Magnetization Reversal by Tuning Rashba Spin-Orbit Interaction and Josephson Phase in a Ferromagnetic Josephson Junction
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Magnetization Reversal by Tuning Rashba Spin-Orbit Interaction andJosephson Phase in a Ferromagnetic Josephson Junction
Shin-ichi Hikino
National Institute of Technology, Fukui College, Sabae, Fukui 916-8507, Japan
We theoretically investigate the magnetization inside a normal metal containing theRashba spin-orbit interaction (RSOI) induced by the proximity effect in an s -wave super-conductor/normal metal /ferromagnetic metal/ s -wave superconductor ( S/N/F/S ) Josephsonjunction. By solving the linearized Usadel equation taking account of the RSOI, we find thatthe direction of the magnetization induced by the proximity effect in N can be reversed bytuning 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 RSOIand Josephson phase can be applied to superconducting spintronics.
1. Introduction In s -wave superconductor/ferromagnetic metal ( S/F ) junctions, it is well known that thepair amplitude of a spin-singlet Cooper pair (SSC) penetrating into F due to the proximityeffect shows damped oscillatory behavior as a function of the thickness of F . One of theinteresting phenomena resulting from the damped oscillatory behavior of the pair amplitudeis a π -state in an S / F / S junction, where the current–phase relation in the Josephson currentis shifted by π from that of ordinary S / I / S and S / N / S junctions. Another notable phenomenon in
S/F junctions is the appearance of odd-frequency spin-triplet Cooper pairs (STCs) induced by the proximity effect.
19, 20)
In an S / F junction havinga uniform magnetization, the STC composed of opposite spin electrons ( S z = 0) and theSSC penetrate into F owing to the proximity effect.
19, 21)
The penetration length of the STCwith S z = 0 and the SSC into F is determined by ξ F = p ~ D F /h ex , which is typically of nmorder. 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 = p ~ D F / πk B T in F ( T is temperature). The feature of this STC is approximately two orders of magnitudelarger than the penetration length of the SSC and STC with S z = 0. Thus, the proximityeffect 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 . Phys. Soc. Jpn. Full Paper of Josephson current in ferromagnetic Josephson junctions (FJJs). The Josephson currentcarried 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 Josephsoncurrent flowing through the F has been observed and established experimentally in FJJs. The detection of long-range Josephson current is a piece of evidence for the presence of theSTC.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 directlymeasure the spin of the STC.
For this purpose,
S/F/N and
S/F/N/F/S junctions con-taining the Rashba spin-orbit interaction (RSOI) have been of considerable interest in recentyears, since the proximity effect coupled with the RSOI in
S/F/N and
S/F/N/F/S junctionsexhibits many fascinating phenomena which are not observed in
S/F/N and
S/F/N/F/S junc-tions without the RSOI.
For
S/F/N junctions, it has been theoretically found that thepair amplitude of the STC penetrating into N with the RSOI is modulated as a function of thethickness of N and the magnitude of the RSOI .
69, 70)
By utilizing the modulation of the pairamplitude of the STC, it is possible to freely control 0- and π -states by tuning the RSOI inJosephson junctions.
68, 70)
Moreover, some authors have theoretically predicted that the directevidence of the STC can also be obtained by detecting the spin Hall effect and magnetoelectriceffect induced by the STC in Josephson junctions containing ferromagnetic metals and theRSOI.
71, 72, 74)
The RSOI is advantageous for studying spin-dependent transport phenomenasince it can be freely controlled by an external electric field.
However, understanding of thespin transport of the STC in consideration of the RSOI is still lacking, since the studies inthis research field have been rare and limited.
71, 72, 74)
Therefore, it is expected that a goodunderstanding of the spin transport of the STC will provide proof for the STC and expeditethe development of superconducting spintronics.
