Experimental and Theoretical Investigation of the Elastic Properties of HfV₂O₇

Abstract

We investigated the elastic properties of the HfV₂O₇ high-temperature phase, exhibiting negative thermal expansion, in a synergetic strategy of first-principle calculations and nanoindentation experiments performed on sputtered films. Self-consistent results were obtained for the measured elastic modulus (73 ± 14 GPa) and dispersion-corrected density functional theory calculations. The elastic properties of HfV₂O₇ are affected by long-range dispersion interaction, which may be induced by severe modification in the second-nearest neighbor O-O bond distance as obtained upon compression. HfV₂O₇ is composed of HfO₆, VO₄, and V₂O₇ building blocks, whereby the latter is characterized by an increasing V-O(-V) bond length upon compression.

1. Introduction

Near-zero expansion composites are contrived for applications in environments with rapidly changing temperatures to guarantee unaltered material performance [1,2,3,4]. Negative thermal expansion (NTE) phases are implemented in these composites as thermal expansion compensators [3,4,5,6]. The cubic high-temperature (HT) phase of HfV₂O₇ (space group: Pa$\overline{3}$, 40 atoms per cell) [6,7,8], which is stable at temperatures above ~370 K [6,9], constitutes a promising candidate due to its isotropic NTE [6,7,9] with a thermal expansion coefficient of −7.2 × 10⁻⁶ 1/K [10]. Upon cooling, the HT phase transforms into an incommensurate structure between 369 K and 340 K [9]. Below 340 K, HfV₂O₇ exhibits a cubic 3 × 3 × 3 superstructure [6,7,9,11] (space group: Pa$\overline{3}$, 1080 atoms per cell). However, NTE is only reported for the HT phase [6,7,9], which is reasoned by phonon modes involving translational and librational movements of the quasi-rigid octahedral HfO₆ and tetrahedral VO₄ units [7,9]. These translations and librations substantially soften upon compression of the lattice which leads to its thermal compression [7,9]. For the application of HfV₂O₇ in a composite, understanding its elastic behavior is important for multiple reasons: the elastic properties of an NTE phase used in a composite not only determine the overall elastic properties of the composite [12], but also its mechanical response in rapid heating or cooling processes which induce thermal stress due to the different coefficients of thermal expansion [13]. Additionally, the ability of the NTE phase to counteract positive thermal expansion depends on its elastic properties [14]. However, the elastic behavior of HT HfV₂O₇, with its quasi-rigid HfO₆ and tetrahedral VO₄ building blocks, has not been investigated yet.

In this work, a synergetic strategy of theory and experiments was devised to examine the elastic behavior of the HfV₂O₇ HT phase. Since conventional density functional theory (DFT) [15] calculations do not accurately describe dispersion interaction [16,17,18,19,20,21] which might influence the elastic properties of HT HfV₂O₇, various DFT approaches, including dispersion-corrected DFT, were employed to calculate the elastic constants. Based on the analysis of the atomic rearrangement upon compression of first (metal-oxide) and second (oxygen-oxygen) nearest neighbors, the various theoretical results were critically appraised. Finally, HfV₂O₇ thin films were synthesized and nanoindentation experiments were carried out to validate the theoretical predictions.

2. Computational and Experimental Details

Ground state electronic structure calculations were performed employing the Vienna Ab initio Simulation Package (VASP) [22] with projector augmented-wave potentials within the framework of DFT [15]. The generalized gradient approximation (GGA), as introduced by Perdew, Burke, and Ernzerhof (PBE) [23], was applied. Integration in the Brillouin zone was performed on a 12 × 12 × 12 k-point grid for the cubic HfV₂O₇ unit cell (40 atoms), according to Monkhorst and Pack [24]. The convergence criterion for the total energy was set to 0.01 meV within a 500 eV cut-off. The equilibrium volume and energy were determined by the Birch–Murnaghan equation of state [25,26], applying two procedures for relaxation: (i) structural relaxation only at equilibrium volume, denoted as static calculation (SC), and (ii) full relaxation (FR) of the cell at each volume and deformation (minimization of interatomic forces) for determination of lattice parameter and elastic constants. These two approaches are referred to as DFT-SC and DFT-FR, respectively, throughout this paper. It has been shown that DFT-SC, as a computationally less expensive approach compared to FR, yields accurate bulk modulus data for isotropic cubic materials [27]. To explicitly include dispersion interaction (weak correlations), additional calculations were performed employing (ii-1) the VASP DFT-D3 method with zero damping, as introduced by Grimme et al. [16], and (ii-2) the self-consistent screening (non-local correlations) method by Tkatchenko and Scheffler [28] (DFT-TS). The cubic shear elastic constants C’ = (C₁₁–C₁₂)/2 and C₄₄ were determined by employing the strain-energy method for all four approaches, using volume-conserving distortions of ±1% and ±2% [29,30], which are well within the commonly employed range for elastic constant calculations [29,31,32]. Elastic (Young’s) modulus (E) and shear modulus (G), as well as Poisson’s ratio (v) were calculated from the elastic constants using the Hill method [33].

