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Theoretical investigation on the high-pressure physical properties of ZnN in cubic zinc blende, rock salt, and cesium chloride structures
Accepted Manuscript Theoretical investigation on the high-pressure physical properties of ZnN in cubic zinc blende, rock salt, and cesium chloride structures J.H. Tian, T. Song, X.W. Sun, T. Wang, G. Jiang PII:
S0022-3697(16)30955-6
DOI:
10.1016/j.jpcs.2017.05.028
Reference:
PCS 8078
To appear in:
Journal of Physics and Chemistry of Solids
Received Date: 23 October 2016 Revised Date:
8 April 2017
Accepted Date: 29 May 2017
Please cite this article as: J.H. Tian, T. Song, X.W. Sun, T. Wang, G. Jiang, Theoretical investigation on the high-pressure physical properties of ZnN in cubic zinc blende, rock salt, and cesium chloride structures, Journal of Physics and Chemistry of Solids (2017), doi: 10.1016/j.jpcs.2017.05.028. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Theoretical investigation on the high-pressure physical properties of ZnN in cubic zinc blende, rock salt, and
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cesium chloride structures
b c
Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China
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a
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J.H. Tian a, b, T. Song b, c, X.W. Sun b, T. Wang b, G. Jiang a, d, *
School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China
School of Material Science and Engineering, Lanzhou University of Technology, Lanzhou 730050, China d
The Key Laboratory of High Energy Density Physics and Technology, Ministry of Education, Chengdu
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610065, China
*
Corresponding author at: Institute of Atomic and Molecular Physics, Sichuan University, Chengdu
610065, China. Tel.: +86 028 85408810. E-mail addresses: [email protected] (J.H. Tian), [email protected] (G. Jiang) 1
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Abstract In the current work, the aim is to report systematic results from first-principles calculations with density functional theory (DFT) on three cubic structures, rock salt (NaCl-type), zinc blende (ZnS-type),
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and cesium chloride (CsCl-type), of ZnN under high pressure. From the enthalpy versus pressure relations, we find that the NaCl-type phase of ZnN is more stable than the ZnS-type phase when the pressure higher
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than 2.55 GPa and high-pressure NaCl-type phase will stabilize up to 150 GPa. Through the careful evaluation with the quasi-harmonic Debye model, a complete set of thermodynamic data up to 2000 K,
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including PVT equation of state, isothermal bulk modulus, Debye temperature, Grüneisen parameter, thermal expansivity, heat capacity, and entropy for the ZnN with high-pressure NaCl-type structure is achieved. This set of data is considered as the useful information to understand the high-temperature and
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high-pressure properties of ZnN.
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Keywords: ZnN; Phase transition; Thermodynamic property; High pressure
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1. Introduction The transition metal nitrides are of great interest and importance for their unique properties and have attracted considerable attention due to their interesting combination of high strength, high hardness, high
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ultra-incompressibility, high melting point, high thermal conductivity, good chemical stability and low electrical resistivity [1-5]. Recently, a lot of studies involving a systematic search throughout the 3d, 4d and
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5d transition metal mono-nitrides in many different structures have provided thorough data to better explain and predict material properties. For 4d transition metal mono-nitrides in rock salt (B1, NaCl-type) and zinc
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blende (B3, ZnS-type) structures, the cohesive energies, atomic and electronic structures of MN (M=Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd) were studied by using the linear muffin-tin orbital (LMTO) method [6] and full-potential linearized augmented plane-wave (FLAPW) method [7, 8]. Zhao et al. have studied the structural, elastic, electronic and mechanical properties of 4d transition metal mono-nitrides which were
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extended to CdN by first-principles calculations considering the NaCl-type, NiAs-type and WC-type structures [4] and they found that the NaCl structure is the most stable for some compounds. They have
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also studied the NaCl, ZnS, cesium chloride (B2, CsCl-type), wurtzite, NiAs and WC structures of 5d transition metal mono-nitrides from LaN to AuN and found that NaCl structure is the most stable for LaN,
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HfN and AuN [9]. Patil et al. [10] have investigated the 5d transition metal nitrides with ZnS, NaCl, fluorite and pyrite structures, not limited to mono-nitrides. For 3d transition metal mono-nitrides, the corresponding studies were mostly on early transition metal elements such as Sc, Ti, V and Cr, and solely on NaCl phase, by Zaoui et al.
[11], Brik and Ma [12] and Holec et al. [13].
Compared with other 3d, 4d and 5d transition metal mono-nitrides, the zinc nitride (ZnN) from the group of 3d transition mono-nitrides has received little attention [14]. Until very recently, Rajeswarapalanichamy et al. [15] have reported that ZnS phase is the lowest energy phase for ZnN, which 3
Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT are some inconsistent with another transition metal mononitride CdN. It is well known that both the Zn and Cd atoms belong to group 12 and CdN has a stable NaCl structure by many studies [4, 15, 16]. The contradictions in the structural stability between ZnN and CdN and the lack of thermodynamic data have
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motivated us to investigate the corresponding properties for ZnN. The layout of this paper is given as follows: The detailed calculations and employed methods, namely first-principles calculations and the quasi-harmonic Debye approximation are given in Section 2. Results and discussions of the structural phase
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transition and thermodynamic properties will be presented in Section 3. Conclusions are given in Section 4.
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2. Computational methods
In this paper the total energy calculations and crystal structure relaxations by force minimizations for ZnN have been performed using ab initio method with the CASTEP code [17] based on density functional
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theory through a plane-wave expansion of the wave functions. The Vanderbilt-type ultrasoft pseudopotentials [18] were utilized to describe the interactions between valence electrons and ions, in which the valence electronic configurations for Zn and N atoms are 3d10 4s2 and 2s2 2p3, respectively. The
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generalized gradient approximation (GGA) of the PBESOL formalism [19], which is dependent on both the
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electron density and its gradient at each space point, was employed for the estimation of the structural and thermodynamic properties in this work. The integrations over the first Brillouin-zone were replaced by a discrete summation on special set of k-points using Monkhorst-Pack method [20] and the 15×15×15 grids were employed for cubic ZnN. The kinetic energy cut-off value was selected as 650 eV, of course, which was sufficient to obtain reliable results. A quasi-Newton variable metric minimization method using the Brodyden–Fletcher–Goldfarb–Shanno update scheme [21] was applied to search the ground state, which provides a very efficient and fast way to explore the optimizing crystal structure.
