Majorana Fermions in Condensed Matter Physics: The 1D Nanowire Case

Philip Haupt, Hirsh Kamakari, Edward Thoeng, Aswin Vishnuradhan

Department of Physics and Astronomy, University of British Columbia, Vancouver, B.C., V6T 1Z1, Canada
(Dated: November 24, 2018)
Majorana fermions are fermions that are their own antiparticles. Although they remain elusive
as elementary particles (how they were originally proposed), they have rapidly gained interest in
condensed matter physics as emergent quasiparticles in certain systems like topological supercon-
ductors. In this article, we briefly review the necessary theory and discuss the “recipe” to create
Majorana particles. We then consider existing experimental realisations and their methodologies.

I. MOTIVATION A).
Kitaev used a simplified quantum wire model to show
Ettore Majorana, in 1937, postulated the existence of how Majorana modes might manifest as an emergent
an elementary particle which is its own antiparticle, so phenomena, which we will now discuss. Consider 1-
called Majorana fermions [1]. It is predicted that the neu- dimensional tight binding chain with spinless fermions
trinos are one such elementary particle, which is yet to and p-orbital hopping. The use of unphysical spinless
be detected via extremely rare neutrino-less double beta- fermions calls into question the validity of the model,
decay. The research on Majorana fermions in the past but, as has been subsequently realised, in the presence
few years, however, have gained momentum in the com- of strong spin orbit coupling it is possible for electrons
pletely different field of condensed matter physics. Arti- to be approximated as spinless in the presence of spin-
ficially engineered low-dimensional nanostructures which orbit coupling as well as a Zeeman field [9]. We require
show signatures characteristic of Majorana bound states spinless fermions since we want to end up with single
have been shown to exist in the system of semiconduc- unpaired Majorana fermions (and so must get rid of all
tor nanowires [2–5], topological insulators [6], magnetic degeneracies, including spin degeneracy). We can write
atom chains [7],etc., just to name a few. The outcome a non-interacting tight binding Hamiltonian with super-
of these results shows that it is possible to simulate el- conducting gap ∆ = |∆| exp(iθ), hopping integral t, and
ementary particles using their quasiparticle counterpart chemical potential µ as
in condensed matter systems.
[−µa†j aj − t(a†j aj+1 + a†j+1 aj )+
X
Another big motivation for realizing Majorana H=
fermions is the fact that they make ideal candidates j (1)
for topological quantum computation circumventing the ∆aj aj+1 + ∆∗ a†j+1 a†j ]
need for quantum error corrections and for minimizing
the interactions with the environment . Quantum al-
gorithms achieved via exchange of Majorana fermions As usual, aj and a†j denote annihilation and creation op-
(called ‘braiding’), and qubit registers stored in spatially erators respectively.
separated Majorana fermions are topologically protected We define the Majorana operators, with superconduct-
from noise and decoherence. This means that small dis- ing phase absorbed into their definitions, as
turbances cannot decohere the qubit registers without    
inducing a topological phase transition. This unique ad- θ θ
c2j−1 = exp i aj + exp −i a†j ,
vantage, combined with much lower error rates result- 2 2
ing from ‘braiding’ operations makes quantum computing  
θ

θ

with Majorana fermion networks the choice of companies c2j = −i exp i aj + i exp −i a†j ,
2 2
such as Microsoft in the race to build the first universal
quantum computer.
for j = 1, . . . , N for an N atom chain. From the definition
we immediately see that ci = c†i for i = 1, . . . , 2N and
therefore create particles which are their own antiparti-
II. THEORY
cles as required. It can also be shown that {ci , cj } = 2δij .
Let us now consider the case where |∆| = t > 0, µ = 0.
A. Kitaev Toy Model Here, equation (1) reduces to (using our new Majorana
operators):
Although Majorana fermions were originally predicted
in the context of elementary particle physics, they can N
X −1
also emerge in solid state systems as emergent quasipar- H = it c2j c2j+1 .
ticles as shown originally by Kitaev [8]. These are spin- 12 j=1
particles which are their own antiparticles, and can be
seen as a solution of the Dirac equation (see appendix Now we can construct new creation and annihilation op-
 2

