Exotic Andreev bound states in topological...

Exotic Andreev bound states in topological superconductors

Yukio Tanaka (Nagoya University)

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

Yukawa Institute (2014)

Main collaborators A. Yamakage (Nagoya) K. Yada (Nagoya)

M. Sato（Nagoya）

T. Hashimoto（Nagoya）

T. Mizushima （Osaka）

S. Takami（Nagoya）

Andreev bound state

Classical example of Andreev bound state

High transparent interface

ABS has been known since 1963.

P.G. de Gennes and D. Saint-James Phys. Lett. 4 151 (1963).

Energy levels of Andreev bound state (nonzero energy)

McMillan(1966), Tomash(1965), Rowell(1966)

magnitude of pair potential x component of Fermi velocity

ABS with nonzero energy

Andreev bound state (non-topological and topological)

Andreev bound state with non zero energy （de Gennes, Saint James 1963) Mid gap (zero energy) Andreev bound state

Surface Andreev bound state Not edge state Edge state Topological Non topological L. Buchholtz & G. Zwicknagl (81):, J. Hara & K. Nagai : Prog. Theor. Phys. 74 (86)

C.R. Hu : (94)

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

Alff, Kashiwaya PRB (1998)

Advance in understanding of Andreev bound state

　 Solution of BdG equation (quasi classical approximation)

　 Topological aspects Analogy to Quantum Hall, Qauntum spin Hall, Topological insulators

conventional

Bulk edge correspondence X. L. Qi, S.C. Zhang, PRL 102, 187001 (2009), Schnyder et. al, PRB 78 195125 (2008) M. Sato and Fujimoto, PRB 79 094504 (2009), Y. Tanaka et. al, PRB 79 060505 (2009)

Flat Andreev bound state and topology

Hamiltonian

Winding number for fixed kｙ

Superconductor

x

Surface

y

Sato, Tanaka, et al, PRB 83 224511 (2011) 　

Midgap Andreev bound state (Winding number)

If we consider a simple Fermi surface

Conventional condition

Sato, Tanaka, et al, PRB 83 224511 (2011) 　

Topological invariant defined in bulk

Winding number & Index theorem (1)

BdG Hamiltonian has a symmetry (chiral symmetry)

Zero energy ABS is an eigenstate of σy.

Number of ZES where the eigenvalue of σy is 1

Number of ZES where the eigenvalue of σy is -1

Index Theorem

From the bulk-edge correspondence, there exists the gapless states on the edge only when integer w1d is nonzero.

Superconductor y

x

dxy-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)

Helical Chiral Flat

-wave p+s -wave

（Topological）Andreev bound states

Andreev bound state (topological edge state) and topological invariant

Tanaka, Sato, Nagaosa J. Phys. Soc. Jpn 81 011013 (2012)

Andreev bound state

Topological invariant

Flat

Chiral

Helical

Cone

Time reversal symmetry

Materials

Theory of tunneling

Insulator （semi-metal）

1d winding Number Z

○ Cuprate pｘ-wave

QHS

QSHS (2D Topological insulator)　

○

○

2d winding Number Z

3d winding Number Z

×

s＋ｐ-wave （NCS）

Topological insulator

Z2

Graphene (zigzag edge)

Sr2RuO4 3He A

3He B

PRL (1995)

JPSJ(1998)

PRB (1997)

PRB (2007)

PRB (2003)

CuxBi2Se3

Topological insulator Bi2Se3

・Strong spin-orbit coupling

・Nonzero topological number Z2

・Helical Dirac Cone as a surface state

Electronic band structure of Bi2Se3 measured by ARPES

Bi2Se3

Y. L. Chen et al. Science 329, 659 (2010)

Crystal structure Bi2Se3

Electronic states of Bi2Se3

15

energy levels of the atomic orbitals in Bi2Se3

two low-‐energy effec;ve orbitals

Se1

Se3

Se2 Bi1

Bi2

unit cell of Bi2Se3

Zhang et al, Nature 09

bonding crystal field

SO coupling

Superconducting topological insulator

Y.S.Hor et al, PRL 104, 057001 (2010)