In this paper, we theoretically propose another setup and way to detect the STC byusing an S N / F / S S N / F / S N only appears when the product of theanomalous Green’s functions of the spin-triplet odd-frequency Cooper pair and spin-singleteven-frequency Cooper pair in the N has a finite value. It is found that the magnetizationshows damped oscillatory behavior as a function of the thickness of N . It is also found that thedirection 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 theperiod of oscillation of magnetization can be changed by tuning θ . This result clearly showsthat the direction of the magnetization can also be controlled by tuning θ as well as the . Phys. Soc. Jpn. Full Paper xyz S d - d S L F L d d = +
S1 S2F
N with RSOI
Fig. 1. (Color online) Schematic illustration of the S N / F / S N is a normalmetal with the RSOI, F is a ferromagnetic metal, and S s -wave superconductor. Thearrow in F indicates the direction of the ferromagnetic magnetization where the magnetization in F is fixed along the z direction. d S , d F , and d are the thicknesses of S , F , and N , respectively,with L = d + d F . We assume that the magnetization is uniform in the F layer and that d S is muchlarger than ξ S . magnitude of the RSOI. Therefore, we expect that these results can be applied to the researchfield of superconducting spintronics.The rest of this paper is organized as follows. In Sect. 2, we introduce an S N / F / S N containing the RSOI of this junction by solving the Usadel equation. In Sect. 3, the nu-merical results of the magnetization are given. The thickness and RSOI dependences of themagnetization are discussed. Moreover, we present the Josephson phase ( θ ) dependence ofthe magnetization. Finally, the θ -magnetization relation is discussed and the magnetizationinduced by the proximity effect is estimated for a typical set of realistic parameters in Sect. 4.A summary of this paper is given in Sect. 5. The detailed calculation of the magnetization isgiven in the Appendices.
2. Magnetization in Normal Metal Induced by Proximity Effect in an S1/N/F/S2Junction with Rashba Spin-Orbit Interaction
We consider a Josephson junction composed of s -wave superconductors ( Ss ) separated bya normal metal/ferromagnetic metal ( N/F ) junction as depicted in Fig. 1. Here, we includethe RSOI in the N and assume uniform magnetization in F in the S /N/F/S ξ T . In this situation, a one-dimensional(1D) model may be a good approximation. Therefore, we adopt the 1D model to analyze themagnetization induced by the proximity effect in the S /N/F/S N with the RSOI inducedby the proximity effect is evaluated by solving the linearized Usadel equation including theSU(2) gauge field in each region ( j = N, F ), i ~ D j ˜ ∂ x ˆ f j ( r ) − i ~ | ω n | ˆ f j ( r ) − sgn( ω n ) h ex ( x ) h ˆ τ z , ˆ f j ( r ) i = ˆ0 , (1) . Phys. Soc. Jpn. Full Paper ˜ ∂ x = ∂ x • − i ~ h ˆ A x , • i , ˆ A x = α R ˆ τ y , < x < d∂ x • , otherwise , where r = ( x, ω n ), ˆ A x is the SU(2) gauge field, which describes the RSOI, and α R is theRSOI constant. D j is the diffusion coefficient in region j , ω n = (2 n + 1) πk B T / ~ ( n : integer) isthe fermion Matsubara frequency, sgn( X ) = X/ | X | is the sign function, and ˆ τ y ( z ) is the y ( z )component of the Pauli matrix. [ ˆ Q, ˆ R ] = ˆ Q ˆ R − ˆ R ˆ Q is the anticommutation relation and ˆ0 isthe zero matrix. The anomalous part ˆ f j ( r ) of the quasiclassical Green’s function is givenby ˆ f j ( r ) = f j ↑↑ ( r ) f j ↑↓ ( r ) f j ↓↑ ( r ) f j ↓↓ ( r ) ! = − f jtx ( r ) + if jty ( r ) f js ( r ) + f jtz ( r ) − f js ( r ) + f mtz ( r ) f jtx ( r ) + if jty ( r ) ! . (2) f js ( r ) is the anomalous Green’s function for the SSC, and f jtx ( ty ) ( r ) and f jtz ( r ) represent theanomalous Green’s functions for