Topological Superconductivity

Annica Black-Schaffer annica.black-schaffer@physics.uu.se

Materials Theory

6th MaNEP winter school, Saas Fee, January 2015

Contents •  Introduction to superconductivity

–  BCS theory –  BdG formulation

•  Unconventional superconductivity

•  Topological matter

•  Topological superconductivity –  p+ip and d+id superconductors –  “Spinless” superconductors – Majorana fermions

Topological Superconductors

“Spinless” superconductors and Majorana fermions

BdG System Classification Topological classes for BdG systems

Schnyder et al., PRB 78, 195125 (2008), [1]: Tewari et al., PRL 109, 150408 (2012)

In addition: Spinless p+ip-wave in 1D belongs to BDI (not D) because effective TRS [1]

“Spinless” p+ip Superconductor •  No known intrinsic “spinless” p+ip SC

•  Multiple proposals for engineered “spinless” p+ip superconductor structures in 1, 2D last ~7 years

•  Hosts Majorana fermions at surfaces and vortices (topological boundary states) in 1, 2D [1]

Read and Green PRB 61, 10267 (2000); Kitaev, arXiv:cond-mat/0010440 (2001)

Schrödinger, Dirac, and Majorana

Schrödinger (1925) � ~22m

r2 = i~ @@t

4X

µ=0

i~�µ@µ = mc

Dirac (1928) relativistically

correct

•  Spin-1/2 •  Electron & positron (hole)

Majorana (1937) 4x4 complex matrices

4x4 imaginary matrices

•  Particle = Antiparticle:

•  Electron “=“ 2 Majorana fermions:

3X

µ=0

i~�µ@µ = mc

c† 6= c

�† = �

c† = �1 + i�2

Majorana Fermions •  Majorana fermion

–  Real (fermionic) solution to a Dirac-like equation –  Its own antiparticle –  1 electron “=“ 2 Majorana fermions ! electron splitting

–  Occurrence •  Neutrino? •  Supersymmetry partners? •  Superconductors?!

–  emergent excitation: Majorana bound/zero state/mode

The Return of the Majorana?

Majorana in BdG How can we get ½ electron in the BdG formalism?

Never if

! not in spinfull p+ip or d+id superconductors

But if

! 1 electron represented by 2 vector components

! ½ electron states if eigenstate has no degeneracy

! Majorana fermion possible (necessary but not sufficient criterion)

ha†k0�

ak0�

i = (1 + eEk0/kBT

)

�1

(c†c† � F †)(cc� F )

�k =

X

k0

Vk,k0Fk0=

X

k0

Vk,k0hc�k0#ck0"i

�k = �1

2

X

k0

Vk,k0�k0

Ek0

�k = �1

2

X

k0

Vk,k0�k0

Ek0tanh[Ek0/(2k

B

T )]

˜H = a†k

✓"(k)�

0

ˆ

�(k)ˆ

�

†(k) �"(k)�

0

◆ak

H = (� ⇥ k) · z

H = ~vF

(� ⇥ k) · z

Uc

⇢(EF

) = 1

Vpot

= V �(r)

Eres

/ 1

V

2

Excitations in Superconductors Quasiparticles in a superconductor are: •  Part electron and part hole

•  Mixed spin-up and spin-down

!  E = 0 states are Majorana fermions if we ignore spin

But … •  Superconductors often have an energy gap

–  Topological SCs have E = 0 boundary states

•  E = 0 states are often spin-degenerate –  2 Majorana ! 1 electron

! Need a “spinless” topological superconductor

Additional formulas

January 9, 2015

|uk|2 + |vk|2 = 1

Fk = hc�k#ck"i

|uk|2 + |vk|2 = 1

|uk|2 = |vk|2 = 1

2

a = uc†" + vc#

a† = u⇤c" + v⇤c†#

ak"| BCS

i = 0

a†�k#| BCS

i = c†�k#

Y

k0 6=k

(uk0+ vk0c†k0"c

†�k0#)|0i

X

k,k0

Vk,k0c†k"c†�k#c�k0#ck0" =

X

k,k0

Vk,k0[(c†k"c

†�k# � F †

k) + F †k][(c�k0#ck0" � Fk0

) + Fk0 ⇡X

k,k0

Vk,k0(Fk0c†k"c

†�k# + F †

kc�k0#ck0" � F †kFk0

]

