Revisited phonon assignment and electro-mechanical properties of chromium disilicide

Abstract

We report a complete study of the lattice dynamics, dielectric, elastic and piezoelectric properties of hexagonal semiconducting chromium disilicide (CrSi₂). From a combined experimental and theoretical study, we have revisited the phonon mode assignments at the zone-center, so that the contradictions met in previous experimental studies between 250 and 300 cm⁻¹ are now explained and understood. We found that the temperature dependence of the Raman frequencies is mainly due to an implicit volume contribution and related to the large Grüneisen parameter. This explains why CrSi₂ has a moderate thermal conductivity although its Debye temperature is quite large. Optic and static dielectric constants have also been analyzed and discussed. The elastic constants of CrSi₂ are large, but this compound is quite brittle. In addition, the relatively low Poisson coefficient associated to the large negative Cauchy pressure of CrSi₂ indicate the angular nature of its bonding. The calculation of its piezoelectric coefficient shows a sizable value with a magnitude similar to that reported for α-quartz. This prediction requires, however, experimental confirmation.

---

I. Introduction

Chromium disilicide (CrSi₂) is a p-type degenerate semiconductor with a narrow and indirect energy bandgap measured to be about 0.4 eV.^1,2 It crystallizes in a hexagonal C40 structure within the P6₂22 space group with Cr and Si atoms in 3d and 6j special Wyckoff positions, respectively. This compound is interesting for high temperature applications due to its refractory properties, its relatively high melting point (1750 K), and its good resistance to oxidation.³ Moreover, its high Seebeck coefficient (α) associated with its low electrical resistivity (ρ) make this compound promising for thermoelectric applications or even as a photoelectric source and in detection in the near infrared region.¹ In addition, it is possible to obtain a high-quality strain-free epitaxial growth on Si (111) surfaces (mismatch of −0.14%), which is highly interesting for microelectronics.³

CrSi₂ alloys have been intensively studied over the past 15 years, mainly for their thermoelectric (TE) properties. The TE potential of a material is evaluated by its reduced figure of merit: where κ is the thermal conductivity. Thus, ZT has to be as high as possible. Pure CrSi₂ has a ZT ≈ 0.2, which is rather low with respect to the best TE materials based on silicon alloys (ZT ≈ 1 in the 600–900 K range).^4–8 Recent papers have also shown the possibility for improving the power factor by alloying with Al, V, Mn, Fe, P and Ti.^9–11 However, CrSi₂ has a moderate thermal conductivity, κ ≈ 10 W m⁻¹ K⁻¹ at room temperature,⁷ which could be additionally reduced by adding defects or by controlling the nanostructure. In a previous work,¹¹ we have shown that the combination of porosity and nanostructure leads to a decrease in κ by a factor of ∼10. This result is particularly interesting as it could lead to an increase in ZT. Unfortunately, this texturing also increases the electrical resistivity and is consequently not sufficiently beneficial in strongly improving its TE properties. In this context, the careful study and understanding of the CrSi₂ phonon dynamics is fundamental to finding the best way to scatter the phonons, and therefore increase ZT. Up to now, relatively little experimental information concerning the dynamical properties of CrSi₂ are available: Raman and infrared spectra were measured on the bulk^12,13 and thin films,^2,12,14 but these spectra do not always agree and mode assignments remain difficult due to the lack of relevant theoretical support.

In this paper, we perform a complete study of the dynamical, dielectric, elastic and piezoelectric properties of CrSi₂ from first principles-based methods. Our study therefore covers all the linear couplings between the applied static homogeneous electric field, strain, and periodic atomic displacements. Our theoretical results are compared to the experimental ones when available. The main objective of this work is the complete investigation of the microscopic physical properties of CrSi₂ for its potential industrial applications and more notably in the field of TE. We also report theoretical infrared and Raman spectra, providing benchmark theoretical data directly useful for the assignments of experimental spectra and in clarifying the previously proposed assignments.

This paper is organized as follows: the next section describes the experimental procedure to characterize the CrSi₂ sample and the details of the Raman measurements. The theoretical framework of our calculations is given in Sec. III. Sec. IV is the heart of the article. First, the experimental study of the anharmonicity of the Raman lines shows that their temperature dependence is dominated by the volume contribution, leading to a large Grüneisen parameter. The latter can also explain why CrSi₂ has a moderate thermal conductivity, although the calculation of its band structure gives a quite large Debye temperature (766 K). Then, we revisit the zone-center phonon assignments of CrSi₂ from a combined experimental and theoretical study, so that the contradictions met in the previous experimental studies concerning the vibration modes between 250 and 300 cm⁻¹ are now explained and understood. Its static dielectric tensor is also discussed. Finally, we compute the piezoelectric coefficient of CrSi₂ and we predict a sizable value with a magnitude similar to that reported for α-quartz. This prediction needs, however, to be experimentally supported. Sec. V concludes the paper with the most important results of this work.

II. Experimental details

Polycrystalline samples were synthesized by arc melting of a stoichiometric amount of high purity Cr (99.995%, Alfa Aesar) and Si lumps (99.999%, Alfa Aesar). Sample homogeneity was improved by re-melting the samples twice. X-ray diffraction experiments reveal that the CrSi₂ phase is almost pure. From Rietveld refinement, the lattice parameters were found to be: a = 4.4317(3) Å and c = 6.3655(2) Å in fair agreement with previous experimental data.³

Temperature dependent micro-Raman experiments were measured on homogeneous sample areas with a Labram Aramis confocal microprobe analyzer (Horiba Jobin-Yvon) equipped with an Olympus microscope and a CCD cooled by a thermoelectric Peltier device. We used the 633 nm He–Ne laser line and a power of 0.7 mW. The resolution of the spectra was 1 cm⁻¹. For high temperature experiments, the samples were placed in a Microvision furnace HFS600Pb4. The counting time at room temperature was twice as long as that at higher temperature. The resonant Raman effect was checked using the 531 nm Krypton ion laser line and a T64000 spectrometer (Horiba Jobin-Yvon). The latter was equipped with an Olympus microscope and a CCD cooled by liquid nitrogen. The resolution of the spectra was between 2 and 3 cm⁻¹. Special care was taken to avoid heating of the samples with too-high laser power. This was especially true for the experiment with the krypton ion laser in which we have used power as low as 0.3 mW.

