Cubic spin-orbit coupling and anomalous Josephson effect in planar junctions
CCubic spin-orbit coupling and anomalous Josephson effect in planar junctions
Mohammad Alidoust, Chenghao Shen, and Igor ˇZuti´c Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway University at Buffalo, State University of New York, Buffalo, NY 14260-1500, USA (Dated: January 22, 2021)Spin-orbit coupling in two-dimensional systems is usually characterized by Rashba and Dressel-haus spin-orbit coupling (SOC) linear in the wave vector. However, there is a growing class ofmaterials which instead support dominant SOC cubic in the wave vector (cSOC), while their super-conducting properties remain unexplored. By focusing on Josephson junctions in Zeeman field withsuperconductors separated by a normal cSOC region, we reveal a strongly anharmonic current-phaserelation and complex spin structure. An experimental cSOC tunability enables both tunable anoma-lous phase shift and supercurrent, which flows even at the zero-phase difference in the junction. Afingerprint of cSOC in Josephson junctions is the f -wave spin-triplet superconducting correlations,important for superconducting spintronics and supporting Majorana bound states. Spin-orbit coupling (SOC) and its symmetry break-ing provide versatile opportunities for materials designand brining relativistic phenomena to the fore of the con-densed matter physics [1–6]. While for decades SOC wasprimarily studied to elucidate and manipulate normal-state properties, including applications in spintronics andquantum computing [7–15], there is a growing interest toexamine its role on superconductivity [16–21].Through the coexistence of SOC and Zeeman field,a conventional spin-singlet superconductivity can ac-quire spin-dependent long-range proximity effects [20, 23,24, 96] as well as support topological superconductiv-ity and host Majorana bound states, a building blockfor fault-tolerant quantum computing [25–27]. In bothcases, Josephson junctions (JJs) provide a desirable plat-form to acquire spin-triplet superconductivity throughproximity effects [28–38]. In contrast, even seeminglywell-established intrinsic spin-triplet superconductivityin Sr RuO [39] is now increasingly debated [40, 41].Extensive normal-state studies of SOC in zinc-blendeheterostructures usually distinguishing the resultingspin-orbit fields due to broken bulk inversion symme-try, Dresselhaus SOC, and surface inversion asymmetry,Rashba SOC, and focus on their dominant linear depen-dence in the wave vector, k [10, 15]. In this linear regime,with a matching strengths of these SOC it is possible tostrongly suppress the spin relaxation [42] and realize apersistent spin helix (PSH) [43, 44] with a controllablespin precession over long distances [45–47].While typically k -cubic SOC contributions (cSOC) inheterostructures are neglected or considered just detri-mental perturbations, for example, limiting the stabil-ity of PSH [45–47], a more complex picture is emerging.Materials advances suggest that such cSOC, shown inFig. 1(a), not only has to be included, but may also dom-inate the normal-state properties [48–57]. However, therole of cSOC in superconducting heterostructures is un-explored. It is unclear if cSOC is detrimental or desirableto key phenomena such as Josephson effect, spin-tripletsuperconductivity, or Majorana bounds states. To address this situation and motivate further cSOCstudies of superconducting properties, we consider JJsdepicted in Fig. 1(b), where s -wave superconductors (S)are separated by a normal region with cSOC which isconsistent with the two-dimensional (2D) electron or holegas, confined along the z-axis [48, 53]. We find that theinterplay between Zeeman field and cSOC results in ananomalous Josephson effect with a spontaneous supercur-rent. While the commonly-expected current-phase rela-tion (CPR) is I ( ϕ ) = I c sin( ϕ + ϕ ) [19, 58], where I c is the JJ critical current and ϕ the anomalous phase( ϕ (cid:54) = 0 , π ), we reveal that CPR can be strongly anhar- (b)(a) S ScSOC !" h !$ s (c) !(' ! )' " ' !(' ! )' " ' wd FIG. 1. (a) Spin-orbit fields in k-space for Rashba cubicspin-orbit coupling (cSOC) ( α c = − β c = −
