Magnetocaloric effect in „Gd
x
Tb
1-x
…
5
Si
4
by Monte Carlo simulations
E. P. Nobrega,
1
N. A. de Oliveira,
2
P. J. von Ranke,
2
and A. Troper
1,2
1
Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, RJ, Brazil
2
Universidade do Estado de Rio de Janeiro, Rua São Francisco Xavier 524, Rio de Janeiro 20550-013, RJ, Brazil
Received 24 April 2006; revised manuscript received 31 August 2006; published 27 October 2006
In this work, we calculate the magnetocaloric effect in the series of compounds Gd
x
Tb
1-x
5
Si
4
. We use the
classical Monte Carlo simulation to deal with the 4 f spin-spin interaction as well as the disorder at the
rare-earth sites. The calculated isothermal entropy change and the adiabatic temperature change upon varia-
tions of the magnetic field are in good agreement with the available experimental data.
DOI: 10.1103/PhysRevB.74.144429 PACS numbers: 75.30.Sg, 75.10.Dg, 75.20.En
I. INTRODUCTION
The magnetocaloric effect in rare-earth metals and their
alloys has been intensively studied in the literature.
1–5
The
series of compounds Gd
5
Si
x
Ge
1-x
4
for x 0.5 undergo first
order transition from the ferromagnetic to the paramagnetic
phase, together with a crystallographic transition from the
orthorhombic phase temperature below T
c
to the mono-
clinic one temperature above T
c
. In the compounds
Gd
5
Si
x
Ge
1-x
4
with x 0.5, the isothermal entropy changes
upon magnetic field variation around the magnetic ordering
temperature
6–9
are very large as compared with that observed
for metallic gadolinium. For x 0.5 the compounds
Gd
5
Si
x
Ge
1-x
4
always exhibit the orthorhombic phase and
undergo a second order magnetic phase transition. The iso-
thermal entropy changes S upon magnetic field variation
in the compounds Gd
5
Si
x
Ge
1-x
4
for x 0.5 are comparable
with the one found in metallic gadolinium. The structural,
magnetic, and thermodynamical properties in the series of
compounds
10
Tb
5
Si
x
Ge
1-x
4
are very similar to those ob-
served in the series of compounds Gd
5
Si
x
Ge
1-x
4
. In the
doped compounds Gd
x
Tb
1-x
5
Si
4
the orthorhombic phase
and the second order magnetic phase transition occur in the
whole range of Gd concentration. The magnetic ordering
temperature in the compounds Gd
x
Tb
1-x
5
Si
4
decreases
from 336 K in Gd
5
Si
4
to 225 K in Tb
5
Si
4
. The magnetoca-
loric effect in the compounds Gd
x
Tb
1-x
5
Si
4
has been ex-
perimentally studied
11
and experimental data of the isother-
mal entropy change S and the adiabatic temperature
change T
ad
upon magnetic field variations, as a function
of Gd concentration, are available.
Despite the great deal of theoretical papers found in the
literature, the theoretical description of the magnetocaloric
effect in doped rare-earth compounds with disorder at the
rare-earth sites has not yet been properly addressed. In order
to calculate the magnetocaloric effect in rare-earth doped
compounds such as Gd
x
Tb
1-x
5
Si
4
, which involve two types
of rare-earth ions, we should go beyond the conventional
molecular field theory. This is because the molecular field
theory replaces the 4 f spin-spin interaction by an interaction
of the local spin with a mean field generated by the first
nearest neighbors. Thus in the usual treatment of the 4 f spin-
spin interaction within the molecular field theory neither
short range interactions, which are very important near the
magnetic phase transition, nor the possibility of having
neighboring sites occupied by different types of rare-earth
ions are considered. Very recently, we have used the classical
Monte Carlo simulation
12,13
to calculate the magnetocaloric
effect in rare-earth compounds with only one type of rare-
earth ions. In those papers, our results show that the Monte
Carlo simulations explain quite well the experimental data of
the magnetocaloric quantities S and T
ad
. Moreover, the
Monte Carlo calculations provides a good description of the
specific heat capacity, even around the magnetic ordering
temperature where the molecular field theory does not work
well.
In the present work, we discuss the magnetic and thermo-
dynamics properties as well as the magnetocaloric effect in
the doped rare-earth compounds Gd
x
Tb
1-x
5
Si
4
, which in-
volve more than one type of rare-earth ions. In the particular
case of such compounds the 4 f spin-spin interaction should
be treated beyond the conventional molecular field theory.
Here, we use the classical Monte Carlo simulations
14,15
where the occupation of a given rare-earth site by a Gd or Tb
ion is randomly determined according to the Gd concentra-
tion. In order to calculate the energy of the system, we use a
model of 4 f -interacting spins and consider the z components
of total angular momentum as quantum quantities, which can
assume discrete values in the interval -J J
z
J. For a given
J
z
, the transverse components J
x
and J
y
were randomly cho-
sen under the condition J
x
2
+ J
y
2
= J
2
- J
z
2
. Within this
approach we restrict the number of available states so that
the upper limit of the magnetic entropy S
mag
= R ln2J +1 is
reproduced, where R is the gas constant. In order to calculate
the magnetocaloric quantities, i.e., the isothermal entropy
change S and the adiabatic temperature change T
ad
upon variation of the magnetic field, we take the electronic
part of the entropy proportional to the temperature and con-
sider the crystalline lattice entropy in the Debye approxima-
tion. The calculated magnetocaloric quantities S and T
ad
for the compounds Gd
x
Tb
1-x
5
Si
4
are in good agreement
with the available experimental data.
11
II. THE METHOD
In order to calculate the magnetocaloric effect in the com-
pounds Gd
x
Tb
1-x
5
Si
4
, on the basis of Monte Carlo calcula-
tions, we start with the following energy:
PHYSICAL REVIEW B 74, 144429 2006
1098-0121/2006/7414/1444296 ©2006 The American Physical Society 144429-1