Curie temperatures and modified de Gennes factors of rare earth nitrides

Article abstract of DOI:10.1016/j.ssc.2011.07.021

The Curie temperatures T_C of the nitrides of the rare earths (Gd, Tb, Dy, Ho, and Er), including binary systems, were investigated. T _C was found to be approximately proportional to the de Gennes factor, ξ=(g-1)²J(J+1), where g is the Lande? g-factor and J is the total angular momentum quantum number of a trivalent rare earth (RE). This proportionality was significantly improved by introducing a modified de Gennes factor, ξ_bi. The conventional de Gennes factor ξ indicates the exchange interaction given by the inner product of the effective spin components of ions of the same kind, whereas our modified de Gennes factor ξ_bi also considers interactions between different kinds of ions and statistical factors calculated on the basis of the binomial distribution. The good proportionality obtained between T_C and ξ_bi indicates that the spin components of RE ions interact with each other. This interaction is considered to be responsible for the ferromagnetism of these nitrides (including binary systems). These considerations were supported by the synthesis of and magnetization measurements on Gd_xEr_1-xN (x=0, 0.25, 0.5, 0.75, 1) samples.

Full text of DOI:10.1016/j.ssc.2011.07.021

Solid State Communications 151 (2011) 1602–1604
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Solid State Communications
journal homepage: www.elsevier.com/locate/ssc
Curie temperatures and modified de Gennes factors of rare earth nitrides
∗
Yusuke Hirayama , Takashi Nakagawa, Takao A. Yamamoto
Graduate School of Engineering, Osaka University, Yamadaoka 2-1, Suita, Osaka 565-0871, Japan
a r t i c l e i n f o
a b s t r a c t
Article history:
The Curie temperatures TC of the nitrides of the rare earths (Gd, Tb, Dy, Ho, and Er), including binary
systems, were investigated. TC was found to be approximately proportional to the de Gennes factor,
ξ = (g −1)2J(J +1), where g is the Landé g-factor and J is the total angular momentum quantum number
of a trivalent rare earth (RE). This proportionality was significantly improved by introducing a modified
de Gennes factor, ξbi. The conventional de Gennes factor ξ indicates the exchange interaction given by the
inner product of the effective spin components of ions of the same kind, whereas our modified de Gennes
factor ξbi also considers interactions between different kinds of ions and statistical factors calculated on
the basis of the binomial distribution. The good proportionality obtained between TC and ξbi indicates that
the spin components of RE ions interact with each other. This interaction is considered to be responsible
for the ferromagnetism of these nitrides (including binary systems). These considerations were supported
by the synthesis of and magnetization measurements on GdxEr1−xN (x = 0, 0.25, 0.5, 0.75, 1) samples.
© 2011 Elsevier Ltd. All rights reserved.
Received 16 March 2011
Received in revised form
11 June 2011
Accepted 14 July 2011
by E.V. Sampathkumaran
Available online 21 July 2011
Keywords:
A. Rare earth nitride
D. Exchange interaction
D. Curie temperature
D. de Genne factor
1. Introduction
Bozorth [25] reported that the TMT values of RE metals and RE al-
loys exhibit a ξ2/3 dependence (ξ values of alloys were obtained
by averaging the ξ values of the alloy components weighted by
their atomic fractions); however, no theoretical explanation was
provided for this relationship.
Rare earth (RE) nitrides form a class of materials with inter-
esting electronic and magnetic properties [1–3], which originate
from their open 4f shells. Recent theoretical calculations predict
that they have variable electronic states (metallic, semiconducting,
insulating, and half-metallicity) and magnetic properties (param-
agnetism, ferromagnetism, and antiferromagnetism) [4–9].
We have investigated the magnetic properties of the RE nitrides
LnN (Ln = Gd, Tb, Dy, Ho, or Er), including their binary systems,
with the aim of using them in magnetic refrigeration for liquefying
hydrogen [27–34]. Magnetic refrigeration generates a cooling heat
of Q = T1S, where 1S is the magnetic entropy change [35].
A large 1S can be extracted from a ferromagnet by sweeping
an externally applied field and using a temperature close to the
TC . We found that these nitrides are ferromagnets with Curie
temperatures below liquid nitrogen temperature, 77 K. To produce
an efficient magnetic refrigeration system for cooling hydrogen
from liquid nitrogen temperature to its liquefaction temperature
(77–20 K), it is desirable to use several materials whose TC values
cover this wide range.
In this paper, we present the results of precise measurements
of the correlation between TC and the de Gennes factor. This is
important not only for deriving a universal relation between TC and
ξ for RE nitrides (including binary systems), as has been done for
RE metals and alloys, but also for gaining a deeper understanding
of the magnetism of this class of materials.
The magnetism of RE nitrides has been investigated since the
1960s. NdN, GdN, TbN, DyN, HoN, and ErN exhibit ferromag-
netism with Curie temperatures TC of 28 [10], 58 K–75 [11–18],
42 [19], 26 [20], 18 [19], and 5 K [10], respectively. However,
some controversy still remains about the magnetism of GdN. Some
reports claim that GdN is ferromagnetic with a TC of 65–75 K
[11–17]. Li et al. reported that stoichiometric GdN is ferromag-
netic [18]. Theoretical studies by Kuznietz [21] and Kasuya [22]
indicate that the magnetism of GdN originates from the RKKY in-
teraction, which is related to the cross-process between f–d mix-
ing and f–d exchange interactions. The transition temperatures of
RE intermetallic compounds [23,24] and RE alloys [25] have been
described in terms of the de Gennes factor ξ, which indicates the
exchange interaction of the spin component of the total quantum
number [26]. On the basis of the variation in the magnetic transi-
tion temperature TMT with the de Gennes factor ξ, the exchange
interaction has been attributed mainly to the R–R exchange inter-
action in RNi2Mn (R = Tb, Dy, Ho, and Er) [23], RNi2, and RMn2 [24].
2. Experimental
GdxEr1−xN (x = 0, 0.25, 0.5, 0.75, 1) samples were synthesized
by carbothermic reduction in a nitrogen gas stream. Powders of
Gd2O3 and Er2O3 (99.99% purity) and an appropriate amount of
∗
Corresponding author. Fax: +81 6 6879 7886.
E-mail address: y-hirayama@mit.eng.osaka-u.ac.jp (Y. Hirayama).
0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ssc.2011.07.021

