Introduction

The spin caloritronics has now constructed the subject of many researches in recent years1,2,3. Spin caloritonics combines the spintronics4,5,6 and thermoelectrics to study the interactions among spin and charge in the presence of temperature bias. It can also provide many different approaches for the thermoelectric waste heat recovery, future information, and device technologies7,8,9. The spin caloritronics usually show different innovative effects such as the SDSE10, spin Seebeck diode (SSD) effect11, thermal SFE12, and the thermal giant magnetoresistance phenomenon13. According to the experimental results, the spin Seebeck effect (SSE) initially showed an interaction between the spin and heat currents in the magnetic metals, simultaneously14. Thus, there is SSE in different magnetic phases such as ferromagnetic metals, semiconductors, and insulators15,16,17, paramagnetic and antiferromagnetic materials18,19, and nonmagnetic materials if a magnetic field is applied20. It is worth mentioning that SDSE and SSE phenomena are both induced due to the interaction between the spin and heat currents. However, the magnons are the carriers in SSE, whereas they are the electrons in SDSE.

In SDSE, the temperature bias can generate a spin current, as an essential physical quantity in spintronics, and provide an efficient way for controlling the electron spins in the presence of the temperature gradient. Since the electron current is often accompanied by Joule heating, the spin current in the low-power-consumption nanodevices is required to be used while the electron current is decreased as much as possible. This is now achievable by SDSE. As shown in the previous studies21, the spin current may be produced by the opposite current direction for different spins. An almost non-dissipative SDSE can be obtained if the electrons in two spin channels, spin-up and spin-down flow in opposite directions with equal values, which is called the perfect SDSE22. It is noted that an essential factor in producing the spin current using a temperature gradient is to find a suitable spin-thermoelectric material, which can keep the spin thermopower with opposite signs for different spins23.

The Graphene (GE), as the first isolated two-dimensional (2D) material, is composed of a honeycomb structure of carbon atoms, and have widely become popular among researchers. GE often shows different thermal24,25, mechanical26,27 and optical28 properties. However, GE has uncommon transport properties, which are related to its unusual electronic structure29,30. This is especially true around the Fermi level, where the charge carriers behave similar to massless particles. Because of the long spin relaxation time and length properties of the GE, it is identified as a promising candidate for the future nanoelectronic and spintronic applications31. The magnetic and thermoelectric properties of GE have also attracted great interest in recent years. Some of the applications of this material have already been proposed and discussed in32,33.

Owing to successful studies of GE, some serious attempts have been made by researchers to develop new forms of low-dimensional materials. More recently, Silicene (SE), a hexagonal atomic structure with silicon atoms two-dimensionally bonded together, was first theoretically developed in the literature34. SE was then fabricated by depositing Si on Ag35, and Ir36 surfaces in the laboratory environment. The structural stability of the SE was then confirmed by other researchers using the phonon spectra calculations37. SE and GE have similar electronic structures near the Fermi level and hence can result in the massless Dirac fermions38. This concept is now widely used for developing the high-performance field-effect transistors39. One of the properties of SE is that it has a larger bandgap, induced by the spin–orbit interaction (SOI). This establishes the quantum spin Hall effect40, and has a significant role in spin transport and spintronic devices. The bandgap in SE can be opened and controlled by applying an external electric field normal to the atomic plane41. However, GE has not such a property. There is also the interaction between the electromagnetic field and spin–orbit coupling in the SE, a feature that can be used to study physics in quantum phase transition42. More recently, several studies have been carried out by researchers to investigate the charge and spin thermal transport properties of the SE43,44,45.

