The body-centered tetragonal chalcopyrite structure of ZnSnP₂ is shown in Fig. 1. Each unit cell contains four formula units and the point group is . The chalcopyrite structure is deduced from that of the zinc-blende by the replacement of the cationic sublattice by two different atomic species. In general, II–V and IV–V bond lengths, denoted by d_II–V and d_IV–V, respectively, are not equal, which is mentioned in the substitution results in two different structural deformations: the first one is the relocation of anions in the x–y plane which is characterized by parameter . Here, a is the lattice constant in the x or y direction. The second consequence of differing anion-cation bond lengths is a deformation of the unit cell along the z direction to a length c, which is generally different from 2a. This tetragonal distortion is characterized by the quantity c/a. For real compounds of pnictide II–IV–V₂ groups in most cases c/a = 1.769–2.016 and u = 0.214–0.304.^[25] The unit vectors of the primitive cell are (a, 0, 0), (0, a, 0), (a/2, a/2, c/2). The cations are located at 4a and 4b while anions are located at 8d Wyckoff positions. Ion positions can be generated using the following minimum set of (x, y, z) coordinates expressed in units of the a and c constants.

Group II: (0, 0, 0), (0, 1/2, 1/4);

Group IV: (0, 0, 1/2), (0, 1/2, 3/4);

Group V: (u, 1/4, 1/8), (−u, 3/4, 1/8), (3/4, u, 7/8), (1/4,−u, 7/8).

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (color online) Crystal structure of ZnSnP2 (chalcopyrite-type; body-centered tetragonal- .

The ground state structural properties are obtained by minimization of the total energy with respect to the unit cell volume. The structural parameters are obtained by optimizing lattice constants and atomic coordinates until all force components are below 5× 10⁻⁵ hartree/Bohr (1 hartree=27.2114 eV). Table 1 summarizes our results obtained after relaxation of the lattice constants, as well as the available calculated^[8,26,27] and experimental values.^[5,28] For ZnSnP₂, optimization of unit cell geometry within LDA leads to a = 5.622 Å, c = 11.233 Å. The calculated lattice constants are underestimated with the maximal error of 0.7%, which is typically the expected precision for the LDA. Even comparison with experimental values at the lowest temperatures available is, strictly speaking, not correct, since it neglects the anharmonic effect of the zero-point vibrations. Even so, the calculated lattice parameters are found to be in good agreement with the experimental ones. The optimized results show our calculation method is feasible and we will adopt the optimized structure parameters to calculate other properties.

Table 1.

Table 1.

Table 1. The equilibrium lattice constant a (c) (in unit Å), axial ratio c/a and internal parameter for ZnSnP2. . Method a c c/a u ABINIT-LDA (Present work ) 5.622 11.233 1.998 0.231 VASP-LDA (Ref. [8]) 5.606 11.089 1.978 0.227 VASP-GGA+U (Ref. [26]) 5.706 11.443 2.005 0.228 VASP-HSE06 (Ref. [26]) 5.671 11.352 2.002 0.232 CASTEP-LDA (Ref. [27]) 5.638 11.228 1.991 0.25 Exp. (Ref. [5]) 5.651 11.302 2.000 0.239 Exp. (Ref. [28]) 5.652 11.305 2.000 0.229

Table 1. The equilibrium lattice constant a (c) (in unit Å), axial ratio c/a and internal parameter for ZnSnP2. .

The Born effective charge tensor quantifies the macroscopic electric response of a crystal to the internal displacements of its atoms. They are important quantities in obtaining the LO–TO phonon splitting, also they provide some information about the ionicity of the material. For atom κ, the Born effective charge tensor is defined as the proportionality coefficient relating, at linear order, the polarization per unit cell, created along the direction β, and the displacement along the direction α of the atoms belonging to the sublattice κ, under the condition of a zero electric field.^[14] In this work, the Born effective charge tensors for each atom of ZnSnP₂ are calculated by DFPT.

Values of the calculated Born effective charge of ZnSnP₂ are presented in Table 2. To the best of our knowledge, no other first-principles calculations of the Born effective charge for ZnSnP₂ exist. The form of for atom κ depends on the site symmetry of the ions. The point symmetries at the site of Zn ion and Sn ion are S₄ (4a sites) and S₄ (4b sites), respectively. For P ion, the point symmetry at the site is C₂ (8d sites). The effective charge tensors for cations are diagonal and obey , as required by symmetry for a tetragonal crystal. According to the calculation results, cations have diagonal Born effective charges with , , and , . Z^* of cations are almost diagonal with an anisotropy of 2% for Zn and 5% for Sn, respectively. The effective charge tensors have nearly spherical symmetry, with small tetragonal distortion caused by . P ions, located at lower symmetry sites, have nondiagonal and anisotropic Z^*. For all anions, while and take the value −1.99 or −2.27 depending on the distortion of u. Also, depending on the u distortion being along the x or y direction, the nondiagonal components , or , are different than zero. We can see that the shape of Z^* for P ions is far from a sphere. This behavior has also been observed in the case of other chalcopyrite semiconductors.^[29–32]

Table 2.

