Electronic structure, first and second order physical properties of MPS₄: a theoretical study

Linear optical properties described in this work are the absorption coefficient, refractive index and dielectric constant. All these quantities are presented as a function of frequency of the electromagnetic wave that propagates through the crystal. The most important optical property of a material is the complex dielectric constant ε. The CASTEP code can calculate its imaginary part ε₂(ω), using the following relation [26]: (1)ε2(ω)=2e2πΩε0∑k,v,c|〈ψkc|u→.r→|ψkv〉2δ(Ekc−Ekv−E)$${{\varepsilon }_{2}}\left( \omega \right)=\frac{2{{e}^{2}}\pi }{\Omega {{\varepsilon }_{0}}}\overset{k, v, c}{\mathop \sum }\,|{{\langle \psi _{k}^{c}\left| \vec{u}.\vec{r} \right|\psi _{k}^{v}\rangle }^{2}}\delta \left( E_{k}^{c}-E_{k}^{v}-E \right)$$ where u→$\overset{\to }{\mathop{\text{u}}}\,$ is the vector defining the polarization of the incident electric field, r→$\overset{\to }{\mathop{\text{r}}}\,$ is the position operator, ψkc$\text{ }\!\!\psi\!\!\text{ }_{\text{k}}^{\text{c}}$ and ψkv$\text{ }\!\!\psi\!\!\text{ }_{\text{k}}^{\text{v}}$ represent the valence band and the conduction band states in which the direct transitions are possible, Ekc−Ekv=ħω$\text{E}_{\text{k}}^{\text{c}}-\text{E}_{\text{k}}^{\text{v}}=\hbar \text{ }\!\!\omega\!\!\text{ }$ is the transition energy, ε₀ is the dielectric constant of free space, e is the electronic charge, Ω is the volume of unit cell and the integral is over the first Brillouin zone (BZ). The real and imaginary parts are linked by the Kramers-Kronig relation [27].

The other optical constants such as refractive index n(ω) and extinction coefficient k(ω) can be derived from the dielectric function using the relations: (2)ε1(ω)=n2(ω)−k2(ω)ε2(ω)=2n(ω).k(ω)$$\begin{array}{*{35}{l}} {{\varepsilon }_{1}}\left( \omega \right)={{n}^{2}}\left( \omega \right)-{{k}^{2}}\left( \omega \right) \\ {{\varepsilon }_{2}}\left( \omega \right)=2n\left( \omega \right).k\left( \omega \right) \\ \end{array}$$

In the Nunes and Vanderbilt formalism [28], the total energy of the electronic system perturbed by the electric field is a functional of the Wannier functions. It is given by the following relation [29, 30]: (3)E(wnE→)=E0(wn)−Ω0.E→.P→$$E\left( {{w}_{n}}\vec{E} \right)={{E}_{0}}\left( {{w}_{n}} \right)-{{\Omega }_{0}}.\vec{E}.\vec{P}$$ where E is the total energy of the system. E₀ is the energy of Kohn-Sham of the fundamental state, Ω₀ is the volume of cell, p→$\overset{\to }{\mathop{\text{p}}}\,$ is the macroscopic polarization, W_n is the Wannier function and E→$\overset{\to }{\mathop{\text{E}}}\,$ is the electric filed. The second term on the right depends explicitly on polarization. It is known as “energy of polarization” and noted as: (4)Epol=−Ω0.E→.P→$${{E}_{pol}}=-{{\Omega }_{0}}.\vec{E}.\vec{P}$$

E→$\overset{\to }{\mathop{\text{p}}}\,$. P→$\overset{\to }{\mathop{\text{E}}}\,$ acts as an external potential as well as the ionic potential; it is linear with the electric field and depends only on it. One can admit that the electric field plays the part of a perturbation. The second order dielectric susceptibility is calculated from the third order derivative of the energy of polarization with respect to electric field components [31]. Using the Voigt contracted notation: (5)dij=12xij(2)$${{d}_{ij}}\,=\,\frac{1}{2}x_{ij}^{\left( 2 \right)}$$ the tetragonal and orthorhombic (P222) crystal systems have four (d₁₄,d₁₅,d₃₁,d₃₆) and three (d₁₄,d₁₅,d₃₆) independent non-zero elements, respectively.

