Investigation of electronic,elastic,and optical properties of topological electride Ca3Pb via first-principles calculations*

Chang Sun(孙畅), Xin-Yu Cao(曹新宇), Xi-Hui Wang(王西惠), Xiao-Le Qiu(邱潇乐),Zheng-Hui Fang(方铮辉), Yu-Jie Yuan(袁宇杰), Kai Liu(刘凯), and Xiao Zhang(张晓),†

1State Key Laboratory of Information Photonics and Optical Communications&School of Science,Beijing University of Posts and Telecommunications,Beijing 100876,China

2Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials&Micro-nano Devices,Renmin University of China,Beijing 100872,China

Keywords: density functional theory(DFT),density of states(DOS),work function,elastic constant

1. Introduction

The discovery of novel materials with excellent performance will not only promote the fundamental research in condensed matter physics and material science but also find great value in applications. It is electrides that are one of the paradigms. Electrides represent a special class of compounds whose electrons are partially confined in the structural cavities (e.g., zero-dimensional (0D) cages, one-dimensional(1D) channels, or two-dimensional (2D) interlayer spacings),rather than attached to atoms, and these electrons behave as anions.[1–3]The history of electrides began with the observation of alkali metal ammonia solutions. Later, the organic electride crystal Cs+(18C6)2e−composed of alkali metals and organic components was synthesized.[1]However,the organic electrides are thermally unstable, easy to decompose at room temperature, and sensitive to water and air. In 2003,the first air-stable inorganic electride [Ca24Al28O64]4+(4e−)(C12A7) was synthesized, making electride materials firmly established in chemistry.[2]Since then, many electrides with various anionic electron confinement topologies have been discovered, such as 0D,[3,4]1D,[5,6]and 2D electrides.[7,8]Unlike ionic compounds, anionic electrons in electrides are usually loosely bound. Such loosely bound nature of anionic electrons leads to their relatively high mobilities and low work functions, which are key ingredients for various applications, such as electron emitters,[9]anode materials,[10]and high-performance catalysts.[11,12]Especially,the 0D electrides where the anionic electrons reside in the cage space, such as C12A7[2]and LaH2,[13]have high chemical reactivities and broad applications.[11]Recently, Ca3Pb was reported to be a 0D electride with a topologically nontrivial band structure.[3]The combination of topological electronic structure and features of anionic electrons in topological electride might lead to exotic physical phenomena,such as floating surface state and the coexistence of 2D surface electride states and topological surface state.[14,15]In order to further understand the physical properties of 0D topological electride Ca3Pb,in this work,we calculate the electronic structure, work function, mechanical and optical properties of Ca3Pb in detail. The present work provides theoretical support for experiments and industrial applications.

2. Calculation method

3. Results and discussion

Figures 1(a) and 1(b) show the crystal structures of Ca3PbO and Ca3Pb, respectively. Ca3PbO has an antiperovskite structure that Pb, Ca and O atoms occupy the corners, face centers and body center of the cubic unit cell, respectively.In addition,the OCa6octahedra connect each other by corner sharing. The structure of Ca3Pb is closely related to that of Ca3PbO, and the former can be formed by simply removing the O atoms from the latter. Thus, Ca3Pb also has the cubic symmetry with the space group of Pm¯3m(No.221),same as Ca3PbO,and the anionic electrons are trapped at the center of the Ca6octahedra.[3]The calculation of lattice relaxation for Ca3Pb indicates that the optimized lattice constant is 4.950 ˚A, in good agreement with the experimental result(a=4.853 ˚A).[21]

The existence of anionic electrons has significant influences on the physical properties of Ca3Pb. The anionic electrons can lead to a low work function, which is a unique feature of electrides. The work function ΦWFis the minimum energy required to move an electron from the Fermi level to the vacuum.[8]The ΦWFcan be calculated by the following formula ΦWF=eΦvac−EF,where Φvacis the electrostatic potential of the vacuum region and e is an electron charge. The surfaces of Ca3Pb along (100) and (110) crystal planes have two types of terminations, i.e., CaPb-terminated (100) and(110) surfaces, and Ca-terminated (100) and (110) surfaces.The ΦWFs of Ca3Pb for different terminations and orientations as a function of slab thickness N (= number of layers)converge quickly when N ≥9. The ΦWFs for N=15 are also listed in Tab.1. Comparing our present calculated values with and without SOC,the ΦWFof(100)Ca-terminated surface has the smallest value, which is much lower than those values of ΦWFs for other surfaces and even smaller than the ΦWFof Ca (2.9 eV).[20]It could be partially ascribed to the highest density of anionic electrons for the (100) Ca-terminated surface. The relatively loose bound anionic electrons can lead to a low work function.This explains the lower ΦWFof(100)Caterminated surface compared with the(110)CaPb-terminated surface because the density of anionic electrons of the former is higher than that of the latter even both surfaces pass through the 1b site(the position anionic electrons occupied). Figure 2 shows the calculated ΦWFof(100)Ca-terminated surface for Ca3Pb and Ca3PbO,respectively. When anionic electrons are replaced by O atoms occupying the center of Ca6octahedra,the ΦWFof Ca3PbO increases.This comparison further proves that the relatively loose bound anionic electrons tend to result in a low work function.[7]

