Anisotropic transport and optical birefringence of triclinic bulk and monolayer NbX₂Y₂ (X = S, Se and Y = Cl, Br, I)

NbX₂Y₂ (X = S, Se and Y = Cl, Br, I) crystallize in triclinic symmetry with space group $P\overline{1}$ [22]. It has two formula units with 10 atoms in the unit cell and the ionic charge is given as Nb⁴⁺(X₂)²⁻(Y₂)²⁻. The bulk structure of NbX₂Y₂ is shown in figure 1(a). The structure contains octahedral cage shaped units of Nb₂X₄ (X = S, Se) which consist of a pair of Nb atoms lying perpendicular to the plane of two X₂ groups. These Nb₂X₄ cages are linked by halogen atoms forming a layer in 'a–b' plane stacked along 'c' axis, making the bulk layered structure [34]. We have done the geometry optimization using GGA and LDA exchange–correlation functionals which is given in table 1. We found a large discrepancy between the theory and experimental volume. So, we have included van der Waal's corrections and observed that the parameters calculated with zero damping DFT-D3 method of Grimme are in good agreement with experiment, which confirms that the layers are bound by weak van der Waal's interaction. The calculated lattice parameters and volume along with experimental values are given in table 1. Figure 1(c) illustrate the Brillouin zone (BZ) and associated 2D BZ for monolayer is considered for further calculation.

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Table 1. Experimental and optimized lattice parameters of NbX₂Y₂ (X = S, Se and Y = Cl, Br, I) and inter-layer distance 'd'(${\text{A}^\circ}$). The experimental values were taken from reference [22].

		NbS2Br2	NbS2I2	NbSe2Cl2	NbSe2Br2	NbSe2I2
Expt.	a(${\text{A}^\circ}$)	6.52	6.77	6.37	6.66	7.06
	b(${\text{A}^\circ}$)	6.58	6.80	6.53	6.72	7.20
	c(${\text{A}^\circ}$)	7.25	7.23	7.26	7.08	7.75
	V(${\text{A}^\circ}^{3}$)	242.70	282.36	236.80	256.11	302.60
LDA	a(${\text{A}^\circ}$)	6.47	7.07	6.15	6.47	6.96
	b(${\text{A}^\circ}$)	6.92	6.58	5.66	6.63	6.89
	c(${\text{A}^\circ}$)	6.92	7.38	6.37	7.19	7.56
	V(${\text{A}^\circ}^{3}$)	231.92	267.01	222.77	243.70	281.78
GGA	a(${\text{A}^\circ}$)	6.71	7.02	6.47	6.76	7.25
	b(${\text{A}^\circ}$)	6.63	6.84	5.77	6.80	7.09
	c(${\text{A}^\circ}$)	8.10	7.82	7.44	8.16	8.59
	V(${\text{A}^\circ}^{3}$)	283.86	287.99	274.39	299.11	347.17
DFT-D2	a(${\text{A}^\circ}$)	6.67	7.10	6.33	6.20	7.19
	b(${\text{A}^\circ}$)	6.59	6.84	6.55	6.79	7.05
	c(${\text{A}^\circ}$)	7.30	7.82	6.50	7.35	7.67
	V(${\text{A}^\circ}^{3}$)	249.30	287.99	241.59	264.13	303.15
DFT-D3	a(${\text{A}^\circ}$)	6.61	7.10	6.31	6.66	7.17
	b(${\text{A}^\circ}$)	6.58	6.84	6.56	6.76	7.02
	c(${\text{A}^\circ}$)	7.28	7.84	7.30	7.70	7.73
	V(${\text{A}^\circ}^{3}$)	246.76	287.88	239.18	262.19	302.43
Inter-layer distance	d(${\text{A}^\circ}$)	2.48	2.83	2.21	2.31	2.58

Elastic constants play very important role in determining the mechanical stability of the system. The elastic tensor is calculated for all the investigated systems and the same is given in supplementary table 1. The elastic constants C₁₁ and C₂₂ are larger than C₃₃ which indicates that these systems are easily compressible along [0 0 1] direction because of the weak van der Waal's force between layers along 'c' axis. According to the crystallographic symmetry, the triclinic structure has 21 independent elastic constants [35]. Due to the low symmetry of the triclinic structure, the off-diagonal elements are negative [36]. The calculated elastic constants of NbX₂Y₂ satisfy the born stability criteria and the eigenvalues of the diagonal matrix is positive which confirms that the compounds are mechanically stable [37]. We have calculated bulk aggregate mechanical properties such as young's modulus (E), bulk modulus (B), shear modulus (G), which are presented in table 2. To calculate those mechanical properties, we have used Hill's approximation which is the averaged results of Viogt and Ruess approximations [38]. We have observed that the Young's modulus is the dominating factor in NbX₂Y₂ crystalline material than the shear and bulk moduli. Moreover, we observe that the stiffness of the material decreases as the size of the atoms increases. Pugh's ratio B_H/G_H indicates whether the material is ductile or brittle [39]. For a ductile material B_H/G_H > 1.75 and for a brittle material, B_H/G_H < 1.75. The B_H/G_H ratio of all the compounds are greater than 1.75 which indicates that they are ductile in nature. We have also investigated the elastic anisotropy by computing the universal anisotropy factor A^U which is given by [40],

