Transport through a Majorana island: strong tunneling regime
TTransport through a Majorana island: strong tunneling regime
Roman M. Lutchyn and Leonid I. Glazman Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA Department of Physics, Yale University, New Haven, CT 06520, USA (Dated: October 22, 2018)In the presence of Rashba spin-orbit coupling, magnetic field can drive a proximitized nanowireinto a topological superconducting phase [1, 2]. We study transport properties of such nanowiresin the Coulomb blockade regime. The associated with the topological superconductivity Majoranamodes significantly modify transport and lead to single-electron coherent transmission through thenanowire - a non-local signature of topological superconductivity. In this work, we focus on the caseof strong hybridization of the Majorana modes with the normal leads. The induced by hybridizationbroadening of the Majorana zero-energy states competes with the charging energy, leading to aconsiderable modification of the Coulomb blockade in a nanowire contacted by two normal leads.We evaluate the two-terminal conductance as a function of the gate voltage, junctions transmissioncoefficients, the geometric capacitance of and the induced superconducting gap in the nanowire.
Topological superconductors provide a promising plat-form for fault-tolerant quantum computation [3–8].These exotic electronic phases of matter are predicted tohost defects binding Majorana zero-energy modes whichobey non-Abelian braiding statistics [9–11]. Theory pre-dicts that Majorana zero modes may be realized at theends of proximitized nanowires [1, 2, 12], and there ismounting experimental evidence for their existence inthese systems [13–22].Most of the proposals for Majorana-based topologicalquantum computation involve mesoscopic islands with asizable charging energy which contain two or more Ma-jorana modes (Majorana islands) [23–31]. Therefore, itis important to understand the interplay of topologicaldegrees of freedom and charging energy of these islands.Another motivation comes from the recent experiment byAlbrecht et al. [20] investigating the dependence of two-terminal conductance through a Majorana island in theCoulomb blockade regime, see Fig. 1a for the device lay-out. The existing theory [32, 33] allows one to evaluatethe conductance of a Majorana island in the weak tunnel-ing regime, g i (cid:28)
1, using resonant level approximation(here g i is the dimensionless normal-state conductanceof i -th junction, G i = g i G , and G = e /h is the con-ductance quantum for spin-polarized electrons). In thatapproximation, only the resonant level comprised of thetwo degenerate ground states of the island is involved inthe formation of narrow Coulomb blockade conductancepeaks, see Fig. 1b.The resonant-level approximation, however, is inappli-cable to the strong-tunneling regime, corresponding toone or both junctions approaching the reflectionless limit(i.e. 1 − g i (cid:28) P . Under this condition, the quasi-continuum ofexcited states with energies above ∆ P also contributesto the electron transport across the island. The problemat hand is rather non-trivial. A similar setting inthe absence of superconductivity and in the limit of zero spacing between the levels of quasi-continuumwas investigated in Ref. [34]; in a symmetric device( g = g ), the maximum conductance reaches only half of the conductance quantum G , and the widthof the Coulomb blockade peak scales proportionally totemperature T . Below we demonstrate that, on thecontrary, the maximum conductance through a Majo-rana island (∆ P (cid:54) = 0) equals G . Thus, upon loweringthe temperature below ∆ P , the maximum two-terminalconductance should increase. We also show that thesuperconductivity modifies the off-peak conductance,which remains finite in the limit T →
