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d-wave superconductor junction
Journal of Magnetism and Magnetic Materials 362 (2014) 36–41
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Spin-dependent barrier effects on the transport properties of graphene-based normal metal/ferromagnetic barrier/d-wave superconductor junction Y. Hajati a,n, A. Heidari b, M.Z. Shoushtari a, G. Rashedi c a
Department of Physics, Faculty of Sciences, Shahid Chamran University of Ahvaz, Ahvaz, Iran Department of mathematics, Izeh branch, Islamic Azad University, Izeh, Iran c Department of Physics, Faculty of Sciences, University of Isfahan, Isfahan, Iran b
art ic l e i nf o
a b s t r a c t
Article history: Received 18 January 2014 Received in revised form 26 February 2014 Available online 13 March 2014
Using the extended Blonder–Tinkham–Klapwijk formalism, the spin-dependent transport properties in a graphene-based normal metal/ferromagnetic barrier/d-wave superconductor (NFBd-wave) junction have been studied theoretically. Here, we have mainly studied the influences of spin-dependent barrier and rotation angle of d-wave superconducting order parameter (α) on the charge and spin conductance. It is found that the rotation angle has a strong effect on the amplitude and phase of the charge conductance oscillations. As a remarkable result, we obtained that because of the spin-dependent barrier (FB), the rotation angle cannot suppress the zero-bias charge conductance and for the maximum rotation angle α ¼ π=4, the charge conductance shows oscillatory behavior which is different from similar nonspin-dependent barrier junctions. We have also shown that the spin filtering application of this junction is drastically changed by the rotation angle α. As α increases, the spin filtering application enhances, being strongest for α ¼ π=4. At last, we propose an experimental setup to detect our predicted effects. & 2014 Elsevier B.V. All rights reserved.
Keywords: Graphene Ferromagnetic barrier Rotation angle Spin filtering application
1. Introduction Graphene, a two-dimensional honeycomb lattice of carbon atoms, has attracted enormous attention in condensed matter physics after its successful synthesis at 2004 [1]. The conduction and valence bands of graphene form conically shaped valleys, touching each other at a point called the Dirac point. There are two inequivalent Dirac points at the Brillouin zone. The low energy excitations in graphene are governed by the Dirac equation and electrons and holes in graphene are massless relativistic particles with a large energy independent velocity 106 m/s [2]. Unusual electronic properties of Dirac fermions in graphene such as very high carrier mobility [3], ambipolar electric field effect [4] and large spin-relaxation length and long lifetime of the spin-polarized current [5,6] make graphene an important material for application purposes specially in the spintronics devices. Graphene is not naturally superconductor nor ferromagnetic, but recent experimental and theoretical studies show that superconductivity and magnetism can be induced in graphene via the proximity effect by a superconductor or ferromagnetic film [4,6–10].
n
Corresponding author. Tel.: +98 93 68148729. E-mail address: [email protected] (Y. Hajati).
http://dx.doi.org/10.1016/j.jmmm.2014.03.018 0304-8853/& 2014 Elsevier B.V. All rights reserved.
Beenakker showed that in a graphene-based normal metal/ superconductor (NS) junction, when the Fermi energy in the normal metal is smaller than the pair potential of superconductor, the specular Andreev reflection occurs instead of the usual retroreflection in the standard (not graphene-based) NS junction [7]. The existence of the novel specular Andreev reflection (AR) process in the graphene-based superconducting junctions leads to fundamentally different tunneling properties occur compare to the standard superconducting junctions [7,11–13]. It was shown that the tunneling charge conductance in a graphene-based normal metal/insulator/superconductor (NIS) junction shows oscillatory behavior versus the barrier strength [11]. Proximity effects in a graphene-based ferromagnetic/superconductors junction have recently attracted much attention in experimental and theoretical studies [6,9,10,14–20]. For the first time, Zareyan showed that when the exchange energy in a graphene-based ferromagnetic/superconductor (FS) junction is greater than Fermi energy EF, the charge conductance of the junction increases [14]. We know that this behavior is drastically different from the standard FS junction. In a graphene-based normal metal/ferromagnetic barrier/superconductor (NFBS) junction, the electrical conductance does not depend on the barrier strength χ G for superconductor having a higher effective Fermi energy but it depends on the ferromagnetic barrier strength χ H
Y. Hajati et al. / Journal of Magnetism and Magnetic Materials 362 (2014) 36–41
and the conductance is an oscillatory function of χ H [21]. We know that this behavior is because of the quantum nature of interference between quasiparticles in the FB region. Deposition of the ferromagnetic insulator (barrier) on the graphene allows one to make new types of magnetic tunneling junctions (MTJ). Recently, these MTJs attracted much attention because of their applications in the spintronics devices [10,22–26]. In the above discussions of the transport property in the graphene-based superconductor junctions, the superconductor considered was conventional superconductor (s-wave). Nowadays, the major candidate symmetry of the superconductivity in high-Tc cuprates (unconventional superconductors) is d-wave. One of the clear qualitative signatures of d-wave symmetry is the so-called zero-bias conductance peak (ZBCP) observed in the I–V characteristics of electron tunneling into thin films of YBCO by Geerk et al. [27–29]. In unconventional superconductor the sign reversal of the order parameter creates the zero energy states (ZES) [30]. Then, the formation of zero energy states produces ZBCP in the electron tunneling experiments. In addition, the proximity effect of graphene-based normal metal/d-wave superconductor [12,13] and ferromagnetic/d-wave superconductor junctions has been studied [31]. In the d-wave superconductors, the order parameter is angle dependent, ΔðθÞ ¼ Δ0 cos ð2θ 2αÞ, where α is the rotation angle of the order parameter relative to the surface metal. In this paper, we study the transport properties of a graphenebased normal metal/ferromagnetic barrier/d-wave superconductor (NFBd-wave) junction including charge and spin conductance as a new case. To this end, we have employed the Dirac–Bogoliubov– de Gennes (DBdG) equation and Blonder–Tinkham–Klapwijk (BTK) formalism. Here, the effect of the rotation angle of d-wave superconducting order parameter on the charge and spin conductance has been studied theoretically. Actually, our main focus is to study the effect of rotation angle on the amplitude and phase of the charge oscillation in the presence of spin-dependent barrier in the middle region of this junction.
