Asymmetric Andreev resonant state with a magnetic exchange field in spin-triplet superconducting monolayer

Asymmetric Andreev resonant state with a magnetic exchange field in spin-triplet superconducting monolayer H. Goudarzi,. Khezerlou, and S.F. Ebadzadeh epartment of Physics, Faculty of Science, Urmia University, Urmia, P.O.Box: 65, Iran National Elites Foundation, Iran Abstract Featuring spin-valley degree of freedom by a magnetic exchange field-induction to gain transport of charge carriers through a junction based on superconducting subgap tunneling can provide a new scenario for future electronics. Transmission of low-energy irac-like electron (hole) quasiparticles through a ferromagnetsuperconductor (FS) interface can be of noticeable importance due to strong spin-orbit coupling in the valence band of monolayer (L-S). The magnetic exchange field (EF) of a ferromagnetic section on top of L-S may affect the electron (hole) excitations for spin-up and spin-down electrons, differently. Tuning the EF enables one to control either electrical properties (such as band gap, SOC and etc.) or spin-polarized transport. We study the influence of EF on the chirality of Andreev resonant state (AS) appearing at the relating FS interface, in which the induced pairing order parameter is chiral -wave symmetry. The resulting normal conductance is found to be more sensitive to the magnitude of EF and doping regime of F region. Unconventional spin-triplet -wave symmetry features the zero-bias conductance, which strongly depends on -doping level of F region in the relating NFS junction. PACS: 5..Ak.6.-b 4.45.+c Keywords: magnetic exchange field monolayer molybdenum disulfide triplet superconductivity Andreev process INTOUCTION Two-dimensional layered transition metal dichalcogenides (TC) including irac-like charge carries, like graphene, present distinct peculiar physical and dynamical properties[, 2]. Particularly, in monolayer molybdenum disulfide the charge carriers demonstrate either electron-like or hole-like quasiparticles belonging to two inequivalent degenerate valleys[, 4]. L-S has a direct band gap and, therefore, is capable for electronics and optical applications. The charge carrier mobility is over at room temperature. Specifically, the inversion symmetry breaks in monolayer, and two inequivalent valleys are interconnected by time reversal symmetry [5, 6]. These features and also strong spin-orbit coupling (SOC) are responsible to the spin-valley degree of freedom in charge transport. The coexistence of valley and spin Hall effect is resulted from coupling of spin and momentum space (valley) at the valence band edges [6]. Consequently, the strong SOC originated from the heavy atom of molybdenum may play an essential role in spin-related investigations. In this regard, presence a magnetic exchange field via the proximity-induction significantly influences the phenomena related to the spin-splitting band structure. This is very useful, since it provides accumulations of spin and valley-polarized carriers with long relaxation times. The valley and spin polarization owing to the EF-induction may involve valleytronic and quantum computing applications [, 8, 9]. Furthermore, it is shown that the control of valley polarization can be possible in L-S by removing the valley degeneracy [,, 2]. The applied exchange splitting energy to the L-S results in a large spin splitting in!#" valley in the valence band, leading to a novel behavior of pseduo-relativistic

