Reentrant superconductivity in proximity to a topological insulator
T. Karabassov, A. A. Golubov, V. M. Silkin, V. S. Stolyarov, A. S. Vasenko
RReentrant superconductivity in proximity to a topological insulator
T. Karabassov, A. A. Golubov,
2, 3
V. M. Silkin,
4, 5, 6
V. S. Stolyarov,
3, 7 and A. S. Vasenko
1, 8, ∗ HSE University, 101000 Moscow, Russia Faculty of Science and Technology and MESA + Institute for Nanotechnology,University of Twente, 7500 AE Enschede, The Netherlands Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4,San Sebasti´an/Donostia, 20018 Basque Country, Spain Departamento de F´ısica de Materiales, Facultad de Ciencias Qu´ımicas,UPV/EHU, 20080 San Sebasti´an, Basque Country, Spain IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain Dukhov Research Institute of Automatics (VNIIA), 127055 Moscow, Russia I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute,Russian Academy of Sciences, 119991 Moscow, Russia (Dated: December 4, 2020)In the following paper we investigate the critical temperature T c behavior in the two-dimensionalS/TI (S denotes superconductor and TI - topological insulator) junction with a proximity induced in-plane helical magnetization in the TI surface. The calculations of T c are performed using the generalself-consistent approach based on the Usadel equations in Matsubara Green’s functions technique.We show that the presence of the helical magnetization leads to the nonmonotonic behavior of thecritical temperature as a function of the topological insulator layer thickness. PACS numbers: 74.25.F-, 74.45.+c, 74.78.Fk
I. INTRODUCTION
Topological state of matter has been receiving a lotof attention for the past decade.
Particularly, three-dimensional topological insulators (3D TI) have large po-tential for fault tolerant quantum computation.
Thisis possible due to strong spin-orbit coupling (SOC) andtime reversal symmetry that take place in such materi-als. There are special topologically protected states onthe surface of the 3D topological insulator. These sur-face states are Dirac helical states, i.e., their spin andmomentum are coupled in a well defined way resulting ina spin-momentum locking effect. Interesting transportproperties are revealed when topological insulator andsuperconductor are in proximity to each other, forminga hybrid structure. In such hybrids in the presence ofa magnetic field or a magnetic moment of an adjacentferromagnet, zero energy Majorana modes can arise.
The proximity effect that takes place in supercon-ducting hybrid structures can lead to various phenomenaoccurring near interfaces. For instance, critical temper-ature T c behaves nonmonotonically as a function of dif-ferent system parameters in S/F bilayers with uniformmagnetization and multilayered S/F spin-valves with amagnetization misalignment in F layers . Particularly, T c demonstrates reentrant behavior under certain rangeof parameters. Such behavior originates from nontrivialdependence of Cooper pair wavefunction, which can alsoresult in oscillating Josephson critical current , den-sity of states and critical temperature in S/F/Sjunctions.According to the theory developed in Refs. 10 and 37,there are no Josephson critical current oscillations in hy- brid S/TI/S structures with a proximity induced uniformin-plane field in the TI layer. At the same time, as pre-dicted in Ref. 37, the critical current demonstrates oscil-latory behavior in case when the TI surface with helicalmagnetization serves as a weak link. Following the re-sults of Zuyzin et al. , the observation of 0 − π phasetransitions in the critical supercurrent may imply non-trivial critical temperature behavior and in particular thereentrant T c behavior in the S/TI junction. Therefore,investigation of the T c in the hybrids with both spin-orbit coupling and helical magnetization is essential forfurther understanding of the underlying physics and po-tential future applications. As far as we know, the criticaltemperature in S/TI structures has not been studied yet.Spin-orbit effects have been discussed actively in theframework of the quasiclassical Green’s functions ap-proach in layered structures. Recently, the gen-eralized quasiclassical theory was developed for a two-dimensional system with SOC and an exchange fieldboth much greater than the disorder strength. It hasbeen shown that spontaneous supercurrent can flow in aJosephson junction, where magnetized superconductorsare weakly coupled through the surface of 3D TI. The goal of this work is to provide a quantitative in-vestigation of the critical temperature in the S/TI hy-brid structure as a function of its parameters applyingthe quasiclassical Green’s function approach. The helicalmagnetization pattern considered in this paper is similarto one previously studied in S/F bilayers with nonuni-form spiral magnets.
Particularly, the superconduct-ing spin valve consisting of a superconducting layer anda spiral magnetic was proposed for the spintronic appli-cation, using re-orientation of the spiral direction as a a r X i v : . [ c ond - m a t . s up r- c on ] D ec method of the spin-valve control. However, natureof the effects that appear in our structure is different,since they are caused not only by in-plane helical mag-netization pattern, but also by the spin-orbit coupling.The paper is organized as follows. In Sec. II, we for-mulate the theoretical model and basic equations for thecases of h ( y ) and h ( x ) helical magnetizations. In Sec. IIIwe present the results of the critical temperature calcula-tions using the single-mode approximation. The resultsare concluded in Sec. IV. II. MODEL
In this work we consider the 2D nanostructure, which isdepicted in the Fig. 1. It consists of superconductor S ofthickness d s and topological insulator (TI) of thickness d n with a proximity induced helical magnetization patternof the following types: h ( y ) = h (cos Qy, sin
Qy, , (1) h ( x ) = h (cos Qx, sin
Qx, , (2)where Q = 2 π/λ and λ determines the actual pattern ofhelical magnetization. It is important to note that weconsider the variations of the magnetization h in the x-y plane. Similar helical pattern with a period λ ≈ The orientation of the structure is alongthe x direction. In order to observe the inverse proxim-ity effect the superconductor must be two-dimensional.Such disordered homogeneous superconducting 2D filmscan be obtained with the help of modern depositiontechniques .To calculate the critical temperature T c ( d n ) of thisstructure we assume the diffusive limit, when the elas-tic scattering length (cid:96) is much smaller than the coher-ence length, and use the framework of the linearizedUsadel equations for the S and TI layers in Matsub-ara representation. . We perform the calculations inthe low proximity limit expanding the Green’s functionaround the bulk solution,ˆ g = (cid:18) sgn ω n f − ¯ f − sgn ω n . (cid:19) (3)Such limit is experimentally feasible and can be easilyachieved in the vicinity of the superconducting criticaltemperature T c or in a hybrid structures with low trans-parent interfaces. A. Helical magnetization h(y)
In this subsection we establish the equations for themagnetization pattern evolving along the S/TI interfaceindicated in Eq. (1), i. e. in y direction. Since the lowproximity limit is assumed, near T c the normal Green’sfunction in a superconductor is g s = sgn ω n , and the Usadel equation for the anomalous Green’s function f s take the following form. In the S layers (0 < x < d s ) itreads ξ s πT cs (cid:18) ∂ ∂x + ∂ ∂y (cid:19) f s − | ω n | f s + ∆ = 0 . (4)In the TI layer we consider the Usadel equation derivedin Ref. 37, (cid:18) ∂∂x − iα h y ( y ) (cid:19) f T + (cid:18) ∂∂y + 2 iα h x ( y ) (cid:19) f T = | ω n | ξ n πT cs f T (5)Since we consider the dirty limit, the spinless Green’sfunction matrix ˆ g s is used in our calculations, whereasthe spin texture is contained in the matrix ˇ g ( n F ) =ˆ g (1 + ˆ η · n F ) /
2, where n F = p F /p F and ˆ η = ( − σ , σ ).The spin-momentum locking effect can be seen from thespin matrix ˇ g , so that spin and momentum are alwaysfixed at the right angle.Finally, the self-consistency equation reads, ∆ ln T cs T = πT (cid:88) ω n (cid:18) ∆ | ω n | − f s (cid:19) . (6)In Eqs. (4)-(6) ξ s = (cid:112) D s / πT cs , ξ n = (cid:112) D n / πT cs , ω n = 2 πT ( n + ), where n = 0 , ± , ± , . . . are the Mat-subara frequencies, T cs is the critical temperature of thesuperconductor S, and f s ( T ) denotes the singlet compo-nents of anomalous Green function in the S(TI) region(we assume (cid:126) = k B = 1).As far as our 2D system is periodic in y directionand large values of helical magnetization parameter Q are considered such that λ (cid:28) W f , we can expand theanomalous Green functions using the Fourier series. Thefunction f T then can be written as, f T ( x, y ) = + ∞ (cid:88) p = −∞ f ( p ) T ( x ) e ipQy . (7)The Usadel equation in the TI layer for the amplitudes f ( p ) T then takes the following form, (cid:18) ∂∂x − iα h y ( y ) (cid:19) f ( p ) T − p Q f ( p ) T − pQh x ( y ) α f ( p ) T = (cid:18) | ω n | ξ n πT cs + 4 h x ( y ) α − ih (cid:48) x ( y ) α (cid:19) f ( p ) T , (8)where, h (cid:48) x ( y ) is a derivative of h x along the y direction.Whereas, in the S layer the singlet function f s as well as∆ can also be expanded into a Fourier series, f s ( x, y ) = + ∞ (cid:88) p = −∞ f ( p ) s ( x ) e ipQy , (9)∆( x, y ) = + ∞ (cid:88) p = −∞ ∆ ( p ) ( x ) e ipQy . (10) FIG. 1. (Color online) (a) Schematic of a 3D topological insulator (TI) - superconductor (S) junction with a proximityinduced helical magnetization pattern. The magnetization vector is given by h ( y ) = h (cos Qy, sin
Qy,
0) (b) and h ( x ) = h (cos Qx, sin
Qx,
0) (c). The junction resides in the x-y plane and the S/TI interface lie in the y direction at x = 0. d n and d s are the thicknesses of TI and S layers respectively, while W f is the width of the junction. γ B is a transparency parameterwhich is proportional to the interface resistance. The amplitudes f ( p ) s obey the following equation, ξ s (cid:32) ∂ f ( p ) s ∂x − p Q f ( p ) s − | ω n | ξ s πT cs f ( p ) s (cid:33) + ∆ ( p ) πT cs = 0 . (11)The self-consistency equation for the Fourier amplitudesin the superconductor can be written as,∆ ( p ) ln T cs T = πT (cid:88) ω n (cid:18) ∆ ( p ) | ω n | − f ( p ) s (cid:19) . (12)From the equations above, it is clear that the ampli-tudes of the Fourier series are decoupled in the vicinity ofthe critical temperature. Therefore, each Fourier compo-nent p satisfies certain Usadel equation and the bound-ary conditions. Moreover, every single Fourier harmonic p of anomalous Green function f ( p ) s and pair amplitude∆ ( p ) determines particular T c through the correspondinggap equation. However, the physical solution is the one,which gives the highest critical temperature T c , i. e. thesolution is energetically favorable.We also need to supplement the equations above withproper boundary conditions to solve the problem. Weassume low transparency limit of the interface betweentopological insulator (TI) and superconducting layer (S).It is also assumed that spin is conserved when the elec-trons tunnel across the interface, whereas momentum isnot conserved. For the Fourier harmonics of the solution f ( p ) that we have introduced above taking all the simpletransformations into account, the boundary conditions at x = 0 take the form, γ B ξ n (cid:18) ∂∂x − ih y ( y ) α (cid:19) f T (0) = f s (0) − f T (0) , (13) γξ n (cid:18) ∂∂x − ih y ( y ) α (cid:19) f T (0) = ξ s ∂f s (0) ∂x . (14)Here we omitted the component index p . The param-eter γ B = R b σ n /ξ n is transparency parameter which is the ratio of resistance per unit area of the surface of thetunneling barrier to the resistivity of the TI layer and de-scribes the effect of the interface barrier . In (14) thedimensionless parameter γ = ξ s σ n /ξ n σ s determines thestrength of suppression of superconductivity in the S lay-ers near the S/TI interface compared to the bulk (inverseproximity effect). No suppression occurs for γ = 0, whilestrong suppression takes place for γ (cid:28)
1. Here σ s ( n ) isthe normal-state conductivity of the S(TI) layer. Theseboundary conditions should also be supplemented withvacuum conditions at the edges ( x = − d n and x = + d s ), ∂f s ( d s ) ∂x = 0 , (cid:18) ∂∂x − ih y ( y ) α (cid:19) f T ( − d n ) = 0 . (15)The solution of the equation (8) can be found in theform, f ( p ) T = C ( ω n ) cosh κ p,y ( x + d n ) exp (cid:20) i h y ( y ) α ( x + d n ) (cid:21) , (16)where, k p,y = (cid:115) | ω n | ξ n πT cs + 4 α h x ( y ) − ih (cid:48) x ( y ) α + Q p ,Q p = p Q + 4 pQh x ( y ) α . Here C ( ω n ) is the coefficient, which is found from theboundary conditions and the wavevector acquires ad-ditional imaginary term due to fast oscillations of theanomalous Green’s function along the y direction com-pared to the case of uniform magnetization ( Q = 0). Theintroduced solution to the equation automatically satis-fies the vacuum boundary conditions (15).As far as ∆ is assumed to be real valued function, wewrite our equations for anomalous Green’s functions inreal form. Also we consider only positive Matsubara fre-quencies ω n . Following the standard procedure we obtainfinal set of equations which are sufficient to calculate crit-ical temperature as a function of d n .Using the boundary conditions (13)-(14) we would liketo write the problem in a closed form with respect to theGreen’s function f s . At x = 0 the boundary conditionscan be written as: ξ s ∂f s (0) ∂x = γγ b + A pT ( ω n ) f s (0) , (17)where A pT ( ω n ) = 1 k p,y coth k p,y d n . The boundary condition (17) is complex. In order torewrite it in a real form, we use the following relation, f ± = f ( ω n ) ± f ( − ω n ) . (18)According to the Usadel equation (4), there is a symme-try relation f ( − ω n ) = f ∗ ( ω n ) which implies that f + is areal while f − is a purely imaginary function. Then, werewrite the Usadel equation in the S layer in terms of f + s and f − s utilizing symmetry relation (18). Since the pairpotential ∆ is considered to be real valued function, wecan find the solution analytically in the Usadel equationfor the imaginary function f − s . Using the solution foundanalytically, it is possible to derive the complex boundarycondition (17) in real form for the function f + s , ξ s ∂f + s (0) ∂x = W ( p ) ( ω n ) f + s (0) , (19)where we used the notations, W ( p ) ( ω n ) = γ A ps ( γ b + Re A pT ) + γA ps | γ b + A pT | + γ ( γ b + Re A pT ) , (20) A ps = κ ps tanh κ ps d s , A pT ( ω n ) = 1 k p,y coth k p,y d n ,κ ps = (cid:115) Q p + | ω n | ξ s πT cs . In the same way we rewrite the self-consistency equa-tion for ∆ in terms of symmetric function f + s consideringonly positive Matsubara frequencies,∆ ( p ) ln T cs T = πT (cid:88) ω n > (cid:18) ( p ) ω n − f ( p )+ s (cid:19) , (21)as well as the Usadel equation in the superconducting Slayer, ξ s (cid:32) ∂ f ( p )+ s ∂x − κ ps f ( p )+ s (cid:33) + 2∆ ( p ) πT cs = 0 . (22)To calculate the critical temperature in the system con-sidered, we use the equations (19)-(22), together with thevacuum boundary condition (15) for the Fourier compo-nents f ( p )+ s . B. Helical magnetization h(x)
