## 1 Introduction

Modern technological developments require more research on the physical properties of materials such as optical and mechanical properties, which are closely linked to crystalline and electronic structures of the materials. In a recent survey [1], it has been pointed out that chalcogenide materials are potential candidates for nonlinear optical applications. They possess high nonlinear optical (NLO) coefficients and large transparency domains extending in the middle-IR region above 5 µm. Furthermore, chalcogenide materials show rich structural and compositional diversity and excellent IR transparency [2].

In the MPS_{4} (M = Al, B, Ga and In) family, InPS_{4} and AlPS_{4} crystallize in a noncentrosymmetric structure and hence, they are good candidates for piezoelectric and electro-optic devices. Furthermore, they are anisotropic which makes them also valuable candidates for second order NLO applications. Their large band gaps, of the order of 3.5 eV, enables them to have good optical transparency in the visible and ultraviolet region of the spectrum without suffering from the double photon absorption in NLO processes [3]. Their crystallographic structure belongs to the so called twice defective chalcopyrite [4]. Several groups of researchers have made these compounds the object of their experimental studies. However, to the authors’ knowledge, no theoretical calculations of electronic band structure or elastic, piezoelectric, electro-optic and NLO properties of the compounds have been reported.

In 1970, the synthesis and growth of small single crystals of InPS_{4} from vapor phase by iodine transport have been reported [5]. The crystallo-graphic structure was determined by Carpentier et al. [6]. It belongs to the non-centrosymmetric space group I-4 with a = 5.60 Å and c = 9.02 Å. It is a three dimensional structure made of corner-connected [PS_{4}] and [InS_{4}] tetrahedrons (Fig. 1a). Bridenbaugh [7] reported that InPS_{4} has relatively large nonlinear second order optical coefficients. In 1975, Bubenzer et al. [8] synthesized crystals of InPS_{4} up to 10 mm using improved vapor growth procedure. They also measured their piezoelectric constants by a non-resonant dynamic method and found that this compound has good coefficients with d_{36} = 21.0 pC/N. Huard et al. [9] synthesized large InPS_{4} single crystals with good optical transparency and used Brillouin scattering method to measure their elastic constants. A complete survey of the elastic and optical properties of InPS _{4}was reported by Jantz et al. [10]. They also evaluated the elastic tensor and found similar results to those reported in the literature [9]. In addition, they reported dispersion curves of the ordinary and extraordinary refraction indices in the entire transparency range of the investigated compound and found a birefringence of the order of 0.04. Furthermore, they measured the second order NLO coefficients and found d_{31} = 36 pm/V and d_{36} = 28 pmV. On the other hand, in 1993, Bolcatto et al. [11] used tight-binding method to calculate the electronic structures of the MPS_{4} (M = B, Al, Ga, and In) thiophosphate family. In 2000, Lavrentyev et al. [12], investigated both experimentally and theoretically X-ray absorption near edge structure (XANES) of K-absorption spectra of P and S atoms in InPS_{4} and in 2003 they focused their research on the electronic structure and chemical bonding in InPS_{4} compound together with Tl_{3}PS _{4}and Sn_{2}P_{2}S_{6} [13]. They presented the partial electronic density of states from the experimental measurements using X-ray spectroscopy and theoretical quantum mechanical calculations, using abinitio multiple scattering FEFF8 code. The same group in 2006 presented their results on the influence of pressure on the birefringence of ZnS, CuGaS_{2} and InPS_{4} compounds using the modified method of augmented plane waves and the code WIEN2k [14]. As far as AlPS_{4} is concerned, no experimental or theoretical study has been reported except its crystal structure [11]. This study revealed that this compound crystallizes in the orthorhombic non-centrosymmetric space group P222 with the following cell parameters: a = 5.61 Å, b = 5.67 Å, c = 9.05 Å. AlPS_{4} crystal structure consists of edge-connected [AlS_{4}] and [PS_{4}] forming helicoidal chains along x and y axis as shown Fig. 1b. These chains are linked together by week Van der Waals interactions.

The aim of this work is the theoretical calculation of electronic structure and physical properties of aluminum and indium thiophosphates by means of DFT and modern theory of polarization.

The remaining part of this paper is organized as follows: after this Introduction, Section 2 gives the computational details. Section 3 presents the calculated results and thorough discussions on the structural, elastic and optical properties of the materials. A brief summary and conclusion are presented in Section 4.

