Effects of high pressure on ScMN
2
‐type (M = V,
Nb, Ta) phases studied by density functional
theory
Robert Pilemalm, Sergey Simak and Per Eklund
The self-archived postprint version of this journal article is available at Linköping
University Institutional Repository (DiVA):
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-159276
N.B.: When citing this work, cite the original publication.
Pilemalm, R., Simak, S., Eklund, P., (2019), Effects of high pressure on ScMN2-type (M = V, Nb, Ta)
phases studied by density functional theory, RESULTS IN PHYSICS, 13, 102293.
https://doi.org/10.1016/j.rinp.2019.102293
Original publication available at:
https://doi.org/10.1016/j.rinp.2019.102293
Copyright: Elsevier (Creative Commons Attribution License)
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Results in Physics
journal homepage:www.elsevier.com/locate/rinp
Effects of high pressure on ScMN
2
-type (M = V, Nb, Ta) phases studied by
density functional theory
Robert Pilemalm
a,⁎, Sergei Simak
b, Per Eklund
a,⁎aThin Film Physics Division, Department of Physics, Chemistry and Biology (IFM), Linköping University, SE-581 83 Linköping, Sweden bTheoretical Physics Division, Department of Physics, Chemistry and Biology (IFM), Linköping University, SE-581 83 Linköping, Sweden
A R T I C L E I N F O
Keywords:
ScTaN2
Inverse MAX phase High pressure Density functional theory
A B S T R A C T
ScMN2-type (M = V, Nb, Ta) phases are layered materials that have been experimentally reported for M = Ta
and Nb, but their high-pressure properties have not been studied. Here, we have used first-principles calculations to study their thermodynamic stability, elastic and electronic properties at high-pressure. We have used density functional theory to calculate the formation enthalpy relative to the competing binary phases, electronic density of states and elastic constants (cij), bulk (B), shear (G) and Young’s (E) modulus as the pressure is varied from 0 to
150 GPa. Our results show that when the pressure increases from 0 to 150 GPa, elastic constants, bulk, shear and elastic moduli increase in the range 53–216% for ScTaN2,72–286% for ScNbN2, and 61–317% for ScVN2.
Introduction
High-pressure studies on materials are important in contributing to increased fundamental understanding of the behavior of materials. For example, high pressure-studies allow explanations of the properties of materials at the Earth’s core[1–3], high-temperature superconductivity occurring at high pressure[4]or in technological applications at high pressure. Thus, ab initio studies of material stability at high pressure are important to determine stability and can lead to prediction of new phases.
The MAX phases are a family of ternary carbides and nitrides de-scribed with the general formula Mn+1AXn, where M is a transition
metal, A is an A-group element, X is carbon or nitrogen and n = 1,2 or 3. In the case of n = 1 a MAX phase is referred to as a “211” MAX phase, for n = 2, “312” and for n = 3, “413” [5,6]. Their high-pressure properties have been investigated both theoretically and experimen-tally and these have concerned phase transformations and their me-chanical properties [7–13]. For example, MAX phases exhibit poly-morphism; one type occurs because of shearing of the A-layer at high pressure in Ti3GeC2[12,14]. A second type of polymorphism is due to
different stacking sequences in Ta4AlC3, where the transition between
the two different stacking sequences is induced by high pressure[10]. The MAX phases have relatively low densities at the same time as they are quite stiff but relatively stiff, and they are damage and thermal shock resistant. This unusual combination of properties makes the MAX
phases good candidates for high pressure applications, especially taking into account that they are readily machinable[8,10].
