Abstract

This work has applied density functional theory (DFT) based calculations to investigate the structural and vibrational properties of FeMnP_1−xA_x (A= Si, Se, Sn and In, x = 0.33) within the first-principles pseudopotential technique. The exchange correlation potentials were treated within generalized gradient approximation (GGA), in the Quantum ESPRESSO code. The Perdew, Burke, Ernzerhof (PBE) functional as implemented in Vanderbilt's ultra-soft pseudo potential (USPP) was used for all the calculations. Vibrational properties were calculated using phonopy code with 1 × 1 × 2 supercell of the conventional unit cell. Thermodynamic properties were predicted using the phonon density of states. The dependence of lattice thermal conductivity on temperature was determined using Debye theory. The optimized structural parameters and corresponding graphical values fit within available experimental data and other theoretical reports. There were no imaginary phonon modes in the phonon dispersion curves revealing that these materials are dynamically stable for magnetic refrigeration.

1. Introduction

Magnetic refrigeration, a cooling technology that relies on magnetocaloric effect (MCE) is currently attracting research interest from researchers all over the world. Studies have shown that this technology has several advantages over the conventional refrigeration technique. Gutfleisch et.al ¹ in their work, found out that the cooling efficiency of magnetic refrigeration is about 60% of the theoretical limit which is higher than that of the conventional cooling method estimated to be 45%. In addition, magnetic refrigeration uses a solid state, water-based coolants which are environmentally friendly as opposed to hydro chlorofluorocarbons (HCFC) used by the conventional cooling methods resulting into both the ozone layer depletion and global warming ². It is therefore being considered as a favorable cooling technology that is likely to succeed the traditional vapor compression refrigeration technique. Magnetocaloric effect works only near a materials transition temperature and is experienced when a changing magnetic field causes a reversible change in temperature of a suitable material ^{3, 4}. Magnetocaloric effect MCE is maximum at the material's Curie temperature (Tc) that is, the temperature above which a ferromagnetic material becomes paramagnetic due to the noise generated by atomic vibrations ^{5, 6}.

Brown ⁷, reported the first near room temperature magnetic refrigerator where he used gadolinium as the magnetic refrigerant material. This great discovery encouraged researchers to study properties of other materials with the aim of developing magnetic materials with exceptional temperature changes when adiabatically magnetized around the room temperature. A major breakthrough was reported when Vinh, et al ⁸, discovered giant MCE in (Mn, Fe)₂(P, As) compounds which were based on abundantly available materials. This compound however had a toxic element arsenic which was later replaced by the nontoxic silicon, while maintaining the outstanding magnetocaloric properties ⁹. Tunable working temperature, low hysteresis, corrosion resistance, compositional stability and low material cost are some of the properties that make (Mn, Fe)₂(P, Si)-type compounds attractive and promising for near-room temperature refrigeration and energy conversion applications ¹⁰.

In the last decade, Höglin, et al ¹¹, synthesized and characterized samples with different amounts of Si content in the FeMnP_1-xSi_x (0.00 < x < 1.00) system. Their main focus was on phase analysis, mechanical, structural, magnetic and dynamical properties. They concluded that this hexagonal Fe₂P-type for the range 0.24 < x < 0.50 material is ferromagnetic with Tc tunable from 215 K (x = 0.24) to 392K (x = 0.48). They also found out that FeMnP_1-xSi_x had a weak mechanical stability across phase transition. A good candidate for magnetic refrigeration must be both mechanically and dynamically stable. To improve the mechanical properties of FeMnP_1-xSi_x (x=0.33), several studies are being conducted to replace Si atom with other atoms ¹². However, no work has been reported on the vibrational properties of this material. In this study the Si atom in FeMnP_1-xSi_x (x = 0.33) was substituted with In, Se, and Sn atoms and the structural and vibrational properties investigated.

