Alteration of tunneling mechanism due to ferromagnetic insulator on Andreev spectroscopy for ferromagnet/superconductor junctions

We have theoretically elucidated the mechanism of the peak splitting of the tunneling conductance arisen from the effect of ferromagnetic insulator in ballistic ferromagnetic metal/superconductor junctions. On this peak splitting of the tunneling conductance, we have found that the exchange potential in the insulator gives crucial changes to the tunneling conductance through the coherence factor in superconductor. The peak splitting exists even though ferromagnetic metal is replaced by normal-metal. Moreover, in such junctions, the magnetization in ferromagnetic metal gives asymmetric tunneling conductance spectra between electron and hole injections. We have also clarified the difference of resultant influences of ferromagnetic insulator on tunneling conductance between spin-band asymmetry ferromagnet and standard Stoner ferromagnet junctions.


Introduction
Tunneling spectroscopy gives us many insights of the electronic properties of metals and superconductors. For instance, the symmetry of superconducting pair potential has been reflected sensitively on the differential conductance spectra in normal metal/unconventional superconductor junctions [1][2][3][4][5][6][7][8][9][10][11]. The junctions including magnetically active interface has also been an important subject from the view point of fundamental physics as well as from an application perspective [12][13][14][15][16] on both the magnetism and the superconductivity in materials. Especially, ferromagnetic metal/superconductor (FM/S) junction has attracted much attention on investigations of the proximity effect, the transport phenomena, and so on [12,[17][18][19][20]. For example, in the FM/ S junctions, the Andreev reflection (AR) is suppressed by the exchange potential in FM [12,17,18,[26][27][28][29]. Indeed, this modulation of AR has been utilized to measure the polarization of FM in the FM/S point contacts [21][22][23][24]. Furthermore, as one of the interesting magnetic effects, there is a spin active interface by the ferromagnetic insulator (FI). It has been shown theoretically that the double conductance peak around a bias voltage corresponding to the superconducting energy gap in semiconductor/S junction [25] appears due to the spin-flip scattering. In the FM/FI/d-wave (FM/FI/d) junction [26][27][28][29], it has been revealed that splitting of zero-bias conductance peak (ZBCP) occurs by the spin filtering effect of FI. Additionally, the spin filtering effect of FI can give electron-refrigeration in the N/FI/s-wave superconductor (N/FI/s) junctions [30]. In these theoretical investigations of FM/S junctions, ferromagnetic metals have been described by the Stoner model referred to as STF below. However, STF has not totally covered itinerant electron ferromagnetism in real materials. On the other hand, there is the ferromagnetism kinetically driven by a spin-dependent bandwidth asymmetry, or, equivalently, by an effective mass splitting between and -spin particles [31][32][33][34][35][36]. We also consider this spin-dependent bandwidth asymmetry ferromagnet (SBAF) as well as STF to obtain more general results than the case of STF only. For SBAF/S junction, some results similar to STF case have been obtained [37][38][39][40]. However, the effect of FI in SBAF/FI/S junctions have not been studied yet. Accordingly, the study of the effect of FI on the tunneling conductance in SBAF/FI/S junctions becomes important to clarify the difference with those in STF/FI/S junctions.
In this paper, we apply our previous formula [40] based on the Blonder-Thinkham-Klapwijk theory [41] to the FM/FI/s-wave and d-wave superconductor junctions, in which the polarization of FM can be hybridized between STF and SBAF. We show a significant role of the exchange potential in FI on the tunneling conductance. It is found that the FI gives specific difference in tunneling conductance between SBAF/FI/S and STF/FI/S junctions. Furthermore, we study in detail how the exchange potential in FI generates the spin filtering effect. As a result, a term leading the spin filtering effect that have not been discussed so far will be presented. And, we also investigate the effect of FI on tunneling conductance when the superconducting pair potential is a breaking time-reversal symmetry [42].

