Magnetization Reversal by Tuning Rashba Spin-Orbit Interaction and Josephson Phase in a Ferromagnetic Josephson Junction
Shin-ichi Hikino
cond-mat.supr-con cond-mat.mes-hall

 
 



National Institute of Technology, Fukui College, Sabae, Fukui 916-8507, Japan
 




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. 




Magnetization Reversal by Tuning Rashba Spin-Orbit Interaction and Josephson Phase in a Ferromagnetic Josephson Junction
    Shin-ichi Hikino
    December 30, 2023
========================================================================================================================








§ INTRODUCTION


In s-wave superconductor/ferromagnetic metal (S/F) junctions, 
it is well known that the pair amplitude of a spin-singlet Cooper pair (SSC) penetrating into F due to the proximity effect shows damped oscillatory 
behavior as a function of the thickness of F <cit.>. 
One of the interesting phenomena resulting from the damped oscillatory behavior of the pair amplitude is a π-state in 
an S/F/S junction, where the current–phase relation in the Josephson current is shifted by π from that of 
ordinary S/I/S and S/N/S junctions <cit.>. 

Another notable phenomenon in S/F junctions is the appearance of odd-frequency spin-triplet Cooper pairs (STCs) 
induced by the proximity effect<cit.>. 
In an S/F junction having a uniform magnetization, 
the STC composed of opposite spin electrons (S_z=0) and the SSC penetrate into F owing to the 
proximity effect <cit.>. 
The penetration length of the STC with S_z=0 and the SSC into F is determined by ξ_ F=√(ħ D_ F/h_ ex), which is typically 
of nm order <cit.>. 
Here, D_ F and h_ ex are the diffusion coefficient and exchange field in F, respectively. 

Moreover, the STC formed by electrons of equal spin (|S_z|=1) can be induced inside F owing to the proximity effect 
when the magnetization in F is nonuniform in S/F junctions. 
The penetration length of the STC with spin |S_z|=1 is determined by ξ_ T=√(ħ D_ F/2 π k_ B T) in F 
(T is temperature). 
The feature of this STC is approximately two orders of magnitude larger than the penetration length of the SSC and STC with S_z=0 <cit.>. 
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 monotonically decreases as a function of the thickness of F and the 
decay length of the STC is roughly determined by ξ_ T. 
The long-range Josephson current flowing through the F has been observed and established experimentally 
in FJJs<cit.>. 
The detection of long-range Josephson current is a piece of evidence for the presence of the STC. 

Another means of obtaining evidence for the presence of the STC is to measure the spin-dependent transport of the STC in S/F junctions, 
since such transport can be used to directly measure the spin of the STC <cit.>. 
For this purpose, S/F/N and S/F/N/F/S junctions containing the Rashba spin-orbit interaction (RSOI) have 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 <cit.>. 
For S/F/N junctions, it has been theoretically found 
that the pair amplitude of the STC penetrating into N with the RSOI is modulated as a function of the thickness of N and the magnitude of the RSOI 
<cit.>. 
By utilizing the modulation of the pair amplitude of the STC, 
it is possible to freely control 0- and π-states by tuning the RSOI in Josephson junctions <cit.>. 
Moreover, some authors have theoretically predicted that the direct evidence of the STC can also be obtained 
by detecting the spin Hall effect and magnetoelectric effect induced by the STC 
in Josephson junctions containing ferromagnetic metals and the RSOI <cit.>. 
The RSOI is  advantageous for studying spin-dependent transport phenomena since it can be freely controlled by an external electric field <cit.>. 
However, understanding of the spin transport of the STC in consideration of the RSOI is still lacking, 
since the studies in this research field have been rare and limited <cit.>. 
Therefore, it is expected that a good understanding of the spin transport of the STC will provide proof for the STC and 
expedite the development of superconducting spintronics <cit.>. 

In this paper, we theoretically propose another setup and way to detect the STC 
by using an S1/N/F/S2 junction differently from Refs. 71, 72, and 74. 

The 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 the 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 the RSOI. 
Moreover, we examine the Josephson phase (θ) dependence of the magnetization. 
It is found that the period of oscillation of magnetization can be changed by tuning θ. 
This result clearly shows that the direction of the magnetization can also be controlled by tuning θ as well as the magnitude of the RSOI. 
Therefore, 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 Sect. <ref>, 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 the Usadel equation. 
In Sect. <ref>, the numerical results of the magnetization are given. 
The thickness and RSOI dependences of the magnetization are discussed. 
Moreover, we present the Josephson phase (θ) dependence of the magnetization. 
Finally, the θ-magnetization relation is discussed and 
the magnetization induced by the proximity effect is estimated for a typical set of realistic parameters in Sect. <ref>.
A summary of this paper is given in Sect. <ref>. 
The detailed calculation of the magnetization is given in the Appendices. 




