Supplementary Materials

Transcription

1 Supplementary Materials Contents I. Experimental Methods and Supplementary Data 1 A. Setup of the Sagnac interferometer 1 B. RHEED patterns of the substrate and the sample C. Supplementary data of the Bi(5 nm)/ni( nm) sample D. Thickness dependence on both Bi and Ni layer of the critical temperature 3 E. Kerr signal of the Bi(40 nm)/ni( nm) sample 4 F. Optical signal from the MgO substrate 5 II. Theory of Superconducting Pairing in Epitaxial Bismuth/Nickel Bilayers 6 A. Introduction 6 B. Description of the model 6 1. Electronic surface states of bismuth 6. Magnetic fluctuations 7 3. Coupling between electronic surface states in Bi and magnetic fluctuations in Ni 7 4. Electron interaction mediated by magnetic fluctuations 8 C. Superconducting gap and pairing symmetry 10 D. Conclusions 11 I. EXPERIMENTAL METHODS AND SUPPLEMENTARY DATA A. Setup of the Sagnac interferometer Fiber Light Source 1550 nm 1 3 PM-Circulator Fiber Polarizer Fiber Phase Modulator PM fiber Optical Detector Collinating & Focusing Optics ω ω Lock-in 1 Lock-in ω Optical Cryostat fig. S1. Schematic of Sagnac interferometer. The Kerr signal is measured using a Sagnac interferometer. Two orthogonal polarizations of light a phase-shift modulated and reflected from the surface of the sample after passing through a quarter-wave plate.

2 B. RHEED patterns of the substrate and the sample fig. S. RHEED patterns of the substrate and the sample. (A) RHEED patterns of annealed MgO(001) substrate. (B) RHEED patterns of the Bi(110)/Ni(001)/MgO(001) sample. C. Supplementary data of the Bi(5 nm)/ni( nm) sample For presentational reasons, in the main text we just show part of the data from the Bi(5 nm)/ni( nm) sample. In this section, we provide supplementary data of that sample. Because the critical temperature is around 4.1 K, we only collected signal below 7.0 K. As shown in fig. S3A, the signal between 4.1 K and 7.0 K is almost constant and the abrupt intensity increase occurs only below the critical temperature. Figure S3B is a zoom-in version of the blue curve in Fig. 1C. We subtract the temperature-independent background of 0.75 µrad contributed by the remanent momentum of Ni and plot the observed signal on the same scale as that of fig. S3A. By comparing these two panels, we can conclude that the spontaneous TRS breaking observed on the Bi side is not due to any unknown transition of Ni layer around 4.1 K.

3 3 A B Bi Ni MgO Bi Ni MgO C D fig. S3. Supplementary data of the Bi(5 nm)/ni( nm) sample. (A) Temperature dependence of the Kerr signal detected observed from the Bi side between 3 K and 7 K. (B) Temperature dependence of the Kerr signal observed from the Ni side with the 0.75 µrad background subtracted.(c) Kerr effect θ K measured during zero-field warmup, after cooling down at zero field. (D) Standard deviation σ(θ K ) between experiments that contains two random contributions: σ 0 due to chiral domains and σ app from the apparatus. In fig. S3C, we show other 1 measured curves of the Bi(5 nm)/ni( nm) sample without training. From these curves we can derive that the standard deviation coming from the chiral domains random distribution is around 58 nrad, as shown in fig. S3D. Using the same method as used in main text, we estimate the domain size d 1.1µm at.7 K for this sample. D. Thickness dependence on both Bi and Ni layer of the critical temperature One important feature of this Bi/Ni bilayer system is that the T c can be systematically tuned by changing either the Bi or the Ni layer thickness. In the fig. S4, we present several samples critical temperatures [11], which is defined when the film resistance drops to half of its normal resistance. The highest T c is around 4.1 K (e.g. when the sample thickness is Bi 5nm/Ni nm). If we increase the Ni thickness with keeping the Bi thickness fixed, the T c will be significantly suppressed, as shown in fig. S4A. But if we fix the Ni layer and increase the Bi layer thickness, the T c increases back close to 4.1 K and then slightly decreases, as shown in fig. S4B. This feature provides the direct evidence that although the Cooper pairs are triggered at the interface, but survive in the Bi interior or near the top surface. Thus the possibility that the superconductivity in Bi/Ni comes from any Bi-Ni compound (e.g. Bi 3 Ni) at the interface can be naturally excluded. This unique thickness dependence has never been found in any other superconducting heterostructure. A possible explanation is that (1) superconductivity is triggered at the interface via ferromagnetic fluctuation, as the theory of this paper suggests; () because of finite size effect, the exchange field at

4 the interface will be significantly enhanced as the Ni layer grown thicker, and thus become strong enough to suppress the pairing near the interface; (3) growing additional Bi on the top actually provides spare space for Cooper pairs to survive near the top surface that is relatively far away from the ferromagnetic interface. This assumption is consistent with the well known unusually long Fermi wavelength ( 30 nm) and extremely long mean free path of electrons (up to mm) in Bi, as well as extremely long decay of the inter-surface interaction in Bi films. 4 A B fig. S4. Dependence of the critical temperature T c on Bi and Ni layer thicknesses. E. Kerr signal of the Bi(40 nm)/ni( nm) sample Because a small increase in the Ni layer thickness would significantly suppress the critical temperature, it is not easy to observe such small signal in samples with thick Ni layers. Therefore, in this experiment we changed the thickness of the Bi layer and fixed the thickness of the Ni layer. In fig. S5, we show the temperature dependence of the Kerr signal of a Bi(40 nm)/ni( nm) sample. The spontaneous TRS breaking happens around 3.7 K, slightly lower than that of the Bi(5 nm)/ni( nm) sample and the Bi(0 nm)/ni( nm) sample. This is consistent with our transport measurements results shown in fig. S4 and also consistent with the pairing picture we proposed in this report. fig. S5. Kerr Signal of the Bi(40 nm)/ni( nm) sample.

5 5 F. Optical signal from the MgO substrate The fig. S6A is a back view of the Bi(5 nm)/ni( nm) sample. After polishing, the 0.5 mm thick MgO substrate becomes completely transparent. To confirm that the observed Kerr signal is irrelevant to the MgO substrate, we measured a 0.5 mm thick pristine MgO substate only coated with nonmagnetic Au reflection layer, as shown in fig. S6B. We cooled the substrate to the lowest temperature in a field of 3000 Oe and measured it upon warming in zero field. The data points are plotted on the same scale as that of Fig. 1B and no sign of spontaneous TRS breaking was observed. A B Au MgO fig. S6. Sample picture and Optical signal from the MgO substrate. (A) Back view of the Bi(5 nm)/ni( nm) sample. (B) Optical signal from the an empty MgO substrate coated with nonmagnetic Au reflection layer.

6 6 II. THEORY OF SUPERCONDUCTING PAIRING IN EPITAXIAL BISMUTH/NICKEL BILAYERS A. Introduction In order to understand the experimental observation of superconductivity in epitaxial Bi/Ni bilayers that breaks time-reversal symmetry spontaneously, we adopt a simple model incorporating the exchange coupling between magnetic moments of nickel (Ni) and itinerant electron states of bismuth (Bi). Due to strong spin-orbit coupling, the surface states of Bi(110) are basically described by nondegenerate Fermi pockets with the largest one centered around the Γ point of the surface Brillouin zone [18-]. The spin and momentum are strongly locked on the Fermi surface as in the surface states of 3D topological insulators. Without loss of generality, here we only consider the Fermi surface around Γ. As we will discuss below, the spin-orbit coupling in Bi and magnetic fluctuations arising from Ni cooperatively induce pairing channels classified by definite total angular momentum J z. The superconducting order parameter breaks the time-reversal symmetry selecting either J z = + or J z =, which can be controlled by a weak training magnetic field applied perpendicular to the surface. B. Description of the model Details of the surface states in Bi/Ni epitaxial bilayers are not known exactly at this time, so we use a simple minimal model to describe a hybrid system of a ferromagnet Ni and the surface states of Bi. The corresponding action consists of three terms: S = S e + S M + S em, (1) where the action S e describes the electronic surface states of Bi coupled to the in-plan magnetization of Ni, S M includes the magnetic fluctuations produced by Ni, and S em accounts for the coupling between magnetic fluctuations and electron surface states of Bi. We describe each term in details below. 1. Electronic surface states of bismuth There are indications [17] that the bulk states in a Bi film are gapped, and the only metallic states are the surface states. Moreover, we only consider the Bi surface exposed to vacuum, because the surface exposed to Ni does not contribute to superconductivity [11]. We consider the Fermi pocket enclosing the center of surface Brillouin zone Γ on the (110) surface of Bi. Due to strong spin-orbit coupling, the electron spin and momentum are strongly locked on this pocket, bearing resemblance to the Dirac electron states at the surface of 3D topological insulators; see Fig. 3(a) for a schematic representation. Therefore, for the (x,y) surface of Bi, we consider the following model describing the helical nature of Dirac electrons subject to the in-plane magnetic field generated by Ni. The corresponding action reads as S e = [ ] dx dy dτ Ψ τ + v F (ˆk σ) z µ F M σ Ψ, Ψ = (Ψ, Ψ ) T, () where σ is a vector of Pauli matrices representing spin, v F and µ F > 0 stand for the Fermi velocity and Fermi energy, respectively, and ˆk = i is the momentum operator. The last term in the action above describes coupling of the electron spin to the in-plane magnetization of nickel M = Mn y where n y is the unit vector along y direction. We eliminate this term from the action () by making the gauge transformation Ψ = e iw r Ψ of the electron fields, which corresponds to redefinition of the momentum p = k w, where w = (M/v F )n x. The effect of such transformation is just a shift of the Fermi surface in the momentum space as shown in Fig. 3(a): The center of the Fermi surface is shifted from Γ point to Γ by the vector w. The fact that increasing Ni thickness suppresses superconductivity [11] leads us to conclusion that the in-plane magnetic field is not strong enough to fully polarize the spin texture of the Fermi surface. The helical spinor eigenstate near the Fermi energy in the shifted basis is p = 1 1 ie iφp, ε(p) = v F p, (3)

