Abstract
We report highpressure Ramanscattering measurements on the transitionmetal dichalcogenide (TMDC) compound HfS_{2}. The aim of this work is twofold: (i) to investigate the highpressure behavior of the zonecenter optical phonon modes of HfS_{2} and experimentally determine the linear pressure coefficients and mode Grüneisen parameters of this material; (ii) to test the validity of different density functional theory (DFT) approaches in order to predict the latticedynamical properties of HfS_{2} under pressure. For this purpose, the experimental results are compared with the results of DFT calculations performed with different functionals, with and without Van der Waals (vdW) interaction. We find that DFT calculations within the generalized gradient approximation (GGA) properly describe the highpressure lattice dynamics of HfS_{2} when vdW interactions are taken into account. In contrast, we show that DFT within the local density approximation (LDA), which is widely used to predict structural and vibrational properties at ambient conditions in 2D compounds, fails to reproduce the behavior of HfS_{2} under compression. Similar conclusions are reached in the case of MoS_{2}. This suggests that large errors may be introduced if the compressibility and Grüneisen parameters of bulk TMDCs are calculated with bare DFTLDA. Therefore, the validity of different approaches to calculate the structural and vibrational properties of bulk and fewlayered vdW materials under compression should be carefully assessed.
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Introduction
Transition metal dichalcogenides (TMDCs) have attracted a great deal of attention during the last few years due to their remarkable properties and great potential for photonics and optoelectronics applications^{1}. TMDCs are van der Waals layered semiconductors of the type MX_{2}, where M stands for a transition metal atom such as Mo, W, or Hf, while X is a group16 atom like S, Se, or Te. Due to their semiconducting character, TMDC monolayers emerge as an alternative to graphene, which does not normally exhibit an electronic bandgap. The optoelectronic properties (optical gaps and spinorbit splittings) of TMDCs can indeed be tailored through strategically selecting their composition, i.e., alloying with different chalcogen or transition metal atoms^{2}.
Despite a great deal of research has been devoted to the study of Mo and Wbased TMDCs, relatively little attention has been paid to Hfcontaining layered compounds. The latter are semiconductor materials with band gaps in the visible or infrared spectral range (the band gap decreases with increasing the chalcogen atomic number)^{3}, which makes them attractive for a wide variety of optoelectronic devices like thirdgeneration solar cells^{4}. Recent advances on Hfbased TMDCs include the investigation of their promising thermoelectric properties^{5,6,7}, their epitaxial growth on graphite or MoS_{2} substrates^{8}, or the realization of different devices such as tunnel fieldeffect transistors (TFETs)^{9} or fewlayer transistors^{10}.
Among HfX_{2} compounds, hafnium disulphide (HfS_{2}) has been found to exhibit particularly high and fast electrical responses^{9,11}. It has been calculated that the mobility of his compound might reach values as high as 1800 cm^{2}V^{−1}s^{−1}, much higher than the value of 400 cm^{2}V^{−1}s^{−1} of MoS_{2}^{11}. Also, it has been found that the sheet current density of tunneling fieldeffect transistors (TFETs) based on HfS_{2} is two orders of magnitude larger than that of MoS_{2}^{12}. It is thus necessary to further investigate the fundamental properties of HfS_{2} and explore its possible use in future device applications.
Raman spectroscopy is a powerful analytical tool which provides, in a nondestructive manner, valuable information about several important aspects of semiconductor materials and structures, such as their lattice dynamics, crystal quality, strain state, composition, or electronic structure. In the case of layered compounds such as graphene or TMDCs, Raman scattering is being widely used to determine flake thicknesses of fewlayer systems and also to probe the strain, stoichiometry, or polytypism and stacking order of the TMDC layers^{13,14}. However, in order to fully understand the vibrational properties of monolayer and fewlayered TMDCs, it is essential to throroughly investigate the lattice dynamics of the buk materials.
