1、高压下 LaB6 的弹性和热力学性质的第一性原理计算（英文） 何熹 傅敏 于白茹 四川大学物理科学与技术学院 四川大学原子与分子物理研究所 摘 要： 运用平面波赝势密度泛函理论, 研究了 CsCl 结构的 LaB6在高压下的弹性和热力学性质.计算中使用了广义梯度近似, 得到在零温零压下 LaB6的晶格常数和已知的实验及其它理论结果相符.同时, 我们还得到了 LaB6的弹性常数 Cij, 体弹模量 B, 剪切模量 G, 杨氏模量 E, 德拜温度 E, 泊松系数 , 压缩波速 VL和剪切波速 VS 与压强的关系.计算发现 LaB6在压强低于 14GPa 时具有力学稳定性.根据准谐德拜模型, 我们还
2、预测了 CsCl 结构 LaB6的热力学性质, 对 014GPa和 01500K 范围内热膨胀系数和比热容的变化进行了研究.最后分析了 LaB6在零温零压和高压下的电子态密度图.关键词： 密度泛函理论; 弹性性质; 电子性质; 热力学性质; 作者简介：何熹 (1992-) , 男, 主要研究领域是第一性原理计算.作者简介：于白茹.E-mail:收稿日期：2017-10-19基金：国家自然科学基金 (11204192) Elastic and thermodynamic properties of LaB6 under pressure:a first-principles studyHE Xi
3、 FU Min YU Bai-Ru College of Physical Science and Technology, Sichuan University; Institute of Atomic and Molecular Physics, Sichuan University; Abstract： The elastic and thermodynamic properties of CsCl-type structure LaB6 under high pressure are investigated by first-principles calculations based
4、on plane-wave pseudopotential density functional theory method within the generalized gradient approximation (GGA) .The calculated lattice parameters of LaB6 under zero pressure and zero temperature are in good agreement with the existing experimental data and other theoretical data.The pressure dep
5、endences of the elastic constants, bulk modulus B (GPa) , shear modulus G, Youngs modulus E, elastic Debye temperatureE, Poisson ratio, compressional wave velocity VLand shear wave velocity VSare also presented.An analysis for the calculated elastic constants has been made to reveal the mechanical s
6、tability of LaB6 up to 14 GPa.The thermodynamic properties of the CsCl-type structure LaB6 are predicted using the quasi-harmonic Debye model.The variations of thermal expansion coefficientand the specific heat capacity Cvare obtained systematically in the ranges of 014 GPa and 01500 K.At last, the
7、pressure dependences of the density of states are also investigated.Keyword： Density functional theory; Elastic properties; Electronic properties; Thermodynamic properties; Received： 2017-10-191 IntroductionThe rare-earth hexaborides RB6 have attracted extensive experimental and theoretical interest
8、 due to their intriguing physical properties.For example, CeB6is a dense Kondo compound and has interesting low-temperature magnetic phases1.SmB6is also an exemplary Kondo insulator which features an energy gap in the electronic density of states (DOS) whose magnitude is strongly temperature depende
9、nt and only fully developed at low temperatures2.EuB6is a ferromagnetic semiconductor with a transition temperature TC=15K.Below TCthe electrical resistivity is drastically reduced, above TC a very large negative magnetoresistance is observed3.Among these compounds, lanthanum hexaboride (LaB6) has a
10、 special place.LaB6, which is metallic at room temperature and becomes a superconductor atTC=0.45K4,5, is a hard, refractory and stable material owing to strong B-B covalent bonds6.And it is a thermionic electron emitter with a low work function, high brightness and long life compared with conventio
11、nal tungsten filaments7.The structural, elastic, and thermodynamic properties of LaB6 have been investigated experimentally and theoretically by several groups.Early in 1977, Tanaka et al.6studied the elastic constants of LaB6for the first time by the measurements of the transit time of pules of lon
12、gitudinal and transverse ultrasonic wave propagating in single crystal.An early electronic structure calculation was estimated by Kubo et al.8, using the three-dimensional Lock-Crisp-West (LCW) folded momentum densities (3DLCW FMDs) within local-density approximation (LDA) method.Its indicated that
13、the Fermi surface topology plays an important role in the determination of structures.Mandrus et al.9explained the temperature dependence of the specific heat and resistivity of LaB6 well by using a model of La ions as independent Einstein oscillators embedded in a Debye framework of boron ions.Xu e
14、t al.10investigated the elastic and thermal properties of LaB6in the framework of density-functional theory (DFT) with a quasi-harmonic Debye model.Bai et al.11achieved the structure and chemical bond characteristics of LaB6 by means of the density functional theory using the state-of-the-art fullpo
15、tential linearized augmented plane wave (FPLAPW) method.In addition, Grel et al.12performed an ab initio study of structural, elastic, lattice-dynamical, and thermodynamic properties of rare-earth hexaborides LaB6 within the density functional theory and linear-response formalism using pseudopotenti
16、als and a plane-wave basis.There have been many other works to investigate the LaB6crystal and its properties13,14.What attracts us most is the pressure induced phase transition of LaB6, which has recently provoked a great deal of controversy.By using the Raman and angle dispersive X-ray diffraction
17、 (ADXRD) , Teredesai et al.15proposed that the pressure induced structural phase transition from CsCl-type structure to the orthorhombic structure occurs at around 10GPa.While Godwal et al.16also using the Raman and ADXRD, proposed that there is no structural or electronic phase transition up to at