37
we should apply the spin-polarized Dirac–Bogoliubov–de Gennes (DBdG) equation: ! H 0 UðxÞ HðxÞ ΔðxÞ ψ ¼ Eψ ð1Þ n HðxÞ þ UðxÞ H 0 Δ ðxÞ where H 0 ¼ iℏvF ðsx δx 7 sy δy Þ; UðxÞ ¼ EF Θð x LÞ þ ðEF þ V G ÞΘðx þ LÞΘð xÞ þ ðEF þ U 0 ÞΘðxÞ HðxÞ ¼ ηs hΘð x LÞ ΔðxÞ ¼ ΔðθÞ expð iφÞΘðxÞ; ΔðθÞ ¼ Δ0 cos ð2θ 2αÞ
ð2Þ
Here sx and sy are Pauli matrices in the pseudospin space of the sublattices and Θ is the Heaviside step function. The potential UðxÞ is the shift of the Fermi energies in the ferromagnetic (EF), ferromagnetic barrier (EF þVG) and superconducting regions (EF þU0) of graphene, respectively. The H 0 , HðxÞ and ΔðxÞ are Dirac Hamiltonian, exchange energy, and order parameter in the ferromagnetic and superconductor region, respectively. The exchange energy in the middle ferromagnetic barrier region is h. The potential U0 can be used to tune the mismatch between Fermi surface of the ferromagnetic and superconducting regions. The DBdG equation is derived within the mean-field approximation [32]. So, U0 should be very large compared to all other energy scales in our calculations: ðEF þ U 0 Þ⪢Δ0 . By solving the spin-dependent DBdG equation (Eq. (1)), we obtain the wavefunctions in each region of the junction. In the normal graphene (NG) region, the wavefunction for incident electron and retro-Andreev reflected hole is given by 00
0 1 1 1 BB iθ C B BB e C ik þ x B e iθ B C N þ rs B ψ N ðxÞ ¼ B BB 0 Ce B 0 @@ @ A 0 0
0
1
B C B C ik þ x Ce N þ r As B B C @ A
0 0 1 e iθA
1
1
C C C ik x C iqy Ce N Ce C C A A
ð3Þ where þðÞ
2. Scattering process and conductance formalism Here we consider a two-dimensional graphene-based NFBdwave junction in which two interfaces is parallel to the y-axis and they are located at x¼ L and x ¼0. The left ferromagnet and right d-wave superconductor in this junction are separated by a ferromagnetic barrier by a thickness of L and effective gate potential VG to control its chemical potential, as seen in Fig. 1. To study a problem concerning superconductivity and relativity, we should consider the Dirac–Bogoliubov–de Gennes (DBdG) equation. The DBdG equation for a graphene-based NS junction was derived in Ref. [7]. In a graphene-based NFBd-wave junction because of real spin degree of freedom in the ferromagnetic region,
kN
¼
ðE þð ÞEF Þ cos ðθðθA ÞÞ; ℏvF
sin ðθðθA ÞÞ ¼
ℏvF q ðE þ ð ÞEF Þ
ð4Þ
The θ is the incident angle of the electron and θA is the Andreev reflection angle of the hole in the N region. Also r s and r As are the normal and Andreev reflection amplitudes in the normal region. The spin-dependent wavefunction in the ferromagnetic barrier graphene (FBG) region is 0 1 B B iθF B þ B Be B ψ F B ðxÞ ¼ B Bps B 0 @ @ 0
0 1 iθF B B C þ B e C ikF x Ce B þ þqs B B C 0 @ A 0
0
0
B B þ ms B B @
1
1
0
iθF B
e
C C ikF x Bþ Ce C A
0
C B C ikF x B Ce B þ n s B C B A @
0 1
1
1
0 0 1 e
iθF B
1
C C C ikF x C iqy Ce B Ce C C A A
ð5Þ
where ℏvF q ; ðE þ ð ÞðEF V G Þ þ ηs hÞ ðE þ ð ÞðEF V G Þ þ ηs hÞ ¼ cos ðθF B þ ð Þ Þ ℏvF
sin ðθF B þ ð Þ Þ ¼ kF B þ ð Þ
Fig. 1. The proposed experimental setup to study the transport properties of a graphene-based NFBd-wave junction. In this plot NG, FBG and d-wave are normal graphene, ferromagnetic barrier graphene and d-wave superconductor graphene regions, respectively. The top and bottom gate allow for the chemical potential in the ferromagnetic region.
ð6Þ
Also, θF B þ ð Þ is the incident angle electron (hole) with respect to the normal direction to the interfaces, in the θF B region and ηs ¼ ηs ¼ 1 ðs ¼ ↑ s ¼ ↓Þ. On the superconducting (SG) region the wavefunctions for transmission of a right moving quasiparticles with a given excitation energy E 40 reads þ
þ
ψ eS þ ðxÞ ¼ ðuðθ þ Þ; uðθ þ Þe iθ ; vðθ þ Þe iϕ ; vðθ þ Þe iðθ
þ
e ϕ þ Þ T iks cos ðθ þ Þx iqy
Þ e
e
38
Y. Hajati et al. / Journal of Magnetism and Magnetic Materials 362 (2014) 36–41
Fig. 2. The normalized charge conductance versus the bias energy, eV =Δ0 , for different values of the d-wave rotation angle. Here, we set EF ¼ 100Δ0 , U 0 ¼ 10EF , χ G ¼ 0, G0 ¼ G↑ þ G↓ : (a) χ H ¼ π=3 and (b) χ H ¼ 0.