0. Therefore,the two-terminal conductance G ( N g ) = G ( N g , T → N g . In this paper, we study theevolution of the G ( N g ) function with the conductances g i and ratio ∆ P /E C .In the strong tunneling regime, it is convenient to usethe bosonization technique which allows one to accountfor charging energy and superconducting pairing non-perturbatively [35]. Weak reflection at the junctions canbe then taken into account within perturbation theory.The effective model for a proximitized nanowire in thetopological regime ( i.e. , spinless p-wave superconductingstate [36]) in the presence of Coulomb blockade can bewritten as H = H NW + H C + H P + H B , (1) H NW = v π (cid:90) ∞−∞ dx (cid:2) ( ∂ x θ ) + ( ∂ x φ ) (cid:3) , (2) H C = E C ( N −N g ) = E C (cid:18) φ ( x ) − φ ( x ) π −N g (cid:19) , (3) H P = − ∆ P D πv (cid:90) x x dx cos 2 θ , (4) H B = − (cid:88) i =1 , Dr i cos 2 φ ( x i ) . (5)Here v , ∆ P , and D are the Fermi velocity in the nanowire, a r X i v : . [ c ond - m a t . s up r- c on ] J a n FIG. 1. (Color online) Panel (a): Schematic plot of the device.Panel (b): Conductance G as a function of the dimensionlessgate voltage N g . In a symmetric device, g = g ≡ g , con-ductance reaches G . Solid (black) curve: Coulomb blockadepeak at g (cid:28) G ( N g ) at intermediate values of g such that ∆ P /E C (cid:28) − g (cid:28)
1; see Eq. (23) for the widthof the maximum and Eq. (24) for the crossover to the weak-tunneling limit. Dot-dashed (blue) curve: G ( N g ) of a sym-metric device in the strong-tunneling limit, E C (1 − g ) (cid:28) ∆ P .Conductance approaches the unitary limit, exhibiting weak N g -dependent oscillations, see Eq. (22). induced superconducting gap, and UV cutoff energy, re-spectively. Charging energy H C depends on the chargetransferred into the Majorana island via the two junc-tions, N = [ φ ( x ) − φ ( x )] /π , with the bare chargingenergy E C = e / C Σ being determined by the geometri-cal capacitance of the proximitized nanowire C Σ (includ-ing its superconducting shell). The barriers at x , aredescribed by the reflection amplitudes r , , respectively.Here we implicitly assume that the superconducting shellrenormalizes level spacing in the nanowire so that thespacing becomes negligibly small. In this respect, ourmodel is similar to the one of Refs. [34, 37]. The term H P accounts for the superconducting proximity effect.Let us now consider the case E C (cid:29) ∆ P and r , (cid:28)
1. In this limit, term H C of Eq. (3) pins the mode φ ( x ) − φ ( x ) responsible for changing the charge of theisland. Integrating out this massive mode, one obtainsan effective boundary Hamiltonian [37] valid in energyband E (cid:28) E C , H B = − (cid:112) c E C D r ( N g ) cos [ φ ( x )+ φ ( x ) − α ( N g )] . (6) Here α ( N g ) is unimportant phase, parameter r ( N g ) is r ( N g ) = (cid:113) r + r + 2 r r cos(2 π N g ) sgn(cos π N g ) , (7) c = e C / π , and C = 0 . r ( N g ) is relevant and grows underthe Renormalization Group (RG) procedure accordingto dr/d(cid:96) = r/ D reaches∆ P or the boundary perturbation H B reaches the strong-coupling limit H B ∼ D . The latter occurs at D ∼ D c defined as D c ∼ Γ ( N g ) = 2 e C π E C r ( N g ) , (8)where we chose the numerical coefficient in accordancewith Ref. [34]. The linear conductance strongly dependson the gate voltage as long as Γ ( N g ) (cid:29) ∆ P . Inthe opposite limit, ∆ P (cid:29) Γ ( N g ), conductance onlyweakly depends on N g , and approaches the unitary limit.We start by considering the limit Γ ( N g ) (cid:29) ∆ P which(at sufficiently large r and r ) is realized far away fromthe charge degeneracy points. Upon reducing the bandwidth D in the course of RG flow to D ∼ D c , the com-bination of fields φ ( x ) + φ ( x ) becomes pinned by thebackscattering term Eq. (6). At smaller energy scales, D (cid:28) Γ ( N g ), the dynamics of φ ( x ) and φ ( x ) con-sists of hops between the equivalent minima of energyEq. (6) which defines the two-dimensional “landscape”in the plane of φ ( x ) , φ ( x ). The least-irrelevant hop-ping term in the effective low-energy Hamiltonian shifts φ ( x ) + φ ( x ) by 2 π ,˜ H B = − λ ( D ) D cos (cid:2) θ ( x +2 ) − θ ( x − )+ θ ( x +1 ) − θ ( x − ) (cid:3) . (9)Here the fields θ ( x − , τ ) and θ ( x +1 , τ ) refer to the proxim-itized nanowire whereas points x − and x +2 belong to theleads, see Fig. 1a. At the crossover energy scale, D ∼ D c ,the running constant λ ( D c ) ∼
