Tunable superconducting critical temperature in ballistic hybrid structures with strong spin-orbit coupling
TTunable superconducting critical temperature in ballistic hybrid structureswith strong spin-orbit coupling
Haakon T. Simensen and Jacob Linder
Center for Quantum Spintronics, Department of Physics,Norwegian University of Science and Technology, NO-7491 Trondheim, Norway (Dated: February 28, 2018)We present a theoretical description and numerical simulations of the superconducting transition in hybridstructures including strong spin-orbit interactions. The spin-orbit coupling is taken to be of Rashba type forconcreteness, and we allow for an arbitrary magnitude of the spin-orbit strength as well as an arbitrary thicknessof the spin-orbit coupled layer. This allows us to make contact with the experimentally relevant case of enhancedinterfacial spin-orbit coupling via atomically thin heavy metal layers. We consider both interfacial spin-orbitcoupling induced by inversion asymmetry in an S / F-junction, as well as in-plane spin-orbit coupling in theferromagnetic region of an S / F / S- and an S / F-structure. Both the pair amplitudes, local density of states andcritical temperature show dependency on the Rashba strength and, importantly, the orientation of the exchangefield. In general, spin-orbit coupling increases the critical temperature of a proximity system where a magneticfield is present, and enhances the superconducting gap in the density of states. We perform a theoretical derivationwhich explains these results by the appearance of long-ranged singlet correlations. Our results suggest that T c inballistic spin-orbit coupled superconducting structures may be tuned by using only a single ferromagnetic layer. I. INTRODUCTION
In recent years, di ff erent ways to exert spin-control over thesuperconducting state and properties has garnered increasinginterest [1, 2]. This includes phenomena such as spin-polarizedsupercurrents [3, 4] where, traditionally, magnetic inhomo-geneities have played a key role in this endeavour as they pro-vide a source to spin-polarized Cooper pairs [5–7]. However,more recently the focus has shifted to exploiting spin-orbitinteractions as a way to achieve a spin-dependent coupling tothe superconducting state. E ff ects such as magnetoanisotropicsupercurrents [8–10], anisotropic and paramagnetic Meissnere ff ects [11], thermospin e ff ects [12, 13], and spin-galvaniccouplings [14, 15] have very recently been investigated inthis context. We note in particular that a recent experiment[16] reported a spin-valve e ff ect on the superconducting tran-sition temperature T c in a layered Nb / Pt / Co / Pt structure. Thiscontrasts previous superconducting spin-valve measurementswhere two ferromagnets were used [17–19] instead of a singlemagnetic layer. The spin-valve e ff ect is made possible due tothe thin Pt layers which provide Rashba spin-orbit interactionsdue to interfacial inversion symmetry breaking.Motivated by this experiment and the interesting physicsarising in spin-orbit coupled hybrid structures including super-conducting elements, we here present a study of the criticaltemperature, local density of states, and the induced pairingcorrelations in such systems. We use a fully quantum mechani-cal treatment and solve the BdG-equations in the ballistic limit.With a non-zero exchange field in the ferromagnetic region,both the pair amplitude, local density of states and criticaltemperature show dependency on the strength and, importantly,the orientation of the exchange field. In general, spin-orbitcoupling increases the critical temperature of the system, andstrengthens the superconducting gap in the density of states.We also present results for the same observable quantities forin-plane spin-orbit coupling in the ferromagnetic region ofan S / F / S- and an S / F-structure. The results are similar to in-terfacial spin-orbit coupling, although the e ff ect is in general stronger. Additionally, this type of spin-orbit coupling givesrise to a stronger anisotropy in the dependence on the exchangefield direction. Our results demonstrate how T c may be con-trolled with a single ferromagnetic layer in ballistic spin-orbitcoupled superconducting hybrids. II. THEORY AND METHODSA. Spin-orbit coupling
A commonly used model for the spin-orbit Hamiltonian insystems where structural inversion asymmetry is broken, forinstance by interfaces, is the Rashba Hamiltonian given by[20, 21] H SO = α R ( ˆ n × ˆ σ ) · k , (1)where α R is the Rashba parameter, ˆ n is the unit vector point-ing in the direction of the broken inversion symmetry, and ˆ σ is the vector of Pauli matrices. We will refer to ˜ h SO ≡ α R ( ˆ n × k ) as the SOC-induced field. To ensure that weuse a Hermitian Hamiltonian, we symmetrize it by letting α R ( x ) k x → { α R ( x ) , k x } , and the Hamiltonian thus becomes H SO = { α R , k }· ( ˆ n × ˆ σ ) , where { . . . } is an anticommutator.This procedure is necessary in hybrid structures, as consideredin this paper, where SOC exists only in certain layers. In orderto test the physical validity of this Hamiltonian in an actualsystem, one could for instance do spin-resolved ARPES mea-surements to test how the crystal momentum of the electronscorrelate with their spin orientation. B. Psuedospin and Cooper pairs
Let us briefly explain how Cooper pairs are formed in sys-tems with both SOC and a magnetic field present. For simplic-ity, we start by defining a Hamiltonian in which the magnetic a r X i v : . [ c ond - m a t . s up r- c on ] F e b field is perpendicular to the SOC-induced fields. Let the mag-netic field be h = h ˆ z , and restrict the SOC-induced field to beparallel to the x -axis and proportional to k y . This type of SOCmay be physically realized by for instance interfacial SOCbetween two regions in a two-dimensional system spanningthe yz -plane. By writing out the Pauli matrices explicitly, theHamiltonian follows as H = − (cid:32) h α R ˆ k y α R ˆ k y − h (cid:33) . (2)As the Hamiltonian includes SOC, spin is no longer conserved,and we refer to pseudospin, σ (cid:48) , as the new well-defined quan-tum number. It can be shown that the singlet s -wave Cooperpair projects onto the new eigenstate basis as (cid:12)(cid:12)(cid:12) k y , ↑ (cid:69) (cid:12)(cid:12)(cid:12) − k y , ↓ (cid:69) − (cid:12)(cid:12)(cid:12) k y , ↓ (cid:69) (cid:12)(cid:12)(cid:12) − k y , ↑ (cid:69) = cos ( θ SO ) (cid:34) (cid:12)(cid:12)(cid:12) k y , ↑ (cid:48) (cid:69) (cid:12)(cid:12)(cid:12) − k y , ↓ (cid:48) (cid:69) − (cid:12)(cid:12)(cid:12) k y , ↓ (cid:48) (cid:69) (cid:12)(cid:12)(cid:12) − k y , ↑ (cid:48) (cid:69) (cid:35) − sin ( θ SO ) (cid:34) (cid:12)(cid:12)(cid:12) k y , ↑ (cid:48) (cid:69) (cid:12)(cid:12)(cid:12) − k y , ↑ (cid:48) (cid:69) + (cid:12)(cid:12)(cid:12) k y , ↓ (cid:48) (cid:69) (cid:12)(cid:12)(cid:12) − k y , ↓ (cid:48) (cid:69) (cid:35) . (3)where cos( θ SO ) = h ˜ h , sin( θ SO ) = α R k y ˜ h , and ˜ h = (cid:113) h + α | k y | .In the presence of only a magnetic field, the singlet state doesnot transform at all since we in this limit have σ = σ (cid:48) and θ SO =