7 where φ p is the azimuthal angle indicating the direction of the momentum p as shown in Fig. 3(a) geometrically, so tan(φ p ) = p y /p x. It follows from the same geometry that φ p = φ p + π. The time-reversal partner of spinor (3) is produced by the time-reversal symmetry operation Θ = iσ y K, where K is complex conjugation. It is a spinor belonging to momentum p as follows p = Θ p = η p p, η p = ie iφp. (4) The phase factor η p can not be gauged away due to Θ = 1. We use eigenstate (3) to project the original spin-full electron operators, i.e., Ψ pα, into the effective spinless electron operators ψ p creating/annihilating electrons in the band near the Fermi energy. The transformation is given as follows Ψ p Ψ p 1 1 ie iφp ψ p = p ψ p. (5) An immediate consequence of relation (4) is that the corresponding time-reversal partner [3] of the effective spinless fermion operator ψ p is ψ p = Θψ p Θ 1 = η pψ p. (6) Following Ref.[3], we use fermionic fields ψ p and ψ p to construct superconducting pairing between the states that are time-reversal partners of each other. It is also straightforward to see that the original electron fields Ψ pα are projected to (6) as follows Ψ p Ψ p p ψ p = 1 ie iφp 1 ψ p = p ψ p. (7) 7. Magnetic fluctuations The important role played by the nickel thin film would be to generate a fluctuating magnetic field. We assume that the ferromagnet moments of Ni are locally aligned along a unit vector n(r, τ). Since the critical temperature of superconductivity is well below the ferromagnetic critical temperature (unlike in the ferromagnets with very low critical temperature such as UCoGe [31], the longitudinal fluctuations of the Ni ferromagnetic moments are not essential. Therefore, we decompose n n y + l, where l describes the transverse fluctuations in (x,z) plane with n y l = 0, which are essential to our discussion below. The low-energy action of these ferromagnet fluctuations is [9] S M = ρ s dx dy dτ [ i(l τ l) z + κ( l) ], (8) where ρ s is the magnetic moment density in Ni, and κ characterizes the spin waves. Precise form of the action is not crucial to our discussion below. Thus, we assume a general form in momentum space as follows S M = 1 β D 1 (q)b qb q, (9) q where b q is creation operator of bosonic fields describing the fluctuations (specifically the out-of-plane fluctuations as discussion below), and D(q) = b( q)b(q) is the propagator of bosonic fields. We use the four-momentum notation q = (q, q n ), where q n = nπ/β with β = 1/T, and the sum runs over momentum q and Matsubara frequencies q n throughout. 3. Coupling between electronic surface states in Bi and magnetic fluctuations in Ni We assume that the helical surface states of Bi interact with the magnetic fluctuations in Ni via a minimal exchange coupling as follows S em = g dx dy dτ Ψ (l σ)ψ, (10)