§ MAGNETIZATION IN NORMAL METAL INDUCED BY PROXIMITY EFFECT IN AN S1/N/F/S2 JUNCTION WITH RASHBA SPIN-ORBIT INTERACTION










 §.§ Setup of junction and anomalous Green's functions

We consider a Josephson junction composed of s-wave superconductors (Ss) 
separated by a normal metal/ferromagnetic metal (N/F) junction as depicted in Fig. <ref>. 
Here, we include the RSOI in the N and assume 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, a one-dimensional (1D) model may be a 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 the RSOI induced by the proximity effect is 
evaluated by solving the linearized Usadel equation including the SU(2) gauge field in each region (j=N, F),<cit.> 


    i ħ D_j∂̃^2_xf̂^j(r) 
    -i2ħ |ω_n| f̂^j(r) 
    
    - sgn(ω_n) h_ ex(x) 
    [
    τ̂_z, f̂^j(r) 
    ]
    = 0̂,



    ∂̃_x = {[ ∂ _x∙  - i1/ħ[ Â_x, ∙],      Â_x = α _Rτ̂_y,   0 < x < d;    ∂ _x∙                                   ,   otherwise ].,

where r=(x,ω_n), Â_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)π k_ BT/ħ (n: integer) is the fermion Matsubara frequency, 
sgn(X)=X/|X| is the sign function, and τ̂_y (z) is the y(z) component of the Pauli matrix.
[Q̂, R̂]=Q̂R̂-R̂Q̂ is the anticommutation relation and 0̂ is the zero matrix. 
The anomalous part f̂^j(r) of the quasiclassical Green's function <cit.> is given by 



    f̂^j( r)    =   ( [ f_↑↑^j( r) f_↑↓^j( r); f_↓↑^j( r) f_↓↓^j( r) ]) 
       =   ( [ - f_tx^j( r) + if_ty^j( r)     f_s^j( r) + f_tz^j( r);   - f_s^j( r) + f_tz^m( r)   f_tx^j( r) + if_ty^j( r) ])
    .

f_s^j(r) is the anomalous Green's function for the SSC, and f_tx(ty)^j(r) and f_tz^j(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 



    h_ex( x ) = {[ h_ex e_z ,    d < x < L;       0,      otherwise ].,

where e_z is a unit vector in the z direction. 
We assume that h_ ex is positive. 

To obtain solutions of Eq. (<ref>), we employ appropriate boundary conditions, i.e., 

    . f̂^S1(r) |_x = 0   =   . f̂^N(r) |_x = 0, 
    
    . f̂^N(r)|_x = d   =   . f̂^F(r) |_x = d, 
    
    . f̂^F(r) |_x = L   =   . f̂^S2(r) |_x = L, 
    
    σ_ S. ∂ _xf̂^S1( r) |_x=0   =   σ_ N. ∂ _xf̂^ N( r) |_x=0, 
    
    
    . ∂_x f̂^F(r) |_x = d   =   . 1/γ _F∂_x f̂^N(r) |_x = d, 
    
    σ_ S. ∂ _xf̂^S2( r) |_x=L   =   σ_ F. ∂ _xf̂^F( r) |_x=L,

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 
σ_ F(N)/σ_ S≪ξ_ F(N)/ξ_ S,
where σ_ S is the conductivity of S in the normal state <cit.>. 
ξ_ F=√(ħ D_ F/h_ ex ) and ξ_ N=√(ħ D_ N/2π k_ B T).  
Notice that the left-hand side of Eqs. (<ref>) and (<ref>) is zero as will be shown later. 
Assuming that d_ S≫ξ_ S, the anomalous Green's function in the Ss attached to N and F 
can be approximately given as 


    f̂_s^ S1(2) (r) = - τ̂_yΔ_ L (R)/√((ħω)^2 + |Δ_ L (R)|^2)≡F̂^ S1(2)
    ,
 
where Δ_ L(R) = Δ e^i θ_ L(R) (Δ: real) and θ_ L(R) 
is the superconducting phase in the left (right) side of the S (see Fig. <ref>). 
The s-wave superconducting gap Δ̂(x) is finite only in the S and is assumed to be constant as follows: 



    Δ̂(x)  = {[ ( [      0 - Δ_ L;   Δ_ L      0 ]), - d_ S < x < 0;     ( [      0 - Δ_ R;   Δ_ R      0 ]), L < x<L_ S;          0̂,                              otherwise ]. .


From Eqs. (<ref>) and (<ref>), it is immediately found that the left-hand side of Eqs. (<ref>) and (<ref>) becomes zero, 
since the rigid boundary condition is assumed in the present calculation<cit.>. 

Assuming d_ F/ξ_ F≪ 1, we can perform the Taylor expansion with x for f̂^ F(r) as follows:<cit.> 



    f̂^ F( r)    ≈   f̂^ F( d,ω_n) 
    	+ . ( x - d)∂ _xf̂^ F( r) |_x=d
        +    ( x - d)^2/2. ∂ _x^2f̂^ F( r)|_x=d.


