studying topological insulators

Studying Topological Insulators via

time reversal symmetry breaking and proximity effect

Roni Ilan UC Berkeley

Joel Moore, Jens Bardarson, Jerome Cayssol, Heung-Sun Sim

Topological phases

Insulating phases with metallic surface states.

Properties/existence of surface states protected by bulk properties.

Topological invariants of the bulk are immune to continuous deformations.

Bulk is insulating but non-trivial!

Probes of topological properties

Many features are attributed to surface states as mirrors of topological properties of the bulk

How can we observe this in experiment?

Topological phases and symmetry

Some topological phases require no symmetry:

Quantum Hall effect

Bulk topological invariant

= Chern number

Chiral edge modes protected due to spatial separation

�xy

=

Topological phases and symmetry

Quantum Spin Hall effect (2DTI)

TI surface states are protected by time reversal symmetry

An opportunity to study surface states by breaking that symmetry

Surface states of 2 and 3D TI’s

Figure from Y. Ando J. Phys. Soc. Jpn. 82, 102001 (2013)

2D 3D

Spin-momentum locking

Dirac fermions

Time reversal symmetry

Half integral spin

[H,T ] = 0 | i |T i

hT | i = hT 2 |T i⇤ = �h |T i⇤ = �hT | i

Absence of backscattering

T 2 = �1

h |V |T i = 0

Experimental detection: 2DTI

[Konig et al 07]

CdTe

CdTe

HgTe

I. Trivial II-IV. Non trivial !L=20 microns (L,W) =(1,1) micron (L,W) = (1,0.5) micron

!

Experimental detection: 3DTI

Bi2X3

Figure from Moore and Hasan, Annu. Rev. Condens. Matter Phys. 2011. Experiments by Xia et al. 2008, 2009 and Hsieh et al. 2009

Interesting questions

2D TI: Surface states are 1D.

!

!

Helical states.

Evidence of luttinger liquid physics? Impurity physics?

Exact solutions for non-equilibrium transport?

G = 2e2/h

Interesting questions

3D TI: Surface states are 2D.

!

Dirac Fermions.

Transport? Topological effects?

Majorana fermion physics?

Luttinger liquid physics on the edge of a 2DTI

Does mean edge is non-interacting?

No!

2e2

h

contactcontact

QSHE

[Maslov and Stone 95, Safi and Schulz 95,97; Furusaki and Nagaosa 96; Oreg and Finkel’stein 96; Alekseev et al. 96; Egger and Grabert 98..... Wen 90, Milliken et al 96]

G = ge2

h

contactcontact

QHE G = ⌫e2

h

Need to break time reversal symmetry:

Local Magnetic impurity

Global magnetic field + impurity

[Maciejko et al 09; Tanaka et al 11; Beri and Cooper 12; Soori et al. 12; Timm. 12, Lezmy Oreg and Berkooz, 12]

Local breaking of TR

Luttinger liquid physics on the edge of a 2DTI

To observe interaction effects we must have backscattering.

Goal

To find a controlled way to induce local backscattering

Use the special properties of 2DTI to obtain exact solution for noneq. conductance

designing a tunable impurity

Point contact in the QHE

Point contact in the QSHE

?

magnetic field

Breaks time reversal symmetry

H = �i~vF

�z

@x

+ µB

ge

2~B · ~�

hz

Eg = 2h?

hz

Eg = 2h?

E2 = (vp� hz)2 + h2?

p

E

Rashba spin orbit coupling

H = �i~vF

�z

@x

� i~2 {↵, @x}�y

v ! v↵ =p↵2 + v2

✓ = tan�1(↵/vF )

tunable impurity

↵ = 0 ↵ = 0↵ 6= 0

[Nitta et al 97;Hinz et al 06]

Rashba SO controllable by gate

Eg = 2↵h/vF

H = �i~vF

�z

@x

+ h�z

� i~2{↵, @

x

}�y

(x) =

8><

>:

R

e

ipRx + r

L

e

ipLx

x < 0

a+ +eip+x + a� �e

ip�x 0 < x < d

t

R

e

ipRx

x > d

Non interacting electrons

0 50 100 150 200M/E0

0.0

0.2

0.4

0.6

0.8

1.0

R

E/E0

3

10

15

R ⇠ (h↵/v2F )2

h/(~vF /d)

Interacting edge

H0 = iv( †"@x " � †

#@x #)

Hint = u2 †" "

†# # + u4

h( †

" ")2 + ( †# #)2

i

"= R #= L

HBS = � †L(0) R + h.c

Helicity reduces the number of degrees of freedom by locking the spin to the momentum

Interacting edge

H = v

4⇡g ((@x�R

)2 + (@x

�L

))2

HBS = � cos(�R � �L)HBS = 1

2

Pn �ne

in�R(0)e�in�L(0) + h.c.