Klein tunneling. In our previous work [], the effect of an exchange field on Klein tunneling and resulting magnetoresistance was studied in a ferromagnetinsulatorferromagnet junction. Newly, utilizing a magnetic field is shown to Zeeman split the band edge states in different valleys [4, 5, 6]. In TC, utilizing interfacial EF can overcome the small valley splitting, and it results in breaking time reversal symmetry[, 8]. Zhao and et al. [] showed an enhanced valley splitting in monolayer utilizing the EF from a ferromagnetic EuS substrate. Also, they have found that the magneto-reflectance measurement shows at magnetic field a valley splitting of. The proximity-induced superconductivity and ferromagnetism have been experimentally shown in L-S [8, 9, 2, 2, 22, 2, 24, 25, 26, 2]. ecently, several investigations have been reported on the L-S superconductor junctions regarding Andreev process at the interface [28, 29,,, 2]. In this paper, we reveal the formation of Andreev resonant energy and resulting tunneling conductance at the FS interface, where the superconducting order parameter is taken to be spin-triplet -wave symmetry. However, there is an essential physics in this system. Specifically, in similar situation in topological insulators, the magnetization induction to the surface states leads to open a band gap at irac point, and resulting chiral ajorana mode energies appear at the FS interface [, 4, 5, 6]. The chirality of ajorana modes is provided by the perpendicular component of induced magnetization. This is considered as an interesting feature of magnetization effect in two-dimensional newly discovered materials. We find a new effect of EF on the Andreev resonant state (AS). The chirality symmetry breaking of AS happens in the presence of EF. We consider the essential dynamical band parameters of L- S contributions (SOC interaction, electron-hole mass asymmetry term and topological term ) to the Andreev process. oreover, due to valence band spin-splitting in L-S caused by strong SOC, the doping regime is a notable aspect to control the transport of charge carriers (for example, see efs. [28, 29,,, 2]). This feature presents more considerable, when spin-splitting is highlighted by a EF. Indeed, it needs, for experimental applications, to determine the range of permissible doping in F region. This paper is organized as follows. Sec. 2 is devoted to present the proposed model and formalism to obtain the exact form of superconducting dispersion energy and corresponding wavefunctions. The normal and Andreev reflection coefficients are found by matching the wavefunctions at the interface. The numerical results of AS and resulting tunneling conductance considering the strong spin-valley effect caused by ferromagnetic exchange field and also chiral superconducting order parameter are presented, and their main characteristics are discussed in sections. Finally, we close with a brief summary. 2 THEOETICAL FOALIS The low energy band structure of L-S can be described by the modified irac Hamiltonian []. This Hamiltonian in addition to first order term of momentum for 2 massive fermions, contains the quadratic terms originated from the difference mass between electron and hole, and also topological characteristics. The strong spin-orbit coupling leads to distinct spin splitting at the valence band for different valleys. In the presence of an exchange field and superconducting gap induced by proximity effect, the irac-bogoliubov-de Gennes (BdG) Hamiltonian is given by: where ( ) + ( )! " #$% &!' #*$ &!',! ".-? @? 24 &,5 6%8#9:. 9: 2> 9'F AB IH, and valley index C E 9: is the full effective Hamiltonian of 9'G are the Pauli matrices. The spin-up and spin-down is labeled by monolayer, and IH 6 denotes the! and!#" valleys. The bare electron mass is, J +, and topological and mass difference band parameters are evaluated by ' and K L, respectively. # is the direct band gap, N O! and LJ P denote the spinorbit coupling and Fermi velocity, respectively. The electrostatic potential! gives the relative shift to the Fermi energy in normal N, ferromagnet F and superconductor S regions as Q $ SN!, 2 ()

) 4 $ $ @ > + 4 4 which denotes the chemical potential in each region. #N$% & is the superconducting order parameter. The globally broken symmetry in the superconductor is characterized with phase. Obviously, the spatial part of triplet order parameter is an odd function under exchange of momentum of the two particles, while the spin part is even. For a spin-triplet symmetry the order parameter is expressed using the -vector as: # & & 5 G (2) where,, and G are an odd-parity function of & and Pauli matrices, which describes the real electron spin ", respectively. The direction of -vector is perpendicular to total spin of a Cooper pair. This order symmetry has a off-diagonal components. Without loss of generality, let us consider the case of & #$ &, which means. The symmetry of L-S lattice plays, of course, a central role in & -dependency of. In L-S, the promising pairing symmetry is F and chiral 'F% 'G -wave symmetries [8]. iagonalizing Eq. () produces the following energy-momentum dispersion around irac point [29]: where %$ $ #$ K 4$ #$! " ()? @ #4?? @ #4? # C.I!, # 4 C 8! and A B A B 8 & +(' &) '. The index IH denotes the electron-like and hole-like excitations, while H distinguishes between the conduction and valence bands. Straightforwardly, it is shown that the superconducting order parameter # $% in Eq. () is renormalized by chemical potential Q$, and also appears as an ordinary gap. This electron-hole superconducting excitation is qualitatively different from that obtained for conventional singlet superconductivity [], so that it seems to remain semigapless. The mean-field conditions are satisfied as long as # $+* Q$. In this condition, the exact form of superconductor wavevector of quasiparticles can be acquired from the relation 4$ 4 where can be responsible to exponentially decaying. In particular, we retain the contribution of and terms representing one of the essential physics of monolayer. Hamiltonian () can be solved to obtain the wave function for superconductor region. The wavefunction, which includes a contribution from both electron-like and hole-like quasiparticles are found as. $ where we introduce >? 2 24 5 # + ( 6 + ( )' 5 # + ( 6 + ( ) 8 9 9 A@ B #, % : @ # F. $? B #? +, 4 -, + ( 5 # + ( 6 + ( + ( 5 # : + ( + ( 6 + ( ) B # ' 6? B # > E? C & ' > E? 2 24 )' &), 8 9 9 @ # F (4)?. Fermi level of each region can be tuned by the magnitude of chemical potential. The momentum of ferromagnetic electrons is coupled with the exchange field in F region. Therefore, the corresponding Fermi wavevector needs to be acquired from its eigenstate:! IH where we define G region are given by:. IH G C G. I" 5 J K L% @ G G J F G C I G C G (5) C A B.#. The wave functions in ferromagnetic. 5 J N L + ( @ J F