18、least 25GPa in CsCl-type structure.The most common assessment of mechanical properties can be made by the determination of its elastic constants.Especially, the elastic constants of materials at high pressure are essential in order to predict and understand material response, strength, mechanical st
19、ability, and phase transition.The comprehensive analysis of elastic constants can provide a deeper insight into the hardness of materials.Furthermore, elastic properties are also related thermodynamically to the specific heat, thermal expansion, Debye temperature, melting point, and so on.Thus in th
20、is work, we put our investigation emphases mainly on the elastic and thermodynamic properties of CsCl-type structure LaB6 (space group Pm 3m) under pressure.From the calculated elastic constants, we will study its mechanical stabilities and anisotropic behaviors, as well as the bulk modulus, shear m
21、odulus, Youngs modulus, Poissons ratio, elastic Debye temperature of LaB6 at diverse pressures.Because the mechanical properties of this substance are studied in detail for the first time, we hope that our work can provide useful help for future research in both experimental and theoretical studies.
22、The rest of the paper is organized as follows.The theoretical method is introduced and the computation details are given in Section 2.Some results and discussion are presented in Section 3.Finally, the summary of our main results and conclusions are given in Section 4.2 Theoretical method and calcul
23、a-tion details2.1 Total energy electronic structure calculationsIn the electronic structure calculations, we employ the plane-wave pseudopotential density functionaltheory method17through the Cambridge Serial Total Energy Package (CASTEP) 18code together with both the generalized gradient approximat
24、ion (GGA) proposed by Perdewet al.19and the local density approximation (LDA) proposed by Vosko et al.20for exchange-correlation potentials.A plane-wave basis set with energy cut-off 390eV is applied.Pseudo-atom calculations are performed for La5d6sand B 2s2p.For the Brillouin-zone sampling, we use
25、the 666 Monkhorst-Pack mesh21, where the self-consistent convergence of the total energy is at 5.010eV/atom.The tolerance for geometry optimization is set to within 5.010eV/atom, the maximum ionic force within 0.01eV/, the maximum ionic displacement within 5.010, and the maximum stress within 0.02GP
26、a.The tolerance for elastic constants is set to within 1.010eV/atom, the maximum force within 0.0002 eV/, and the maximum strain amplitude within 0.003 GPa.These parameters are carefully tested.It is found that these parameters are sufficient to lead to a well-converged total energy.2.2 Elastic prop
27、ertiesTo calculate the elastic constants under hydrostatic pressure p, we use the symmetry-dependent strains that are non-volume conserving.The elastic constants, Cijkl, with respect to the finite strain variables are defined as22where ijand eklare the applied stress and Eulerian strain tensors, and
28、 X, xare the coordinates before and after deformation, respectively.Under the hydrostatic pressure p, we havewhere Cijkl denote the second-order derivatives with respect to the infinitesimal strain (Eulerian) , is the finite strain variable.The fourthrank tensor Cgenerally greatly reduces when takin
29、g into account the symmetry of the crystal.In a cubic crystal, it is reduced to three components, i.e.C11, C12, and C44.The bulk modules Band the shear modules G of the LaB6are taken as23where R and V represent Reuss and Voigt boundaries, respectively.The polycrystalline Youngs modulus E and the Poi
30、ssons ratioare then calculated from these elastic constants using the following relations24.The elastic Debye temperature E may be estimatedfrom the average sound velocity Vm25where h is Plancks constants, k Boltzmanns cons tant, NAAvogadros number, nthe number of atoms per formula unit, M the molec
31、ular mass per formula unit, the density, and V mis obtained from25where VS and VL are the shear and longitudinal sound velocities, respectively.The probable values of the average shear and longitudinal sound velocities can be calculated b2.3 Thermal propertiesTo investigate the thermal properties, w
32、e change the cell volume to obtain the corresponding energy, and then export them into the quasiharmonic Debye model27to calculate the thermal properties.In this model, the non-equilibrium Gibbs function G (V;p, T) has the following from:where E (V) is the total energy as a function of the call volu
33、me V, pis the hydrostatic pressure, (V) is the Debye temperature as a function of V, and Avibis the vibrational Helmholtz free energy.Based on this model, the specific heat Cv, Cp, and the thermal expansion coefficientcan be deduced from the following expressions:Through the quasi-harmonic Debye mod