1, and it decreases uponreducing the band width. The RG flow for λ in the do-main Γ ( N g ) (cid:29) D (cid:29) ∆ P is controlled by dλ/d(cid:96) = − λ and yields λ ( D ) ∼ λ ( D c ) D/D c ∼ D/D c . The dynam-ics of fields θ ( x , ) on energy scales E (cid:46) D is governedby Eqs. (2), (4), and (9) with the boundary conditions ∂ x θ ( x ± , ) = 0, compatible with Eq. (9).Hamiltonian (9) corresponds to an electron transferinto one end of the proximitized wire, while another elec-tron is taken out from the opposite end. This way, asingle electron charge e is transferred between the leads.The corresponding current operator reads I = eλ ( D ) D sin (cid:2) θ ( x +2 ) − θ ( x − )+ θ ( x +1 ) − θ ( x − ) (cid:3) . (10)Using it, one may evaluate the two-terminal conductanceat temperatures T (cid:28) Γ ( N g ) by means of Kubo for-mula [38], G = 12 T (cid:90) ∞−∞ dt Π (cid:18) it + 12 T (cid:19) , Π( τ ) = (cid:104) I ( τ ) I (0) (cid:105) (11)(here τ is imaginary time). In the intermediate range oftemperatures, ∆ P (cid:28) T (cid:28) Γ ( N g ), one may ignore thepairing interaction Eq. (4) and use the free-field actionto determine the time evolution of the current operatorEq. (10). The result for the conductance G ( N g , T ) is G ( N g , T ) G = c T Γ ( N g ) . (12)Finding the numerical coefficient c here is beyond theaccuracy of the RG treatment, but it is known from theexact solution, c = π [34].At lower temperatures, T (cid:28) ∆ P , fluctuations of thefield θ ( x, τ ) within the proximitized wire ( x < x < x )are suppressed by the superconducting pairing term,Eq. (4). To evaluate the conductance, we may reduce theband width down to D ∼ ∆ P , yielding λ (∆ P ) ∼ ∆ P D c inEq. (9), where now fields θ ( x +1 , τ ) and θ ( x − , τ ) are pinnedto a minimum of pairing energy. With these fields beingpinned, Eq. (9) describes tunneling of an electron be-tween points x − and x +2 belonging to the opposite leads.The corresponding tunneling action takes the form S B = (cid:90) dω π | ω | π | θ − | − (cid:90) /T ∆ − P dτλ ( D ) D cos √ θ − , (13)where θ − = [ θ ( x +2 ) − θ ( x − )] / √
2. Note that the bound-ary perturbation term in Eq.(13) becomes marginal now( i.e. , dλ/d(cid:96) = 0), and the problem at hand maps ontoweak tunneling of a free fermion across an impurity. Us-ing Kubo formula (11), one can readily calculate two-terminal conductance to find G ( N g ) G = c · ∆ P Γ ( N g ) , c ≈ π . (14)This temperature-independent conductance is due toelastic cotunneling processes in which an electron entersthe BCS condensate at one end of the wire with anotherelectron exiting the condensate from its opposite end,leaving no excitations behind. The results obtained inthe adjacent temperature intervals, Eqs. (12) and (14),match each other at T ∼ ∆ P .To find the numerical coefficient c in Eq. (14) we usedthe following re-fermionization procedure. First, we writethe low-energy Hamiltonian (9) in the fermion represen-tation:˜ H B = c v Γ ( N g ) ψ † ( x +2 ) ψ † ( x +1 ) ψ ( x − ) ψ ( x − ) + h . c . . (15)Note that Eq. (15) conserves the number of electrons inthe proximitized wire, so it commutes with the chargingenergy Eq. (3). To find the numerical coefficient c , weuse Eq. (15) to derive the current operator I = i c ev Γ ( N g ) (cid:0) ψ † ( x +2 ) ψ † ( x +1 ) ψ ( x − ) ψ ( x − ) − h . c . (cid:1) . (16) Then, applying