The ScMN2-type structure is a closely related structure to a 211
MAX phase. These structures have been experimentally observed in the ScTaN2 and ScNbN2 systems [15–17], but has been relatively little
studied. As determined by Niewa at al. using Rietveld refinement of X-rays and neutron diffraction results[17], the structure of ScMN2
be-longs to the space group P63/mmc (#194). In the archetypical case of
ScTaN2, it is comprised of alternating layers of ScN6/3octrahedra and
TaN6/3prisms. Sc occupies 2a Wyckoff positions, Ta the 2d positions
and N the 4f positions.[17,18]The structure can be termed “inverse MAX phase”, because the Wyckoff positions are the inverted positions of a 211 MAX phase [19]. The structure of ScTaN2is visualized in Fig. 1. We have previously[20]shown from density functional theory (DFT) calculations that ScTaN2, ScNbN2 and ScVN2 are
thermo-dynamically stable at 0 K and 0 GPa[18,20]. Our previous results also suggest that these materials could be interesting for thermoelectric applications and the thermoelectric properties could be tuned by doping. Thus, there is an interest to synthesize these materials and this motivates that the fundamental understanding of these materials needs to be increased. So far, no information on either thermodynamic sta-bility or elastic properties at high pressure of these materials is avail-able. Here, we here present a high-pressure study on physical properties of ScTaN2, ScNbN2and ScVN2to investigate if any phase decomposition
occurs and how the elastic properties change with pressure.
https://doi.org/10.1016/j.rinp.2019.102293
Received 6 March 2019; Received in revised form 7 April 2019; Accepted 15 April 2019
⁎Corresponding authors.
E-mail addresses:robert.pilemalm@liu.se(R. Pilemalm),per.eklund@liu.se(P. Eklund).
Computational details
In order to determine the phase stability of ScMN2(where M is Ta,
Nb or V) at different pressures, P, and the volume, V, corresponding to each pressure, the formation enthalpy relative the competing binary phases in each state was calculated according to
=
H HScMN2 HScN H ,MN (1)
where the enthalpy, H, of each phase with the internal energy, U, is calculated according to
= +
H U PV (2)
For relaxation the energy and force tolerance was 0.0001 eV and 0.001 eV/Å, respectively. It is well known that NaCl structured ScN, or here referred as c-ScN, is the ground state stable phase and that the binary remains in this phase up to a pressure about 366 GPa [21], which motivates using only this phase for ScN in the calculation of formation enthalpy relative to the competing binary phases. For the MN binary, we choose in first instance the NaCl structure for TaN, NbN and VN as they have been used in calculations of formation enthalpy re-lative to the competing binary phases [18,20]. We also considered hexagonal TaN. In this system there are two binary phases, ε-TaN and π-TaN that only differ by a shift in N position, with the consequence that the two phases cannot be readily distinguished with x-ray diffraction. They can, however, be distinguished by neutron diffraction[22]. For
this reason, there are some discrepancies in the descriptions of the structure of the ε-TaN phase in literature [22–24]. π-TaN (P-62 m, #189) is the most stable phase of the two[22], which we therefore use in the calculations. It is here referred as h-TaN. For NbN the most stable phase has the anti-TiP structure (P63/mmc, #194)[25], which here is
referred as h-NbN. For VN at 0 K, the most stable phase has the WC structure[26,27], here referred as h-VN and at low temperatures also a tetragonal phase can exist[28,29], here referred as t-VN. However, for VN, the cubic phase is known to be stabilized by atomic vibration at higher temperatures [22,29], which implies that h-VN and t-VN are stable only at low temperature. For discussion about properties near room temperature, it is therefore more accurate to consider c-VN as the competing phase. We have included all of these phases.
Density functional theory (DFT) calculations were performed with the help of the Vienna Ab initio Simulation Package (VASP)[30–34]. The cutoff energy was set to 650 eV, projector augmented wave (PAW) basis sets[35]were used and the exchange-correlation potential was modeled with the generalized gradient approximation according to Perdew, Burke and Ernzerhof (PBE-GGA) [36]. The unit cells of the considered systems consisted of 8 atoms each and an 8 × 8 × 8 k-point mesh was used for the energy calculations. The elastic tensor for each pressure was determined in VASP by first introducing finite distortions in the lattice followed by calculations on strain-stress relationships
[37]. For these calculations 25 × 25 × 11 k-points mesh was used. The magnitude of the strains was in the order of 0.015 Å. Furthermore, density of states (DOS) was calculated for selected pressures with a plane wave cutoff of 650 eV and with a 25 × 25 × 11 k-point mesh. The PBE-GGA functional was used together with the tetrahedron method with Blöchl correction[38]. The value of the level broadening was 0.2 eV.