Hexagonal Fe2P-type magnetocaloric materials have been attracting a lot of research interest lately because of their promising application in magnetic refrigeration. They have proved to be efficient in energy consumption and are environmentally friendly as opposed to the traditional gas compressor-based refrigeration methods. These materials work under repeated magnetic and thermal cycles which results into a large local strains in the polycrystalline samples and so they need to be both structurally and dynamically stable across the phase transition. For most magnetocaloric materials that have so far been studied as possible candidates for magnetic refrigeration, limitations such as high cost, toxicity poor dynamical stability and poor structural stability have been reported. There is therefore a need to conduct extensive investigations to obtain materials with better performance in magnetic refrigeration that may overcome the above stated limitations. This work therefore has analyzed the structural and vibrational properties of FeMnP_1−xA_x (A= Si, Se, Sn and In, x = 0.33) and found out that FeMnP0.66 In_0.33 is a better candidate for magnetic refrigeration than other three compounds.

2. Computational Details

For the proposed magnetocaloric materials, structural relaxation and optimization was done using Broyden, Fletcher, Goldfarb and Shannon (BFGS) approach ¹³ considering ferromagnetic (FM) arrangement to obtain the most energetic stable states. Thereafter, energy-volume calculation was implemented and the data obtained fitted to the third-order Birch-Murnaghan equation of state (EOS) to extract the structural parameters. A realistic phonon calculation on the grid of FeMnP_1−xA_x (A= Si, Se, Sn and In, x = 0.33) was carried out using phonopy code interfaced with quantum espresso ¹⁴. The plane-wave basis projector augmented wave (PAW) method ¹⁵ was employed in the framework of density functional theory (DFT) within the generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE) form ¹⁶ as implemented in the phonopy-QE code. A plane-wave energy cutoff of 90 Ry and an energy convergence criterion of the order of 10^-10 eV were used. A 14 ×14 ×16 k-point sampling mesh was used for the unit cell and the equivalent density mesh was used for the supercells together with a 0.05 Ry smearing width. For the phonon calculations, supercell and finite displacement approaches were used, with 1× 1× 2 supercell of the conventional unit cell and the atomic displacement distance of 0.01 Å. The size of the phonon calculation used is 3N = 27 × q linear-response calculations. Figure 1 shows an optimized schematic representation of the host crystal FeMnP_1-x Si_x (x = 0.33) with all the atomic positions being indicated.

3. Results and Discussion

The FeMn-based magnetocaloric materials of the form FeMnP_1-xA_x (A = Si, Se, Sn and In) are solids described to have a hexagonal structure of space group P62m. The transition metals (Fe and Mn) occupy 3f and 3g sites, Phosphorous occupy the 2c site and (A= Si, Se, Sn and in) occupy 1b site. It has been observed that close to the room temperature, this group of materials, undergoes first order FM to PM transition. FeMnP_0.66Si_0.33 is characterized by primitive lattice vectors a, b, c, whose lengths are 5.811, 5.811, 3.529 Angstrom respectively. The angle γ between the lattice vectors a and b = 120°, angle α between lattice vectors a and c =90°, angle β between lattice vectors b and c =90°. The unit cell contains 4 formula units and 9 atoms.

Magnetocaloric materials function under repeated magnetic and thermal cycles and so need to be structurally stable. One way to optimize the mechanical stability of FeMnP_1−x A_x (A = Si, x = 0.33) is to alloy it with other viable nontoxic elements. The main aim of alloying the atoms at A site is to mimic the bonding between atoms without altering the structure of the compound. Xiaoxia et al ¹⁷ observed that close to room temperature, this group of materials, undergoes first order magneto elastic transition leading to changes in the ratio of the lattice parameters (c/a) without significant changes in the volume.

Bond angle together with bond length are considered to be among the key structural properties of materials. Bond angle accounts for the three dimensional set up of atoms which represents a molecule. It dictates various properties of a substance such as biological activity, polarity, magnetism, color, phase of matter, and reactivity ¹⁸.