the STC with | S z | = 1 and | S z | = 0, respectively. Theexchange field h ex ( x ) in the F is given by h ex ( x ) = ( h ex e z , d < x < L , otherwise , (3)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.,ˆ f S1 ( r ) (cid:12)(cid:12)(cid:12) x =0 = ˆ f N ( r ) (cid:12)(cid:12)(cid:12) x =0 , (4)ˆ f N ( r ) (cid:12)(cid:12)(cid:12) x = d = ˆ f F ( r ) (cid:12)(cid:12)(cid:12) x = d , (5)ˆ f F ( r ) (cid:12)(cid:12)(cid:12) x = L = ˆ f S2 ( r ) (cid:12)(cid:12)(cid:12) x = L , (6) σ S ∂ x ˆ f S ( r ) (cid:12)(cid:12)(cid:12) x =0 = σ N ∂ x ˆ f N ( r ) (cid:12)(cid:12)(cid:12) x =0 , (7) ∂ x ˆ f F ( r ) (cid:12)(cid:12)(cid:12) x = d = 1 γ F ∂ x ˆ f N ( r ) (cid:12)(cid:12)(cid:12)(cid:12) x = d , (8) σ S ∂ x ˆ f S ( r ) (cid:12)(cid:12)(cid:12) x = L = σ F ∂ x ˆ f F ( r ) (cid:12)(cid:12)(cid:12) x = L , (9)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 thenormal state. ξ F = p ~ D F /h ex and ξ N = p ~ D N / πk B T . Notice that the left-hand side ofEqs. (7) and (9) is zero as will be shown later. Assuming that d S ≫ ξ S , the anomalous Green’sfunction in the S s attached to N and F can be approximately given asˆ f S1(2) s ( r ) = − ˆ τ y ∆ L(R) q ( ~ ω ) + | ∆ L(R) | ≡ ˆ F S1(2) , (10) . Phys. Soc. Jpn. Full Paper where ∆
L(R) = ∆ e iθ L(R) (∆: real) and θ L(R) is the superconducting phase in the left (right)side of the S (see Fig. 1). The s -wave superconducting gap ˆ∆( x ) is finite only in the S and isassumed to be constant as follows:ˆ∆( x ) = − ∆ L ∆ L ! , − d S < x < − ∆ R ∆ R ! , L < x < L S ˆ0 , otherwise . (11)From Eqs. (10) and (11), it is immediately found that the left-hand side of Eqs. (7) and (9)becomes zero, since the rigid boundary condition is assumed in the present calculation. Assuming d F /ξ F ≪
1, we can perform the Taylor expansion with x for ˆ f F ( r ) as fol-lows:
31, 80) ˆ f F ( r ) ≈ ˆ f F ( d, ω n ) + ( x − d ) ∂ x ˆ f F ( r ) (cid:12)(cid:12)(cid:12) x = d + ( x − d ) ∂ x ˆ f F ( r ) (cid:12)(cid:12)(cid:12) x = d . (12)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 f F ( r ) asˆ f F ( r ) ≈ − d F γ F ∂ x ˆ f N ( r ) (cid:12)(cid:12)(cid:12) x = d + ( x − d ) γ F ∂ x ˆ f N ( r ) (cid:12)(cid:12)(cid:12) x = d + ˆ F S2 + i sgn( ω n ) h ex d ~ D F h ˆ τ z , ˆ F S2 i − i sgn( ω n ) ( x − d ) h ex ~ D F h ˆ τ z , ˆ F S2 i . (13)The general solutions of Eq. (1) in the N are given by f N s ( r ) f N tx ( r ) f N tz ( r ) = A e k N x + A e − k N x , + B i e i ˜ αx e k α x + C i e i ˜ αx e − k α x + F i e − i ˜ αx e k α x + G i e − i ˜ αx e − k α x , (14)where k α = q α + k and p | 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 alsousing the result in Eq. (12), we can obtain the anomalous Green’s functions in the N as f N s ( r ) = (cid:20) − i ∆ L E ω n (cid:18) − k N d F γ F (cid:19) sinh[ k N ( x − d )] + i ∆ R E ω n sinh( k N x ) (cid:21) Q ω n ( d ) , (15) f N tx ( r ) = if N tz ( r ) , (16) . Phys. Soc. Jpn. Full Paper and f N tz ( r ) = sgn( ω n ) h ex d ~ D F ∆ R E ω n Φ ω n ( d ) t ω n ( x, d ) , (17)where E ω n = p ( ~ ω n ) + ∆ , Q − ω n ( d ) = sinh( k N d ) + k N d F γ F cosh( k N d ) , (18)Φ − ω n ( d ) = ( iα R + k α ) d F γ F h e ( iα R + k α ) L + e − ( iα R + k α ) L i C ω n ( d )+ d F γ F h ( iα R − k α ) e ( iα R − k α ) L + ( iα R + k α ) e − ( iα R + k α ) L i C ω n ( d ) − d F γ F h ( iα R − k α ) e ( − iα R + k α ) L − ( iα R + k α ) e − ( iα R + k α ) L i C ω n ( d ) , (19)and t ω n ( x, d ) = i (cid:2) C ω n ( d ) + 2 C ω n ( d ) (cid:3) × [cos ( α R d ) sinh ( k α x ) + i sin ( α R d ) cosh ( k α x )] − (cid:2) C ω n ( d ) + 2 C ω n ( d ) (cid:3) sin ( α R x ) e − k α x + i (cid:2) C ω n ( d ) + 2 C ω n ( d ) (cid:3) sinh ( k α x ) e − iα R x , (20)where the explicit formulae of the functions C ijω n ( d ) ( i, j = 1 , ,
