Valence electron concentration- and N vacancy-induced elasticity in cubic early transition metal nitrides

Motivated by frequently reported deviations from stoichiometry in cubic transition metal nitride (TMNx) thin films, the effect of N-vacancy concentration on the elastic properties of cubic TiNx, ZrNx, VNx, NbNx, and MoNx (0.72<x<1.00) is systematically studied by density functional theory (DFT) calculations. The predictions are validated experimentally for VNx (0.77<x<0.97). The DFT results indicate that the elastic behavior of the TMNx depends on both the N-vacancy concentration and the valence electron concentration (VEC) of the transition metal: While TiNx and ZrNx exhibit vacancy-induced reductions in elastic modulus, VNx and NbNx show an increase. These trends can be rationalized by considering vacancy-induced changes in elastic anisotropy and bonding. While introduction of N-vacancies in TiNx results in a significant reduction of elastic modulus along all directions and a lower average bond strength of Ti-N, the vacancy-induced reduction in [001] direction of VNx is overcompensated by the higher stiffness along [011] and [111] directions, resulting in a higher average bond strength of V-N. To validate the predicted vacancy-induced changes in elasticity experimentally, close-to-single-crystal VNx (0.77<x<0.97) are grown on MgO(001) substrates. As the N-content is reduced, the relaxed lattice parameter a0, as probed by X-ray diffraction, decreases from 4.128 A to 4.096 A. This reduction in lattice parameter is accompanied by an anomalous 11% increase in elastic modulus, as determined by nanoindentation. As the experimental data agree with the predictions, the elasticity enhancement in VNx upon N-vacancy formation can be understood based on the concomitant changes in elastic anisotropy and bonding.


Introduction
When it comes to material systems suitable for hard protective coatings [1,2] and diffusion barrier layers [3,4], group IV, V, and VI transition metal nitrides, or TMNs, are often the materials of choice in industrial applications, due to their excellent thermal stability [1,5], corrosion resistance [6], and mechanical properties [7,8].Additionally, the properties of binary TMN compounds can be enhanced by modifying the occupancy of metal and nonmetal sublattices via alloying constituents.For example, metastable ternary (Ti,Al)N [9] shows age hardening at elevated temperatures before decomposing into its stable constituents [10], and metastable quaternary (Ti,Al)(O,N) exhibits significant thermal stability enhancement in comparison to the (Ti,Al)N counterpart [11].
With respect to elasticity, vacancy-induced changes in the elastic modulus seem to follow a different trend for early TMN compounds.For TiNx (x = N/Ti) [31,33], an increase in N vacancy concentration, from TiN1.00 to TiN0.67, has been experimentally shown to continuously reduce the elastic modulus, from ~428 to 325 GPa, respectively.The vacancyinduced reduction in elasticity of TiNx was attributed to a reduction in the bond density of the compound [31].Similar behavior is also reported for HfNx [34], another compound from group IV TMNs.However, reports on group V NbNx indicated an anomalous increase in elastic modulus via both N and Nb vacancy incorporation, as observed both theoretically [16] and experimentally [35].For TaNx thin films [36], a reduction in elastic modulus is measured as x is increased from stoichiometric x = 1.00 to overstoichiometric x = 1.35, which is the opposite of the behavior observed for NbNx, another group V TMN [16].Apart from the scattered literature on the vacancy-induced changes in mechanical properties of TMNs, there is no systematic study on the effect of vacancies in conjunction with the increase in valence electron concentration (VEC) on the elastic properties of group IV to V and VI TMNs.Vivid morphological differences, which are typically not considered in the structural models used in ab initio simulations, make correlative experimental and theoretical studies challenging with respect to elastic properties.In order to deconvolute the intrinsic properties of the material systems from morphological effects such as grain bouondaries, an effective approach is to carry out property measurements on epitaxial close-to-single-crystal layers, as previously done for ScN(001) [37], TiN(001) [4,38], HfN(001) [14], VN(001) [39], NbN(001) [16], and CrN(001) [18], to name a few.