Table 2.

Table 2. Calculated Born effective charge tensors, Z*, eigenvalues of the symmetric part of , average of eigenvalues , and force-field model effective charges of Ref. [10]. . Atom Z* Point charge of Ref. [10] 1.75 0.9 2.55 0.4 –2.15 –0.65 –2.15 –0.65

Table 2. Calculated Born effective charge tensors, Z*, eigenvalues of the symmetric part of , average of eigenvalues , and force-field model effective charges of Ref. [10]. .

The average eigenvalues for cations and anions are comparable to the values in Ref. [10]. The calculated average eigenvalues ( in our work for Zn, Sn, and P are 1.75, 2.55, and −2.15, respectively. Obviously, the three atoms have smaller effective charges than their formal charges, +2, +4, and −3 for Zn, Sn, and P, respectively. Such difference can be derived from the strong covalent characteristic for bonds. To the best of our knowledge, no other first-principle calculations and experimental data of Z^* for ZnSnP₂ exist. We have compared our results with other literature data calculated with a different force-field model.^[10] This discrepancy is pronounced for the effective charges in Ref. [10] obtained by a least square fit of the phonon energies and Coulomb splittings, probably because of the use of approximate eigenvectors and assumptions about the dynamical charges. Earlier results by Akdogan et al.^[31] gave Born effective charges of CuGaS₂ with the same linear response as ours, and average of eigenvalues displayed a good agreement with experimental data.

The crystal symmetry of ZnSnP₂ is body-centered tetragonal with a centro-symmetric space group and a corresponding point group. Positions of nonequivalent atoms are indicated in Fig. 2. There are four distinct P, two Zn, and two Si atoms. Transformation properties of these atoms under symmetry operations of the space group are noted in Table 3. As a tetragonal crystal, ZnSnP₂ has phonon dispersion relations consisting of 24 branches whose group theoretical analysis at the BZ center ( yields a decomposition into

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) (001) projection of unit cell showing positions of four nonequivalent P atoms, the two nonequivalent Zn atoms and the two nonequivalent Sn atoms. Symmetry coordinates for A modes ( ; displacements in the plane are represented by arrows and displacements in the z direction by open and closed circles.

Table 3.

Table 3.

Table 3. Character table for the D2d point group (tetragonal). . Number D2d E 2S4 C2 2 2σd Symmetry Activity Optical Acoustic A1 1 1 1 1 1 , z2 R 1 0 A2 1 1 1 –1 –1 – silent 2 0 B1 1 –1 1 1 –1 R 3 0 B2 1 –1 1 –1 1 xy R, IR 3 1 E 2 0 –2 0 0 (x, , (xz, yz) R, IR 6 1

Table 3. Character table for the D2d point group (tetragonal). .

The A₁ and B₁ modes are Raman-active, B₂ and E modes are both Raman- and IR-active, and the A₂ mode is silent. These IR modes are polar modes and subject to an LO–TO splitting. Symmetry coordinates for the onefold A and B representations are given in Figs. 2 and 3, respectively, and for the twofold E representations in Fig. 4. A detailed discussion can be found in Ref. [33].

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (color online) Symmetry coordinates for B modes ( ; displacements in the plane are represented by arrows and displacements in the z direction by open and closed circles.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. (color online) Symmetry coordinates for E modes (7 ; displacements in the plane are represented by arrows and displacements in the z direction by open and closed circles.

Complete phonon branches of ZnSnP₂ are plotted for the high-symmetry lines in the BZ together with the corresponding phonon density of states (DOS) in Fig. 5. An interesting feature of acoustic branches around the point is observed, especially along –X and –Z directions. In order to obtain the phonon dispersion curves throughout the BZ, the dynamical matrices are obtained 4× 4× 4 grid of q points, and real space force constants are then found by Fourier transform of the dynamical matrices. The acoustic sum rule is applied to force the three acoustic phonon frequencies at the point equal to zero strictly as being implied by translation symmetry.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. Calculated phonon dispersion curves along symmetry lines in the Brillouin zone and the corresponding phonon density of states (DOS) for ZnSnP2.

In Table 4, we compare calculated phonon frequencies of ZnSnP₂ at the point with the measured Raman^[7] and infrared^[6,7] values. For comparison, we cite also the phonon frequencies from theoretical calculations through direct methods^[8] and a classical model.^[7,9,10] As shown in Fig. 5 and Table 4, three groups of modes can be identified. A clear gap exists between the twelve upper energy branches and the remaining twelve branches. The low frequency group of modes has a range of values from 82 cm⁻¹ to 115 cm⁻¹ and the medium frequency group of modes has a range of values from 180 cm⁻¹ to 207 cm⁻¹. The highest frequency group of modes can be up to 370 cm⁻¹. Unfortunately, some vibration modes have not been observed in previous experiments, including infrared and Raman measurements. To get more accurate measurements, inelastic neutron scattering experiments should be adopted, but it is not available in previous works. Otherwise, for some of the modes there are large differences among the reported theoretical frequencies, for example the highest frequency B₂ mode ranges from 350 cm⁻¹ to 375 cm⁻¹. Our calculation shows that the mode is at 365 cm⁻¹, which is in very good agreement with experiments.^[6,7] It makes us very confident in the prediction of the frequencies for the others.

