s-wave SC

Topological phases in superconductor/noncollinear

magnet interfaces with strong spin–orbit coupling

Henri Menke Alexander Toews Andreas P. SchnyderMax Planck Institute for Solid State Research, Stuttgart, Germany

DPG-Frühjahrstagung 2016 — TT 68.3

Motivation

STM tip

superconductor

• p-wave superconductors are rare in nature

• Topological phase of a superconductor

• This phase can be induced in tailored materialsSee talk by A. Yazdani, TT 26.4

• Topological superconductors host Majorana edge states

• Majorana edge states are robust against disorder

• Candidate for fault tolerant quantum storage

� 1/10

Noncollinear Magnets• Collinearity of two vectors means that they are linearly dependent

a = γb , γ is scalar

• Example of a noncollinear spin texture

θ

Bn = B(sinθn cosφn, sinθn sinφn, cosθn)>

θ = θn+1 − θn = 2π3 , φn = 0� 2/10

Noncollinear Magnets• Collinearity of two vectors means that they are linearly dependent

a = γb , γ is scalar

• Example of a noncollinear spin texture

θ

Bn = B(sinθn cosφn, sinθn sinφn, cosθn)>

θ = θn+1 − θn = 2π3 , φn = 0� 2/10

Hamiltonian

s-wave SC

• Tight-binding approximation

• superconductive s-wave pairing

• sd coupling

• Rashba-type spin–orbit coupling lk = sinkxy

H = t∑〈i,j〉,σ

c†i,σcj,σ − µ∑i,σc†i,σci,σ +∆

∑i

(c†i,↑c

†i,↓ + ci,↓ci,↑

)+∑i,α,β

(Bi ·σ)αβc†i,αci,β + λ∑i

[(c†i−δx ,↓ci,↑ − c

†i+δx ,↓ci,↑

)+ h.c.]

P. M. R. Brydon et al., Phys. Rev. B 91, 064505 (2015)

S. Nadj-Perge et al., Phys. Rev. B 88, 020407 (2013)

� 3/10

Hamiltonian

s-wave SC

• Tight-binding approximation

• superconductive s-wave pairing

• sd coupling

• Rashba-type spin–orbit coupling lk = sinkxy

H = t∑〈i,j〉,σ

c†i,σcj,σ − µ∑i,σc†i,σci,σ +∆

∑i

(c†i,↑c

†i,↓ + ci,↓ci,↑

)+∑i,α,β

(Bi ·σ)αβc†i,αci,β + λ∑i

[(c†i−δx ,↓ci,↑ − c

†i+δx ,↓ci,↑

)+ h.c.]

P. M. R. Brydon et al., Phys. Rev. B 91, 064505 (2015)

S. Nadj-Perge et al., Phys. Rev. B 88, 020407 (2013)

� 3/10

sd vs. Rashba

• It can be shown that the effects of spin–orbit coupling can be

mapped to a helical spin texture by a unitary transformation in spin

space.

• Ferromagnetic spin texture

Bn = B =

BxByBz

• Spin–orbit coupling

λ ≠ 0

vs.

• Helical spin texture

Bn = B

sinθn cosφnsinθn sinφn

cosθn

• No spin–orbit coupling

λ = 0

� 4/10

Topological Invariant

• Transform Hamiltonian to Majorana representation

b2n−1,σ = fn,σ + f †n,σ , b2n,σ = −i(fn,σ − f †n,σ )

H = i4

∑qb†qA(q)bq

• Bulk Z2 invariant ν ∈ {0,1}

(−1)ν =∏a

Pf[A(Λa)]√det[A(Λa)]

=∏a

sgn(Pf[A(Λa)]

) = ±1Λa ∈ {0, π} are the time-reversal invariant momenta.

� 5/10

Topological Invariant

• Transform Hamiltonian to Majorana representation

b2n−1,σ = fn,σ + f †n,σ , b2n,σ = −i(fn,σ − f †n,σ )

H = i4

∑qb†qA(q)bq

• Bulk Z2 invariant ν ∈ {0,1}

(−1)ν =∏a

Pf[A(Λa)]√det[A(Λa)]

=∏a

sgn(Pf[A(Λa)]

) = ±1Λa ∈ {0, π} are the time-reversal invariant momenta.

� 5/10

Comparison• Helical spin texture, α = cosθ/2 and β = sinθ/2.

Pf[A(q)] = B2 − [µ − 2tα cos(q)]2 − [∆0 − 2itβ sin(q)]2

S. Nadj-Perge et al., Phys. Rev. B 88, 020407 (2013)

• Ferromagnetic spin texture with spin–orbit coupling

Pf[A(q)] = |B|2 − [µ − 2t cos(q)]2 − [∆0 − 2iλ sin(q)]2

Parameter matching

t ← tα , λ← tβ• Helical spin texture with spin–orbit coupling

Pf[A(q)] = B2 − [µ − 2tα cos(q)+ 2λβ cos(q)]2

− [∆0 − 2itβ sin(q)− 2λα sin(q)]2

� 6/10

Comparison• Helical spin texture, α = cosθ/2 and β = sinθ/2.

Pf[A(q)] = B2 − [µ − 2tα cos(q)]2 − [∆0 − 2itβ sin(q)]2

S. Nadj-Perge et al., Phys. Rev. B 88, 020407 (2013)

• Ferromagnetic spin texture with spin–orbit coupling

Pf[A(q)] = |B|2 − [µ − 2t cos(q)]2 − [∆0 − 2iλ sin(q)]2

Parameter matching

t ← tα , λ← tβ• Helical spin texture with spin–orbit coupling

Pf[A(q)] = B2 − [µ − 2tα cos(q)+ 2λβ cos(q)]2

− [∆0 − 2itβ sin(q)− 2λα sin(q)]2

� 6/10

Numerical Evidence• Helical spin texture, θ = 2π/3, t = 2∆

1 12 24 36 480

0.1

0.2

0.3

Site index

LDO

S

λ = 0

−1 −0.5 0 0.50

0.2

0.4

0.6

E/∆

λ = 0

• Ferromagnetic spin texture with spin–orbit coupling, t = ∆

1 12 24 36 480

0.1

0.2

0.3

Site index

LDO

S

λ = √3

−1 −0.5 0 0.50

0.2

0.4

0.6

E/∆

λ = √3

� 7/10

Edge State Fragility• Can we spoil the topological phase with strong spin–orbit coupling?

0 1 2 3 4 5

−1

−0.5

0

0.5

1

λ/∆

E/∆

� 8/10

Edge State Fragility• Can we spoil the topological phase with strong spin–orbit coupling?

� 9/10

Outlook & Ongoing Research• Two-dimensional systems with noncollinear and noncoplanar spin

textures S. Nakosai et al., Phys. Rev. B 88, 180503 (2013)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

kx/π

E/∆

noncollinear

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

kx/π

noncoplanar

0

0.2

0.4

• Computation of the edge currents

� 10/10

References & Further Reading

[1] P. M. R. Brydon, S. Das Sarma, H.-Y. Hui, and J. D. Sau, Phys. Rev. B 91, 064505

(2015).

[2] S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A. Yazdani, Phys. Rev. B 88,

020407 (2013).

[3] S. Nakosai, Y. Tanaka, and N. Nagaosa, Phys. Rev. B 88, 180503 (2013).