Chiral topological superconductivity in Josephson junction
CChiral topological superconductivity in Josephson junction
Shao-Kai Jian and Shuai Yin Condensed Matter Theory Center, Department of Physics,University of Maryland, College Park, Maryland 20742, USA School of Physics, Sun Yat-Sen University, Guangzhou 510275, China (Dated: September 8, 2020)We consider a heterostructure of semiconductor layers sandwiched between two superconductors,forming a two-dimensional Josephson junction. Applying a Zeeman field perpendicular to the junc-tion can render a topological superconducting phase with chiral Majorana edge mode. We showthat the phase difference between two superconductors can efficiently reduce the magnetic fieldrequired to achieve the chiral topological superconductivity, providing an experimentally feasiblesetup to realize chiral Majorana edge modes. We also construct a lattice Hamiltonian of the setupto demonstrate the chiral Majorana edge mode and the Majorana bound state localized in vortices.
Introduction.—
Since the discovery of topological insu-lators, intensive theoretical and experimental studies onvarious kinds of symmetry protected topological orderhave arised [1, 2], among which the pursuit of Majo-rana fermions and Majorana zero modes is one of themost interesting and important issues. Owing to its non-Ablian statistics, the Majorana zero mode is of great po-tential to implement topological qubits in fault-toleranttopological quantum computations [3–7]. The Majoranazero mode can be realized in 5/2 fractional quantumHall states [8, 9], vortices of spinless p + ip superconduc-tors [10] and superconductor-semiconductor heterostruc-tures [11–14] in two dimensions, the edge of a ferromag-netic atomic chain on a superconductor [15–17], and theplanar Josephson junction [18–20], etc. leading to ongo-ing interest among both condensed-matter and quantumcomputation communities.On the other hand, one-dimensional chiral Majoranafermion—the simplest solution satisfying the real versionof Dirac equation proposed originally by Ettore Majo-rana in 1937 [21]—also attracts lots of attentions in re-cent years. Realization of the chiral Majorana fermions inreal world is not only of great theoretical importance, butmore importantly also opens up a new avenue for topo-logical quantum computation [22]. The chiral Majoranafermion can be realized at the edge of two-dimensionalchiral topological superconductors in class D [2]. Chi-ral topological superconductor is a real analog of inte-ger quantum Hall insulator, where the number of chiralMajorana edge modes reflects the BdG Chern numberof the occupied bands. A series of theoretical proposalsarises including the superconductor-semiconductor het-erostructure [12], and superconductor-quantum anoma-lous Hall insulator heterostructure [23], and so on. Al-though experimental works try to realize the later pro-posal [24], debates about whether true chiral Majoranaedge modes are observed remain [25–28].Here, we propose a different strategy to realize chi-ral Majorana edge mode by taking the advantage ofthe phase difference in Josephson junction. As shownin Fig. 1(a), we consider a few layers of semiconduc- !" !" ' SSsemiconductor ) * (a) � ���� � / � - � / � � / � - � / � Δ � � χ / � Δ � - � χ / � S SSemiconductorLayers (b)
FIG. 1. (a) A few layers of semiconductor thin film is sand-wiched between two superconductors (S). The heterostructureform a Josephson junction. The order parameters of two su-perconductors are given by ∆ e ± iχ/ . A out-of-plane magneticfield is also applied to tune the topological superconductivity.