3) are presented in AppendixA. From Eqs. (15)–(17), it is immediately found that f N s ( r ) representing the SSC is an evenfunction 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.Note that f N ty ( r ) = 0 since we assume that the present junction is a 1D model and themagnetization in the F has only a z component. Also note that f N tx ( tz ) ( r ) is exactly zerowhen 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 magneticlayer is needed to produce the STC. The role of the RSOI is to induce a finite f N tx ( r ) in the N of the present junction. On the basis of the quasiclassical Green’s function theory, the magnetization M ( d, θ )induced by the proximity effect is given by
23, 60) M ( d, θ ) = ( M x ( d, θ ) , M y ( d, θ ) , M z ( d, θ ))= AV Z d m ( x, θ ) dx, (21)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 )) . Phys. Soc. Jpn. Full Paper = − gµ B πN F k B T X ω n sgn( ω n )Im (cid:2) f N s ( r ) f N ∗ t ( r ) (cid:3) (22)with f t N ( r ) = ( f N tx ( r ) , − f N ty ( r ) , f N tz ( r )) . (23) m ( x, θ ) is the local magnetization density in the N , g is the g factor of an electron, and µ B isthe Bohr magneton. A and V = Ad are the cross-section area of the junction and the volumeof N , respectively. N F is the density of states per unit volume and per electron spin at theFermi energy.It is obvious from Eq. (22) that f N s ( r ) and f N t ( r ) must both be nonzero to induce afinite m ( x, θ ). However, as described in Sect. 2.1, a nonzero f N t ( r ) occurs only when the F layer is involved in the junction. Therefore, the origin of the magnetization in the N isconsidered to be the STCs induced by the proximity effect.
19, 60, 63, 66)
Because f N ty ( r ) = 0 (seeSect. 2.1), m y ( x, θ ) and M y ( d, θ ) are always zero. It is also noticeable that M (2) x ( d, θ ) and M (3) x ( d, θ ) are only induced in the N when the RSOI and the Josephson coupling are finitein the S/N/F/S junction. This result is in sharp contrast to Josephson junctions composedof a metallic multilayer system without the RSOI.
63, 66, 67)
Therefore, in what follows, we onlyconsider the x and z components of M ( d, θ ).Substituting Eq. (15) and the complex conjugate of Eq. (23) into Eq. (22), and integrat-ing Eq. (22) with respect to x from 0 to d , we can obtain the x and z components of themagnetization given by Eq. (21). The x component M x ( d, θ ) is decomposed into three parts, M x ( d, θ ) = M (1) x ( d ) + M (2) x ( d, θ ) + M (3) x ( d, θ ) , (24)where M (1) x ( d ) = − gµ B πN F k B T h ex d ~ D F d × X ω n ∆ E ω n Q ω n ( d )Im [Φ ω n ( d ) w ω n ( d )] , (25) M (2) x ( d, θ ) = gµ B πN F k B T h ex d ~ D F d X ω n ∆ E ω n (cid:18) − k N d F γ F (cid:19) × Q ω n ( d )Im [Φ ω n ( d ) u ω n ( d )] cos θ, (26)and M (3) x ( d, θ ) = gµ B πN F k B T h ex d ~ D F d X ω n ∆ E ω n (cid:18) − k N d F γ F (cid:19) × Q ω n ( d )Re [Φ ω n ( d ) u ω n ( d )] sin θ. (27) . Phys. Soc. Jpn. Full Paper
Similarly, the z component M z ( d, θ ) is also decomposed into three parts, M z ( d, θ ) = M (1) z ( d ) + M (2) z ( d, θ ) + M (3) z ( d, θ ) , (28)where M (1) z ( d ) = gµ B πN F k B T h ex d ~ D F d × X ω n ∆ E ω n Q ω n ( d )Re [Φ ω n ( d ) w ω n ( d )] , (29) M (2) z ( d, θ ) = − gµ B πN F k B T h ex d ~ D F d X ω n ∆ E ω n (cid:18) − k N d F γ F (cid:19) × Q ω n ( d )Re [Φ ω n ( d ) u ω n ( d )] cos θ, (30)and M (3) z ( d, θ ) = gµ B πN F k B T h ex d ~ D F d X ω n ∆ E ω n (cid:18) − k N d F γ F (cid:19) × Q ω n ( d )Im [Φ ω n ( d ) u ω n ( d )] sin θ. (31)The explicit formulae of the functions w ω n ( d ) and u ω n ( d ) in Eqs. (25)–(31) are given inAppendix B. From Eqs. (25)–(27) and Eqs. (29)–(31), it is immediately found that the mag-netizations 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 (1) z ( d ), is due to the proximityeffect common in the S/F multilayer systems.
