Majorana fermions in a tunable semiconductor device
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec Majorana fermions in a tunable semiconductor device
Jason Alicea Department of Physics, California Institute of Technology, Pasadena, California 91125 (Dated: December 10, 2009)The experimental realization of Majorana fermions presents an important problem due to their non-Abelian nature and potential exploitation for topological quantum computation. Very recently Sau et al .[arXiv:0907.2239] demonstrated that a topological superconducting phase supporting Majorana fermions can berealized using surprisingly conventional building blocks: a semiconductor quantum well coupled to an s -wavesuperconductor and a ferromagnetic insulator. Here we propose an alternative setup, wherein a topological su-perconducting phase is driven by applying an in-plane magnetic field to a (110)-grown semiconductor coupled only to an s -wave superconductor. This device offers a number of advantages, notably a simpler architectureand the ability to tune across a quantum phase transition into the topological superconducting state, while stilllargely avoiding unwanted orbital effects. Experimental feasibility of both setups is discussed in some detail. PACS numbers:
I. INTRODUCTION
The problem of realizing and manipulating Majoranafermions in condensed matter systems is currently a topicof great theoretical and experimental interest. Roughly, Ma-jorana fermions constitute ‘half’ of a usual fermion. Thatis, creating an ordinary fermion f requires superposing twoMajorana modes γ , —which can be separated by arbitrarydistances—via f = γ + iγ . The presence of n well-separated Majorana bound states thus allows for the construc-tion of n ordinary fermions, producing (ideally) a manifold of n degenerate states. Braiding Majorana fermions around oneanother produces not just a phase factor, as in the case of con-ventional bosons or fermions, but rather transforms the statenontrivially inside of this degenerate manifold: their exchangestatistics is non-Abelian . Quantum information encoded inthis subspace can thus be manipulated by such braiding op-erations, providing a method for decoherence-free topologi-cal quantum computation . Majorana fermions are thereforeclearly of great fundamental as well as practical interest.At present, there is certainly no dearth of proposals forrealizing Majorana fermions. Settings as diverse as frac-tional quantum Hall systems at filling ν = 5 / , stron-tium ruthenate thin films , cold atomic gases , su-perfluid He-3 , the surface of a topological insulator ,semiconductor heterostructures , and non-centrosymmetricsuperconductors have all been theoretically predicted tohost Majorana bound states under suitable conditions. Never-theless, their unambiguous detection remains an outstandingproblem, although there has been recent progress in this di-rection in quantum Hall systems .Part of the experimental challenge stems from the fact thatstabilizing topological phases supporting Majorana fermionscan involve significant engineering obstacles and/or extremeconditions such as ultra-low temperatures, ultra-clean sam-ples, and high magnetic fields in the case of the ν = 5 / fractional quantum Hall effect. The proposal by Fu andKane noted above for realizing a topological superconduct-ing state by depositing a conventional s -wave superconduc-tor on a three-dimensional topological insulator surface ap-pears quite promising in this regard. This setting should in principle allow for a rather robust topological superconduct-ing phase to be created without such extreme conditions, al-though experiments demonstrating this await development.Moreover, Fu and Kane proposed methods in such a setupfor creating and manipulating Majorana fermions for quan-tum computation. The more recent solid state proposals notedabove involving semiconductor heterostructures and non-centrosymmetric superconductors utilize clever ways ofcreating an environment similar to the surface of a topolog-ical insulator ( i.e. , eliminating a sort of fermion doublingproblem ) in order to generate topological phases supportingMajorana modes.The present work is inspired by the semiconductor pro-posal of Sau et al . in Ref. 12, so we briefly elaborate on ithere. These authors demonstrated that a semiconductor withRashba spin-orbit coupling, sandwiched between an s -wavesuperconductor and a ferromagnetic insulator as in Fig. 1(a),can realize a topological superconducting phase supportingMajorana modes. The basic principle here is that the fer-romagnetic insulator produces a Zeeman field perpendicularto the semiconductor, which separates the two spin-orbit-splitbands by a finite gap. If the Fermi level lies inside of this gap,a weak superconducting pair field generated via the proximityeffect drives the semiconductor into a topological supercon-ducting state that smoothly connects to a spinless p x + ip y superconductor. Sau et al . also discussed how such a de-vice can be exploited along the lines of the Fu-Kane proposalfor topological quantum computation. The remarkable aspectof this proposal is the conventional ingredients it employs—semiconductors benefit from many more decades of studycompared to the relatively nascent topological insulators—making this a promising experimental direction.The main question addressed in this paper is largely a prac-tical one—can this proposed setup be further simplified andmade more tunable, thus (hopefully) streamlining the route to-wards experimental realization of a topological superconduct-ing phase in semiconductor devices? To this end, there aretwo obvious modifications that one might try. First, replacingthe ferromagnetic insulator with an external magnetic field ap-plied perpendicular to the semiconductor certainly simplifiesthe setup, but unfortunately induces undesirable orbital effectswhich change the problem significantly and likely spoil thetopological