Chemical bonding and Born charge in 1T-HfS$_2$

We combine infrared absorption and Raman scattering spectroscopies to explore the properties of the heavy transition metal dichalcogenide 1T-HfS$_2$. We employ the LO-TO splitting of the $E_u$ vibrational mode along with a reevaluation of mode mass, unit cell volume, and dielectric constant to reveal the Born effective charge. We find $Z^*_{\rm{B}}$ = 5.3$e$, in excellent agreement with complementary first principles calculations. In addition to resolving controversy over the nature of chemical bonding in this system, we decompose Born charge into polarizability and local charge. We find $\alpha$ = 5.07 \AA$^3$ and $Z^{*}$ = 5.2$e$, respectively. Polar displacement-induced charge transfer from sulfur $p$ to hafnium $d$ is responsible for the enhanced Born charge compared to the nominal 4+ in hafnium. 1T-HfS$_2$ is thus an ionic crystal with strong and dynamic covalent effects. Taken together, our work places the vibrational properties of 1T-HfS$_2$ on a firm foundation and opens the door to understanding the properties of tubes and sheets.

1T-HfS 2 is a layered material with a P3m1 (#164) space group at 300 K [20]. Each Hf 4+ ion has D 3d site symmetry and is located at the center of a S 2− octahedron. The van der Waals gap is 3.69Å, and the sheet thickness is 2.89Å. Photoemission studies reveal an indirect band gap of 2.85 eV between Γ and M/L, which varies slightly from the ≈ 2 eV optical gap [21]. 1T-HfS 2 forms a high performance transistor with excellent current saturation [22]. The carrier mobility is on the order of 1800 cm 2 V −1 s −1 -much higher than MoS 2 and thickness dependent as well [23,24]. Group theory predicts that at the Γ point 1T-HfS 2 has vibrational modes with symmetries of A 1g + E g + A 2u + E u . The A 1g + E g modes are Raman-active, and the A 2u + E u modes are infrared-active [20,25]. Despite many years of work, there are a surprising number of unresolved questions about 1T-HfS 2 -even in single crystal form. In the field of vibrational spectroscopy, there is controversy about mode assignments, the role of resonance in creating hybrid modes, the presence or absence of surface phonons, and the use of this data to reveal the Born effective charge (Z * B ). As an example, Born effective charges between 3.46e and 5.5e have been reported by various experimental [26,27] and theoretical [28] groups. Evidence for the degree of ionicity (or covalency) is both interesting and important because 5d orbitals tend to be more diffuse than those of their 3d counterparts. Within this picture, 1T-HfS 2 has the potential to sport significant covalency. High pressure Raman scattering spectroscopy reveals a first-order phase transition near 11 GPa and different ∂ω/∂P 's (and thus mode Gruneisen parameters) for the hybrid E u and fundamental A 1g modes [29]. At the same time, variable temperature Raman scattering spectroscopy shows a systematic blueshift of the spectral features down to 100 K, except for the large A 1g mode near 330 cm −1 which redshifts [24,29]. In few-and single-layer form, 1T-HfS 2 is suitable for high-performance transistors [22,30,31], displays a direct gap (rather than indirect as in the bulk) [32], exhibits photocatalytic behavior appropriate for water splitting [33], reveals applications in photodetection [34], is susceptible to strain effects [35], and is useful in N, C, and P surface adsorption [36]. This system can be integrated into van der Waals heterostructures and grown vertically as well [31,37,38].
In order to explore the vibrational properties of 1T-HfS 2 , we measured the infrared absorption and Raman scattering response and employed the results to evaluate Born effective charge. We find that Z * B = 5.3e, in excellent agreement with our complementary first principles calculations. In order to understand how Z * B relates to the nominal 4+ charge of the Hf center, we employ a Wannier function analysis to project out the different orbital contributions. This analysis reveals that the sulfur p orbital transfers charge to the cation and that this contribution enhances the Born charge beyond the nominal value. Decomposing Z * B into polarizability and local charge, we find that there is strong ionicity as well as significant covalency in this system. Both are quite different than in 2H-MoS 2 -probably on account of spin-orbit coupling. We also identify two weak structural distortions near 210 and 60 K evidenced by subtle frequency shifts of the E g and A 1g vibrational modes as well as changes in the phonon lifetimes. Taken together, these findings resolve controversy over the nature of chemical bonding in 1T-HfS 2 and clarify the role of the 5d center in the process.