4
Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT The thermodynamic properties of the cubic ZnN in the present investigation have been predicted by the quasi-harmonic Debye approximation model based on GIBBS code [22]. In the quasi-harmonic Debye model, the non-equilibrium Gibbs function G*(V; P, T) is taken in the form of
G * (V ; P , T ) = E (V ) + PV + Avib [Θ (V ); T ] ,
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(1)
where E(V) is the total energy per primitive cell of ZnN, P is pressure, V is volume and PV corresponds to
expressed in terms of the Debye temperature ΘD:
Θ D , − D T
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−Θ D 9 ΘD Avib (Θ D ; T ) = nk BT + 3ln 1 − e T 8 T
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the constant hydrostatic pressure condition, and Avib is the vibrational Helmholtz free energy which can be
(2)
where D (Θ D / T ) represents the Debye integral, n is the number of atoms per formula unit. The value of ΘD is expressed by:
1
(3)
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KS h 2 12 3 Θ D = 6π V n f (σ ) , k M
where M is the molecular mass per formula unit, KS is the adiabatic bulk modulus, σ is the Poisson’s ratio, and f(σ) is formulated by Ref. [23].Therefore, the non-equilibrium Gibbs function G*(V; P, T) as a function
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of (V; P, T) can be minimized with respect to volume V:
∂G *(V ; P, T ) = 0 . ∂V P ,T
(4)
By solving Eq. (4) we obtain the various thermodynamic properties. The isothermal bulk modulus KT which is approximately presented by the static compressibility is given together with the Grüneisen parameter γ, constant volume heat capacity CV, volumetric thermal expansion coefficient αV, and entropy S as follows:
d 2 E (V ) KT ≈ K S = V , 2 dV
γ =−
d ln Θ(V ) . d ln V 5
(5)
(6)
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αV = Θ S = nk 4 D T
3Θ D / T , − Θ /T e D − 1
γ CV KT V
,
(8)
−Θ T − 3ln 1 − e
.
(9)
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3. Results and discussion
(7)
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Θ CV = 3nk 4 D D T
In this paper, we consider three different cubic structures, namely zinc blende (ZnS-type, space group
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F-43m), rock salt (NaCl-type, space group Fm-3m) and cesium chloride (CsCl-type, space group Pm-3m) structures for ZnN, as shown in Figure 1. In order to check the stability of the structures predicted from a thermodynamic perspective and show how the structures of ZnN vary under pressure, for each proposed structure, the equilibrium geometry and zero-temperature Gibbs free energy, i.e. enthalpy, at more than 15
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pressure points in the range from 0 to 150 GPa are obtained by the complete optimization of the lattice constants and the atomic positions, as shown in Figure 2 (a). A phase transition is said to occur when the
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changes in structural details of the phase are caused by such variations of free energy. For a given pressure at 0 K, a stable structure is the one for which the enthalpy has the lowest value. From Figure 2 (a) we can
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see that the ZnS-type structure is stable under low pressure and the NaCl-type structure is the high-pressure phase, where the CsCl-type structure is always unstable in the whole pressure range. The transition pressure Pt is a pressure at which H(P) curves for both phases cross. Figure 2 (b) gives the Pt from the ZnS-type to NaCl-type structure based on the enthalpy difference calculations and the value is about 2.55 GPa. Rajeswarapalanichamy et al. [15] have reported that ZnS-type structure is the lowest energy phase for ZnN, but there is no phase transition pressure Pt from ZnS-type to NaCl-type structure reported. Take into account the phase transition pressure, the calculated pressure and volume relations of 6
Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT ZnS-type and NaCl-type structures for ZnN are given in Figure 3. At 2.55 GPa, the corresponding volume reduction ∆V/V is about 16.5%, as marked in the diagram. Figure 4 evaluates the energy and stability of both the structures. It can also be found from this figure that at ambient conditions, ZnS-type structure with
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the lowest energy is more stable than the NaCl-type structure in the present study. By fitting the E-V curve for ZnS-type structure of ZnN using 3rd-order Birch-Murnaghan equation [24], the equilibrium lattice constant a0, isothermal bulk modulus K0 and corresponding pressure derivative K'0 are obtained
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successfully with DFT-GGA-PBESOL calculations. These important equation of state parameters a0, K0
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and K'0, along with the transition pressure Pt and volume collapse -∆V/V0 from ZnS-type to NaCl-type structure for ZnN are all shown in Table 1. Unfortunately, to the best of our knowledge, the experimental values for these are not yet available for comparison. Our results are in good agreement with the DFT-GGA-PBE and DFT-LDA-CA-PZ calculated results 4.595 and 4.4854 Å, 139 and 144 GPa, and 5.011
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and 5.066, respectively [15], which suggesting that our calculations are reliable and accurate. In the following, we will focus on the thermodynamic properties predictions for the high-pressure NaCl-type
account.
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structure of ZnN through the quasi-harmonic Debye model in which the phononic effects are taken into
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The pressure dependences of the relative volume V/V0 for ZnN in the high-pressure NaCl-type structure at three selected temperature of 0, 1000, and 2000 K is exhibited in Figure 5. It is seen that, at the same pressure, the relative volume V/V0 of higher temperature is less than that of lower temperature. As the pressure increases, the relative volume decreases at a given temperature. This effect of increasing pressure on ZnN is just the same as decreasing temperature of ZnN. The conclusions are very common for many materials but the knowledge of the thermal equation of state is important as it is a tool to describe the structural behavior over a vast range of temperature and pressure. For example, the isothermal bulk modus 7
Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT KT can easily be obtained from the isothermal equation of state, that is it depends on volume V and pressure P at constant temperature T, and usually defined as follows [25]:
∂P KT = −V . ∂V T
(10)
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As we known, in the quasi-harmonic Debye model based on the GIBBS code [22], the resulting E-V data is fitted by a polynomial, to calculate the bulk modulus as a function of volume from equation (5), and
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the bulk modulus values are then used to calculate the Debye temperature for each unit cell volume of ZnN from equation (3). Using this model, the pressures effect on the isothermal bulk modulus KT at
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temperatures of 0, 1000 and 2000 K is plotted for the NaCl-type structure of ZnN in Figures 6. It is easily seen from Figure 6 that the isothermal bulk modulus is linearly increasing with pressure in the range of 0 to 150 GPa, and has a slightly decrease with temperature. That is to say, KT is greatly affected by the pressure compared with the temperature in the whole pressure and temperature ranges for ZnN.
