Pure thermal spin current and perfect spin-filtering with negative differential thermoelectric resistance induced by proximity effect in graphene/silicene junctions

The spin-dependent Seebeck effect (SDSE) and thermal spin-filtering effect (SFE) are now considered as the essential aspects of the spin caloritronics, which can efficiently explore the relationships between the spin and heat transport in the materials. However, there is still a challenge to get a thermally-induced spin current with no thermal electron current. This paper aims to numerically investigate the spin-dependent transport properties in hybrid graphene/silicene nanoribbons (GSNRs), using the nonequilibrium Green’s function method. The effects of temperature gradient between the left and right leads, the ferromagnetic exchange field, and the local external electric fields are also included. The results showed that the spin-up and spin-down currents are produced and flow in opposite directions with almost equal magnitudes. This evidently shows that the carrier transport is dominated by the thermal spin current, whereas the thermal electron current is almost disappeared. A pure thermal spin current with the finite threshold temperatures can be obtained by modulating the temperature, and a negative differential thermoelectric resistance is obtained for the thermal electron current. A nearly zero charge thermopower is also obtained, which further demonstrates the emergence of the SDSE. The response of the hybrid system is then varied by changing the magnitudes of the ferromagnetic exchange field and local external electric fields. Thus, a nearly perfect SFE can be observed at room temperature, whereas the spin polarization efficiency is reached up to 99%. It is believed that the results obtained from this study can be useful to well understand the inspiring thermospin phenomena, and to enhance the spin caloritronics material with lower energy consumption.

along the y direction to match the lattice of GE. The distance between the C and Si atoms at the interface is equal to 1.80 Å 57 . Figure 1 shows the atomic configuration of the hybrid GSNR selected for this study. As shown in Fig. 1, the hybrid GSNR is constructed using three different parts, including the semi-infinite metallic armchair GE nanoribbons (AGNRs) on the left and right electrodes and a ZSNR in the central region 64 . Some dangling carbon bonds also remain at the GE/SE interface, which can be passivated by an additional hydrogen atom. It is assumed that the length of central region is almost equal to M = 8 unit cells. Each unit cell is composed of N S = 16 atoms, leading to a length and width of 29.85 and 25.27 Å, respectively, for the central region. Besides, each unit cell of the left and right leads includes N G = 46 atoms.

Tight-binding and Green's function method.
In this subsection, the electronic quantum transport in the quasi-one-dimensional hybrid GSNR structure is studied. As explained earlier, the hybrid GSNR system is in the xy plane, and consists of three different regions, including the AGNRs on the left and right leads and a ZSNR in the middle scattering region (see Fig. 1a). The generalized Hamiltonian for the system can be described as follows: in which, H R(L) and H C represent the Hamiltonian for the isolated right (left) lead and central region, respectively. Using the TB approximation model 40,42,65 , the Hamiltonian of the right (left) lead and the central region are obtained as follows: (1) H T = H L + H R + H C + H CL + H CR , Figure 1. (a) A schematic representation of the hybrid graphene/silicene nanoribbon (GSNR) adopted for this study, the central region consists of about M = 8 unit cells, each has N S =16 atoms; each unit cell of the leads also has N G = 46 atoms, (b) the external electric and ferromagnetic exchange fields applied to the system, where T L and T R define the temperature of the left and right leads, respectively. www.nature.com/scientificreports/ where t R (L) and t C are the hopping energies between the nearest-neighbor atoms in the right (left) lead, and the central region, so their values are equal to 2.66 and 1.60 eV, respectively. c † iα (c iα ) creates (annihilates) an electron with spin α at atom i. i, j and i, j represent the nearest-neighbor and next-nearest-neighbor pairs, respectively. λ so is the effective SOI parameter and is equal to 3.9 meV for the central region 42 . Due to the weak SOI in the GE 66 , the values of λ so and λ r are assumed equal to zero for the left and right leads. σ = (σ x , σ y , σ z ) is the Pauli matrix. υ ij is defined as υ ij = (d i × d j )/|d i × d j | where d i and d j are the two nearest bonds connecting the next-nearest-neighbors. ξ i = +1, −1 for the sublattices A and B, respectively; and 2 l is the buckling distance for the SE. E z is also the perpendicular electric field and produces a voltage difference of 2elE z between the two sublattices, where e is the electron charge. This causes the electrons to experience a staggered potential when jumping from a site to its nearest-neighbor. It is worth mentioning that the external electric field can efficiently differentiate the SE from GE. M z is the ferromagnetic exchange field, which can be produced by the proximity effect of a ferromagnetic material 42 . E yG and E yS are the inhomogeneous transverse electric field components along the x direction, applied to the leads and central region, respectively. y iG is also the normal distance of atom i from the middle of the ribbon.
