Controlling the Superconducting Transition by Rotation of an Inversion Symmetry-Breaking Axis
CControlling the Superconducting Transition by Rotation of an Inversion Symmetry-Breaking Axis
Lina G. Johnsen, Kristian Svalland, and Jacob Linder
Center for Quantum Spintronics, Department of Physics,Norwegian University of Science and Technology, NO-7491 Trondheim, Norway (Dated: August 20, 2020)We consider a hybrid structure where a material with Rashba-like spin-orbit coupling is proximity coupled toa conventional superconductor. We find that the superconducting critical temperature T c can be tuned by rotatingthe vector n characterizing the axis of broken inversion symmetry. This is explained by a leakage of s -wavesinglet Cooper pairs out of the superconducting region, and by conversion of s -wave singlets into other types ofcorrelations, among these s -wave odd-frequency pairs robust to impurity scattering. These results demonstrate aconceptually different way of tuning T c compared to the previously studied variation of T c in magnetic hybrids. Introduction .— Over the last years, research on combiningsuperconducting and magnetic materials has shown that thephysical properties of the resulting hybrid structure may bedrastically altered compared to those of the individual mate-rials [1–3]. In a conventional superconductor (S), electronscombine into s -wave singlet Cooper pairs [4]. A decrease inthe s -wave singlet amplitude leads to a loss of superconduct-ing condensation energy, and thus also a suppression of thesuperconducting critical temperature, T c . Such a decrease canbe obtained by leakage of Cooper pairs into a nonsupercon-ducting material in proximity to the superconductor, and byconversion of s -wave singlets into different singlet and tripletCooper pairs. For the latter to happen, the nonsuperconduct-ing material must introduce additional symmetry-breaking.This is the case in superconductor-ferromagnet hybrids wherethe spin splitting of the energy bands of the homogeneousferromagnetic material (F) leads to creation of opposite-spintriplets [2, 3, 5].A single, homogeneous ferromagnet cannot alone causevariation in the s -wave singlet amplitude under rotations ofthe magnetization m . However, experiments [6–10] havedemonstrated that the critical temperature of F/S/F and S/F/Fstructures can be modulated by changing the relative orienta-tion of the magnetization of the ferromagnets. The misalign-ment opens all three triplet channels, leading to a strongerdecrease in the superconducting condensation energy asso-ciated with the singlet amplitude. Recent work [11–15] hasshown that the rotational invariance of the S/F structure canalso be broken by adding thin heavy normal metal layers thatboost the interfacial Rashba spin-orbit coupling. Spin-orbitcoupling (SOC) introduces inversion symmetry-breaking per-pendicular to an axis, here characterized by the vector n .While ferromagnetism only leads to spin splitting of the en-ergy bands of spin-up and spin-down electrons, Rashba SOCis in addition odd under inversion of the momentum compo-nent perpendicular to n . This raises an interesting question.While the proximity effect and accompanying change in T c ina S/F bilayer is invariant under rotations of m , is it possiblethat T c in a S/SOC bilayer is not invariant under rotations of n (see Fig. 1)?Motivated by this, we explore the possibility of T c modula-tion under reorientations of the inversion symmetry-breakingvector n in a bilayer consisting of a conventional supercon- T c θ π /2 θ n SOCS F T c θ π /2 S θ m Figure 1. In a S/F bilayer (left), T c is invariant under a rotation of m .In a S/SOC bilayer (right), the inversion symmetry perpendicular to n is broken. This opens up the possibility for a variation in T c undera rotation of n . ductor and a material with Rashba-like SOC in the bulk.We also include interfacial Rashba SOC with an inversionsymmetry-breaking vector n int perpendicular to the interface.This simple model illustrates the concept of tuning T c via ro-tation of n .When the bulk SOC is stronger than than the interfacialcontribution, we discover a suppression of T c when rotating n from an out-of-plane (OOP) to an in-plane (IP) orientation.This effect is enhanced by increasing the interfacial SOC, pro-vided that n || n int when n is OOP. The difference in T c for IPand OOP orientations of n can at least partly be accounted forby the absence of s -wave odd-frequency triplets for an OOPorientation of n . Since s -wave triplets are robust with respectto impurity scattering, we expect our prediction of an IP sup-pression of T c to be observable not only in the ballistic limitcovered by our theoretical framework, but also in the diffusivelimit. When interfacial SOC dominates, the T c modulationchanges qualitatively. The critical temperature is instead sup-pressed for anti parallel compared to parallel n and n int . Thisis explained by a reduced leakage of s -wave singlets into thenonsuperconducting region when the total SOC magnitude isincreased. Moreover, we demonstrate a variation in T c evenwhen n is varied solely in the plane of the SOC layer. The lattice Bogoliubov–de Gennes framework .— We con-sider a 3D cubic S/SOC lattice structure of size N x × N y × N z with interface normal along the x axis. We assume peri-odic boundary conditions along the y and z axes. The in-version symmetry-breaking in the nonsuperconducting layeris accounted for by the existence of a Rashba SOC term in theHamiltonian, with a constant magnitude λ . In addition, weinclude a perpendicular Rashba contribution with n int = x and magnitude λ int at the atomic layer closest to the interface.Our Hamiltonian thus accounts for both a Rashba-like SOC a r X i v : . [ c ond - m a t . s up r- c on ] A ug field in the bulk of the nonsuperconducting material, and in-terfacial Rashba SOC. We use the ballistic-limit tight-bindingBogoliubov–de Gennes framework, following a similar ap-proach to that in Refs. [15–17]. Our Hamiltonian is givenby H = − t (cid:88) (cid:104) i , j (cid:105) ,σ c † i ,σ c j ,σ − (cid:88) i ,σ µ i c † i ,σ c i ,σ − (cid:88) i U i n i , ↑ n i , ↓ − i (cid:88) (cid:104) i , j (cid:105) ,α,β c † i ,α ( λ n + λ int n int ) · (cid:110) σ × (cid:104)
12 (1 + ζ )( d i , j ) x + ( d i , j ) || (cid:105)(cid:111) α,β c j ,β . (1)Above, t is the hopping integral, µ i is the chemical potentialat lattice site i , U i > is the attractive on-site interaction giv-ing rise to superconductivity, σ is the vector of Pauli matrices, d i , j is the vector from site i to site j , and ( d i , j ) x and ( d i , j ) || are its projections onto the x axis and yz plane, respectively.If site i and j are both inside the SOC layer, ζ = 1 . Other-wise, ζ = 0 . c † i ,σ and c i ,σ are the second quantization electroncreation and annihilation operators at site i with spin σ , and n i ,σ ≡ c † i ,σ c i ,σ is the number operator. The Rashba term [18]has been symmetrized in order to allow for IP components of n while ensuring a Hermitian Hamiltonian. The supercon-ducting term is treated by a mean-field approach, assuming c i , ↑ c i , ↓ = (cid:104) c i , ↑ c i , ↓ (cid:105) + δ and neglecting terms of second orderin the fluctuations δ . The terms of the Hamiltonian are onlynonzero in their respective regions.We diagonalize the Hamiltonian numerically and computethe physical quantities of interest as outlined in the Supple-mental Material. The superconducting gap ∆ i ≡ U i (cid:104) c i , ↑ c i , ↓ (cid:105) is treated iteratively. We calculate T c by a binomial search[19] where we for each of the N T temperatures considereddecide whether the gap increases toward a superconductingstate or decreases toward a normal state from an initial guessmuch smaller than the zero-temperature gap. In this way, wedo not calculate the exact value for the gap, and we can thusget high accuracy in T c for a low number of iterations N ∆ .In order to confirm that the modulation of T c is caused byconversion of s -wave even-frequency singlets into other sin-glet and triplet correlations, we consider the even-frequency s -wave singlet amplitude S s, i ≡ (cid:104) c i , ↑ c i , ↓ (cid:105) − (cid:104) c i , ↓ c i , ↑ (cid:105) . Asa measure of the total s -wave singlet amplitude of the su-perconductor, we introduce the quantity S ≡ N x,S (cid:80) i | S s,i | ,where the sum is taken over the superconducting region only.We also define the opposite- and equal-spin odd-frequency s -wave triplet amplitudes S , i ( τ ) ≡ (cid:104) c i , ↑ ( τ ) c i , ↓ (0) (cid:105) + (cid:104) c i , ↓ ( τ ) c i , ↑ (0) (cid:105) , and S σ, i ( τ ) ≡ (cid:104) c i ,σ ( τ ) c i ,σ (0) (cid:105) [17], wherethe time-dependent electron annihilation operator is given by c i ,σ ( τ ) ≡ e iHτ c i ,σ e − iHτ [20]. The s -wave triplet amplitudeis of particular interest as it is the only triplet amplitude robustto impurity scattering. Other superconducting correlations,such as p -wave and d -wave correlations, also appear due tothe presence of SOC, as will be discussed later in this work. The superconducting critical temperature .— By followingthe above approach, we plot the critical temperature and the (a) (b) (c) T c / T c , S T c / T c , S T c / T c , S λ int = λ /4 λ /2 λ λ φ = φ = π /2 φ = π /2 π /2 π /4 π /2 π /4 π /2 π π π /2 S SOC λ n λ int n int xyz θ φ 𝒮 / 𝒮 S 𝒮 / 𝒮 S Figure 2. The T c modulation under rotation of n between IP andOOP orientations (a) is qualitatively different for λ int < λ and λ int >λ . The dashed line marks T c for λ = λ int = 0 . Depending on thematerial parameters, T c can have either its IP maxima (b) or minima(c) along the cubic axes. Notice the strong correlation between T c and the total s -wave singlet amplitude at T = T − c . Above, T c,S and S S corresponds to when the superconductor is without proximity tothe SOC layer. We have used parameters N x,S = 7 , N x,HM = 3 , N y = N z = 85 , µ S = 1 . , µ HM = 1 . , U = 2 . , λ = 0 . , N T = 20 , and N ∆ = 35 for panels (a) and (b), and N x,S = 5 , N x,HM = 2 , N y = N z = 100 , µ S = 1 . , µ HM = 1 . , U = 1 . , λ = 0 . , N T = 25 , and N ∆ = 40 for panel (c), corresponding tocoherence lengths ξ = 4 and ξ = 7 , respectively. In panels (b) and(c), λ int = 0 . total s -wave singlet amplitude in Fig. 2. To ensure that theeffect is robust, we use two different parameter sets. Theparameters are given in the figure caption. All length scalesare scaled by the lattice constant a , the SOC magnitudes arescaled by ta , and the remaining energy scales are scaled by t . For t ∼ eV and a ∼ ˚A, the order of magnitude of λ is − eVm, which corresponds well to Rashba parametersfound in several materials [21]. In order to make the systemcomputationally manageable, the lattice size and coherencelength ξ ∝ ∆ − must be scaled down, leading to an overesti-mation of ∆ and thus T c . The results in Fig. 2 must thereforebe seen mainly as qualitative.For both sets of