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

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 3.00 μB, 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 × 10⁻¹⁵ s. The room temperature PF values of FeRhCrSi and FeRhCrGe compounds are 2.3 μW/cm K² and 0.83 μW/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.

Full text

J. Appl. Phys. 127, 165102 (2020); https://doi.org/10.1063/1.5139072 127, 165102
© 2020 Author(s).
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); 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

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 CrossMar
k
Submitted: 18 November 2019 · Accepted: 5 April 2020 ·
Published Online: 22 April 2020
Shakeel Ahmad Khandy
1
and Jeng-Da Chai
1,2,a)
AFFILIATIONS
1
Department of Physics, National Taiwan University, Taipei 10617, Taiwan
2
Center 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 struc-
ture 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 3.00 μB, 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
−15
s. The room temperature PF values of FeRhCrSi and
FeRhCrGe compounds are 2.3 μW/cm K
2
and 0.83 μW/m K
2
, 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 proper-
ties 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-
tion, or large integral magnetic moments. Half-metallic ferromag-
nets (HMFs) represent a new class of materials that exhibit
semiconducting properties in one spin (down) channel and behave
as a conductor in the other spin (up) channel. They find
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 antici-
pated: the oxide compounds such as CrO
2
,
6
TiO
2
,andVO
2
,
7
some
ternary compounds (specifically, spinels with the general formula
AB
2
O
4
such as Fe
3
O
4
and LiMn
2
O
48–10
), single or double perovskites
(e.g., BaPaO
3
and Sr
2
SnMnO
611,12
), and dilute magnetic semicon-
ductors [DMSs, e.g., Cu-doped ZnO,
13
Cr-doped CdZ (Z = S, Se,
and Te),
14
Mn-doped GaN,
15
etc.]. In addition, the Heusler materials
that include half Heuslers [e.g., CoCrZ (Z = S and Se)],
16
full
Heuslers [Co
2
TaZ (Z = Si and Ge)],
17
and quaternary Heuslers
(FeVRuSi
18
) have also accounted for the integral magnetic moment,
spin polarization, and half-metallic properties.
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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,
19–23
and 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),
31
and many others, have been discovered
experimentally or predicted theoretically. This research work is
anticipated to investigate the structural, electronic, elastic, thermo-
electric, 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 theo-
retically,
33
but there is no data available regarding its thermody-
namic, 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 code
34
is used to accomplish the spin-
polarized density functional calculations on FeRhCrZ EQH alloys.
Full-potential linearized augmented plane wave (FP-LAPW)
method
35
[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 elec-
trons. For the exchange-correlation energy functional, we adopt the
Perdew–Burke–Ernzerhof (PBE) functional
36
and the Tran–Blaha
modified Becke–Johnson (TB-mBJ) potential.
37
For strongly local-
ized 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 func-
tional. Later, the on-site Hubbard correction (PBE + U with
U
eff
= 1.36 eV and 0.68 eV for Fe and Cr, respectively)
38
and spin–
orbit coupling
39
are 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 code
40
is utilized with uniform hydrostatic pressure
applied in all directions. Additionally, the thermodynamic amounts
of melting temperature (T
m
) and Debye temperature (θ
D
)from
elastic constants are tallied by means of the following equations:
41
Tm(K) ¼[553(K) þ(5:911)C11GPa] +300 K, (1)
θD¼h
k
3n
4π
NAρ
M