...
a1 a2 aN−1 aN
c1 c2 c3 c3 c2N−3 c2N−2 c2N−1 c2N

...
ã 1 ã 2 ã N−2 ã N−1

FIG. 1: Illustration for Kitaev’s toy model, i.e. p-wave superconducting tight-binding chain. Each square represents
an electron and the circles are Majorana fermions. Upper diagram: each Majorana operator c2i and c2i−1 can be
obtained by splitting fermion operator ai . Lower diagram: case |∆| = t > 0, µ = 0, the diagonalised Hamiltonian
can be obtained by combining Majorana operators on neighbouring sites instead: this gives two unpaired operators
c1 and c2N , which can be combined to give a non-local fermion operator ãM .

erators by combining Majorana operators on neighbour- B. Mapping Kitaev Model in Semiconductors

ing sites.
Kitaev’s toy-model’s key ingredient is spinless nearest
1 neighbour p-wave superconductivity which has not been
ãj = (c2j + ic2j+1 ),
2 realised in real materials. In 2010, however, two seminal
† 1 papers show how to map the Kitaev p-wave quantum
ãj = (c2j − ic2j+1 ).
2 wire to an s-wave quantum wire in the presence of strong
spin orbit coupling and a magnetic field. [10, 11]. The
The Hamiltonian from equation (1) now becomes resulting Hamiltonian, without superconductivity, is
−1 
Hk,k0 ,σ,σ0 a†kσ ak0 σ0 ,
N  X
1 H=
ã†j ãj
X
H = 2t − . k,k0 ,σ,σ 0
j=1
2
p2
Hk,k0 ,σ,σ0 = hkσ| − µ + αn̂ · (σ × p) + Bσz |k 0 σ 0 i.
For an illustration of the discussion so far, see FIG. 1. 2m
Here we can see that the ãj operators correspond to Here the magnetic field is aligned along the wire (in
Fock . Notice, however, that the Majorana operators the positive z direction), n̂ is perpendicular to the plane
c1 and c2N are completely absent from this diagonalised in which the wire lies, and σ is the vector of Pauli matri-
Hamiltonian. These can be combined to a single, highly ces. In our case the term n̂ · (σ × p) simplifies to σx pz .
non-local fermionic operator This Hamiltonian is simply diagonlized, with the result-
ing energy spectrum being
1
ãM = (c1 + c2N )
2 ~2 kz2 p
E(kz ) = − µ ± α2 kz2 + B 2 .
2m
Occupying this state requires zero energy (since it does
not appear in the Hamiltonian), and thus we can have an If we now introduce BCS superconductivity with the
odd number of quasiparticles at zero energy cost (unlike gap parameter ∆, the new Hamiltonian can be diago-
the superconductors we are used to, where we require an nalized using the Bogoliubov-de-Gennes transformation
even number, i.e. Cooper pair condensates). This even- [12], resulting in the new dispersion relation
ness/oddness is called parity and can be determined by  2 2 2
~ kz
the eigenvalue of ã†M ãM (0 for even or 1 for odd parity). E 2 (kz )± = − µ + (αkz )2 + B 2 + ∆2
2m
Although we only showed this for a special case, s 2
namely |∆| = t > 0, µ = 0, Kitaev showed that the Ma-
 2 2
2 2 2
~ kz
jorana edge states (called Majorana zero modes, MZMs) ±2 (B∆) + [B + (αkz ) ] −µ .
2m
exist as long as |µ| < 2t [8] (i.e. µ is inside the gap).
More generally, these Majorana states may not be lo- The effects of the different components of the Hamilto-
calised, but instead decay exponentially away from the nian is shown in FIG. 2 as a function of increased mag-
ends. netic field. Magnetic field induces topological transitioni
 3