H CR(CL) also shows the Hamiltonian for the coupling between the central region with the right (left) lead and using the TB approximation can be written as where t CR(CL) represents the hopping energy between the central region and right (left) lead. The contact hopping energy can also be determined by averaging the values obtained from the Harrison's scaling law 67,68 . In this study, namely, the hopping energies are re-calculated for the GE and SE at the interface when the C and Si atoms bond-lengths are changed to 1.8 Å. The geometric mean of the obtained hopping energy values (i.e., 2.07 eV) will then be used as the t CR(CL) parameter, which is required to be the same order of t R(L) and t C 55 . The electronic transport is assumed ballistic. Such an assumption is valid when the mean free path of the carriers, which is in the order of the micron in the SE and GE at room temperature, is larger than the sample dimensions. This is due to the high mobility of charge carriers in the SE and GE, and thus they can simply move long distances without inelastic scattering. In the central region, the spin direction of the electron is assumed preserved, and the spinflip scattering is ignored. Thus, the spin-up and spin-down electron transports can individually be studied. This assumption is correct because the spin diffusion length in the SE is in the order of several micrometers 29,69 .
To determine the spin-dependent current, it is first required the spin-dependent electron transmission function to be calculated. The Green's function of the central region can be written as follows 70 : where � R(L),α is the right (left) self-energy matrix and is computed by Eq. (6). � R(L),α includes the effect of two semi-infinite AGNRs on the central region.
in which, g R(L),α is the surface Green's function of the right (left) lead and is computed by where H 00 R(L) defines the TB Hamiltonian matrix of the unit cell studied in this research for the right (left) lead. H 01( †) R(L) is the hopping matrix between two adjacent unit cells in the right (left) lead. T R(L) is the transfer matrix of the right (left) lead and can be determined using an iterative methodology, as proposed in Reference 70 . The electron transmission function can then be obtained using the following equations 70 : where Ŵ R(L),α is the broadening function and describes the coupling between the right (left) lead with the central region.Ŵ R(L),α is given by where f R(L) is the Fermi-Dirac distribution. e, h, and T R(L) are also the electron charge, Plank constant, and temperature of the right (left) lead, respectively. As shown in Eq. (10), the temperature gradient ΔT = T L -T R between the leads produces a nonzero value of f L -f R , and thus the spin-dependent current is simply identified as a function of ΔT and T R(L) . The net spin and charge currents can then be obtained by I S = I up -I dn and I C = I up + I dn , respectively. To compute the other spin-dependent thermoelectric quantities, an intermediate function can be defined as follows 71 : where E F and T are the Fermi Energy and temperature values, respectively. Assuming a linear response regime, ΔT ∼ = 0, the other spin-dependent thermoelectric parameters such as the spin-dependent thermopower, electrical conductance, and the electron's contribution to the thermal conductance can be calculated 72 . Based on Eq. (11), the spin-dependent Seebeck coefficient can then be calculated as follows: where S S = S up -S dn and S C = (S up + S dn )/2 are the spin and charge Seebeck coefficients, respectively 72 .

Results and discussion
In this section, the results of the current study are presented and discussed. For this purpose, Eqs. (8) and (10), along with the intermediate function, have been used to evaluate the obtained results. A hybrid nanoribbon system is first defined and then subjected to different local external fields, including the perpendicular electric (E Z ) and ferromagnetic exchange (M z ) fields, which are applied to the central region (see Fig. 1). The application of the local external electric 73 and ferromagnetic exchange fields are now experimentally feasible. For example, a ferromagnetic exchange field can be created by the proximity with a ferromagnetic insulator EuO as suggested for GE 74 . An inhomogeneous transverse electric field with the magnitude of E yS and E yG are also applied to the device, the left and right leads, respectively 75 . The electric current obtained by the temperature difference (ΔT) without any external bias voltage, i.e., the difference between the temperature of the left, T L , and the right, T R , leads are calculated.
