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Ashani et al. Energy Mater. 2025, 5, 500111 https://dx.doi.org/10.20517/energymater.2025.10 Page 3 of 13

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
[38]
implemented in the VAMPIRE simulation package 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
[39]
is used. Our directional spin-dependent transport properties were explored using the BoltzTraP code .
Since most of the transport properties are carrier relaxation time (τ) dependent, we considered the
scattering contributions from the impurity effects (τ ) polar optical phonon (τ ), and acoustic phonon
imp
opt
(τ ). Therefore, the total τ, which depends on energy and temperature, can be expressed as:
aco
(1)

In addition, we also computed the lattice part of the thermal conductivity with phono3py code using a
[40]
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 V S O. The red and black colors denote vanadium
2 2
atoms with spin-up and spin-down, while the blue and yellow colors represent the oxygen and sulfur atoms.
The monolayer V S O has a tetragonal crystal structure and a space group of P4mmm (123). We obtained
2 2
the lattice constants of a = b = 3.84 Å after structure optimization. The monolayer V S O is made of a V O
2
2 2
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 V S O monolayer exhibits high-degree symmetry along in-plane and out of plane. Note that
2 2
the thermal stability, cleavage energy, and dynamic stability were already reported elsewhere . To
[41]
investigate the magnetic ground state of monolayer V S O, we considered ferromagnetic (FM) and three
2 2
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 ) between this state and the FM state (E = E - E ) was 536 meV/unit cell.
AFM
ex
FM
ex
Each V atom had an atomic magnetic moment of magnitude 1.7 μ . However, the net magnetic moment of
B
the unit cell vanishes due to the anti-parallel coupling between the V atoms. The AFM-Neel ground state in
the V S O monolayer is consistent with other theoretical reports [6,41] . Figure 1E reveals the electronic band
2 2
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 V S O
2 2
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 based
[42]
on:
(2)

(3)

Here, S and S are the spin moment direction of atoms at the neighboring sites i and j, J is the exchange
j
i
ij
interaction between atoms at the neighboring sites, and k is the anisotropy energy per atom. Meanwhile, N α
u
and n represent the number of atoms in the sublattice and the mean of sublattice magnetization of each
α
atom. Using E , the difference between the total energies of the FM and AFM spin states, we extracted the
ex
exchange interaction via the relation: