Density Functional Study of Mechanical, Electronic and Pressure Induced Phase Transition Properties of CaFé2As2 ()
1. Introduction
The exposure of iron Pnictide materials superconducting at high temperatures has renewed research on their superconductivity mechanisms [1] . The superconducting iron Pnictide class of 122 compounds, are structurally simple as compared to the 111 family of Pnictides [2] . 122 class of iron pnictides has atomic arrangement of AFe2As2 which is stable with divalent (A = Ba, Ca, Sr, and Eu) atoms. They crystalize at room temperature into
type tetragonal structure [3] with space group of 14/mmm [4] . CaFe2As2 as one of the 122 classes, when subjected to either conditions of pressure or temperature transits from the tetragonal stable phase to orthorhombic [5] . As our motivation, Iron Pnictide compounds have been identified to possess superconducting properties when they are either doped or subjected to the external pressure to phase transition point [6] . Electron or hole doping suppresses magnetic and structural instabilities inducing superconductivity. Application of pressure on the parent compounds suppresses magnetic order [6] and enhances superconductivity in CaFe2As2 [6] [7] .
Electrons in 3d orbitals in CaFe2As2 enables electrons in low-lying electronic state with many hole and electron bands pair at low temperatures, which is one of the conditions that leads to superconductivity [8] . At the superconductivity state, CaFe2As2 is very sensitive to applied pressure and this causes it to have several phases at low temperatures. In the collapsed tetragonal shape, the electron correlation and superconductivity disappear [9] and CaFe2As2 transits to orthorhombic phase at room temperature [10] . If pressure is applied by a non-hydrostatic medium, CaFe2As2 becomes a superconductor, but small uniaxial pressure parts overturns its superconductivity [11] .
CaFe2As2 on cooling through 170 K, undergoes first-order magnetic phase transition [12] . Superconductivity appears above 40 K when the single crystal of the compound is electron doped [13] . This compound contains oxide blocks on the c-axis and two-dimensional iron (Fe) and arsenic (As) tetrahedron layers which act as charge reservoirs and denote the structural phase transition. The presence of iron in this compound contributes to its magnetic properties which when tuned can turn on and off its superconducting behavior [14] .
Na-substitution in between Ca and Fe with composition
induce superconductivity at 20 K. Upon application of pressure, it lacks any sign of superconductivity but has a reduction of its unit cell volume [15] . Reduction of the unit cell volume and hole doping are very essential for superconductivity.
The objectives of this paper first are to present an elaborate calculation on density functional investigations on the electronic and mechanical properties of CaFe2As2. Secondly to study pressure induced phase transition of CaFe2As2. The rest of this paper is organized such that Section 2 is the computational methodology, Section 3 presents results, and the conclusion is in Section 4.
2. Computational Method
The first principle calculation study of CaFe2As2 under induced pressure was undertaken in the framework of density functional theory (DFT) based on plane wave self-consistent field (PWscf) and ultrasoft pseudopotential (USPP) method as treated in the Perdew-Burke Ernzerhof (PBE) [16] generalized gradient approximation and local density approximation. The computational calculations were performed using Quantum Espresso simulation package and the optimized cell dimensions were fitted using Murnaghan fitting methodology of second order given by Equation (1). The equation gives the relationship which exists between the volume of a body and pressure upon which it is subjected. This shows that when bodies are compressed more, it becomes more difficult to compress them beyond some limit. The Murnaghan equation models experimental data with satisfactory accuracy.
(1)
The k-points, and the kinetic energy cut-off values were properly checked through graphing and accurate values were obtained for the convergence of the ground state energy at minimum convergence threshold in the calculation using the proper basis sets [17] . Minimum point of convergence is essential in finding a proper initial setting in making decision on reliability and accuracy of simulation. Self-Consistency Field in Density Functional Theory determines ground state electron density through energy change which brings a slight change in electron density. Convergence is achieved when the energy reaches the global minimum. Convergence is achieved by a set threshold whereby a smaller threshold produces more accurate results. The minimization is given by the relation in Equation (2):
(2)
The valence configuration used for CaFe2As2 was 3s23p64s2 for Ca, 3s24s23p6 for Fe and 4s24p3 for As. The convergence threshold was estimated at 10−8 eV which is sound for accuracy. The Brillouin sampling was based on the Monkhost scheme [18] where K-point mesh of the irreducible high symmetry points in the Brillouin zone used was 6 × 6 × 4. For the elastic constant calculation, the ‘quasi-static’ approximation was used, where the elastic constants were computed at absolute temperature and saved as elastic constants.