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Debye temperature ΘD is an important parameter in the study of the materials thermodynamic properties and it is not only closely correlated with many physical properties of solids, such as thermal
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expansivity, melting points, and specific heat but also it can be used to characterize the strength of covalent bonds in solids. The dependence of the Debye temperature ΘD on pressure and temperature for ZnN in
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NaCl-type structure is depicted in Figure 7 (a) and (b). The calculated Debye temperature ΘD of ZnN with NaCl-type structure at zero temperature and pressure is 617.11 K, where the value is 563.45 K for the ZnS-type structure, and no other results are supplied for comparison. Take into account the effects of pressure and temperature, we will find from Figure 7 (a) and (b) that the trend of ΘD is consistent with the variation of the isothermal bulk modulus. At low pressure, θD decreases significantly when the temperature changes from 0 to 2000 K. As the pressure goes higher, the decreased magnitude of becomes small. In addition, as we known, the larger the ΘD is, the stronger the covalent bonds are [26]. Hence, with 8
Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT increasing pressure, the covalent bonds of ZnN with NaCl-type structure become stronger. The Grüneisen parameter γ is a physical quantity of central importance related to thermal and elastic properties of materials [27]. In order to estimate γ, we need to plot lnΘD versus lnV, which indicated by
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equation (6), and the slope of the yielded straight line equals minus of the γ. In Figure 8 (a) and (b), we present the effects of pressure and temperature on the Grüneisen parameter γ at temperatures of 0, 1000 and 2000 K and at pressures of 0, 50, 100 and 150 GPa for the NaCl-type structure of ZnN, respectively. From
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Figure 8 (a), one can find that when the temperature keeps constant, the Grüneisen parameter decreases
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with the increasing pressures, but the trend will become smooth under high pressure. When the pressure keeps constant, the Grüneisen parameter increases with the increasing temperatures, but the trend will become rapid under low pressure, as shown in Figure 8 (b). In virtue of the fact that the effect of increasing pressure on the material is the same as decreasing temperature on the material. In this study, we obtained γ0
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= 2.183 using quasi-harmonic Debye model, and the value will decrease to 1.528 at 150 GPa. Within the quasi-harmonic approximation, the anharmonicity is restricted to the thermal expansion. The temperature dependences of the volumetric thermal expansion coefficient αV for the NaCl-type
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structure of ZnN at 0, 50, 100 and 150 GPa are plotted in Figure 9. It is evident that at a constant pressure,
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the αV increases rapidly at the temperatures below 300 K. Then, except for the zero-pressure data, it switches to a linearly and smoothly increasing trend at the temperatures above 300 K. Besides, it can be find that the effect of pressure on αV is larger at higher temperatures, and obviously smaller at lower temperatures. It well known that there are certain difficulties associated with the measurement of αV at high temperatures which lead to considerable uncertainties in the experimental values [28], so the theoretical calculation is a good estimate of the errors for thermal expansion investigations of materials. Another thermodynamic parameter, heat capacity, is also very important. The heat capacity of a 9
Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT substance is not only providing essential insight into its vibrational properties, but is also mandatory for many applications. The relationship between temperature and constant volume heat capacity CV for ZnN in NaCl-type structure under different pressures from 0 to 150 GPa with an interval of 50 GPa is curved in
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Figure 10. The calculated specific heat capacity CV is 40.97 J mol-1 K-1 at zero pressure and ambient temperature. For ZnN, CV is increasing with the elevating temperature. At intermediate temperatures, the temperature dependence of CV is governed by the details of vibrations of the atoms. However, under higher
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temperature, as the anharmonic effect on CV is suppressed, CV is close to a limited value when temperature
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keeps increasing, namely, Dulong-Petit limit [29].
Another thermodynamic parameter, entropy S, is also predicted at different pressures of 0, 50, 100 and 150 GPa for the NaCl-type structure of ZnN in the temperature range of 0 to 2000 K, as shown in Figure 11. In thermodynamics, S is a measure of the number of specific ways in which a thermodynamic
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system may be arranged, often taken to be a measure of disorder, or a measure of progressing towards thermodynamic equilibrium. From the curves in the Figure 11, the entropy is zero at 0 K and it changes increase rapidly as temperature increases at low temperature. As the temperature increases the vibrational
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contribution to the entropy increases, therefore the entropy increases with temperature increasing [30].
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Furthermore, the pressure has much weaker effect on S than the temperature. The calculated values of enthalpy at room temperature is 36.04 J mol-1 K-1 for ZnN.
4. Conclusions
The pressure-induced ZnS-type, NaCl-type and CsCl-type structural phase transitions and thermodynamic properties including the PVT equation of state, isothermal bulk modulus, Debye temperature, Grüneisen parameter and entropy for high-pressure NaCl-type structure of ZnN have been
10
Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT studied systematically by first-principles calculations and the quasi-harmonic Debye model in the present work. From the enthalpy versus pressure relations, we find that the ZnS-type structure is stable under low pressure, which in accordance with the theoretical prediction by Rajeswarapalanichamy et al., and the
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NaCl-type structure is the high-pressure phase, where the CsCl-type structure is always unstable in the whole pressure range. The phase transition pressure between ZnS-type and the NaCl-type structures for ZnN is about 2.55 GPa. Prediction of the thermodynamic properties for ZnN shows us that when the
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temperature increases, the volume compression ratio keeps the trend of reducing, and the reduced
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amplitude becomes larger at higher pressure conditions. On the contrary of the isothermal bulk modulus and Debye temperature, which are more sensitive to pressure than temperature, the volume, Grüneisen parameter, thermal expansion coefficient, and entropy of ZnN with NaCl-type structure are proportional to the temperature and inversely proportional to the pressure. Furthermore, the pressure has much weaker
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effect on entropy than the temperature. At sufficiently low temperatures, the constant-volume heat capacity is proportional to T3, and under higher temperature, the constant-volume heat capacity is close to a Dulong-Petit limit. All the properties of cubic ZnN in the whole pressure range from 0 to 150 GPa and
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temperature range from 0 to 2000 K are summarized and discussed. Our calculated results can be seen as a
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prediction for future investigations.
Acknowledgements The authors would like to thank the supports by the National Natural Science Foundation of China under Grant Nos. 51562021, 11464027, and 11164013, the Program for Longyuan Youth Innovation Talents of Gansu Province of China, the Foundation of A Hundred Youth Talents Training Program of Lanzhou Jiaotong University, the National Natural Science Foundation of Gansu Province under Grant Nos.