Here, we systematically investigate the effect of N vacancy concentration on the elastic properties of binary cubic TiNx, ZrNx, VNx, NbNx, and MoNx (0.72    1.00) by density functional theory (DFT) calculations.The predictions are validated experimentally for epitaxially grown VNx(001) (0.77    0.97) thin films.The calculated bulk moduli, Poisson's ratios, and elastic moduli of the binary compounds with varying N vacancy concentration exhibit dependency on the VEC in groups IV, V, and VI.It is demonstrated that while for TiNx an increase in N vacancy concentration results in a reduction in elastic modulus, a N vacancy-induced elasticity enhancement is observed in the VNx.The results of crystal orbital Hamilton population (COHP) analyses and elastic anisotropy calculations reveal that the stiffness increase (decrease) upon vacancy introduction into VNx (TiNx) is caused by direction-dependent bond-strengthening (bond-softening).
For all ab initio simulations, 2 × 2 × 2 supercells with cubic B1 symmetry containing 64 atoms were constructed for TiN, ZrN, VN, NbN, and MoN.Three (corresponding to 4.7%), six (9.4%), or nine (14.1%) vacancies on random positions of the N sublattice were then introduced, yielding a total of 20 different structures.The stiffness tensors of these systems were determined with the strain-stress method as implemented in Ref. [50] and projected onto the cubic symmetry as described in the literature [51].From the stiffness tensor, the bulk, shear, and elastic modulus, as well as Poisson's ratio, were calculated with Hill's approximation [52].Additionally, temperature-dependent equilibrium volume and elastic moduli were calculated by the Debye-Grüneisen model [53] as described elsewhere [54].
Furthermore, the directional elastic moduli were calculated with the help of equation (1), as derived by Nye [55].
with [ℎ ̅  ̅  ̅ ] as the normalized vector along the direction [ℎ] and   as elements from the compliance tensor (S).
In addition to the elastic properties, the electronic structure and bonding characteristics were analyzed for two of the systems, VNx and TiNx, with an ideal vacancy-free structure and the three N vacancy concentrations described above.The optimized structures used for the calculation of elastic properties were utilized as input structures.Single-point (static) DFT simulations using VASP were performed, employing a slightly reduced k-mesh of 5 × 5 × 5 and the Blöchl tetrahedron method for Brillouin zone integration [56], with other key settings similar to the preceding simulation series described above.The valence electron configurations of the PAW potential files were 3p 6 4s 1 3d 4 for V, 3s 2 3p 6 4s 1 3d 3 for Ti and 2s 2 2p 3 for N. The wavefunctions of the simulated systems were generated by VASP and post-processed with the LOBSTER package (version 4.0.0,Institute of Inorganic Chemistry, RWTH Aachen University) [57][58][59][60] in order to project the delocalized, plane-wave-based information onto local orbitals.This enables the calculation of a more precise, atom-and orbital-resolved density of states, as well as the (integrated) crystal orbital Hamilton population ([I]COHP) [61] in order to estimate the bonding character and strength of individual interatomic bonds.It should be noted that, while the COHP is not a direct descriptor of the bond strength, it is strongly correlated with the latter property and is routinely used to assess it in various systems [62][63][64][65].

Experimental details
Vanadium nitride (VNx) thin films from stoichiometric to understoichiometric were synthesized using reactive direct current magnetron sputtering (DCMS) in a load-locked ultra-high-vacuum chamber.The base pressure of the system was below 1  10 −5 Pa at respective deposition temperatures (Ts) of 230, 430, 600, and 700 °C.An elemental V target ( 99.5% purity) with a diameter of 50 mm was powered in DC at a constant averaged-power of 100 W, which resulted in a power density of ~5 W/cm 2 .Initial growth experiments performed in Ar/N2 working gas mixture (not shown here) resulted in extremely N-deficient thin films, hence, near-stoichiometric cubic VNx could not be achieved even at the lowest Ts = 230 °C employed here.This has also been reported by Mei et al. [39], where they systematically investigated the effect of N2 partial pressure on the chemical composition of VNx.In addition, high energetic Ar + irradiation-induced defects [23,66,67] could hinder the growth of high crystalline quality, epitaxial VNx thin films.Therefore, the working gas was pure N2 (5.0 purity) with a deposition pressure of 2.6 Pa.Such a high discharge pressure was employed to thermalize sputtered atoms and neutralize the majority of ions [68].