phase. The second obvious modification, then,would be applying an in-plane magnetic field. While thissidesteps the problem of unwanted orbital effects, unfortu-nately in-plane fields do not open a gap between the spin-orbit-split bands in a Rashba-coupled semiconductor. Physi-cally, opening a gap requires a component of the Zeeman fieldperpendicular to the plane in which the electron spins orient;with Rashba coupling this always coincides with the semicon-ductor plane. (See Sec. III for a more in-depth discussion.)Our main result is that a topological superconducting statesupporting Majorana fermions can be generated by in-planemagnetic fields if one alternatively considers a semiconduc-tor grown along the (110) direction with both Rashba and Dresselhaus coupling [see Fig. 1(b)]. What makes this pos-sible in (110) semiconductors is the form of Dresselhaus cou-pling specific to this growth direction, which favors aligningthe spins normal to the semiconductor plane. When Rashbacoupling is also present, the two spin-orbit terms conspire torotate the plane in which the spins orient away from the semi-conductor plane. In-plane magnetic fields then do open a fi-nite gap between the bands. Under realistic conditions whichwe detail below, the proximity effect can then drive the sys-tem into a topological superconducting phase supporting Ma-jorana modes, just as in the proposal from Ref. 12.This alternative setup offers a number of practical advan-tages. It eliminates the need for a good interface between theferromagnetic insulator (or magnetic impurities intrinsic tothe semiconductor ), reducing considerably the experimentalchallenge of fabricating the device, while still largely elimi-nating undesired orbital effects. Furthermore, explicitly con-trolling the Zeeman field in the semiconductor is clearly ad-vantageous, enabling one to readily sweep across a quantumphase transition into the topological superconducting stateand thus unambiguously identify the topological phase exper-imentally. We propose that InSb quantum wells, which enjoysizable Dresselhaus coupling and a large g -factor, may pro-vide an ideal candidate for the semiconductor in such a de-vice. While not without experimental challenges (discussed insome detail below), we contend that this setup provides per-haps the simplest, most tunable semiconductor realization ofa topological superconducting phase, so we hope that it willbe pursued experimentally.The rest of the paper is organized as follows. In Sec. II weprovide a pedagogical overview of the proposal from Ref. 12,highlighting the connection to a spinless p x + ip y supercon-ductor, which makes the existence of Majorana modes in thissetup more intuitively apparent. We also discuss in some de-tail the stability of the topological superconducting phase aswell as several experimental considerations. In Sec. III we in-troduce our proposal for (110) semiconductor quantum wells.We show that the (110) quantum well Hamiltonian maps ontothe Rashba-only model considered by Sau et al . in an (unphys-ical) limit, and explore the stability of the topological super-conductor here in the realistic parameter regime. Experimen-tal issues related to this proposal are also addressed. Finally,we summarize the results and discuss several future directionsin Sec. IV. (b) s−wave SCQuantum well with RashbaFM insulator M (a) (110) Quantum well withRashba & Dresselhaus B s−wave SC FIG. 1: (a) Setup proposed by Sau et al . for realizing a topologicalsuperconducting phase supporting Majorana fermions in a semicon-ductor quantum well with Rashba spin-orbit coupling. The s -wavesuperconductor generates the pairing field in the well via the proxim-ity effect, while the ferromagnetic insulator induces the Zeeman fieldrequired to drive the topological phase. As noted by Sau et al. , theZeeman field can alternatively be generated by employing a mag-netic semiconductor quantum well. (b) Alternative setup proposedhere. We show that a (110)-grown quantum well with both Rashbaand Dresselhaus spin-orbit coupling can be driven into a topologicalsuperconducting state by applying an in-plane magnetic field. Theadvantages of this setup are that the Zeeman field is tunable, orbitaleffects are expected to be minimal, and the device is simpler, requir-ing neither a good interface with a ferromagnetic insulator nor thepresence of magnetic impurities which provide an additional disor-der source. II. OVERVIEW OF SAU-LUTCHYN-TEWARI-DAS SARMAPROPOSAL
To set the stage for our proposal, we begin by pedagogi-cally reviewing the recent idea by Sau et al . for creating Ma-jorana fermions in a ferromagnetic insulator/semiconductor/s-wave superconductor hybrid system [see Fig. 1(a)]. Theseauthors originally proved the existence of Majorana modesin this setup by explicitly solving the Bogoliubov-de GennesHamiltonian with a vortex in the superconducting order pa-rameter. An index theorem supporting this result was subse-quently proven . We will alternatively follow the approachemployed in Ref. 19 (see also Ref. 8), and highlight the con-nection between the semiconductor Hamiltonian (in a certainlimit) and a spinless p x + ip y superconductor. The advan-tage of this perspective is that the topological character of theproximity-induced superconducting state of interest becomesimmediately apparent, along with the existence of a Majoranabound state at vortex cores. In this way, one circumvents thecumbersome problem of solving the Bogoliubov-de Gennesequation for these modes. The stability of the superconduct-ing phase, which we will also discuss in some detail below,becomes more intuitive from this viewpoint as well. A. Connection to a spinless p x + ip y superconductor Consider first an isolated zincblende semiconductor quan-tum well, grown along the (100) direction for concreteness.Assuming layer (but not bulk) inversion asymmetry and re-taining terms up to quadratic order in momentum , the rele-vant Hamiltonian reads H = Z d r ψ † (cid:20) − ∇ m − µ − iα ( σ x ∂ y − σ y ∂ x ) (cid:21) ψ, (1)where m is the effective mass, µ is the chemical potential, α is the Rashba spin-orbit coupling strength, and σ j are Paulimatrices that act on the spin degree of freedom in ψ . (We set ~ = 1 throughout.) The Rashba terms above can be viewedas an effective magnetic field that aligns the spins in the quan-tum well plane, normal to their momentum. Equation (1) ad-mits two spin-orbit-split bands that appear ‘Dirac-like’ at suf-ficiently small momenta where the ∇ / m kinetic term can beneglected. The emergence of Majorana modes can ultimatelybe traced to this simple fact.Coupling the semiconductor to a ferromagnetic insulatorwhose magnetization points perpendicular to the 2D layer isassumed to induce a Zeeman interaction H Z = Z d r ψ † [ V z σ z ] ψ (2)but negligible orbital coupling. Orbital effects will presum-ably be unimportant in the case where, for instance, V z arisesprimarily from exchange interactions rather than direct cou-pling of the spins to the field emanating from the ferromag-netic moments. With this coupling, the spin-orbit-split bandsno longer cross, and resemble a gapped Dirac point at smallmomenta. Crucially, when | µ | < | V z | the electrons in thequantum well then occupy only the lower band and exhibit asingle Fermi surface. We focus on this regime for the remain-der of this section.What differentiates the present problem from a conven-tional single band (without spin-orbit coupling) is the struc-ture of the wavefunctions inherited from the Dirac-likephysics encoded in H at small momenta. To see this, it isilluminating to first diagonalize H + H Z by writing ψ ( k ) = φ − ( k ) ψ − ( k ) + φ + ( k ) ψ + ( k ) , (3)where ψ ± annihilate states in the upper/lower bands and φ ± are the corresponding normalized wavefunctions, φ + ( k ) = (cid:18) A ↑ ( k ) A ↓ ( k ) ik x − k y k (cid:19) (4) φ − ( k ) = (cid:18) B ↑ ( k ) ik x + k y k B ↓ ( k ) (cid:19) . (5)The expressions for A ↑ , ↓ and B ↑ , ↓ are not particularly enlight-ening, but for later we note the following useful combinations: f p ( k ) ≡ A ↑ A ↓ = B ↑ B ↓ = − αk p V z + α k (6) f s ( k ) ≡ A ↑ B ↓ − B ↑ A ↓ = V z p V z + α k . (7)In terms of ψ ± , the Hamiltonian becomes H + H Z = Z d k [ ǫ + ( k ) ψ † + ( k ) ψ + ( k )+ ǫ − ( k ) ψ †− ( k ) ψ − ( k )] , (8) with energies ǫ ± ( k ) = k m − µ ± p V z + α k . (9)Now, when the semiconductor additionally comes into con-tact with an s-wave superconductor, a pairing term will begenerated via the proximity effect, so that the full Hamilto-nian describing the quantum well becomes H = H + H Z + H SC (10)with H SC = Z d r [∆ ψ †↑ ψ †↓ + h.c. ] . (11)(We note that H is a continuum version of the lattice modeldiscussed in Ref. 9 in the context of topological superfluidsof cold fermionic atoms.) Rewriting H SC in terms of ψ ± andusing the wavefunctions in Eqs. (4) and (5) yields H SC = Z d k (cid:20) ∆ + − ( k ) ψ † + ( k ) ψ †− ( − k )+ ∆ −− ( k ) ψ †− ( k ) ψ †− ( − k )+ ∆ ++ ( k ) ψ † + ( k ) ψ † + ( − k ) + h.c. (cid:21) , (12)with ∆ + − ( k ) = f s ( k )∆ (13) ∆ ++ ( k ) = f p ( k ) (cid:18) k y + ik x k (cid:19) ∆ (14) ∆ −− ( k ) = f p ( k ) (cid:18) k y − ik x k (cid:19) ∆ . (15)The proximity effect thus generates not only interband s-wavepairing encoded in the first term, but also intra band p x ± ip y pairing with opposite chirality for the upper/lower bands. Thisis exactly analogous to spin-orbit-coupled superconductors,where the pairing consists of spin-singlet and spin-triplet com-ponents due to non-conservation of spin .We can now immediately understand the appearance of atopological superconducting phase in this system. Consider ∆ much smaller than the spacing | V z − µ | to the upper band.In this case the upper band plays essentially no role and cansimply be projected away by sending ψ + → above. Theproblem then maps onto that of spinless fermions with p x + ip y pairing, which is the canonical example of a topologicalsuperconductor supporting a single Majorana bound state atvortex cores . (The dispersion ǫ − ( k ) is, however, somewhatunconventional. But one can easily verify that the dispersioncan be smoothly deformed into a conventional k / m − µ form, with µ > , without closing a gap.) Thus, in this limitintroducing a vortex in the order parameter ∆ must produce asingle Majorana bound state in this semiconductor context aswell.We emphasize that in the more general case where ∆ is notnegligible compared to | V z − µ | , the mapping to a spinless p x + ip y superconductor is no longer legitimate. Neverthe-less, since the presence of a Majorana fermion has a topolog-ical origin, it can not disappear as long as the bulk excitationgap remains finite. We will make extensive use of this fact inthe remainder of the paper. Here we simply observe that thetopological superconducting state and Majorana modes willpersist even when one incorporates both bands—which we dohereafter—provided the pairing ∆ is sufficiently small that thegap does not close, as found explicitly by studying the full un-projected Hamiltonian with a vortex in Ref. 12.It is also important to stress that when ∆ greatly exceeds V z ,it is the Zeeman field that essentially plays no role. A topolog-ical superconducting state is no longer expected in this limit,since one is not present when V z = 0 . Thus as ∆ increases,the system undergoes a quantum phase transition from a topo-logical to an ordinary superconducting state, as discussed bySau et al . and Sato et al . in the cold-atoms context. Thetransition is driven by the onset of interband s -wave pairingnear zero momentum. B. Stability of the topological superconducting phase