Y. Hirayama et al. / Solid State Communications 151 (2011) 1602–1604
1603
calculated from Arrott plots generated from data sets for magnetic
field strengths of 5 T, 4 T, and 3 T in Fig. 2(b) and were 11 K, 26 K,
and 37.5 K for x = 0.25, 0.5, and 0.75 in GdxEr1−xN, respectively.
Fig. 3(a) shows TC –ξ plots for the RE nitrides (including binary
systems) that we have so far synthesized; the plots assume that the
RE nitrides have trivalent states. The de Gennes factors of binary RE
nitrides were obtained by weighting each ξ by the atomic fraction
of the RE. All the data points lie relatively close to a line passing
through the origin. The LMS method was applied and a correlation
coefficient of R2 = 0.965 was obtained. However, in view of its
physical meaning and algebraic expression, we considered that
it is not appropriate to apply the averaging method to binary RE
nitrides for the following reason. The defining expression, ξ =
(g − 1)2J(J + 1), corresponds to the inner product of two parallel
0.5
vectors of the same kind with magnitudes of (g − 1){J(J + 1)}
,
Fig. 1. Lattice parameter as a function of average radius for trivalent rare earth ions.
which is the effective spin component along the direction of the
total angular momentum J. Thus, the de Gennes factor indicates
the exchange interaction between the spin components of the
same kinds of ions. However, binary nitrides (e.g., GdxEr1−xN)
have three kinds of neighboring pairs (e.g., Gd–Gd, Gd–Er, and
Er–Er) and thus three kinds of interactions. A more suitable
averaging method is required for such systems. Although Fig. 3(a)
exhibits a reasonable proportionality, careful observation reveals a
discernible tendency to deviate from the LMS line, showing slight
concavity; this deviation is most obvious in the plots of the present
binary GdxEr1−xN samples.
The line was determined by the least mean square method.
amorphous carbon were thoroughly mixed with a mortar and
pestle. The powder mixtures were shaped into pellets and heated
at 1500 °C in a reaction tube through which 99.9995% purity
nitrogen gas was flowed for 8 h. The phases in the products
were identified from powder X-ray diffraction (XRD) patterns
obtained using a diffractometer (RINT Ultima+; Rigaku Corp.) with
Cu Kα radiation. Their magnetizations were measured using a
magnetometer (PPMS or MPMS system; Quantum Design Corp.)
under various applied fields H and at various temperatures T. The
ferromagnetic Curie temperature, TC , was calculated by drawing an
Arrott plot [36].
We introduce the following three terms to account for the three
interactions in binary nitrides:
ξGd–Gd = (g − 1)J · (g − 1)J = (g − 1)2J(J + 1)
3. Results and discussion
′
′
′
′
′
′
′
ξEr–Er = (g − 1)J · (g − 1)J = (g − 1)2J (J + 1)
(1)
′
′
The XRD patterns obtained for GdxEr1−xN (x
=
0.25, 0.5,
ξGd–Er = (g − 1)J · (g − 1)J