Table 4.

Table 4.

Table 4. A comparison of calculated phonon frequencies (in unit cm−1) at the point with Raman and infrared data as well as with other theory values. Two numbers in a row correspond to LO/TO frequencies. . Mode Theory/present Experiment Theory IR[6] IR[7] R[7] Ab initio[8] Estimated[9] Keating’s model[10] Rigid ion model[7] A1 R 313 309 298 309 304 307 A2 silent 311 295 308 292 silent 345 333 332 331 B1 R 105 106 106 138 124 R 207 205 191 214 208 R 362 353 358 333 346 352 B2 R, IR 101/100 105/101 98 149/148 140/139 R, IR 330/328 –/327 347/– 353/321 331 345/335 329/324 R, IR 365/364 –/368 365/– 360/359 350 375/354 372/348 E R, IR 82/82 77/77 85 125/125 115/115 R, IR 115/115 119/116 108 165/164 154/153 R, IR 180/180 182/179 157 188/187 190/189 R, IR 328/325 318/313 317/– 325/321 318 312/311 317/315 R, IR 340/335 –/327 339/328 328/– 331/327 321 329/328 334/333 R, IR 353/343 –/368 360/342 342/– 354/331 337 375/345 378/350

Table 4. A comparison of calculated phonon frequencies (in unit cm−1) at the point with Raman and infrared data as well as with other theory values. Two numbers in a row correspond to LO/TO frequencies. .

It should be noted that the direct method was performed to calculate the phonon frequencies of ZnSnP₂ by Lazewski et al.^[8] Although the calculated frequencies show a better agreement with ours, the LO–TO splittings are not discussed. This is because the splitting depends on the effective charges and dielectric tensors of the system, which are not directly accessible to the direct method. The IR modes group into modes with displacements either in the x, y plane or along the z direction. The E mode has a displacement pattern in the x, y plane, and the B₂ mode has displacements along the z direction. The LO–TO splittings for B₂ and E modes are presented in Table 4 and the values of LO–TO splitting agree well the experimental results.^[6,7] The missing modes in the experiment are predicted successfully. For ZnSnP₂, the LO–TO splitting is small, ranging from 0 cm⁻¹ to 10 cm⁻¹. The splitting is even zero in the low frequency region, because the low frequency modes are essentially the folded acoustic modes, they correspond to whole molecular units moving relative to each other. The largest difference appears on the E mode with the highest frequency. The large LO–TO splittings of these modes suggest they involve large effective charges and make large contributions to the static dielectric tensor of ZnSnP₂. A similar splitting case occurs for CuGaSe₂ and the results can be found in Ref. [29].

The dielectric tensor for a tetragonal system such as this will be diagonal with two distinct elements. For ZnSnP₂, the calculated electronic ( and static ( dielectric tensors are diagonal and have two independent components and perpendicular to and along the c axis, respectively. In Table 5 we compare the values calculated for the dielectric tensor of ZnSnP₂ with theoretical and experimental data available. The electronic dielectric constants are calculated to be and . The static dielectric tensor can be decomposed into contributions of different modes and calculated by the generalized Lyddane–Sachs–Teller (LST) relation:^[34]

Table 5.

Table 5.

Table 5. Static and high frequency dielectric tensor components of ZnSnP2. . ε0 This work 12.01 11.91 11.94 13.86 13.74 13.78 Exp. (Ref. [6]) 8.1±0.2 10.0±0.4 Calc. (Ref. [35]) 9.90 Calc. (Ref. [36]) 12.68 12.75 12.73 Calc. (Ref. [27]) 10.08

Table 5. Static and high frequency dielectric tensor components of ZnSnP2. .

The dielectric properties of ZnSnP₂ have not been studied very much experimentally. We are only aware of the measurement of the dielectric constant of ZnSnP₂ in which a value of (or has been reported in Ref. [6]. On the theoretical side, the dielectric constant has been calculated by Verma et al.^[34] using a plasma oscillations theory. ε₀ from Ref. [35] is 9.90 and seems to be more consistent with the experimental value 10.0. Our calculated values and previous theoretical values^[27,36] are calculated by using the DFT and larger than the experimental results. The overestimation of the calculation is a well-known fact of the DFT due to the underestimation of the band gap.^[34,37,38] We note that a smaller energy gap yields a larger ε₀ value. This can be explained on the basis of the Penn model.^[39] The Penn model is based on the expression . It is clear that ε₀ is inversely proportional to . The dielectric constants calculated in our work deviate from other theoretical data^[27,35] because a small band gap was adopted in the calculation.