ψ hS ðxÞ ¼ ðuðθ Þ; uðθ Þe iθ ; vðθ Þe iϕ ; vðθ Þe iðθ h
eiks
ϕ Þ T
Þ
cos ðθ Þx iqy
ð7Þ
e
where the wave vector for an electron-like quasiparticle (ELQ) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e kS ¼ ðEF þ U 0 þ E2 jΔðθ þ Þj2 Þ=ℏvF and for a hole-like quasiparticle qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h (HLQ) kS ¼ ðEF þU 0 E2 jΔðθ Þj2 Þ=ℏvF . The coherent factors are vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 u 0 u 0 u u 2 2 E jΔðθÞj E2 jΔðθÞj2 u1 u1 A; vðθÞ ¼ t @1 A uðθÞ ¼ t @1 þ 2 2 E E ð8Þ þ
θeS ,
We have also defined θ ¼ θ ¼ π θhS and e iϕ0 ðΔðθ 7 Þ=jΔðθ 7 ÞjÞ. The transmission angles θðiÞ S for the i e HLQ quasiparticles are given by kS sin ðθiS Þ ¼ kN sin ðθÞ;
iϕ 7
e ¼ ELQ and i ¼ e; h: The total wavefunction for quasiparticles in the superconducting region is ψ S ¼ tψ eS þ þ t 0 ψ hS , where the t and tʹ are the transmission coefficients of the electron-like and hole-like quasiparticles in the SG region, respectively. Because of the translational invariance in the y-direction, all transverse wave vectors (q) during the scattering process are conserved. Using the continuity of wavefunction at the boundaries (boundary conditions), the amplitude of spin-dependent Andreev reflection r As and normal reflection r s can be calculated by ψ N ðx ¼ LÞ ¼ ψ F B;s ðx ¼ LÞ;
ψ F B;s ðx ¼ 0Þ ¼ ψ S ðx ¼ 0Þ ð9Þ In the limit of a thin barrier V G þ ηs h⪢E þ EF , θF B þ ð Þ -0 and
ηs hL V GL þ ð Þ ¼ ηs X H þ ð ÞX G ℏvF ℏvF
ð10Þ
Using the BTK formalism, the differential charge and spin conductance at zero temperature for incident s spin electron in the ferromagnetic barrier region in this junction are given by [33] Z π=2 2e2 cos θAs GQs ðeVÞ ¼ N ∑ dθs cos θs 1 þ jr As j2 jr s j2 h s ¼ ↑;↓ 0 cos θs Z π=2 2 2e cos θ As N ∑ GSs ðeVÞ ¼ dθs cos θs 1 þ jr As j2 jr s j2 h s ¼ ↑;↓ 0 cos θs ð11Þ where in a graphene sheet with width W, NðEÞ ¼ ðjE þ EF jWÞ=πℏvF is the density of states (DOS).
3. Numerical results and discussion First we plot the normalized charge conductance versus the bias energy, eV=Δ0 , for doped graphene in a graphene-based NFBS junction to see how they change upon the rotation angle α (see Fig. 2(a)). It is found that the peak of the charge conductance is
Fig. 3. The normalized charge conductance versus the bias energy, eV=Δ0 , for different values of (a) χ H for χ G ¼ 0 (b) χ G for χ H ¼ 0. Here, we set α ¼ π=4, EF ¼ 100Δ0 and U 0 ¼ 10EF .
replaced by lower eV by increasing α from 0 to π/4. For the maximum rotation angle α ¼ π=4, the formation of ZBCP is clear, but this curve does not start from common value of 2 due to the presence of the spin-dependent barrier in the FB region (χ H ¼ π=3). In Fig. 2(b), we can see the ZBCP for this junction which has been started from 2 by putting χ H ¼ 0[13]. In Fig. 3 we study the effects of the exchange energy χ H and the barrier strength χ G on the charge conductance with respect to bias energy eV =Δ0 for α ¼ π=4. As indicated in Fig. 3(a), the ZBCPs start from lower values by increasing exchange energy and the curves are oscillating by a period of π. From Fig. 3(b) it is clear that the variation of the barrier strength does not have any effect on the formation of ZBCP in all curves and the charge conductance oscillates by a period of π. Now, let us study the effect of α on the charge conductance versus the barrier strength (χ G ). In a graphene-based NId-wave junction (for χ H ¼ 0), the charge conductance is an oscillatory function of the barrier strength by a period of π for α ¼ 0, as seen in Fig. 4(a). We found that by increasing α from 0 to π=4, the amplitude of charge conductance oscillation decreases. For α ¼ π=4, the charge conductance is insensitive to the barrier strength, this result coincides with Ref. [31]. We have also studied the effect of exchange energy on the charge conductance versus the barrier strength by putting χ H ¼ π=3 (see Fig. 4(b)). We can see that the χ H has a strong effect on the charge conductance.
Y. Hajati et al. / Journal of Magnetism and Magnetic Materials 362 (2014) 36–41
Fig. 4. Plot of the zero-bias charge conductance as a function of χ G for different values of the d-wave rotation angle for EF ¼ 100Δ0 , U 0 ¼ 0: (a) χ H ¼ 0 and (b) χH ¼ π=3.
Fig. 5. Plot of the zero-bias charge conductance as a function of χ H for different values of the d-wave rotation angle for EF ¼ 100Δ0 , U 0 ¼ 0: (a) χ G ¼ 0 and (b) χ G ¼ π=3.