(b) The proximitized SC state forms a domain wall in thesemiconductor layers. The width of the domain wall and ofthe semiconductor layers are denoted by L and W , respec-tively. tor sandwiched between two superconductors, forminga two-dimensional Josephson junction with a supercon-ducting phase difference χ . In the presence of a Zeemanfield perpendicular to the junction, the phase difference χ provides a useful knob to tune the topological transi-tion from a trivial superconductor to a chiral topologicalsuperconductor with a Majorana edge mode. The advan-tage of this setup is that (a) the critical Zeeman field toachieve the topological transition is efficiently reduced bythe phase difference χ without destroying the supercon-ducting order and (b) the transition can be realized with-out carefully gating the system. The continuous topolog-ical transition is described by two-dimensional Majoranacone at Γ point, the center point of Brillouin zone. Wealso construct a lattice model to show the chiral Majo-rana edge mode and Majorana zero bound states withinthe vortices—the two essential signatures of chiral topo-logical superconductors. Josephson junction and phase diagram.—
We considera two-dimensional Josephson junction shown in Fig. 1(a), a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p where layers of semiconductor are sandwiched betweentwo superconductors with a superconducting phase dif-ference χ . Due to the proximity effect, the two super-conducting orders will penetrate into the semiconductorlayers and form a domain wall in the semiconductor lay-ers. As shown in Fig. 1(b), the width of the domain walland the semiconductor layers are L and W , respectively.The superconducting order in the semiconductor layersis given by∆( z ) = ∆ e − iχ/ Θ( z + L/
2) + ∆ e iχ/ Θ( z − L/ , (1)where ∆ ( χ ) is the magnitude (phase difference) of thesuperconductivity, and Θ is the step function. z denotesthe direction perpendicular to the junction. We assumethat inside the domain wall | z | ≤ L/ ± L/ W/L → ∞ .The BdG Hamiltonian of the semiconductor layers is H = Ψ † H BdG
Ψ, where Ψ = ( ψ ↑ , ψ ↓ , ψ †↓ , − ψ †↑ ), and H BdG ( k , z ) = (cid:16) k − ∂ z m − µ (cid:17) τ z + H SOC ( k ) τ z (2)+ (∆( z ) τ + + H.c. ) + E Z σ z , (3)where k = ( k x , k y ) denotes the momenta in the junctionplane, τ ( σ ) is the Pauli matrix acting on the Nambu(spin) space and τ ± ≡ ( τ x ± iτ y ). E Z denotes the Zee-man energy induced by the Zeeman field perpendicularto the junction (along z direction). H SOC is the Dressel-haus spin-orbit coupling which will be discussed later.The phase transition between trivial and topologicalsuperconductors is characterized by a gap closing at k = and consequently a change of BdG Chern num-ber. As a result, we calculate the junction spectrumin the k = subspace by the scattering theory [29].As the spin is conserved during the scattering, we canfocus on σ z = 1 subspace.The transmission amplitudeof the Josephson junction is T = diag( e ik e L , e − ik h L ),where k e/h = (cid:112) m ( µ ± ( (cid:15) − E Z )) is the wave-vector ofthe normal state wavefunction. By matching the wave-function at the superconducting-normal state interface at z = ± L/ S ± L/ = e ± iχ/ τ z Se ∓ iχ/ τ z , where S = e − iφ S (cid:18) re iφ N √ − r √ − r − re − iφ N (cid:19) , (4)where r is the normal reflection amplitude, and φ S,N are two phases given in the Supplemental Materials [30].Using the scattering and transmission matrix, the spec-trum of the bound state is determined by [29] det(1 − π π χ E Z / Δ C = = FIG. 2. The phase diagram of Josephson junction as a func-tion of the junction phase difference and the Zeeman energy. C denotes the BdG Chern number, where C = 0 ( C = 1)implies trivial (topological) superconductivity. The dashed(solid) line is the phase boundary without (with) normal re-flection. S − L/ T S + L/ T ) = 0, i.e.,cos[( k e − k h ) L − φ S ]= (1 − r ) cos χ + r cos[( k e + k h ) L + 2 φ N ] . (5)Assuming weak superconducting pairing E Z (cid:46) ∆ (cid:28) µ ,the zero mode is given byarccos E Z ∆ = ˜ χ π E Z E T + nπ, (6)where n ∈ Z , ˜ χ is a function of χ ,˜ χ ( χ ) = arccos (cid:104) (1 − r ) cos χ + r cos 2 k F L (cid:105) , (7)and E T ≡ ( π/ v F /L is the Thouless energy of the junc-tion. Here k F = √ mµ and v F = (cid:112) µ/m . Note thedomain wall width is in the order of the lattice constant,thus, the Thouless energy is in the order of the chemicalpotential, E T ∼ µ , which is the largest energy scale inthe question.In the weak superconducting pairing limit, ∆ (cid:28) µ ,the normal reflection and the effect from Thouless en-ergy may be neglected. Then phase diagram is simplygiven by E Z ∆ = | cos χ | , as shown by the dashed line inFig. 