Applying the boundary conditions of Eqs. (<ref>) and (<ref>) to Eq. (<ref>) and substituting Eq. (<ref>) into Eq. (<ref>), 
we can approximately obtain the anomalous Green's function of f^ F(r) as 




    f̂^ F(r)    ≈   
    		-d_ F/γ_ F. ∂ _xf̂^ N(r)|_x=d
    		+(x-d)/γ_ F. ∂ _xf̂^ N(r)|_x=d
    		+ F̂^ S2
    		+ i  sgn(ω_n) h_ ex d_ F^2/2 ħ D_ F[ 
    		τ̂_̂ẑ, F̂^ S2] 
       -    i  sgn(ω_n) (x-d)^2 h_ ex/2ħ D_ F[
    		τ̂_̂ẑ, F̂^ S2]
    .





The general solutions of Eq. (<ref>) in the N are given by


    [ [  f_s^N(  r); f_tx^N(  r); f_tz^N(  r) ]]    =   [ [ A_1;   0;   0 ]]e^k_Nx + [ [ A_2;   0;   0 ]]e^ - k_Nx, 
       +    B[ [ 0; i; 1 ]]e^iα̃xe^k_αx + C[ [ 0; i; 1 ]]e^iα̃xe^ - k_αx + F[ [ 0; i; 1 ]]e^ - iα̃xe^k_αx + G[ [ 0; i; 1 ]]e^ - iα̃xe^ - k_αx,


where k_α = √(3 α_ R^2 + k_ N^2) and √(2|ω_n|/D_ N). 
Here, we assume that α_ R≠0 to obtain Eq. (<ref>). 
Applying the boundary conditions given in Eqs. (<ref>), (<ref>), (<ref>), and (<ref>) to Eq. (<ref>) 
, and also using the result in Eq. (<ref>), we can obtain the anomalous Green's functions in the N as 



    f^ N_s (r)    =   [
    -i Δ_ L/E_ω_n(
    1-k_ N d_ F/γ_ F)
    sinh[k_ N(x-d)] 
    +
    i Δ_ R/E_ω_nsinh(k_ Nx)
    ] 
    Q_ω_n(d), 
    
    
    f^ N_tx(r)    =    i f^ N_tz(r),



and

    f^ N_tz(r)    =    sgn (ω_n) h_ exd_ F^2/ħ D_ FΔ_ R/E_ω_nΦ_ω_n(d) t_ω_n(x,d),

where E_ω_n=√((ħω_n)^2 + Δ^2), 


    Q_ω_n^-1(d) = 
    sinh(k_ Nd) + k_ Nd_ F/γ_ Fcosh(k_ Nd),




    Φ_ω_n^-1(d)    =   ( iα _R + k_α)d_F/γ _F[ e^( iα _R + k_α)L 
    + e^ - ( iα _R + k_α)L]C^31_ω_n( d ) 
       +   d_F/γ _F[ ( iα _R 
    - k_α)e^( iα _R - k_α)L 
    + ( iα _R + k_α)e^ - ( iα _R 
    + k_α)L]C^32_ω_n( d ) 
       -   d_F/γ _F[ ( iα _R - k_α)e^(  - iα _R 
    + k_α)L - ( iα _R + k_α)
    e^ - ( iα _R + k_α)L]C^33_ω_n( d )
    ,


and



    t_ω _n( x,d)    =    
    	i2[ C_ω _n^21( d ) 
    	+ 2C_ω _n^31( d )] 
       ×   [ cos( α _Rd)sinh( k_αx) 
    	+ isin( α _Rd)cosh( k_αx)] 
       -    2[ C_ω _n^22( d ) + 2C_ω _n^32( d )]sin( α _Rx)e^ - k_αx
       +    i2[ C_ω _n^23( d ) + 2C_ω _n^33( d )]sinh( k_αx)e^ - iα _Rx,



where the explicit formulae of the functions C_ω_n^i j(d) (i,j=1,2,3) are presented in Appendix A. 

From Eqs. (<ref>)–(<ref>), it is immediately found that 
f_s^ N( r) representing the SSC is an even function with ω_n, 
whereas f_tx(tz)^ N(r) representing the STC is an odd function with ω_n 
since f_tx(tz)^ N(r) is proportional to sgn(ω_n). 
Therefore, f_tx(tz)^ N(r) describes the odd-frequency STC. 
Note that f_ty^ N(r) = 0 since we assume that the present junction is a 1D model and the magnetization in the F has only a z component <cit.>. 
Also note that f_tx(tz)^ N(r) is exactly zero when h_ ex=0, which corresponds to no magnetic layer in the present junction studied here. 
This result shows that the RSOI does not induce the STC in the N, and thus the magnetic layer is needed to produce the STC. 
The role of the RSOI is to induce a finite f_tx^ N(r) in the N of the present junction<cit.>. 




 §.§ Magnetization induced by proximity effect in normal metal with RSOI

On the basis of the quasiclassical Green's function theory, 
the magnetization M(d,θ) induced by the proximity effect is given by<cit.>


    M(d,θ)    =   
    		(M_x(d,θ),M_y(d,θ),M_z(d,θ)) 
       =   A/V∫_0^dm(x,θ) dx
    ,

where θ=θ_ R-θ_ L is the Josephson phase in the junction and 

    m (x,θ)    =    (m_x(x,θ),m_y(x,θ),m_z(x,θ)) 
       =   - g μ_ Bπ N_ F k_ B T 
    	∑_ω_n sgn(ω_n)
    	 Im[
    	f_s^ N(r) f_t^   N*(r) 
    	]

with 

    f_t^   N(r) = 
    	(f_tx^ N(r),-f_ty^ N(r),f_tz^ N(r))
    .

m(x,θ) is the local magnetization density 
in the N, g is the g factor of an electron, and μ_ B is the Bohr magneton. 
A and V=Ad are the cross-section area of the junction and the volume of N, respectively. 
N_ F is the density of states per unit volume and per electron spin at the Fermi energy. 

It is obvious from Eq. (<ref>) that  f_s^ N( r) and f_t^   N( r) must both be nonzero to induce a finite 
m (x,θ). However, as described in Sect. <ref>, a nonzero f_t^   N( r) 
occurs only when the F layer is involved in the junction. 
Therefore, the origin of the magnetization in the N is considered to be the STCs induced 
by the proximity effect <cit.>.  
Because f_ty^ N( r)=0 (see Sect. <ref>), m_y(x,θ) and M_y (d,θ) are always zero. 

It is also noticeable that M_x^(2)(d,θ) and M_x^(3)(d,θ) are 

only induced in the N when the RSOI and the Josephson coupling are finite in the S/N/F/S junction. 
This result is in sharp contrast to Josephson junctions composed of a metallic multilayer system without the RSOI<cit.>.
Therefore, in what follows, we only consider the x and z components of M(d,θ). 

Substituting Eq. (<ref>) and the complex conjugate of Eq. (<ref>) into Eq. (<ref>), 
and integrating Eq. (<ref>) with respect to x from 0 to d, we can obtain the x and z components of the magnetization given by Eq. (<ref>). 
The x component M_x(d,θ) is decomposed into three parts, 


    M_x(d,θ)    =   
    M_x^(1)(d) + M_x^(2)(d,θ) + M_x^(3)(d,θ),

where



    M_x^(1)(d)    =   
    - g μ_ Bπ N_ F k_ B T
    h_ ex d_ F^2/ħ D_ Fd
       ×   ∑_ω_nΔ^2/E_ω_n^2
    Q_ω_n(d)
     Im[
    Φ_ω_n(d) w_ω_n(d)
    ]
    , 
    
    M_x^(2)(d,θ)    =   
    g μ_ Bπ N_ F k_ B T
    h_ ex d_ F^2/ħ D_ Fd∑_ω_nΔ^2/E_ω_n^2(
    1-k_ Nd_ F/γ_ F) 
       ×    Q_ω_n(d)
     Im[
    Φ_ω_n(d) u_ω_n(d)
    ] cosθ,



and 


    M_x^(3)(d,θ)    =   
    g μ_ Bπ N_ F k_ B T
    h_ ex d_ F^2/ħ D_ Fd∑_ω_nΔ^2/E_ω_n^2(
    1-k_ Nd_ F/γ_ F) 
       ×    Q_ω_n(d)
     Re[
    Φ_ω_n(d) u_ω_n(d)
    ] sinθ.



Similarly, the z component M_z(d,θ) is also decomposed into three parts, 


    M_z(d,θ)    =   
    M_z^(1)(d) + M_z^(2)(d,θ) + M_z^(3)(d,θ),


where 



    M_z^(1)(d)    =   
    g μ_ Bπ N_ F k_ B T
    h_ ex d_ F^2/ħ D_ Fd
       ×   ∑_ω_nΔ^2/E_ω_n^2
    Q_ω_n(d)
     Re[
    Φ_ω_n(d) w_ω_n(d)
    ]
    , 
    
    M_z^(2)(d,θ)    =   
    - g μ_ Bπ N_ F k_ B T
    h_ ex d_ F^2/ħ D_ Fd∑_ω_nΔ^2/E_ω_n^2(
    1-k_ Nd_ F/γ_ F) 
       ×    Q_ω_n(d)
     Re[
    Φ_ω_n(d) u_ω_n(d)
    ] cosθ,



and


    M_z^(3)(d,θ)    =   
    g μ_ Bπ N_ F k_ B T
    h_ ex d_ F^2/ħ D_ Fd∑_ω_nΔ^2/E_ω_n^2(
    1-k_ Nd_ F/γ_ F) 
       ×    Q_ω_n(d)
     Im[
    Φ_ω_n(d) u_ω_n(d)
    ] sinθ.