MgO(001) substrates with a dimension of 10  10  0.5 mm 3 were mounted at a defined target-to-substrate distance of 6 cm.Prior to the depositions, the unpolished backsides of MgO(001) substrates were coated with TiN to optimize heat conduction and avoid localized heating effects.Moreover, in order to optimize the surface of MgO(001) substrates, wetcleaning was conducted by using isopropanol and acetone in sonication [69], accompanied by a N2 blow-drying procedure.Lastly, an annealing step was performed at 800 °C for one hour in the deposition chamber prior to the deposition of the desired thin films.The substrate holder was kept at floating potential for all depositions and the thin films were grown to a thickness of approximately ~400 nm.
Time-of-flight elastic recoil detection analysis (ToF-ERDA) at the Tandem Laboratory of Uppsala University [70] was used for depth profiling of the film composition.A primary ion beam of 36 MeV 127 I 8+ was employed and the detection telescope, including a solid-state detector was located at 45° with respect to the primary ion beam.The incident and exit angle of the ions and detected recoils with respect to the specimen surface were both 22.5°.Depth profiles were obtained from time-energy coincidence spectra by using CONTES [71] and a TiN reference sample [72], which has been characterized by Rutherford backscattering spectrometry, was also probed.All resulting depth profiles were found homogeneous and the maximum oxygen impurity content was 1 at.%.The total maximum measurement uncertainties were 3% relative deviation of the deduced values for V and N.
For the X-ray photoelectron spectroscopy (XPS) measurements, samples were inserted in an AXIS Supra instrument (Kratos Analytical Ltd.) equipped with a monochromatic Al-Kα X-ray source.The base pressure of the system during acquisition was < 5.0 × 10 −6 Pa.Highresolution N 1s spectra were obtained using a pass energy of 10 eV and a step size of 0.04 eV (6 sweeps, dwell time 1000 ms).The measurement spot size was 700 × 300 µm 2 .The binding energy (BE) scale of the spectrometer was calibrated using a sputter-cleaned Ag standard (Ag 3d5/2 signal at 368.2 eV).No charging effect of the VNx samples was observed.
Structural analysis of the thin films was performed with a Siemens D5000 X-ray diffraction (XRD) system (Munich, Germany) using a Cu Kα radiation source, operated at a Reciprocal space maps (RSM) were acquired using a PANalytical Empyrean diffractometer with Cu Kα radiation for the phase structure and growth orientation analysis.
Symmetric and asymmetric RSMs were recorded around MgO(002) and MgO(204), respectively using a four-axis goniometer and a primary optics consisting of a parabolic graded multilayer mirror, collimator, and a channel-cut 2-bounce Ge(220) monochromator.
The in-plane coherence lengths ( ∥ ), corresponding to average mosaic domain sizes [73], are determined from the widths of (002) diffraction peaks parallel to the diffraction vector [74]: where   is the full-width at half-maximum of the peak intensities of the VNx (002) diffractions along the  direction.
The surface topography and morphology of the thin films were characterized using scanning electron microscopy (SEM) at an acceleration voltage of 10 kV and a current of 50 pA within an FEI Helios Nanolab 660 dual-beam microscope (Hillsboro, OR, USA).
The elastic modulus  was determined by nanoindentation using a Hysitron (Minneapolis, MN, USA) TI-900 TriboIndenter equipped with a Berkovich geometry diamond tip with 100 nm radius.At least 25 quasistatic indents with a maximum load of 0.8 mN were performed.Indentation depths were less than 40 nm, which is ~5% of the film thickness, to minimize the substrate effect [75].The tip area function was determined with a fused silica reference before each measurement series and verified to remain unchanged thereafter.
The reduced modulus was acquired from the unloading segment of load-displacement curves using the method of Oliver and Pharr [76].The elastic moduli of the films were then obtained from the reduced moduli data using the composition-dependent Poisson's ratios calculated in this study and the isotropic approximation.

