Odd-even rule for zero-bias tunneling conductance in coupled Majorana wire arrays
OOdd-even rule for zero-bias tunneling conductance in coupled Majorana wire arrays
Deepti Rana and Goutam Sheet ∗ Department of Physical Sciences, Indian Institute of Science Education and Research Mohali, Mohali, Punjab, India
A semiconducting nanowire with strong Rashba coupling and in proximity of a superconductorhosts Majorana edge modes. An array of such nanowires with inter-wire coupling gives an approxi-mate description of a two-dimensional topological superconductor, where depending on the strengthof the magnetic field and the chemical potential, a rich phase diagram hosting trivial and differenttypes of non-trivial phases can be achieved. Here, we theoretically consider such a two-dimensionalassembly of spin-orbit coupled superconducting nanowires and calculate the collective tunnelingconductance between normal electrodes and the wires in the topological regime. When the numberof wires in the assembly is N , as a consequence of the way the Majorana bonding and anti-bondingstates form, we find that N conductance peaks symmetric about the bias V = 0 appear, for even N . When N is odd, a ZBCP also appears. Such an assembly can be realized by standard nano-fabrication techniques where individual nanowires can be turned ON or OF F by using mechanicalswitch (or local top gating) to make N either even or odd – thereby switching the ZBCP OF F or ON , respectively. Hence, our results can be used to realize and detect topological superconductivityefficiently, unambiguously and in a controlled manner. A topological superconductor is a superconductor forwhich a non-zero superconducting (pairing) gap exists inthe bulk while the boundary hosts gapless self-hermitianmodes [1–5]. Because of their self-hermitian proper-ties, such boundary modes are termed as the ”Majoranamodes”. In a one-dimensional (1D) topological supercon-ducting system, like a superconducting, spin-orbit cou-pled quantum wire (a Majorana wire) [6–14], these zero-energy modes are bound to the ends of the wire [15]. Inthis context these modes are also often referred to as Ma-jorana zero modes or Majorana bound states. In a twodimensional (2D) topological superconductor, the bound-ary hosts gap-less Majorana chiral modes which propa-gate along the edge [16]. On the other hand, the coresof quantum vortices in the bulk of such 2D superconduc-tors host Majorana bound states [17–19]. Like all su-perconductors, such topological superconductors also re-spect robust particle-hole symmetry – as a consequence ofthis, the boundary modes are also extremely robust. Thephysics of Majorana modes have attracted substantial at-tention of the contemporary condensed matter physicscommunity for their exotic fundamental properties like,their non-abelian exchange statistics, which makes thempotentially important ingredients for building a topolog-ical quantum computer [20–26].In this paper, we theoretically considered an assemblyof Majorana nanowires placed on an s -wave supercon-ductor [27]. We calculated the tunneling conductancebetween normal electrodes mounted on the wires and theassembly (array) of the wires. Our key observations are:(a) when the number of wires in the assembly is N , and N is even, N conductance peaks symmetric about thebias V = 0 appear, (b) when N is odd, a ZBCP, alongwith the conductance peaks symmetric about V = 0 alsoappears. These are subject to certain condition that will ∗ [email protected] be discussed in detail later. It is known that mere obser-vation of a ZBCP does not provide a solid signature ofMajorana bound states. This is mainly because a ZBCPin tunneling spectroscopy involving superconductors canappear for a number of reasons other than Majoranabound states [28–30]. As per our calculations, simplyby changing the number of transport-active wires (e.g.,through mechanical switch or by local top gating) it willbe possible to probe the Majorana states