The electro-optic coefficients r_ijk characterize the linear electro-optical effect i.e. when the variation of refractive index is linearly proportional to the applied static or low frequency electric field. It is also known as the Pockels effect. These coefficients result from the electronic rijkel$\text{r}_{\text{ijk}}^{\text{el}}$ and ionic rijkion$\text{r}_{\text{ijk}}^{\text{ion}}$ contributions plus the contribution due to the opposite piezoelectric effect rijkpiezo$\text{r}_{\text{ijk}}^{\text{piezo}}$ [32]. There are four independent non-zero electro-optic elements (r₃₁, r₄₁, r₅₁, r₆₃) for tetragonal I-4 class crystals and three elements (r₄₁, r₅₂, r₆₃) for orthorhombic P222 class crystals.

Elastic properties of a material are very important because they describe its response to an applied stress, or conversely the stress required to maintain a given deformation, and they are related to various physical phenomena, such as transmission of acoustic and thermal waves through the material.

Properties such as the bulk modulus (response to an isotropic compression), Poisson ratio, Lame constants, and so forth may be computed from the values of elastic stiffness constants C_ij. Those constants are the relationships that determine the deformations produced by a given stress acting on a particular material. In tetragonal and orthorhombic crystal systems, the elastic tensor contains 7 (C₁₁, C₃₃, C₄₄, C₆₆, C₁₂, C₁₃, C₁₆) and 9 (C₁₁, C₂₂, C₃₃, C₄₄, C₅₅, C₆₆, C₁₂, C₁₃, C₂₃) independent elements, respectively. For small elastic deformations ε_i along the direction i of the crystal, its potential energy has the form: (6)Up=12∑i,j6Cijεiεj$${{U}_{p}}\,=\,\frac{1}{2}\underset{i,j}{\overset{6}{\mathop \sum }}\,{{C}_{ij}}{{\varepsilon }_{i}}{{\varepsilon }_{j}}$$ and the stability condition is to have a determinant [C_ij]> 0. This condition results in the following stability relations: for tetragonal and orthorhombic system, respectively: (7)Cij>0(i=1,3,4,6)C11−C12>0C11+C33−2C13>02C11+C33+2C12+4C13>0$$\begin{array}{*{35}{l}} {{C}_{ij}}\,>\,0\left( i\,=\,1,3,4,6 \right) \\ {{C}_{11}}\,-\,{{C}_{12}}\,>\,0 \\ {{C}_{11}}\,+\,{{C}_{33}}\,-\,2{{C}_{13}}\,>\,0 \\ 2{{C}_{11}}\,+\,{{C}_{33}}\,+\,2{{C}_{12}}\,+\,4{{C}_{13}}\,>\,0 \\ \end{array}$$(8)Cii>0i=1to6∑i≠j≠13Cii+2Cij>0Cii+Cjj−2Cij>0i≠j=1,2,3$$\begin{array}{*{35}{l}} {{C}_{ii}}\,>\,0\left( i\,=\,1\,\text{to}\,6 \right) \\ {\sum \limits_{i\ne j\ne 1}^3} {{C}_{ii}} +\,2{{C}_{ij}}\,>\,0 \\ {{C}_{ii}}\,+\,{{C}_{jj}}\,-\,2{{C}_{ij}}\,>\,0\left( i\,\ne \,j\,=\,1,2,3 \right) \\ \end{array}$$

Several mechanical properties can be calculated through these coefficients, most important are: bulk B and shear G modulus, from which one can calculate the elastic and Young’s E modulus, and Poisson’s ratio ˛ [33].

From the response function capabilities of ABINIT, it is possible to estimate the piezoelectric coefficients dij by calculating the second derivatives of the total energy (2DTE) with respect to electric fields, and Cartesian strains [19]. In this method strains are treated as perturbations. The piezoelectric tensor, contains 3 × 6 elements. The total number of independent piezoelectric tensor members is determined by the point group of the material. In tetragonal and orthorhombic crystals systems, the matrix of piezoelectric tensor d contains four (d₁₄, d₁₅, d₃₁, d₃₆) and 3 (d₁₄, d₁₅, d₃₆) non zero elements, respectively; they have the same form and number of independent constants as nonlinear optical tensor [34].

Full structure relaxation was carried out for both compounds for which the Hellman-Feynman forces acting on sulphur atom passed from few eV/ Å prior to relaxation to lower than 10⁻3 eV/ Å after the process. The calculated lattice parameters and atomic positions of sulphur atom in InPS₄ and AlPS₄ are presented in Table 1. Compared to the experimental values the calculated cell parameters are on average 0.83 % underestimated by the LDA and 6.92 % overestimated by the GGA. It should be noted here that the calculated a and b cell parameters of AlPS₄ have almost the same values which makes us think that its crystalline structure is tetragonal instead of the orthorhombic one. Our theoretical calculations also show a significant increase of c cell parameter of this compound.