Bond length is the average distance between nuclei of 2 bonded atoms in a molecule. It determines the bond strength and bond dissociation energy in that it is inversely proportional to both. This implies that when all other factors are constant, a shorter bond distance results into a stronger bond and a higher dissociation energy ¹⁹. It is evident from Table 1 that the percentage deviations of the present theoretical results as outlined in columns 5 and 6 are between 0.3% and 2.0%. This shows that doping the A site of FeMnP_0.66Si_0.33 with Se, In and Sn does not affect the structure of the compounds significantly.

To obtain a relaxed structure, variable cell relaxation calculation and optimization of the K-points, cutoff energy and lattice parameters was done. Plots of calculated total energies of the proposed compounds as functions of volume in FM phase are presented in Figure 2, Figure 3, Figure 4 and Figure 5. These energies were then fitted to the Birch Murnaghan equation of state to obtain the equilibrium lattice constants and bulk modulus.

From these graphs it is observed that there is a clear resemblance in the energy-volume curves, with similar shape but different minima which again shows that doping the A site does not significantly alter the structure of the compounds.

The pressure derivatives of the bulk modulus for FeMnP_0.66Si_0.33, FeMnP_0.66Se_0.33, FeMnP_0.66Sn_0.33, FeMnP_0.66In_0.33 were calculated at 1.00, 1.00, 1.10 and 2.45, respectively. These results suggests that FeMnP_0.66In_0.33 is the most rigid and less prone to compressive deformation.

Table 2 shows the calculated values of the lattice parameters of FeMnP_1−xA_x (A= Si, Se, Sn and In, x = 0.33) along with the available experimental and theoretical values. It can be seen that for FeMnP_0.66Si_0.33 the percentage deviation of the structural parameters a, c and V from experimental results were 0.892%, 0.486% and 3.054% respectively. This is a general trend inherent to GGA calculations showing that the obtained lattice parameters are in good agreement with experimental data. The percentage deviations for FeMnP_1−xA_x (Se, Sn and In, x = 0.33) were not reported due to unavailability experimental data for these materials.

The displacement of an atom from its equilibrium point causes a rise in the forces on all the atoms in the crystal. A series of phonon frequencies result from the analysis of these forces that are associated with a systematic set of displacements. This provides very useful information in accounting for a number of properties and behaviour of crystalline materials, for example superconductivity, mechanical properties, phase transition and thermal properties ²⁰. At Γ (q=0), the equilibrium point, a crystal is said to be dynamically stable if its potential energy always increases against any combinations of atomic displacements. This is equivalent to the condition that all phonons have real and positive frequencies in the harmonic approximation ²⁰. However, under virtual thermodynamic conditions, imaginary frequency or negative eigenvalues can appear in the band structure. This indicates dynamical instability of the system, implying that the corrective atomic displacements reduce the potential energy in the vicinity of the equilibrium atomic positions ²¹.

Figure 6, Figure 7, Figure 8 and Figure 9 show the phonon dispersion spectra along high symmetry points and phonon densities of states graphs for FeMnP_0.66Si_0.33, FeMnP_0.66Se_0.33, FeMnP_0.66Sn_0.33 and FeMnP_0.66In_0.33 respectively. No imaginary frequencies were observed in dispersion spectra for both the acoustic and optical modes confirming the dynamical stability of these compounds. The phonon dispersion curves for the four compounds in the Figure 6, Figure 7, Figure 8 and Figure 9 show some resemblance and this can only imply that the bonding characteristics for these compounds are similar.

The phonon dispersion curves are represented only by acoustic branches for one atom per unit cell. It is already well established that solids with more than one atom in the smallest unit cell exhibit two types of phonons: acoustic phonons which are the lower mode and optical phonons which constitute the upper mode ²². The difference between acoustic and optical branches arises because of more degrees of freedom of vibrations for atoms in the unit cell. Generally, for N atoms per unit cell there will be 3 acoustic branches (1 longitudinal and 2 transverse) and 3N-3 optical branches that is, N-1 longitudinal and 2N-2 transverse modes ²³. This is exhibited in Figure 2, Figure 3, Figure 4 and Figure 5, whereby, they all show 27 modes of vibration out of which 3 are acoustic and 24 are optical branches. Of the optical branches, 8 are longitudinal and 16 are transverse.

