Spin Seebeck effect in bipolar magnetic semiconductor: A case of magnetic MoS2 nanotube.

Bipolar magnetic semiconductors (BMSs) are a new member of spintornic materials. In BMSs, one can obtain 100% spin-polarized currents by means of the gate voltage. However, most of previous studies focused on their applications in spintronics instead of spin caloritronics. Herein, we show that BMS is an intrinsic model for spin Seebeck effect (SSE). Without any gate voltage and electric field, currents with opposite spin orientation are generated and flow in opposite directions with almost equal magnitude when simply applying a temperature bias. This is also due to the special electronic structure of BMS where the conduction and valence bands near the Fermi level belong to opposite spin orientation. Based on density function theory and non-equilibrium Green's function methods, we confirm the thermal-induced SSE in BMS using a case of magnetic MoS2 nanotube. The magnitude of spin current in zigzag tube is almost four times higher than that in armchair tube. BMS is promising candidates for spin caloritronic applications.

Introduction

The observation of spin Seebeck effect (SSE) is a remarkable achievement of spin caloritronics, a subject which combines spintronics and calorifics in a material [1], [2]. To achieve SSE, the spin-polarized currents should be generated and flow in opposite directions in the device by a temperature gradient instead of any other external field, this requires apposite conduction and valence spin channels approaching to the Fermi level (EF) [3], [4]. The Joule heating is inhibited in the device under a pure spin currents, providing alternative strategy for low-power-consumption technology. Notably SSE has been discovered in such materials as silicene nanoribbon [5], graphyne nanoribbon [6] and boron-nitrogen nanotube [7]. However, the spin channel seems uncontrollable since it depends on the configuration and dimension of these nanostructures. Thus, a temperature induced SSE with pure and stable spin-polarized currents is still challenging.

In this work, we show that BMS is an intrinsic and stable model for SSE, which completely meets the requirements of SSE. BMS was proposed by Yang et al. [8] to achieve 100% spin-polarized current utilizing a gate voltage. The schematic density of states of BMS is given in Fig. 1c. From this figure, one can see that BMS are actually belonging to spin-polarized semiconductors (SPSs). However, different from common SPSs, the conduction bands (CBs) and valence bands (VBs) in BMSs near the EF are belonging to carriers with opposite spin orientation. According to Yang et al. [8], adjusting the position of the EF using a gate voltage realizes reversible half-metallicity in BMS. This can be applied to a spintronic device with bipolar field-effect spin-filter. For practical application, the band gap (BG) of BMS should have small value to ensure the feasibility of manipulating the spin-polarized currents. From another point of view, the electronic structure of BMS rightly meets the generation of SSE, which also provides a alternative applications in spin caloritronics. While it is worthwhile to note that not all BMS materials are ideal candidates for perfect SSE since the two spin channels may not completely symmetrical. In some cases, this could be optimized by tuning the position of the Fermi level using a gate voltage.

Fig. 1

Device models based on zigzag (a) and armchair (b) magnetic MoS2 NT, TL and TR denote the temperatures of source and drain, respectively. (c) shows the schematic density of states of BMS.

To clearly show how SSE occurs in BMS, we study the thermal spin transport (TST) characters of magnetic MoS2 nanotube (NT) using first-principles density function theory (DFT) and non-equilibrium Green’s function (NEGF) methods. Controlling synthesis of MoS2 NT is already a proven technique [9], and various field-effect transistors based on MoS2 NT show good performance [10], [11]. DFT calculations reveal the semiconducting feature of MoS2 NT [12]. However, to control the electrical, optical, and even magnetic properties for more potential applications, doping the semiconducting MoS2 NT with minute amounts of foreign atoms is desirable, and this has been demonstrated by Tenne et al. [13]. Based on Kim et al. [14], substitution on Mo site with 3d transition metals (V, Mn, Fe, Co and Cr) can induce notable magnetic moments, which transforms the semiconducting MoS2 NT to one-dimensional magnetic semiconductor. Herein, our calculations are based on magnetic MoS2 nanotube with Cr as the impurity.

Device model and computational methods

In term of the description of carbon NTs [15], [16], the MoS2 NT can also be distinguished according to the factor n and m, these are n = m “armchair” nanotube and n ≠ 0, m = 0 “zigzag” nanotube. Fig. 1a and Fig. 1b are the device models based on magnetic MoS2 NT with zigzag and armchair configurations, respectively, and in the right panels of Fig. 2a and 2b are the side views. Seifert et al [12]. have studied the structure and electronic properties of MoS2 NTs with the diameter ranging from 8 to 26 Å, corresponding to the indices (n, n)/(n, 0) from (6, 6)/(10, 0) to (14, 14)/(22, 0), respectively. The experimental tube usually has larger diameter ~ 25 nm, and the largest diameters are in consistent with the smallest experimentally discovered tubes based on Seifert et al [12]. Here, our calculations are based on (14, 14) zigzag and (22, 0) armchair MoS2 NTs. To induce magnetism, one Mo in unit tube is substituted by Cr.