As shown in the previous studies (e.g.,46), the SDSE can emerge in the graphene-based nanodevices. However, because GE shows an extremely low figure of merit47, it was introduced as an inefficient thermoelectric material in the next researches48. Hence, GE rarely shows a perfect SDSE. On the other hand, the SE may be a more promising material than GE in thermoelectrics. The previous works showed that SE could significantly improve the Seebeck coefficient49 due to its nonzero energy gap. SE also has some special properties in thermoelectric. For instance, the spin-dependent thermoelectric transport properties of the zigzag SE nanoribbons (ZSNRs) have been studied. The results showed that these ZSNRs could show a high spin-filter efficiency50. Another type of ZSNRs was then studied10 and a perfect SDSE has been observed. The theoretical studies showed that the quasi-one-dimensional wire can provide more significant thermoelectric properties than those of 2D structures51. This may be related to the significant changes in the electronic and thermal properties of the material. It is worth mentioning that the thermoelectric properties of the nanostructures can be further improved by different modifications such as hybridization52, doping53, absorption54, and so on. The past theoretical studies showed that the hybrid nanostructures such as hybrid MoS2/WS255 and BN/GE nanoribbons56 have higher thermoelectric properties than single nanostructures. In recent years, lateral and vertical GE/SE heterojunctions have theoretically57,58,59 and experimentally60,61 been studied. The results showed that the hybrid systems composed of GE and a different 2D material62,63 (e.g., SE) could remarkably help to produce a structural type with different properties and applications. However, there is now lack of data about the spin transport, electronic behavior, and thermoelectric properties of the hybrid GSNRs, and further studies are still required in this field.

In this paper, the electronic and thermal spin-dependent transport properties of the hybrid GSNRs are numerically investigated. Besides, in this study, the tight-binding (TB) approach based on the nonequilibrium Green’s function (NEGF) method is used. The effects of ferromagnetic exchange field, together with the local external electric fields, are also included. A practical way is then proposed to achieve the SDSE, SFE, and negative differential thermoelectric resistance (NDTR) in the hybrid GSNRs. The main findings of the current study are reported and discussed in two separate sections, including (1) SDSE and (2) SFE. In the SDSE case, the spin-up and spin-down currents with almost equal magnitudes are produced in opposite directions (i.e., nearly perfect SDSE) if a temperature gradient is created between the left and right leads. It is noted that because the spin-polarized conduction electrons are the carriers, thus the results reported in this research are limited to SDSE the phenomenon. A nearly pure spin with very small charge thermopower is also observed. However, the spin polarization has reached up to 99% in the SFE case. It is noted that because of the competition between the spin-dependent currents, some interesting transport features such as the change of the flowing direction and NDTR is observed. This evidently confirms its potential thermoelectric device applications by selecting various device temperature sets.

Models and theoretical calculations

Device structure

The SE atoms have an in-plane distance of 2.30 Å, which is much larger than the bond length of GE, i.e., 1.42 Å. In addition, the next-nearest-neighbor distance in the GE is equal to 2.46 Å, which is almost 7% larger than the in-plane atomic distance of SE. Thus, the construction of a lattice-matched one-dimensional interface between the GE and SE is impossible along the same chirality58. On the other hand, there is a small out-of-plane buckling in the SE, and thus it is not located in a plane42. To consider this buckling feature, the Si atoms in the GE/SE nanocomposite are first oriented in an out-of-plane mode with two parallel sublattices A and B vertically separated by a distance of 0.46 Å. The interface is then established via extending the SE by 7% 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 Å57. 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 region64. 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 NS = 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 NG = 46 atoms.

Figure 1
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 NS =16 atoms; each unit cell of the leads also has NG = 46 atoms, (b) the external electric and ferromagnetic exchange fields applied to the system, where TL and TR define the temperature of the left and right leads, respectively.

Full size image

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:

$${H}_{\mathrm{T}}= {H}_{\mathrm{L}}+ {H}_{\mathrm{R}}+ {H}_{\mathrm{C}}+ {H}_{\mathrm{CL}}+ {H}_{\mathrm{CR}},$$
(1)

in which, HR(L) and HC represent the Hamiltonian for the isolated right (left) lead and central region, respectively. Using the TB approximation model40,42,65, the Hamiltonian of the right (left) lead and the central region are obtained as follows:

