Anisotropy of grain in nanoscaled magnetic materials
Y. Sun, G. B. Han, B. P. Han, M. Liu, and R. W. Gao
School of Physics and Microelectronics, Shangdong University, Jinan, Shangdong 250100, China
͑Received 10 May 2006; accepted 12 December 2006; published online 26 March 2007͒
The inter-grain exchange-coupling interaction and the anisotropy of grain in nanoscaled magnetic
materials have been studied by putting forward an expression of anisotropy at grain boundary,
K
1
ij
͑r͒, suitable for different coupling conditions. The value of anisotropy at grain interface K
1
ij
͑0͒
has a great effect on ͗K
ij
͘, especially on the average anisotropy of soft-hard phase grains ͗K͘. When
K
1
sh
͑0͒ is smaller, the effect of the hard grain on the anisotropy of soft grain is weak, and ͗K͘
increases monotonously with increasing D. When K
1
sh
͑0͒ is larger, the effect of the hard grain on the
anisotropy of soft grain is strong, and ͗K͘ appears maximum at a certain grain size. When the value
of K
1
ij
͑0͒ is in a certain range, the calculated variations of anisotropy with D are consistent with that
of coercivity given by other authors. © 2007 American Institute of Physics.
͓DOI: 10.1063/1.2435813͔
I. INTRODUCTION
Nanoscaled magnets have attracted much attention for
potential magnetic properties due to the exchange-coupling
interaction between grains. The theoretical energy product of
nanocomposite magnets can be as high as 1 MJ/m
3
.
1
How-
ever, experimental studies have shown that the energy prod-
uct of nanocomposite magnets is far below the theoretical
value. The severe decrease of coercivity is thought to be the
main reason for the low energy product, and the coercivity is
mostly determined by the anisotropy and microstructure of
nanoscaled materials. The anisotropy of the grain is influ-
enced by the inter-grain exchange-coupling interaction.
Herzer
2
pointed out that when the grain size D is smaller
than the exchange correlation length L
ex
, the anisotropy con-
stant should be written as ͗K͘= K
1
͑D /L
ex
͒
3/2
, where K
1
is the
first anisotropy constant, when D Ͼ L
ex
, ͗K͘ = K
1
. Arcas et
al.
3
put forward the theory of the partial exchange-coupling
interaction, which insisted that the anisotropy at the grain
boundary could be expressed by K
1
/N
1/2
, where N is the
average neighbor grain number that is close to six, and the
anisotropy in the inner part of the grain still to be K
1
. Both of
them regard the anisotropy of the coupled part of the grain as
a fixed constant smaller than K
1
. Han et al.
4
considered the
anisotropy at the grain boundary which should vary with the
distance r to the grain surface, and described it as K
1
͑r͒
=K
1
͑2r/ L
ex
͒
3/2
. However, the anisotropy at the soft-hard
grain boundary is expressed as K
1
͑r͒= K
1
h
−͑K
1
h
−K
1
s
͒
ϫ͓͑L
ex
/2ϯr͒ / L
ex
͔
3/2
, where K
1
s
and K
1
h
are the common first
anisotropy of soft and hard grains, respectively. Although
Ref. 4 pointed out the character of continuous variation of
anisotropy at the grain boundary, however, the anisotropy at
the hard-hard grain interface is smaller than that of the soft-
hard grain interface according to the equations given by Han
et al., which is not consistent with fact. In our opinion, the
anisotropy at the grain boundary should vary continuously,
the anisotropy at the hard-hard grain interface should be
larger than that of the soft-hard grain interface, and the an-
isotropy of the soft grain should be enhanced and that of the
hard grain be reduced due to the exchange-coupling interac-
tion between soft and hard grains. Based on the points men-
tioned above, we put forward an expression of anisotropy at
grain boundary, K
1
ij
͑r͒, suitable for a different coupling con-
dition, which may be helpful for investigations on coercivity.
II. ANISOTROPHY AT GRAIN BOUNDARY AND
AVERAGE ANISOTROPHY OF GRAIN
When the grains of two phases ͑i , j͒ couple each other,
the anisotropy at the grain boundary is supposed to vary in a
similar manner as that in the inhomogeneous region of the
grain proposed by Kronmüller et al.,
5
and can be expressed
as
K
1
ij
͑r͒ = K
1
i
− ⌬K
ij
͑͑L
ex
ij
−2r/L
ex
ij
͒͒
3/2
͑i, j = s, h͒͑1͒
and
⌬K
ij
= K
1
i
− K
1
ij
͑0͒, ͑2͒
where K
1
ij
͑r͒ denotes the anisotropy at i phase grain boundary
and r is the distance to the grain surface. ⌬K
ij
and L
ex
ij
have
similar physical meanings as to K
1
ij
͑r͒. Corresponding to a
different grain interface ͑soft-soft, hard-hard, soft-hard, and
hard-soft͒, ⌬K
ij
can be, respectively, expressed as
⌬K
ss
= K
1
s
− K
1
ss
͑0͒,
⌬K
hh
= K
1
h
− K
1
hh
͑0͒,
⌬K
sh
= K
1
s
− K
1
sh
͑0͒,
⌬K
hs
= K
1
h
− K
1
hs
͑0͒. ͑3͒
The anisotropies at grain interfaces, K
1
ij
͑0͒, determined by
the exchange-coupling degree and microstructure of the
grain boundary satisfy the following relations:
K
1
ss
͑0͒ Ͻ K
1
sh
͑0͒ = K
1
hs
͑0͒ Ͻ K
1
hh
͑0͒. ͑4͒
The variations of anisotropies at different grain boundaries
are shown in Fig. 1.
Assuming that grains are a sphere with diameter D as
illustrated in Fig. 2. When D ϾL
ex
, the exchange-coupling
JOURNAL OF APPLIED PHYSICS 101, 063908 ͑2007͒
0021-8979/2007/101͑6͒/063908/4/$23.00 © 2007 American Institute of Physics101, 063908-1