>. >? >? & >? & >? >? where.4 2!N#E 8" 8 9 : Having established the states that participate in the scattering, the total wave function for a right-moving electron with angle of incidence,a left-moving electron by the substitution and a left-moving hole by angle of reflection may then be written as:, where we define and & 2 2 5 4 8 9 9 : @ F, 2 2, + ( 5 2 2 + ( 5 8 9 9 : @ F!#E 4 > @ 8 9 9 AB : + ( @ F are amplitude of normal and Andreev reflections, respectively. The normalization factor? ensure that the particle current density is conserved, which has been given in previous work []. It is instructive to consider the effect of Fermi vector mismatch on the Andreev reflection amplitude. To explore how the Fermi wave vector mismatch influences the scattering processes, the Fermi momentum in normal and superconducting regions of the system may be controlled by means of tuning the chemical potential. We proceed to study the Andreev reflection and resulting conductance in FS and NFS L-S junctions. The normal and ferromagnetic regions are extended from and to, respectively. L-S is covered by superconducting electrodes in the region to. The strategy for calculating the scattering coefficients in the junction is to match the wave functions at boundaries. F. F!,. F. $ F, where. $. $ C ". (6). $. The coefficients and correspond to transmission of electron and hole, receptively. We find the following solution for the normal and Andreev reflection coefficients: - + ( 6 + ( 6 P + ( 6 + ( 6 N + ( 6 P + ( 6 N The parameters of are introduced in Appendix. The above expressions demonstrate exactly the Andreev process at the interface, leading to the spin-valley polarized transport of charge carriers through relevant junctions. NUEICAL ESULTS AN ISCUSSION In this section, we analyze in detail the dynamical transport properties of FS and NFS junctions when the -wave symmetry pairing is deposited on top of L-S. Coupling the magnetic exchange field in F region with strong SOC of L-S influences the Andreev resonant states at the FS interface. On the other hand, the unconventional superconducting order with nonzero orbital angular momentum plays a crucial role in materials with strong SOC. ecently, it is shown that the quasiparticle superconducting excitations are influenced by the triplet component of -wave pairing symmetry [29]. 4

