A generic new platform for topological quantum computation using semiconductor heterostructures
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n A generic new platform for topological quantum computation using semiconductor heterostructures
Jay D. Sau , Roman M. Lutchyn , Sumanta Tewari , , and S. Das Sarma Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics,University of Maryland, College Park, Maryland 20742-4111, USA Department of Physics and Astronomy, Clemson University, Clemson, SC 29634
We show that a film of a semiconductor in which s -wave superconductivity and a Zeeman splitting are in-duced by proximity effect, supports zero-energy Majorana fermion modes in the ordinary vortex excitations.Since time reversal symmetry is explicitly broken, the edge of the film constitutes a chiral Majorana wire. Theheterostructure we propose – a semiconducting thin film sandwiched between an s -wave superconductor and amagnetic insulator – is a generic system which can be used as the platform for topological quantum computationby virtue of the existence of non-Abelian Majorana fermions. PACS numbers: 03.67.Lx, 71.10.Pm, 74.45.+c
Introduction.
In two spatial dimensions, where permuta-tion and exchange are not necessarily equivalent, particles canhave quantum statistics which are strikingly different fromthe familiar statistics of bosons and fermions. In situationswhere the many body ground state wave-function is a lin-ear combination of states from a degenerate subspace, a pair-wise exchange of the particle coordinates can unitarily ro-tate the ground state wave-function in the degenerate sub-space. In this case, the exchange statistics is given by a multi-dimensional unitary matrix representation (as opposed to justa phase factor) of the 2D braid group, and the statistics is non-Abelian [1]. It has been proposed that such a system, wherethe ground state degeneracy is protected by a gap from lo-cal perturbations, can be used as a fault-tolerant platform fortopological quantum computation (TQC) [2].Recently, the ν = 5 / FQH state at high magnetic fieldsand at low temperature has been proposed as a topologicalqubit [2]. This theoretical conjecture, however, awaits ex-perimental verification [3, 4]. An equivalent system, in whichthe ordered state is in the same universality class as the / FQH state, is the spin-less (spin-polarized) p x + ip y super-conductor/superfluid [5]. In a finite magnetic field, a vor-tex excitation in such a superconductor traps a single, non-degenerate, zero-energy bound state. The key to non-Abelianstatistics is that the second-quantized operator for this zero-energy state is self-hermitian, γ † = γ , rendering γ a Majoranafermion operator. If the constituent fermions have spin, thespin-degeneracy of the zero energy excitation spoils the non-Abelian statistics, because the mutual statistics of the vortex-fermion composites becomes trivial. To circumvent this prob-lem in a realistic, spinful, p x + ip y superconductor such asstrontium ruthenate, it has been proposed that the requisiteexcitations are the exotic half-quantum vortices, which can bethought of as ordinary vortex excitations in only one of thespin sector in the condensate [6].Even though quenching the spin-degeneracy by either theapplication of a magnetic field [7] or by using spin-lessatomic systems [8] is possible in principle, it is practicallyvery difficult. Therefore, it is desirable to have systems whosemost natural excitations themselves follow non-Abelian statis-tics in spite of the fermions carrying, as they do in a realistic system, a spin quantum number. The recent proposal by Fuand Kane [9] points out one such system – the surface of astrong TI in proximity to an s -wave superconductor – whichsupports a non-degenerate Majorana fermion excitation in thecore of an ordinary vortex. In this paper, we propose a simplegeneric TQC platform by showing that it is possible to re-place the TI with a regular semiconductor film with spin-orbitcoupling, provided the time-reversal symmetry is broken byproximity of the film to a magnetic insulator. The three ingre-dients of non-Abelian statistics – spin-orbit coupling, s -wavesuperconductivity, and Zeeman splitting – are experimentallyknown to occur in many solid state materials. It is encourag-ing that the s -wave proximity effect has already been demon-strated in 2D InAs heterostructures which additionally alsohave a substantial spin-orbit coupling [10]. Thus, the struc-ture we propose is one of the simplest to realize non-AbelianMajorana fermions in the solid-state context. Theoretical Model.
