Giant spin Seebeck effect in two-dimensional V₂S₂O altermagnet

INTRODUCTION

Altermagnetic (AM) materials are symmetry-compensated antiferromagnets whose band structures are characterized by spin splitting^[1]. This implies that over the entire Brillouin zone, the up and down spin bands of the AM materials do not entirely overlap. Rather, the band structure exhibits orientational-dependent splitting. The spin-orbit coupling (SOC)-induced magnetic property has been the cornerstone of spintronic applications such as spin-orbit torque, spin transistors, topological insulators, and the spin Hall effect^[2,3]. Nevertheless, the swift decoherence of spin-polarized electrons caused by SOC restricts the broad applications of these materials^[4]. In contrast, spin splitting has been achieved in AM materials without applying SOC^[5-7]. The source of the spin splitting in AM materials is the magnetic space group, which is protected by crystal symmetry^[8,9]. This spin-splitting allows the realization of highly spin-polarized currents^[8,10]. The study of AM materials is still in its infancy; however, it may be a fertile field for theory and potential applications. For example, the unique band feature in AM materials may make them potential materials for thermal transport devices, valleytronics, field-effect electronics, photo-magnetism, spin caloritronics, spin transfer torque, superconductivity, and spintronic applications^[9-12]. The AM materials also have the potential for high-speed device operation because of their ultra-high dynamic speed and zero net magnetic moment, allowing resistance to external magnetic disturbance^[13]. Several bulk materials, such as RuO₂, FeSb₂, FeF₂, MnTe, Ve₂Te₂O, MnF₂, MnO, CoF₃, MnO₂, NaFeO₂, NdB₂C₂, and some GdFeO₃-type perovskites, have been predicted to be AM materials^[13-16].

In parallel with the discovery of the AM behavior in RuO₂ and other bulk materials, it has been predicted that two-dimensional (2D) materials such as Mn₂ClI, V₂S₂O, MnTeMoO₆, V₂F₇Cl, Cr₂SO, V₂SeTeO, Cr₂O₂, CrMoC₂S₆, Mn₂ClF, RuF₄, and Cr₂SeO can also display an AM behavior^[4-6,17,18]. However, most studies have concentrated on basic unconventional properties with minimal reports on thermoelectricity in AM materials^[19-24]. Recently, Sukhachov et al. suggested that AM materials could be employed for effective thermoelectricity^[25]. Using the AM spin-splitting effect, Bai et al. also reported efficient spin-charge conversion in RuO₂ altermagnet^[26]. Besides, Lyu and Li pointed out that the transport properties in AM materials were directional dependent^[27]. Indeed, the AM behavior has been experimentally confirmed in RuO₂ and MnTe using angle-resolved photoemission spectroscopy^[28,29]. Fan et al. also reported experimental work that the AM spin splitting effect could produce anisotropic spin currents in AM material with polarization, which depends on the crystal orientation of the material^[30]. Moreover. Bai et al. also reported a few microvolts in altermagnet RuO₂ spin-charge conversion efficiency using spin Seebeck effect measurement^[26]. Motivated by the predicted potentials of unconventional spin-splitting effect in AM materials, we propose to leverage directional spin-dependent thermoelectric properties and predict a massive spin-Seebeck coefficient in V₂S₂O monolayer AM material.

Indeed, thermoelectricity is a fertile field for testing and predicting high-performance transport materials via theory-based material search techniques. Although precise prediction of these high-performance transport materials computationally entails vigorous optimization, the experimental approach is even more complicated. In our work, we inspect the orientational spin-dependent transport properties at finite temperatures and aim to propose a giant spin Seebeck effect in the V₂S₂O altermagnet for the spin-polarized current generation in spintronics and thermoelectric device applications.