$ >? & >? & Also, an exchange field induction to L-S is found to give rise a notable spin-splitting energy in valence band between spin-up and spin-down charge carriers, leading to the spin-valley polarized Klein tunneling []. In this regards, if we consider a FS interface, the above features can straightforwardly determine the exhibition of Andreev process and resulting subgap tunneling conductance. A typical FS interface can be of significant importance owing to the appearance of Andreev resonant states and, especially, formation of chiral ajorana mode in topological insulators [, 4, 5, 6].. Andreev reflection and resonant state In one-dimensional limit, transport of tunneling electrons is in -direction with wavevector F incident angle. Conservation of momentum in -direction enables one to find incident electron (hole) angle from S region to the interface $ in terms of, which is given by 4$. We show in Fig. the angle-resolved Andreev and normal reflection probabilities in zero-bias, where the EF affects the Andreev process. Similar to previously obtained results [28, 29,, 2], the band parameters and of L-S make a significant effect on the reflection of quasiparticles from FS interface. In the presence of EF, there is no perfect A at normal incidence. However, the density of probability of reflections is conserved, i.e., %. The signature of exchange field is believed to decline A in NFS junction, since the incident electrons from N region experience a high spin-splitting energy depending on their spin-polarization. In this system, the effective superconductor subgap is adjusted by a factor including the characteristic band parameters of L-S. Next, we proceed to look for the possibility to form chiral bound energy mode resulted from Andreev resonant state at the FS interface. Occurring the perfect A, the electron reflected to N region may vanish. Thus, the AS reads: where + ( 6 #$, 8, P + ( 6 N - and. Parameters are given in Appendix. In Fig. 2, we present these states in terms of incident electron from N region for several values of EF. As a remarkable point, we find the chirality symmetry of AS to be conserved in the absence of EF, whereas it is broken in the presence of EF, so that, the magnitude of AS differs in its positive and negative values (left and right sides of curves # ) for nonzero incident angles. Comparing to the gapless surface state of topological insulator, where chiral ajorana mode energy appears at the FS interface[, 4, 5, 6], the direct band gap of L-S can be responsible to disappearing the chirality symmetry of AS. However, in topological insulator, the perpendicular component of magnetization (,: ) causes to open a gap in irac point, and the sign of magnetization provides chiral ajorana mode of Andreev bound state..2 Subgap normal conductance We now proceed to calculate the subgap tunneling conductance in NFS and FS junctions. According to the Blonder-Tinkham-Klapwijk formalism [9], we can calculate the tunneling conductance by 5 - N () where ' is the ballistic conductance of spin and valley-dependent transverse modes in a sheet of of width that denotes the bias voltage. The upper limit of integration in Eq. () needs to obtain exactly based on the fact that the incidence angle of electron-hole in the three regions may be less than. 5

H H ependence of the resulting normal Andreev conductance 4 on EF is presented in Figs. for three different values of EF 4 for -doped F region. For both FS and NFS junctions the appearance of zero-bias conductance is found. This is the main feature of unconventional superconducting order parameter [4]. We observe that,at a determined bias energy, the conductance curve asymptotically vanishes and presents a peak. This can be described by the fact that the effective superconducting gap in electron-hole energy excitations is renormalized by a coefficient. In NFS junction, the valence band spin-splitting difference between N and F regions gives rise to momentum difference of Andreev reflected electron (hole), and thus, it leads to diminish A probability and resulting subgap conductance, as seen in Fig. (b). Whereas, in FS junction, the conductance peak grows up with the increase of EF (see, Fig. (a)), because the SOC is dominated, here. This is in contrast to the similar situation in graphene junction when Q [4]. The above result can be understood by the fact that, in one hand, the effective superconducting gap with spin triplet-wave symmetry is enhanced by the SOC interaction and, on the other hand, coupling EF with momentum of Andreev reflected electron (hole) leads to decrease the wavevector mismatch. Fig. 4(b) shows that the zero-bias conductance (ZBC) strongly depends on -doping level of F region in NFS junction, while it is almost constant L in FS junction, as seen in Fig. 4(a). For a constant exchange field, the conductance peak enhances with the increase of chemical potential of F region Q! for #$, in NFS structure. In FS structure, we observe a contrary behavior herein. For -doped F region Q, the valence band spin-splitting is strongly influenced by the EF, and there are four possible critical values for band energy # 4, which allows appropriately to occur A in the range of ferromagnetic chemical potential # % N EQ # N. Indeed, this antithetical outcomes are directly resulted from Andreev process at the NS or FS interfaces. To more clarify, one can investigate Andreev bound stats in a SFS Josephson junction as a future work. In FS junction, the switching behavior of normal conductance can be controlled by tuning the bias energy. In low -doping of F region, the switching is shifted toward zero bias. With the increase of bias energy up to effective superconducting gap '#N$, the tunneling conductance results in a constant in FS and in NFS junctions. In the latter case, the doping of F region has no effect on the conductance. Finally, we focus on the influence of dynamical band parameters of L- S such as electron-hole mass asymmetry term, topological term and SOC term in the resulting subgap conductance in NFS junction, as seen in Fig. 5. In the case of EF, the term and -doped N and F regions Q Q has no effect, as in agreement with previous works [28, 29,, 2]. The absence of topological term gives rise to enhancing the conductance peak. ore importantly, taking the SOC term to neglect, we find a sharp switching conductance in zero bias. 4 CONCLUSION In summary, we have considered transition metal dichalcogenide in layered structure, where irac-like electrons (holes) have experienced proximity-induced a magnetic exchange field or a superconducting pair potential. The unconventional spin-triplet -wave order parameter has been found to be more effective in our proposed structure owing to the existence of a strong SOC in band structure of monolayer. Spin-valley degree of freedom is a key point to control the transport of charge carriers by tuning the valence band locked-spin-valley splitting via the applying a EF. A key finding of the present work is that the Andreev resonant energy at the relating FS interface exhibits an asymmetric behavior with the presence of EF. Thus, we can not consider the chirality symmetry of AS, comparing to the similar situation in topological insulator FS interface with inducing a magnetization perpendicular to the surface. Andreev process at the FS interface has led to the tunneling conductance, which was controlled by tuning the EF and also doping regime. The wavevector mismatch has been studied to obtain the relating normal conductance in FS and NFS junctions. A sharp switching conductance in zero bias has been achieved in the absence of SOC. Acknowledgment 6