Results and discussion
Eqs.(3)–(9) define the formation enthalpies relative to the com-peting binary phases ΔHi (i = 1,2,…7) that were calculated with the
enthalpies H of the previously defined phases, H at a given pressure. The chosen pressure values were 0, 30, 50, 100, and 150 GPa. V is the volume, corresponding to each P, per formula unit and was obtained by first fitting energies and volumes to the Birch-Murnaghan equation of state (EOS)[39]. = H1 H ScTaN( 2) H c( ScN) H c( TaN) (3) = H H ScTaN2 ( 2) H c( ScN) H h( TaN) (4) = H3 H ScNbN( 2) H c( ScN) H c( NbN) (5) = H4 H ScNbN( 2) H c( ScN) H h( NbN) (6) = H5 H ScVN( 2) H c( ScN) H c( VN) (7) = H6 H ScVN( 2) H c( ScN) H h( VN) (8) = H7 H ScVN( 2) H c( ScN) H t( VN) (9)
Fig. 2a–c shows the calculated formation enthalpies relative to the competing binary phases as a function of pressure for the three different material systems. The formation enthalpy relative to the competing binary phases of each pressure is shown with markers. The continuous lines are cubic spline interpolations.Fig. 2a shows the enthalpies of formation for ScTaN2as defined in Eqs.(3) and (4). At 0 GPa, ScTaN2is
stable. Pressure increases the thermodynamic stability of this phase, as can be seen by comparing the enthalpies of formation with those of c-TaN as well as h-c-TaN. For pressures in the range 0–90 GPa, the enthalpy of formation relative to c-TaN tends to be lower than relative to h-TaN. At pressures > 90 GPa there is an intersection between the two,
Fig. 1. The structure of ScTaN2viewed along the [1 1 0] zone axis. The lattice c
vector points upwards.
R. Pilemalm, et al. Results in Physics 13 (2019) 102293
indicating that h-TaN is more stable than c-TaN at high pressure.
Fig. 2b shows the formation enthalpies relative to the competing binary phases of ScNbN2as a function of pressure as defined in Eq.(5) and (6). Similarly to ScTaN2, ScNbN2is stable at 0 GPa and the effect of
pressure is to increase the thermodynamic stability of this phase. It can be noted that the two curves inFig. 2b are similar and have almost the same slope for all pressures. The curves do not intersect, in contrast to the case of ScTaN2.
Fig. 2c shows the formation enthalpies relative to the competing binary phases as a function of pressure for ScVN2as defined in Eqs. (7)–(9). It can first be noted that ScVN2exhibits a positive formation
enthalpy at 0 relative to c-ScN and h-VN. For VN, the cubic phase is known to be stabilized by atomic vibration at higher temperatures
[22,29], which implies that h-VN is stable only at very low tempera-tures. It is therefore more reasonable to consider c-VN as the competing phase. Relative to c-VN, the ScVN2phase is stable at 0 GPa. Thus, it is
reasonable to conclude that ScVN2is thermodynamically stable at room
temperature and pressure, even though the formation enthalpy relative to h-VN at 0 K is positive. However, even when considering h-VN as competing phase, ScVN2 is thermodynamically stable at pressures
higher than 40 GPa this phase becomes thermodynamically stable.
Tables 1–3show the calculated elastic constants, the bulk modulus (B), shear modulus (G) and elastic modulus (E) of each material system.