Here we consider the system consisting of a supercon-ductor and topological insulator with helical magnetiza-tion pattern presented in the Eq. (2). In this case the Us-adel equation should be rewritten in terms of magnetiza-tion h ( x ). We assume that the anomalous Green’s func-tion does not depend on y coordinate and thus the corre-sponding derivatives are neglected. The Usadel equationin the TI layer then takes the following form, (cid:18) ∂∂x − ih y ( x ) α (cid:19) f − h x ( x ) α f = | ω n | ξ n πT cs f (23)In order to rewrite the Eq. (23) in real form we introducethe following anzatz, f ( x ) = f L ( x ) exp (cid:20) − i h αQ cos Qx (cid:21) . (24)Inserting this substitution into the Eq. (23), we obtainthe equation for real valued function in the TI layer, ∂ f L ∂x = (cid:18) | ω n | ξ n πT cs + 4 h cos Qxα (cid:19) f L (25)For this system we utilize the same boundary conditionsas in previous subsection and express them in real formusing symmetry relation (18). After the substitutions theboundary conditions take the form, γ B ξ n ∂f L (0) ∂x = C f + s (0) − f L (0) , (26) γξ n ∂f L (0) ∂x = ξ s C ∂f + s (0) ∂x . (27)where C = cos (2 h /αQ ). Finally, the boundary condi-tions at the free edges at x = d s and x = − d n , ∂f s ( d s ) ∂x = 0 , ∂f T ( − d n ) ∂x = 0 . (28)Similarly, we introduce the self-consistency equationfor ∆ in terms of symmetric function f + s treating onlypositive Matsubara frequencies,∆ ln T cs T = πT (cid:88) ω n > (cid:18) ω n − f + s (cid:19) , (29)and the Usadel equation in the S layer, ξ s πT cs ∂ f + s ∂x − ω n f + s + 2∆ = 0 , (30)Since the Eq. (23) can not be solved analytically, to ob-tain the critical temperature T c the whole set of equations(25)-(30) must be calculated numerically. C. Single-mode approximation
In this subsection we present the single mode approx-imation method. The solution of the problems (19)-(22)and (25)-(30) can be searched in the form of the followinganzatz, f + s ( x, ω n ) = f ( ω n ) cos (cid:18) Ω x − d s ξ s (cid:19) , (31a)∆( x ) = δ cos (cid:18) Ω x − d s ξ s (cid:19) , (31b)where δ and Ω do not depend on ω n . The above solutionautomatically satisfies boundary condition (15) at x = d s .
1. Case of h(y)
Substituting expression (31) into the (22) we obtain, f ( ω n ) = 2 δ Ω πT cs + πT cs ξ s Q p . (32)To determine the critical temperature T c we have to sub-stitute the Eqs. (31)-(32) into the self-consistency equa-tion (21) at T = T c . Then it is possible to rewrite theself-consistency equation in the following form,ln T cs T c = ψ (cid:18)
12 + Ω + Q p T cs T c (cid:19) − ψ (cid:18) (cid:19) , (33)where ψ is the digamma function, ψ ( z ) ≡ ddz ln Γ( z ) , Γ( z ) = (cid:90) ∞ η z − e − η dη. (34)Boundary condition (19) at x = 0 yields the followingequation for Ω,Ω tan (cid:18) Ω d s ξ s (cid:19) = W ( p ) ( ω n ) . (35)Generally, in order to calculate the critical temperature T c , the problem is put on the grid with finite numberof the Fourier harmonics N and the following conditionshould be used, T c = max (cid:16) T ( p ) c (cid:17) p = 0 , , ...N. (36)The critical temperature behavior is found from the so-lution of the transcendental equations (33), (35) as wellas Eq. (36). Thus, the solution that gives the highestcritical temperature T c is the only one, which is realizedphysically.However, we find that to calculate the critical temper-ature it is sufficient to use the zeroth ( p = 0) harmonic ofthe full Fourier solution for the certain parameter range. -4 -3 -2 -1 d n =0.5 n T c / T cs Q n p=0 p=1 a FIG. 2. (Color online). T c ( Q ) dependencies for two harmonicsolutions at ξ n h /α = 0 . T c . The parameters usedin the calculations: γ B = 0 . W f = 100 ξ n In Fig. 2 we demonstrate the parameter regime, for whichthe T c calculation requires consideration of only p = 0Fourier component. Such situation is possible due torapid decay of the p > Q . From the plot it can be noticed thatfor Qξ n > . p = 1 is notonly lower than T c for p = 0 but rapidly drops to zero at Qξ n ≈ . p > Q (cid:29)
1, where the p = 0 harmonic is sufficient fordescription of the T c in the bilayer. Since the functionoscillates quickly ( Qξ n (cid:29) T c along the y direction.
2. Case of h(x)
As far as the solution of the Eq. (25) can not be foundin analytical form, we calculate the function f L numeri-cally and solve the problem (25)-(30) incorporating singlemode approximation (31). III. RESULTS AND DISCUSSION
In this section we present the results of the criticaltemperature calculations using the single-mode approx-imation. Some of the parameters are set to the certainvalues and are not changed throughout the paper, oth-erwise it is indicated. Such parameters are: γ = 0 . ξ s = ξ n and the width of the junction W f = 20 ξ n . A. Case of h(y)
In Fig. 3 (a) the critical temperature dependencies areplotted for different values of the transparency parameter γ B . The helical magnetization parameter ξ n h /α = 0 . T c / T cs d n / n B =0.3 B =0.2 B =0.1 B =0.0 b T c / T cs d n / n =2 n =1.5 n =0.9 n a FIG. 3. (Color online). Behavior of the critical temperature T c as a function of d n . (a) Each plot corresponds to particularvalue of the transparency parameter γ B : blue solid line to γ B = 0, red dashed line to γ B = 0 . γ B = 0 . γ B = 0 .
3. (b) Effect of λ on T c ( d n )dependence. Each curve corresponds to particular value of λ :blue solid line to λ = 0 . ξ n , red dashed line to λ = 1 . ξ n andblack dash-dotted line to λ = 2 ξ n . The parameters used inthe calculations: ξ n h /α = 0 . Q = 2 π/λ , λ = ξ n (for plota), d s = 1 . ξ s and λ = ξ n ( λ = 2 π/Q ). We normalize T c by its max-imum value T cs in the absence of the proximitized TIlayer and the TI thickness d n by the coherence length ξ n . As expected, for perfectly transparent S/TI interface(blue solid line) the critical temperature decreases signifi-cantly, showing nonmonotonic behavior with a minimumat d n ≈ ξ n and eventually saturates at T c ≈ . T cs . Forlarger values of γ B or at moderate and high resistancesof the interface T c ( d n ) saturates at larger temperaturesand what is more interesting, the position of the T c min-imum shifts towards smaller values of d n . Unlike T c ( d n )dependencies in ordinary S/F systems with uniform aswell as out of plane spiral magnetization, here the crit-ical temperature does not demonstrate completely reen-trant behavior, i. e. the T c does not vanish in a certaininterval of d n . The impact of different λ on the criticaltemperature behavior is depicted in Fig. 3 (b). Here we n h / =0.3 n h / =0.2 n h / =0.1 T c / T cs uniform h = n bca T c / T cs d n / n n h / =0.4 d d n / n FIG. 4. (Color online). Comparison of the critical tempera-ture behavior between the S/TI bilayer with uniform magne-tization and S/TI bilayer with helical magnetization patternintroduced in Eq. (1). The curves were calculated for differ-ent values of the h /α : plot (a) corresponds to ξ n h /α = 0 . ξ n h /α = 0 .
2, plot (c) to ξ n h /α = 0 . ξ n h /α = 0 .
4. The general parameters in the bilayershave been set to the identical values such as γ and the coher-ence lengths. The transparency parameter is also same forthe both systems γ B = 0 . took γ B = 0 . ξ n h /α = 0 .