## 2 Computational

The electronic structure, linear optical properties and elastic constants were calculated using the pseudopotential plane wave method, within density functional theory (DFT) as implemented in CASTEP computer code [15]. The generalized gradient approximation of Predew-Burke-Ernzerhof (GGA) and local density approximation of Ceperley-Alder (LDA) were used as functional for exchange and correlation effects [16]. Ultra-soft pseudopotentials (USP) [17] were used with the following valence electronic configurations In (4d^{10} 5s^{2} 5p^{1}), Al (3s^{2} 3p^{1}), P (3s^{2} 3p^{3}), and S(3s^{2} 3p^{4}). In the elementary cells of the compounds the atomic positions are: In (0, 1/2, 3/4), P (0, 0, 0) for InPS_{4} and Al (0, 0, 0), (1/2, 0, 1/2), P (0, 1/2, 0), (0, 0, 1/2) for AlPS_{4}. These atomic positions are not affected by the optimization because they are fixed by symmetry, however, it was necessary to relax the position of sulphur atoms in both compounds because of its general position (x, y, z).

The second order optical susceptibilities, piezoelectric and electro-optic coefficients were calculated within the framework of modern theory of polarization and density-functional perturbation theory DFPT as implemented in the open-source ABINIT code package [18, 19]. This theory has been shown to give successful description of the dielectric and piezoelectric properties of a wide range of materials in which electronic correlation is not too strong [20–22]. These calculations were carried out using Troullier-Martins norm-conserving pseudopotentials [23]. The exchange-correlation effects were treated within the (LDA) approximation [24] and Perdew-Wang 92 functional parameterizations [25]. For the crystal structure relaxation, electronic structure and all physical properties calculations, the convergence has been reached for both codes with an energy tolerance of 10^{−}6 eV with E_{cut} = 600 eV and a 6 × 6 × 4 mesh grid for Brillouin zone sampling.

### 2.1 Linear optical properties

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 ε_{2}(ω), using the following relation [26]:

_{0}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:

### 2.2 Nonlinear optical properties

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]:

_{0}is the energy of Kohn-Sham of the fundamental state, Ω

_{0}is the volume of cell,

_{n}is the Wannier function and

_{14},d

_{15},d

_{31},d

_{36}) and three (d

_{14},d

_{15},d

_{36}) 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 _{31}, r_{41}, r_{51}, r_{63}) for tetragonal I-4 class crystals and three elements (r_{41}, r_{52}, r_{63}) for orthorhombic P222 class crystals.

### 2.3 Elastic properties

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 (C_{11}, C_{33}, C_{44}, C_{66}, C_{12}, C_{13}, C_{16}) and 9 (C_{11}, C_{22}, C_{33}, C_{44}, C_{55}, C_{66}, C_{12}, C_{13}, C_{23}) independent elements, respectively. For small elastic deformations ε_{i} along the direction i of the crystal, its potential energy has the form:

_{ij}]

*>*0. This condition results in the following stability relations: for tetragonal and orthorhombic system, respectively:

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].

### 2.4 Piezoelectric coefficients

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 (d_{14}, d_{15}, d_{31}, d_{36}) and 3 (d_{14}, d_{15}, d_{36}) non zero elements, respectively; they have the same form and number of independent constants as nonlinear optical tensor [34].

## 3 Results and discussion

### 3.1 Electronic structure

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 InPS_{4} and AlPS_{4} 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 AlPS_{4} 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.

Structural parameters of relaxed lattice and sulphur coordinates.

InPS_{4} | AIPS_{4} | |||||||||

Abinit | Castep | Castep | Exp. | Abinit | Castep | Castep | Exp. | |||
---|---|---|---|---|---|---|---|---|---|---|

(LDA) | (LDA) | (GGA) | [11] | [5] | (LDA) | (LDA) | (GGA) | [11] | ||

a [ Å] | 5.521 | 5.554 | 5.802 | 5.623 | 5.60 | 5.554 | 5.585 | 5.700 | 5.61 | |

b [ Å] | 5.521 | 5.554 | 5.802 | 5.623 | 5.60 | 5.613 | 5.582 | 5.702 | 5.67 | |

c [ Å] | 8.895 | 8.789 | 9.193 | 9.058 | 9.02 | 8.960 | 8.899 | 11.567 | 9.05 | |

x | 0.3110 | 0.3114 | 0.3068 | 0.3057 | [11] | 0.2159 | 0.2148 | 0.1988 | 0.200 | |

