Odd-Frequency Superconductivity in Topological Insulators and Multiband Superconductors

Odd-Frequency Superconductivity in Topological Insulators and

Multiband Superconductors

Annica Black-Schaffer

Nordita July 25^th, 2016

Outline

•  Introduction to odd-frequency pairing

•  Odd-frequency pairing in topological insulator-superconductor hybrid structures

–  Spin-singlet s-wave superconductor –  Spin-triplet p-wave superconductor

•  Odd-frequency pairing in multiband superconductors

Outline

•  Introduction to odd-frequency pairing

•  Odd-frequency pairing in topological insulator-superconductor hybrid structures

–  Spin-singlet s-wave superconductor –  Spin-triplet p-wave superconductor

•  Odd-frequency pairing in multiband superconductors

Superconducting Symmetries

The superconducting order parameter is fermionic:

spin-singlet s-wave ^or spin-triplet p-wave

The order parameter can also be odd in time/frequency: ^[1]

odd-frequency spin-triplet s-wave

orbital

spin

[1]: Berezinskii, JETP Lett. 20, 287 (1974)

Odd-frequency (ω) Pairing

BCS order parameter:

vanishes for an odd-frequency component

Equal-time odd-frequency order parameter: ^[1,2]

Theory proposals for odd-frequency bulk superconductors exists ^[1,2]

but only found so far at interfaces

[1]: Abrahams et al, PRB 52, 1271 (1995), [2]: Dahal et al, NJP 11, 065005 (2009)

S|F Interface

Spin-singlet s-wave pairing in SC converted into odd-frequency spin-triplet s-wave pairing in FM

•  Long-range superconducting proximity effect in the FM

•  s-wave = robust against impurities

singlet SC FM

Bergeret et al, RMP, 77, 1321 (2005), [1]: Eschrig, Phys. Today 64, 43 (2011)

[1]

S|N Interface

Spin-singlet s-wave pairing in SC converted into odd-frequency spin-singlet p-wave pairing

•  Only high-transparency junctions

•  p-wave = only ballistic systems

singlet SC Normal metal

[1]: Tanaka et al, PRL 99, 037005 (2007)

[1]

Outline

•  Introduction to odd-frequency pairing

•  Odd-frequency pairing in topological insulator-superconductor hybrid structures

–  Spin-singlet s-wave superconductor –  Spin-triplet p-wave superconductor

•  Odd-frequency pairing in multiband superconductors

Topological Insulator (TI)

Surface state of a topological insulator

–  Dirac spectrum

–  Momentum locked to spin: H ~ k ^Ÿ σ

E

kx ky

TI – SC Hybrid Structure

SC

TI _H

TI = X

k,↵,

c^†_↵,kk · ^↵ c _,k H_SC = X

k,↵,

"(k)d^†_↵,kd_↵,k + X

i,↵,

(i)_↵ d^†_↵,id^†_,i + H.c.

H_T = X

↵

T_ic^†_↵,id_↵,i + H.c.

Conventional s-wave SC with position (i) dependent order parameter

Local tunneling

TI surface state

ABS and Balatsky, PRB 86, 144506 (2012)

Analytic Derivation

Anomalous Green’s function in TI:

Order parameter for s-wave odd-frequency pairing:

à Odd-frequency spin-triplet s-wave pairing:

–  Spatially inhomogeneous SCs

@_⌧Fˆ_TI(⌧|i)|⁰ ⇠ @

@x

ABS and Balatsky, PRB 86, 144506 (2012)

= + + ...

TI TI TI SC TI

T T

S|N Junction in a 2D TI

TI SC y x

Kane-Mele 2D TI

F_s(⌧|i) = (hcⁱ#(⌧ )c_i"(0) c_i"(⌧ )c_i#(0)i)/2

F_t(⌧|i) = (hcⁱ#(⌧ )c_i"(0) + c_i"(⌧ )c_i#(0)i)/2

Spin-singlet s-wave ^pairing:

Spin-triplet s-wave pairing:

−20

0

20

10 0 0 0.1

|Fs|

x y

t = 0

−20

0

20

10 0 0 0.01

x y

t = 1

|Ft|

−20

0

20 10 0

0 0.01

t = 6

|Ft|

−10 0 10

0

0.01 ⇥ : @^⌧F_t|⁰ : @F_s

@x

Δ = 0.16 Δ= 0.3

ABS and Balatsky, PRB 86, 144506 (2012)

In-surface Supercurrent

In-surface supercurrent: ⁼ _{| |e}^ıkx

I

⇥ : @^⌧F_t|⁰ : @F_s

@x

0 0.1 0.2 0.3

0 0.02 0.04

0 0.3

0.19 0.2

k

|∂F|

Ratio

ABS and Balatsky, PRB 86, 144506 (2012)

Gradient-Induced Odd- ω Pairing

•  Electric field induced sublattice staggering in silicene and stanene

•  Linear k-dependence of the pairing in a p-wave superconductor:

Odd-frequency s-wave spin-triplet pairing

Kuzmanovski and ABS (in preparation), ABS and Balatsky, PRB 87, 220506(R) (2013),

Effective d-vector

Odd-frequency pairing in TIs

Odd-frequency pairing in TI-SC hybrid structures

•  Spin-singlet s-wave SC with in-plane gradient à

Odd-frequency spin-triplet s-wave pairing

•  SN junctions

•  Supercurrents

•  Sublattice staggering

•  Spin-triplet p-wave SC à

Odd-frequency spin-triplet s-wave pairing

Outline

•  Introduction to odd-frequency pairing

•  Odd-frequency pairing in topological insulator-superconductor hybrid structures