RESULTS AND DISCUSSION
First-principles predictions of charge and bonding Figure 1(a) displays the projected density of states of 1T-HfS 2 computed using density functional theory and atom centered local projectors [39]. The bands can be assigned Hf and S character easily, and the degree of hybridization between the atoms is not dominant (albeit nonzero). This reveals the strong ionic nature of this system: The valence band is composed mainly of S-p orbitals whereas the conduction band is predominantly Hf-d in character. We find a band gap of 2.05 eV using the HSE06 functional. This is consistent with the small electronegativity of the Hf (1.3 on the Pauling scale) compared to that of S (2.6 on the Pauling scale).
Despite the apparent ionicity of the density of states, the dynamical Born effective charges in 1T-HfS 2 are anomalous. We find an in-plane value for the Hf ions as Z * B,xx = +6.4e. By contrast, the out-of-plane value for Hf Z * B,zz is only 2.0e. This reveals that either (i) the Hf ions are strongly polarizable, or (ii) the small degree of covalency is strongly dependent on ionic displacements [40]. Figure 1(b) displays the phonon dispersions of 1T-HfS 2 . While the spectrum is highly dispersive within the plane (for instance, in the Γ-M direction) it is much less so in the out-of-plane direction (for instance, along Γ-A). This difference is a natural consequence of the layered crystal structure and is the origin of the spikes in the phonon density of states [right panel, Fig. 1(b)]. One aspect of these predictions that will be important for later discussion is the mode order around the A 1g fundamental. Notice that the E u feature is predicted to be below the A 1g mode, whereas the A 2u mode is predicted to be above the A 1g fundamental. These features are labeled in Fig. 1(b). Theoretical phonon frequencies are in excellent agreement with our measured results [ Table 1].
Lattice dynamics can be used to gain information about the chemical bonding in crystals.
In 1T-HfS 2 , the E u optical mode is due to the in-plane vibrations of the Hf cations against the S anions [ Fig. 1(c)]. The frequency difference between the longitudinal and transverse optical modes (the LO-TO splitting) depends on the permittivity, as well as the dynamical charges of the ions. Formally, the LO-TO splitting stems from a non-analytic term added to the dynamical matrices of ionic insulators at the zone center [2]: Here the s, t are atomic indices and α, β are cartesian directions. Z * B is the Born effective charge and q is the wavevector. With this extra term present in the dynamical matrix, the two-fold degeneracy of the E u optical modes is lifted: where u i is the displacement vector of E u (T O) mode. By considering the fact that Z * Hf,xx is always equal to 2 · Z * S,xx because of charge neutrality, and the symmetry imposed form of the E u displacement pattern, we can write this equation for q x as: where the m * is the effective mass, determined by: In this case, the effective mass is 47.02 u. A general expression for effective mass is provided in Supplementary Table I] [24,25,29,42]. There are two fundamental infrared-active phonons. The E u symmetry mode is extremely broad and centered at 155 cm −1 . It is ascribed to an in-plane, out-of-phase motion of the sulfur layers against the hafnium. As we will discuss below, the maximum corresponds to the transverse This feature is due to an out-of-plane + in-phase sulfur layer stretching, and although it is polarized in the c direction, it appears in nascent form here probably due to surface flakiness or slight misalignment. In any case, it is very small. There is also a minute structure near 310 cm −1 which, as we shall see below, corresponds to the weakly activated longitudinal optical (LO) frequency of the E u mode. Of course, each polar phonon has separate LO and T O components, and the LO frequency is always higher than that of the T O frequency due to the polarizability of the surrounding medium. Displacement patterns for each phonon are shown in Fig. 1(c) and summarized in Table 1. What is striking about these results is the overall lack of temperature-induced change over the 300 -8 K range -in both peak position and intensity. Traditionally, modeling of frequency vs. temperature effects provides important information on anharmonicity in a solid as well as the various force constants. The idea is to bring together frequency vs. temperature plots along with one of several different equations that depend upon the situation. The absence of mode hardening with decreasing temperature suggests, however, that anharmonic effects in 1T-HfS 2 are modest and that energy scales are high. Oscillator strength sum rules are obeyed, as expected. Below 100 K, there is a small resonance in the form of a dip that develops on top of the 155 cm −1 E u symmetry phonon. This structure is one of the spectroscopic signatures of a weak local lattice distortion. Because evidence for the effect is stronger in the Raman response, we will defer this discussion until later. Partial sum rules on the E u vibrational mode are obeyed, as expected.