0.75) indicate that they are single phases of the mononitride with
the NaCl structure. The lattice parameters of GdxEr1−xN were
determined by Cohen’s method, which demonstrated the validity
of Vegard’s rule, as we have previously reported [30,31]. Fig. 1
shows the lattice parameter plotted against the average radius
of the trivalent ions of the constituent elements, together with
previously reported data. The data for the binary nitrides are
plotted against the effective ionic radius, which was calculated by
weighting the radii of the two constituent elements by their atomic
ratios. All the data points lie close to the line determined by the
least mean square (LMS) method, indicating that all the nitrides
belong to the same class of material. GdN and ErN have a lattice
constant mismatch of about 3%. Fig. 2(a) shows the temperature
′
′
′
= (g − 1) J(J + 1) · (g − 1) J (J + 1)
′
′
where J (J ) is the quantum number and g (g ) is the Lande g-
factor of Gd3+ (Er3+). For binary RE nitrides, the de Gennes factor
was modified by assuming that it consists of three components,
ξGd–Gd, ξEr–Er, and ξGd–Er. Assuming that the Gd3+ and Er3+ ions
are randomly arranged, the modified de Gennes factor ξbi will be
the expected value obtained by summing these three terms with
statistical factors based on the binomial distribution. Note that
there are 12 nearest-neighbor sites in the NaCl structure.

12
−
n
ξbi = x
12Cnxn(1 − x)12−nξGd–Gd
12
n=0
dependences of the magnetization of the present Gd0.5Er0.5
N

12 − n
sample for various applied magnetic fields. The samples with other
+
12C12−nxn(1 − x)12−nξGd–Er
compositions exhibited similar behavior. Their TC values were
12
a
b
Fig. 2. (a) Magnetization data for Gd0.5Er0.5N under different magnetic fields. (b) Arrott plot of data obtained under fields of 5, 4, and 3 T at seven different temperatures.

1604
Y. Hirayama et al. / Solid State Communications 151 (2011) 1602–1604
a
b
Fig. 3. Curie temperature of RE nitrides as a function of (a) the de Gennes factor ξ = (g − 1)2J(J + 1) and (b) the modified de Gennes factor ξbi = x2ξGd–Gd + 2x(1 −
x)ξGd–Er + (1 − x)2ξEr–Er
.

12
−
n
ions and statistical factors due to their random arrangement. It was
concluded that the spin components of nearest-neighbor RE ions
mainly interact and give rise to the ferromagnetism of RE nitrides
(including binary systems).
+ (1 − x)
12Cnxn(1 − x)12−nξGd–Er
12
n=0