2. It is interesting to note that at χ = π , the criticalZeeman field is reduced to zero, namely, an infinitesimalZeeman field can turn the system into the chiral topolog-ical superconducting phase. Including corrections fromthe normal reflection to the lowest nontrivial order of∆ /µ , r = ∆ / (2 µ ) + O (∆ /µ ), the phase diagram is cor-rected to the solid line in Fig. 2. Especially at χ = π , thecorrection is given by ˜ χ ( χ = π ) = π − ∆ /µ cos k F L . Thecritical Zeeman energy at χ = π is modified to be ∆ /µ ,which is still small at weak pairing limit, making an ex-perimentally achievable way to realize chiral topologicalsuperconductivity. Topological phase transition.—
In the following, we willshow that the topological phase transition is described bya two-dimensional Majorana cone—a real version of theDirac fermion. It is well known that spin-orbit couplingis crucial in the realization of topological phases. Directsemiconductors with an inversion-asymmetric zinc blendestructure (point group T d ), such as GaAs, InSb, CdTeetc, have considerable size of spin-orbit coupling and arecommon in making quantum structures [31]. Moreover,many of these materials have a similar band structurewith the smallest gap at Γ point. To give a concrete ex-ample of spin-orbit couplings, we consider a heterostruc-ture consisting of a few layers of such semiconductorsgrown in (001) direction. Because of the phase differencein the Josephson junction, the heterostructure breaks in-version symmetry. However, such a structure inversionasymmetry cannot induce Rashba spin-orbit coupling,since the system respects the composite S and timereversal symmetry. As a result, the symmetry allowedspin-orbit coupling is the Dresselhouse term [31, 32], H SOC ( k ) = α ( k x σ x − k y σ y ), where α is the strengthof spin-orbit coupling.To simplify the question, we set the width of domainwall L to be zero. At the k = subspace of the nor-mal semiconductor without the proximitized supercon-ducting order and the external magnetic field, a continu-ous kinetic term of the one Kramers pair near the Fermipoints, ψ , ↑ and ψ , ↓ , is described by one-dimensionalDirac Hamiltonian − iv F ∂ z σ z . Adding the proximitizedSC order and the Zeeman field, the BdG Hamiltonianin Nambu basis Ψ = ( ψ , ↑ , ψ , ↓ , ψ † , ↑ , ψ † , ↓ ) T is given by(note the basis difference in Eq. (3)), H BdG ( k , z ) = H ( z ) + H ( k ), where H ( z ) = − iv F ∂ z σ z − ∆ (cid:48) σ y τ y − ∆ (cid:48)(cid:48) ( z ) σ y τ x + E Z σ z τ z , (8) H ( k ) = k x + k y m + α ( k x σ x − k y σ y τ z ) , (9)and ∆ (cid:48) = ∆ cos( χ/
2) and ∆ (cid:48)(cid:48) ( z ) = ∆ sin( χ/ z ) referto the real and imaginary part of the superconductingorder, respectively. We have separated the Hamiltonianinto an unperturbed part H and a perturbation H , re-spectively.In Eq. (8), because of the domain wall formed in∆ (cid:48)(cid:48) ( z ) [33], we get two bound states with eigen-energy E ± = ± ( E Z − ∆ cos χ/
2) that are related by particle-hole transformation, ψ + ( z ) = f ( z )(1 , , , − T , ψ − ( z ) = f ( z )(0 , , − , T , (10)where f ( z ) = ( N ) − / exp[ − (cid:82) z dz (cid:48) ∆ (cid:48)(cid:48) ( z (cid:48) ) v F ], and N is thenormalization factor.Now we consider the perturbation Eq. (9) to the boundstate ψ ± . Using first-order perturbation theory, it isstraightforward to get the effective Hamiltonian in thebound states subspace ( ψ + , ψ − ) T , H eff = (cid:18) δm − α ( k x + ik y ) − α ( k x − ik y ) − δm (cid:19) , (11) π π χ E Z / Δ C = = (a) - π π k - Δ '0 Δ ' ϵ k χ = E Z / Δ = (b) - π π k - Δ '0 Δ ' ϵ k χ = E Z / Δ = (c) - π π k - Δ '0 Δ ' ϵ k χ = E Z / Δ = (d) FIG. 3. (a) The phase diagram as a function of ( χ, E z / ∆). C = 0 ( C = 1) is the Chern number indicating the normalSC (the chiral TSC). The solid line is the phase boundarycalculated by lattice Hamiltonian. The parameters are givenby W = 25, L = 2, t = t (cid:48) = 1, µ = − . α = 0 .
6, and ∆ =0 .