unambiguouslythrough controlled appearance and disappearance of theZBCP by making N odd and even, respectively.Our model setup is shown schematically in Figure 1(a). The setup consists of an array (in the x − y plane)of N parallel Rasbha nanowires, lying in close proxim-ity to an s -wave superconductor, with Zeeman field ( V z )applied along the ˆ z direction. As a consequence of su-perconducting proximity effect, a superconducting gap(∆) is induced in the wires. The model also includesinter-wire tunneling which is facilitated by the underly-ing superconductor.The Hamiltonian for the system described above canbe written as : H = H || + H ⊥ (1)where H || is the Hamiltonian describing intra-wire dy-namics while and H ⊥ describes the inter-wire dynamics.In the mean field approximation, these two Hamiltonianscan be expressed as: H || = − t x (cid:88) i,δ,σ [ c † i + δ x σ c iσ ] − µ (cid:88) iσ c † iσ c iσ − V z (cid:88) i [ c † i ↑ c i ↓ + h.c. ]+∆ (cid:88) i [ c † i ↑ c † i ↓ + c i ↓ c i ↑ ]+ iα (cid:88) i,δ [ c † i + δ x σ y c i + h.c. ] (2) H ⊥ = − t ⊥ (cid:88) i,δ,σ [ c † i + δ y σ c iσ + h.c. ] − iβ (cid:88) i,δ [ c † i + δ y σ x c i + h.c. ](3) a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n where c † iσ ( c iσ ) is the fermionic operator that creates (an-nihilates) a particle at lattice site i = { i x , i y } with spin σ . δ x and δ y represent the nearest neighbour lattice vectorsin x and y directions respectively. t x gives the hoppingmatrix element along the wires (chosen to be along x ), α is the Rasbha spin-orbit coupling , µ is the chemical po-tential, t ⊥ is the weak inter-wire hopping matrix element, β is the weak spin-orbit coupling term between the wiresin transverse direction and that is linked to inter-wirehopping. It is understood that the effective inter-wirecoupling between the wires can be significant only whenthe distance between the wires is less than a coherencelength in the superconductor underneath.The Hamiltonian (1) can be written in the momentumspace as : H = 12 (cid:88) Ψ † k h ( k )Ψ k ; k = ( k x , k y ) h ( k ) = (cid:15) k τ z + αsin ( k x ) τ z σ y − V z σ z + ∆ τ x + βsin ( k y ) τ z σ x (4)where (cid:15) k = − t x cos ( k x ) − t ⊥ cos ( k y ) − µ , σ and τ are Pauli matrices in spin space and particle-hole ba-sis respectively. The uncoupled wires ( t ⊥ =0; β =0)can be tuned into two phases: (a) trival phase, when V z < (cid:112) ∆ + ( µ + 2 t x ) and (b) topological phase, when V z > (cid:112) ∆ + ( µ + 2 t x ) . Here, µ + 2 t x is the chemicalpotential measured w.r.t bottom of the conduction band.We have chosen t x = t as the energy unit for our calcu-lations.Now, the Hamiltonian can be written as: H = (cid:88) j (cid:90) dk x [ (cid:15) j ( k x )Ψ † k x ,j Ψ k x ,j + αsin ( k x )Ψ † k x ,j σ y Ψ k x ,j − V z Ψ † kx,j σ z Ψ k x ,j + ∆Ψ † k x ,j ( iσ y )Ψ †− k x ,j + h.c. ]+ (cid:88) j (cid:90) dk x [ − t ⊥ Ψ † k x ,j Ψ k x ,j +1 − iβ Ψ † k x ,j σ x Ψ k x ,j +1 + h.c. ](5)where (cid:15) j ( k x ) = − t x cos ( k x ) − µ From this, the spectrum of the system is obtained as : E = V z + ∆ + (cid:15) k + | γ k | ± (cid:113) ( V z ∆) + ( V z + | γ k | ) (cid:15) k (6)where γ k = αsin ( k x ) i + βsin ( k y ) . The energy dispersion as a function of k x is calculatedfor the trivial case when V z = 0 for an assembly of fiveuncoupled (Figure 1 (b)) and weakly coupled (Figure 1(d)) wires. For V z = 0.85 t the system is in topologicalregime. In this regime, the dispersion of five uncoupled(Figure 1 (c)) and weakly coupled wires (Figure 1(e))are also shown. Periodic boundary conditions along x direction and open boundary conditions along y direc-tion are enforced that makes the momentum along x a (d (a)(b) (c) (d) (e) FIG. 1. (a) Schematic illustrating an array of parallel Rasbhananowires lying in proximity to s-wave superconductor andwith Zeeman field V z applied along z direction. Energy dis-persion for an assembly of five Majorana wires as a functionof the momentum along the wires, ( k x ): for uncoupled (inred) (b) trivial regime ( V z = 0, t ⊥ = 0, β = 0) (c) topologicalregime ( V z = 0.85 t , t ⊥ = 0, β = 0) and coupled (in blue) (d)trivial regime ( V z = 0, t ⊥ = 0.3 t , β = 0.3 t ) (e) topologicalregime ( V z = 0.85 t , t ⊥ = 0.3 t , β = 0.3 t ) is shown. The otherparameters used for calculations are α = 1 t , µ = -2 t and ∆= 0.6 t good quantum number. The results presented in Figure1 are consistent with the earlier calculations on coupledMajorana wires [31].In Figure 2, we show the variation of the energy spec-tra with the Zeeman field V z . Figure 2 (a) and 2 (b)show the variation for an assembly of 3 and 4 uncoupledwires respectively while Figure 2 (c) and 2 (d) show thevariation for an assembly of 3 and 4 coupled wires respec-tively, in the topological regime. When the wires don’thave any inter-wire coupling ( t ⊥ = 0; β = 0) , as ex-pected, it is seen that for V z > ∆ (∆= 0.6 t in this case),we obtain zero energy states indicating the topologicalregime. Now, when there is non-zero interwire coupling( t ⊥ (cid:54) = 0; β (cid:54) = 0), the spectra get modified due to mix-ing of states and the topological regime is achieved fora higher value of Zeeman field ( V z > . t in this case).Futhermore, it is also observed that for three wires, statesat E = 0 appear along with states that emerge at finiteenergy near zero energy symmetric about E = 0. On theother hand, for a 4-wire assembly no zero energy statescan be seen but 4 peaks around E = 0 are seen. We willlater see that these lead to odd-even rule in the conduc-tance which is the focal theme of this article. (b)(a)(c) (d) FIG. 2. Plot of energy spectrum as a function of Zeeman fieldin the topological regime for : (a), (b) N = 3 and 4 uncoupled(in red) wires respectively, both hosting zero energy statesbeyond V z = 0.6t (c), (d) N = 3 and 4 coupled (in blue) wireswhere, for N = 3, a zero energy state can be seen beyond V z > . t while no such state appears for N =4. Number ofstates near zero energy is equal to N in both the cases. Lowest48 levels have been plotted. The parameters used for plottingfor both uncoupled and coupled case in the topological regimeare mentioned in Figure 1. (b) (d)(c) (a) FIG. 3. (a) Schematic for measuring tunneling conductance ofarray of Majorana wires. The normal leads (in blue) are con-nected to Majorana wires (red) with a tunnel barrier (green)at the interface of the junction. A bias V is applied to thenormal leads. Using mechanical switch, the number of trans-port active wires can be controlled. Plot of zero temperaturedifferential conductance vs applied bias V of different numberof wires for (b) trivial uncoupled case (c) trivial coupled case(d) topological uncoupled case.. Now, we design a thought tunneling experiment to in-vestigate the possible role of inter-wire coupling in trans-port through the aforementioned Majorana wire assem-bly. A schematic representation of the said set-up isshown in Figure 3 (a). A normal lead is attached toone end of each semi-infinite semiconducting nanowirewith strong Rasbha coupling proximitized by s-wave su-perconductor in the presence of an external applied mag-netic field. The interface between the metal electrodeand each wire falls in the tunneling regime of transport.The normal lead has the same Hamiltonian (equation2) as the nanowire except for the superconducting term(∆). Also, chemical potential of the normal lead is chosengreater than superconducting gap so that the normal leadremains topologically trivial. The tunnel barrier at theinterface of each N-S junction is modelled by adding anadditional onsite energy of strength 10 t on one end (left)of each wire. We have assumed semi-infinite nanowiresfor our calculations to exclude finite size effects [33]. (a) (b) (c)(d) (e) (f) (g) (h) (i) FIG. 4. Plot of differential conductance vs applied bias for N = 2-10 Majorana wires for coupled topological regime ( V z =0.85 t ; t y = 0.3 t ; β = 0.3 t ). The ZBCP peak appears only