0. With SOC present however, it is evident that thesinglet state projected onto the eigenbasis results in both apseudospin-singlet and a pseudospin-triplet component.Since Cooper pairs are comprised of electrons of approx-imately equal energy, Eq. (3) must be modified in order toreflect the real pairings. We have to pair electrons of di ff erentmomenta, such that the momentum shift cancels the energydi ff erence caused by the SOC and the magnetic field. Aspseudospin by definition defines the two possible eigenstatesof the Hamiltonian in Eq. (2) for a given momentum k y , itis apparent that electrons with equal pseudospin and | k y | arefound at the same energy level, while the ones with oppositepseudospin and equal | k y | are found at di ff erent energy levels.We thus treat pseudospin just as we treat spin with only mag-netic fields present. That is, we define a shifted momentum k ± y = k y + ( ∆ k ) ± , where the ± applies for pseudospin up / down.( ∆ k ) ± is defined such that the di ff erent single-particle pseu-dospin states involved in the two-particle states have equalenergy. By using the notation (cid:12)(cid:12)(cid:12) k ± y , σ (cid:48) (cid:69) = (cid:12)(cid:12)(cid:12) k y , σ (cid:48) (cid:69) e i ( ∆ k ) ± x , wecan express the s -wave singlet Cooper pair wave function as ψ ⊥ ( x ) ∼ cos ( θ SO ) (cid:40) (cid:12)(cid:12)(cid:12) k + y , ↑ (cid:48) (cid:69) (cid:12)(cid:12)(cid:12) − k − y , ↓ (cid:48) (cid:69) e i [( ∆ k ) + − ( ∆ k ) − ] x − (cid:12)(cid:12)(cid:12) k − y , ↓ (cid:48) (cid:69) (cid:12)(cid:12)(cid:12) − k + y , ↑ (cid:48) (cid:69) e − i [( ∆ k ) + − ( ∆ k ) − ] x (cid:41) − sin ( θ SO ) (cid:40) (cid:12)(cid:12)(cid:12) k + y , ↑ (cid:48) (cid:69) (cid:12)(cid:12)(cid:12) − k + y , ↑ (cid:48) (cid:69) + (cid:12)(cid:12)(cid:12) k − y , ↓ (cid:48) (cid:69) (cid:12)(cid:12)(cid:12) − k − y , ↓ (cid:48) (cid:69) (cid:41) , (4) where we have neglected the change in θ SO due to momen-tum shift, and where the ⊥ indicates that the magnetic fieldand SOC-induced fields are orthogonal to each other. Onemay observe that the pseudospin-singlet component of the sin-glet state gains a phase shift, whereas the pseudospin-tripletcomponent does not. We choose to adapt the terminologywhich is frequently used on spin-triplet pairs, and name thepseudospin-triplet pair which is not subject to a pair-breakingphase a long-ranged pair. From this analysis, it is evidentthat a fraction of the s -wave Cooper singlet pair can adapt along-ranged behaviour in a system featuring both Rashba SOCand a magnetic exchange field. It therefore follows that thesinglet pair in such a system is partly long-ranged and partlyshort-ranged. In contrast, in the absence of SOC, the s -wavesinglet pair would have adapted only short-ranged behaviour.If we define the system such that the magnetic field andSOC-induced fields are parallel, spin would still be a good (con-served) quantum number. As a consequence, no pseudospin-triplet component of the singlet state will appear, and the sin-glet state remains purely short-ranged. If we had made the anal-ysis completely general, that is include all intermediate angles,the algebra would have become messy. However, the analysiswould have revealed that the transition between orthogonaland parallel setup happens smoothly and gradually, and theparallel and orthogonal setup therefore represents two extrema.Another consequence of an arbitrary magnetization alignmentis that both the singlet states and all the triplet states wouldhave been projected onto the eigenbasis as linear combinationsof both the pseudospin-singlet and all the pseudospin-triplets.A consequence of this is that mixing between the Cooper pairspin-states may occur. C. Solving the BdG equations
The Hamiltonian for a system which includes a magneticexchange field h , and Rashba SOC reads [20, 22, 23] H = (cid:88) σ (cid:90) d r ˆ ψ † ( r , σ ) (cid:104) H e − h ( r ) · ˆ σ + i α R ( r )( ˆ σ × ˆ n ) · ∇ (cid:105) ˆ ψ ( r , σ ) + (cid:90) d r (cid:26) ∆ ∗ ( r ) ˆ ψ ( r , ↓ ) ˆ ψ ( r , ↑ ) + ∆ ( r ) ˆ ψ † ( r , ↑ ) ˆ ψ † ( r , ↓ ) (cid:27) . , (5)where ˆ ψ † and ˆ ψ are electronic creation and annihilation opera-tors respectively, and H e = (cid:126) k m + V ( r ), that is the combinedkinetic energy and non-magnetic potential energy. We haveassumed that the SOC-term of the Hamiltonian is Hermitian.If this is not the case, the symmetrization procedure presentedin the last section must be applied. We will now specializethis Hamiltonian to a two-dimensional system spanning the xy -plane. We follow closely the technical procedure presentedin many papers by K. Halterman and various coauthors, forinstance in Ref. [24]. The system is translationally invariant inthe y -direction, and of length d in the x -direction.We first perform a Bogoliubov transformation of the elec-tronic operators,ˆ ψ ( x , ↑ ) = (cid:88) n (cid:20) u n , ↑ ( x ) γ n − v ∗ n , ↑ ( x ) γ † n (cid:21) , ˆ ψ ( x , ↓ ) = (cid:88) n (cid:20) u n , ↓ ( x ) γ n + v ∗ n , ↓ ( x ) γ † n (cid:21) , (6)where u n , ↑ ( x ) and v n , ↑ ( x ) are quasielectron and quasihole wavefunctions, respectively. We have assigned a label n to eachstate, denoting they are energy eigenstates. This operator trans-formation should by definition transform the Hamiltonian intothe form H = (cid:80) n ,σ E n ,σ γ † n ,σ γ n ,σ . These quasiparticle ampli-tudes are found by solving the Bogoliubov-de Gennes (BdG)equations. It is however convenient to first expand these am-plitudes as Fourier series. With ψ n ≡ [ u n , ↑ , u n , ↓ , v n , ↑ , v n , ↓ ], theFourier expansion follows as ψ n ( x ) = (cid:114) d ∞ (cid:88) q = ˆ ψ nq sin (cid:16) k q x (cid:17) , (7) where k q = q π/ d , and where the components of ˆ ψ nq = [ˆ u ↑ nq , ˆ u ↓ nq , ˆ v ↑ nq , ˆ v ↓ nq ] T are the Fourier components of the expan-sion. By expanding the wave functions in sine-functions, theboundary conditions are automatically satisfied. It is moreoveruseful to define new spin-orbit operators, h i SO ,˜ h i SO u n ,σ ≡ h i SO u n ,σ , ˜ h i SO v n ,σ ≡ − h i SO v n ,σ . (8)where the particle / hole-dependence is isolated in the sign con-vention. With these definitions, the BdG equations in Fourierspace are given by ˆ H e − ˆ h z − ˆ h z SO − ˆ h x + i ˆ h y − ˆ h x SO + i ˆ h y SO ∆ − ˆ h x − i ˆ h y − ˆ h x SO − i ˆ h y SO ˆ H e + ˆ h z + ˆ h z SO ˆ ∆