Another way to reveal temperature effects is to examine phonon lifetimes [ Fig. 2(b)].
These values, which are expressions of the Heisenberg uncertainty principle, can be calculated from the linewidth of each vibrational mode as τ = Γ , where Γ is the full width at half maximum and is the reduced Planck's constant [1]. We find that the phonon lifetime of the E u mode is 0.03 ps -exceptionally short compared to that of the A 2u mode. The lifetime of the E u mode is nearly insensitive to temperature as well, meaning that this mode dissipates energy very well -even at low temperature. In other words, τ is large because there are many scattering events. On the other hand, the lifetime of the 336 cm −1 A 2u feature is on the order of 0.4 ps. Overall, these phonon lifetimes are shorter than those of 2H-MoS 2 (τ = 5.5 and 2.3 ps for the E 1u and A 2u modes, respectively) as well as those of polar semiconductors like GaAs, ZnSe, and GaN (which tend to be between 2 and 10 ps) [44]. Likewise, silicon has a phonon lifetime between 1.6 and 2 ps depending on the carrier (hole) density [45]. This means that carrier-phonon scattering is of greater importance in 1T-HfS 2 than in 2H-MoS 2 or the traditional semiconductors. Employing a characteristic phonon velocity of 4700 m/s [32], we find mean free paths in 1T-HfS 2 of 143 pm and 1740 pm for the E u and A 2u modes, respectively. The mean free path for the E u mode is slightly smaller than all of the characteristic length scales in the system including the 369 pm van der Waals gap, the 289 pm sheet thickness, and the 253 pm Hf-S bond length whereas the mean free path for the A 2u mode is 4 or 5 times larger than these characteristic length scales.

Revealing the Born effective charge via infrared spectroscopy
The Born effective charge of transition metal dichalcogenides has been of sustained interest [20,26,27,44,46]. This is because Born charge can be calculated from first principles as summarized in the previous section and revealed directly from spectroscopic data by taking into account the relationship between the longitudinal and transverse optic phonon frequencies as indicated in Equation 3. In the absence of a robust experimentally determined volume, we used a theoretically predicted value [27]. Z * B is extremely sensitive to the value of m k and the choice of ε(∞). Employing these numbers, we find Z * B = 5.3e. Interestingly, prior studies have led to a variety of Born effective charges for HfS 2 with values from 3.46e to 5.5e [26][27][28]. These findings are summarized in Table 2 and Here, η is the depolarization constant. For two-dimensional systems, η = 1/3 [48,49]. We find α = 5.07Å 3 and Z * = 5.2e from our experimental value of the Born charge in the ab plane. These values compare well with those obtained from our theoretical value of Z * B : α = 6.85Å 3 and Z * = 6.2e. These results along with literature values are summarized in Table   3.
Polarizability entails the sum of all cationic and anionic electron cloud volume contributions whereas local effective charge is related to short range interactions indicative of ionic displacement and chemical bonding. These values reveal the nature of chemical bonding.
Typically, covalent systems such as silicon have Z * close to zero whereas ionic materials have larger local effective charges. For example, Z * = 0.15e for MoS 2 [46], Z * = 1.14e for MnO [50], and Z * = 0.8e for NaCl [51]. Higher effective and ionic charge in MnO is consistent with the distinction of a Reststrahlen band. 1T-HfS 2 clearly has the largest local charge in this series.

Origin of anomalous Born effective charge
In a completely ionic crystal where electrons are attached to ions and displaced along with them by the same exact amount, the dynamical Born effective charge is equal to the formal ionic charge -which would have given Z * B for Hf. 1T-HfS 2 is closer to this limit due to the low electronegativity of the cation (1.3 in the Pauling scale). This is lower than any 3d transition metal, including Mo which has an electronegativity of 2.2. The electronegativity of S is 2.6, so the Mo-S bonds in MoS 2 are highly covalent. Nevertheless, we note that the Born effective charge of Hf in the in-plane direction is more than 50% larger than the nominal charge of +4. This anomalous Born charge signals either (i) covalency between the cations and anions or (ii) cation polarizability [40]. Uchida explored the issue in terms of static and dynamic charge [26]. We can address the question more robustly with contemporary tools.