Model and formulation
In the present paper, we employ a two-dimensional ballistic FM/FI/S junction(see figure 1) with semi-infinite electrodes as in our previous paper [40]. The interface located at x=0 is described by V x , V 0 and V ex are the δ function, a non-magnetic and a ferromagnetic barrier potentials, respectively.
Here, s =  or  is spin index and 1 1 r = + -( )for   ( )-spin. The Bogoliubov-de Gnnes (BdG) equation describing the system is given by Here, r u s ( ) and r v s ( ) are the wave functions for electronlike and holelike quasiparticles (ELQ and HLQ) with the eigenvalue E, respectively. And r H 0 s ( ) is the single particle Hamiltonian.
In the ferromagnet side (x 0 < ), the single particle Hamiltonian is given by where m σ is the effective mass for σ-spin band particle, U ex is the exchange potential and E FM is the Fermi energy. Here, we assume a hybrid ferromagnet between SBAF and STF for FM. The pure SBAF and STF can be described by The magnetization M of the ferromagnet is given by M P P = -  where the polarization P s for σ-spin is expressed as P 1 1 1 where m S and E FS are the effective mass of the quasiparticle and the Fermi energy, respectively. In this paper, we treat singlet superconductors. And the spatial dependence of the pair potential is simply taken as r Q( ) is the Heaviside step function. In the following, we apply the quasiclassical approximation where E F  (E, Δ). In this approximation, the pair potential Δ can be described by  For -spin electron injection with angle q  to the interface normal, there are four scattering trajectories; AR with angle AR q , normal reflection (NR), transmission to superconductor as ELQ, and transmission as HLQ (see figure 1).
For these scattering processes, the probability coefficients a s and b s for AR and NR are obtained under appropriate boundary conditions for the BdG-wave functions )and the barrier parameters Z s is defined as The angle resolved conductance G S,s for σ-spin at zero temperature can be calculated by using AR and NR coefficients as . In the calculation, we should take into account the critical angle C q for injected particles as sin 1 cos 1

Differential conductance
We begin with the , under the fixed value of magnetization M. ). It is noted that the double peak is different from those indicated by Zutic and Sarma [25] since any spin-flip scattering processes are not considered in our model. Thus, the double peak will be owing to the spin filtering effects of FI. In figure 3, we show G eV T ( )in FM/FId junction for M 0.75 = . For pure STF, the results by Kashiwaya et al [26] have been reproduced. For 0 a = ( figure 3(a)), G eV T ( )shows the double peak around eV 0 = D similar to those in s-wave superconductor case. For α = 8 p and 4 p (figures 3(b) and (c)), the asymmetric splitting of ZBCP has been obtained. As in the case of s-wave superconductor, the height of peaks decreases as FM approaches pure STF. From these results, we can notice for both s-and d-wave cases that SBAF gives relatively larger double peaks than those in STF. As shown in our previous paper [40], the height of the peak can be attribute to the difference in the critical angle C q between SBAF and STF.
To see the constitution of the splitting peak in the tunneling conductance, each spin components of the conductance, G T↑ and G T↓ are shown in G T↑ and G T↓ are shifted to the negative eV and to the positive eV, respectively. Then, the resulting ZBCP is asymmetrically suppressed. As will be described below, these behaviors of the tunneling conductance can be interpreted by the impact of FI on the Andreev and NR coefficients.