$${H}_{\mathrm{R}(\mathrm{L})}=-{t}_{\mathrm{R}(\mathrm{L})}\sum_{\langle i,j\rangle,\alpha }{c}_{i\alpha }^{\dagger}{c}_{j\alpha }+e{E}_{y\mathrm{G}} \sum_{i,\alpha }{y}_{i\mathrm{G}}{c}_{i\alpha }^{\dagger}{c}_{i\alpha }+\mathrm{H}.\mathrm{c}.,$$
(2)
$${H}_{\mathrm{C}}=-{t}_{\mathrm{C}}\sum_{\langle i,j\rangle,\alpha }{c}_{i\alpha }^{\dagger}{c}_{j\alpha }+i\frac{{\lambda }_{so}}{3\sqrt{3}}\sum_{\ll i,j\gg ,\alpha ,\beta }{\nu }_{ij}{c}_{i\alpha }^{\dagger}{{{(\sigma }_{z})}_{\alpha \beta }c}_{j\beta }+{M}_{z} \sum_{i,\alpha }{c}_{i\alpha }^{\dagger}{\sigma }_{z}{c}_{i\alpha }+ e{lE}_{z} \sum_{i,\alpha }{\xi }_{i}{c}_{i\alpha }^{\dagger}{c}_{i\alpha }+e{E}_{y\mathrm{S}} \sum_{i,\alpha }{y}_{i\mathrm{S}}{c}_{i\alpha }^{\dagger}{c}_{i\alpha }+\mathrm{H}.\mathrm{c}.,$$
(3)

where t R (L) and tC 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. \({c}_{i\alpha }^{\dagger}({c}_{i\alpha })\) creates (annihilates) an electron with spin α at atom i. \(\left\langle {i,j} \right\rangle\) and \(\left\langle {\left\langle {i,j} \right\rangle } \right\rangle\) represent the nearest-neighbor and next-nearest-neighbor pairs, respectively. λso is the effective SOI parameter and is equal to 3.9 meV for the central region42. Due to the weak SOI in the GE66, the values of λso and λr are assumed equal to zero for the left and right leads. σ = (σx, σy, σz) is the Pauli matrix. \({\upsilon }_{ij}\) is defined as \({\upsilon }_{ij}=({{\varvec{d}}}_{i}\times {{\varvec{d}}}_{j})/{|{\varvec{d}}}_{i}\times {{\varvec{d}}}_{j}|\) where \({{\varvec{d}}}_{i}\) and \({{\varvec{d}}}_{j}\) are the two nearest bonds connecting the next-nearest-neighbors. \({\xi }_{i}=+1,-1\) for the sublattices A and B, respectively; and 2 l is the buckling distance for the SE. Ez is also the perpendicular electric field and produces a voltage difference of 2elEz 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. Mz is the ferromagnetic exchange field, which can be produced by the proximity effect of a ferromagnetic material42. EyG and EyS are the inhomogeneous transverse electric field components along the x direction, applied to the leads and central region, respectively. yiG 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

$${H}_{\mathrm{CR}(\mathrm{CL})}=-{t}_{\mathrm{CR}(\mathrm{CL})}\sum_{\langle i,j\rangle,\alpha }{c}_{i\alpha }^{\dagger}{c}_{j\alpha }+\mathrm{H}.\mathrm{c}.,$$
(4)

where tCR(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 law67,68. 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 tCR(CL) parameter, which is required to be the same order of tR(L) and tC55. 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 spin-flip 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 micrometers29,69.

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 follows70:

$${G}_{\mathrm{C},\mathrm{\alpha }}={\left[\left(E+\mathrm{i}{0}^{+}\right)\mathrm{I}-{H}_{\mathrm{C}}-{\sum }_{\mathrm{L},\mathrm{\alpha }} -{\sum }_{\mathrm{R},\mathrm{\alpha }} \right]}^{-1},$$
(5)

where \({\Sigma }_{\mathrm{R}\left(\mathrm{L}\right),\mathrm{\alpha }}\) is the right (left) self-energy matrix and is computed by Eq. (6). \({\Sigma }_{\mathrm{R}\left(\mathrm{L}\right),\mathrm{\alpha }}\) includes the effect of two semi-infinite AGNRs on the central region.