25 and d s = 1 . ξ s . From thegraph one can notice that T c becomes more suppressedfor smaller values of spatial period λ ( which is expressedin terms of Q as λ = 2 π/Q ), which means that λ acts asan additional cause of the superconducting correlationsdepairing. It is worth mentioning that rather oppositeeffect has been observed in the S/F hybrid bilayers without of plane spiral magnetization ,where T c experiencedenhancement as Q increased. B. Uniform and helical magnetization
In this subsection we compare the T c ( d n ) behavior inS/TI bilayers with the uniform and helical magnetisationinduced on the TI surface. In Fig. 4 the comparisonbetween S/TI with uniform h and with h ( y ) is shown.From the figure one can see that there is a significantdifference in the T c ( d n ) dependence for both cases. First,let us discuss the origin of T c suppression in the case ofuniformly magnetized TI surface. The wavevector of thepair correlations in topological insulator can be writtenas, κ = (cid:114) ω n D + 4 α h x , (37)where h x is the magnetization component along the x direction. Here h x is responsible for depairing of thesuperconducting correlations and suppresses the criti-cal temperature T c with the decay length ξ = 1 /κ ≈ min (cid:104)(cid:112) ω n /D, α/ h x (cid:105) . However, h y component of themagnetization does not play role in suppression of super-conducting correlations but introduces a phase shift inthe wavefunction, which has no quantitative effect on T c .Thus, in Fig. 4 the critical temperature in case ofuniform magnetization (red solid lines) expresses mono-tonic decay due to h x component. Other type of behav-ior appears when large enough values of Q are consid-ered in the system. In this case the wavevector acquiresadditional imaginary term of the form (16) and decaylength squared now becomes inverse proportional to Q as ξ = 1 /κ ≈ min (cid:104)(cid:112) ω n /D, α/ h , (cid:112) α/ h Q (cid:105) . In fact, T c demonstrates nonmonotonic behavior due to fast oscil-lations of helical magnetization along y axis. This behav-ior is indicated by black dashed lines (Fig. 4) and it canbe seen that T c ( d n ) loses its nonmonotonicity as h /ξ n grows from clearly pronounced (plots a and b) to hardlyrecognizable minimum (plots c and d) in the dependence. C. Case of h(x)
Now we turn to the case of S/TI hybrid structure withthe TI layer magnetized along the x -axis [Fig. 1 (c)]. InFig. 5 the critical temperature dependencies as functionsof the TI layer thickness d n are shown. The effect of vary-ing magnetization strength h with parameter Q fixed to Q = 2 . T c behavior.For small values of h /α the critical temperature demon-strates slightly nonmonotonic behavior with a kink ataround d n ≈ ξ n and eventual saturation (a black dot-ted line). This nonmonotonic feature becomes more pro-nounced as h /α is increased (a blue solid line). However,for certain value of magnetization strength h the crit-ical temperature drops to zero gradually (a red dashedline). Finally, at relatively large h the critical temper-ature drops sharply down to zero without any bend (ablack dash-dotted line).The origin of such T c ( d n ) curves is a coupling of heli-cal magnetization and momentum of the quasiparticles.However, unlike the magnetization pattern h ( y ), here thetopological insulator TI is magnetized by h ( x ) along thedirection of d n . Hence, the effects on the critical temper-ature are more explicit and clearer to understand. As itwas discussed above, h y component has no quantitativeimpact on the magnitude of T c , therefore, the observedeffects are purely due to variation of h x and namely be-cause of its periodicity. Obviously, the number of kinksdemonstrated in the Fig. 5(a), where we observed justone, depends on magnetization parameter Q = 2 π/λ . InFig. 5(b) the critical temperature behavior for different Q is shown. It can be seen that the smaller spatial magne- T c / T cs d n / n n h / =0.9 n h / =1.1 n h / =1.2 n h / =1.3 b T c / T cs d n / n =0.8 n =1.1 n =1.6 n =3.1 n a FIG. 5. (Color online). T c ( d n ) dependencies for the config-uration of helical magnetization introduced in Eq. (2). (a)Each curve corresponds to particular value of h /α with fixedhelical magnetization parameter Q = 2. Black dotted linecorresponds to ξ n h /α = 0 .
9, blue solid line to ξ n h /α = 1 . ξ n h /α = 1 . ξ n h /α = 1 . λ and fixed ξ n h /α = 1 .
4. Blue solid line - λ = 0 . ξ n ,red dashed line - λ = 1 . ξ n , black dotted line - λ = 1 . ξ n anddash-dotted line - λ = 3 . ξ n The rest of the parameters usedin the calculations: γ B = 0, d s = 1 . ξ s tization period λ the more kinks are produced in the T c .In the calculations above we assumed that the magnetiza-tion pattern h ( x ) at x = 0 reduces to h (0) = h (1 , , T c decays significantly in our system asa function of the TI thickness d n . We can take into ac-count φ simply by rewriting the magnetization pattern(2) as, h ( x ) = h (cos( Qx + φ ) , sin( Qx + φ ) , . (38)In Fig. 6 (a) the effect of various initial φ on T c ( d n ) forfixed h /α = 1 . λ = ξ n is illustrated. From the plotwe observe that while φ = 0 and φ = π/ T c as a function of d n (red dotted and blue a n =2 n T c / T cs n h / =1 n h / =1.4 T c / T cs d n / n d n = n T c / T cs Q n =0, = /4, = /2 b FIG. 6. (Color online). Influence of the arbitrary initialphase φ in the magnetization pattern h ( x ) = h (cos( Qx + φ ) , sin( Qx + φ ) , φ : red dotted line to φ = 0, blue solid line to φ = π/ φ = π/
2. (a) T c ( d n ) dependen-cies calculated for ξ n h /α = 1 . λ = ξ n . The inset plotshows T c behavior as a function of phase φ for fixed thick-ness d n = 2 ξ n and two different values of ξ n h /α = 1 , .
4. (b) T c ( Q ) curves calculated for d n = ξ n and ξ n h /α = 1 .