Effect of FI
In this subsection, to see and to emphasize more clearly the effect of FI, we treat mainly N/   . As a result, the difference between SBAF and STF will be difficult to identify when Z ex is close to Z 0 . For d-wave cases(figure 7), it is also clear that the asymmetric splitting of ZBCP is due to the influence of M. In contrast to s-wave cases, asymmetric splitting of ZBCP is kept except Z Z ex 0 = . Although the height of peaks is suppressed as Z ex increases, the magnitude of peaks in SBAF case is larger than that in STF case. In order to clarify the origin of the double peak as a result of direct influence of the Z ex , we show the Andreev and NR coefficients because the conductance consists of these coefficients within the BTK model.
It is found in figure 8 that the behavior of the coefficients for and -spin particles are inverted with respect to eV 0 = for eV Therefore, for eV 0 0  -D < , the probability of AR for -spin electron is lower than that for -spin electron.   In the case of d-wave superconductor, as shown in figure 9, since AR is depressed by Z ex , a 2 does not appear in G eV T ( ) [26], since the conservation of the probability density flow is satisfied. These obtained results indicate that the split of the conductance peak has not been a straightforward shift in bias voltage.
We can approach to the origin of the split of the conductance peak (figures 2-5) by analyzing the denominator of the conductance. Subtracting the denominators between G S, and G S, , we have the term D . Thus, we can conclude that D leads the spin filtering effect. Therefore, it is notable that the balance between reflections for -spin and -spin particle injections is broken when D 0 ¹ . Conversely, the spin filtering effect dose not occur for D=0. It is obvious from equation   The reason for this NR is that the AR holes with -spin caused by -spin electrons completely transmitted interface are converted to the AR electrons with -spin again after being reflected by Z 0, . The probabilities of NR for both -spin and -spin electrons injections have the same weight, because these NR are arisen from Z 0, . It is also noted that G  becomes real number in eV . As a result, D becomes 0 for are injected from N to the interface in figure 10. In addition to the usual NR ((a) in figure 10) by Z ↑ , there is the NR ((b) in figure 10) through AR by Z ↓ as explained above.
From the above discussion, we define D res and res G to quantitatively describe the effect of Z ex as In order to simply grab the energy dependence of D res and res G , we integrated The eV dependence of res G and D res for G S, are shown in figure 11. For the s-wave case, since D res has a minimum value at a certain eV nearly 0 D , the resulting G T↑ has the conductance peak at eV 0 » D ( figure 4(a)). On the other hand, for G S, , since D res has opposite sign, G T↓ has the peak at The sign of res G is reversed with comparing to s-wave case as shown in figure 11. Thus, D res has minimum value at a certain eV 0 < , the resulting conductance peak of G T↑ emerges at a certain negative eV nearly 0. In contrast to this, the peak of G T↓ is at a certain positive eV close to 0 ( figure 4(b)) because D res has opposite sign. From these  obtained results, it can be interpreted that the spin-filtering effect is due to the imaginary part of the coherence factor G  induced by Z ex . Additionally, for the emergence of the splitting of the conductance peak for both s-and d-wave superconductors, it seems that there is optimal combination of Z 0 and Z ex . Indeed, it can be easily found that Z Z Z For example, as shown in figure 5, it is found that the splitting peak is sharper when Z ex close to Z 3 0 . It will be an interesting problem to compare the splitting of ZBCP by the broken time-reversal symmetry (BTRS) state [26,38,42]  --wave junction (see figures 12(a) and 13(a)). This means that the -spin and -spin components of G eV T ( )shift in opposite directions on eV by the change of the minimum value of D res owing to Z ex . Therefore, as shown in figure 12(b), the double peak structure of G eV T ( )originated from the imaginary s-wave component collapses with increasing Z ex . On the other hand, as shown in figure 13 13(b)). G eV T ( )around eV 0 = will be useful not only to investigate the difference between BTRS and Z ex in the origin of ZBCP splitting but also to distinguish between BTRS pair potentials. In figure 14, we show the Z ex -dependence of G eV We can see that the height of splitting peak due to the BTRS is suppressed with change from pure SBAF to pure STF. With the increase of Z ex , it can be seen for all FM cases that the position of splitting peak is shifted to negative eV direction and the structure of peak is broken with the suppression of height. For d d i

Conclusion
In this paper, we have studied the tunneling conductance in two-dimensional ballistic ferromagnet metal/FI/ superconductor (FM/FI/S) junctions. For both s-wave and d-wave superconductor cases, it has been unveiled that the ferromagnetic barrier potential Z ex has given rise to a split of the conductance peak reflecting AR bound states. It is also found that the height of the conductance peak becomes larger with increasing the magnitude of Z ex for SBAF/FI/S junction in contrary to those for STF/FI/S junction. From these obtained results, we can speculate that differences between STF and SBAF may be more clearly observed in the tunneling conductance for junctions including FI rather than non-magnetic insulator case. Furthermore, as a most important result, it has been clarified that the exchange potential in FI induces the imaginary part of superconducting coherence factor as a cause of conductance peak splitting. The resulting symmetrical splitting peak are made asymmetric by the magnetization of FM. It is a scenario of asymmetric splitting peak in FM/FI/S junction. In accordance with this scenario, we have investigated the double peak structure in a junction including the d s i   d d i x y xy 2 2 + + -wave superconductor. Although it is an example of a specific magnitude of subdominant paring potential, the tunneling conductance in junctions including the BTRS is also effective for discriminating between SBAF and STF. In this study, we have never considered other magnetic effects such as spin mixing effects and proximity effects. Inclusion of these effects would be necessary for a realistic theory and be an important future problems.

Appendix. Denominator of G S,s
Here, we give the denominator of G S,s for N/FI/S junction.