Philosophical Magazine, 2016 VOL. 96, NO. 35, 3654–3670 http://dx.doi.org/10.1080/14786435.2016.1234081
Structural and anisotropic elastic properties of hexagonal MP (M = Ti, Zr, Hf) monophosphides determined by first-principles calculations Runyue Li and Yonghua Duan School of Material Science and Engineering, Kunming University of Science and Technology, Kunming, China
ABSTRACT
First-principles calculations were performed to investigate the structural properties, phase stabilities, elastic properties and thermal conductivities of MP (M = Ti, Zr, Hf ) monophosphides. These monophosphides are thermodynamically and mechanically stable. Values for the bulk modulus B, shear modulus G, Young’s modulus E and Poisson’s ratio ν were calculated by Voigt–Reuss–Hill approximation. The mechanical anisotropy was discussed via several anisotropy indices and three-dimensional (3D) surface constructions. The order of elastic anisotropy is ZrP > HfP > TiP. The minimum thermal conductivities of these monophosphides were investigated using Clarke’s model and Cahill’s model. The results revealed that these monophosphides are suitable for use as thermal insulating materials and that their minimum thermal conductivities are anisotropic.
ARTICLE HISTORY
Received 31 March 2016 Accepted 5 September 2016 KEYWORDS
First-principles calculations; monophosphides; phase stability; anisotropic elasticity
1. Introduction Compounds composed of transition metals and main group elements are of interest in a variety of scientific areas including surface science [1], catalysis [2], astrophysics [3,4] and organometallic chemistry [5,6]. A large number of transition metal borides, carbides, nitrides and silicides have been extensively studied due to their excellent mechanical, chemical and electrical properties [7–9]. Some phosphides have remarkable physical and chemical properties; however, only a few reports of transition metal phosphides have been presented in the literature to date [10–12]. Among the transition metal phosphides, the metallic MP (M = Ti, Zr, Hf) monophosphides feature a high hardness [13–15]. TiP has a hexagonal structure [16], whereas ZrP and HfP possess cubic and hexagonal structures [17], respectively. The hexagonal Ti, Zr and Hf phosphides, termed β-(Ti, Zr and Hf)P, have a superstructure of NiAs that can be thought of as interpenetrating slabs of NaCl-like (ABC) and NiAs-like (BCB) sandwiches [18]. The cubic ZrP and HfP, termed α-ZrP and α-HfP, possess NaCl-type lattices [19]. α-ZrP has been successfully synthesised via a sodium co-reduction of ZrCl4 and PCl3 at 350 °C for 6 h [20]. In addition, an α-ZrP single crystal was obtained by chemical vapour deposition from CONTACT Yonghua Duan
[email protected]
© 2016 Informa UK Limited, trading as Taylor & Francis Group
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a system of ZrC14, PCl3, H2 and Ar on a quartz substrate at 1050–1250 °C [21]. Whiskers of β-HfP, α-ZrP and β-TiP have been prepared from HfC14/ZrCl4/TiC14 + PC13 + H2 + Ar gas mixtures at 1050–1250 °C using the mixed-metal impurity-activated chemical vapour deposition process [22]. β-ZrP could transform into α-ZrP at 1425 °C, and the relatively high temperature stabilities of the hexagonal phosphides follow the order TiP 0. As shown in Table 2, the elastic constants Cij of TiP, ZrP and HfP meet the mechanical stability criteria, indicating that these MP monophosphides are mechanically stable. The elastic constant C11 refers to the x directional (a-axis) resistance to linear compression, while the elastic constant C33 means the resistance to linear compression along the z direction (c-axis) [35]. Obviously, in Table 2, TiP possesses the largest C11, suggesting that TiP is the most incompressible along the a-axis. ZrP has the smallest C11, which corresponds to a large compressibility of ZrP under