Mechanical properties play significant role in the crystal structure studies. Empirically, mechanical properties of materials are determined by elastic properties which include the shear modulus (G), bulk modulus (B), Young modulus (E), and Poisson’s ratio (n).
Specifically, elastic properties are achieved by the elastic constants (Cij), which are identified as the stress tensor versus small strain. Additionally, the mechanical stability of materials is calculated by elastic constants. For cubic crystal structure, there exist three independent elastic constants; C11, C12, and C44, which determine the mechanical stability given by Born-Huang’s stability criteria [19] [20] in Equation (3):
(3)
Taking into account the structural symmetry, the shear and bulk moduli are evaluated in compliance with Voigt-Reuss-Hill (VRH) approximation. Additionally, the Young’s modulus is identified as the ratio of linear strain versus linear stress, which gives the degree of elastic stiffness. Poisson’s ratio indicates the advances of the covalent bond. A high ratio proves the presence of ionic and metallic bonds. Values of Cij can be used in the evaluation of Poisson’s ratio and elastic moduli of CaFe2As2. As per Voigt approximation, the bulk and shear moduli isotropy can be acquired by doing linear combination of elastic constants [20] . With a different format, Reuss obtained estimates for bulk and shear moduli isotropy by employing the use of single crystal elastic constants. Hill confirmed that Voigt and Reuss estimates were lower and upper polycrystalline elastic moduli limits respectively, hence the averages became realistic as shown in Equation (4).
(4)
Elastic anisotropy calculation is an important aspect as it gives important information on the likelihood of the introduction of micro cracks in the compounds in trying to improve their durability [21] . We calculate the elastic anisotropy factor (A) by making use of the following Equations (5-7). The shear anisotropic factor for the {1 0 0} shear planes between the <011> and <010> directions is:
(5)
For the {0 1 0} shear planes between <101> and <001> directions it is:
(6)
For the {0 0 1} shear planes between <110> and <010> directions it is
(7)
But since our crystal does not have C22 and C55 our calculations of shear anisotropic factors is only for A1 which is 1.0914 showing that our compound is isotropic.
Another important mechanical property of materials is the Vickerhardness
. The Vicker hardness test determines the hardness of a material. It also helps in testing hardness in thin layers in trying to overcome the problem of cracking in brittle materials. It can be predicted by Equation (8) [22] :
(8)
Materials with Vicker’s hardness beyond 40 GPa are considered to be super-hard materials. The Vicker’s hardness to our compound is 18.1193 GPa an indication that the compound is not hard enough to resist being dented.
Debye temperature is another basic physical property which distinguishes between low and high temperature regions for a given material. We calculated the Debye temperature for this compound by the use of the following formula [23] :
(9)
where V is the unit cell volume, n number of atoms in a unit cell,
bulky modulus, M molar mass,
poisson ratio. The calculated Debye temperature is 3.8707 K hence we expect frozen high frequency modes [24] .
Pressure, which is the main basis for phase transition, was calculated for a good range of volume by the relationship of Equation (10):
(10)
Structural transition is associated with the atomic packing and orientation in the crystal, which brings about the change of the related properties of the compound. This change is due to different conditions, for instance, temperature and pressure. Given that our calculations are done at ground state temperature, only the pressure is varied in these calculations.
3. Results and Discussion
Superconductivity of this material which is the main focus of this study appears at structural phase transition point brought about by the application of external pressure. This brings the need to understand structural properties of CaFe2As2.
3.1. Structural and Mechanical Properties
Optimization of the cut-off energy convergence was achieved at 45 Ry. Figure 1 shows the optimized structure of CaFe2As2.
The optimized cell parameter a = 3.97 Å and c = 12.84 Å, which agrees well with experimental and other theoretical work as shown in Table 1 below.
Optimization of parameters such as lattice constants and cut-off energy was done to obtain a relaxed structure free from stress [25] .
Table 2 gives results of the elastic constants of CaFe2As2 as calculated in this work and reported for the first time.