5. The dashed line is determined by Eq. (6) and the dottedline indicates gapless points. The spectra of the Josephsonjunction at the three red points are shown in (b-d). where the mass is δm = E Z − ∆ cos χ/
2. At critical point δm = 0, the Hamiltonian describes a massless Majoranacone, corresponding to the phase transition from a trivialto a topological superconductor.Similar analysis can be applied to the other Kramerspair ψ , ↓ and ψ , ↑ , which shows that the critical point oc-curs at E Z / ∆ = − cos χ/
2. Combined with above results,the phase boundary is E Z / ∆ = | cos χ/ | , (we assume E Z > Topological gap.—
The spectrum at k = is given by (cid:15) = ± (∆ cos χ/ ± E Z ), thus away from the transitionpoint, the gap is in the order of ∆. Here, we analyze thetopological gap at generic k , especially, at k (cid:29) E Z /α , k = (cid:113) k x + k y . Using the scattering theory [30], thespectrum at k (cid:29) E Z /α is determined byarccos (cid:15) ∆ = ( k eσ − k hσ ) L ± χ , (12)where k e/hσ = (cid:112) m ( µ − E kσ ± (cid:15) ), E kσ = k / m − σαk ,with σ = ±
1. In the weak superconducting pairing limit,∆ (cid:28) µ , the spectrum is (cid:15) ∆ = cos (cid:104) χ ± k F L (cid:112) − E kσ /µ ∆ µ × (cid:15) ∆ (cid:105) . (13)We can see that the dependence on k is suppressed by∆ /µ for E Z /α (cid:28) k (cid:28) k F . As k increases such that E kσ ∼ µ , the prefactor of ∆ /µ diverges. Nevertheless,notice that the bound state exists only if k e/hσ is a realnumber, i.e., E kσ ≤ µ − (cid:15) . So when E kσ = µ − (cid:15) , thespectrum changes to (cid:15) ∆ = cos (cid:104) χ ± k F L (cid:115) ∆ µ × (cid:114) (cid:15) ∆ (cid:105) . (14)Combining the results of Eqs. (13) and (14), we concludethat the topological gap is (cid:15) ∼ ∆ cos χ up to a smallcorrection of ∆ /µ . Notice that at χ = π , the systemhas a small gap, so the better place to get the chiraltopological superconductor is in between χ = 0 and π ,where the Zeeman field is small while the gap is sizable. Tight binding model.—
To demonstrate the topologicalphase, we construct a tight binding Hamiltonian to modelthe setup in Fig. 1. The lattice model is given by H = H + H SOC + H Z + H SC , where H = − t W (cid:88) j z =1 (cid:88) i ;ˆ µ =ˆ x, ˆ y ( c † i ,j z c i +ˆ µ,j z + H.c. ) − t (cid:48) W − (cid:88) j z =1 (cid:88) i ( c † i ,j z c i ,j z +1 + H.c. ) − µ W (cid:88) j z =1 (cid:88) i c † i ,j z c i ,j z , (15) H SOC = iα W (cid:88) j z =1 (cid:88) i ;ˆ µ =ˆ x, ˆ y [( c † i ,j z σ x c i +ˆ x,j z − c † i ,j z σ y c i +ˆ y,j z ) − H.c. ] , (16) H Z = E Z W (cid:88) j z =1 (cid:88) i c † i ,j z σ z c i ,j z , (17) H SC = (cid:88) i (cid:104) [ W/ − [ L/ (cid:88) j z =1 ∆ e − iχ/ c i ,j z ( iσ y ) c i ,j z (18)+ W (cid:88) j z = W +1 − [ W/ − [ L/ ∆ e iχ/ c i ,j z ( iσ y ) c i ,j z (cid:105) + H.c.. (19)Here, i denotes sites in square lattice in the xy plane,while j z is the layer index along the z direction. Thenumber of semiconductor layer is W . t ( t (cid:48) ) is the in-plane ( z -direction) nearest neighbor hopping amplitude. µ is the chemical potential, α is the strength of in-planespin-orbit coupling, and ∆ ( χ ) is the amplitude (phasedifference) of the proximatized superconducting orders.The phase diagram of the lattice model is shown inFig. 3(a). The solid line in Fig. 3(a) is the phase bound-ary calculated by the lattice model, while the dashed lineis determined by Eq. (6). One can see that they matcheach other well. The spectrum of the junction at differ-ent phases indicated by the red points in Fig. 3(a) arealso plotted in Figs. 3(b-d), where one can see that bytuning the phase difference and/or Zeeman field, the en-ergy gap will close at phase boundary and reopen in thetopological superconducting phase. - π π k x - Δ � Δ � E (a) - π π k x - Δ � Δ � E (b) ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● n - Δ � / � Δ � / � E (c) (d) FIG. 4. (a,b) The spectrum of the system in k x direction(open boundary condition in y direction) in the topological su-perconducting phase and the trivial phase, respectively. Theparameters are given by W = 5, L = 2, t = t (cid:48) = 1, µ = − . α = 1, ∆ = 1, φ = 0 . π . (c) The spectrum of the systemwith two vortices. The disk (square) points correspond tothe topological (trivial) phase. The parameters are given by W = 6, L = 2, t = t (cid:48) = 1, µ = − . α = 1, ∆ = 1, φ = 0 . π .(d) The local density of the ground state wavefunction inthe topological phase. Two Majorana zero modes appear inthe vortices. The topological (trivial) phase corresponds to E z = 0 .
