# An Ab Initio Study of Pressure-Induced Reversal of Elastically Stiff and Soft Directions in YN and ScN and Its Effect in Nanocomposites Containing These Nitrides

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

## 3. Results

_{11}/$\delta p$, $\delta $C

_{12}/$\delta p$ and $\delta $C

_{44}/$\delta p$ according to Equations (1)–(3). Regarding the calculated values of second-order elastic constants ${C}_{ij}$(p = 0 GPa), Table 1 shows that they are in an excellent agreement with previously published theoretical results for both YN and ScN when we selected GGA calculations [24,25,38] (LDA predicts both materials to be metallic [23]).

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Bacon, D.; Barnett, D.; Scattergood, R. Anisotropic continuum theory of lattice defects. Prog. Mater. Sci.
**1979**, 23, 51–262. [Google Scholar] [CrossRef] - Ting, T.C.T. Anisotropic Elasticity; Oxford University Press: New York, NY, USA, 1996. [Google Scholar]
- Udyansky, A.; von Pezold, J.; Bugaev, V.N.; Friák, M.; Neugebauer, J. Interplay between long-range elastic and short-range chemical interactions in Fe-C martensite formation. Phys. Rev. B
**2009**, 79, 224112. [Google Scholar] [CrossRef] - Lothe, J. Dislocations in Continuous Elastic Media, in Elastic Strain Fields and Dislocation Mobility; Series of Modern Problems in Condensed Matter Physics; Elsevier: Amsterdam, The Netherlands, 1992; Volume 31. [Google Scholar]
- Mouhat, F.; Coudert, F.-X. Necessary and sufficient elastic stability conditions in various crystal systems. Phys. Rev. B
**2014**, 90, 224104. [Google Scholar] [CrossRef] - Kraut, E.A. Advances in the theory of anisotropic elastic wave propagation. Rev. Geophys.
**1963**, 1, 401–448. [Google Scholar] [CrossRef] - Ting, T.C.T. Longitudinal and transverse waves in anisotropic elastic materials. Acta Mech.
**2006**, 185, 147–164. [Google Scholar] [CrossRef] - Thurston, R.; Brugger, K. Third-order elastic constants + velocity of small amplitude elastic waves in homogeneously stressed media. Phys. Rev.
**1964**, 133, A1604. [Google Scholar] [CrossRef] - Brugger, K.; Thurston, R. Sound velocity in stressed crystals + 3-order elastic coefficients. J. Acoust. Soc. Am.
**1964**, 36, 1041. [Google Scholar] [CrossRef] - Brugger, K. Pure modes for elastic waves in crystals. J. Appl. Phys.
**1965**, 36, 759. [Google Scholar] [CrossRef] - Brugger, K. Generalized Gruneisen parameters in anisotropic Debye model. Phys. Rev.
**1965**, 137, 1826. [Google Scholar] [CrossRef] - Körmann, F.; Dick, A.; Grabowski, B.; Hallstedt, B.; Hickel, T.; Neugebauer, J. Free energy of bcc iron: Integrated ab initio derivation of vibrational, electronic, and magnetic contributions. Phys. Rev. B
**2008**, 78, 033102. [Google Scholar] [CrossRef] - Zhao, J.; Winey, J.M.; Gupta, Y.M. First-principles calculations of second- and third-order elastic constants for single crystals of arbitrary symmetry. Phys. Rev. B
**2007**, 75, 094105. [Google Scholar] [CrossRef] - Ledbetter, H.; Naimon, E. Elastic properties of metals and alloys. II. Copper. J. Phys. Chem. Ref. Data
**1974**, 3, 897. [Google Scholar] [CrossRef] - Lincoln, R.C.; Koliwad, K.M.; Ghate, P.B. Morse-Potential Evaluation of Second- and Third-Order Elastic Constants of Some Cubic Metals. Phys. Rev.
**1967**, 157, 463–466. [Google Scholar] [CrossRef] - De Jong, M.; Winter, I.; Chrzan, D.C.; Asta, M. Ideal strength and ductility in metals from second- and third-order elastic constants. Phys. Rev. B
**2017**, 96, 014105. [Google Scholar] [CrossRef] - Kim, K.Y.; Sachse, W.; Every, A.G. On the determination of sound speeds in cubic crystals and isotropic media using a broadband ultrasonic point-source/point-receiver method. J. Acoust. Soc. Am.