1
3
Vm, (2)
Vm¼1
3

1
32
V3
s
þ1
V3
l

1
3, (3)
Vs¼ffiffiffiffi
G
ρ
sand Vl¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3Bþ4G
3ρ
s:(4)
Here, the symbols have their usual meanings and v
m
is the average
sound velocity in terms of compressional (v
l
) and shear (v
s
) sound
velocities.
42
Phonon spectra are calculated by the pseudopotential-
based Quantum Espresso package
43
within the framework of
PBE.
36
The 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 the supplementary 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 (E
0
) of both these alloys as mentioned in Table I. In addi-
tion to equilibrium lattice constant and ground-state energy, the
calculated values of Bulk modulus and its derivative for all the
three configurations are listed in Table 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.
33
To further guarantee the stability of the Fe–
Si alloy, we determine the dynamic stability from the phonon dis-
persion curve and phonon density of states as displayed in Fig. 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
.Itis
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, mag-
netic, transport, and thermodynamic properties of these alloys.
Electronic properties and magnetism
In Figs. 2 and 3, 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.
33
pre-
dicted 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.
32
A 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.
45,46
This can be further argued from the
similar results of EQH alloys like CoFeCrGe and CoMnCrAl
reported by the same group,
46
where 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 3represent the overall d-state contributions from Fe/
Rh/Cr elements in the whole FeRhCrZ molecule and are particu-
larly 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, B05.95 4.03 5.00 ……
Total energy, E
0
(eV) −201 337.68 −201 337.07 −201 336.01 ……
FeRhCrGe
Lattice constant, a
o
(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, B06.10 8.78 5.00 ……
Total energy, E
0
(eV) −250 563.95 −250 562.76 −250 561.27 ……
a
Venkateswara et al., Phys. Rev. B 100, 180404(R) (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.
b
Feng et al., Appl. Sci. 8, 2370 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution (CC BY) license.
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
and 5, we can argue that our results are more reliable and compara-
ble 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) and 4(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.
32
At the same time, TB-mBJ [see Figs. 5(a)
and 5(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
z
2
,d
x
2
-y
2
) 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. 47 and 48. 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 (e
g
and t
2g
) of the
transition-metal (Fe/Rh/Cr) atoms can be viewed in reference to
the possible d–dbandgap 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 t
2g
and e
g
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 cal-
culations reflect the octahedral symmetry where the much lower t
2g
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 poten-
tial 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–dintermixing 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-t
2g
+ Rh-t
2g
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 the supplementary 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 the supplemen-
tary 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–
0.1 μB. 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,
see Figs. 4 and 5) 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 3.0 μB is, thus, reserved, which is also supported by the
experimental value of 2.90 μB/f.u. in the case of Fe–Ge alloy. The
integrity in the magnitude of moments is theoretically well estab-
lished 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 3.0 μB.
48–51
In the Fe–Ge
alloy, the negligible discrepancy of 0.1 μB 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 direc-
tions 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 [C
12
<B<C
11
;
(C
11
−C
12
)>0; (C
11
+2C
12
)>0; and C
44
> 0] described for cubic
materials.
52–54
The calculated bulk, shear, Young’smoduli,andother
related parameters establish the hardness viz-a-viz the tensile
strength, ductile or brittleness, plastic or elastic behavior, etc., of any
material. Table III enlists the elastic coefficients computed from the
equations mentioned elsewhere.
55,56
B/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.
55
Same phenomenon is
supported by the negative value of Cauchy pressure (C
12
–C
44
)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 func-
tional and TB-mBJ potential (black
green bands represent the d-band con-
tributions 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 C
12
–C
44
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 reli-
able 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.
56
Thus, 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 model
57
to evaluate the thermal heat capacity
(C
V
), expansion coefficient (α), and the effect of pressure or tempera-
ture 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 in Table IV. Conclusively, the large θ
D
and T
m
values 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 μ
B
): Fe-magnetic moment (M
Fe
), Rh-magnetic moment (M
Rh
), Cr-magnetic
moment (M
Cr
), Si/Ge magnetic moment (M
Z
), magnetic moment in the interstitial region (M
Int
), and total magnetic moment (M
Total
).
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
a
Feng et al., Appl. Sci. 8, 2370 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution (CC BY) license.
b
Venkateswara et al., Phys. Rev. B 100, 180404(R) (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.
Journal of
Applied Physics ARTICLE scitation.org/journal/jap
J. Appl. Phys. 127, 165102 (2020); doi: 10.1063/1.5139072 127, 165102-6
Published under license by AIP Publishing.

FIG. 6. (a) Heat capacity (C
V
) vs tem-
perature and (b) C
V
vs pressure; (c)
thermal expansion coefficient (α)vs
temperature and (d) αvs pressure for
FeRhCrZ alloys calculated by quasi-
harmonic Debye approximation.
TABLE IV. Calculated values of average sound velocity (v
m
), compressional velocity (v
l
), shear sound velocity (v
s
), Debye temperature (θ
D
), and melting temperature (T
m
) for
the FeCrRhGe alloy and its comparison with previously studied EQH alloys.
Parameter v
s
(m/s) v
l
(m/s) v
m
(m/s) θ
D
(K) T
m
(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
a
The 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.
b
Paudel and Zhu, J. Magn. Magn. Mater. 453, 10 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution (CC BY) license.
TABLE III. Calculated values of elastic (C
11
,C
12
,C
44
), bulk (B), Shear (G), Young’s (Y) moduli (in GPa), Poisson’s ratio (υ), B/G ratio, and Cauchy’s pressure (C
12
–C
44
) for
the FeRhCrGe alloy.
Method C
11
C
12
C
44
C
12
-C
44
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
a
Feng et al., Appl. Sci. 8, 2370 (2018). Copyright 2018 Author(s), licensed under a Creative Commons Attribution (CC BY) license.
Journal of
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J. Appl. Phys. 127, 165102 (2020); doi: 10.1063/1.5139072 127, 165102-7
Published under license by AIP Publishing.

Heat capacity furnishes the information about the lattice
vibrations of a material. Therefore, we calculated the heat capacity at
constant volume (C
V
)asdisplayedinFigs. 6(a) and 6(b) with
varying temperatures and pressures. Noticeably, the sharp increase in
C
V
plot is observed up to 300 K, and then its increment raises
slowly. Furthermore, it (C
V
) 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 T
3
law (C
V
α
T
3
).
60
However, the pressure increase has less significance but
opposite influence on C
V
. Its room temperature value for the Fe–Si
alloy is ∼75 J Mol
−1
K
−1
. Since Cv= Cp (specific heat at constant
pressure), we can argue that the experimental value of
C
p
∼100 J Mol
−1
K
−1
at 300 K for the Fe–Ge alloy is roughly under-
estimated by our theoretically predicted value of ∼80 J Mol
−1
K
−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
In Figs. 6(c) and 6(d), the thermal expansion coefficient (α)is
plotted against temperature and pressure gradients. It seems that α
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, αagrees with the T
3
law,
FIG. 7. Transport coefficients, viz, (a)
Seebeck coefficient (S), (b) electrical
conductivity (σ), and (c) thermopower
(S
2
σ) as a function of temperature for
FeRhCrSi and FeRhCrGe alloys at an
optimal doping concentration of
10
18
cm
−3
.
Journal of
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J. Appl. Phys. 127, 165102 (2020); doi: 10.1063/1.5139072 127, 165102-8
Published under license by AIP Publishing.

