Research Article  Open Access
SiHua Li, CuiE Hu, XiaoLu Wang, Yan Cheng, "FirstPrinciples Study of Structure, Elastic Properties, and Thermal Conductivity of Monolayer Calcium Hydrobromide", Advances in Condensed Matter Physics, vol. 2021, Article ID 6619252, 7 pages, 2021. https://doi.org/10.1155/2021/6619252
FirstPrinciples Study of Structure, Elastic Properties, and Thermal Conductivity of Monolayer Calcium Hydrobromide
Abstract
In recent years, some laboratories have been able to prepare calcium hydrobromide (CaHBr) by melting hydride and anhydrous bromide or metal and bromide in a hydrogen atmosphere at 900°C and have studied some of its properties. But there are few theoretical studies, especially the theoretical studies of monolayer CaHBr. We use the firstprinciples method to calculate the structure, elastic properties, and lattice thermal conductivity of the monolayer CaHBr based on the Boltzmann transport equation. We obtain a stable crystal structure by the optimization of monolayer CaHBr. By calculating the elastic constant of monolayer CaHBr, its mechanical stability is proved, and the elastic limit of monolayer CaHBr is obtained by biaxial tensile strain on monolayer CaHBr. And the corresponding phonon spectra show no imaginary frequency, indicating the dynamic stability of the monolayer CaHBr. By the ShengBTE code, we calculate the lattice thermal conductivity of the monolayer CaHBr, the iterative solution of BTE and RTA at 300 K–1200 K is obtained, and the lattice thermal conductivity at room temperature is and , respectively. It can be seen that the lattice thermal conductivity of monolayer CaHBr is low. And by analyzing the phonon spectrum, the scattering rate, and the mean free path of the phonons, the lattice thermal conductivity of monolayer CaHBr mainly depends on the acoustic modes. We hope this study can provide theoretical guidance for the experiments and practical application of monolayer CaHBr.
1. Introduction
Graphene’s discovery broke the prediction that twodimensional (2D) crystalline materials could not exist stably at finite temperatures [1–6]. Various 2D materials are limited in 2D plane because of carrier migration and heat diffusion. It has made the materials exhibit many strange properties and has attracted extensive attention [7–11]. In recent years, some scholars began to pay attention to the monolayer calcium hydrobromide (CaHBr) [12–14], which can be prepared by melting hydride with anhydrous bromide [15] or metal with bromide in a hydrogen atmosphere at 900°C [14].
The hydride ionconducting electrolytes can be used for the electrochemical detection of hydrogen in the liquid sodium coolant of fast reactors. Some laboratories discovered that the CaBr_{2}CaHBr system has a solid electrolyte that conducts hydride ions, which can be used to develop an electrochemical hydrogen meter, and the system will show a eutectic reaction between CaBr_{2} and CaHBr at 576°C [13]. In 1996, de Castro Vítores et al. [12] obtained the bond energy of CaHBr complex from independent cross molecular beam and van der Waals spectroscopy experiments. Very recently, Kumar et al. [15] measured the molar heat capacities of bulk CaHBr via the differential scanning calorimetry. It can be seen that some properties of bulk CaHBr have been studied in experiments, but there are few theoretical studies on them, especially the thermal transport properties of monolayer CaHBr.
The monolayer CaHBr is a nonmagnetic wideband gap semiconductor whose heat carriers are electrons and phonons, in which phonons dominate the heat transfer. The lattice thermal conductivity is a key parameter for the thermal transport of semiconductors and insulators [16–19]. Peierls proposed the lattice thermal conductivity of semiconductors [20] and insulators could be described at the micros level using the phonon Boltzmann transport equation (BTE) [21]. Since then, many methods have been found to calculate the thermal conductivity of materials, such as relaxation time approximation (RTA) [22] and Callway’s model [23, 24], where RTA contradicts inelastic scattering processes in principle, and the required parameters in the Callway model can only be obtained by fitting the experimental data [25]. Therefore, these methods have corresponding limitations.
