Topological Superconductor: A Review Theoretically...

Preprint typeset in JHEP style - HYPER VERSION

Topological Superconductor: A Review Theoretically

& Experimentally

Ze-Yang Li 1,2†, 1300011317

1State Key Laboratory for Mesoscopic Physics and School of Physics, Peking University;

Collaborative Innovation Center of Quantum Matter, Beijing 100871, P. R. China2International Center for Quantum Materials, School of Physics, Peking University,

Beijing 100871, P. R. China

†[email protected]

Abstract: There is a saying that theoretical constructions never go beyond experimen-

tal works in superconductor field. However, with recent year’s tremendous development

of topological material, scientists realized that topological states of matter can actually

be widespread, and hence the idea of topological superconductor became much more in-

teresting and hence numerous theoretical works within the framework of well-established

description of topological material seems to have a chance to gain the opportunity to get

a step forward. Recent experiments, which in great agreement with theoretical predic-

tions, confirm that. In this review, we first briefly review topological insulators’ theory

and get some intuition what topology state means, and then construct basic theoretical

description of topological superconductors with consideration of Majorana fermions, then

finally briefly analyze and discuss the current status and further perspective of topological

superconductors, both experimentally and theoretically.

Keywords: Topological Superconductors, Majorana Fermions, Topological Material.

http://jhep.sissa.it/stdsearch

Ze-YangLi’sAssignment(Draft)

Contents

1. Introduction and A Quick Review of Topological Insulator Theory 2

1.1 Berry Phase, Quantum Spin Hall Effect as the Inspiration of Topological

Materials 2

1.2 Theoretical Description for Topological Insulator 6

2. Basic Theoretical Model 7

2.1 Introducing the Bogoliubov-de-Gennes (BdG) Formalism for s-Wave Super-

conductors 8

2.2 Majorana Description for a Condensed Matter System 8

2.3 Why p-wave Superconductors are Topologically Non-trivial 10

3. Random-Matrix-Theory (RMT) Approach 12

4. Experimental Confirmation 12

4.1 [5]’s Science: Signatures of majorana fermions in hybrid superconductor-

semiconductor nanowire devices 12

4.2 [6]: Zero-bias peaks and splitting in an Al-InAs nanowire topological super-

conductor as a signature of Majorana fermions 14

4.3 [7]: Observation of superconductivity induced by a point contact on 3D

Dirac semimetal Cd3As2 crystals 16

4.4 Fractional Josephson Effect 17

5. Qubit based on Majorana Fermions: a Start 17

6. Perspective 18

7. Acknowledgements 18

Appendix A 2-state Berry Phase Calculation I

– 1 –


1. Introduction and A Quick Review of Topological Insulator Theory

With the explosion of interest in unconventional superconductivity in the past two decades,

there have been two primary research foci: (1) the microscopic mechanism that produces

the unconventional superconducting pairing potential and (2) new quasiparticle phenomena

(Bogoliubov-like excitation) [1].

As well demonstrated in the course presentation, the theoretical description of topo-

logical material goes first at quantum spin hall effect. Actually, quantum spin hall effect

can be regarded as a two-dimensional version of topological insulator for only its edge

state has non-zero conductance contribution while its bulk state doesn’t. Like our starting

expectation of topological material, quantum spin hall effect gives a dissipationless trans-

port in both side both direction. It’s strong against impurities due to the additional berry

phase introduced by the impurity just interference to cancel backward scattering. In this

subsection, we first review what’s berry phase, followed by Bernivig’s book [1], and then

construct theoretical model for QSH/TI, then show the computational simulation of such

systems.

1.1 Berry Phase, Quantum Spin Hall Effect as the Inspiration of Topological

Materials

Berry phase is a geometry phase accumulated when the Hamiltonian is slowly changed in

the space of parameter. We consider a adiabatic dynamical evolution under a parametric

Hamiltonian H(R) on its n-th eigenstate, for a closed loop in parameter space:

|n(R(t))〉 = eiθ|n(R(0))〉,

where the additional phase is partially due to dynamical term,

θ =1

~

∫ t

0En(R(t′))dt′ − i

∫ t

0

⟨n(R(t′))

∣∣∣∣ ddt′∣∣∣∣n(R(t′))

⟩

An additional topological term comes from the second term if the system is topological

nontrivial. A straight forward way of such nontrivial Hamiltonian is H(R) = ε(R) + R ·σfor a 2-state system. (which would be regarded as an example at Appendix A).

In general, as we transform the time-differentiate to parameter-differentiate, i.e., d/dt⇒∇RR, we have the additional geometry phase can be calculated through a path in param-

eter space:

γn = i

∫c〈n(R)|∇R|n(R)〉dR =

∫cR

dR ·An(R), An(R) = i〈n(R)|∇R|n(R)〉 (1.1)

– 2 –

http://laserroger.github.io/LaserPublic/Academic_and_Research/QSHtalk.pdf


Figure 1: Topological transport at edge of QSH system. Spinful state can transport towards each

direction at each edge.

