Full text of DOI:10.1002/(SICI)1521-3951(200008)220:2<959::AID-PSSB959>3.0.CO;2-K

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G. M. Buendia and N. Hurtado: Numerical Study of a 3D Mixed Ising Ferrimagnet
959
phys. stat. sol. ꢀb) 220, 959 ꢀ2000)
Subject classification: 75.10.Hk; 75.40.Mg; 75.50.Gg
Numerical Study
of a Three-Dimensional Mixed Ising Ferrimagnet
in the Presence of an External Field
Â
G. M. Buendia and N. Hurtado
Â
Department of Physics, Universidad Simon BolÂõvar, Apartado 89000, Caracas 1080, Vene-
zuela
ꢀReceived January 24, 2000; in revised form April 11, 2000)
We present a numerical study based on a Monte Carlo algorithm of the magnetic properties of a
mixed Ising ferrimagnetic model on a cubic lattice where spins s ˆ Æ1=2 and spins S ˆ 0; Æ1 are
in alternating sites on the lattice. We carried out exact ground state calculations and employ a
Monte Carlo simulation to obtain the finite-temperature phase diagram of the model. A compen-
sation point appears when the next-nearest-neighbor interaction between the spins s ˆ Æ1=2 ex-
ceeds a minimum value. We found a strong dependence of the compensation temperature with the
interactions in the Hamiltonian, particularly the crystal and the external field. An applied field can
change the range of values of the compensation temperature from zero up to a maximum value
that depends on the field.
1.Introduction
Ferrimagnetic alloys have been the object of intense experimental studies because their
role in high-density magneto-optical recording [1, 2]. In a ferrimagnet the different tem-
perature dependences of the sublattice magnetizations raise the possibility of the exis-
tence of compensation temperatures: temperatures below the critical point where the
total magnetization is zero [3]. It has been shown that the coercivity is a function of the
temperature and that it has a peak at the compensation point, favoring the creation of
small, stable, magnetic domains [4, 5]. In magneto-optical recording devices the coerciv-
ity is changed by local heating of the media with a focused beam. Direct overwrite
capability has been demonstrated in amorphous ferrimagnetic films with compensation
temperatures higher than room temperatures [2, 6]. Also, the strong and continuous
effort to synthesize low-density, transparent magnets, with spontaneous moments at
room temperature, has reopened the interest in studying the ferrimagnetic ordering that
plays a fundamental role in these materials [7 to 9]. In this work we describe a three-
dimensional, spin 1=2±spin 1, mixed Ising model with ferrimagnetic ordering that, un-
der certain conditions, exhibits compensation points. Mixed Ising systems have been
studied before with mean-field [10, 11] and nonperturbative techniques [12 to 14] pro-
viding interesting information about the interactions responsible for the existence of
compensation points. In the present study we extend previous Monte Carlo simulations
of a two dimensional model to a three-dimensional case and include the effect of exter-
nal fields. We are interested particularly in the ground state, the finite-temperature
phase diagrams, and the location and characterization of the compensation tempera-
tures.

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G. M. Buendia and N. Hurtado
960
2.The Model and Its Ground States
We study a three-dimensional Ising system with spins s ˆ Æ1=2 and S ˆ Æ1; 0 located
in alternating sites of a cubic lattice. A Hamiltonian for this model that includes nearest
neighbor interactions between the S and the s spins, next-nearest neighbor interactions
between the s spins, crystal field and external magnetic field can be written as
ꢀ
ꢁ
P
P
P
P
P
H ˆ ÀJ1
siSj À J2
sisk À D
S2j À H
si ‡
Sj
;
ꢀ1†
j
i
j
hnni
hnnni
where J1 and J2 are the exchange-interaction parameters, D is the crystal field and H is
the external field, all in energy units.
The models to be considered will be labeled by enumerating the parameters different
from zero in the Hamiltonian. For example the J1 ±D one is the model with all the
parameters in the above Hamiltonian zero except J1 and D. Since we are interested in
studying the possible existence of compensation points, all our numerical results are
obtained for the ferrimagnetic coupling, J1 < 0, between nearest neighbors. We include
only the next-nearest neighbor interaction, J2, between the s spins because previous
work done in two dimensional models shows that it is responsible for the existence of
compensation temperatures [12, 13].
Since for our Hamiltonian, the ground state is translationally invariant, to obtain the
ground state diagrams we only have to calculate the energy of the configurations of a
2 Â 2 Â 2 unit cell [15]. This cell has 24 Â 34 configurations. Taking into account rota-
tional symmetries we found that there are 75 different configurations. The ground state
of the model depends on the particular choice of parameters in the Hamiltonian, as can
be seen in Figs. 1 and 2. In Fig. 1 we show the ground state diagram for the J1 ÀJ2ÀD
Fig. 1. Ground state diagram for the J1 ÀJ2 ÀD model ꢀJ1 ˆ À1=2). The configurations of the unit
cell in each of the four regions are labeled as in Table 1