**1993**, 93, 1393–1406. [Google Scholar] [CrossRef] - Tasnádi, F.; Abrikosov, I.A.; Rogström, L.; Almer, J.; Johansson, M.P.; Odén, M. Significant elastic anisotropy in Ti
_{1−x}Al_{x}N alloys. Appl. Phys. Lett.**2010**, 97, 231902. [Google Scholar] [CrossRef] - Saha, B.; Sands, T.D.; Waghmare, U.V. Electronic structure, vibrational spectrum, and thermal properties of yttrium nitride: A first-principles study. J. Appl. Phys.
**2011**, 109, 073720. [Google Scholar] [CrossRef] - Yang, J.W.; An, L. Ab initio calculation of the electronic, mechanical, and thermodynamic properties of yttrium nitride with the rocksalt structure. Phys. Status Solidi (b)
**2014**, 251, 792–802. [Google Scholar] [CrossRef] - Mancera, L.; Rodriguez, J.A.; Takeuchi, N. Theoretical study of the stability of wurtzite, zinc-blende, NaCl and CsCl phases in group IIIB and IIIA nitrides. Phys. Status Solidi (b)
**2004**, 241, 2424–2428. [Google Scholar] [CrossRef] - Zerroug, S.; Ali Sahraoui, F.; Bouarissa, N. Ab initio calculations of yttrium nitride: Structural and electronic properties. Appl. Phys. A
**2009**, 97, 345–350. [Google Scholar] [CrossRef] - Stampfl, C.; Mannstadt, W.; Asahi, R.; Freeman, A.J. Electronic structure and physical properties of early transition metal mononitrides: Density-functional theory LDA, GGA, and screened-exchange LDA FLAPW calculations. Phys. Rev. B
**2001**, 63, 155106. [Google Scholar] [CrossRef] - Liu, Z.T.Y.; Zhou, X.; Khare, S.V.; Gall, D. Structural, mechanical and electronic properties of 3d transition metal nitrides in cubic zincblende, rocksalt and cesium chloride structures: A first-principles investigation. J. Phys. Condens. Matter
**2014**, 26, 025404. [Google Scholar] [CrossRef] [PubMed] - Mattesini, M.; Magnuson, M.; Tasnádi, F.; Höglund, C.; Abrikosov, I.A.; Hultman, L. Elastic properties and electrostructural correlations in ternary scandium-based cubic inverse perovskites: A first-principles study. Phys. Rev. B
**2009**, 79, 125122. [Google Scholar] [CrossRef] - Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. Rev.
**1964**, 136, B864–B871. [Google Scholar] [CrossRef] - Kohn, W.; Sham, L.J. Self-consistent equations including exchange and correlation effects. Phys. Rev.
**1965**, 140, A1133–A1138. [Google Scholar] [CrossRef] - Kresse, G.; Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B
**1993**, 47, 558–561. [Google Scholar] [CrossRef] - Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B
**1996**, 54, 11169–11186. [Google Scholar] [CrossRef] - Blöchl, P.E. Projector augmented-wave method. Phys. Rev. B
**1994**, 50, 17953–17979. [Google Scholar] [CrossRef][Green Version] - Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B
**1999**, 59, 1758. [Google Scholar] [CrossRef] - Holec, D.; Friák, M.; Neugebauer, J.; Mayrhofer, P.H. Trends in the elastic response of binary early transition metal nitrides. Phys. Rev. B
**2012**, 85, 064101. [Google Scholar] [CrossRef] - Perdew, J.P.; Wang, Y. Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B
**1992**, 45, 13244–13249. [Google Scholar] [CrossRef] - Zhou, L.; Holec, D.; Mayrhofer, P.H. First-principles study of elastic properties of Cr-Al-N. J. Appl. Phys.
**2013**, 113, 043511. [Google Scholar] [CrossRef] - Togo, A.; Tanaka, I. First principles phonon calculations in materials science. Scr. Mater.
**2015**, 108, 1–5. [Google Scholar] [CrossRef][Green Version] - Mancera, L.; Rodríguez, J.A.; Takeuchi, N. First principles calculations of the ground state properties and structural phase transformation in YN. J. Phys. Condens. Matter
**2003**, 15, 2625. [Google Scholar] [CrossRef] - Pearson’s Handbook of Crystallographic Data for Intermetallic Phases; American Society for Metals: Metals Park, OH, USA, 1985.
- Brik, M.; Ma, C.G. First-principles studies of the electronic and elastic properties of metal nitrides XN (X=Sc, Ti, V, Cr, Zr, Nb). Comput. Mater. Sci.