For closed loop, An(R) is invariance for any gauge transformation. In that way,

according to Stokes theorem, we have

γn = −Im

∫cR

dS · (∇× 〈n(R)|∇|n(R)〉)

= −Im

∫cR

dSiεijk∇j 〈n(R)|∇k|n(R)〉

= −Im

∫cR

dSiεijk 〈∇jn(R)|n(R)〉〈n(R)|∇kn(R)〉

− Im

∫cR

dSiεijk∑m 6=n〈∇jn(R)|m(R)〉〈m(R)|∇kn(R)〉

where the first term is simply a zero for the both are purely imaginary. Hence we have

γn = −Im

∫cR

dSiεijk∑m6=n〈∇jn(R)|m(R)〉〈m(R)|∇kn(R)〉 (1.2)

Apply a mathematical simplification

En〈m|∇n〉 = 〈m|∇H|n〉+ Em〈m|∇n〉 ⇒ 〈m|∇n〉 =〈m|∇H|n〉En − Em

(1.3)

we can define the Berry Curvature,

Vni = Im∑m 6=n

〈n(R)|(∇RH(R))|m(R)〉 × 〈m(R)|(∇RH(R))|n(R)〉(Em(R)− En(R))2

and hence the nontrivial term, is

γn = −∫∫

cdS ·Vn (1.4)

Chern Number is also derived based on this. Chern number is simply integration over

a closed surface of such vector and divided by 2π. Note the relationship between Chern

Number and Hall Conductance:

– 3 –


σ =e2

h· CN (1.5)

Based on our consideration, we can theoretical explain the robust of edge state, as

illustrated in Figure. 2. Time reversal (Non-magnetic) impurities protected scattering

with additional geometry phase 2π and interference cancel it (for spin, V± = ± R2R3 , which

gives a circle rotation of Ω = 2π yields γ = π, i.e., e−iγ = −1).

To describe such a system, we consider the

Figure 2: TR protected impurity scatter-

ing with additional geometry phase 2π and

interference cancel it (for spin, V± = ± R2R3 ,

which gives a circle rotation of Ω = 2π

yields γ = π, i.e., e−iγ = −1). For a more

specific consideration, see [2]

4-band theory. Define new basis for a soc(Spin-

Orbital Coupling) system: |E1, ↑〉, |E1, ↓〉, |H1, ↑〉, |H1, ↓〉. Under these 4 basis, with requirements

that the system should obey time-reversal sym-

metry and inversion symmetry, we can expand

any 4 × 4 Hamiltonian for a particular momen-

tum by Gamma matrix:

H(k) = ε(k)I4×4 +∑

di(k)Γi + dij(k)Γij

A 2-D Quantum Well system with fixed kzhas precise form

H(k) =ε(k)I4×4 + (M − 2B)Γ0 − 2B cos(kxa)Γ0

− 2B cos(kya)Γ0 +A sin(kxa)Γ1

+A sin(kya)Γ2

(1.6)

Define M(k) = M − 2B(2 − cos(kxa) − cos(kya)), and then we get the form of full

Hamiltonian with any possible Hamiltonian:

H =∑k

(M(k)Γ0 +A sin(kxa)Γ1 +A sin(kya)Γ2) (1.7)

Square gives

H2(k) =(M2(k) +A2 sin2(kxa) +A2 sin2(kya)

)I4×4 (1.8)

which gives bulk energy-band equation, see Figure 3 for HgTe. However, this only

works for infinite system size (i.e., bulk state). For finite size, say at y direction, the

description is a little bit different. An straight forward consideration is to take a discrete

rather than continuous momentum possible value for the creation and annihilation could

be fourier transformed by the following form (x direction labeled by momentum kx while

y direction by the site j):

ck =1

L

∑j

eikyjckx,j ,

– 4 –


and we have the form of Hamiltonian (for edge state mainly)

H =1

L

∑k,j

(M(k)c†k,jck,j + T c†k,jck,j+1 + T †c†k,j+1ck,j

)≡∑k,k′

H(k)δk,k′ ,

where T = − iA2

Γ2 +BΓ5

(1.9)

which is decoupled in momentum space. Hence, we can have the decoupled Schrodinger

equation for different component of momentum, i.e., when we decompose the state to

different momentum components |ψ〉 = ⊗k|ψ(k)〉, where |ψ(k)〉 =∑

j ψ(k, j)c†k,j |0〉, with

the following ansatz

ψ(k, j) = λ−jψ(k, 0) ≡ λ−jφ(k) (1.10)

we got two eigenvalue function gives both edge state dispersion and the “skin length” λ:

E(k) = ±A sin k

λ±,1,2 = 1(2B∓A)

[4B −M − 2B cos(k)] + +,−

√[4B −M − 2B cos(k)]2 +A2 − 4B2

(1.11)

A physical meaning of λ requires |λ| > 1, so

−2B < (4B −M − 2B cos(k)) < 2B, |k|max = cos−1(1−M/2B) (1.12)

This particular dispersion is exactly the red part in Fig-

Figure 3: Bulk and Edge

state dispersion curve.

ure 3. So far we established a description for quantum spin

hall system, which is exactly what topological insulator per-

forms in 2-D. Similar to HgTe band inversion predicted in alloy

BixSb1−x material. Also, techniques can handle this for Angle-

Resolved Photoemission Spectroscopy (ARPES) can indepen-

dently image surface and bulk spectrum, see Figure 4. Let

us re-consider the band theory, the most fabulous and beau-

tiful theory in 20-th century. The fact of topology is strongly

related to the shape of the energy band of a particular ma-

terial. In an insulator an energy gap separates the occupied

valence band states from the empty conduction-band states.