1, middle), and both ( α c = β c = −
1, bottom).(b) Schematic of the Josephson junction. The middle re-gion hosts cSOC and an effective Zeeman field, h , betweenthe two s -wave superconductors (S). (c) Spin textures in thecSOC region resulting from the normal-incident electrons within-plane spin orientations [see Fig. 1(b)] when S is at normal-state, the upper (lower) panel α c = 1, β c = 0 ( α c = β c = 1).The in-plane spin orientations of the incident electrons φ s arefrom 0 (bottom row) to π/ a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n monic and host Majorana bound states. Instead of the p -wave superconducting correlations for linear SOC, their f -wave symmetry is the fingerprint of cSOC.To study cSOC, we consider an effective Hamiltonian H = 12 (cid:90) d p ˆ ψ † ( p ) H ( p ) ˆ ψ ( p ) , (1)where H ( p ) = p / m ∗ + σ · h + H cSOC ( p ), with momen-tum, p = ( p x , p y ,
0) [see Fig. 1(b)], effective mass, m ∗ ,Pauli matrices, σ , effective Zeeman field, h , realized froman externally applied magnetic field or through magneticproximity effect [6, 59], and cSOC term [48, 49, 53, 54] H cSOC ( p ) = iα c (cid:126) ( p − σ + − p σ − ) − β c (cid:126) ( p − p + σ + + p p − σ − ) , (2)expressed using cSOC strengths α c and β c , for Rashbaand Dresselhaus terms, where p ± = p x ± ip y , and σ ± = σ x ± iσ y . The field operator in spin space is given byˆ ψ ( p ) = [ ψ ↑ ( p ) , ψ ↓ ( p )] T , with ↑ , ↓ spin projections.To describe S regions in Fig. 1(b), we use an s -waveBCS model with a two-electron amplitude in spin-Nambuspace ∆ (cid:104) ψ †↑ ψ †↓ (cid:105) + H.c., given by the effective Hamiltonianin particle-hole space H ( p ) = (cid:18) H ( p ) − µ ˆ1 ˆ∆ˆ∆ † − H † ( − p ) + µ ˆ1 (cid:19) , (3)where µ is the chemical potential and ˆ∆ is a 2 × ψ = ( ψ ↑ , ψ ↓ , ψ †↓ , − ψ †↑ ) T .To calculate the charge current, we use its quantumdefinition where no charge sink or source is present.Therefore, the time variation of charge density van-ishes, ∂ t ρ c ≡ r → r (cid:48) (cid:80) στσ (cid:48) τ (cid:48) [ ψ † στ ( r (cid:48) ) H στσ (cid:48) τ (cid:48) ( r ) ψ σ (cid:48) τ (cid:48) ( r ) − ψ † στ ( r (cid:48) ) H † στσ (cid:48) τ (cid:48) ( r (cid:48) ) ψ σ (cid:48) τ (cid:48) ( r )]. H στσ (cid:48) τ (cid:48) is the componentform of H , with spin (particle-hole) label σ ( τ ), andand r ≡ ( x, y, J = (cid:82) d r { ˆ ψ † ( r ) −→H ( r ) ˆ ψ ( r ) − ˆ ψ † ( r ) ←−H ( r ) ˆ ψ ( r ) } , where H ( r ) is obtained by substitut-ing p ≡ − i (cid:126) ( ∂ x , ∂ y , H operateson. By an exact diagonalization of H , we obtainspinor wavefunctions ˆ ψ l,r,m ( p ) within the left ( x < x > d ) S region and the middle normal region(0 < x < d ) in Fig. 1(b). The wavefunctions and general-ized velocity operators v l,r,mx are continuous at the junc-tions, i.e., ˆ ψ l = ˆ ψ m | x =0 , ˆ ψ m = ˆ ψ r | x = d , v lx ˆ ψ l = v mx ˆ ψ r | x =0 ,and v mx ˆ ψ m = v rx ˆ ψ r | x = d . The spinor wavefunctions aregiven in the Supplmental Material [60].The complexity of H precludes simple solutions andwe evaluate the wavefunctions and supercurrent numer-ically. To reduce the edge effects, we consider Fig. 1(b)geometry with W/d (cid:29) I = 2 | e ∆ | / (cid:126) , where e is the electron charge, and∆ the energy gap in S. The energies are normalized