As is known, ShengBTE code [26] can be used to obtain the crystal lattice thermal conductivity by solving the Boltzmann transport equation based on the force constants between harmonic and anharmonic atoms calculated from the firstprinciples [27]. To date, ShengBTE code has been successful to obtain the thermal conductivity and related physical quantities of many materials [28–33]. In this work, we will use the ShengBTE code to calculate the thermal conductivity of monolayer CaHBr, hoping to provide the reference value for the future experiments and theories.
2. Theoretical Methods and Calculation Details
The crystal structure is optimized by using the Vienna Ab initio Simulation Package (VASP) [34, 35] based on density functional theory. The Perdew–Burke–Ernzerhof (PBE) functional under the generalized gradient approximation (GGA) is selected as the exchange correlation functional [36, 37]. In order to eliminate the layer to layer interaction, we use a vacuum layer. We use the cutoff energy of the plane wave to be 600 eV, the energy convergence of the electron relaxation to 10^{−8} eV, and a Monkhorst–Pack grid of kpoint sampling for structural optimization. The optimized unit lattice is expanded to a supercell of , and then, the secondorder harmonic force constants (harmonic IFCs) and the thirdorder anharmonic force constants (anharmonic IFCs) are calculated by using Phonopy software and ShengBTE, respectively. And using the Phonopy software package can also get the phonon frequency.
There have been many studies detailing the calculation of lattice thermal conductivity using ShengBTE code [25, 26, 30, 31]; so here, we just briefly introduce this method. The resulting linearized phonon BTE when the scattering source is only twophonon and threephonon processes can be written aswhere is the relaxation time of mode , as obtained from perturbation theory. As a matter of fact, setting all to zero is equivalent to working within the RTA. The threephonon scattering rates can be expressed as ; , where is the phonon frequency of mode , and normal process correspond to , while Umklapp processes correspond to .
The lattice thermal conductivity can be obtained in terms of aswhere is the unit cell volume. In the approach implemented in ShengBTE, equation (2) is starting with a zerothorder approximation . The stopping criterion is that the relative change in the calculated conductivity tensor is less than a configurable parameter. Stopping at the zeroth iteration is equivalent to operating under the RTA. In addition, many physical quantities can also be calculated by ShengBTE code, such as the scalar mean free path for mode .
3. Results and Discussion
3.1. Structure and Elastic Properties
The initial structure of the monolayer CaHBr is obtained from the bulk CaHBr belonging to the orthogonal structure (P4) crystal. In order to obtain the equilibrium geometric structure of monolayer CaHBr, we use the GGA method to calculate the lattice constant and obtain the total energy E and the corresponding cell volume V; the energyvolume (EV) data are then fitted to the Vinet equation [29]. Thus, we obtain the equilibrium lattice constant , which is well consistent with another theoretical value [38].
The elastic constants of materials are the critical significance physical quantity to measure the mechanical energy and important parameter to reflect the mechanical properties of materials [39–41]. Thus, the calculation of the elastic constant is of great significance for the measurement of the elastic limit of the lattice under external stress. We can calculate the elastic constants of twodimensional materials by using the related theory of bulk elastic constants. In Table 1, we list the calculated elastic constants , , , and , which can be used to obtain the layer modulus , Young’s modulus under the Cavendish coordinates (its directions [10] and [01] in 2D materials), and Poisson’s ratio [42, 43].

The monolayer CaHBr satisfies the stability criterion, which can be expressed by four necessary and sufficient conditions for the determination of mechanical stability [40, 41]: , (the bulk material is ) > 0, [44, 45]. Therefore, the monolayer CaHBr is mechanically stable. And from the elastic constant of monolayer CaHBr, it can be seen that it is smaller than that of graphene and [43, 46], indicating that it is a twodimensional material with weaker hardness.
The stressstrain curve can be used to estimate the elastic limit of monolayer CaHBr. In the calculation, since the lattice constants of CaHBr on the xaxis and yaxis are the same, the corresponding biaxial tensile strain is made, as shown in Figure 1. It is our biaxial stressstrain curve. It can be seen from Figure 1 that the maximum stress that monolayer CaHBr can withstand under biaxial conditions is 4.11 N/m, which corresponds to a strain of 28%. Compared with other twodimensional materials, monolayer CaHBr has weaker tensile capacity [32, 42].
