Theory of the spin-filtering effect in ferromagnet/ferromagnetic insulator/superconductor junctions

Although it is well known that the ferromagnetic insulator at the interface of the superconducting junctions works as a spin-filter, its detailed mechanism has not yet been clarified. Here we present a simple factor inducing the spin-filtering effect in ferromagnet/ferromagnetic insulator/superconductor junctions by applying the extended Blonder-Thinkham-Klapwijk theory. From the factor, one can see easily how the spin-filtering effect is induced by the ferromagnetic insulator. Furthermore, it is also shown that the spin-filtering effect can be caused only by ferromagnet without ferromagnetic insulator. As an interesting application, we study the spin-filtering effect for a chiral p-wave superconductor. In ferromagnet/insulator/chiral p-wave superconductor junction, we find that the zero-bias conductance peak is shifted in opposite bias-voltage directions for Stoner model and spin bandwidth asymmetry ferromagents. It is also shown that the additional shift by ferromagnetic insulator may be useful for identifying the dominant mechanism of the Stoner and spin bandwidth asymmetry mixed ferromagnet.


Introduction
Heterostructures consisting of ferromagnetic and superconducting materials have received much attention from not only the fundamental physics in condensed matter but also applications such as spintronics devices [1][2][3][4][5][6][7][8][9]. As is well known in such structures, the Andreev reflection(AR) is an essential physics [10]. The AR is the scattering process occurs at the nonsuperconducting metal/superconductor(NSCM/SC) interface for which an incoming electron(hole) from NSCM to SC is retroreflected as a hole(electron) with opposite spin. In other words, the AR reflects electron-hole coherence effect of superconductor. Therefore, the AR provides us some information about superconducting states. Indeed, in the junction consisting of unconventional superconductors such as d-wave [11][12][13][14][15][16] and chiral p-wave [17,18] pairing states, a zero-bias conductance peak (ZBCP) due to the Andreev bound state(ABS) formed at the interface of the junctions have been observed [19][20][21][22]. On the other hand, the AR in the ferromagnet/superconductor(FM/SC) junction shows mutually exclusive a feature of FM and SC. About the Stoner ferromagnet(Stoner-FM) /conventional SC junctions, de Jong and Beenaker clarified theoretically that the AR is affected by the exchange interaction in the FM and showed a possibility of the measurement of the polarization of FM using the FM/SC point contact AR spectroscopy [23]. After the proposal by de Jong and Beenaker, the polarization of FM was obtained from the conductance measurements in the FM/SC point contacts by Soulen et al, [24] and Upadhyay et al [25]. The de Jong and Beenaker theory was also extended to the Stoner-FM/unconventional SC junctions [26][27][28][29] by extending the BTK theory [30] within the quasiclassical Andreev approximation. In the Stoner-FM/insulator (I)/d-wave SC junctions, the suppression of ZBCP due to the breaking of the retroreflectivity of AR called virtual AR(VAR) [26,27] has been confirmed experimentally after the theoretical prediction [31,32]. As an interesting magnetic effect other than FM, there are problems of spin-active interface [33][34][35][36]. The problem of the spinactive interface can be thought separately into a simple spin-filtering(SF) effect and a more complicated spinmixing effect. Here,the simple SF effect means that the strength of interface potential is only different between -and -spin particles without the rotation of exchange field in insulator and spin flip scattering at the interface. For the spin-mixing effect, a spin-dependent ABS was clarified firstly by Fogelström [36]. Moreover, the spinmixing effect leading to the spin singlet-triplet mixing revealed by Eschrig and co-workers [4,37,38] is currently one of the most important themes in this field [39][40][41][42][43][44][45][46][47]. For a simple SF effect, it has been investigated theoretically that the ferromagnetic insulator(FI) switches the Josephson junction from 0-to π-junction [48]. In Normal metal(NM) or Stoner-FM/FI/d-wave SC junctions, the splitting of ZBCP owing the FI been shown [26]. Furthermore, it has been also clarified that the effective cooling power can be obtained by the AR suppression due to FI in NM/FI/s-wave SC junctions compared to that in NM/I/s-wave SC junctions [49]. As shown in these results, even a simple SF effect would expect an interesting phenomenon. Especially, more recently, for NM/FI/s-wave, d-wave SC junctions, we find the detailed mechanism of the influence of FI on AR which results the double conductance peaks at gag-edge for s-wave SC and the ZBCP-splitting for d-wave SC [50].
On the other hand, as a model of ferromagnet different from Stoner-FM, there is a ferromagnet by a spindependent bandwidth asymmetry or, equivalently, by an difference of effective mass between and -spin particles [51][52][53][54][55]. In the spin bandwidth asymmetry ferromagent(SBA-FM), the magnetization is given by the ratio of effective mass between and -spin particles. For SBA-FM/I/ d-wave [56], p-wave [57] SC junctions, the ZBCP is suppressed due to the VAR increasing the magnetization, as in Stoner-FM case. Furthermore, as an interesting influence of the SBA-FM on charge transport, the conductance below the superconducting energy gap for minority spin larger than that for majority spin in SBA-FM/I/s-wave SC junction and the asymmetric ZBCP splitting in SBA-FM/I/ -d x y 2 2-wave SC junction are theoretically calculated [56]. These results mean that the SBA-FM produces the SF effect.
However,the mechanism of SF effect due to SBA-FM has not been elucidated at all. Thus, it is one of the important problems to clarify the mechanism and the difference of the SF effect between FI and SBA-FM. In addition to this, also related to superconducting Sr 2 RuO 4 [58][59][60][61], it is an interesting problem to investigate the SF effect in the NM, Stoner-and SBA-FM/FI/chiral p-wave SC junctions which has not been studied sufficiently yet.
In this paper, we will clarify a mechanism of the SF effect which differs from cause by the singlet-triplet spinmixing, spin-flip scattering and spin-orbit coupling at the interface of juctions. It is shown clearly that the imaginary part of superconducting coherence function induced due to the exchange potential of FI and the wavenumber difference between and -spin particles in FM is an essential for the SF effect. As an application, we will also study the SF effect on the Stoner-and SBA-FM/FI/chiral p-wave SC junctions. For the both Stoner and SBA-FMs, we can find the ZBCP shift similar to that in NM/FI/SC structures formed on the surface of three-dimensional topological insulator [62]. It has been shown that the conductance in each case for the Stoner-FM and SBA-FM shifts to opposite energy direction by the SF effect.
This paper is organized as follows. In section 2, the SF factor is presented together with the reflection coefficients and the tunneling conductance. In section 3, at first, the bias voltage dependences of the coherence function for the chiral p-wave superconductor induced by the SF effect are shown for the several incident angles. Then, we present the results for the NM/FI/chiral p-wave SC junction and FM/I(FI)/chiral p-wave SC junctions. In section 4, we summarize our results.