1

+ = +1 + -2 = -1

h(-E) e(E) †

=

h e

Kitaev’s 1D Toy Model •  1D chain of spinless electrons with superconducting pairing

i = 1 2 3 …. i-1 i i+1 … N

Nearest neighbor hopping

Spinless pairing Chemical potential

Kitaev, arXiv:cond-mat/0010440 (2001)

Kitaev’s 1D Toy Model

Change basis Majorana fermions

i = 1 2 3 …. i-1 i i+1 … N

A:

B:

Trivial Phase

i = 1 2 3 …. i-1 i i+1 … N

A:

B:

Topological trivial phase: Δ = t = 0, µ < 0

Unique ground state •  Vacuum state for the electrons •  Bulk gap (|µ| lowest excitation energy)

Non-Trivial Phase

i = 1 2 3 …. i-1 i i+1 … N

A:

B:

Topological non-trivial phase: µ = 0, Δ = t ≠ 0 Majorana

Majorana

Degenerate ground state •  Bulk gap (t) •  Zero-energy Majorana modes at the boundary

Spinless Superconductors Need effective spinless topological superconductors •  Spinless ~ spin-triplet ! p-wave

–  1D spinless topological p-wave state has Δ real (class BDI)

–  2D spinless topological p-wave state has Δ ~ kx+iky (p+ip) (class D)

How do we get rid of the spin DoFs?

Superconducting TI Surface TIs surface states: • 

•  Spin set by momentum ! “spinless”

Add conventional s-wave superconductivity by proximity effect:

E

kx ky

Ek =

p"(k)2 + |d(k)|2 ± |q(k)|

H = (� ⇥ k) · z

H = ~vF

(� ⇥ k) · z

Uc

⇢(EF

) = 1

Vpot

= V �(r)

Eres

/ 1

V

V ! 1

Himp

= VX

�

c†A�

cA�

H = ~vF

(ky

�x

� kx

�y

) +m�z

H = ~vF

k · �

�xy

= ne2

hn 2 Z

3

SC

Bi2Se3 H.c.) ~

with

Spinless px + ipy superconductor Fu and Kane, PRL 100, 096407 (2008)

Bi2Se3 – SC Hybrid Structure

SC

Bi2Se3

2D superconductor (D4h symmetry):

Bi2Se3 on a cubic lattice (2 orbitals per site (τ)): [1]

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

Local tunneling:

Superconductivity in Bi2Se3

Proximity-induced superconducting pairing 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) Projected on the surface state: spinless p+ip-wave

Majoranas in TI-SC Structures

TI-SC hybrid structures: spinless p+ip-wave in the TI surface state But how to get a boundary in the surface? •  Break TRS with a magnetic field/domain with canted

spins ! Boundary between different topo. phases

2D TI:

3D TI:

Alicea, Rep. Prog. Phys. 75, 076501 (2012)

SOC Semiconductors Spin-orbit coupled (SOC) semiconductors + magnetic field

Add conventional s-wave superconductivity by proximity effect ! spinless px + ipy superconductor

Semiconductor, spin degenerate

Spin-orbit (Rashba) coupling (e.g. InAs)

Zeeman split bands

Spinless superconductor

2D: Sau et al. PRL 104, 040502 (2010). 1D: Lutchyn et al. PRL 105, 077001 (2010), Oreg et al. PRL 104, 077002 (2010)