Phonon dispersions are computed along a given line of high symmetry points. Acoustic modes converge at the gamma high symmetry point. The frequency of acoustic phonons tends to zero and exhibit a linear relationship with phonon wave-vector for long wavelengths. Acoustic modes vibrate at a slower frequency with two adjacent atoms vibrating together in the same direction. Optical phonons have a non-zero frequency at the Brillouin zone center and show no dispersion near that long wavelength limit. Optical modes of vibration have a higher frequency compared to acoustic modes and two neighboring atoms vibrate in a direction opposite to each other. This occurs if the lattice basis consists of two or more atoms. They are called optical because in ionic fluctuations in displacement they create an electrical polarization that couples to the electromagnetic field ²⁴. A phonon bandgap between the optical and acoustic modes affects the phonon scattering process and thus lattice thermal conductivity. Gaps indicate a higher rate of acoustic – optical phonon - phonon scattering leading to high thermal conductivity. No band gap was observed in the four compounds studied in this work hence their lattice thermal conductivity was relatively low.

According to study done by Mackinnon et al ²⁵, presence of phonon anomaly (pseudo gap) is essentially used to measure of superconductivity of structures. From Figure 6, Figure 7, Figure 8 and Figure 9, it is clear that the upper and lower phonon branches are well separated which shows that the structures are superconductors. However, the structure in Figure 9 is a more superior super conductor due to a wider gap.

At non-zero temperature, the energy of ions due to their vibration affects a solid’s thermal properties such as the heat capacity, entropy and its free energy. It is therefore necessary to investigate these properties for various applications, and in this case for materials to be used in near-room-temperature refrigeration. Using the thermodynamic relations, the heat capacity at constant volume (C_v), Helmholtz free energy (F) and entropy (S), can be computed as functions of temperature as shown in equations 1, 2 and 3 ²⁶.

(1)

(2)

(3)

where ω; and are the phonon frequency, the temperature, the Boltzmann constant, and the reduced Planck constant, respectively.

The temperature dependence graphs of free energy, entropy, and specific heat capacity at constant volume of FeMnP_0.66Si_0.33, FeMnP_0.66Se_0.33, FeMnP_0.66Sn_0.33 and FeMnP_0.66 In_0.33 are presented in Figure 10, Figure 11, Figure 12, Figure 13 respectively. The Debye model estimates the phonon contribution to specific heat capacity of a solid. It predicts that at low-temperatures (temperatures below Debye temperature), the specific heat of crystals is proportional to the cube of temperature. This model also recovers the Dulong- petit law at high temperatures which states that above the Debye temperature, the gram-atomic specific heat capacity of solids is constant ²⁷. From the Figure 10, Figure 11, Figure 12 and Figure 13, it can be seen that for the four compounds, the heat capacity curves show sharp increase in the low temperature region, and at high temperatures, C_v approaches a constant value (Dulong-Petit limit). This result is in good agreement with the Debye model of specific heat capacity both for the low and high temperature range.

The classical equipartition law shows that that C_v = 3R at high temperatures, where C_v is the specific heat capacity at constant volume and R is the universal gas constant with the value of 8.314 J K^-1 mol^{-1 28}. In this study, there are 9 atoms in one formula, which means the theoretical value of C_v should be 224.478 J K^-1 mol^-1. From all the four graphs it is observed that the calculated C_v becomes a constant of 224 JK^-1mol^-1 at high temperatures. This gives a negligible deviation of 0.213% implying that the calculated and theoretical C_v show very good agreement. According to Fourier’s law of heat conduction, the thermal conductivity of a solid is inversely proportional to its temperature. A high Debye temperature therefore implies that a crystals thermal conductivity is low. This study established that the calculated Debye temperatures were 505.262 K, 476.259 K, 354.424 K and 291.571 K for FeMnP_0.66Si_0.33, FeMnP_0.66Se_0.33, FeMnP_0.66Sn_0.33 and FeMnP_0.66 In_0.33 respectively. From these results it is clear that _FeMnP0.66 In_0.33 is a better thermal conductor than the other compounds.

