tunneling conductance and surface states transition in superconducting topological insulators yukio...
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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)
Main collaborators
Theory
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 state
Surface zero energy stateL. 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
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)
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
D
D
0.95
1
1.05
1.1
0 0.5 1 1.5 2
/q p
E/D
D
/q p
E/D
Dtheory
expt.
S.Kashiwaya, 1995
-D
-D
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
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)
Andreev bound state in the presence of spin-orbit coupling
Iniotakis, Tanaka et al, Phys. Rev. B 76, 012501 (2007)
Spin-singlet ( s-wave ) Ds spin-triplet(p-wave ) Dp
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 closes
No Andreev bound state
Calculated conductance
Feature of the Andreev bound states
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)
HelicalChiral Flat
-wave p+s -wave
Flat dispersion of ABS in NCS superconductor
P. M. R. Brydon et al, PRB11
3d case LaAlO3
SrTiO3
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)
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)
ABS in B-phase of superfluid 3He
21Y. Asano et al, PRB ’03
tunnelin
g c
on
duct
an
ce
bias-voltage barr
ier
no zero-bias peakdue to linear dispersionof surface states
BW state (B-phase in 3He)full gap superconductor
Metalz
xy
z=0
BW
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 structuresurface
statetunneling conductance
2Dnodal flat band
zero-bias peak
full 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 states
spin-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
Se2
Bi1
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
Cu
Intra-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
L. Hao and T. K. Lee, PRB 2011, T. H. Hsieh and L. Fu, PRL 2012
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
L. Hao and T. K. Lee, PRB 2011, T. H. Hsieh and L. Fu, PRL 2012
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
energy
large chemical potentialcone
Structural transition of ABS
33AY, 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
energyat transitiongroup velocity=0
Structural transition of ABS
34A.Yamakage, K. Yada, M. Sato, and Y. Tanaka, 2012
L. Hao and T. K. Lee, PRB ’11 T. H. Hsieh and L. Fu, PRL ’12
energy
momentum
energy
small chemical potentialcaldera
Structural transition of ABS
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/D
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
ph
son
cu
rren
t
Fu and Berg, PRL 10
Josephson effect in d-wave/N/STI
Jose
phso
n c
urr
en
t
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 coupling
Zeeman
Proximity 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)
ESE
No
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 energyIn 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 topological
topological
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 2p
Non-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 D2 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
Yamakage et al., arXiv: 1112.5035(2011)
64
Structural transition of Andreev bound state
Transition line
Yamakage et al., arXiv: 1112.5035(2011)
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)