THEORETICAL METHODS

All our calculations were conducted in this work using the spin-polarized density functional theory as implemented in the Vienna ab initio simulation package (VASP)^[31,32]. We used the Perdew, Burke, and Ernzerhof parametrization within the generalized gradient approximation for an exchange functional^[33], including the van der Waals interaction^[34] as a correlation functional. A plane wave basis set is employed with an energy cutoff of 600 eV. The energy and force convergence criteria are set to 10^-6 eV and 10^-3 eV/Å, respectively. A well-converged Monkhorst-Pack scheme-generated k-point mesh of 10 × 10 × 1, which also ensures the convergence of the energy and force, was used^[35]. Furthermore, we applied a vacuum distance of 25 Å along the z-axis to prevent adjacent units’ interaction. The Heyd-Scuseria-Ernzerhof screened hybrid functional was used to calculate the spin-polarized electronic band structure^[36,37]. To estimate the critical temperature, we applied a simulation of the field cooling and Landau-Lifshitz-Gilbert Huen method as implemented in the VAMPIRE simulation package^[38] for the temperature-dependent sub-lattice magnetization calculation. To this end, a supercell of 100 × 100 × 1 with total time steps of 100,000 and 0.1 fs is used. Our directional spin-dependent transport properties were explored using the BoltzTraP code^[39]. Since most of the transport properties are carrier relaxation time (τ) dependent, we considered the scattering contributions from the impurity effects (τ_imp) polar optical phonon (τ_opt), and acoustic phonon (τ_aco). Therefore, the total τ, which depends on energy and temperature, can be expressed as:

In addition, we also computed the lattice part of the thermal conductivity with phono3py code^[40] using a 4 × 4 × 1 supercell, 6 × 6 × 1 K-mesh, and energy cutoff 600 eV.

RESULTS AND DISCUSSION

Figure 1A reveals the side and top views of monolayer V₂S₂O. The red and black colors denote vanadium atoms with spin-up and spin-down, while the blue and yellow colors represent the oxygen and sulfur atoms. The monolayer V₂S₂O has a tetragonal crystal structure and a space group of P4mmm (123). We obtained the lattice constants of a = b = 3.84 Å after structure optimization. The monolayer V₂S₂O is made of a V₂O plane embedded by two S planes with interatomic distances of 1.921 Å and 2.479 Å from V to O and S. This shows that the V₂S₂O monolayer exhibits high-degree symmetry along in-plane and out of plane. Note that the thermal stability, cleavage energy, and dynamic stability were already reported elsewhere^[41]. To investigate the magnetic ground state of monolayer V₂S₂O, we considered ferromagnetic (FM) and three different antiferromagnetic (AFM) spin configurations in a supercell of dimension 2 × 2 × 1. In Figure 1B-D, we show the different AFM spin configurations. The most stable AFM-Neel state was obtained [Figure 1B], and the energy difference (E_ex) between this state and the FM state (E_ex = E_FM - E_AFM) was 536 meV/unit cell. Each V atom had an atomic magnetic moment of magnitude 1.7 μ_B. However, the net magnetic moment of the unit cell vanishes due to the anti-parallel coupling between the V atoms. The AFM-Neel ground state in the V₂S₂O monolayer is consistent with other theoretical reports^[6,41]. Figure 1E reveals the electronic band structure without SOC. This electronic band structure shows a spin splitting around the X and Y points, while in other parts of the Brillouin zone, both spin-down (↓) and spin-up (↑) bands were completely overlapped. The spin splitting at X and Y points was 0.51 eV at the valence band maximum. AM V₂S₂O monolayer also showed a semiconducting behavior with a direct band gap of 1.15 eV at the Y and X points. We also computed the critical (Neel) temperature via the model of Heisenberg spin Hamiltonian^[42] based on:

Figure 1. (A) Schematic illustration of top and side views of V₂S₂O monolayer. Magnetic configurations of (B) AFM-Neel, (C) AFM-Stripy, and (D) AFM-Zigzag in a 2 × 2 × 1 supercell; (E) Band structure of monolayer V₂S₂O with First Brillouin zone with high symmetry points as inset (red lines for spin-up and blue dashed-lines for spin-down); (F) Temperature-dependent sublattice magnetization of V₂S₂O monolayer. AFM: Antiferromagnetic.