III. Computational details

First-principles calculations are performed within the density functional theory (DFT) framework as implemented in the ABINIT package.¹⁵ The exchange–correlation energy functional is evaluated using the local density approximation (LDA) parametrized by Perdew and Wang.¹⁶ The all-electron potentials are replaced by norm-conserving pseudopotentials generated according to the Troullier–Martins scheme.¹⁷ The Cr 3d⁵ 4s¹ and Si 3s² 3p² electronic states are considered as the valence states. The electronic wave functions are expanded in plane-waves up to a kinetic energy cutoff of 55 Ha and integrals over the Brillouin zone are approximated by sums over an 8 × 8 × 8 mesh of special k-points according to the Monkhorst–Pack scheme.¹⁸ Atomic relaxation is performed at the experimental lattice parameters (see previous section) until the maximum residual forces on each atom were less than 5 × 10⁻⁶ Ha per Bohr. This leads to an electronic band gap, E_g = 0.49 eV, which is consistent with the experimental values^1,2,19 reported between 0.35 and 0.55 eV in the literature. Thus, the well-known DFT band-gap problem is less severe in CrSi₂ because the states on both sides of the band gap are mainly of 3d-Cr character with only a small admixture of p and s states.¹³

Dynamical matrices, dielectric constants, Born effective charges, elastic and piezoelectric constants are calculated within a variational approach to density functional perturbation theory.²⁰ Phonon dispersion curves are interpolated according to the scheme described by Gonze et al.²¹ In this scheme, the dipole–dipole interactions are subtracted from the dynamical matrices before the Fourier transformation, so that only the short-range part is handled in real space. A 4 × 4 × 4 q-points grid in the irreducible Brillouin zone is employed for the calculation of the vibrational band structure while a denser 100 × 100 × 80 grid is used for the calculation of the phonon density-of-states and the thermodynamic functions. The Raman susceptibility tensors are obtained within a nonlinear response formalism, making use of the 2n + 1 theorem.²² The infrared transmittance, infrared reflectivity and Raman spectra are calculated as described in ref. 23–25, respectively.

In our calculations, the orthogonal reference system (x, y, z) is chosen such as z is aligned along the C₆-axis and x is aligned along the crystallographic a-axis.

IV. Results and discussion

A. Raman measurements on polycrystalline samples

Unlike infrared spectroscopy, relative band intensities within a Raman spectrum are largely unaffected by powder particle size and packing density. They do have some influence on the overall intensity, hence the signal to noise ratio, of a Raman spectrum. The position of the Raman lines is also influenced by the particle size. However, according to Table 1, the frequency shift of the Raman lines is weak between polycrystalline powders, single crystals, or thin films.

Table 1 Zone-center phonon frequencies (in cm⁻¹) reported in the literature using Raman and infrared spectroscopies, together with their symmetry when available. Frequencies supposed to have a wrong assignment are in bold

Raman				Infrared
Present	Ref. 14	Ref. 2	Ref. 12	Ref. 12	Ref. 14	Ref. 2 and 19	Ref. 13
Powder	Film^c	Film^d	Single crystal^d	Film	Film	Film	Single crystal
a 531 nm laser line.b 633 nm laser line.c Laser line not reported.d 514.5 nm laser line.
233^b			231 E1	229 E1	228 E1	231	230
251^b				252 A2	250 A2	253	252
	295–305 E1	300	290 E1	290 E1	295 E1	295	296
301^a^,^b		300	300 E1	297 E1
307^b–311^a		310	305 E2
						342
			354 E2
357^a^,^b	350–360 E1	350	355 E1	355 E1	351 E1	362	352
				382 A2	375 A2	381	370
399^a^,^b		400	397 E2
413^a^,^b		410	412 A1

---

Fig. 1 shows the Raman spectra of our polycrystalline CrSi₂ sample at room temperature using two laser line excitations: λ = 531 nm (green) and λ = 633 nm (red). These spectra are normalized to the highest intensity line at 307 cm⁻¹. We observe that the intensity of the doublet in the 395–415 cm⁻¹ range is strongly dependent on the laser line excitation. This phenomenon has been checked on a large number of samples and could be attributed to a resonant (or quasi-resonant) effect. In addition, this effect highlights the two lines centered at 233 and 251 cm⁻¹ measured using the red laser. Only the first line has been reported in previous Raman works^2,12,14 as they were performed using a green laser. So, these findings call in question the mode assignment reported in the literature. This will be further discussed in Section IV.D.

Fig. 1 Raman spectra of CrSi₂ polycrystals at room temperature and using two different laser line excitations.

In order to understand the anharmonicity of the lattice vibrations, we have performed Raman experiments between room temperature and 500 °C (see Fig. 2). The temperature dependence of the most intense Raman lines is displayed in Fig. 3. Their frequency decreases linearly as a function of temperature. There are two different contributions to the temperature dependence of the Raman shifts:²⁶ the explicit (ω_expl.) and the implicit (ω_impl.) volume contributions. The first one comes from the 3- and 4-phonon interactions, while the second one can be calculated using the volume thermal expansion (β_V) and the Grüneisen parameter (γ_m) of a specific Raman mode m. At high temperature and under the isotropic approximation, the implicit contribution varies linearly with the temperature as follows:

ω_impl. = ω_0m(1 − γ_mβ_VT), (1)

where ω_0m is the Raman frequency shift of the mode m at 0 K. Here, we are interested in comparing the slopes of the different Raman modes with the expected slope for the implicit contribution alone. The accurate determination of γ_m requires Raman experiments under pressure and on single crystals. However, because these experiments are not available, one makes the approximation to use, instead of γ_m, the thermodynamic Grüneisen parameter defined as: where B, V and C_p are the bulk modulus, the volume and the experimental heat capacity, respectively. Using the experimental measurements of β_V and C_p from Dasgupta et al.⁷ and the elastic constants measured by Nakamura et al.,²⁷ we found Γ ≈ 2 at room temperature. This value decreases to Γ ≈ 1.8 at 600 K. Substituting γ_m by Γ in eqn (1) and using β_V = 3.62 × 10⁻⁵ K⁻¹, one gets a slope normalized to ω_0m of 6.5 to 7.2 × 10⁻⁵ K⁻¹. As we can see in Fig. 3, these values are very close to the experimental values obtained for the modes at 230 and 310 cm⁻¹, but are 20% larger than those for the three modes at the highest energies.²⁸ Consequently, the present results could suggest that the temperature dependence of the Raman shift is dominated by the volume contribution. This result is related to the quite large Grüneisen parameter found in CrSi₂ and, as we will see later, it will have a strong impact on explaining its thermal conductivity.