parameters, we see a qualitatively similarbehavior for rotations of n in the xz plane, (see Fig. 2(a) forthe first set of parameters). When λ int = 0 , we find a sup-pression of T c for an IP n compared to an OOP n . When < λ int < λ , there are still maxima at the OOP directions n || n int and ( − n ) || n int , but when increasing λ int the magni-tude of the former increases while the magnitude of the latterdecreases. As long as n is parallel to n int in the OOP config-uration, the T c modulation from IP to OOP is thus enhancedby the additional interface contribution. For λ int > λ , T c ismaximal for n || n int and minimal for ( − n ) || n int . The changein T c from the parallel to the anti parallel configuration in-creases with an increasing λ int . The results presented hereonly depend on the relative orientations of n and n int , and areindependent of whether n int is directed out of or into the non-superconducting material. Notice that in all cases, nonzeroSOC increases T c compared to when λ = λ int = 0 . This isexplained by a decreased leakage of conventional singlets intothe nonsuperconducting region.From panels (b) and (c), we see that there is also an IP vari-ation in T c , that may give the strongest in-plane suppressioneither when n is oriented at a π/ angle with respect to thecubic axes, or when n is oriented along the cubic axes. As wefind a similar variation in the normal-state free energy, whichonly depends on the eigenenergy spectrum of the system, thisvarying modulation of the IP component of T c is likely to becaused by band-structure effects due to the crystal structure ofthe cubic lattice. In order to demonstrate the IP modulation,the interfacial SOC should preferably be as small as possible.To demonstrate that the T c modulation can be attributed tothe variation of the s -wave singlet amplitude in the supercon-ducting region, we plot the total s -wave singlet amplitude asa function of the IP angle of n (panels (b) and (c)). As ex-pected, it is of a similar form as the variation in T c . The slightdeviation between T c and S is caused by S being calculatedat a temperature T − c slightly below T c . We have verified thatthe variation in S and T c is similar also for panel (a).The variation in the s -wave singlet amplitude inside the su-perconducting region is caused by a reduced leakage of s -wave singlets out of the superconducting region, and con-version of s -wave singlets into other singlet and triplet cor-relations. When λ int is nonzero, the length of λ n + λ int n int changes under rotations of n , leading to an effective changein the magnitude of the SOC. Increased SOC causes an in-crease in the Fermi vector mismatch [22], due to a change inthe Fermi surface in the nonsuperconducting material. Sincethe overlap between the Fermi surfaces of the two materialsdecreases, there is an increase in the normal reflection at theinterface, as our analytical results verify. For large λ int , the T c modulation is dominated by variation in the Fermi vector mis-match. If we further investigate the triplet amplitudes presentfor different orientations of n , we find that the s -wave odd-frequency triplet amplitude is absent for n = x , i.e. when n has no IP component. For all other orientations of n , the s -wave odd-frequency anomalous triplet amplitude is nonzero.This suggests that the OOP to IP change in T c is at leastpartly caused by the increase in the s -wave triplet amplitudefrom zero when n points OOP to an increasing finite valueas the IP component of n increases. When λ int is small, sothat the length of λ n + λ int n int is approximately constant, wemay therefore expect an IP suppression of T c not only in theballistic-limit materials covered by our theoretical framework,but also in diffusive materials. Below, we perform analyticalcalculations which prove that odd-frequency pairing is absentwhen n points OOP. The continuum Bogoliubov–de Gennes framework .— In or-der to explain the absence of s -wave odd-frequency tripletswhen n is OOP, we consider two 2D continuum systems thatcan be treated analytically within the Bogoliubov–de Gennes framework [23–30]: a SOC/S bilayer with an OOP n = x ,and a F/S bilayer with magnetization m (cid:107) z . We use con-ventions similar to those in Refs. [29, 30]. Our systems arelocated in the xy plane, with interface normal along x and theinterface at x = 0 .We find the scattering wave functions Ψ n ( x ) , and ˜Ψ m ( x ) that we will use to construct the Green’s functions in the sys-tem from the time-independent Schr¨odinger equations [29–31] H ( p y )Ψ n ( x ) = ( ω + iδ )Ψ n ( x ) ,H ∗ ( p y ) ˜Ψ m ( x ) = ( ω + iδ ) ˜Ψ m ( x ) , (2)respectively, where H ( p y ) = ( − ∂ x /η + p y /η − µ )ˆ τ ˆ σ + ∆ i ˆ τ + ˆ σ y − ∆ ∗ i ˆ τ − ˆ σ y + h x ˆ τ ˆ σ x + h y ˆ τ ˆ σ y + h z ˆ τ ˆ σ z − λ ( n x p y + n y i∂ x )ˆ τ ˆ σ z + iλn z ∂ x ˆ τ ˆ σ y + λn z p y ˆ τ ˆ σ x . (3)Above, δ > is real and infinitesimal, η ≡ m/ (cid:126) , p y isthe momentum in the y direction, and h = ( h x , h y , h z ) isthe magnetic exchange field. The terms are only nonzero intheir respective regions. The four components of the scat-tering wave functions correspond to spin-up and spin-downelectrons, and spin-up and spin-down holes, respectively. Thespins are defined with respect to the z axis. The indices n and m refer to the eight possible wave functions describing scat-tering of quasiparticles incoming from the left and right. Inthe continuum