8∆ ( E z = 0 . To explore the chiral edge state of topological super-conducting phase, we use open boundary condition in the y direction, and plot the energy spectrum as a functionof k x as shown in Figs. 4(a) and 4(b) for the topologicalphase and the trivial phase, respectively. The two chi-ral Majorana states along + y and − y edges are clearlyshown in Fig. 4(a) in the topological superconductingphase. Moreover, when vortices are created via externalmagnetic field, the chiral topological superconductor isexpected to host Majorana zero mode localized in thevortex core. In Fig. 4(c), we observe the Majorana zeromode in the topological phase with two vortices, whilein trivial phase there is no Majorana zero mode. Thelocal density of ground state wavefunction in the topo-logical phase is plotted in Fig. 4(d), corresponding to thelocalized Majorana zero mode at the vortex core. Conclusions.—
In this paper, we show that a two-dimensional Josephson junction of the superconductor-semiconductor-superconductor sandwich heterostructurecan provide a useful setup to realize the chiral topologi-cal superconductor with chiral Majorana edge mode. Thetopological superconducting phase appears in a large por-tion of the phase diagram spanned by Zeeman field andthe superconducting phase difference of the Josephsonjunction. Compared to the previous setup [12], the Zee-man field is reduced within the critical strength and gat-ing is not longer necessary. These advantages facilitatepossible experimental reaches.
Acknowledgement:
We thank Hong Yao, Sankar DasSarma, Ady Stern, Zhongbo Yan, and Yingyi Huang forhelpful discussions. S.-K. J. is supported by the SimonsFoundation via the It From Qubit Collaboration. [1] A. Kitaev, AIP Conference Proceedings , 22 (2009)[2] S. Ryu, A. Schnyder, A. Furusaki, and A. Ludwig, NewJ. Phys. , 065010 (2010).[3] A. Y. Kitaev, Ann. Phys. , 2 (2003).[4] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, M. P. A.Fisher, Nature Physics , 412 (2011).[5] B. I. Halperin, Y. Oreg, A. Stern, G. Refael, J. Alicea,and F. von Oppen, Phys. Rev. B , 144501 (2012).[6] S. D. Sarma, M. Freedman, and C. Nayak, npj QuantumInformation , 15001 (2015).[7] S. R. Elliott and M. Franz, Rev. Mod. Phys. , 137(2015).[8] A. Stern, Nature , 187 (2010).[9] N. Read and D. Green, Phys. Rev. B , 10267 (2000).[10] A. Kitaev, Ann. Phys. , 2 (2006).[11] L. Fu and C. L. Kane, Phys. Rev. Lett. , 096407(2008).[12] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma,Phys. Rev. Lett. , 040502 (2010).[13] J. Alicea, Phys. Rev. B , 125318 (2010). [14] J. D. Sau, S. Tewari, R. M. Lutchyn, T. D. Stanescu, andS. Das Sarma, Phys. Rev. B , 214509 (2010).[15] A. Kitaev, Physics-Uspekhi , 131 (2001).[16] S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A.Yazdani, Phys. Rev. B , 020407 (2013).[17] B. Braunecker and P. Simon, Phys. Rev. Lett. ,147202 (2013).[18] F. Pientka, A. Keselman, E. Berg, A. Yacoby, A. Stern,and B. I. Halperin, Phys. Rev. X , 021032 (2017).[19] A. Fornieri et al, Nature , 89 (2019).[20] H. Ren et al, Nature , 93 (2019).[21] E. Majorana, Nuovo Cim. , 171 (1937).[22] B. Liana, X.-Q. Sun, A. Vaezi, X.-L. Qi, and S.-C. Zhang,PNAS , 184516 (2010).[24] Q. L. He et al., Science , 294 (2017).[25] W. Ji and X.-G. Wen, Phys. Rev. Lett. , 107002(2018).[26] Y. Huang, F. Setiawan, and J. D. Sau, Phys. Rev. B ,100501 (2018).[27] M. Kayyalha et al., Science , 64 (2020).[28] P. Zhang, L. Pan, G. Yin, Q. L. He, and K. L. Wang,arXiv:1904.12396.[29] C. W. J. Beenakker, Phys. Rev. Lett, , 3836 (1991).[30] See the Supplemental Materials for details.[31] R. Winkler, Spin-orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (Springer-VerlagBerlin Heidelberg 2003).[32] G. Dresselhaus, A. F. Kip, and C. Kittel, Phys. Rev. ,568 (1954).[33] R. Jackiw and C. Rebbi, Phys. Rev. D, , 3398 (1976). SUPPLEMENTAL MATERIALA. Spectrum in Josephson junction