00 ˆ ∆ ∗ − ( ˆ H ∗ e − ˆ h z + ˆ h z SO ) − ˆ h x − i ˆ h y + ˆ h x SO + i ˆ h y SO ˆ ∆ ∗ − ˆ h x + i ˆ h y + ˆ h x SO − i ˆ h y SO − ( ˆ H ∗ e + ˆ h z − ˆ h z SO ) ˆ u ↑ n ˆ u ↓ n ˆ v ↑ n ˆ v ↓ n = E n ˆ u ↑ n ˆ u ↓ n ˆ v ↑ n ˆ v ↓ n , (9)where we have defined ˆ u σ n = [ˆ u σ n , ˆ u σ n , ˆ u σ n , ... ] and ˆ v σ n = [ˆ v σ n , ˆ v σ n , ˆ v σ n , ... ]. The BdG equations determine the quasipar-ticle amplitudes, as well as the energy spectrum. The matrixelements appearing above are defined asˆ H e ( q , q (cid:48) ) = d (cid:90) d dx sin (cid:16) k q (cid:48) x (cid:17)(cid:34) (cid:126) m (cid:18) π qd (cid:19) + V ( x ) + E ⊥ − E F (cid:35) × sin (cid:16) k q x (cid:17) , ˆ ∆ ( q , q (cid:48) ) = d (cid:90) d dx sin (cid:16) k q (cid:48) x (cid:17) ∆ ( x ) sin (cid:16) k q x (cid:17) , ˆ h i ( q , q (cid:48) ) = d (cid:90) d dx sin (cid:16) k q (cid:48) x (cid:17) h i ( x ) sin (cid:16) k q x (cid:17) , ˆ h i SO ( q , q (cid:48) ) = d (cid:90) d dx sin (cid:16) k q (cid:48) x (cid:17) h i SO ( x ) sin (cid:16) k q x (cid:17) , where i ∈ { x , y , z } in the two last definitions.One of the main goals of solving the BdG equations isfinding the superconducting energy gap, ∆ . It is defined as ∆ ( r ) = V SC ( x ) (cid:68) ˆ ψ ( x , ↑ ) ˆ ψ ( x , ↓ ) (cid:69) , (10) where V SC is a coupling strength between electrons inside theenergy interval [ E F − (cid:126) ω D , E F + (cid:126) ω D ]. By insertion of the Bo-goliubov transformation in Eq. (6), and V SC ( x ) = λ ( x ) / D ( E F ),where the weak-coupling constant λ is finite inside supercon-ductors and zero elsewhere, while D ( E F ) = m π (cid:126) is the energy-independent density of states per area in two dimensions, weobtain ∆ ( x ) = λ ( x ) E F k F (cid:80) (cid:48) n [ u n , ↑ ( x ) v ∗ n , ↓ ( x ) + u n , ↑ ( x ) v ∗ n , ↓ ( x )] tanh( E n / k B T ) , (11)The sum over n is a sum over all energy eigenstates, whichformally is a sum over all eigenstates of Eq. (9) for everypossible value of E ⊥ . The primed summation indicates thatthis is a constrained sum over energy levels within the energyinterval where s -wave singlet Cooper pairing occurs. Thisopens up for using a self-consistent approach. We start outby guessing an initial ∆ . The closer the initial guess is tothe actual ∆ , the fewer iterations through the BdG equationsare necessary. Before starting this procedure, we make ∆ adimensionless quantity by letting ∆ ( x ) / ∆ → ∆ ( x ), where ∆ is the bulk value of the superconducting energy gap within aclean superconductor. ∆ ( x ) should therefore presumably beconstrained to | ∆ ( x ) | ≤
1. For most situations, using a zerothorder approach by guessing ∆ = ∆ = ffi ciently accurate startingpoint. Solve the BdG equations in (9) for this ∆ , and obtain aset of eigenvectors ψ n . Use this set of eigenvectors to definea new ∆ using Eq. (11), and repeat this procedure until ∆ converges towards the true superconducting gap. In this paper,we stopped the procedure when ∆ at no point had a relativechange of more than 10 − between two consecutive iterations. D. Pair amplitudes
The singlet energy gap captures the singlet correlation withinthe superconducting regions of a system. Outside of these re-gions, the amplitude is by definition identically zero due to λ ( x )being zero. To provide information on the proximity e ff ect, thatis how far into non-superconducting regions superconductingorder penetrates, we define the s -wave singlet pair amplitude f ( x ) = ∆ ( x ) λ ( x ) . (12)This amplitude is chosen to be normalized to | f | ≤
1. Wefurthermore define the s -wave triplet amplitudes, that is theodd-frequency triplets, as [25] f ( x , τ ) = (cid:68) ˆ ψ ( x , τ, ↑ ) ˆ ψ ( x , , ↓ ) + ˆ ψ ( x , τ, ↓ ) ˆ ψ ( x , , ↑ ) (cid:69) , (13) f ( x , τ ) = (cid:68) ˆ ψ ( x , τ, ↑ ) ˆ ψ ( x , , ↑ ) − ˆ ψ ( x , τ, ↓ ) ˆ ψ ( x , , ↓ ) (cid:69) , (14) f ( x , τ ) = (cid:68) ˆ ψ ( x , τ, ↑ ) ˆ ψ ( x , , ↑ ) + ˆ ψ ( x , τ, ↓ ) ˆ ψ ( x , , ↓ ) (cid:69) , (15)(16)where τ is the relative time coordinate. We name the tripletscaptured by the f -amplitude ( s z = z -axis, thus being σ z -eigenstates. We name the Cooper pairs responsible for the f - and f -amplitudes ( s z = ± σ z -eigenstates, and hence have no well defined s z as they are linear combinations of two-particle states with s z = ±
1. By insertion of the Bogoliubov transformations inEq. (6), and utilizing the identities which were used in thederivation of Eq. (11), we obtain f ( x , τ ) = (cid:88) n (cid:104) u n , ↑ ( x ) v ∗ n , ↓ ( x ) − u n , ↓ ( x ) v ∗ n , ↑ ( x ) (cid:105) ζ n ( τ ) , (17) f ( x , τ ) = − (cid:88) n (cid:104) u n , ↑ ( x ) v ∗ n , ↑ ( x ) + u n , ↓ ( x ) v ∗ n , ↓ ( x ) (cid:105) ζ n ( τ ) , (18) f ( x , τ ) = − (cid:88) n (cid:104) u n , ↑ ( x ) v ∗ n , ↑ ( x ) − u n , ↓ ( x ) v ∗ n , ↓ ( x ) (cid:105) ζ n ( τ ) , (19)where ζ n ( t ) = sin (cid:16) E n τ (cid:126) (cid:17) − i cos (cid:16) E n τ (cid:126) (cid:17) tanh (cid:16) E n k B T (cid:17) . In this paper,the triplet pair amplitudes have been normalized with the sameprefactor as f . Additionally, we only plot the real part of thepair amplitudes. E. LDOS
The local density of states (LDOS), N ( E , x ), provides infor-mation on the distribution of states as a function of energy and position. Its interpretation is that N ( E , x ) dE equals the num-ber of quantum states within the infinitesimal energy interval[ E , E + dE ] at position x . It can be expressed as [23] N ( E , x ) = (cid:88) n (cid:88) σ (cid:110) | u n σ ( x ) | δ ( E − E n ) + | v n σ ( x ) | δ ( E + E n ) (cid:111) , (20)where the δ -function is the Dirac delta function. As all theenergy levels are discretized, N ( E , x ) will be a discrete dis-tribution function. To smoothen out the density of states, weperform a convolution with a Gaussian of width 0 . ∆ . Inthis paper, the LDOS is normalized to be 1 in the normal metallimit, that is several times ∆ away from E F . F. Critical temperature