1/4 < g < 1 One relevant term

Maps to a spinless Luttinger Liquid

d�d` = (1� n2g)�

[Kane and Fisher 92, Kane and Teo 09, and many others]

G(T ) ⇠ e2

h

⇣TTB

⌘2(1/g�1)TB/T ! 1

TB/T ! 0G(T ) ⇠ e2

h

h1�

�TBT

�2(1�g)i

TB ⇠ �1/(1�g)

Spinless luttinger liquid with impurity

�

e(x, t) = 1p2(�L(x, t) + �R(�x, t))

�

o(x, t) = 1p2(�

L

(x, t)� �

R

(�x, t))

[Fendley, Ludwig, Saleur 95; Fendley, Lesage, Saleur 96]

HSG

=

18⇡g

R10 [⇧

2+ (@

x

�o

)

2] + � cos�o

(0)

Integrable model, can be solved exactly

�R

�L

�

R

(x, t) ⌘ �

o(x+ t)

�

L

(x, t) ⌘ �

o(�x+ t)

Integrability[Ghoshal and Zamolodchikov 94; Fandley Ludwig and Saleur 95; Fendley Lesage and Saleur 96]

Infinite set of conserved quantities with impurity

find “quasiparticle” basis

One body S matrices

Additive energies H = vP

i pi

Particles scatter one-by-one

Two body S matrices in the bulk

Particles scatter elastically off each other

Does not mean scattering is trivial! Permute quantum numbers, phase shifts

-4 -2 2 4

0.5

1.0

1.5

-4 -2 2 4

0.5

1.0

1.5

-4 -2 2 4

0.2

0.4

0.6

0.8

fj =1

1+e�µj/T+✏jTBA equations for ✏j(✓)

HSG

=

18⇡g

R10 [⇧

2+ (@

x

�o

)

2] + � cos�o

(0)

[Fendley Lesage and Saleur 96, Koutouza, Siano and Saleur 01]

EXACT SOLUTION AT g = 1� 1/m

I(V, TB , T ) =T (m� 1)

2

Zd✓

cosh

2[✓ � ln(TB/T )]

ln

✓1 + e(m�1)V/2T�✏+(✓)

1 + e�(m�1)V/2T�✏+(✓)

1 + e�(m�1)V/2T�✏+(1)

1 + e(m�1)V/2T�✏+(1)

◆

V = �✓1� 1

g

◆I(W ) +W

G(V, TB , T ) =dIBS

dV= G(V/2T, TB/T )

G(V/2T, 0) = g

Include contact correction

Conductance vs. temperature

10�4 10�3 10�2 10�1 100 101 102 103

TB/T

0.0

0.2

0.4

0.6

0.8

1.0

G/(

e2 /h)

m

V/T = 1.0

345

g = 1� 1m

G(T ) ⇠ e2

h

⇣TTB

⌘2(1/g�1)

G(T ) ⇠ e2

h

h1�

�TBT

�2(1�g)i

VOLTAGE DEPENDENCE

10�1 100 101 102 103 104

V/T

0.0

0.2

0.4

0.6

0.8

1.0

G/G

1

m

TB/T = 0.5

345

10�1 100 101 102 103 104

V/T

0.0

0.2

0.4

0.6

0.8

1.0

G/G

1

m

TB/T = 2.0

345

W/O FL contact (shifted down)

With FL contact

Conclusion

A setup for transport experiment on the QSHE edge: controlled TR symmetry breaking

+ controlled SO coupling = “point contact”

Interacting edge theory is integrable model: Exact solutions for conductance with FL leads

From experiment: estimate LL parameter Confirm integrability

Confirm effects of leads

Perfect transmission and Majorana fermions in 3DTI

Detect a perfectly transmitted mode and signature of Majorana fermions

Problem with current setups

Large number of modes and finite chemical potential

Large contribution to transport from bulk states

Dirac fermion on a curved surface- TI nanowires

x

[Ando and Suzuura 2002, Ran, Vishwanath, and Lee 2008, Rosenberg, Guo, and Franz 2010, Ostrovsky, Gornyi, and Mirlin 2010, Bardarson and Moore 2013]

(x, y +W ) = e

i⇡ (x, y)

x

(x, y +W ) = e

i⇡+i2⇡�/�0 (x, y)

� = �0/2odd number of modes

T 2 = �1ST = �S

Perfectly transmitted mode

[Ando and Suzuura 2002, Ran, Vishwanath, and Lee 2008, Rosenberg, Guo, and Franz 2010, Ostrovsky, Gornyi, and Mirlin 2010, Bardarson and Moore 2013]

Dirac fermion on a curved surface- TI nanowires

“Whereas bulk transport tends to obscure the observation of surface state transport in the normal state, the supercurrent is found to be carried mainly by the surface states”

Majorana states �� = ⇡

[Veldhorst et al. Nature Mater. 2012]

[Fu and Kane, PRL 2008]

[ Alicea Rep Prog Phys (2012), Beenakker, Annu. Rev. Cond. Mat. Phys. 2013, Grosfeld and Stern 2011, Potter and Fu 2013, Weider et al 2013]

Proximity effect - Josephson Junctions

3DTI and Majorana fermions

En(�) = �q1� ⌧n sin

2 (�/2)

� = ⇡

⌧ = 1

E = 0

+

Zero energy Majorana state requires a perfectly transmitted mode

L ⌧ ⇠Short JJ [Beenakker 2006, Titov and Beenakker 2006]

