Thermoelectric properties, phonon, and mechanical stability of new half-metallic quaternary Heusler alloys: FeRhCrZ (Z = Si and Ge)

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quaternary Heusler alloys: FeRhCrZ (Z = Si and Ge)

Cite as: J. Appl. Phys. 127, 165102 (2020); https://doi.org/10.1063/1.5139072

Submitted: 18 November 2019 . Accepted: 05 April 2020 . Published Online: 22 April 2020 Shakeel Ahmad Khandy , and Jeng-Da Chai

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Thermoelectric properties, phonon, and mechanical stability of new half-metallic

quaternary Heusler alloys: FeRhCrZ (Z = Si and Ge)

Cite as: J. Appl. Phys. 127, 165102 (2020);doi: 10.1063/1.5139072

View Online Export Citation CrossMark

Submitted: 18 November 2019 · Accepted: 5 April 2020 · Published Online: 22 April 2020

Shakeel Ahmad Khandy¹ and Jeng-Da Chai^1,2,a) AFFILIATIONS

1Department of Physics, National Taiwan University, Taipei 10617, Taiwan

2Center for Theoretical Physics and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan

a)Author to whom correspondence should be addressed:jdchai@phys.ntu.edu.tw

ABSTRACT

Computer simulations within the framework of density functional theory are performed to study the electronic, dynamic, elastic, magnetic, and thermoelectric properties of a newly synthesized FeRhCrGe alloy and a theoretically predicted FeRhCrSi alloy. From the electronic structure simulations, both FeRhCrZ (Z = Si and Ge) alloys at their equilibrium lattice constants exhibit half-metallic ferromagnetism, which is established from the total magnetic moment of 3.00μB, and that the spin moment of FeRhCrGe is close to the experimental value (2.90 μB).

Their strength and stability with respect to external pressures are determined by simulated elastic constants. The Debye temperatures of FeRhCrSi and FeRhCrGe alloys are predicted to be 438 K and 640 K, respectively, based on elastic and thermal studies. The large power factors (PFs) of the two investigated alloys are in contour with those of the previously reported Heusler compounds. Besides, the conservative estimate of relaxation time speculated from the experimental conductivity value is 0.5 × 10⁻¹⁵s. The room temperature PF values of FeRhCrSi and FeRhCrGe compounds are 2.3μW/cm K²and 0.83μW/m K², respectively. Present investigations certainly allow the narrow bandgap, spin polarization, and high PF values to be looked upon for suitable applications in thermoelectrics and spintronics.

Published under license by AIP Publishing.https://doi.org/10.1063/1.5139072

INTRODUCTION

Exploration of new materials, particularly the Heusler alloys and their offshoots accompanied by tunable properties, has attained significant attention from the material scientists worldwide.

Modern technologies ranging from superconductivity to energy conversion and data storage to contactless sensing are typically boosted by Heusler alloys. This class of materials emerged as the ground-breaking area of research due to multi-dimensional properties like compatible thin film interfaces, large Curie temperatures, magnetoresistance, etc.^1,2 The scientific community accomplished sufficient research for the prediction of new materials with some predefined properties such as half-metallicity, high spin polariza-

applications in spintronics for developing basic computer units, data storage devices, magnetic sensors, high-tech electronic devices, spin valves, and tunnel junctions. The phenomenon of half- metallicity in Heusler alloys was first predicted by Groot et al. in 1983.^3–5 For several years, great effort was put forth to study the HMF character originating from the d-orbitals of transition elements in such materials. Until today, five kinds of HMFs have been anticipated: the oxide compounds such as CrO2,⁶TiO2, and VO2,⁷some ternary compounds (specifically, spinels with the general formula AB₂O₄such as Fe₃O₄and LiMn₂O₄^8–10), single or double perovskites (e.g., BaPaO₃ and Sr₂SnMnO₆^11,12), and dilute magnetic semiconductors [DMSs, e.g., Cu-doped ZnO,¹³ Cr-doped CdZ (Z = S, Se, and Te),¹⁴Mn-doped GaN,¹⁵etc.]. In addition, the Heusler materials