Due to the proximity effect, the superconducting (SC) order will penetrate into semiconductor layers. Becausethe two superconductors have different SC order (most importantly, the different SC phase), there is a domain wallin the semiconductor layers. Assuming the size of the domain wall is L as shown in Fig. 1(b), the SC order in thesemiconductor layers is given by ∆( z ) = ∆ e − iχ/ Θ( − z + L/
2) + ∆ e iχ/ Θ( z − L/ , (S1)where ∆ is the magnitude of superconductivity, and χ denotes the phase difference between two superconductors. Θis the step function. Note the difference between the width of semiconductor layers denoted by W , and the widthof domain wall denoted by L , as shown in Fig. S1. Because the proximity length of SC order is quite large, and theproximitized SC order in semiconductors opens up a finite energy gap, the amplitude of wavefunctions will concentratenear the domain wall. As a result, one focus on the interface at ± L/ W/L → ∞ .The BdG Hamiltonian in the semiconductor layers is H = Ψ † H BdG
Ψ, where Ψ = ( ψ ↑ , ψ ↓ , ψ †↓ , − ψ †↑ ), and H BdG ( k x , k y , z ) = (cid:16) k x + k y − ∂ z m − µ (cid:17) τ z + H SOC ( k x , k y ) τ z + (∆( z ) τ + + H.c. ) + E Z σ z , (S2)where τ ( σ ) is the Pauli matrix acting on the Nambu (spin) space and τ ± = ( τ x ± iτ y ). E Z denotes the Zeemanenergy. Note that the Zeeman field is along z direction. H SOC is the Dresselhaus spin-orbit coupling which will bediscussed in the next Sector.Owing to the translational symmetry in xy -plane, k x and k y are good quantum numbers. Let’s first focus on k x = k y = 0 subspace in this Sector to determine the phase diagram, and then in later Sector on generic k x , k y toexplore the topological gap. The Hamiltonian in the k x = k y = 0 subspace reduces to H BdG (0 , , z ) = (cid:16) − ∂ z m − µ (cid:17) τ z + (∆( z ) τ + + H.c. ) + E Z σ z . (S3)For the normal state, i.e., | z | < L/
2, the eigenstates with eigen-energy (cid:15) are ψ ± eσ ( z ) = 1 (cid:112) k eσ (cid:18) (cid:19) ⊗ | σ (cid:105) e ± ik eσ z , ψ ± hσ ( z ) = 1 (cid:112) k hσ (cid:18) (cid:19) ⊗ | σ (cid:105) e ± ik hσ z , (S4)where σ = ± | + 1 (cid:105) = (1 , T and | − (cid:105) = (0 , T . The wavevector is k e/hσ = (cid:112) m ( µ ± ( (cid:15) − σE Z )). For theproximitized SC state at | z | > L/
2, the eigenstates with eigen-energy (cid:15) is give by φ ± eσ ( z ) = 1 √ q eσ [( (cid:15) − σE Z ) / ∆ − / (cid:18) e iη eσ / e − iη eσ / (cid:19) ⊗ | σ (cid:105) e ± iq eσ z , (S5)and φ ± hσ is given by replacing the index e to h in Eq. (S5). Here η e/hσ = χ ± η σ , and η σ = arccos (cid:15) − σE Z ∆ . Notice thatfor t >
1, arccos t = − i ln( t + √ t − q e/hσ = [2 m ( µ ± (cid:112) ( (cid:15) − σE Z ) − ∆ )] / , and Re q e/hσ > q eσ >
0, Im q hσ <
0. The wavefunctions are normalized to have same current.We are ready to evaluate the scattering matrix. To simplify the problem, it is easy to observe that the spin isconserved during the scattering. So let’s focus on σ = 1 and neglect the index σ . By matching the wavefunctionEq. (S4) and Eq. (S5) at the superconducting-normal state interface at z = ± L/
2, the scattering matrix is S ± L/ = e ± iχ/ τ z Se ∓ iχ/ τ z , where S = e − iφ S (cid:18) re iφ N √ − r √ − r − re − iφ N (cid:19) , (S6)where r = r n /r d is normal reflection amplitude. r n and r d are positive numbers, and r d e iφ S = − e − iη ( k h − q e )( k e − q h ) + e iη ( k e + q e )( k h + q h ) , (S7) r n e iφ N = − e − iη ( k h − q e )( k e + q h ) + e iη ( k e − q e )( k h + q h ) . (S8) � ���� � / � - � / � � / � - � / � Δ � � χ / � Δ � - � χ / � S SSemiconductorLayers
FIG. S1. Schematic plot of Josephson junction.