a-axis uniaxial stress. Additionally, it is noted that the order of C33 is similar to that of C11. As a result, the order of incompressibility along the c-axis is also ZrP 0.26) is ductile. Otherwise, it is brittle [41,42]. As shown in Table 3, it is obvious that TiP and HfP are brittle due to their small values of B/G and ν, less than 1.75 and 0.26, respectively. ZrP has critical values of exactly B/G = 1.75 and ν = 0.26. However, it can be further explored using the classical criteria of the Cauchy pressure (C12−C44). A crystal with a negative Cauchy pressure is brittle; otherwise, it is ductile [43]. The negative values of the Cauchy pressure for TiP (−57.9 GPa), ZrP (−25.3 GPa) and HfP (−58.8 GPa) confirm their brittle nature. HfP is the most brittle phase, whereas ZrP is the least brittle phase. The hardness H is a basic parameter reflecting the elastic and plastic properties of a solid. The hardness of polycrystalline materials can be evaluated using a theoretical model [44]:
( )0.583 H = 2 k2 G − 3,
(11)
where k refers to Pugh’s modulus ratio (k = G/B). The hardness calculated from the theoretical model may be not very accurate because of the macroscopic concepts of bulk and shear moduli. However, it can provide a guideline to reveal basal factors controlling the hardness of solids. The calculated results of hardness for the considered monophosphides are also listed in Table 3. It can be seen that the calculated and experimental values of the hardness of TiP are 14.97 and 13.31 GPa [10], respectively, indicating that the calculated value agrees with the experimental one. Figure 3 plots the values of the elastic moduli (B, G and E) and hardness H for TiP, ZrP and HfP. For better comparison with the elastic moduli, the values of H are multiplied by a factor of 10. In Equation (11), the hardness is related to both the bulk and shear moduli. However, from Figure 3, the shear modulus is more pertinent to hardness than the bulk modulus due to the different natures of bulk and shear moduli. TiP has the highest hardness compared to the other monophosphides, which is in good agreement with the result predicted from shear modulus. There are no available experimental and theoretical data of the elastic moduli and the hardness for TiP, ZrP and HfP. However, a useful guide for future work can be provided by our calculations. 3.3. Anisotropy of elastic moduli The elastic anisotropy is of importance to the possibility of including micro-cracks in crystals. There are several methodologies used to indicate the elastic anisotropy of hexagonal crystals. First, the elastic anisotropy of MP monophosphides can be characterised by the values of the bulk modulus along the a-axis (Ba) and c-axis (Bc), and the expressions are as follows [45],
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Figure 3. (colour online) Calculated bulk, shear and Young’s moduli and hardness of TiP, ZrP and HfP. The values of hardness are multiplied by the factor of 10 for a better comparison.
Ba = a
Λ dP = , da 2+𝛼
(12)
B dP = a, dc 𝛼
(13)
Bc = c
Λ = 2(C11 + C12 ) + 4C13 𝛼 + C33 𝛼 2 ,
𝛼=
C11 + C12 − 2C13 . C33 − C13
(14)
(15)
Second, the elastic anisotropy can be described by the universal anisotropy index AU and by the anisotropy in compression and shear Acomp and Ashear, respectively. The elastic anisotropy indexes AU, Acomp and Ashear for a crystal with any symmetry can be expressed as follows [46,47]:
AU = 5
Acomp =
GV BV + − 6 ≥ 0, GR BR
B V − BR , BV + BR
Ashear =
G V − GR GV + GR
(16)
(17)
where BV (GV) and BR (GR) are the bulk moduli (shear moduli) in the Voigt and Reuss approximations, respectively. A crystal with AU = 0 is isotropic. The large deviation of AU from zero suggests a high mechanical anisotropy. The values of Acomp and Ashear ranging from zero to 1 represent isotropy and the maximum elastic anisotropy [48].