**2012**, 51, 380–388. [Google Scholar] [CrossRef] - Gall, D.; Petrov, I.; Hellgren, N.; Hultman, L.; Sundgren, J.E.; Greene, J.E. Growth of poly- and single-crystal ScN on MgO(001): Role of low-energy ${\mathrm{N}}_{2}^{+}$ irradiation in determining texture, microstructure evolution, and mechanical properties. J. Appl. Phys.
**1998**, 84, 6034–6041. [Google Scholar] [CrossRef] - Birch, F. Finite Elastic Strain of Cubic Crystals. Phys. Rev.
**1947**, 71, 809. [Google Scholar] [CrossRef] - Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett.
**1996**, 77, 3865–3868. [Google Scholar] [CrossRef] [PubMed] - Titrian, H.; Aydin, U.; Friák, M.; Ma, D.; Raabe, D.; Neugebauer, J. Self-consistent Scale-bridging Approach to Compute the Elasticity of Multi-phase Polycrystalline Materials. MRS Proc.
**2013**, 1524, mrsf12-1524-rr06-03. [Google Scholar] [CrossRef] - Friák, M.; Counts, W.; Ma, D.; Sander, B.; Holec, D.; Raabe, D.; Neugebauer, J. Theory-Guided Materials Design of Multi-Phase Ti-Nb Alloys with Bone-Matching Elastic Properties. Materials
**2012**, 5, 1853–1872. [Google Scholar] [CrossRef][Green Version] - Zhu, L.F.; Friák, M.; Lymperakis, L.; Titrian, H.; Aydin, U.; Janus, A.; Fabritius, H.O.; Ziegler, A.; Nikolov, S.; Hemzalová, P.; Raabe, D.; Neugebauer, J. Ab initio study of single-crystalline and polycrystalline elastic properties of Mg-substituted calcite crystals. J. Mech. Behav. Biomed. Mater.
**2013**, 20, 296–304. [Google Scholar] [CrossRef] [PubMed] - Mayrhofer, P.H.; Fischer, F.D.; Boehm, H.J.; Mitterer, C.; Schneider, J.M. Energetic balance and kinetics for the decomposition of supersaturated Ti1-xAlxN. Acta Mater.
**2007**, 55, 1441–1446. [Google Scholar] [CrossRef] - Wu, L.; Chen, M.; Li, C.; Zhou, J.; Shen, L.; Wang, Y.; Zhong, Z.; Feng, M.; Zhang, Y.; Han, K.; et al. Ferromagnetism and matrix-dependent charge transfer in strained LaMnO
_{3}-LaCoO_{3}superlattices. Mater. Res. Lett.**2018**, 6, 501–507. [Google Scholar] [CrossRef] - Koutná, N.; Holec, D.; Friák, M.; Mayrhofer, P.H.; Šob, M. Stability and elasticity of metastable solid solutions and superlattices in the MoN–TaN system: First-principles calculations. Mater. Des.
**2018**, 144, 310–322. [Google Scholar] [CrossRef] - Jiang, M.; Xiao, H.Y.; Peng, S.M.; Yang, G.X.; Liu, Z.J.; Zu, X.T. A comparative study of low energy radiation response of AlAs, GaAs and GaAs/AlAs superlattice and the damage effects on their electronic structures. Sci. Rep.
**2018**, 8. [Google Scholar] [CrossRef] [PubMed] - Wen, Y.N.; Gao, P.F.; Xia, M.G.; Zhang, S.L. Half-metallic ferromagnetism prediction in MoS2-based two-dimensional superlattice from first-principles. Mod. Phys. Lett. B
**2018**, 32. [Google Scholar] [CrossRef] - Friák, M.; Tytko, D.; Holec, D.; Choi, P.P.; Eisenlohr, P.; Raabe, D.; Neugebauer, J. Synergy of atom-probe structural data and quantum-mechanical calculations in a theory-guided design of extreme-stiffness superlattices containing metastable phases. New J. Phys.
**2015**, 17, 093004. [Google Scholar] [CrossRef][Green Version] - Dai, Q.; Eckern, U.; Schwingenschlog, U. Effects of oxygen vacancies on the electronic structure of the (LaVO
_{3})_{6}/SrVO_{3}superlattice: A computational study. New J. Phys.**2018**, 20. [Google Scholar] [CrossRef] - Jiang, M.; Xiao, H.; Peng, S.; Qiao, L.; Yang, G.; Liu, Z.; Zu, X. First-Principles Study of Point Defects in GaAs/AlAs Superlattice: the Phase Stability and the Effects on the Band Structure and Carrier Mobility. Nanoscale Res. Lett.