For the calculation of the critical temperature, we followclosely the procedure of Ref. [26], where T c is found bytreating ∆ as a small first-order perturbation. We can thensolve the BdG equations once to zeroth order, that is with ∆ =
0, and use perturbation theory to define a finite ∆ from theeigenvectors. This first-order ∆ will be T -dependent, and wefind T c by identifying the point where ∆ = ∆ cannot be made arbitrarily close to zero. The resultis a matrix eigenvalue problem, ∆ (1) l = (cid:88) k J lk ( T ) ∆ (1) k , (21)where the matrix elements J lk are defined by the formula J lk ( T ) = λ E F k F d (cid:88) n (cid:88) m (cid:107) (cid:88) p , q K pql (cid:40) v (0) † mq J u (0) np (cid:80) i , j u (0) † ni J v (0) m j K i jk E pn − E hm tanh (cid:32) E pn k B T (cid:33) + v (0) † nq J u (0) mp (cid:80) i , j u (0) † mi J v (0) n j K i jk E hn − E pm tanh (cid:32) E hn k B T (cid:33)(cid:41) , (22)where v nj = [ v ↑ n j , v ↓ n j ] T and u n j = [ u ↑ n j , u ↓ n j ] T are vec-tors of quasiholes and quasielectrons, (0)-superscipt de-notes zeroth order, J is the (2 ×
2) exchange matrix (de-fined in Appendix A), and E hn and E pn are the zeroth or-der energy spectra of quasiholes and quasielectrons, re-spectively. To simplify notation, we have introduced K i jk = (cid:82) d dx Θ ( x − x ) sin( k i x ) sin (cid:16) k j x (cid:17) sin( k k x ). The sumsover i , j , p and q go over the Fourier wave numbers. Theconstrained sum over n goes over the kinetic energy contribu-tions from all directions. The sum over m (cid:107) goes over all kineticenergy contributions from the x -direction, with m ⊥ = n ⊥ im-plied.Eq. (21) is a matrix eigenvalue equation. It has one obvioussolution, the trivial solution, that is ∆ ( x ) =
0. This solution isof no particular interest, since it implies that superconductivityis absent. If we assume ∆ ( x ) (cid:44) J ( T ) has an eigenvalue whichis 1. Since superconductivity is sensitive to temperature, oneshould therefore expect only the trivial solution to remain if T > T c , where T c is the critical temperature where supercon-ductivity breaks down. This involves that all the eigenvalues of J ( T ) falls below 1. The critical temperature is therefore foundby identifying at which temperature the largest eigenvalue of J ( T ) drops below 1. III. RESULTS AND DISCUSSION
SOC lifts spin-rotational symmetry, and thus the simulta-neous presence of SOC and a magnetic field should revealspin-anisotropic behaviour of superconductivity. This is whatmotivates us to explore SOC in F / S-structures. We will lookat two types of spin-orbit coupling. First, we will explore thehighly localized interfacial SOC, of which results are givenin Sec. III A. We will thereafter look at in-plane SOC insideF-regions, of which results are given in Secs. III B and III C.We use (cid:126) ω D / E F = .
04 for all calculations, as well as T = h / E F = .
3, mostrealistically realized by placing transition metal ferromagnetssuch as Fe, Ni, or Co in the ferromagnetic region. The Rashbastrength is varied between up to α R k F / E F = .
5. Large SOCe ff ects are probably easiest to realize at the surface of heavymetals such as Au or Pt, as it has been reported that thesestructures give a Rashba e ff ect two orders of magnitude largerthan in semiconducting 2DEGs [21]. As we in this paper wouldlike to explore the general e ff ects of combining exchange fieldsand the Rashba e ff ect, we vary α R over a relatively wide range.The dimensions used in this paper coincide with the routinelyachieved experimental dimensions in heterostructures.In all systems, we assume that the Fermi level E F is equaland constant in all regions. Furthermore, we assume that thee ff ective masses, work functions and densities are equal in allregions, in addition to there being no scattering potential in theinterface between the regions. This is clearly a crude simplifi-cation, and in order to provide a more a realistic description ofspecific materials one should rather use parameters obtainedfrom experiments. In this paper, we have chosen not to includethese parameters, as doing so would result in more undeter-mined parameters that would complicate the analysis. Themain purpose of this paper is to show the qualitative e ff ect ofcombining exchange fields and SOC, and we therefore choosethe simplest approximation, namely that all of these param-eters are constant throughout the system. Despite the crudeapproximation, similar routines (excluding SOC) have previ-ously provided results which coincide well with experimentalresults [27]. For future work, it is straightforward to includeFermi level mismatch and interface scattering potentials in thenumerical method, and thereby obtain results which are closerto specific materials. A. Interfacial SOC in an F / S-structure
We start by looking at interfacial SOC in an F / S-stucture.The system is two-dimensional, of length d = . ξ in the x -direction, and is translationally invariant in the y -direction.The length of the F-region has been set to 0 . ξ , and the super-conductor’s length has been set to ξ , where ξ is the coherencelength of the superconductor. In between these regions, thereis a SOC-layer of width 0 . ξ . The system is illustrated inFig. 1. The SOC-potential has been Gaussian distributed in-side this region, that is α R ( x ) ∼ N ( x ; λ SO , σ SO ), where theexpectation value of the distribution, λ SO , is in the middleof the SOC-region. The variance of the distribution is σ ,and 4 σ SO = . ξ covers most of the distribution. We use aRashba coupling strength of α R k F / E F = .
5. In the F-region,we define a magnetic field h = h (cid:0) sin ( θ h ) ˆ x + cos ( θ h ) ˆ z (cid:1) , inwhich we set h / E F = . z x − y θ h h d d S σ SO d F ˆn FIG. 1. An illustration of the F / S-structure with SOC in the junction.The system considered is in reality not of restricted length along the y -axis, but is of infinite extent in this direction. Moreover, the structureis of zero height, that is of no extent in the z -direction.
1. Pair amplitudes
The s -wave singlet amplitude is plotted for five di ff erentmagnetization angles, θ h , in Fig. 2. The upper plot shows theresults for θ h =
0, and the magnetization angle is increasedby π/ θ h =
0, itsmaximum before the oscillations at the boundary is approx-imately 0 .