HfV₂O₇ thin films were deposited on 50 mm diameter sapphire (0001) substrates by direct current magnetron sputtering of elemental Hf and V targets (50 mm diameter, 99.95% purity) at base pressures below 1 × 10⁻⁴ Pa. The target-to-substrate distance was set to 100 mm. The V and Hf target power was set to 200 W and 63 W, respectively, in order to achieve a stoichiometry of 2:1 on the metal sublattice. A mixture of argon (99.9999% purity) and oxygen (99.999% purity) was used as sputtering gas. Depositions were carried out at an oxygen partial pressure of 4.7 × 10⁻² Pa and an overall deposition pressure of 8.6 × 10⁻¹ Pa. The deposition temperatures were selected to be 200 °C, 350 °C and 450 °C. The deposited thin films were characterized using X-ray diffraction (XRD), carried out in a Bruker AXS D8 Discover XRD (Bruker Corporation, Billerica, MA, USA), which was equipped with a general area detector diffraction system, using a Cu K_α X-ray source at an acceleration voltage of 40 kV and a current of 40 mA with an incidence angle of 15°. The chemical composition of the as-deposited thin films was analyzed in terms of the Hf to V ratio by energy dispersive X-ray spectroscopy (EDX) in a JEOL JFM-6480 scanning electron microscope (JEOL Ltd., Tokyo, Japan) equipped with an EDAX Genesis 2000 (EDAX Inc., Mahwah, NJ, USA) and operated at an acceleration voltage of 20 kV. To measure non-conducting HfV₂O₇ thin films by EDX, the films were capped with approximately 10 nm Nb, which had been sputtered at 200 W subsequent to the cooling process without intentional heating. Nanoindentation was carried out in a Hysitron TriboIndenterTM (Hysitron Inc., Eden Prairie, MN, USA) with a 100 nm radius Berkovich diamond tip at the loads of 0.15 mN, 0.2 mN and 0.3 mN, all corresponding to a penetration depth below 10% of the thickness of the uncapped sample. For each load, a grid of 6 × 6 points was measured for sufficient statistics. The reduced modulus E_R was determined using the Oliver–Pharr method [34]. E_R can be converted into the elastic modulus of the specimen according to

$$\frac{1}{E_R}=\frac{1-v_i^2}{E_i}+\frac{1-v_s^2}{E_s},$$

(1)

where i and s refer to the indenter and sample, respectively. E_i and v_i were taken to be 1140 GPa and 0.07, respectively.

3. Results and Discussion

Energy-volume curves for the HT phase of cubic HfV₂O₇ in the case of static calculation (SC) and full structural relaxation (FR) are shown in Figure 1.