Highpressure experiments are commonly employed in semiconductor physics studies in order to further probe the optical, electronic and vibrational properties of semiconductors. In particular, highpressure techniques provide a highly useful benchmark to test the results of existing theoretical models, like for instance density functional theory (DFT), for the calculation of the electronic and latticedynamical properties of semiconductors. This may be critical in the case of TMDCs, whose properties are strongly affected by the weak van der Waals bonds between layers. In the case of Ramanscattering investigations, the pressure behavior of the optical phonons provides useful information about phase stability and anharmonic effects. In combination with theoretical calculations, it may also help one to assign the features that show up in the Raman spectra.
Firstprinciple calculations have recently shown that, contrary to bulk HfS_{2}, its monolayer form exhibits a phononic gap which can be tailored by applying biaxial strain^{15}. So far, however, the highpressure effects on the vibrational properties of bulk HfS_{2} have been investigated neither theoretically nor experimentally. With regard to ambient pressure conditions, several works dealing with the resonant and nonresonant Raman spectrum of bulk 1THfS_{2} have been published in the literature^{16,17,18}. All the previous studies report the firstorder A_{1g} and E_{g} Ramanactive modes, but there is some discrepancy with regard to a weak feature that appears at ~325 cm^{−1} as a lowfrequency shoulder of the intense A_{1g} peak. While some authors^{16,18} assign this feature to a Ramanforbidden E_{u}(LO) mode, Roubi and coworkers^{17} attribute it to a forbidden A_{2u}(LO) mode. It should be noted, however, that half the frequency of this mode (~162 cm^{−1}) is very close to the frequency of the zonecenter E_{u}(TO) measured by Lucovsky and coworkers (166 cm^{−1}) with infrared (IR) reflectance spectroscopy^{19}. Highpressure measurements, in combination with latticedynamical calculations, could help to clarify the origin of all the Raman features and better assign these modes.
In the present work we report on Ramanscattering measurements under high hydrostatic pressures on bulk HfS_{2}. From the pressure dependence of the Ramanactive modes we obtain the corresponding mode Grüneisen parameters. The experimental pressure coefficients are compared with theoretical values obtained with latticedynamical ab initio calculations performed with density functional theory (DFT), using different functionals and methodologies. We find that the best agreement between theory and experiment is found with DFTGGA calculations including Van der Waals interaction. In contrast, we show that bare DFTLDA, which is widely used to calculate the properties of 2D (bulk and fewlayer) compounds at ambient conditions, does not correctly predict the behavior of bulk HfS_{2} upon compression. Similar conclusions are reached for the case of 2HMoS_{2}.
Results and Discussion
Three different types of calculations were performed with ABINIT in order to find the best agreement between theoretical and experimental lattice constants of HfS_{2} at ambient pressure: (i) calculations within the generalized gradient approximation (GGA), using fullyrelativistic projector augmentedwave (PAW) potentials with PerdewBurkeErnzerhof (PBE) exchangecorrelation functionals^{20}; (ii) PAWPBE calculation including Grimme’s D3 dispersion correction^{21} to take into account longrange van der Waals (vdW) interactions; (iii) calculations within the PerdewWang local density approximation (LDA); DFTLDA usually provides successful structural results in layered compounds because of a compensating effect between the overestimated covalent part of the interlayer bonding and the neglected vdW forces^{14}.
At ambient pressure, we obtained the following lattice constants for GGA calculations with (and without) vdW corrections: a = 3.616 (3.650) Å and c = 5.801 (7.021) Å. While the bare GGAPBE calculations clearly overestimate the c parameter, the vdWcorrected GGA results are in very good agreement with the experimental values (a = 3.630 Å and 3.622, c = 5.854 and 5.88 Å)^{19,22}, thus reflecting the importance of vdW interactions in the structural properties of these layered materials. On the other hand and, as expected, the LDA calculations provide somewhat underestimated values (a = 3.556 Å and c = 5.677 Å), although in reasonable agreement with the experimental data.