the Kubo formula Eq. (11) and ignor-ing the pairing interaction, we re-derive the conductancein the re-fermionized scheme and match the result withEq. (12). That procedure fixes the coefficient c = π inEqs. (15) and (16). After that, we return to the evalu-ation of conductance at T (cid:28) ∆ P . We are interested inthe dominant, elastic contribution to the conductance;in a long proximitized wire segment, only the Majoranastates contribute to the elastic amplitude [39]. The corre-sponding part of the Hamiltonian (15) can be written interms of Majorana fermion operators γ and γ localized,respectively, at x and x ,˜ H B ≈ − πv ∆ P Γ ( N g ) ψ † ( x +2 ) ψ ( x − ) γ γ + h . c . . (17)Finally, the calculation of the two-terminal conductanceby means of the Kubo formula (11) applied to thelow-energy tunneling processes described by Eq. (17)yields constant c in Eq. (14).We now consider weak-reflection case, Γ ( N g ) (cid:28) ∆ P ,which is realized in a symmetric device at a gate voltageclose to a charge degeneracy point, or at any gate volt-age if the reflection amplitudes r , are sufficiently small(and not necessarily equal each other). At intermediateenergy scale, E C (cid:29) E (cid:29) ∆ P , the pairing interactionEq. (4) and the boundary Hamiltonian Eq. (6) can betreated perturbatively. Thus, the only constraint on fluc-tuations of φ ( x ) and θ ( x ) within the proximitized wire isthe pinning of the combination of fields φ ( x ) − φ ( x ) bycharging energy. As follows from Ref. [34], the conduc-tance in the regime T (cid:29) ∆ P (cid:29) Γ is G ≈ G / P , the pair-ing interaction (4) suppresses the fluctuations of θ ( x )within the proximitized segment, i.e. ∂ τ θ ( x, τ ) = 0.Thus, the condition ∂ x φ ( x , , τ ) = 0 is enforced at theends of the segment. To evaluate conductance in thelimit of no backscattering (Γ ( N g ) → x and x to obtain the bound-ary action in terms of the relevant degree of freedom φ + = ( φ ( x ) + φ ( x )) / √ S = (cid:90) ∆ P dω π | ω | π | φ + | . (18)The dc conductance is obtained then by using Kubo for-mula Eq. (11); current operator in this limit is given by I = e π [ ∂ t φ ( x ) + ∂ t φ ( x )] = e π √ ∂ t φ + . (19)Upon evaluating Π( τ ) = e π (cid:104) ∂ τ φ + ( τ ) ∂ τ (cid:48) φ + ( τ (cid:48) ) (cid:105) τ (cid:48) =0 us-ing Eq. (18), we find that G ( N g ) = G in the absence ofbackscattering. The full quantized value of the conduc-tance G ( N g ) is in agreement with the notion of single-electron resonant tunneling via a Majorana state [32, 33].One may notice that the conductance grows by a factorof 2 once temperature is lowered across the scale set by∆ P . This prediction can be easily verified in current ex-periments on proximitized nanowires [19, 20].To account for backscattering, we use Eq. (6) with thebandwidth D ∼ ∆ P , H B = − (cid:112) c E C ∆ P r ( N g ) cos( φ ( x )+ φ ( x ) − α ( N g )) (20)with c ∼
1. At E (cid:46) ∆ P the long-wavelength fluctu-ations within the proximitized wire ( i.e. , in the interval x < x < x ) are gapped out by the pairing term (4).As a result, the boundary term (20) becomes marginal( dr/d(cid:96) = 0) and remains small. The backscattering termEq. (20) augments the free-field Hamiltonian and modi-fies the boundary action, S = S − (cid:90) T − ∆ P − dτ (cid:112) c E C ∆ P · r ( N g ) cos( √ φ + ) . (21)One can notice that the problem at hand maps onto thesingle-impurity model in the weak-backscattering limit,characterized by strong fluctuations φ + of charge passingthrough the nanowire. This is to be contrasted with thestrong-pinning limit, Eq. (13).The evaluation of the correction to the conductance δG within the second-order perturbation theory in r ( N g ),see, e.g. , Refs. [41, 42], yields G ( N g ) − G G ∼ − Γ ( N g )∆ P , (22)where Γ ( N g ) is defined in Eq. (8). The numericalprefactor in Eq. (22) is beyond the accuracy of theRG procedure. The maximal value of Γ( N g ) equalsΓ max = (2 e C /π ) E C | r + r | and is reached at everyinteger N g . If the reflection amplitudes r , are smallenough so that Γ max (cid:28) ∆ P , then Eq. (22) is applicableat all gate voltages. In the opposite case, Eq. (22)may be applicable in the vicinity of the half-integervalues of N g , provided the setup is almost symmetric, E C | r − r | (cid:28) ∆ P .The developed scaling theory allows us to establish theevolution of the G ( N g ) function upon increase of the re-flection amplitudes. The two-terminal conductance os-cillations with N g are fully washed out by quantum fluc-tuations if r or r is zero. At small but finite ampli-tudes, Γ max (cid:28) ∆ P , oscillations are weak, see Eq. (22)and Fig. 1b. We will sketch further evolution of G ( N g )assuming a symmetric setup, r = r ≡ r . Once r be-comes large enough so that Γ max (cid:29) ∆ P , the applicabilityof Eq. (22) is confined to the vicinities of the half-integervalues of N g . One may use Eq. (14) to estimate con-ductance away from these degeneracy points. MatchingEqs. (22) and (14) with each other, we find η ∼ [∆ P /E C (1 − g )] / (23) for the width of the conductance maxima, see Fig. 1b.Further increase of the reflection amplitudes eventu-ally results in the crossover to a weak-tunneling regime, g , (cid:28)
1. Considering it, we will still concentrate on asymmetric setup, g = g ≡ g . At ∆ P = 0, quantumfluctuations of charge of the island result in the loga-rithmic renormalization of the transmission amplitudesof the two junctions connecting it with the leads [34].Due to this “charge Kondo” renormalization, the trans-mission amplitudes reach value ∼ T K ≈ E C exp( − π / √ g ), if N g is tuned to a narrow re-gion, |N g − / | (cid:46) T K /E C . The presence of ∆ P does notprevent the aforementioned logarithmic renormalizationas long as ∆ P (cid:28) T K . At energy scales below T K we mayuse the strong-tunneling RG theory developed above,with the proper replacement of the parameters. Namely,in Eq. (6) we change E C → T K and modify r ( N g ) fromthe one given in Eq. (7) to r ( N g ) ∼ ( E C /T K )( N g − / ( N g ) of Eq. (8) is changedto ˜Γ ( N g ) ∼ ( E C /T K )( N g − / . We may use ˜Γ ( N g )to estimate the conductance with the help of Eqs. (22)and (14). It easy to see that the maxima of G ( N g ) underthe considered conditions have width η ∼ (cid:113) ∆ P T K /E C , ∆ P (cid:46) T K . (24)At even smaller g , the gap ∆ P exceeds T K and cuts offthe logarithmic renormalization of the transmission am-plitudes before those reach the strong-tunneling limit. Asthe result, G ( N g ) corresponds to a conventional Breit-Wigner resonance [33], with the width defined by theproperly renormalized tunneling amplitudes [34], η ∼ ∆ P E C g/ π cos (cid:104) π E C / ∆ P )ln( E C /T K ) (cid:105) , ∆ P (cid:38) T K . (25)Notice that Eq. (24), valid at an intermediate range ofconductances (defined by the ratio ∆ P /E C ), matchesthe strong- and weak-tunneling results, Eqs. (23) and(25), at T K ∼ E C and T K ∼ ∆ P , respectively [43].Coulomb blockade of electron transport across anormal-state metallic island results in oscillations of theconductance G ( N g ) with the gate voltage. The pe-riodicity of oscillations corresponds to the increment e of the charge N g induced on the island by thegate. Conductance accross a proximitized wire in thetopologically-nontrivial superconducting state exhibitsoscillations with the same period. Yet, the behavior ofthe function G ( N g ) is drastically different. This becomesespecially clear in the case of a symmetric device with twoidentical single-channel junctions. In the normal state, G ( N g ) is controlled by an unstable two-channel Kondofixed point. The conductance maxima scale linearly withtemperature T , reaching value G = G / T →
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