Though the gap in an atomic insulator, such as solid argon, is

much larger than that of a semiconductor, there is a sense in

which both belong to the same phase. One can imagine tuning

the Hamiltonian so as to interpolate continuously between the

two without closing the energy gap. Such a process defines a

topological equivalence between different insulating states. If one adopts a slightly coarser

– 5 –


“stable” topological classification scheme, which equates states with different numbers of

trivial core bands, then all conventional insulators are equivalent. Indeed, such insulators

are equivalent to the vacuum, which according to Dirac’s relativistic quantum theory also

has an energy gap for pair production, a conduction band electrons, and a valence band

positrons. Are all electronic states with an energy gap topologically equivalent to the

vacuum? The answer is no, and the counterexamples are fascinating states of matter[3].

Our previously discussing quantum

Figure 4: ARPES on Bi2Se3. Courtesy of Fisher

Group (Stanford)

spin hall effect system is a great coun-

terexample; actually, every QHE is such

a counterexample. What is the difference

between a quantum Hall state and an or-

dinary insulator? Explained by TKNN,

it is a matter of topology. A 2-D band

structure consists of a mapping from the

crystal momentum k defined on a torus

to the Bloch Hamiltonian. Gapped band

structures can be classified topologically

by considering the equivalence classes of

Hamiltonian that can be continuously de-

formed into one another without closing

the energy gap. These classes are distin-

guished by a topological invariant n ∈ Zcalled the Chern number, as we demon-

strated at the berry phase part.

1.2 Theoretical Description for Topo-

logical Insulator

The theory of 3D topological insulator is pretty similar to the previously constructed QSH.

To make a clear introduction, we here presents the Z2 invariance first and then consider

the Bi1−xSbx topological insulators.

Z2 Invaciance

Since the Hall conductivity is odd under T , the topologically nontrivial states can only

occur when T symmetry is broken. However, the spin-orbit interaction allows a different

topological class of insulating band structures when T symmetry is unbroken. Hence we

have to examine the role of T symmetry for spin 1/2 particles, and hence the operator

form is simple:

T = eiπSy/~K, K : complex conjugate operator

Obviously, T 2 = −1. This leads to Kramer’s theorem that all eigenstates of T invariant

Hamiltonian are at least two-fold degenerate for T |χ〉 = c|χ〉 is not exist. In the absence

of soc, Kramers’s degeneracy is simply the degeneracy between up and down spins; in the

– 6 –


Bi: Class (0;000) Sb: Class (1;111)

Λa Symmetry Label δa Λa Symmetry Label δa

1Γ Γ+6 Γ−6 Γ+

6 Γ+6 Γ+

45 −1 1Γ Γ+6 Γ−6 Γ+

6 Γ+6 Γ+

45 −1

3L Ls La Ls La La −1 3L Ls La Ls La La +1

3X Xa Xs Xs Xa Xa −1 3X Xa Xs Xs Xa Xa −1

1T T−6 T+6 T−6 T+

6 T−45 −1 1T T−6 T+6 T−6 T+

6 T−45 −1

Table 1: Symmetry labels for the Bloch states at the 8T invariant momenta Λa a for the five

valence bands of Bi and Sb.

presence of soc, however, it has nontrivial consequences. First it requires

T H(k)T −1 = H(−k) (1.13)

One can classify the equivalence classes of Hamiltonians satisfying this constraint that

can be smoothly deformed without closing the energy gap. With careful theoretical con-

sideration, the number of surface states crossing the Fermi energy’s oddity is determined

to be the number of Kramers Z2 invariance:

NK = ∆ν mod2 (1.14)

Group Representation Approach to Bi1−xSbx

The first 3D topological insulator to be identified experimentally was the semicon-

ducting alloy Bi1−xSbx whose unusual surface bands were mapped in an angle resolved

photoemission spectroscopy ARPES experiment by a Princeton University group led by

Hasan.

2. Basic Theoretical Model

In this section, we briefly review the topological superconductor theoretically. Considera-

tions of topological band theory can also be used to topologically classify superconductors.

This is a subject that has seen fascinating recent theoretical developments.

Before going any further, here we give firstly about what is topological superconduc-

tor(TSC). Like the concept of TI, TSCs are gapped superconductors with “gapless and

topologically robust quasiparticles” propagating on the boundary. What’s anything differ-

ent if such a topological non-trivial property is obtained? We all know that in BCS theory,

the Cooper-pair violates the conservation of charge (for considering the large charge back-

ground) but only the charge %2 is conserved. This is really similar to Majorana fermions

for the excitation is a superposition of electron and hole. So this might be something

we would like to consider. However, normal superconductors are far away from Majo-

rana fermions for they are simple s-wave superconductors and the creation operator for an

– 7 –


excitation is still distinct from its hermitian conjugate for different spin index:

d = uc†↑ + vc↓, d† = v∗c†↓ + u∗c↑ (2.1)

which is not what we like for Majorana fermions. However, if the system is spinless super-

conductors, i.e., paired systems with only one active fermionic species rather than two, is

the best test platforms for Majorana fermions, which takes place with odd parity for Pauli

exclusion. Hence the p-wave superconductor is desired. To achieve p-wave superconductor,

the low dimension is required. Interestingly and agree with our intuition, they are all nec-

essarily topological non-trivial (which is exactly topological superconductors), which we

are going to analyze specifically in the later subsection.