$${\sum }_{\mathrm{R},\alpha } = {H}_{\mathrm{CR}} {g}_{\mathrm{R},\alpha }(E) {H}_{\mathrm{CR}}^{\dagger},$$
(6a)
$${\sum }_{\mathrm{L},\alpha } = {H}_{\mathrm{LC}}^{\dagger} {g}_{\mathrm{L},\alpha }(E) {H}_{\mathrm{LC}},$$
(6b)

in which, gR(L),α is the surface Green’s function of the right (left) lead and is computed by

$${g}_{\mathrm{R}\left(\mathrm{L}\right),\alpha }(E) ={\left[\left(E+\mathrm{i}{0}^{+}\right)\mathrm{I}-{H}_{\mathrm{R}\left(\mathrm{L}\right)}^{00}-{H}_{\mathrm{R}(\mathrm{L})}^{01({\dagger})}{T}_{\mathrm{R}(\mathrm{L})}\right]}^{-1},$$
(7)

where \({H}_{\mathrm{R}(\mathrm{L})}^{00}\) defines the TB Hamiltonian matrix of the unit cell studied in this research for the right (left) lead. \({H}_{\mathrm{R}(\mathrm{L})}^{01({\dagger})}\) is the hopping matrix between two adjacent unit cells in the right (left) lead. \({T}_{\mathrm{R}(\mathrm{L})}\) is the transfer matrix of the right (left) lead and can be determined using an iterative methodology, as proposed in Reference70. The electron transmission function can then be obtained using the following equations70:

$${T}_{\alpha }\left(E\right)={Tr}\left[{\Gamma }_{\mathrm{L},\alpha }\left(E\right){\mathrm{G}}_{\mathrm{C},\alpha }\left(E\right){\Gamma }_{\mathrm{R},\alpha }\left(E\right){G}_{\mathrm{C},\alpha }^{\dagger}\left(E\right)\right],$$
(8)

where \({\Gamma }_{\mathrm{R}\left(\mathrm{L}\right),\alpha }\) is the broadening function and describes the coupling between the right (left) lead with the central region.\({\Gamma }_{\mathrm{R}\left(\mathrm{L}\right),\alpha }\) is given by

$${\Gamma }_{\mathrm{R}(\mathrm{L}),\alpha }=\mathrm{i}\left({\sum }_{R\left(L\right),\alpha } -{\sum }_{R\left(L\right),\alpha }^{\dagger} \right).$$
(9)

Spin-dependent thermoelectric

The thermally-induced spin-dependent electric current can be computed by using the Landauer-Büttiker expression, as follows71:

$${I}_{\alpha }=\frac{e}{h}{\int }_{-\infty }^{+\infty }{T}_{\alpha }\left(E\right)[{f}_{\mathrm{L}}\left(E,{T}_{\mathrm{L}}\right)-{f}_{\mathrm{R}}\left(E,{T}_{\mathrm{R}}\right)]dE,$$
(10)

where f R(L) is the Fermi–Dirac distribution. e, h, and TR(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 = TL—TR between the leads produces a nonzero value of fL—fR, 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 IS = Iup – Idn and IC = Iup + Idn, respectively. To compute the other spin-dependent thermoelectric quantities, an intermediate function can be defined as follows71:

$${L}_{n,\alpha }\left({E}_{\mathrm{F}},\mathrm{T}\right)=-\frac{1}{h}\int {\left(E-{E}_{\mathrm{F}}\right)}^{n}\frac{\partial f\left(E,{E}_{\mathrm{F}},T\right)}{\partial E}{T}_{\alpha }\left(E\right)dE,$$
(11)

where EF and T are the Fermi Energy and temperature values, respectively. Assuming a linear response regime, ΔT \(\cong\) 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 calculated72. Based on Eq. (11), the spin-dependent Seebeck coefficient can then be calculated as follows:

$${S}_{\alpha }=-\frac{1}{eT}\left(\frac{{L}_{1\alpha }}{{L}_{0\alpha }}\right).$$
(12)

where SS = Sup—Sdn and SC = (Sup + Sdn)/2 are the spin and charge Seebeck coefficients, respectively72.