$$\begin{equation}{A}^{\text{U}}=5\frac{{G}_{\text{V}}}{{G}_{\text{R}}}+\frac{{B}_{\text{V}}}{{B}_{\text{R}}}-6\end{equation} \tag{ 3 }$$

Here, G_V and G_R are shear modulus, and B_V and B_R are bulk modulus under Voigt and Ruess approximations respectively. When A^U = 0 the material is isotropic, otherwise it turns out to be anisotropic. All of our structures possess huge anisotropy factor. Especially, the elastic anisotropy of NbS₂I₂ is 8.18. This value is higher than the elastic anisotropy of some other previously reported triclinic compounds such as Mn₄Al₁₁ (0.16), Si₄P₄Ru (0.57), TlSbS₂ (6.64) etc [41].

We have calculated the Debye temperature (θ_D) for the investigated compounds and the values are given in table 2 [42]. The values of θ_D are very low for all the compounds. It is known that softer solids have low Debye temperature and the calculated values of B, G and Y indicates the softness of NbX₂Y₂ [43]. Also, as size of the atom increases θ_D decreases. We expect that the intrinsic lattice thermal conductivity of NbX₂Y₂ would be low, due to it is low debye temperature (θ_D) which is favourable for thermoelectric applications [44]. Especially, the value of θ_D for NbSe₂I₂ is 149.8 K, which is lower than several previously reported thermoelectric materials such as MoS₂ (262 K) [45], K₂Cu₂GeS₄ (373 K) [46], Cu₃NbS₄ (332 K) [47],TaCoSn (375 K) [48] etc.

The phonon dispersion is calculated and the phonon bandstructure of NbS₂Br₂ is given in figure 2. For the remaining compounds the same is given supplementary figure 1 (https://stacks.iop.org/JPCM/33/485501/mmedia). The absence of imaginary phonon modes confirms the dynamical stability in the investigated compounds. The phonon frequency is decreasing with higher atomic masses which is a common trend in phonon dispersion and the same is observed here. The maximum range of phonon frequency is observed for NbS₂Br₂ which is 17.45 THz. For all the compounds, the acoustic modes are degenerate at the high symmetry point (Γ). The phonon dispersion curves of investigated compounds reveals flat phonon bands and also the optical modes lying in low frequency region are highly interacting with acoustic modes. These features also indicates the low thermal conductivity of NbX₂Y₂.

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The electronic structure properties of NbX₂Y₂ (X = S, Se and Y = Cl, Br, I) compounds were calculated using PBE functional. An accurate calculation of band gap is very important for evaluating thermoelectric and optical properties. Since the PBE functional underestimates the band gap in comparison to the reported experimental band gap values, we have employed TB-mBJ functional in our calculations which improve the band gap values [22]. The bandstructure of NbX₂Y₂ using TB-mBJ is showing in figure 3(a) and in supplementary figure 2 and the comparative band gap values of PBE and TB-mBJ functionals are given in figure 3(b). From the calculated electronic structure, we can conclude that these compounds are semiconductors and consistent with experiment as well [22]. For all the compounds, the valence band maximum (VBM) is positioned at the high symmetry point 'T' and conduction band minimum (CBM) is positioned at the high symmetry point 'Y', confirming these compounds as indirect band gap semiconductors, and we have also observed direct band gap at the high symmetry point 'T' [49]. The band gap is decreasing with increasing size of the atom. The partial and total density of states (DOS) of NbX₂Y₂ are shown in figures 4(a) and (b) respectively. We can observe that the Nb-d states are contributing more together with X-p and Y-p states in valence band region. Nb-d, X-p and Y-p states are contributing more in the conduction band region. Also, the presence of flat nature in the valence band and dispersive conduction band reveals the possibilities for a promising thermoelectric materials, and the same is discussed in upcoming section.