in thecase of conductance of odd number of wires (shown in green)while no such peak is observed in the case of conductance ofeven number of wires (shown in black). The zero temperature tunneling conductance of the ar-ray is calculated using S -matrix method. By computingthe reflection matrix at the N-S junction, tunneling con-ductance can be found; where the reflection matrix ( r ) isexpressed in terms of electron and hole scattering chan-nels at energy E as : r = (cid:18) r ee r eh r he r hh (cid:19) (7)where r ee ( r eh ) refers to the normal (Andreev) reflec-tion submatrix. For N conducting channels in the leadthe differential conductance can be evaluated using theBlonder-Tinkham-Klapwijk (BTK) formula [34] (in theunits of e /h ): G = dI/dV = [ N − T r ( r ee r † ee − r eh r † eh )] E = V (8)We have employed KWANT, a numerical transport pack-age in Python, to calculate the components of the reflec-tion matrix [35].Figure 3 (b) shows the differential conductance vs ap-plied bias V , when there is no inter-wire coupling in thenon-topological (trivial) regime . The differential conduc-tance resembles the zero temperature density of statesof a conventional superconductor with coherence peaksappearing at induced superconducting gap energy ± ∆.As expected, the magnitude of conductance at ± ∆ in-creases monotonically with increase in number of wires.Figure 3 (c) depicts the differential conductance whenthere is inter-wire coupling in the non-topological (triv-ial) case for N number of wires. Unlike the trivial uncou-pled case, enhancement of magnitude of conductance at ± ∆ is not monotonic with increase in number of wires.Figure 3 (d) shows the plot of conductance vs appliedbias V for uncoupled topological phase where each Ma-jorana bound state contributes a conductance of 2 e /h to the total differential conductance. Upto this part, thenumber of wires only contributes to the overall scaling ofthe absolute magnitude of the differential conductance.However, more interesting physics emerges when weakcoupling between the wires is also introduced in the topo-logical regime.As it can be seen in Figure 4, in the topological regime,turning on weak coupling between the wires has resultedin N conductance peaks, symmetric about bias V =0.The most important feature here is the emergence of a ZBCP when N is odd (Figure 4 (b), 4 (d), 4 (f) and4 (h)). On the other hand, for even values of N (Fig-ure 4 (a), 4 (c), 4 (e), 4 (g) and 4 (i)), ZBCP doesn’tappear. This is due to Majoranas at the edges of wireshybridizing into bonding and anti-bonding orbitals whencoupled. While, in even number of cases all the Majo-ranas are coupled pairwise leading to only satellite peaksin the conductance (around V = 0), in the case of oddnumber of wires one Majorana is left unpaired that con-tributes a conductance of 2 e /h at zero bias.It has been earlier proposed that ZBCP in tunnelingexperiments can be a smoking gun signature of Majo-rana modes. However, as it is also known, a ZBCPcan originate due to other possible factors as well. Thatmakes ZBCP based detection of Majorana modes some-what ambiguous. In this context, our proposed exper-imental scheme is very important. In this scheme, theconfirmation of Majorana is not based on the appear-ance of ZBCP alone, but our calculations provide a de-tailed scheme based on the number of transport-activenanowires in a single device where depending on whetherthe number of wires is odd or even, the ZBCP can beswitched ON or switched OF F respectively, in a con-trolled fashion. Hence, the proposed scheme would pro-vide direct, unambiguous signature of Majorana boundstates and topological superconductivity.We thank Subhro Bhattacharjee and Tanmoy Das forfruitful discussions. DR thanks DST INSPIRE for finan-cial support. GS acknowledges financial support fromSwarnajayanti Fellowship awarded by the Department ofScience and Technology, Govt. of India (grant number:
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