Topological superconducting phases in disordered quantum wires with strongspin-orbit coupling

Piet W. Brouwer, Mathias Duckheim, Alessandro Romito, and Felix von OppenDahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany

(Dated: March 15, 2011)

Zeeman fields can drive semiconductor quantum wires with strong spin-orbit coupling and inproximity to s-wave superconductors into a topological phase which supports end Majorana fermionsand offers an attractive platform for realizing topological quantum information processing. Here,we investigate how potential disorder affects the topological phase by a combination of analyticaland numerical approaches. Most prominently, we find that the robustness of the topological phaseagainst disorder depends sensitively and non-monotonously on the Zeeman field applied to the wire.

PACS numbers: 74.78.Na,73.63.Nm,03.67.Lx,71.23.-k

Introduction.—Topological quantum information pro-cessing promises to go a long way towards alleviatingthe problem of environmental decoherence in quantumcomputers [1, 2]. In this scheme, information is storedand processed by nonlocal qubits based on quasiparticleswith nonabelian quantum statistics [3, 4]. These nonlo-cal qubits are topologically protected against local per-turbations. Qubit operations are based on quasiparticleexchanges which are themselves topological in nature andthus insensitive to small variations in the operation.

The simplest type of quasiparticles obeying nonabelianquantum statistics are zero-energy Majorana fermions[5]. Such Majorana bound states are carried by vor-tices of weak-pairing superconductors with spinless p-wave pairing symmetry, a phase which is widely expectedto be realized by the ν = 5/2 fractional quantum Hallstate [6]. Starting with the seminal suggestion of Fuand Kane [7] that zero-energy Majorana fermions canin principle be engineered to exist in hybrid structures ofconventional s-wave superconductors and topological in-sulators, various superconductor hybrids have been pre-dicted to support Majorana fermions. These hybrids in-volve, e.g., standard two-dimensional electron systemswith strong spin-orbit coupling [8], metallic surface states[9], semiconductor quantum wires with strong spin-orbitcoupling [10, 11], or half-metallic wires and films [12–14].

Realizations based on quantum wires are perhaps par-ticularly promising for realizing topological quantum in-formation processing since they allow for relatively de-tailed scenarios [15] of how to manipulate (i.e., create,transport, or fuse) Majorana fermions. At the same time,disorder is known to often have drastic consequences forthe electronic properties of one-dimensional electron sys-tems [16]. This motivates us to address the importantquestion of how robust these proposed realizations of Ma-jorana fermions are against potential disorder.

The effects of disorder on topological phases in one di-mension which support Majorana fermions were alreadyinvestigated by Motrunich et al. [17] in a paper whichpredates the current wave of interest by almost a decade.Motrunich et al. consider a disordered p-wave supercon-

ductor in one dimension within a renormalization groupapproach. Their central finding is that disorder causesa sharp transition to a non-topological phase at a criti-cal disorder strength. More recent work on disorder ef-fects on Majorana fermions focuses on the strong-disorderlimit where the wire breaks up into topological and non-topological domains [18, 19].

It is the central purpose of the present paper to em-phasize that the physics of disorder in proximity-coupledsemiconductor wires with strong spin-orbit coupling isconsiderably richer. It reduces to the model studied byMotrunich et al. (sometimes refered to as Kitaev’s toymodel [20]) only in the limit of a large Zeeman field. Incontrast, we find that the semiconductor quantum wiresexhibit additional regimes, depending on the strength ofthe Zeeman coupling, which display characteristic differ-ences in the effectiveness of disorder. Most importantly,our results can give significant guidance to efforts to re-alize Majorana fermions in the laboratory.Semiconductor wires.—We consider a semiconductor

quantum wire with strong Rashba spin-orbit coupling uin an external Zeeman field B. For definiteness, we takethe spin-orbit field and the Zeeman field to point alongthe x- and z-directions, respectively. If the wire is inproximity to an s-wave superconductor, the correspond-ing Bogoliubov-de Gennes (BdG) Hamiltonian takes theform [10, 11]