which ends (in the figure) with the topological supercon- as a hole (|reh|2 ) by:
ducting bulk state with Majorana fermions at the edge
of the nanowire. dI
G(V ) = = 2G0 |reh |2 , (2)
dV
2
where G0 = eh is the conductance quanta. If a zero
III. EXPERIMENTAL REALIZATION
energy mode is present in the superconductor, the reflec-
tion amplitude is maximized |reh |2 = 1 similar to the
A. Material Choice and Device Fabrication resonant tunneling from equal double barriers which re-
sults to perfect Andreev reflection and G = 2G0 . Res-
As shown in the previous section, Majorana Zero onant tunneling measurement provides the local density
Modes (MZMs) can be obtained by tuning chemical po- of states of this interface where the MZMs are expected
tential or magnetic field to drive the system towards to reside.
topological superconductor phase. The other compo- The InAs/Al device tunneling schematic is shown in
nents of the ’recipe’ (spin-orbit coupling, proximitized lower part of fig.4, and the differential conductance re-
superconductivity) are intrinsic material properties. The sults are shown in fig. 6. The conductance spectrum
common choice for the 1D nanowire with strong spin- shows the topological transition from trivial supercon-
orbit interaction so far has been the heavy element semi- ductor (the normal proximited superconductivity) into
conductors InSb and InAs[10]. The two criteria, super- the topological superconductivity
p with MZMs at critical
conductivity and magnetic field, compete in a way that magnetic field, Bc = ∆2 + µ2 ≈ 0.7 Tesla. Zero bias
large magnetic field can destroy the triplet pairing in peak (ZBP) is not unique to MZMs, but further investi-
the induced superconductivity. A large Zeeman splitting, gations have eliminated the false positives coming from
however, is required in order to prevent interaction be- e.g. local Andreev bound states, disorder-induced zero-
tween the pairs of Majorana fermions (at the same edge bias states, etc. Furthermore, the measured ZBP was
location) of the two spin channels, which combines into shown not to depend on the tunneling barrier height as
fermionic mode at zero energy. Therefore, nanowire with in the case of local Andreev bound states and is the char-
a large Lande g-factor is desired to obtain large Zeeman acteristic of robust topological MZMs[5, 10].
splitting at fields below the critical field of s-wave super-
conductor.
The choice of superconductors, correspondingly, re- IV. FUTURE DIRECTIONS: TOWARDS
quire a large superconducting gap and high critical field QUANTUM COMPUTING WITH MAJORANA
to withstand the applied in-plane magnetic fields. In FERMIONS
the experiments so far, two different superconductors
have been used: NbTiN (Type-II superconductor) and Al The results published in [5] shows a very convincing
(Type-I). The first generation of the device used NbTiN evidence of the Majorana bound states (MZMs) in the
due to its much higher critical field. It was discovered, semiconductor nanowire devices. The next step would
however, that Al has two main advantages in terms of be to prove the possibility of creating a nanowire junc-
higher interface quality and a type-I hard superconduct- tion and readout for ’braiding’ operation. As mentioned
ing bandgap as compared to NbTiN. Higher interface re- in the introduction, quantum computing operation is ob-
sults from capability of in-situ deposition of Al, which tained via exchaging the adjacent Majorana fermions and
suppress undesired sub bandgap density of states. Al as this operations forms a ’weave’ pattern unique to that
type-I superconductor has an additional benefit of not particular operation. In 1D nanowire, however, there
having issues with vortices disturbances created by mag- is only one channel for the Majorana fermion to move
netic field in type-II superconductor. This vortices are around. Therefore, a junction is required to allow one
suspected to turn the band gap into a ’soft gap’ which Majorana to exit the channel, before switching its loca-
degrades the conductance signal of MZMs[5, 10]. The de- tion to the neighboring Majorana fermions (fig. 7. By
vice schematics of the first and latest generation of MZMs forming ’trenches’ on the substrate, network of nanowire
nanowire are shown in fig. 4. can be grown from the bottom-up to form what is called
a ’hashtag’ circuit (fig.8). Preliminary measurements of
this ’hashtag’ have shown phase coherent transport, and
B. Signature of Majorana Fermions: Zero Bias therefore shows a very promising development for real-
Peak world topological quantum computing in the near future.