The single-particle effective Hamilto-nian H for the conduction band of a spin-orbit coupled semi-conductor in contact with a magnetic insulator is given by (weset ¯ h = 1 henceforth) H = p m ∗ − µ + V z σ z + α ( ~σ × ~p ) · ˆ z. (1)Here, m ∗ , V z and µ are the conduction-band effective mass ofan electron, effective Zeeman coupling induced by proximityto a magnetic insulator (we neglect the direct coupling of theelectrons with the magnetic field from the magnetic insulator),and chemical potential, respectively. The coefficient α de-scribes the strength of the Rashba spin-orbit coupling and σ α are the Pauli matrices. Despite the similarity in the spin-orbit-coupling terms, H and the Hamiltonian for the TI surface inRef. [9] differ by the existence of a spin-diagonal kinetic en-ergy term in Eq. (1). Because of the spin-diagonal kinetic en-ergy, there are in general two spin-orbit-split Fermi surfaces inthe present system, in contrast to the surface of a TI in whichan odd number of bands cross the Fermi level [9]. In Eq. (1),for out-of-plane Zeeman coupling such that | V z | > | µ | , a sin-gle band crosses the Fermi level. Thus, analogous to a strongTI surface (but arising from qualitatively different physics),the system has a single Fermi surface, which is suggestive of FIG. 1: (Color online) Schematic picture of the proposed heterostruc-ture exhibiting Majorana zero-energy bound state inside an ordinaryvortex. non-Abelian topological order if s -wave superconductivity isinduced in the film. We show below that this is indeed the caseby analyzing the Bogoliubov de Gennes (BdG) equations fora vortex in the superconductor in the heterostructure shown inFig. 1.The proximity-induced superconductivity in the semicon-ductor can be described by the Hamiltonian, ˆ H p = Z d r { ∆ ( r )ˆ c †↑ ( r )ˆ c †↓ ( r ) + H . c } , (2)where ˆ c † σ ( r ) are the creation operators for electrons with spin σ and ∆ ( r ) is the proximity-induced gap. The correspondingBdG equations written in Nambu space become, (cid:18) H ∆ ( r )∆ ∗ ( r ) − σ y H ∗ σ y (cid:19) Ψ( r ) = E Ψ( r ) , (3)where Ψ( r ) is the wave function in the Nambu spinor basis, Ψ( r ) = ( u ↑ ( r ) , u ↓ ( r ) , v ↓ ( r ) , − v ↑ ( r )) T . Using the solutionsof the BdG equations, one can define Bogoliubov quasiparti-cle operators as ˆ γ † = R d r P σ u σ ( r )ˆ c † σ ( r )+ v σ ( r )ˆ c σ ( r ) . Thebulk excitation spectrum of the BdG equations with ∆( r ) =∆ has a gap for non-vanishing spin-orbit coupling. BdG equations for a vortex.
We now consider the vortexin the heterostructure shown in Fig.1, and take the vortex-likeconfiguration of the order parameter: ∆ ( r, θ ) = ∆ ( r ) e ıθ .Because of the rotational symmetry, the BdG equations canbe decoupled into angular momentum channels indexed by m with the corresponding spinor wave-function, Ψ m ( r,θ ) = e ımθ (cid:0) u ↑ ( r ) , u ↓ ( r ) e iθ ,v ↓ ( r ) e − iθ , − v ↑ ( r ) (cid:1) T . (4)Because of the particle-hole symmetry of the BdG equations,if Ψ( r ) is a solution with energy E then ıσ y τ y Ψ ∗ ( r ) is alsoa solution at energy − E . Here τ y is defined to be the Paulimatrix in Nambu spinor space. In particular, zero-energysolutions of the BdG equations must come in pairs, Ψ( r ) and ıσ y τ y Ψ ∗ ( r ) , unless these two wave-functions refer tothe same state. Thus, a zero-energy solution in an angularmomentum channel m is always paired with another zero-energy solution in the channel − m and, therefore, can be non-degenerate only if it corresponds to the m = 0 angular mo-mentum channel. The radial BdG equations describing the zero-energy statein the m = 0 channel can be written as (cid:18) H ∆ ( r )∆ ( r ) − σ y H ∗ σ y (cid:19) Ψ( r ) = 0 , (5) H = − η ( ∂ r + r ∂ r )+ V z − µ α ( ∂ r + r ) − α∂ r η ( − ∂ r − r ∂ r + r ) − V z − µ with η = m ∗ . Since the BdG matrix is real, there are two so-lutions Ψ( r ) and Ψ ∗ ( r ) with E = 0 . Furthermore, it followsfrom the particle-hole symmetry of the BdG equations that ıσ y τ y Ψ( r ) is also a solution. Thus, any non-degenerate E =0 solution must satisfy the property ıσ y τ y Ψ( r ) = ıλ Ψ( r ) .Moreover, because ( ıσ y τ y ) = − , the possible values of λ are λ = ± . The value of λ sets a constraint on the