It can be seen that the curves are affected significantly depending on the calculation approach employed. Enabling full structural relaxation at every volume has a two-fold effect compared to the static calculation. First, the minimum (equilibrium lattice parameter at 0 K) is shifted to larger values. Comparison with the experimental lattice parameter of 8.7862 Å, determined at 480 K, [6] yield deviations below 2% for both employed calculations and they are, thus, considered to be in reasonable agreement with the experiment. DFT calculations of equilibrium lattice parameters using PBE functionals tend to overestimate experimentally determined values within 2% [35]. Moreover, given a thermal expansion coefficient of −7.2 × 10⁻⁶ 1/K [10], the calculated lattice parameter at 0 K should be decreased by 0.35% at 480 K. Second, the energetic response to deviations from equilibrium volume is significantly stronger for the static calculation than for the full relaxation case. This shape of the curves is directly related to the bulk modulus (B), representing the energetic response to uniform compression (hydrostatic pressure with uniform uniaxial strains), and is, therefore, indicating a considerably smaller B in the case of full structural relaxation at every volume. Furthermore, the curve becomes more asymmetric which has implications for the pressure derivative of the bulk modulus (B’), as elaborated below. The obtained differences between these two approaches suggest dynamic changes in the internal free parameters upon deformation having a pronounced impact on the elastic behavior. Thus, the structure of HT HfV₂O₇, as depicted in Figure 2a, is examined in more detail.

HfV₂O₇ is composed of HfO₆ octahedra and VO₄ tetrahedra, which are connected by corner-sharing oxygen atoms, forming an openly-packed network structure [8]. While all oxygen atoms in a HfO₆ octahedron are shared with neighboring VO₄ tetrahedra, only three of four oxygen atoms in VO₄ are shared with HfO₆ octahedra, whereas one is forming another VO₄ tetrahedron, therefore creating a V₂O₇ unit. Within this V₂O₇ unit, the V-O-V bonds maintain an angle of 180°, as set by the space group. In contrast to the NTE material ZrW₂O₈, in which one W-O bond is non-bridging, all oxygen atoms in HfV₂O₇ exhibit a two-fold coordination lowering the flexibility of the structure [36].

To study the structural changes in HfV₂O₇ upon compression in the case of full structural relaxation at every volume (FR), the change in the bond length of the first (metal-oxygen) and second (oxygen-oxygen) nearest neighbors is examined and summarized in

Table 1. Comparison of individual bond lengths in equilibrium and upon uniform compression (5 GPa), that is, hydrostatic pressure with uniform uniaxial strains, for the full relaxation at every volume (FR) approach.

Neighbor	Bond	Equilibrium [Å]	Compressed at 5 GPa [Å]	Change [%]
1st	V-O(-Hf)	1.703	1.685	−1.0
	V-O(-V)	1.776	1.781	0.3
	Hf-O	2.077	2.043	−1.6
2nd	O-O(1)	2.791	2.710	−2.9
	O-O(2)	2.830	2.872	1.5
	O-O(3)	2.932	2.819	−3.9
	O-O(4)	2.944	2.959	0.5

. Examples of the individual bonds formed within the structure are illustrated in Figure 2b.

Uniform compression without structural relaxation at every volume (SC) pushes all atoms closer together so that all bond lengths decrease. In contrast, allowing for full structural relaxation at every volume (FR) reveals a complex structural response of cubic HfV₂O₇ to the imposed compression, as shown in Table 1. While the Hf-O and V-O(-Hf) (shared oxygen between HfO₆ octahedra and VO₄ tetrahedra) bond distances are decreasing by ≥ 1% for a compression of 5 GPa, an increase in bond length of 0.3% for the rigid V-O(-V) bonds, present in the formed V₂O₇ unit (see Figure 2b), is obtained. Thus, static calculations fail to capture the pronounced atomic rearrangement and are consequently considered to be inappropriate to describe the elastic behavior of HfV₂O₇.

It has been shown that full structural relaxation is also required for systems with dominant dispersion (van der Waals) interaction [27,37,38,39], suggesting that dispersion interaction may be important in HfV₂O₇. An investigation of the second-nearest neighbor interaction (see Table 1) revealed severe modifications in the O-O distance. The change in bond distance is ranging from a reduction of 3.9%, obtained within the HfO₆ octahedra, to an increase of 1.5%, as observed within the VO₄ tetrahedra. It seems that long-range O-O interactions are subtle and may affect the elastic properties. However, conventional DFT calculations with GGA do not accurately describe dispersion interaction [16,17,18,19,20,21]. Hence, two additional dispersion-corrected DFT calculations were performed (DFT-D3 and DFT-TS). In general, these calculations are reported to describe a variety of compounds with dispersion interaction more accurately, for example, graphene [17,40], graphite [17,18,19,41], hexagonal boron nitride [17], MoS₂ [19,20,40,42,43], and V₂O₅ [19,44,45]. The calculated elastic properties and lattice parameters using the various approaches are summarized in