**2018**, 13. [Google Scholar] [CrossRef] - Chen, H.; Millis, A.J.; Marianetti, C.A. Engineering Correlation Effects via Artificially Designed Oxide Superlattices. Phys. Rev. Lett.
**2013**, 111. [Google Scholar] [CrossRef][Green Version] - Mottura, A.; Janotti, A.; Pollock, T.M. A first-principles study of the effect of Ta on the superlattice intrinsic stacking fault energy of L1
_{2}-Co_{3}(Al,W). Intermetallics**2012**, 28, 138–143. [Google Scholar] [CrossRef] - Rosengaard, N.; Skriver, H. Ab-initio study of antiphase boundaries and stacking-faults in L1
_{2}and D0_{22}compounds. Phys. Rev. B**1994**, 50, 4848–4858. [Google Scholar] [CrossRef] - Torres-Pardo, A.; Gloter, A.; Zubko, P.; Jecklin, N.; Lichtensteiger, C.; Colliex, C.; Triscone, J.M.; Stephan, O. Spectroscopic mapping of local structural distortions in ferroelectric PbTiO
_{3}/SrTiO_{3}superlattices at the unit-cell scale. Phys. Rev. B**2011**, 84. [Google Scholar] [CrossRef] - Chawla, V.; Holec, D.; Mayrhofer, P.H. Stabilization criteria for cubic AlN in TiN/AlN and CrN/AlN bi-layer systems. J. Phys. D Appl. Phys.
**2013**, 46. [Google Scholar] [CrossRef] - Cooper, V.R.; Rabe, K.M. Enhancing piezoelectricity through polarization-strain coupling in ferroelectric superlattices. Phys. Rev. B
**2009**, 79. [Google Scholar] [CrossRef][Green Version] - Chen, B.; Zhang, Q.; Bernholc, J. Si diffusion in gaas and si-induced interdiffusion in gaas/alas superlattices. Phys. Rev. B
**1994**, 49, 2985–2988. [Google Scholar] [CrossRef] - Schmid, U.; Christensen, N.; Cardona, M.; Lukes, F.; Ploog, K. Optical anisotropy in GaAs/AlSs(110) superlattices. Phys. Rev. B
**1992**, 45, 3546–3551. [Google Scholar] [CrossRef] - Gibson, Q.D.; Schoop, L.M.; Weber, A.P.; Ji, H.; Nadj-Perge, S.; Drozdov, I.K.; Beidenkopf, H.; Sadowski, J.T.; Fedorov, A.; Yazdani, A.; et al. Termination-dependent topological surface states of the natural superlattice phase Bi
_{4}Se_{3}. Phys. Rev. B**2013**, 88. [Google Scholar] [CrossRef] - Park, C.; Chang, K. Structural and electronic-properties of gap-alp (001) superlattices. Phys. Rev. B
**1993**, 47, 12709–12715. [Google Scholar] [CrossRef] - Romanyuk, O.; Hannappel, T.; Grosse, F. Atomic and electronic structure of GaP/Si(111), GaP/Si(110), and GaP/Si(113) interfaces and superlattices studied by density functional theory. Phys. Rev. B
**2013**, 88. [Google Scholar] [CrossRef] - Abdulsattar, M.A. SiGe superlattice nanocrystal pure and doped with substitutional phosphorus single atom: Density functional theory study. Superlattices Microstruct.
**2011**, 50, 377–385. [Google Scholar] [CrossRef] - Botti, S.; Vast, N.; Reining, L.; Olevano, V.; Andreani, L. Ab initio and semiempirical dielectric response of superlattices. Phys. Rev. B
**2004**, 70. [Google Scholar] [CrossRef] - Rondinelli, J.M.; Spaldin, N.A. Electron-lattice instabilities suppress cuprate-like electronic structures in SrFeO
_{3}/OSrTiO_{3}superlattices. Phys. Rev. B**2010**, 81. [Google Scholar] [CrossRef] - Momma, K.; Izumi, F. VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr.
**2011**, 44, 1272–1276. [Google Scholar] [CrossRef]