1. Growing steadily by increasing magnetizationangle, this maximum doubles as θ h approaches π/
2. Hence, itseems as though a magnetization perpendicularly aligned tothe SOC-induced fields results in best conditions for supercon-ductivity to exist.If the magnetic field and SOC-induced fields are aligned inthe z -direction, s z is still a conserved quantum number. For the s -wave singlet Cooper pairs, the SOC then mimics a potential x/ h = 0 x/ h = : /8 x/ h = : /4 x/ h = 3 : /8 -0.23 0 0.25 0.5 0.75 1 x/ h = : /2 f ( x ) FIG. 2. The singlet pair amplitude plotted for five di ff erent magne-tization angles, θ h , for an F / S-structure with SOC in a thin layer atthe interface. The SOC-layer is Gaussian distributed within the bluedotted lines (which cover a width of 4 σ SO ). barrier, causing the F- and S-regions to be partly decoupled.In this case, we would therefore in general expect SOC to pro-tect the superconducting state to some extent by damping theproximity e ff ect. If the magnetic field and SOC-induced fieldsare not aligned however, we cannot precisely make qualitativepredictions by evaluating s z -states. We therefore turn to thepseudospin eigenstates, derived in Sec. II B. The main resultof this section was that if the SOC-induced field is perpen-dicular to a magnetic field, a component of the singlet statebecomes long-ranged. That is, if we project the singlet stateonto the eigenbasis, it will in general be a linear combinationof a pseudospin-singlet and a ( s (cid:48) = ± ff ect,the leakage of singlets is reduced, allowing for a larger singletamplitude to sustain. This e ff ect of SOC is θ h -dependent, andwill therefore increase as θ h increases. The results obtained bynumerical calculations seem to support this analysis.Another prediction from Sec. II B was that mixing betweenthe triplet pairs should occur at intermediate angles, due tothey using a common set of pseudospin channels through thesystem. This is verified by Fig. 3, where all triplet amplitudesare plotted for five di ff erent relative times τ = ω D t . In theabsence of SOC, the f -amplitude would not have appearedby rotating the magnetic field in the xz -plane. With SOChowever, it clearly appears, and this must therefore be due toCooper pair spin-mixing caused by SOC.s At θ h = s z isa conserved quantum number, and no ( s z = ± ff ect seem to grow. At θ h = π/ h = 0 h = 0 h = 0 h = : /8 h = : /8 h = : /8 h = : /4 h = : /4 h = : /4 h = 3 : /8 h = 3 : /8 h = 3 : /8 -0.23 0 0.25 0.5 0.75 1 x/ -1.5-0.75 0 0.75 1.5 h = : /2 -0.23 0 0.25 0.5 0.75 1 x/ -1.5-0.75 0 0.75 1.5 h = : /2 -0.23 0 0.25 0.5 0.75 1 x/ -7.5-3.75 0 3.75 7.5 h = : /2 = = 0 = = 3 = = 6 = = 9 = = 12 f (x) f (x) f (x) ( -2 ) ( -2 ) ( -4 ) FIG. 3. The triplet amplitudes for five di ff erent magnetization angles, θ h , for the F / S-structure with a SOC-layer in the junction. Each plotcontains triplet correlations for five di ff erent relative times, τ . shift of spin basis does not cause spin mixing, as predicted bythe discussion in Sec. II B.
2. LDOS
As a consequence of the analysis so far, we expect the bandgap to be more developed for higher magnetization angles, θ h . This is due to the creation of long-ranged singlets, whichshould imply fewer triplet states relative to singlet states, thusreducing the number of states within the band gap. When θ h =
0, this e ff ect does not occur, and the plots should bequalitatively rather equal to a clean F / S-junction. For θ h = π/
2, the e ff ect should be at its maximum, creating the mostprominent band gap. The LDOS at four di ff erent positions areplotted in Fig. 4, both inside the F- and S-region.The plots show very clearly that the superconducting gapbecomes much more prominent for higher magnetization an-gles. For θ h =
0, one can in fact almost not spot any gap atall. As we rotate θ h further towards π/
2, this gap grows, andit is almost a complete gap for θ h = π/
2. This applies to allpositions in the system, both inside the F-region and inside theS-region. As the energy gap grows with θ h , this indicates thatthe fraction of singlet states grows, and that superconductivityis thus being strengthened. This is just in accordance withthe analytical derivation in Sec. II B, where the existence oflong-ranged singlets was predicted. N ( E ) x/ = -0.115 x/ = 0.115x/ = 0.515 x/ = 0.915 -1 -0.5 0 0.5 100.51 -1 -0.5 0 0.5 100.511.52-1 -0.5 0 0.5 1 E/ " E/ " h = 0 h = : /8 h = : /4 h = 3 : /8 h = : /2 FIG. 4. The LDOS for the F / S-structure with SOC in the interfaceplotted at four di ff erent positions, as indicated above each plot. Ateach position, the LDOS is plotted for five di ff erent magnetizationangles, θ h . The results are obtained with N ⊥ =
3. Critical temperature
We have so far seen that the closer θ h comes to π/
2, thestronger is the enhancing e ff ect on superconductivity. In orderto reveal the exact angular dependence, we have plotted thecritical temperature with respect to the magnetization angle inFig. 5. The analysis is done for three di ff erent Rashba couplingstrengths. The magnetic field is as before, h / E F = .
3. Firstly, h / : T c / T c , R k F /E F = 0.5 , R k F /E F = 0.3 , R k F /E F = 0.1 FIG. 5. The critical temperature of an F / S-structure with SOC in theinterface plotted for three di ff erent Rashba parameters, as function ofthe magnetization angle, θ h . these results confirm that the Hamiltonian is invariant underthe transformation θ h → π − θ h , as the plot is symmetric about π/
2. Furthermore, the results clearly indicate that the closerthe magnetization angle is to π/
2, the more robust is the super-conducting state. This is an interesting result, as we are able tocontrol the critical temperature by adjusting a macroscopic pa-rameter. Although not directly comparable, these results showsimilar behaviour as obtained by a quasiclassical approach inthe di ff usive limit in Ref. [28, 29]. In these works, it wasshown that for equal weights of Rashba and Dresselhaus SOC,rotating the magnetic field over an interval of π/ / S-structure with interfacial SOCstudied here, with a fully quantum mechanical approach.In summary, the F / S-structure with interfacial SOC showsinteresting properties. Firstly, it allows for controlling thecritical temperature by adjusting macroscopic factors such asthe magnetic field. Secondly, it may be used to control tripletamplitudes. Such a structure therefore serves as a promisingalternative to magnetic multilayers for the purpose of achievingsuperconducting spin-valve e ff ects. B. In-plane SOC in an S / F / S-structure
Our observations from the previous section, and the predic-tions made in Sec. II B, motivates us to look at the case ofin-plane SOC. That is, the combined presence of SOC and mag-netic fields gives rise to long-ranged singlet pairs. The analysisso far predicts that the closer these interactions are in space,the more prominent the e ff ects will be. In-plane SOC inside aferromagnet maximizes the spatial copresence of the spin-orbitinteraction and the exchange field’s e ff ect on the electrons,and we thus expect a larger relative amount of long-rangedsinglet pairs as compared to with interfacial SOC. We startout by looking at a S / F / S-structure, with in-plane SOC in theF-region. One way to realize such a setup, a so-called Rashbaferromagnet, is to use a thin film of a strong transition metalferromagnet, e.g.
Fe, Co or Ni. The Rashba e ff ect could be fur-ther enhanced by adding a thin layer ( ∼ . ξ . The S-regions are of length d S1 /ξ = d S2 /ξ = . d F /ξ = . ff erence between the superconductors.As opposed to in the case of interfacial SOC, where a spin-rotational symmetry remained in the system, the spin-rotationalsymmetry is completely broken by the combined e ff ect of themagnetic field and SOC. We have to keep the magnetic fieldcompletely general, and write it as h = h (cid:18) cos( φ ) sin( θ ) ˆ x + sin( φ ) sin( θ )ˆ z + cos( θ )ˆ y (cid:19) , (23)where φ is the azimuthal angle and θ is the polar angle ofslightly modified spherical coordinates, that is with y and z having changed roles. Moreover, the SOC Hamiltonian be-comes H SO = α R (cid:104) k x σ z − k z σ x (cid:105) + α R σ z i (cid:104) δ ( x − x L ) − δ ( x − x R ) (cid:105) , (24)where x L and x R are the x -coordinates of the left and rightboundaries of the SOC-region respectively, and where the posi-tion dependence of the Rashba parameter has been suppressed y xz θ h dd F d S2 d S1 φ ˆn FIG. 6. An illustration of the S / F / S-structure with in-plane SOC in theF-region. The system considered is in reality not of restricted lengthalong the z -axis, but is of infinite extent in this direction. Moreover,the structure is of zero height, that is of no extent in the y -direction. in the notation. Interestingly, we see that the requirement ofrendering the Hamiltonian Hermitian leads to an e ff ective bar-rier term at the interfaces which looks like a spin-dependentscattering potential with an imaginary amplitude. This termmay seem like an unwanted term. It introduces complex num-bers on the diagonal of H SO , which in general could causecomplex eigenvalues, resulting in complex, unphysical ener-gies. However, the actual matrix elements entering the diagonalof the BdG equations in Eq. (9) remain purely real even in thepresence of the additional δ -function term, as can be verifiedby direct insertion. For this analysis, we set magnetic fieldstrength to h / E F = .