Elastic constants C11 and C33 portray the level of deformation resistance along the a-axis and c-axis, respectively. Evaluated elastic constant C11 for CaFe2As2 is greater than that of elastic constant C33 by around 39.9%, a clear indication that this compound contains strong deformation resistance along the a-axis compared to c-axis.
Figure 1. Crystal structure of
as visualized from quantum espresso Xcrysden package,
.
Table 1. Comparison of experimental and theoretical cell dimensions.
Table 2. Elastic constants of tetragonal CaFe2As2.
On the other side, the elastic constant C44 describes the extent of shear distortion in the [1 0 0] plane, while the elastic constant C66indicates the shear resistance in the [1 1 0] direction [28] . This compound’s elastic constants satisfy the requirement for mechanical stability conditions given by Equation (5). Elastic constants C44 and C66 of CaFe2As2 obtained are 25.95 GPa and 31.73 GPa which are far less than other iron pnictides, a clear indication that CaFe2As2 has a lower deformation resistance compared to other iron pnictides. Deformation resistance is extensively accepted as a crucial engineering property of asphalt used for surfacing and construction of roads [29] . A higher deformation resistance indicates reduced potential for asphalt shearing, rutting and shoving mainly under heavy and slow moving loads making CaFe2As2 non effective.
The Young’s modulus E and Poisson’s ratio n and their relationships with bulk modulus and shear modulus are as tabulated in Table 3 below [30] .
Shear modulus indicates strength of the material unlike the bulk modulus. We have determined that G > B, hence CaFe2As2 mechanical failure should be corrected by application of the shear component. Bulk modulus B reveals the extent to which a material is compressible in relation to another. The higher the value of B, the incompressible the material is [31] . BaFe2As2 is more incompressible than CaFe2As2 with a value of 71 GPa. The Poisson’s ratio [32] determines ductility or brittleness of the material. The higher the ratio, the more ductile the
material is, whereas a low value indicates less brittleness. For
indicates ductility otherwise the material is brittle.
shows that the hardness is related inversely, meaning that the smaller the ratio, the harder the material. CaFe2As2 is hard and brittle with
. Poisson’s ratio (n) also helps us to assess the
mechanical properties of crystalline solids. Its low value indicates stability against shear [33] . Poisson’s ratio at the same time reveals the nature of interatomic forces where a range of 0.25 and 0.50 indicates central force interaction and outside this range for non-central force interactions. Moreover according to Poisson’s ratio, materials whose ratio is less than 0.26 undergo brittle failure, but above this ratio they undergo ductile failure. Poisson’s ratio of CaFe2As2 shows that it is brittle [34] . Resistance to compressive or expansive forces is measured by Young’s modulus. From Table 3, the value of E is small, even smaller than that of BaFe2As2 of 101.6 GPa, indicating that CaFe2As2 definitely cannot withstand large tensile stress [35] .
Cauchy pressure [36] is the difference between C12 and C44 elastic constants. This parameter reveals more about the elastic response and large density of solids. Cauchy pressure will indicate ductility or brittleness failure of crystalline solids. A positive and negative Cauchy pressure indicates ductility and brittleness, respectively. It also reveals chemical bonds. Positive value indicates metallic bonds while the negative one indicates covalent bonds. In our study, the Cauchy pressure of CaFe2As2 is −3.37, indicating that our compound is brittle with covalent bonding characteristics. Brittle materials like the ceramics which have a wide range of properties hence used for multiple applications.
3.2. Electronic Structure Properties
We present the band structure and projected density of states of CaFe2As2. The band structure of CaFe2As2 shown in Figure 2 was computed along high symmetry points; Γ-X-N-X-N-Γ.
Fermi level is at the center between the conduction band and the valence band, separating the particles in each band with specific quantum states from
Table 3. Mechanical properties of Bulk, Shear, Young’s moduli and Poisson’s ratioin GPa.
Figure 2. Band structure of CaFe2As2 showing that it is metallic in nature.
interacting [37] . From Figure 2, it is seen that the conduction band and valence band are overlapping, which is a clear indication that the compound is metallic and compares well with
[38] . The projected density of states of CaFe2As2 is shown in Figure 3.