In the context of topological superconductors, our presentation will deal only with

the quasiparticle physics, and we do not consider any microscopic origin of the unconven-

tional superconductivity. In our discussion we assume that there exists some finite pairing

strength, induced by interactions or occasionally through the proximity effect, and that

the quasiparticle physics is well described using a mean-field formulation. Thus, we are

interested in noninteracting quasiparticles that are coupled to a well-defined background

pairing potential, and we ignore the (possibly important) effects that would result from

considering a fully self-consistent solution.

2.1 Introducing the Bogoliubov-de-Gennes (BdG) Formalism for s-Wave Su-

perconductors

In the mean-field theory of a superconductor, the Hamiltonian for a system of spinless

electrons (conventional s-wave Bardeen- Cooper-Schrieffer (BCS) superconductor) may be

written in the form

H = µN +1

2

∑k

(c†kc−k)HBdG(k)

(ckc†−k

)(2.2)

where HBdG is a 2× 2 block matrix and can be expanded like

HBdG = [ε(p)− µ]σz + ∆1(k)σx + ∆2(k)σy, ∆ = ∆1 + i∆2

This gives the excitation spectrum for the system, see Figure 5.

2.2 Majorana Description for a Condensed Matter System

In this part, we simply consider what is Majorana fermions1.

Let us consider one fermion mode ψ, the 2d Hilbert space spanned by unoccupied

and occupied states |0〉, |ψ〉. The creation/annihilation operatorsψ†, ψ has widely known

anti-commutation form. In Majorana case, we define two linear-independent operator for

one fermion:

σ1 = ψ + ψ†, σ2 = i(ψ† − ψ)

with “real-valuedness”: σ†i = σi. It still satisfies anti-commutation relationship that

σi, σj = 2δij . The fermion number could be rewritten by n = ψ†ψ = (1− iσ2σ1)/2.

1Ref: Fa Wang’s Notes

– 8 –


Two fermion case si similar,

Figure 5: Plot of the dispersion re-

lation for an s-wave superconductor.

The curves in the figures are plots of

the energies E±(p) =√ε(p)2 + |∆|2,

dotted m = 1, µ = 0.1,∆ = 0 and solid

m = 1, µ = |∆| = 0.1

ψ1 = |0〉〈ψ1|+ |ψ2〉〈ψ1ψ2|, ψ2 = |0〉〈ψ2| − |ψ1〉〈ψ1ψ2|

If let the basis to be |0〉, |ψ2〉, |ψ1〉, |ψ1ψ2〉, the matrix

form is

ψ1 =

(0 1

0 0

)⊗ σ0, ψ2 = σ3 ⊗

(0 1

0 0

)We define two times two Majorana operator

γ1 ≡ ψ1 + ψ†1 = σ1 ⊗ σ0, γ2 ≡ i(ψ†1 − ψ1) = σ2 ⊗ σ0γ3 ≡ ψ2 + ψ†2 = σ3 ⊗ σ1, γ4 ≡ i(ψ†2 − ψ2) = σ3 ⊗ σ2

while these operator also have “real-valuedness” and

obey anti-commutation relation. Fermion number is

n1 = (1− iγ2γ1)/2, n2 = (1− iγ4γ3)/2.

Hence it’s really straightforward to generalize to N fermion mode ψ1, · · · , ψN , basis

are |n1, n2, · · · , nN 〉, with the 2N Majorana operator:

γ2i−1 = ψi + ψ†i = σ3 ⊗ · · · ⊗ σ3(i−1)⊗σ3

⊗ σ1 ⊗ σ0 ⊗ · · · ⊗ σ0(N−i)⊗σ0

(2.3)

γ2i = i(ψi − ψ†i ) = σ3 ⊗ · · · ⊗ σ3(i−1)⊗σ3

⊗ σ2 ⊗ σ0 ⊗ · · · ⊗ σ0(N−i)⊗σ0

(2.4)

The property is also similar:

• “real-valuedness” γi = γ†i .

• Anti-commutation γi, γj = 2δ − ij.

– No vacuum for majorana while γγ = 1 s.t. There is no non-trivial bosonic

hermitian operator from a single Majorana, and

γγ|·〉 = |·〉 6= 0⇒ |·〉 6= 0

• Particle-hole transformation of fermion if switch two γ operator:

ψi =γ2i−1 + iγ2i

2→ γ2i + iγ2i−1

2= iψ†

• Non-locality:

– γj affects/depends on many sub-Hilbert spaces.

– Hamiltonian can only contain products of even number of Majorana fermion

operators.

– Hamiltonian preserves fermion parity

[P, H] = 0, P = (−1)∑i ni

– 9 –


– Non-trivial observables must contain two or more Majoranas (information is

stored non-locally).

• Non-Abelian statistics

– Abelian statistics: with certain number of fermions at fixed positions

∗ Hilbert space is 1-D.