1, and the Rashba coupling strength isset to α R k F / E F = .
1. Pair amplitudes
The singlet amplitudes for magnetization along x -, y - and z -axis are plotted in Fig. 7. The qualitative behaviour of thesinglet amplitudes in these magnetization setups are all approx-imately the same. There is however a significant quantitativedi ff erence between the di ff erent setups. The superconductingstate seem to prefer the y -alignment of the magnetic field, andis most suppressed by an x -aligned field. Hence, as with inter-facial SOC, in-plane SOC introduces a prominent dependenceupon the direction of the magnetic field. If SOC was switchedo ff , the singlet amplitudes would drop to zero no matter themagnetization direction, implying that SOC once again showsan enhancing e ff ect on superconductivity.The triplet amplitudes for the same magnetization setupsare plotted in Fig. 8. Note that the axes are scaled di ff erently,and the graphical amplitudes are thus not directly compara-ble between the di ff erent plots. If SOC was switched o ff , therotation of the magnetic field would only cause the triplet am-plitudes to rotate between each other. An x -aligned, y -alignedand z -aligned magnetization would have given a non-zero f - x/ h k ^x x/ h k ^y x/ h k ^z f ( x ) FIG. 7. The singlet amplitude plotted for the S / F / S-structure within-plane SOC in the F-region, for magnetization along the x -, y - and z -axis. The dotted blue lines indicate the junctions between the F- andS-regions. The Rashba parameter has been set to α R k F / E F = . amplitude, f -amplitude and f -amplitude, respectively. Foreach magnetization configuration, all other than the mentionedtriplet amplitude would have been identically zero. With SOCswitched on however, all triplet amplitudes appear for the x -and y -aligned fields, while only the f -amplitude remains non-zero for the z -aligned field.There are two reasons to the appearance of other triplet am-plitudes. Firstly, as explained in Sec. II B, SOC induces mag-netic impurities in the junction between the S- and F-regions.These cause an inhomogeneous magnetization configurationwhen the magnetic field points along the x - or y -axis, whichalone would result in two triplet amplitudes to appear. Sec-ondly, SOC introduces spin-mixing when the magnetic fieldis not either orthogonal or parallel to the SOC-induced field.These e ff ects combined generally cause all triplet amplitudesto be present, except for h (cid:107) ˆ z , where no spin mixing occursand thus only one non-zero triplet amplitude appears.
2. LDOS
As magnetization in either the x -, y - or z -directions clearlygive di ff erent pair amplitudes, it makes an interesting analysisto take a closer look at the configuration of states around theFermi energy for each case. The LDOS at four di ff erent posi-tions have therefore been plotted in Fig. 9. The upper two plotsshow the density of states at two di ff erent positions inside theleft S-region, while the two lower plots do the same for insidethe F-region. Inside the S-region, there is a fully developed h k ^x x/ -2-1 0 1 2 h k ^x x/ -2-1 0 1 2 h k ^x h k ^y h k ^y h k ^y x/ h k ^z x/ -4-2 0 2 4 h k ^z x/ -4-2 0 2 4 h k ^z = = 0 = = 3 = = 6 = = 9 = = 12 f (x) f (x) f (x) ( -2 ) ( -2 ) ( -2 ) FIG. 8. The triplet amplitudes for the S / F / S-structure with in-planeSOC in the F-region for magnetization along the x -, y - and z -axis.The results are plotted for five di ff erent relative times τ , as indicatedby the legend. The black dotted lines indicate the junctions betweenthe di ff erent regions. Note that the axes are scaled di ff erently, and thegraphical amplitudes are thus not directly comparable. -1 -0.5 0 0.5 1 E/ " h k ^xh k ^yh k ^z N ( E ) x/ = 0.2 x/ = 0.4x/ = 0.525 x/ = 0.55 -1 -0.5 0 0.5 10123 -1 -0.5 0 0.5 10123-1 -0.5 0 0.5 1 E/ " FIG. 9. The LDOS at four di ff erent positions inside the S / F / S-structurewith in-plane SOC in the F-region. The positions are indicated in theplots, and the colour coding is indicated by the legend. The resultsare obtained with N ⊥ = energy gap for magnetization in the y - and z -directions, with h (cid:107) ˆ y giving the largest gap. The gap is much less developedfor the x -aligned magnetic field. Inside the F-region, the am-plitudes are being suppressed for all system setups, with anaverage of about 0 .
25 outside the band gap region. This isboth an e ff ect of the magnetic field, which suppresses certainspin-configurations, as well as due to SOC suppressing statesdependent upon both their momentum and spin. The band gap is fully developed at both positions insidethe F-region for both the y - and z -aligned fields. Once again,the gap is widest for the y -aligned field. These results areconsistent to the results obtained for the singlet amplitudes. Ingeneral, the band gap seems to be wider and more prominentfor h (cid:107) ˆ y , and weakens for h (cid:107) ˆ z and h (cid:107) ˆ x , in that order.
3. Critical temperature
The analysis so far has been performed with a constantRashba parameter. We follow up this analysis by investigatinghow varying the Rashba parameter a ff ects the physics of thesystem. This analysis is once again performed for the magneticfield pointing in both the x -, y - and z -directions. The resultsare plotted in Fig. 10. R k F /E F T c / T c FIG. 10. The critical temperature plotted with respect to the Rashbaparameter for the S / F / S-structure with in-plane SOC in the F-region.The magnetic field strength is set to h / E F = .
1. Each line representsa magnetic field orientation along an axis, orthogonal to the others, asindicated by the legend.
The critical temperature increases with increasing α R forall three magnetic field configurations. The degree to whichthe temperature rises di ff er between all configurations, as weshould expect after the analysis so far. The critical tempera-ture is generally higher for a magnetic field pointing in the y -direction. For magnetic field configurations in the xz -plane,that is in the plane which is spanned by the physical system, thecritical temperature is di ff erent for small Rashba parameters.The critical temperature is in general higher when the magneticfield is pointing in the z -direction, but this di ff erence vanishesalmost entirely as α R k F approaches 0 . E F . These results areconsistent with what observed for the singlet amplitude andfor the LDOS. However, this analysis also brings some newand interesting observations, which can help us understand thephysics better.We start by looking into the easily visible di ff erence be-tween magnetization along the y -direction and in the xz -plane.It is obvious from Fig. 10 that superconductivity is most re-sistant to thermal e ff ects with a magnetic field along the y -axis. This result is rather straightforward to understand. Thesuperconductivity-enhancing e ff ect caused by interfacial SOC0was observed to be most prominent at θ h = π/
2, as is alsopredicted by theory. For the S / F / S-system considered here, theSOC-induced fields are always parallel to the xz -plane, perpen-dicular to the y -axis. Thus, for magnetization in the y -direction,the requirement for maximal e ff ect of SOC is always satisfied,which explains the higher T c .In a pure F / F / S-structure, a perpendicular relative orientationof neighbouring magnetic field regions causes a lower criticaltemperature than a parallel alignment [30]. This is due to thelong-range triplet production, which e ff ectively causes anotherchannel of triplet leakage to occur. We can use this result toexplain the φ -dependence of T c , that is the di ff erence betweenthe x - and z -directions. When φ = π/
2, all magnetic fields inthe system are either parallel or antiparallel. Hence, in this con-figuration, only the short-range triplet channel is open. When φ is decreased however, the production of long-range tripletpairs is increased. This e ff ect reaches its maximum at φ = ≤ φ < π/
2, which implies lowercritical temperature for an x -aligned magnetic field than for a z -aligned field. For an increasing Rashba parameter however,the amount of short-ranged triplets are reduced, which furtherimplies less leakage into other triplet channels. Thus for in-creasing α R , T c should be less sensitive to changes in φ . Asis evident from the results, full φ -invariance seems to occur at α R k F / E F ≈ . φ, θ ) is plotted in Fig. 11. Theplot contains three graphs, each of which corresponds to rota-tion in either the xy -, yz - or zx -plane. The Rashba parameterhas been set to α R k F / E F = .
