Theoretical Investigation on Electronic and Magnetic Structures of FeRhHidekazu Takahashi, Masaaki Araidai*, Susumu Okada and Kenji Shiraishi*
Graduate School of Pure and Applied Sciences,University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan*Institute of Materials and Systems for Sustainability, Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan

In order to clarify the mechanism behind antiferromagnetic (AFM)-ferromagnetic (FM) phase transition, weinvestigate the electronic and magnetic structures of FeRh by using first principles calculations with the GGA + Umethod. By choosing the appropriate values of the on-site Coulomb interaction (U) of Fe3d and Rh4d electrons, we
succeed in explaining the reported AFM-FM phase transition experiments for the first time by obtaining the totalenergy difference between the AFM and FM states (ΔE). Other physical quantities such as the density of states (DOS)are also consistent with experimental reports.
Key words: magnetic refrigeration, inverse magnetocaloric material, antiferromagnetic-ferromagnetic phasetransition, GGA + U method, on-site Coulomb interaction
1. Introduction
Room-temperature magnetic refrigeration technology
using magnetocaloric materials (MCMs) has recently
gained attention because it is expected to eliminate
refrigerants that are greenhouse gases and improve thesystemefficiencyofrefrigerators.1)
Variousperformance-enhancing
MCMs
suchasLa(FexSi1-x)13Hy,2) MnFe(P,Si),3) and so on have beenfound so far. Inverse magnetocaloric materials (IMCMs)
such as FeRh4) and Mn2-xCrxSb5) have also been the focus
of attention because of the appearance of the ‘giant’
inverse magnetocaloric effect (IMCE) under a lowmagnetic field. In particular, FeRh shows a first-orderantiferromagnetic (AFM)-ferromagnetic (FM) phasetransition at Ttr ≈ 320 ~ 370 K without a magnetic fieldand exhibits a giant adiabatic temperature change ΔTad(= 13 K) under 1.95 T. Recent studies suggest that ΔTad
can be increased up to 18 K per 1 T.6) The appearance of
new MCMs and IMCMs that are superior to FeRh is now
expected. It is quite important to understand the why
ΔTad is large in FeRh. Moreover, it is essential to clarify
the microscopic mechanism behind the AFM-FM phasetransition in order to develop new magnetocaloric
materials.
Since FeRh was discovered by Fallot in 1938,7)
various experiments8)-19) on its physical parameters havebeen performed, such as the difference between theminimum total energy of the FM state and that of theAFM state (ΔE) related to Ttr.10)
Various theoretical investigations have also been
carried out. For example, Kittel proposed the exchange
inversion model.20) There have also been various theories
proposed. However, all of these currently remain thesubject of debate. Several investigations into electronic
and magnetic structures using various band-structure
calculation methods have been reported to date.21)-28) Weshould point out here that recent advanced
first-principles calculations26)-28) are not able toreproduce important physical quantities such as ΔE. In
particular, if ΔE calculations are inconsistent withexperiments, its mechanism as an IMCM cannot beclarified. Generally, in alloys containing 3d and 4dtransition metals, regard for electronic correlation isknown to be important. However, this is not consideredin all of the reported calculations, including Refs.
(26)-(28). Therefore, we treat this electronic correlation
as the on-site Coulomb interaction (U) and examine theeffect of U on various physical quantities, particularly
ΔE. In this paper, we investigate the electronic andmagnetic structures of FeRh and have succeeded in
reproducing
theAFM-FM
phasetransition
quantitatively for the first time.
2. Calculation Method
In FeRh, which is a CsCl crystal with a simple cubicstructure, the magnetic structures in the AFM and FM
states are shown in Fig. 1. In order to treat FeRh in both
the AFM and FM states, the crystal is considered aface-centered cubic structure with lattice constant a’= 2a,where a is the lattice constant in the case of a simple
cubic structure. The atomic positions of two Fe atoms
are (0.0, 0.0, 0.0) and (0.5a’, 0.5 a’, 0.5 a’). The atomic
positions of two Rh atoms are (0.25 a’, 0.25 a’, 0.25 a’)and (0.75 a’, 0.75 a’, 0.75 a’).

(a)

(b)

Fig. 1 Magnetic structure of (a) AFM and (b) FM
states of FeRh. The blue and violet circles
indicate Fe and Rh atoms, respectively. ↑ and ↓indicate the directions of the magnetic
moments.
The Vienna ab initio simulation package (VASP) 29),30)
with the projector augmented wave (PAW) method31),32)
was utilized for performing the first principles

calculations. We used the Perdew-Burke-Ernzerhof
exchange-correlation functional33) with the Hubbard U
method (GGA + U)34) including U and the exchange
parameter (J ) for each atomic orbital in the materials,
as our GGA calculation results such as ΔE (= 31.5meV/atom) and the magnitude relation of Fe magnetic
moment values between the AFM and FM states (3.12μB,3.18μB) were inconsistent with experimental results. In
the calculations, the Dudarev approach, which is onlydependent on Ueff = U – J35), was adopted. The
magnitude of U and J for the Fe3d electron (UFe, JFe) andfor the Rh4d electron (URh, JRh) were set to (2.0 eV, 1.0eV) and (1.95 eV, 1.0 eV), respectively so that theobtained physical quantities of the Fe and Rh crystals(lattice constant, bulk modulus and density of states(DOS)) using these U and J were consistent with thereported experiments. The Methfessel-Paxton method36)
was adopted by choosing the width of the smearing Δσ =0.05 eV. A plane-wave basis set with an energy cutoff(Ecut) of 830 eV and an 11×11×11 Monkhorst-Pack kpoints mesh37) was used.38)
3. Results and Discussion
A comparison of the physical quantities obtained in
the present analysis by using the above GGA + U
calculation method and those reported in experiments isgiven in Table I. The calculation results are detailed asfollows. First, the relationship between the totalenergies of the AFM and FM states and the lattice
constant is investigated by choosing appropriatemagnitudes of UFe and URh to estimate ΔE. The obtained
results are shown in Fig. 2. As shown, the AFM state isstable when the lattice constant is small. With
increasing the lattice constant, the total energy of theAFM state has a minimum at lattice constant (aAFM) (=2.99 Å) and the total energy difference between the AFM
and FM states becomes smaller. When the lattice
constant is increased further, the total energy of the FM
state has a minimum at lattice constant (aFM) (= 3.01 Å).
Finally, the FM state becomes stable. The obtainedvalues for aAFM and aFM are consistent with theexperiment (aAFM = 2.981 Å and aFM = 2.999 Å).9) It isnoted that the obtained value for ΔE (= 2.71 meV/atom)

Fig. 2 Total energy versus lattice constant. On the basisof the minimum of the AFM state, the closed
squares and the diamonds indicate the totalenergies of the AFM and the FM states,
respectively.

is consistent with the experimental value ΔE (= 2.80
meV/atom).10)
The other physical quantities that were obtained aredetailed as follows. The magnetic moments of the Fe andRh atoms (mFe, mRh) in the AFM and FM states areinvestigated. The obtained magnetic moments arealmost identical to those in the experiments.11) The
obtained mFe of the AFM state (= 3.31μB) becomes equal
to mFe of the FM state (= 3.31μB), which is slightly
different from the tendency obtained by the experiment(mFe of the AFM state > mFe of the FM state). In all of thereported calculations21)-28), the relationship whereby mFeof the AFM state < mFe of the FM state is satisfied.
However, in the present work, the magnitude relation of
the obtained mFe values between the AFM and FM
states is approaching that of the experiment.
The obtained relationship between the obtained
values of mFe and mRh of the AFM and FM states and thelattice constant is shown in Fig. 3. With increasing thelattice constant, (i) the change in mFe is much larger
than that in mRh, (ii) the change in mFe in the AFM stateis larger than that in the FM state, and (iii) themagnitude relation of mFe values between the AFM andFM states is reversed near the observed lattice constant.

Fig. 3 mFe and mRh of the AFM and FM states versus
lattice constant.
The critical magnetic field at 0 K (Hc(0)) which canprovide us with useful information to understand theIMCE of FeRh can be obtained by22)
Hc (0) = ΔE / (MFM –MAFM),(1)
where MFM and MAFM (= 0 μB) are the magnetization per
atom of the FM and AFM states, respectively. The
obtained Hc(0) (= 21.6 T) is consistent with theexperiments (= 21.2 T ~ 29.7 T).12),13),14)
The DOS and the partial density of states (PDOS) near
the Fermi level (EF) of the Fe and Rh atoms of the FM
and AFM states were investigated and are shown in Figs.
4(a)-(d) and Figs. 5(a)-(d), respectively. The distributionof DOS near EF of the FM state shown in Figs. 4(a) and(b) and of the AFM state shown in Figs. 4(c) and (d) aresimilar to the reported photoemission spectra.15),16) In
particular, the main peak positions of DOS near EF arearound –5.0 eV to –4.0 eV and around –3.0 eV to –2.0 eV.
Moreover, the obtained ratio of DOS at EF of the FM
state and that of the AFM state (D(EF)FM / D(EF)AFM) (=4.72) is also consistent with the experiments.17),18) The
reason the obtained value of D(EF)FM / D(EF)AFM is large