by∆, lengths by ξ S = (cid:126) / √ m ∗ ∆, cSOC strengths by ∆ ξ .The junction length is set at d = 0 . ξ S .To investigate the role of cSOC on the ground-stateJosephson energy, E GS , and the CPR obtained from thesupercurrent I ( ϕ ) ∝ ∂E GS /∂ϕ , we first consider a sim-ple situation with only Rashba cSOC ( α c (cid:54) = 0, β c = 0)and effective Zeeman field h x ( h y = h z = 0). The evo-lution of E GS with | h x | , where its minima are denotedby dots in Fig. 2(a), shows a continuous transition from ϕ = 0 to π state (blue to green dot). For ϕ (cid:54) = 0, E GS minima come in pairs at ± ϕ [69]. The correspondingCPR reveals in Fig. 2(b) a competition between the stan-dard, sin ϕ , and the next harmonic, sin 2 ϕ , resulting in I ( − ϕ ) = − I ( ϕ ). There is no spontaneous current ex-pected in a Josephson junction with SOC, I ( ϕ = 0) = 0,but only I c reversal with h x . Such a scenario of a continu-ous and symmetric 0- π transition is well studied withoutSOC in S/ferromagnet/S JJs due to the changes in theeffective magnetization or a thickness of the magnetic re-gion [70–77].While our previous results suggest no direct cSOC in-fluence on CPR, a simple in-plane rotation of h , h x = 0, h y (cid:54) = 0, drastically changes this behavior. This is shownin Figs. 3(b) where, at fixed | h y | = 2 . I ( ϕ = 0) (cid:54) = 0, and strong anharmonic CPR thatcannot be described by I ( ϕ ) = I c sin( ϕ + ϕ ). Unlike inFig. 3(a), a relative sign between α c and h alters the CPR -1.5 -1 -0.5 0 0.5 1 1.5-0.0700.07 FIG. 2. (a) Josephson energy and (b) associated supercurrentevolution with the superconducting phase difference ϕ . Zee-man field values, h x , are chosen near a 0- π transition. Theother parameters are α c = ± . β c = 0, µ = ∆, h y = 0. -1.5 -1 -0.5 0 0.5 1 1.5-202 0 0.5 1 1.5 2-1012 ExactN=1,Err=0.0995N=2,Err=0.0574N=3,Err=0.0358
FIG. 3. (a) Josephson energy and (b) related supercurrentevolution with the superconducting phase difference ϕ Zee-man field, h y , at a fixed magnitude and varying Rashba cSOCstrength α c are considered. The other parameters are β c = 0, µ = ∆, h x = 0. (c) Three fits to the green curve in (b) usingthe generalized CPR from Eq. (4) with N = 1 , , and Josephson energy, where the ground states ϕ appearat single points [green, red dots in Fig. 3(a)], consistentwith ϕ ∝ α c h y .If instead of µ = ∆, we consider a regime µ (cid:29) ∆, theevolution of Josephson energy from Fig. 2(a) changes.While 0- π transitions with | h x | remain, there are nolonger global minima with ϕ (cid:54) = 0 , π and the CPR revealsa stronger anharmonicity. In contrast, for µ (cid:29) ∆, theanomalous Josephson effect from Fig. 3 remains robustand similar ϕ states are accessible (see Ref. [60]).Simple harmonics used to describe anharmonic CPR inhigh-temperature superconductors [78, 79] here are notvery suitable. By generalizing a short-junction limit forCPR [77, 78, 80], we identify a much more compact formwhere only a small number of terms gives an accuratedescription. To recognize the importance of SOC andtwo nondegenerate spin channels, σ , we write I ( ϕ ) ≈ N (cid:88) n =1 (cid:88) σ = ± I σn sin( nϕ + ϕ σ n ) (cid:113) − τ σn sin ( nϕ/ ϕ σ n / , (4) where τ σn is the normal region transparency for spin chan-nel σ . With only few lowest terms in this expansion( N = 1 , , N = 3 expansion inEq. (4), in a standard { sin , cos } expansion, with the cor-responding phase shifts as extra fitting parameters, re-quires N >
