Upload
others
View
8
Download
0
Embed Size (px)
Citation preview
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
Se1 Bi1 Se2 Bi1’ Se1’
orbital 1
orbital 2
Se1 Bi1 Se2 Bi1’ Se1’
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.
Mizushima, Yamakage, Sato, Tanaka, PRB 90 184516(2014)
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
Se1 Bi1 Se2 Bi1’ Se1’
Se1 Bi1 Se2 Bi1’ Se1’
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
Se1 Bi1 Se2 Bi1’ Se1’
orbital 1
orbital 2
surface
orbital 2
orbital 1
enhancement of
Mizushima, Yamakage, Sato, Tanaka, PRB 90 184516(2014)
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
Mizushima, Yamakage, Sato, Tanaka, PRB 90 184516(2014)
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)
Mizushima, Yamakage, Sato, Tanaka, PRB 90 184516(2014)
Bulk topological odd-parity pairing
A1u state
There is no mixing of additional components at the surface.
Mizushima, Yamakage, Sato, Tanaka, PRB 90 184516(2014)
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.
Mizushima, Yamakage, Sato, Tanaka, PRB 90 184516(2014)
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.
Mizushima, Yamakage, Sato, Tanaka, PRB 90 184516(2014)
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
-
+
-
+
Bo Lu, K. Yada, M. Sato and Y. Tanaka arXiv:1406.3804
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
Bo Lu, K. Yada, M. Sato and Y. Tanaka arXiv:1406.3804
Obtained conductance
Bo Lu, K. Yada, M. Sato and Y. Tanaka arXiv:1406.3804
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