Spin caloritronics, concentrating on the interaction between the spin and heat currents, is one of the hot topics in condensed matter physics because it plays an essential role in the growth of primary sciences and advanced low-power-consumption technologies (1–6). Recently, Slachter et al. (7) experimentally discovered the spin-dependent Seebeck effect (SDSE) through the heat transfer within the interface between a ferromagnet and a non-magnetic metal. After that, various studies have been reported concerning the SDSE in one-dimensional nanoribbons with armchair or zigzag edges, such as graphene, silicene, black phosphorus and germanene nanoribbons (8–10). For example, Majidi et al. (11) demonstrated that spin-up and spin-down currents flow in reverse orientations with two various threshold temperatures in hydrogen-terminated zigzag-edge germanene nanoribbons with a temperature difference among the source and the drain. Zheng et al. (12) found electric fields to improve spin thermoelectric efficiency of germanene nanoribbon. As we know, germanene honeycomb lattice can be sliced along <1 1 ¯ 10> and <2 1 ¯   1 ¯ 0 > orientations to generate armchair and zigzag/Klein edges, respectively. However, the works performed on the electronic structure and SDSE on zigzag GeNRs (ZGeNRs) with Klein edge are rare. Indeed, these two kinds of reconstructed edges for zigzag graphene nanoribbons (GNRs) have been found through experimenters (13, 14), and their significant impact on the ZGNRs’ band structures and magnetic states have been proved theoretically (15, 16). Therefore, in the current research, ab initio computations are incorporated with the nonequilibrium Green’s function technique to investigate the electronic structures and thermal spin-related transport characteristics of ZKGeNRs. Moreover, it is indicated that the ZKGeNRs can have stable ferromagnetic states, and the SDSE can also be achieved. The obtained results confirm that ZKGeNRs can be utilized in spin caloritronic devices.

Here we turn to introduce the device designs and theoretical method briefly. The N-ZKGeNRs in the present designs have one zigzag edge and one Klein edge (see Figure 1A,B), while two H atoms saturate both edges. N stands for the number of Ge atoms rows through the ZGeNR width, which changes from four to 9. Two probe spin caloritronic devices were then constructed using N-ZKGeNRs (say N = 4), as presented in Figure 1C. The left and right contacts are semi-infinite ZKGeNRs, while the mid scattering area involves five modules of ZKGeNRs. We concentrate on the spin currents generated through a temperature gradient, ΔT, between the left temperature T_L and the right 1 T_R, i.e., ΔT = T _L -T _R .

The computations were accomplished with the Atomistix Toolkit (ATK) package (17, 18), incorporating the spin density functional theory with the nonequilibrium Green’s function strategy. The Double-Zeta-Polarized (DZP) basis set was utilized to accomplish the geometry optimization and electronic structure computations, while the Generalized-Gradient-Approximation method (19, 20) adopts the exchange-correlation potential. The cut-off energy was 120 Hartree, while a Monkhorst–Pack 1 × 1×100 k-mesh was adopted. In the LandauerBüttiker formulation, the spin-related currents of the devices were determined through the upcoming Eq. 21: I ↑ ( ↓ ) = e h ∫ − ∞ ∞ { T ↑ ( ↓ ) ( E ) [ f L ( E , T L ) − f R ( E , T R ) ] } d E (1) where e stands for the electron charge, h describes the Planck constant, and T _L(R) stands for the left (right) electrode’s temperature. f _{L(R) (} E,T _L(R)) indicates the left (right) electrode’s mean Fermi–Dirac distribution: f L ( R ) ( E , T L ( R ) ) = { 1 + exp [ E − μ L ( R ) / k B T L ( R ) ] } − 1 (2) where μ L ( R ) describe the left (right) electrode’s chemical potential, k B stands for the Boltzmann constant, and T ^{↑(↓) (} E) describes the spin-related transport coefficient: T ↑ ( ↓ ) ( E ) = T r [ Γ L G r Γ R G a ] (3) where G ^r(a) stands for the Green’s function retarded in the mid area, while Γ _L(R) describes the left (right) electrode’s coupling matrix. In addition, the calculation methods of the spin figures of merit (Z_sT) can be found in our and others previous studies (22, 23).

At first, the band structures of the N-ZKGeNRs (N = 4–9) were verified, as presented in Figure 2. The nanoribbons’ band structures vary significantly by increasing the value of N. For N = 4, the conduction band minimum (CBM) is related to the spin-down state and located at the Γ point, which is above the Fermi level (E _F ), while the spin-up state below E_F was considered the valence band maximum (VBM) and placed in the line of Γ-Z. The valence and conduction bands have reverse spin polarities while approaching E_F. Furthermore, the spin-related bands of 4-ZKGeNRs are equal to 0.158 eV. These characteristics indicate that the 4-ZKGeNR is an indirect-gap BMS (24–26). The ferromagnetic shape is kept unchanged by increasing the nanoribbon width, while the band structures around the Fermi level vary significantly. For N = five to six, the CBM and VBM of the spin-down and spin-up channels are located at Γ and Γ-Z points, respectively, while the bandgaps are equal to 0.071 and 0.018 eV. Based on Wang and Hu’s works (27–29), the bandgap of about 0.1 eV or lower than 0.1 eV can be described as “gapless”. Therefore, the band structure can be considered gapless, indicating that 5-ZKGeNR and 6-ZKGeNR exhibit a BSGS behavior. For N = seven to nine, the spin-up and spin-down bands are across the E_f. These ZKGeNRs are intrinsically FM. In short, ZKGeNRs can have three states while increasing the width, including indirect-gap BMS state, BSGS state, and FM state.