Spin-dependent seebeck effect. In order to study the thermal spin transport properties of the considered hybrid GSNRs, the values of the spin-up (I up ) and spin-down (I dn ) currents are determined as a function of T R for three different temperature gradients ΔT = 10, 20, and 40 K (see Fig. 2). The values of M z = 0.181 eV, E z = 0.081 V/Å, E yS = 0.127 V/Å, and E yG = 0.913 V/Å are selected, and a positive spin-up and negative spin-down currents are evidently observed. These values have been selected based on the previous studies (e.g., [76][77][78][79], and it is believed that they can provide relatively large symmetric spin currents. A nearly perfect SDSE is identified in this hybrid GSNRs 14,15,80 . Because the spin-up and spin-down currents have only produced due to the temperature www.nature.com/scientificreports/ gradient, and they flow in opposite directions with almost equal magnitudes. The GSNR evidently shows a good insulating behavior without any charge or spin current for the low-temperature values. As shown in Fig. 2(a), the threshold temperatures of T th = 60, 50, and 40 K are almost obtained for ΔT = 10, 20, and 40 K, respectively; where the spin-up and spin-down currents are almost zero for T R < T th , whereas the spin currents proportionally increase with respect to T R for T R > T th . The spin currents also increase in opposite directions as ∆T increases (see Fig. 2b) such that they vary linearly and symmetrically respect to the zero-current axis for the entire range of ∆T. The SDSE is further confirmed by all these issues. It is worth mentioning that the studied hybrid GSNRs can significantly provide larger thermally-induced spin currents compared to those given by the pristine AGNRs and ZSNRs.
In order to well understand the SDSE phenomenon in the considered hybrid GSNRs, the Landauer Büttiker formula (Eq. 10) is studied in detail. As can be seen from Eq. (10), the transport coefficients and the difference between the Fermi-Dirac distributions of the left (f L ) and right (f R ) electrodes provide the two factors affecting the thermal spin-dependent currents. Because the leads are composed of similar material and density of states, the difference in carrier concentrations between these two leads is due to the temperature difference and is calculated using the Fermi-Dirac distribution. It is noted that f L − f R is an odd function around the Fermi level, and thus the sign of the current function is determined by the slope of the transmission coefficients for this region. f L − f R is almost equal to zero for the higher energy values (see the inset of Fig. 3d) 10 . As the carriers with the energies www.nature.com/scientificreports/ higher (lower) than the Fermi level flow from the left (right) lead towards the right (left) lead, the electron, I e < 0 (hole, I h > 0) currents are generated, respectively. A net zero thermal current is obtained when the transmission spectrum is symmetric, and I e and I h currents neutralize each other. The variations of the spin-dependent transmission coefficients (T up and T dn ) versus the energy E−E F around the Fermi level (here E F is set to zero) are shown in Fig. 3(a). As shown in Fig. 3(a), because the spin-up and spin-down transmission spectra have been located below and above the Fermi level, I up and I dn will also have opposite signs. Two narrow transmission bands for the spin-up and spin-down electrons almost occur within the range of − 0.2 eV < E-E F < − 0.04 eV and 0.04 eV < E − E F < 0.2 eV, respectively (see Fig. 3a). It is worth mentioning that the electron-hole symmetry is often disturbed by these transmission peaks and result in a nonzero net spin current 81 . For further explanation, the peak value of the spin-down electron transmission has been located above the Fermi level, and thus the electrons can be transported from the right lead to the left one. This leads to a negative spin-down current. On the contrary, because the peak value of the spin-up electron transmission occurs below the Fermi level, the transport of holes produces a positive spin-up current from the left lead to the right one (see Fig. 3a). There also exist two peak values for the spin-up and spin-down electrons in the above and below the Fermi level. However, compared to the former peak values, they are very small. Accordingly, a nearly perfect SDSE is observed in the hybrid GSNRs, because the transmission peaks for the spin-up and spin-down electrons are almost symmetric relative to the Fermi level. In this study, a relatively moderate threshold temperature T th (~ 50 K) for each spin current is needed to extend the Fermi distribution and cover the transmission peaks. This causes a nonzero spin current to be produced between the left and right leads. All these issues may be attributed to the fact that the Fermi distribution is exponentially decreased, and the transmission spectrum shows a relatively medium energy gap for the spin-up and spin-down electrons.