Fig. 1. (a) and (b) Crystal structures of Ca3PbO and Ca3Pb, respectively. The blue and gray balls represent Ca and Pb atoms, respectively.The small red ball occupying the body center of the unit cell represents the O atom. (c) and (d) Band structures of Ca3PbO and Ca3Pb,respectively. The fat-band representation shows the contributions of atomic orbitals on the projected band structure with Ca-3d orbitals in pink,Pb-6p orbitals in blue,and Pb-6s orbitals in green. The projected band structure from −5 to −4 eV(red color)in panel(c)originates from the O-2p orbitals. In panel(d),the band marked by the red dots is mainly composed of anionic electrons at the interstitial sites.

Table 1. The work function of Ca3Pb with different surface terminations.

Fig. 2. Work functions for the Ca-terminated (100) surfaces of (a) Ca3Pb and(b)Ca3PbO.

The elastic constants characterize the resistance of the material to external stress and deformation. The elastic constants of cubic Ca3Pb in the linear strain range have only three independent components: C11, C12, and C44, which are 61.89 GPa, 21.32 GPa, and 34.61 GPa, respectively.The elastic properties were calculated by Voigt–Reuss–Hill approximation.[24]Bulk modulus B = (BV+BR)/2, shear modulus G=(GV+GR)/2,where

Then, Young’s modulus E and Poisson’s ratio σ can be obtained by the following formulas:[25]

The calculated elastic constants satisfy the criterion of crystal structure stability: C11−C12＞0, C11+2C12＞0,C44＞0.[25]The calculated B, G, E, σ, and B/G are shown in Tab. 2. It can be seen that Ca3Pb has good mechanical properties and can maintain its stable performance when subjected to mechanical distortion. The result with considering the SOC effect on elastic calculation is also shown. The result with SOC calculation has changed B (4.4%), G (0.9%),E (1.7%), σ (4.9%), and B/G (4.0%) compared with the results without considering SOC, indicating that the effect of SOC of heavy atom Pb on elastic properties is not obvious.The bulk modulus is an important indicator reflecting the fracture strength,and the shear modulus is an important indicator reflecting the strength against plastic deformation. If the ratio of B to G is less than 1.75, the material is brittle. Otherwise,the material has a better ductility.[26]Different from other typical electrides such as Ca2N, C12A7 and Y2C, Ca3Pb has a strong brittleness.[27–29]The stiffness of the material can be measured by Young’s modulus E. The E value of Ca3Pb is 66.13 GPa,much smaller than these of C12A7 and Y2C.The ratio of quantities of ionic bonds to covalent bonds in certain material and the ability to resist deformation can be characterized by Poisson’s ratio σ. For the materials containing only covalent bonds,the value of σ is less than or equal to 0.1,and for materials that only contain ionic bonds, the value of σ is greater than or equal to 0.25.[30]The σ of Ca3Pb is 0.184,and it suggests that the chemical bonds in Ca3Pb are partially covalent,possibly due to the smaller electronegativity difference between Ca and Pb compared with other electrides listed in Tab.2.

Table 2.Mechanical properties of Ca3Pb in comparison with other electrides.

The degree of elastic anisotropy of a solid can be presented by the elastic anisotropic factor A.[25]It means complete isotropy when the value of A is 1, and anisotropy when A deviates from 1. In the cubic structure,A is given by

The calculated A is 1.706,indicating that Ca3Pb shows a little elastic anisotropy. Three-dimensional surface structures of Young’s modulus E and shear modulus G are plotted in Fig. 3. The relationship between E and G and direction can be described by the following formulas:[31]

where

Among them, l1, l2, and l3are the directional cosines in the system,and Sijis the elastic compliance constant. The parameters of θ and ϕ are Euler angles. The minimum value of E is 50.97 GPa,and the maximum value is 70.80 GPa;the minimum value of G is 20.19 GPa,and the maximum value is 34.61 GPa.

Fig.3. Three-dimensional surface constructions of(a)Young’s modulus E and(b)shear modulus G of Ca3Pb.