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We now discuss the calculated transport properties for bulk NbX₂Y₂ based on the semi-classical Boltzmann transport theory. Thermoelectric properties of the investigated compounds are being calculated as a function of carrier concentration. Our results show that these compounds have large thermopower along with appreciable electrical conductivity, which is essential for a thermoelectric material. Also, all the compounds exhibit anisotropic nature in thermoelectric properties, which can be attributed to the low symmetry of triclinic systems. So, it is very important to analyse the properties in all the three crystallographic axes. The values of the band gaps are in the optimal range for a good thermoelectric performance. From the DOS plots, we can observe that all the five compounds have large DOS just below Fermi level, which is also reflected in the more flat nature of valence band in electronic band structure. Such an observed trend in the DOS is expected to result in larger thermopower [50].

Here, we have discussed the transport properties for the fixed optimal doping concentraion of 10¹⁹ cm⁻³ at room temperature T = 300 K. The calculated values of thermoelectric coefficients are shown in table 3. All the investigated compounds exhibit quite a large thermopower for both the carriers. For example, NbS₂Br₂ has thermopower value of around 484 μV K⁻¹ for holes and NbSe₂Cl₂ has thermopower value of around 490 μV K⁻¹ for electrons. The thermopower is decreasing as a function of carrier concentration and the same is increasing with temperature.

We have plotted the electrical conductivity scaled by relaxation time (σ/τ), due to the assumption of CSTA. The conductivity increases with the carrier concentration. The maximum electrical conductivity is observed along 'c' axis for holes and for the electrons along the 'a' axis. For instance, in NbSe₂I₂, the hole conductivity is 2.06 × 10¹⁷ Ω⁻¹ m⁻¹ s⁻¹ along 'c' axis and the electron conductivity is 4.17 × 10¹⁷ Ω⁻¹ m⁻¹ s⁻¹ along 'a' axis. In general, the electrical conductivity due to electrons is higher than that of holes, which is due to the more dispersive nature of the conduction band. The anisotropy of thermoelectric behaviour is an important highlight for the present series. Though the anisotropic nature of thermopower is less prominent along different directions, the anisotropy in electrical conductivity (σ/τ) is quite large which is reflected in the power factor (S² σ/τ) as well. Also, the anisotropy in conductivity is more prominent for electrons compared to holes. The electrical conductivity of NbS₂I₂ for different directions and the ratio of electrical conductivity along different axes for NbS₂I₂ is shown in figures 5(a) and (b), from which we can observe that σ_a is 40 times higher than σ_c and σ_b is 25 times higher than σ_c which is huge anisotropy compared to earlier reports [51].

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The calculated power factor for bulk NbX₂Y₂ is shown in figure 6 and 7. In general, the powerfactor is high for electrons compared to holes along 'a' and 'b' axes. But, we observed an opposite trend along 'c' axis where the power factor is high for hole carriers than electrons. For example in NbS₂Br₂, the S² σ/τ for electrons along 'a' and 'b' axes are 0.54 × 10¹¹ (W mK⁻² s⁻¹) and 0.26 × 10¹¹ (W mK⁻² s⁻¹) respectively. But, for holes, along the same corresponding axes, it is 0.10 × 10¹¹ (W mK⁻² s⁻¹) and 0.13 × 10¹¹ (W mK⁻² s⁻¹). On the otherhand, the S² σ/τ along 'c' axis is 0.30 × 10¹¹ (W mK⁻² s⁻¹) for holes and 0.017 × 10¹¹ (W mK⁻² s⁻¹) for electrons. Also, the maximum power factor of 0.77 × 10¹¹ (W mK⁻² s⁻¹) is observed in NbSe₂Cl₂, along 'a' axis for electron concentration. These results suggest that n-type carriers have better thermoelectric performance in 'a' axis for NbX₂Y₂. Also the average power factor for n-type is high compared to p-type. Therefore, comparing the behaviour of both electron and holes, we can conclude that the our compounds are more favourable for n-type doping.

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For a better thermoelectric performance, the material should have low lattice thermal conductivity. As mentioned earlier, the lattice thermal conductivity of NbX₂Y₂ is expected to be very low since they possess very low Debye temperature. Materials with low Debye temperature have more modes activated at room temperature. This causes high scattering rate of phonons due to the increasing phonon population. Moreover, the strong coupling between the acoustic branches and low lying optical branches can also induce more phonon scattering, thereby reducing the lattice thermal conductivity [52]. In order to estimate the lattice thermal conductivity of investigated compounds, we used Slack's model, which captures the experimental results more correctly [53]. The predicted values of lattice thermal conductivity and Gruneisen parameter of investigated compounds are shown in table 4. In particular, the κ_l of NbS₂I₂ is nearly 0.688 Wm⁻¹ K⁻¹. We observed that the calculated lattice thermal conductivity is very low [43, 54, 55]. Such a low thermal conductivity is desirable for any thermoelectric applications. Since the investigated compounds have high power factor and expected to have low lattice thermal conductivity, these compounds could be good candidates for future thermoelectric applications.