Fig. 2 Raman spectra of CrSi₂ polycrystals at different temperatures recorded using the 633 nm laser line. The dotted lines highlight the frequency shift of the main lines.

Fig. 3 Temperature dependence of the most intense Raman lines. Line equations are also given.

B. Optical dielectric tensor

The electronic dielectric permittivity tensor, ε^∞, describes the response of the electron gas to a homogeneous electric field if the ions are taken as fixed at their equilibrium positions. No scissor correction has been used in the calculation of this tensor. Due to the hexagonal symmetry, ε^∞ is diagonal with two independent components labelled xx = yy = ⊥ and zz = ∥. Our calculations give: ε^∞_⊥ = 31.16 and ε^∞_∥ = 26.50, corresponding to an ordinary optical refractive index n_o = 5.58, and an extraordinary refractive index n_e = 5.14. CrSi₂ is therefore a negative uniaxial crystal with a moderately isotropic electronic response to a homogeneous electric field. Their calculated mean values, and n̄ = 5.4, are consistent with the corresponding experimental ones: (ref. 13 and 2) and n̄ = 5.3 (ref. 1). This is a consequence of the good agreement between the calculated and measured band gap energy.

C. Phonon dispersion curves and density-of-states

Phonon dispersion curves give a criterion for the crystal stability and indicate, through the prediction of soft modes, the possible phase transitions. Indeed, if all phonon frequencies are positive, the crystal is locally stable. However, if it appears that some frequencies are imaginary (soft modes), then the system is unstable. The phonon dispersion curves for CrSi₂ are displayed in Fig. 4 along high-symmetry directions. No soft mode is predicted by our calculations in the whole Brillouin zone, suggesting that the compound is thermodynamically stable at ambient pressure and 0 K. The contribution of each kind of atom to each phonon branch is also displayed in this figure using a color code. We observe that pure atomic motions of Si atoms dominate the modes above 350 cm⁻¹, while pure atomic motions of Cr atoms contribute around 300 cm⁻¹. For the other frequencies, collective atomic motions of Cr and Si atoms are involved. The acoustic branches show a significant dispersion and they exhibit a noticeable mixing with the first low-frequency optical branches in the whole Brillouin zone. Thus, this compound has interesting thermal expansion properties, which have been measured by Dasgupta et al.⁷ The atomic contributions to the phonon acoustic mode come from collective motions of Cr and Si atoms. In contrast, the optical phonon branches are quite dispersionless except at high frequency. The maximum value of the phonon frequencies is close to the K high-symmetry point. We do not observe a clear gap between the optical branches. As a consequence, the phonon density-of-states spectrum has a fairly continuous profile with a complex multipeak structure between 100 and 470 cm⁻¹ (see Fig. 5). The low density of modes at low frequencies up to the first peak around 170 cm⁻¹ suggests stiffness of the CrSi₂ lattice.²⁹ This peak is mainly due to the lowest energy transverse acoustic (TA) mode at the Brillouin zone boundaries (at the K and M high-symmetry points). The constant-volume heat capacity (C_v) has been also derived from the knowledge of the phonon density-of-states (inset of Fig. 5). At high temperatures, the specific heat approaches the classical Dulong and Petit asymptotic limit: C_v(T → ∞) = 74.83 J mol⁻¹ K⁻¹. The linear fit of C_v with respect to T³ at very low temperatures (T < 10 K) gives access to an estimation of the Debye temperature. As a rule of thumb, a higher Debye temperature implies a higher associated thermal conductivity. We find a calculated Debye temperature, θ^calc_D = 766 K, in fair agreement with the experiment²⁷ (θ^exp_D = 793 K). This Debye temperature is relatively high and similar to that of diamond silicon and could suggest good thermal conductivity for CrSi₂. However, the thermal conductivity of CrSi₂ is about one order of magnitude smaller⁷ (∼10 W m⁻¹ K⁻¹) than that of diamond silicon³⁰ (∼156 W m⁻¹ K⁻¹). This can not be easily explained by the larger average mass and the larger number of atoms in the CrSi₂ primitive cell. In order to explain this difference, the anharmonicity should be higher in CrSi₂ than in diamond silicon. A good measure of the crystal anharmonicity comes from their mode Grüneisen parameter.³¹ As previously discussed, the Grüneisen parameter of CrSi₂ is much larger than in silicon where it is 0.56 at high temperatures.³¹ The Umklapp scattering model for calculating the thermal conductivity at high temperatures as proposed by Slack (see eqn (7.2) in ref. 31), associated with the experimental Debye temperature²⁷ and the Grüneisen parameter of about 2, gives at room temperature κ = 24 W m⁻¹ K⁻¹ for CrSi₂. This value is the same order of magnitude as in the experiments and one order of magnitude lower than the thermal conductivity of silicon. Note that it is not unusual to overestimate the thermal conductivity using the Slack model.^31,32 Some other thermoelectrics, such as the antifluorite Mg₂X compounds (with X = Si, Ge or Sn) with similar thermal conductivities, have much lower Debye temperatures than CrSi₂ but also lower Grüneisen parameters (≈1.3–1.4).³¹ Clearly, the relatively low thermal conductivity of CrSi₂ is due to the compensation of its high Debye temperature by its relatively high Grüneisen parameter. Therefore, an efficient way to decrease the thermal conductivity of CrSi₂ would be to reduce its Debye temperature by alloying it with heavier atoms. In addition, alloying will increase short wavelength phonon scattering. As discussed by Vineis et al.,³³ combination of alloying and a wide size distribution of nanoparticles is preferable in effectively scattering different phonon modes and reducing the thermal conductivity. We suggest that this approach will be even more efficient in CrSi₂ when it is alloyed with heavy atoms because not only will alloying increase the phonon scattering but it will also decrease the sound velocity and therefore will further reduce the thermal conductivity.

