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Magnetization Reversal by Tuning Rashba Spin-Orbit Interaction and Josephson Phase in a Ferromagnetic Josephson Junction

We theoretically investigate the magnetization inside a normal metal containing the Rashba spin-orbit interaction (RSOI) induced by the proximity effect in an s-wave superconductor/normal metal/ferromagnetic metal/s-wave superconductor (S/N/F/S) Josephson junction. By solving the linearized Usadel equation taking account of the RSOI,we find that the direction of the magnetization induced by the proximity effect in N can be reversed by tuning the RSOI.Moreover, we also find that the direction of the magnetization inside N can be reversed by changing the superconducting phase difference, i.e., Josephson phase. From these results, it is expected that the dependence of the magnetization on the RSOI and Josephson phase can be applied to superconducting spintronics.


I. INTRODUCTION
It is well known that the Cooper pair penetrates into a normal metal due to the proximity effect in the s-wave superconductor/normal metal (S/N) hybrid junctions [1,2]. One of important phenomena generated by the proximity effect is Josephson effect, which is characterized DC current flowing without a voltage-drop between two Ss separated by N [1,2]. The Josephson effect is a macroscopic quantum-mechanical phenomenon to preserve superconducting phase (Josephson phase) coherence between two superconductors. Such a phase coherence monotonically decreases with the thickness of N [2].
Another notable phenomenon in the S/F junctions is the appearance of odd-frequency spin-triplet Cooper pairs (STCs) induced by the proximity effect [21,22]. Where the anomalous Green's function of the oddfrequency spin-triplet Cooper pair becomes odd function with respect to the fermion Matsubara frequency ω n . In an S/F junction having a uniform magnetization, the STC composed of opposite spin electrons (i.e., total spin projection on z axis being S z = 0) and SSCs penetrates into the F due to the proximity effect [21,23]. The penetration length of STC with S z = 0 and SSC into the F is very short and the amplitude of STC inside the F exhibits a damped oscillatory behavior as a function of the thickness of F. The characteristic penetration length is determined by ξ F = D F /h ex , which is typically a order of few nanometers [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Where, D F and h ex are the diffusion coefficient and the exchange field in the F, respectively.
The STC formed by electrons of equal spin (|S z | = 1) can be also induced inside the F due to the proximity effect when a magnetization in the F is non-uniform in S/F junctions. The feature of this STC is approximately 2 orders of magnitude larger than the penetration length of the SSC . The penetration length of STC with spin |S z | = 1 is determined by ξ T = D F /2πk B T in F s (T is temperature). Thus, the proximity effect of the STC with |S z | = 1 is referred to as the long-range proximity effect (LRPE).
The STC with |S z | = 1 induced by the proximity effect can be detected by the observation of Josephson current in ferromagnetic Josephson junctions (FJJs). The Josephson current carried by the STC with |S z | = 1 shows monotonically decrease as a function of the thickness of F and its decay length of STC is about determined by ξ T . The long-range Josephson current flowing through the F has been observed and established experimentally in FJJs [55][56][57][58][59][60][61]. The detection of long-range Josephson current is a piece of evidence for the presence of STC.
Another candidate to evidence the presence of the STC is to measure the spin dependent transport of the STC in S/F junctions, since such a transport directly mea-sures the spin of the STC [62][63][64][65][66][67][68][69]. For this purpose, recently S/F/N and S/F/N/F/S junctions containing the Rashba spin-orbit interaction (RSOI) has been of considerable interest in recent years, since the proximity effect coupled with the RSOI in S/F/N and S/F/N/F/S junctions exhibits many fascinating phenomena which are not observed in S/F/N and S/F/N/F/S junctions without the RSOI [70][71][72][73][74][75][76]. In the S/F/N junctions, it has been theoretically reported that the pair amplitude of the STC penetrating into N with the RSOI modulates as functions of the thickness of N and the magnitude of the RSOI [71,72]. By utilizing the modulation of the pair amplitude of STC, it is possible to freely control 0-and π-states by tuning the RSOI in Josephson junctions [70,72]. Moreover, some authors have theoretically predicted the direct evidence of the STC by detecting the spin Hall effect and magnetoelectric effect induced by the STC in Josephson junctions containing Fs and the RSOI [73,74,76]. The RSOI has an advantage to study spin-dependent transport phenomena, since this interaction can be freely controlled by an external electric field [80]. However, understanding of spin transport of STC with the RSOI has been insufficiency, and thus how to manipulate and detect the spin of the STC has the space to study. It is expected that well understanding of spin transport of STC gives us the credence of STC and expedites the development of superconducting spintronics [77][78][79].