Fig. 2

The left, middle, and right panels respectively represent the calculated band structure, transmission spectrum, and structure in side view of the zigzag (a) and armchair (b) NTs.

We carry out the structural relaxation and electronic band structure calculations via DFT method within the generalized gradient approximation (GGA) Perdew-Burke-Eenzerhof (PBE) correlation functional implemented in VASP [17], [18]. We use projector augment wave potential and a plane wave energy cutoff of 450 eV. A Monkhorst-Pack k-points of 1 × 1 × 21 is used. The lattice and atom’s positions are totally relaxed until the maximum force becomes less than 0.01 eV/Å. We set a precision of 10-6 eV for the criterion of the self-consistent total energy. A vacuum spacing of 15 Å is imposed to avoid the interaction between neighboring NTs. Considering the energy gap underestimation of GGA function, we also adopt the modified Becke-Johnson (MBJ) [19] function in this work to ensure that we can achieve more accurate band structures. The TST calculations are then carried out via NEGF/DFT methods within Atomistix ToolKit (ATK) code [20], [21], [22]. We set the cutoff energy and the Monkhorst-Pack k-points as 150 Ry and 1 × 1 × 100, respectively. The infinitesimal energy for the iteration of transmission spectrum is set as 10-6 eV, and we increase the sampling points to 201 in order to obtain the accurate transmission curves. Additionally, we consider a Double-Zeta-Polarized (DZP) basis set to obtain the accurate results. These computational details are common and reliable for the theoretical investigation of TST properties. According to the Landauer-Büttiker formula, the spin-dependent current is given by the following equation [23], [24],where TL(R) and fL(R) are the temperature and Fermi-Dirac distribution of left and right electrodes, respectively. is the spin-dependent transmission function, which is determined by the retarded/advanced Green’s functions of the central scattering region and also the coupling between the central scattering region and the left (right) electrode. Herein, we pay our attentions to the spin-dependent currents in the NT induced by temperature bias without any external field or gate voltage.

Results and discussion

Calculated band structures of (14, 14) zigzag and (22, 0) armchair MoS2 NTs indicate that they are ideal BMS materials, as shown in the left panel of Fig. 2a and 2b, respectively. Based on the investigation of Yang et al. [8], MoS2 NTs hold the advantages in spintronic applications due to its half-metallic behaviors are very robust and its opposite spin polarization can be obtained via the effect of electron/hole doping. Instead of focusing on their applications in spintronics, we here discuss the potential application in spin caloritronics. Both the zigzag NT and armchair NT features obvious indirect BGs around the EF, and values of BGs are about 0.5 eV, while larger BG is found in armchair NT. Moreover, zigzag NT exhibits remarkable band dispersion as compared to armchair NT, which indicates the smaller effective mass and higher mobility of carriers in zigzag NT. These band features are important because of their non-negligible influence on the magnitude of current through the device. It is important in BMS that CBs and VBs, above and below the EF, belong to opposite spin orientation. For magnetic MoS2 NTs, there are spin-up conduction and spin-down valence bands approaching to the EF, and these spin-up CBs and spin-down VBs occupy a remarkable energy span, which means that two opposite spin transport channels could be generated in the device under an external field. This is beneficial to the emergence of notable SSE when a temperature bias is applied, i.e., the spin-up and spin-down currents arise and flow in opposite directions, as long as the two opposite spin channels are nearly symmetrical [5].

Calculated transmission spectrum of zigzag and armchair NTs devices without a gate voltage are shown in the middle panel of Fig. 2a and b. Consistent with the electronic band characteristics, transmission channels belonging to opposite spin orientation can be distinguished above and below the EF. It is easy to obtain a spintronic device by adjusting the EF up or down through a gate voltage, which produces 100% spin polarization. In addition, the spin-dependent currents primarily depend on the net transmission around the EF according to the formula of spin currents described in Eq. (1). Hence, when a temperature bias is applied to the device, it is desirable to generate spin-dependent currents flowing in opposite directions, and thus occur the SSE [5], [25]. To confirm these analysis, we have to discuss the TST through the devices.

