Tunneling Conductance and Surface States Transition in Superconducting Topological Insulators

Tunneling Conductance and Surface States Transition in Superconducting Topological

Insulators

Yukio Tanaka (Nagoya University)

http://www.topological-qp.jp/english/index.html

Chernogolovka June 17 (2012)

http://www.topological-qp.jp/english/index.html

Main collaboratorsTheory

Y. Asano （ Hokkaido ）A. Golubov (Enshede)A. Yamakage (Nagoya)K. Yada (Nagoya)M. Sato （ Nagoya ）T. Yokoyama （ Tokyo ）N. Nagaosa （ Tokyo ）M. Ueda （ Tokyo ）Y. Tanuma(Akita ）Y. Nazarov(Delft)M. Sigrist (ETH)Y. Fominov (Landau Institute)J. Linder (Tronheim)S. Kawabata(AIST)

Experiment

S. Kashiwaya （ AIST ）Y. Maeno (Kyoto)Y. Ando (Osaka)M. Koyanagi (AIST)

(1) Theory of Tunneling Conductance in Superconducting Topological Insulator

A. Yamakage, K. Yada, M. Sato and Y. Tanaka

(2) Majorana fermion and odd-frequency Cooper pair

Y. Asano and Y. Tanaka

Phys. Rev. B 85 180509(R) 2012

arXiv: 1204.4226

Surface Andreev bound state (ABS) up to now

(1)d-wave (cuprate)(2)chiral p-wave (Sr2RuO4) (3)helical (NCS superconductor) (4)3d superconductor (superfluid 3He)

The presence of ABS is supported by the bulk topological invariant.

Y. Tanaka, M. Sato and N. Nagaosa, J. Phys. Soc. Jpn. 81 011013 (2012)

Tunneling effect in unconventional superconductors

s-wave

Normal metal

Cuprate

Unconventional superconductor

？Important issue ofcuprate in the 90s.

Tunneling conductance in d-wave junction

Normal metal d-wave superconductor

angle between the normal to the interface and the lobe directionBulk ldos (blue line ）

Zero bias conductance peak

Andreev bound stateSurface zero energy state

L. Buchholtz & G. Zwicknagl : Phys. Rev. B 23 (1981) 5788.J. Hara & K. Nagai : Prog. Theor. Phys. 74 (1986) 1237.

C.R. Hu : Phys. Rev. Lett. 72 (1994) 1526.

Y. Tanaka & S. Kashiwaya: Phys. Rev. Lett. 74 (1995) 3451.

Conductance formula in unconventional superconductor

Condition for ABS

surfaceFlat zero energy band C.R. Hu : Phys. Rev. Lett. 72 (1994) 1526.

transparency

（ Tanaka and Kashiwaya PRL 74 3451) Bruder (1990)Blonder TinkhamKlapwijk (1982)

Surface

Phase change of pair potential is π

Tanaka Kashiwaya PRL 74 3451 (1995), Kashiwaya, Tanaka, Rep. Prog. Phys. 63 1641 (2000) Hu(1994) Matsumoto Shiba(1995)

ー

ー＋＋

Well known example of Andreev bound states in d-wave superconductor

ky

ABS in d-wave

y

Flat dispersion!!Zero energy

(110)direction





Extension to spin-triplet superconductor

–1 0 10

1

2

3

py

px

eV/

T(e

V)

px

+ipy

J. Phys. Soc. Jpn. 67, 3224 (1998)Phys. Rev. B. 56, 7847 (1997)

Normal metal superconductor

L. Buchholtz & G. Zwicknagl : Phys. Rev. B 23 (1981) 5788.J. Hara & K. Nagai : Prog. Theor. Phys. 74 (1986) 1237

Condition for ABS

chiral p

px

surface

surface

flat dispersion

linear dispersion

Chiral superconductor Sr2RuO4

Maeno (1994)

yx ipp

Similar structure to cuprate

Edge surface current

Recent experiment of Sr2RuO4

Experiment

Sr2RuO4 Au

S/I/N

SiO2

It is possible to fit experimental data taking into account of anisotropy of pair potential.

Phys. Rev. Lett. 107, 077003 (2011)S. Kashiwaya, et al,

Tunneling spectrum in two-dimensional topological superconductors

dx2-y2-wavenodal gap

chiral p-wavefull gapchiral edge state

broad zero-bias peak due to linear dispersion

0.95

1

1.05

1.1

0 0.5 1 1.5 2

q/p

E/

q/p

E/

theory

expt.

