Valley–spin Seebeck effect in heavy group-IV monolayers

Akin to electron spin, the valley has become another highly valued degree of freedom in modern electronics, specifically after tremendous studies on monolayers of group-IV materials, i.e. graphene, silicene, germanene and stanene. Except for graphene, the other heavy group-IV monolayers have observable intrinsic spin–orbit interactions due to their buckled structures. Distinct from the usual electric or optical control of valley and spin, we here employ a temperature difference to drive electron motion in ferromagnetic heavy group-IV monolayers via designing a caloritronic device locally modulated by an interlayer electric (Ez) field. A unique valley–spin Seebeck (VSS) effect is discovered, with the current contributed only by one (the other) valley and one (the other) spin moving along one (the opposite) direction. This effect is suggested to be detected below the critical temperature about 18 K for silicene, 200 K for germanene and 400 K for stanene, arising from the characteristic valley–spin nondegenerate band structures tuned by the Ez field, but cannot be driven in graphene without spin–orbit interaction. Above the critical temperature, the VSS effect is broken by overlarge temperature broadening. Besides the temperature, it is also found that the Ez field can drive a transition between the VSS effect and the normal spin Seebeck effect. Further calculations indicate that the VSS effect is robust against many realistic perturbations. Our research represents a conceptually but substantially major step towards the study of the Seebeck effect. These findings provide a platform for encoding information simultaneously by the valley and spin quantum numbers of electrons in future thermal-logic circuits and energy-saving devices.


Introduction
The study of Dirac electrons, as inspired by the rise of graphene since its fabrication in 2004, has made fruitful achievements in the most recent decade [1][2][3][4][5]. Particularly, two-dimensional (2D) heavy group-IV materials as graphene-like materials, including silicene, germanene and stanene, have attracted considerable research in the last five years owing to their controllable Dirac nature [6][7][8][9][10][11][12][13][14]. Distinct from graphene, electrons in the other 2D heavy group-IV monolayers have an observable intrinsic spin-orbit interaction because of their buckled structures [9][10][11]15]. Significantly, the low-energy Dirac electrons in these materials can feel multiple degrees of freedom, involving charge, spin, valley and sublattice-pseudospin. Here, the 'valley' benefits from the fact that the Dirac electrons locate around two inequivalent points in the first Brillouin region. Up to now, many spinvalley related phenomena have been explored theoretically in heavy group-IV monolayers, such as the hightemperature quantum spin Hall effect [6], valley-polarized quantum anomalous Hall effect [11], topological superconducting effect [13], spin-valley filtering effect [16][17][18] and bipolar spin-valley diode effect [19]. Any of these predictions, if verified in experiment, can provide giant opportunities not only for the development of physics and material science but also for future technology applications.
In spite of many previous findings related to spin and valley, we notice that no one has ever proved whether the spin Seebeck effect and valley Seebeck effect could coexist in a real condensed-matter system. By studying the temperature-driven valley-spin transport in a designed ferromagnetic device locally modulated by an interlayer electric (E z ) field, we here discover a unique phenomenon we call the valley-spin Seebeck (VSS) effect, for which the current is contributed only by one (the other) valley and one (the other) spin moving along one (the opposite) direction. This effect surely reflects the coexistence of the spin Seebeck effect and valley Seebeck effect, and is suggested to be detected below the critical temperature 18K for silicene, 200K for germanene and 400K for stanene, attributed to the specific valley-spin nondegenerate band structures, but cannot be driven in graphene without spin-orbit interaction. Above the critical temperature, the VSS effect disappears because of overlarge source-temperature broadening, independent of temperature difference. Besides the temperature, the VSS effect can also be turned on and off by the E z field. It is further demonstrated that the VSS effect is robust against many realistic perturbations. Our findings pave the way for 2D heavy group-IV materials applied in future thermal-logic circuits and energy-saving devices.

