Abstract

Hexagonal Fe₂P-type magnetocaloric materials have been attracting a lot of research interest lately as a result of their promising application in magnetic refrigeration. These materials work under repeated magnetic and thermal cycles which results to large local strains in the polycrystalline samples and so they need to be mechanically stable across the phase transition. Hence, there is a need to conduct extensive investigations in order to obtain materials which may have better performance in magnetic refrigeration. In this study the elastic properties of FeMnP_1−x A_x (A= Si, Se, Sn, In, x = 0.33) were investigated using first principles density functional theory within the generalized gradient approximations as expressed in Quantum Espresso code. The work conclusively shows that FeMnP_0.66 In_0.33 has the highest Poisson’s ratio, Pugh’s and machinability index hence most ductile of the selected materials. Moreover, it had the highest anisotropic ratio further proving that of the four compounds, it is the most suitable for sustainable operation as a magnetocaloric refrigerant.

1. Introduction

Countries all over the world are currently directing a major part of their resources towards search for clean energy that may solve the problems of energy crisis and global warming. Several meetings have taken place all over the world whose main agenda is to find clean energy that is efficient, reliable, affordable, renewable and sustainable ^{1, 2, 3}. Home and office appliances such as air conditioners and refrigerators are some of the heavy consumers of energy. The running of these appliances depends entirely on hydrofluorocarbons, chlorofluorocarbons and some other chemicals which are used to complete vapor – compression refrigeration cycle. Consequently, the reaction of these gases with oxygen produces CO_2, a greenhouse gas that creates a harmful effect on human livelihood and ecosystem ^{4, 5}. The need to replace the present-day refrigeration technologies with environmentally friendly and more energy efficient ones has led to major researches being done all over the world ⁶.

This work has therefore investigated the elastic properties of fe₂P-type materials for near-room-temperature refrigeration in order to identify the one that is most suitable as a candidate for modelling a magnetic refrigerant that is affordable, environmentally friendly and efficient.

The study of elastic properties of solids helps researchers to analyze some of their vital properties such as anisotropy, ductility and brittleness. The Elastic constants are used to show how dynamic and mechanical properties are connected in regard to the type of forces present in the solids. Much emphasis is put on the stiffness and stability of the material. The elastic constant also plays a major role in predicting the mechanical nature in solid-state physics. A Hexagonal crystal has five elastic constants which are independent, these are given as C₁₁, C₁₂, C₁₃, C₃₃ and C₄₄. These constants are obtained by fixing the full energies of a strained crystal to a fourth-order polynomial strain.

Elastic constants are in many cases used to measure the ability of a material to resist deformation from a stress that is externally applied. The study of elastic properties for the polycrystalline magnetocaloric materials is very important since they are related to the structural stability of the system ^{7, 8}. At the temperature T_C, the elastic and magnetic properties transit simultaneously while the material remains stable by exhibiting minimal change in crystal parameters and greater elastic constants ^{9, 10}. It is therefore necessary to have a deeper and clearer understanding of how the structural stability and elastic constants are related and also how some of the materials used as dopants positively affects these properties.

The bulk modulus B, which is usually used to measure the ability to resist deformation upon the application of pressure, also measures how the material resists to change of volume upon subjecting it to change of pressure. A greater value of bulk moduli results into a greater capacity to resist deformation ^{7, 11, 12}. Shear modulus G, measures the resistance to shear deformation on shear pressure.

Poisson’s ratio which is a measure of stability of a crystal against shear ranges between -1 to 0.5. The larger the Poisson’s ratio, the better the plasticity ¹³. The Young’s modulus, which is given as a ratio of stress to strain, measures the stiffness of solid materials. The larger the value, the stiffer the material ¹⁴.

The prediction of how the brittle and ductile of materials behaves is done by considering the ratio of shear modulus to bulk modulus (G/B) ¹⁵. A larger G/B value shows brittleness while a lower G/B value indicates ductility. The separation of ductility from brittleness is done by a critical value given as 0.57. The elastic constants C₁₁ to C₄₄ can also be used to define ductility or brittleness of a crystal. A positive value shows that the polycrystalline phases are ductile, otherwise it is brittle ¹⁶.

Being that FeMnP_1-x A_x systems have a hexagonal crystal structure, the full elastic-constant matrix is therefore constructed using five independent elastic constants, that is, C₁₁, C₁₂, C₁₃, C₃₃ and C₄₄. Nye ¹³, noted that the corresponding stability conditions for the hexagonal system are given by C₁₁ > 0; C₁₁ - C₁₂ > 0; C₄₄ > 0; (C₁₁+C₁₂) C₃₃-2C₁₃² > 0. The shear moduli G and bulk moduli B are given by equations 1 and 2 ¹⁷.

(1)

(2)

The shear moduli and bulk moduli are also approximated by Voigt-Reuss-Hill (VRH) ¹⁷, where the Voigt bounds of B and G are given by equations 3 and 4

(3)

(4)

And the Reuss bounds are given as

(5)

(6)

Using VGH, B and G are expressed as

(7)

(8)

where the subscripts V and R are the Voigt and the Reuss forms respectively.

The Young’s moduli E, Poisson’s ratio ʋ and anisotropic coefficient A are obtained according to equations 9, 10 and 11 respectively ^{17, 18}

(9)

(10)

(11)

Figure 1 below is a schematic representation of the host crystal FeMnP_1-x Si_x (x = 0.33) showing all the atomic positions.

2. Methodology

This work applied First principle Density function Theory and all calculations were done using the Quantum-opEn-Source Package for Research in Electronics, Simulations and Optimization (Quantum Espresso) computer code which is a multi-purpose software for ab initio calculations of periodic and disordered condensed matter systems. The electron-ion potential was described by means of Vanderbilt's ultra-soft pseudo potentials (USPP), this ensured rapid convergence in the calculated total energy of the system ¹⁹. To treat the exchange and correlation energies, the Perdew, Burke, Ernzerhof (PBE) form of the Generalized Gradient Approximation (GGA) pseudo potential was used ²⁰. Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm was used to optimize the atomic positions. Calculations of Poisson ratios, Young, Bulk and Shear Modulus (all in Voigt-Reuss-Hill approximations), Elastic constants, average Debye sound velocity, solid density and Debye temperature were done using Thermo_PW code interfaced with Quantum Espresso ²¹. The wave functions (W.F) Kinetic Energy cut-off points were all set at 90 Ry and ecutrho set at 1080 Ry. A smearing width of 0.05 Ry and convergence threshold of 1.0E-6 Ry with a mixing beta of 0.7 were applied. The K points separation of 3 x 3 x 5 was used for all the compounds. The systems were visualized using XcrysDen in Linux operating system. The graphs were plotted using xmgrace and GNU plot.

3. Results and Discussion

The elastic stability of a hexagonal crystal structure is determined using the elastic potential free energy equation

(12)

where . When, we have equilibrium state and when , there is a state of mechanical deformation. This therefore requires that the principal minors of matrix given by equation (13) below must all be positive -definite.

(13)

where.

The matrix in equation (13) is symmetrical about its leading diagonal, that is, ²².

Crystals of hexagonal class have 5 independent elastic constants but due to added relation in the matrix we get the element X which now defines a tetragonal crystal. The calculated eigenvalues of the stiffness matrix from equation (13) leads to the following four basic conditions called Born stability conditions for elastic stability of a hexagonal crystal: C₁₁ > 0; C₁₁ > C₁₂; C₄₄ > 0; (C₁₁+C₁₂) C₃₃ > 2C₁₃^{2 23}.

From Table 1 below, it can be noted that the values of elastic constants calculated satisfy the four conditions stated above. This therefore shows that the four structures studied in this work are elastically stable. Stability ratios is shown in the last two columns of Table 1 from which it can be concluded that all the four components are elastically stable.