Model and formulation
In this section, we present the SF factor with reconstructed reflection coefficients and tunneling conductance based on extended BTK theory formulated in previous paper [57]. The FM/FI/SC heterostructure considered in here is a two-dimensional ballistic junction as shown in figure 1, where semi-infinite FM and SC correspond to region x<0 and x>0, respectively. A flat interface in the y-direction is located at x=0. Assuming a nonmagnetic V 0 and magnetic V ex potentials at x=0, the simple spin-filtering FI barrier for s =   ( ) or -spin can be described by is the Dirac delta function. The single particle Hamiltonian for σ-spin in the FM is given by where m σ is the effective mass for σ-band particles, U ex is the exchange potential, and E FM is the Fermi energy.
. The spatial dependence of the pair potential is taken as D = DQ ( ) ( ) r x for simplicity, where Δ is the 2×2 matrix in spin-space and Θ(x) is the Heaviside step function. Here  where ε is the energy of the quasiparticle and s is the inversion of σ-spin. The BdG equation can be solved under the quasiclassical Andreev approximation where the wave vector dependence of Δ is replaced by the angle θ S between the direction of the trajectory of quasiparticles in the SC and the interface normal [11,12]. Then, a chiral p-wave pair potential can be taken as D = D q with the magnitude of pair potential Δ 0 at zero temperature T=0 [63]. In the quasiclassical Andreev approximation, the magnitude of wave vectors k σ in the FM is given by In the SC, the magnitude of wave vectors of the electronlike quasiparticles(ELQ) and the holelike quasiparticles(HLQ) can be denoted by Since the translational symmetry holds along the flat interface, the y-component of all wave vectors is conserved In this situation, for example, there are four scattering processes for the injection of -spin electrons from the FM side at an angle q  to the interface as shown in figure 1: normal reflection(NR) with angle q  , AR as holes with angle q  , ELQ and HLQ transmitted with angle q S to the SC side. Considering < <  , and θ S are injection, Andreev reflection and transmission angles, respectively. Since the normal reflection at the interface is totally specular, the normal reflection angle is also q  . The dispersion relations represent the Stoner-and spin bandwidth asymmetry-, Stoner-spin bandwidth asymmetry mixed-ferromagets from the top to the bottom.
where the probability coefficients   b ee 2 reflection probabilities for the injection of electron with   ( )-spin can be obtained as The coherence function Γ ± is given by Using the real part of the coherence function product * * G G = G G + G G 2, the term of denominator  is given by Similarly, the imaginary part using Here, we consider the spin dependence of terms  and    ( ) . The product of coherence function Γ + Γ − of  and    ( ) dose not contain the effect of FM and FI because the magnetic proximity effect is not considered in our model. It can be seen obviously that the effect of FM and FI appears in the   To clarify an origin of the SF effect, we focus on the terms   ( ) a eh 2 , only    ( ) in the denominator changes. Therefore, we choose the    ( ) as the SF factor. In calculating the conductance for a situation < <   k k k S set here, we consider the critical angle θ C for -spin particles injection. The θ C is originated from the translational symmetry along the interface for the particle with -spin and can be expressed by q For the -spin injection q q >  C , the x-component of the wave vector of AR with -spin becomes purely imaginary. That is, the evanescent mode exists in the AR process for -spin injection, which is called VAR [26]. With taking care of the critical angle θ C and the probability conservation of quasiparticle flow [57], the bias voltage V dependence of angle resolved conductance   ( ) G S, for   ( )-spin of the system at T=0 where ε=eV can be calculated by the extended BTK formula = + - where   ( ) g fm is the group velocity ratio between σ and s -spins in FM and is given by The group velocity ratio   ( ) g fm not only guarantees the probability conservation but also works as a SF factor. Here we note that the evanescent mode of AR for q q >  C does not contribute to the conductance in our extentded BTK formula, since the corresponding  g fm on + - , and the total conductance is given by where * * G G = G G -G G + -+ -+ - [˜˜]˜˜˜Ĩm and * * G G = G G + G G + -+ -+ - [˜˜]˜˜˜Re . Here we note that * * G G + -G G = - 2for several incident angles θ S . At θ S =0 where the ZES exists, 2(a)). For finite angles except θ S =±π/2, it is found that in first terms becomes dominant since the q cos S decreases. Thus, at θ S =±π/2, the G G 2reverses and shows the same behavior as s-wave pair potential(see figure 2(f)). However, since the angle θ S =±π/2 corresponds to the incident parallel to the interface, 2does not appear. The SF effect for each incident angles is determined by  s . Although the  s cannot actually integrate independently in the denominator of AR and NR probabilities, for easy understanding, we introduce the angle averaged SF factor   ò q q =