2. Theparameters used in the calculations: γ B = 0, ξ n h /α = 0 . d s = 1 . ξ s solid line), the critical temperature has higher values atalmost every d n for φ = π/ T c as a function of φ for fixed d n = 2 ξ n .Another interesting result can be noticed in Fig. 6 (b)illustrating T c ( Q ) dependencies for the same values of φ and fixed TI layer thickness d n = ξ n . One can recognize that depending on φ the critical temperature behavesdifferently as Q changes. For φ = 0 (red dotted line)there is no superconductivity in the Qξ n interval [0 , . T c is completely suppressed by slowly evolving nearextremum h x magnetization component at the vicinityof the S/TI interface. However, for φ = π/ T c decays rapidly and vanishes at Qξ n ≈ Qξ n ≈ .
4. Finally in the case of φ = π/ Q , but decaysgradually as Q is further increased. IV. CONCLUSION
In this work we have formulated a theoretical approachand presented the results of a quantitative investigationof the superconducting critical temperature in the S/TIhybrid structure, where an in-plane helical magnetizationis induced at the TI surface. The calculations are basedon the quasiclassical Usadel equations, taking into ac-count SOC at the surface of the topological insulator. Wehave found that in the case of in-plane helical magnetiza-tion h ( y ), evolving along the interface, the calculationsreveal nonmonotonic behavior of the critical temperatureas a function of the TI layer thickness with a well pro-nounced minimum, the effect which is absent in the caseof uniform magnetization. Moreover, in the case of he-lical magnetization, evolving perpendicular to the inter-face h ( x ), the critical temperature demonstrates highlynonmonotonic behavior as well. However, this depen-dence has been shown to be qualitatively different fromthe case of h ( y ), showing number of kinks, which dependson helical magnetization parameters.The results are important for further understanding ofthe underlying physics and potential future applicationsof superconductor - TI hybrid systems. ACKNOWLEDGMENTS
T.K. and A.S.V. acknowledge support of the Mir-ror Laboratories project of the HSE University andthe Bashkir State Pedagogical University. V.S.S. ac-knowledges support of the joint French (ANR) / Rus-sian (RSF) Grant “CrysTop” (20-42-09033). A.A.G.acknowledges support by the European Union H2020-WIDESPREAD-05-2017-Twinning project SPINTECHunder Grant Agreement No. 810144. ∗ [email protected] L. Fu, C. L. Kane and E. J. Mele Phys. Rev. Lett. ,106803 (2007). M. Z. Hasan and C. L. Kane, Rev. Mod. Phys., , 3045(2010). M. Sato and Y. Ando, Reports on Progress in Physics ,076501 (2017). S.-Q. Shen,
Topological Insulators Dirac Equation in Con-densed Matters (Berlin: Springer, 2012). G. Tkachov
Topological Insulators: The Physics of Spin
Helicity in Quantum Transport (Singapore: Pan Stanford,2015). S. D. Sarma, M. Freedman and C. Nayak, Phys. Today ,32 (2006). R. Aguado and L. P. Kouwenhoven, Phys. Today , 44(2020). X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011). L. Fu, and C. L. Kane, Phys. Rev. Lett. , 096407(2008). Y. Tanaka, T. Yokoyama, and N. Nagaosa, Phys. Rev.Lett. , 107002 (2009). M. Sato and S. Fujimoto, Phys. Rev. B , 094504 (2009). J. Alicea, Rep. Prog. Phys. , 076501 (2012). C. W. J. Beenakker, Annu. Rev. Condens. Matter Phys. , 113 (2013). G. Tkachov and E. N. Hankiewicz, Phys. Rev. B ,075401 (2013). E. A. Demler, G. B. Arnold, and M. R. Beasley, Phys. Rev.B , 15174 (1997). A. Ozaeta, A. S. Vasenko, F. W. J. Hekking, and F. S.Bergeret, Phys. Rev. B , 060509(R) (2012). F. S. Bergeret and I. V. Tokatly, Phys. Rev. Lett. ,117003 (2013). I. V. Bobkova and A. M. Bobkov, Phys. Rev. B , 184518(2017). F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod.Phys. , 1321 (2005). A. I. Buzdin, Rev. Mod. Phys. , 