The calculated energy band structures, total and partial density of states of InPS₄ and AlPS₄ are presented in Fig. 2. From these figures, one can draw several important remarks concerning the electronic properties of the two materials. Both compounds have indirect wide band gap: 2.535 eV for InPS₄ (M to G) and 2.706 eV for AlPS ₄(Y to S). Certainly these values are underestimated compared to the experimental ones, as in the case of InPS₄ which has an experimental gap of 3.44 eV [12]. This is a known failure of DFT attributed to the fact that this theory treats the ground state only and to the discontinuity in the exchange-correlation potential [16]. Furthermore, we notice an appreciable similarity between the band structures and density of states of our compounds. This has been much expected because of the presence of [PS₄]³− and [MS₄]³+ tetrahedrons in both compounds. The valence bands of the materials are dominated by the contribution of p states of sulfur with a large contribution of p states of phosphorus and s states of the metal in the lower subband around − 5 eV. The conduction band of the compounds is dominated by p electron states of phosphorus and sulfur and s and p states of the metal atoms. Furthermore, the d states of indium are localized around − 14 eV and have no contribution to the valence or conduction bands. They are expected to play no role in the optical properties of InPS₄.

Fig. 2

We also carried out Mulliken population analysis that reflects the nature of bond between two atoms. It has a value ranging between 0 and 1. The tendency towards 0 indicates a dominant ionic character whereas the tendency towards the unity indicates the domination of covalent character. The intermediate interval shows a mixing character and a negative value indicates that the bond does not exist.

The results of Mulliken population analysis for both the compounds are presented in Table 2. These data indicate the existence of two types of mixed bonds in each compound. The P–S bonds are slightly skewed towards the covalent character whereas the M–S bonds have a rather ionic nature. The sign of DZ in Table 3 informs us on the donor or acceptor role played by each element in the compound. These results reveal a donor character of phosphorus and an acceptor character of sulfur atoms in both the compounds. The metal atoms are donors with higher activity of aluminum over indium. We also notice the absence of indium d electrons activity in donor-acceptor character.

Graphs in Fig. 3 present the calculated real and imaginary parts of the complex dielectric constant ε(ω) = ε₁(ω)+ iε₂(ω) the refractive index η(ω) and the absorption coefficient (ω). These quantities are plotted along the electric field polarization directions 〈1 0 0〉, 〈0 1 0〉 and 〈0 0 1〉.

Fig. 3

We can also follow the electronic transition from the valence band to the conduction band through the variations of imaginary part of ε(ω) which presents several peaks. There are three essential peaks in the imaginary part ε2001(ω)$\text{ }\!\!\varepsilon\!\!\text{ }_{2}^{001}(\text{ }\!\!\omega\!\!\text{ )}$ of InPS₄. The first one, A at 3.40 eV, corresponds to transitions from S 3p (VB) to In 5s and S 3p (CB). The next peak, B at 5.77 eV, is mainly due to transitions from S 3p (VB) to s and p states of the three elements in the CB. The third peak at 7.70 eV is due to the electronic transitions from S 3p and P 3p (VB) to In 5s, S 3p and P 3s of the CB. In case of AlPS₄ we notice a dominant peak at 8.27 eV which mainly corresponds to transitions from 3p states of sulfur and phosphorus to s and p states of three constituent elements in the CB. It should be noted that in the previous discussion only vertical interband transitions have been taken into consideration, since many direct and indirect electronic transitions may occur with an energy corresponding to the same peak.