8 where g stands for the exchange coupling between electrons and magnetic fluctuations. While the fluctuations of the magnetic field created by magnetic moments can have both in-plane (l x ) and out-of-plane components (l z ), here we only explore the effects caused by the out-of-plane fluctuations, i.e., l = b(r, τ)z. In particular, we show that such fluctuations are responsible for pairing with definite total angular momenta J z of the Cooper pairs. The in-plane (l x ) fluctuations may give rise to more exotic superconducting states with Amperean pairing [9], which we do not explore here. In the momentum space, the coupling to the out-of-fluctuations reads as where S em = 1 β [ bq s q + b ] qs q, (11) q 8 s q = g p Ψ pσ z Ψ p+q, s q = g p Ψ pσ z Ψ p q. (1) 4. Electron interaction mediated by magnetic fluctuations The magnetic fluctuations coupled linearly to electrons, as represented in (11), mediate an effective spin-spin interaction between conducting electrons. Integrating out the magnetic fluctuations, the latter reads as S int = g Aβ q,p,p D(q) Ψ p qσ z Ψ p Ψ p +q σz Ψ p, (13) where A is the area of the system, and the fermion four-momentum notation p = (p, p n ), where p n = (n + 1)π/β, is used. To simplify presentation, from now on we ignore the frequency dependence of bosonic propagator and only consider its momentum dependence, i.e., D(q, q n ) = D(q). The appearance of only σ z in the effective interaction (13) is due to the fact that we only considered the out-of-plane magnetic fluctuations. We argue that this interaction leads to the electron pairing in the Cooper channel with the appropriate symmetry relevant to the experiments. Superconductivity sets in by condensing the pairs of electrons with momenta p and p on the shifted Fermi surface as shown in Fig. 3(b). However, note that, with respect to the original momentum coordinates k, the Cooper pairs have a non-zero center-of-mass momentum w resembling the famous Fulde-Ferrel (FF) [3] state, as opposed to Larkin- Ovchinnikov (LO) [33] state in the presence of a parallel magnetic field [34-38]. Our focus, here, is on classification of the paired states formed on the shifted Fermi surface with zero net momentum. Rearranging the electron momenta to the Cooper pairing channel, the interaction takes the following form S int = 1 Aβ V αβγδ (p p )Ψ p γ Ψ p α Ψ pβψ pδ, (14) p,p αβγδ V αβγδ (p p ) = g D(p p )σ z αβσ z γδ. (15) The interaction (14) involves spinfull states. Using projections described by (5) and (7), the expression for interaction between spinless fermions, i.e., ψ p and ψ p, reads as S int = 1 U(p, p ) Aβ ψ p ψ p ψ ψ p p, (16) p,p U(p, p ) = 1 (K p,p + K p,p ), (17) K p,p = g D(p p )Λ s p,p, (18) where the vertex function Λ s p,p is Λ s p,p = p σ z p p σ z p = 1 ( [ 1 + cos(φ p φ p )] = sin φp φ p ). (19)