To assess the combined effects of external fields on the hybrid GSNRs, the band structures of the leads and central region are also shown in Fig. 3(b),(c). The lowest-energy subbands for each band structure are related to the electron and hole for E − E F > 0 and E − E F < 0, respectively. As shown in Fig. 3(b), the spin-dependent subbands of the leads are almost matched, whereas the lowest-energy subbands of the central region belong to two different spin states (see Fig. 3c). The effects of each external field are (1) the ferromagnetic exchange field (M z = 0.181 eV) causes different spins to shift in opposite directions and lead to a huge spin splitting 74 . In other words, the proximity effect causes the gapless spin edge states are damaged (2) the semiconducting gaps are also generated between the hole and electron subbands when a perpendicular electric field (E z = 0.081 V/Å) is applied 41 . This is due to the fact that the inversion symmetry is broken by the staggered sublattice potential (3) the transverse electric field (E yS = 0.127 V/Å and E yG = 0.913 V/Å) produces the gaps between the hole and electron subbands, and causes the electron states to shift to the lower energies, whereas the hole states shift to the higher ones 82 . The band structure shown in Fig. 3(b),(c) is obtained when all these fields are applied simultaneously. Figure 3(d)-(f) show the current spectrum J = T(f L − f R ) of the spin currents, plotted for different temperature sets. The area limited to the current spectrum curve and the energy axis (E − E F ) defines the spin current (i.e., I up and I dn ). As shown in Fig. 3(d)-(f), the spin-up current spectra are relatively symmetric with respect to the spin-down ones about the Fermi level and with almost equal areas for the entire different temperature sets. A comparison between the exact areas obtained from the current spectrum (J) for the spin-up and spin-down clearly shows that there is a very small difference between the spin-up and spin-down areas in each plot. This again confirms the nearly perfect SDSE. For I up , the peak value of the current spectrum at ∆T = 20 K is larger than that of ∆T = 10 K when T R is assumed equal to 350 K. This clearly shows that the spin current grows as ∆T increases. Nonetheless, the area under the J curves for T R = 300 K is smaller than that of T R = 350 K when ∆T is assumed to be 20 K. This indicates that the spin currents grow with the increase of T R . Because, there is a bandgap in the transmission spectrum, a zero J value is also observed around the Fermi level.
The thermal-driven net spin currents, I S (= I up − I dn ), and the total charge currents, I C (= I up + I dn ), are also calculated. In Fig. 4, I S and I C are plotted versus T R and ∆T. As shown in Fig. 4(a),(c), I S value is generally increased as T R or ∆T values are increased. For example, I S value is almost 70, 86, and 100 times larger than those of I C for T R = 400 K and ∆T = 10, 20, and 40 K, respectively, which are much greater than those reported in 10,83 . Thus, the carrier transport through the hybrid nanostructure is controlled by the spin current.I C also shows some interesting transport properties when T R or ∆T increases. For example, I C is zero for T R < T th when ∆T is equal to 20 K (see Fig. 4c). I C also drops to the negative values when T R is larger than T th . This indicates the appearance of a thermoelectric switcher. As T R is further increased, I C reaches to its peak value where the negative differential thermal resistance emerges 84 . Indeed, the NDTR occurs due to the competition between I up and I dn with opposite flowing directions. I C also decreases to a zero value as T R increases to a critical temperature value of T R = 358 K. This clearly confirms the emergence of the thermal-induced pure spin current. The flowing direction of I C changes for T R > 358 K, because its sign gets reverse. The variations of I C versus ∆T are also computed and plotted in Fig. 4(d). As shown in Fig. 4(d), I C is negative for the smaller values of T R = 200 and 250 K, and different values of ∆T, whereas I C is almost equal to zero for the relatively large values of T R = 350 K. The observed behaviors clearly confirm that the current of the hybrid GSNRs is appropriate for different device applications by selecting various device temperature sets.
