Jens Hjörleifur Bárðarson - DIPC

Transport in topological insulators

1

Jens Hjörleifur Bárðarson

Max Planck Institute PKS, Dresden

KTH Royal Institute of Technology, Stockholm

Brief intro

Topological insulators are bulk insulators with metallic surface

3

0

0 +1

-1

Vacuum Topological insulator

The periodic table of topological insulators

Ryu, Schneider, Furusaki, Ludwig NJP (2010), Ryu et al PRB, Kitaev …4

Cartan T C S H NLsM Name d=0 d=1 d=2 d=3

A 0 0 0 U(N) U(2n)/U(n)xU(n)

Unitary 0 0

AIII 0 0 1 U(N+M)/U(N)xU(M)

U(n) Chiral Unitary

0 0

AI +1 0 0 U(N)/O(N) Sp(2n)/Sp(n)xSp(n)

Orthogonal 0 0 0

BDI +1 +1 1 O(N+M)/O(N)xO(M)

U(2n)/Sp(2n) Chiral orthogonal

0 0

D 0 +1 0 SO(2N) O(2n)/U(n) BdG 0

DIII -1 +1 1 SO(2N)/U(N) O(2n) BdG 0

AII -1 0 0 U(2N)/Sp(2N) O(2n)/O(n)xO(n)

Symplectic 0

CII -1 -1 1 Sp(N+M)/Sp(N)xSp(M)

U(2n)/O(2n) Chiral symplectic

0 0

C 0 -1 0 Sp(2N) Sp(2n)/U(n) BdG 0 0 0

CI +1 -1 1 Sp(2N)/U(N) Sp(2n) BdG 0 0 0


Ryu, Schneider, Furusaki, Ludwig NJP (2010), Ryu et al PRB, Kitaev …5


A 0 0 0 U(N) U(2n)/U(n)xU(n)

Unitary 0 0


U(n) Chiral Unitary

0 0


Orthogonal 0 0 0

BDI +1 +1 1 O(N+M)/O(N)xO(M)


0 0

D 0 +1 0 SO(2N) O(2n)/U(n) BdG 0

DIII -1 +1 1 SO(2N)/U(N) O(2n) BdG 0


Symplectic 0



0 0

C 0 -1 0 Sp(2N) Sp(2n)/U(n) BdG 0 0 0

CI +1 -1 1 Sp(2N)/U(N) Sp(2n) BdG 0 0 0

Symmetries — Wigner’s theorem

Haake, Quantum Signatures of Chaos6

Any symmetry acts as a unitary or antiunitary transformation in Hilbert space

unitary

antiunitary (anti-linear!)

Can always write with complex conjugation

Time reversal symmetry — Examples

Haake, Quantum Signatures of Chaos7

spinless fermions (or integral spin):

half-integral spin fermions:

note: basis dependent!

2D topological insulator

Topological insulators in the unitary class (A) — Quantum Hall effect

9

Landau levelssurface chiral — no backscattering

Topological insulators in the symplectic class (AII) — Quantum spin Hall effect

Kane and Mele, PRL 200510

Absence of backscattering

?

G =2e2

h

Quantum Spin Hall effect in HgTe quantum wells

König et al, Science (2007)11

Non-local conductance

12

Ii =e2

h

X

j

Tij(Vi � Vj)I1 = I = �I4 I2 = I3 = 0

V2 = 0 V3 = V

I

V

1

2 3

4

Non-local conductance

13

I = I1 = (V1 � V2) + (V1 � V4) = 2V1 � V4

0 = I2 = (V2 � V1) + (V2 � V3) = �V1 � V

0 = I3 = (V3 � V2) + (V3 � V4) = 2V � V4

�I = I4 = (V4 � V3) + (V4 � V1) = 2V4 � V � V1

I

V

1

2 3

4

) V1 = �V

V4 = 2V

) I = �4V

) G =4e2

h

Ex. Find the conductance in the H-bar

3D topological insulator

3D topological insulators

Hasan and Kane RMP 201015

Scaling theory of localization

Abrahams, Anderson, Licciardello, and Ramakrishnan PRL 197916

Drude

0

Anderson


Abrahams, Anderson, Licciardello, and Ramakrishnan PRL 197917

0?

Conductance as transmission (Landauer)

18

Weak (anti) localization

19

?

Spin-momentum locking and Berry’s phase

20


21


22

Field theory of diffusion (nonlinear sigma model)

23

Scaling theory of localization — 2D

24

0

WAL

WL

Metal—insulator transition

Absence of backscattering and perfectly transmitted mode

25

Absence of backscattering and perfectly transmitted mode

26

if N odd


27

0

WAL

WL

Absence of localisation

28

0.3

0.6

0.9

1.2

1.5

1.8

1 10 100 103

�m� [

4e2 /h

]

L*/d

m

`

0.3

0.01.50.5

JHB, Tworzydlo, Brouwer, Beenakker, PRL 2007Nomura, Koshino, Ryu PRL 2007

Surface of a topological insulator can not be localized

29

�(�)

�

JHB, Tworzydlo, Brouwer, Beenakker, PRL 2007Nomura, Koshino, Ryu PRL 2007

Note: stronger condition than being gapless

Topology of energy bands

30

-0.3

-0.2

-0.1

0

0.1

0.2

0.2

0.25

0.3

Ene

rgy

Ene

rgy

(b) random SO model(a) massless Dirac model

Nomura, Koshino, Ryu PRL 2007

Symmetric spaces and topological terms

31

Symmetric space with two disconnected components

Topological term gives a different sign in the action to different components

A topological insulator in d dimensions has a d-1 dimensional surface that can not be localized. The non-linear sigma model describing the surface correspondingly has a

topological term


Ryu, Schneider, Furusaki, Ludwig NJP (2010)32


A 0 0 0 U(N) U(2n)/U(n)xU(n)