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Among Heuslers, the equiatomic quaternary Heusler (EQH) alloys are very recent and only few materials of this class are studied or predicted today. From first-principles simulations, various materials are being studied or discovered continuously for this purpose,^19–23and their stability is considered via Monte Carlo simulations and other methods.^24,25 Quaternary half-metallic or ferromagnetic Heuslers, such as YCoTiZ (Z = Si and Ge),²⁶ CoFeCrZ (Z = Al, Ga, and Ge),²⁷ FeCrRuSi,²⁸ CoMnCrZ (Z = Al, As, Si, and Ge),²⁹ ZrFeVZ (Z = Al, Ga, and In),³⁰ CoFeMnZ (Z = Al, Ga, Si, and Ge),³¹and many others, have been discovered experimentally or predicted theoretically. This research work is anticipated to investigate the structural, electronic, elastic, thermoelectric, and magnetic properties of the recently synthesized FeRhCrGe alloy with the help of density functional theory (DFT) calculations. This material has been experimentally reported to possess the Curie temperature of 550 K.³² Another material, FeCrRhSi, has been reported to be a ferromagnetic half-metal theoretically,³³ but there is no data available regarding its thermodynamic, mechanical, and transport properties. Therefore, in a very first attempt, we investigate and compare the detailed ground-state properties of FeRhCrZ (Z = Si and Ge) alloys with keen interest on the electronic structure, mechanical/dynamical stability, and thermoelectric properties. The rest of this paper is arranged as Computational Methodology, Results and Discussion, and Conclusion. This work inspires the consideration of the d-state transition element based ferromagnetic EQH alloys for the applica- tion in future spintronic devices.

COMPUTATIONAL METHODOLOGY

WIEN2k simulation code³⁴ is used to accomplish the spin- polarized density functional calculations on FeRhCrZ EQH alloys.

Full-potential linearized augmented plane wave (FP-LAPW) method³⁵[with the muffin-tin radii: 2.5 (Fe), 2.4 (Rh), 2.3 (Cr), 1.5 (Si), and 1.9 (Ge)] is employed to treat the core and valence electrons. For the exchange-correlation energy functional, we adopt the Perdew–Burke–Ernzerhof (PBE) functional³⁶ and the Tran–Blaha modified Becke–Johnson (TB-mBJ) potential.³⁷For strongly localized d-orbital systems, it is well known that the PBE functional often underestimates the size of the bandgap. Therefore, we also calculate the electronic structure with the TB-mBJ potential.

However, owing to the lack of an energy functional associated with the TB-mBJ potential, properties related to the total energies (e.g., relaxed geometries) of systems cannot be directly obtained from the TB-mBJ potential and, hence, are obtained from the PBE functional. Later, the on-site Hubbard correction (PBE + U with U_eff= 1.36 eV and 0.68 eV for Fe and Cr, respectively)³⁸and spin– orbit coupling³⁹are also employed to calculate the electronic band structures of these alloys. The convergence criterion for self- consistent calculations is set at a value of less than 0.1 mRy for energy. The cutoff energy is chosen as−6.0 Ry for the separation of valence and core states. A dense mesh of 10 × 10 × 10 k-points is used for the Brillouin-zone integration. For elastic properties, the cubic elastic code⁴⁰ is utilized with uniform hydrostatic pressure applied in all directions. Additionally, the thermodynamic amounts of melting temperature (Tm) and Debye temperature (θD) from

elastic constants are tallied by means of the following equations:⁴¹ Tm(K)¼ [553(K) þ (5:911)C11GPa]+ 300 K, (1)

θD¼ h k

3n 4π

NAρ M

¹₃

Vm, (2)

Vm¼ 1 3

¹₃ 2 V_s³þ 1

V_l³

¹₃

, (3)

Vs¼ ffiffiffiffi G ρ s

and Vl¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3Bþ 4G

3ρ s

: (4)

Here, the symbols have their usual meanings and vmis the average sound velocity in terms of compressional (vl) and shear (vs) sound velocities.⁴²Phonon spectra are calculated by the pseudopotential- based Quantum Espresso package⁴³ within the framework of PBE.³⁶The cutoff for the kinetic energy is fixed at 50 Ry for the plane-wave expansion of the electronic wave functions, keeping the charge-density cutoff at 300 Ry and the Marzari–Vanderbilt cold smearing at 0.001 Ry.