Josephson junction on a 3DTI nanowire

L/W ⇠ 1

Hosts a PTM

Hosts Majorana fermions

� = �0/2

�� = ⇡

[Ando and Suzuura 2002, Ran, Vishwanath, and Lee 2008, Rosenberg, Guo, and Franz 2010, Ostrovsky, Gornyi, and Mirlin 2010, Bardarson and Moore 2013, Fu and Kane PRL 2008, Cook and Franz 2011, Cook at al 2012]

Current Phase relation - clean wire

� = �0/2� = 0

µW/~v

102030

0

En(�) = �q1� ⌧n sin

2 (�/2)

I /X

n

@En/@�

⌧n =

k2nk2n cos

2(knL) + (µ/~v)2 sin2(knL) kn =

p(µ/~v)2 � q2n

[Titov and Beenakker 2006]

[RI, Jens Bardarson, H.S. Sim, Joel Moore,2013]

µW/~v

102030

0


� = �0/2� = 0


Parity anomaly

�

p

p̄

�† = �

�1

�2 c† = �1 � i�2c = �1 + i�2

E(�) = ± cos(�/2)

� = �0/2� = 0

µW/~v

102030

0

Parity anomaly


�† = �

�1

�2 c† = �1 � i�2c = �1 + i�2


�

p

p̄

E(�) = ± cos(�/2)

Current Phase relation - disorder

� = �0/2 µW/~v = 20

hV (r)V (r0)i = g~v

2⇡⇠2De|r�r0|/2⇠2D [Bardarson et al. PRL 2007]


Critical current - disorder

0.0 0.5 1.00.0

0.5

1.0I c

[e�

/h̄]

g

0.20.51.02.00.2 0.8

0.0

0.5

µ⇠D/~v = 0.5


µ = 0

�/�0

0.0 0.5 1.0�/�0

0.0

0.5

1.0I c

[e�

/h̄]

µ⇠D/h̄v = 2.0

1.5

1.0

0.5

0.20

g = 2


Critical current - disorder

Conclusion

Josephson junctions on 3DTI nanowires are a great place to look for signatures of surface

physics:

PTM , Majorana fermions

At finite chemical potential, enhanced by disorder (and at finite temperature)

Not constrained by parity conservation

Thanks

• Paul Fendley (UVA)

• David Goldhaber Gordon Group (Stanford)

• Kam Moler Group (Stanford)

• Fernando De Juan (Berkeley)

E(�)

�

p

p̄

Ic(�)

1/2

e�/2~

�/�0

e�/2~(1� (T/�)2)

T ⇡ 30mK

� ⇡ 1K

Effects of temperature

Some numbers

B = 10mT

vF = 5.5⇥ 105m/s

Eg ⇡ 100� 300mK

E = ±q

(v↵p+ vFh/v↵)2 + ↵20h

2/v2↵

↵0 ⇡ 5⇥ 104m/s

Temperature in experiment ~30mK (taken from Konig et al)


H = �i~vF

�z

@x

� i~2 {↵, @x}�y

R

(x) = (v

2F

+ ↵

2)

�1/4e

i�(x)

✓cos ✓

i sin ✓

◆

L

(x) = (v

2F

+ ↵

2)

�1/4e

�i�(x)

✓i sin ✓

cos ✓

◆

�(x) =

Zx

0

Epv

2F

+ ↵(x)2.

2✓(x) = tan�1(↵/vF )

(x�0 ) =

✓vF

v↵

◆1/2

e

i✓�x

(x+0 )

(x1) = T

x1,x0 (x0) Tx1,x0 = P

x

ei

Rx1x0

dx

(vF

�

z

+↵�

y

)

v

2F

+↵

2 [E+ i

2 (@x

↵)�y

]


(x�0 ) =

✓vF

v↵

◆1/2

e

i✓�x

(x+0 )

(x1) = T

x1,x0 (x0) Tx1,x0 = P

x

ei

Rx1x0

dx

(vF

�

z

+↵�

y

)

v

2F

+↵

2 [E+h�

z

+ i

2 (@x

↵)�y

]

H = �i~vF

�z

@x

+ h�z

� i~2{↵, @

x

}�y

Experimental feasibility

Short junction limit L ⌧ ⇠

L/W ⇠ 1Mode suppression

Flux through wire � > �0/2

Distance from Dirac point µ⇠D/~v

B ⇡ 0.2T

Disorder strength

g ⇡ 1 n ⇠ 8⇥ 1010 cm�2

[Sacepe et al. Nature Commun 2011, Kong et al. Nano Letters 2010, Beidenkophf et al. 2011, ,Williams et al. PRL 2012, Tian et al. Sci. Rep. 2012, Veldhorst et al Nature Mater. 2012, Kim et al Nature Phys. 2012, Cho et al. Nature Commun 2013...]

W ⇠ 400nm

g

⇠D ⇠ 10nm

What’s the big deal about Majorana fermions?

�† = � �1

�2 c† = �1 � i�2c = �1 + i�2

Ground state degeneracy

Topological protection

Potential for the perfect qubit

Non Abelian statistics

studying topological insulators

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