The transmission spectrums of ZKGeNRs should be verified to identify the induced spin-related current in the presented ZKGeNRs and illustrate the transport carriers’ behavior. Figure 3 presents the spin-related transmission spectrums. It can be observed that all the ZKGeNRs have similar characteristics, except for different transport channels. In the energy interval of [-0.3, 0.3] eV, the transmission coefficient is 2, while there exist peak values for spin-up and spin-down electrons in transmission spectrums. The mentioned peak values are caused by the band structure, as Figure 2 shows. Because the right and left electrodes have similar material and density of state, the Fermi-Dirac distribution (f_R (E, T_R) − f_L (E, T_L)) determines the carrier behavior and concentration using the constructed device, depending on the electrons temperature at two leads. Fermi-Dirac distribution function is described by Eq. 2, in which µ stands for the chemical potential and is chosen as zero in the performed computations. The number of electrons with energies higher than the Fermi level, flow from the hotter electrode (left) to the lower one (right), because the electron distribution of hotter electrode is higher than that of the lower one, causing in a spin-down current. For the same reason, the number of holes with energies lower than the Fermi energy flow from the hotter electrode to the lower one, too, causing the spin-up current. Figure 3 shows that since the transmission for spin-down electrons is more than spin-up ones in a domain of energies greater than the Fermi level, there exists a spin-down negative current for the mentioned domain. Moreover, since the spin-up current’s behavior is precisely opposite to the spin-down one, a spin-up positive current can be generated.

The thermal spin transport features of the proposed ZKGeNRs should be verified to go through their spin-dependent currents. Figure 4 shows the spin-related currents through the N-ZKGeNRs devices versus T _L . For the 4-ZKGeNRs device, there are no spin-up currents (I _up ) when T _L < 100 K and no spin-down currents (I _dn ) when T _L < 120 K for three values of ΔT, demonstrating that no thermal-induced spin-related currents can be produced in these ranges of T _L , while the temperature difference (ΔT) is not important. This means that there exists a threshold temperature Tth at around 100 K for I _up and 120 K for I _dn , respectively. When T _L > Tth, both I _up and I _dn grow significantly with the increase of T _L . Nevertheless, they move in the reverse orientations, i.e., I _up is negative, while I _dn is positive. Certainly, this is induced by the SDSE (10). Moreover, the greater the ΔT, the higher the spin-related currents. The spin-related currents in terms of ΔT curves are presented in Figure 5, for T _L = 300, 350, and 400 K. These curves demonstrate that the spin-related currents are nearly symmetric around the zero-current axis and robust within a wide domain of temperature gradients. This confirms the generation of the SDSE through the spin-related currents in terms of ΔT curves. The devices of 5-ZKGeNRs and 6-ZKGeNRs have the same rules with 4-ZKGeNRs, but have smaller Tth and lager I _up and I _dn, as shown in Figures 4B,C. In addition, as shown in Figures 4D–F, I _dn of N-ZKGeNRs (N = 7–9) devices have similar rules with 4-ZKGeNRs, while Tth for I _up is equal to zero.

Finally, we turn to investigate the thermoelectric conversion efficiency of these ZKGeNRs devices. First, we plot the calculated results of spin thermopower (S_s) versus the chemical potential (μ) at T = 300 K for the ZKGeNRs (N = 4–9) in Figure 6A. It is obvious that the S_s decrease with increasing N for N = four to six, while the S_s are approximately equal for N = 7–9. Next, we plot the calculated results of spin figures of merit (Z_sT) versus the chemical potential (μ) at T = 300 K for the ZKGeNRs (N = 4–9) in Figure 6B. The maximum values related to N-ZKGeNRs are about 47.9, 25.3, 18.7, 9.6, 6.5, and 4.7 for N = four to nine, respectively. Apparently, the 4-ZKGeNRs has the largest value of Z_sT because it has the largest spin thermopower and near-linear increase of thermal conductance due to the increasing the ribbon width (30, 31). Interesting, these values are comparable with the results obtained for devices based on GeNRs (12), much larger than them at room temperature. Similar behavior was also found in sawtooth GNRs (31), edge-defected GNRs (32) and armchair GNRs with triangular antidots (33). Generally, a large Z_sT value supports a higher thermoelectric conversion performance in materials. These results confirm that ZKGeNRs are suitable candidate materials for spin caloritronic devices.

The current research incorporated the first-principles computations with the nonequilibrium Green’s function to evaluate the electronic structures and thermally-induced spin-related transport features of various N-ZKGeNRs. Firstly, it was demonstrated that the N-ZKGeNRs move from an indirect-gap BMS to SGS state and finally to FM state by increasing the nanoribbon width variable N. Secondly, the SDSE could be observed by generating a temperature gradient across the mentioned ZKGeNRs. In addition, the threshold temperature decreases as N increases and then disappears, while the spin currents increase as N increases. Generally, the mentioned discoveries strongly demonstrate the potential of N-ZKGeNRs for application in thermal spin nanodevices.