4. The results are consistent with / T c / T c FIG. 11. The critical temperature in the S / F / S-structure with in-planceSOC plotted with respect to di ff erent magnetization angles, with α R k F / E F = . θ and φ have been rotated between 0 and π/ xy -plane (blue), zy -plane (red) and zx -plane (green). The angle χ represents either φ or θ , and is specified by the legend for eachindividual line. the analysis made in the discussion of magnetization in the x -, y - or z -direction. We also observe that the graphs are all strictlyincreasing or decreasing, and contain thus no local minima ormaxima. The transition between the di ff erent extrema, namely magnetization along the coordinate axes, happens smoothly.There are no intermediate angles at which e ff ects other thanthose discussed up until now occur.Fig. 11 shows that the largest change in T c by rotating themagnetic field happens for rotation in the xy -plane. The di ff er-ence between θ = θ = π/ . T . This structurethus has a great potential in controlling T c by adjusting boththe SOC-strength and the magnetization angles. It also servesa candidate for controlling the triplet production, as magnetiza-tion along the x -, y - and z -axis all give di ff erent properties forthe triplet amplitudes. Additionally, these e ff ects are obtainablefor a structure of only 1 . ξ , which is generally shorter thanrequired for clean ferromagnet-superconductor-structures. C. In-plane SOC in an S / F-structure
We will now look at in-plane SOC in an S / F-structure. Thestructure will have equal dimensions as the previous S / F / S-structure, only with the F-region to the far right side of thesystem. That is, the S-region is of length d S = ξ , while theF-region is of length d F = . ξ . We still use α R k F / E F = . h / E F = .
1. The system is illustrated in Fig. 12. The
F + S OC S hˆn φy z xθd S d F d FIG. 12. An illustration of the S / F-structure with in-plane SOC in theF-region. The system considered is in reality not of restricted lengthalong the z -axis, but is of infinite extent in this direction. Moreover,the structure is of zero height, that is of no extent in the y -direction. qualitative di ff erence between this structure and the S / F / S-structure is that this is a bilayer structure rather than a trilayer,and that the SOC-region now forms a boundary region. Theresults are very similar to the S / F / S-structure, and we willtherefore not give a complete treatment of this structure. Wewill restrict the analysis to include the critical temperature plotsanalogous to the ones given for the S / F / S-structure, and thediscussion will mainly focus on the di ff erences.1
1. Critical temperature
The critical temperature with respect to the Rashba couplingstrength, α R , is plotted in Fig. 13. There are several qualitativesimilarities to the corresponding plot for the S / F / S-structure,given in Fig. 10. Firstly, we observe that the x -aligned mag-netic field has a clearly visible suppressed critical temperaturecompared to with the magnetic field pointing in either the y -or z -directions. With an increasing Rashba parameter, we alsoobserve that the critical temperature is strictly increasing for allof the three magnetization configurations. The most interesting R k F /E F T c / T c FIG. 13. The critical temperature plotted with respect to the Rashbaparameter for the S / F-structure with in-plane SOC in the F-region.The magnetic field strength is set to h / E F = .
1. Each line representsa magnetic field orientation along an axis, orthogonal to the others, asindicated by the legend. observations might however be the di ff erences. We see thatfor all configurations, the critical temperature is non-zero ata lower Rashba parameter than in the S / F / S-structure. Thismay be explained from the fact that the superconductor lengthhere is twice the size of the lengths of the two S-regions in theS / F / S-structure. With no SOC, few singlet pairs may tunnelthrough the F-region, and the two regions are in that sensemore or less decoupled. In the S / F-structure in comparison,the doubled length of the S-region makes superconductivityarise at an earlier stage. As we increase the Rashba couplingstrength however, the critical temperature of the S / F / S-systemeventually passes that of the S / F-system. At this point, morepairs may pass through the F-region in the S / F / S-structure,and we might say that the F-region gradually adapts normalmetal properties. While in the S / F-structure, the F-region isat the boundary, and the singlet pairs cannot simply tunnelthrough into a new S-region, but rather has to be reflected andpass through the F-region once more before re-entering the S-region. This results in more triplet conversion, which explainswhy the critical temperature of the S / F / S-region eventuallybecomes larger than that of the S / F-region.In Fig. 14, the critical temperature is plotted with respectto the magnetization angles ( φ, θ ) in the xy - xz - and yz -plane.This should be compared to the results for the S / F / S-structureplotted in Fig. 11. We observe that for the rotations in the xz - and yz -plane, the local extrema of T c seem generally not / T c / T c FIG. 14. The critical temperature in the S / F-structure with in-planceSOC plotted with respect to di ff erent magnetization angles, with α R k F / E F = . θ and φ have been rotated between 0 and π/ xy -plane (blue), zy -plane (red) and zx -plane (green). The angle χ represents either φ or θ , and is specified by the legend for eachindividual line. to be where the magnetization is along one of the coordinateaxes, but rather shifted somewhat from this point. This isinterestingly just what we observe when rotating the relativemagnetization direction in a clean F / F / S-structure. In the S / F / S-structure with SOC, there was no such e ff ect. This is likely dueto the fact this e ff ect occurs on both sides of the F-region in theS / F / S-structure, e ff ectively adding together to produce resultswhere the extrema are found at φ and θ equal to 0 and π/ / F-structure with in-plane SOC may therefore, at least tosome extent, serve the same purpose as an F / F / S-structure. Putin other words, the F-region with in-plane SOC may to someextent serve as a substitute for an F / F-region.
IV. CONCLUSION
The e ff ects of strong Rashba spin-orbit coupling inferromagnet-superconductors-structures have been analyzedwith both an analytical and a numerical approach. We first dida theoretical analysis in which the s -wave spin-singlet state wasprojected onto the pseudospin eigenbasis in a system where aspin-orbit field and an exchange field coexist throughout theentire non-superconducting material. The analysis showed thatthe spin-singlet state is projected onto the eigenbasis as a linearcombination of a of a short-ranged pseudospin-singlet state anda long-ranged pseudospin-triplet state, depending upon boththe relative orientation and the strengths of the exchange andspin-orbit fields. The spin-singlet therefore gains a long-rangedcomponent, which can traverse through the system with a slowdecay. The theoretical analysis predicts that the copresence ofspin-orbit coupling and magnetic fields can raise the criticaltemperature compared to if spin-orbit coupling is absent.The numerical calculations support the predictions of thetheoretical analysis. We explored interfacial spin-orbit cou-pling between a ferromagnet and a superconductor, as wellas in-plane spin-orbit coupling in the ferromagnetic region of2an S / F / S- and an S / F-structure. Both the pair amplitudes, lo-cal density of states and the critical temperature showed to bestrongly dependent upon the direction of the exchange field. Byrotating the relative orientation between the spin-orbit couplingand exchange fields, or by adjusting either the magnetic field orRashba coupling, one may therefore control the superconduct-ing properties of the system, making these structures possiblecandidates for use in cryogenic spintronics components.