Results and discussion
The calculated bulk moduli, together with the Poisson's ratios, and elastic moduli of the binary nitrides in the ground state, as a function of the N/TM ratio x are depicted in Figure 1 (a-c).The introduction of N vacancies in the cubic binary TMNx results in a reduction of the bulk modulus, independent of the TM element, see Figure 1 (a).The decrease is ~17% for group IV nitrides (TiNx and ZrNx), as x changes from 1.00 to 0.72.A less significant vacancyinduced reduction in bulk modulus of ~9% is calculated for the group V nitrides (VNx and NbNx).Moreover, for the group VI binary MoNx, the stoichiometric cubic structure is not mechanically stable, as already reported by Balasubramanian et al.As the elastic modulus of binary TMN is known to be anisotropic [77], different directional elastic responses for TiNx and VNx, with and without vacancies, were analyzed.
The results are depicted in Figure 2 showing the elastic modulus in the three main planes, (001), (011), and (111).In both stoichiometric compounds the directional elastic modulus is the strongest along the [100] direction, see Figure 2   to an electronic stabilization of the cubic structure of VNx [78].For both TiNx and VNx, the emergence of peaks at -2 eV is attributed to the vacancy-derived states, in good agreement with other reported electronic structures for the understoichiometric cubic TiNx [33] and VNx [78].Nonetheless, there is no other significant change in the DOS between the stoichiometric and understoichiometric compositions, which is why we will in the following focus on the differences in local electronic structure in the vicinity of a N vacancy and compare the DOS to that of the pristine non-vacancy-containing system.The results of total and partial differential density of states (dDOS) are illustrated in Figure 3 (a) and (b) for TiNx and VNx, respectively.The electronic ICOHP value, correlated with a weakening of the bond, a more negative ICOHP for VNx with respect to increasing N vacancy concentration suggests that bond strengthening seems to be the primary reason behind the anomalous stiffness increase.These findings are consistent with the calculated data by Rueß et al. [54], who showed that V vacancy-induced bond strengthening is the origin of the stiffness increase in overstoichiometric VNx.
Another important aspect of the presence of N vacancies in cubic binary TMNx is the local lattice relaxation.Therefore, we systematically analyze changes in the bonding character with respect to N vacancy concentration for both TiNx and VNx, with the results summarized in Figure 4.In an effort to experimentally validate the N vacancy-induced bond strengthening, VNx thin films were grown epitaxially on single crystal MgO(001) substrates as a function of Ts in a pure N2 atmosphere.The resulting thin film compositions, as obtained by ToF-ERDA, are plotted in Figure 5 (a), showing the N/V ratio x with respect to Ts.A steep reduction in x is evident as a function of substrate temperature, where an increase from 230 to 700 °C leads to a reduction in the  value from near-stoichiometric 0.97 ± 0.06 to understoichiometric 0.77 ± 0.04.The reduction in the N content of the layers at higher temperatures is evidently attributed to N2 desorption due to the thermally-activated mechanisms on the surface of the growing film, as has been shown theoretically by Sangiovanni et al. [79].It can be also learned from Figure 5 (a), that near-stoichiometric VN(001) thin films can be synthesized at temperatures below ~430 °C under the here stated deposition conditions, as also reported by Mei et al. [80].The significant influence of the deposition temperature upon the crystal structure evolution of the thin films has direct consequences for the morphological evolution, as probed by surface SEM imaging in Figure 6 (c-f).The presence of the faceted structures within the (002) matrix of the VN0.97 deposited at low Ts (230 °C) can be correlated with limited adatom surface mobility [84].On the other hand, very smooth featureless surfaces have been obtained as Ts is increased to above 600 °C, see Figure 6 (e) and (f).The results of high-resolution XRD reciprocal space maps (HR-RSM) of the thin films grown at Ts = 430, 600, and 700 °C with  = 0.93, 0.86, and 0.77, respectively, acquired over symmetric 002 and asymmetric 204 reflections are shown in Figure 7.Moreover, the in-plane coherence length ( ∥ ), in-plane strain (ɛ‖), and relaxed lattice parameter (a0) were determined as a function of Ts from the HR-RSM results and are listed in Table 1.