Table 1 shows a summary of these results, together with bulk modulus (B_{0}) values obtained after structural relaxation at different pressures and subsequent fitting of the resulting volumepressure data, using a 3^{rd} order BirchMurnaghan equation of state. The Table also shows data for an additional calculation performed with the Quantum Espresso package using PBEsol functionals, i.e., the revised version of GGAPBE for the solid state^{23}. As can be seen in the table, a large compressibility (low bulk modulus) is obtained with the bare GGA calculations for HfS_{2}, which is clearly a consequence of the overestimated c parameter at zero pressure. In turn, the PBEsol approach provides significantly improved results for the c parameter. However, a relatively large compressibility is still obtained with this functional. In contrast, similar bulk modulus values are found with GGAvdW and LDA, although the latter yields a somewhat lower compressibility. This result may be attributed to the fact that the overestimation of the interlayer covalent bonding in LDA calculations becomes larger upon compression. Unfortunately, no Xray diffraction measurements as a function of pressure have been so far reported for HfS_{2}.
With regard to the vibrational properties, group theoretical analysis of the zonecenter phonons of HfS_{2} with the ideal 1Tstructure (D_{3d} point group, with 3 atoms in the primitive cell) shows that 9 vibrational modes are present in the total representation: Γ = A_{1g} + E_{g} + 2A_{2u} + 2E_{u}, where the odd modes (u, ungerade) are infraredactive while the even modes (g, gerade) are Ramanactive. Infrared and Raman activities are not compatible because the crystal structure has a center of inversion symmetry (and one of the modes for each of the usymmetries correspond to acoustic modes). The atomic displacements for the optical modes can be found in previous works^{15,17,19}.
The zeropressure frequency values (ω_{i0}), phonon pressure coefficients (a_{i}) and mode Grüneisen parameters (γ_{i}) obtained in this work with the Finite Displacement (FD) and Density Functional Perturbation Theory (DFPT) methods are shown in Tables 2 and 3. Table 2 shows the results for the gerade (g) modes of HfS_{2} (see below), while Table 3 shows the data for the TO and LO ungerade (u) phonons. The Grüneisen parameters were obtained using the expression γ_{i} = B_{0}a_{i}/ω_{i0}, where, for the sake of comparison between different DFT methodologies, a constant value of B_{0} = 30.6 GPa as obtained from DFT calculations within GGA + vdW has been employed. These data will be later discussed in reference to the experimental results.
Figure 1 shows the calculated phonon dispersion and onephonon density of states (1PDOS) over the whole Brillouin zone as obtained with Phonopy for HfS_{2} at 0 and 6 GPa, using the finite displacement (FD) method and a PBE functional including the vdW correction. The figure includes splitting between LO and TO modes, which was not taken into account in previous works^{15}. It should be recalled, however, that DFT calculations tend to yield large errors in the case of LO frequencies as a consequence of the bandgap problem and the computation of the dielectric constants. This is particularly important in the case of DFTLDA, although it is not substantially improved within GGA^{24}.
The corresponding 1PDOS curves are displayed in the right panel of the figure. As expected, all phonon branches show an overall upward frequency shift upon compression, without any softening of the modes. The results at zero pressure are consistent with those obtained by DFPT within PBEGGA as reported in ref.^{5}. As can be seen in the figure, no phononic gap exists in bulk HfS_{2}, which suggests that reduced optical phonon lifetimes may be expected in this compound for all the firstorder modes due to an increased number of pathways for phonon decay. Interestingly, many of the phonon branches, like for instance the acoustic phonons along ΓKMΓ or the optical phonons along the ΓA direction, exhibit very low dispersion and therefore increased twophonon density of states. As a consequence of this, secondorder Raman scattering processes may be favored in HfS_{2}.
In order to evaluate the validity of different functionals and methodologies based on DFT calculations, Raman measurements were performed at different pressures and temperatures. Figure 2 shows ambientpressure Raman spectra at room temperature and low temperature (100 K) of HfS_{2}, acquired with 532nm excitation radiation. An additional roomtemperature spectrum excited with λ = 785 nm has also been included in the figure for comparison purposes. Four different modes around 137, 264, 326 and 338 cm^{−1} clearly show up in the lowtemperature spectrum. The Raman peaks at ~338 and 264 cm^{−1} can be unambiguously assigned to the A_{1g} and E_{g} modes, respectively^{16,17,18}. It is interesting to note that these modes only involve S vibrations, and as a consequence they are not affected by isotopic anharmonic effects due to the strong isotopic variability of natural hafnium. In contrast, strong isotopic broadening may be expected for the uphonons, since these do involve Hf vibrations.