34、el, one could obtain the thermodynamic quantities of LaB6under pressure and high temperature.By applying the method, we have investigated the thermodynamic properties of several materials successfull.3 Results and discussion3.1 Structural propertiesTo investigate the elastic and thermodynamic proper
35、ties, we must determine the structures of LaB6at first.LaB6has a bcc-like structure (space group Pm m) with La at the position (0, 0, 0) and B at the position (0.5, 0.5, x) , where xis the positional parameter of the B atoms.The structure information can be absolutely described by lattice parameter
36、aand positional parameter x.To determine the ground state structure of LaB6, we use the following steps.Firstly, we fix the lattice parameter aand take a series of different values of positional parameters xto calculate the total energies E, so that we can obtain an E-x curve and find a lowest energ
37、y Emin.The positional parameter x with the energy Eminis what we require.Secondly, with the obtained positional parameter x, we take a series of different values of lattice parameter aand repeat the above steps, the lattice parameter aalso can be obtained.And for each a, we can calculate its corresp
38、onding primitive cell volume V, and then obtained the energy-volume (E-V) curve of LaB6.By fitting the calculated E-V data to the third-order Birch-Murnaghan equation of state (EOS) 33, the bulk modulus B0at p=0and T=0can be obtained.All the equilibrium structure parameters and bulk moduli are liste
39、d in Tab.1.It can be seen that our results of lattice parameter a, positional parameter x and bulk modulus B0from GGA calculations are well consistent with the experimental data34,35and other theoretical data11,14.And the errors of lattice parameter a are less than 0.1%, respectively.On the other ha
40、nd, our LDA results are not satisfactory, which are a little small when compared with the experimental34and other theoretical data11, except for bulk modulus B0.Therefore, in this work, the GGA functional forms are applied in the following calculations.Tab.1 Calculated equilibrium lattice parameters
41、 of LaB6, together with the experiment data and other theoretical results 下载原表 The pressure and temperature dependence of the relative volume V/V0of LaB6are illustrated in Fig.1.It is shown that, as the applied pressure increases from 0to 14GPa, the volume of LaB6decrease linear at the giving temper
42、ature, and the relative volume V/V0of higher temperature is less than that of lower temperature at the same pressure.This means, under higher temperature, LaB6is easier to be compressed, as temperature could make LaB6soft.Fig.1 Normalized primitive unit cell volume V/V0as a function of pressure 下载原图
43、3.2 Elastic propertiesWelist our calculated elastic constants and aggregate elastic modulus Bof the cubic structure LaB6at 0Kand 0GPa in Tab.2.It can be seen clearly that our results are in agreement with the other theoretical12,14and experimental data36, which indicates that our results are reasona
44、ble.In addition, from the Eq. (3) , the bulk modulus B (inTab.2) obtained byour elastic constants is177.9GPa, which is consistent with the value estimated by fitting the E-V data mentioned above.Tab.2 Calculated elastic constants Cijof LaB6at 0K and 0 GPa, in comparison with the experi-mental data a
45、nd other theoretical results 下载原表 Fig.2 Pressure dependencies of elastic constants of LaB6at 0K 下载原图Tab.3 Calculated elastic constants Cij (GPa) , bulk modulus B (GPa) , shear modulus G (GPa) , B/G, acoustic veloci-ties, and VLand VS (km/s) , and elastic Deybe temperatureE (K) of LaB6under pressure
46、p (GPa) 下载原表 In Tab.3, we also list the bulk modulus and shear modulus, which can easily describe the hardness of a crystal in an indirect way.It is found that both bulk modulus Band shear modulus Gincrease gradually with the increasing pressure.This implies that the compressibility of LaB6 becomes
47、lower as the pressure increases.From the ratio of B/G, one can distinguish the ductility and brittleness of metals.The threshold is around 1.7538.When B/G1.75, the material behaves in a ductile manner, otherwise the material behaves in a brittle manner.Duan et al.14obtained the B/Gis 1.33at 0 GPa an
48、d 0K.The B/Gas a function of pressure is displayed in Tab.3.It can be seen that the value of B/Gincreases with the increasing pressure, indicating that it becomes much harder with the increasing pressure, and it is brittle in nature up to 14GPa.Youngs modulus is defined as the ratio of stress to sta
49、in, and is used to provide a measure of the stiffness of the solid, i.e.the larger the value of E, the stiffer the material.Tab.3illustrates that Youngs modulus increases with pressure when p14 GPa, indicating that the pressure can, to some extent, improve the stiffness of this material.Poissons ratio is defined as the absolute value of ratio of transverse strain to longitudinal strain, when materials subject to longitudinal stress.Poissons ratio=0.25is