Fig. 4 Calculated phonon dispersion curves for CrSi₂. A color has been assigned to each point based on the contribution of each kind of atom to the associated dynamical matrix eigenvector: red for the Si atoms and green for the Cr atoms.

Fig. 5 Phonon density-of-states for CrSi₂. Inset: calculated constant-volume heat capacity.

D. Infrared and Raman spectroscopies

Despite several experimental investigations into the CrSi₂ lattice dynamics, the assignment of its zone-center phonon modes is not yet unambiguously established in the literature, especially below 320 cm⁻¹. This controversy is mainly due to (i) the difficulty in making high-quality single crystals and (ii) the fact that the assignments are made by analyzing the counterpart modes in the experimental infrared and Raman spectra (CrSi₂ is a noncentrosymmetric structure). The experimental phonon frequencies reported in the literature using infrared and Raman spectroscopies are listed in Table 1.

At the zone-center, the optical phonon modes of CrSi₂ can be classified, according to the irreducible representations of the D₆ point group, into: Γ_opt = A₁ ⊕ 2A₂ ⊕ 4E₂ ⊕ 4E₁ ⊕ 3B₂ ⊕ 2B₁. The E₁ modes are both infrared and Raman active and they are polarized in the x–y plane. The A₁ and E₂ modes are Raman active, whereas the A₂ modes are infrared active and polarized along the z-direction. The B₁ and B₂ modes are silent. Their frequencies are calculated to be 204 and 300 cm⁻¹ (B₁ modes), and 238, 284 and 393 cm⁻¹ (B₂ modes). They can not however be compared to experimental values since no experimental inelastic neutron scattering data are presently available. Close to the Γ-point, the macroscopic electric field splits the polar active modes into transverse (TO) and longitudinal (LO) modes (see Table 2). The LO frequencies are calculated²⁰ from the TO frequencies with the additional knowledge of the Born effective charges and ε^∞. Thus, their predictions should be reliable.

Table 2 Revisited zone-center phonon mode assignment

	Calc.^a (cm^−1)	Exp. (cm^−1)
		Raman	Infrared
a Present.b Ref. 2 and 19.c Ref. 14.d Ref. 12.e Ref. 13.
E1(TO1 + LO1)	225	231–233^a^,^d	228–231^b^,^c^,^d^,^e
E1(TO2)	249	251^a	250–253^b^,^c^,^d^,^e
E1(LO2)	268
A2(TO2)	280		290–296^b^,^c^,^d^,^e
E1(TO3)	282	295–300^a^,^c^,^b	297^d
E1(LO3)	291	290^d
E2	311	305–311^a^,^b^,^d
A2(LO1)	320		342^b
E2	338	354^d
E1(TO4)	336	352–362^a^,^c^,^b^,^d	350–360^b^,^c^,^d^,^e
A2(TO2)	361		370–382^b^,^c^,^d^,^e
E2	388	397–400^a^,^b^,^d
A1	403	410–413^a^,^b^,^d
E1(LO4)	440
E2	448
A2(LO2)	466

---

The analysis of the infrared reflectivity spectra allows us to quantify the LO–TO splitting strength. These spectra are calculated at normal incidence and they are displayed in Fig. 6 (left panels). The reflectivity saturates to unity because our formalism neglects the damping of the phonon modes. We observe a large LO–TO splitting (more than 50 cm⁻¹) for the E₁(TO) modes centered at 336 cm⁻¹ and for the A₂(TO) mode centered at 361 cm⁻¹. The eigenvectors of the polar TO modes do not necessarily correspond to those of their corresponding LO modes due to the long-range Coulomb interactions. We calculated the overlap matrix between the eigenvectors of the TO and LO as follows:

where M_κ is the mass of the κ^th-atom, and u^LO_m and u^TO_n are the eigendisplacement vectors of the m^th LO and n^th TO modes, respectively. This overlap is strong for the E₁(LO2) mode (see Table 3). As a consequence, the eigenvector of the E₁(LO2) is a mixing of those of the E₁(TO2) and E₁(TO3) modes. In contrast, the eigenvector of the E₁(LO1), E₁(LO3) and E₁(LO4) modes can be associated at the first order to the E₁(TO1), E₁(TO3) and E₁(TO4) eigenvectors, respectively. Similarly, the overlap matrix between the A₂(TO) and A₂(LO) modes shows that their mixing is strong (see Table 4).

Fig. 6 Calculated infrared reflectivity spectra (left) of a CrSi₂ single crystal and calculated infrared transmittance spectrum (right) of a CrSi₂ polycrystalline powder. We assume a Lorentzian line shape and a constant linewidth fixed at 5 cm⁻¹ in the calculated infrared transmittance spectrum. Inset A: experimental infrared transmittance spectrum on a polycrystalline film recorded by Chaix-Pluchery et al.¹² Inset B: calculated infrared spectrum in the 278–283 cm⁻¹ range using a smaller constant linewidth fixed at 1 cm⁻¹.