19, 63, 66, 67)
The other θ -independent part of themagnetization, i.e., M (1) x ( d ), is due to not only the proximity effect but also the presenceof the RSOI, since the magnetization of the F is oriented along the z axis in the presentjunction. The θ -dependent part of M (2) z ( d, θ ) is induced by the coupling between the twosuperconductors only when S/F multilayer systems compose the Josephson junction.
63, 66, 67)
The θ -dependent part of M (2) x ( d, θ ) is induced by the coupling between the two superconduc-tors and the finite RSOI. M (3) x ( d, θ ) and M (3) z ( 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. (24)–(31) are rather complicated. Therefore, we will present numericalresults of magnetizations calculated here in the next section.
3. Results
In this section, we numerically evaluate the magnetizations of Eqs. (24) and (28) in-duced by the proximity effect in the
S/N/F/S junction. In order to perform the numericalcalculation of M x ( d, θ ) and M z ( d, θ ), the temperature dependence of ∆ is assumed to be∆ = ∆ tanh(1 . p T C /T − is the superconducting gap at zero temperature . Phys. Soc. Jpn. Full Paper and T C is the superconducting transition temperature. The thicknesses of N and F are nor-malized by ξ D = p ~ D N / πk B T C and the magnetizations of the x and z components arenormalized by M = ( gµ B N F ∆ ). Figure 2 shows the x and z components of the magnetization induced by the proximity ef-fect inside the N as a function of d . The solid (black) and dashed (red) lines are magnetizationsfor ¯ α = 0 .
2, and 0.5, respectively. In Fig. 2, P.R. and N.R. are abbreviations for positive andnegative regions, respectively. A.L. denotes the auxiliary line separating the positive and neg-ative regions of magnetization in Fig. 2. We find that M x ( d, θ ) and M z ( d, θ ) exhibit dampedoscillatory behavior as a function of d . From Fig. 2, it is found that the magnetizations can bereversed by changing the thickness of N . Moreover, it is also clearly found that the period ofoscillation 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 bytuning α R .Figures 3(a) and 3(b) show the x and z components of the magnetization as a function Fig. 2. (Color online) (a) x component M x ( d, θ ) and (b) z component M z ( d, θ ) of magnetization inthe N as a function of d for ¯ α = 0.2 and 0.5. M x ( d, θ ) and M z ( d, θ ) show the damped oscillatorybehavior with d , where P.M and N.R. are, respectively, positive and negative regions of M x ( d, θ )and M z ( d, θ ). Here we set T /T C = 0 . γ F = 0 . θ = π/ d F /ξ D = 0 .
01, and h ex = 30. ¯ α = α R ξ D and ξ D = p ~ D N / πk B T C . . Phys. Soc. Jpn. Full Paper
Fig. 3. (Color online) (a) x component M x ( d, θ ) and (b) z component M z ( d, θ ) of magnetization inthe N as a function of ¯ α for d/ξ D = 1 .
3. Here we set
T /T C = 0 . γ F = 0 . d F /ξ D = 0 . h ex = 30. ¯ α = α R ξ D and ξ D = p ~ D N / πk B T C . It is clearly found that magnetizations arereversed (a) from negative to positive values then from positive to negative values and (b) frompositive to negative values then from negative to positive values by increasing ¯ α . of α R , respectively. Here, we set the thickness of N as d/ξ D = 1 .
3. From Fig. 3(a), it isfound that the sign of M x ( d, θ ) is changed from negative to positive with increasing α R within0 . ξ D . α R . . ξ D . The sign of M x ( d, θ ) is then changed from positive to negative withfurther increasing α R . From Fig. 3(b), it is found that the sign of M z ( d, θ ) is changed fromnegative to positive with increasing α R within 0 . ξ D . α R . . ξ D . By further increasing α R , the sign of M z ( d, θ ) is changed from positive to negative. From these results, it is clearlyfound that the direction of the x and z components of the magnetization can be reversed bytuning α R . Figure 4 shows the magnetization as a function of d . Figures 4(a) and 4(b) show the x and z components of the magnetization, respectively. In Fig. 4, the solid (black) and dashed(red) lines are the magnetizations for θ = 0 and π/
2, respectively. ¯ α is set to 0.5. From Fig. 4,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 > d/ξ D near the third minimum of M x ( z ) ( d, θ ), we canperform 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., the magnetization–phase rela- . Phys. Soc. Jpn. Full Paper
Fig. 4. (Color online) (a) x component M x ( d, θ ) and (b) z component M z ( d, θ ) of magnetizationin the N as a function of d for θ = 0 and π/