20 [60].Key insights into the CPR and an explicit functionaldependence for the ϕ state is obtained by a system-atic I ( ϕ ) symmetry analysis with respect to the cSOC( α c , β c ) and Zeeman field or, equivalently, magnetization( h x,y,z ) parameters [60]. We find that h z plays no rolein inducing the ϕ state, it only produces I ( ϕ ) reversals,explaining our focus on h z = 0 [Figs. 2 and 3].These properties are expressed as an effective phaseshift to the a sinusoidal CPR, sin( ϕ + ϕ ), extracted fromEq. (4). We again distinguish small- and large- µ regime( µ = ∆ v.s. µ = 10∆). In the first case, for the JJgeometry from Fig. 1, we obtain ϕ ∝ Γ y (cid:16) α + Γ β (cid:17) h x β c + Γ x (cid:16) α − Γ β (cid:17) h y α c , (5)where the parameters Γ , ,x,y are introduced throughtheir relations, Γ > Γ , Γ <
1, Γ >
1, Γ y ( h y = 0) =Γ x ( h x = 0) = 1, Γ y ( h y (cid:54) = 0) <
1, Γ x ( h x (cid:54) = 0) <
1. Theserelations are modified as µ and h change. For µ (cid:29) ∆,the functional dependence for the ϕ state is simplified ϕ ∝ (cid:16) α − Γ β (cid:17) h x β c + (cid:16) α − Γ β (cid:17) h y α c , (6)where Γ > Γ and Γ , >
1. Therefore, ϕ state occurswhen h shifts p ⊥ to I ( ϕ ) and thus alters the SOC [60].Taken together, these results reveal that cSOC in JJsupports a large tunability of the Josephson energy, an-harmonic CPR, and the anomalous phase, key to manyapplications, from post-CMOS logic, superconductingspintronics, quiet qubits, and topological quantum com-puting. Realizing π states in JJs is desirable for im-proving rapid single flux quantum (RSFQ) logic, withoperation >
100 GHz [81, 82] and enhancing coherenceby decoupling superconducting qubits from the environ-ment [83]. However, common approaches for π statesusing JJs combining s - and d -wave superconductors orJJs with ferromagnetic regions [78, 79] pose variouslimitations. Instead, extensively studied gate-tunableSOC [10, 38, 45, 53, 54, 84], could allow not only a fasttransformation between 0 and π states in JJs with cSOC,but also an arbitrary ϕ state to tailor desirable CPR.An insight to the phase evolution and circuit operationof JJs with cSOC is provided by generalizing the classi-cal model of resistively and capacitively shunted junc-tion (RSCJ) [85]. The total current, i , is the sum of thedisplacement current across the capacitance, C , normalcurrent characterized by the resistance, R , and I ( ϕ ), φ π C d ϕdt + φ πR dϕdt + I ( ϕ ) = i, (7)where φ is the magnetic flux quantum and I ( ϕ ) yields agenerally anharmonic CPR, as shown from Eq. (4), whichcan support 0, π , and turnable ϕ states. As we have seenfrom Figs. 2 and 3, this CPR tunability is accompaniedby the changes in Josephson energy, which in turn is re-sponsible for the changes in effective values of C , R , andthe nonlinear Josephson inductance. This JJ tunabilitycomplements using voltage or flux control [86, 87].In JJs with ferromagnetic regions, I c is the tunable I c by changing the underlying magnetic state [32, 88, 89]. InJJs with cSOC, tuning I c could be realized through gatecontrol by changing the relative strengths of α c and β c ,even at zero Zeeman field. This is shown in Fig. 4 by cal-culating Max[ I ( ϕ )] with ϕ ∈ [0 , π ]. In the low- µ regime,the maximum I c occurs at slightly curved region near thesymmetry lines | α c | = | β c | . For the high- µ regime, the re-gion of maximum I c evolves into inclined symmetry lines, | α c | = A| β c | , A <