Here, S_i and S_j are the spin moment direction of atoms at the neighboring sites i and j, J_ij is the exchange interaction between atoms at the neighboring sites, and k_u is the anisotropy energy per atom. Meanwhile, N_α and n_α represent the number of atoms in the sublattice and the mean of sublattice magnetization of each atom. Using E_ex, the difference between the total energies of the FM and AFM spin states, we extracted the exchange interaction via the relation:

where m and N are the average magnetic moment and the number of magnetic atoms in the system. Figure 1F shows the temperature-dependent sublattice magnetization curve. Overall, we obtained a Neel temperature of 746 K for the V₂S₂O monolayer.

Now, we explore the spin-dependent transport properties such as spin-dependent Seebeck coefficients, electrical conductivity, electronic thermal conductivity (K_e), and lattice thermal conductivity (K_L). First, the spin-dependent Seebeck coefficient [S^(↑↓)] is obtained using:

Here, q, T, and L are the elementary charge, the temperature, and the generalized linear spin-dependent transport coefficient expressed as:

The conductivity tensor over the bands (i) and the k-points (k) can be expressed as σ^(↑↓)(i,k) = e²τi,kV², where V represents group velocity. Figure 2 reveals the spin-dependent Seebeck coefficients for the spin-up channel (S^↑) and spin-down channel (S^↓). In the x-direction, the Seebeck coefficient of the spin-down channel comes first before the spin-up channel Seebeck coefficient in both doped systems. Meanwhile, in the y-direction, the Seebeck coefficient of the spin-up channel appears first before the spin-down component in both the electron and hole-doped systems. This feature is the direct consequence of the directional spin-dependent band structure in Figure 1E.

Figure 2. Spin-dependent Seebeck coefficient (S^↑ and S^↓) along x-direction at (A) 100 K vs. chemical potentials; (B) 200 K vs. chemical potentials; (C) 300 K vs. chemical potentials; and y-direction at (D) 100 K vs. chemical potentials; (E) 200 K vs. chemical potentials; and (F) 300 K vs. chemical potentials.

Note that the spin Seebeck coefficient does not depend on τ. However, other transport properties, such as electronic thermal and electrical conductivities, strongly depend on it. Hence, it is crucial to estimate τ accurately. Nonetheless, in most studies, the energy-independent constant relaxation time approach has been adopted with limitations in accounting for some scattering effects. This usually results in an overestimation of the thermoelectric performance. To remedy this, in our study, we applied the temperature-energy dependent relaxation time considering the contributions from the acoustic and optical phonons as implemented in the works of Casu et al. and Marfoua et al.^[43,44]. Besides, we also considered the impurity scattering effect using the relation of Brooks-Herring. Note that the impurity scattering effect can be written as:

where the Z_I, n_I, ϵ_s, ϵ₀ and m represent the impurity charge, the ionized impurity concentration, the relative dielectric constant, the vacuum’s permittivity, and the effective mass, respectively. Also, $$ x=\frac{\hbar^{2} q_{0}^{2}}{8 m E} $$ where q₀ denotes Debye screening wave vector. The optical polar scattering can be estimated by:

where the sum is over all longitudinal optical polar phonons with energy $$ e_{i}^{L O} $$ according to the Ridley model^[39]. The contribution from the acoustic phonon scattering is obtained from:

where D, ρ, v, E, and m are the band energy deformation potential, mass density, the sound average velocity, the energy of the electron, and the effective mass, respectively. To estimate the energy and temperature dependent τ, the intrinsic material quantities such as the dominant longitudinal optical phonon frequency (ħω_LO), high-frequency dielectric and lattice dielectric constants (ε_∞ and ε_L), deformation potential (DP), mass density (ρ), the average velocity of sound (v), and the effective mass ratio (m*/m) of the system are required. We present all these parameters in Tables 1 and 2 for the spin-up and spin-down channels, respectively.