The transmission amplitude is simply T = (cid:18) e ik e L e − ik h L (cid:19) . (S9)Using the scattering and transmission matrix, the spectrum of the bound state is determined by [29] det(1 − S − L/ T S + L/ T ) = 0, i.e.,cos[( k e − k h ) L − φ S ] = (1 − r ) cos χ + r cos[( k e + k h ) L + 2 φ N ] . (S10)To proceed, we assume weak pairing E Z (cid:46) ∆ (cid:28) µ , then we have r = ∆ / (2 µ ) + O (∆ /µ ), φ S = η , and φ N = 0.The zero mode is given by arccos E Z ∆ = ˜ χ π E Z E T + nπ, (S11)where ˜ χ is a function of χ : ˜ χ ( χ ) = arccos (cid:16)(cid:2) − ∆ (2 µ ) (cid:3) cos χ + ∆ (2 µ ) cos 2 k F L (cid:17) , (S12)and E T ≡ ( π/ v F /L is the Thouless energy of the junction. Here k F = √ mµ and v F = (cid:112) µ/m . Note the domainwall width is of order of lattice constant, as a result, the Thouless energy is of order of chemical potential, E T ∼ µ ,which is the largest energy scale in the question.As we will show in the next Section, the appearance of zero mode implies a gap closing that separates trivial andtopological superconducting phases. At the zeroth order of ∆ /µ , there is no normal reflection and ˜ χ = χ , the phasediagram is simply given by E Z ∆ = | cos χ | , as shown by the dashed line in Fig. 2. It is interesting to see that at χ = π ,an infinitesimal magnetic field can turn the system into a chiral topological superconducting phase.Including correction from the normal reflection to the first order of ∆ /µ , the relation between χ and ˜ χ is given inEq. (S12) and the phase diagram is corrected to the solid line in Fig. 2. At χ = 0 and χ = π , the correction is in thelinear order of ∆ /µ , i.e., ˜ χ ( χ = 0) = ∆ /µ sin k F L , and ˜ χ ( χ = π ) = π − ∆ /µ cos k F L , respectively. While at othergeneric values of χ , the correction is in the quadratic order of ∆ /µ , i.e., ˜ χ ( χ ) = χ + (∆ /µ ) (cos χ − cos 2 k F L ) / sin χ .It is useful to note that when the normal reflection is considered, the critical Zeeman energy at χ = π is given by∆ /µ , which is a still a small number at weak pairing. B. Spin-orbit coupling and topological phase transition
It is well known that spin-orbit coupling is crucial in the realization of topological phases. Direct semiconductorswith an inversion-asymmetric zinc blende structure (point group T d ), such as GaAs, InSb, CdTe etc, have considerablesize of spin-orbit coupling and are common in making quantum structures. Moreover, many of these materials havea similar band structure with the smallest gap at Γ point. To give a concrete example of spin-orbit couplings, weconsider a heterostructure consisting of a few layers of such semiconductors grown in (001) direction. Because of thephase difference in the Josephson junction, the heterostructure breaks inversion symmetry. However, such a structure k z ϵ k ψ ψ ●● FIG. S2. The dispersion of k x = k y = 0 subspace along k z direction. Because in k x = k y = 0 subspace the SOC vanishes, theband is doubly degenerate. We denote left/right mover by ψ / , respectively. inversion asymmetry cannot induce Rashba spin-orbit coupling, since the system is invariant under the composite S and time reversal operations. As a result, the symmetry allowed spin-orbit coupling is the Dresselhouse term [31, 32] H SOC = α ( k x σ x − k y σ y ) , (S13)where α is the strength of spin-orbit coupling.In the following, we will show that the gap closing in the pervious Section is described by two-dimensional Majoranacone. To simplify the question, we set the width of domain wall L to be zero, and neglect the normal reflection. Inthe k x = k y = 0 subspace of the semiconductor (without proximitized SC order), the dispersion along k z direction isshown in Fig. S2. We make a continuous model near the Fermi point, i.e., ψ , , for right and left movers. Because theproximitized SC order couples the Kramers pair, we first analyze the Kramers pair ψ , ↑ and ψ , ↓ with kinetic termgiven by one-dimensional Dirac Hamiltonian H = − iv F ∂ z σ z .Owing to the proximitized SC order, the BdG Hamiltonian in Nambu space Ψ = ( ψ , ↑ ,z , ψ , ↓ ,z , ψ † , ↑ ,z , ψ † , ↓ ,z ) T isgiven by (note the basis difference in Eq. (S2)), H BdG ( k x , k y , z ) = H ( z ) + H ( k x , k y ), where H ( z ) = − iv F ∂ z σ z − ∆ (cid:48) ( z ) σ y τ y − ∆ (cid:48)(cid:48) ( z ) σ y τ x + E Z σ z τ z , (S14) H ( k x , k y ) = k x + k y m + α ( k x σ x − k y σ y τ z ) , (S15)and ∆ (cid:48) ( z ) = ∆ cos( χ/
2) and ∆ (cid:48)(cid:48) ( z ) = ∆ sin( χ/ z ) refer to the real and imaginary part of SC order given inEq. (S1). Here, we have separated the Hamiltonian into two parts, H and H , which will be treated as unperturbedpart and perturbation, respectively.In H , because of the domain wall formed in ∆ (cid:48)(cid:48) ( z ), similar to domain wall in one-dimensional Dirac Hamilto-nian [33], we get two related bound states that are related by particle-hole transformation, ψ + ( z ) = f ( z )(1 , , , − T , ψ − ( z ) = f ( z )(0 , , − , T , (S16)where f ( z ) = ( N ) − / exp[ − (cid:82) z dz (cid:48) ∆ (cid:48)(cid:48) ( z (cid:48) ) v F ], and N is the normalization factor. The eigen-energies of these boundstates are E ± = ± ( E Z − ∆ cos χ/ E Z / ∆ = cos χ/ H to the bound state ψ ± . Using first-order perturbation theory, it is straight-forward to get the effective Hamiltonian in the bound states subspace ( ψ + , ψ − ) T , H eff = (cid:18) δm − α ( k x + ik y ) − α ( k x − ik y ) − δm (cid:19) , (S17)where the mass is δm = E Z − ∆ cos χ/
2. At critical point δm = 0, the Hamiltonian describes a Majorana cone,indicating a topological transition. Namely, the gap closing realizes a transition from a trivial to a topologicalsuperconductor.Similar analysis can be applied to the other Kramers pair ψ , ↓ and ψ , ↑ in Fig. S2, and the transition appearsat E Z / ∆ = − cos χ/
2. Combined with above results, the phase boundary is E Z / ∆ = | cos χ/ | , consistent with theresults from the previous Section. C. Topological gap
In previous sections, we have analyzed the spectrum at k x = k y = 0. Away from the transition point, the gapat k x = k y = 0 is in the order of ∆. Here, we analyze the topological gap at generic k x , k y , especially, when k x , k y (cid:29) E Z /α . Since we are interested in αk (cid:29) E Z , k = (cid:113) k x + k y , we neglect the Zeeman energy, and theHamiltonian is H BdG ( E Z = 0) = (cid:16) k x + k y − ∂ z m − µ + H SOC ( k x , k y ) (cid:17) τ z + (∆( z ) τ + + H.c. ) . (S18)where the proximitized SC order is again given by Eq. (S1). In the following, we will use the same notation in SectionA. For the normal state in | z | < L/
2, the eigenstates with eigen-energy (cid:15) are given by ψ ± eσ ( z ) = 1 (cid:112) k eσ (cid:18) (cid:19) ⊗ | σ (cid:105) e ± ik eσ z , ψ ± hσ ( z ) = 1 (cid:112) k hσ (cid:18) (cid:19) ⊗ | σ (cid:105) e ± ik hσ z , (S19)where σ = ± | + 1 (cid:105) = 1 / √ e iφ k / , e − iφ k / ) T and | − (cid:105) = 1 / √ e iφ k / , − e − iφ k / ) T , φ k = arctan k y /k x . Thewavevector is k e/hσ = (cid:112) m ( µ − E kσ ± (cid:15) ), where E kσ = k / m − σαk . For the proximitized SC state in | z | < − L/ (cid:15) is give by φ ± eσ ( z ) = 1 √ q eσ [ (cid:15) / ∆ − / (cid:18) e iη e / e − iη e / (cid:19) ⊗ | σ (cid:105) e ± iq eσ z , (S20)and φ ± hσ is given by replacing the index e to h in Eq. (S20). Here η e/h = χ ± η , and η = arccos (cid:15) ∆ . The wavevector is q e/hσ = [2 m ( µ − E kσ ± √ (cid:15) − ∆ )] / , and Re q e/hσ >
0, Im q eσ >
0, Im q hσ < z = ± L/
2, we can obtain the scattering matrix, and then getthe bound state spectrum. The spectrum is determined byarccos (cid:15) ∆ = ( k eσ − k hσ ) L ± χ , (S21)here we have neglected the normal reflection which is a small correction in the weak pairing. To proceed, we againassume the weak pairing ∆ (cid:28) µ , and get (cid:15) ∆ = cos (cid:104) χ ± k F L (cid:112) − E kσ /µ ∆ µ × (cid:15) ∆ (cid:105) . (S22)The dependence on k is suppressed by ∆ /µ . Thus, to the lowest order, the gap is given by (cid:15) = ∆ cos χ/ k increases such that E kσ ∼ µ , above approximation may break down. Nevertheless, notice that thebound state exists when k e/hσ is a real number, i.e., E kσ ≤ µ − (cid:15) . At E kσ = µ − (cid:15) , we have (cid:15) ∆ = cos (cid:104) χ ± k F L (cid:115) ∆ µ × (cid:114) (cid:15) ∆ (cid:105) , (S23)where the suppression is sublinear in ∆ /µ . At generic momentum, the topological gap is (cid:15) = ∆ cos χ up to a smallcorrection of ∆ /µ/µ