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Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT 148RJZA027, the Colleges and Universities Scientific Research Program of Gansu Province under Grant Nos. 2015B-040 and 2015B-048, and the Young Scholars Science Foundation of Lanzhou Jiaotong University under Grant No. 2014022.
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[10] S.K.R. Patil, N.S. Mangale, S.V. Khare, S. Marsillac, Thin Solid Films 517 (2008) 824. [11] A. Zaoui, S. Kacimi, B. Bouhafs, A. Roula, Physica B 358 (2005) 63. [12] M. Brik, C.-G. Ma, Comp. Mater. Sci. 51 (2012) 380. [13] D. Holec, M. Friák, J. Neugebauer, P.H. Mayrhofer, Phys. Rev. B 85 (2012) 064101. [14] Z.T.Y. Liu, X. Zhou, D. Gall, S.V. Khare, Comp. Mater. Sci. 84 (2014) 365. [15] R. Rajeswarapalanichamy, G. Sudha Priyanga, A. Jemmy Cinthia, K. Iyakutti, Comp. Mater. Sci. 99 (2015) 117.
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Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT [16] S.D. Gupta, P.K. Jha, A. Pandya, Solid State Sci. 21 (2013) 66. [17] M. Segall, P.J. Lindan, M.a. Probert, C. Pickard, P. Hasnip, S. Clark, M. Payne, J. Phys.: Condens. Matter 14 (2002) 2717.
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[18] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [19] J.P. Perdew, A. Ruzsinszky, G.I. Csonka, O.A. Vydrov, G.E. Scuseria, L.A. Constantin, X. Zhou, K.
[20] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188.
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Burke, Phys. Rev. Lett. 100 (2008) 136406.
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[25] X.W. Sun, Y.D. Chu, Z.J. Liu, T. Song, P. Guo, Q.F. Chen, Mater. Chem. Phys. 116 (2009) 34. [26] Q. Chen, Z. Huang, Z. Zhao, C. Hu, Comp. Mater. Sci. 67 (2013) 196. [27] T.P. Kaloni, Y.C. Cheng, U. Schwingenschlögl, J. Appl. Phys. 113 (2013) 104305.
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[28] O.L. Anderson, K. Zou, Phys. Chem. Miner. 16 (1989) 642.
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[29] P. Debye, Annalen der Physik 344 (1912) 789. [30] K.R. Babu, C.B. Lingam, S. Auluck, S.P. Tewari, G. Vaitheeswaran, J. Solid State Chem. 184 (2011) 343.
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Table Captions:
Table 1. Calculated equilibrium lattice constant a0, isothermal bulk modulus K0 and its pressure derivative K'0 from the 3rd-order Birch-Murnaghan equation for ZnS-type structure, along with the
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transition pressure Pt and volume collapse -∆V/V0 from ZnS-type to NaCl-type structure for ZnN,
Present work 4.515
K0 (GPa)
144.73
K'0
4.747
Pt (GPa)
2.55
-∆V/V0
16.5%
Other calculation 4.588 a, 4.472 b 4.595 c, 4.4854 b
Ref. [5] Ref. [15]
139 a, 144 b
Ref. [15]
5.011 a, 5.066 b
Ref. [15]
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a0 (Å)
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compared with the available theoretical data.
DFT-GGA-PW91 calculation
b
DFT-LDA-CA-PZ calculation
c
DFT-GGA-PBE calculation
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a
Table 1
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Figure Captions: Figure 1. Crystal structures of (a) the zinc blende type, (b) the rock salt type, and (c) the cesium chloride type ZnN phases. The large and small spheres represent zinc and nitrogen atoms, respectively.
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Figure 2. (a) Enthalpies per formula unit of the ZnS-type, NaCl-type, and CsCl-type structures as function of pressure at zero temperature and (b) estimation of phase transition pressure from ZnS-type to NaCl-type
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structure for ZnN.
Figure 3. Volume versus pressure curves from ZnS-type to NaCl-type structure for ZnN.
Figure 4. Energy versus volume curves of ZnN with both the ZnS-type and NaCl-type structures.
1000, and 2000 K.
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Figure 5. The normalized volume V/V0 as a function of pressure for the NaCl-type structure of ZnN at 0,
Figure 6. The variations of isothermal bulk modulus BT with the pressure for the NaCl-type structure of
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ZnN at 0, 1000, and 2000 K.
Figure 7. (a) The variations of Debye temperature ΘD with the pressure at 0, 1000, and 2000 K and (b) the
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variations of ΘD with the temperature at 0, 50, 100, and 150 GPa for the NaCl-type structure of ZnN.
Figure 8. (a) The variations of Grüneisen parameter γ with the pressure at 0, 1000, and 2000 K and (b) the variations of γ with the temperature at 0, 50, 100, and 150 GPa for the NaCl-type structure of ZnN.
Figure 9. The dependences of volumetric thermal expansion coefficient αV on the temperatures for the NaCl-type structure of ZnN at 0, 50, 100, and 150 GPa.
Figure 10. The dependences of constant volume heat capacity CV on the temperatures for the NaCl-type structure of ZnN at 0, 50, 100, and 150 GPa. 15
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Figure 11. The dependences of enthalpy S on the temperatures for the NaCl-type structure of ZnN at 0, 50,
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EP
TE D
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100, and 150 GPa.
16
Figure 1
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-1959
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ZnS structure NaCl structure CsCl structure
-1962
-1968
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H / eV
-1965
-1974
(a) -1977 0
15
30
45
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-1971
60
75
90
105
120
135
150
P / GPa
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0.2 0.1
ZnS structure
0.0
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-0.2 -0.3
NaCl structure
2.55 GPa
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∆H / eV
-0.1
-0.4 -0.5
(b)
-0.6
0
5
10
15
20
25
P / GPa
Figure 2 17
30
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24
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ZnS structure
23 22 21
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20 19
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V/Å
3
- ∆V / V0 = 16.5%
NaCl structure
18
2.55 GPa 17 0
2
4
6
8
10
12
14
16
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P / GPa
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ZnS structure NaCl structure
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ACCEPTED MANUSCRIPT
1. Phase transition and thermodynamic properties of cubic ZnN are reported. 2. The pressure up to 150 GPa.
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3. The temperature up to 2000 K.