Entropy is another important thermal property that is commonly used in modeling the thermodynamic behavior of solids. Figure 10, Figure 11, Figure 12 and Figure 13 also show the calculated entropy as a function of temperature for the compounds FeMnP_0.66Si_0.33, FeMnP_0.66Se_0.33, FeMnP_0.66 Sn_0.33 and FeMnP_0.66In_0.33 respectively. It is an expected behavior that the vibrational entropy curves increase as temperature increases due to the vibrational frequencies increase with temperature ²⁹. This is seen from the shape of the entropy graphs presented in the Figure 10, Figure 11, Figure 12 and Figure 13. High entropy alloys (HEAs) are gaining significant attention in material science and engineering. This is attributed to their desirable properties such as better strength to weight ratio, high tensile strength, high degree of fracture resistance as well as high resistance to corrosion and oxidation. From this work the entropies in J/K/Mol recorded for FeMnP_0.66Si_0.33, FeMnP_0.66Se_0.33, FeMnP_0.66 Sn_0.33 and FeMnP_0.66In_0.33 are approximately 470, 500,510 and 530 respectively. This makes FeMnP0.66In0.33 the best candidate for magnetic refrigeration owing to its high entropy value.

Helmholtz free energy is a quantity of concern to physicists because it defines the amount of energy available to do work. Helmholtz free energy decreases as temperature increases and this is confirmed by the formula F = U - TS where F is the Helmholtz free energy, U is the internal energy, T is the absolute temperature and S is the entropy. Following this equation, it is expected that if both T and S increase then F must decrease. The maximum amount of work done by a system is directly proportional to the decrease in the free energy. From Figure 10, Figure 11, Figure 12 and Figure 13, FeMnP_0.66In_0.33 has the highest decrease in Helmholtz free energy hence the highest amount of energy available to do work.

4. Conclusion

First principles calculation based on density functional theory was used to investigate the effect of alloying the A site of FeMnP_1−x A_x (A = Si, x = 0.33) with Se, Sn and In respectively on the structural and vibrational properties of the compounds. The structural properties including bond length, lattice parameters, bulk modulus and first-order pressure derivative of bulk modulus were calculated. The structural properties for FeMnP_0.66Si_0.33 were in agreement with available theoretical and experimental results with a deviation of around 2%. The comparison of structural parameters of FeMnP_1−x A_x (A = Se, Sn, In; x = 0.33) with experimental results was not done since there is no previous data available. However, the percentage deviation of the bond lengths of these compounds from those of FeMnP_0.66Si_0.33 confirmed that alloying did not alter the structure of the material. The lack of negative frequencies on phonon dispersion curves of these compounds revealed that they are dynamically stable. The temperature dependence graphs of the thermodynamic properties of FeMnP_0.66Si_0.33, FeMnP_0.66Se_0.33, FeMnP_0.66Sn_0.33 and FeMnP_0.66 In_0.33 were calculated using quasiharmonic approximation. The calculated values of specific heat capacity, entropy and free energy are in excellent agreement with theoretical predictions. This study shows that alloying with Indium had better results of structural, thermodynamic and dynamical properties of the magnetocaloric material. This work can therefore be used as reference to motivate further experimental research on vibrational properties of FeMnP_0.66 In_0.33 for use in magnetic refrigeration.

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 all my Labmates: Otieno Vincent and Chirchir for giving me company during the computational time. 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