Although the charge conductance is an oscillatory function of the barrier strength over a period of π, all curves do not start from common value of 2. By increasing α the charge conductance for χ G ¼ 0 increases and for α ¼ π=4, the charge conductance does not depend on χ G and starts from lower value as compared to Fig. 4(a) (χ H ¼ 0). We found that the retro and specular Andreev reflections in graphene-based NFBd-wave junction are modified in the presence of exchange energy and when the pair potential signs are opposite to each other (for α ¼ π=4), they do not depend on the barrier strength. So the charge conductance does not depend on χ G . The more interesting case is to study the effect of α on the zerobias charge conductance versus exchange energy in this junction (specially for α ¼ π=4). For more systematically study of the effect of the spin-dependent barrier, we have plotted the zero-bias charge conductance versus χ H for different values of the rotation
39
angle α, as seen in Fig. 5(a). In this curve χ G ¼ 0 has been considered. For α ¼ 0, the charge conductance curve shows oscillatory behavior by a period of π. By increasing α from 0 to π/4, all oscillatory curves start from 2 and the amplitude of their oscillations decrease. Interestingly, for α ¼ π=4 the charge conductance shows oscillatory behavior and the period of oscillations becomes half as compared to α ¼ 0 curve, which has not been reported before as we know. In Fig. 5(b) we have also studied the effects of the barrier strength on the charge conductance versus the exchange energy by putting χ G ¼ π=3. For the small values of the rotation angle, the χ G has a strong effect on the starting point of the charge conductance and also the shape of the curves. By approaching the rotation angle to α ¼ π=4, this effect decreases and for α ¼ π=4 there are not any differences between Fig. 5(a) and (b), by which this result confirms our earlier findings (see Fig. 4 and also Ref. [31]). As a check of the validity of our calculations, we emphasize that the result for α ¼ 0 in Figs. 4 and 5 coincides with Fig. 5 in Ref. [21]. This also indicates that our formalism is more general. It is obtained that the Fermi wave vector mismatch, rotation angle and exchange energy suppress the amplitude of charge conductance oscillation in the graphene-based FIS and NIS junctions. Also it is found that in a graphene-based NFBS-wave junction, Fermi wave vector cannot suppress the amplitude of charge conductance oscillation [21]. This is due to the presence of the spin-dependent barrier and quantum nature of interference between electron-like and hole-like quasiparticles in the middle region of this junction. As a remarkable result, we show that in a graphene-based NFBd-wave junction, the amplitude of charge conductance oscillations with respect to χ H could not be suppressed by rotation angle and for α ¼ π=4, we still have the charge conductance oscillation (Fig. 5(a) and (b)). This means that at the maximum value of the rotation angle, the Dirac type quasiparticles can transmit through a thin barrier and the relativity is not destroyed by the rotation angle. The quantum nature of interference between electron-like and hole-like quasiparticles in the FB region is responsible for this behavior. This behavior is in contrast to the graphene-based NId-wave junction in which the rotation angle can suppress the amplitude of the charge conductance oscillations [31]. Also this behavior is in sharp contrast to the similar nonrelativistic junctions in which the motion of quasiparticles is described by the Schrodinger equation. In similar nongraphene-based junctions, by increasing χ G and χ H the charge conductance does not show oscillatory behavior [34–41]. The oscillatory behavior in graphene-based junctions was interpreted in terms of transmission resonance property of the Dirac Bogoliubov quasiparticles [42–44]. The zero-bias charge conductance versus the rotation angle α has been plotted in Figs. 6 and 7 for different values of the barrier strength and exchange energy in this junction, respectively. The periodicity of the charge conductance with respect to α (by a period of π=2) and the barrier strength (by a period of π) are clear in both curves of Fig. 6 and also one can see that the G(0) does not depend on the barrier strength at α ¼ π=4 (also see Fig. 4(a)) and the curve starts from common value of 2. From Fig. 6(b), it can be seen that by introducing the exchange energy, G(0) shows oscillatory behavior for χ G ¼ 0 and by varying the barrier strength from 0 to χ H , the amplitude of charge conductance oscillation reduces and for χ G ¼ χ H the charge conductance does not depend on α. Also for χ G 4χ H , the G(0) starts to oscillate again, but the concavity of the curve reverses. Interestingly, in the graphene-based Nd-wave junction (by putting χ G ¼ 0; χ H ¼ 0) the G(0) is insensitive to the rotation angle of order parameter (see Figs. 6(a) and 7(a)). Also the alternative form of the charge conductance with respect to α (by a period of π=2) and the exchange energy (by a period of π) has been shown in both curves of Fig. 7. In contrast
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Y. Hajati et al. / Journal of Magnetism and Magnetic Materials 362 (2014) 36–41
Fig. 8. Plot of the normalized spin conductance versus the bias energy, eV=Δ0 , for different values of the d-wave rotation angle. Here, we set EF ¼ 100Δ0 , U 0 ¼ 10EF , χ G ¼ 0 and χ H ¼ π=3.
Fig. 6. Plot of the zero-bias charge conductance as a function of the d-wave rotation angle for different values of χ H for EF ¼ 100Δ0 , U 0 ¼ 0: (a) χ G ¼ 0 and (b) χ G ¼ π=3.
for the subgap energies. For α ¼ π=4, we can observe the formation of the zero-bias spin conductance dip (ZBSCD) and the spin current starts from zero. By increasing bias energy, the spin current increases monotonically and becomes flat finally. So upon varying α from 0 to π/4, the spin filtering application of this system can be tuned from eV ¼ Δ0 to 0, respectively. Hence, the spin filtering application of this junction is drastically changed by the rotation angle α. As α increases, the spin filtering application enhances, being strongest for α ¼ π=4.
4. Conclusion In summary, the influences of the spin-dependent barrier and rotation angle of d-wave superconducting order parameter (α) on the charge and spin conductance in a graphene-based NFBd-wave junction have been studied within the BTK formalism. We analytically obtained that due to the spin-dependent barrier and quantum interference of quasiparticles in the FB region, the rotation angle cannot suppress the zero-bias charge conductance and for α ¼ π=4, this junction shows oscillatory behavior which is different from similar non-spin-dependent barrier junctions. As another remarkable result, we have shown that the spin filtering application of this junction is drastically changed by the rotation angle α. As α increases, the spin filtering application enhances, being strongest for α ¼ π=4. Hence for α ¼ π=4, we can observe the formation of the zero-bias spin conductance dip (ZBSCD) and the spin current starts from zerobias energy. It is expected that the theoretical results obtained may be confirmed by future experiments. References
Fig. 7. Plot of the zero-bias charge conductance as a function of the d-wave rotation angle for different values of χ G for EF ¼ 100Δ0 , U 0 ¼ 0: (a) χ H ¼ 0 and (b) χ H ¼ π=3.