S0022-3697(16)30955-6
DOI:
10.1016/j.jpcs.2017.05.028
Reference:
PCS 8078
To appear in:
Journal of Physics and Chemistry of Solids
Received Date: 23 October 2016 Revised Date:
8 April 2017
Accepted Date: 29 May 2017
Please cite this article as: J.H. Tian, T. Song, X.W. Sun, T. Wang, G. Jiang, Theoretical investigation on the high-pressure physical properties of ZnN in cubic zinc blende, rock salt, and cesium chloride structures, Journal of Physics and Chemistry of Solids (2017), doi: 10.1016/j.jpcs.2017.05.028. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Theoretical investigation on the high-pressure physical properties of ZnN in cubic zinc blende, rock salt, and
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cesium chloride structures
b c
Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China
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a
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J.H. Tian a, b, T. Song b, c, X.W. Sun b, T. Wang b, G. Jiang a, d, *
School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China
School of Material Science and Engineering, Lanzhou University of Technology, Lanzhou 730050, China d
The Key Laboratory of High Energy Density Physics and Technology, Ministry of Education, Chengdu
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610065, China
*
Corresponding author at: Institute of Atomic and Molecular Physics, Sichuan University, Chengdu
610065, China. Tel.: +86 028 85408810. E-mail addresses: [email protected] (J.H. Tian), [email protected] (G. Jiang) 1
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Abstract In the current work, the aim is to report systematic results from first-principles calculations with density functional theory (DFT) on three cubic structures, rock salt (NaCl-type), zinc blende (ZnS-type),
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and cesium chloride (CsCl-type), of ZnN under high pressure. From the enthalpy versus pressure relations, we find that the NaCl-type phase of ZnN is more stable than the ZnS-type phase when the pressure higher
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than 2.55 GPa and high-pressure NaCl-type phase will stabilize up to 150 GPa. Through the careful evaluation with the quasi-harmonic Debye model, a complete set of thermodynamic data up to 2000 K,
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including PVT equation of state, isothermal bulk modulus, Debye temperature, Grüneisen parameter, thermal expansivity, heat capacity, and entropy for the ZnN with high-pressure NaCl-type structure is achieved. This set of data is considered as the useful information to understand the high-temperature and
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high-pressure properties of ZnN.
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Keywords: ZnN; Phase transition; Thermodynamic property; High pressure
2
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1. Introduction The transition metal nitrides are of great interest and importance for their unique properties and have attracted considerable attention due to their interesting combination of high strength, high hardness, high
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ultra-incompressibility, high melting point, high thermal conductivity, good chemical stability and low electrical resistivity [1-5]. Recently, a lot of studies involving a systematic search throughout the 3d, 4d and
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5d transition metal mono-nitrides in many different structures have provided thorough data to better explain and predict material properties. For 4d transition metal mono-nitrides in rock salt (B1, NaCl-type) and zinc
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blende (B3, ZnS-type) structures, the cohesive energies, atomic and electronic structures of MN (M=Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd) were studied by using the linear muffin-tin orbital (LMTO) method [6] and full-potential linearized augmented plane-wave (FLAPW) method [7, 8]. Zhao et al. have studied the structural, elastic, electronic and mechanical properties of 4d transition metal mono-nitrides which were
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extended to CdN by first-principles calculations considering the NaCl-type, NiAs-type and WC-type structures [4] and they found that the NaCl structure is the most stable for some compounds. They have
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also studied the NaCl, ZnS, cesium chloride (B2, CsCl-type), wurtzite, NiAs and WC structures of 5d transition metal mono-nitrides from LaN to AuN and found that NaCl structure is the most stable for LaN,
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HfN and AuN [9]. Patil et al. [10] have investigated the 5d transition metal nitrides with ZnS, NaCl, fluorite and pyrite structures, not limited to mono-nitrides. For 3d transition metal mono-nitrides, the corresponding studies were mostly on early transition metal elements such as Sc, Ti, V and Cr, and solely on NaCl phase, by Zaoui et al.
[11], Brik and Ma [12] and Holec et al. [13].
Compared with other 3d, 4d and 5d transition metal mono-nitrides, the zinc nitride (ZnN) from the group of 3d transition mono-nitrides has received little attention [14]. Until very recently, Rajeswarapalanichamy et al. [15] have reported that ZnS phase is the lowest energy phase for ZnN, which 3
Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT are some inconsistent with another transition metal mononitride CdN. It is well known that both the Zn and Cd atoms belong to group 12 and CdN has a stable NaCl structure by many studies [4, 15, 16]. The contradictions in the structural stability between ZnN and CdN and the lack of thermodynamic data have
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motivated us to investigate the corresponding properties for ZnN. The layout of this paper is given as follows: The detailed calculations and employed methods, namely first-principles calculations and the quasi-harmonic Debye approximation are given in Section 2. Results and discussions of the structural phase
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transition and thermodynamic properties will be presented in Section 3. Conclusions are given in Section 4.
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2. Computational methods
In this paper the total energy calculations and crystal structure relaxations by force minimizations for ZnN have been performed using ab initio method with the CASTEP code [17] based on density functional
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theory through a plane-wave expansion of the wave functions. The Vanderbilt-type ultrasoft pseudopotentials [18] were utilized to describe the interactions between valence electrons and ions, in which the valence electronic configurations for Zn and N atoms are 3d10 4s2 and 2s2 2p3, respectively. The
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generalized gradient approximation (GGA) of the PBESOL formalism [19], which is dependent on both the
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electron density and its gradient at each space point, was employed for the estimation of the structural and thermodynamic properties in this work. The integrations over the first Brillouin-zone were replaced by a discrete summation on special set of k-points using Monkhorst-Pack method [20] and the 15×15×15 grids were employed for cubic ZnN. The kinetic energy cut-off value was selected as 650 eV, of course, which was sufficient to obtain reliable results. A quasi-Newton variable metric minimization method using the Brodyden–Fletcher–Goldfarb–Shanno update scheme [21] was applied to search the ground state, which provides a very efficient and fast way to explore the optimizing crystal structure.
