Gate controlled Majorana Zero Modes on 2D heterostructures
GGate Controlled Majorana Zero Modes of a Two-Dimensional TopologicalSuperconductor
Nima Djavid, Gen Yin, Yafis Barlas, and Roger K. Lake ∗ Department of Electrical engineering, University of California Riverside, Riverside, California 92521, USA Department of Electrical engineering, University of California at Los Angeles, Los Angeles, California 90095, USA Department of Electrical engineering, University of California Riverside, Riverside 92521, USA
Half-integer conductance, the signature of Majorana edge modes, was recently observed in a thin-film magnetic topological insulator / superconductor bilayer. This letter analyzes a scheme for gatecontrol of Majorana zero modes in such systems. Gating the top surface of the thin-film magnetictopological insulator controls the topological phase in the region underneath the gate. The voltageof the transition depends on the gate width, and narrower gates require larger voltages. Relativelylong gates are required, on the order of 2 µ m, to prevent hybridization of the end modes and toallow the creation of Majorana zero modes at low gate voltages. Applying voltage to T-shaped andI-shaped gates localizes the Majorana zero modes at their ends. This scheme may provide a facilemethod for implementing quantum gates for topological quantum computing. I. INTRODUCTION
Majorana fermions are charge-neutral fermionic parti-cles that are their own antiparticles originally proposedby Ettore Majorana [1]. Prior theoretical work suggestedthat Majorana fermions could exist in topological super-conductors as elementary excitations [2–5]. The first ex-perimental demonstration was the zero-bias anomaly ob-served in a III-V semiconductor nanowire coupled to ans-wave superconductor [6, 7], a material system that isa physical implementation of Kitaev’s one dimensionaltopological superconductor model [2]. In the middle ofthe superconducting gap, zero-energy localized states ap-pear at the ends of such nanowires. These states are Ma-jorana zero modes (MZMs). MZMs obey non-Abelianstatistics [8], and they can be used for fault-tolerant,topological quantum computing [9, 10]. This requires theprecise control of the position of the MZMs in a nanowirenetwork. Gate control has been proposed and analyzed[10, 11], and recent experiments demonstrated prototypesof such nanowire networks [12].Although previous experiments using III-V nano-wireshave shown exciting possibilities, an implementation intwo-dimensional (2D) thin films would be more compat-ible with conventional semiconductor device fabrication.Such a scheme in the InAs / superconducting system hasbeen proposed [13]. Recently, quantized half–integer con-ductance ( e h ) was observed in a different material sys-tem consisting of a thin film magnetic topological insu-lator (MTI) capped by an s–wave superconductor (Nb)[14]. The half–integer conductance suggested the exis-tance of a chiral Majorana mode propagating along theedge[14–17]. In this system, the MTI consisted of Crdoped, epitaxial, thin film (Bi,Sb) Te . Recently, signa-tures of MZMs were observed in a similar material sys-tem by scanning probe measurements, where the anti- ∗ [email protected] Super ConductorTI Keyboard Gates
FIG. 1. A quantum anomalous Hall insulator/ superconduc-tor heterostructure. The crossbar shaped gates at the top, canchange the electro-static potential of the top surface locally. periodic boundary condition is induced by a supercon-ducting vortex[18]. Prior theoretical studies on the MTI–superconductor system focused on the topological phasediagram [19], and the most recent theoretical studies pro-pose gate control of MZMs in ribbon geometries withlarge aspect ratios [20, 21].In this letter, we build upon that recent work. Wetheoretically demonstrate the micron–scale gate dimen-sions required for creating MZMs, and we analyze howgate geometry effects the gate voltage required to createthe MZMs. The system under consideration is an array of‘keyboard’ gates [20] on top of the MTI / superconductorbilayer as illustrated in Fig. 1. The effect of the geomet-ric shape of the gated area on unwanted hybridizationand the topological band gap is analyzed. Fundamentalbuilding blocks of the crossbar gate-array, the I-shapedand the T-shaped gates, are demonstrated. To ensurethat the MZMs are not trivial low energy modes, thesymmetry of the MZM wave functions are analyzed toshow that the the wave function is its own complex con-jugate. a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l SC TI (a)
FIG. 2. (a) Phase diagram of the system with ∆ t = 0. V g is applied to the top surface. (b) and (c) show the spectral function A ( k y , E ) at the edge site ( x = 0) of a semi–infinite plane ( −∞ < x ≤ −∞ < y < ∞ ) at different gate voltages. (b) V g =0.0 mV corresponding to the red star in (a) and (c) V g = -20 mV corresponding to the yellow star in (a). (d) The half-integerplateau in conductivity of a 100 nm wide by 40 nm long MTI / superconductor bilayer with topological insulator leads for twodifferent values of ∆ t as shown in the legend. E f = 0 . V g = 0, and ∆ b = 0. Inset: Illustration of the structure. The system, as shown in Fig. 1, consists of a thin-film MTI placed on the top of an s-wave superconductor.Electric gates on top of the MTI control the top-surfaceelectrostatic potential. The Hamiltonian of the system is[22] H BdG = (cid:18) H ( k ) − µ ∆ k ∆ † k − H ∗ ( − k ) + µ, (cid:19) . (1)where ∆ k = (cid:18) i ∆ t σ y i ∆ b σ y (cid:19) , (2)and H ( k ) = (cid:126) v f ( k y σ x − k x σ y ) ˜ τ z + m ( k ) ˜ τ x + J H σ z + V g τ z + I ) . (3)Ψ = [( ψ ↑ t ψ ↓ t ψ ↑ b ψ ↓ b ) , ( ψ †↑ t ψ †↓ t ψ †↑ b ψ †↓ b )] is the basisof the Hamiltonian where ψ ↑ t ( k ) corresponds to an up-spin electron on the top surface and ψ †↓ b ( k ) correspondsto a down-spin hole at the bottom. σ and ˜ τ are Paulimatrices corresponding to the spin and the top-bottomsurfaces, respectively. ∆ t and ∆ b are the proximity-induced Cooper-pairing interactions at the top and bot-tom surfaces, respectively. The pairing interaction ofthe bottom surface that is in contact with the super-conductor is ∆ b = 1 . t = 0. m ( k ) = m + m k represent the hybridizationof the top and bottom surfaces of the thin–film MTI. Thevalue of these terms for 3 quintuple layers of Bi Se are m = 70 meV and m = 18 eV˚A [24, 25]. The quan-tity (cid:126) v F = 3 .
29 eV˚A is consistent with DFT results [26]. J H is the Hund’s rule coupling from the ferromagneticexchange interaction induced by the Cr dopants. Forcalculations using a fixed J H , the value is J H = 65 meV.The chemical potential µ = 0. The last term in Eq. (3)represents the gate voltage V g applied at the top surface of the MTI, with the bottom surface adjacent to the su-perconductor at ground. An equivalent approach wouldbe to shift µ by − V g / − V g / V g /
2. This latter approach is theway the gate voltage was included in recent work [20].To model finite and spatially varying structures, theHamiltonian is transformed into a tight–binding modelon a square lattice by substituting k x → − i ∂∂x and k y → − i ∂∂y in Eq. (1) and discretizing the deriva-tives using a 1 nm discretization length [27]. Each sitein the tight-binding model is then represented by an8 × − iη with η = 0 . H BdG to assist convergenceof the surface Green’s function. The zero–temperature,two–terminal conductance is then σ xx = e h T ( E F ) where T ( E F ) is the transmission coefficient at the Fermi energy.To investigate a single edge mode of a semi-infiniteplane ( −∞ < x ≤ −∞ < y < ∞ ) as shown in Fig.2(b,c), only the substitution k x → − i ∂∂x is made, and thederivative is discretized on the 1 nm grid. Since the edgeof the half-plane is parallel to ˆ y , k y remains a good quan-tum number. The Hamiltonian then becomes a semi-infinite, one-dimensional chain model, where each site ofthe chain is represented by a 8 × k y –dependent matrix.The 8 × G R ( k y , E ) is calculatedusing the decimation method [29, 30]. Note that this istraditionally referred to as the ‘surface Green’s function,’however, for this system, the ‘surface’ is an ‘edge.’ To re-solve the edge spectrum, the energy broadening η used inthe calculation of the surface Green’s function is 1 meV,which is chosen to be five times larger than the energydiscretization step size. The spectral function at the edgesite is A ( k y , E ) = − (cid:8) tr (cid:2) G R ( k y , E ) (cid:3)(cid:9) .In the superconducting Nambu space, the topologicalsuperconducting Chern number (TSC), N , is allowed tobe − , − , , , N characterizes the number ofchiral edge modes [16]. One practical way to tune theTSC number is to apply an out-of-plane external elec-tric field to modify the top-surface electrostatic potentialenergy [16, 20]. The topological phase diagram of thesystem represented by Eq. (1) is plotted in Fig. 2(a) asa function of the top-gate potential V g and the magnitudeof the exchange energy J H . The phase boundaries are ob-tained by the gap closing in the BdG Hamiltonian (Eq.(1)) at k = 0. To determine the the TSC number in eachregion, we evaluate the number of edge states from thebandstructure calculation of a 150 nm wide ribbon thatis periodic along x . N is the number of the degeneracyof the edge states along one edge. The ribbon width ischosen to be sufficiently wide such that the hybridizationof the edge states is negligible. The blue area belongs tothe trivial phase ( N = 0) of a normal insulator. The pur-ple regions correspond to N = 2, which is topologicallyequivalent to a non-superconducting quantum anomalousHall insulator with Chern number C = 1. In the grey ar-eas, N = 1, and a single Majorana edge mode propagatesalong the edges. As shown in Fig. 2(a), when V g is zero, N = 1 only occurs over a narrow range of exchange po-tentials. Therefore, gating the top surface can controlthe transition between different topological phases.To demonstrate the voltage-controlled topologicaltransition, we numerically calculate the edge-state spec-trum of the semi-infinite plane at different values of V g .A semi-infinite plane is chosen to ensure that the edgestate hybridization is zero since the opposite edge is at x = −∞ . Choosing the parameters for N = 0 and N = 1as shown by the two points in Fig. 2(a), the edge spec-tral function is plotted versus k y and E in Figs. 2(b)and (c), respectively. In Fig. 2(b) the applied voltage iszero, N = 0, and a trivial gap opens at the Dirac point.Applying a −
20 mV potential to the top surface, a topo-logical transition occurs, and a gapless Majorana edgemode appears as shown in Fig. 2(c).For further verification of the model, we constructa 2-terminal, finite-width device consisting of a centralsuperconducting / MTI bilayer region with two non-superconducting, topological insulator leads mimickingthe experimental setup recently reported [16]. The struc-ture is illustrated in the inset of Fig. 2(d) where thelength of the superconductor area is 40 nm and the width is 100 nm. As seen in Fig. 2(d), a half-integer plateauin conductivity appears during a scan of the Hund’s-ruleexchange energy J H , which emulates a scan of an exter-nally applied magnetic field. This plateau is the result ofa combination of normal reflection and Andreev reflec-tion [16]. FIG. 3. (a) The lowest positive–energy state at V g = − V g = −
50 mV. (c)Ground state energy as a function of the applied voltage fordifferent widths. The length is fixed at 1.6 µ m. Compo-nents of the MZM used to verify that the zero–energy modeis indeed a Majorana mode: (d) Re[ (cid:104) γ | ψ ↑ b (cid:105) ] (e)Re[ (cid:104) γ | ψ †↑ b (cid:105) ] (f)Im[ (cid:104) γ | ψ ↑ b (cid:105) ] (g) Im[ (cid:104) γ | ψ †↑ b (cid:105) ] We now show that a voltage applied to a gate witha large aspect ratio can create localized Majorana zeromodes at the ends. Fig. 3 shows the simulation geometrythat consists of a long, thin, gated region within a rect-angular supercell. The dimensions of the gated regionare 28 nm × µ m, and the dimensions of the supercellare 150 nm × µ m. Fig. 3(a) is a color map of of thelowest positive–energy ( E ≥
0) state | ψ i | at each site i at a gate voltage of V g = −
20 mV. The thin width of thegated area, 28 nm, is less than the penetration depth of aMajorana edge mode. This hybridizes the states on theopposing long edges of the gated region, so that a gapis opened in the energy spectrum and there is no zero–energy mode along the edges. Further decreasing V g to-50 mV, a pair of bound states appear at the ends of thegate as shown in Fig. 3(b), and the energy of these boundstates drops 5 to 6 orders of magnitude from 5 meV to ∼ − eV, suggesting that they are MZMs. The hy-bridization of the MZMs at the ends of the gated regionsis negligible since they are 1.6 µ m apart.The voltage at which the MZMs appear depends onthe geometry of the gated region. Fig. 3(c) shows acalculation of the ground state energy as a function ofthe gate voltage for 4 different gate widths. The gatelengths are fixed at 1.6 µ m. For each gate width, there isa critical gate voltage at which the ground-state energygoes to zero. The magnitude of V g required to achieve thezero-energy state increases as the gate width decreases.To confirm that the localized end-modes are indeedMZMs and not simply very low-energy states, the eigen-vectors Ψ of the zero-modes are analyzed to determineif they satisfy the property Ψ = Ψ † . The eight coeffi-cients of each mode at each site j can be divided intofour groups with each of the groups containing a pair ofcoefficients that are complex-conjugate, as shown in Eq.(4). Ψ =( A − A i ) ψ ↑ t + ( A + A i ) ψ †↑ t + ( A − A i ) ψ ↑ b + ( A + A i ) ψ †↑ b + ( − B + B i ) ψ ↓ t + ( − B − B i ) ψ †↓ t + ( B − B i ) ψ ↓ b + ( B + B i ) ψ †↓ b (4)Ψ is the wave function of a MZM, and A , , B , are thesite–dependent normalization coefficients. The real andimaginary parts of (cid:104) Ψ | ψ ↑ b (cid:105) and (cid:104) Ψ (cid:12)(cid:12)(cid:12) ψ †↑ b (cid:69) are shown inFig. 