is examined by using PDOSs, as shown in Figs. 5(a)-(d),respectively. PDOS at EF (PD(EF)) of a Fe atom of theFM state for the minority spin state is much higher thanthat of the AFM state for the minority spin state, while
PD(EF) of an Fe atom for both the AFM and FM statesfor the majority spin state is low and PD(EF) of an Rh
atom of both the AFM and FM states for the majority
spin and minority spin state are also low. Therefore,
D(EF)FM / D(EF)AFM becomes large due to the largechange in the contribution of Fe3d electrons to D(EF) inthe AFM-FM phase transition.
The effect of U on ΔE and mFe in the AFM and FM
states is investigated using PDOS. Comparing Fig. 5(b)with Fig. 5(d), magnetic states of Rh for the AFM andFM states are nonmagnetic and FM states, respectively.
When the magnitude of URh is increased to 1.95 eV, thenonmagnetic state becomes unstable and the FM state isenergetically advantageous. As a result, ΔE becomes
smaller compared with the GGA calculation result. FromFig. 5(a) and (c), we can see that the Fe3d band in themajority spin state for the AFM and FM states is almost
filled, while that in the minority spin state for the AFMand FM states is inadequately filled. Similar results (notshown) were obtained by the GGA calculation. When themagnitude of UFe is increased to 2.0 eV, Fe3d band at EFin the minority spin state moves to the upper energyside, which results in a decrease of the occupied Fe3d
PDOS in the minority spin state. This leads to anincrease of mFe in the AFM and FM states andimprovement of the magnitude relation of mFe values
between the AFM and FM states. From the above results,
the effect of U plays a crucial role in discussing theAFM-FM phase transition.

Fig. 4 DOS near EF of (a) the FM state, (b) the FM state
for both the majority spin and the minority spinstates, (c) the AFM state, and (d) the AFM statefor each of the majority spin and the minority spinstates.

Fig. 5 PDOS near EF of (a) an Fe atom of the FM state,(b) an Rh atom of the FM state, (c) an Fe atom of
the AFM state and (d) an Rh atom of the AFM
state.

Table I Comparison of physical quantities obtained in the present analysis using the GGA+ U method andthose reported in experiments.

presentwork
Ref. (8)Ref. (9)Ref. (10)Ref. (11)Ref. (12)Ref. (13)Ref. (14)Ref. (17)Ref. (18)
Ref. (19)

AFM

FMaAFM

aFM

Å

Å

2.99

3.01

2.987
2.981

2.997
2.999

ΔE

AFMFMHc(0)

mFe

mRh

mFe

mRhmeV

μB

μB

μB

μB

T

2.71

3.31

0

3.31

1.04

21.6

3.30

0

3.17

0.97

2.80

D(EF)FM /D(EF)AFM

γFM/γAFMi)

4.72

2.41
23.4
29.7
21.2

3.75
5.95i) Ratio of Sommerfeld coefficient γ of the FM state and that of the AFM state (γFM/γAFM),ii) Bulk modulus of the AFM state (BAFM), iii) Bulk modulus of the FM state (BFM)

AFMFMBAFMii)

BFMiii)

GPa

GPa

195.7

210.6

142±14133±20142158

The band structures of the AFM and FM states areinvestigated and are shown in Figs. 6(a)-(c), respectively.
The two band structures are considerably different. In
particular, in the FM state, the number of branches
crossing the Fermi surface in the minority spin state ishigher than that in the majority spin state. In the AFMstate, a hole pocket appears at the center of the Г point.

Fig. 6 Obtained band structures of (a) the FM state forthe majority spin state, (b) the FM state for theminority spin state, and (c) the AFM state.
4. Conclusion
In summary, the electronic and magnetic structures ofFeRh are investigated by using first principles
calculations including appropriately chosen values of U
for the Fe3d and Rh4d electrons, and we succeed inexplaining the reported AFM-FM phase transition
experiments for the first time by obtaining ΔE consistent
with the experimental reports, also for the first time.
Moreover, other obtained physical quantities are also
consistent with the experimental reports. We willattempt to clarify the mechanism behind the AFM-FM
phase transition by using the knowledge obtained in thepresent analysis.
Acknowledgements The authors are indebted tocolleagues on our team for advising us on computer
system support.
References
1) K. G. Sanderman: Scripta Mater., 67, 566 (2012).
2) A. Fujita, S. Fujieda, Y. Hasegawa, and K. Fukamichi: Phys.
Rev. B, 67, 104416 (2003).
3) N. H. Dung, Z. Q. Ou, L. Caron, L. Zhang, D. T. C. Thanh, G.
A. de Wijs, R. A. de Groot, K. H. J. Buschow, and E. Brück:Adv. Energy Mater., 1, 1215 (2011).
4) M. P. Annaorazov, K. A. Asatryan, G. Myalikgulyev, S. A.
Nikitin, A. M. Tishin, and A. L. Tyurin: Cryogenics, 32, 867(1992).
5) L. Caron, X. F. Miao, J. C. P. Klaasse, S. Gama, and E. Brück:Appl. Phys. Lett., 103, 112404 (2013).
6) A. M. Tishin and Y. I. Spichkin: Int. J. Refrig., 37, 223 (2014).
7) M. Fallot: Ann. Phys., 10, 291 (1938).