The iron pnictide materials are formed in layered structures with charge reservoirs insulating the layers which contribute a lot in enhancing superconductivity. Iron based superconductors have a formation of constructive 2-D structure with irregular charge reservoir. For instance, CaFe2As2 has a tetragonal structure at room temperature and transit to orthorhombic upon subjection to pressure.
From the angle resolved photoemission spectroscopy studies together with electronic structure calculations, it is quite evident that the neighborhood Fermi level is made up of Fe d states (Figure 3(d)). Fe 3d states that have a photo-electronic cross-section of around 10 times that of As p states in between the phonon energy. In Figure 3(b), Ca, p also exhibits the higher binding energy between the range of −3 ev and 3 ev. CaFe2As2 splits at Ca layer keeping half of the calcium atoms on each side of the split surface which shows the reconstruction hence the electronic properties of Ca are different from that of bulk Ca.
The contributions of Ca, d states are basically above the energy region with a nominal contribution at or below the energy region. In the Fe-d and As-d, the contribution below the energy region shows a shift to the lower energies in the
tetragonal phase that keeps the PDOS energy region approximately at the same energy range for both tetragonal and orthorhombic structural phases. From Figure 1, Ca layers are packed among two layers and any adjustment in Ca state below the energy region is as a result of the hybridization of Ca s-states with the electronic states attributed to As in the valence band [38] . The compression at the c-axis as a result of the transition to the tetragonal state brought about by the subjection to an external pressure, which brings about reduction of the separation distance between As and Ca atomic layers which results in the enhancement of Ca s and As p. Even though it is evident that the contribution of Ca in the valence band is not much, its role in the structural change and electronic structure cannot be ignored. The S electrons in the cations play a key role in standardizing disorder effects and pairing interactions.
3.3. Phase Transition
CaFe2As2 compound has a tetragonal phase as the stable phase with five atoms; one Ca, two Fe, and two as which upon subjection to external pressure undergoes a phase transition to the orthorhombic phase, Figure 4 below. CaFe2As2 tetragonal stable phase undergoes a phase transition at an external pressure of 0.2 GPa to the orthorhombic phase.
Phase transition of this compound from the stable phase tetragonal to orthorhombic occurs at ground state temperature and external pressure of 0.2 GPa as shown in Figure 5. Phase transition occurs when the enthalpy and pressure changes are the same or nearly the same [39] . These two changes are indicated by two curves of different colors. Where the two lines coincide is the phase transition pressure and in this case is the best at 0.2 GPa.
Figure 4. Orthorhombic crystal structure with cell parameters
.
(a) Pressure = 0.1 GPa (b) Pressure = 0.2 GPa (c) Pressure = 0.3 GPa
Figure 5. Graphs of Volume against Energy for applied pressure of 0.1 GPa, 0.2 GPa, and 0.3 GPa. The best fit occurs at 0.2 GPa where the two lines properly coincide.
Figure 6. Graph of internal pressure against internal volume at an external pressure of 0.2 GPa.
Heat is a form of energy and the internal energy of this material changes as heat is transferred into or out of it as shown in Figure 6. Before the phase transition and after, the minimum energy and the enthalpy are quite different. Figure 6 is in agreement with Figure 5 with the phase transition at 0.2 GPa. This is in good agreement with the related studies showing that the pressure suppresses structural phase transition at low temperature stabilizing superconductivity [39] . The product of pressure and volume enables the measurement of energy within the material even when the system does not work with the surrounding. In Figure 6, clearly shows the estimated transition pressure which is 0.2 GPa and internal volume at about 730 a.u3.
4. Conclusion
We have here reported density functional theory calculations of electronic, mechanical and pressure induced phase transition of CaFe2As2. The electronic structure calculations show that this material is metallic based on its band structure. From calculations of elastic constants, mechanical properties have been analyzed indicating that this material is brittle. Elastic constants show that CaFe2As2 is isotropic just like its parent compound
. The material undergoes a structural phase change at an external pressure of 0.2 GPa from the tetragonal stable phase to orthorhombic at ground state temperature. Further studies could be made on the superconducting properties of CaFe2As2 and especially phase transition temperature Tc and magnetic transition properties.
Acknowledgements
I want to extend my gratitude to the Center for High-Performance Computing (CHPC), Cape Town, the Republic of South Africa, for the provision of HPC cluster resources that was necessary in this computational study.