The stability of the topological superconducting state wasbriefly discussed in Ref. 12, as well as Ref. 9 in the cold-atomssetting. Here we address this issue in more detail, with the aimof providing further intuition as well as guidance for experi-ments. Given the competition between ordinary and topolog-ical superconducting order inherent in the problem, it is use-ful to explore, for instance, how the chemical potential, spin-orbit strength, proximity-induced pair field, and Zeeman fieldshould be chosen so as to maximize the bulk excitation gap inthe topological phase of interest. Furthermore, what limits thesize of this gap, and how does it decay as these parameters aretuned away from the point of maximum stability? And howare other important factors such as the density impacted by thechoice of these parameters?Solving the full Bogoliubov-de Gennes Hamiltonian as-suming uniform ∆ yields energies that satisfy E ± = 4 | ∆ ++ | + ∆ − + ǫ + ǫ − ± | ǫ + − ǫ − | r ∆ − + ( ǫ + + ǫ − ) . (16)We are interested in the lower branch E − ( k ) , in particularits value at zero momentum and near the Fermi surface. Theminimum of these determines the bulk superconducting gap, E g ≡ ∆ G ( µV z , mα V z , ∆ V z ) .To make the topological superconducting state as robust aspossible, one clearly would like to maximize the p -wave pair-ing at the Fermi momentum, k F = r m h mα + µ + p V z + mα ( mα + 2 µ ) i . (17)Doing so requires mα /V z ≫ . In this limit we have | ∆ ++ ( k F ) | ∼ ∆ / while the s -wave pairing at the Fermi m α /V z ∆ / V z Topological SCOrdinary SC (a) µ /V z ∆ / V z −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.20.40.60.81 0.10.20.30.40.5 Topological SC Ordinary SCOrdinarySC (b)
FIG. 2: Excitation gap E g normalized by ∆ in the proximity-inducedsuperconducting state of a Rashba-coupled quantum well adjacent toa ferromagnetic insulator. In (a), the chemical potential is chosen tobe µ = 0 . For ∆ /V z < the system realizes a topological supercon-ducting phase supporting a single Majorana mode at a vortex core,while for ∆ /V z > an ordinary superconducting state emerges. Inthe topological phase, the gap is maximized when ∆ /V z = 1 / and mα /V z ≫ , where it is given by E g = V z / . In contrast, the gapvanishes as mα /V z → because the effective p -wave pair field atthe Fermi momentum vanishes in this limit. In (b), we have taken mα /V z = 0 . to illustrate that V z can exceed mα by more thanan order of magnitude and still yield a sizable gap in the topologicalsuperconducting phase. momentum is negligible, ∆ + − ( k F ) ∼ . We thus obtain E − ( k F ) ∼ ∆ , (18)which increases monotonically with ∆ . At zero momentum,however, we have E − ( k = 0) = | V z − p ∆ + µ | . (19)This initially decreases with ∆ as interband s -wave pairingbegins to set in, and vanishes when ∆ = p V z − µ signalingthe destruction of the topological superconducting state . Itfollows that for a given V z , the topological superconductor ismost robust when mα /V z ≫ , µ = 0 , and ∆ = V z / ;here the bulk excitation gap is maximized and given by E g = V z / .As will become clear below, for practical purposes it is alsouseful to explore the limit where V z is much larger than both ∆ and mα . Here the gap is determined solely by the p -wavepair field near the Fermi surface [except for µ very close to V z , where it follows from Eq. (19)]. This pairing will cer-tainly be reduced compared to the mα /V z ≫ limit, be-cause the lower band behaves like a conventional quadrati-cally dispersing band in the limit mα /V z → . To leadingorder in mα /V z and ∆ /V z , the gap is given by E g ≈ s mα V z (cid:18) µV z (cid:19) ∆ . (20)There are two noteworthy features of this expression. First,although the gap indeed vanishes as mα /V z → , it doesso very slowly; V z can exceed mα by more than an order ofmagnitude and still yield a gap that is a sizable fraction of thebare proximity-induced ∆ . Second, in this limit the gap canbe enhanced by raising µ near the bottom of the upper band.These results are graphically summarized in Fig. 2, whichdisplays the gap E g normalized by ∆ . Figure 2(a) assumes µ = 0 and illustrates the dependence on mα /V z and ∆ /V z ;Fig. 2(b) assumes mα /V z = 0 . and illustrates the depen-dence on ∆ /V z and µ/V z . Note that despite the relativelysmall value of mα /V z chosen here, the gap remains a siz-able fraction of ∆ over much of the topological superconduc-tor regime. C. Experimental considerations
The quantity mα comprises a crucial energy scale regard-ing experimental design. Ideally, this should be as large aspossible for at least two reasons. First, the scale of mα lim-its how large a Zeeman splitting V z is desirable. If mα /V z becomes too small, then as discussed above the effective p -wave pairing at the Fermi surface will eventually be stronglysuppressed compared to ∆ , along with the bulk excitation gap.At the same time, having a large V z is advantageous in that thetopological superconductor can then exist over a broad rangeof densities. This leads us to the second reason why large mα is desired: this quantity strongly impacts the density inthe topological superconductor regime, n = ( mα ) π µmα + s (cid:18) V z mα (cid:19) + 2 µmα . (21)One should keep in mind that if the density is too small, dis-order may dominate the physics . Experimental values for the Rashba coupling α dependstrongly on the properties of the quantum well under con-sideration, and, importantly, are tunable in gated systems (see also Ref. 25). In GaAs quantum wells, for instance, α ≈ . eV ˚A and α ≈ . eV ˚A have been mea-sured. Using the effective mass m = 0 . m e ( m e is the bareelectron mass), these correspond to very small energy scales mα ∼ mK for the former and a scale an order of magni-tude smaller for the latter. In the limit mα /V z ≪ , Eq. (21)yields a density for the topological superconductor regime of n ∼ cm − and ∼ cm − , respectively. Disorder likelydominates at such low densities. Employing Zeeman fields V z which are much larger than mα can enhance these densities by one or two orders of magnitude without too dramaticallyreducing the gap (the density increases much faster with