4
Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT The thermodynamic properties of the cubic ZnN in the present investigation have been predicted by the quasi-harmonic Debye approximation model based on GIBBS code [22]. In the quasi-harmonic Debye model, the non-equilibrium Gibbs function G*(V; P, T) is taken in the form of
G * (V ; P , T ) = E (V ) + PV + Avib [Θ (V ); T ] ,
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(1)
where E(V) is the total energy per primitive cell of ZnN, P is pressure, V is volume and PV corresponds to
expressed in terms of the Debye temperature ΘD:
Θ D , − D T
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−Θ D 9 ΘD Avib (Θ D ; T ) = nk BT + 3ln 1 − e T 8 T
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the constant hydrostatic pressure condition, and Avib is the vibrational Helmholtz free energy which can be
(2)
where D (Θ D / T ) represents the Debye integral, n is the number of atoms per formula unit. The value of ΘD is expressed by:
1
(3)
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KS h 2 12 3 Θ D = 6π V n f (σ ) , k M
where M is the molecular mass per formula unit, KS is the adiabatic bulk modulus, σ is the Poisson’s ratio, and f(σ) is formulated by Ref. [23].Therefore, the non-equilibrium Gibbs function G*(V; P, T) as a function
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of (V; P, T) can be minimized with respect to volume V:
∂G *(V ; P, T ) = 0 . ∂V P ,T
(4)
By solving Eq. (4) we obtain the various thermodynamic properties. The isothermal bulk modulus KT which is approximately presented by the static compressibility is given together with the Grüneisen parameter γ, constant volume heat capacity CV, volumetric thermal expansion coefficient αV, and entropy S as follows:
d 2 E (V ) KT ≈ K S = V , 2 dV
γ =−
d ln Θ(V ) . d ln V 5
(5)
(6)
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αV = Θ S = nk 4 D T
3Θ D / T , − Θ /T e D − 1
γ CV KT V
,
(8)
−Θ T − 3ln 1 − e
.
(9)
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3. Results and discussion
(7)
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Θ CV = 3nk 4 D D T
In this paper, we consider three different cubic structures, namely zinc blende (ZnS-type, space group
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F-43m), rock salt (NaCl-type, space group Fm-3m) and cesium chloride (CsCl-type, space group Pm-3m) structures for ZnN, as shown in Figure 1. In order to check the stability of the structures predicted from a thermodynamic perspective and show how the structures of ZnN vary under pressure, for each proposed structure, the equilibrium geometry and zero-temperature Gibbs free energy, i.e. enthalpy, at more than 15
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pressure points in the range from 0 to 150 GPa are obtained by the complete optimization of the lattice constants and the atomic positions, as shown in Figure 2 (a). A phase transition is said to occur when the
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changes in structural details of the phase are caused by such variations of free energy. For a given pressure at 0 K, a stable structure is the one for which the enthalpy has the lowest value. From Figure 2 (a) we can
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see that the ZnS-type structure is stable under low pressure and the NaCl-type structure is the high-pressure phase, where the CsCl-type structure is always unstable in the whole pressure range. The transition pressure Pt is a pressure at which H(P) curves for both phases cross. Figure 2 (b) gives the Pt from the ZnS-type to NaCl-type structure based on the enthalpy difference calculations and the value is about 2.55 GPa. Rajeswarapalanichamy et al. [15] have reported that ZnS-type structure is the lowest energy phase for ZnN, but there is no phase transition pressure Pt from ZnS-type to NaCl-type structure reported. Take into account the phase transition pressure, the calculated pressure and volume relations of 6
Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT ZnS-type and NaCl-type structures for ZnN are given in Figure 3. At 2.55 GPa, the corresponding volume reduction ∆V/V is about 16.5%, as marked in the diagram. Figure 4 evaluates the energy and stability of both the structures. It can also be found from this figure that at ambient conditions, ZnS-type structure with
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the lowest energy is more stable than the NaCl-type structure in the present study. By fitting the E-V curve for ZnS-type structure of ZnN using 3rd-order Birch-Murnaghan equation [24], the equilibrium lattice constant a0, isothermal bulk modulus K0 and corresponding pressure derivative K'0 are obtained
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successfully with DFT-GGA-PBESOL calculations. These important equation of state parameters a0, K0
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and K'0, along with the transition pressure Pt and volume collapse -∆V/V0 from ZnS-type to NaCl-type structure for ZnN are all shown in Table 1. Unfortunately, to the best of our knowledge, the experimental values for these are not yet available for comparison. Our results are in good agreement with the DFT-GGA-PBE and DFT-LDA-CA-PZ calculated results 4.595 and 4.4854 Å, 139 and 144 GPa, and 5.011
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and 5.066, respectively [15], which suggesting that our calculations are reliable and accurate. In the following, we will focus on the thermodynamic properties predictions for the high-pressure NaCl-type
account.
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structure of ZnN through the quasi-harmonic Debye model in which the phononic effects are taken into
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The pressure dependences of the relative volume V/V0 for ZnN in the high-pressure NaCl-type structure at three selected temperature of 0, 1000, and 2000 K is exhibited in Figure 5. It is seen that, at the same pressure, the relative volume V/V0 of higher temperature is less than that of lower temperature. As the pressure increases, the relative volume decreases at a given temperature. This effect of increasing pressure on ZnN is just the same as decreasing temperature of ZnN. The conclusions are very common for many materials but the knowledge of the thermal equation of state is important as it is a tool to describe the structural behavior over a vast range of temperature and pressure. For example, the isothermal bulk modus 7
Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT KT can easily be obtained from the isothermal equation of state, that is it depends on volume V and pressure P at constant temperature T, and usually defined as follows [25]:
∂P KT = −V . ∂V T
(10)
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As we known, in the quasi-harmonic Debye model based on the GIBBS code [22], the resulting E-V data is fitted by a polynomial, to calculate the bulk modulus as a function of volume from equation (5), and
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the bulk modulus values are then used to calculate the Debye temperature for each unit cell volume of ZnN from equation (3). Using this model, the pressures effect on the isothermal bulk modulus KT at
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temperatures of 0, 1000 and 2000 K is plotted for the NaCl-type structure of ZnN in Figures 6. It is easily seen from Figure 6 that the isothermal bulk modulus is linearly increasing with pressure in the range of 0 to 150 GPa, and has a slightly decrease with temperature. That is to say, KT is greatly affected by the pressure compared with the temperature in the whole pressure and temperature ranges for ZnN.