In this paper, we theoretically propose an another setup and way to detect the STC by using a simple S1/N/F/S2 junction differently from Ref [73,74,76]. The simple S1/N/F/S2 junction can be easily achieved by using recent device fabrication techniques. Based on the linearized Usadel equation including the RSOI, we formulate the magnetization induced by the proximity effect. It is shown that the magnetization in the N only appears when the product of anomalous Green's functions of the spin-triplet odd-frequency Cooper pair and spin-singlet even-frequency Cooper pair in the N has a finite value. It is found that the magnetization shows damped oscillatory behavior as a function of the thickness of N. It is also found that the direction of magnetization can be controlled by tuning the magnitude of RSOI. Moreover, we examine the Josephson phase (θ) dependence of magnetization. As a result, the period of oscillation of magnetization is changed by tuning θ. This result clearly shows that the direction of magnetization can be also controlled by θ as well as the magnitude of RSOI. We expect that these results can be applied to the research field of superconducting spintronics.
The rest of this paper is organized as follows. In Sec. II, we introduce an S1/N/F/S2 Josephson junction and formulate the magnetization induced by the proximity effect in the N containing the RSOI of this junction by solving Usadel equation. In Sec. III, the numerical results of magnetization are given. The thickness and the RSOI dependences of magnetization are discussed. Moreover, we present the Josephson phase (θ) dependence of mag- (Color online) Schematic illustration of the S1/N /F /S2 junction studied, where N is a normal metal with the Rashba spin-orbit interaction(RSOI), F is a ferromagnetic metal, and S1(2) is an s-wave superconductor. Arrow in F indicates the direction of ferromagnetic magnetization. While the magnetization in F is fixed along the z direction. dS, dF and d are the thicknesses of S, F , and N , respectively, with L = d + dF. We assume that the magnetization is uniform in F layer, and that dS is much larger than ξS.
netization. Finally, θ-magnetization relation is discussed and the magnetization induced by the proximity effect is estimated for a typical set of realistic parameters in Sec. IV. The summary of this paper is given in Sec. V. We consider the Josephson junction composed of s-wave superconductors (Ss) separated by normal metal/ferromagnetic metal (N/F ) junction as depicted in Fig. 1. Here, we include the Rashba spin-orbit interaction (RSOI) in the N and assume the uniform magnetization in F in the S1/N/F/S2 junction. We assume that the width of the junction is smaller than ξ T . In this situation, one dimensional (1D) model may be good approximation. Therefore, we adopt the 1D model to analyze the magnetization induced by the proximity effect in the S1/N/F/S2 junction.
In the diffusive transport limit, the magnetization inside the N with RSOI induced by the proximity effect is evaluated by solving the linearized Usadel equation including SU(2) gauge field in each region (j = N, F ) [71] i D j∂ where r = (x, ω n ) andÂ x is the SU(2) gauge field, which describes the RSOI and α R is the RSOI constant.
D j is the diffusion coefficient in region j, ω n = (2n + 1)πk B T / (n : integer) is the fermion Matsubara frequency. sgn(X) = X/|X| is the sign function, andτ y(z) is the y(z) component of Pauli matrix. [Q,R] =QR −RQ is anticommutation relation and0 is zero matrix. The anomalous partf j ( r) of the quasiclassical Green's function [47] is given bŷ . (2) f j s ( r) is the anomalous Green's function for the SSC, and f j tx(ty) ( r) and f j tz ( r) represent the anomalous Green's functions for the STC with |S z | = 1 and |S z | = 0, respectively. The exchange field h ex (x) in the F is given by where e z is a unit vector in the z direction. We assume that h ex is positive.
To obtain solutions of Eq.(1), we employ appropriate boundary conditions, i.e., where γ F = σ F /σ N and σ F(N) is the conductivity of F (N ). Moreover, in the present calculation, we adopt the rigid boundary condition ξS , where σ S is the conductivity of S in the normal state [20].