Table 2. Comparison of the calculated lattice parameter a and the elastic properties (B: Bulk modulus, B’: pressure derivative of B, C_ij: Elastic constants, ν: Poisson’s ratio, E: Elastic modulus, G: Shear modulus) of HfV₂O₇ using DFT. Calculations were performed with structural relaxation only at equilibrium volume (SC), at every volume without dispersion correction (FR), as well as with dispersion correction, by applying methods according to Grimme et al. [16] (D3) and Tkatchenko and Scheffler [28] (TS).

Property	DFT-SC	DFT-FR	DFT-D3	DFT-TS
a [Å]	8.856	8.956	8.918	8.926
B [GPa]	140	77	73	56
B’	4.3	−3.2	−5.6	−5.2
C11 [GPa]	277	209	130	99
C12 [GPa]	72	11	44	34
C44 [GPa]	72	69	28	24
v	0.25	0.11	0.30	0.29
E [GPa]	208	178	87	69
G [GPa]	83	80	33	27

.

The introduced dispersion interaction (DFT-D3 and DFT-TS) has only a minor effect on the resulting lattice parameter, whereas the elastic properties are altered significantly. For dispersion-corrected DFT according to Tkatchenko and Scheffler [28], B is reduced to 40% and 73% of the initial value obtained for static calculation and full structural relaxation, respectively. Furthermore, for all calculations with enabled full structural relaxation at every volume and deformation (DFT-FR, DFT-D3, and DFT-TS), the first derivative of the bulk modulus with respect to pressure, B’, is predicted to be negative, which is common for NTE compounds and is known as pressure-induced softening [1,46]. For comparison, B’ calculated by DFT-SC is positive, confirming that the elastic behavior of NTE HfV₂O₇ is not properly captured within this static approach, which is consequently neglected in the following discussion.

Despite the reasonable agreement in B between DFT-FR and dispersion-corrected DFT, pronounced deviations for the single-crystal elastic constants C_ij and the resulting elastic constants E, G, and v of a polycrystalline aggregate, approximated according to Hill [33], are obtained. The elastic moduli E (shear moduli G) are about 2.0 (2.4) and 2.6 (3.0) times lower for DFT-D3 and DFT-TS, respectively, compared to the full relaxation case. Between both dispersion-corrected DFT calculations, however, a maximum deviation (predicted for C₁₁) of 27% was obtained. At this stage, it is not confirmed that dispersion-corrected DFT, which also considers long-range interaction [16,17,18,19,20,21], yields a more realistic description of the elastic behavior of NTE HfV₂O₇.

To evaluate the reliability of the predicted elasticity data, results are to be compared to experimental values. However, experimental data about the elastic properties of the HT phase, exhibiting NTE, are missing. Experimental room temperature studies on the 3 × 3 × 3 superstructure of HfV₂O₇ revealed a surprisingly low B of 12.8 ± 0.4 GPa [8]. The experimentally determined bulk modulus of the HfV₂O₇ 3 × 3 × 3 superstructure deviates, as expected, from the calculated bulk moduli of HT HfV₂O₇, ranging between 56 GPa and 77 GPa, as shown in Table 2. The structural difference between the low-temperature 3 × 3 × 3 superstructure and the HT phase may account for the obtained difference in bulk moduli since similar densities for both structures are expected, according to Baran [11]. At the example of isostructural ZrV₂O₇, it has been found that some of the V-O(-V) bonds are bent away from 180° in the 3 × 3 × 3 superstructure [47]. For the HT phase of HfV₂O₇, these rigid V-O(-V) bonds resist the imposed compression and even elongate upon compression, as discussed above (see Table 1). Hence, it may be speculated that the higher resistance to compression of the HT phase originates from the 180° V-O(-V) bonds. Furthermore, it needs to be considered that elastic properties are generally calculated in the ground state (0 K), whereas experiments are often conducted at ambient temperatures. While the bulk moduli of common compounds show only small temperature dependency [48], the bulk moduli of NTE phases, as HfV₂O₇, are reported to be highly temperature dependent [1,46,48,49]. For example, for NTE ZrW₂O_8, an exceptional increase in bulk modulus of 40% upon cooling from 300 K to 0 K is reported [48].