Cu doped topological insulator

M. Kriener et al., PRL 106, 127001 (2011)

CuxBi2Se3

Resistivity Specific heat

Tc 3.8K

Model Hamiltonian　(Normal state )

Model Hamiltonian 　(superconducting state)

H.Zhang et al, Nature Phys. 5, 438 (2009)

BdG Hamiltonian

Pauli matrix 　σ : orbital, s : spin, τ : particle-hole

8 × 8 matrix

unit cell Se Bi Se Bi Se

Cu

pz orbital Bi2Se3 Cu0.05Bi2Se3

ARPES

Effective Hamiltonian of CuxBi2Se3　

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 pair potentials

Inter-orbital

Energy gap irreducible representation

spin Orbital parity

Matrix form

Δ1a Δ1b

full gap singlet even

Δ2 full gap triplet odd

Δ3 point node singlet odd

Δ4 point node triplet odd

orbital 1

orbital 2

orbital 1

orbital 2

Pair potential Irreducible representation spin orbital Gap

structure

Parity (orbital

inversion)

Topological number

A1g� Singlet Intra inter full gap even No

�A1u� triplet inter full gap odd DIII Z

A2u� singlet intra Point node (z-direction) odd DIII Z2

Eu� triplet inter

Point node

[x(y)-direction]

odd DIII Z2

Topological natures of four pairings 　

Supplementary materials in S. Sasaki et al PRL 107 217001 (2011)

Energy Gap function Full Gap

Point Node

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

Yamakage et al., PRB Rapid (2012)

spin-singlet orbital-evenspin-triplet inter-orbital

spatial inversion odd

spatial inversion odd

-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

Bulk local density of state

full gap

point node

E2

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

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

Energy (E/Δ)

2 -2

-22

Surface state generated at z=0

STI (Superconducting topological insulator)

vacuum

z-axisSTI

Obtained Dispersions of Andreev bound state

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

Normal Cone Caldera Cone Deformed Cone

(Only positive spin helicity kx sy – ky sx = +k states are shown.)

(Only negative energy states are shown.)

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

A. Yamakage, PRB, 85, 180509(R) (2012)

spin-triplet inter-orbital spatial inversion odd-parity

A. Yamakage, K. Yada, M. Sato, and Y. Tanaka, PRB 2012

transition

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

energy

transition point: group velocity = 0

Structural transition of the dispersion of ABS

(similar to B phase of 3He)

Although original Hamiltonian is a 8 × 8　matrix, is it possible to make a compact analytical formula

of Andreev bound state?

Yes. If we concentrate on the low energy excitation near the Fermi energy, it is possible.

Fermi surface direction of d-vector

Quasi-classical theory Yip(PRB 2013) Y. Nagai ( 2013)

Bulk Analytical formula of ABS

No

No

No (along z-direction) (In other directions, yes)

Bulk Energy spectrum and ABS based on quasi-classical approximation

Takami, Yada, Yamakage, Sato, and Tanaka, J. Phys. Soc. Jpn. 83 064705 (2014)

Charge transport in normal metal / STI junctions

STI (Superconducting topological insulator)

Normal metal

z-axisSTI

Conductance between normal metal / superconducting topological insulator junction (numerical calculation)

Similar to conventional spin-singlet s-wave superconductor

Zero bias conductance peak is possible even for Δ2 case with full gap Hsieh and Fu PRL 108 107005(2012); arXiv: 1109.3464 A. Yamakage, PRB, 85, 180509(R) (2012)

Conductance between normal metal / STI junction

Point node case

Tunneling conductance strongly depends on the direction of nodes.A. Yamakage, PRB, 85, 180509(R) (2012)

(Spatial inversion odd-parity)(numerical calculation)

Conductance based on quasi-classical approximation

(Tansmissivity ) （Tansmissivity in normal state）

Conductance formula available in (single band) unconventional superconductors

（Tanaka and Kashiwaya PRL 74 3451 1995)

transparency

conventional s-wave

Calculcated tunneling spectroscopy of STI

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

U-‐shaped gap structure

ZBCP or ZBCP spliJng

No ZBCP (ZBCP is possible from in-‐plane junc;on)

ZBCP

Experiments by Sasaki can be explained by Δ2 or Δ4 pairing.