We present the temperature and spin-dependent carrier relaxation time along the y-direction (τ_y) and x-direction (τ_x) in Supplementary Figures 1A-F and 2A-D shows the contributions to the carrier relaxation time from different scattering effects. We found that the spin-down (spin-up) carrier had a longer τ along the x-direction (y-direction) in both electron and hole-doped systems. This can be attributed to the effect of the effective mass. As displayed in Tables 1 and 2, the spin-up (spin-down) carrier possessed relatively lower effective mass along the y-direction (x-direction). We further analyzed the scattering mechanisms due to impurity, optical polar, and acoustic phonon scatterings. As revealed in Supplementary Figure 2A-D, the τ was mainly affected by optical polar scattering. We now discuss the spin and temperature-dependent electrical conductivity [σ^(↑↓)]. Here, σ^↓ and σ^↑ are the electrical conductivities of carriers of the up and down spin channels. The spin-dependent electrical conductivity is computed using:

Figure 3A and B reveals the temperature and spin-dependent electrical conductivity as a function of chemical potential. In the x-direction, the electrical conductivity was entirely influenced by the carriers of spin-down (blue color) because the contribution from the spin-up channel carriers (red color) was negligible. Meanwhile, in the y-direction, the opposite behavior was found. This behavior originates from the electronic band structure where the spin-down (spin-up) channel controls the bandgap in the x direction (y-direction). Besides, we found that the hole-doped system showed higher electrical conductivity than the electron-doped structure. For instance, in the x-direction and in room temperature, the σ of the spin-down carriers was 3.93 × 10⁴ (Ω m)^-1 and 1.5 × 10⁴ (Ω m)^-1 for the hole and electron-doped systems at a chemical potential of ±0.49. Similarly, in the y-direction these values became 3.22 × 10⁴ and 1.23 × 10⁴ (Ω m)^-1, respectively, for the spin-up carriers at the same conditions. We attribute this carrier type dependency to the effect of the effective mass as displayed in Tables 1 and 2.

Figure 3. Spin-dependent electrical conductivities along (A) x-direction and (B) y-direction at different temperatures.

Next, we calculate the total effective spin-dependent Seebeck coefficient (S_eff-spin) and effective charge-dependent Seebeck coefficient (S_eff-charge) by associating the spin-dependent Seebeck coefficients with spin-dependent σ. These Seebeck coefficients are expressed as:

Figure 4A-D reveals the computed results. Along the x-direction (y-direction), the S_eff-spin and S_eff-charge are determined by the spin-down carriers (spin-up carrier) since σ^↑(σ^↓) is negligible. Due to this feature, we expect the generation of a pure spin-polarized current in a particular direction when a heat gradient is applied along that direction. At 300 K, we obtained a maximum S_eff-spin of 1.8 mV/K in both electron and hole-doped systems along the x and y directions. Note that both S_eff-spin and S_eff-charge have approximately the same magnitude at all temperatures along the x and y directions. This feature results from the negligible value of either σ^↑ or σ^↓ along each direction as discussed earlier. Indeed, we achieved a giant effective Seebeck coefficient in the V₂S₂O monolayer altermagnet compared with previously reported structures. For instance, the maximum spin Seebeck coefficients reported for GaMnAs, Pt/YIG, Cr₂O₃, HfMnGe, EuTiO₃ and V₂SeTeO monolayers are ~5 μV/K, ~6 μV/K, 20 μV/K, ~80 μV/K, 0.1 mV/K, and 0.64 mV/K respectively at room temperature^[45-49].

Figure 4. Effective spin Seebeck coefficients at different temperatures in (A) x-direction vs. chemical potentials and in (B) y-direction vs. chemical potentials, and effective charge Seebeck coefficients in (C) x-direction vs. chemical potentials, and in (D) y-direction vs. chemical potentials.

We also investigated the electronic and lattice thermal conductivities. Figure 5A and B shows the K_e for the spin-up (red color) and spin-down (blue color) carriers along the x-direction and y-direction as a function of temperature and chemical potential. Similar to the σ, the electronic part of the thermal conductivity is also entirely determined by either of the spin carriers depending on the direction under consideration. To be precise, the spin-up (spin-down) carriers were wholly responsible for the Ke in the y-direction (x-direction). Also, the K_e in the hole-doped system was greater than that of the electron-doped system. This may also be linked to the relaxation time disparity between the two regions. The spin-down carriers had K_e of 2.24 Wm^-1K^-1 in the hole-doped system, while it became 0.49 Wm^-1K^-1 in the electron-doped system at 300 K with chemical potentials of ±0.49. Meantime, the spin-up channel carriers had Ke of 1.83 W.m^-1K^-1 in the hole-doped system and 0.41 Wm^-1K^-1 in the electron-doped system at the same conditions.