model, the symmetrization of the Rashba termenters through the boundary conditions of the wave functionsat x = 0 rather than through the Hamiltonian [32]. From thescattering wave functions, we construct the retarded Green’sfunction in Nambu ⊗ spin space for x > x and x < x ,and apply boundary conditions at x = x .The even-(odd-)frequency singlet and triplet retardedanomalous Green’s functions can be written in terms of thecenter of mass coordinate X ≡ ( x + x ) / and the relativecoordinate x ≡ x − x as [29, 30] F r,E ( O )0 ( X, x, p y ; ω ) = [ F r ( X, x, p y ; ω ) + ( − ) F r ( X, − x, − p y ; ω )] / ,F r,E ( O ) i ( X, x, p y ; ω ) = [ F ri ( X, x, p y ; ω ) − (+) F ri ( X, − x, − p y ; ω )] / , (4)where i = { , , } , and F r ( X, x, p y ; ω ) = [ F r ↑↓ ( X, x, p y ; ω ) − F r ↓↑ ( X, x, p y ; ω )] / ,F r ( X, x, p y ; ω ) = F r ↑↑ ( X, x, p y ; ω ) ,F r ( X, x, p y ; ω ) = F r ↓↓ ( X, x, p y ; ω ) ,F r ( X, x, p y ; ω ) = [ F r ↑↓ ( X, x, p y ; ω ) + F r ↓↑ ( X, x, p y ; ω )] / (5)represents the singlet amplitude, the equal-spin triplet am-plitudes ( i = 1 , ), and the opposite-spin triplet amplitude( i = 3 ), respectively. The retarded anomalous Green’s func-tions F rσσ (cid:48) ( X, x, p y ; ω ) are anomalous elements of the re-tarded Green’s function in Nambu ⊗ spin space. Odd (even)frequency refers to the symmetry of the Green’s function un-der inversion of relative time, corresponding to ω changing(not changing) sign.The analytical expressions obtained for the even- and odd-frequency singlet and triplet retarded anomalous Green’sfunctions are given in the Supplemental Material. Their spa-tial symmetries are determined by their parities under inver-sion of x and p y . Although the s -wave and d x − y -wavetriplets have the same parities along the x and y axis, the pres-ence of the s -wave triplet is proven by a nonzero result whenintegrating over all spatial coordinates. Singlet and triplet amplitudes .— For the 2D SOC/S struc-ture with n = x , we find that s - and p x -wave singlets, and p y -and d xy -wave opposite-spin triplets are present. At the firstglance, it might seem strange that the odd-frequency s -wavetriplet amplitude is zero, when it is nonzero for a 2D F/S struc-ture with magnetization along the z axis. Although the Hamil-tonians of these systems are of a similar form, they allow forthe existence of different triplet amplitudes. The crucial dif-ference leading to a generation of p y - and d xy -wave tripletsin the SOC/S system rather than s - and p x -wave triplets as inthe F/S system, is the momentum dependence of the Rashbaterm.We have also investigated a 2D SOC/S structure for an IPorientation n = z numerically and find additional equal-spintriplets with an odd-frequency symmetry. For a 3D SOC/Ssystem with n OOP, the Rashba term depends on the momen-tum both along the y and z axes. Similarly as in 2D, we expectthis to allow for triplets that are odd under inversion of p y and p z . This is ultimately the reason for the absence of s -wavetriplets. Experimental realization .— We finally comment on thepossibilities of an experimental realization of the predicted T c variation upon redirecting n . We suggest cleaving a noncentrosymmetric metal, such as BiPd [33–35], in different di-rections and growing a superconductor (with a higher T c ) onthe surface, see Fig. 3(a). This requires a material that canbe cleaved along at least two axes. Alternatively, one coulddeposit superconductors on the surface of a curved non cen-trosymmetric material with a long edge (several mm), seeFig. 3(b) [36]. In both scenarios, different samples wouldhave their inversion symmetry-breaking axis in different di-rections, corresponding to a systematic rotation of n from IPto OOP. We underline that although n rotates along with thelattice in the nonsuperconducting region, the difference in T c as n changes from IP to OOP is robust. The reason is thatthe corresponding change in the proximity effect exists evenin our continuum model without the underlying lattice.In order to observe IP variations, we suggest growing a nor-mal metal (N) with a cubic lattice structure at different anglescompared to a transition metal dichalcogenide (TMDC) withIP inversion symmetry-breaking [37], see Fig. 3(c). This cor-responds to an effective IP rotation of n compared to the lat- S S SOCSOC nn SNTMDC n n
SSOC SSOC nn (a) (b)(c) n SSOC n Figure 3. For the experimental observation of the IP to OOP T c modulation, we suggest growing the superconductor on (a) differentsurfaces of a non centrosymmetric material or (b) on a curved noncentrosymmetric material. For observing IP variations, we suggest(c) growing a normal metal with a cubic lattice structure at differentangles compared to a TMDC with IP inversion symmetry-breaking,and then growing the superconductor on top. The N/TMDC bilayereffectively enables a rotation of n compared to the lattice. tice. The superconductor is grown on top of the normal metal,which should be a light element with as little interfacial SOCas possible. The ideal scenario, albeit challenging, would beto induce an in situ rotation of n in the nonsuperconducting re-gion via electric gating in different directions, that induces aninversion-symmetry-breaking field. However, since n is ro-tated inside the non centrosymmetric material, λ may in prin-ciple vary. This is not the case for our previous suggestions,since we do not rotate n inside the non centrosymmetric ma-terial, but instead change the position of the superconductor.Concluding, we have shown that the superconducting tran-sition temperature T c can be altered by rotating the inversionsymmetry-breaking axis n in a proximate material, providinga conceptually different way of controlling T c compared toprevious studies. Moreover, we have shown that when in ad-dition an interfacial spin-orbit coupling perpendicular to theinterface is present and substantial, the behavior of T c as n isvaried can change qualitatively.The authors would like to thank J. A. Ouassou, and J. W. Wellsfor helpful discussions. This work was supported by the Re-search Council of Norway through its Centres of Excellencefunding scheme Grant No. 262633 QuSpin. Supplemental Material