H =

(p2

2m+ upσx + V (x)− µ

)τz −Bσz + ∆τx. (1)

Here, p and x denote the momentum and position alongthe wire and ∆ is the proximity-induced gap. The Paulimatrices σi (τi) operate in spin (particle-hole) space. Dis-order is modeled through a Gaussian white noise poten-tial V (x) with zero mean and correlator 〈V (x)V (x′)〉 =γδ(x− x′) [21]. The disorder strength γ is related to themean scattering time τ0 = u/γ at µ = B = 0.

Eq. (1) is written in a basis which corresponds to thefour-component Nambu operator Ψ = [ψ↑, ψ↓, ψ

†↓,−ψ

†↑]

in terms of the electronic field operator ψσ(x). In this ba-sis, the time-reversal operator takes the form T = iσyK,

arX

iv:1

103.

2746

v1 [

cond

-mat

.mes

-hal

l] 1

4 M

ar 2

011

2

(d)E

p

B + ∆2∆

∆B −∆pF

(b) E

p2B pF

(a)E

p

�so

pF

(c) E

p2BpF

Figure 1: (Color online) Dispersion relations in the absence of disorder for: (a) the quantum wire in the normal phase andzero Zeeman field; (b) the quantum wire in the normal phase and finite Zeeman field B � εso, including an illustration (redarrows) of a second-order process contributing to the effective disorder potential in the regime of intermediate Zeeman fields;(c) the quantum wire in the normal phase and strong Zeeman field B � εso. (d) Quasiparticle excitation spectrum near (fullline) and at (dashed line) the topological phase transition ∆ = B. The red (dashed) arrows illustrate a second-order processto the high-energy subspace near p = 0, which contributes to the random Zeeman field in the low-energy model.

where K denotes complex conjugation. The BdG Hamil-tonian (1) obeys the symmetry {H, CT} = 0, withC = −iτy. Thus, if |ψ〉 is an eigenspinor ofH with energyE, CT |ψ〉 is an eigenspinor of energy (−E). Zero-energyMajorana fermions are characterized by spinors satisfy-ing CT |ψ0〉 = |ψ0〉, Majorana modes by CT |ψp〉 = |ψ−p〉.Regimes and effective low-energy models.—In the ab-

sence of disorder and of the proximity-induced gap, i.e.,for ∆ = V (x) = 0, the Hamiltonian Eq. (1) has theparticle dispersion E

(0)p,± = p2/2m ± [(up)2 + B2]1/2. A

topological superconducting phase requires that there areonly two Fermi points, cp. Figs. 1(a)-(c). In this case, thechirality dependence of the spin orientation induced bythe Rashba coupling turns the normal state into a helicalliquid [10, 11]. The superconducting phase for B < ∆is a conventional, non-topological s-wave superconduc-tor. A topological superconducting phase can be real-ized when the Zeeman field is larger than ∆. For strongspin-orbit coupling εso � ∆ (with the spin-orbit energyεso = mu2/2), it is natural to distinguish three regimes:(i) Weak Zeeman fields (εso � B > ∆ with B−∆� ∆):The gap at p = 0 is given by b = B−∆ in the presence ofboth Zeeman field and proximity-induced superconduc-tivity. Thus, as long as |B −∆| � ∆, the gap at p = 0is much smaller than the proximity-induced gap at theFermi points which is equal to ∆. In this case, the rel-evant low-energy degrees of freedom for µ ' 0 are thosewith momenta p near p = 0 [11]. (Note that here we take∆ and B to be positive.)

We can develop a low-energy model by focusing onsmall p where p2/2m can be neglected relative to thespin-orbit term and expanding about the critical pointB = ∆ at which the p = 0 gap closes. The low-energy subspace is then spanned by the eigenspinors〈x|p,+〉 = (1/2

√L)[1,±1, 1,∓1]T eipx of the BdG Hamil-

tonian, and the Hamiltonian evaluated in this subspacetakes the form

H =

(up− v −b+ w−b+ w −up+ v

)(2)

Remarkably, all matrix elements of the disorder potentialin the low-energy subspace vanish so that to this order,

we find that the disorder fields v = w = 0. Note thatthe eigenspinors of the low-energy subspace correspondto Majorana modes.

The leading contributions to the disorder fields v andw are quadratic in the bare disorder potential, emergingfrom virtual excitations of high-energy states shown inFig. 1(d). Adding the contributions from the three rel-evant high-energy subspaces near p = 0 and p = ±pF(pF = 2mu), we obtain

v = Vup

u2p2 + 4∆2V −

∑±V

u(p± pF )/2

u2(p± pF )2 + ∆2V,(3)

w = V2∆

u2p2 + 4∆2V +

∑±V

∆/2

u2(p± pF )2 + ∆2V.(4)

In the low-energy limit, the random gauge field v(x)vanishes, while w(x) is a Gaussian white noise Zeemanfield with 〈w(x)〉 = γ/u and 〈w(x)w(x′)〉 − 〈w(x)〉2 =(γ2/2u∆)δ(x − x′). The effective Hamiltonian Eq. (2)remains a valid approximation for γ � γmax ∼ u∆.(ii) Intermediate Zeeman fields (εso � B � ∆): Whenthe Zeeman field is much larger than the proximity-induced gap ∆, but still small compared to the spin-orbitenergy εso, the gap at p = 0 is of order B and thus muchlarger than the gap at the Fermi points of order ∆. Inthis case, the relevant low-energy degrees of freedom forµ ' 0 are those near the Fermi points ±pF . It is im-portant to realize that the spin orientation at the Fermipoints is dominated by the spin-orbit coupling so thatelectrons at the two Fermi points have almost antipar-allel spins. Thus, the proximity effect is not attenuatedby spin effects, while disorder-induced backscattering isstrongly suppressed.

This can be made explicit by projecting the orig-inal Hamiltonian Eq. (1) onto the lower band ofthe normal-state Hamiltonian of the clean wire.The corresponding electron and hole eigenspinors are|p, e〉 = [cos(α/2),− sin(α/2), 0, 0]T eipx/

√L and |p,h〉 =

[0, 0,− sin(α/2), cos(α/2)]T eipx/√L with tanα = up/B.

Evaluating the matrix elements of the full Hamiltonianin this low-energy subspace and linearizing the disper-sion about the Fermi points, we find a spinless p-wave

3

superconductor

H = veff(pλz − pF )τz + ∆effλyτx + Veff(x)λxτz, (5)

with Fermi momentum pF = 2mu, Fermi velocity veff =u, and gap function ∆eff = ∆. The disorder potentialhas the correlation function 〈Veff(x)Veff(x′)〉 = γeffδ(x −x′), corresponding to the scattering time τeff = veff/γeff .The Pauli matrices λi operate in the space of left- andright-movers. In Eq. (5), we ignore forward scattering bydisorder, which is not expected to affect the results.

Unlike for weak Zeeman fields, the effective dis-order potential Veff(x) has a contribution Veff(x) =(B/4εso)V (x) [such that γeff = (B/4εso)2γ] which is lin-ear in the bare disorder potential. However, this contri-bution includes a strong suppression by spin effects andfor strong enough disorder γ � γlin ∼ (Bu)(B/εso)2,the dominant contribution to Veff remains quadratic inthe bare disorder potential, involving virtual intermedi-ate states near p = 0, cp. Fig. 1(b). We find that thecorresponding magnitude of the effective disorder poten-tial is γeff = γ2/2Bu. Note that for intermediate Zeemanfields, the projection to the lower subband remains welldefined for γ � Bu.(iii) Strong Zeeman fields (B � εso � ∆): The low-energy theory in this limit has been derived previouslyfor the clean case and can be obtained as in the case ofintermediate Zeeman fields [15]. The relevant degrees offreedom remain those in the vicinity of the Fermi points[cp. Fig. 1(c)], but the spin orientation is now dominatedby the Zeeman field and electrons at the two Fermi pointshave essentially parallel spin. Consequently, the proxim-ity effect is suppressed, while backscattering is controlleddirectly by the bare disorder potential V (x). Indeed, theprojection [valid for γ � Bu(B/εso)1/2] readily yields aspinless p-wave superconductor Eq. (5). Unlike in theregimes of weak and intermediate Zeeman fields, the de-tailed parameters differ for the cases of fixed chemical po-tential µ ' 0 and fixed density n = 2mu/π. For fixed µ,one finds pF = (2mB)1/2, veff = (2B/m)1/2, and ∆eff =2∆(εso/B)1/2. For fixed n, we have pF = 2mu, veff = u,and ∆eff = 4∆(εso/B). In both cases, Veff(x) = V (x)Phase diagram.—Equation (5) is the continuum ver-

sion of the lattice model studied by Motrunich et al. [17].It has a sharp transition to a non-topological phase when∆effτeff = 1/2. At the critical disorder strength the den-sity of states ν(ε) is singular [17, 22–24]

ν(ε) ∝ |(vFε/∆eff) ln3(ε/∆eff)|−1, (6)

whereas for weaker disorder ν(ε) has a power-law tailwith an exponent that depends on τeff [17, 24],

ν(ε) ∝ ε4∆effτeff−3. (7)

Although both the topological and the non-topologicalphases have a gapless excitation spectrum in the pres-ence of disorder, the topological phase distinguishes it-self by the presence of Majorana end states at an energy

that is exponentially small in the wire length L [17]. Al-ternatively, for a wire that is coupled to normal-metalleads, the topological phase is characaterized by a neg-ative sign of the determinant of the reflection matrix r,whereas det r is positive for the nontopological phase [25].

These results allow us to derive the phase diagram ofthe quantum wire as function of Zeeman coupling anddisorder in some detail. Consider first the regime (iii)of large Zeeman fields at fixed µ, for which 1/τeff =γ(m/2B)1/2 and ∆eff = 2∆(εso/B)1/2. Hence, the phaseboundary between the topological and non-topologicalphases is at the critical disorder strength γcr = 4∆u.The critical disorder strength decreases with the Zee-man field due to the concurring suppression of the p-wave gap ∆eff when considering the case of fixed den-sity, where 1/τeff = γ/u and ∆eff = 4∆(εso/B) so thatγcr = 8∆u(εso/B).

In the intermediate regime (ii), the contribution toVeff(x) which is linear in V (x) gives rise to an elas-tic scattering rate 1/τeff = γ(B/4εso)2/u. The associ-ated critical disorder strength is γcr = 2u∆(4εso/B)2.This result is appropriate as long as γcr � γlin, whichholds for B � (∆/εso)1/5εso. The expression for γcr de-creases with the Zeeman field, so that the critical disorderstrength does not increase with B throughout the entirerange of B where the phase boundary between topologi-cal and non-topological phases is governed by the disor-der contribution linear in the bare disorder potential. Incontrast, the critical disorder strength increases with Bwhen B � (∆/εso)1/5εso, where the quadratic contribu-tion to Veff is dominant. The quadratic contribution toVeff gives an elastic scattering rate of 1/τeff = γ2/2Bu2.This yields γcr = 2(∆Bu2)1/2 for the critical disorderstrength.

In the regime (i) of weak Zeeman fields, the low-energymodel in Eq. (2) maps onto a random-hopping chain withstaggered mean hopping [22]. In this model, an infinitedisorder strength is needed to tune to the critical point ifa finite gap is present in the non-disordered chain [26, 27],indicating that any finite disorder strength preserves thetopological phase. However, we still find a dependenceof the phase boundary on disorder due to a more basiceffect: Since the effective disorder potential is quadraticin the bare disorder V (x), the disorder-induced Zeemanfield w(x) has not only a random component, but alsoa finite average 〈w(x)〉 = γ/u. Due to this average, thephase boundary is no longer given by b = 0, but ratherdetermined by b−γ/u = 0, or B = ∆+γ/u. The densityof states shows a power-law dependence on both sides ofthe topological phase transition, with

ν(ε) ∝ εα−1, with α = γ−2|4u∆(bu− γ)|. (8)

Exactly at the phase transition, ν(ε) has the Dyson sin-gularity (6) [22, 26, 27], with ∆eff = γ2/4u2∆. Of course,these conclusions are restricted to disorder strengthssmaller than the maximal disorder strength γmax ∼ u∆,

4

γmax

γlin

NONTOPOLOGICAL

γ B�soBopt∆

∆u

TOPOLOGICAL

∝ B1/2 ∝ B−2

∝ B−1u(B −∆)

(a)(b)

detr

−1

0

1

0.01 0.1 1 10

B/2εSO

10−5

10−4

10−3

10−2

10−1

γ

(B −∆)u

2(∆u2B)1/2

2u∆(4εSO/B)2

γcr,ii4∆u

Figure 2: (Color online) Phase diagrams of the quantum wireas function of Zeeman field B and disorder strength γ in alog-log plot (a) for fixed density n and (b) for fixed chemicalpotential including numerical results. In (a), the long-dashedline delineates the limit of validity of the low-energy models;at the short-dashed line, Veff(x) changes between linear andquadratic dependence on the bare disorder V (x). In (b), thenumerics is based on calculating the determinant of the reflec-tion matrix for the full Hamiltonian (1), cp. Ref. [25]. γcr,ii(dashed line) interpolates between the linear and quadraticcontributions to γeff for intermediate Zeeman fields.

where the mapping to the Hamiltonian in Eq. (2) is valid.In fact, the matching with the regime of intermediateZeeman fields shows that the topological phase is lostonce disorder becomes of the order of γmax.Conclusion.—Our results for the critical disorder

strength vs Zeeman field are summarized in the phasediagrams in Fig. 2, where we also include a numericalphase diagram based on a scattering approach to the fullmodel (1). This phase diagram emphasizes the importantimplication that the topological superconducting phase ismost stable against disorder at the optimal Zeeman fieldBopt ∼ (∆/εso)1/5εso. This is valid for fixed chemical po-tential [Fig. 2(b)], but most pronounced for fixed density[Fig. 2(a)], where the critical disorder strength γcr → 0as B →∞.

As discussed above, the cause of this effect is thevanishing of the effective p-wave gap ∆eff in the limitB →∞. It is instructive to compare the large-B limit ofthe semiconductor model considered here with that of ahalf-metallic ferromagnet, brought into electrical contactwith a superconductor with spin-orbit coupling [13, 14].In both models, charge carriers are fully spin polarized.Yet, the induced gap ∆eff is independent of the exchangefield for the half metals of Refs. 13, 14, the reason beingthat ∆eff is set by the strength of the spin-orbit couplingin the superconductor. Since ∆eff directly determines thecritical disorder strength γcr, we are thus lead to expectthat even at fixed density, the critical disorder strengthγcr for a semiconductor wire can be made to saturate inthe limit B →∞, if the helical state in the semiconductorwire is attributed to spin-orbit coupling in the supercon-ductor, rather than spin-orbit coupling intrinsic to the

semiconductor.

We acknowledge discussions with Yuval Oreg and fi-nancial support by the Deutsche Forschungsgemeinschaftthrough SPP 1285 as well as by the Alexander von Hum-boldt foundation. Upon completion of this manuscriptRef. [28] appeared, which discusses the influence of im-purities in the superconductor on the proximity effect.

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