Table 3 Overlap matrix between the eigenvectors of the E₁(LO) and E₁(TO) modes of CrSi₂. Values between parentheses are the frequencies (in cm⁻¹) of the different modes whereas the values between square brackets are the mode effective charges

		TO1	TO2	TO3	TO4
E1		[4.670]	[7.895]	[4.293]	[11.173]
		(225)	(249)	(282)	(336)

---

LO1	(230)	0.937	0.336	0.058	0.071
LO2	(268)	0.226	0.776	0.525	0.267
LO3	(291)	0.151	0.369	0.820	0.411
LO4	(440)	0.218	0.386	0.222	0.869

---

Table 4 Overlap matrix between the eigenvectors of the A₂(LO) and A₂(TO) modes of CrSi₂. Values between parentheses are the frequencies (in cm⁻¹) of the different modes whereas the values between square brackets are the mode effective charges

		TO1	TO2
A2		[11.241]	[9.050]
		(280)	(361)

---

LO1	(320)	0.808	0.589
LO2	(466)	0.589	0.808

---

Fig. 7 displays the calculated non-resonant spectrum of a CrSi₂ polycrystalline compound. In practice, this average is performed by evaluating the Raman tensor components for an arbitrary orientation in space using Euler angles.³⁴ The experimental spectrum, recorded at room temperature and using the 633 nm laser line excitation, is also reported in this figure. As expected, we observe a poor agreement between the calculated and experimental relative intensity of the Raman lines due to a resonant (or quasi-resonant) effect. This effect does not alter the frequency position of the Raman lines but enhances their intensities as highlighted in Fig. 1, rendering the relative intensity of the calculated Raman lines meaningless. However, assignment of the Raman lines is still possible using first principles-based methods because the number of experimental Raman active modes and their frequency position are in qualitative good agreement with the calculated ones. This assignment is symbolized by the dotted lines in the figure. The observed frequency shift of the lines between the experiment and the calculation is related to the anharmonicity of the modes that is not included in our calculations.

Fig. 7 Calculated (non-resonant) and experimental (resonant, using a 633 nm laser line) Raman spectrum of a CrSi₂ polycrystalline powder. We assume a Lorentzian line shape and a constant linewidth fixed at 7 cm⁻¹ in the calculated spectrum.

The calculated and experimental infrared transmittance spectra are displayed in Fig. 6 (right panel) for a polycrystalline compound. In this configuration only the TO modes are sensitive. The experimental infrared spectrum is dominated as expected by six infrared bands at 229, 252, 290, 297, 355 and 382 cm⁻¹. The strong band at 355 cm⁻¹, with a shoulder on its high-frequency side at 382 cm⁻¹, is correctly calculated both in position and relative intensity. The doublet around 295 cm⁻¹ is not distinctly observed in the calculated spectrum because we use a constant linewidth (5 cm⁻¹) to represent the bands, but it appears when we decrease this linewidth at 1 cm⁻¹ (see inset B of Fig. 6). While this doublet is always observed by the authors, its relative intensity can change. Indeed, the relative intensity of the high-frequency component appears smaller on the calculated spectrum than on the experimental one recorded by Chaix-Pluchery et al.¹² However, it is well-reproduced by our calculation when we consider the experimental spectrum recorded by Borghesi et al.¹⁴ The bands centered at 252 and 229 cm⁻¹ are well-reproduced by our calculation and the latter has the smallest intensity. Thus, this good calculation–experiment agreement suggests a reliable assignment of the infrared modes.

In agreement with the literature, the experimental bands centered at 229, 297 and 355 cm⁻¹ are assigned as E₁, while the line at 382 cm⁻¹ is assigned as A₂. For the two remaining infrared experimental bands (290 and 252 cm⁻¹), we suggest a revisited assignment to the one reported in the literature (see Tables 1 and 2). Indeed, Chaix-Pluchery et al.¹² and Borghesi et al.¹⁴ have assigned the band at 290 cm⁻¹ as E₁ because a weak and broad Raman line can be observed close to this frequency. However, our calculations support that this mode is in fact two distinct modes: the first one is infrared active (calculated to be at 280 cm⁻¹) and assigned as A₂, while the second one is Raman active (calculated to be at 291 cm⁻¹) and assigned as E₁(LO). This E₁(LO) line should undergo a frequency dispersion as a function of angle between the phonon wavevector and the CrSi₂ basal plane (estimated to be ∼10 cm⁻¹ by our calculation). As a consequence, this revisited assignment could be unambiguously checked with the measurement of this frequency angular dependence using Raman spectroscopy on a single crystal. Similarly, Chaix-Pluchery et al.¹² and Borghesi et al.¹⁴ have assigned the infrared band at 252 cm⁻¹ as A₂ because it has no counterpart in the Raman spectrum. However, we assigned this band as E₁(TO) because (i) a weak line can be observed at 252 cm⁻¹ in the experimental Raman spectrum of Chaix-Pluchery et al.¹² recorded using the 514.5 nm laser line in the configuration with 0.2 ≤ ≤ 0.3 (Fig. 5c in their paper), and (ii) a weak and broad line at 252 cm⁻¹ can be observed in our experimental Raman spectra using the 633 nm laser line (see also Fig. 1). The four Raman active E₂ modes and the A₁ mode are unambiguously assigned by our calculations and this assignment is consistent with the literature. Our revisited assignment of the CrSi₂ phonon modes is summarized in Table 2.

E. Static dielectric constant

The static dielectric constant of CrSi₂ can be estimated theoretically, adding to the purely electronic response, the contribution coming from the response of the crystal lattice. To estimate this last contribution, one can use a model that assimilates the solid into a system of undampened harmonic oscillators. By doing so, the static dielectric constant, ε⁰, appears as the sum of an electronic contribution (ε^∞) and a contribution arising from each individual polar phonon mode (ε^ph_m) such as:^20,24where the sum runs over all modes m, α and β denote the Cartesian components, Ω₀ is the unit cell volume and S is the infrared oscillator strength. Results of this calculation are reported in Table 5. The electronic and phonon contributions each contribute about half of the total static dielectric constant. The phonon contribution to the E₁ modes is mainly governed by the modes centered at 249 and 336 cm⁻¹, and each contribute around 20% of the total ε⁰_⊥. Similarly, the mode centered at 280 cm⁻¹ dominates ε⁰_∥ by contributing about 37% of the total.