THE LATTICE BOGOLIUBOV–DE GENNES FRAMEWORK
If the inversion symmetry-breaking axis directed along n has an in-plane component, a Rashba Hamiltonian of the form − i (cid:88) (cid:104) i , j (cid:105) ,α,β λ i c † i ,α n · ( σ × d i , j ) α,β c j ,β (6)is in general non-Hermitian. This term is the second quantizedform of ˆ h = ( n × σ ) · λ ( x )ˆ p , where σ is the vector of Paulimatrices, λ ( x ) is the x dependent Rashba spin-orbit couplingstrength, and ˆ p = (ˆ p x , ˆ p y , ˆ p z ) = − i (cid:126) ∇ is the momentumoperator [18]. Above, d i , j is the vector from lattice site i tosite j . More generally, the symmetrized version of the firstquantized Rashba spin-orbit coupling operator is ˆ h = 12 ( n × σ ) · { λ ( x ) , ˆ p } . (7)We write this on a second quantized form as H λ = (cid:80) i , j ,α,β (cid:10) i , α (cid:12)(cid:12) ˆ h (cid:12)(cid:12) j , β (cid:11) c † i ,α c j ,β . The spatial part of the over-lap integral can be written (cid:10) i (cid:12)(cid:12) ˆ h (cid:12)(cid:12) j (cid:11) = 12 ( n × σ ) · x (cid:2)(cid:10) i (cid:12)(cid:12) λ ( x )ˆ p x (cid:12)(cid:12) j (cid:11) + (cid:10) j (cid:12)(cid:12) λ ( x )ˆ p x (cid:12)(cid:12) i (cid:11) ∗ (cid:3) + ( n × σ ) · y (cid:10) i (cid:12)(cid:12) λ ( x )ˆ p y (cid:12)(cid:12) j (cid:11) + ( n × σ ) · z (cid:10) i (cid:12)(cid:12) λ ( x )ˆ p z (cid:12)(cid:12) j (cid:11) , (8)where (cid:10) i (cid:12)(cid:12) λ ( x )ˆ p m (cid:12)(cid:12) j (cid:11) = (cid:90) ∞−∞ dm φ ∗ i ( r ) λ ( x )ˆ p m φ j ( r ) (9)for m = x, y, z . Here, φ j ( r ) ≡ φ ( r − R j ) , where R j de-scribes the position of lattice site j . We assume each φ j tobe highly localized. Then λ ( x ) can be approximated to beconstant inside each Wigner-Seitz cell, the derivative can bediscretized as ∂ m φ j ( r ) = 12 [ φ j − ˆ m ( r ) − φ j + ˆ m ( r )] , (10)and (cid:90) d r φ ∗ i ( r ) φ j ( r ) = δ i , j . (11)We also assume that λ ( x ) = λ is constant and nonzero insidethe material with spin-orbit coupling, and that λ ( x ) acts as astep function at the interface. It follows that the symmetrizedspin-orbit coupling contribution to the Hamiltonian is H λ = − i (cid:88) (cid:104) i , j (cid:105) ,α,β λc † i ,α n · (cid:110) σ × (cid:104)
12 (1 + ζ )( d i , j ) x + ( d i , j ) || (cid:105)(cid:111) α,β c j ,β . (12)Above, d i , j is decomposed into a part ( d i , j ) x perpendicularto the interface, and a part ( d i , j ) || parallel to the interface. Ifsite i and j are both inside the material with spin-orbit cou-pling, ζ = 1 , while if site i and site j are on opposite sides ofthe interface, ζ = 0 .Using the symmetrized Rashba contribution, our Hamilto-nian is given by Eq. (1) in the letter. In the following, we usea similar approach to that in Refs. [15–17]. For brevity of no-tation, we introduce i ≡ i x , i || ≡ ( i y , i z ) , and k ≡ ( k y , k z ) . We assume periodic boundary conditions in the y and z direc-tions, and introduce the Fourier transform along the y and z axes, c i ,σ = 1 (cid:112) N y N z (cid:88) k c i, k ,σ e i ( k · i || ) , (13)where the sum is taken over the allowed k inside the first Bril-louin zone. In the following, we also use the relation N y N z (cid:88) i || e i ( k − k (cid:48) ) · i || = δ k , k (cid:48) . (14)By choosing the basis B † i, k ≡ [ c † i, k , ↑ c † i, k , ↓ c i, − k , ↑ c i, − k , ↓ ] , (15)and applying the Fourier transform as well as Eq. (14), werewrite the Hamiltonian as H = H + 12 (cid:88) i,j, k B † i, k H i,j, k B i, k . (16)The constant term H is of no importance for our further cal-culations. Above, H i,j, k = (cid:15) i,j, k ˆ τ ˆ σ + (∆ i ˆ τ + − ∆ ∗ i ˆ τ − ) i ˆ σ y δ i,j − { ( λn x + λ int | n int | )[sin( k y )ˆ τ ˆ σ z − sin( k z )ˆ τ ˆ σ y ] − λ [ n z sin( k y ) − n y sin( k z )]ˆ τ ˆ σ x } δ i,j + iλ (1 + ζ )( n y ˆ τ ˆ σ z − n z ˆ τ ˆ σ y ) · ( δ i,j +1 − δ i,j − ) / , (17)where ˆ τ ± ≡ (ˆ τ ± i ˆ τ ) / , ˆ τ i ˆ σ j ≡ τ i ⊗ σ j is the Kroneckerproduct of the Pauli matrices spanning Nambu and spin space, (cid:15) i,j, k ≡ {− t [cos( k y ) + cos( k z )] − µ i } δ i,j − t ( δ i,j +1 + δ i,j − ) , (18)and ∆ i is the superconducting gap at site i . By rewriting theHamiltonian as H = H + 12 (cid:88) k W † k H k W k (19)in terms of the basis W † k ≡ [ B † , k , ..., B † i, k , ..., B † N x , k ] , (20)the Hamiltonian can be diagonalized numerically as H = H + 12 (cid:88) n, k E n, k γ † n, k γ n, k . (21)This yields eigenenergies E n, k , and eigenvectors Φ n, k givenby Φ † n, k ≡ [ φ † ,n, k · · · φ † N x ,n, k ] ,φ † i,n, k ≡ [ u ∗ i,n, k v ∗ i,n, k w ∗ i,n, k x ∗ i,n, k ] . (22)The new quasiparticle operators introduced above are relatedto the old operators by c i, k , ↑ = (cid:88) n u i,n, k γ n, k ,c i, k , ↓ = (cid:88) n v i,n, k γ n, k ,c † i, − k , ↑ = (cid:88) n w i,n, k γ n, k ,c † i, − k , ↓ = (cid:88) n x i,n, k γ n, k . (23)The eigenenergies and eigenvectors are used for computingthe singlet and triplet amplitudes and the superconducting crit-ical temperature. In finding the eigenenergies and eigenvec-tors, the superconducting gap must be calculated iteratively.The superconducting gap is defined by ∆ i ≡ U i (cid:104) c i , ↑ c i , ↓ (cid:105) .By Fourier transforming along the y and z axes, rewriting tothe new quasi-particle operators, and using that (cid:104) γ † n, k γ m, k (cid:105) = f (cid:0) E n, k / (cid:1) δ n,m , we find that the gap is given by ∆ i = − U i N y N z (cid:88) n, k v i,n, k w ∗ i,n, k [1 − f ( E n, k / , (24)where f (cid:0) E n, k / (cid:1) is the Fermi-Dirac distribution.We define the even-frequency s -wave singlet amplitude as S s, i ≡ (cid:104) c i , ↑ c i , ↓ (cid:105) − (cid:104) c i , ↓ c i , ↑ (cid:105) . The even-frequency s -wavesinglet amplitude inside the superconducting region is relatedto the superconducting gap by S s,i = 2∆ i /U i . By the samemethod as used for finding the expression for the supercon-ducting gap, we find that the odd-frequency s -wave triplet am-plitudes are given by S ,i ( τ ) = 1 N y N z (cid:88) n, k [ u i,n, k x ∗ i,n, k + v i,n, k w ∗ i,n, k ] · e − iE n, k τ/ [1 − f ( E n, k / ,S ↑ ,i ( τ ) = 1 N y N z (cid:88) n, k u i,n, k w ∗ i,n, k · e − iE n, k τ/ [1 − f ( E n, k / ,S ↓ ,i ( τ ) = 1 N y N z (cid:88) n, k v i,n, k x ∗ i,n, k · e − iE n, k τ/ [1 − f ( E n, k / . (25)Our binomial search algorithm [19] for the superconductingcritical temperature, is as follows. We divide our temperatureinterval N T times. For each of the N T iterations, we recal-culate the gap N ∆ times from an initial guess with a magni-tude ∆ / , where ∆ is the zero-temperature supercon-ducting gap. If the gap has increased towards a superconduct-ing solution after N ∆ iterations, we conclude that the currenttemperature is below T c . If the gap has decreased towards anormal-state solution, we conclude that the current tempera-ture is above T c . We measure the magnitude of the gap in the middle of the superconducting region. The advantage of thisalgorithm, is that we are not dependent upon recalculating thegap until it converges. The parameter N ∆ must only be largeenough that the increase or decrease in ∆ i at site i = N x,S / reflects the overall behavior of the gap under recalculation.When we choose an initial guess so that the gap as a functionof lattice site has a similar shape as for the gap very close to T c , it more likely that the gap increases for all lattice sites, ordecreases for all lattice sites, under recalculation. We can thenget a high accuracy with a low N ∆ .The superconducting coherence length is given by ξ ≡ (cid:126) v F /π ∆ [4], where v F ≡ (cid:126) dE k dk (cid:12)(cid:12) k = k F is the normal-stateFermi velocity [4], E k is the normal-state eigenenergies whenintroducing periodic boundary conditions along all three axes,and k F is the corresponding Fermi momentum averaged overthe Fermi surface. We round ξ down to the closest integernumber of lattice points. THE CONTINUUM BOGOLIUBOV–DE GENNESFRAMEWORK
The continuum Bogoliubov–de-Gennes framework [23–30]allows us to obtain analytical expressions for the singlet andtriplet retarded anomalous Green’s functions of the 2D SOC/Ssystem with n = x and the 2D F/S system with m || z . Wehave not given these expressions in the letter, as we are mainlyinterested in their symmetries under spatial inversion. Here,we provide the analytical expressions for the wave functionsand the singlet and triplet retarded anomalous Green’s func-tions for these two systems, as well as the wave functions forthe 2D SOC/S system with n = z . The scattering wave functions
We find expressions for the scattering wave func-tions Ψ n ( x ) and ˜Ψ m ( x ) by using the time-independentSchr¨odinger equations given in Eq. (2) in the letter. The in-dices n and m refers to the different possible scattering pro-cesses. These contain reflection and transmission coefficientsthat are determined by the boundary conditions at the inter-face. The wave functions Ψ n ( x ) satisfies the boundary con-ditions [32] [Ψ n ( x )] x =0 + = [Ψ n ( x )] x =0 − [ˆ v Ψ n ( x )] x =0 + = [ˆ v Ψ n ( x )] x =0 − (26)where ˆ v ≡ ∂H ( p y ) /∂ ( − i∂ x ) is the velocity operator. Theconjugate wave functions ˜Ψ m ( x ) satisfies a similar set ofboundary conditions with ˆ v ≡ ∂H ∗ ( p y ) /∂ ( − i∂ x ) .In the following, we give expressions for the scatteringwave functions inside a 2D superconductor, a 2D materialwith Rashba-like spin-orbit coupling for n = x and n = z ,and a 2D ferromagnet with h = h z , treating each materialseparately. We choose the superconducting region to be lo-cated at x > , while the non superconducting region is lo-cated at x < . The superconducting region