When calculating the TST properties, the spin-dependent current is driven by simply a temperature bias (ΔT) without any external voltage. The temperature bias is defined as ΔT = TL-TR, where TL and TR denote the temperature of the source (left-electrode) and drain (right-electrode), respectively. During the calculation, the drain is set as high temperature. Because the source and drain belong to same material and have the same electronic structure, carrier concentration between source and drain depends on the Fermi-Dirac distribution (fL(E, TL)-fR(E,TR)) [5], [7], which is primarily determined by the temperature bias. When the temperature bais is applied, electrons and holes with energy higher and lower than the EF will simultaneously transport from the drain to source, giving rise to electron current Ie and hole current Ih flowing in the opposite directions. For BMS, the spin-polarized transport channels leads to the spin-dependent currents, i.e., the spin-up Ie↑ and spin-down Ih↓. If the spin channels are highly symmetrical, the charge current is inhibited since Ie↑ and Ih↓ are counteracted, which called perfect SSE [5], [7].

Calculated spin-dependent currents versus the source temperature (TL) and temperature bias (ΔT) are shown in the left and right panels of Fig. 3, respectively. Results in Fig. 3a and 3b belong to the devices based on zigzag and armchair NTs, respectively. Since fL(E, TL)-fR(E,TR) is an odd function, the sign of current depends on the slop of the transmission coefficient T(E) near the EF according to Eq (1). We expect opposite signs for spin-up current (Iup) and spin-down one (Idn) as the spin-up and spin-down transmission channels are located above and below the EF, respectively. As shown in Fig. 3, the Iup > 0 while Idn less than 0. It is clearly found that Iup and Idn are generated via the ΔT without any external voltage, which arises individually from the spin-splitting electrons and holes flowing from the drain to source. Remarkably, the amplitudes of Iup and Idn are almost the same in the whole temperature region, which indicates that an ideal SSE is achieved by BMS characteristic. As the increase of ΔT, the spin currents increase. The threshold temperatures (Tth) arise from the BGs, and the higher Tth of armchair NT is ascribed to its larger BG, as found in Fig. 3a and b. Moreover, it is compelling that currents in zigzag device is higher than that in armchair one, primarily due to the smaller BG and higher carrier mobility of zigzag NT as discussed before. In general, an ideal SSE, characterized by the nearly same amplitudes and threshold temperature in Iup and Idn, is achieved in BMS using the case of magnetic MoS2 NTs.

Fig. 3

Calculated spin-dependent currents (Iup and Idn) of (a) zigzag magnetic NT (ZMN) and (b) armchair magnetic NT (AMN), the left and right panels denote the currents versus source temperature (TL) and temperature bias (ΔT), respectively.

To illuminate the generation of SSE, we also perform the calculations of the total charge current (Ic) and spin current (Is) versus ΔT, as plotted in Fig. 4a. One can notice that the charge current is suppressed while the spin current increases with the increase of ΔT, which indicates that the spin current dominates the thermal-induced transport in the device. The suppressed charge current contributes to reduced Joule heating, hence, the SSE devices can be applied to waste heat recovery and low-power-consumption technology. Calculated current spectra J(E) shown in Fig. 4b also points to the emergence of SSE. The magnitude of current is determined by the area covered under the curve associated with axis J(E) = 0. Obviously, two areas of current spectra with opposite spin channels are nearly symmetrical about the EF, also suggesting the generation of SSE. However, it is worthwhile to note that the SSE achieved in magnetic MoS2 NT is not completely perfect since the a little higher spin-down current leads to the incompletely inhibited charge current as found in Fig. 4a. Even so, it is desirable to explore other BMS materials as candidates for excellent SSE. In addition, the SSE can be further optimized by adjusting the dopants or the position of the Fermi level. To date, many BMS materials have been reported, such as carbon nanotube [8], boron-nitrogen nanotube [7] and SiN-SiC nanofilm [26], etc. Wu et al. [7] also discovered the robust SSE in magnetic boron-nitrogen nanotube. Thus, our calculations and results can be easily extended to other BMS materials.

Fig. 4

(a) The calculated total spin current (Is) and charge current (Ic) as a function of temperature bias (ΔT) for zigzag magnetic NT (ZMN) and armchair magnetic NT (AMN). (b) The spin-dependent current spectra at fixed ΔT = 250 K with different TL.

Conclusion

In summary, we have shown that BMS material is an ideal candidate for SSE, which is expected to generate nearly pure spin current simply via a temperature bias. Based on First-principles DFT combined with NEGF methods, the concept is clarified via a case of magnetic MoS2 NT. No matter if it is the zigzag or armchair configuration of the NT, the SSE is remained. Zigzag tube shows better transport performance, as its spin current is four times higher than that in armchair tube, which leads to the higher total spin current in zigzag tube, i.e., ~65nA at TL = 350 K/ΔT = 300 K. Our study reveals the certainly application of BMS materials in spin caloritronic devices. Thus, through electrical control or heat management, BMS based multifunctional devices exhibit great flexibility of applications in not only spintronics but also spin caloritronics.

Compliance with ethics requirements

This article does not contain any studies with human or animal subjects.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.