We can also see similarities and differences between ε¹⁰⁰(ω) and ε⁰⁰¹(ω). The similarities are as follows: for low frequencies, which correspond to energies less than band gap energy (E_g(InPS₄) = 2.33 eV, E_g(AlPS₄) = 2.70 eV), ε2100(ω)$\text{ }\!\!\varepsilon\!\!\text{ }_{2}^{100}(\text{ }\!\!\omega\!\!\text{ )}$ and ε2001(ω)$\text{ }\!\!\varepsilon\!\!\text{ }_{2}^{001}(\text{ }\!\!\omega\!\!\text{ )}$ are negligible and the material behaves as a transparent dielectric. The energy absorption range starts from E_g and extends up to about 15 eV. In this range the material behaves like a dielectric absorber for energies lower than the energy of resonance, i.e. 5.13 eV (InPS₄), 8.53 eV (AlPS₄) which corresponds to point D in Fig. 3, and as metallic absorber in the rest of the range, where it becomes reflective. Beyond this range (E > 15 eV) the material becomes transparent with the dielectric constant tending to 1 as shown in Fig. 3. Most notable is the small difference between ε1100(ω)$\text{ }\!\!\varepsilon\!\!\text{ }_{1}^{100}(\text{ }\!\!\omega\!\!\text{ )}$ and ε1001(ω)$\text{ }\!\!\varepsilon\!\!\text{ }_{1}^{001}(\text{ }\!\!\omega\!\!\text{ )}$ which confirms the anisotropy and birefringence nature of the medium. This is best illustrated by the static limits of the calculated refraction indices of the compounds presented in Table 4 together with available experimental data. These values show a positive birefringence (n_e(ω)> n₀(ω)) for the uniaxial compound (InPS₄) which is in good agreement with the experimental data of Jantz et al. [10] (Fig. 4). We should emphasize that no experimental or previous theoretical data for AlPS₄ are available in the literature to make a meaningful comparison.

Fig. 4

InPS₄ has four nonzero second order NLO tensor elements (d₁₄, d₁₅, d₃₁, d₃₆) imposed by the point group of the crystal and four electro-optic coefficients (r₁₃, r₄₁ r₅₁, r₆₃). Assuming Kleinmann symmetry conditions [35], these elements reduce to two independent elements d₁₄ = d₃₆ and d₁₅ = d₃₁. On the other hand, AlPS₄ second order NLO tensor reduces to three non-vanishing elements (d₁₄ = d₂₅ = d₃₆).

The calculated NLO tensor elements d_ij and electro-optical coefficients r_ij are presented in Table 5. The calculated d_ij for InPS₄ are in fairly good agreement with the experimental counterparts with less than 10 % difference, whereas no experimental or theoretical data for AlPS₄ are available. Since our calculations procedures gave good results for InPS₄, we believe that NLO coefficients for AlPS ₄are reliable.

Regarding the electro-optic coefficients, to our knowledge no theoretical or experimental results have been published for these two materials. Therefore, our results show that InPS₄ is better candidate for electro-optic applications with higher E-O coefficients than AlPS₄. Nevertheless, we note that both compounds have low electro-optic coefficients compared to the standard ones.

The calculation of the piezoelectric tensors by Abinit code enabled us to evaluate four independent elements for InPS₄ and three for AlPS₄. These elements are presented in Table 6.

The calculated piezoelectric coefficients for InPS₄ are in good agreement with the experimental data except for d₁₄. Zhang et al. [36] explained this discrepancy by the fact that the piezoelectric coefficients are very sensitive to crystal quality, design of the samples and data processing method. On the other hand, these results show that AlPS ₄has higher piezoelectric coefficients than InPS₄ and better coefficients than reference materials such as quartz (d₁₄ = 0.727 pC/N, d₁₁ = 2.310 pC/N). It should be noted that no experimental or theoretical data are available to make a meaningful comparison.

The calculated elastic coefficients of the compounds are presented in Table 7. It is clear that the calculated C_ij of InPS₄ are in good agreement with the available experimental data. Moreover, the elastic coefficients calculated by both the codes are very close. They also show that elastic coefficients of InPS₄ are twice smaller than those of quartz for both longitudinal and transverse elements. The elastic properties of AlPS₄ have not been reported in the literature yet. Since our calculations for InPS₄ gave good results, we believe that the present elastic coefficients for AlPS₄ can be taken as a reference for future investigations.

However, these theoretical results show that the compound (AlPS₄) has lower C₃₃ as well as trans-verse coefficients compared to InPS₄. All stability criteria equation7 and equation8 are satisfied for both compounds except for Abinit calculated C₆₆ of AlPS₄ that shows small negative value. As pointed out earlier and shown in Fig. 1, the one dimensional alternating AlS₄-PS₄ chains along x and y directions are hold together in the z direction by weak Van der Waals interactions. This configuration of atoms explains the high values of C₁₁ and C₂₂ and the small values of z oriented C₃₃, C₆₆ coefficients. It also explains the unexpected negative values of C₆₆ (Abinit) and C₁₂ (Abinit) since Van der Waals interactions are not well accounted by standard DFT calculations. This is due to the fact they are long-range forces between fragments across lower density regions which are not accounted in local (LDA) or semi local (GGA) approximations of the exchange-correlation functional [37].