spin-degree of freedom of Ψ( r ) , such that v ↑ ( r ) = λu ↑ ( r ) and u ↓ ( r ) = λv ↓ ( r ) . This allows one to eliminate the spin de-grees of freedom in Ψ( r ) and define a reduced spinor Ψ ( r ) =( u ↑ ( r ) , u ↓ ( r )) T . The corresponding reduced BdG equationstake the form of a × matrix differential equation: − η ( ∂ r + r ∂ r )+ V z − µ λ ∆( r )+ α ( ∂ r + r ) − λ ∆( r ) − α∂ r − η ( ∂ r + r ∂ r − r ) − V z − µ Ψ ( r ) = 0 . (6)We now approximate the radial dependence of ∆ ( r ) by ∆ ( r ) = 0 for r < R and ∆ ( r ) = ∆ for r ≥ R . Inview of the stability of the putative Majorana zero-energy so-lution to local changes in the Hamiltonian [5], such an ap-proximation can be made without loss of generality. For r < R the analytical solution of Eq. (6) is given by Ψ ( r ) =( u ↑ J ( zr ) , u ↓ J ( zr )) T with the constraint (cid:18) ηz + V z − µ zααz ηz − V z − µ (cid:19) (cid:18) u ↑ u ↓ (cid:19) = 0 . (7)Here J n ( r ) are the Bessel functions of the first kind. Thecharacteristic equation for z is ( ηz − µ ) − V z − α z = 0 . (8)In the case µ > V z the roots of Eq.(8) are real: z , = ±√ w and z , = ±√ w with w , > being the solu-tions for z . In the opposite limit, < µ < V z , thereare two real solutions z , = ± p | w | and two imaginarysolutions z , = ± i p | w | . Since the Bessel functions aresymmetric, we find two solutions which are well-behaved atthe origin: Ψ ( r ) = ( u ↑ J ( z r ) , u ↓ J ( z r )) T and Ψ ( r ) =( u ↑ J ( z r ) , u ↓ J ( z r )) T . Therefore, the full solution at r
We have shown above thata non-degenerate Majorana state exists in a vortex in the su-perconductor only in the parameter regime ( µ + ∆ ) < V z .This suggests that there must be a quantum phase transition(QPT) separating the parameter regimes ( µ + ∆ ) < V z and ( µ +∆ ) > V z , even though the system in both regimes is an s -wave superconductor. A non-degenerate zero-energy solu-tion cannot disappear unless a continuum of energy levels ap-pears around E = 0 . Such a continuum of states at E = 0 canonly appear if the bulk gap closes, which can be used to definea topological quantum phase transition. In the present system,such a phase transition can be accessed by varying either theZeeman splitting or the chemical potential. A similar topolog-ical quantum phase transition has already been predicted forultra-cold atoms with vortices in the spin-orbit coupling [12].The bulk gap of the present system can be calculated fromthe bulk excitation spectrum, E = V z + ∆ + ˜ ǫ + α k ± q V z ∆ + ˜ ǫ ( V z + α k ) (11)where ˜ ǫ = ηk − µ . As seen in Fig. 3, the excitation gapfirst increases as a function of ∆ (proximity-induced pair-potential) and then decreases and vanishes at a critical point, ∆ c = p V z − µ , before re-opening and increasing with ∆ E g , r e d µ red =0.5, α red =0.2 µ red =0.0, α red =1.0 µ red =0.5, α red =1.0 µ red =0.8, α red =1.0 µ red =1.2, α red =1.0 FIG. 3: (Color online) Quasiparticle gap versus pairing potential forvarious values of the chemical potential µ . Here E g, red = E g /V z , ∆ , red = ∆ /V z , µ red = µ/V z and α red = α/ √ ηV z . The max-imum value of the gap in the topologically non-trivial superconduc-tor, and the corresponding area in the phase diagram, decreases withincreasing values of µ . For large negative µ , the system makes atransition to a semiconductor. The phase to the right of the criticalpoint is the topologically trivial s -wave superconductor. ∆ . The critical point marks the phase transition betweena topologically non-trivial (left) and a topologically trivial(right) s -wave superconducting phases. The scale of the gapin the topologically non-trivial phase is set by the strength ofthe spin-orbit coupling α and the position of the critical point.The fact that the phase on the right-side of the critical pointdoes not support a non-degenerate Majorana mode can be ver-ified by observing that, for these values of ∆ , it is possibleto reduce V z such that | V z | < | µ | without the gap vanishing atany point. This is the phase without Majorana Fermion excita-tions. In fact, this phase can be reached from the conventional s -wave superconductor with V z = 0 and α = 0 without cross-ing a phase transition. Majorana edge modes and TQC.
In analogy with Ref. [9],we find that an interface between two superconductor layers,which can be deposited on the semiconductor thin film, sup-ports a pair of zero-energy excitations when the phase differ-ence between the superconductors is π . This geometry can beanalyzed in a way that closely follows our derivation of the lo-calized state in a vortex in the m = 0 channel, since the BdGHamiltonian can again be reduced to a real matrix. In thiscase, we find that, in the parameter regime ( µ + ∆ ) < V z ,there are 3 linearly independent solutions on each side ofthe interface. Since the number of constraints to be satisfiedat the interface (we assume the interface to be of negligiblewidth) remains 5 as before, one expects a pair of indepen-dent zero-energy solutions. The interface, therefore, consti-tutes a non-chiral Majorana wire, which can be exploited forbraiding in a way completely analogous to Ref. [9] to performTQC. Majorana bound states as well as Majorana edge modesin our system can be studied experimentally using non-localAndreev reflection [13] and electrically detected interferome-try [14, 15] experiments. The experimental implementation of this proposal involvesa heterostructure of a magnetic insulator ( e.g.
EuO), a strongspin-orbit coupled semiconductor ( e.g.
InAs) and an s-wavesuperconductor with a large T c ( e.g. Nb). Using these mate-rials, it is possible [11] to induce an effective superconduct-ing pairing potential ∆ ∼ . meV and a tunneling-inducedeffective Zeeman splitting V z ∼ meV. Additionally, thestrength of spin-orbit interaction α in InAs heterostructuresis electric-field tunable and can be made as large as α ≈ meV- ˚A [16]. With these estimates, the quasiparticle gap E g isof the order of 1 K. Given that the chemical potential is gate-tunable and can be of the of the order of ∆ , we numericallyestimate the magnitude of the excitation energy for the boundstates in a vortex core of size ∼ nm to be of the order of . K [11], which sets the temperature scale for TQC in thissystem.
Conclusion.
Our proposed TQC platform should be sim-pler to implement experimentally than any of the TQC candi-dates proposed in the literature so far, since it involves a stan-dard heterostructure with a magnetic insulator, a semiconduc-tor film, and an ordinary s -wave superconductor. We believethat the proposed scheme provides the most straightforwardmethod for the solid-state realization of non-Abelian Majo-rana fermions. A significant practical advantage of the pro-posed TQC scheme is its generic simplicity: it requires nei-ther special samples or materials nor ultra-low temperaturesor high magnetic fields.This work is supported by DARPA-QuEST, JQI-NSF-PFC,and LPS-NSA. We thank M. Sato and S. Fujimoto for dis-cussion. ST acknowledges DOE/EPSCoR Grant [1] C. Nayak et al. , Rev. Mod. Phys. , 1083 (2008).[2] S. Das Sarma, M. Freedman, and C. Nayak, Phys. Rev. Lett. ,166802 (2005).[3] A. Stern and B. I. Halperin, Phys. Rev. Lett. , 016802 (2006).[4] P. Bonderson, A. Kitaev, and K. Shtengel, Phys. Rev. Lett. ,016803 (2006).[5] N. Read and D. Green, Phys. Rev. B , 10267 (2000).[6] D. A. Ivanov, Phys. Rev. Lett. , 268 (2001).[7] S. Das Sarma, C. Nayak, and S. Tewari, Phys. Rev. B ,220502 (R) (2006).[8] S. Tewari et al. , Phys. Rev. Lett. , 010506 (2007).[9] L. Fu and C. L. Kane, Phys. Rev. Lett. , 096407 (2008).[10] A. Chrestin, T. Matsuyama, and U. Merkt, Phys. Rev. B ,8457 (1997)[11] J. D. Sau, R. Lutchyn, S. Tewari, S. Das Sarma (in preparation).[12] M. Sato, Y. Takahashi, and S. Fujimoto, Phys. Rev. Lett. ,020401 (2009).[13] J. Nilsson, A. R. Akhmerov, and C. W. J. Beenakker, Phys. Rev.Lett. , 120403 (2008).[14] L. Fu and C. L. Kane, Phys. Rev. Lett. , 216403 (2009).[15] A. R. Akhmerov, J. Nilsson, and C. W. J. Beenakker, Phys. Rev.Lett. , 216404 (2009).[16] C. Yang et al. , Phys. Rev. Lett.96