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

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Majorana Fermions in Condensed Matter Physics: the 1D Nanowire Case 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 ∆ = j∆j exp(iθ), hopping integral t, and ementary particles using their quasiparticle counterpart chemical potential µ as in condensed matter systems. X y y y Another big motivation for realizing Majorana H = [−µajaj − t(ajaj+1 + aj+1aj)+ fermions is the fact that they make ideal candidates j (1) for topological quantum computation circumventing the ∆a a + ∆∗ay ay] need for quantum error corrections and for minimizing j j+1 j+1 j the interactions with the environment . Quantum al- y gorithms achieved via exchange of Majorana fermions As usual, aj and aj 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 y c = exp i a + exp −i a ; vantage, combined with much lower error rates result- 2j−1 2 j 2 j ing from `braiding' operations makes quantum computing θ θ y with Majorana fermion networks the choice of companies c2j = −i exp i aj + i exp −i a ; 2 2 j 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 y we immediately see that ci = ci for i = 1;:::; 2N and therefore create particles which are their own antiparti- II. THEORY cles as required. It can also be shown that fci; cjg = 2δij. Let us now consider the case where j∆j = 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−1 X also emerge in solid state systems as emergent quasipar- H = it c2jc2j+1: 1 ticles as shown originally by Kitaev [8]. These are spin- 2 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 c c1 c2 c3 c3 c2N−3 c2N−2 2N−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 j∆j = 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 operatora ~M . 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 a~j = (c2j + ic2j+1); neighbour p-wave superconductivity which has not been 2 realised in real materials. In 2010, however, two seminal y 1 papers show how to map the Kitaev p-wave quantum a~ = (c2j − ic2j+1): j 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 N−1 X y X 1 H = Hk;k0,σ,σ0 a ak0σ0 ; H = 2t a~ya~ − : kσ j j 2 k;k0,σ,σ0 j=1 2 p 0 0 Hk;k0,σ,σ0 = hkσj − µ + αn^ · (σ × p) + Bσzjk σ i: For an illustration of the discussion so far, see FIG. 1. 2m Here we can see that thea ~j operators correspond to Here the magnetic field is aligned along the wire (in Fock . Notice, however, that the Majorana operators the positive z direction), n^ 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 n^ · (σ × p) simplifies to σxpz. non-local fermionic operator This Hamiltonian is simply diagonlized, with the result- ing energy spectrum being 1 a~ = (c + c ) M 2 1 2N 2k2 E(k ) = ~ z − µ ± pα2k2 + B2: z 2m z 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 y 2 ~ kz 2 2 2 the eigenvalue ofa ~ a~ (0 for even or 1 for odd parity). E (kz)± = − µ + (αkz) + B + ∆ M M 2m Although we only showed this for a special case, s namely j∆j = t > 0, µ = 0, Kitaev showed that the Ma- 2k2 2 ±2 (B∆)2 + [B2 + (αk )2] ~ z − µ : jorana edge states (called Majorana zero modes, MZMs) z 2m exist as long as jµj < 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 (jrehj2 ) by: ducting bulk state with Majorana fermions at the edge dI of the nanowire. G(V ) = = 2G jr j2; (2) dV 0 eh 2 where G = e is the conductance quanta. If a zero III. EXPERIMENTAL REALIZATION 0 h energy mode is present in the superconductor, the reflec- 2 tion amplitude is maximized jrehj = 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 with MZMs at critical p 2 2 conductivity and magnetic field, compete in a way that magnetic field, Bc = ∆ + µ ≈ 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.
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