∗ Eexchanges of fermion pairs just change the phase of wavefunction. Different

fermion pair exchanges commute.

– Non-Abelian statistics: with 2N Majoranas at fixed positions,

∗ Hilbert space is 2N -D

∗ Ddifferent Majorana pair exchange/braiding do not commute: represented

as non-commuting 2N × 2N matrices.

– Braiding of Majorana fermion: a unitary transformation on Hilbert spaceρ[σi,j ] :

γi → γj , γj → −γi.

ρ[σij ] =1− γiγj√

2

– Example: ρ[σ2,3]ρ[σ1,2] 6= ρ[σ1,2]ρ[σ2,3], see Figure 6.

Our goal is to realize well-separated localized Majorana zero

Figure 6: Non-Abelian

Braiding

modes in a system with bulk gap. We know the Majorana zero

modes that [γ,H] = 0. Actions of these Majoranas do not change

energy. If we have 2n Majorana zero modes we have 2n-fold de-

generate ground states. Majorana zero modes act non-trivially in

this subspace. Clearly, it requires bulk gap so that there is clear

separation between ground & excited states. Localized well well

separated so that the local perturbations will not lift the “topo-

logically protected” ground state degeneracy for it cannot involve more than one Majorana

mode.

For detailed modeling, we can see the next part.

2.3 Why p-wave Superconductors are Topologically Non-trivial

Here we consider the 1-D p-wave superconductor. It might sound weird about the idea

of 1-D superconductors; however it’s okay for the boundary - the end of a chain - has a

gapless dispersion.

We use the following parameter to describe: t the hopping energy, µ the chemical

potential, N the number of site (assumed to be even) and ∆ the p-wave interaction. Based

on this, we can write down the Hamiltonian like

H = H0 −∆

N−1∑i

(ψiψi+1 + ψ†i+1ψ†i ), H0 = −t

N−1∑i

(ψ†iψi+1 + ψ†i+1ψi)− µN∑i

(ψ†iψi −1

2)

(2.5)

– 10 –


(a)

(b)

(c)

Figure 7: 1-D Majorana illustration. (a) the general case, (b) trivial phase for local coupling, (c)

non-trivial for distance coupling

In terms of Majoranas, it becomes a tight-bind model of Majorana fermions. This is exact

the 1-D Kitaev Chain.

H =i

2

N−1∑i

[(∆− t)γ2i+1γ2i + (∆ + t)γ2i+2γ2i−1] + µN−1∑i

γ2iγ2i−1

(2.6)

There are two solvable special case. First , t = ∆ = 0, µ < 0, the Hamiltonian

is a simple sum over N mutally commuting terms and has a unique ground state, i.e.,

∀i, ψ†iψi = 0. The bulk excitations of energy is −µ when iγ2iγ2i−1 = −1. Second ,

t = −∆ > 0, µ = 0.1. In this case, H = −t∑N−1

i iγ2i+1γ2i. It’s a sum over N −1 mutually

commuting terms iγ2i+1γ2i. Ground state is all iγ2i+1γ2i = 1, and bulk excitations of

energy wt is one iγ2i+1γ2i = −1. Hamiltonian is not relates to γ1, γ2N , i.e., the Majorana

zero mode exists: there exists a Two-fold degeneracy: iγ1γ2N = ∓1 . Action of γ1(2N)

switches between the two degenerate states.

– 11 –


In this case, the fermion parity is either 1/0. For

εk

∆k

εk

∆k

(a)

(b)

|µ| < |2t|, nontrivial

|µ| > |2t|, trivial

Figure 8: Illustration of trivial or

nontrivial case for odd/even winding

around (0, 0.

a intuitive consideration, the ground state degeneracy

change from 1 to 2 is really nontrivial. Use the idea of

previously mentioned BdG form,

H =1

2

∑k

(ψ†k, ψ−k)(εkσz + ∆kσy)

(ψkψ†−k

)

with εk = −µ− 2t cos k,∆k = 2∆ sin k, the dispersion

is simply E(k) = ±√ε2k + ∆2

k. Under a mapping from

k → (εk,∆k), the image of Brillouin zone is a closed

loop; whether it contains the (0, 0) is the distinction of

trivial or not. I.e., winding around the origin odd(non-

trivial) or even(trivial) number of times.

This universal idea is so far the most widely used

way to achieve Majorana fermion. For experiment, see

section 4

3. Random-Matrix-Theory (RMT) Approach

The application of RMT to superconductivity is based on a connection between the quasi-

particle excitation spectrum and the eigenvalues of a real antisymmetric matrix; for a

review, see [4]. Due to the limit of time, this part is omitted (and might could be found at

my website laserroger.github.io/LaserPublic/academic_research.html

4. Experimental Confirmation

We briefly take a look at three fabulous experiments: [5], [6] and [7].

4.1 [5]’s Science: Signatures of majorana fermions in hybrid superconductor-

semiconductor nanowire devices

This part we talk about the experimental work by Mourik et at. It just the same as what we

mentioned at previously Majorana fermion consideration that realize a 1-D spinless fermion

with p-wave pairing. The approach is semiconductor wire with spin-orbital coupling plus

Zeeman field and proximity to s wave superconductor. The energy band after soc term is

no-longer degenerate, for the orbital part is strongly related to the orbital part and hence

the different spin index represents different band and has only a degeneracy point at zero

momentum; with additional Zeeman field, the degeneracy is lifted further and is gaped at

all momentum space, as in Figure 9.

This method is proposed at Fu Liang & Xu Cenke’s work [9]. Edge states of 2D topo-

logical insulator (i.e., Quantum Spin Hall system) is to some extent in proximity of s-wave

superconductor. They described it as a novel superconductor-ferromagnet-superconductor

(SC-FM-SC) Josephson array deposited on top of a two-dimensional quantum spin Hall

insulator. The observable phenomenon rather than intuition is that The Majorana bound

– 12 –


Figure 9: The energy band shaping by SOC and Zeeman field.

Figure 10: Braiding/exchange in “1D” without “collision” by sidetracks.

state at the interface between SC and FM leads to charge-e tunneling between neighboring

superconductor islands, in addition to the usual charge-2e Cooper pair tunneling. That’s

exactly the Majorana fermion phenomena that one can consider Majorana fermion as ‘half

a fermion’. More specifically, it’s a braiding/exchange in “1D” without “collision” by side-

tracks. By gating the local µ, trivial large µ to nontrivial small µ transport is derived, see

Figure 10.

The transport measurement can show

Figure 12: Color-scale plot of dI/dV versus V and

B. The ZBP is highlighted by a dashed oval; green

dashed lines indicate the gap edges. At ∼ 0.6T ,

a non-Majorana state is crossing zero bias with a

slope equal to ∼ 3meV/T (indicated by sloped yel-

low dotted lines).

the phase transition in the bulk super-

conductor. When there is a transition

from the trivial to the topological non-

trivial superconducting states by turning

on a magnetic field, the bulk gap would

collapse and then reopen as the field strength

increase would provide strong but indi-

rect evidence for the onset of topologi-

cal superconductivity and, by extension,

the appearance of Majorana and states in

the wire, see Figure 12. In (a), the triv-

ial case, only available low-energy degree

of freedom reside in the metallic region

lie along x < 0. Using the Green func-

tion and S matrix method for scattering

in transport experiment, the differential

conductance G(V ) = dI/dV is derived

by

– 13 –


G(V ) =2e2

h|SPH(eV )|2 (4.1)

where |SPH(eV )|2 is the probability that an incident electron at energy E Andreev reflects

as a hole at the junction, passing charge 2e into the superconductor, a bias voltage V

applied across the junction generates a current. However, particle-hole symmetry of the

modeling Hamiltonian (BdG) shows that

S(E) = σxS∗(−E)σx

requires S(0) to be completely diagonal or completely ofF-Diagonal. The purely diagonal

case corresponds to the onset of perfect normal reflection?with unit probability an electron

reflects as an electron and similarly for holes – and hence a vanishing zero-bias conductance.

In contrast, the ofF-Diagonal possibility yields perfect Andreev reflection; here electrons

scatter perfectly into holes and vice versa, yielding a quantized zero-bias conductance.

4.2 [6]: Zero-bias peaks and splitting in an Al-InAs nanowire topological su-

perconductor as a signature of Majorana fermions

This Nature Physics paper is much more clear in delivering its information. It contains not

only a precise introduction of TSC, Majorana fermion, but also a good numerical method.

Considering this paper also measure Zero Bias Conductance, I here mainly express how

this universal paper is derived based on numerical simulation.

While the system is likely to be in the diffusive regime, we work with a model of an

infinite ballistic wire divided into M segments, each segment is described by a 4× 4 BdG

Hamiltonian. The Hamiltonians of the segments depend on three parameters µ,Ez,∆. For

example, a segment with ∆ = 0 represents a normal wire (such as the leads of the device),

and a segment with Ez > ∆ and µ = 0 represents a topological superconductor. Total

Hamiltonian is H(x, p) =∑L

l=1 Πl(x)Hl(p). Πl(x) is a boxcar function which equals when is

inside segment l and zero otherwise (segments 1 and M are semi-infinite). Given an energy

E , we find the momenta pElm of the modes in each segment, as well as the corresponding

eigenvectors:

H(pElm)vElm = EvElm (4.2)

After discarding divergent modes in the left-most and right-most segments, we write

a wavefunction of a scattering state which is a general linear combination of the modes in

each segment (the index j goes over all the scattering states):

wEj (x) =

∑l,m

Πl(x)aEjlmvElme

ipElmx

We then determine the coefficients aEjlm by requiring that the wavefunction and its

derivative are continuous at the interfaces between the segments (we can also add barriers

– 14 –


Figure 13: Comparison between experiments(Left) and simulations(Right). Left: Color plot of

the ZBP with equal height contours lines, from 0.106e2/h to 0.197e2/h. The arrows indicate the

transition from a single ZBP to split peaks. Right: Simulated behaviour using analytical expressions

for the wire spectrum. Contours lines of constant-size Majorana wavefunction, ξ = hvF /Eg ∼ 1.5L,

3L and 10L are blue, red and black, respectively. The simulation of ξ < 3L (red line contour) is

similar to the contours of the data Left

in the form of delta-function potentials, which will impose jumps in the wavefunction

derivative). The modes with real momenta and positive (negative) group velocity in the

left-most (right-most) segment are incoming modes, and it turns out that choosing the

coefficients of the incoming modes determines the coefficients of the rest of the modes. We

can therefore choose a basis for the scattering states such that each basis state will have

only one incoming mode, and separate them into right-incoming and left-incoming states.