Thermal spin-filtering effect. The thermal SFE without any threshold temperatures in the hybrid GSNRs is studied in this subsection. The effects of an electric field of E z = 0.030 V/Å and an exchange field of M z = 0.157 eV, applied perpendicularly to the central region of the hybrid GSNR are also investigated. The inhomogeneous transverse electric fields equal to E yS = 0.575 V/Å and E yG = 0.029 V/Å are also considered. Figure 5(a) shows the variations of the thermally-induced currents versus T R for different values of ΔT. As shown in Fig. 5(a), larger values of I up are obtained for the high temperatures, whereas I dn is zero for the entire range of temperature as T R increases. The nearly perfect thermal SFE is valid 11,12,83 , and this evidently illustrates that the spin-up www.nature.com/scientificreports/ transport channels are opened, whereas the spin-down transport channels always remain close. For example, I up reaches to its maximum value and is then decreased to zero at a critical temperature value of T R = 172 K and ΔT = 20 K. However, I up sign is reverse and the flowing direction is changed for T R > 172 K. It is noted that when the I up is maximum, the NDTR also reaches its peak value. Again, the NDTR occurs for the larger values of temperature and ΔT. As a result, the hybrid GSNR can be used as a thermal spin device with various multiple attributes. Figure 5(b) also shows the variations of I up and I dn versus ΔT for different values of T R = 200, 250, and 350 K. As shown in Fig. 5(b), I up increases when T R and ΔT are increased, whereas I dn remains almost zero for different values of ΔT. This further confirms that the SFE has been produced. The spin-dependent transmission spectrum is also displayed in Fig. 6(a). As shown in Fig. 6(a), the first peak value of the spin-up transmission occurs in the energy values ranging from 0 to 0.03 eV, whereas the second peak value occurs in the energy values ranging from − 0.03 to − 0.13 eV. Hence, the spin-up dominates the transport  www.nature.com/scientificreports/ properties and yields a nearly perfect SFE. However, the spin-down transmission is almost zero within this range of energies. Like in SDSE case, the combined effects of external fields on the hybrid GSNRs are also evaluated for the SFE case and similar results are obtained. Figure 6(b),(c) shows the band structures of the leads and central region for the considered hybrid GSNRs, respectively. As shown in Fig. 6(b), the spin-dependent subbands of the leads are almost matched, whereas the lowest-energy subbands of the central region belong to two different spin states (see Fig. 6c). As can be seen from Fig. 6(c), the ferromagnetic exchange field (M z = 0.157 eV) causes different spins to shift in opposite directions. The semiconducting gaps are also generated between the hole and electron subbands when a perpendicular and/or transverse electric fields (E z = 0.030 V/Å, E yS = 0.575 V/Å and E yG = 0.029 V/Å) are applied. However, the transverse electric field causes the electron states to shift to the lower www.nature.com/scientificreports/ energies, whereas the hole states shift to the higher ones. It is noted that the spin splitting induced by the external electric field is very small and negligible, as shown in inset of Fig. 6(b). Figure 6(d),(f) show the current spectrum of the spin currents, plotted for different temperature sets. A comparison between the current spectrum of spin-up and spin-down clearly shows that the spin-down spectra are almost zero. However, the spin-up current spectra are dominant in all cases. This again confirms the nearly perfect SFE. For I up , the peak value of the current spectrum at ∆T = 20 K is larger than that of ∆T = 10 K when T R is assumed equal to 350 K. This confirms that the spin current is increased as ∆T increases. The J area for T R = 300 K is smaller than that of T R = 350 K when ∆T is assumed as 20 K. This shows that the increase of T R increases the spin currents. Figure 7 show the variations of spin polarization efficiency, SPE (%) = (|I up | −|I dn |)/ (|I up | +|I dn |) × 100, versus T R and ΔT, respectively. The results clearly show that a high SPE is achieved for the selected T R values. For example, the SPE is almost equal to 100% for the low temperature values, whereas it is measured about 99% for a wide range of T R and ΔT values 12,85 . It is noted that a numerical fluctuation for the T R values ranging from 140 to 190 K is also observed. This basically is related to the reverse sign of I up and the mutual competition between I up and I dn in this range of temperatures, as shown in inset of Fig. 5(a). It is noted that the spin channels of the pristine AGNRs and ZSNRs are partly conductive and the magnitude of the associated SPE is much less than that of the considered hybrid structure.