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Third, for hexagonal crystals, the shear anisotropic factors A1, A2 and A3 are calculated according to the following expressions [49],
A1 =
4C44 C11 + C33 − 2C13
for the {1 0 0} plane
(18)
A2 =
4C55 C22 + C33 − 2C23
for the {0 1 0} plane
(19)
A3 =
4C66 C11 + C22 − 2C12
for the {0 0 1} plane
(20)
A crystal with A1, A2 and A3 values equal to 1.0 is isotropic. Otherwise, it is anisotropic. Finally, by solving the Christofel equation for a hexagonal lattice, the anisotropy of the compression wave (P) is obtained by [50]
ΔP =
C33 C11
(21)
The anisotropies of the shear wave polarised perpendicular to the basal plane (S1) and that polarised in the basal plane (S2) are calculated by
ΔS1 = ΔS2 =
C11 + C33 − 2C13 4C44 2C44
.
(22)
C11 − C12
For S2 and P waves, the extreme values occur along the c-axis; for S1, the extreme value occurs at an angle of 45° from the c-axis in the ac plane. A crystal with ΔP = ΔS1 = ΔS2 = 1 is isotropic. Several elastic anisotropy indexes are listed in Table 4. It can be observed that Bc is larger than Ba for these monophosphides. The values of Bc/Ba are 1.398, 1.450 and 1.425 for TiP, ZrP and HfP, respectively. This indicates that the compression along the c-axis is more difficult than that along the a-axis. This result agrees well with the comparison of C11 and C33. This can be explained by the presence of the ionic metal–phosphorus bonds in the [0 0 1] direction in these monophosphides. In Table 4, it is clear that the order of the value of AU is TiP ZrP > HfP. These MP monophosphides are ideally qualified for use as thermal insulating materials because of their relatively low thermal conductivity.
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Because the total thermal conductivity is treated as the summation of three acoustic branches (one is the longitudinal mode and the others are transverse modes), Cahill’s model should fit better than Clarke’s model for a discussion of the anisotropy in thermal conductivity [62]. According to the symmetry of hexagonal crystal, the results listed in Table 6 are confined to two low index crystallographic directions, ie the [1 0 0] and [0 0 1] directions. The values of vl, vt1 and vt2 in the [1 0 0] and [0 0 1] directions in Table 5 are significantly different. This results in the difference of the calculated thermal conductivities (λmin[1 0 0] and λmin[0 0 1]) in Table 6. λmin[0 0 1] always being larger than λmin[1 0 0] in these MP monophosphides suggests that the dependences of thermal conductivities in the [0 0 1] direction are more prominent than those in the [1 0 0] direction. In hexagonal monophosphides, M atoms are aligned mainly along the [0 0 1] direction, whereas P atomic layers are parallel to the (1 0 0) plane. The phonon collision probability along the [1 0 0] direction is smaller than that along the [0 0 1] direction. Thus, the phonon mean free path along the [1 0 0] direction is larger than that along the [0 0 1] direction. As a result, the thermal conductivity in the [0 0 1] direction is larger than that in the [1 0 0] direction.
4. Conclusions First-principles methods have been utilised to systematically investigate the phase stability, elastic moduli, hardness, elastic anisotropy properties, Debye temperatures and thermal conductivity of MP (M = Ti, Zr, Hf) monophosphides. The obtained formation enthalpies indicate that HfP is the most stable phase. The calculated elastic constants reveal that these monophosphides are mechanically stable. The bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio and hardness were calculated and discussed. The values of C33 for MP are very large among the elastic constants, which correspond to an incompressibility of MP under z direction (c-axis) uniaxial stress. According to several anisotropy indexes and 3D surface constructions, the order of the elastic anisotropy of MP monophosphides is ZrP > HfP > TiP. Furthermore, the order of the Debye temperature and the minimum thermal conductivity for these monophosphides is TiP > ZrP > HfP, and these monophosphides are suitable for use as thermal insulating materials. The thermal conductivity and the sound velocity are anisotropic in these crystal structures based on calculated sound velocities and thermal conductivities. We hope our results can provide guidance for further theoretical and experimental studies on these monophosphides.
Disclosure statement No potential conflict of interest was reported by the authors.
Funding This work was supported by the Reserve Talents Project of Yunnan Province [grant number 2015HB019], the National Natural Science Foundation of China [grant number 51201079] and the Scientific Research Fund of the Department of Education of Yunnan Province [grant number 2015Z038].
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