V z than the gap decreases), though this may still be insufficientto overcome disorder effects.Due to their stronger spin-orbit coupling, quantum wellsfeaturing heavier elements such as In and Sb appear morepromising. A substantially larger α ≈ . eV ˚A has beenmeasured in InAs quantum wells with effective mass m ≈ . m e , yielding a much greater energy scale mα ∼ . K.The corresponding density in the mα /V z ≪ limit is now n ∼ cm − . While still small, a large Zeeman field cor-responding to mα /V z = 0 . raises the density to a morereasonable value of n ∼ cm − . As another example, theRashba coupling in InGaAs quantum wells with m ≈ . m e was tuned over the range α ∼ . − . eV ˚A with a gate ,resulting in a range of energy scales mα ∼ . − . K. Thedensities here are even more promising, with n ∼ − cm − in the limit mα /V z ≪ ; again, these can beenhanced significantly by considering V z large compared to mα .To conclude this section, we comment briefly on the setupsproposed by Sau et al ., wherein the Zeeman field arises eitherfrom a proximate ferromagnetic insulator or magnetic impu-rities in the semiconductor. In principle, the Rashba couplingand chemical potential should be separately tunable in eithercase by applying a gate voltage and adjusting the Fermi levelin the s -wave superconductor. The strength of the Zeemanfield, however, will largely be dictated by the choice of mate-rials, doping, geometry, etc . Unless the value of mα can begreatly enhanced compared to the values quoted above, it maybe advantageous to consider Zeeman fields which are muchlarger than this energy scale, in order to raise the density atthe expense of suppressing the bulk excitation gap somewhat.A good interface between the ferromagnetic insulator and thequantum well will be necessary to achieve a large V z , if thissetup is chosen. Allowing V z to arise from magnetic impuri-ties eliminates this engineering challenge, but has the draw-back that the dopants provide another disorder source whichcan deleteriously affect the device’s mobility . Nevertheless,since semiconductor technology is so well advanced, it is cer-tainly worth pursuing topological phases in this setting, espe-cially if alternative setups minimizing these challenges can befound. Providing one such alternative is the goal of the nextsection. III. PROPOSED SETUP FOR (110) QUANTUM WELLS
We now ask whether one can make the setup proposed bySau et al . simpler and more tunable by replacing the ferro-magnetic insulator (or magnetic impurities embedded in thesemiconductor) responsible for the Zeeman field with an ex-perimentally controllable parameter. As mentioned in the in-troduction, the most naive possible way to achieve this wouldbe to do away with the magnetic insulator (or magnetic im-purities) and instead simply apply an external magnetic fieldperpendicular to the semiconductor. In fact, this possibilitywas pursued earlier in Refs. 19 and 13. It is far from obvious,however, that the Zeeman field dominates over orbital effectshere, which was a key ingredient in the proposal by Sau et al .Thus, these references focused on the regime where the Zee-man field was smaller than ∆ , which is insufficient to drivethe topological superconducting phase. (We note, however,that a proximity-induced spin-triplet order parameter, if largeenough, was found to stabilize a topological state .) An ob-vious alternative would be applying a parallel magnetic field,along the quantum well plane, since this (largely) rids of theunwanted orbital effects. This too is insufficient, since replac-ing V z σ z with V y σ y in Eq. (2) does not gap out the bands at k = 0 , but only shifts the crossing to finite momentum. A. Topological superconducting phase in a (110) quantum well
We will show that if one alternatively considers azincblende quantum well grown along the (110) direction, atopological superconducting state can be driven by applica-tion of a parallel magnetic field. What makes this possiblein (110) quantum wells is their different symmetry comparedto (100) quantum wells. Assuming layer inversion symmetryis preserved, the most general Hamiltonian for the well up toquadratic order in momentum is H = Z d r ψ † " − ∂ x m x + ∂ y m y ! − µ − iβ∂ x σ z ψ (22)Here we allow for anisotropic effective masses m x,y due toa lack of in-plane rotation symmetry, and β is the Dressel-haus spin-orbit coupling strength. Crucially, the Dressel-haus term favors alignment of the spins normal to the plane, incontrast to the Rashba coupling in Eq. (1) which aligns spins within the plane. Although we did not incorporate Dressel-haus terms in the previous section, we note that in a (100)quantum well they, too, favor alignment of spins within theplane.As an aside, we note that the above Hamiltonian has beenof interest in the spintronics community because it preservesthe S z component of spin as a good quantum number, result-ing in long lifetimes for spins aligned normal to the quantumwell . ( H also exhibits a ‘hidden’ SU(2) symmetry whichfurthered interest in this model, but this is not a microscopicsymmetry and will play no role here.) We are uninterested inspin lifetimes, however, and wish to explicitly break layer in-version symmetry by imbalancing the quantum well using agate voltage and/or chemical means. The Hamiltonian for the(110) quantum well then becomes H (110) = H + H R , where H R = Z d r ψ † [ − i ( α x σ x ∂ y − α y σ y ∂ x )] ψ (23)represents the induced Rashba spin-orbit coupling terms upto linear order in momentum. While one would naivelyexpect α x = α y here, band structure effects will generi-cally lead to unequal coefficients, again due to lack of ro-tation symmetry. We can recast the quantum well Hamilto-nian into a more useful form by rescaling coordinates so that ∂ x → ( m x /m y ) / ∂ x and ∂ y → ( m y /m x ) / ∂ y . We thenobtain H (110) = Z d r ψ † (cid:20) − ∇ m ∗ − µ − iλ D ∂ x σ z − iλ R ( σ x ∂ y − γσ y ∂ x ) (cid:21) ψ. (24)The effective mass is m ∗ = √ m x m y and the spin-orbit pa-rameters are λ D = β ( m x /m y ) / , λ R = α x ( m y /m x ) / ,and γ = ( α y /α x ) p m x /m y .With both Dresselhaus and Rashba terms present, the spinswill no longer align normal to the quantum well, but ratherlie within the plane perpendicular to the vector λ D ˆ y + γλ R ˆ z .Consider for the moment the important special case γ = 0 and λ D = λ R . In this limit, H (110) becomes essentially identical to Eq. (1), with the important difference that here the spinspoint in the ( x, z ) plane rather than the ( x, y ) plane. It followsthat a field applied along the y direction, H Z = Z d r ψ † [ V y σ y ] ψ, (25)with V y = gµ B B y / , then plays exactly the same role as theZeeman term V z in Sau et al .’s proposal discussed in the pre-ceding section—the bands no longer cross at zero momentum,and only the lower band is occupied when | µ | < | V y | . In thisregime, when the system comes into contact with an s -wavesuperconductor, the proximity effect generates a topologicalsuperconducting state supporting Majorana fermions at vor-tex cores, provided the induced pairing in the well is not toolarge .The full problem we wish to study, then, corresponds to a(110) quantum well with both Dresselhaus and Rashba cou-pling, subjected to a parallel magnetic field and contacted toan s -wave superconductor. The complete Hamiltonian is H = H (110) + H Z + H SC , (26)with H SC the same as in Eq. (11). Of course in a real system γ will be non-zero, and likely of order unity, and λ R generallydiffers from λ D . The question we must answer then is how farthe topological superconducting phase survives as we increase γ from zero and change the ratio λ R /λ D from unity. Certainlyour proposal will be viable only if this state survives relativelylarge changes in these parameters. B. Stability of the topological superconducting phase in (110)quantum wells
To begin addressing this issue, it is useful to proceed asin the previous section and express the Hamiltonian in termsof operators ψ †± ( k ) which add electrons to the upper/lowerbands: H = Z d k [˜ ǫ + ( k ) ψ † + ( k ) ψ + ( k ) + ˜ ǫ − ( k ) ψ †− ( k ) ψ − ( k )]+ (cid:20) ˜∆ + − ( k ) ψ † + ( k ) ψ †− ( − k ) + ˜∆ −− ( k ) ψ †− ( k ) ψ †− ( − k )+ ˜∆ ++ ( k ) ψ † + ( k ) ψ † + ( − k ) + h.c. (cid:21) . (27)The energies ˜ ǫ ± are given by ˜ ǫ ± ( k ) = k m − µ ± δ ˜ ǫ ( k ) δ ˜ ǫ ( k ) = q ( V y − γλ R k x ) + ( λ D k x ) + ( λ R k y ) , (28)while the interband s - and intraband p -wave pair fields nowsatisfy | ˜∆ + − ( k ) | = ∆ (cid:20) − ( λ D + γ λ R ) k x + λ R k y − V y δ ˜ ǫ ( k ) δ ˜ ǫ ( − k ) (cid:21) | ˜∆ ++ ( k ) | = | ˜∆ −− ( k ) | (29) = ∆ (cid:20) λ D + γ λ R ) k x + λ R k y − V y δ ˜ ǫ ( k ) δ ˜ ǫ ( − k ) (cid:21) . Increasing γ from zero to of order unity affects the abovepair fields rather weakly. The dominant effect of γ , which canbe seen from Eq. (28), is to lift the k x → − k x symmetry ofthe ∆ = 0 bands. Physically, this symmetry breaking arisesbecause when γ = 0 the spins lie within a plane that is not per-pendicular to the magnetic field. This, in turn, suppresses su-perconductivity since states with k and − k will generally havedifferent energy. While in this case the Bogoliubov-de Gennesequation no longer admits a simple analytic solution, one cannumerically compute the bulk energy gap for the uniform su-perconducting state, E g ≡ ∆ G ( µV y , mλ D V y , ∆ V y , λ R λ D , γ ) , to ex-plore the stability of the topological superconducting phase.Consider first the illustrative case with µ = 0 , mλ D /V y =2 , and ∆ /V y = 0 . . The corresponding gap as a functionof λ R /λ D and γ appears in Fig. 3(a). At λ R /λ D = 1 and γ = 0 , where our proposal maps onto that of Sau et al ., the gapis E g ≈ . , somewhat reduced from its maximum valuesince we have taken ∆ /V y > / . Remarkably, as the figuredemonstrates this gap persists unaltered even beyond γ = 1 ,provided the scale of Rashba coupling λ R /λ D is suitably re-duced . Throughout this region, the lowest-energy excitationis created at zero momentum, where the energy gap is sim-ply E g = V y − ∆ . This clearly demonstrates the robustnessof the topological superconducting state well away from theRashba-only model considered by Sau et al ., and supports thefeasibility of our modified proposal in (110) quantum wells.Let us understand the behavior of the gap displayed in Fig.3(a) in more detail. As described above, the plane in whichthe spins reside is tilted away from the ( x, z ) plane by anangle θ = cos − [1 / p γλ R /λ D ) ] . Non-zero θ givesrise to the anisotropy under k x → − k x , which again tendsto suppress superconductivity. One can see here that reducing λ R /λ D therefore can compensate for an increase in γ , leading λ R / λ D γ Gapless SCTopological SCp x SC (a) λ R / λ D ∆ / V y Ordinary SCTopological SC Gapless SCp x SC (b) µ /V y λ R / λ D −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 100.20.40.60.81 00.10.20.30.40.5 Topological SCGapless SC
OrdinarySC OrdinarySC (c)
FIG. 3: Excitation gap E g normalized by ∆ in the proximity-inducedsuperconducting state of a (110) quantum well, with both Rashbaand Dresselhaus spin-orbit coupling, in a parallel magnetic field. In(a) we set µ = 0 , mλ D /V y = 2 , and ∆ /V y = 0 . , and illustratethe dependence of the gap on the Rashba coupling anisotropy γ aswell as λ R /λ D . When γ = 0 and λ R /λ D = 1 , the problem mapsonto the Rashba-only model considered by Sau et al . Remarkably,the gap survives unaltered here even in the physically relevant casewith γ of order one, provided the Rashba coupling is reduced. In (b)and (c), we focus on the realistic case with γ = 1 to illustrate thestability of the topological phase in more detail. We take µ = 0 and mλ D /V y = 2 in (b), and allow ∆ /V y as well as λ R /λ D to vary.In (c), we fix ∆ /V y = 0 . and mλ D /V y = 2 , allowing µ/V y and λ R /λ D to vary. to the rather robust topological superconducting phase evidentin the figure.On the other hand, at fixed λ R /λ D which is sufficientlylarge ( & . in the figure), increasing γ eventually resultsin the minimum energy excitation occurring at k y = 0 and k x near the Fermi momentum. Further increasing γ thenshrinks the gap and eventually opens pockets of gapless ex-citations, destroying the topological superconductor. Con-versely, if λ R /λ D is sufficiently small ( . / in the fig-ure), the gap becomes independent of γ . In this region theminimum energy excitations are created at k x = 0 and k y near the Fermi momentum. As λ R /λ D → , the lowerband transitions from a gapped topological p x + ip y super-conductor to a gapless nodal p x superconductor. This followsfrom Eq. (29), which in the limit λ R = 0 yields a pair field ˜∆ −− = ∆ λ D k x / [2 q V y + λ D k x ] that vanishes along theline k x = 0 . While a gapless p x superconducting phase isnot our primary focus, we note that realizing such a state ina (110) quantum well with negligible Rashba coupling wouldbe interesting in its own right.To gain a more complete picture of topological supercon-ductor’s stability in the physically relevant regime, we furtherillustrate the behavior of the bulk excitation gap in Figs. 3(b)and (c), fixing for concreteness γ = 1 and mλ D /V y = 2 . Fig-ure 3(b) plots the dependence of the gap on ∆ /V y and λ R /λ D when µ = 0 , while Fig. 3(c) displays the gap as a function of λ R /λ D and µ/V y when ∆ /V y = 0 . . C. Experimental considerations for (110) quantum wells
The main drawback of our proposal compared to theRashba-only model discussed by Sau et al . can be seen inFig. 3(b). In the previous section, we discussed that it maybe desirable to intentionally suppress the gap for the topolog-ical superconducting state by considering Zeeman splittingswhich greatly exceed the Rashba energy scale mα , in or-der to achieve higher densities and thereby reduce disordereffects. Here, however, this is possible to a lesser extent sincethe desired strength of V y is limited by the induced pairingfield ∆ . If ∆ /V y becomes too small, then the system entersthe gapless regime as shown in Fig. 3(b).Nevertheless, our proposal has a number of virtues, suchas its tunability. As in the proposal of Sau et al . , thestrength of Rashba coupling can be controlled by applying agate voltage , and the chemical potential in the semiconduc-tor can be independently tuned by changing the Fermi levelin the proximate s -wave superconductor. In our case the pa-rameter γ ∝ p m x /m y can be controlled to some extent byapplying pressure to modify the mass ratio m x /m y , althoughthis is not essential. More importantly, one has additional con-trol over the Zeeman field, which is generated by an externallyapplied in-plane magnetic field that largely avoids unwantedorbital effects. Such control enables one to readily tune thesystem across the quantum phase transition separating the or-dinary and topological superconducting phases [see Fig. 3(b)].This feature not only opens up the opportunity to study thisquantum phase transition experimentally, but also provides anunambiguous diagnostic for identifying the topological phase.For example, the value of the critical current in the quantumwell should exhibit a singularity at the phase transition, whichwould provide one signature for the onset of the topologicalsuperconducting state. We also emphasize that realizing therequired Zeeman splitting through an applied field is tech-nologically far simpler than coupling the quantum well to a ferromagnetic insulator, and avoids the additional source ofdisorder generated by doping the quantum well with magneticimpurities.Since the extent to which one can enhance the density inthe topological superconducting phase by applying large Zee-man fields is limited here, it is crucial to employ materialswith appreciable Dresselhaus coupling. We suggest that InSbquantum wells may be suitable for this purpose. Bulk InSb en-joys quite large Dresselhaus spin-orbit interactions of strength760eV ˚A (for comparison, the value in bulk GaAs is 28eV ˚A ;see Ref. 33). For a quantum well of width w , one can crudelyestimate the Dresselhaus coupling to be λ D ∼ eV ˚A /w ;assuming w = 50 ˚A, this yields a sizable λ D ∼ . eV ˚A. BulkInSb also exhibits a spin-orbit enhanced g -factor of roughly50 (though confinement effects can substantially diminish thisvalue in a quantum well ). The large g -factor has importantbenefits. For one, it ensures that Zeeman energies V y of ordera Kelvin, which we presume is the relevant scale for ∆ , canbe achieved with fields substantially smaller than a Tesla. Theability to produce Zeeman energies of this scale with relativelysmall fields should open up a broad window where V y exceeds ∆ but the applied field is smaller than the critical field for theproximate s -wave superconductor (which can easily exceed1T). Both conditions are required for realizing the topologicalsuperconducting state in our proposed setup. A related bene-fit is that the Zeeman field felt by the semiconductor will besignificantly larger than in the s -wave superconductor, sincethe g -factor for the latter should be much smaller. This fur-ther suggests that s -wave superconductivity should thereforebe disturbed relatively little by the required in-plane fields. IV. DISCUSSION
Amongst the proposals noted in the introduction, theprospect for realizing Majorana fermions in a semiconductorsandwiched between a ferromagnetic insulator and s -wave su-perconductor stands out in part because it involves rather con-ventional ingredients (semiconductor technology is extraordi-narily well developed). Nevertheless, this setup is not withoutexperimental challenges, as we attempted to highlight in Sec.II above. For instance, a good interface between a ferromag-netic insulator and the semiconductor is essential, which posesan important engineering problem. If one employs a magneticsemiconductor instead, this introduces an additional source ofdisorder (in any case magnetic semiconductors are typicallyhole doped).The main goal of this paper was to simplify this setup evenfurther, with the hope of hastening the experimental realiza-tion of Majorana fermions in semiconductor devices. Weshowed that a topological superconducting state can be drivenby applying a (relatively weak) in-plane magnetic field to a(110) semiconductor quantum well coupled only to an s -wavesuperconductor. The key to realizing the topological phasehere was an interplay between Dresselhaus and Rashba cou-plings; together, they cause the spins to orient within a planewhich tilts away from the quantum well. An in-plane mag-netic field then plays the same role as the ferromagnetic insu-lator or an applied perpendicular magnetic field plays in theRashba-only models considered in Refs. 12,13,19, but im-portantly without the detrimental orbital effects of the per-pendicular field. This setup has the virtue of simplicity—eliminating the need for a proximate ferromagnetic insulatoror magnetic impurities—as well as tunability. Having con-trol over the Zeeman field allows one to, for instance, readilysweep across the quantum phase transition from the ordinaryto the topological superconducting state. Apart from funda-mental interest, this phase transition can serve as a diagnosticfor unambiguously identifying the topological phase experi-mentally ( e.g. , through critical current measurements). As amore direct probe of Majorana fermions, a particularly sim-ple proposal for their detection on the surface of a topolog-ical insulator was recently put forth by Law, Lee, and Ng .This idea relies on ‘Majorana induced resonant Andreev re-flection’ at a chiral edge. In a topological insulator, such anedge exists between a proximity-induced superconducting re-gion and a ferromagnet-induced gapped region of the surface.In our setup, this effect can be realized even more simply,since the semiconductor will exhibit a chiral Majorana edgeat its boundary, without the need for a ferromagnet. Finally,since only one side of the semiconductor need be contactedto the s -wave superconductor, in principle this leaves openthe opportunity to probe the quantum well directly from theother.The main disadvantage of our proposal is that if the Zee-man field in the semiconductor becomes too large comparedto the proximity-induced pair field ∆ , the topological phasegets destroyed [see Fig. 3(b)]. By contrast, in the setup pro-posed by Sau et al ., the topological superconductor surviveseven when the Zeeman field greatly exceeds both ∆ and mα .Indeed, we argued that this regime is where experimentalistsmay wish to aim, at least initially, if this setup is pursued.Although the gap in the topological phase is somewhat sup-pressed in the limit mα /V z ≪ , large Zeeman fields allowthe density in this phase to be increased by one or two ordersof magnitude, thus reducing disorder effects. (We should note,however, that the actual size of Zeeman fields that can be gen-erated by proximity to a ferromagnetic insulator or intrinsicmagnetic impurities is uncertain at present.) Since one is notafforded this luxury in our (110) quantum well setup, it is es-sential to employ materials with large Dresselhaus spin-orbit coupling in order to achieve reasonable densities in the semi-conductor. We argued that fairly narrow InSb quantum wellsmay be well-suited for this purpose. Apart from exhibitinglarge spin-orbit coupling, InSb also enjoys a large g -factor,which should allow for weak fields (much less than 1T) todrive the topological phase in the quantum well while disturb-ing the proximate s -wave superconductor relatively little.There are a number of open questions which are worthexploring to further guide experimental effort in this direc-tion. As an example, it would be worthwhile to carry outmore accurate modeling, including for instance cubic Rashbaand Dresselhaus terms and (especially) disorder, to obtain amore quantitative phase diagram for either of the setups dis-cussed here. Exploring the full spectrum of vortex boundstates (beyond just the zero-energy Majorana mode) is an-other important problem. The associated ‘mini-gap’ providesone important factor determining the feasibility of quantumcomputation with such devices. We also think it is usefulto explore other means of generating topological supercon-ducting phases in such semiconductor settings. One intrigu-ing possibility would be employing nuclear spins to produce aZeeman field in the semiconductor . More broadly, the pro-posals considered here can be viewed as examples of a rathergeneral idea discussed recently for eliminating the so-calledfermion-doubling problem that can otherwise destroy the non-Abelian statistics necessary for topological quantum com-putation. Very likely, we have by no means exhausted the pos-sible settings in which Majorana fermions can emerge, evenwithin the restricted case of semiconductor devices. Mighthole-doped semiconductors be exploited in similar ways togenerate topological superconducting phases, for instance, orperhaps heavy-element thin films such as bismuth? Acknowledgments
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Spin-Orbit Coupling Effects in Two-DimensionalElectron and Hole Systems (Springer, Berlin, 2003). K. T. Law, P. A. Lee, and T. K. Ng, Phys. Rev. Lett. , 237001(2009). D. L. Bergman and K. L. Hur, Phys. Rev. B , 184520 (2009). Lowering the density does, however, lead to a smaller Fermi en-ergy and thus a larger ‘mini-gap’ associated with the vortex-corebound states. Large mini-gaps are ultimately desirable for topo-logical quantum computation. In this paper we are concerned witha more modest issue—namely, simply finding a stable topologicalsuperconducting phase in the first place. With this more limitedgoal in mind, large densities are clearly desired to reduce disordereffects.37