At Ts = 430 °C, a significant broadening is observed for the in-plane direction (qx) along the θ-2θ direction is observed, see Figure 7 (a), which accounts for a large mosaicity in this sample.The mosaicity is significantly reduced at higher deposition temperatures, as denoted by a smaller in-plane diffraction broadening, Figure 7 (b) and (c).In-plane coherence length  ∥ increases from 17 to 93 nm as Ts is increased from 230 to 600 °C, respectively.The here reported  ∥ = 93 nm for understoichiometric VN0.86 is among the largest in-plane coherency lengths obtained for the epitaxially grown binary transition-metal nitrides reported so far [13,18,34,[36][37][38][39], which  reflects the high crystalline quality of this thin film.Further increase in Ts to 700 °C for the growth of VN0.77 resulted in a reduction of  ∥ to 62 nm, which can be correlated to a higher concentration of N vacancies in this highly understoichiometric film [39].
Furthermore, from the asymmetric maps, a fully-relaxed thin film on the single crystal substrate has the 204 diffraction peaks centered on the dashed line extending from the origin along the 204 diffraction peaks of the substrate.Here, the lateral displacement of the 204 diffraction peaks of the film deposited at Ts = 430 °C to the right side of the line indicates the presence of compressive strain with an in-plane biaxial strain ɛ‖ of −1.46% in this layer.
Relaxed lattice parameters  0 ()obtained from the HR-RSMs are summarized in Table 1.The  0 () decreases from 4.128 Å for VN0.97 to 4.096 Å for VN0.77, following a nearly linear trend.This negative slope with respect to  is also observed in other works for VNx [39] and TiNx [31].The relaxed lattice parameters from the thin films are plotted in Figure 8 (a) together with the lattice parameters obtained from the DFT calculations by considering N vacancies as the origin of understoichiometry in VNx.The maximum deviations between the calculated and measured values are below 1.0%, signifying good agreement.In general, the calculated lattice parameters are found to be smaller than the experimentally measured values, however, showing a similar negative trend with respect to .The good agreement between the experimental data and the predictions in turn suggests that the formation of understoichiometric VNx is governed by N vacancies.
By considering the high crystalline quality of the synthesized thin films, and therefore neglecting the morphological differences, the chemical composition-induced changes in elastic modulus of VNx (0.77    0.97) are evaluated using nanoindentation experiments and the results are depicted in Figure 8 (b).Additionally, the calculated elastic moduli of VNx (0.72    1.00) are included, together with the measured and calculated elastic modulus of the stoichiometric VN1.00 from literature.For the near-stoichiometric VN0.97 thin film, the measured elastic modulus is 400 ± 25 GPa, which is in good agreement with other reported values for bulk [85], thin film [15], and calculated VN1.00 [86].As  is reduced from 0.97 to 0.77, the elastic modulus increases continuously to 444 ± 24 GPa.This anomalous 11% increase in the stiffness of VNx due to the N vacancy presence is consistent with the predicted elasticities, as evident in Figure 8 (b).As shown in Figure 1 and Figure 4, this anomalous elasticity increase can be rationalized by the N vacancy-induced elastic anisotropy and bond strengthening, as a characteristic of the group IV binary cubic VNx compound.Based on the above presented theoretical predictions on the effect of VEC and N-vacancies on the elastic properties of group IV (TiNx) and V (VNx) cubic transition metal nitrides the different trends in the mechanical properties reported in literature can readily be rationalized: On one hand, experimental reports on N vacancy-induced reduction in elasticity of TiNx by Jhi et al. [33] and Shin et al. [31] are consistent with the here presented predictions.On the other hand, also the reported elasticity of the group V NbNx thin films exhibited an anomalous behavior with respect to N vacancy concentration [16].[9]), DFT VN1.00 at 0 K (Fulcher et al. [86]), and thin film VN1.00 (Mei et al. [15]).