3.2. Phonon Spectra and Scattering Rates
We calculated the phonon dispersion curves of monolayer CaHBr in Figure 2(a) together with the main view and side view of the primitive unit cell for the monolayer CaHBr (Figure 2(b)). It is shown that there is no imaginary frequency in the high symmetry directions of the phonon spectra, which indicates its dynamic stability [25, 47].
(a)
(b)
(c)
In Figure 3, we illustrate the contributions of the phonon modes to total lattice thermal conductivity at room temperature. The phonon acoustic branches clearly dominate the lattice thermal conductivity of the monolayer CaHBr, while the contribution from the optical branches is quite small. Although the contribution of the optical branch is small, the optical branch provides a scattering channel for the acoustic mode, resulting in the threephonon scattering. So, the contribution of the optical branches cannot be ignored [29, 48].
The total converged phonon scattering rates of the monolayer CaHBr at room temperature are illustrated in Figures 4 and 5, which are corresponding to the acoustic modes and the optic modes, respectively. We notice that there is a gap between the acoustic and optical branches, consistent with that of the phonon spectra in Figure 2(a). The phonon scattering rate of the three acoustic branches is much smaller than that of the optical branch, from which it can be seen that the acoustic branch mainly contributes to the thermal conductivity of this material.
3.3. Phonon Mean Free Path and Thermal Conductivity
By calculating the phonon mean free path (MFP), we can understand how the material size affects the thermal conductivity. In Figure 6, we show the functional relationship between the cumulative lattice thermal conductivity and the maximum mean free path (MFP) at room temperature. When drawing a curve with a logarithmic scale as the horizontal axis, we can find the similarity of the curve to a logistic function, indicating that the form is suitable for the following nonparametric function:
It is found from Figure 6 that the acoustic phonons with a length of 0–3.5 nm contribute to the thermal conductivity, while the optical phonons larger than 3.5 nm contribute little to the thermal conductivity.
For the efficiency and reliability of devices, the thermal transport property of materials is very important. At present, there are no experimental data and theoretical data for the thermal conductivity of the monolayer CaHBr. By testing the sensitivity of the thermal conductivity to the temperature, the functional relationship between the lattice thermal conductivity and the temperature can be obtained in Figure 7, where we show the lattice thermal conductivities of the monolayer CaHBr in the temperatures ranging from 30 K to 1200 K with a kgrid in the scalebroad = 1.0. It can be seen that the lattice thermal conductivity increases exponentially with the increase of temperature at low temperatures and tends to be proportional to at high temperatures. The lattice thermal conductivity of monolayer CaHBr at room temperature is and for BTE and RTA, respectively.
4. Conclusions
Based on density functional theory and Boltzmann transport equation, we study the structure, elastic properties, and lattice thermal conductivity of the monolayer CaHBr. The obtained equilibrium lattice constant is well consistent with another theoretical value . The mechanical and thermodynamic stability of monolayer CaHBr is proved by the obtained elastic properties and phonon spectra without imaginary frequency. And the elastic limit of monolayer CaHBr is obtained by biaxial tensile strain. The secondorder and thirdorder interatomic force constants are obtained by using the finite difference method. The thermal conductivity of the monolayer CaHBr at room temperature is and , respectively, by BTE and RTA iterations. The two methods can obtain good results for the calculation of thermal conductivity of monolayer CaHBr. It can be seen that the lattice thermal conductivity of the obtained monolayer CaHBr is low, and the result shows that the lattice thermal conductivity mainly depends on the acoustic modes. We hope that our results can provide theoretical guidance for the experimental exploration and application of the relevant properties of layered monolayer CaHBr.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no known conflicts of interest.
Acknowledgments
This work was supported by the Guizhou SciTech Fund of China (Grant No. [2017] 1125). The authors also acknowledge the support for the computational resources by the Chinese Academy of Engineering Physics and the State Key Laboratory of Polymer Materials Engineering of China in Sichuan University.