to Fig. 6(a), the G(0) depends on the barrier strength at α ¼ π=4 (also see Fig. 5) and the curves do not start from 2. Finally, we have plotted the effect of the rotation angle on the spin conductance as a function of the bias energy, in Fig. 8. As we see for α ¼ 0, there is no spin current for the subgap energies (eV r Δ0 ) and we have just the spin current for energy eV 4 Δ0 . The lack of the spin current follows the fact that the total spin of the copper pair is zero for the subgap energies. This is one of the clear applications of this system for usage in the spintronics devices as a spin filter. As another remarkable result, we obtained that by increasing the rotation angle from 0 to π/4, there is a spin current
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Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Spin-dependent barrier effects on the transport properties of graphene-based normal metal/ferromagnetic barrier/d-wave superconductor junction Y. Hajati a,n, A. Heidari b, M.Z. Shoushtari a, G. Rashedi c a
Department of Physics, Faculty of Sciences, Shahid Chamran University of Ahvaz, Ahvaz, Iran Department of mathematics, Izeh branch, Islamic Azad University, Izeh, Iran c Department of Physics, Faculty of Sciences, University of Isfahan, Isfahan, Iran b
art ic l e i nf o
a b s t r a c t
Article history: Received 18 January 2014 Received in revised form 26 February 2014 Available online 13 March 2014
Using the extended Blonder–Tinkham–Klapwijk formalism, the spin-dependent transport properties in a graphene-based normal metal/ferromagnetic barrier/d-wave superconductor (NFBd-wave) junction have been studied theoretically. Here, we have mainly studied the influences of spin-dependent barrier and rotation angle of d-wave superconducting order parameter (α) on the charge and spin conductance. It is found that the rotation angle has a strong effect on the amplitude and phase of the charge conductance oscillations. As a remarkable result, we obtained that because of the spin-dependent barrier (FB), the rotation angle cannot suppress the zero-bias charge conductance and for the maximum rotation angle α ¼ π=4, the charge conductance shows oscillatory behavior which is different from similar nonspin-dependent barrier junctions. We have also shown that the spin filtering application of this junction is drastically changed by the rotation angle α. As α increases, the spin filtering application enhances, being strongest for α ¼ π=4. At last, we propose an experimental setup to detect our predicted effects. & 2014 Elsevier B.V. All rights reserved.
Keywords: Graphene Ferromagnetic barrier Rotation angle Spin filtering application
1. Introduction Graphene, a two-dimensional honeycomb lattice of carbon atoms, has attracted enormous attention in condensed matter physics after its successful synthesis at 2004 [1]. The conduction and valence bands of graphene form conically shaped valleys, touching each other at a point called the Dirac point. There are two inequivalent Dirac points at the Brillouin zone. The low energy excitations in graphene are governed by the Dirac equation and electrons and holes in graphene are massless relativistic particles with a large energy independent velocity 106 m/s [2]. Unusual electronic properties of Dirac fermions in graphene such as very high carrier mobility [3], ambipolar electric field effect [4] and large spin-relaxation length and long lifetime of the spin-polarized current [5,6] make graphene an important material for application purposes specially in the spintronics devices. Graphene is not naturally superconductor nor ferromagnetic, but recent experimental and theoretical studies show that superconductivity and magnetism can be induced in graphene via the proximity effect by a superconductor or ferromagnetic film [4,6–10].
n
Corresponding author. Tel.: +98 93 68148729. E-mail address: [email protected] (Y. Hajati).
http://dx.doi.org/10.1016/j.jmmm.2014.03.018 0304-8853/& 2014 Elsevier B.V. All rights reserved.
Beenakker showed that in a graphene-based normal metal/ superconductor (NS) junction, when the Fermi energy in the normal metal is smaller than the pair potential of superconductor, the specular Andreev reflection occurs instead of the usual retroreflection in the standard (not graphene-based) NS junction [7]. The existence of the novel specular Andreev reflection (AR) process in the graphene-based superconducting junctions leads to fundamentally different tunneling properties occur compare to the standard superconducting junctions [7,11–13]. It was shown that the tunneling charge conductance in a graphene-based normal metal/insulator/superconductor (NIS) junction shows oscillatory behavior versus the barrier strength [11]. Proximity effects in a graphene-based ferromagnetic/superconductors junction have recently attracted much attention in experimental and theoretical studies [6,9,10,14–20]. For the first time, Zareyan showed that when the exchange energy in a graphene-based ferromagnetic/superconductor (FS) junction is greater than Fermi energy EF, the charge conductance of the junction increases [14]. We know that this behavior is drastically different from the standard FS junction. In a graphene-based normal metal/ferromagnetic barrier/superconductor (NFBS) junction, the electrical conductance does not depend on the barrier strength χ G for superconductor having a higher effective Fermi energy but it depends on the ferromagnetic barrier strength χ H
Y. Hajati et al. / Journal of Magnetism and Magnetic Materials 362 (2014) 36–41
and the conductance is an oscillatory function of χ H [21]. We know that this behavior is because of the quantum nature of interference between quasiparticles in the FB region. Deposition of the ferromagnetic insulator (barrier) on the graphene allows one to make new types of magnetic tunneling junctions (MTJ). Recently, these MTJs attracted much attention because of their applications in the spintronics devices [10,22–26]. In the above discussions of the transport property in the graphene-based superconductor junctions, the superconductor considered was conventional superconductor (s-wave). Nowadays, the major candidate symmetry of the superconductivity in high-Tc cuprates (unconventional superconductors) is d-wave. One of the clear qualitative signatures of d-wave symmetry is the so-called zero-bias conductance peak (ZBCP) observed in the I–V characteristics of electron tunneling into thin films of YBCO by Geerk et al. [27–29]. In unconventional superconductor the sign reversal of the order parameter creates the zero energy states (ZES) [30]. Then, the formation of zero energy states produces ZBCP in the electron tunneling experiments. In addition, the proximity effect of graphene-based normal metal/d-wave superconductor [12,13] and ferromagnetic/d-wave superconductor junctions has been studied [31]. In the d-wave superconductors, the order parameter is angle dependent, ΔðθÞ ¼ Δ0 cos ð2θ 2αÞ, where α is the rotation angle of the order parameter relative to the surface metal. In this paper, we study the transport properties of a graphenebased normal metal/ferromagnetic barrier/d-wave superconductor (NFBd-wave) junction including charge and spin conductance as a new case. To this end, we have employed the Dirac–Bogoliubov– de Gennes (DBdG) equation and Blonder–Tinkham–Klapwijk (BTK) formalism. Here, the effect of the rotation angle of d-wave superconducting order parameter on the charge and spin conductance has been studied theoretically. Actually, our main focus is to study the effect of rotation angle on the amplitude and phase of the charge oscillation in the presence of spin-dependent barrier in the middle region of this junction.