the reduction in lattice parameter of VNx with increasing N vacancy concentration goes in hand with an anomalous 11% increase in elastic modulus and is in very good agreement with DFT predictions.Based on the theoretical and experimental data presented here, it is evident that the elastic behavior of early transition metal nitrides is defined by the N vacancy concentration and the transition metal valence electron concentration.COHP analyses and elastic anisotropy calculations reveal that this behavior can be rationalized by considering the direction-dependent bond-strengthening (bond-softening) upon vacancy introduction into VNx (TiNx).This study offers a strategy to design nitride thin films with defined elastic properties by controlling the valence electron-and vacancy-concentration and is expected to be relevant for other isostructural binary and ternary nitrides as well.
voltage and current of 40 kV and 40 mA, respectively.The X-ray source and the detector were coupled in θ-2θ scans (Bragg-Brentano geometry), scanning a 2θ range from 41° to 46° to obtain the (200) diffractions from VNx thin films and the MgO substrate.A step size of 0.05° and dwell time of 2 s per step were used for the measurements.θ-rocking curves along (200) diffraction plane of VNx were acquired with an incident parallel beam within the same diffractometer.
Figure 1 (b).Here, the Poisson's ratio values follow completely different trends with respect

Figure 1 :
Figure 1: (a) Bulk modulus, (b) Poisson's ratio, and (c) elastic modulus at ground state for cubic TMNx (TM = Ti, V, Zr, Nb, and Mo) as well as (d) the average ICOHP v alues determined for TM-N bonds in cubic TiNx and VNx as a function of x.
(a) and (d), consistent with[77].The introduction of N vacancies results in a significant reduction of the direction-dependent TiNx elastic modulus, see Figure2(a), (b), and (c).In contrast, for VNx, the vacancy-induced reduction in elastic modulus along the [100] direction is smaller and overcompensated by concomitant changes along the [110] and [111] directions.Therefore, as vacancies are introduced in TiNx, marginal changes in directionality are accompanied by a reduced elastic modulus in all directions, while in VNx, N vacancies cause changes in the directional elastic moduli leading to an overall increase in stiffness.

Figure 3 .
Figure 3.Total and partial differential density of states (dDOS) analysis in the vicinity of a N vacancy for the vacancy-containing systems and a similar region in the fully-occupied systems for (a) TiNx and (b) VNx as a function of x.Ef designates the Fermi energy.

Figure 4 (
a) and (d) show the local lattice relaxations evident in the structural model, projected in the (100) plane, which are induced by the presence of a N vacancy in TiN0.81 and VN0.81, respectively.For TiN0.81 the distance between two Ti atoms along [100] is increased by ca.2.6% from 4.24 Å in pristine TiN to 4.35 Å near the vacancy site, which shows an outward displacement of the atoms.However, the local relaxation along the vacancy site of VN0.81 shows an inward displacement of V atoms from 4.13 Å to 3.86 Å by 6.5% towards the vacancy site.Figure 4 (b), (c), (e), and (f) depict the effect of the local lattice relaxations upon bond length and bond strength (expressed via the ICOHP) distributions within TiNx and VNx as a function of the vacancy concentration.For both binary compounds, the introduction of a N vacancy leads to a distribution in the overall bond lengths and energies, in contrast to the stoichiometric systems.However, the

Figure 4 .
Figure 4. Projections of the structural models in the (001) plane, depicting the lattice close to the vacancy site for (a) TiN0.81 and (d) VN0.81.(b)Ti-N ICOHP values vs. Ti-N bond length.(c) Ti-N ICOHP histogram.(e) V-N ICOHP values vs. V-N bond length.(f) V-N ICOHP histogram.The average ICOHP for the TM-N bond of each structure is indicated with dashed lines and the corresponding average values are included in figures (c) and (f).vacancy-induced distribution is much larger in the case of VNx, as evident from the scattering

Figure 5 .
Figure 5. (a) N/V ratio x of VNx/MgO(001) thin films deposited by DCMS as a function of substrate temperature determined by TOF-ERDA.(b) XPS N 1s spectra of (a) VNx/MgO(001) with respect to N/V ratio x.The presence of N vacancies has been further indicated by XPS measurements, with