As expected, all the features in Fig. 2 are broadened and shifted to lower frequencies in the roomtemperature spectra. In particular, the mode at ~325 cm^{−1} only appears as a broad lowfrequency shoulder of the A_{1g} mode. In turn, it is worth noting that the E_{g} mode at ~264 cm^{−1} is significantly enhanced, relatively to the rest of peaks, in the spectrum excited with λ = 785 nm. As in other layered compounds^{25,26,27}, this intensity enhancement may be attributed to resonance effects arising from excitonphonon coupling involving particular exciton states of HfS_{2}. Given that atoms in the E_{g} phonon vibrate inplane, the observed intensity increase in Fig. 2 may be ascribed to electronic transitions involving excitonic states with strong inplane character. This is what occurs, for instance, is the case of the C exciton in MoS_{2}^{25}. However, given that the combination modes in TMDCs exhibit strong resonance enhancement due to coupling between phonons of nonzero momentum and excitonic transitions^{27,28}, it cannot be completely ruled out that the strong peak that appears at ~264 cm^{−1} for 785nm excitation has contribution from a secondorder Raman band. More work is required to fully understand excitonic resonance Raman effects in HfS_{2}.
In spite of the Raman inactivity of umodes, two different works^{16,18} attribute the weak peak below the A_{1g} mode around 326 cm^{−1} to an E_{u}(LO) mode. In contrast, ref.^{17} assigns this feature to an A_{2u}(LO) mode. In that work, a weak, broad band around 155 cm^{−1} is also attributed to an E_{u}(TO) mode. However, such feature in not reported in any other previous work, whereas the E_{u}(TO) phonon was found at 166 cm^{−1} by means of infraredreflectance spectroscopy^{19}. Note that the peak at 326 cm^{−1} could also correspond to a 2E_{u}(TO) mode, since this feature is not far from twice the frequency of the E_{u}(TO) mode reported by Lucovsky and coworkers.
Leaving aside the previous works on HfS_{2}, several studies on layered compounds like MoS_{2}, MoSe_{2} or WS_{2} have also reported Raman inactive umodes^{27,28,29}. In those works, ungerade modes were observed amid a large number of secondorder modes excited resonantly, and were attributed to symmetry breaking of the selection rules through disorder or the participation strongly localized electronic states. Other works on TMDCs reported bulkinactive gmodes in fewlayer compounds like MoTe_{2}^{30}, which was attributed to a loss of periodicitiy along the caxis. In the case of HfS_{2}, it cannot be ruled out that the umodes show up in the Raman spectra due to a contribution of fewlayer domains in the investigated samples. Regardless of their origin, highpressure Ramanscattering measurements, in combination with DFT latticedynamical calculations, may provide highly valuable information to identify the different features that appear in the Raman spectra of vdW compounds.
Figure 3 shows selected Raman spectra at different pressures, up to 8.5 GPa, for bulk HfS_{2}. As expected, all modes are found to blueshift with increasing pressure. Around 4 GPa, a very weak, broad feature is found to show up around 120 cm^{−1}. This mode has never been reported in previous works and probably emerges due to a particular resonant excitation at this pressure range (note that, as can be seen in Fig. 1, 1THfS_{2} has no firstorder optical modes below ~150 cm^{−1}). As already mentioned, secondorder resonant Raman scattering in TMDCs may yield a large number of bands at some given excitation conditions. In this case, resonance enhancement may be achieved by pressureinduced bandgap shifts. In contrast, the mode at 137 cm^{−1} progressively weakens upon compression and is hardly visible above 6 GPa. Around 10 GPa, as shown in Fig. 4, a new peak appears above the A_{1g} mode. This peak becomes dominant at 12.7 GPa and can be tentatively attributed to a highpressure phase of HfS_{2}. The study of this possible new phase is beyond the scope of the present work.