–  Spin-singlet s-wave superconductor –  Spin-triplet p-wave superconductor

•  Odd-frequency pairing in multiband superconductors

Bi

Se

– SC Hybrid Structure

SC Bi₂Se₃

2D superconductor

Bi₂Se₃ (τ orbital Pauli matrix) ^[1]

ABS and Balatsky, PRB 87, 220506(R) (2013), [1]: Rosenberg and Franz, PRB 85, 195119 (2012)

Local tunneling

Superconductivity in Bi

Se

Proximity-induced superconductivity in Bi₂Se₃ SC

Classification of all superconducting symmetries in Bi₂Se₃

–  Spin-singlet/triplet, spatial (s/d/p-wave), even/odd-frequency, even/odd orbital

ABS and Balatsky, PRB 87, 220506(R) (2013)

Frequency and Interband Index

Complete reciprocity between oddness in frequency and orbital index

Generic property for multiband superconductors

ABS and Balatsky, PRB 87, 220506(R) (2013)

Multiband Superconductors

•  S: Spin (spin-singlet: S = 0 or spin-triplet: S = 1)

•  P: Spatial parity (even: s-,d-wave or odd: p-,f-wave)

•  T: Time (even or odd-frequency)

•  O: Orbital/band parity

Spin-singlet s-wave: TO = 1

ABS and Balatsky, PRB 88, 104514 (2013)

Two-Band SC with Band Hybridization

Bands (orbitals) a & b with finite interband hybridization/scattering Γ:

ABS and Balatsky, PRB 88, 104514 (2013), Komendova, Balatsky, and ABS, PRB 92, 094517 (2015)

Interband pairing F₁₂:

Interband Pairing

Perturbation theory to infinite order in Γ:

(using a geometric series)

Odd-interband:

Even-interband:

D =

Komendova, Balatsky, and ABS, PRB 92, 094517 (2015)

Interband pairing: Γ ≠ 0

Odd-frequency, odd-interband pairing: Γ ≠ 0, Δ₁ ≠ Δ₂

Interband Frequency Dependence

Odd-frequency Even-frequency

Δ₂ = 2.5 meV, Γ = 3 meV

Komendova, Balatsky, and ABS, PRB 92, 094517 (2015)

Odd-frequency, odd-interband pairing: Γ ≠ 0, Δ₁ ≠ Δ₂

0 300 600 900 1200

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 0.5 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

0 500 1000 1500 2000 2500

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 2 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

0 700 1400 2100 2800 3500

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 2.5 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

0 500 1000 1500 2000 2500

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 2.8 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

0 400 800 1200 1600

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 4.5 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

0 500 1000 1500 2000

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 7.5 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

0 500 1000 1500 2000 2500

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 10 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

0 500 1000 1500 2000

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 15 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

(a) (b)

(c) (d)

(e) (f)

(g) (h)

0 300 600 900 1200

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 0.5 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

0 500 1000 1500 2000 2500

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 2 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

0 700 1400 2100 2800 3500

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 2.5 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

0 500 1000 1500 2000 2500

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 2.8 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

0 400 800 1200 1600

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 4.5 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

0 500 1000 1500 2000

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 7.5 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

0 500 1000 1500 2000 2500

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 10 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

0 500 1000 1500 2000

-20 -15 -10 -5 0 5 10 15 20 DOS [states/eV/nm3 ]

E - E_F[meV]

∆₁= 15 meV,∆₂= 2.5 meV, Γ = 3 meV NN₁

N₂

(a) (b)

(c) (d)

(e) (f)

(g) (h)

DOS for Two-Band Superconductor

Komendova, Balatsky, and ABS, PRB 92, 094517 (2015)

Additional gaps with coherence peaks at high energies Only appears with odd-frequency pairing

Hybridization Gaps

Komendova, Balatsky, and ABS, PRB 92, 094517 (2015)

Intraband SC gaps Hybridization

SC gaps

hole electron

Hybridization gaps

•  High-energy gaps with pronounced coherence peaks

•  Only appears for finite odd-frequency pairing

Two bands Two bands + Superconductivity

Multiband Superconductors

Odd-frequency pairing in multiband superconductors

–  Odd-frequency, odd-interband pairing if there exist interband pairing

•  Finite interband hybridization (+ non-identical intraband pairing)

•  Hybridization gaps only if odd-frequency pairing is present

•  TI-SC hybrid structures

•  Iron-based superconductors, heavy fermion superconductors, Sr₂RuO₄, MgB₂, …

ABS and Balatsky, PRB 88, 104514 (2013), Komendova, Balatsky, and ABS, PRB 92, 094517 (2015)

Summary

•  Odd-frequency pairing in TI-SC hybrid structures

–  Spin-singlet s-wave SC with in-plane gradient à

Odd-frequency spin-triplet s-wave pairing –  Spin-triplet p-wave SC à

Odd-frequency spin-triplet s-wave pairing

•  Odd-frequency pairing in multiband superconductors

–  Odd-frequency, odd-interband pairing if there is interband pairing –  Gives hybridization gaps

Acknowledgements

Collaborators:

Alexander Balatsky (LANL/Nordita) Jacob Linder (NTNU)

In Uppsala:

Lucia Komendova Dushko Kuzmanovski Kristofer Björnson

Fariborz Parhizgar (IPM)

The Carl Trygger Foundation

Funding: