Control of Majorana edge modes by a g-factor engineerednanowire spin transistor

Amrit De a,n, Alexey A. Kovalev b

a Department of Physics & Astronomy, University of California, Riverside, CA 92521, USAb Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience, University of Nebraska-Lincoln, Lincoln, NE 68588, USA

a r t i c l e i n f o

Article history:Received 17 June 2013Accepted 9 August 2013Communicated by Sebastian T.B.Goennenwein

Keywords:A. SemiconductorsC. Nanowire quantum wellD. Majorana modesD. Spin transistor

a b s t r a c t

We propose the manipulation of Majorana edge states via hybridization and spin currents in a nanowirespin transistor. The spin transistor is based on a heterostructure nanowire comprising of semiconductorswith large and small g-factors that form the topological and non-topological regions respectively. Thehybridization of bound edge states results in spin currents and 4π-periodic torques, as a function ofthe relative magnetic field angle – an effect which is dual to the fractional Josephson effect. We establishrelation between torques and spin-currents in the non-topological region where the magnetic field isalmost zero and spin is conserved along the spin–orbit field direction. The angular momentum transfercould be detected by sensitive magnetic resonance force microscopy techniques.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

It is believed that nanowires with strong spin–orbit interac-tions can realize topologically protected quantum bits (qubits)[1–3] based on Majorana zero energy modes [4–8]. Some of theother proposals to realize topologically protected qubits includeschemes based on topological insulators [9,10], fractional quantumHall states [11], cold atom systems [12,13], p-wave superconduc-tors [14] and superfluids in 3He–B phase [15].

Typically proposals for observing these Majorana zero energymodes are based on quantum tunneling and transport type phe-nomena [16–21]. Some exciting recent proposals for observing theseedge modes are based on the unconventional Josephson effect witha 4π periodicity [22–25]. A dual effect has also been suggested inwhich case a torque between magnets exhibits 4π periodicity in thefield orientations [26,27,48].

It is this dual of the Josephson effect that could in fact beemployed in spintronic devices that can in principle be employedin spintronic devices. In particular, it is important to understand therole of mechanical torques that should inevitably accompany Major-ana hybridization due to conservation of angular momentum. It hasbeen predicted that conservation of angular momentum in macro-spin molecules can result in quantum entanglement of a tunnelingspin with mechanical modes [28,29]. A flow of spin current betweentwo magnets has been demonstrated to induce spin-transfer torqueeffect [30,31] and mechanical torques [32,33], also by conservation of

angular momentum. A flux qubit has been shown to decohere dueto exchange of angular momentum between the qubit and elasticmodes of a solid [34].

In this paper we propose the manipulation of Majorana edgestates via hybridization and spin currents using a gated nanowirespin transistor. For our calculations we consider a spin transistorthat comprises of GaSb–GaAs–GaSb type-II quantum well withstrong spin–orbit interactions. Our choice of semiconductors isbased on a number of factors. First the formation of topologicaland non-topological regions of the wire is determined by therelative g-factors of the wires. While GaSb has a large g � �9, therelatively smaller g � �0:3 for (GaAs) gives an excellent g-factorcontrast of about 30. GaSb–GaAs nanowires and other Sb contain-ing hetero-junction nanowires have been grown [35–38]. ThoughInSb has a much higher g-factor and InSb–InAs heterojunctionnanowires have been grown [39,40], their g-factor contrast is onlyabout 4. Second, the relatively small conduction band offset ofthese two materials leads to the formation of a shallow type-IIquantum well, thereby only requiring a small gate voltage to raisethe chemical potential in the well region.

Typically strong quantum confinement effects can stronglyalter the electronic g-factors and drive it towards the bare electronvalue due to effects of orbital angular momentum quenching[41,42] in materials with sufficiently strong spin–orbit coupling.In the case of InAs–InP nanowire quantum dots it has been shownthat the g-factor can be tuned through zero [43,44]. Therefore inthe case of GaAs (which has a small negative bulk g), a sufficientlysmall quantum well should drive g towards 0, thus making it aperfect non-topological region.

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Solid State Communications

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n Corresponding author.E-mail address: amritde@gmail.com (A. De).

Please cite this article as: A. De, A.A. Kovalev, Solid State Commun (2014), http://dx.doi.org/10.1016/j.ssc.2013.08.004i

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The topological region semiconductor should have strong spin–orbit interactions and large Zeeman splitting. The schematic of ourproposal is shown in Fig. 1. The wire is proximity-coupled to ans-wave superconductor which results in proximity induced pairingin the wire. The spin transistor allows the bound edge statesto hybridize thus resulting in spin-current induced 4π-periodictorque, as a function of the relative magnetic field angle. As anexample of application for our proposal, arrays of nanowires withzero energy edge modes could be used as quantum memory – inwhich case there arises a need to efficiently read out informationfrom memory elements. The spin and angular momentum flowsdiscussed here could be employed for that. In general the nano-wire architecture allows the combination of various lattice mis-matched materials and have attracted much attention due totheir potential electronic and optoelectronic applications such assingle electron transistors, field sensors, and low-power electro-nics [38,45,46].

We also establish relation between torques and spin-currentsin the non-topological region where the magnetic field is almostzero and spin is conserved along the spin–orbit field direction.Sensitive magnetic resonance force microscopy measurements canprovide further evidence for the existence of these edge modesand their hybridization. Finally, we show that this non-dissipativespin current can be controlled by the external gate voltage (seeFig. 1) which leads to similar functionalities as the Datta and Dasspin-field-effect transistor [47].

2. Tight binding calculations for spin currents andedge hybridization

Consider a semiconducting quantum wire, with the Rashbaspin-splitting, placed on top of a superconducting substrate (asper the coordinates shown in Fig. 1). In this solid state system,Majorana modes are charge less, localized zero-energy collectivequasiparticle excitations of the superconducting ground state thatsatisfy the Bololiubov-de Gennes (BdG) Hamiltonian:

H¼ k2

2mτzþ iαsokyτzszþΔ0 � τþB � r: ð1Þ

where we have used the Nambu spinor basis Ψ T ¼ ðψ ↑;ψ ↓; ψ†↓;

�ψ †↑Þ, r and τ are Pauli vectors that respectively act on particle

and hole sectors. Here αso is the strength of Rashba spin-splittingterm, B¼ ½Bo cos θ; �Bo sin θ; �Bz� is the magnetic field vector,Δ0 ¼ ½Δ cos ϕ;Δ sin ϕ; �μ�, μ is the chemical potential and Δeiϕ

is the superconducting pairing potential. In general, the energyspectrum of the BdG Hamiltonian supports gapped and gaplessphases. The overall phase diagram is more complicated thanTI edge systems [26,27] due to the presence of the k2τz term.Here, we limit our consideration to the Δ24B2

z part of the phase

diagramwhere the energy bands are always gapped. There are twogaped phases, – topological (T) for Δ2�B2

z oB2o�μ2 and non-

topological (N ) for Δ2�B2z 4B2

o�μ2,separated by a quantum phasetransition at Δ2�B2

z ¼ B2o�μ2.

The coupling between a magnetic field and the spin of anelectron is determined by the g-factor, which would thereforedetermine whether the semiconductor is in the N or T phase.Hence it is possible to engineer a nanowire quantum wellstructure that can form T jNjT or NjT jN regions even when placedin a uniform magnetic field. For a T jNjT type system, the hybridi-zation across the N region (which forms the well) can be gatecontrolled. Our proposed spin transistor that comprises of GaSb–GaAs–GaSb type-II quantum well within a nanowire is shownin Fig. 1.

In order to treat arbitrary 1D heterostructures and non-uniformmagnetic fields, we transform the BdG Hamiltonian, Eq. (1), ontothe following tight binding model with nearest neighbor hopping:

H¼ ∑j;s;s0

c†jþ1s �t0s0þ iαj

2sz

� �ss0

cjs0 þH:c:� �

þ∑j;sð2t0�μiÞc†jscjsþ∑

jð ~Δ ic

†j↑c

†j↓þH:c:Þ

þ∑jð ~Bjc

†j↑cj↓þH:c:Þ ð2Þ

where we have used the complex parameters ~Δ ¼Δ expðiϕÞ,~B ¼ Bo expð� iθÞ and c†jsðcjsÞ creates (annihilates) an electron ofspin s on site j. The proximity induced gap Δ¼ 0:5 meV. Heret0 ¼ ℏ=2mna2 is the hopping strength, αj ¼ αðjÞ

so=a where a is thelattice spacing. In our calculations, we use the following parametersfor GaSb and GaAs, mGaSb ¼ 0:041me, mGaAs ¼ 0:067me, αGaSb

so ¼0:187 eV A

˚and αGaAs

so ¼ 0:024 eV A˚. The magnetic field at each site

is given by ~Bj ¼ gjμBB=2, where μB is the Bohr magneton, gj is theLande g-factor of the semiconductor at that given lattice site and Bis the applied field. Although quantum confinement effects can alterthe electronic g-factors in materials with sufficiently strong spin–orbit coupling [41,42,44], as we are considering non-topologicalwell regions that are fairly large – we use the bulk g-factors:gGaAs ¼ �0:32 and gGaSb ¼ �8:72.

The overall length of the wire was taken to be 3 μm corre-sponding to 300 grid sites for a grid spacing of a¼10 nm. Typicallythe GaSb section of the nanowires is about 60 nm in diameter,while the GaAs sections are about 40 nm wide [36]. Taking theeffective masses, the transverse quantization, the bandgaps andthe valance band offset of these semiconductors into account,we estimate that the barrier height of the quantum well is about52 meV.

In Fig. 2 we show the edge states from our tight bindingcalculations as a function of bias voltage applied to the non-topological GaAs well region for two different well widths. Therespective energies of these states are shown in Fig. 3. It is clearlyseen that the shorter non-topological well region results in higherhybridization energies due to more dominant finite size effects. Inthe absence of any bias, the quantum well prevents the hybridiza-tion of the edge modes as indicated by the separate red and blueedge states. This separation of the edge modes persists till thethreshold Vb � 30 mV is reached at which point a split in theenergy spectrum is seen due to hybridization of the edge modes.As Vb is further increased, the edge modes abruptly return to theirunhybridized state as the bias voltage now acts as a barrierpreventing any tunneling effects. The unhybridized Majorana edgemodes (γ1 and γ4) are formed at the ends of the structure and theyhave nearly zero energy. The hybridized edge modes (γ2 and γ3)are formed in the middle and they have non-zero energy due tofinite size effects. This non-zero energy edge mode can now bemanipulated by spin currents and magnetic field gradients.

Fig. 1. (Color online) A semiconductor nanowire with an embedded type-II GaSb–GaAs–GaSb quantum well placed on top of s-wave superconductor in the presenceof a magnetic field. The relatively much smaller g-factor of GaAs makes it a non-topological region. The gate voltage, Vb, can be used in order to control thehybridization between the edge modes formed at the hetero-junction interface.

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In general the three components of the spin current operatorare as follows:

JjðrÞ ¼Re Ψ †ðrÞ sj∂H∂p

þ∂H∂psj

" #Ψ ðrÞ

!ð3Þ

where j¼ x; y; z and Ψ ðrÞ is the position dependent Nambu spinorwhich can be calculated by diagonalizing the tight bindingHamiltonian. The spin currents were numerically calculated usingfinite differences and the tight binding wavefunctions.

The spatial distribution of the z component of the spin currentfor the hybridized edge states as a function of various gate biasvoltages is shown in Fig. 4. The Rashba spin-splitting is alongthe z-direction and only the Jz spin current is non-zero in the non-topological GaAs well region (spin is conserved along thez-direction for the non-topological region). It is seen that the gatebias voltage can strongly affect the spin current. Overall the spincurrent is nearly constant in the N region and is thereforeconserved.

In Fig. 5(a), we plot hybridization energies of the edge states asa function of relative angle, θ, between the magnetic fields on theleft and right topological GaSb region. Separate magnetic tips withlocalized dipole type fields could be used to realize this. Notice the4π periodic behavior. The corresponding spin torque ∂EnðθÞ=∂θagrees with spin current in Fig. 5(b). It is the θ dependence ofthe hybridization energy that leads to mechanical torques. Suchtorques could be detectable by the exertion of a mechanical torqueon a nano-magnetic tip [26,27]. In the coming cases we observedifference between the spin current and the torque which is the

result of additional torques produced by the coupling between theorbital and spin degrees of freedom.

3. Analytic results

The numerical results presented in the previous section can bebetter understood by the analytical results presented here.

In general, an effective two level system or a Majorana qubitcan be formed by four Majorana edges where three of the edgesare hybridized (in Fig. 1 we assume that edges γ1 and γ2 areseparated from each other). The effective low energy Hamiltoniancan be written as

H¼ iEnðθÞγ2γ3þ iEtγ3γ4 ð4Þwhere θ is the angle between the magnetic fields, γi describeMajorana edge states that can form two fermions b0 ¼ ðγ1þ iγ2Þ=2and b″¼ ðγ3þ iγ4Þ=2, where b†ðbÞ are ferminoic creation (annihila-tion) operators.

We analyze the hybridization of the edge modes and resultingspin-currents and torques on T jNjT structures. In order to calculateEnðθÞ, we consider a semiconductor nanowire with two infinite Tregions (GaSb) separated by a finite N region (GaAs). We introduceparameters fΔL;BL;μL;θLg for the left T region, fΔM ;μMg forthe middle N region and fΔR;BR;μR;θRg for the right T region.The phase of superconducting pairing is assumed to be constant(i.e. ϕ¼ 0) throughout the wire. In order to determine the boundstate at a single T jN boundary, we need to find the 4-componentzero energy solution to the Hamiltonian (Eq. (1)). We use ansatzΨ ðxÞ ¼ expðκxÞΨ ðκÞ, where κ is complex. In doing so we arrive at

0 20 40 60−50

0

50

Ene

rgy

(μeV

)

Ene

rgy

(μeV

)

0 20 40 60−50

0

50

Vb (mV)Vb (mV)

E1

E2E3

E4

Fig. 3. (Color online) The hybridization energies as a function of bias voltage corresponding to Fig. 2 for well widths of (a) 20 nm and (b) 40 nm.

0123

0

50

0

0.1

0.2

0123

0

50

0

0.1

0.2

Length (μm)Length (μm)

Vb (mV)V

b (mV)

|Ψ|2

|Ψ|2

Fig. 2. (Color online) Edge modes as a function of hetero-junction bias voltage Vb with the wire subjected to constant magnetic field for the non-topological GaAs well widthsof (a) 200 nm and (b) 400 nm. Hybridization of the Majorana edge modes is seen at Vb ¼ 30 meV.

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four solutions that decay into the topological region, i.e. withReðκÞ40, and four solutions that decay into the non-topologicalregion, i.e. with ReðκÞo0. A linear combination of these solutionson each side has to be continuous and has continuous derivative atthe T jN boundary which leads to unique solution for the edge state.

We denote such solutions as jψ L⟩¼ exp½iθLsz=2�jψ0L ⟩ for the left

Majorana edge and as jψR⟩¼ exp½iθRsz=2�jψ0R⟩ for the right Major-

ana edge. Next we employ lowest order perturbation theory tofind the hybridization energy of Majorana modes and spin currentat the boundary when the solutions for the left and right edgesweakly overlap. For the hybridization energy we obtain

EnðθÞ � j⟨ψ LjHjψR⟩jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi⟨ψ0

L jψ0L ⟩⟨ψ

0Rjψ0

R⟩q ; ð5Þ

which becomes

EnðθÞ � En0 exp½�Reðκn2Þℓn� cosθ2þΦ0þImðκn2Þℓn

� �; ð6Þ

where κn2 ¼mn=ℏðiαso� iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðiΔþμÞℏ=mnþα2

so

pÞ, En0 and Φ0 depend

on the parameters of the T and N regions and not on the wirelength, ℓn, and θ.

Similarly for a semiconductor nanowire with two infinite Nregions separated by a finite T region (GaAs), the hybridizationenergy for the topological region is

Et � Et0ðexp½�κt2ℓt �þjA0j exp½�Reðκt1Þℓt � cos ½arg A0þImðκt1Þℓt �Þ;ð7Þ

where κt1 and κt2 are the roots offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2�½κ2ðℏ=2mÞ2þμ�2

q¼Δþαsoκ

such that RðκÞ40. Here Et0 and A0 depend on parameters of the Tand N regions and not on ℓt and θ.