S.Kashiwaya, 1995

-

-

Injected angle

Angle resolved conductance

Injected angleKashiwaya et al, Phys. Rev. Lett. 107, 077003 (2011)

YBCO(110)

zero energy flat band of surface states

Sr2RuO4





Andreev bound state in the presence of spin-orbit coupling

Iniotakis, Tanaka 　 et al, Phys. Rev. B 76, 012501 (2007)

Spin-singlet （ s-wave ） s spin-triplet(p-wave ） p

Andreev bound state

CePt3Si

Zero bias conductance peak by Andreev bound state

No Andreev bound state

Bulk energy gap

Bulk energy gap

Helical superconductor

Gap closesNo Andreev bound state

Calculated conductance

Feature of the Andreev bound statesdxy-wave Chiral p-wave NCS (Helical)

Hu(94)

Tanaka Kashiwaya (95)

Tanaka Kashiwaya (97)Sigrist Honerkamp (98)

Non-centrosymmetric superconductor (NCS)

Iniotakis (07)Eschrig(08)Tanaka (09)

HelicalChiral Flat

-wave p+s -wave

Flat dispersion of ABS in NCS superconductor

P. M. R. Brydon et al, PRB11

3d case LaAlO3SrTiO3

Edge 　

(mixing of d and p-wave pairing)

2d case

K. Yada, et al, Phys. Rev. B Vol. 83 064505 (2011)

Flat ABS one of the Fermi surface is absent by SO coupling

Superconducting Materials where zero bias conductance peak by ABS is observed

YBa2CuO7-d (Geerk, Kashiwaya, Iguchi, Greene, Yeh,Wei..)

Bi2Sr2CaCu2Oy (Ng, Suzuki, Greene….)

La2-xSrxCuO4 (Iguchi)

La2-xCexCuO4 (Cheska)

Pr2-xCexCuO4 (R.L.Greene)

Sr2RuO4 (Mao, Maeno, Laube,Kashiwaya)

k-(BEDT-TTF)2X, X=Cu[N(CN)2]Br (Ichimura)

UBe13 (Ott)

CeCoIn5 (Wei Greene)

PrOs4Sb12 (Wei)

PuCoGa5 (Daghero)

Superfluid 3He (Okuda, Nomura, Higashitani, Nagai)





ABS in B-phase of superfluid 3He

21Y. Asano et al, PRB ’03

tunn

elin

g co

nduc

tanc

e

bias-voltage barri

er

no zero-bias peakdue to linear dispersionof surface states

BW state (B-phase in 3He)full gap superconductor

Metalz

xy

z=0BW

Chung, S.C. Zhang (2009)Volovik (2009)

Salomaa Volovik (1988) Schnyder (2008) Roy (2008) Nagai (2009)Qi (2009) Kitaev(2009)

perpendicular injection ZES: Buchholtz and Zwicknagle (1981)

Dirac Cone type ABS

ABS and tunneling conductance

spacedimension gap structure surface

statetunneling conductance

2Dnodal flat band

zero-bias peakfull chiral/helical

3D

nodal flat band

fullBW

helicaldouble peak

superconductingtopological insulator ?

To clarify tunneling conductance in new type of three-dimensional topological superconductor (superconducting topological insulator).

Motivation

Superconducting topological insulatortopological insulator……metallic surface states

surfacestates

Y. S. Hor et al, PRL ’10

23

superconducting topological insulatorCuxBi2Se3

S. Sasaki et al, PRL ’11

tunneling conductance(point contact)

zero-bias peak⇒gapless surface states

new type of three-dimensionaltopological superconductor

L. A. Wray et al, Nature Phys. 10

Superconductivity on the surface statesspin-triplet superconducting gap in bulk not in surface

L. Hao and T. K. Lee, PRB 2011, T. H. Hsieh and L. Fu, PRL 2012

energy bulk

surface

momentum

Electronic states of Bi2Se3

25

energy levels of the atomic orbitalsin Bi2Se3

two low-energy effective orbitals

Se1

Se3

Se2Bi1

Bi2

unit cell of Bi2Se3

Zhang et al, Nature 09

Hamiltonian of a superconducting topological insulator

26

Hamiltonian of a superconducting topological insulator

： orbital (spin) ： spin

full gap point nodes

L. Fu and E. Berg, PRL ’10

s-wave spin-triplet (orbital-singlet) superconductor（ supporting gapless surface states ）