2. Here we set ¯ α = 0 . T /T C = 0 . γ F = 0 . d F /ξ D = 0 .
01, and h ex = 30. ¯ α = α R ξ D and ξ D = p ~ D N / πk B T C . The insets show the behaviorof magnetizations from d/ξ D = 0.9 to d/ξ D =1.6. It is clearly found that the period of oscillationin M x ( d, θ ) and M z ( d, θ ) can be controlled by θ . tion. Figures 4(a) and 4(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 functionas shown in Fig. 5(a). On the other hand, the behavior of M z ( d, θ ) with θ is a sine functionas shown in Fig. 5(b). From Fig. 5, it is immediately found that M x ( d, θ ) and M z ( d, θ ) can bevaried from positive to negative values and vice versa by changing θ . This result indicates thatthe magnetizations can be reversed by changing θ when the thickness of N is appropriatelyset as mentioned above.
4. Discussion
Here, we discuss why the magnetization–phase relation of M x ( d, θ ) is shifted by π/ M z ( d, θ ). From Eq. (22), the x component of the magnetizationis proportional to Im[ f s ( r ) f tx ( r )]. From Eq. (16), f tx ( r ) = if tz ( r ), Im[ f s ( r ) f tx ( r )] =Im[ f s ( r ) f tz ( r ) e iπ/ ]. Therefore, the magnetization–phase relation of M x ( d, θ ) is shifted by π/ M z ( d, θ ).We approximately estimate the amplitude of the magnetization induced by the proximityeffect. As shown in Figs. 2 and 3, the magnetization in the N has a finite value in the lengthscale of ξ D . In the dirty limit, ξ D is on the orders of 10–100 nm. Therefore, the magnetizationinduced by the proximity effect has a finite value in this length scale. To estimate the amplitude . Phys. Soc. Jpn.
Full Paper
Fig. 5. (Color online) (a) x component M x ( d, θ ) and (b) z component M z ( d, θ ) of magnetization in the N as a function of θ for d/ξ D = 1 .
45. Here we set ¯ α = 0 . T /T C = 0 . γ F = 0 . d F /ξ D = 0 . h ex = 30. ¯ α = α R ξ D and ξ D = p ~ D N / πk B T C . For d/ξ D = 1 .
45, the magnetizations arereversed from positive to negative then from negative to positive by increasing θ . of the magnetization, we evaluate the normalized factor of the magnetization, i.e., M = gµ B N F ∆ (for instance, see Fig. 2). When we use a typical set of parameters,
84, 85) M isapproximately 100 A/m. Therefore, the order of the magnetization amplitude is between 10and 10 (see Fig. 4). It is expected that this order of magnetization amplitude can be detectedby magnetization measurement utilizing a SQUID. Finally, we approximately estimate the thickness ( d ) of N , the thickness ( d F ) of F , andthe magnitude of α R . We estimate d and d F by considering realistic materials. When an In-GaAs/InAlAs quantum well as a normal metal and CuNi as a ferromagnetic metal are chosen,from Fig. 2 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 effectstudied 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 calculationis easily achieved by performing realistic experiments. For instance, InGaAs/InAlAsand InAs/AlSb quantum wells are good candidates as the N in the present junction studiedhere. Therefore, it is expected that the magnetization induced by the proximity effectcan be easily reversed by tuning α R . . Phys. Soc. Jpn. Full Paper
5. Summary
We have theoretically studied the magnetization reversal by tuning the RSOI ( α R ) andJosephson phase in an S/N/F/S junction. The magnetizations of the x and z components areinduced by the appearance of the odd-frequency spin-triplet and even-frequency spin-singletCooper pairs in the N . We have shown that the magnetizations exhibit damped oscillatorybehavior as a function of the thickness of N for finite α R . The period of oscillation of themagnetizations induced by the proximity effect is varied by changing α R and becomes shortwith 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 dependon the Josephson phase ( θ ). As a result, the amplitude and the oscillation period of themagnetizations can be controlled by tuning θ . It has also been found that the direction of themagnetizations in the N can be reversed by changing θ as well as α R . These results clearlyshow that the variation of the magnetization by tuning α R and θ is a good means of observingthe 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 ), whichis the θ -independent part of the magnetization, is due to the proximity effect, the exchangefield 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, theexchange field in the F , and the RSOI in the N in the S/N/F/S junction. (ii) M (1) z ( d ), whichalways appears in the S/F junctions, is induced by the proximity effect. The θ -dependent part M (2) z ( d, θ ) results from the finite Josephson coupling between two superconductors and theexchange field in the F . M (3) z ( d, θ ) appears only when the Josephson coupling, the exchangefield, and the RSOI are finite. For the θ -dependence of the magnetizations, we have foundthat M (2) x ( z ) ( d, θ ) is a cosine function of θ and M (3) x ( z ) ( d, θ ) is sinusoidal with θ .We have also shown that the magnetization induced by the proximity effect can be largeenough to be detected in typical experiments. Therefore, it is expected that a Josephsonjunction including the F and the RSOI in the N , such as the one studied here, has a potentialfor 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.