The effects of central region length. In this subsection, the effects of central region length (L) on the thermally-induced current for the considered hybrid ZGSNRs is studied. The variations of the spin currents (I up and I dn ) versus T R at ∆T = 40 K for M = 7-13, and selected external fields has been shown in Fig. 8(a), in the SDSE case. Indeed, the L parameter proportionally changes with M, in all cases. According to Fig. 8(a), SDSE occurs for M = 8, 9, 12, and 13 for a wide range of T R values around the room temperature (see the filled darker points). However, M = 8 was selected in this research because it can provide a perfect SDSE and stronger spin currents. Figure 8(b) also shows the changes of I S and I C versus M at T R = 300 K and ∆T = 40 K. As shown in Fig. 8(b), I S is two order of magnitude (i.e., they differ by a factor ranging from 10 to 100) larger than I C for some M values (see darker filled points). This again confirms that M = 8 can suitably provide the largest I S value and a stronger SDSE for the selected external fields. Figure 8(c) shows the variations of the spin currents versus T R at ∆T = 40 K for M = 7-13 and the adopted external fields, in the SFE case. It is realized that the I up is relatively enhanced for M = 8 and leads to a larger thermal-induced spin current and a stronger thermal SFE in the hybrid ZGSNRs. Figure 8(d) also provides evidence that the I up (I dn ) is two order of magnitude larger than I dn (I up ) for some M values at T R = 300 K and ∆T = 40 K, in the SFE case (see darker filled points). It is noted that the results reported in this section are all limited to the considered range of parameters and thus they cannot be generalized for various values of external fields.
Thermoelectric performance. The Seebeck coefficients for two different configurations of ferromagnetic exchange and local external electric fields, as discussed above, are studied in this section. Figure 9 shows the variations of the spin-up (S up ) and spin-down (S dn ) Seebeck coefficients, namely, the spin (S S ) and charge (S C ) Seebeck coefficients versus the Fermi energy. The SDSE and SFE are obtained for the hybrid GSNRs for the selected values of exchange and electric fields, as shown in Fig. 9(a),(b), respectively. In order to balance the thermal forces acting on the charge carriers, a larger bias for the lower electric conductance is often required. This consequently produces larger S S values 86,87 . Different behaviors are generally observed for the S up and S dn of the two studied configurations. This is especially true for the E F values ranging from − 0.2 to 0.2 eV. In the SFE case,  www.nature.com/scientificreports/ the peak values of − 3.67 and 3.77 for the S up and S dn are almost obtained for the E F = 0.20 eV and E F = − 0.06 eV values, respectively (see Fig. 9), whereas these values for the SDSE are equal to 1.73 and − 2.0, and occur around the Fermi level. As can be seen from Fig. 9, when S up is equal to zero for some values of the Fermi energy, S dn is nonzero and vice versa. This is related to the fact that the electron and hole currents are canceled out by each other in one spin channel, whereas there is a thermally-induced spin-polarized current in the another one. For example, according to Fig. 9(b), in the SFE case, S up values are almost equal to zero for 0.2 < E F < 0.3 eV, whereas S dn has nonzero values in this range. It is worth mentioning that S up equals to zero at three different points for the E F values ranging from − 0.2 to 0.2 eV, whereas S dn equals to zero at only one point in this region. As shown in Fig. 9, S up and S dn also have equal values with different signs at several E F values. This results in zero S C and nonzero S S values in SDSE and SFE cases. It is noted that S C slightly changes around the S C = 0 line in the SDSE case (see Fig. 9a), whereas in the SFE case, it occurs at some different points (see Fig. 9b). This may be attributed to the fact that the S up and S dn are almost symmetric respect to this line. This shows that ΔT can produce a pure spin current without any charge current. Hence, a zero-charge voltage and a nonzero spin voltage (V S = S S T) are generated by the hybrid GSNRs 88 . The transport is created in such an issue by the spin-up holes and spin-down electrons with similar magnitudes but different current directions. Thus, ΔT can generate a pure spin current and a near-perfect SDSE. It is also observed that S S is almost flat for the larger values of E F . This may be due to the inversely symmetric relationship of the spin-dependent Seebeck coefficients and their linear dependence on the E F . This leads to a constant difference between the S up and S dn . The Seebeck polarization, P S = (|S up |-|S dn |) / (|S up | +|S dn |) is also shown in Fig. 9 (see the thicker colored line) to separately specify the effect of each spin channel on the S S . This is performed by changing the color of S S as the P S value is changed. As shown in Fig. 9, because S up (S