9 Notice that if we were to consider the charge fluctuations instead of spin fluctuations, the σ z in (13) has to be replaced with identity, which then leads to the following expression for the vertex function Λ c p,p = p p p p = 1 ( ) [1 + cos(φ p φ p )] = cos φp φ p. (0) For momenta near the Fermi surface the propagator D(p p ) depends on the p p cos(φ p φ p ), implying that it can be expanded in terms of cylindrical harmonics as follows D(p p ) = D m cos(m(φ p φ p )), (1) m=0 where D m are coefficients of expansion. In expression (16) the interaction kernel U(p, p ) has been symmetrized, which is traced back to the anti-commutation relation of electron fields, i.e., ψ p ψ p = ψ p ψ p. The latter implies that upon flipping the momentum p p (φ p = φ p + π) the operator product ψ p ψ p remain unchanged due to η p = η p : ψ p ψp = η pψ p ψ p = η pψ p ψ p = η pη p ψ p ψ p = ψ p ψ p. () Therefore, in the expansion of U(p, p ) only the even harmonics survive: U(p, p ) = g [ m=even D m cos(m(φ p φ p )) cos(φ p φ p ) m=odd D m cos(m(φ p φ p )) ] 9, (3) implying that a typical term in the interaction has a separable structure in terms of harmonics, i.e. we can rewrite it in a compact form as follows U(p, p ) = m=even U m χ m (p)χ m(p ), χ m (p) = e i mφp, U m = U m, (4) where U m recollects the corresponding coefficients. The harmonics appearing in the interaction (4) has important impact on the pairing symmetry, which is the subject of the next section. Of particular importance is the lowest order harmonics given as where U(p, p ) u s + u s cos((φ p φ p )), (5) ( u s = g D 0 D ) ( 1, u s = g D D ) 1. (6) For charge density fluctuations with vertex function (0) we obtain the same expression (5) for the effective interaction but with different coefficients as ( u c = g D 0 + D ) ( 1, u c = g D + D ) 1. (7) Given a mechanism of superconductivity, the pairing instability in a particular Cooper channel occurs when the corresponding coefficient in (6) or (7) becomes negative giving rise to an attractive interaction between electrons. In particular, the coefficients u s and u s, or u c and u c for charge fluctuation-mediated interaction, in (5) control the magnitude of pairing in m = 0 and m = ± channels, respectively. The spectrum of magnons has an energy gap ξ due to uniaxial magnetic anisotropy [39]. The corresponding propagator reads as D(q) 1/(κ q + ξ). Thus, the coefficients D m in the expansion (1) are positive and decrease by increase of m, namely D 0 D 1 D > 0. This hierarchy of energy scales implies that u c u c 0 for charge fluctuations and u s 0 and u s 0 for spin fluctuations. Therefore, the pairing instability in m = 0 channel is dominant for charge fluctuations. On the other hand, the spin fluctuations favor the pairing instability in m = ± channels. These latter harmonics play a crucial role in our classification of pairing symmetry below. This is the reason behind the importance of the magnetic fluctuations as a mechanism of superconductivity in the Bi/Ni system. It was shown in [8] that phonons cannot possibly produce time-reversal-breaking superconductivity. Moreover, the absence of superconductivity in the bilayers with nonmagnetic substrates [11] further supports this idea.

10 10 C. Superconducting gap and pairing symmetry To determine the pairing symmetry of the superconducting state, let us introduce condensate of spinless fermions between electron fields ψ p and its time-reversal partner ψ p using the Gor kov anomalous function [3] f(p, τ τ ) = T τ [ψ p (τ) ψ p (τ )]. (8) Notice that f(p, τ τ ) = f( p, τ τ ); see the relation in (). Then, effective interaction (16) gives the mean-field gap equation as follows [ (p ) = 1 U(p, p ) T ] f(p, p n ), (9) A p n where f(p, p n ) stands for the Fourier transform of f(p, τ τ ) to Matsubara frequencies and has the following expression: f(p, p n ) = (p) p n + ε p + (p). (30) The symmetry of the pairing wave function can be inferred from harmonics appearing in the effective interaction. Indeed, the form of interaction (4) implies the pairing gap and anomalous function have a phase dependence with winding around the Fermi surface in a given Cooper channel m: m (p) e imφp, f m (p, p n ) e imφp, m = even integer. (31) The pairing gap is an even function of momentum m (p) = m ( p), as discussed for noncentrosymmetric superconductors with strong spin-orbit coupling [3,30]. Notice that if we used a condensate comprised of pairing between electron fields ψ p and ψ p, the spineless pairing ψ p ψ p = η p ψ p ψp e i(m 1)φp, m = even integer (3) would have odd parity [6,7]. The pairing between time-reversed states as used here, however, is more convenient and symmetric [40]. This completes the classification of the pairing symmetry of the spinless fermions constituting the orbital angular momentum part of the condensate of spinfull system. The latter is described as the anomalous function in original basis F αβ (p, τ τ ) = T τ [Ψ pα (τ)ψ pβ (τ )]. (33) Using relations in (5) and (7), we express the condensate (33) in terms of (8). It reads as F (p) F (p) F (p) F (p) = 1 ie iφp 1 1 ie iφp where F αβ (p) = F αβ (p, 0 + ) and f(p) = f(p, 0 + ) should be understood as f(p), (34) F αβ (p, 0 + ) = T p n F αβ (p, p n ), f(p, 0 + ) = T p n f(p, p n ). (35) Our objective now is to show that the superconducting order parameter in the original basis can be classified by total angular momentum. That is to show that all components of the pairing matrix, i.e., F αβ, have the same J z = L z + S z where L z (S z ) stands for z component of the orbital angular momentum (spin) of the pair. Pairing electrons in a given Cooper channel labeled by an even integer m, we obtain (i) for F the orbital angular momentum is L z = m 1 and S z = 1 giving rise to J z = m; (ii) for F we have L z = m + 1 and S z = 1 giving rise to J z = m; and (iii) for off-diagonal components F and F we obtain L z = m and S z = 0 leading to J z = m. Hence, the pairing states are classified by the total angular momentum J z = m. The states J z = m and its time