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 (EZ) and ferromagnetic exchange (Mz) fields, which are applied to the central region (see Fig. 1). The application of the local external electric73 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 GE74. An inhomogeneous transverse electric field with the magnitude of EyS and EyG are also applied to the device, the left and right leads, respectively75. The electric current obtained by the temperature difference (ΔT) without any external bias voltage, i.e., the difference between the temperature of the left, TL, and the right, TR, 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 (Iup) and spin-down (Idn) currents are determined as a function of TR for three different temperature gradients ΔT = 10, 20, and 40 K (see Fig. 2). The values of Mz = 0.181 eV, Ez = 0.081 V/Å, EyS = 0.127 V/Å, and EyG = 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 GSNRs14,15,80. Because the spin-up and spin-down currents have only produced due to the temperature 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 Tth = 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 TR < Tth, whereas the spin currents proportionally increase with respect to TR for TR > Tth. 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.

Figure 2
figure 2

The thermally-induced spin currents for the considered hybrid GSNRs (a) the variation of the spin currents versus the right lead temperature, TR, for different temperature gradients, ∆T = 10, 20, and 40 K. The spin-up (Iup) and spin-down (Idn) currents flow in positive and negative directions, respectively (i.e., SSE) (b) the variation of the spin currents versus ∆T for the right lead temperatures TR = 200, 250, and 350 K.

Full size image

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 (fL) and right (fR) 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 fL − fR 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. fL − fR is almost equal to zero for the higher energy values (see the inset of Fig. 3d)10. As the carriers with the energies higher (lower) than the Fermi level flow from the left (right) lead towards the right (left) lead, the electron, Ie < 0 (hole, Ih > 0) currents are generated, respectively. A net zero thermal current is obtained when the transmission spectrum is symmetric, and Ie and Ih currents neutralize each other. The variations of the spin-dependent transmission coefficients (Tup and Tdn) versus the energy E−EF around the Fermi level (here EF 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, Iup and Idn will also have opposite signs.

Figure 3
figure 3

(a) The spin-dependent transmission spectra (Tup and Tdn) versus E − EF, (b) the band structures of the right (left) lead with an applied electric field of EyG = 0.913 V/Å, (c) the band structures of the ferromagnetic central region including an exchange field of Mz = 0.181 eV, electric fields of EZ = 0.081 V/Å and EyS = 0.127 V/Å, and the variation of the spin-dependent current spectrum J(E) versus E − EF for (d) TR = 350 and ∆T = 10 K, (inset shows the difference between the Fermi–Dirac distributions for the left and right leads, fL(E)-fR(E), as a function of E − EF), (e) TR = 350 and ∆T = 20 K, and (f) TR = 300 and ∆T = 20 K.