References
 Z. Sun and H. Chang, “Graphene and graphenelike twodimensional materials in photodetection: mechanisms and methodology,” ACS Nano, vol. 8, no. 5, pp. 4133–4156, 2014. View at: Publisher Site  Google Scholar
 X. Tong, H. Cheng, and R. Zeng, “Synthesis of graphene nanosheets via hydrogen reduction of thermal exfoliated graphite oxide,” Materials Review, vol. 45, no. 7, pp. 1558–1565, 2014. View at: Google Scholar
 A. Gupta, T. Sakthivel, and S. Seal, “Recent development in 2D materials beyond graphene,” Progress in Materials Science, vol. 73, pp. 44–126, 2015. View at: Publisher Site  Google Scholar
 S. Z. Butler, S. M. Hollen, L. Cao et al., “Progress, challenges, and opportunities in twodimensional materials beyond graphene,” ACS Nano, vol. 7, no. 4, pp. 2898–2926, 2013. View at: Publisher Site  Google Scholar
 R. Buchanan, “Introduction to graphene special edition,” Journal of Applied Toxicology, vol. 37, no. 11, pp. 12861287, 2017. View at: Publisher Site  Google Scholar
 F. Pulizzi, O. Bubnova, S. Milana, D. Schilter, D. Abergel, and A. Moscatelli, “Graphene in the making,” Nature Nanotechnology, vol. 14, no. 10, pp. 914–918, 2019. View at: Publisher Site  Google Scholar
 A. Bablich, S. Kataria, and M. C. Lemme, “Graphene and twodimensional materials for optoelectronic applications,” Electronics, vol. 5, no. 13, 2016. View at: Publisher Site  Google Scholar
 S. Ulstrup, A. G. Čabo, J. A. Miwa et al., “Ultrafast band structure control of a twodimensional heterostructure,” ACS Nano, vol. 10, no. 6, pp. 6315–6322, 2016. View at: Publisher Site  Google Scholar
 J. Tao, T. Luttrell, and M. Batzill, “A twodimensional phase of TiO2 with a reduced bandgap,” Nature Chemistry, vol. 3, no. 4, pp. 296–300, 2011. View at: Publisher Site  Google Scholar
 S. J. Kim, K. Choi, B. Lee, Y. Kim, and B. H. Hong, “Materials for flexible, stretchable electronics: graphene and 2D materials,” Annual Review of Materials Research, vol. 45, no. 1, pp. 63–84, 2015. View at: Publisher Site  Google Scholar
 A. A. Balandin, S. Ghosh, W. Bao et al., “Superior thermal conductivity of singlelayer graphene,” Nano Letters, vol. 8, no. 3, pp. 902–907, 2008. View at: Publisher Site  Google Scholar
 M. de Castro Vítores, R. Candori, F. Pirani, V. Aquilanti, M. Garay, and A. González Ureña, “Determination of D00 (Ca…HBr) from full and half collision studies. Van der Waals and charge transfer contributions to the formation of Ca…HX (X = Cl, Br) complexes,” Chemical Physics Letters, vol. 263, no. 34, pp. 456–462, 1996. View at: Publisher Site  Google Scholar
 C. V. Vishnu Vardhan, S. Ghosh, S. Nagaraj, R. Sridharan, and T. Gnanasekaran, “Phase diagram study of CaBr2CaHBr system,” Journal of Thermal Analysis And Calorimetry, vol. 112, no. 1, pp. 127–131, 2013. View at: Publisher Site  Google Scholar
 P. Ehrlich, B. Alt, and L. Gentsch, “Uber die Hydridchloride der Erdalkalimetalle,” Zeitschrift fur anorganische und allgemeine Chemie, vol. 283, no. 16, p. 58, 1956. View at: Publisher Site  Google Scholar
 S. S. Kumar, S. Ghosh, M. Sahu, and R. Ganesan, “Molar heat capacity measurement of CaHCl and CaHBr,” Journal of Chemical Sciences, vol. 131, no. 9, p. 100, 2019. View at: Publisher Site  Google Scholar
 G. Qin, Z. Qin, S.Y. Yue, Q.B. Yan, and M. Hu, “External electric field driving the ultralow thermal conductivity of silicene,” Nanoscale, vol. 9, no. 21, p. 7227, 2017. View at: Publisher Site  Google Scholar
 H. Wang, G. Qin, G. Li, Q. Wang, and M. Hu, “Low thermal conductivity of monolayer ZnO and its anomalous temperature dependence,” Physical Chemistry Chemical Physics, vol. 19, no. 20, p. 12882, 2017. View at: Publisher Site  Google Scholar
 G. Qin, Z. Qin, H. Wang, and M. Hu, “Lonepair electrons induced anomalous enhancement of thermal transport in strained planar twodimensional materials,” Nano Energy, vol. 50, p. 425, 2018. View at: Publisher Site  Google Scholar
 H. Liu, G. Qin, Y. Lin, and M. Hu, “Disparate strain dependent thermal conductivity of twodimensional pentastructures,” Nano Letters, vol. 16, no. 6, p. 3831, 2016. View at: Publisher Site  Google Scholar
 R. Peierls, “Zur kinetischen theorie der Wärmeleitung in kristallen,” Annalen der Physik, vol. 395, no. 8, pp. 1055–1101, 1929. View at: Publisher Site  Google Scholar
 M. Omini and A. Sparavigna, “Beyond the isotropicmodel approximation in the theory of thermal conductivity,” Physical Review B, vol. 53, no. 14, pp. 9064–9073, 1996. View at: Publisher Site  Google Scholar