RESULTS AND DISCUSSION Structural properties

The XX⁰YZ type quaternary Heuslers are reported to have three possible configurations, viz., type-I (with X at 4c, X⁰at 4d, Y at 4b, and Z at 4a), type-II (with X at 4b, X⁰at 4d, Y at 4c, and Z at 4a), and type-III (with X at 4c, X⁰at 4b, Y at 4d, and Z at 4a).⁴⁴ The detailed structures with corresponding lattice sites are shown in Fig. S1 of thesupplementary material. Among them, the ground- state structure is determined by standardized energy minimization techniques. The experimental lattice constant (5.90 Å for the Fe– Ge alloy) and theoretical lattice constant (5.80 Å for the Fe–Si alloy) are set in the calculations to establish the total energy vs volume for all the three configurations. The crystal structure opti- mization performed through the variation of total energy with volume establishes the type-I configuration to be the ground-state structure for both these alloys (see Fig. S2 in the supplementary material). This can also be confirmed from the magnitude of total energy (E₀) of both these alloys as mentioned inTable I. In addition to equilibrium lattice constant and ground-state energy, the calculated values of Bulk modulus and its derivative for all the three configurations are listed inTable I. Since the Fe–Ge alloy is synthesized experimentally, its stability is definite. From the forma- tion and cohesive energy data, the Fe–Si alloy is presumed to be stable by Feng et al.³³To further guarantee the stability of the Fe– Si alloy, we determine the dynamic stability from the phonon dispersion curve and phonon density of states as displayed inFig. 1.

The total of 12 phonon branches with no negative frequencies results from the four atoms of the FeRhCrSi unit cell. Among them, three acoustic branches comprise of two transverse (TA) and one longitudinal (LA) branches, whereas the nine optical branches comprise of three longitudinal optical (LO) and six transverse

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optical (TO) branches. The optical phonons of FeRhCrSi are not coupled, creating a gap between the optical and acoustic phonon modes. The higher atomic masses of the Rh atom produce the major amplitude contribution from 0–170 cm⁻¹, Fe from 170– 200 cm⁻¹, Cr from 260–300 cm⁻¹, and Si from 310–450 cm⁻¹. It is noteworthy to mention that the lattice constant of the Fe–Ge alloy calculated by PBE is equal to the reported experimental value, and hence from onward, these optimized values in type-I configuration are further used to calculate the band structure and elastic, magnetic, transport, and thermodynamic properties of these alloys.

Electronic properties and magnetism

InFigs. 2 and3, the spin-resolved electronic band structures obtained with the PBE functional and TB-mBJ potential have been put forward. The localization of d-bands (green color) near the Fermi level in the spin-up phase simply presents a metal-like picture for both these alloys. On the other hand, the spin-down band structures obtained with PBE display an energy gap of 0.8 eV for Fe–Si and an energy gap of 0.5 eV for Fe–Ge. Feng et al.³³predicted the FeRhCrSi alloy as a half-metal with a semiconducting band structure in the spin-up channel rather than in the spin-down

channel, which contradicts with our PBE results. To overcome the issue of underestimation by PBE, we employed the more advanced TB-mBJ potential. The use of TB-mBJ potential clears up all ambi- guities and displays a bandgap (0.9 eV for Fe–Si and 0.6 eV for Fe–

Ge) in the spin-down state. At the same time, the metallic character in the spin-up channel is retained, where the d-band distribution is sufficiently large in magnitude. For the Fe–Ge system, this also contradicts with the spin semi-metallic (SSM) argument claimed by previous investigations.³²A material can specifically be called as SSM, when a semi-metallic band structure is observed in the spin-up channel, provided that the bandgap is strictly present in the spin-down channel.^45,46This can be further argued from the similar results of EQH alloys like CoFeCrGe and CoMnCrAl reported by the same group,⁴⁶where this behavior in the spin-up channel is claimed as metallic only with densities of states (DOS)∼ 5.0 states/eV f.u. The green-colored localized bands in Figs. 2 and 3 represent the overall d-state contributions from Fe/

Rh/Cr elements in the whole FeRhCrZ molecule and are particularly localized at the Fermi level in the spin-up case only. Yet, a little contribution from Si/Ge-p states cannot be neglected. The indirect spin-down gap calculated by TB-mBJ in both the cases is observed between the Г-point of BZ in the valence band and TABLE I. DFT simulated lattice parameters of FeRhCrZ alloys in possible configurations within F-43m space group.

