Topological protection - kitpc

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2 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

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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


Transport in non
-
interacting topological insulators


Liouville

equation


kinetic
equation


Current
-
induced spin polarization


Electron
-
electron interactions


M
ean
-
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?


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
σ
xy
=
n (e
2
/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?


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)
]


M
ixes conduction, valence bands in bulk


S
urface states now connect conduction, valence bands


Effective Hamiltonian on next slide


Bulk
conduction


Bulk valence

E
g


Boring
semiconductor

Why are some materials TI?





This is all
k.p

theory


Set
k
x

=
k
y

= 0


S
olve for bound states in the z
-
direction:
k
z

=
-
i

d/
dz


Next consider
k
x
,
k
y

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
-

Bi
2
Se
3



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

More on Bi
2
Se
3


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 Bi
2
Te
3



H
as warping term in dispersion


Fermi surface not circle but hexagon


Bulk dielectric constant ~ 200


S
urface 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


S
cattering
off surface
defects


initial state interferes with
final state


S
tanding
-
wave interference pattern


S
patial
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


M
ap 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 cm
2
/
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 = H
0

+ H
E

+ U


H
0
= band



H
E

= Electric field


U = Scattering potential


Impurity average



ε
F

τ
p

>> 1


τ
p

= momentum relaxation time


ε
F

in bulk gap


electrons


T=0


no phonons, no
ee
-
scattering



Bulk
conduction


Bulk valence

Surface


states

ε
F


TI vs. Familiar Materials





Unlike
graphene



σ

is
pseudospin



N
o valleys


Unlike
semiconductors


SO is weak in semiconductors


No
spin
precession in TI



Semiconductor with SO

Effective magnetic field


k
x



k
y


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


A
pply electric field ~ study density matrix


Starting point:
Liouville

equation




M
ethod 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 1
st

Born approximation


Fermi Golden Rule

Spin precession


Scattering

Driving term due to the
electric field

Scattering in

Scattering out

Scattering term


Density matrix = Scalar + Spin




Spin






S
cattering 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/(
Ak
F
τ
)


Ak
F
τ

~ (Fermi energy)
x
(momentum
scattering
time)


Assumes
(
Ak
F
τ
) >> 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


C
onserved spin ~ like
Drude

conductivity







Precessing

spin ~ extra contribution


Needs some care


Produces a singular contribution to the conductivity


C
f.
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


D
isorder 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 Bi
2
Se
3

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


k
x


k
x



k
y


k
y

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


e
e
-
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

E
F


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 2k
F

the dielectric function is (RPA)



Effective scattering potential


All potentials renormalized


ee
, impurities (below)




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


r
s

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


r
educe to one
-
particle using Wick’s theorem


Interactions give a mean
-
field correction
B
MF



Think of it as an exchange term



B
MF



effective
k
-
dependent
ee
-
Hamiltonian




Spin polarization generates new spin
polarization


self
-
consistent


Renormalization (B
MF

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 S
ee






C
omes 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


k
x


k
y


k
x


k
y


Non
-
interacting




Interacting


Electron
-
electron interactions


First
-
order correction




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


Because of linear screening


k
TF



k
F


Not
observable by
itself


Embedded

as it were in original
result


Kinetic equation solved analytically to all orders in
r
s



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
o
nly 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


r
s

is small (but result holds even if
r
s

made artificially large)


Coulomb potential strongly screened


Interaction effects expected to be weak


For example Bi
2
Se
3


Relative permittivity ~ 100


Interactions account for up to 15% of conductivity


Bi
2
Te
3

has relative permittivity ~ 200


This
is only the beginning


first generation TI



D. Culcer, PRB
84, 235411 (
2011)

I
nteractions 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


I
s 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)

ε
F


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


W
eak 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
)