935 (2005). Y. V. Fominov, N. M. Chtchelkatchev, and A. A. Golubov,Phys. Rev. B , 014507 (2002). Ya. V. Fominov, A. A. Golubov, T. Yu. Karminskaya, M.Yu. Kupriyanov, R. G. Deminov, L. R. Tagirov, JETPLett. , 308 (2010) [Pis’ma ZhETF , 329 (2010)]. A. I. Buzdin, L. N. Bulaevskii, and S. V. Panyukov, JETPLett. , 178 (1982) [Pis’ma ZhETF , 147 (1982)]. A. I. Buzdin and M. Yu. Kupriyanov, JETP Lett. , 321(1991) [Pis’ma ZhETF , 308 (1991)]. V. V. Ryazanov, V. A. Oboznov, A. Yu. Rusanov, A. V.Veretennikov, A. A. Golubov, and J. Aarts, Phys. Rev.Lett. , 2427 (2001). V. A. Oboznov, V. V. Bol’ginov, A. K. Feofanov, V. V.Ryazanov, and A. I. Buzdin, Phys. Rev. Lett. , 197003(2006). A. V. Vedyayev, N. V. Ryzhanova, N. G. Pugach, Journalof Magnetism and Magnetic Materials, , 53 (2006). A. S. Vasenko, A. A. Golubov, M. Yu. Kupriyanov, andM. Weides, Phys. Rev. B , 134507 (2008). S. V. Bakurskiy, V. I. Filippov, V. I. Ruzhickiy, N. V.Klenov, I. I. Soloviev, M. Yu. Kupriyanov, A. A. Golubov,Phys. Rev. B , 094522 (2017). T. Kontos, M. Aprili, J. Lesueur, and X. Grison, Phys.Rev. Lett. , 304 (2001). A. S. Vasenko, S. Kawabata, A. A. Golubov, M. Yu.Kupriyanov, C. Lacroix, F. S. Bergeret, and F. W. J.Hekking, Phys. Rev. B , 024524 (2011). J. S. Jiang, D. Davidovi`c, D. H. Reich, and C. L. Chien,Phys. Rev. Lett. , 314 (1995). L. R. Tagirov, Physica C , 145 (1998). Yu. N. Proshin, Yu. A. Izyumov, and M. G. Khusainov,Phys. Rev. B , 064522 (2001). Yu. N. Khaydukov, A. S. Vasenko, E. A. Kravtsov, V. V.Progliado, V. D. Zhaketov, A. Csik, Yu. V. Nikitenko, A.V. Petrenko, T. Keller, A. A. Golubov, M. Yu. Kupriyanov,V. V. Ustinov, V. L. Aksenov, and B. Keimer, Phys. Rev.B , 144511 (2018). T. Karabassov, V. S. Stolyarov, A. A. Golubov, V. M.Silkin, V. M. Bayazitov, B. G. Lvov, and A. S. Vasenko,Phys. Rev. B , 104502 (2019). A. Zyuzin, M. Alidoust, and D. Loss, Phys. Rev. B ,214502 (2016). F.S. Bergeret and I. V. Tokatly, Phys. Rev. B , 134517(2014). S. H. Jacobsen and J. Linder, Phys. Rev. B , 024501(2015). B. Bujnowski, R. Biele, and F.S. Bergeret, Phys. Rev. B , 224518 (2019). J. R. Eskilt, M. Amundsen, N. Banerjee, and Jacob Linder,Phys. Rev. B , 224519 (2019). M. Nashaat, I. V. Bobkova, A.M. Bobkov, Y.M. Shukrinov,I.R. Rahmonov, and K. Sengupta, (2019). I. V. Bobkova, A. M. Bobkov, A. A. Zyuzin, and M. Ali-doust, Phys. Rev. B , 134506 (2016). M. Alidoust, Phys. Rev. B , 245418 (2018). M. Alidoust, Phys. Rev. B , 155123 (2020). Y. Lu and T.T. Heikkil¨a, Phys. Rev. B , 104514 (2019). M. Alidoust and H. Hamzehpour, Phys. Rev. B , 165422(2017). T. Champel and M. Eschrig, Phys. Rev. B , R220506(2005). T. Champel and M. Eschrig, Phys. Rev. B , 054523(2005). T. Champel, T. L¨ofwander, and M. Eschrig, Phys. Rev.Lett. , 077003 (2008). N.G. Pugach, M. Safonchik, T. Champel, M.E. Zhito-mirsky, E. L¨ahderanta, M. Eschrig, and C. Lacroix, Appl.Phys. Lett. , 162601 (2017). N.G. Pugach and M.O. Safonchik, JETP Lett. , 302(2018). N.G. Pugach, M.O. Safonchik, D.M. Heim, and V.O.Yagovtsev, Phys. Solid State , 2237 (2018). M. Bode, M. Heide, K. Von Bergmann, P. Ferriani, S.Heinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch, S. Bl¨ugel,and R. Wiesendanger, Nature , 190 (2007). C. Brun, T. Cren, and D. Roditchev, Supercond. Sci. Tech-nol. W. Belzig, F. K. Wilhelm, C. Bruder, G. Sch¨on, and A. D.Zaikin, Superlattices Microstruct. , 1251 (1999). K. D. Usadel, Phys. Rev. Lett. , 507 (1970). M. Yu. Kuprianov and V. F. Lukichev, JETP , 1163(1988) [ZhETF , 139 (1988)]. E. V. Bezuglyi, A. S. Vasenko, V. S. Shumeiko, and G.Wendin, Phys. Rev. B , 014501 (2005); E. V. Bezug-lyi, A. S. Vasenko, E. N. Bratus, V. S. Shumeiko, and G.Wendin, ibid. , 220506(R) (2006). A. A. Golubov, M. Yu. Kupriyanov, V. F. Lukichev, andA. A. Orlikovskii, Sov. J. Microelectron. , 191 (1984)[Mikroelektronika12