We have plotted in Fig. 5 the pressure dependence of all the observed Raman features up to 10 GPa, below the phase transition that presumably occurs in bulk HfS_{2}. The results of linear fits to all the experimental curves as a function of pressure (P) are also shown in the figure. These where obtained by using the expression ω_{i}(P) = ω_{i}(0) + a_{i}P, where ω_{i}(0) and a_{i} stand for the zeropressure frequency and pressurecoefficient for the ith mode, respectively. The experimental data are summarized in Tables 2 and 3, together with theoretical values obtained within DFTbased latticedynamical calculations of the zonecenter phonons performed with the FD and DFPT methods. The corresponding mode Grüneisen parameters (γ_{i} = B_{0}a_{i} /ω_{i0}, with B_{0} = 30.6 GPa as obtained with the present DFTGGA calculations including vdW interactions) are also given.
In the case of the A_{1g} mode, excellent agreement is found between the experimental pressure coefficients and most of the approaches considered in this work. For this phonon mode, with atoms vibrating outofplane, all methods provide similar results because interlayer forces have a low effect on the resulting frequencies of vibration. As expected, among the DFPT values (note that vdW corrections are not yet implemented for DFPT phonon calculations in Quantum Espresso), LDA provides the best zeropressure frequency value. In the case of FD calculations, LDA and GGA + vdW provide comparable results. The bare GGA value, however, yields a too low value as a consequence of the overestimated c parameter at zero pressure.
Similar conclusions can be reached for the IRactive A_{2u}(TO) mode, although in this case the bare GGA calculations tend to yield slightly reduced pressure coefficients. In turn, DFPTLDA predicts a zeropressure frequency that is ~6–7% larger than the rest of values. Unfortunately, no experimental results are available to perform a comparison between theory and experiment for this mode.
Interestingly, when the zonecenter modes involve inplane atomic displacements, as is the case of E_{g} and E_{u} modes, the LDA calculations predict fairly low pressure coefficients, below 2 cm^{−1}/GPa, both with the FD and DFPT methods. In contrast, the GGA and GGAvdW results predict much larger values. In particular, the GGAvdW pressure coefficient obtained with the FD method for the E_{g} mode (2.37 cm^{−1}/GPa) is in remarkable agreement with the experimental value (2.33 cm^{−1}/GPa). This result indicates that the GGAvdW calculations are better suited to model the highpressure latticedynamical properties of 1THfS_{2}.
Such a conclusion is not limited to the case of HfS_{2}, but can be extended to other layered compounds. To further test this point, we have performed additional latticedynamical calculations for the case of the archetypical TMDC compound 2HMoS_{2} (see the results in Supplementary Table 1). In this case we also find that, although the LDA functional predicts remarkable zeropressure Raman frequencies, the theoretical pressure coefficients for the Esymmetry modes, including the lowfrequency shear mode, are significantly lower than those obtained with PBE functionals (with and without vdW corrections). Bearing in mind the relatively large dispersion of the experimental data reported in the literature (Supplementary Table 1), only the GGAvdW calculations provide good agreement between theoretical and experimental values for both the zeropressure Raman frequencies and their corresponding phonon pressure coefficients in bulk MoS_{2}. Similar conclusions were also reached elsewhere^{31} for the case of black phosphorous. In that work, although no direct comparison was shown regarding different methods to calculate the lattice dynamics of this layered material as a function of pressure, it was found that only the PBEvdW calculations are able to simultaneously predict the structural and vibrational properties of black phosphorous under compression.