Appendix A: Coefficients C ijω n ( d ) In Eqs. (19) and (20), the coefficients C ijω n ( d ) are given by C ω n ( d ) = iα R d F (cid:20) ( iα R − k α ) d F γ F e ( − iα R + k α ) L − ( iα R + k α ) d F γ F e − ( iα R + k α ) L (cid:21) . Phys. Soc. Jpn. Full Paper + k α d F (cid:20) ( iα R − k α ) d F γ F e ( iα R − k α ) L + ( iα R + k α ) d F γ F e − ( iα R + k α ) L (cid:21) , (A · C ω n ( d ) = − ( iα R + k α ) d F (cid:20) ( iα R − k α ) d F γ F e ( − iα R + k α ) L − ( iα R + k α ) d F γ F e − ( iα R + k α ) L (cid:21) − k α d F ( iα R + k α ) d F γ F h e ( iα R + k α ) L + e − ( iα R + k α ) L i , (A · C ω n ( d ) = − ( iα R + k α ) d F (cid:20) ( iα R − k α ) d F γ F e ( iα R − k α ) L + ( iα R + k α ) d F γ F e − ( iα R + k α ) L (cid:21) + iα R d F ( iα R + k α ) d F γ F h e ( iα R + k α ) L + e − ( iα R + k α ) L i , (A · C ω n ( d ) = iα R d F (cid:26)(cid:20) − ( iα R − k α ) d F γ F (cid:21) e k α d − (cid:20) iα R + k α ) d F γ F (cid:21) e − k α d (cid:27) e − i ˜ αd − k α d F (cid:26)(cid:20) iα R − k α ) d F γ F (cid:21) e iα R d − (cid:20) iα R + k α ) d F γ F (cid:21) e − i ˜ αd (cid:27) e − k α d , (A · C ω n ( d ) = − ( iα R + k α ) d F (cid:26)(cid:20) − ( iα R − k α ) d F γ F (cid:21) e k α d − (cid:20) iα R + k α ) d F γ F (cid:21) e − k α d (cid:27) e − iα R d + k α d F (cid:20) iα R + k α ) d F γ F (cid:21) h e ( iα R + k α ) d − e − ( iα R + k α ) d i , (A · C ω n ( d ) = ( iα R + k α ) d F (cid:26)(cid:20) iα R − k α ) d F γ F (cid:21) e i ˜ αd − (cid:20) iα R + k α ) d F γ F (cid:21) e − iα R d (cid:27) e − k α d − iα R d F (cid:20) iα R + k α ) d F γ F (cid:21) h e ( iα R + k α ) d − e − ( iα R + k α ) d i . (A · Appendix B: Integration with respect to x in Eq. (21) In this Appendix, we will first provide the analytical form of the local magnetization den-sity 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 compo-nent 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) x ( x, θ ) , (B · m (1) x ( x, θ ) = − gµ B πN F k B T h ex d ~ D F X iω n ∆ E ω n Q ω n ( d ) × sinh( k N x )Im [Φ ω n ( d ) t ω n ( x, d )] , (B · m (2) x ( x, θ ) = gµ B πN F k B T h ex d ~ D F X iω n ∆ E ω n (cid:18) − k F d F γ F (cid:19) × Q ω n ( d ) sinh[ k N ( x − d )] × Im [Φ ω n ( d ) t ω n ( x, d )] cos θ, . Phys. Soc. Jpn. Full Paper (B · m (3) x ( x, θ ) = gµ B πN F k B T h ex d ~ D F X iω n ∆ E ω n (cid:18) − k F d F γ F (cid:19) × Q ω n ( d ) sinh[ k N ( x − d )] × Re [Φ ω n ( d ) t ω n ( x, d )] sin θ, (B · Q ω n ( d ), Φ ω n ( d ), and t ω n ( x, d ) are respectively given in Eqs. (18)–(20).Substituting Eqs. (B · ·