Spin-filtering factor of ferromagnetic insulator
In the NM/FI/SC junction, i.e., γ=1.0, χ=0.0,    ( ) is reduced to a simple form given by We can see easily that the FI does not work as a spin-filter for Z ex =Z 0 . In this case, for the injection of particles with -spin transmitting perfectly through the =  ( ) Z 0 -interface, the NR occurs through the Andreev reflection of the AR particles with -spin once reflected normally at the =  ( ) Z Z 2 0 -interface. This implies that probabilities of the AR and NR for the injection of particles with -spin are equal to those of the particles with -spin, respectively. As the result, for Z ex =Z 0 , the SF effect does not occur. For 0<Z ex <Z 0 , the shift-like behavior of ABS owing to the SF factor is expected. Figure 3 shows the Z ex dependence of  s for Z 0 =5.0. All curves are sinusoidal symmetrical with respect to eV=0 reflecting G G . With the increase of Z ex , the peak of  s increases and decreases after reaching a maximum value. The Z ex giving a maximum peak is calculated by  2 ). From the numerator, it can be found that the FM induces a spin-filter owing to the wavenumber difference. It is also obvious that the SF effect do not occur in the metallic limit Z 0 =0. Therefore, in addition to suppression of AR, the SF effect originated in polarization of FM should be expected in the FM/I/ 2for several incident angles θ S . We have set (a) θ S =0 (dash line for s-wave superconductor as comparison), SC junctions as reported for the SBA-FM [56]. However, this SF effect seems to be overlooked for the Stoner-FM.