37
we should apply the spin-polarized Dirac–Bogoliubov–de Gennes (DBdG) equation: ! H 0 UðxÞ HðxÞ ΔðxÞ ψ ¼ Eψ ð1Þ n HðxÞ þ UðxÞ H 0 Δ ðxÞ where H 0 ¼ iℏvF ðsx δx 7 sy δy Þ; UðxÞ ¼ EF Θð x LÞ þ ðEF þ V G ÞΘðx þ LÞΘð xÞ þ ðEF þ U 0 ÞΘðxÞ HðxÞ ¼ ηs hΘð x LÞ ΔðxÞ ¼ ΔðθÞ expð iφÞΘðxÞ; ΔðθÞ ¼ Δ0 cos ð2θ 2αÞ
ð2Þ
Here sx and sy are Pauli matrices in the pseudospin space of the sublattices and Θ is the Heaviside step function. The potential UðxÞ is the shift of the Fermi energies in the ferromagnetic (EF), ferromagnetic barrier (EF þVG) and superconducting regions (EF þU0) of graphene, respectively. The H 0 , HðxÞ and ΔðxÞ are Dirac Hamiltonian, exchange energy, and order parameter in the ferromagnetic and superconductor region, respectively. The exchange energy in the middle ferromagnetic barrier region is h. The potential U0 can be used to tune the mismatch between Fermi surface of the ferromagnetic and superconducting regions. The DBdG equation is derived within the mean-field approximation [32]. So, U0 should be very large compared to all other energy scales in our calculations: ðEF þ U 0 Þ⪢Δ0 . By solving the spin-dependent DBdG equation (Eq. (1)), we obtain the wavefunctions in each region of the junction. In the normal graphene (NG) region, the wavefunction for incident electron and retro-Andreev reflected hole is given by 00
0 1 1 1 BB iθ C B BB e C ik þ x B e iθ B C N þ rs B ψ N ðxÞ ¼ B BB 0 Ce B 0 @@ @ A 0 0
0
1
B C B C ik þ x Ce N þ r As B B C @ A
0 0 1 e iθA
1
1
C C C ik x C iqy Ce N Ce C C A A
ð3Þ where þðÞ
2. Scattering process and conductance formalism Here we consider a two-dimensional graphene-based NFBdwave junction in which two interfaces is parallel to the y-axis and they are located at x¼ L and x ¼0. The left ferromagnet and right d-wave superconductor in this junction are separated by a ferromagnetic barrier by a thickness of L and effective gate potential VG to control its chemical potential, as seen in Fig. 1. To study a problem concerning superconductivity and relativity, we should consider the Dirac–Bogoliubov–de Gennes (DBdG) equation. The DBdG equation for a graphene-based NS junction was derived in Ref. [7]. In a graphene-based NFBd-wave junction because of real spin degree of freedom in the ferromagnetic region,
kN
¼
ðE þð ÞEF Þ cos ðθðθA ÞÞ; ℏvF
sin ðθðθA ÞÞ ¼
ℏvF q ðE þ ð ÞEF Þ
ð4Þ
The θ is the incident angle of the electron and θA is the Andreev reflection angle of the hole in the N region. Also r s and r As are the normal and Andreev reflection amplitudes in the normal region. The spin-dependent wavefunction in the ferromagnetic barrier graphene (FBG) region is 0 1 B B iθF B þ B Be B ψ F B ðxÞ ¼ B Bps B 0 @ @ 0
0 1 iθF B B C þ B e C ikF x Ce B þ þqs B B C 0 @ A 0
0
0
B B þ ms B B @
1
1
0
iθF B
e
C C ikF x Bþ Ce C A
0
C B C ikF x B Ce B þ n s B C B A @
0 1
1
1
0 0 1 e
iθF B
1
C C C ikF x C iqy Ce B Ce C C A A
ð5Þ
where ℏvF q ; ðE þ ð ÞðEF V G Þ þ ηs hÞ ðE þ ð ÞðEF V G Þ þ ηs hÞ ¼ cos ðθF B þ ð Þ Þ ℏvF
sin ðθF B þ ð Þ Þ ¼ kF B þ ð Þ
Fig. 1. The proposed experimental setup to study the transport properties of a graphene-based NFBd-wave junction. In this plot NG, FBG and d-wave are normal graphene, ferromagnetic barrier graphene and d-wave superconductor graphene regions, respectively. The top and bottom gate allow for the chemical potential in the ferromagnetic region.
ð6Þ
Also, θF B þ ð Þ is the incident angle electron (hole) with respect to the normal direction to the interfaces, in the θF B region and ηs ¼ ηs ¼ 1 ðs ¼ ↑ s ¼ ↓Þ. On the superconducting (SG) region the wavefunctions for transmission of a right moving quasiparticles with a given excitation energy E 40 reads þ
þ
ψ eS þ ðxÞ ¼ ðuðθ þ Þ; uðθ þ Þe iθ ; vðθ þ Þe iϕ ; vðθ þ Þe iðθ
þ
e ϕ þ Þ T iks cos ðθ þ Þx iqy
Þ e
e
38
Y. Hajati et al. / Journal of Magnetism and Magnetic Materials 362 (2014) 36–41
Fig. 2. The normalized charge conductance versus the bias energy, eV =Δ0 , for different values of the d-wave rotation angle. Here, we set EF ¼ 100Δ0 , U 0 ¼ 10EF , χ G ¼ 0, G0 ¼ G↑ þ G↓ : (a) χ H ¼ π=3 and (b) χ H ¼ 0.