Takami, Yada, Yamakage, Sato, and Tanaka, J. Phys. Soc. Jpn. 83 064705 (2014)

Summary (1) Theory of tunneling spectroscopy of STI

1. Δ2 and Δ4 are consistent with point-contact experiment by Ando’s group. 2. Zero-bias conductance peak is possible even in full-gap topological 3d superconductors, differently from the case of BW in 3He. 3. Structural transition of the energy dispersion of ABS.

4. Analytical formula of ABS and conductance for fitting experiment is available now.

Yamakage, Yada, Sato, and Tanaka, Physical Review B 85 180509(R) 2012 Takami, Yada, Yamakage, Sato, and Tanaka, J. Phys. Soc. Jpn. 83 064705 (2014)

Tunneling spectroscopy SEM image of an actual sample

(~50 nm)

Sn CuxBi2Se3

meV75.0468.0Z

=Γ

=

S. Sasaki et al PRL 107 217001 (2011)Ando’s group (Osaka)

Conventional BCS

Status of tunneling experiments

•  G. Koren, et al, Phys. Rev. B 84, 224521 (2011). •  T. Kirzhner, et. al, Phys. Rev. B 86, 064517 (2012). •  G. Koren and T. Kirzhner, Phys. Rev. B 86, 144508 (2012).

35

Consistent with Ando’s group with ZBCP

Contradict with Ando’s group with full gap (STM)•  N. Levy, et al, Phys. Rev. Lett. 110 117001 (2013) •  (Peng et al, Phys. Rev. B 88 024515 (2013) )

We must need further experimental and theoretical research.

Composition and crystal structures of the actual samples have not fully clarified yet.

Conflicting report which does not support topological superconductivity

•  Self-consistent determination of pair potential •  Surface-Dirac cone stemming topological

insulator •  Evolution of Fermi surface

Levy et al., PRL 110, 117001 (2013)

T. Mizushima How to resolve this discrepancy?

Mizushima, Yamakage, Sato, Tanaka, arXiv:1311.2768

orbital 1

orbital 2

Se1 Bi1 Se2 Bi1’ Se1’

orbital 1

orbital 2


orbital 1

orbital 2


V V U

Short-‐range electron density-‐density interac;on

intra-‐orbital spin-‐singlet inter-‐orbital spin-‐singlet/triplet

Fu and Berg, PRL 105, 097001 (2010)

We have determined the spatial dependence of the pair potential.

Mizushima, Yamakage, Sato, Tanaka, PRB 90 184516(2014)

Electron interaction and pair potentials in doped TI

Bulk: non-‐topological s-‐wave pairing

Inevitable mixture of additional component due to the orbital polarization.

The separation between the conduction band and Dirac Fermi momentum of the Dirac cone kFD.

The large magnitude of δ induces the splitting of coherence peak of SDOS.


Spatial dependence of pair potential and LDOS

surface

CB

VB

Dispersion of surface Dirac cone

LDOS for orbital 1 LDOS for orbital 2

Orbital polarization within the penetration depth of the surface Dirac fermions

wave func;ons of surface Dirac cone

“Penetra;on depth” of the Dirac cone

surface surface



orbital 1

orbital 2

surface

Orbital Polarization of surface Dirac cone (normal state)

Wave func;ons of surface Dirac cone

LDOS: orbital 1 LDOS orbital 2

orbital 1: bulk orbital 2: bulk + surface Dirac cone

orbital polarization ⇒ drives a surface parity mixing

Orbital polarization at the surface (superconducting) Total LDOS


orbital 1

orbital 2

surface

orbital 2

orbital 1

enhancement of


from SCF calculation at surface

Levy et al., PRL 110, 117001 (2013)

Bulk s-‐wave + Dirac cone Bulk s-‐wave without Dirac cone

Orbital Parity-‐mixing by the surface Dirac fermions Extra coherence peak besides a conven;onal peak at the bulk gap

Surface superconducting gap (splitting of single coherent peak) results from the synergy effect between the orbital polarization of the surface Dirac cone and the parity mixing of the pair potential.

