Junction of two topological insulators and a p-wave superconductor

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Junction of two topological insulators and a p-wave superconductor

Oindrila Deb

Centre for High Energy Physics Indian Institute of Science

Bangalore, India

September 3, 2014

Soori, Deb, Sengupta and Sen, Phys. Rev. B 87, 245435 (2013)

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Organization of the talk

•

Introduction

•

Scattering at the junction

•

Hamiltonian and energy dispersion

•

Probability and charge current

•

A general boundary condition

•

Charge conductance

•

Spin conductance

•

Summary

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Introduction

•

Topological insulators are gapped in the bulk but have gapless states on the surface. So the surface states are metallic.

•

Described by massless Dirac equation, edge states are

robust against static (Time-reversal-invariant) disorder.

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• Transmission across a junction of a normal metal (NM) and a superconductor (SC) has been extensively studied

(Blonder et al, Phys. Rev. B 25, 4515 (1982)1982, K. Sengupta et al,Phys. Rev. B 63, 144531 (2001))

• The sub-gap transport in such junctions is governed by Andreev reflection

• A junction between a topological insulator (TI) and

superconductor (SC) is different and more complex because : 1. Surface states of a TI displayspin-momentum locking 2. Particle and hole are coupledin a superconductor

• Four component spinorformalism to describe both spin and particle-hole degrees of freedom

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Scattering at the junction

• Schematic diagram (3D-pic) of the set-up. Normal or Andreev scatterings of the incident electron on TI-1 and TI-2 and BdG quasiparticle (evanescent) transmissions on the SC side

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Hamiltonian and energy dispersion

• Hamiltonian on a TI-surface:

H1= Z Z

dki dkj ψ_~^†

k [~vFnˆ· (σ ×−→

k) − µTII] ψ~k

whereψis a two component spinor,v_F is the Fermi velocity ,ˆn points normal to the 2-D surface ,k_i, k_j are momenta in the 2-D plane

• E = −µTI± ~vF

qk_i²+k_j²

• Wave fns on the SC side are described by a four component

BdG spinor: ΨSC=





 Ψ↑

Ψ↓

Ψ^⋆_↑ Ψ^⋆_↓







• SC Hamiltonian :

H_SC = Z∞

−∞

dx Z∞

0

dyΨ^†(x,y)h

−~²∇~² 2m − µSC

τ^z+ ∆(x,y)i Ψ(x,y),

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• SC pair potential∆(x,y)can be s-wave (singlet) or p-wave (triplet)

• SC pair potential∆(x,y)for p-wave (triplet) SC:

∆(x,y) = 0 ∆0f(~k)(~d· ~σ)iσ^y

−∆0f^∗(~k)(~d· ~σ^∗)iσ^y 0

!

where f(~k) = −i(ax∂x+ay∂y)

We choose ax =0 and ay =1 making the pair potential anisotropic

• The choice of~d (= ˆx, ˆy, ˆz)determines the spin-state of the Cooper pair

• E = ± q

(_2m^~²(k_x²+k_y²) − µSC)²+ ∆²₀(−→ a ·−→

k)(−→ a^∗·−→

k)

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Probability and charge current

• Let’s assume :Ψ =ψ φ

whereψandφare the upper two and lower two components of the BdG spinorΨ

• Probability density :ρp= ψ^†ψ + φ^†φ = Ψ^†Ψ Charge density :ρc =e(ψ^†ψ − φ^†φ) =eΨ^†τ^zΨ

• Corresponding currents−→ J_p and−→

J_c satisfy the equation of continuity : ∂tρ +−→

∇ ·−→ J =0

• Using the equations of motion and continuity we can find the currents :

−

→JpTI =vF[ψ^†nˆ× −→σ ψ + φ^†nˆ×−→ σ^∗φ]

−

→JpSC= _m^~Im(Ψ^†τ^z−→

∇Ψ) +−→ Jpair

• −→

Jpair = ˆx^2∆~⁰Re[axψ^†(−→

d · −→σ )iσ^yφ] + ˆy^2∆~⁰Re[ayψ^†(−→

d · −→σ )iσ^yφ]

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A general boundary condition

• To get the boundary condition we use conservation ofprobability currentnormal to the junction

• (ˆy·−→

Jp1)y→0₋− (ˆz·−→

Jp2)z→0₋ = (ˆy·−→ Jp3)y→0+

• We find that a general boundary condition involves three time-reversal invariant barriers near the junction

• Physically these barriers can arise due to gate voltages and also a lattice mismatch between TIs and SC

• General boundary condition :

1 Ψ3=M(χ1)Ψ1+ βM^†(χ2)Ψ2 2 _mv^~

1∂yΨ3−2χ3Ψ3+_~^∆_v⁰

1

0 −ay(−→ d · −→σ )σ^y a^∗_y(−→

d · −→σ^∗)σ^y 0

!