Full size image

Two narrow transmission bands for the spin-up and spin-down electrons almost occur within the range of − 0.2 eV < E–EF <  − 0.04 eV and 0.04 eV < E − EF < 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 current81. 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 Tth (~ 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 − EF > 0 and E − EF < 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 (Mz = 0.181 eV) causes different spins to shift in opposite directions and lead to a huge spin splitting74. 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 (Ez = 0.081 V/Å) is applied41. This is due to the fact that the inversion symmetry is broken by the staggered sublattice potential (3) the transverse electric field (EyS = 0.127 V/Å and EyG = 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 ones82. 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(fL − fR) of the spin currents, plotted for different temperature sets. The area limited to the current spectrum curve and the energy axis (E − EF) defines the spin current (i.e., Iup and Idn). 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 Iup, the peak value of the current spectrum at ∆T = 20 K is larger than that of ∆T = 10 K when TR 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 TR = 300 K is smaller than that of TR = 350 K when ∆T is assumed to be 20 K. This indicates that the spin currents grow with the increase of TR. 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, IS (= Iup − Idn), and the total charge currents, IC (= Iup + Idn), are also calculated. In Fig. 4, IS and IC are plotted versus TR and ∆T. As shown in Fig. 4(a),(c), IS value is generally increased as TR or ∆T values are increased. For example, IS value is almost 70, 86, and 100 times larger than those of IC for TR = 400 K and ∆T = 10, 20, and 40 K, respectively, which are much greater than those reported in10,83. Thus, the carrier transport through the hybrid nanostructure is controlled by the spin current.IC also shows some interesting transport properties when TR or ∆T increases. For example, IC is zero for TR < Tth when ∆T is equal to 20 K (see Fig. 4c). IC also drops to the negative values when TR is larger than Tth. This indicates the appearance of a thermoelectric switcher. As TR is further increased, IC reaches to its peak value where the negative differential thermal resistance emerges84. Indeed, the NDTR occurs due to the competition between Iup and Idn with opposite flowing directions. IC also decreases to a zero value as TR increases to a critical temperature value of TR = 358 K. This clearly confirms the emergence of the thermal-induced pure spin current. The flowing direction of IC changes for TR > 358 K, because its sign gets reverse. The variations of IC versus ∆T are also computed and plotted in Fig. 4(d). As shown in Fig. 4(d), IC is negative for the smaller values of TR = 200 and 250 K, and different values of ∆T, whereas IC is almost equal to zero for the relatively large values of TR = 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.

Figure 4
figure 4

(a) The variation of the net spin current (IS = Iup − Idn) versus TR for ∆T = 10, 20, and 40 K, (b) the variation of IS versus ∆T for TR = 200, 250, and 350 K, (c) the total electron current (IC = Iup + Idn) as a function of TR for ∆T = 10, 20, and 40 K, and (d) the variation of IC versus ∆T for TR = 200, 250, and 350 K.

Full size image

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 Ez = 0.030 V/Å and an exchange field of Mz = 0.157 eV, applied perpendicularly to the central region of the hybrid GSNR are also investigated. The inhomogeneous transverse electric fields equal to EyS = 0.575 V/Å and EyG = 0.029 V/Å are also considered. Figure 5(a) shows the variations of the thermally-induced currents versus TR for different values of ΔT. As shown in Fig. 5(a), larger values of Iup are obtained for the high temperatures, whereas Idn is zero for the entire range of temperature as TR increases. The nearly perfect thermal SFE is valid11,12,83, and this evidently illustrates that the spin-up transport channels are opened, whereas the spin-down transport channels always remain close. For example, Iup reaches to its maximum value and is then decreased to zero at a critical temperature value of TR = 172 K and ΔT = 20 K. However, Iup sign is reverse and the flowing direction is changed for TR > 172 K. It is noted that when the Iup 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 Iup and Idn versus ΔT for different values of TR = 200, 250, and 350 K. As shown in Fig. 5(b), Iup increases when TR and ΔT are increased, whereas Idn remains almost zero for different values of ΔT. This further confirms that the SFE has been produced.

Figure 5
figure 5

(a) The variations of the spin currents versus TR for ∆T = 10, 20, 40, the spin-up currents (Iup) have finite values and the spin-down currents (Idn) are nearly zero (i.e., SFE), and (b) the spin currents versus ∆T for TR = 200, 250, and 350 K.

Full size image

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 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 (Mz = 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 (Ez = 0.030 V/Å, EyS = 0.575 V/Å and EyG = 0.029 V/Å) are applied. However, the transverse electric field causes the electron states to shift to the lower 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 Iup, the peak value of the current spectrum at ∆T = 20 K is larger than that of ∆T = 10 K when TR is assumed equal to 350 K. This confirms that the spin current is increased as ∆T increases. The J area for TR = 300 K is smaller than that of TR = 350 K when ∆T is assumed as 20 K. This shows that the increase of TR increases the spin currents.

Figure 6
figure 6

(a) The spin-dependent transmission spectra (Tup and Tdn) versus E − EF, (b) the band structures of the right (left) lead with an applied electric field of EyG = 0.029 V/Å (inset shows the spin splitting induced by the electric field is very small and negligible) (c) the band structures of the ferromagnetic central region including an exchange field of Mz = 0.157 eV, electric fields of EZ = 0.030 V/Å, and EyS = 0.575 V/Å, and the variation of the spin-dependent current spectrum J(E) for (d) TR = 350 and ∆T = 10 K, (e) TR = 350 and ∆T = 20 K, and (f) TR = 300 and ∆T = 20 K.