ACKNOWLEDGMENTS
M. Amundsen, N. Banerjee, S. Jacobsen, J. A. Ouassou,and V. Risingård are thanked for useful discussions. We thankin particular K. Halterman for useful correspondence. J.L.acknowledges funding via the Outstanding Academic Fellowsprogram at NTNU, the NV-Faculty, and the Research Councilof Norway Grant numbers 216700 and 240806. This work waspartially supported by the Research Council of Norway throughthe funding of the Center of Excellence ”QuSpin” project no.262633.
Appendix A: Finding T c with perturbation theory We start by defining particle- / hole-amplitude vectors u n ( x ) = (cid:32) u n , ↑ ( x ) u n , ↓ ( x ) (cid:33) , v n ( x ) = (cid:32) v n , ↑ ( x ) v n , ↓ ( x ) (cid:33) , (A1)and the matrices¯ H e = (cid:32) H e H e (cid:33) , ¯ ∆ = J ∆ , (A2)where J is the (2 ×
2) exchange matrix, sometimes also referredto as the backward identity matrix, J = (cid:32) (cid:33) . (A3)This matrix will also be of use later to express cross-couplingterms like u n ,σ v n , − σ . Furthermore define σ as the vector ofPauli matrices, and a slightly altered vector of Pauli matrices˜ σ = [ − σ x , σ y , σ z ]. By using this notation, the BdG equationstake the form (cid:32) ¯ H e − h · σ − h SO · σ ¯ ∆ ¯ ∆ ∗ − (cid:104) ¯ H e − h · ˜ σ + h SO · ˜ σ (cid:105)(cid:33) (cid:32) u n v n (cid:33) = E n (cid:32) u n v n (cid:33) , (A4)where we remind ourselves that h SO is a momentum-dependentoperator, while σ and ˜ σ act in spin space. We now do aperturbation expansion u n = u (0) n + δ u (1) n + O ( δ ) , (A5) v n = v (0) n + δ v (1) n + O ( δ ) , (A6) E n = E (0) n + δ E (1) n + O ( δ ) , (A7)¯ ∆ = + δ ¯ ∆ (1) + O ( δ ) , (A8)where δ is an arbitrary perturbation parameter, which eventu-ally will be set to 1. u (1) n is conventionally assumed to be anorthogonal function to u (0) n , that is (cid:82) d dx u (1) † n ( x ) u (0) n ( x ) =
0, and v (1) n is likewise assumed to be or-thogonal to v (0) n . We have defined the superconducting bandgap such that it first enters the equations at order O ( δ ). Tozeroth order, Eq. (A4) is diagonal, meaning u n and v n are com-pletely decoupled. This implies that u (0) n and v (0) n have separateenergy spectra, E pn and E hn respectively, where p and h denoteparticle and hole, and are found by solving the zeroth orderBdG equations: (cid:16) ¯ H e − h · σ − h SO · σ (cid:17) u (0) n ( x ) = E pn u (0) n ( x ) , (A9) − (cid:16) ¯ H e − h · ˜ σ + h SO · ˜ σ (cid:17) v (0) n ( x ) = E hn v (0) n ( x ) . (A10)To order first order, O ( δ ), the BdG equations read (cid:16) ¯ H e − h · σ − h SO · σ (cid:17) u (1) n + ¯ ∆ (1) v (0) n = E (1) n u (0) n + E (0) n u (1) n , (A11) − (cid:16) ¯ H e − h · ˜ σ + h SO · ˜ σ (cid:17) v (1) n + ¯ ∆ (1) ∗ u (0) n = E (1) n v (0) n + E (0) n v (1) n . (A12)Now operate on Eq. (A11) with (cid:80) m (cid:107) (cid:44) n (cid:107) (cid:82) d dx (cid:48) u (0) m ( x ) u (0) † m ( x (cid:48) ),and on Eq. (A12) with (cid:80) m (cid:107) (cid:44) n (cid:107) (cid:82) d dx (cid:48) v (0) m ( x ) v (0) † m ( x (cid:48) ). Use theorthogonality and completeness relations of u n and v n , andthe following formulas for the first order corrections are thenobtained: u (1) n ( x ) = (cid:88) m (cid:107) (cid:44) n (cid:107) (cid:82) d dx (cid:48) u (0) † m ( x (cid:48) ) ¯ ∆ (1) ( x (cid:48) ) v (0) n ( x (cid:48) ) E (0) n − E pm u (0) m ( x ) , (A13) v (1) n ( x ) = (cid:88) m (cid:107) (cid:44) n (cid:107) (cid:82) d dx (cid:48) v (0) † m ( x (cid:48) ) ¯ ∆ (1) ∗ ( x (cid:48) ) u (0) n ( x (cid:48) ) E (0) n − E hm v (0) m ( x ) , (A14)where it is implied in the notation that the perpendicular energyquantum number is equal for all involved wave functions, thatis m ⊥ = n ⊥ . The sum over m (cid:107) (cid:44) n (cid:107) is a sum over a completeset of one-dimensional eigenfunctions, with the exception of m (cid:107) = n (cid:107) , which is not included due to the assumption thatthe first order corrections are orthogonal to the zeroth orderfunctions. Keep in mind that for the perturbation expansionto be valid, the fractions in Eqs. (A13) and (A14) have to be (cid:28) ∆ ( x ), that is ∆ (1) ( x ), by using the first order re-sults for the wave functions. First expand ∆ ( x ) in its Fouriercomponents, ∆ ( x ) = (cid:88) q ∆ q sin (cid:16) k q x (cid:17) , (A15)where, as previously, k q = q π/ d . Equivalently, we may write ∆ l = d (cid:90) d dx ∆ ( x ) sin( k l x ) . (A16) ∆ ( x ) is non-zero only inside intrinsic superconductors. Thusfor a system in the x -direction with a non-superconductingregion on the interval [0 , x ), and a superconducting materialon the interval [ x , d ], we may also write ∆ ( x ) = Θ ( x − x ) ∆ ( x ) = Θ ( x − x ) (cid:88) q ∆ q sin (cid:16) k q x (cid:17) , (A17)where Θ ( x − x ) is the unit step function. It seems as thoughintroducing this step function is unnecessary, but it will comeof use quite soon. We now insert the definition of ∆ ( x ), givenin Eq. (11), into Eq. (A16), and obtain ∆ l = λ E F k F d (cid:88) n (cid:90) d dx v † n ( x ) J u n ( x ) sin( k l x ) tanh( E n / k B T ) , (A18)where J is the exchange matrix defined in Eq. (A3). Wenow do a perturbation expansion of the Fourier coe ffi cients of ∆ . Since we are working to first order, we want to find ∆ (1) l ,and ∆ (0) l = ∆ l to first order on the left hand side, and theperturbation expansions of u n and v n in Eqs. (A5) and (A6) onthe right hand side of Eq. (A18), we obtain δ ∆ (1) l = λ E F k F d (cid:88) n (cid:90) d dx (cid:16) v (0) † n ( x ) + δ v (1) † n ( x ) (cid:17) J (cid:16) u (0) n ( x ) + δ u (1) n ( x ) (cid:17) sin( k l x ) tanh( E n / k B T ) . (A19)We observe that there is no term of order O ( δ ) on either sideof the equation, which is consistent. Now insert the first orderresults from Eqs. (A13) and (A14) into (A19), expand u n and v n as in Eq. (7), and expand ∆ (1) ( x ) as in Eq. (A17). 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