8) A. I. Zakharov, A. M. Kadomtseva, R. Z. Levitin, and E. G.
Ponyatovskii: Sov. Phys. JETP, 19, 1348 (1964).
9) T. Miyanaga, T. Itoga, T. Okazaki, and K. Nitta: J. Phys. Conf.
Ser., 190, 012097 (2009).
10) D. W. Cooke, F. Hellman, C. Baldasseroni, C. Bordel, S.
Moyerman, and E. E. Fullerton: Phys. Rev. Lett., 109,255901 (2012).
11) G. Shirane, R. Nathans, and C. W. Chen: Phys. Rev., 134,A1547 (1964).
12) J. B. McKinnon, D. Melville, and E. W. Lee: J. Phys. C, 3,S46 (1970).
13) B. K. Ponomarev: Sov. Phys. JETP, 36, 105 (1973).
14) T. Zhou, M. K. Cher, L. Shen, J. F. Hu, and Z. M. Yuan: Phys.
Lett. A, 377, 3052 (2013).
15) J.-S. Lee, E. Vescovo, L. Plucinski, C. M. Schneider, andC.-C. Kao: Phys. Rev. B, 82, 224410 (2010).
16) A. X. Gray, D. W. Cooke, P. Krüger, C. Bordel, A. M. Kaiser,S. Moyerman, E. E. Fullerton, S. Ueda, Y. Yamashita, A.
Gloskovskii, C. M. Schneider, W. Drube, K. Kobayashi, F.
Hellman, and C. S. Fadley: Phys. Rev. Lett., 108, 257208(2012).
17) P. Tu, A. J. Heeger, J. S. Kouvel, and J. B. Comly: J. Appl.
Phys., 40, 1368 (1969).
18) J. Ivarsson, G.R. Pickett, and J. Tóth: Phys. Lett. A, 35, 167(1971).
19) J. A. Ricodeau and D. Melville: J. Phys. F, 2, 337 (1972).
20) C. Kittel: Phys. Rev., 120, 335 (1960).
21) V. L. Moruzzi and P. M. Marcus: Phys. Rev. B, 46, 2864
(1992).
22) V. L. Moruzzi and P. M. Marcus: Solid State Commun., 83,735 (1992).
23) A. Szajek and J. A. Morkowski: Physica B, 193, 81 (1994).
24) M. E. Gruner, E. Hoffmann, and P. Entel: Phys. Rev. B, 67,064415 (2003).
25) R. Y. Gu and V. P. Antropov: Phys. Rev. B, 72, 012403 (2005).
26) L. M. Sandratskii and P. Mavropoulos: Phys. Rev. B, 83,174408 (2011).
27) Y. Kaneta, S. Ishino, Y. Chen, S. Iwata, and A. Iwase: Jpn. J.
Appl. Phys., 50, 105803 (2011).
28) A. Deák, E. Simon, L. Balogh, L. Szunyogh, M. dos SantosDias, and J. B. Staunton: Phys. Rev. B, 89, 224401 (2014).
29) G. Kresse and J. Furthmüller: Comput. Mat. Sci., 6, 15(1996).
30) G. Kresse and J. Furthmüller: Phys. Rev. B, 54, 11169(1996).
31) P. E. Blöchl: Phys. Rev. B, 50, 17953 (1994).
32) G. Kresse and D. Joubert: Phys. Rev. B, 59, 1758 (1999).
33) J. P. Perdew, K. Burke, and M. Ernzerhof: Phys. Rev. Lett.,
77, 3865 (1996).
34) V. I. Anisimov, J. Zaanen, and O.K. Andersen: Phys. Rev. B,44, 943 (1991).
35) S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys,and A. P. Sutton: Phys. Rev. B, 57, 1505 (1998).
36) M. Methfessel and A. T. Paxton: Phys. Rev. B, 40, 3616(1989).
37) H. J. Monkhorst and J. D. Pack: Phys. Rev. B, 13, 5188
(1976).
38) ΔE obtained with Ecut 830 eV and an 11×11×11 mesh iscompared with that obtained with Ecut 680 eV and an
11×11×11 mesh. The difference in ΔE for the two conditions
is less than 0.01 meV. ΔE obtained with Ecut 830 eV and an
11×11×11 mesh is compared with that obtained with Ecut
830 eV and a 15×15×15 mesh. The difference in ΔE for thetwo conditions is about 0.1 meV.
Received Mar. 04, 2016; Accepted Apr. 23, 2016