[111] // zfor Bi2Se3

Hamiltonian of the parent topological insulator

Pair potential proposed by Fu and BergEnergy gap spin Orbit

Δ1 full gap singlet intra

Δ2 full gap triplet inter

Δ3 point node along kz direction

singlet intra

Δ4 point node along kx direction

triplet inter

orSeBiSeBiSe

unit cell

SeBiSeBiSe

CuxBi2Se3 Effective orbital pz orbital (No momentum dependence)

Cu

CuIntra-orbital

Liang Fu, Erez Berg, PRL,105, 097001 (2010)

pz orbital

Candidate of CuxBi2Se3

Inter-orbital(orbital triplet) (orbital singlet)

Pairing function in superconducting topological insulator

28L. Fu and E. Berg, PRL ’10

spin singlet

s-wave pairing

topological insulator: two orbitals

spin triplet (orbital singlet)

no surface states gapless surface states

full gap nodal gapfull gap nodal gap

Surface states in topological insulatorsin the normal phase

29

surface statesat the Fermi level


Orbital degrees of freedomis quenched.s-wave spin-triplet superconducting gap is impossible

J. Linder et al, PRL 10 (momentum-dependent case)

on the surface

helical surface states

Superconductivity on the surface states

energy spectrum of topological insulator


energy bulk

surface

momentum

Superconductivity on the surface states

31

spin-triplet superconducting gap in bulk not in surface

L. Hao and T. K. Lee, PRB ’11, T. H. Hsieh and L. Fu, PRL ’12

energy bulk

surface

energy bulk

surface

spin-tripletsuperconductor

twisted spectrum

momentum

Structural transition of ABS

32A. Yamakage, Y, K. Yada, M. Sato, and Y. Tanaka, PRB 12 L. Hao and T. K. Lee, PRB ’11 T. H. Hsieh and L. Fu, PRL ’12

energy

momentum

energylarge chemical potential

cone


33AY, K. Yada, M. Sato, and Y. Tanaka, PRB 12L. Hao and T. K. Lee, PRB ’11 T. H. Hsieh and L. Fu, PRL ’12

energy

momentum

energyat transitiongroup velocity=0


34A.Yamakage, K. Yada, M. Sato, and Y. Tanaka, 2012L. Hao and T. K. Lee, PRB ’11 T. H. Hsieh and L. Fu, PRL ’12

energy

momentum

energysmall chemical potential

caldera


35AY, K. Yada, M. Sato, and Y. Tanaka, 2012

transition

L. Hao and T. K. Lee, PRB ’11 T. H. Hsieh and L. Fu, PRL ’12

energy

transition point:group velocity = 0

Tunneling conductance in full-gap superconducting topological insulators

36

structural transition -> group velocity ~ zero -> large surface DoS

full-gap case

eV/

Metalz

xy

z=0

STI

zero-bias peak even in the full gap case

A. Yamakage , K. Yada, M. Sato, and Y. Tanaka, PRB2012

Summary: Theory of tunneling spectroscopy of

superconducting topological insulators

1. Zero-bias conductance peak is possibleeven in full-gap topological 3d superconductors, differently from the case of BW states.

2. This originates from the structural transition of energy dispersion of ABS.

Yamakage, Yada, Sato, and Tanaka, Physical Review B 85 180509(R) 2012

Josephson effect in s-wave/STI

s-wavesinglet

STIfull gaptriplet

Jose

phso

n cu

rrent

Fu and Berg, PRL 10

Josephson effect in d-wave/N/STI

Jose

phso

n cu

rrent

irrespective of anisotropic pairings

(1) Theory of Tunneling Conductance in Superconducting Topological Insulator

A. Yamakage, K. Yada, M. Sato and Y. Tanaka

(2) Majorana fermion and odd-frequency Cooper pair

Y. Asano and Y. Tanaka

Phys. Rev. B 85 180509(R) 2012

arXiv: 1204.4226

Majorana Fermion and odd-frequency pairing

Kitaev(01); Lutchyn(10), Oleg(10)Beenakker(11), …

Kouwnehoven(12) Science

Nature, News, March(2012)

Spin-orbit couplingZeemanProximity coupling to s-wave

Superconductivity on Nanowire in topological phase is similar to spin-triplet p-wave Kitaev 01