0.7736 | 0.7757 | 0.7778 | 0.740 | |||||||

y | 0.2496 | 0.2461 | 0.2289 | 0.2389 | [11] | 0.2749 | 0.2768 | 0.2779 | 0.260 | |

0.7908 | 0.7907 | 0.8013 | 0.800 | |||||||

z | 0.1286 | 0.1294 | 0.1327 | 0.1302 | [11] | 0.1206 | 0.1204 | 0.1000 | 0.125 | |

0.6248 | 0.6242 | 0.6003 | 0.630 |

The calculated energy band structures, total and partial density of states of InPS_{4} and AlPS_{4} 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 InPS_{4} (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 InPS_{4} 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 [PS_{4}]^{3}− and [MS_{4}]^{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 InPS_{4}.

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.

Mulliken population analysis.

InPS_{4} | AlPS_{4} | ||||||
---|---|---|---|---|---|---|---|

Bond | Population | Bond | lengths[ Å] | Bond | Population | Bond | lengths [ Å] |

This work | Exp. [11] | This work | Exp. s[11] | ||||

P–S | 0.56 | 2.045 | 2.53 | P–S | 0.55 | 2.056 | 2.1 |

In–S | 0.43 | 2.534 | 2.47 | Al–S | 0.47 | 2.266 | 2.1 |

In–P | – | 3.701 | 3.60 | Al–P | ..63.00 | 2.851 | 2.8 |

Electronic contribution of each atom.

InPS_{4} | AlPS_{4} | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Element | s | p | d | Total | DZ | Element | s | p | Total | DZ |

S | 1.89 | 4.52 | 0.00 | 6.41 | −0.41 | S | 1.89 | 4.59 | 6.48 | −0.48 |

P | 1.46 | 2.85 | 0.00 | 4.30 | +0:70 | P | 1.46 | 2.82 | 4.28 | +0:72 |

In | 0.96 | 1.11 | 9.99 | 12.06 | +0:94 | Al | 0.76 | 1.02 | 1.78 | +1:22 |

### 3.2 Linear optical properties

Graphs in Fig. 3 present the calculated real and imaginary parts of the complex dielectric constant ε(ω) = ε_{1}(ω)+ iε_{2}(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〉.

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 _{4}. 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 AlPS_{4} 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 ε^{100}(ω) and ε^{001}(ω). The similarities are as follows: for low frequencies, which correspond to energies less than band gap energy (E_{g}(InPS_{4}) = 2.33 eV, E_{g}(AlPS_{4}) = 2.70 eV), _{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 (InPS_{4}), 8.53 eV (AlPS_{4}) 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 _{e}(ω)> n_{0}(ω)) for the uniaxial compound (InPS_{4}) 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 AlPS_{4} are available in the literature to make a meaningful comparison.

Refractive indices of InPS_{4} and AlPS_{4}.

InPS_{4} | AlPS_{4} | |||||
---|---|---|---|---|---|---|

This work | Exp. [9] | |||||

Castep (GGA) | Abinit (LDA) | Castep (GGA) | Abinit (LDA) | |||

Low ω (≈1.7 eV) | ||||||

n_{(100)}(ω) | 2.32 | 2.537 | 2.446 | n_{(100)}(ω) | 2.07 | 2.135 |

n(001)(ω) | – | – | – | n(001)(ω) | 2.07 | 2.176 |

n(001)(ω) | 2.38 | 2.609 | 2.546 | n(001)(ω) | 1.95 | 2.114 |

### 3.3 Nonlinear optical properties

InPS_{4} has four nonzero second order NLO tensor elements (d_{14}, d_{15}, d_{31}, d_{36}) imposed by the point group of the crystal and four electro-optic coefficients (r_{13}, r_{41} r_{51}, r_{63}). Assuming Kleinmann symmetry conditions [35], these elements reduce to two independent elements d_{14} = d_{36} and d_{15} = d_{31}. On the other hand, AlPS_{4} second order NLO tensor reduces to three non-vanishing elements (d_{14} = d_{25} = d_{36}).

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

Electro-optic and second order optical coefficients of InPS_{4} and AlPS_{4} (in pm/V).

InPS_{4} | AlPS_{4} | |||
---|---|---|---|---|

This work | Exp.[10] | This work | ||

d_{14} | 31.54 | 28 | d_{14} −1.918 | |

d_{15} | 34.84 | 36 | r_{41} 1.170 | |

r_{13} | −4.12 | r_{52} 0.620 | ||

r_{41} | −4.38 | r_{63} 0.260 | ||

r_{51} | −3.99 | |||

r_{63} | −4.13 |

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 InPS_{4} is better candidate for electro-optic applications with higher E-O coefficients than AlPS_{4}. Nevertheless, we note that both compounds have low electro-optic coefficients compared to the standard ones.