The explicit formulae of the functions w_ω_n(d) and u_ω_n(d) in Eqs. (<ref>)–(<ref>) are given in Appendix B. 
From Eqs. (<ref>)–(<ref>) and Eqs. (<ref>)–(<ref>), 
it is immediately found that the magnetizations of the 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. 
Note that M_x(d,θ) is always zero without the RSOI as mentioned above. 


One of the θ-independent parts of the magnetization, i.e., M_z^(1)(d), 
is due to the proximity effect common in the S/F multilayer systems<cit.>. 
The other  θ-independent part of the magnetization, i.e., M_x^(1)(d), is due to not only the proximity effect but also the presence of the RSOI, 
since the magnetization of the F is oriented along the z axis in the present junction<cit.>. 
The θ-dependent part of M_z^(2)(d,θ) is induced by the coupling between the two superconductors 
only when S/F multilayer systems compose the Josephson junction<cit.>. 
The θ-dependent part of M_x^(2)(d,θ) is induced by the coupling between the two superconductors and the finite RSOI. 
M_x^(3)(d,θ) and M_z^(3)(d,θ) appear 
when the RSOI, the exchange field, and the Josephson coupling are finite. 
The expressions for the magnetizations M_x(d,θ) and M_z(d,θ) given in Eqs. (<ref>)–(<ref>) are rather complicated. 
Therefore, we will present numerical results of magnetizations calculated here in the next section. 








§ RESULTS

In this section, we numerically evaluate the magnetizations of Eqs. (<ref>) and (<ref>) induced by the proximity effect in the S/N/F/S junction. 
In order to perform the numerical calculation of M_x(d,θ) and M_z(d,θ), 
the temperature dependence of Δ is assumed to be Δ=Δ_0tanh(1.74√(T_ C/T-1)), 
where Δ_0 is the superconducting gap at zero temperature and T_ C is the superconducting transition temperature. 
The thicknesses of N and F are normalized by ξ_ D=√(ħ D_ N/2π k_ B T_ C) and 
the magnetizations of the x and z components are normalized by M_0=(g μ_ B N_ FΔ_0). 




 §.§ Thickness dependence of magnetizations induced by the proximity effect


Figure 2 shows the x and z components of the magnetization induced by the proximity effect inside the N as a function of d. 
The solid (black) and dashed (red) lines are magnetizations for α̅ =0.2, and 0.5, respectively. 
In Fig. <ref>, P.R. and N.R. are abbreviations for positive and negative regions, respectively. 
A.L. denotes the auxiliary line separating the positive and negative regions of magnetization in Fig. <ref>. 
We find that M_x(d,θ) and M_z(d,θ) exhibit damped oscillatory behavior as a function of d. 
From Fig. <ref>, 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 for which M_x(z)(d,θ)≈ 0, the magnetizations can be easily reversed by tuning α_ R. 






Figures <ref>(a) and <ref>(b) show the x and z components of the magnetization as a function of α_ R, respectively. 
Here, we set the thickness of N as d/ξ_ D=1.3. 
From Fig. <ref>(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 then changed from positive to negative with further increasing α_ R. 
From Fig. <ref>(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 the x and z components of the magnetization can be reversed by tuning α_ R. 




 §.§ RSOI dependence of magnetization and magnetization-phase relation


Figure <ref> shows the magnetization as a function of d. 
Figures <ref>(a) and <ref>(b) show the x and z components of the magnetization, respectively. 
In Fig. <ref>, the solid (black) and dashed (red) lines are the magnetizations for θ =0 and π/2, respectively. 
α̅ is set to 0.5. 
From Fig. <ref>, it is found that the periods of oscillation of M_x(d,θ) and M_z(d,θ) can be changed by tuning θ. 
Note that the variation of the oscillation period of the magnetizations becomes large when d/ξ_ D >1 as shown in Fig. <ref>. 
By setting d/ξ_ D near the third minimum of M_x(z)(d,θ), 
we can perform magnetization reversal by changing θ as well as α_ R (see Figs. <ref> and <ref>). 

Figure <ref> shows the magnetization as a function of θ, i.e., the magnetization–phase relation<cit.>. 
Figures <ref>(a) and <ref>(b) show the x and z components of the magnetization, respectively. 
The thickness of N is set to d/ξ_ D=1.45. 
The behavior of M_x(d,θ) with θ is a cosine function as shown in Fig. <ref>(a). 
On the other hand, the behavior of M_z(d,θ) with θ is a sine function as shown in Fig. <ref>(b). 
From Fig. <ref>, it is immediately found that M_x(d,θ) and M_z(d,θ) can be varied 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. 