Table 5 Calculated transverse optical phonon contributions (ω_m) to the static dielectric constant (ε⁰) and mode oscillator strengths (S_m). Note: 1 a.u. = 253.2638413 m³ s⁻²

	E1 modes			A2 modes
	ωm (cm^−1)	Sm (× 10^−4 a.u.)	ε^ph⊥	ωm (cm^−1)	Sm (× 10^−4 a.u.)	ε^ph∥
	225	3.893	6.36	280	20.373	21.53
	249	10.384	13.90	361	14.969	9.51
	282	2.586	2.70
	336	19.892	14.66
Total (phonons)			37.61			31.04
ε^∞			31.16			26.50
ε^0			68.77			57.54

---

F. Elastic constants and related mechanical properties

Elastic constants give interesting information on the nature of the forces operating in solids. In particular, they provide information on the stability and the stiffness of materials. Within the D₆ point group, the elastic tensor has five independent elements to be determined: C₁₁, C₁₂, C₁₃, C₃₃ and C₄₄. These independent elements have been calculated at 0 K according to the linear response formalism and using a full relaxation (lattice parameters and atomic positions) of the CrSi₂ primitive unit cell. Our relaxed lattice parameters are: a = 4.34 Å and c = 6.27 Å. The elastic constants are listed in Table 6 with the experimental ones measured at room temperature by Nakamura et al.,²⁷ and an overall qualitative agreement is observed between the two sets. The discrepancies between the calculations and the experiment could be related to (i) the calculated equilibrium lattice constants being slightly smaller then the experimental ones, (ii) the temperature effect, and (iii) the LDA because the calculation of the elastic constants is observed to be dependent on the used functional. CrSi₂ is mechanically stable because the elastic constants satisfy Born mechanical stability restrictions for hexagonal structures, which are given by the following system of inequations:^35,36 C₁₁ − |C₁₂| > 0, (C₁₁ + 2C₁₂)C₃₃ − 2C₁₃² > 0 and C₄₄ > 0.

Table 6 Calculated and experimental relaxed-ion elastic constants (C) and their corresponding bulk (B), Young (E) and shear (G) moduli (in GPa), B_X/G_X ratios and Poisson coefficients (ν) using the Voigt (V), Reuss (R) and Hill (H) approaches^a

	Calc.	Exp.		Calc.	Exp.		Calc.	Exp.		Calc.	Exp.
	Present	Ref. 27		Present	Ref. 27		Present	Ref. 27		Present	Ref. 27
a C66 = (C11 − C12)/2.
C11	454.0	372.2	BV	219.99	172.29	BR	219.68	171.73	BH	219.84	172.01
C12	72.6	45.3	GV	177.31	153.59	GR	176.44	153.03	GH	176.88	153.31
C13	118.5	82.6	BV/GV	1.24	1.12	BR/GR	1.25	1.12	BH/GH	1.24	1.12
C33	453.0	385.2	νV	0.18	0.16	νR	0.18	0.16	νH	0.18	0.16
C44	172.7	149.1	EV	419.28	355.22	ER	417.54	353.95	EH	418.41	354.585

---

From the calculated elastic constants, we derived the macroscopic mechanical parameters, namely the bulk, shear and Young's moduli, using the Voigt–Reuss–Hill (H) approach.³⁷ It is known that the Voigt (V) bound is obtained from the average polycrystalline moduli based on an assumption of uniform strain throughout a polycrystal and is the upper limit of the actual effective moduli, while the Reuss (R) bound is obtained by assuming a uniform stress and is the lower limit of the actual effective moduli.^36,37 The explicit expressions of the bulk (B) and shear (G) moduli as a function of elastic constants for hexagonal structures are given in ref. 36.

At the same time, we derived the Young modulus and Poisson ratio. All of these mechanical properties are listed in Table 6. We observe that the parameter limiting the stability of CrSi₂ is the shear modulus, as G_X < B_X, where X = R, V, H. The typical value of the Poisson ratio is about 0.2 for covalent materials such as silicon whereas it is about 0.3–0.4 for ionic materials and 0.5 for the purely-ionic limit.³⁸ In the present case, the calculated value of the Poisson ratio is 0.18 (0.16 from the experiment), whatever the chosen approach. This suggests that the atomic bonding of CrSi₂ has a strong covalent character. The large negative Cauchy pressures (see ref. 39 for the case of low symmetry systems), P^Cauchy_z = C₁₂ − C₆₆ (≈−118.1 GPa both from our calculations and from experiments²⁷) and P^Cauchy_x = C₁₃ − C₄₄ (≈−54.2 GPa from our calculations and −66.5 GPa from experiments²⁷), associated to this relatively small Poisson ratio indicate that the bondings have highly directional character (this is also the case for compounds with a diamond structure^38,39). We note however that the Cauchy pressure is much more negative than in Si or Ge. This is certainly related to the very large values of the elastic constants in CrSi₂. From the above discussion, it is therefore not surprising to find that CrSi₂ is very brittle because its B/G ratio is only around 1.25. Indeed, the brittle or ductile behavior of a material can be estimated according to the value of the B/G ratio, as proposed by Pugh.⁴⁰ If B/G > 1.75, a ductile behavior is predicted, otherwise the material behaves in a brittle manner. This result agrees well with the experiments.

G. Piezoelectric properties

Finally, we report the piezoelectric constants of CrSi₂ to complete its mechanical properties. Within the D₆ point group, there is only one independent piezoelectric coefficient to be calculated. We found a piezoelectric stress of e₄₁ = 0.31 C m⁻² and a piezoelectric strain of d₄₁ = 1.79 pC N⁻¹. There is no experimental value of these coefficients reported for CrSi₂ in the literature. However, for comparison with other binary compounds, we underline that α-quartz is presently the most-used piezoelectric material with a piezoelectric strain of d₁₁ = 2.31 pC N⁻¹.⁴¹ Nevertheless, its α–β phase transition at 846 K restricts its applications. The α-quartz phase of GeO₂ does not have a α–β phase transition⁴² and its piezoelectric strain d₁₁ = 5.72 pC N⁻¹, is more than twice that of α-quartz.⁴³ This piezoelectric activity is observed until 1273 K. Thus, the sizable piezoelectric response of CrSi₂, associated with its thermal stability (i.e. no phase transition observed up to its melting temperature at about 1750 K), could make this compound an interesting piezoelectric material. This prediction requires experimental confirmations.