We normalize the scattering states such that:

〈wEj |wE′

k 〉 = δ(E − E′)δjk

The charge density and charge current carried by a scattering state:

ρ[w] = ew†τzw,J[w] = eRe[w†( pm

+ αsocσzτz

)w]

where αsoc is the soc constant. The charge continuity equation gives

∂ρ[w]

∂t+ ∇ · J[w] = 4e∆Im[w†σxτxw]

From this equation we see that there is a source (or drain) term due to the supercon-

ductor (where ∆ 6= 0). This is an important property of the BTK model, which implicitly

accounts for the current that goes through the superconductor and into the ground. To

obtain the current we need to integrate over the energy, using the Fermi-Dirac distribution

as the occupation:

I =

∫dE

fF-D(E − eV )∑j∈inc

J [wEk ]

– 15 –


Herewehaveattributedtheentirevoltagedroptotheelectronsincomingfromtheleft. It is also

possible to divide the voltage drop differently between electrons incoming from left and

right, to better capture the real capacitance relations of the contacts. The differential

conductance is:

G =dI

dV= −e

∫dEf ′F-D(E − eV )

∑j∈left-inc

J [wEk ]

To account for the appearance of the

Figure 14: (a) Basic experimental setup re-

quired to observe the fractional Josephson effect

stemming from Majorana modes fused across a

superconductor-insulator-superconductor junction.

Purple regions indicate s-wave superconductors that

drive the green regions into a 1D topological state;

dashed regions are trivially gapped. The Majorana

γ1,2 mediate a component of the Josephson current

that is 4π periodic in φR − φL. When the barrier

in the junction is replaced by a superconductor with

phaseφM as in (b), two Majoranas mediate a second

type of unconventional current that is 4π periodic

in both side and can be isolated with Shapiro step

measurements, while in (c), yields only conventional

Josephson physics with 2π periodicity.

two superconducting gaps (∆ind,∆Al), we

have added up the conductance of two

different structures. The main channel

is simulated by a wire with a topological

segment that has the proximity-induced

pairing potential of ∆ind. A second chan-

nel is simulated by a wire which ends in

a superconducting segment with the Alu-

minum pairing potential of ∆Al.

This two-channel technic is a prac-

tical way to simulate the main features

of the experiment within the limitations

of the numerical model. In the exper-

imental devices (when one of the elec-

trodes is pinched off) all the electrons

eventually go into the Aluminum, and

when their energy is below the Aluminum

superconducting gap, do so almost ex-

clusively by Andreev reflections. To ac-

count for this we only take the contribu-

tion of Andreev reflection into account

when calculating the conductance in the

main channel.

4.3 [7]: Observation of superconductivity induced by a point contact on 3D

Dirac semimetal Cd3As2 crystals

This paper mainly consider the new proposed Cd3As2 Dirac semimetal crystals. Using

point-contact spectroscopy measurements, exotic superconductivity around the point is

obtained and the ZBCP is derived which could be a signal for topological p-wave supercon-

ductor. The experiment is also similar, but the different is the material and the method.

The superconductivity is derived via a point contact equipment, which is also called the

‘needle-anvil’ configuration. This group proposed an theoretical explanation of the exotic

feature based TSC. Personally, I think this is a first step to 3-D TSC material, though still

within the transport measurement method.

– 16 –


Figure 15: a, Point Contact resistance versus temperature curve of Sample 2, showing an onset

Tc of 7.1K. Inset: Temperature dependence of the bulk resistivity measured by the standard

four-probe method, showing non-superconducting behaviour. b, Normalized PC spectrum at a PC

resistance of 65 for temperatures from 0.28 to 4.0K. c, ZBCP at different temperatures

4.4 Fractional Josephson Effect

This design contains a trivial gap separating two topological superconductors. Cooper

pairs and Majoranas aside the gap both contribute to the conducting current I = IC + IMwhich is different in periodic to the SC phase due to different charge carry. Considering

the difficulties to detect, the only possible way is SC-Insulator-SC junction, as illustrated

in Figure 14. Due to the passage limit, the details is not listed here.

5. Qubit based on Majorana Fermions: a Start

This section is a re-statement of our previously discussed Majorana case and demon-

strate why it’s probable to make quantum computation.

– 17 –


The idea to store quantum informa-

Figure 16: Top view of a 2D topological insulator,

contacted at the edge by two superconducting elec-

trodes separated by a magnetic tunnel junction. A

pair of Majorana fermions is bound by the super-

conducting and magnetic gaps. The tunnel split-

ting of the bound states depends on the supercon-

ducting phase difference φ, as indicated in the plot

[≈ cos(f/2)]. The crossing of the levels at φ = π is

protected by quasiparticle parity conservation. Fig-

ure from [10].

tion in Majorana fermions originates from

Alexei Kitaev[11], as in Figure 16. The

massless Dirac fermions now propagate

along a 1D edge state, again with the

spin pointing in the direction of motion.