12 − n
+
12C12−nxn(1 − x)12−nξEr–Er
= x2ξGd–Gd + 2x(1 − x)ξGd–Er + (1 − x)2ξEr–Er
12
References
.
(2)
[1] J.M. Jakobsen, et al., Solid State Commun. 121 (2002) 447.
[2] D.X. Li, Y. Haga, H. Shida, T. Suzuki, Y.S. Kwon, Phys. Rev. B 54 (1996) 10483.
[3] D.X. Li, et al., Physica B 199–200 (1994) 631.
[4] C.G. Duan, et al., Phys. Rev. Lett. 94 (2005) 237201.
[5] P. Larson, W.R.L. Lambrecht, Phys. Rev. B 74 (2006) 085108.
[6] P. Larson, W.R.L. Lambrecht, A. Chantis, M. van Schilfgaarde, Phys. Rev. B 75
(2007) 045114.
[7] F. Leuenberger, A. Parge, W. Felsch, K. Fauth, M. Hessler, Phys. Rev. B 72 (2005)
014427.
[8] C.M. Aerts, et al., Phys. Rev. B 69 (2004) 045115.
[9] A.G. Petukhov, W.R.L. Lambrecht, B. Segall, Phys. Rev. B 53 (1996) 4324.
[10] O. Vogt, et al., J. Alloys Compd. 223 (1995) 226.
[11] D.X. Li, et al., J. Phys.: Condens. Matter. 9 (1997) 10777.
[12] G. Busch, et al., J. Appl. Phys. 38 (1967) 1386.
[13] P. Junod, et al., Phys. Kondens. Mater. 8 (1969) 323.
[14] J.J. Veyssie, et al., Phys. Lett. 13 (1964) 29.
[15] W. Stutius, et al., Phys. Kondens. Mater. 10 (1969) 152.
[16] T.R. McGuire, et al., J. Appl. Phys. 40 (1969) 1009.
[17] R.J. Gambino, et al., J. Appl. Phys. 41 (1970) 933.
[18] D.X. Li, et al., Physica B 199–200 (1994) 631.
[19] A. Vendl, et al., J. Nucl. Mater. 79 (1979) 246.
[20] G. Busch, et al., Phys. Lett. 6 (1963) 79.
[21] N. Kuznietz, et al., J. Appl. Phys. 42 (1971) 1470.
[22] T. Kasuya, et al., J. Magn. Magn. Mater. 167 (1997) L1–L6.
[23] K. Inoue, et al., J. Phys. Soc. Japan 64 (1995) 2175.
[24] J.L. Wang, C. Marquina, M.R. Ibarra, G.H. Wu, Phys. Rev. B 73 (2006) 094436.
[25] R.M. Bozorth, J. Appl. Phys. 38 (1967) 1366.
In Fig. 3(b), TC is replotted against ξbi; the data points for the
binary samples are clearly closer to the LMS line. The correlation
coefficient R2 is 0.978, which is significantly higher than that
(0.965) determined from the TC –ξ plots in Fig. 3(a). The data
points for GdxEr1−xN are much closer to the LMS line. Since
Er and Gd have the greatest difference in the values of (g −
0.5
1){J(J + 1)} of all of the binary systems shown in Fig. 3(b),
their values should be changed the most when the de Gennes
factor is replaced by the modified de Gennes factor (i.e., ξ →
ξbi). Therefore, TC will also consist of three components of the
exchange interaction between nearest-neighbor RE ions; this
is supported by theoretical calculations that predict that the
exchange interaction is mainly the result of nearest-neighbor
magnetic ions [37]. TMT values for RE metals and alloys vary with
ξ2/3 [25], which indicates that in addition to nearest-neighbor
ions, second-nearest-neighbor and third-nearest-neighbor ions
that have higher metallicities may contribute to the magnetism
produced by indirect RKKY interactions [37]. The ferromagnetism
and TC of RE nitrides are determined by the statistically averaged
strength of interaction between randomly distributed nearest-
neighbor pairs (e.g., Gd–Gd, Gd–Er, and Er–Er).
[26] S. Chikazumi, Physics of Ferromagnetism, OXFORD, pp. 181–182.
[27] T.A. Yamamoto, et al., J. Alloys Compd. 17 (2004) 376.
[28] T. Nakagawa, et al., J. Alloys Compd. 408–412 (2006) 191.
[29] S. Nishio, et al., J. Appl. Phys. 99 (2006) 08K901.
[30] T. Nakagawa, et al., J. Alloys Compd. 364 (2004) 53.
[31] T. Nakagawa, et al., J. Alloys Compd. 408–412 (2006) 187.
[32] Y. Hirayama, et al., J. Alloys Compd. 462 (2008) L12–L15.
[33] Y. Hirayama, et al. Material Research Society, Boston, MA, 2008, p. 152.
[34] Y. Hirayama, et al., IEEE. Trans. Magn. 44 (2008) 2997.
[35] V.K. Pecharsky, et al., J. Magn. Magn. Mater. 200 (1999) 44.
[36] A. Arrott, Phys. Rev. 108 (1957) 1394.
4. Conclusions
The TC values of binary RE nitrides are found to have a slight
concave deviation from linearity when they are plotted against
the de Gennes factor ξ. This deviation was removed and a good
proportionality was obtained by adopting a modified de Gennes
factor ξbi that accounts for interactions between different kinds of
[37] C.G. Duan, et al., Appl. Phys. Lett. 88 (2006) 182505.