Full size image

Figure 7 show the variations of spin polarization efficiency, SPE (%) = (|Iup| −|Idn|)/ (|Iup| +|Idn|) × 100, versus TR and ΔT, respectively. The results clearly show that a high SPE is achieved for the selected TR 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 TR and ΔT values12,85. It is noted that a numerical fluctuation for the TR values ranging from 140 to 190 K is also observed. This basically is related to the reverse sign of Iup and the mutual competition between Iup and Idn 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.

Figure 7
figure 7

(a) The variation of the spin polarization efficiency (SPE) versus TR for ∆T = 10, 20, and 40 K (b) the SPE variation versus ΔT for TR = 200, 250, and 350 K.

Full size image

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 (Iup and Idn) versus TR 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 TR 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 IS and IC versus M at TR = 300 K and ∆T = 40 K. As shown in Fig. 8(b), IS is two order of magnitude (i.e., they differ by a factor ranging from 10 to 100) larger than IC for some M values (see darker filled points). This again confirms that M = 8 can suitably provide the largest IS value and a stronger SDSE for the selected external fields. Figure 8(c) shows the variations of the spin currents versus TR at ∆T = 40 K for M = 7–13 and the adopted external fields, in the SFE case. It is realized that the Iup 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 Iup (Idn) is two order of magnitude larger than Idn (Iup) for some M values at TR = 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.

Figure 8
figure 8

The variations of (a) Iup and Idn versus TR at ∆T = 40 K for the lengths of M = 7–13; Note that the plots shown in the positive and negative regions of the Y-axis are related to Iup and Idn, respectively, (b) IS and IC versus M at TR = 300 K and ∆T = 40 K, and the selected external fields, in the SSE case, (c) Iup and Idn versus TR at ∆T = 40 K for M = 7–13; Note that the plots on the X axis are related to Idn, and the others to the Iup (d) Iup and Idn versus M at TR = 300 K and ∆T = 40 K, and the adopted external fields, in the SFE case.

Full size image

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 (Sup) and spin-down (Sdn) Seebeck coefficients, namely, the spin (SS) and charge (SC) 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 SS values86,87. Different behaviors are generally observed for the Sup and Sdn of the two studied configurations. This is especially true for the EF values ranging from − 0.2 to 0.2 eV. In the SFE case, the peak values of − 3.67 and 3.77 for the Sup and Sdn are almost obtained for the EF = 0.20 eV and EF = − 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 Sup is equal to zero for some values of the Fermi energy, Sdn 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, Sup values are almost equal to zero for 0.2 < EF < 0.3 eV, whereas Sdn has nonzero values in this range. It is worth mentioning that Sup equals to zero at three different points for the EF values ranging from − 0.2 to 0.2 eV, whereas Sdn equals to zero at only one point in this region. As shown in Fig. 9, Sup and Sdn also have equal values with different signs at several EF values. This results in zero SC and nonzero SS values in SDSE and SFE cases. It is noted that SC slightly changes around the SC = 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 Sup and Sdn 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 (VS = SS T) are generated by the hybrid GSNRs88. 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 SS is almost flat for the larger values of EF. This may be due to the inversely symmetric relationship of the spin-dependent Seebeck coefficients and their linear dependence on the EF. This leads to a constant difference between the Sup and Sdn. The Seebeck polarization, PS = (|Sup|—|Sdn|) / (|Sup| +|Sdn|) is also shown in Fig. 9 (see the thicker colored line) to separately specify the effect of each spin channel on the SS. This is performed by changing the color of SS as the PS value is changed. As shown in Fig. 9, because Sup (Sdn) is almost dominant for some values of EF, SS color tends to magenta (turquoise) color in this region. However, because Sup and Sdn have almost equal values, but with different signs, the SS color, almost tends to blue for the entire EF values. It is worth mentioning that the Sup or Sdn signs are related to the p- or n-type nature of the device. SS and SC have the same sign when the Sup is dominant, whereas their signs are opposite as Sdn is dominant.