=iσ^x ⊗ τ^z[M(χ1)Ψ1− βM^†(χ2)Ψ2] β =q

v2

v1 andM(χ) =cos(χ) −i sin(χ)σ^x⊗ τ^z

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Charge conductance

• We want to solve the scattering problem and find the conductance

• Consider an electron incident on the junction from TI-1 with energy E which lies in the SC gap

• The wavefunctions on TI and SC sides will look like :

Ψ1= ψ1p+r_Nψ1p+r_Aψ1h, Ψ2=t_Nψ2p+t_Aψ2h. Ψ3=t1ψ^↑+_sc +t3ψ^↑−_sc +t2ψ_sc^↓++t4ψ^↓−_sc ,

• The scattering amplitudes r_N(A), t_N(A), t_1,2,3,4can be determined using the boundary conditions

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• Conservation of the probability current at the junction implies : v1sinθ1=v1(sinθ1RN+sinθ1hRA) +v2(sinθ2TN+sinθ2hTA)

• Charge current conservation perpendicular to the junction : J3=J1,in−J2,out

• Incoming charge current alongy on TI-1 :ˆ J_1,in=ev₁[sinθ1(1−R_N) +sinθ1hR_A]

• Outgoing charge current along−ˆz on TI-2 : J2,out =ev2[sinθ2TN−sinθ2hTA]

• Hence using conservation of probability current we get : J3=2e(v1sinθ1hRA+v2sinθ2hTA)

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• A voltage bias V is applied on the TI-1 side maintaining the TI-2 and SC at the same Fermi energy.

• The differential conductance G3=dI3/dV on SC-side at an applied bias voltage V .

• Integrating over all angles of incidenceθ1, the differential conductance is:

G3(V) =G0(V) Z π

0

dθ1[sinθ1hRA+sinθ2hTA],

where RN(A)= |rN(A)|², TN(A) = |tN(A)|²and G0(V) =e²W(µ_TI+eV)/(hv1µ_TI)

and W is the width of the sample

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Zero bias peak

Figure: The sub-gap conductance of a py-wave SC in units of G0, for~d = ˆz andχ1= χ²=0

• The zero bias peak is due to themid-gap bound statei.e E=0 state on SC side

• At E=0 transmission isindependentofχ3

• Peak is sharper at largeχ3

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• Conductance is different for different spin triplet states i.e for different−→

d

• For−→

d = ˆz at E=0: rA=tN=0 and tAand hence the conductance depends only on(χ1− χ2)

• For−→

d = ˆy we found the similar result as−→ d = ˆz

• For−→

d = ˆx we found tA=tN =0 , while|rA|²= (sinθ1)²and

|rN|²= (cosθ1)²areindependent ofχ1,χ2

• Electrons and holes seeopposite barrier strengthsχi and−χi

respectively

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Figure: Conductance of a p_y-wave SC at E=0 for different spin pairings:~d= ˆz (blue dashed line),~d= ˆx (green solid line) and~d= ˆy (red dot-dashed line)

•

This can be used to distinguish between different

spin-triplet states of the p-wave SC.

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• Typically, in a Schr ¨odinger system, the conductance decays with the barrier strength, while in a Dirac system, conductance is a periodic function of the barrier strength.

• In our setup, the TI-sides are Dirac-like while the SC is Schr ¨odinger-like.

• So, the conductance depends onbarriersχ1,2periodicallyand decays with barrierχ3on SC side

Figure:Conductance of a py-wave SC with~d= ˆz at different energies in the SC gap

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Spin conductance

•

For a

=

x

,

y, z, the a component of the spin density is given by

ρ^a = ~

2 Ψ^†₃τ^z⊗ σ^aΨ3

•

The spin current

~J^a

corresponding to the spin density

ρ^a

can be calculated on the SC side using the equations of motion.