### 3.4 Elastic and piezoelectric properties

The calculation of the piezoelectric tensors by Abinit code enabled us to evaluate four independent elements for InPS_{4} and three for AlPS_{4}. These elements are presented in Table 6.

Piezoelectric coefficients of InPS_{4} and AlPS_{4}

InPS_{4} | AlPS_{4} | ||
---|---|---|---|

This work | Exp. [8] | This work | |

d_{14} | −1.16 | 8.7±1.5 | d_{14} −73.82 |

d_{15} | 0.71 | 0.3±1.0 | d_{25} −10.69 |

d_{31} | 6.73 | −7.3±1.0 | d_{36} −28.19 |

d_{36} | 12.58 | 21±1.5 |

The calculated piezoelectric coefficients for InPS_{4} are in good agreement with the experimental data except for d_{14}. 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 _{4}has higher piezoelectric coefficients than InPS_{4} and better coefficients than reference materials such as quartz (d_{14} = 0.727 pC/N, d_{11} = 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 InPS_{4} 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 InPS_{4} are twice smaller than those of quartz for both longitudinal and transverse elements. The elastic properties of AlPS_{4} have not been reported in the literature yet. Since our calculations for InPS_{4} gave good results, we believe that the present elastic coefficients for AlPS_{4} can be taken as a reference for future investigations.

Elastic constants of InPS_{4} and AlPS_{4} (GPa).

InPS_{4} | AlPS_{4} | ||||||
---|---|---|---|---|---|---|---|

This work | Exp. | This work | |||||

C_{ij} | Castep LDA | Abinit LDA | Literature [9] | Literature [10] | C_{ij} | Castep LDA | Abinit LDA |

C_{11} | 44.15 | 44.03 | 43.80 | 43.68 | C_{11} | 72.11 | 67.84 |

C_{33} | 39.58 | 39.44 | 40.70 | 40.56 | C_{22} | 73.79 | 68.02 |

C_{44} | 23.41 | 21.97 | 22.80 | 22.80 | C_{33} | 14.06 | 11.82 |

C_{66} | 16.52 | 15.78 | 13.60 | 13.92 | C_{44} | 10.41 | 6.68 |

C_{12} | 8.98 | 10.31 | 8.90 | 9.02 | C_{55} | 8.76 | 4.48 |

C_{13} | 25.22 | 25.55 | 26.60 | 23.80 | C_{66} | 2.36 | −1.46 |

C_{16} | 0.89 | 0.87 | 0.30 | 0.80 | C_{12} | 0.54 | −0.30 |

C_{13} | 9.01 | 7.93 | |||||

C_{23} | 10.62 | 8.50 |

However, these theoretical results show that the compound (AlPS_{4}) has lower C_{33} as well as trans-verse coefficients compared to InPS_{4}. All stability criteria equation__7__ and equation__8__ are satisfied for both compounds except for Abinit calculated C_{66} of AlPS_{4} that shows small negative value. As pointed out earlier and shown in Fig. 1, the one dimensional alternating AlS_{4}-PS_{4} 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 C_{11} and C_{22} and the small values of z oriented C_{33}, C_{66} coefficients. It also explains the unexpected negative values of C_{66} (Abinit) and C_{12} (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].

## 4 Conclusions

In summary, we have calculated the electronic structure, first and second order optical properties and electro-optical, elastic and piezoelectric tensors of the metal thiophosphates ternary compounds InPS_{4} and AlPS_{4} by means of density functional theory. The piezoelectric, electro-optic and second order optical tensors have been calculated using plane wave pseudopotentials method with local-density approximation (LDA) for exchange and correlation effects coupled with the modern theory of polarization. The calculated elastic constants, electro-optic, piezoelectric and second order optical tensors of InPS_{4} are in agreement with the available experimental results. Furthermore, these results show that AlPS_{4} exhibits higher piezoelectric coefficient in comparison with quartz and is a promising candidate for piezoelectric applications.

The authors wish to thank the Laboratory of Physics of Materials funding this work. For the author A.H. Reshak the result was developed within the CENTEM Project, Reg. No. CZ.1.05/2.1.00/03.0088, co-funded by the ERDF as part of the Ministry of Education, the Youth and Sports OP RDI Program.

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