ψ hS ðxÞ ¼ ðuðθ Þ; uðθ Þe iθ ; vðθ Þe iϕ ; vðθ Þe iðθ h
eiks
ϕ Þ T
Þ
cos ðθ Þx iqy
ð7Þ
e
where the wave vector for an electron-like quasiparticle (ELQ) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e kS ¼ ðEF þ U 0 þ E2 jΔðθ þ Þj2 Þ=ℏvF and for a hole-like quasiparticle qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h (HLQ) kS ¼ ðEF þU 0 E2 jΔðθ Þj2 Þ=ℏvF . The coherent factors are vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 u 0 u 0 u u 2 2 E jΔðθÞj E2 jΔðθÞj2 u1 u1 A; vðθÞ ¼ t @1 A uðθÞ ¼ t @1 þ 2 2 E E ð8Þ þ
θeS ,
We have also defined θ ¼ θ ¼ π θhS and e iϕ0 ðΔðθ 7 Þ=jΔðθ 7 ÞjÞ. The transmission angles θðiÞ S for the i e HLQ quasiparticles are given by kS sin ðθiS Þ ¼ kN sin ðθÞ;
iϕ 7
e ¼ ELQ and i ¼ e; h: The total wavefunction for quasiparticles in the superconducting region is ψ S ¼ tψ eS þ þ t 0 ψ hS , where the t and tʹ are the transmission coefficients of the electron-like and hole-like quasiparticles in the SG region, respectively. Because of the translational invariance in the y-direction, all transverse wave vectors (q) during the scattering process are conserved. Using the continuity of wavefunction at the boundaries (boundary conditions), the amplitude of spin-dependent Andreev reflection r As and normal reflection r s can be calculated by ψ N ðx ¼ LÞ ¼ ψ F B;s ðx ¼ LÞ;
ψ F B;s ðx ¼ 0Þ ¼ ψ S ðx ¼ 0Þ ð9Þ In the limit of a thin barrier V G þ ηs h⪢E þ EF , θF B þ ð Þ -0 and
ηs hL V GL þ ð Þ ¼ ηs X H þ ð ÞX G ℏvF ℏvF
ð10Þ
Using the BTK formalism, the differential charge and spin conductance at zero temperature for incident s spin electron in the ferromagnetic barrier region in this junction are given by [33] Z π=2 2e2 cos θAs GQs ðeVÞ ¼ N ∑ dθs cos θs 1 þ jr As j2 jr s j2 h s ¼ ↑;↓ 0 cos θs Z π=2 2 2e cos θ As N ∑ GSs ðeVÞ ¼ dθs cos θs 1 þ jr As j2 jr s j2 h s ¼ ↑;↓ 0 cos θs ð11Þ where in a graphene sheet with width W, NðEÞ ¼ ðjE þ EF jWÞ=πℏvF is the density of states (DOS).
3. Numerical results and discussion First we plot the normalized charge conductance versus the bias energy, eV=Δ0 , for doped graphene in a graphene-based NFBS junction to see how they change upon the rotation angle α (see Fig. 2(a)). It is found that the peak of the charge conductance is
Fig. 3. The normalized charge conductance versus the bias energy, eV=Δ0 , for different values of (a) χ H for χ G ¼ 0 (b) χ G for χ H ¼ 0. Here, we set α ¼ π=4, EF ¼ 100Δ0 and U 0 ¼ 10EF .
replaced by lower eV by increasing α from 0 to π/4. For the maximum rotation angle α ¼ π=4, the formation of ZBCP is clear, but this curve does not start from common value of 2 due to the presence of the spin-dependent barrier in the FB region (χ H ¼ π=3). In Fig. 2(b), we can see the ZBCP for this junction which has been started from 2 by putting χ H ¼ 0[13]. In Fig. 3 we study the effects of the exchange energy χ H and the barrier strength χ G on the charge conductance with respect to bias energy eV =Δ0 for α ¼ π=4. As indicated in Fig. 3(a), the ZBCPs start from lower values by increasing exchange energy and the curves are oscillating by a period of π. From Fig. 3(b) it is clear that the variation of the barrier strength does not have any effect on the formation of ZBCP in all curves and the charge conductance oscillates by a period of π. Now, let us study the effect of α on the charge conductance versus the barrier strength (χ G ). In a graphene-based NId-wave junction (for χ H ¼ 0), the charge conductance is an oscillatory function of the barrier strength by a period of π for α ¼ 0, as seen in Fig. 4(a). We found that by increasing α from 0 to π=4, the amplitude of charge conductance oscillation decreases. For α ¼ π=4, the charge conductance is insensitive to the barrier strength, this result coincides with Ref. [31]. We have also studied the effect of exchange energy on the charge conductance versus the barrier strength by putting χ H ¼ π=3 (see Fig. 4(b)). We can see that the χ H has a strong effect on the charge conductance.
Y. Hajati et al. / Journal of Magnetism and Magnetic Materials 362 (2014) 36–41
Fig. 4. Plot of the zero-bias charge conductance as a function of χ G for different values of the d-wave rotation angle for EF ¼ 100Δ0 , U 0 ¼ 0: (a) χ H ¼ 0 and (b) χH ¼ π=3.
Fig. 5. Plot of the zero-bias charge conductance as a function of χ H for different values of the d-wave rotation angle for EF ¼ 100Δ0 , U 0 ¼ 0: (a) χ G ¼ 0 and (b) χ G ¼ π=3.