Figure 5. Electronic thermal conductivity along (A) x-direction and (B) y-direction for spin-up and spin-down at different temperatures (red color for spin-up and blue color for spin-down); (C) The lattice thermal conductivity vs. temperature along x-direction (color black) and y-direction (color red).

The K_L is given by:

where τ_i, v_i, and C_i are the phonon lifetime, group velocity, and mode heat capacity. Besides, the phonon mode, number of unit cells, and volume of the system are represented with i, N, and V. Figure 5C shows the lattice part of the thermal conductivity of the V₂S₂O monolayer along the x and y directions. The K_L was decreased with increasing temperature. This behavior agrees with the evolution law of K_L. For instance, at 100 K we obtained an isotropic K_L of 0.5 Wm^-1K^-1 in both directions. This value decreased to 0.2 Wm^-1K^-1 at 300 K. The low K_L at 300 K may be an indication of relatively stronger phonon-phonon interaction leading to substantial anharmonic scattering. Due to this stronger scattering, the phonon mean free path will be decreased, and this may result in low K_L. Overall, we found that the total thermal conductivity of the system was mostly governed by the K_e. For comparison, we present the total thermal conductivity of the V₂S₂O monolayer with other well-studied MoS₂ in Supplementary Table 1.Finally, we evaluated the spin and charge dimensionless figure of merit (ZT). We obtained the dimensionless ZT using the transport parameters in the relations:

where σ = σ^↑ + σ^↓, and K_e is the sum of the electronic part of the thermal conductivities of the spin-down and spin-up carriers. Figure 6A-D reveals the computed results. Due to the directional spin-dependence of the σ, we also obtained ZT_Charge = ZT_spin along the y and x directions. At 300 K, we obtained a maximum ZT of 0.86 in the hole-doped systems, while it became 0.63 in the electron-doped systems.

Figure 6. The dimensionless effective spin figure of merit in (A) x and (B) y-directions and the dimensionless charge figure of merit in (C) x and (D) y-directions vs. chemical potentials.

This carrier type dependent ZT feature comes from the electrical conductivity in which σ^↑(σ^↓) is greater in the hole-doped system along the y-direction (x-direction). Overall, the ZT is controlled by only one spin component in each direction. Our findings may suggest that the V₂S₂O altermagnet system may be used for potential pure spin current generation in spintronic and thermoelectricity.

CONCLUSIONS

We probed the spin-dependent transport properties of the altermagnet V₂S₂O monolayer using the spin-polarized density functional theory and Boltzmann transport theory. This V₂S₂O monolayer had a direct band gap of 1.15 eV and a uniform spin splitting of 0.51 eV at the X and Y points in the valence band maximum. This band gap was determined by the spin-down (spin-up) channels at the X-point (Y-point). We obtained a high critical temperature of about 746 K. Due to the directional spin-dependent feature in the band structure, the Seebeck coefficients, electrical conductivity, effective spin and charge Seebeck coefficients, electronic part of the thermal conductivity, and ZT are influenced by spin-down (spin-up) carriers in the x-direction (y-direction). The electrical conductivity obtained in the hole-doped systems is more than twice its value in the electron-doped systems in each direction at 300 K. We also found that the maximum magnitude of the effective spin and charge Seebeck coefficients was almost the same at all temperatures due to the orientational spin-dependent nature of the electrical conductivity. In the hole-doped systems, the electronic part of the thermal conductivities had higher values than in the electron-doped systems. We obtained a low isotropic K_L of about 0.2 Wm^-1K^-1 at 300 K. Essentially, the altermagnet V₂S₂O monolayer displayed a giant spin-dependent effective Seebeck coefficient of about 1.8 mVK^-1 at 300 K and at a small electron or hole doping. This value is multiple times greater than reported values for most bulk and 2D materials. Besides, we also obtained ZT_Charge = ZT_spin. The maximum ZT was 0.86 for the hole-doped systems and 0.63 for the electron-doped systems. Overall, we suggest that V₂S₂O altermagnet may be used for potential pure spin-polarized current generation in spintronics and thermoelectric device applications.