Design principle for the valley-spin Seebeck effect
For any temperature-driven caloritronic device, two heat reservoirs are requisite: one is a high-temperature source (S), the other is a low-temperature drain (D). We here use ( ) T S D to describe the temperature of S (D). We define the direction of the current from S to D as positive. Because electrons in heavy group-IV monolayers have valley and spin degrees of freedom, the current should be valley-spin dependent. Below we use h ( ) s to denote the valley (spin) index. According to the generalized Laudauer-Büttiker approach [18,[29][30][31][32] .  h s represents the number of valley-spin dependent transmission modes, depending on the detailed device structure. Because silicene, germanene and stanene are semiconductors with small spin-orbit band gaps less than 0.2eV, the temperature primarily plays the role of exciting thermoelectrons here [31][32][33], different from that of exciting magnons in ferromagnetic insulators [41]. From the distribution differencef f S D in equation (1), we judge that electrons with energy higher than the Fermi energy e F flow from S to D on account of - . This happens in undoped pristine graphene and its derivatives with electron-hole symmetry.
To create electron-hole asymmetry near the Fermi energy it is a necessary condition to generate nonzero current, but needs to far from enough to effectively drive the VSS effect in heavy group-IV monolayers. Based on band theory, we conclude that the detailed form of  h s in equation (1) depends on the band-to-band tunneling mechanism, and dictates whether the valley or spin Seebeck effect happens. On the one hand, to drive the normal spin Seebeck (NSS) effect, the transmission should be electron-dominated for one spin but holedominated for the other spin. This requires that the band structure near the Fermi energy must be nondegenerate for two spins besides electron-hole asymmetry. This is relatively easy to realize in the ferromagnetic state of semiconductors [10,19,31]. On the other hand, to generate the valley Seebeck effect, one must ensure that the transmission near the Fermi energy is electron-dominated for one valley but holedominated for the other valley [29,30]. This condition cannot be satisfied in a sample of silicene, germanene or stanene only in the presence of the E z field, although the band structure is both valley-polarized and spinpolarized [10]. Conceivably, to realize the VSS effect, the transmission should be electron-dominated for one spin in one valley but hole-dominated for the other spin in the other valley. In this case, there might exist a particular energy region, inside which one valley is insulating for two spins but the other valley is conductive for only one spin. Luckily, we notice that, for heavy group-IV monolayers, the coexistence of ferromagnetic field and E z field could satisfy this criterion [29]. Additionally, to observe the VSS effect, the energy broadening induced by the source temperature cannot exceed the particular insulating region for two spins, or otherwise the deep-energy bands outside the band gap can break the transport feature of the VSS effect.
Here, we need to make clear how to obtain the transmission modes  h s in equation (1). For simplicity, the reduced Planck constant ÿ is set as 1, and the Fermi velocity u F (5.5 10 5 m s −1 for silicene,4.6 10 5 m s −1 for germanene,4.9 10 5 m s −1 for stanene [10,15]) is set as 1. For a sample of width W, the total transmission modes are generally obtained by [19,29]  where k, f are the modulus and angle of the wave-vector, represents the reduced constant with the dimension of density of states, and e f h ( ) T , s describes the angle-dependent transmission for each mode. Then the spin-valley dependent current is calculated from equation (1) as

Realization of the valley-spin Seebeck effect
At the top of figure 1(a), we show our proposed caloritronic field effect transistor device, including S, D and G (the central gate region controlled by an E z field). A ferromagnetic field with strength M is uniformly applied in the whole sample. From previous first-principles calculations, it is known that the low-energy electrons in heavy group-IV monolayers can be described by an effective Hamiltonian which reads [20][21][22][23], where λ (3.9 meV for silicene, 43 meV for germanene, 100 meV for stanene) represents the spin-orbit coupling, h = + - describe the Pauli matrices of sublattice pseudospin and electron real-spin respectively. Although Rashba-type spin-orbit couplings as perturbations in these heavy group-IV monolayers exist in principle, they play little role even under external fields in studying the band structures and transport properties [6,10]. Thus the low-energy effective Hamiltonian here can be expressed as where the second term indicates the ferromagnetic field, the third term represents the E z field with = Q -Q - Herein, Q( ) x is the Heaviside function, and ℓ denotes the interlayer distance. After matrix diagonalization, the energy bands are solved as The energies at valleys . In regions S and D, it is found that the transport-forbidden regions for spin-up and spin-down electronic states read, respectively, , identical for two different valleys. In region G, the valley degeneracy is destroyed by locally applying ℓE z , and the band gap regions for spin-up and spin-down bands are obtained, respectively, as l h l l h l -- To obtain information on h I s , it is still needed to solve h T s in equation (3). Based on quantum mechanics, we derive the transmission amplitude by matching the wave functions at the interfaces = x L 0, as