Notice that the left-hand side of Eqs. (7) and (9) is zero as will be later shown. Assuming that d S ≫ ξ S , the anomalous Green's function in the Ss attached to N and F can be approximately given aŝ where ∆ L(R) = ∆e iθ L(R) (∆: real) and θ L(R) is the superconducting phase in the left (right) side of Ss (see Fig. 1). The s-wave superconducting gap∆(x) is finite only in the S and assume to be constant as follows, From Eqs. (10) and (11), it is immediately that the left-hand side of Eqs. (7) and (9) becomes zero, since the rigid boundary condition is assumed in the present calculation [20].
Assuming d F /ξ F ≪ 1, we can perform the Taylor expansion with x forf F ( r) as follows [33,81], Applying the boundary conditions of Eqs. (5) and (8) to Eq. (12) and substituting Eq. (12) into Eq. (1), we can approximately obtain the anomalous Green's function of The general solutions of Eq. (1) in the N are given by where k α = 3α 2 R + k 2 N and 2|ω n |/D N . Here, we assume that α R = 0 to obtain Eq. (14). Applying the boundary conditions given in Eqs. (4), (6), (7), and (9) to Eq. (14) , and also using the result in Eq. (12), we can obtain the anomalous Green's functions in the N as and and Where the explicit formulae of the functions C ij ωn (d) (i, j = 1, 2, 3) are presented in APPENDIX A. From Eqs. (15) -(17), it is immediately found that f N s ( r) representing the SSC is an even function with ω n , whereas f N tx(tz) ( r) representing the STC is an odd function with ω n since f N tx(tz) ( r) is proportional to sgn(ω n ). Therefore, f N tx(tz) ( r) describes the odd-frequency STC. It should be noticed that f N ty ( r) = 0 since we assume that the present junction is the 1D model and the magnetization in the F has only z component [71]. It should be noticed that f N tx(tz) ( r) is exactly zero when h ex = 0, which corresponds to no magnetic layer in the present junction studied here. This result represents that the RSOI does not induce the STC in the N , and thus the magnetic layer is needed to produce the STC. The interplay of RSOI is to induce a finite f N tx ( r) in the N of present junction [82].

B. Magnetization induced by proximity effect in normal metal with RSOI
Based on the quassiclassical Green's function theory, the magnetization M (d, θ) induced by the proximity effect is given by [25,62] where θ = θ R − θ L is the Josephson phase in the junction and with m(x, θ) is the local magnetization density in the N , g is the g factor of electron, and µ B is the Bohr magneton. A and V = Ad are the cross-section area of junction and the volume of N , respectively. The density of states N F per unit volume and per electron spin at the Fermi energy is assumed to be approximately the same for up and down electrons of the F in the quasiclassical Green's function theory [19][20][21]. It is obvious in Eq. (22) that f N s (x) and f N t (x) are both required to be nonzero to induce finite m(x, θ). However, as described in Sec. II A, nonzero f N t (x) occurs only when F layer is involved in the junction and f N t (x) = 0 whenever f N s (x) = 0 for ∆ L = ∆ R = 0. Therefore, the origin of the magnetization in the N is considered to be due to the STCs induced by the proximity effect [21,62,65,68]. Because of f N ty (x) = 0 (see Sec. II A), m y (x, θ) and thus M y (d, θ) are always zero. It is also noticeable that M (2) x(z) (d, θ) and M where and Similarly, the z component M z (d, θ) is also decomposed into three parts, where and The explicit formulae of the functions w ωn (d) and u ωn (d) in Eqs. , it is immediately found that the magnetizations of x and z components are exactly zero when the exchange field h ex is zero. Therefore, the F is indeed required to induce the magnetization inside the N . Here, it should be noticed that M x (d, θ) is always zero without the RSOI as mentioned above. One of the θ-independent part of the magnetization, i.e., M (1) z (d), is due to the proximity effect inducing STCs common in the S/F and S/F multilayer systems [21,65,68,69]. The other θ-independent part of the magnetization, i.e., M x (d), appears due to not only the proximity effect but also the presence of RSOI, since the magnetization of the F points to z axis in the present junction [21]. The θ-dependent part of M (2) z (d, θ) is induced by the coupling between two Ss only when S/F multilayer systems compose the Josephson junction [65,68,69]. The θ-dependent part of M (2) x (d, θ) is induced by the coupling between two Ss and the finite RSOI. M (3) x (d, θ) and M (3) z (d, θ) appear when the RSOI in N , the exchange field in the F , and the Josephson coupling are finite. The expressions of the magnetizations M x (d, θ) and M z (d, θ) given in Eq. (24) -(31) are rather complicate. Therefore, we will present numerical results of magnetizations calculated here in the next section.