and its value (at 0 GPa and 300 K) for both these alloys is about
∼1.50 × 10
−5
K
−1
. The increase in pressure tends to decrease the α
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.
61
These approximations hold
good for low doping levels and when the variation of the scattering
time is confined within the energy range of k
B
T, i.e., if the scatter-
ing 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 coeffi-
cients 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 coeffi-
cients are conveniently expressed theoretically in terms of Fermi
energy.
65
In 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 con-
ventional (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. 66 and 67. Within the above limits, the electri-
cal conductivity and Seebeck coefficient take up the following
forms:
σ¼e2ðΞ(ε)
@f0
@ε

dε(5)
and
S¼e
TσðΞ(ε)
@f0
@ε

(εμ)dε:(6)
Here, dϵis band energy, Tis the temperature, eis the elec-
tronic charge, μis the chemical potential, Ξis the transport kernel,
and f
0
is the distribution function. We make use of the two-current
model
68,69
to sum the individual values of transport coefficients in
the spin-up and spin-down states. Later, the electrical conductivity,
Seebeck, and thermopower (PF = S
2
σ) are plotted in Figs. 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 mag-
nitude can be seen from −2.5 μV/K in the case of Fe–Si and
−1.3 μV/K in Fe–Ge (at 50 K) to a maximum of −24.9 μV/K and
−21.2 μV/K (at 800 K), respectively. The negative value of total S
for the present alloys is an indication of n-type carriers, which is
reflected from the electronic structure as well. From the band
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
−15
s for both
the alloys. Then, we figure out the electrical conductivity (σ) and
its variation with temperature as depicted in Fig. 7(b). The value of
σis 4.45 × 10
5
(Ωm)
−1
for Fe–Si and 4.50 × 10
5
(Ωm)
−1
for Fe–Ge
at room temperature. Interestingly, while using the constant τ
value, the experimental value of σ[4.56 × 10
3
(S cm)
−1
] for
FeRhCrGe is comparable to the present simulated data. Finally, the
power factor (S
2
σ) plotted in Fig. 7(c) is observed to reach a
maximum value of ∼22.0 μW/cm K
2
(for Fe–Si) and ∼16.0 μW/
cm K
2
(for Fe–Ge) at 800 K. The value of PF is 2.3 μW/cm K
2
for
the FeRhCrSi compound and 0.83 μW/m K
2
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 compara-
ble and competitive enough with the existing conventional thermo-
electric materials like CoTiSb (23.2 μW/cm K
2
at 1100 K
70,71
) and
FeMnTiSb (10.6 μW/cm K
2
at 300 K
72
) 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
22.0 μW/cm K
2
(at 800 K), which is equal to that of the experimen-
tally reported value of FeNbSb (22.7 μW/cm K
2
at 700 K).
72,73
In Fig. 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 16 μV/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-
mopower (S
2
σ) as a function of chemical potential for the FeRhCrSi alloy. (d)
Seebeck coefficient (S), (e) electrical conductivity (σ), and ( f) thermopower
(S
2
σ) as a function of the chemical potential for the FeRhCrGe alloy at three dif-
ferent temperatures (300 K, 600 K, and 900 K).
Journal of
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J. Appl. Phys. 127, 165102 (2020); doi: 10.1063/1.5139072 127, 165102-9
Published under license by AIP Publishing.

16 μV/K through 0 in both sides of the doping region at 700 K. It
can be seen from Fig. 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 attrib-
uted 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 from Fig. 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
74,75
as well as the
thermoelectric figure of merit (ZT), and the observed plots are dis-
played in Figs. 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 com-
parison to available thermoelectric materials, and this can be attrib-
uted 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 func-
tional calculations:
•FeRhCrZ alloys are strictly stable in type-I configuration, agree-
ing well with the experiment as well. Here, the metallic proper-
ties in the spin-up state are exhibited, whereas the spin-down
state reflects a maximum semiconducting gap.
•Magneto-electronic calculations decisively confirm the ferromag-
netic and half-metallic nature with a net magnetic moment of
3.0 μB 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 tem-
perature variations.
•FeRhCrSi displays a high PF of 22.0 μW/cm K
2
at higher temper-
atures, which is comparable to that of the experimentally
reported PF of FeNbSb (22.7 μW/cm K
2
).
•The figure of merit reaches a maximum of 0.45 at higher temper-
atures. 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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J. Appl. Phys. 127, 165102 (2020); doi: 10.1063/1.5139072 127, 165102-12
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