What is odd-frequency pairing

spin- 　 singlet 　 triplet

orbital 　 even- 　 odd

Time (frequency) even- odd

Preexisting Cooper pair　　 (even-frequency ）

Spin-singlet even-parity(BCS , Cuprate )

Odd-frequency Cooper pair

Spin-triplet odd-parity(3He,Sr2RuO4,UPt3 )

Spin-triplet even-parityBerezinskii (1974)

Spin-singlet odd-parity Balatsky Abraham(1992)

Generation of odd-frequency pairing by symmetry breaking

(1)Translational invariance (inversion symmetry) is broken

　　 ESE 　　　 OSO ETO 　　　 OTE (inhomogeneous system, junction, vortex..)(2)Spin rotational symmetry is broken (exchange field) (Efetov, Volkov, Bergeret, Eschrig)

　 ESE 　　　 OTE ETO 　　　 OSO

Fermi Dirac statisticsESE (Even-frequency spin-singlet even-parity)ETO (Even-frequency spin-triplet odd-parity)OTE (Odd-frequency spin-triplet even-parity)OSO (Odd-frequency spin-singlet odd-parity)

(1)(2)(3)

(4)

• ESE (Even-frequency spin-singlet even-parity)• ETO (Even-frequency spin-triplet odd-parity)• OTE (Odd-frequency spin-triplet even-parity)Berezinskii• OSO (Odd-frequency spin-singlet odd-parity)Balatsky,Abraham

Bulk state

ESE (s,dx2-y2 -wave)

ESE (dxy-wave)

ETO (px-wave)

ETO (py-wave)

Sign change(MABS)

No

Yes

Interface-induced symmetry(subdominant component )

Yes

No

ESE + (OSO)OSO +(ESE)OTE + (ETO)ETO + (OTE)

Symmetry of the Cooper pair in junctions(No spin flip)

Phys. Rev. Lett. 99 037005 (2007)

(1) (2) (3) (4)

Mid gap Andreev bound state 　（ MABS ）

Surface

＋ー

ー

ー＋

Odd-frequency pairing＋

MABS

　Low transparent limit(Surface state)

Y. Tanaka, et al Phys. Rev. Lett. 037005 (2007)

(1)(2)(3)(4)

Proximity into 　 DN 　 (Diffusive normal metal)even-parity (s-wave)○ 　 Odd-parity 　 ×　

Bulk state

ESE(s,dx2-y2 -wave)

ESE (dxy-wave)ETO (px-wave)

ETO (py-wave)

Sign change

No

Yes

Interface-induced state(subdominant) Proximity into DN

Yes

No

ESE + (OSO)OSO +(ESE)OTE + (ETO)ETO + (OTE)

ESENo

No

Proximity effect into DN (No spin flip)

Y. Tanaka, et al Phys. Rev. Lett. 037005 (2007)

OTE

Y. Tanaka and Golubov, PRL. 98, 037003 (2007)

(1) (2) (3) (4)

ESE (Even-frequency spin-singlet even-parity)ETO (Even-frequency spin-triplet odd-parity)OTE (Odd-frequency spin-triplet even-parity)OSO (Odd-frequency spin-singlet odd-parity

Case (3) is very interesting!!

Density of states in DN

Conventional proximity effect with Even-frequency Cooper pair in DN

Unconventional proximity effect with Odd-frequency Cooper pair in DN

Tanaka, Kashiwaya PRB 70 　 012507 (2004)

Peak(dip) width, Thouless energy In the actual calculation, DN is attached to normal electrode.

Anomalous proximity effect expected in chiral p-wave superconductor

Asano PRL 99, 067005 (2007)

DN

RD

Odd-frequency triplet s-wave in diffusive normal metal (DN)

LDOSin DN Tanaka

PRB(2005)

Majorana fermion in Nano-wire

Topological(Majorana)

Non Topological

Nano wire on the insulator (diffusive)

(a): non topological (b): topological

Robust zero bias conductance peak independent of disorder

normal superconductor

Charge conductance in nano wire

Similar anomalous charge transport has been clarified in Diffusive normal metal/px-wave superconductor junction in 2004.

Tanaka and Kashiwaya, PRB 2004

arXiv: 1204.4226

(Conventional proximity effect)

0 1 20

1

2

3

RD

R/R

B

(1)

/RB

(3)

(2)

Zero voltage resistanceof the junction

R is independent of RD

(3) px-wave(No proximity effect)

(Anomalous proximity effect)

Anomalous proximity effect in DN/px-wave junction

Tanaka and Kashiwaya PRB (2004)

Majorana fermion in Nano-wire

Topological

Non Topological

non topologicaltopological

normal superconductor

Local density of state in nano wire

Similar anomalous charge transport has been clarified in diffusive normal metal/p-wave superconductor junction in 2004.