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Debye temperature ΘD is an important parameter in the study of the materials thermodynamic properties and it is not only closely correlated with many physical properties of solids, such as thermal
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expansivity, melting points, and specific heat but also it can be used to characterize the strength of covalent bonds in solids. The dependence of the Debye temperature ΘD on pressure and temperature for ZnN in
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NaCl-type structure is depicted in Figure 7 (a) and (b). The calculated Debye temperature ΘD of ZnN with NaCl-type structure at zero temperature and pressure is 617.11 K, where the value is 563.45 K for the ZnS-type structure, and no other results are supplied for comparison. Take into account the effects of pressure and temperature, we will find from Figure 7 (a) and (b) that the trend of ΘD is consistent with the variation of the isothermal bulk modulus. At low pressure, θD decreases significantly when the temperature changes from 0 to 2000 K. As the pressure goes higher, the decreased magnitude of becomes small. In addition, as we known, the larger the ΘD is, the stronger the covalent bonds are [26]. Hence, with 8
Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT increasing pressure, the covalent bonds of ZnN with NaCl-type structure become stronger. The Grüneisen parameter γ is a physical quantity of central importance related to thermal and elastic properties of materials [27]. In order to estimate γ, we need to plot lnΘD versus lnV, which indicated by
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equation (6), and the slope of the yielded straight line equals minus of the γ. In Figure 8 (a) and (b), we present the effects of pressure and temperature on the Grüneisen parameter γ at temperatures of 0, 1000 and 2000 K and at pressures of 0, 50, 100 and 150 GPa for the NaCl-type structure of ZnN, respectively. From
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Figure 8 (a), one can find that when the temperature keeps constant, the Grüneisen parameter decreases
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with the increasing pressures, but the trend will become smooth under high pressure. When the pressure keeps constant, the Grüneisen parameter increases with the increasing temperatures, but the trend will become rapid under low pressure, as shown in Figure 8 (b). In virtue of the fact that the effect of increasing pressure on the material is the same as decreasing temperature on the material. In this study, we obtained γ0
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= 2.183 using quasi-harmonic Debye model, and the value will decrease to 1.528 at 150 GPa. Within the quasi-harmonic approximation, the anharmonicity is restricted to the thermal expansion. The temperature dependences of the volumetric thermal expansion coefficient αV for the NaCl-type
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structure of ZnN at 0, 50, 100 and 150 GPa are plotted in Figure 9. It is evident that at a constant pressure,
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the αV increases rapidly at the temperatures below 300 K. Then, except for the zero-pressure data, it switches to a linearly and smoothly increasing trend at the temperatures above 300 K. Besides, it can be find that the effect of pressure on αV is larger at higher temperatures, and obviously smaller at lower temperatures. It well known that there are certain difficulties associated with the measurement of αV at high temperatures which lead to considerable uncertainties in the experimental values [28], so the theoretical calculation is a good estimate of the errors for thermal expansion investigations of materials. Another thermodynamic parameter, heat capacity, is also very important. The heat capacity of a 9
Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT substance is not only providing essential insight into its vibrational properties, but is also mandatory for many applications. The relationship between temperature and constant volume heat capacity CV for ZnN in NaCl-type structure under different pressures from 0 to 150 GPa with an interval of 50 GPa is curved in
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Figure 10. The calculated specific heat capacity CV is 40.97 J mol-1 K-1 at zero pressure and ambient temperature. For ZnN, CV is increasing with the elevating temperature. At intermediate temperatures, the temperature dependence of CV is governed by the details of vibrations of the atoms. However, under higher
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temperature, as the anharmonic effect on CV is suppressed, CV is close to a limited value when temperature
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keeps increasing, namely, Dulong-Petit limit [29].
Another thermodynamic parameter, entropy S, is also predicted at different pressures of 0, 50, 100 and 150 GPa for the NaCl-type structure of ZnN in the temperature range of 0 to 2000 K, as shown in Figure 11. In thermodynamics, S is a measure of the number of specific ways in which a thermodynamic
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system may be arranged, often taken to be a measure of disorder, or a measure of progressing towards thermodynamic equilibrium. From the curves in the Figure 11, the entropy is zero at 0 K and it changes increase rapidly as temperature increases at low temperature. As the temperature increases the vibrational
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contribution to the entropy increases, therefore the entropy increases with temperature increasing [30].
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Furthermore, the pressure has much weaker effect on S than the temperature. The calculated values of enthalpy at room temperature is 36.04 J mol-1 K-1 for ZnN.
4. Conclusions
The pressure-induced ZnS-type, NaCl-type and CsCl-type structural phase transitions and thermodynamic properties including the PVT equation of state, isothermal bulk modulus, Debye temperature, Grüneisen parameter and entropy for high-pressure NaCl-type structure of ZnN have been
10
Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT studied systematically by first-principles calculations and the quasi-harmonic Debye model in the present work. From the enthalpy versus pressure relations, we find that the ZnS-type structure is stable under low pressure, which in accordance with the theoretical prediction by Rajeswarapalanichamy et al., and the
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NaCl-type structure is the high-pressure phase, where the CsCl-type structure is always unstable in the whole pressure range. The phase transition pressure between ZnS-type and the NaCl-type structures for ZnN is about 2.55 GPa. Prediction of the thermodynamic properties for ZnN shows us that when the
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temperature increases, the volume compression ratio keeps the trend of reducing, and the reduced
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amplitude becomes larger at higher pressure conditions. On the contrary of the isothermal bulk modulus and Debye temperature, which are more sensitive to pressure than temperature, the volume, Grüneisen parameter, thermal expansion coefficient, and entropy of ZnN with NaCl-type structure are proportional to the temperature and inversely proportional to the pressure. Furthermore, the pressure has much weaker
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effect on entropy than the temperature. At sufficiently low temperatures, the constant-volume heat capacity is proportional to T3, and under higher temperature, the constant-volume heat capacity is close to a Dulong-Petit limit. All the properties of cubic ZnN in the whole pressure range from 0 to 150 GPa and
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temperature range from 0 to 2000 K are summarized and discussed. Our calculated results can be seen as a
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prediction for future investigations.
Acknowledgements The authors would like to thank the supports by the National Natural Science Foundation of China under Grant Nos. 51562021, 11464027, and 11164013, the Program for Longyuan Youth Innovation Talents of Gansu Province of China, the Foundation of A Hundred Youth Talents Training Program of Lanzhou Jiaotong University, the National Natural Science Foundation of Gansu Province under Grant Nos.
11
Revision to PCS_2016_707_R1 ACCEPTED MANUSCRIPT 148RJZA027, the Colleges and Universities Scientific Research Program of Gansu Province under Grant Nos. 2015B-040 and 2015B-048, and the Young Scholars Science Foundation of Lanzhou Jiaotong University under Grant No. 2014022.
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[25] X.W. Sun, Y.D. Chu, Z.J. Liu, T. Song, P. Guo, Q.F. Chen, Mater. Chem. Phys. 116 (2009) 34. [26] Q. Chen, Z. Huang, Z. Zhao, C. Hu, Comp. Mater. Sci. 67 (2013) 196. [27] T.P. Kaloni, Y.C. Cheng, U. Schwingenschlögl, J. Appl. Phys. 113 (2013) 104305.
