field correction BMF
- Think of it as an exchange term

- Renormalization (BMF goes into driving term)

D. Culcer, PRB 84, 235411 (2011)

Electron-electron interactions Comes from precessing term – i.e. rotation Remember the current operator is proportional to the spin

- Renormalization of spin density due to interactions
- Correction to density matrix called See

- This is the bare correction
- How can spin rotation give a renormalization of the spin density?

- 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

- 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)

Electron-electron interactions Kinetic equation solved analytically to all orders in rs

- First-order correction
- Same form as the non-interacting case, same density dependence
- Because of linear screening – kTFkF
- Not observable by itself
- Embedded as it were in original result

D. Culcer, PRB 84, 235411 (2011)

Reduction of the conductivity

D. Culcer, PRB 84, 235411 (2011)

Why reduction? Another way of looking at the problem

- Interactions lower Fermi velocity
- They enhance the density of states

- 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.

TI

Rashba

Minority spin subband, spins gain energy. Polarization reduced.

D. Culcer, PRB 84, 235411 (2011)

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 Reduction is density independent No Stoner-like divergence Is TI phenomenology robust against interactions out of equilibrium? This is an exact result (within HF/RPA)

- T = 0 conductivity of interacting system
- Same form as non-interacting TI
- But renormalized – reduction factor

- Peculiar feature of linear dispersion – linear screening
- The only thing that can be `varied’ is the permittivity

- YES

D. Culcer, PRB 84, 235411 (2011)

Bilayer graphene Chirality As Dirac point is approached

εF

- Quadratic spectrum
- Perhaps renormalization is observable

- But pseudospin winds twice around FS
- Gapless
- Gap can be induced by out-of-plane electric field

- 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 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 Interactions in non-equilibrium TI – other aspects

- 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?

- 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)

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