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Numerical Study of a Three-Dimensional Mixed Ising Ferrimagnet
961
Fig. 2. Ground state diagram for the J1 ÀJ2 ÀDÀH model ꢀJ1 ˆ À1=2, D ˆ 1, H > 0). The config-
urations of the unit cell in each of the six regions are labeled as in Table 1
Ta ble 1
Ground-state configurations, degeneracies and energies of the 2 Â 2 Â 2 unitary spin cells
indicated in Figs. 1 and 2. The symbol convention is: S ˆ *, +, 0 and s ˆ ", #. The
upper line gives the clockwise alignment of the spins at the top face of the cube, and the
lower one the clockwise alignment at the bottom face
configuration
degeneracy
energy per site
3
1
2
*
#
+
"
0
"
0
#
0
#
*
"
*
"
*
#
*
#
*
"
#
*
"
+
"
0
#
0
"
0
#
+
"
*
"
*
"
*
"
*
*
#
+
"
0
"
0
#
0
#
*
"
*
"
*
#
*
#
*
#
#
*
"
+
"
0
#
0
"
0
#
+
"
*
#
*
"
*
"
*
1
1
1
1
6
6
1
4
6
4
E ˆ 6J1 À J2 À 4D À 2H
2
3
2
E ˆ 6J1 À J2 À 4D ‡ 2H
3
2
3
E ˆ À J2 À 2H
3
2
4
E ˆ À J2 ‡ 2H
1
E ˆ J2
2
5
1
2
6
E ˆ 2J1 ‡ J2 À 4D
3
2
7
E ˆ À6J1 À J2 À 4D À 6H
8
E ˆ 3J1 À 4D À 3H
E ˆ 12 J2 À 4D À 4H
E ˆ À3J1 À 4D À 5H
9
10

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G. M. Buendia and N. Hurtado
962
model ꢀH ˆ 0 and J1 < 0), there are four different regions. The ground state diagram
for the J1 ÀJ2 ÀDÀH model ꢀJ1 < 0, H > 0 and D > 0) is divided in six regions as seen
in Fig. 2. The configurations of the unit cell in each region are labeled as in Table 1.
The equations of the boundaries between regions are obtained by pairwise equating the
ground-state energies.
3.The Monte Carlo Simulations
We used standard importance sampling methods [16], to simulate the model described
by equation ꢀ1) on a cubic lattice of volume L Â L Â L with periodic boundary condi-
tions. Configurations were generated by sequentially traversing the lattice and making
single-spin flip attempts at each site. The flips are accepted or rejected according to a
heat-bath algorithm. One Monte Carlo Step ꢀMCS) was defined as L Â L Â L at-
tempted spin moves, 104 MCS were used to obtain each data point in lattices with
L ˆ 40, after discarding the first 103 steps. We define b ˆ 1=kBT and take Boltzmann's
constant kB ˆ 1. Our program calculates the internal energy per site,
1
L3
E ˆ
hHi ;
ꢀ2†
the specific heat per site,
b2
L3
2
C ˆ
‰hH2i À hHi Š ;
ꢀ3†
ꢀ4†
the sublattice magnetizations per site,
ꢂ
ꢃ
DP
E
P
2
L3
2
L3
M1 ˆ
Si
;
M2 ˆ
sj
;
i
j
the total magnetization per site, M ˆ ꢀM1 ‡ M2†=2, and the susceptibility,
2
c ˆ bꢀhM2i À hMi † :
ꢀ5†
In order to locate the compensation point, we define an order parameter per spin as
ꢂ
ꢃ
P
1
L3
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
ꢄ
O ˆ
ꢀSi ‡ sj†
;
ꢀ6†
ij
which is equivalent to the average of the absolute value of the total magnetization.
At the compensation temperature, Tcomp, the sublattice magnetizations have equal
magnitude and opposite signs
jM1ꢀTcomp†j ˆ jM2ꢀTcomp†j
ꢀ7†
and
sign ‰M1ꢀTcomp†Š ˆ Àsign ‰M2ꢀTcomp†Š ;
ꢀ8†
such that the total magnetization is zero. The Tcomp is always lower than the critical
temperature, Tc.