**Figure 1.**Schematic visualization of the 2-atom primitive (

**a**) and 8-atom conventional cube-shape (

**b**) unit cells of NaCl-structure of YN (some atoms are shown together with their periodic images).

**Figure 2.**Computed changes in the elasticity of rock-salt structure YN visualized as directional dependencies of the Young’s modulus. The zero-pressure case based on the second-order elastic constants computed by the stress-strain method is shown in part (

**a**). The estimated changes in the Young’s modulus due to 1 GPa of hydrostatic pressure are shown for different directions in absolute terms (in GPa) in part (

**b**) and relatively (divided by the value for this direction in the zero-pressure case) in part (

**c**). The visualized changes (in the second-order elasticity at the hydrostatic pressure of 1 GPa) are predicted using the second-order and third-order elastic constants computed for the zero-pressure state according to Equations (1)–(3). Finally, the directional dependence of the second-order elasticity computed at the 1.6 GPa is shown in part (

**d**). Mind the change in the scale between the parts (

**a**) and (

**d**).

**Figure 3.**The same as in Figure 2 but for ScN. Part (

**d**) is computed at the hydrostatic pressure of 8.0 GPa. The parts (

**a**) and (

**d**) were visualized by the SC-EMA [42,43,44] library (scema.mpie.de) based on our ab initio computed elastic constants.

**Figure 4.**Quantum-mechanically calculated second-order elastic constants of YN (

**a**) and ScN (

**b**) for different hydrostatic pressures. The vertical dash-dotted lines indicate the zero hydrostatic pressure.

**Figure 5.**Quantum-mechanically computed (

**a**) dependence of the Zener elastic anisotropy ratio ${A}_{\mathrm{Z}}$ and the band-gap energy (

**b**) as a function of hydrostatic pressure for both YN and ScN. The horizontal dashed line for ${A}_{\mathrm{Z}}$ = 1 represents the border value of the elastic anisotropy, the vertical dash-dotted line corresponds to zero hydrostatic pressure.

**Figure 6.**Quantum-mechanically calculated phonon dispersions at zero pressure for YN (

**a**) and ScN (

**b**) and for YN also for the hydrostatic pressure p = 1.6 GPa (

**c**) and for ScN for p = 6.5 GPa (

**d**).

**Figure 7.**The calculated directional dependences of the Young’s modulus of two tetragonally deformed states of YN with $a=b=0.990\phantom{\rule{4pt}{0ex}}{a}_{\mathrm{eq}}$ (

**a**) $a=b=1.010\phantom{\rule{4pt}{0ex}}{a}_{\mathrm{eq}}$ (

**b**), respectively. The tetragonal lattice c parameters are equal to the values corresponding to the minimum energy (and zero stress ${\sigma}_{c}$ = 0).