Parameter Y-I Y-II Y-III Expt. Theory

FeRhCrSi

Lattice constant, a_o(Å) 5. 80 5.83 5.911 … 5.82^a

Bulk modulus, B (GPa) 236.88 209.09 300.71 … …

Derivative of B, B⁰ 5.95 4.03 5.00 … …

Total energy, E0(eV) −201 337.68 −201 337.07 −201 336.01 … …

FeRhCrGe

Lattice constant, ao(Å) 5.90 5.91 5.91 5.90^b 5.85^b

Bulk modulus, B (GPa) 212.13 219.99 298.20 … …

Derivative of B, B⁰ 6.10 8.78 5.00 … …

Total energy, E₀(eV) −250 563.95 −250 562.76 −250 561.27 … …

FIG. 1. Dynamical stability of FeRhCrSi compound governed by (a) phonon band dispersion and (b) partial phonon DOS of individual atoms.

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X-point of BZ in the conduction band. Thus, the present class can be classified as half-metallic EQH materials.

With the help of densities of states (DOS) shown in Figs. 4 and5, we can argue that our results are more reliable and comparable to experimental data. These plots clearly demonstrate that the spin-up states in both Fe–Si and Fe–Ge systems are metallic due to the more significant occupation and amount of DOS at the Fermi level. For both the FeRhCrZ materials,Figs. 4(a)and4(b)show the magnitude of the total DOS obtained with PBE at the Fermi level up to∼4.00 states/eV f.u. in the spin-up channel, which is fairly in agreement with the experimental data of 5.05 states/eV f.u. in the case of Fe–Ge system.³²At the same time, TB-mBJ [seeFigs. 5(a) and5(b)] reduces the magnitude of total DOS up to 1.05 eV for Fe–Ge, which can still be classified as a metal and 3.60 eV/f.u. for Fe–Si being of the same nature. This can be further simplified by the projected densities of states (pDOS) contributed by individual atoms. Both the PBE and TB-mBJ calculations clearly indicate that the doubly degenerate e_g(d_z², d_x²_-y²) and triply degenerate t_2g(d_xy, d_yz, d_xz) states of all the three transition-metal atoms are active at the Fermi level in the spin-up channel. The maximum contribution comes from Cr peaks, which is responsible for its large magnetic moment in both these alloys. In the present case, the origin of spin- down energy gap can be linked to the Slater Pauling rule (Zt-24);

the details of which can be understood from Refs.47and48. The materials under study have 27 valence electrons each, among which 12 pairs of spin-down states are fully occupied in the spin-down channel and the remaining three electrons are partially filled in the antibonding states. Here, the spin-down d-states (egand t2g) of the

transition-metal (Fe/Rh/Cr) atoms can be viewed in reference to the possible d–d bandgap mechanism. For simplicity, Fe–Rh hybridization can be taken into consideration first and then the Fe–Rh hybrid orbitals intermix with Cr-orbitals and later the Z atomic orbitals add sequentially.^49,50 The individual t2g and eg

states (from pDOS) of these transition-metal atoms are shown in Figs. S3 and S4 in the supplementary material, where we can see the octahedral splitting in both the PBE and TB-mBJ calculations.

The conduction bands of the spin-down channel from the PBE calculations reflect the octahedral symmetry where the much lower t2g

states of Fe/Rh/Cr and the higher e_g states are separated by the Fermi level in the energy gap region. However, the TB-mBJ potential preserves the same situation with more prominent peaks of Fe and Cr rather than Rh states, keeping the bandgap nearly constant.

Thus, the orbital sketch of the FeRhCrZ molecule clues the possible d–d intermixing with octahedral symmetry, which leads to the exhibition of a down-spin energy gap in these alloys. This gap arises between the occupied hybrid triplet states (Fe-t2g+ Rh-t2g

and Cr-t_2g) and the unoccupied (Fe-e_g+ Rh-e_g and Cr-e_g) states which are localized at the A, B, and C sites. In addition, we tried the PBE + U (see Fig. S5 in the supplementary material) and PBE + SOC (see Fig. S6 in thesupplementary material) methods to describe the intricate behavior of transition-metal d-states, where it has been established that the half-metallicity is persistent within these effects also. When the PBE + U method is applied, we can observe that the spin-down channels (see Fig. S5 in thesupplementary material) are still semiconducting with a bandgap of 0.79 eV and 0.67 eV for Si and Ge systems, respectively, whereas the SOC calculated gap is 0.49 eV for the Fe–Si system and 0.50 eV for the Fe–Ge system. However, the spin-up channel shows the metallic character in both these approximations. Hence, the half-metallic behavior of FeRhCrZ alloys is strictly established.