In order to critically appraise the reliability of the DFT predictions, more experimental elasticity data of HfV₂O₇ are required. Therefore, HfV₂O₇ samples were synthesized using magnetron sputtering and were subsequently characterized by nanoindentation to study the elastic modulus. For this purpose, HfV₂O₇ phase formation studies were conducted. The influence of temperature was investigated by depositing stoichiometric thin films at 200 °C, 350 °C and 450 °C. The resulting diffractograms are shown in Figure 3.

The thin film deposited at 200 °C is X-ray amorphous, showing only a peak at 39° which can be attributed to bcc Nb [50], originating from the Nb capping layer. With an increase in temperature to 350 °C, additional peaks can be observed which can all be assigned to the HfV₂O₇ HT structure [6]. Characteristic peaks for the low-temperature 3 × 3 × 3 superstructure [6,7,11] were not detected. Hence, based on these XRD results it was assumed that the HT phase is present. With a further increase in temperature to 450 °C, no structural changes based on XRD and no change in chemical composition based on EDX were observed. The lattice parameter was determined to be 8.76 ± 0.01 Å, deviating by less than 2% from the dispersion-corrected DFT results. This value is in very good agreement with the experimental lattice parameter of 8.7862 Å reported in the literature [6]. The reduced modulus of HfV₂O₇ was subsequently measured by nanoindentation to be 74 ± 14 GPa. Employing the calculated Poisson’s ratio based on DFT-FR of 0.11 in Equation (1) yields an elastic modulus of HfV₂O₇ of 78 ± 16 GPa which is inconsistent with the predicted value of 178 GPa (see Table 2). On the other hand, using the Poisson’s ratios of dispersion-corrected DFT (DFT-D3 and DFT-TS) reveal experimentally determined elastic moduli of HfV₂O₇ of 72 ± 14 GPa and 73 ± 14 GPa, respectively. The DFT-TS calculated modulus of 69 GPa, based on the Hill average, deviates by 5% from this experimentally determined value. The upper and lower bound of this value, applying the Voigt [51] and Reuss [52] method, were calculated to lie in the narrow range between 70 GPa and 68 GPa, respectively. Hence, dispersion-corrected DFT, using the self-consistent screening (non-local correlations) method by Tkatchenko and Scheffler [28], is suggested to yield the most accurate estimate of the elastic properties of HfV₂O₇ which is self-consistent with the experimental results obtained. Nevertheless, it still needs to be considered that the proposed reliability of the predicted ground state properties is based on the comparison to experimental values determined at room temperature. As already discussed, the bulk modulus of NTE ZrW₂O₈ exhibits an exceptional increase of 40% upon cooling from 300 K to 0 K [48], while the elastic modulus and Poisson’s ratio only increase by 12% and 13%, respectively, [48] implying that the elastic modulus for the comparison of theory and experiment is less prone to temperature influence. Consequently, the results suggest that the elastic properties of HfV₂O₇ are affected by the long-range dispersion interaction, which may be ascribed to the severe modification in the second-nearest neighbor O-O bond distance obtained upon compression.

4. Conclusions

The work aimed to study the elastic properties of the HT phase of HfV₂O₇, exhibiting NTE, in a synergetic strategy using both, ground state DFT calculations and nanoindentation experiments on sputtered HfV₂O₇ thin films. Employing dispersion-corrected DFT calculations according to Tkatchenko and Scheffler [28], the elastic properties of HT HfV₂O₇ were determined, proving to be self-consistent with the elastic modulus data obtained by experiment (deviations within 5%). The structural analysis revealed that the V-O(-V) bonds elongate upon compression possibly explaining the higher resistance of the HT phase to compression compared to the 3 × 3 × 3 superstructure. Furthermore, dispersion interaction plays a major role in HfV₂O₇, which may be attributed to the pronounced modifications found for the O-O bond distances upon compression. The good agreement between the theoretical and experimental data, presented in this work, may encourage additional experimental studies on the elastic properties of HfV₂O₇ and other NTE materials.