§ 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. (<ref>), the x component of the magnetization is proportional to Im[f_s( r) f_tx( r)]. 
From Eq. (<ref>), f_tx( r) = i f_tz( r), Im[f_s( r) f_tx( r)] =  Im[f_s( r) f_tz( r) e^i π/2]. 
Therefore, the magnetization–phase 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 the magnetization induced by the proximity effect. 
As shown in Figs. <ref> and <ref>, the magnetization in the N has a finite value in the length scale of ξ_ D. 
In the dirty limit, ξ_ D is on the orders of 10–100 nm<cit.>. 
Therefore, the magnetization induced by the proximity effect has a finite value in this length scale. 
To estimate the amplitude of the magnetization, 
we evaluate the normalized factor of the magnetization, i.e., M_0=g μ_ B N_ FΔ_0 (for instance, see Fig. <ref>). 
When we use a typical set of parameters<cit.>, M_0 is approximately 100 A/m. 
Therefore, the order of the magnetization amplitude is between 10 and 10^4 (see Fig. <ref>). 
It is expected that this order of  magnetization amplitude can be detected by magnetization measurement utilizing a SQUID<cit.>. 

Finally, we approximately estimate the thickness (d) of N, the thickness (d_ F) of F, and the magnitude of α_ R. 
We estimate d and d_ F by considering realistic materials. 
When an InGaAs/InAlAs quantum well as a normal metal and CuNi as a ferromagnetic metal are chosen, 
from Fig. <ref> and Refs. 5 and 86, the suitable d and d_ F are of 100 nm and nm orders, respectively. 
The total thickness between the two superconductors is of 100 nm orders. 
For this thickness, the Josephson coupling is still finite and thus the magnetization induced by proximity effect studied here 
can be controlled by the Josephson phase.
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 performing realistic experiments
<cit.>. 
For instance, InGaAs/InAlAs and InAs/AlSb quantum wells are good candidates as the N in the present junction studied here<cit.>. 
Therefore, it is expected that the magnetization induced by the proximity effect can be easily reversed by tuning α_ R. 




§ SUMMARY

We have theoretically studied the magnetization reversal by tuning the RSOI (α_ R) and Josephson phase in an S/N/F/S junction. 
The magnetizations of the x and z components are induced by the appearance of the odd-frequency spin-triplet 
and even-frequency spin-singlet Cooper pairs in the N. 
We have shown that the magnetizations exhibit 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 is varied 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 also been 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 the magnetization by tuning α_ R and θ is a good means of observing the spin of STCs. 

We have theoretically shown that the magnetizations are decomposed into 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) The appearance of M^(1)_x(d), which is the θ-independent part of the 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) 
result from the finite Josephson coupling between the two superconductors, the exchange field in the F, and the RSOI in the N in the S/N/F/S junction. 
(ii) M_z^(1)(d), which always appears in the S/F junctions, is induced by the proximity effect. 
The θ-dependent part M_z^(2)(d,θ) results from the finite Josephson coupling between two superconductors and the exchange field in the F. 
M_z^(3)(d,θ) appears only when the Josephson coupling, the exchange field, and the RSOI are finite. 
For the θ-dependence of the magnetizations, 
we have found that M_x(z)^(2)(d,θ) is a cosine function of θ and M_x(z)^(3)(d,θ) is sinusoidal with θ. 

We have also shown 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, has 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. 



§ ACKNOWLEDGMENTS
 
The authors would like to thank M. Mori for useful discussions and comments. 





§ COEFFICIENTS C^IJ_Ω_N(D)

In Eqs. (<ref>) and (<ref>), the coefficients C^ij_ω_n(d) are given by



    C_ω _n^21( d )    =    
    iα _Rd_F[ ( iα _R - k_α)d_F/γ _Fe^(  - iα _R + k_α)L 
    - ( iα _R + k_α)d_F/γ _Fe^ - ( iα _R + k_α)L] 
       +   k_αd_F[ ( iα _R - k_α)d_F/γ _Fe^( iα _R 
    - k_α)L + ( iα _R + k_α)d_F/γ _Fe^ - ( iα _R + k_α)L], 
    
    C_ω _n^22( d ) 
       =     - ( iα _R + k_α)
    d_F[ ( iα _R - k_α)d_F/γ _Fe^(  - iα _R + k_α)L 
    - ( iα _R + k_α)d_F/γ _Fe^ - ( iα _R + k_α)L] 
       -   k_αd_F( iα _R + k_α)d_F/γ _F[ e^( iα _R + k_α)L 
    + e^ - ( iα _R + k_α)L], 
    
    C_ω _n^23( d ) 
       =     - ( iα _R + k_α)d_F[ ( iα _R 
    - k_α)d_F/γ _Fe^( iα _R - k_α)L + ( iα _R 
    + k_α)d_F/γ _Fe^ - ( iα _R + k_α)L] 
       +    iα _Rd_F( iα _R + k_α)d_F/γ _F[ e^( iα _R + k_α)L + e^ - ( iα _R + k_α)L], 
    