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[29] P. Debye, Annalen der Physik 344 (1912) 789. [30] K.R. Babu, C.B. Lingam, S. Auluck, S.P. Tewari, G. Vaitheeswaran, J. Solid State Chem. 184 (2011) 343.
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Table Captions:
Table 1. Calculated equilibrium lattice constant a0, isothermal bulk modulus K0 and its pressure derivative K'0 from the 3rd-order Birch-Murnaghan equation for ZnS-type structure, along with the
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transition pressure Pt and volume collapse -∆V/V0 from ZnS-type to NaCl-type structure for ZnN,
Present work 4.515
K0 (GPa)
144.73
K'0
4.747
Pt (GPa)
2.55
-∆V/V0
16.5%
Other calculation 4.588 a, 4.472 b 4.595 c, 4.4854 b
Ref. [5] Ref. [15]
139 a, 144 b
Ref. [15]
5.011 a, 5.066 b
Ref. [15]
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a0 (Å)
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compared with the available theoretical data.
DFT-GGA-PW91 calculation
b
DFT-LDA-CA-PZ calculation
c
DFT-GGA-PBE calculation
AC C
EP
a
Table 1
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Figure Captions: Figure 1. Crystal structures of (a) the zinc blende type, (b) the rock salt type, and (c) the cesium chloride type ZnN phases. The large and small spheres represent zinc and nitrogen atoms, respectively.
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Figure 2. (a) Enthalpies per formula unit of the ZnS-type, NaCl-type, and CsCl-type structures as function of pressure at zero temperature and (b) estimation of phase transition pressure from ZnS-type to NaCl-type
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structure for ZnN.
Figure 3. Volume versus pressure curves from ZnS-type to NaCl-type structure for ZnN.
Figure 4. Energy versus volume curves of ZnN with both the ZnS-type and NaCl-type structures.
1000, and 2000 K.
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Figure 5. The normalized volume V/V0 as a function of pressure for the NaCl-type structure of ZnN at 0,
Figure 6. The variations of isothermal bulk modulus BT with the pressure for the NaCl-type structure of
EP
ZnN at 0, 1000, and 2000 K.
Figure 7. (a) The variations of Debye temperature ΘD with the pressure at 0, 1000, and 2000 K and (b) the
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variations of ΘD with the temperature at 0, 50, 100, and 150 GPa for the NaCl-type structure of ZnN.
Figure 8. (a) The variations of Grüneisen parameter γ with the pressure at 0, 1000, and 2000 K and (b) the variations of γ with the temperature at 0, 50, 100, and 150 GPa for the NaCl-type structure of ZnN.
Figure 9. The dependences of volumetric thermal expansion coefficient αV on the temperatures for the NaCl-type structure of ZnN at 0, 50, 100, and 150 GPa.
Figure 10. The dependences of constant volume heat capacity CV on the temperatures for the NaCl-type structure of ZnN at 0, 50, 100, and 150 GPa. 15
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Figure 11. The dependences of enthalpy S on the temperatures for the NaCl-type structure of ZnN at 0, 50,
AC C
EP
TE D
M AN U
SC
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100, and 150 GPa.
16
Figure 1
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-1959
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ZnS structure NaCl structure CsCl structure
-1962
-1968
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H / eV
-1965
-1974
(a) -1977 0
15
30
45
M AN U
-1971
60
75
90
105
120
135
150
P / GPa
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0.2 0.1
ZnS structure
0.0
EP
-0.2 -0.3
NaCl structure
2.55 GPa
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∆H / eV
-0.1
-0.4 -0.5
(b)
-0.6
0
5
10
15
20
25
P / GPa
Figure 2 17
30
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24
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ZnS structure
23 22 21
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20 19
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V/Å
3
- ∆V / V0 = 16.5%
NaCl structure
18
2.55 GPa 17 0
2
4
6
8
10
12
14
16
EP
TE D
P / GPa
AC C
Figure 3
18
18
20
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-1971
-1972
ZnS structure NaCl structure
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-1974
-1975
-1976 10
15
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E / eV
-1973
20
25
30
3
EP
TE D
V/Å
AC C
Figure 4
19
35
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1.00 0.95
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0K 1000 K 2000 K
0.90
0.80
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V / V0
0.85
0.70 0.65 0.60 0
15
30
45
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0.75
60
75
90
105
120
AC C
EP
TE D
P / GPa
20
Figure 5
135
150
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800 700 600
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0K 1000 K 2000 K
400
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KT / GPa
500
200 100 0
15
30
45
M AN U
300
60
75
90
105
120
AC C
EP
TE D
P / GPa
21
Figure 6
135
150
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1200
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1100 1000
800
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ΘD / K
900
700
0K 1000 K 2000 K
500
(a) 400 0
15
30
45
M AN U
600
60
75
90
105
120
135
150
TE D
P / GPa
1200
1000
ΘD / K
AC C
900
EP
1100
800
0 GPa 50 GPa 100 GPa 150 GPa
700 600 500
(b)
400 0
250
500
750
1000
T/K
22
1250
1500
1750
2000
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2.65
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2.50 2.35
0K 1000 K 2000 K
2.20
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γ
2.05 1.90
1.60
(a) 1.45 0
15
30
45
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1.75
60
75
90
105
120
135
150
TE D
P / GPa
2.7
0 GPa 50 GPa 100 GPa 150 GPa
2.3
γ
AC C
2.1
EP
2.5
1.9
1.7
(b) 1.5 0
250
500
750
1000
1250
1500
1750
T/K
Figure 8 23
2000
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12
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6
4
2
0 0
250
500
M AN U
-5
αV / 10 K
-1
8
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0 GPa 50 GPa 100 GPa 150 GPa
10
750
1000
1250
1500
1750
AC C
EP
TE D
T/K
24
Figure 9
2000
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55
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35
0 GPa 50 GPa 100 GPa 150 GPa
-1
CV / J mol K
-1
45
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25
5 0
250
500
M AN U
15
750
1000
1250
1500
1750
AC C
EP
TE D
T/K
25
Figure 10
2000
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150
90
-1
S / J mol K
-1
120
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0 GPa 50 GPa 100 GPa 150 GPa
30
0 0
250
500
M AN U
SC
60
750
1000
1250
1500
1750
AC C
EP
TE D
T/K
26
Figure 11
2000
ACCEPTED MANUSCRIPT
1. Phase transition and thermodynamic properties of cubic ZnN are reported. 2. The pressure up to 150 GPa.
AC C
EP
TE D
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3. The temperature up to 2000 K.