V. Conclusions

In this article, we have reported a complete study of the lattice dynamics, dielectric, elastic and piezoelectric properties of hexagonal semiconducting CrSi₂. First, we observed that the experimental Raman spectrum of CrSi₂ shows a resonant character and that the temperature dependence of the Raman frequencies is mainly due to an implicit volume contribution, leading to large Grüneisen parameter. The latter can also explain why CrSi₂ has a moderate thermal conductivity, although it has quite a large Debye temperature. Consequently, an efficient way for reducing its thermal conductivity could be to reduce its Debye temperature by alloying it with large amounts of heavy atoms as not only this will increase the scattering of the phonons but will also decrease their velocity. This kind of alloying is expected to be more efficient than in cubic Mg₂Si where a small percent of Ge substitution for Si leads to a decrease in the thermal conductivity by a factor of 2 to 3 (Ref. 4). Indeed, even if CrSi₂ and Mg₂Si have a similar thermal conductivity, Mg₂Si has a lower Debye temperature and Grüneisen parameter.

We have also revisited the zone-center phonon assignments for CrSi₂ from the combination of our Raman measurements and our calculations, so that the contradictions met in the previous experimental studies concerning the vibration modes between 250 and 300 cm⁻¹ are now explained and understood. We suggest that the experimental Raman line centered at 290 cm⁻¹ assigned as E₁ in the literature is in fact two distinct modes: the first one is infrared active (calculated to be 280 cm⁻¹) and assigned as A₂, while the second one is Raman active (calculated to be 291 cm⁻¹) and assigned as E₁(LO). This E₁(LO) line should undergo a frequency dispersion as a function of angle between the phonon wavevector and the CrSi₂ basal plane (estimated to be ∼10 cm⁻¹ by our calculation). As a consequence, this revisited assignment could be unambiguously checked with the measurement of this frequency angular dependence using Raman spectroscopy on a single crystal. Similarly, we suggest that the experimental infrared band at 252 cm⁻¹ should be assigned as E₁(TO) instead of A₂ as reported in the literature. The static dielectric constant of CrSi₂ was also analyzed and our calculations show that the electronic and the phonon contributions each contribute about half of the total.

Finally, the calculation of the piezoelectric coefficient of CrSi₂ shows a sizable value with a magnitude similar to that reported for α-quartz. This prediction requires, however, experimental confirmation. Thus, compounds with the same hexagonal symmetry as CrSi₂ (i.e. P6₂22) could be interesting for piezoelectric applications, especially those with an energy band gap larger than in CrSi₂.

Acknowledgements

This work was realized with the support of HPC@LR, a Center of Competence in High-Performance Computing from the Languedoc-Roussillon region, funded by the Languedoc-Roussillon region, Europe and Université Montpellier 2.

References

1. M. C. Bost and J. E. Mahan, An investigation of the optical constants and band gap of chromium disilicide, J. Appl. Phys., 1988, 63, 839–844.
2. H. Lange, M. Giehler, W. Henrion, F. Fenske, I. Sieber and G. Oertel, Growth and Optical Characterization of CrSi₂ Thin Films, Phys. Status Solidi B, 1992, 171, 63–75.
3. H. Lange, Electronic Properties of Semiconducting Silcides, Phys. Status Solidi B, 1997, 201, 3–65.
4. V. K. Zaitsev, M. I. Fedorov and E. A. Gurieva, Thermoelectrics on the Base of Solid Solutions of Mg₂B^IV Compounds (B^IV = Si, Ge, Sn), Thermoelectric Handbook - macro to nano, CRC Press, Taylor & Francis, 2006, ch. 29 and ref. therein.
5. M. I. Fedorov and V. K. Zaitsev, Thermoelectrics of Transition Metal Silicides, Thermoelectric Handbook – Macro to Nano, CRC Press, Taylor & Francis, 2006, ch. 31 and ref. therein.
6. W. Liu, X. Tan, K. Yin, H. Liu, X. Tang, J. Shi, Q. Zhang and C. Uher, Convergence of Conduction Bands as Means of Enhancing Thermoelectric Performance of n-Type Mg₂Si_1−xSn_x Solid Solutions, Phys. Rev. Lett., 2012, 108, 166601.
7. T. Dasgupta, J. Etourneau, B. Chevalier, S. F. Mater and A. M. Umarji, Structural, thermal, and electrical properties of CrSi₂, J. Appl. Phys., 2008, 103, 113516.
8. T. Dasgupta and A. M. Umarji, Role of milling parameters and impurity on the thermoelectric properties of mechanically alloyed chromium silicide, J. Alloys Compd., 2008, 461, 292–297.
9. T. Pandey and A. K. Singh, Origin of enhanced thermoelectric properties of doped CrSi₂, RSC Adv., 2014, 4, 3482–3486.
10. D. Parker and D. J. Singh, Very heavily electron-doped CrSi₂ as a high-performance high-temperature thermoelectric material, New J. Phys., 2012, 14, 033045.
11. S. Karuppaiah, M. Beaudhuin and R. Viennois, Investigation on the thermoelectric properties of nanostructured Cr_1−xTi_xSi₂, J. Solid State Chem., 2013, 199, 90–95.
12. O. Chaix-Pluchery and G. Lucazeau, Vibrational Study of Transition Metal Disilicides, MSi₂ (M = Nb, Ta, V, Cr), J. Raman Spectrosc., 1998, 29, 159–164.
13. V. Bellani, G. Guizzetti, F. Marabelli, A. Piaggi, A. Borghesi, F. Nava, V. N. Antonov VI, N. Antonov, O. Jepsen, O. K. Andersen and V. V. Nemoshkalenko, Theory and experiment on the optical properties of CrSi₂, Phys. Rev. B: Condens. Matter Mater. Phys., 1992, 46, 9380–9388.
14. A. Borghesi, A. Piaggi, A. Franchini, G. Guizzetti, F. Nava and G. Santoro, Far-Infrared Vibrational Spectroscopy in CrSi₂, Europhys. Lett., 1990, 11, 61–65.
15. X. Gonze, B. Amadon, P. M. Anglade, J. M. Beuken, F. Bottin, P. Boulanger, F. Bruneval, D. Caliste and R. Caracas, et al., ABINIT: First-principles Approach to Material and Nanosystem Properties, Comput. Phys. Commun., 2009, 180, 2582–2615.
16. J. P. Perdew and Y. Wang, Accurate and Simple Analytic Representation of the Electron-gas Correlation Energy, Phys. Rev. B: Condens. Matter Mater. Phys., 1992, 45, 13244–13249.
17. N. Troullier and J. L. Martins, Efficient Pseudopotentials for Plane-wave Calculations, Phys. Rev. B: Condens. Matter Mater. Phys., 1991, 43, 1993–2006.
18. H. J. Monkhorst and J. D. Pack, Special Points for Brillouin-zone Integrations, Phys. Rev. B: Solid State, 1976, 13, 5188–5192.
19. W. Henrion, H. Lange, E. Jahne and M. Giehler, Optical Properties of Chromium and Iron Disilicide Layers, Appl. Surf. Sci., 1993, 70/71, 569–572.
20. X. Gonze and C. Lee, Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory, Phys. Rev. B: Condens. Matter Mater. Phys., 1997, 55, 10355–10368.
21. X. Gonze, J.-C. Charlier, D. C. Allan and M. P. Teter, Interatomic Force Constants From First Principles: The Case of α-quartz, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 50, 13035–13038.
22. M. Veithen, X. Gonze and P. Ghosez, First-Principles Study of the Electro-Optic Effect in Ferroelectric Oxides, Phys. Rev. Lett., 2004, 93, 187401.
23. P. Hermet, J.-L. Bantignies, A. Rahmani, J.-L. Sauvajol, M. R. Johnson and F. Serein, Far- and Mid-Infrared of Crystalline 2,2′-Bithiophene: Ab Initio Analysis and Comparison with Infrared Response, J. Phys. Chem. A, 2005, 109, 1684–1691.
24. P. Hermet, L. Gourrier, J.-L. Bantignies, D. Ravot, T. Michel, S. Deabate, P. Boulet and F. Henn, Dielectric, Magnetic, and Phonon Properties of Nickel Hydroxide, Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 84, 235211.
25. P. Hermet, M. Veithen and P. Ghosez, First-principles Calculations of the Nonlinear Optical Susceptibilities and Raman Scattering Spectra of Lithium Niobate, J. Phys.: Condens. Matter, 2007, 19, 456202.
26. G. Lucazeau, Effect of pressure and temperature on Raman spectra of solids: anharmonicity, J. Raman Spectrosc., 2003, 34, 478–496.
27. M. Nakamura, Elastic Constants of Some Transition-Metal-Disilicide Single-Crystals, Metall. Mater. Trans. A, 1994, 25, 331–340.
28. As can be seen in Fig. 3, one can note that the energy of the Raman mode at 300 cm⁻¹ decreases linearly as a function of temperature with a slope normalized to ω_0m of about 5.9 × 10⁻⁵ K⁻¹. However, the error bar for this mode is large due to its overlapping with the Raman mode at 308 cm⁻¹. We estimate the error bar for the slope to be about 20% and therefore we are not able to conclude anything about the behavior of this mode with the temperature.
29. P. Hermet, K. Niedziolka and P. Jund, A first-principles investigation of the thermodynamic and mechanical properties of Ni-Ti-Sn Heusler and half-Heusler materials, RSC Adv., 2013, 3, 22176–22184.
30. C. J. Glassbrenner and G. A. Slack, Thermal Conductivity of Silicon and Germanium from 3 K to the Melting Point, Phys. Rev., 1964, 134, A1058–A1069.
31. G. A. Slack, The Thermal Conductivity of Nonmetallic Crystals, Solid State Phys., 1979, 34, 1–71.
32. L. Bjerg, B. B. Iversen and G. K. H. Madsen, Modeling the thermal conductivities of the zinc antimonides ZnSb and Zn₄Sb₃, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 89, 024304.
33. C. J. Vineis, A. Shakouri, A. Majundar and G. Kanatzidis, Nanostructured Thermoelectrics: Big Efficiency Gains from Small Features, Adv. Mater., 2010, 22, 3970–3980.
34. H. Djani, P. Hermet and P. Ghosez, First-Principles Characterization of the P2₁ab Ferroelectric Phase of Aurivillius Bi₂WO₆, J. Phys. Chem. C, 2014, 118, 13514–13524.
35. M. Born, On the stability of crystal lattices I, Proc. Cambridge Philos. Soc., 1940, 36, 160–172.
36. Z. J. Wu, E. J. Zhao, H. P. Xiang, X. F. Hao, X. J. Liu and J. Meng, Crystal structures and elastic properties of superhard IrN₂ and IrN₃ from first principles, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 76, 054115.
37. H. Ledbetter, Monocrystal-polycrystal elastic constant models, Handbook of Elastic Properties of Solids, Liquids, and Gases, vol. III: Elastic Properties of Solids: Biological and Organic Materials, Earth and Marine Sciences, 2001, pp. 313–324.
38. H. Ledbetter, Poisson ratio and covalency-ionicity tetrahedral semiconductors, Handbook of Elastic Properties of Solids, Liquids, and Gases, vol. II: Elastic Properties of Solids: Theory, Elements and Compounds, Novel Materials, Technological Materials, Alloys, and Building Materials, 2001, pp. 281–287.
39. S. Ganeshan, S. Shang, H. Zhang, Y. Wang, M. Mantina and Z. K. Liu, Elastic constants of binary Mg compounds from first-principles calculations, Intermetallics, 2009, 17, 313–318.
40. S. F. Pugh, Relations between the elastic moduli and the plastic properties of polycrystalline pure metals, Philos. Mag., 1954, 45, 823–843.
41. P. Armand, S. Clement, D. Balitsky, A. Lignie and P. Papet, Large SiO₂-substituted GeO₂ single-crystals with the α-quartz structure, J. Cryst. Growth, 2011, 316, 153–157.
42. G. Fraysse, A. Lignie, P. Hermet, P. Armand, D. Bourgogne, J. Haines, B. Ménaert and P. Papet, Vibrational origin of the thermal stability in the highly distorted α-quartz-type material GeO₂: an experimental and theoretical study, Inorg. Chem., 2013, 52, 7271–7279.
43. A. Lignie, W. Zhou, P. Armand, B. Rufflé, R. Mayet, J. Debray, P. Hermet, B. Ménaert, P. Thomas and P. Papet, High-Temperature Elastic Moduli of Flux-Grown α-GeO₂ Single Crystal, ChemPhysChem, 2014, 15, 118–125.

---