Maximally localized Wannier functions can be utilized to explain the origin of anomalous Born effective charges [52]. The macroscopic electronic polarization can be expressed in terms of the center of localized Wannier functions as where W n (r) is the Wannier function and the sum is over the filled Wannier orbitals. By displacing the Hf atoms in the in-plane and out-of-plane directions, it is possible to calculate the shift of the center of each Wannier function, and hence get a orbital-by-orbital or bandby-band decomposition of the Born effective charges.   Table 1] [25,29,42]. The two Raman-active fundamentals at 259 and 336 cm −1 are even symmetry vibrational modes. We attribute these features to E g symmetry (in-plane, out-of-phase sulfur layer shearing) and the A 1g symmetry (out-of-plane opposing sulfur motions leading to layer breathing), respectively. There is also a weak overtone mode near 650 cm −1 that is slightly less than twice the frequency of the A 1g fundamental. This feature has also been referred to as a second order mode [25]. There are several hybrid features as well. For instance, the E u (T O) mode appears weakly in the Raman spectrum near 132 cm −1 due to an in-plane motion of the sulfur against the hafnium. Finally, the shoulder near 325 cm −1 has been the subject of some controversy in the literature. It was previously described as an in-plane shearing (E u ), an out-of-plane sulfur translation (A 2u ), or even a surface phonon [24,25,29,42]. Based upon prior pressure studies, the position of this phonon diverges from that of the A 1g mode near 325 cm −1 as pressure is increased [29]. If this feature was a surface phonon it would likely track parallel to the much larger mode. Because it does not, it is unlikely to be a surface phonon. With the help of prior literature as well as the common position with the E u mode in the infrared studies, we assign this structure as an E u symmetry phonon [24,25,29,42]. Additional justification for this assignment comes from our phonon density of states calculations and the predicted order of the hybrid modes around the A 1g fundamental [ Fig. 1(b)]. We do not see the hybrid A 2u mode in our spectra -even at low temperature, although different laser wavelengths should reveal it [25]. we see that the linewidth narrows considerably with decreasing temperature, with slight broadening across T 1 ≈ 60 K and T 2 ≈ 210 K. There is also a slight redshift across T 1 .
Analysis of the E g sulfur-layer stretching mode in Fig. 4(c), again shows linewidth narrowing as base temperature is approached, also with noticeable broadening across the two crossover regimes. A slight redshift below T 1 is again present. We attribute the 60 and 210 K transitions in 1T-HfS 2 to local lattice distortions involving a slight motion of the S centers with respect to the Hf ions so as to change the bond lengths and angles a little while maintaining the same overall space group.
Using these linewidth trends and the technique described previously, we calculated phonon lifetimes for the Raman-active vibrational modes of 1T-HfS 2 [ Fig. 4(d)]. As a reminder, the A 1g and E g modes are the fundamentals. The lifetime of the A 1g phonon rises steadily with decreasing temperature. The behavior of the E g symmetry mode is different.
It rises gradually below T 2 and dramatically across T 1 . This suggests that carrier-phonon scattering is reduced with decreasing temperature. Overall the lifetimes of the Raman-active even symmetry modes are similar to those of the infrared-active phonon modes in 1T-HfS 2 [ Fig. 2(b)] -with the exception of the E u symmetry vibrational mode which is very lossy and therefore sports an extremely short lifetime. This is one surprising feature in 1T-HfS 2 that is not replicated in more traditional systems like 2H-MoS 2 .
where the polarization is calculated using the implementation of the the Perturbation Expression After Discretization (PEAD) approach in the VASP package [56,57]. Electric fields of 2 meV/Å, 2 meV/Å, and 10 meV/Å are applied separately along a, b, and c axes to calculate the derivatives. Similarly, the Born effective charge is calculated through the derivatives of the polarization with respect to the ionic displacements. At this step, using the Hellman-Feynman forces enables the use of the more computationally efficient formula Here F is the electric enthalpy which is the sum of the Kohn-Sham energy and the energy gain due to the interaction between the polarization and the external electric field: F = E KS − ΩP · E. The Hellman-Feynman force F is given by F i = ∂F ∂u i . F solely depends on the ground state wavefunction, and hence is easier to calculate than the polarization. As a reliability check, density functional perturbation theory (DFPT) [58] combined with multiple functionals (including PBEsol [59] and revised-TPSS meta-GGA [60]) with spin-coupling is also performed to get the Born effective charge, which can be found in the Supplementary Information. Both approaches provide similar results for the dielectric constant and Born effective charge. We chose the latter method for its compatibility with Hatree-Fock method, and as a result, the hybrid functionals.