Surface density of state


Simple U-shaped STM spectrum in CuxBi2Se3 is not originated from bulk s-wave pairing in the presence of surface Dirac cone existing in normal state.

STS data does not exclude the possible topological superconductivity.

Levy et al., PRL 110, 117001 (2013)


Bulk topological odd-parity pairing

A1u state

There is no mixing of additional components at the surface.


Evolution of the Fermi surface Odd parity pairing

QCP T. Hashimoto et al 2014

Surface DOS of A1u state (Evolution of Fermi surface)

U-shaped Fermi LDOS is possible by the evolution of the Fermi surface from spheroidal to cylinder

Lahoud, PRB2013

Calculation of conductance odd parity pairing by changing Fermi surface

Evolution of Fermi surface can produce U shaped gap like spectra in odd parity pairing.

CuxBi2Se3 is still a strong candidate of topological superconductor.


Summary of SCF calculation

(1)If a topologically trivial bulk s-wave pairing symmetry is realized, the resulting surface density of state hosts an extra coherent peak at the surface induced gap besides a conventional peak.

(2)No such surface parity for topological odd-parity superconductors

(3) The simple U-shaped scanning tunneling microscope spectrum in CuxBi2Se3 does not originate from s-wave superconductivity.

(4)Bulk odd-parity pairing gives a consistent understanding on the ZBCP observed in the point-contact tunneling spectroscopy and the U-shaped form of STS in CuxBi2Se3.


Crossed surface flat bands of doped Weyl semimetal superconductors

Lu Bo K. Yada

Lu Bo, K. Yada, M. Sato and Y. Tanaka arXiv:1406.3804

M. Sato

Wyle semimetal ・Zero-gap semiconductor caused by band crossing ・Linear dispersion near the band crossing point ・Breaking the time reversal symmetry or the inversion symmetry is necessary

Nodal point is stable against the perturbation

・Topological insulator/normal insulator superlattice(Burkov and Balents, PRL(2011)) ・Pyrochlore iridates (X. Wan, et al., PRB(2011)) ・HgCr2Se4 (G. Xu, et al., PRL(2011))

Weyl semimetal (candidate system)

X. Wan, et al., PRB(2011)

•  Key feature of Weyl semimetal: Fermi arc

Pyrochlore Iridates

Hamiltonian of Wyle superconductor

Weyl points at (0, 0,±Q) where M=tzcosQ

On the north pole and south pole, the directions of spin are the same, and therefore, s-wave pair potential can not open a gap. point nodes

G. Y. Cho, J. H. Bardarson, Y.-M. Lu, and J. E. Moore PRB(2012)

M term: Time reversal symmetry breaking

Surface states of superconducting Weyl semimetal

Dispersion of the surface state Quasiparticle spectra at E=0

Bo Lu, K. Yada, M. Sato and Y. Tanaka arXiv:1406.3804

Chern Number C1

-

+

-

+


3He A-phase

Weyl semimetal

magnetic mirror reflection symmetry

+ particle-hole

symmetry 3He A-phase (single fermi surface)

Doped Weyl semimetal≠ 3He A-phase×2 The existence of the nontrivial state (Fermi arc from normal state Weyl semimetal)

Surface Andreev bound states of superconducting Weyl semimetal

magnetic mirror reflection symmetry

+ particle-hole symmetry

Surface Andreev bound states of superconducting Weyl semimetal

Can not connect

particle-hole symmetry


Obtained conductance


Summary of topological superconductor in doped Wyel semimetal

(1)We have obtained anomalous dispersion of Andreev bound state in superconducting doped Weyl semimetal with Fermi arcs in normal state.

(2)The Fermi arcs enable to support crossed flat bands of Andreev bound state.

(3)This flat band produces large Zero bias conductance peak.

arXiv:1406.3804

Exotic Andreev bound states in topological...

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