Figure 9
figure 9

The variations of the spin-up (Sup), spin-down (Sdn), spin (Ss) and charge (Sc) Seebeck coefficients versus the Fermi energy and at room temperature for (a) the SSE and (b) the SFE cases. The SS color changes in accordance with the Seebeck polarization value.

Full size image

Figure 10 shows the variations of the spin Seebeck (Ss) and spin polarization (Ps) versus Ez and Mz for different values of the transverse electric fields. Various behaviors can be seen for the Ss and Ps 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 Ss and Ps, respectively. Ss and Ps are also considered as odd functions with respect to Mz for the entire plots89. This results in IS direction can be tuned by changing the magnetization direction in the central region. This shows that IS can magnetically be manipulated. Unlike Ez, Ss and Ps can significantly be changed by varying the exchange field. This further illustrates that Ss and Ps can magnetically be employed. It is noted that Ss and Ps are reach their peak values when Mz 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 Ez when EyG = EyS = 0 (Fig. 10(a),(b)). The results show that SS increases when the absolute value of Ez is increased and Ez < 0 (see Fig. 10c), whereas the Ps can be large for the entire Ez values. However, the Ss value can almost be reached to ± 1 for the larger magnitudes of Ez. The maximum value of Ss almost occurs at Mz = 0.1 eV, as illustrated in Fig. 10(c). As shown in Fig. 10, PS and SS reach their peak values for a limited number of areas as the inhomogeneous transverse electric fields are adopted. SS absolute maximum value is increased in SDSE and SFE cases as compared to the two previous cases (see Fig. 10e–h).

Figure 10
figure 10

The combined effects of ferromagnetic exchange fields, MZ, and external electric fields (EZ, EyG and EyS) on the spin Seebeck coefficient (SS) and Seebeck polarization (PS) for the considered hybrid GSNR and (a,b) EyG = EyS = 0 (c,d) EyG = EyS = 0.001 V/Å (e,f) EyG = 0.913 V/Å, EyS = 0.127 V/Å (g,h) EyG = 0.029 V/Å, EyS = 0.575 V/Å. The scale of spin Seebeck coefficients in all plots is kB/e.

Full size image

Figure 11 also shows the variations of the thermal-driven net spin current (IS) and the total charge current (IC) versus Mz and Ez at TR = 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 IS and IC 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 IS and IC, respectively. IS is also considered as an odd function with respect to Mz for the entire plots i.e., IS sign is varied by changing Mz sign. It is noted that these functions are even with respect to Ez when EyG = EyS = 0 (Fig. 11a,b). A pure spin current can be observed for a limited range of small and large values of Mz as IS reaches its peak value, and IC = 0. Figure 11(c),(d) also provides evidence that a pure spin current can be obtained as the Mz value is almost equal to 0.181 eV (see also Fig. 4c).

Figure 11
figure 11

The combined effects of ferromagnetic exchange fields, Mz, and external electric fields (Ez and EyG, EyS) on the net spin current (IS) and total electron current (IC) for the considered hybrid GSNR and (a,b) EyG = EyS = 0 (c,d) EyG = 0.913 V/Å, EyS = 0.127 V/Å. The results are for TR = 358 K and ∆T = 20 K. IS and IC have nA units in all plots.

Full size image

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, Mz, and the local external electric fields, Ez and Ey, 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 spin-down currents with the threshold temperature Tth (~ 50 K) were observed for the values of Mz = 0.181 eV, Ez = 0.081 V/Å, EyS = 0.127 V/Å, and EyG = 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, Mz = 0.157 eV, Ez = 0.030 V/Å, EyS = 0.575 V/Å, and EyG = 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 (TR) 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 GSNRs by selecting various device temperature sets. The results reported and discussed in this study are limited to the considered set of parameters and thus they cannot be generalized for various values of external fields.