Tanaka and Kashiwaya, PRB 2004

robust zero energy peak of LDOS

arXiv: 1204.4226

Anomalous current phase relation of Josephson current

52

topological

non-topological

static Josephson current 2pNon-static Josephson current: 4p

Similar anomalous current phase relation appears in d-wave junction (Tanaka 96, Barash 96) and p-wave junction (Yakovenko 04).

arXiv: 1204.4226

Induced odd-frequency pairing in topological phase

53

Non Topological Topological

Odd-frequency pairing is hugely enhanced in topological phase

arXiv: 1204.4226

Summary: Nano wire hosting Majorana fermion

1. Majorana fermion should be always hosting odd-frequency pairing.

2. Anomalous proximity effect, anomalous charge transport are expected similar to spin-triplet p-wave superconductor junctions.

3. Nano wire is an idealistic system to study anomalous proximity effect expected for spin-triplet px-wave superconductor.

Y. Asano and Y. Tanaka arXiv: 1204.4226

Calculation of surface states

55

STIz

xy

z=0

1. construct the wave function in the STI

2. the coefficient t is determined by the confined condition

: wave function of evanescent state with energy E

Energy Gap functionFull Gap

Point Node

Fu and Berg, Phys. Rev. Lett. 105 097001(2010)

Yamakage et al., PRB 85 180509R(2012)

-2 -1 0 1 20

1

2

3

4

-2 -1 0 1 2-2 -1 0 1 20

1

2

3

4

-2 -1 0 1 2

Local density of state

full gap

point node

E2

Δ1:singlet, full gap Δ2:triplet, full gap

Δ3:singlet, point node Δ4:triplet, point node

Ldos

Energy (E/Δ)

2 -2

-22

Surface state generated at z=0

STI (Superconducting topological insulator)

vacuum

z-axisSTI

Andreev bound state

Hsieh and Fu PRL 108 107005(2012); arXiv: 1109.3464

Normal Cone Caldera Cone

Helical Majorana (Surface state)

Deformed Cone(Only positive spin helicitykx sy – ky sx = +k statesare shown.)

(Only negative energystates are shown.)

(solution of confinement condition y(z=0)=0)

Yamakage et al., arXiv: 1112.5035

Charge transport in normal metal / STI junctions

STI (Superconducting topological insulator)

Normal metal

z-axisSTI

Tunneling conductance between normal metal / superconducting topological insulator junction

Similar to conventional s-wave superconductor

Zero bias conductance peak is possible even for 2 case with full gap Hsieh and Fu PRL 108 107005(2012); arXiv: 1109.3464 Yamakage et al., arXiv: 1112.5035(2011)

Tunneling conductance between normal metal / superconducting topological insulator junction (2)

Full gap case

Point node case

Tunneling conductance strongly depends on the direction of nodes.

Yamakage et al., arXiv: 1112.5035(2011)

Tunneling conductance

63

Andreev bound state (Majorana Fermion ）

Full Gap Point Node


64

Structural transition of Andreev bound state

Transition line


65

Velocity of Majorana fermion along x-direction

Transition line

Josephson effect in singlet/triplet junction

first order Josephson currentsinglet triplet

Josephson current

in the absence of spin-dependent H’

Geshkenbein Larkin 88, Y. Asano et al, PRB 03

Josephson effect in s-wave/STI

s-wavesinglet

STIfull gaptriplet

Jose

phso

n cu

rren

t

Fu and Berg, PRL 10

Josephson effect in d-wave/N/STI

Jose

phso

n cu

rren

t

irrespective of anisotropic pairings

Absence of spin-dependent tunneling

STI

Assumption: The left system has the same or the higher symmetry as STI (D3d).

Rotational symmetry Mirror symmetry

Absence of spin-dependent tunneling

STI

Assumption: The left system has the same or the higher symmetry as STI (D3d).

3D TSCs show a robust sin2jprotected by the symmetry

cf. A spin-dependent tunneling is possible in Sr2RuO4 since the electronic state has higher angular momentum in lower point group symmetry . (Asano)

Documents

Tunneling Conductance and Surface States Transition in Superconducting Topological Insulators