•

The corresponding spin conductance is given by

G^a_s =G_s0Rπ

0 dθ₁J_y^a, where G_s0=eW(µ_TI+eV)/(hv₁)².

•

Now, there are nine currents corresponding to three spin

components and three possible choices of the

~

d vector.

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~

d

= ˆ

x

~

d

= ˆ

y

~

d

= ˆ

z J

_s^x

0 non-zero non-zero

J

_s^y

0 non-zero 0

J

_s^z

non-zero non-zero 0

Table: Expressions for y component of spin currents for different spin pairings of the p-wave SC.

~

d

= ˆ

x

~

d

= ˆ

y

~

d

= ˆ

z G

^x_s

0 non-zero non-zero

G

_s^y

0 0 0

G

^z_s

0 0 0

Table: Spin conductances for different spin pairings.

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•

The x -spin conductance is non-zero for the cases

~

d

= ˆ

y and

~

d

= ˆ

z.

•

Let us look at the features of the x -spin conductance for the case

~

d

= ˆ

z .

Left panel: G^x_sas a function of E=eV forχ1= χ2=0,µTI/∆0=5 andµsc/∆0=100.

Right panel: G^x_sas a function ofχ = χ₁= χ₂= χ₃and E .

•

The unusual satellite peak (SP) in addition to the ZBP is a

novel feature.

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•

To understand the SP, we relate the spin-currents on SC side to the physical quantities on TI-1 and TI-2 using the boundary condition.

•

The x -spin current on the SC side of the junction is linearly related to the steady state charge densities on the TI-1 and TI-2 sides, all evaluated at the junction:

yˆ· ~J_s^x = ~v_F

2 [Ψ^†₁τ^zΨ1− Ψ^†₂τ^zΨ2] = 1+R_N+R_N^′ −R_AΓ_1A−T_N+T_AΓ_2A

where

R_N^′ =Re(rN+r_Ne^{−i 2}^θ¹),

Γ_1A= ν²_Ecos²θ₁− ν_Ecosθ₁q

ν_E²cos²θ₁−1,Γ_2A= |ν_Ecosθ₁|

when

νE≡ (µTI+E)/(µTI−E)>1.

•

It is interesting to note that the spin conductance on the SC

side depends on both the phase and magnitude of the

reflection amplitude r

_N

.

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•

The following figure shows the contributions to the spin conductance from the different terms in the previous equation integrated over all incident angles

θ₁

.

−0.5 −0.25 0 0.25 0.5

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

E/∆₀ Different contributions to Gx s in units of Gs0

χ₃ = 0 µ_TI/∆₀ = 5 µ_SC/∆₀=100 G^x

s R‘

N T_AΓ_2A−R_AΓ_1A R_N−T_N

•

From the figure, it is evident that the Andreev scattering

term,

T_AΓ_2A−R_AΓ_1A

, and the phase term

R_N^′

contribute

the most to the SP.

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•

From the contour plot of spin-current y

ˆ·

J

_s^x

as a function of E and angle of incidence

θ₁

, the contribution to the SP from a range of

θ1

is clearly visible.

•

If

ν_E

cos

θ₁>

1, the Andreev modes on TI-1 and TI-2

become evanescent and do not carry any current.

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•

At a given

(E, θ1), momentum of the Andreev reflected hole

is:

kyh=

q

(µTI−E)²− (µTI+E)²cos²θ1/(~v1) =µTI−E

~v1

q

1 − ν²_Ecos²θ1

.

•

The contribution to the SP comes from region where the Andreev modes exist.

•

And therefore, the SP appears at a positive bias.

•

Also, the phase of the reflection amplitude r

_N

changes from

−π

to

π

through 0 when E is varied at a fixed

θ₁

.

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Summary

• We described a formalism to study thetransportacross junctions of topological insulators and superconductors and derived the appropriateboundary conditions.

• A general boundary condition involvesthree time-reversal invariant barriers

• Charge conductance shows azero bias peak

• Tunneling conductance of TI-SC junction provides a novel method for detection ofdifferent directions of~d of SC (spin-state of Cooper pairs in SC)

• Such junctions can inject spin current into SC for certain triplet-pairings of the Cooper pairs

• The spin conductance shows asatellite peakat finite bias voltages in addition to the zero bias peak

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