All the first-principles calculations are performed in the primitive 3 atom unit cell with a 12 × 12 × 6 k-point grid and cut-off energy of 500 eV. The lattice constants and vectors are taken from the experimental literature, but the internal coordinates of the S ions are obtained through structural optimization of forces. The energy tolerance for self-consistency is set to 10 −8 to get a well-converged wavefunction. To reproduce the experimental bandgap more closely, HSE hybrid functional is employed [61]. In the case of 1T-HfS 2 , a energy band gap of 2.05 eV can be achieved by using the screening parameter of 0.2, which is the so-called HSE06 approximation. Reports of the band gap of HfS 2 span values from 1.96 eV (from optical absorption) [62] to 2.85 eV (from combined angle-resolved and inverse photoemission) [21]. Note that band structures calculated using PBEsol or meta-GGA functionals both underestimate the band gap by at least a factor of two, which influences the prediction accuracy of electric field response. A comparison between different functionals is presented in the Supplementary Information. Since Hf is a heavy element with strong spin-orbital coupling (SOC) [63], DFT calculations that take SOC into account were also performed, but no significant change of Born effective charges and phonon frequencies are observed. A detailed comparison of all theoretical results is provided in the Supplementary Information.
To further explain the origin of Born effective charge, we employed the maximum localized Wannier function (MLWF) [64,65] to project the band-by-band contribution. The Wannier90 software package is used for this analysis [66].    Literature results and our own work -both experimental and theoretical -are included.     Fig. 1(a)]. The symmetric nature of the large E u phonon points to a homogeneous system, and therefore a high quality crystal. Supplementary Fig. 1(b) shows the slight red shift (≈ 1 cm −1 ) of the A 1g mode around 340 cm −1 as temperature decreases from 300 K to 8 K. The other Raman-active features shown (E g and E u ) also show a slight redshift to base temperature, with the peaks sharpening at the lowest temperatures.    [1]. Supplementary Fig.   2 shows a general schematic detailing the relationship between FWHM (Γ), frequency (ω), and phonon lifetime (τ ).   Supplementary Fig. 3(a)], where the valence band is mostly of S character, and mixing between Hf and S states is smaller.

Supplementary
The experimentally determined space group of HfS 2 is P3m1 (#164), which is in the trigonal crystal system. There are 2 E u phonon modes in total, of which one is optical and the other one is acoustic. The optical mode is split into two due to the LO-TO splitting.
The form of the dynamical matrix eigenvectors for both the acoustic and the optical E u mode is set by symmetry to be: such that Eu A and Eu O denote acoustic and optical E u modes, respectively. Figure S4 shows a schematic of the optical mode. Normalizing the eigenvectors gives:   is the analytical contribution to the dynamical matrix, which gives the TO frequencies. In order to obtain the LO frequencies one needs to add the so-called nonanalytical contribution to the dynamical matrix as well [2]: normalized E u dynamical matrix eigenvectors as Here γ m is the polarization induced by a unit displacement by the dynamical matrix eigenvalue | u| = 1. In this sense it is like a mode effective charge, however due to normalization of the dynamical matrix eigenvector (not the displacement) it has different units than Z: where u i is the i th component of the dynamical matrix eigenvector. In this case, the eigenvector is the normalized one for the E u mode, which is shown in the first row of Equation S10 . In the specific (diatomic) case of HFS 2 , we have Z Hf,xx = −2Z S,xx , thus the effective mode would be: Here the m * is the effective mass, which has the relationship: In the end, the Born effective charge can be simplified using this effective mass from Equation S13 : Hf,xx m * (S17) HfS 2 is an ionic crystal with a relatively simple band structure: the conduction band is mainly of Hf-d character, and the valence band is mainly of S-p character. This simplifies the Wannierization process as the bands of interest are not entangled with each other. As shown in Supplementary Fig. 5, the band structure reconstructed from the Wannier based tight binding model matches with the DFT one perfectly (except at very high energies).
In the main text, only one S-p orbital (oriented along the z-direction) is shown. The other two S-p orbitals are similar to the p z orbital in the sense that they also hybridize with the nearest neighbor Hf cation [ Supplementary Fig. 6]. The 3 other s-p orbitals centered