On the other hand, and as shown in Table 3, the theoretical pressure coefficients for the E_{u}(LO) mode of bulk HfS_{2} are found to be very small, much lower than 1 cm^{−1}/GPa. This result, together with the theoretical ω_{i0} values found for this mode, allow us to discard the conclusions of refs^{16,18} with regard to the origin of the peak at ~320 cm^{−1} (Fig. 2). In those works, this feature was attributed to an E_{u}(LO) mode. In contrast, the experimental pressure coefficient (zeropressure frequency) for this band is 3.58 cm^{−1}/GPa (321.1 cm^{−1}), much larger than the values predicted by the different calculations for the E_{u}(LO) mode, e. g. 0.71 cm^{−1}/GPa (304.8 cm^{−1}) with GGA + vdW and the FD method. Note also that the E_{u}(LO) mode should involve inplane vibrations and, following the behavior of the E_{g} mode, resonant enhancement of this mode would have been expected for 785nm excitation. In contrast, the theoretical values for the outofplane A_{2u}(LO) [3.25 cm^{−1}/GPa (317.8 cm^{−1}) with GGA + vdW and the FD method] are compatible with the experimental values. Note also that no combinations of firstorder modes at different points of the Brillouin zone, using the phonon dispersion of Fig. 1, provide a satisfactory enough explanation for this feature. In particular, it cannot be attributed to a 2E_{u}(TO) mode, since all the present DFT calculations predict a much larger pressure coefficient for any secondorder combination of optical modes. Therefore, given the good agreement between the experimental and theoretical (GGA + vdW) values, we conclude that the lowfrequency shoulder of the A_{1g} mode can be assigned to the silent A_{2u}(LO) phonon of HfS_{2}. More work would be necessary, however, to fully understand the actual mechanism giving rise to silent (symmetryforbidden) modes in HfS_{2} and, in general, in layered TMDCs.
Finally, we use the theoretical phonon dispersion as a function of pressure (Fig. 1) in combination with the experimental pressure coefficients in order to tentatively assign the remaining Raman feature at 137 cm^{−1} as well as the weak band that shows up above 3 GPa (Fig. 2). First, we note that the experimental pressure coefficient of the latter, and also that of the feature at 137 cm^{−1} is just around 1.7 cm^{−1}/GPa, much smaller than the value predicted for the E_{u}(TO) mode (Table 3). Hence, in contrast to the conclusions of ref.^{17}, it can be ruled out that forbidden firstorder Raman scattering by the E_{u} (TO) mode occurs in HfS_{2}. Indeed, given that numerous phonon branches display low dispersion along relative large regions of the Brillouin zone (see Fig. 1), and bearing in mind the low experimental pressure coefficients of the observed bands, it is very likely that these two modes correspond to secondorder difference modes. For instance, combinations like Eg(L)LA(L) or A_{2u}(TO,L)E_{u}(TO,L) might explain the pressure behavior of the peak at 137 cm^{−1}, while the band that shows up above 3 GPa at 118 cm^{−1} can be explained with a E_{u}(TO,M)LA(M) or a E_{u}(TO,L)LA(L) combination.
Conclusion
We have presented a joint theoretical and experimental highpressure Ramanscattering study of the vibrational properties of bulk 1THfS_{2}. Comparison between the experimental phonon pressure coefficients and the results of DFT calculations with different functionals, with and without vdW interactions, indicate that GGA + vdW properly describes the pressure behavior of bulk HfS_{2}. In contrast, it is found that bare DFTLDA fails to reproduce the structural and vibrational properties of this compound under compression. Similar conclusions can be reached in the case of 2HMoS_{2}. DFTLDA is widely used for calculations of phonons and other properties in layered compounds due to a wellknown compensating effect between the underestimated bond lengths and the overestimated interatomic forces. However, the present results suggest that DFTLDA is not valid to predict the compressibility or the mode Grüneisen parameters of bulk or fewlayered vdW materials. In particular, we have found that DFTLDA gives rise to a sizable underestimation of phonons involving inplane atomic displacements. From the pressure dependence of several Raman features and the theoretical data, it has been possible to shed additional light on the origin of different features that appear in the Raman spectrum of HfS_{2}. In particular, the lowfrequency shoulder below A_{1g} peak has been assigned to a forbidden A_{2u}(LO) mode. More work is required in order to understand the scattering mechanism responsible for the appearance of Ramaninactive ungerade modes in the Raman spectrum of TMDCs.