J N N J J This work was supported by National Elites Foundation of I..Iran. Authors acknowledge the Vice- Presidency for Science and Technology of I..Iran. APPENIX The parameters in Andreev and normal reflections are given as: P + ( 6 P + ( 6 5 J 5 # H 5 J + ( 5 # K H K 5 # 5 + ( 5 # 5 + ( 5 5 5 # N @ J + ( 5 + ( 5 J N + ( @ J + ( 5 5 J + ( 5 # N @ J + ( 5 + ( 5 J N + ( @ J 5 J + ( 5 N + ( @ J + ( 5 J + ( 5 N @ J 5 J + ( 5 N + ( @ J + ( 5 J + ( 5 N @ J + ( @ J @ J + ( @ J @ J + ( @ J I 5 # @ J + ( @ J. + ( 5 # @ eferences [] H. S. atte, et al., Angew. Chem. Int. Ed. 49, 459 (2). [2] W. Zhao, et al., ACS nano, 9 (2). [] B. adisavljevic, A. adenovic, J. Brivio, V. Giacometti, and A. Kis, Nature Nanotechnology 6, 4 (2). [4] E. S. Kadantsev, P. Hawrylak, Solid State Communications 52, 99 (22). [5] Z. Y. Zhu, Y. C. Cheng, U. Schwingenschlogl, Phys. ev. B 84, 542 (2). [6] i Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys. ev. Lett. 8, 9682 (22). [] Q. Y. Zhang, et al., Adv. ater. 28, 959 (26). [8] J. S. Qi, et al., Phys. ev. B 92, 24 (25). [9] N. ohling,. uss, and G. Burkard, Phys. ev. Lett., 68 (24). [] J. Lee, et al., Nature nanotech., 42 (26). [] H. T. Yuan, et al., Nature Phys.9, 56 (2). [2] Y. Ye, et al., Nature Nanotech., 598 (26). []. Khezerlou, H. Goudarzi, Superlattices and icrostructures 86, 24 (25). [4] G. Aivazian, et al., Nature Phys., 48 (25). [5]. acneill, et al., Phys. ev. Lett.4, 4 (25). J

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Figure captions Figure (Color online) The probability of normal (dashed lines) and Andreev (solid lines) reflections as a function of incident angle for several values of the magnetic exchange field in NFS junction when Q! Q and Q!$ %. %. It is seen that the maximum Andreev reflection (green solid line) occurs in Figure 2 (Color online) The plot shows Andreev resonant state as a function of electron incident angle in NFS junction for several values of EF when Q Q! and Q$ %. The solid lines correspond to % and the dashed lines 4 and. Figure (Color online) Plot of the tunneling conductance as a function of the bias voltage for several values, and in (a) FS junction and (b) NFS junction. we set Q and Q!$ L in Fig.(a) and Q O Q! and Q$ in Fig.(b). Figure 4 (Color online) Normalized Andreev conductance of L-S as a function of bias energy for different values of ferromagnetic chemical potential. (a) ependence of tunneling conductance for three diferent values Q Q! O (green solid line), Q (blue dashed line) and % (red dashed line) when and Q$ L in FS junction. (b) ependence of (green solid line), Q (blue (red dashed line) when, Q and Q!$ in tunneling conductance for three diferent values Q dashed line) and Q NFS junction. Figure 5 (Color online) The Plot shows the Andreev conductance as a function of bias voltage in NFS junction. It shows the role of dynamical band parameters of L-S ( and ). We set Q!, Q, Q!$ and. 9

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