A material is said to be anisotropic if it displays properties with different values when measurements are taken along axes in different directions. It can be easily seen in compounds or single crystals of solid elements where atoms, ions or molecules are arranged in regular lattices. The elastic anisotropy, denoted by A, is used to determine the how elastic properties of solids behave towards the direction of the stress ²⁴.

Hexagonal crystals have three shear-type anisotropy ratios defined by equations (14), (15) and (16) given below:

(14)

(15)

And

(16)

The shear-type anisotropy ratios derived from values in Table 1 using the three equations are given in the Table 2 below

When the elastic anisotropy ratio A=1, a crystal is isotropic. Otherwise values of A which are less than or greater than unity indicates anisotropy. The calculated shear anisotropic factors for the studied compounds are listed in Table 2. It can be easily observed that all the materials under consideration are elastically anisotropic. Further, when C₃₃ > C₁₁, the compounds are more incompressible along the C-direction than along the A-direction. From Table 1, it is noted that only FeMnP_0.66In_0.33 satisfies this condition.

Calculations of Poisson ratios ‘ν’, Young modulus ‘E’, Bulk modulus ‘B’ and Shear Modulus ‘G’ under Voigt and Reuss approximations as well as Voigt-Reuss-Hill average of the two approximations are presented in Table 3.

Bulk modulus also known as incompressibility, is a measure of how a substance withstands changes in volume when compressed from all sides. The capacity of a material to resist deformation is directly proportional to its bulk modulus. That is, the larger the bulk modulus, the higher the ability to resist deformation. From Figure 2 and Table 3, it is observed that FeMnP_0.66Si_0.33 has the highest value of bulk modulus while FeMnP_0.66 In_0.33 has the lowest value. The results clearly demonstrate that, of the four materials investigated in this work, FeMnP_0.66Si_0.33 has the strongest capacity to resist fracture.

This a numerical value that measures the ability of materials to resist transverse deformation. Mathematically, it is the ratio of shear stress to shear strain. A larger value of shear modulus indicates that the solid is highly rigid and may require greater force to be deformed. A smaller value of shear modulus shows that the material is either soft or flexible and little force is required to deform it. Note that fluids have zero shear modulus and therefore no force is required to deform them ²⁵.

The calculated results of shear modulus are presented in both Table 3 and Figure 3. Of the 4 compounds studied, FeMnP_0.66 Si_0.33 has the largest value of shear modulus hence, the strongest capacity to resist plastic deformation.

This is a ratio that measures the tensile elasticity of a material. That is, a measure of the ability of a material to withstand variations in length when subjected to compression or lengthwise tension. It is obtained when longitudinal stress is divided by strain, which indicates how stiff a material is. The larger the ratio, the stiffer the material ²⁶. From Figure 4, it is observed that the E values decrease in the following order: FeMnP_0.66 Si_0.33 > FeMnP_0.66 Se_0.33 > FeMnP_0.66 Sn_0.33 > FeMnP_0.66 In_0.33. It can be deduced that FeMnP_0.66 Si_0.33 is the stiffest of all the four materials studied.

Pugh's ratio is a measure of how ductile or brittle a material is. The critical value for this ratio is given as 1.75 which separates the two conditions. A value of less than 1.75 represents brittleness and a value above 1.75 indicates ductility ²⁷. Considering this, the results in Figure 5 shows that the Pugh’s ratios for all the four materials are above the critical value. This therefore implies that they are all ductile though FeMnP_0.66 In_0.33 is the most ductile. Machinability, which is an important property depending on both the elastic constants and bulk modulus indicates how easily a material can be cut. It is the ratio of to C₄₄ ^{28, 29}. The machinability of the four compounds studied, that is, FeMnP_0.66 Si_0.33, FeMnP_0.66 Se_0.33, FeMnP_0.66 In0_.33 and FeMnP_0.66 Sn_0.33 are 1.539, 1.836, 1.933 and 2.042 respectively. The results therefore show that both FeMnP_0.66 In0_.33 and FeMnP_0.66 Sn_0.33 have good machinability compared to the other two.