III. RESULTS
In this section, we numerically evaluate the magnetizations of Eqs. (24) and (28)    A. Thickness dependence of magnetizations induced by the proximity effect Figure 2 shows the x and z components of magnetization induced by the proximity effect inside the N as a function of d. The solid (black) and dashed (red) lines are magnetizations forᾱ = 0.2, and 0.5, respectively. In Fig. 2, the denotation P.R. and N.R. are the abbreviated expressions of positive and negative regions, respectively. The denotation A.L. is the auxiliary line to separate positive and negative regions of magnetization in Fig. 2. We find that M x (d, θ) and M z (d, θ) exhibit damped oscillatory behavior as a function of d. From Fig. 2, it is found that the magnetizations can be reversed by changing the thickness of N . Moreover, it is also clearly found that the period of oscillation of M x (d, θ) and M z (d, θ) becomes short with increasing α R . Therefore, by setting d near the thickness of M x(z) (d, θ) ≈ 0, the magnetizations can be easily reversed by tuning α R . Figures 3 (a) and (b) show x and z components of magnetization as a function of α R . Here, we set the thickness of N as d/ξ D = 1.3. From Fig. 3 (a), it is found that the sign of M x (d, θ) is changed from negative to positive with increasing α R within 0.4ξ D α R 0.6ξ D . The sign of M x (d, θ) is changed from positive to negative with further increasing α R . From Fig. 3 (b), it is found that the sign of M z (d, θ) is changed from negative to positive with increasing α R within 0.4ξ D α R 0.6ξ D . By further increasing α R , the sign of M z (d, θ) is changed from positive to negative. From these results, it is clearly found that the direction of x and z components of magnetization can be reversed by tuning α R .

B. RSOI dependence of magnetization and
Magnetization-phase relation Figure 4 shows the magnetization as a function of d. Figures 4 (a) and (b) are the x and z component of magnetization, respectively. In Fig 4, the solid (black) and dashed (red) lines are magnetizations for θ = 0 and π/2, respectively.ᾱ is set to be 0.5. From Fig. 4, it is found that the period of oscillation of M x (d, θ) and M z (d, θ) change by tuning θ. It should be noticed that the variation of the oscillation period of the magnetizations becomes large when d/ξ D > 1 as shown in Fig. 4. By setting d/ξ D near third minimum of M x(z) (d, θ), we can perform magnetization reversal by changing θ as well as α R (see Figs. 3 and 4). Figure 5 shows the magnetization as a function of θ, i.e., magnetization-phase relation. Figures 4 (a) and (b) are the x and z component of magnetization, respectively. The thickness of N is set to be d/ξ D = 1.45. The behavior of M x (d, θ) with θ is a cosine function as shown in   5 (a). On the other hands, the behavior of M z (d, θ) with θ is a sine function as shown in Fig. 5 (b). From Fig.5, it is immediately found that M x (d, θ) and M z (d, θ) vary from positive to negative values and vice versa by changing θ. This result indicates that the magnetizations can be reversed by changing θ when the thickness of N is appropriately set as mentioned above.

IV. DISCUSSION
Here, we discuss why the magnetization-phase relation of M x (d, θ) is shifted by π/2 compared with that of M z (d, θ). From Eq. (22), the x component of magnetization is proportional to Im[f s ( r)f tx ( r)]. From Eq. (16), i.e., f tx ( r) = if tz ( r), Im[f s ( r)f tx ( r)] = Im[f s ( r)f tz ( r)e iπ/2 ]. Therefore, the magnetizationphase relation of M x (d, θ) is shifted by π/2 compared with that of M z (d, θ).