3(d)-(g). Numerically, Re[ (cid:104) Ψ | ψ ↑ b (cid:105) ] and Re[ (cid:104) Ψ | ψ †↑ b (cid:105) ]are identical, whereas Im[ (cid:104) Ψ | ψ ↑ b (cid:105) ] and Im[ (cid:104) Ψ | ψ †↑ b (cid:105) ] havedifferent signs, which satisfies Eq. (4). Similar resultsare obtained for the other bases. This confirms that thezero–energy states are MZMs.The motivation for an array of crossbar gates is tomimic a 1D network of wires for gate–controlled transferand exchange of MZMs. A fundamental building block ofsuch a network is a T–junction as shown in Fig. 4. Withvoltage applied to the horizontal section of the gate, twoMZMs are created at the ends of the I-shaped gated area.Turning off the voltage of the left side gate and applyingit to the vertical gate results in the MZM at the end ofthe ‘L’. The MZM does not appear at the sharp corner ofthe ‘L’. Controlling the voltages of the gates moves thetopological regions ( N = 1) and the associated MZMs.Such a network of top gates can implement a pixel-by-pixel control of the geometric shape of the topologicalregion, such that more complicated braiding operationscan be achieved within this scheme.All of the calculations presented are for 3 quintuplelayers. In terms of the model Hamiltonian (3), only theinterlayer hybridization terms, m and m , change dueto layer thickness. For example, at 5 quintuple layers,their values become m = 20 . m = 5 eV˚A [25]. The phase diagram of the topological transitionsshown in Fig. 2(a) does not change. This means that theoptimum value for | J H | is approximately m , or, in otherwords, the spin-splitting due to the magnetic exchangeinteraction from the Cr dopants should be close to thehybridization gap induced by the inter-surface couplingof the top and bottom layers.For the two dimensional system represented by Eqs.(1) - (3) with ∆ t = 0, the energy gap at Γ is E Γ =2 | (cid:15) ( V G ) − J H | where (cid:15) ( V G ) = √ (cid:112) m + V G + ∆ b − √ x FIG. 4. Shifting the MZM in the left (a) to the top (b) bychanging the gate electric potential. The gates are set to beon and off inside the dashed and solid lines, respectively. Thewidths of the gated regions are 28nm. with x = V G + 4 m V G + 4 m ∆ b − V G ∆ b + ∆ b . Ig-noring the ∆ b terms since ∆ b << J H , m , at V G =0, the gap E Γ ≈ m − J H ) as seen in Fig. 2(b).The voltage required to close the gap at Γ is V G ≈ m [1 − (cid:112) J H /m ) − ≈
10 mV, which scales with m . When V G is applied in a ribbon geometry, the hy-bridization of the two states along the long edges of theribbon effectively increases the parameter m , which ne-cessitates larger gate voltages to drive the initial energygap to zero, as seen in Fig. 3(c).Once the bands invert, they take on the Mexican hatshape, the energy gap moves away from Γ, and its value isdetermined by the proximity induced Cooper pairing po-tential, ∆ b . This is the energy gap in which the edge stateresides seen in Fig. 2(c), and it is also the gap in whichthe MZMs reside. The MZMs are confined at a topolog-ical domain wall with the energy barriers determined by∆ b , m , and J H . Underneath the gated ribbon, the en-ergy spectrum is gapped by the hybridization energy ofthe two edge modes. Outside of the ribbon in the ungatedregion, the trivial energy gap 2( m − J H ) confines theMZMs. The energy gaps affect the spatial extent of theMZM wavefunctions. As they are reduced, the increasedtunneling necessitates wider and longer gate regions andgreater separation between the gates. As J H is reduced,lower temperatures would be required to maintain themagnetic ordering. Thus, thicker films with lower inter-surface hybridization require smaller exchange coupling,and allow lower voltage operation, but at the cost of lowertemperatures and larger areas.In summary, a gated MTI / superconductor bilayerprovides a platform for 2D spatial control of Majoranazero modes. The phase diagram of the system showsthat a gate voltage can control the topological transitionbetween the N = 0 and N = 1 states. The voltage ofthe transition depends on the gate width, and narrowergates require larger voltages. Relatively long gates arerequired, approximately 2 µ m, to prevent hybridization of the end modes and to allow the creation of MZMs atlow gate voltages. The MZM positions can be controlledby the local gating of the top surface. This scheme mayprovide a facile method for implementing quantum gatesfor topological quantum computing. Acknowledgements:
This work was supported bythe National Science Foundation under Award NSFEFRI-1433395 2-DARE: Novel Switching Phenomenain Atomic Heterostructures for Multifunctional Appli-cations and in part by FAME, one of six centers ofSTARnet, a Semiconductor Research Corporation pro-gram sponsored by MARCO and DARPA.. [1] E. Majorana and L. Maiani, in
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