4) into Eq. (21) and performing the integration with respect to x ,we can obtain the x component of the magnetization, M x ( d, θ ) = d R d m x ( x, θ ) dx = d R d m (1) x ( x ) dx + d R d m (2) x ( x, θ ) dx + d R d m (3) x ( x, θ ) dx = − gµ B πN F k B T d P iω n h ex d ~ D F ∆ E ωn Q ω n ( d ) R d sinh ( k N x ) Im [Φ ω n ( d ) t ω n ( x, d )] dx + gµ B πN F k B T d P iω n h ex d ~ D F ∆ E ωn (cid:16) − k N d F γ F (cid:17) Q ω n ( d ) R d sinh [ k N ( x − d )] Im [Φ ω n ( d ) t ω n ( x, d )] dx cos θ + gµ B πN F k B T d P iω n h ex d ~ D F ∆ E ωn (cid:16) − k N d F γ F (cid:17) Q ω n ( d ) R d sinh [ k N ( x − d )] Re [Φ ω n ( d ) t ω n ( x, d )] dx sin θ = M (1) x ( d, θ ) + M (2) x ( d, θ ) + M (3) x ( d, θ ) , where M (1) x ( d ), M (2) x ( d, θ ), and M (3) x ( d, θ ) are respectively given in Eqs. (25)–(27). The func-tions w ω n ( d ) and u ω n ( d ) in Eqs. (25)–(27) are given as u ω n ( d ) = u (1) ω n ( d ) + u (2)) ω n ( d ) + u (3) ω n ( d ) , (B · u (1) ω n ( d ) = − (cid:2) C ω n ( d ) + 2 C ω n ( d ) (cid:3) k N sin [( α R − ik α ) d ] − ( α R − ik α ) sinh ( k N d )( α R − ik α ) + k ,u (2) ω n ( d ) = 4 α R (cid:2) C ω n ( d ) + 2 C ω n ( d ) (cid:3) e − k α d × k α k N cos ( α R d ) + k N α R sin ( α R d ) − e k α d (cid:2) k α k N cosh ( k N d ) − (cid:0) α + k (cid:1) sinh ( k N d ) (cid:3)(cid:0) α + k α (cid:1) + 2 (cid:0) ˜ α − k α (cid:1) k + k ,u (3) ω n ( d ) = − i α R (cid:2) C ω n ( d ) + 2 C ω n ( d ) (cid:3) × ik α k N cosh ( k α d ) + k N α R sinh ( k α d ) − α R k α [ ik N cosh ( k N d ) + 2 α R sinh ( k N d )] e iα R d h α + ( k α − k N ) i h α + ( k α + k N ) i e − iα R d w ω n ( d ) = w (1) ω n ( d ) + w (2)) ω n ( d ) + w (3) ω n ( d ) , (B · w (1) ω n ( d ) = e − ( iα R + k α ) d ( α R − ik α ) + k h(cid:16) − e ( iα R + k α )2 d (cid:17) k N cosh ( k N d ) − i (cid:16) e ( iα R + k α )2 d (cid:17) ( α R − ik α ) sinh ( k α d ) i ,w (2) ω n ( d ) = e − k α d a (1) ω n ( d ) + a (2) ω n ( d ) − a (3) ω n ( d ) (cid:0) α + k α (cid:1) − (cid:0) α − k (cid:1) k ,a (1) ω n ( d ) = 2 α R k α k N e k α d ,a (2) ω n ( d ) = − α R k N cosh ( k N d ) [ k α cos ( α R d ) + ˜ α sin ( α R d )] , . Phys. Soc. Jpn. Full Paper a (3) ω n ( d ) = 2 α R (cid:2)(cid:0) α + k (cid:1) cos ( α R d ) + 2 k α α R sin ( α R d ) (cid:3) sinh ( k N d ) ,w (3) ω n ( d ) = 12 e − iα R d h b (1) ω n ( d ) + b (2) ω n ( d ) + b (3) ω n ( d ) i , (B · b (1) ω n ( d ) = e iα R d " iα R α + ( k α − k N ) − iα R α + ( k α + k N ) ,b (2) ω n ( d ) = − iα R cosh [( k α − k N ) d ] + ( k α − k N ) sinh [( k α − k N ) d ] α + ( k α − k N ) , and b (3) ω n ( d ) = iα R cosh [( k α + k N ) d ] + ( k α + k N ) sinh [( k α + k N ) d ] α + ( k α + k N ) . We can also obtain the z component of the magnetization given in Eqs. (29)–(31) byfollowing the procedure used to derive the x component of the magnetization. . Phys. Soc. Jpn. Full Paper
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