Stoner ferromagnet
In the Stoner-FM case(γ=1.0), the magnitude of  s fm is calculated by 2χ. Since   c 0 1,  s in a fixed c ¹ 0 will be in the order~Z 1 0 3 for high barrier as tunneling limit. For this reason, it seems that the SF effect owing to Stoner-FM has not been noticed in the tunnel limit. It is also found in equation  decreases with increasing χ. Thus, as shown in figure 6, the resulting  s decreases for each increase in Z 0 and χ. It should be noticed that both results show inverted behavior to those of the FI case. Therefore, the peak shift will be opposite to those of the FI case. However, this difference is derived from the model of Stoner-FM and FI assumed in here. If the magnetizations of Stoner-FM and FI are set in mutually different directions, the expected peak shift will be in the same direction. Figure 7 shows the Z 0 -dependence of the angle averaged s s s | |P g a fm eh 2 and s s | | P b ee 2 for χ=0.5 giving M=0.5. In both AR and NR cases, probabilities for   ( )-spin are shifted in the positive (negative) eV direction. However, as mentioned above, the SF effect weakens as Z 0 increases, and both

Spin bandwidth asymmetry ferromagnet
For the SBA-FM case(χ=0.0), the magnitude of  s fm is calculated as . As shown in figure 9,     ( ) shows the same tendency as that of the Stoner-FM case for the dependency Z 0 and γ. However, its behavior is opposite,i.e., the same tendency as the NM/FI/p-wave SC case where     that the conductance peaks shifts to eV=0 with increasing Z 0 . Since the critical angle of AR θ C for 0.2M0.9 is larger than that of Stoner-FM as demonstrated in our previous paper [57], the conductance peak becomes higher and shifts as compared with those of Stoner-FM case. On the other hand, as already mentioned in section I, the effects of the SBA-FM on the conductance has been calculated for s-andd x y 2 2-wave SC cases [56]. In [56] ,using the wavenumber ratio q g q g = + -+ - between σ and s instead of the group velocity ratio s g fm , the spin-dependent conductance for the injection of    is different, the mechanism of the SF effect owing to  s will be common.
From our results, it may be expected that the measurement of the position and height of the conductance peak for weak barrier Z 0 excluding tunneling and metallic limits give an information on the cause of the SF effect as well as the distinction between the pure Stoner-and pure SBA-FMs.
3.6. Stoner-Spin bandwidth asymmetry mixed ferromagnet From the above results, it is expected that the peak position varies depending on the degree of mixing in Stoner and SBA composite ferromagnet. In figure 13, we show the (χ, γ)-dependence of the angle averaged C S /C N and for M=0.9, Z 0 =2.0. As the proportion of SBA-FM γ increases from the pure Stoner-FM (χ=0.9, γ=1.0), the  C C S N , ) shows that the peak position moves from positive(negative) to negative (positive) eV. Since the M is near the half-metal limit,  C C S N , (figure 13(b)) does not contribute much to the total conductance C S /C N ( figure 13(c)). Hence, the peak position of angle averaged C S /C N (figure 13(c)) moves from positive to negative eV according to  C C S N , ( figure 13(a)) . In such a mixed FM, it may be seen which mechanism of Stoner-χ and SBA-γ is more dominant from the peak position except eV=0.