Here, we discuss the magnetic properties on the basis of pDOS and total and individual spin moment contributions in a molecule from its constituent atoms. The atom-resolved spin moments are listed in Table II. Since Cr with ∼2.0 μB is having maximum half-filled d-orbitals, it is a significant contributor toward the net magnetic moment. Then, Fe with five unpaired d-orbitals accumulates nearly a unit magnetic moment and Rh with two unpaired d-orbitals gives a small moment of ∼0.2–

0.1μB. This small moment of Rh can be observed also from the small peaks of d-states as compared to Fe and Cr atoms. The maximum population in spin-polarized density of states (pDOS, seeFigs. 4and5) of these atoms follow the trend Cr > Fe > Rh, and hence, the magnetic moments also increase in the same pattern. At the same time, Si/Ge atoms couple in a weakly antiparallel direction to balance the spin and charge effects. The net ferromagnetic moment of 3.0μB is, thus, reserved, which is also supported by the experimental value of 2.90μB/f.u. in the case of Fe–Ge alloy. The integrity in the magnitude of moments is theoretically well established by SP rule for half-metals, where the 24 valence electrons are fully compensated with a remnant of three unpaired electrons giving rise to integral moment equal to 3.0μB.^48–51In the Fe–Ge alloy, the negligible discrepancy of 0.1μB from the experiment can be associated with the impurity or anti-site disorder (reported by experiments) in the synthesized samples. To conclude, we can, thus, designate these materials as ferromagnetic half-metals.

FIG. 2. Spin-resolved electronic band profiles of FeRhCrSi in both spin directions calculated by the PBE functional and TB-mBJ potential (green-colored bands represent the d-band contributions from Fe, Rh, and Cr atoms).

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Mechanical properties

The elastic parameters of the FeRhCrZ alloys are computed to predict the mechanical stability within the Born limits [C₁₂< B < C₁₁; (C₁₁−C12) > 0; (C₁₁+ 2C₁₂) > 0; and C₄₄> 0] described for cubic materials.^52–54The calculated bulk, shear, Young’s moduli, and other related parameters establish the hardness viz-a-viz the tensile strength, ductile or brittleness, plastic or elastic behavior, etc., of any

material.Table IIIenlists the elastic coefficients computed from the equations mentioned elsewhere.^55,56B/G or Pugh’s ratio signifies the Fe–Ge material as brittle in nature, whereas Fe–Si as ductile, because if B/G < 1.75, then the material is claimed to be brittle, and if B/G > 1.75, the material is said to be ductile.⁵⁵Same phenomenon is supported by the negative value of Cauchy pressure (C₁₂–C44) for the Fe–Ge alloy, and the positive value of the Fe–Si alloy maintains FIG. 3. Spin-resolved electronic band profiles of FeRhCrGe in both spin directions calculated by the PBE functional and TB-mBJ potential (black green bands represent the d-band contributions from Fe, Rh, and Cr atoms).

FIG. 4. Total DOS and pDOS of (a) FeRhCrSi and (b) FeRhCrGe alloys calculated by the PBE functional.

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the ductile nature because if C12–C44 is positive, the material is ductile and vice versa. This transition from ductile to brittle can be thought as Ge doping could increase the electronic exchange effect between the neighboring atoms which, in turn, decreases the bulk modulus/shear modulus ratio, and both these factors are critical to the deformation capability of a material. The previously reported values of B/G and Cauchy pressure for the Fe–Si alloy are not reliable because they disagree with each other, and hence, our results are more accurate. For any material, the critical value of Poisson’s ratio is 0.25, below which the bonds are said to be non-central, and the values between 0.25 and 0.50 characterize the presence of central forces.⁵⁶Thus, central forces in the Fe–Si alloy are present and non- central bonds can be argued in the Fe–Ge alloy.

Thermodynamic properties

The specific information about the materials response when put under severe constraints (high temperature/pressure) can be achieved by investigating the thermodynamic processes. First, the Debye temperature and sound velocities calculated from elastic constants are put together in Table IV. Later, we applied the

quasi-harmonic Debye model⁵⁷to evaluate the thermal heat capacity (C_V), expansion coefficient (α), and the effect of pressure or temperature on these speculated thermodynamic parameters is discussed accordingly. These properties are described in the temperature range from 0 to 800 K accompanied by pressure variations from 0 to 25 GPa.

The Debye temperature [438 K for Fe–Si and 640 K for Fe–Ge] estimates the highest mode of thermal phonon vibrations, and the participation of these phonons in the thermal conduction processes is critical for heat transfer. These values are quite larger than the different EQH alloys studied previously^28,58 as summa- rized inTable IV. Conclusively, the largeθDand T_mvalues recom- mend the stability of these materials against temperature effects.