    C_ω _n^31( d ) 
       =    iα _Rd_F{[ 1 - ( iα _R 
    - k_α)d_F/γ _F]e^k_αd 
    - [ 1 + ( iα _R + k_α)d_F/γ _F]
    e^ - k_αd}e^ - iα̃d
       -   k_αd_F{[ 1 + ( iα _R - k_α)d_F/γ _F]e^iα _Rd 
    - [ 1 + ( iα _R + k_α)d_F/γ _F]e^ - iα̃d}e^ - k_αd, 
    
    C_ω _n^32( d ) 
       =     - ( iα _R + k_α)d_F{[ 1 - ( iα _R - k_α)d_F/γ _F]e^k_αd 
    - [ 1 + ( iα _R + k_α)d_F/γ _F]e^ - k_αd}e^ - iα _Rd
       +   k_αd_F[ 1 + ( iα _R + k_α)d_F/γ _F][ e^( iα _R 
    + k_α)d - e^ - ( iα _R + k_α)d],



and



    C_ω _n^33( d )    =   ( iα _R + k_α)d_F{[ 1 + ( iα _R - k_α)
    	d_F/γ _F]e^iα̃d - [ 1 + ( iα _R + k_α)d_F/γ _F]
    	e^ - iα _Rd}e^ - k_αd
       -    iα _Rd_F[ 1 + ( iα _R + k_α)d_F/γ _F]
    	[ e^( iα _R + k_α)d - e^ - ( iα _R + k_α)d].







§ INTEGRATION WITH RESPECT TO X IN EQ. (<REF>)

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. (<ref>)–(<ref>) into Eq. (<ref>). 
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_x^( 1 )( x ) + m_x^( 2 )( x,θ) + m_x^( 3 )( x,θ),


where 

    m_x^(1)(x,θ)    =   
    -g μ_ Bπ N_ F k_ B T
    h_ ex d_ F^2/ħ D_ F∑_i ω_nΔ^2/E_ω_n^2 
    Q_ω_n(d)  
       ×   sinh(k_ N x)
     Im[
    Φ_ω_n(d) t_ω_n(x,d)
    ]
    , 
    
    m_x^(2)(x,θ)    =   
    g μ_ Bπ N_ F k_ B T
    h_ ex d_ F^2/ħ D_ F∑_i ω_nΔ^2/E_ω_n^2(
    1-k_ Fd_ F/γ_ F) 
       ×   
    Q_ω_n(d)  
    sinh[k_ N (x-d)] 
       ×    Im[
    Φ_ω_n(d) t_ω_n(x,d)
    ] cosθ,


and

    m_x^(3)(x,θ)    =   
    g μ_ Bπ N_ F k_ B T
    h_ ex d_ F^2/ħ D_ F∑_i ω_nΔ^2/E_ω_n^2(
    1-k_ Fd_ F/γ_ F) 
       ×    Q_ω_n(d)
    sinh[k_ N (x-d)] 
       ×    Re[
    Φ_ω_n(d) t_ω_n(x,d)
    ] sinθ,

where the functions Q_ω_n(d), Φ_ω_n(d), and t_ω_n(x,d) are respectively given in Eqs. (<ref>)–(<ref>). 
Substituting Eqs. (<ref>)–(<ref>) into Eq. (<ref>) and performing the integration with respect to x, 
we can obtain the x component of the magnetization, 



    [                       M_x( d,θ) = 1/d∫_0^d m_x( x,θ)dx = 1/d∫_0^d m_x^( 1 )( x )dx  
    + 1/d∫_0^d m_x^( 2 )( x,θ)dx  + 1/d∫_0^d m_x^( 3 )( x,θ)dx;                            =  - gμ _BπN_Fk_ BT1/d∑_iω _nh_exd_F^2/ħD_FΔ ^2/E_ω _n^2Q_ω _n( d )∫_0^d sinh( k_Nx) Im[ Φ _ω _n( d )t_ω _n( x,d)]dx; + gμ _BπN_Fk_ BT1/d∑_iω _nh_exd_F^2/ħD_FΔ ^2/E_ω _n^2( 1 - k_Nd_F/γ _F)Q_ω _n( d )∫_0^d 
    sinh[ k_N( x - d)] Im[ Φ _ω _n( d )t_ω _n( x,d)]dxcosθ;  + gμ _BπN_Fk_ BT1/d∑_iω _nh_exd_F^2/ħD_FΔ ^2/E_ω _n^2( 1 - k_Nd_F/γ _F)Q_ω _n( d )∫_0^d sinh[ k_N( x - d)] Re[ Φ _ω _n( d )t_ω _n( x,d)]dxsinθ;                                                                                          = M_x^( 1 )( d,θ) + M_x^( 2 )( d,θ) + M_x^( 3 )( d,θ), ]


where M_x^(1)(d), M_x^(2)(d,θ), and M_x^(3)(d,θ) are respectively given in Eqs. (<ref>)–(<ref>). 
The functions w_ω_n(d) and u_ω_n(d) in Eqs. (<ref>)–(<ref>) are given as 