The optical properties of the NbX₂Y₂ compounds is studied for the energy range of 0–10 eV. The calculated real and the imaginary part of dielectric constants, refractive index, and absorption coefficient of NbS₂Br₂ are given in figure 8. The complex dielectric function is (ω) = ₁(ω) + i₂(ω), where ₁ is the real part and ₂ is the imaginary part of the dielectric function. Since it provides intrinsic optical characteristics of materials, the real and imaginary parts of the dielectric function spectra are known as 'optical constants'. Also, if we know imaginary part of the dielectric function, we can find real part of the dielectric function using Kramers–Kronig relation and the other optical properties, i.e., the refractive index n(ω) and absorption coefficient α(ω) etc, can be obtained from ₁(ω) and ₂(ω) [56].

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The dielectric tensor constants are position and symmetry dependent. According to the crystal symmetry, the dielectric tensor of triclinic structure is given as six independent components including off-diagonal elements and three frequency dependent parameters due to orientation of axes given as [14, 57, 58],

$$\begin{equation}{\epsilon}=\left[\begin{array}{ccc}{{\epsilon}}_{xx}& {{\epsilon}}_{xy}& {{\epsilon}}_{xz}\\ {{\epsilon}}_{xy}& {{\epsilon}}_{yy}& {{\epsilon}}_{yz}\\ {{\epsilon}}_{xz}& {{\epsilon}}_{yz}& {{\epsilon}}_{zz}\end{array}\right]\end{equation} \tag{ 4 }$$

The static dielectric constant is an important factor used to estimate the capacitance based on the capacitance formula where, the static dielectric constant is directly proportionate to capacitance [59]. The static dielectric constants ₁ are calculated from real dielectric part. The static dielectric part ₁ is directly related with static refractive index n and is given by $n=\sqrt{{{\epsilon}}_{1}}$. We have observed a strong anisotropy in static dielectric constant between cross plane direction z' and the in-plane directions 'x' and 'y'. The same trend is reflected in the calculated refractive indices, and the same are given in table 5.The maximum imaginary dielectric (₂) response of the material in all the triclinic structures are in y-direction. The interband transition is mainly from the Nb-d states of valence band to the X-p or Y-p states of conduction band. Also, the first transition is observed in visible region and second transition is observed in ultra-violet (UV) region.

The optical absorption is an important factor of the material, considering any optoelectronic applications. The first peak is observed nearly at 2.5–3 eV in the visible region and as energy increases, we can observe the second peak at 5–6 eV in the UV region. The peak emerged from the electronic transition of Nb-d states in valence band to X-p and Y-p in conduction band. There are very few studies which reported the off-diagonal elements in the literature. For example, in Ga₂O₃, _xz = 0.3 and in CdWO₄, _2xy ≈ 2 [15, 16]. But, in all of NbX₂Y₂, the off-diagonal components have high values. For instance, in NbS₂Br₂, the ${{\epsilon}}_{{2}_{xy}}\approx 5$. These huge off-diagonal values reflect in the anisotropy of other properties such as absorption coefficients and refractive indices.

The refractive index of the NbX₂Y₂ is given in table 5. The maximum refractive index observed in NbSe₂I₂ is 3.85 at 2.5 eV in the y-direction and the same is shown in supplementary figure 3. Clearly, the anisotropic nature is observed in the refractive index which can be measured from birefringence [60]. In general, a light ray entering a birefringent crystal splits into two propagating eigenstates with orthogonal linear polarization and different propagation directions. Each of the two polarization states experience a different refractive index. The static value of birefringence is calculated by Δn_xy (n_xx − n_yy ), Δn_xz (n_xx − n_zz ) and Δn_yz (n_yy − n_zz ) and the same is shown in figure 8(d) for NbSe₂I₂ and the birefringence values for all the investigated compounds are given in table 6. The value of Δn_xy is less than Δn_xz and Δn_yz . We have observed the maximum value of static birefringence to be 0.313 for NbSe₂I₂. Also the calculated value is much large compared to some reported common birefringent crystals such as CaCO₃ (0.16) [61], YVO₄ (0.26) [62], LiGaSe₂ (0.045) [63] etc [64]. These interesting optical properties of NbX₂Y₂ compounds may certainly fetch possible applications in polarized devices, beam splitter [19] and scintillators [65].