Hence, the present materials can be regarded as high melting and Debye temperature alloys. More importantly, the experimentally reported large Curie temperature (550 K; Ref. 32) for the Fe–Ge alloy accompanied by large Debye temperature strongly facilitates the possibility of its applications in spintronic devices as well as in magnetic materials. As already discussed, Fe–Ge has been recently synthesized experimentally, and hence, further research in charac- terization of both these alloys for the magneto-electronic and spintronic applications has not yet been realized.

FIG. 5. Total DOS and pDOS of (a) FeRhCrSi and (b) FeRhCrGe alloys calculated by the TB-mBJ potential.

TABLE II. The calculated, total, and atomic magnetic moments of EQH FeRhCrZ alloys (in μB): Fe-magnetic moment (MFe), Rh-magnetic moment (MRh), Cr-magnetic moment (MCr), Si/Ge magnetic moment (MZ), magnetic moment in the interstitial region (MInt), and total magnetic moment (MTotal).

Method M_Int M_Fe M_Rh M_Cr M_Z M_Total

FeRhCrSi

PBE 0.05 0.65 0.21 2.11 −0.02 3.00

TB-mBJ −0.04 0.88 0.17 2.03 −0.04 3.00

PBE + U −0.02 1.11 0.05 1.91 −0.05 3.00

SOC 0.04 0.68 0.22 2.08 −0.02 3.00

Theory^a … −0.26 0.22 3.10 −0.06 3.00

FeRhCrGe

PBE 0.06 0.59 0.18 2.18 −0.03 3.00

TB-mBJ −0.03 1.05 0.08 1.96 −0.06 3.00

PBE + U 0.02 1.10 0.06 1.98 −0.06 3.10

SOC 0.07 0.61 0.18 2.17 −0.03 3.01

Experiment^b … … … 2.90

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FIG. 6. (a) Heat capacity (CV) vs temperature and (b) CV vs pressure; (c) thermal expansion coefficient (α) vs temperature and (d)α vs pressure for FeRhCrZ alloys calculated by quasi- harmonic Debye approximation.

TABLE IV. Calculated values of average sound velocity (vm), compressional velocity (vl), shear sound velocity (vs), Debye temperature (θD), and melting temperature (Tm) for the FeCrRhGe alloy and its comparison with previously studied EQH alloys.

Parameter vs(m/s) vl(m/s) vm(m/s) θD(K) Tm(K) ± 300

FeRhCrSi 3050 6059 3418 438 1981

FeRhCrGe 4596 7288 5052 640 2866

FeRuCrSi^a 3848 7008 4363 565 2687

CoFeZrGe^b 2526 5492 3434 429 1980

CoFeZrSi^b 3251 6327 4328 556 2151

aThe values for FeRuCrSi are calculated from elastic constants taken from Wang et al., Sci. Rep.7, 16183 (2017). Copyright 2017 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

TABLE III. Calculated values of elastic (C11, C12, C44), bulk (B), Shear (G), Young’s (Y) moduli (in GPa), Poisson’s ratio (υ), B/G ratio, and Cauchy’s pressure (C12–C44) for the FeRhCrGe alloy.

Method C11 C12 C44 C12-C44 B G Y B/G υ

FeRhCrSi

PBE (present) 280.52 148.28 79.07 69.21 192.37 73.60 195.84 2.61 0.33

Theory^a 294.70 112.90 106.60 06.30 173.50 100.00 251.70 1.74 …

FeRhCrGe

PBE (present) 434.18 125.67 225.31 −99.64 228.50 193.56 445.33 1.18 0.17

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Heat capacity furnishes the information about the lattice vibrations of a material. Therefore, we calculated the heat capacity at constant volume (C_V) as displayed in Figs. 6(a) and 6(b) with varying temperatures and pressures. Noticeably, the sharp increase in CV plot is observed up to 300 K, and then its increment raises slowly. Furthermore, it (CV) approaches the Dulong–Petit limit, sig- nifying that the total phonon modes in this system are fully excited.⁵⁹ Below this temperature, it simply follows T³ law (C_V α T³).⁶⁰ However, the pressure increase has less significance but opposite influence on CV. Its room temperature value for the Fe–Si alloy is ∼75 J Mol⁻¹K⁻¹. Since Cv = Cp (specific heat at constant pressure), we can argue that the experimental value of