B and G are calculated with the Voigt approximation (Bvand Gv)[40]
and the Reuss approximation (BRand GR)[40,41]. The Voigt
approx-imation is based on the assumption that the strain throughout the polycrystalline aggregate is uniform, while the Reuss approximation is based on the assumption that the stress in uniform [40]. For a hex-agonal lattice, the Voigt shear modulus, GV,and the Voigt bulk
mod-ulus, BV, are: = + + + G 1 c c c c c c c 15(2 2 ) 1 5(2 ( )/2)) v 11 33 12 13 44 11 12 (10) = + + + B 2 c c c c 9( 2 /2) V 11 12 13 33 (11)
The Reuss shear modulus, GR and Reuss bulk modulus, BRfor a
hexagonal lattice are:
= + + + + G s s s s s s 15 4(2 ) 4( 2 ) 3(2 ) R 11 33 12 13 44 66 (12) = + + + B s s s s 1 (2 ) 2( 2 ) R 11 33 12 13 (13)
For the calculation of the Young’s modulus in both cases, i.e., EVand
ER, the following relation was used[40]:
= + E BG B G 9 3 (14)
In order to determine if the systems are elastically anisotropic the universal elastic anisotropy index, AU, was calculated[42]:
= + A G G E E 5 6 U V R V R (15)
At each pressure, it is valid for the elastic constants inTables 1–3
that C11> |C12|, 2C213< C33(C11+ C12), C44> 0 and
C11− C12> 0, which are the necessary and sufficient conditions for
elastic stability (Born criteria) in the case of a hexagonal crystal structure[43]. This means that all three phases are elastically stable at higher pressure. ScTaN2and ScNbN2are stable at ambient pressures
and their stability increases with increasing pressure. No other ternary phases are known to exist in these systems. It is thus likely that these phases that will be stable at higher pressures. The case of ScVN2 is
Fig. 2. Formation enthalpies relative to the competing binary phases of (a)
ScTaN2, (b) ScNbN2, and (c) ScVN2. The phases in parentheses are the phases
that the formation enthalpies are relative to. The pressures for which the cal-culations are performed are marked in the graphs. The lines are cubic spline interpolations.
somewhat more complicated since ΔH6> 0 at 0 GPa; however, that is
assuming the 0 K h-VN is the most competing phase. Assuming instead that c-VN is the competing phase, since this is the stable form of VN at room temperature, ΔH5< 0 already at 0 GPa. This means that the
elastic constants also for pressures below 40 GPa are of interest. For all three material systems (Tables 1–3), it can be noted that the elastic constants Cij, for all indices i,j increase with pressure. That is,
increased pressure makes the materials stiffer. The change is largest for ScTaN2inTable 1. The changes in elastic constants also results in
in-crease of the bulk, shear and elastic moduli as can be noted inTables 1–3. The universal elastic anisotropy indices inTables 1–3for 0 GPa are very low, which is comparable to the indices of other hexagonal systems and indicates that the anisotropy in elastic properties in these materials is limited [42]. It can furthermore be noted inTables 1–3 that the universal anisotropy indices increase as the pressure increases. The change is most pronounced increase, when the pressure is increased from 0 GPa to 150 GPa is in ScTaN2, but it can be seen that ScVN2has
the highest universal anisotropy index at 150 GPa.
Fig. 3shows the calculated total and partial DOS of ScTaN2. The
figure shows two cases, 0 GPa (Fig. 3a) and 150 GPa (Fig. 3b). The basic structure of the DOS does not change significantly, but it can be noted when the two figures are compared that peaks in the high-pressure graphs are shifted towards lower energies and that the states of each energy value are lower. All bands in the DOS have also become broader.