1. Similar to linear SOC, in the dif-fusive regime for cSOC, one expects that the minimumin I c occurs near these symmetry lines because of thepresence of long-range spin-triplet supercurrent [63, 90].We expect that a hallmark of JJs with cSOC goes be-yond CPR and will also influence the spin structure andsymmetry properties of superconducting proximity ef-fects. Linear SOC is responsible for mixed singlet-tripletsuperconducting pairing [16], while with Zeeman or ex-change field it is possible to favor spin-triplet proximityeffects which can become long-range [20, 33] or host Ma-jorna bound states [25, 26]. To explore the proximityeffects in the cSOC region, we calculate superconductingpair correlations using the Matsubara representation forthe anomalous Green function, F ( τ ; r , r (cid:48) ) [92], F ss (cid:48) ( τ ; r , r (cid:48) ) = + (cid:104) T τ ψ s ( τ, r ) ψ s (0 , r (cid:48) ) (cid:105) ( − iσ ys s (cid:48) ) , (8)where s, s (cid:48) , s are spin indices, the summation is impliedover s , τ is the imaginary time, ψ s is the field operator,and T τ denotes time ordering of operators [60]. FIG. 4. Normalized critical supercurrent as a function ofcSOC strength α c and β c for (a) µ = ∆ and (b) µ = 10∆.The Zeeman field is set to zero. FIG. 5. Real and imaginary parts of equal-spin supercon-ducting correlations in the k-space, ξ S = (cid:126) / √ m ∗ ∆ is thecharacteristic length. (a), (b) Linear Rashba, α = 1. (c), (d)cSOC, α c = 1, β c = 0. (e), (f) cSOC, α c = β c = 1. The otherparameters are the same for all panels. For a translationally invariant SOC region, spin-tripletcorrelations in Fig. 5, obtained from Eq. (8), provide astriking difference between linear and cubic SOC. Unlikethe p -wave symmetry for linear Rashba SOC [Figs. 5(a),5(b)], we see that the f -wave symmetry is the fingerprintfor cSOC, retained with only α c (cid:54) = 0 [Figs. 5(c), 5(d)]or both α c , β c (cid:54) = 0 [Figs. 5(e), 5(f)]. Remarkably, unlikethe commonly-sought p -wave symmetry, we confirm thatwith a suitable orientation of the Zeeman field cSOC alsosupports Majorana flat bands [60].While we are not aware of any Josephson effect ex-periments in 2D systems dominated by cSOC, our stud-ied parameters are within the range of already re-ported measurements. Choosing m ∗ of an electron mass,and ∆ = 0 . ξ S ≈
14 nm. The resultingcSOC strength from Fig. 3(b) with α c ∆ ξ ≈
50 eV˚A is compatible with the values in 2D electron and holegases [55, 56]. The Zeeman splitting 2 . × . g -factor ma-terials [10], or from magnetic proximity effects, measuredin 2D systems to reach up to ∼
20 meV [6]. Even thoughwe have mostly focussed on the tunable Rashba SOC, theDresselhaus SOC can also be gate tunable [45, 94], offer-ing a further control of the anomalous Josephson effect.Our results reveal that the cSOC in JJs provides ver-satile opportunities to design superconducting responseand test its unexplored manifestations. The anomalousJosephson effect could serve as a sensitive probe to quan-tify cSOC. While identifying the relevant form of SOCis a challenge even in the normal state [10, 12], in thesuperconducting state already a modest SOC can givea strong anisotropy in the transport properties [24, 95–97] and enable extracting the resulting SOC. Identify-ing SOC, either intrinsic, or generated through magnetictextures, remains important for understanding which sys-tems could host Majorana bound states [37, 98–111].With the advances in gate-tunable structures and novelmaterials systems [38, 53–56, 93, 112], the functional de-pendence of the anomalous phase ϕ and the f -wave su-perconducting correlations could also enable decouplingof the linear and cubic SOC contributions [60]. For thefeasibility of such decoupling, it would be useful to con-sider methods employed in the studies of the nonlinearMeissner effect [113–120]. Even small corrections to thesupercurrent from the magnetic anisotropy of the non-linear Meissner response offer a sensitive probe to distin-guish different paring-state symmetries.C.S. and I.ˇZ. were supported by NSF ECCS-1810266,and I.ˇZ by DARPA DP18AP900007, and the UB Centerfor Computational Research. [1] C. L. Kane and E. J. Mele, Quantum Spin Hall Effectin Graphene, Phys. Rev. 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