The scattering wave functions on the superconducting sideof the interface are Ψ n ( x ) =Ψ R in ,n ( x )+ c n, [ u v ] T e iq + x x + c n, [0 − u v T e iq + x x + d n, [0 − v u T e − iq − x x + d n, [ v u ] T e − iq − x x , x > , (27) ˜Ψ m ( x ) = ˜Ψ R in ,m ( x )+ ˜ c m, [ u v ] T e iq + x x + ˜ c m, [0 − u v T e iq + x x + ˜ d m, [0 − v u T e − iq − x x + ˜ d m, [ v u ] T e − iq − x x , x > , (28)where the quasi-particles incoming from the right are de-scribed by the wave functions Ψ R in , ( x ) = [ u v ] T e − iq + x x Ψ R in , ( x ) = [0 − u v T e − iq + x x , Ψ R in , ( x ) = [0 − v u T e iq − x x , Ψ R in , ( x ) = [ v u ] T e iq − x x , (29)and ˜Ψ R in , ( x ) = [ u v ] T e − iq + x x , ˜Ψ R in , ( x ) = [0 − u v T e − iq + x x , ˜Ψ R in , ( x ) = [0 − v u T e iq − x x , ˜Ψ R in , ( x ) = [ v u ] T e iq − x x . (30) Ψ R in , ( x ) = Ψ R in , ( x ) = Ψ R in , ( x ) = Ψ R in , ( x ) =˜Ψ R in , ( x ) = ˜Ψ R in , ( x ) = ˜Ψ R in , ( x ) = ˜Ψ R in , ( x ) = 0 . Wereserve the indices n, m = { , , , } for scattering pro-cesses with particles or quasi-particles scattering at the inter-face from the left. Above, q ± x = {− p y + η [ µ ± (cid:112) ( ω + iδ ) − | ∆ | ] } / (31)are the allowed k x values, and u ≡
12 [1 + (cid:112) ( ω + iδ ) − | ∆ | / ( ω + iδ )] , (32) v ≡
12 [1 − (cid:112) ( ω + iδ ) − | ∆ | / ( ω + iδ )] . (33) The region with Rashba spin-orbit coupling, n = x The scattering wave functions on the side of the interfacewith Rashba-like spin-orbit coupling are Ψ n ( x ) =Ψ L in ,n ( x )+ a n, [1 0 0 0] T e − ik e, ↑ x x + a n, [0 1 0 0] T e − ik e, ↓ x x + b n, [0 0 1 0] T e ik h, ↑ x x + b n, [0 0 0 1] T e ik h, ↓ x x , x < , (34) ˜Ψ m ( x ) = ˜Ψ L in ,m ( x )+ ˜ a m, [1 0 0 0] T e − ik e, ↑ x x + ˜ a m, [0 1 0 0] T e − ik e, ↓ x x + ˜ b m, [0 0 1 0] T e ik h, ↑ x x + ˜ b m, [0 0 0 1] T e ik h, ↓ x x , x < , (35)if n = x . The particles incoming from the left are describedby Ψ L in , ( x ) = [1 0 0 0] T e ik e, ↑ x x , Ψ L in , ( x ) = [0 1 0 0] T e ik e, ↓ x x , Ψ L in , ( x ) = [0 0 1 0] T e − ik h, ↑ x x , Ψ L in , ( x ) = [0 0 0 1] T e − ik h, ↓ x x , (36)and ˜Ψ L in , ( x ) = [1 0 0 0] T e ik e, ↑ x x , ˜Ψ L in , ( x ) = [0 1 0 0] T e ik e, ↓ x x , ˜Ψ L in , ( x ) = [0 0 1 0] T e − ik h, ↑ x x , ˜Ψ L in , ( x ) = [0 0 0 1] T e − ik h, ↓ x x . (37) Ψ L in , ( x ) = Ψ L in , ( x ) = Ψ L in , ( x ) = Ψ L in , ( x ) =˜Ψ L in , ( x ) = ˜Ψ L in , ( x ) = ˜Ψ L in , ( x ) = ˜Ψ L in , ( x ) = 0 . We re-serve the indices n, m = { , , , } for scattering processeswith particles incoming from the right. Above, k e ( h ) , ↑ ( ↓ ) x = {− p y + η [ µ ± ( ω + iδ ± (cid:48) λp y )] } / (38)are the allowed k x values. ± correspond to electrons andholes, respectively, while ± (cid:48) correspond to spin up and spindown, respectively. The ferromagnetic scattering wave functions
The scattering wave functions for a ferromagnet with h = h z , are on the same form as for a material with Rashba spin-orbit coupling where n = x , and are thus given by Eqs. (34),(35), (36), and (37). The allowed k x values are in this casegiven by k e ( h ) , ↑ ( ↓ ) x = {− p y + η [ µ ± ( ω + iδ ) ∓ (cid:48) h ] } / . (39) ± refers to electrons and holes, respectively, and ∓ (cid:48) refers tospin up and spin down, respectively. The region with Rashba spin-orbit coupling, n = z The scattering wave functions on the side of the interfacewith Rashba-like spin-orbit coupling are Ψ n ( x ) = Ψ L in ,n ( x )+ a n, [ 1 ie iφ T e − ik e, + x x + a n, [ − ie iφ T e − ik e, − x x + b n, [0 0 1 ie − iφ ] T e ik h, − x x + b n, [0 0 − ie − iφ ] T e ik h, + x x , x < , (40) ˜Ψ m ( x ) = ˜Ψ L in ,m ( x )+ ˜ a m, [ 1 ie iφ T e − ik e, + x x + ˜ a m, [ − ie iφ T e − ik e, − x x + ˜ b m, [0 0 1 ie − iφ ] T e ik h, − x x + ˜ b m, [0 0 − ie − iφ ] T e ik h, + x x , x < , (41)if n = z . The particles incoming from the left are describedby Ψ L in , ( x ) = [ 1 ie iφ T e ik e, + x x , Ψ L in , ( x ) = [ − ie iφ T e ik e, − x x , Ψ L in , ( x ) = [0 0 1 ie − iφ ] T e − ik h, − x x , Ψ L in , ( x ) = [0 0 − ie − iφ ] T e − ik h, + x x (42)and ˜Ψ L in , ( x ) = [ 1 ie iφ T e ik e, + x x , ˜Ψ L in , ( x ) = [ − ie iφ T e ik e, − x x , ˜Ψ L in , ( x ) = [0 0 1 ie − iφ ] T e − ik h, − x x , ˜Ψ L in , ( x ) = [0 0 − ie − iφ ] T e − ik h, + x x . (43) Ψ L in , ( x ) = Ψ L in , ( x ) = Ψ L in , ( x ) = Ψ L in , ( x ) =˜Ψ L in , ( x ) = ˜Ψ L in , ( x ) = ˜Ψ L in , ( x ) = ˜Ψ L in , ( x ) = 0 . Above, k e ( h ) , +( − ) x = k e ( h ) , +( − ) cos( φ ) are the allowed k x values,where k e ( h ) , +( − ) = { [( λη/ + η ( µ ± ω + iδ )] / ± ± (cid:48) λη/ } / . (44) ± correspond to electrons and holes, respectively, and ± (cid:48) cor-respond to the two different spin-mixed states. We define k e ( h ) , +( − ) x to be positive by setting φ ∈ [ − π/ , π/ . The singlet and triplet retarded anomalous Green’s functions
From the scattering wave functions, we construct the re-tarded Green’s function in Nambu ⊗ spin space [29, 30], G r ( x > x , p y ; ω ) = (cid:88) n,m =1 α nm Ψ n ( x , p y ) ˜Ψ Tm +4 ( x , p y ) , (45) G r ( x < x , p y ; ω ) = (cid:88) n,m =1 β nm Ψ n +4 ( x , p y ) ˜Ψ Tm ( x , p y ) . (46)The coefficients α nm and β nm are found from the boundaryconditions of the retarded Green’s function at x = x [29,30], [ G r ( x > x , p y ; ω )] x = x = [ G r ( x < x , p y ; ω )] x = x , [ ∂ x G r ( x > x , p y ; ω )] x = x − [ ∂ x G r ( x < x , p y ; ω )] x = x = η ˆ τ ˆ σ , (47)We rewrite to the center of mass coordinate X ≡ ( x + x ) / ,and the relative coordinate x ≡ x − x , and calculate theeven- and odd-frequency singlet and triplet anomalous contri-butions to the retarded Green’s functions according to Eqs. (4)and (5) in the letter. We use that F r ( X, x, p y ; ω ) ≡ [ G r , ( X, x, p y ; ω ) − G r , ( X, x, p y ; ω )] / ,F r ( X, x, p y ; ω ) ≡ G r , ( X, x, p y ; ω ) ,F r ( X, x, p y ; ω ) ≡ G r , ( X, x, p y ; ω ) ,F r ( X, x, p y ; ω ) ≡ [ G r , ( X, x, p y ; ω )+ G r , ( X, x, p y ; ω )] / . (48) The SOC/S system, n = x For simplicity of notation, we define k e ≡ k e, ↑ x , k e ≡ k e, ↓ x , k h ≡ k h, ↓ x , and k h ≡ k h, ↑ x . On the side of the interfacewith Rashba-like spin-orbit coupling, where x , x < , thenonzero even- and odd-frequency singlet and triplet retardedanomalous Green’s functions are given by F r,E ( X, x, p y ; ω ) = η i u v ( q + x + q − x ) (cid:88) l =1 , D l e − i ( k el − k hl ) X cos[( k el + k hl ) x/ ,F r,O ( X, x, p y ; ω ) = η u v ( q + x + q − x ) (cid:88) l =1 , D l e − i ( k el − k hl ) X sin[( k el + k hl ) x/ ,F r,E ( X, x, p y ; ω ) = η i u v ( q + x + q − x ) (cid:88) l =1 , ( − l − D l e − i ( k el − k hl ) X cos[( k el + k hl ) x/ ,F r,O ( X, x, p y ; ω ) = η u v ( q + x + q − x ) (cid:88) l =1 , ( − l − D l e − i ( k el − k hl ) X sin[( k el + k hl ) x/ , (49)where D l ≡ u ( k el + q + x )( k hl + q − x ) + v ( k hl − q + x )( − k el + q − x ) . (50)On the superconducting side of the interface, where x , x > , the nonzero even- and odd-frequency singlet and triplet re-tarded anomalous Green’s functions are given by F r,E ( X, x, p y ; ω ) = − η i u v ( u − v ) (cid:26) e i ( q + x − q − x ) X cos[( q + x + q − x ) x/ (cid:88) l =1 , D l ( k el + k hl ) − (cid:18) q + x e iq + x | x | + 1 q − x e − iq − x | x | (cid:19) − (cid:88) l =1 , (cid:18) E l D l q + x e iq + x X + F l D l q − x e − iq − x X (cid:19)(cid:27) ,F r,O ( X, x, p y ; ω ) = η u v e i ( q + x − q − x ) X sin[( q + x + q − x ) x/ (cid:88) l =1 , D l ( k el + k hl ) ,F r,E ( X, x, p y ; ω ) = − η i u v ( u − v ) (cid:26) e i ( q + x − q − x ) X cos[( q + x + q − x ) x/ (cid:88) l =1 , ( − l − D l ( k el + k hl ) − (cid:88) l =1 , ( − l − (cid:18) E l D l q + x e iq + x X + F l D l q − x e − iq − x X (cid:19)(cid:27) ,F r,O ( X, x, p y ; ω ) = η u v e i ( q + x − q − x ) X sin[( q + x + q − x ) x/ (cid:88) l =1 , ( − l − D l ( k el + k hl ) , (51) where E l ≡ u ( k el − q + x )( − k hl − q − x ) + v ( k hl + q + x )( k el − q − x ) ,F l ≡ u ( k el + q + x )( − k hl + q − x ) + v ( k hl − q + x )( k el + q − x ) . (52)There are no equal-spin triplets in the system. The F/S system
The even- and odd-frequency singlet and triplet retardedanomalous Green’s functions of the F/S system are given bythe same expressions as for the SOC/S system with n = x if we let F r,E ( X, x, p y ; ω ) ↔ F r,O ( X, x, p y ; ω ) in Eqs. (49)and (51). There are no equal-spin triplets in the system. The symmetries of the singlet and triplet retarded anomalousGreen’s functions
Finally, we investigate the spatial symmetries of the sin-glet and triplet retarded anomalous Green’s functions of theSOC/S systems with n = x and n = z and the F/S sys-tem with m || z . P x is inversion of the relative x coordinate, x → − x . P y is inversion of the momentum along the y axis, p y → − p y . P is total spatial inversion, and must be 1 for F E and F O , and -1 for F O and F E , according to the Pauliprinciple. For P = 1 , we may have P x = P y = 1 , which de-scribes an s - or a d x − y -wave amplitude, or P x = P y = − ,which describes a d xy -wave amplitude. For P = − , we mayhave P x = 1 and P y = − , which describes a p y -wave ampli-tude, or P x = − and P y = 1 , which describes a p x -waveamplitude. Considering P, P x , and P y is not sufficient fordetermining whether a Green’s function has an s -wave or a d x − y -wave symmetry. In order to prove the presence of s -wave singlets and triplets, we apply the Fourier transform, F r,E ( O )0(3) ( X, p x , p y ; ω )= (cid:90) ∞∞ dx F r,E ( O )0(3) ( X, x, p y ; ω ) e − ip x x , (53)and set p x and p y to zero, which is equivalent to integratingover all spatial coordinates. The SOC/S system, n = x The symmetries of the Green’s functions in Eqs. (49) and(51) under P x , P y and P are given in Table I. We see fromthe table that F r,E can represent s - and d x − y -wave sin-glets, F r,O represents p x -wave singlets, F r,E represents a p y -wave opposite-spin triplets, and F r,O represents d xy -waveopposite-spin triplets. By integrating over all of space, we findthat s -wave singlets are present.0 P x P y P F r,E F r,O − − F r,E − − F r,O − − Table I. The above table shows the parities of the SOC/S system with n = x under x → − x (P x ), p y → − p y (P y ), and total spatial inver-sion (P) for the nonzero singlet and triplet even- and odd-frequencyretarded anomalous Green’s functions given in Eqs. (49) and (51). The SOC/S system, n = z The symmetries of the Green’s functions of the SOC/S sys-tem for n = z are shown in Table II. These were found nu-merically by the same approach as for the two other systems.We see that the same singlet and opposite-spin triplet ampli-tudes are present as for n = x . In addition, we have nonzeroequal-spin triplet amplitudes, that are a mix of triplet ampli-tudes with different symmetries under P x and P y . F r,E and F r,E are therefore a mix of p x - and p y -wave even-frequencytriplets, while F r,O and F r,O are a mix s - and d -wave triplets. P x P y P F r,E F r,O − − F r,E − − F r,O − − F r,E - - − F r,O - - F r,E - - − F r,O - - Table II. The above table shows the parities of the SOC/S systemwith n = z under x ↔ − x (P x ), p y → − p y (P y ), and total spa-tial inversion (P) for the singlet and triplet even- and odd-frequencyretarded anomalous Green’s functions present in the system. In ad-dition to the singlets and triplets present for n = x shown in Table I,we have nonzero equal-spin triplet amplitudes with mixing (-) of thepossible symmetries in P x and P y . The F/S system
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