Unitary 0 0


U(n) Chiral Unitary

0 0


Orthogonal 0 0 0

BDI +1 +1 1 O(N+M)/O(N)xO(M)


0 0

D 0 +1 0 SO(2N) O(2n)/U(n) BdG 0

DIII -1 +1 1 SO(2N)/U(N) O(2n) BdG 0


Symplectic 0



0 0

C 0 -1 0 Sp(2N) Sp(2n)/U(n) BdG 0 0 0

CI +1 -1 1 Sp(2N)/U(N) Sp(2n) BdG 0 0 0

Weak topological insulators

33

Strong and weak indices

Can think of as stacking of lower dimensional strong topological insulators

1

1.5

2

1 10 100

1

2

10 100

Does not localize unless average translation symmetry broken

Mong, JHB, Moore PRL 2012

Summary of lecture 1

34

�(�)

�

Topological insulators characterized by a surface that doesn’t localize

In 3D top. ins. disorder drives the surface into a symplectic metal, characterized by weak anti-localization

Topological insulator nanowires

Jens Hjörleifur Bárðarson

Max Planck Institute PKS, Dresden

Review: JHB and J. E. Moore, Rep. Prog. Phys. 2013

KTH Royal Institute of Technology, Stockholm

Surface of a 3D topological insulator is a supermetal

JHB, Tworzydlo, Brouwer, Beenakker, PRL 2007

+

�(�)

�

=

Nomura, Koshino, Ryu PRL 2007

Topological insulator nanowires host various interesting modes

Perfectly transmitted mode Chiral modes Majorana modesPerfectly transmitted mode

Dirac fermion on a curved surface

JHB, Brouwer, Moore PRL 2010

E

Ran, Vishwanath, Lee PRL 2008; Rosenberg, Guo, Franz PRB(R) 2010; Ostrovsky, Gornyi, Mirlin PRL 2010

Aharonov-Bohm flux modifies spectrum


� = 0 � = 0.05�0 � = 0.25�0 � = 0.45�0 � = 0.5�0

E

k

From even to odd — the perfectly transmitted mode emerges

8 modes 7 modes

JHB, Brouwer, Moore PRL 2010; JHB J. Phys. A: Math. Theor. 2008Ando, Suzuura J. Phys. Soc. Jpn. 2002;

Number of modes depends on energy and flux

2

4

6

8

10

Number of modes

Energy


Conductance vs. chemical potential


Experimental status

Peng, …, Chui, Nature Mater. 2010

L. Jauregui,…,Y.P. Chen, Nature Nanotechnol. 2016S. Cho,…,N. Mason, Nature Comm. 2015

Dufouleur, Veyrat, Xypakis, JHB,…, Giraud Sci. Rep. 2017

0 100 200 300 400E(meV )

0.0

0.1

0.2

0.3

�G2 (

e4/h

2 )

� = 0

� = ⇡/2

� = ⇡

100 1100.10

0.15

0.20

Theory


Perfectly transmitted mode Chiral modes Majorana modes

Going chiral…

F. de Juan, R. Ilan, JHB, PRL 2014

E

Lee PRL 2009; Sitte et al PRL 2012

Conductance vs. chemical potential @ B = 2T


0 10 20µ (meV)

0

1

2

3G

N(e

2 /h)

sh

w

B¶

disorder strength

Conductance vs. magnetic field at μ = 10 meV


0 1 2 3B? (T )

0

1

2

3 Ax

B¶0 p 2 p

s



Kitaev chain

51Alicea Rep. Prog. Phys. (2012)

twofold degenerate ground state

A recipe for a tunable topological superconductor

Cook and Franz PRB 2011

+

+

=

(proximity)

Detecting topological superconductivity via Béri degenercay

Béri PRB 2009; Wimmer, Akhmerov, Dahlhaus, Beenakker NJP (2011)

trivial

topological

Getting to the single mode regime


Perfectly transmitted mode

Chiral mode

NS Conductance vs. chemical potential

h = 0.0, nV = 0h = 0.5, nV = 1

0 10 200

10

20

m HmeVL

GNSHe2 êhL

B¶ = 0 T


h = 0.0, nV = 0h = 0.5, nV = 1

0 10 200

2

4

6

m HmeVL

GNSHe2 ê

hL

B¶ = 2 T

Majorana interferometer

Fu and Kane, PRL (2009), Akhmerov, Nilsson, and Beenakker PRL (2009)


SNS and p-n junctions

Ilan, JHB, Sim, Moore NJP 2014

0.0 0.5 1.0�/�0

0.25

0.50

I cR

N[2

⇡�

/e]

0.0 0.2 0.5 1.0 1.5 2.0

µ⇠D/h̄v = 4, 10

Ilan, de Juan, Moore PRL 2015



JHB, Brouwer, Moore PRL 2010; Ilan, JHB, Sim, Moore NJP 2014; F. de Juan, R. Ilan, JHB, PRL 2014Review: JHB and Moore, Rep. Prog. Phys. 2013

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Jens Hjörleifur Bárðarson - DIPC