    u_ω_n(d)    =    u_ω_n^(1)(d) + u_ω_n^(2))(d) + u_ω_n^(3)(d), 
    
    u_ω _n^( 1 )( d )    =     
    - 2[ C_ω _n^21( d ) + 2C_ω _n^31( d )]
    k_Nsin[ ( α _R - ik_α)d] 
    - ( α _R - ik_α)
    sinh( k_Nd)/( α _R - ik_α)^2 + k_N^2, 
    
    u_ω _n^( 2 )( d )    =    
    4α _R[ C_ω _n^22( d ) + 2C_ω _n^32( d )]e^ - k_αd
       ×   k_αk_Ncos( α _Rd) + k_Nα _Rsin( α _Rd) - e^k_αd[ k_αk_Ncosh( k_Nd) - ( 2α _R^2 + k_N^2)
    sinh( k_Nd)]/( α _R^2 + k_α ^2)^2 
    + 2( α̃_R^2 - k_α ^2)k_N^2 + k_N^4, 
    
    u_ω _n^( 3 )( d )    =    
    - i4α _R[ C_ω _n^23( d ) + 2C_ω _n^33( d )] 
       ×   ik_αk_Ncosh( k_αd) + k_Nα _Rsinh( k_αd) - α _Rk_α[ ik_Ncosh( k_Nd) + 2α _Rsinh( k_Nd)]e^iα _Rd/[ α _R^2 + ( k_α - k_N)^2]
    [ α _R^2 + ( k_α + k_N)^2]e^ - iα _Rd





    w_ω_n(d)    =    w_ω_n^(1)(d) + w_ω_n^(2))(d) + w_ω_n^(3)(d), 
    
    w_ω _n^( 1 )( d )    =   e^ - ( iα _R + k_α)d/( α _R - ik_α)^2 
    + k_N^2[ (  - 1 + e^( iα _R + k_α)2d)k_Ncosh( k_Nd) 
    - i( 1 + e^( iα _R + k_α)2d)( α _R - ik_α)
    sinh( k_αd)], 
    
    w_ω _n^( 2 )( d )    =   e^ - k_αda_ω _n^( 1 )( d ) + 
    a_ω _n^( 2 )( d ) - a_ω _n^( 3 )( d )/( α _R^2 + k_α ^2)^2 
    - ( 4α _R^2 - k_N^2)k_N^2, 
    
    a_ω_n^(1)(d)    =    2 α_ R k_α k_ N e^k_α d, 
    
    a_ω _n^( 2 )( d )    =     
    - 2α _Rk_Ncosh( k_Nd)[ k_αcos( α _Rd) 
    + α̃sin( α _Rd)], 
    
    a_ω _n^( 3 )( d )    =    
    2α _R[ ( 2α _R^2 + k_N^2)
    cos( α _Rd) + 2k_αα _Rsin( α _Rd)] sinh( k_Nd), 
    
    w_ω _n^( 3 )( d )    =   1/2e^ - iα _Rd[ b_ω _n^( 1 )( d ) 
    + b_ω _n^( 2 )( d ) + b_ω _n^( 3 )( d )], 
    
    b_ω _n^( 1 )( d )    =   e^iα _Rd[ iα _R/α _R^2 + ( k_α - k_N)^2 
    - iα _R/α _R^2 + ( k_α + k_N)^2], 
    
    b_ω _n^( 2 )( d )    =     
    - iα _Rcosh[ ( k_α - k_N)d] 
    + ( k_α - k_N)sinh[ ( k_α - k_N)d]/α _R^2 
    + ( k_α - k_N)^2,



and



    b_ω _n^( 3 )( d )    =   iα _Rcosh[ ( k_α + k_N)d] 
    + ( k_α + k_N)sinh[ ( k_α + k_N)d]/α _R^2 + ( k_α + k_N)^2.




We can also obtain the z component of the magnetization given in Eqs. (<ref>)–(<ref>) 
by following the procedure used to derive the x component of the magnetization. 








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note1
In the S/N/F/S junction without the RSOI, the STC composed of opposite spin electrons 
is only induced by the proximity effect inside the N, and thus only the z component of magnetization is induced<cit.>. 

note

We need to apply the DC bias current to the present junction to change θ. 
Typically, the magnitude of the bias current is about 10 μA and the size of the device is about 1 μm<cit.>. 
In this case, the magnitude of the magnetic field (H) induced by the DC bias current is about 20 A/m based on Ampere's law H=I/2 π r, 
where I and r are the DC bias current and the distance from circuit, respectively. 
This H is much smaller than the magnetization induced by the proximity effect. 
Therefore, the influence of H can be neglected. 




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note7

N_ F=1/4π^2( 2m/ħ^2)^3/2ε_ F^1/2≈ 5.4 ×10^27   eV^-1 m^-3 
per spin with the Fermi energy 
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Δ_0=1 meV for Nb <cit.>.  

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