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 first-principles 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 two-dimensional (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 ion-conducting electrolytes can be used for the electrochemical detection of hydrogen in the liquid sodium coolant of fast reactors. Some laboratories discovered that the CaBr2-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 CaBr2 and CaHBr at 576°C [13]. In 1996, de Castro Vítores et al. [12] obtained the bond energy of Ca-HBr 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 first-principles [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 k-point sampling for structural optimization. The optimized unit lattice is expanded to a supercell of , and then, the second-order harmonic force constants (harmonic IFCs) and the third-order 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 two-phonon and three-phonon 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 three-phonon 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 zeroth-order 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 energy-volume (E-V) 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 two-dimensional 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 two-dimensional material with weaker hardness.
The stress-strain curve can be used to estimate the elastic limit of monolayer CaHBr. In the calculation, since the lattice constants of CaHBr on the x-axis and y-axis are the same, the corresponding biaxial tensile strain is made, as shown in Figure 1. It is our biaxial stress-strain 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 two-dimensional 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 three-phonon 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 k-grid 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 second-order and third-order 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 Sci-Tech 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.