 J. M. Ziman, Electrons and Phonons, Oxford University Press, London, UK, 1960.
 J. Callaway, “Model for lattice thermal conductivity at low temperatures,” Physical Review, vol. 113, no. 4, pp. 1046–1051, 1959. View at: Publisher Site  Google Scholar
 P. B. Allen, “Improved callaway model for lattice thermal conductivity,” Physical Review B, vol. 88, Article ID 144302, 2013. View at: Publisher Site  Google Scholar
 C. He, C.E. Hu, T. Zhang, Y.Y. Qi, and X.R. Chen, “Lattice dynamics and thermal conductivity of cesium chloride via firstprinciples investigation,” Solid State Communications, vol. 254, pp. 31–36, 2017. View at: Publisher Site  Google Scholar
 W. Li, J. Carrete, N. A. Katcho, and N. Mingo, “ShengBTE: a solver of the Boltzmann transport equation for phonons,” Computer Physics Communications, vol. 185, no. 6, pp. 1747–1758, 2014. View at: Publisher Site  Google Scholar
 R. A. Cowley, “The lattice dynamics of an anharmonic crystal,” Advances in Physics, vol. 12, no. 48, pp. 421–480, 1963. View at: Publisher Site  Google Scholar
 N. Guo, K. M. Yam, and C. Zhang, “Light controllable catalytic activity of Au clusters decorated with photochromic molecules,” Nanotechnology, vol. 29, no. 24, Article ID 245705, 2018. View at: Publisher Site  Google Scholar
 N. Guo, K. M. Yam, X. L. Wang, and C. Zhang, “Ndoped ZnO nanosheets: towards high performance two dimensional catalysts,” Nanotechnology, vol. 29, no. 10, Article ID 105707, 2018. View at: Publisher Site  Google Scholar
 Y. Zhou, Y.Q. Zhao, Z.Y. Zeng, X.R. Chen, and H.Y. Geng, “Anisotropic thermoelectric properties of Weyl semimetal NbX (X = P and As): a potential thermoelectric material,” Physical Chemistry Chemical Physics, vol. 21, no. 27, p. 15167, 2019. View at: Publisher Site  Google Scholar
 Y. Zhou, W. L. Tao, Z. Y. Zeng, X. R. Chen, and Q. F. Chen, “Thermoelectric properties of topological insulator lanthanum phosphide via firstprinciples study,” Journal of Applied Physics, vol. 125, no. 4, Article ID 045107, 2019. View at: Publisher Site  Google Scholar
 X. L. Pan, Y. Q. Zhao, Z. Y. Zeng, X. R. Chen, and Q. F. Chen, “Electronic, elastic, optical and thermal transport properties of pentaPdAs2 monolayer: firstprinciples study,” Solid State Communications, vol. 307, Article ID 113802, 2020. View at: Publisher Site  Google Scholar
 X.F. Chen, L. Wang, Z.Y. Zeng, X.R. Chen, and Q.F. Chen, “Straintunable electronic, elastic, and optical properties of CaI2 monolayer: firstprinciples study,” Philosophical Magazine, vol. 100, no. 15, pp. 1982–2000, 2020. View at: Publisher Site  Google Scholar
 P. E. Blöchl, “Projector augmentedwave method,” Physical Review B, vol. 50, no. 24, p. 17953, 1994. View at: Publisher Site  Google Scholar
 G. Kresse and J. Furthmüller, “Efficient iterative schemes forab initiototalenergy calculations using a planewave basis set,” Physical Review B, vol. 54, no. 16, pp. 11169–11186, 1996. View at: Publisher Site  Google Scholar
 J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Physical Review Letters, vol. 77, no. 18, pp. 3865–3868, 1996. View at: Publisher Site  Google Scholar