(This is the helical edge state responsible

for the quantum spin Hall effect.) A Ma-

jorana fermion appears as a zero mode

at the interface between a superconduc-

tor (S) and a magnetic insulator (I). The

Figure shows two zero modes coupled by

tunneling in an SIS junction, forming a

two-level system (a qubit). The two states

|1〉and |0〉 of the qubit are distinguished

by the presence or absence of an unpaired

quasiparticle. For well-separated Majo-

ranas, with an exponentially small tun-

nel splitting, this is a nonlocal encoding

of quantum information: Each zero mode

by itself contains no information on the

quasiparticle parity. Dephasing of the

qubit is avoided by hiding the phase in

much the same way that one would hide

the phase of a complex number by separately storing the real and imaginary parts. The

complex Dirac fermion operator a = 1/2(γ1 + iγ2) of the qubit is split into two real Majo-

rana fermion operators. Whereas two Majoranas encode one qubit, 2n Majoranas encode

the quantum information of n qubits in 2n nearly degenerate states. Without these degen-

eracies, the adiabatic evolution of a state Ψ along a closed loop in parameter space would

simply amount to multiplication by a phase factor, Ψ→ eiαΨ, but now the operation may

result in multiplication by a unitary matrix, Ψ→ UΨ. Because matrix multiplications do

not commute, the order of the operations matters. This produces the non-Abelian statistics

discovered by Gregory Moore and Nicholas Read, in the context of the fractional quantum

Hall effect, and by Read and Dmitry Green, in the context of p-wave superconductors. The

adiabatic interchange (braiding) of two Majorana bound states is a non-Abelian unitary

transformation of the form

Ψ→ exp(iπ

4σz

)Ψ

Two interchanges return the Majoranas to their starting position, but the final state iσzΨ

is in general not equivalent to the initial stateΨ. Such an operation is called topological

because it is fully determined by the topology of the braiding; in particular, the coefficient

in the exponent is precisely π/4. This could be useful for a quantum computer, even though

not all unitary operations can be performed by the braiding of Majoranas. [10].

– 18 –


6. Perspective

Topological superconductor is still a very hot field, both experimentally and theoretically.

Hasan in Princeton recently released an experimental research about this particular subject

[12] which firstly go beyond ZBCP and DCP features to observe TSC2, and Ashvin pub-

lished one theoretical consideration paper [13] with fancy idea and fabulous mathematical

technique aimed at the well-known Non-abelian property of TSC.

TSC, on the one hand provides a fundamentally new way to store and manipulate

quantum information with possible applications in a quantum computer, inspires funda-

mental studies on non-local entanglement and the intrinsic property of space-time on the

other hand.

7. Acknowledgements

I would like to thank Prof. Jian Wang and Teaching Assistant Xi Zhang for giving me an

opportunity to receive such a fabulous lectures even as a undergraduate student. The slides

at course inspire me to think a lot and know much more than expected. I also benefited

when preparing the presentation for the course with Jia-Chen Yu and Shang-Jie Xue. I

also appreciate numerous help out of the course, especially Mr. Hong-Lie Ning who

gave me a great amount of help, both theoretically instruction and paper suggestion.

2PbTaSe2 material via ARPES method, but not so convincing yet.

– 19 –


Appendix

Appendix A 2-state Berry Phase Calculation

Consider a case where

H(R) = ε(R) + R · σ (Ap-1)

Straightforwardly, the energy levels are E± = ε(R)±√

d · d. Employ spherical coor-

dinates and then d = d(sin θ cosφ, sin θ sinφ, cos θ). Hence the two state is

|+〉 =

(cos(θ2

)e−iφ

sin(θ2

) ), |−〉 =

(sin(θ2

)e−iφ

− cos(θ2

) ) (Ap-2)

which are orthogonal and normalized. Two Berry vector potential and corresponding Berry

curvature are

Aθ = 0, Aφ = sin2

(θ

2

), Fθφ = ∂θAφ − ∂φAθ =

sin θ

2(Ap-3)

Immediately,

V−i = εijkFij, ⇒ V− =d

d3

like a monopole in R parameter space of strength. An integral over a loop that its surface

across this monopole gives nontrivial Chern number contribution. It’s the most important

concept and the simples example for Chern number calculating for

γ(C) =

∫∫cdS ·V

This can also be applied to consider a spin in a magnetic field where Hamiltonian is

given by H(B) = B · S. The Berry vector curvature is

Vn(B) = nB

B3, n = −S, · · · , S

and that explains the common arguing that a electron (or fermion) would turn back to

itself not with a 2π rotation but a 4π rotation.

Appendix I


References

[1] B Andrei Bernevig. Topological insulators and topological superconductors. Princeton

University Press, 2013.

[2] Di Xiao, Ming-Che Chang, and Qian Niu. Berry phase effects on electronic properties. Rev.

Mod. Phys., 82:1959–2007, Jul 2010.

[3] M. Z. Hasan and C. L. Kane. Colloquium : Topological insulators. Rev. Mod. Phys.,

82:3045–3067, Nov 2010.

[4] C. W. J. Beenakker. Random-matrix theory of majorana fermions and topological

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