**Figure 8.**Schematic visualization of 16-atom ScN/PdN supercell (

**a**) and the corresponding directional dependence of the Young’s modulus (

**b**). The calculations for this nanocomposite were performed using 7 × 7 × 4 k-point grid. Small black arrows indicate the shifts of N atoms off the transition-metal planes perpendicular to the [001] direction.

**Table 1.**Calculated second-order elastic constants ${C}_{ij}$(p = 0 GPa) (in comparison with selected literature values—when a GGA was used as in our case) and their pressure changes ($\delta $C

_{11}/$\delta p$, $\delta $C

_{12}/$\delta p$, and $\delta $C

_{44}/$\delta p$) as approximatively evaluated for $\delta p$ = 1 GPa from computed third-order elastic constants ${C}_{ijk}$(p = 0 GPa). Theoretical values taken from Ref. [24] are related to GGA-PW91 approximation [33] similarly as in our case (marked by *), GGA-PW91 + U (marked by **), GGA-PBE [41] (marked by

^{†}) or GGA-PBE + U (marked by

^{††}).

C_{11} | C_{12} | C_{44} | $\mathit{\delta}$C_{11}/$\mathit{\delta}\mathit{p}$ | $\mathit{\delta}$C_{12}/$\mathit{\delta}\mathit{p}$ | $\mathit{\delta}$C_{44}/$\mathit{\delta}\mathit{p}$ | |
---|---|---|---|---|---|---|

YN | 318 | 81 | 124 | 7.55 | 1.12 | −0.70 |

(321 [24] *) | (81 [24] *) | (124 [24] *) | ||||

(304 [24] **) | (76 [24] **) | (122 [24] **) | ||||

(317 [24] ${}^{\u2020}$) | (80 [24] ${}^{\u2020}$) | (123 [24] ${}^{\u2020}$) | ||||

(310 [24] ${}^{\u2020\u2020}$) | (81 [24] ${}^{\u2020\u2020}$) | (124 [24] ${}^{\u2020\u2020}$) | ||||

ScN | 388 | 106 | 166 | 7.49 | 1.02 | −0.51 |

(399 [24]) | (96 [24]) | (158 [24]) | ||||

(397 [25]) | (131 [25]) | (170 [25]) | ||||

(354 [38]) | (100 [38]) | (170 [38]) | ||||

C${}_{111}$ | C${}_{112}$ | C${}_{123}$ | C${}_{144}$ | C${}_{166}$ | C${}_{456}$ | |

YN | −4100 | −160 | 180 | 180 | −225 | 185 |

ScN | −5100 | −190 | 260 | 200 | −330 | 215 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Friák, M.; Kroupa, P.; Holec, D.; Šob, M. An Ab Initio Study of Pressure-Induced Reversal of Elastically Stiff and Soft Directions in YN and ScN and Its Effect in Nanocomposites Containing These Nitrides. *Nanomaterials* **2018**, *8*, 1049.
https://doi.org/10.3390/nano8121049

**AMA Style**

Friák M, Kroupa P, Holec D, Šob M. An Ab Initio Study of Pressure-Induced Reversal of Elastically Stiff and Soft Directions in YN and ScN and Its Effect in Nanocomposites Containing These Nitrides. *Nanomaterials*. 2018; 8(12):1049.
https://doi.org/10.3390/nano8121049

**Chicago/Turabian Style**

Friák, Martin, Pavel Kroupa, David Holec, and Mojmír Šob. 2018. "An Ab Initio Study of Pressure-Induced Reversal of Elastically Stiff and Soft Directions in YN and ScN and Its Effect in Nanocomposites Containing These Nitrides" *Nanomaterials* 8, no. 12: 1049.
https://doi.org/10.3390/nano8121049