Debye temperature ΘDis an important parameter in material thermodynamics which is related to specific heat,melting temperature,and lattice vibration. ΘDcan qualitatively reflect the strength of the material’s covalent bond. ΘDcan be obtained from the average sound velocity[32]

where h is the Planck constant, k is the Boltzmann constant,n is the number of atoms in the molecule,NAis the Avogadro constant, M is the molecular mass, and ρ is the density. The average sound velocity vmcan be obtained by the following formula:[32]

where vland vtare the longitudinal and transverse sound velocities, respectively. The calculation results at 0 GPa are shown in Tab.3.The ΘDof Ca3Pb is 262.66 K,which is much smaller than 353 K of Y2C.[28]

Table 3. Calculated thermodynamic properties of Ca3Pb at 0 GPa.

The optical properties of materials enable us to understand their properties and describe their response to electromagnetic radiation. Since optical properties in the infrared region are usually sensitive to the band structure near the Fermi level, the research on optical properties will provide useful information about the characteristics of electronic structures and inspire their applications in optoelectronic devices.A complex dielectric function ε(ω) is the basic data for studying the optical properties, which describes the response of matter to the electromagnetic field. It can be expressed as ε(ω)=ε1(ω)+iε2(ω), where ε1(ω) and ε2(ω) are the real and imaginary parts, respectively. The imaginary part can be obtained by the equation[33]

The integral covers the first Brillouin zone and the momentum dipole element Mvk(k)=(k〈uck|Eelec∇|uvk〉]) describes the interband transition directly from valence bands to conduction bands, where Eelecis the electric field, and the parameters of uckand uvkare the wave functions of the conduction band and the valence band, respectively. The parameter hωcv(k)=Eck−Evkdescribes the excitation energy required for the interband transition. On the other hand,the real part of the dielectric function about intraband information can be calculated from the imaginary part using Kramers–Kronig transformation[33]

where P is the main value. Figure 4(a) shows the ε1(ω)and ε2(ω) of Ca3Pb for the energy range up to 50 eV, and Fig. 4(b) depicts the total and partial densities of state (DOS and PDOS).It is obvious that the Fermi surface is dominated by the Pb-6p orbitals with a little contribution from Ca orbitals.Above Fermi energy,the unoccupied orbitals are composed of Ca-3d and Ca-4s orbitals. As shown in Fig. 4(a), the ε2(ω)spectrum has a high absorption below 10 eV and reaches the peak at 2.01 eV in the infrared region, which originates predominantly from the optical transitions of the Pb-6p states to the Ca-3d states.The value of ε2(ω)becomes zero in the range of 12 eV to 23 eV and at the region above 30 eV, suggesting that Ca3Pb is transparent in these energy ranges. The ε1(ω)spectrum shows that the threshold energy, i.e., the first critical point occurs at 0 eV. From this point, the curve linearly declines to its minimum as the energy increases from 0 eV to 4 eV. This reduction may be due to the intraband transitions,and the large negative values ε1(ω) indicate the metallic nature of this compound. In addition, in the ultraviolet region between 22 eV and 27 eV,Ca3Pb exhibits some small peaks.

Fig. 4. (a) Dielectric function of Ca3Pb, in which the red curve is the imaginary part and the blue one is the real part. (b) Total and partial density of states of Ca3Pb.

Based on the complex dielectric function, we can get some important optical functions, such as absorption coefficient α(ω),refractive index n(ω),reflectivity R(ω),and electron energy loss spectroscopy(EELS,L(ω)).[33]

The calculated results are presented in Fig.5. As shown in Fig.5(a), a broadening absorption peak appears in the low energy region from 0 eV–10 eV,corresponding to the low energy peak of ε2(ω). The absorption spectrum is caused by the continuous band transition near the Fermi surface, which indicates the properties of metals. Another absorption peak emerges in the ultraviolet region, corresponding to the small peak in the ε2(ω) spectrum between 22 eV and 32 eV. This may be due to the transition from the deep energy level to the conduction band. Up to 35 eV, the absorption coefficient approaches zero. Figure 5(b) shows the energy dependence of n(ω). The static refractive index n(0) is 7.1, then it decreases rapidly and has the lowest value at about 10 eV.With the energy increasing further, n(ω) begins to increase again and reaches a peak value at about 22 eV.Figures 5(c)and 5(d)show the R(ω) and L(ω) of Ca3Pb, respectively. The value of reflectivity at zero frequency R(0)is 0.67. When the incident electromagnetic energy is less than 8.5 eV,the reflectivity oscillates around 0.5. With the increase of incident electromagnetic energy to more than 8.5 eV, the reflectivity decreases sharply. This may be related to the switch of intraband to interband absorptions. The EELS is used to describe the energy loss of an electron passing through a solid at a certain speed. The characteristic peaks of L(ω)are related to the oscillation of the plasma. The energy loss is the largest at 10 eV(Fig.5(d)),corresponding to the sharp drop in the R(ω)when the energy is larger than 8.5 eV and the zero crossing of ε1(ω).

Fig. 5. The energy dependence of (a) absorption coefficient, (b) refractive index,(c)reflectivity,and(d)electron energy loss spectroscopy of Ca3Pb.

4. Conclusion