Methods
Samples
For the present study, we used a commercial HfS_{2} bulk sample from 2DSemiconductors Inc. grown by the Bridgman technique. The indirect and direct optical bandgap of the sample, as measured by optical measurements at ambient conditions, are 1.39 and 2.09 eV, respectively^{3}.
High pressure measurements
Small flakes of around 50 × 50 μm^{2} were detached from the bulk HfS_{2} sample and loaded into a gasketed membranetype diamond anvil cell (DAC) with 400 µm culetsize diamonds. A mixture of methanolethanolwater (16:3:1) was employed as pressure transmitting medium, and the ruby fluorescence method was used to evaluate the pressure applied to the flake. Roomtemperature microRaman measurements were acquired during the upstroke cycle, up to 13 GPa, by using a HORIBA JobinYvon LabRamHR spectrometer coupled to a highsensitive CCD camera. The spectra were excited with the second harmonic of a continuouswave Nd:YAG laser (λ = 532 nm), and a ×50 longworking distance objective was employed to focus the laser light and to collect the backscattered radiation. To further probe resonance effects in HfS_{2} at room temperature, additional Raman spectra excited with λ = 785 nm radiation were also acquired.
Firstprinciple calculations
DFT calculations for bulk 1THfS_{2} and 2HMoS_{2} were carried out with the ABINIT^{32} and Quantum Espresso^{33} packages. Details for the case of MoS_{2} can be found in the Supplementary Material. For all calculations on HfS_{2}, the plane wave basis cutoff was 30 Ha and the MonkhorstPack kpoint grid was set to 8 × 8 × 4. The structures were first fully relaxed until the interatomic forces were lower than 10^{−5} eV/Å. Latticedynamical calculations were performed with the Phonopy software^{34} for different pressure values, up to 10 GPa, within the FD method. 2 × 2 × 2 supercells were found to give convergence for all the phonon modes of both compounds. From these calculations, zonecenter phonon frequencies for the Ramanactive A_{1g} and E_{1g} modes and for the infraredactive A_{2u} and E_{u} modes of HfS_{2} were obtained. For the construction of the corresponding dynamical matrices, the forces associated to the selected finite displacements were calculated with ABINIT using the GGA, GGAvdW and LDA methods. In order to calculate the splitting between transverse optical (TO) and longitudinal optical (LO) phonon modes, dielectric tensors and Born effective charges were calculated as a function of pressure with Quantum Espresso, using the DFPT approach implemented in this package. For this purpose, LDA and PBEsol functionals were used. For comparison purposes, zeropressure frequencies and pressure coefficients for the zonecenter modes including TOLO splitting were also calculated within DFPT.
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Acknowledgements
Work supported by the Spanish Government through projects MAT201571035R and FIS201783295P, and by the National Science Centre (NCN) Poland POLONEZ 3 no. 2016/23/P/ST3/04278 and grant OPUS 11 no. 2016/21/B/ST3/00482. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreement No 665778. T.W. acknowledges support within the Maestro grant from NCN (no. 2014/14/A/ST3/0065). F. D. acknowledges the support within the FUGA grant from the NCN (No. 2014/12/S/ST3/00313). DFT calculations were performed at ICM at the University of Warsaw, and at ICTJACSIC.
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J.I. wrote the manuscript and performed firstprinciple calculations, T.W. carried out firstprinciple calculations, F.D. and S.H. realized the highpressure Raman experiments, R.O. realized the data analysis and figures, R.K. provided the samples and overall guidance. All authors discussed the results and commented on the manuscript.
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Ibáñez, J., Woźniak, T., Dybala, F. et al. Highpressure Raman scattering in bulk HfS_{2}: comparison of density functional theory methods in layered MS_{2} compounds (M = Hf, Mo) under compression. Sci Rep 8, 12757 (2018). https://doi.org/10.1038/s4159801831051y
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DOI: https://doi.org/10.1038/s4159801831051y
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