Semiconductor – Superconductor Hybrid Structures

Rashba spin-orbit coupled 2D semiconductor

Conventional s-wave superconductor

Ferromagnetic insulator

Vortex

Rashba SOC + s-wave SC + Zeeman field:

!  Effective spinless p+ip-wave superconductivity

Björnson and ABS, PRB 88, 024501(2013)

Majorana Fermion in a Vortex Core

2 Majorana = 1 electron

Probability density of the lowest energy eigenstates:

Majorana in the vortex core

Majorana on the sample edge

Björnson and ABS, PRB 88, 024501(2013)

Self-Consistent Solution of Vortex Core

Self-consistent solution •  Accurate value of Δ#•  Δ decays quickly in the topo. phase!!

I: Normal region, Δ > Vz II: Topological region (Majorana), Δ < Vz!

Björnson and ABS, PRB 88, 024501 (2013)

Self-Consistent Solution of Vortex Core

Self-consistent solution •  Accurate value of Δ#•  Δ decays quickly in the topo. Phase

•  Additional region I’ -  Local phase transition in the vortex core -  Two low-energy Majorana modes inside

vortex core ! 1 electron state

I: Normal region, Δ > Vz II: Topological region (Majorana), Δ < Vz!

Björnson and ABS, PRB 88, 024501 (2013)

Majoranas in SOC Systems

1D nanowires

Alicea, Rep. Prog. Phys. 75, 076501 (2012)

Semiconductor-SC hybrid structures Localized Majorana modes at: •  Vortices (2D) •  Wire end points (1D)

2D systems

Experimental Hunt 2012 1D InSb nanowire (Semiconductor with strong SOC coupling)

+ Magnetic field

+ s-wave superconductor

Mourik et al., Science 336, 1003 (2012)

E = 0 state Majorana?

Majorana?

Experimental Hunt 2014

Pb substrate (SC with strong SOC)

+ Fe ad-atoms

Localized E = 0 states Majorana?

Nadj-Perge et al., Science 346, 602 (2014)

Non-Abelian Statistics 1 electron ! 2 delocalized Majorana fermions

(non-local property)

Statistical property of Majorana fermions accessed by braiding

Alicea, Rep. Prog. Phys. 75, 076501 (2012)

Non-Abelian Statistics Cont. Combine Majoranas (arbitrarily) into operators

!

! Non-Abelian statistics

Internal rotation, non-trivial phase

Internal rotation, non-trivial phase

Swaps half of f1 with half of f2

Topological Quantum Computer Final state depends only on the braiding •  Majorana braiding = Quantum gate operation

!  Topological quantum computation •  Immune to decoherence = robust

Universal topological quantum computation also requires a π/8 phase shift operation.

Can be implemented with arbitrarily small errors

Topological Superconductors

“Spinless” superconductors and Majorana fermions TI-SC hybrid structures SOC superconductors + magnetism Majorana fermion (½ electron) •  Topological boundary state •  Topological quantum computation

Summary •  Introduction to superconductivity

–  BCS theory –  BdG formulation

•  Unconventional superconductivity

•  Topological matter

•  Topological superconductivity –  p+ip and d+id superconductors –  “Spinless” superconductors – Majorana fermions

Acknowledgements

Collaborators: A. Balatsky (LANL/Nordita) J. Linder (NTNU) J. Fransson (Uppsala) K. Le Hur (Ecole Polytechnique) A.-M. Tremblay (Sherbrooke) C. Honerkamp (Aachen) M. Fogelström (Chalmers) S. Doniach (Stanford)

Uppsala group: K. Björnson, A. Bouhon, T. Löthman, D. Kuzmanovski, L. Komendova The Carl Trygger

Foundation

Funding:

Summary •  Introduction to superconductivity

–  BCS theory –  BdG formulation

•  Unconventional superconductivity

•  Topological matter

•  Topological superconductivity –  p+ip and d+id superconductors –  “Spinless” superconductors – Majorana fermions