11 reversal partner J z = J z are degenerate, so the time reversal symmetry can be broken spontaneously by applying a weak training magnetic field. We argue that the latter situation can be achieved by taking into account the lowest harmonics. The interaction (5) contains m = 0 and m = ± harmonics. The harmonic m = 0 leads to a superconducting order parameter with J z = 0. The second harmonics, however, give rise to a richer structure. Indeed, the harmonics m = ± lead to superconducting order parameters with opposite total angular momentum: J z = for m = and J z = + for m = +. The angular momentum flips sign under time reversal: J z J z. Therefore, the second term in the interaction (5) paves the way for time reversal symmetry to be broken spontaneously. Upon switching the direction of the training magnetic field the total angular momentum J z changes sign as shown schematically in Fig. 3(b). The angular momenta m = ± discussed here correspond to winding numbers m 1 = 3 and m 1 = 1 derived in Ref.[6] for a spin-orbit coupled system with purely repulsive Hubbard interaction due to different choice of basis for the condensate as elaborated in (3). However, the latter model yields even integer values J z in the basis we used here [5]. 11 D. Conclusions In this work we showed that in the presence of spin-orbit coupling and spin-momentum locking the superconducting states are classified by the total angular momentum J z = L z + S z which can only take even integer values. Strictly speaking, we showed that the superconducting order parameter can be labeled as J z = m with m an even integer. The superconducting state discussed in Ref.[4] has J z = 0 and is time-reversal invariant. This state arises from the interaction coefficient u c in (7). The lowest-order state that can break time-reversal symmetry is J z = ±. We showed that the J z = ± state is the case for superconducting state observed in Bi/Ni epitaxial bilayer. This has an important implication for understanding the spontaneously broken time reversal symmetry phenomena observed in this experiment. The effective interaction between electrons mediated by magnetic fluctuations of Ni admits superconducting states with nonzero J z = + and J z =, which are time-reversal partners of each other, in an attractive interaction channel. These states correspond to superconducting states with d ± id symmetries. As shown in Fig. 3(b), a weak training field H training drives the system to a chiral state. One remark here is in order. We considered a continuum model for the surface states of Bi that is invariant under the orthogonal group of rotations G = O(). The real surface of Bi(110) has a lower symmetry group [18]. The only symmetry element is a mirror plane [1], which together with C 1 rotation (π rotation about the normal to the surface) form the dihedral group D 1. The latter has two one-dimensional representations A 1 and A with a basis function 1 and xy, respectively. The A 1 representation yields a symmetric basis function with respect to mirror reflection, while the basis function xy of A representation is anti-symmetric. Examples of these basis could be s and d x y orbitals for symmetric basis and d xy for the anti-symmetric one. The latter representation necessarily must have nodes. Therefore, it is possible to make two degenerate time-reversal-breaking combinations d xy ± id x y. However, because the pairing states with d xy and d x y symmetries belong to different representations, they are not degenerate and, hence, their corresponding transition temperatures are necessarily different. Therefore, our theory predicts that, on lowering temperature, there may be a superconducting phase transition first to the d xy state with nodes and then to a time-reversal-breaking state d xy ± id x y at a lower temperature, eliminating the nodes to gain condensation energy, as in UPt 3 [4]. The transition temperatures could be very close to each other and their diagnoses could be a direction for future study. There is great current interest in realization of topological superconductivity and Majorana fermions. The superconducting Bi/Ni bilayers offer several advantages compared with other proposals in the literature [35,41]. First, subject to confirmation by spin-polarized angle-resolved photoemission measurements, it is likely that the Bi surface states in Bi/Ni bilayers have non-degenerate Fermi surfaces due to strong spin-orbit interaction. Second, superconductivity is intrinsic, so an external s-wave superconductor is not needed for proximity effect. Third, superconductivity spontaneously breaks time-reversal symmetry and is topologically non-trivial with J z = ±, so there are two chiral edge states [3] moving in the same direction.