Spin-filtering effects in the FM/FI/SC junction
Although the effect as the splitting of ZBCP is not seen in the NM/FI/chiral p-wave SC junction, a remarkable FI effect can be expected because the symmetrically shift between  C C S N , is broken by the polarization P σ in the FM/FI/chiral p-wave SC junctions. It can also be predicted from k k ¹ s s in FM that the effect of FI will not disappear even if Z ex =Z 0 . Indeed, for example, the   for -spin incident electrons is given as Thus,the SF effect of FI for Z ex =Z 0 will not disappear by synergy with FM but the increment of Z ex reduces conductance through the suppression of AR. The Z ex dependences of angle averaged C S /C N in the FM/FI/pwave SC junctions are shown for fixed M=0.9, Z 0 =2.0 in figure 14. For pure Stoner-FM case ( figure 14(a)), with increasing Z ex , the position of conductance peak moves to a negative bias voltage direction since the conductance peak shifts for  C C S N , by FI are opposite to that due to Stoner-FM(see figure 5 and figure 8). However, as Z ex approaches Z 0 , becomes dominant in the total conductance C S /C N , and the influence of Stoner-FM is reflected more strongly than that of FI. As a result, a conductance peak for Z ex =Z 0 is seen at a positive voltage. On the other hand, in the case of SBA-FM ( figure 14(b)), it can be seen that the peak position is shifted to a more negative bias voltage with increasing Z ex since the shift direction by FM and FI is the same. Although these behaviors can be seen even in the case of the mixed FM given by c g( ) 0.64, 4.1 ( figure 14(c)), the deviation of the peak to the negative bias voltages is reduced by the effect of Stoner-χ. However, in a mixed FM, the peak shift by FI may provide information on the dominant magnetic factor of FM.

Summary
In this paper, we have presented a general SF factor induced in the FM/FI/SC junctions. In roughly speaking, the SF factor is represented by a simple product of the exchange potential in FI and FM and the imaginary part of superconducting coherence function Im[Γ + Γ − ]. The G G + -

[ ] Im
never emerges in NM/I/SC junction. Therefore, the mechanism of the SF effect could be interpreted as being due to the Im[Γ + Γ − ] induced by the spin dependent tunneling. The SF effect by the Im[Γ + Γ − ] is a new perspective that has been overlooked and has not been discussed so far. As explained below, the SF factor shifts the conductance s C C S N , for σ-spin by a mechanism different from the shift of ABS by the spin-mixing scattering and Zeeman splitting. The SF factor appears commonly in the denominator of the AR and NR reflection probabilities and shows sinusoidal behavior as a function of energy by reflecting the Im[Γ + Γ − ]. And, the sign of SF factor is reversed for  and -spins by the magnetic effect. That is, the positions of the minimum value of the denominator of AR and NR probabilities for  and -spins are shifted in opposite eV direction depending on the sign of the SF factor. As the result, the s C C S N , composed of the AR and NR probabilities is shifted. This is a scenario of the SF effect in the FM(NM)/ FI/SC junctions. Especially, for a simply FI without the spin-mixing effect, the pure Stoner-and SBA-FM cases, we can see easily how the SF effect occurs. For the FI case, the SF factor appears only when 0<Z ex <Z 0 and induces a symmetrical shift with respect to and -spins. However, the total conductance C S /C N in the NM/ FI/chiral p-wave SC junction does not show any ZBCP-splitting or ZBCP-shift like d-wave case except for the suppression of ZBCP since only the incident angle θ S =0 contributes to the formation of ZES. In the pure Stoner-and SBA-FM cases, the SF factor does not appear in the metallic limit Z 0 =0.0 and are weakened in the tunneling limit,e.g.  Z 5.0 0 . Hence, the SF effect in FM/I/chiral p-wave SC junctions have not been shown in many studies targeting the junctions in the metallic limit or the tunneling limit. For both the pure Stoner-and SBA-FM/I/chiral p-wave SC junctions, the ZBCP shifts in mutually opposite directions have been shown in an intermediate barrier potential. Furthermore, it has been shown that these ZBCP-shift may be tunable by FI. From these obtained results, it will be expected that the conductance in FM/FI/chiral p-wave junctions give us a useful information to the mechanism of the ferromagnet and a tips for developing spintronics devices by using the concept of simply SF factor. However, since the electronic band structures of the FM,FI and SC are not taken into consideration, it is difficult to mention control and detect of parameters γ, χ and Z ex for actual experiments in the present model. In addition, other realistic effects such as the singlet-triplet spin mixing scattering, the spin-orbit scattering, the superconducting(or inverse) proximity effect, the interface roughness, and also the detailed effect of evanescent AR process near the interface are ignored. In a future work, inclusion of these effects should be necessary for comparison with actual experiments and applications.