We approximately estimate the amplitude of magnetization induced by the proximity effect. As shown in Figs. 2 -3, the magnetization in the N has a finite value in the length scale of ξ D . In the dirty limit, ξ D is a range of several dozen to several hundred nanometers [83]. Therefore, the magnetization induced by the proximity effect has a finite value in this length scale. To estimate the amplitude of magnetization, we evaluate the normalized factor of magnetization, i.e., M 0 = gµ B N F ∆ 0 (for instance, see Fig. 2). When we use a typical set of parameters [84,85], M 0 is approximately 100 A/m. Therefore, the order of magnetization amplitude is between 10 and 10 6 . It is expected that this order of magnetization amplitude can be detected by the magnetization measurement by utilizing SQUID [86].
Finally, we shall approximately estimate the magnitude of α R . In the present calculation, we chose α R to be one order smaller than ξ D . The magnitude of α R used in the numerical calculation is easily achieved by realistic experiments [87][88][89][90][91]. For instance, InGaAs/InAlAs and InAs/AlSb quantum wells are good candidates as a N in the present junction studied here [87][88][89]. Therefore, it is expected that the magnetization induced by the proximity effect can be easily reversed by tuning α R .

V. SUMMARY
We have theoretically studied the magnetization reversal by tuning the RSOI (α R ) and Josephson phase in the S/N/F/S junction. The magnetizations of x and z components are induced by the appearance of the oddfrequency spin-triplet and the even frequency spin-singlet Cooper pairs in the N. We have shown that the magnetizations exhibit the damped oscillatory behavior as a function of the thickness of N for finite α R . The period of oscillation of the magnetizations induced by the proximity effect varies by changing α R and becomes short with increasing α R . Therefore, the direction of magnetizations can be controlled by tuning α R for a fixed thickness of N. We have found that the magnetizations induced in the N depend on the Josephson phase (θ). As a result, the amplitude and the oscillation period of the magnetizations can be controlled by tuning θ. It has been also found that the direction of the magnetizations in the N can be reversed by changing θ as well as α R . These results clearly show that the variation of magnetization by tuning α R and θ is a good fingerprint to observe the spin of STC.
We have theoretically shown that the magnetizations are decomposed of three parts, i.e., M x(z) (d, θ) = M (1) x(z) (d) + M (2) x(z) (d, θ) + M (3) x(z) (d, θ). (i) Appearance of M (1) x (d), which is the θ-independent part of magnetization is due to the proximity effect, the exchange field in the F, and the RSOI in the N. On the other hand, the θ-dependent parts M (2) x (d, θ) and M (3) x (d, θ) results from the finite Josephson coupling between the two Ss, the exchange field in the F, and the RSOI in the N in the , which always appears in the S/F junctions is induced due to the proximity effect. The θ-dependent part M    x(z) (d, θ) sinusoidal with θ. We have also discussed that the magnetization induced by the proximity effect can be large enough to be detected in typical experiments. Therefore, it is expected that a Josephson junction including the F and the RSOI in the N such as the one studied here can have a potential for low Joule heating spintronics devices, since the direction of the magnetization inside the N can be easily controlled by changing α R and θ. and Appendix B: Integration with x in Eq. (21) In this APPENDIX, we will first provide the analytical form of the local magnetization density in the N . Within quasiclassical Green's function theory, the local magnetization density m(x, θ) in the N is obtained by substituting Eqs. (15) -(17) into Eq. (22). The x component m x (x, θ) of the local magnetization density can be decompose into θ-independent and θ-dependent parts m x (x, θ) = m (1) x (x) + m (2) x (x, θ) + m (3) and where the functions Q ωn (d), Φ ωn (d), and t ωn (x, d) are given in Eqs. (18) - (20). Substituting Eqs. (B2) -(B4) into Eq. (21) and performing the integration with x, we can obtain the x component of magnetization, x (x, θ) dx + 1 where M (1) x (d, θ), and M a (1) ωn (d) = 2α R k α k N e kαd , a (2) ωn (d) = −2α R k N cosh (k N d) [k α cos (α R d) +α sin (α R d)] , a (3) ωn (d) = 2α R 2α 2 R + k 2 N cos (α R d) + 2k α α R sin (α R d) sinh (k N d) , We can also obtain the z component of magnetization given in Eq. (29) -(31) obeying same procedure to derive the x component of magnetization.