Fig. 4shows the calculated total and partial DOS of ScNbN2of the
case 0 GPa (Fig. 4a) and 150 GPa (Fig. 4b).Fig. 5shows the calculated total and partial DOS of ScVN2and at 0 GPa (Fig. 5a) and 150 GPa
(Fig. 5b). In both cases, it can be noted that the peaks in the high-pressure case have shifted towards lower energies that the number of states of each energy value is lower and that each band has become broader, just like for ScTaN2. When the high-pressure effects between Figs. 3–5are compared, the shift of the bands are most pronounced in
Fig. 5and least inFig. 3. This means that broadening of the bands and the shift of bands are largest in the ScVN2case and lowest in the ScTaN2
case. The general trends are that with increasing pressure, the intera-tomic distances decrease, which in turn lead to an overlap of the electronic wave functions. The total of number of electronic states does not change with pressure. Therefore, the number of states of each en-ergy level is decreased, but the enen-ergy bands are broader.
Conclusions
We have used DFT calculations to investigate the formation en-thalpies relative to the competing binary phases, elastic properties and DOS ScTaN2, ScNbN2and ScVN2under high pressure. Increased
pres-sure has a stabilizing effect of the thermodynamics of the three phases relative to their competing phases. The elastic constants and, bulk, shear, and elastic moduli increase in the range 53–216 % for ScTaN2,
72–286% for ScNbN2, and 61–317% for ScVN2, when pressure increases
from 0 to 150 GPa the elastic anisotropy is relatively limited but in-creases in all three phases for increasing pressure. This study suggests that these phases are interesting to study further experimentally at high-pressure conditions.
Table 2
Elastic constants, bulk moduli, shear and elastic moduli of ScNbN2at different pressures. Moduli are estimated with Voigt (subscript V) and Reuss (subscript R)
approximations. Pressure C11 C12 C13 C33 C44 BR BV GR GV ER EV AU 0 522 152 130 546 185 268 268 189 190 460 460 0.012 30 663 217 212 676 223 365 365 232 233 574 576 0.027 50 740 261 264 752 240 423 423 252 255 631 638 0.068 100 904 369 386 916 268 556 556 287 298 735 758 0.211 150 1035 474 502 1057 280 676 676 306 327 798 844 0.396 Table 1
Elastic constants, bulk moduli, shear and elastic moduli of ScTaN2at different pressures. Moduli are estimated with Voigt (subscript V) and Reuss (subscript R)
approximations. Pressure C11 C12 C13 C33 C44 BR BV GR GV ER EV AU 0 551 158 143 552 196 283 283 197 197 479 479 0.005 30 703 225 229 686 239 384 384 244 245 604 606 0.023 50 787 272 283 762 258 446 446 266 268 665 670 0.065 100 962 387 413 933 287 587 587 302 313 774 797 0.212 150 1102 500 536 1077 300 714 714 322 344 840 889 0.400 Table 3
Elastic constants, bulk moduli, shear and elastic moduli of ScVN2at different pressures. Moduli are estimated with Voigt (subscript V) and Reuss (subscript R)
approximations. Pressure C11 C12 C13 C44 C33 BR BV GR GV ER EV AU 0 580 145 121 167 572 255 256 178 179 432 436 0.057 30 624 213 204 206 728 356 357 222 224 551 556 0.064 50 698 257 257 220 812 415 416 241 246 606 616 0.112 100 849 369 384 239 994 548 551 271 285 697 729 0.306 150 972 477 504 247 1153 669 674 286 313 750 813 0.556
R. Pilemalm, et al. Results in Physics 13 (2019) 102293
Fig. 4. Total DOS and DOS projections for ScNbN2at a) 0 GPa b) 150 GPa.
R. Pilemalm, et al. Results in Physics 13 (2019) 102293
Acknowledgements
The authors acknowledge support from the Swedish Research Council (VR) through project grant number 2016-03365, the Knut and Alice Wallenberg Foundation through the Wallenberg Academy Fellows program, the Swedish Government Strategic Research Area in Materials Science on Functional Materials at Linköping University (Faculty Grant SFO-Mat-LiU No. 2009 00971). The calculations were performed using computer resources provided by the Swedish national infrastructure for computing (SNIC) at the National Supercomputer Centre (NSC). From NSC we would furthermore like to thank Weine Olovsson for support by practical guidance to the calculations.
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