From the perturbative solutions we can express spin current atpoint r as

JzðrÞ ¼Re½ψ †

LðrÞ7 iψ †RðrÞ�fsz; υg½ψ LðrÞ7 iψRðrÞ�

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi⟨ψ0

L jψ0L ⟩⟨ψ

0Rjψ0

R⟩q

0B@

1CA ð8Þ

where υ ¼ ∂H=∂p and θ¼ θR�θL. This leads to:

Jz ¼∂EnðθÞ∂θ

ð9Þ

which relates spin current to the hybridization energy over the Nregion. This shows that the torque ∂EnðθÞ=∂θ is generated solely bythe spin current passing through the middle N region.

4. Manipulation of Majorana qubit by gates

Without loss of generality we can consider a fixed electronparity state (number of particles modulo 2). For example, for anodd parity state, the available Hilbert space of two fermionscorresponding to the four edge modes, (b0 ¼ ðγ1þ iγ2Þ=2 andb″¼ ðγ3þ iγ4Þ=2) is αj1;0⟩þβj0;1⟩. This is equivalent to a Hilbertspace of a spin 1/2 system. The Hamiltonian of Eq. (4) thenbecomes

H¼ EnðθÞsxþEtsz; ð10Þwhich shows that by gate tuning En and Et one can performarbitrary rotations of the Majorana qubit. The relative anglebetween magnetic fields, θ, can also be used for qubit manipula-tions. Form the analysis in the previous section it becomes clearthat rotations along the x-axis are accompanied by mechanical

1 1.5 2−40

−20

0

20

40

J z fo

r E3

(μeV

/deg

)

Length (μm)1 1.5 2

−40

−20

0

20

40

J z fo

r E4

(μeV

/deg

)

Length (μm)

Vb = 48 mV, V

b = 50 mV, V

b = 52 mV, V

b = 54 mV

Fig. 4. (Color online) The spatial distribution of the z component of the spin current for the (a) E3 and (b) E4 states as a function of various bias voltages, Vb. The spin currentonly exists in the middle section when the Majorana edge modes are allowed to hybridize over the non-topological region. Note that the real part of x and y components ofthe spin currents is zero.

0 90 180 270 360−50

0

50

J z (μ

eV/d

eg)

θ (deg)0 90 180 270 360

−50

0

50

Ene

rgy

(μeV

)

θ (deg)

E1

E2

E3

E4

Fig. 5. (Color online) (a) Hybridization energy of Majorana bound states as a function of the relative angle, θ, between the magnetic fields on the left and right T region(separate dipolar magnetic tips with more localized fields can be used for this). Note the 4π periodic behavior. (b) The spin current as a function of θ with bias voltage ofVb ¼ 50 mV is applied.

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torques. Nevertheless, we estimate that this should not be a strongsource of decoherence [34].

5. Summary

We have proposed the manipulation of Majorana edge states ina gated nanowire spin transistor that comprises of GaSb–GaAs–GaSb type-II quantum well. The formation of topological and non-topological region of the wire is determined by the large g-factorcontrast of the two semiconductors. In general, larger spin-orbitinteractions and smaller band gaps lead to larger g-factors in bulk.It is possible to obtain a full Bloch sphere rotations of the Majoranaqubit by gates controlling the hybridization energies. The setupcan be easily generalized to a larger number of interchangingtopological/nontopological regions where multi qubit rotations canbe achieved by applying time dependent gates. Spin currents andtorques can be further used in order to couple such a wire to variousread out schemes [49]. The flows of angular momentum, comprisinga signature of Majorana edge states, can be detected by sensitivemagnetic resonance force microscopy techniques.

Acknowledgements

We would like to thank Kirill Shtengel and Leonid Pryadko formultiple helpful discussions. AD was supported in part by the U.S.Army Research Office under Grant No. W911NF-11-1-0027 and byNSF Grant No. 1018935. This research was performed in part at theCentral Facilities of the Nebraska Center for Materials andNanoscience, which is supported by the Nebraska ResearchInitiative.

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