Cp∼ 100 J Mol⁻¹K⁻¹at 300 K for the Fe–Ge alloy is roughly under- estimated by our theoretically predicted value of ∼80 J Mol⁻¹K⁻¹. This small discrepancy can be attributed to the experimentally reported anti-site disorder in the crystal structure. The experiment considered 50% anti-site disorder between the tetrahedral sites, i.e., Fe and Rh or Cr and Rh in type-I and type-II configurations, respectively.³²

InFigs. 6(c)and6(d), the thermal expansion coefficient (α) is plotted against temperature and pressure gradients. It seems thatα increases with increasing temperature but strongly decreases with pressure. However, it sharply increases up to 300 K and then satu- rates with almost a constant slope. Thus,α agrees with the T³law,

FIG. 7. Transport coefficients, viz, (a) Seebeck coefficient (S), (b) electrical conductivity (σ), and (c) thermopower (S²σ) as a function of temperature for FeRhCrSi and FeRhCrGe alloys at an optimal doping concentration of 10¹⁸cm⁻³.

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and its value (at 0 GPa and 300 K) for both these alloys is about

∼1.50 × 10⁻⁵K⁻¹. The increase in pressure tends to decrease theα value very sharply in accordance with the quasi-harmonic Debye model.

Thermoelectric coefficients

We make use of the Boltztrap code, under constant relaxation time approximation (CRTA) and rigid band approximation (RBA), to calculate the transport properties.⁶¹These approximations hold good for low doping levels and when the variation of the scattering time is confined within the energy range of kBT, i.e., if the scattering time varies slowly in the energy scale of thermal agitation.^62–64 The band structure (absolute to the Fermi level) directly forecasts the Seebeck coefficient of a material, which in combination with electrical/thermal conductivity decides the thermoelectric response of that material. Both the Seebeck and electrical conductivity coefficients robustly depend on the Fermi level, which, in turn, depends on the concentration and effective mass of the carriers as well as on the temperature. Therefore, the thermoelectric transport coefficients are conveniently expressed theoretically in terms of Fermi energy.⁶⁵In this section, the determination of the possible trend of PF and ZT for FeRhCrZ alloys is achieved. We also compare our simulated results with the experimentally reported ones and then the materials are crosschecked to find their compatibility with conventional (room temperature) or high temperature TE possibilities or both. The basic understanding of the method of calculation and the approach of semi-classical Boltzmann transport theory can be achieved from Refs.66and67. Within the above limits, the electrical conductivity and Seebeck coefficient take up the following forms:

σ ¼ e² ð

Ξ(ε) @f0

@ε

dε (5)

and

S¼ e Tσ

ð

Ξ(ε) @f0

@ε

(ε μ)dε: (6)

Here, dϵ is band energy, T is the temperature, e is the electronic charge,μ is the chemical potential, Ξ is the transport kernel, and f₀is the distribution function. We make use of the two-current model^68,69to sum the individual values of transport coefficients in the spin-up and spin-down states. Later, the electrical conductivity, Seebeck, and thermopower (PF = S²σ) are plotted inFigs. 7(a)–7(c).

The Seebeck coefficient (S) as shown in Fig. 7(a)is a major descriptor of thermopower, i.e., the ability to produce electric potentials with respect to temperature. Around the Fermi level, the optimum value of S calculated at 300 K is−7.13 μV/K for Fe–Si and−4.31 μV/K for Fe–Ge alloy. An exponential increase in magnitude can be seen from −2.5 μV/K in the case of Fe–Si and

structure calculations of both these materials, the spin-up channel exhibits metallic behavior, and hence, the spin-down channel is of n-type (with electrons as majority carriers). Taking the advantage of the experimental data of electrical conductivity of FeRhCrGe, we make use of the deduced relaxation time τ ∼ 0.5 × 10⁻¹⁵s for both the alloys. Then, we figure out the electrical conductivity (σ) and its variation with temperature as depicted inFig. 7(b). The value of σ is 4.45 × 10⁵(Ω m)⁻¹for Fe–Si and 4.50 × 10⁵(Ω m)⁻¹for Fe–Ge at room temperature. Interestingly, while using the constant τ value, the experimental value of σ [4.56 × 10³(S cm)⁻¹] for FeRhCrGe is comparable to the present simulated data. Finally, the power factor (S²σ) plotted in Fig. 7(c) is observed to reach a maximum value of ∼22.0 μW/cm K² (for Fe–Si) and ∼16.0 μW/