Although the charge conductance is an oscillatory function of the barrier strength over a period of π, all curves do not start from common value of 2. By increasing α the charge conductance for χ G ¼ 0 increases and for α ¼ π=4, the charge conductance does not depend on χ G and starts from lower value as compared to Fig. 4(a) (χ H ¼ 0). We found that the retro and specular Andreev reflections in graphene-based NFBd-wave junction are modified in the presence of exchange energy and when the pair potential signs are opposite to each other (for α ¼ π=4), they do not depend on the barrier strength. So the charge conductance does not depend on χ G . The more interesting case is to study the effect of α on the zerobias charge conductance versus exchange energy in this junction (specially for α ¼ π=4). For more systematically study of the effect of the spin-dependent barrier, we have plotted the zero-bias charge conductance versus χ H for different values of the rotation
39
angle α, as seen in Fig. 5(a). In this curve χ G ¼ 0 has been considered. For α ¼ 0, the charge conductance curve shows oscillatory behavior by a period of π. By increasing α from 0 to π/4, all oscillatory curves start from 2 and the amplitude of their oscillations decrease. Interestingly, for α ¼ π=4 the charge conductance shows oscillatory behavior and the period of oscillations becomes half as compared to α ¼ 0 curve, which has not been reported before as we know. In Fig. 5(b) we have also studied the effects of the barrier strength on the charge conductance versus the exchange energy by putting χ G ¼ π=3. For the small values of the rotation angle, the χ G has a strong effect on the starting point of the charge conductance and also the shape of the curves. By approaching the rotation angle to α ¼ π=4, this effect decreases and for α ¼ π=4 there are not any differences between Fig. 5(a) and (b), by which this result confirms our earlier findings (see Fig. 4 and also Ref. [31]). As a check of the validity of our calculations, we emphasize that the result for α ¼ 0 in Figs. 4 and 5 coincides with Fig. 5 in Ref. [21]. This also indicates that our formalism is more general. It is obtained that the Fermi wave vector mismatch, rotation angle and exchange energy suppress the amplitude of charge conductance oscillation in the graphene-based FIS and NIS junctions. Also it is found that in a graphene-based NFBS-wave junction, Fermi wave vector cannot suppress the amplitude of charge conductance oscillation [21]. This is due to the presence of the spin-dependent barrier and quantum nature of interference between electron-like and hole-like quasiparticles in the middle region of this junction. As a remarkable result, we show that in a graphene-based NFBd-wave junction, the amplitude of charge conductance oscillations with respect to χ H could not be suppressed by rotation angle and for α ¼ π=4, we still have the charge conductance oscillation (Fig. 5(a) and (b)). This means that at the maximum value of the rotation angle, the Dirac type quasiparticles can transmit through a thin barrier and the relativity is not destroyed by the rotation angle. The quantum nature of interference between electron-like and hole-like quasiparticles in the FB region is responsible for this behavior. This behavior is in contrast to the graphene-based NId-wave junction in which the rotation angle can suppress the amplitude of the charge conductance oscillations [31]. Also this behavior is in sharp contrast to the similar nonrelativistic junctions in which the motion of quasiparticles is described by the Schrodinger equation. In similar nongraphene-based junctions, by increasing χ G and χ H the charge conductance does not show oscillatory behavior [34–41]. The oscillatory behavior in graphene-based junctions was interpreted in terms of transmission resonance property of the Dirac Bogoliubov quasiparticles [42–44]. The zero-bias charge conductance versus the rotation angle α has been plotted in Figs. 6 and 7 for different values of the barrier strength and exchange energy in this junction, respectively. The periodicity of the charge conductance with respect to α (by a period of π=2) and the barrier strength (by a period of π) are clear in both curves of Fig. 6 and also one can see that the G(0) does not depend on the barrier strength at α ¼ π=4 (also see Fig. 4(a)) and the curve starts from common value of 2. From Fig. 6(b), it can be seen that by introducing the exchange energy, G(0) shows oscillatory behavior for χ G ¼ 0 and by varying the barrier strength from 0 to χ H , the amplitude of charge conductance oscillation reduces and for χ G ¼ χ H the charge conductance does not depend on α. Also for χ G 4χ H , the G(0) starts to oscillate again, but the concavity of the curve reverses. Interestingly, in the graphene-based Nd-wave junction (by putting χ G ¼ 0; χ H ¼ 0) the G(0) is insensitive to the rotation angle of order parameter (see Figs. 6(a) and 7(a)). Also the alternative form of the charge conductance with respect to α (by a period of π=2) and the exchange energy (by a period of π) has been shown in both curves of Fig. 7. In contrast
40
Y. Hajati et al. / Journal of Magnetism and Magnetic Materials 362 (2014) 36–41
Fig. 8. Plot of the normalized spin conductance versus the bias energy, eV=Δ0 , for different values of the d-wave rotation angle. Here, we set EF ¼ 100Δ0 , U 0 ¼ 10EF , χ G ¼ 0 and χ H ¼ π=3.
Fig. 6. Plot of the zero-bias charge conductance as a function of the d-wave rotation angle for different values of χ H for EF ¼ 100Δ0 , U 0 ¼ 0: (a) χ G ¼ 0 and (b) χ G ¼ π=3.
for the subgap energies. For α ¼ π=4, we can observe the formation of the zero-bias spin conductance dip (ZBSCD) and the spin current starts from zero. By increasing bias energy, the spin current increases monotonically and becomes flat finally. So upon varying α from 0 to π/4, the spin filtering application of this system can be tuned from eV ¼ Δ0 to 0, respectively. Hence, the spin filtering application of this junction is drastically changed by the rotation angle α. As α increases, the spin filtering application enhances, being strongest for α ¼ π=4.
4. Conclusion In summary, the influences of the spin-dependent barrier and rotation angle of d-wave superconducting order parameter (α) on the charge and spin conductance in a graphene-based NFBd-wave junction have been studied within the BTK formalism. We analytically obtained that due to the spin-dependent barrier and quantum interference of quasiparticles in the FB region, the rotation angle cannot suppress the zero-bias charge conductance and for α ¼ π=4, this junction shows oscillatory behavior which is different from similar non-spin-dependent barrier junctions. As another remarkable result, we have shown that the spin filtering application of this junction is drastically changed by the rotation angle α. As α increases, the spin filtering application enhances, being strongest for α ¼ π=4. Hence for α ¼ π=4, we can observe the formation of the zero-bias spin conductance dip (ZBSCD) and the spin current starts from zerobias energy. It is expected that the theoretical results obtained may be confirmed by future experiments. References
Fig. 7. Plot of the zero-bias charge conductance as a function of the d-wave rotation angle for different values of χ G for EF ¼ 100Δ0 , U 0 ¼ 0: (a) χ H ¼ 0 and (b) χ H ¼ π=3.
to Fig. 6(a), the G(0) depends on the barrier strength at α ¼ π=4 (also see Fig. 5) and the curves do not start from 2. Finally, we have plotted the effect of the rotation angle on the spin conductance as a function of the bias energy, in Fig. 8. As we see for α ¼ 0, there is no spin current for the subgap energies (eV r Δ0 ) and we have just the spin current for energy eV 4 Δ0 . The lack of the spin current follows the fact that the total spin of the copper pair is zero for the subgap energies. This is one of the clear applications of this system for usage in the spintronics devices as a spin filter. As another remarkable result, we obtained that by increasing the rotation angle from 0 to π/4, there is a spin current
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