The mutual insulating region for two spins is
. The wave-vector modulus in regions S, D and G are obtained respectively as . 7 s s 2 Note that, for uniformity, the parameters l = M and L=50nm are used below to carry out the calculations and plot figures without special instructions.
In figures 1(b)-(d), we plot the current h I s versus the source temperature T S for silicene, germanene and stanene, respectively, at l = ℓE 2 z . The temperature difference is fixed at = T 20 SD K, except for the inset in figure 1 Two different phases are found, one is marked as I-VSS and the other is marked as II-NSS. It can be seen that the VSS effect can be observed below the critical temperature 18K for silicene, 200K for germanene and 400K for stanene. The increasing critical temperature from silicene to stanene is consistent with the increasing spin-orbit interaction. The critical temperature T c could be approximately determined by , where k B is the Boltzmann constant. Above the critical temperature, the NSS effect is detectable. As is known, the index-change relationship of the Fermi-Dirac distribution function determines that only electronic states near the Fermi energy could contribute to the current. Consequently, the thermodynamical transition from VSS phase to NSS phase is attributed to the fact that, only if the energy broadening k T 5 S B induced by the source temperature does not exceed the gap size l 2 , high-energy subbands cannot break the transport feature of the VSS effect. For the phase I-VSS, it is seen that only spin-up (down) states from valley ¢ ( ) K K contribute to the positive (negative) current, in agreement with the band structures in figure 1(a), where hole (electron) states contribute most to  ¢  ( ) I I K K . As T S increases, the hole-dominated current ¢  I K and the electron-dominated current  I K become nonzero due to enhanced temperature broadening. As T S becomes much larger than 400K, all the values of h | | I s in figures 1(c) and (d) tend to stabilize due to convergence of the integral in equation (3), just as calculated in figure 1(b).
In the driving force tends to zero and thus all the currents h I s tend to zero. Owing to  T T S SD , except for the inset in figure 2(a), the nonzero h I s curves basically increase linearly as T SD increases. Also, such linear relationships can be judged from equation (3) In the low-T S case, the linear relationships disappear although the curves h ( ) I T s SD vary monotonically, as seen from the inset in figure 2(a). In comparison with figures 1(b)-(d), T S and ℓE z here do not change, and hence we have reason to conclude that the presence of the VSS effect is basically independent of T SD , because no new phase is induced by increasing T SD . For germanene, the VSS effect could also be detected if T S is fixed at a small value less than the critical temperature 200K, just as shown by the inset in figure 2(a).
In The main result here is that the VSS and NSS effects could be switched to each other by tuning the E z field. To understand this phenomenon, we need to analyze the variation of band structures in region G induced by the E z field. As ℓE z varies, the mutual insulating region for two spins l h l l h l - I s cross due to the valley degeneracy. Now one may wonder whether inelastic scattering at higher temperatures affects our ballistic results in figures 1-3. It is true that electron-phonon interactions should be enhanced as temperature increases due to more excited phonon modes, and thus electron mobility naturally decreases. For electron-phonon scattering, we divide it into two types: intravalley scattering and intervalley scattering. For intravalley scattering, increasing temperature cannot drive the jump between different states labeled by different valley-spin indexes due to the absence of the spin-valley flipping mechanism, but only reduces the magnitude of h I s . This indicates that the qualitative conclusions in the ballistic regime remain the same. For intervalley scattering, recent first-principles calculations have demonstrated that only electrons with energy more than several tens of meV or larger in silicene and germanene could feel this scattering induced by the stronger electron-phonon coupling [48], even near room temperature. Consequently, it is reasonable to believe that inelastic scattering has little effect on the detectable properties of the VSS effect in our considered range of temperatures. Additionally, note that the net current contributed by spin-up and spin-down electrons always flows in opposite directions (pure spin current) in figures 1-3 and indeed the spin polarizability is 100%, benefiting from the completely spin-polarized subbands near the Fermi energy (see figure 1(a)). Thus it is not necessary to define the degree of spin polarization here. Nevertheless, one can notice that the NSS in figure 2(b) for gemanene is different from that in figure 2(a) for silicene, resulting from different degrees of valley polarization r v . Further considering that the total current contributed by two opposite directions is always zero in figures 1-3 due to symmetry, it is difficult to define an overall effective r v for bidirectional transport, but one can still treat r v separately for positive and negative currents via a definition r = å