This is the most important factor to be considered when selecting any material for use. It is simply given by calculating the ratio of the lateral strain to that of the longitudinal strain in the direction of the stretching force. The ratio usually varies between 0 and 0.5. A material that is perfectly incompressible (incompressible material is one that does not change volume as it is deforms when subjected to stress) has a Poisson ratio of 0.5. From Figure 6, it can be noted that all the four materials are ductile though the Poisson’s ration for FeMnP_0.66 In_0.33 is higher than the rest. This therefore implies that is FeMnP_0.66 In_0.33 is most suitable for use.

This is a key physical quantity of solids, being derived from the atomic thermal vibration of solids. It is not only characterized by the force of binding between atoms but also reflects the level of dynamic distortion of crystal lattice ³⁰. It is closely linked to a number of physical properties of solids like expansion coefficient, elasticity, lattice stability, hardness, specific heat and melting point. It is used to differentiate the regions of solids with low temperature from high temperatures. The elastic Debye temperature is directly proportional to the average elastic velocity as shown in equation (12) below

(12)

where

(13)

and

(14)

From Equations (12), (13) and (14), is the Planck’s constant, is the Boltzmann’s constant, is the atomic volume, is the mean acoustic wave velocity, is transverse sound velocity and is the longitudinal sound velocity.

From Table 4, the Debye temperatures for the following materials FeMnP_0.66 Si_0.33 FeMnP_0.66 Se_0.33, FeMnP_0.66 In_0.33 and FeMnP_0.66 Sn_0.33 were found to be 505.262K, 476.259K, 291.571K and 354.424K respectively. This study was unable to compare these results with any other since none was existing. Nonetheless, it can be easily seen that FeMnP_0.66 Si_0.33 has the highest Debye temperature implying that it the highest interatomic bonding, the highest melting point, greatest hardness and highest thermal conductivity than the rest.

The results found in Table 5 shows that the average total magnetization of the four compounds studied is units. This is in agreement with the work by ³¹. These values show that the polarization of the conduction cannot be easily seen while the magnetization measurements give total magnetization.

4. Conclusion

In this work, first principles DFT using Thermo_PW interfaced with Quantum Espresso Code was used to investigate the elastic properties and the mechanical stability of FeMnP_1−x A_x (A= Si, Se, Sn, In, x = 0.33) as magnetocaloric materials. These materials undergo repeated magnetization and demagnetization cycles and so must exhibit mechanical stability at temperatures around their phase transition. The elastic constants, bulk modulus, shear modulus, young’s modulus, Pugh’s ratio, machinability index, elastic anisotropy factor and Debye temperature, which are essential for determining the long-term applicability of these materials in magnetic refrigeration were obtained. The calculated values for elastic constants for all the materials were found to be in line with the Born and Huang conditions for mechanical stability. In addition, the values of the elastic constants were relatively high indicating that the materials studied had good resistance against deformation. This study conclusively shows that FeMnP_0.66 In_0.33 has the highest Poisson’s ratio, Pugh’s and machinability index hence most ductile of the selected materials. Moreover, it had the highest anisotropic ratio A₂ =1.605 further proving that of the 4 compounds, it was the most suitable for sustainable operation as a magnetocaloric refrigerant.

Acknowledgements

On behalf of the authors, I would like to thank Dr. Onyango Lawrence for taking his time to proofread the document and giving me more support during the time I was writing the manuscript. More thanks also goes to Egerton University for granting me scholarship to pursue my PhD degree. Last but not least, I would like to thank the Editorial team, Reviewers and Administrators of SCIEP for their valuable contributions which have greatly helped in improving the quality of the manuscript.

References