Low-bias transport of a normal metal-superconductor

interface is predominantly determined by the Andreev V. CONCLUSION
reflection, in which incident electron is reflected as a hole
and a Cooper pair is created in the superconductor re- This short report was intended to give a brief overview
sulting in a net charge transfer of 2e. Differential con- of the physics of Majorana fermions, an ever-growing sub-
ductance is related to the probability of electron reflected ject of interest, especially in condensed matter physics.
 4

FIG. 2: Semiconductor nanowire proximitized with s-wave superconductivity as magnetic field is increased.
Left:Trivial (normal s-wave) superconducting phase. Middle: Crossing of energy band occur as a result of
topological transition with delocalized Majorana across nanowire. Right: Re-opening of the gap into the topological
superconducting state with localized Majorana at the edges of nanowire (Majorana Zero Modes). ∆1 and ∆2 are
superconducting gap at k = 0 and kF which magnitude differ in the topological superconducting phase.[13]

Modes (MZMs) have been shown which shows a strong

indication of localized Majorana fermions in the nanowire
edges. This is the unique feature of the topological su-
perconducting phase. Furthermore, the current status of
realizing a scalable quantum computer using nanowires is

FIG. 4: Upper: First generation of InSb/NbTiN

nanowire MZMs device[14]; Lower: Latest generation of
InSb/Al schematics (inset shows false-color electron FIG. 5: Andreev resonant tunneling in
micrograph)[10]. metal/superconductor interface in the presence of
MZMs or zero energy bound states[10].

We illustrated the fundamental principles in a simplified

toy model, first proposed by Kitaev, then discussed one
of the first experimental realisations and the methodolo-
gies used to find Majorana fermions by mapping Kitaev being pursued, showing very promising results for a more
p-wave superconductivity to semiconductor nanowires. robust and fault tolerant topological quantum computer
The devices and signatures resulting from Majorana Zero using Majorana Fermions.

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 5

FIG. 8: Nano-‘hashtag’ network built with

semiconductor nanowires for future topological
FIG. 6: Upper:Tunneling conductance as a function of
quantum computing device (modified from [16, 17]).
in-plane magnetic field. Lower: Horizontal slice of the
zero-bias peak which shows quantized conductance
(2G0 ) on the onset of MZMs transition (≈ 0.7 Tesla).
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 6

Appendix A: Majorana’s Solution to the Dirac 4 component spinors. In the case of charged spin- 21 parti-
Equation cles, which the Dirac equation was initially derived to de-
scribe, the components correspond to the two spin states
The relativistic Dirac equation can be derived [18] by of the electron and the two spin states of the positron.
p̂2
replacing the classical Hamiltonian 2m + V (x̂, t) in the In the Weyl representation, with
Schrödinger
p Equation with the relativistic Hamiltonian
p2 + m2 , in which case the corresponding equation of
   
0 1 0 σi
γ0 = , γi = ,
motion becomes 1 0 −σ i 0
∂ p
i ψ = p2 + m2 ψ.
∂t the equation can be rewritten as (iγ µ ∂µ − m)ψ = 0 [19].
To obtain a Lorentz invariant form of the equation, we The choice of matrices γ µ are not unique. In particu-
can rewrite p2 + m2 as the square of a different quantity, lar, as realised originally by Majorana in 1937, if the γ
p2 +m2 = (α·p+α0 m)2 for some objects α = (α1 , α2 , α3 ) matrices are chosen as
and α0 . Upon squaring the quantity α · p + βm and
equating with p2 + m2 , we obtain the relations
   
0 0 −σ 1 1 0 σ0
γ =i , γ =i
( σ1 0 σ0 0
αi2 = 1
,
{αi , αj } = 0    
σ0 0 0 σ2
γ2 = i , γ 3
=
which can be satisfied by setting 0 −σ 0 −σ 2 0
   
0 σ I 0
α= , α0 = .
σ 0 0 −I the solutions to the Dirac equation in this are real val-
ued and neutral. The spinor ψ now describes a spin- 21
∂
The resulting Dirac equation is i ∂t ψ = (α · p̂ + α0 m)ψ. particle which is its own antiparticle, a Majorana fermion
Since the αi ’s are 4 × 4 matrices, the solutions ψ must be [9].