Robustness of the valley-spin Seebeck effect
Since we have obtained information for how to realize the VSS effect in silicene, germanene and stanene, it is still necessary to discuss the robustness of the VSS effect in some real complex experimental conditions. Below, we take stanene for instance and study the influence of the ferromagnetic field and the Fermi energy on the VSS effect near room temperature. The fixed parameters read = T 340 Note that the conclusions for stanene here are universally valid for germanene and silicene under lower T S .
Firstly, we show the influence of ferromagnetic field M on the VSS effect in figure 4(a). Our results indicate that the VSS effect is always robust at  l M . This result depends on such a fact that the ferromagnetic field does not essentially change the electron-hole asymmetry in figure 1(a) in spite of band variations. Further considering that the ferromagnetic field has been successfully introduced in graphene by the proximity effect of the ferromagnetic insulating substrate [49], it is reasonable to believe that the ferromagnetic field could also be applied in silicene, germanene and stanene. In this sense, one can easily observe the VSS effect in a large range of M-strength. In addition, provably, the length L of region G does also not change the realization of the VSS effect.
Secondly, we consider a stanene sample that is uniformly doped e ¹   0 , and thus the NSF effect is observed. Lastly, we discuss the influence of some complex conditions on the VSS effect. Firstly, in terms of band changes, the Rashba interactions do not change the valley-spin dependent band gap although there is spin splitting outside the valley points [13][14][15][16][17]. Actually, for silicene, germanene and stanene, the spin splitting induced by Rashba perturbations plays little role in affecting the spin-valley transport, as proved by previous first-principles calculations [10]. Secondly, note that our concerned VSS effect is based on bulk states of the materials but not edge states [47] and thus could be detected in experiments more easily due to the stronger signal, although this VSS effect has some common transport features to valley-polarized quantum spin Hall edge states [28]. Thirdly, a real sample of heavy group-IV monolayers also inevitably contains atomic defects in the bulk, such as bits of vacancies [12,49], which induce variations of supercell bands. To realize the VSS effect, the defect ratio cannot exceed 8%, otherwise the VSS effect is broken by defect states that contribute to the current [50]. Fourthly, to avoid intervalley scattering, local gates are suggested to deplete edge carriers and form gatedefined sample edges, near which the variations of edge potential do not change the delectability of the VSS effect owing to the negligible proportion of edge atoms [19]. In short, although there has been little experimental study about the temperature-driven electron motion in heavy group-IV monolayers so far, we still have sufficient reason to believe that the VSS effect discussed here is detectable by developing the latest experimental detection techniques on valley or spin in graphene [44][45][46][47]. To be sure, our results here advance our previous work about the realization of NSS or valley Seebeck effects as well as confirm the rough prediction of the coexistence of these two effects [51].

Conclusion
In summary, based on monolayers of heavy group-IV materials, we have designed a caloritronic device locally modulated by an interlayer electric field to realize a unique valley-spin Seebeck (VSS) effect, which supports bidirectional transport with each net current direction locked by both valley and spin. Associated with the strength of the spin-orbit band gap, such a VSS effect is suggested to be detectable using current experimental technology below the critical temperature 18K for silicene, 200K for germanene and 400K for stanene, arising from the specific valley-spin band structures. We have also found that above the critical temperature, the VSS effect may be broken by overlarge temperature broadening, basically independent of temperature difference. Besides the temperature, the local electric field can also drive a transition between the VSS and normal spin Seepbeck effects by tuning the band structures. Furthermore, we have proved that the VSS effect found here is robust against many realistic perturbations. These findings provide new insights to apply two-dimensional heavy group-IV materials in future information procession (encoding information simultaneously by valley and spin) in thermal logic circuits and low-dissipation devices.