dn ) is almost dominant for some values of E F , S S color tends to magenta (turquoise) color in this region. However, because S up and S dn have almost equal values, but with different signs, the S S color, almost tends to blue for the entire E F values. It is worth mentioning that the S up or S dn signs are related to the p-or n-type nature of the device. S S and S C have the same sign when the S up is dominant, whereas their signs are opposite as S dn is dominant. Figure 10 shows the variations of the spin Seebeck (S s ) and spin polarization (P s ) versus E z and M z for different values of the transverse electric fields. Various behaviors can be seen for the S s and P s parameters obtained from four different values of the transverse electric fields. As can be seen from the color bars in Fig. 10, the red and blue colors represent the large and small values for S s and P s , respectively. S s and P s are also considered as odd functions with respect to M z for the entire plots 89 . This results in I S direction can be tuned by changing the magnetization direction in the central region. This shows that I S can magnetically be manipulated. Unlike E z , S s and P s can significantly be changed by varying the exchange field. This further illustrates that S s and P s can magnetically be employed. It is noted that S s and P s are reach their peak values when M z is almost equal to ± 0.1 eV, and with no transvers (or with homogeneous) electric fields. It is noted that these functions are even with respect to the E z when E yG = E yS = 0 ( Fig. 10(a),(b)). The results show that S S increases when the absolute value of E z is increased and E z < 0 (see Fig. 10c), whereas the P s can be large for the entire E z values. However, the S s value can almost be reached to ± 1 for the larger magnitudes of E z . The maximum value of S s almost occurs at M z = 0.1 eV, as illustrated in Fig. 10(c). As shown in Fig. 10, P S and S S reach their peak values for a limited number of areas as the inhomogeneous transverse electric fields are adopted. S S absolute maximum value is increased in SDSE and SFE cases as compared to the two previous cases (see Fig. 10e-h). Figure 11 also shows the variations of the thermal-driven net spin current (I S ) and the total charge current (I C ) versus M z and E z at T R = 358 K and ∆T = 20 K (see Fig. 4c) for the SDSE case values of the transverse electric fields and in the absence of these fields. Different behaviors can be seen for the I S and I C parameters obtained from two plots. As can be seen from the color bars, the red and blue colors represent the large and small values for I S and I C , respectively. I S is also considered as an odd function with respect to M z for the entire plots i.e., I S sign is varied by changing M z sign. It is noted that these functions are even with respect to E z when E yG = E yS = 0 (Fig. 11a,b). A pure spin current can be observed for a limited range of small and large values of M z as I S reaches its peak value, and I C = 0. Figure 11(c),(d) also provides evidence that a pure spin current can be obtained as the M z value is almost equal to 0.181 eV (see also Fig. 4c).

Conclusions
To realize the SDSE and the thermal SFE, the spin-dependent thermoelectric transport properties of hybrid GSNRs, as spin caloritronics devices, have been studied in this research. The effects of the temperature gradient between the left and right leads, ferromagnetic exchange fields, M z , and the local external electric fields, E z and E y , were also included. The results showed that a nearly perfect SDSE could be observed in the hybrid GSNRs. Because the spin-up and spin-down currents are only produced due to the temperature gradient, and they flow in opposite directions with almost equal magnitudes. In the SDSE case, positive spin-up and negative spindown currents with the threshold temperature T th (~ 50 K) were observed for the values of M z = 0.181 eV, E z = 0.081 V/Å, E yS = 0.127 V/Å, and E yG = 0.913 V/Å. Different charge current behaviors have also been obtained by selecting various device temperature sets. A nearly zero charge thermopower was also obtained, which further demonstrates the emergence of the SDSE. In the SFE case, M z = 0.157 eV, E z = 0.030 V/Å, E yS = 0.575 V/Å, and E yG = 0.029 V/Å were considered. Larger values of the spin-up current were obtained for the high temperatures, whereas the spin-down current is zero for the entire range of temperatures as the right lead temperature (T R ) increases. Thus, a nearly perfect SFE was achieved at room temperature, whereas the spin polarization has reached up to 99%. It is noted that due to the competition between the spin-dependent currents, some interesting transport features such as the change of the flowing direction and negative differential thermoelectric resistance were observed. This evidently confirms the potential thermoelectric device applications of the studied hybrid www.nature.com/scientificreports/ www.nature.com/scientificreports/