 S. H. Vosko, L. Wilk, and M. Nusair, “Accurate spindependent electron liquid correlation energies for local spin density calculations: a critical analysis,” Canadian Journal of Physics, vol. 58, no. 8, p. 1200, 1980. View at: Publisher Site  Google Scholar
 N. Mounet, M. Gibertini, P. Schwaller et al., “Twodimensional materials from highthroughput computational exfoliation of experimentally known compounds,” Nature Nanotechnology, vol. 13, no. 3, pp. 246–252, 2018. View at: Publisher Site  Google Scholar
 Y. Lu, Y. Yang, F. Zheng, B.T. Wang, and P. Zhang, “Electronic, mechanical, and thermodynamic properties of americium dioxide,” Journal of Nuclear Materials, vol. 441, no. 13, pp. 411–420, 2013. View at: Publisher Site  Google Scholar
 P. Lab Guerie, F. Pascale, M. M Rawa, C. ZicovichWilson, N. Makhouki, and R. Dovesi, “Phonon vibrational frequencies and elastic properties of solid SrFCl. An ab initio study,” The European Physical Journal B, vol. 43, no. 4, pp. 453–461, 2005. View at: Google Scholar
 M. Mérawa, Y. Noël, B. Civalleri, R. Brown, and R. Dovesi, “Raman and infrared vibrational frequencies and elastic properties of solid BaFCl calculated with various Hamiltonians: anab initiostudy,” Journal of Physics: Condensed Matter, vol. 17, no. 3, p. 535, 2005. View at: Publisher Site  Google Scholar
 R. Ran, C. Cheng, Z.Y. Zeng, X.R. Chen, and Q.F. Chen, “Mechanical and thermal transport properties of monolayer PbI2 via firstprinciples investigations,” Philosophical Magazine, vol. 99, no. 10, pp. 1277–1296, 2019. View at: Publisher Site  Google Scholar
 R. C. Andrew, R. E. Mapasha, A. M. Ukpong, and N. Chetty, “Mechanical properties of graphene and boronitrene,” Physical Review B: Condensed Matter and Materials Physics, vol. 85, no. 12, Article ID 125428, 2012. View at: Publisher Site  Google Scholar
 Z. Zhang and W. Guo, “Intrinsic metallic and semiconducting cubic boron nitride nanofilms,” Nano Letters, vol. 12, no. 7, pp. 3650–3655, 2012. View at: Publisher Site  Google Scholar
 F. Mouhat and F. Coudert, “Necessary and sufficient elastic stability conditions in various crystal systems,” Physical Review B: Condensed Matter and Materials Physics, vol. 90, no. 22, Article ID 224104, 2014. View at: Publisher Site  Google Scholar
 H. Peelaers and C. G. Van de Walle, “Elastic constants and pressureinduced effects in MoS_{2},” The Journal of Physical Chemistry C, vol. 118, no. 22, p. 12073, 2014. View at: Publisher Site  Google Scholar
 T. Liang, W.Q. Chen, C.E. Hu, X.R. Chen, and Q.F. Chen, “Lattice dynamics and thermal conductivity of lithium fluoride via firstprinciples calculations,” Solid State Communications, vol. 272, pp. 28–32, 2018. View at: Publisher Site  Google Scholar
 X. Y. Hou, Y. Cheng, C. E. Hu, C. G. Piao, and H. Y. Geng, “Thermoelectric properties of strontium sulfide via first principles calculations,” Solid State Communications, vol. 305, Article ID 113755, 2020. View at: Publisher Site  Google Scholar
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Copyright © 2021 SiHua Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.