cm K²(for Fe–Ge) at 800 K. The value of PF is 2.3 μW/cm K²for the FeRhCrSi compound and 0.83μW/m K² for the FeRhCrGe compound at room temperature, which clearly designates the Fe–Si compound as more efficient in thermoelectric conversion than the Fe–Ge alloy. Despite these small values, the PF’s are quite comparable and competitive enough with the existing conventional thermoelectric materials like CoTiSb (23.2μW/cm K² at 1100 K^70,71) and FeMnTiSb (10.6μW/cm K² at 300 K⁷²) The thermopower of FeRhCrZ seems to be increasing with respect to temperature, and we can propose that further experiments can be augmented for their possible thermoelectric applications at higher temperatures.

Remarkably, we observe that FeRhCrSi displays a high PF of 22.0μW/cm K²(at 800 K), which is equal to that of the experimentally reported value of FeNbSb (22.7μW/cm K²at 700 K).^72,73

InFig. 8(a), the variation of S in FeRhCrSi with respect to chemical potential at different temperatures in the p-type doping region reaches a maximum of 16μV/K, and in n-type region, it goes on increasing in magnitude to a maximum of −23 μV/K at 700 K. Similarly, in FeRhCrGe, this value reaches a maximum of

FIG. 8. (a) Seebeck coefficient (S), (b) electrical conductivity (σ), and (c) ther-

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16μV/K through 0 in both sides of the doping region at 700 K. It can be seen fromFig. 8(c)that the maximum increase in PF can be achieved by p-type doping in the Fe–Ge alloy, but in the Fe–Si system, the same can be achieved when the n-type dopants are added in the whole range of chemical potential. This can be attributed to the significant increase in electrical conductivity through the n-type region in the earlier case, while as this parameter shows the reverse trend in the latter case as seen fromFig. 8(b). At the same time, the Fe–Si system shows more significant improvement rather than the Fe–Ge compound. We have also calculated the lattice thermal conductivity via slacks approach^74,75as well as the thermoelectric figure of merit (ZT), and the observed plots are displayed inFigs. 9(a)and 9(b). The observed ZT of Fe–Si reaches a maximum of 0.45 at 800 K and that of Fe–Ge reaches 0.41 at the same temperature. However, these values are quite small in comparison to available thermoelectric materials, and this can be attributed to the small Seebeck coefficients in the considered alloys.

Hence, the present findings suggest the maximum potential of the FeRhCrSi alloy as a high temperature thermoelectric material rather than the FeRhCrGe alloy. Therefore, future studies should be carried out to enhance the thermopower of these materials and to expense the waste heat (temperature gradient) properly into usable electric power.

CONCLUSIONS

The electronic, thermodynamic, elastic, phonon, and magnetic properties of the FeRhCrZ alloys within the LiMgPdSn prototype phase have been investigated using first-principles density functional calculations:

• FeRhCrZ alloys are strictly stable in type-I configuration, agree- ing well with the experiment as well. Here, the metallic properties in the spin-up state are exhibited, whereas the spin-down state reflects a maximum semiconducting gap.

• Magneto-electronic calculations decisively confirm the ferromagnetic and half-metallic nature with a net magnetic moment of 3.0μB at their equilibrium lattice constants.

• The elastic constants and their derivatives profusely establish the brittleness of the Fe–Ge alloy and ductile properties of the Fe–Si system.

• These materials exhibit high Debye and melting temperatures, which guarantee the stability of these materials against large temperature variations.

• FeRhCrSi displays a high PF of 22.0 μW/cm K²at higher temperatures, which is comparable to that of the experimentally reported PF of FeNbSb (22.7μW/cm K²).

• The figure of merit reaches a maximum of 0.45 at higher temperatures. Besides, these results suggest the potential of FeRhCrZ as promising high-temperature thermoelectric materials and promote their experimental realization for future applications.

SUPPLEMENTARY MATERIAL

See the supplementary material for additional figures. The primitive cell configurations and partial density of states have been plotted using different exchange-correlation approximations.

ACKNOWLEDGMENTS

This work was supported by the Ministry of Science and Technology of Taiwan (Grant No. MOST107–2628-M-002-005-MY3), the National Taiwan University (Grant Nos. NTU-108L4000 and NTU-CDP-105R7818), and the National Center for Theoretical Sciences of Taiwan.

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