Field-induced magnetic ordering in NiCl
2
" 4SC„ NH
2
…
2
A. Paduan-Filho,* X. Gratens, and N. F. Oliveira, Jr.
Instituto de Fı ´sica, Universidade de Sa ˜ o Paulo, Caixa Postal 66.318, 05315 – 970 Sa ˜ o Paulo, Sa ˜ o Paulo, Brazil
Received 24 October 2003; published 23 January 2004
We have investigated the low-temperature T field-induced ordering in NiCl
2
• 4SC(NH
2
)
2
, from magnetiza-
tion measurements. This is an S =1 system with a large single-ion uniaxial anisotropy leaving a singlet ground
state. An external magnetic field ( B) parallel to the symmetry axis induces an antiferromagnetically ordered
phase for B
c 1
B B
c 2
. The establishment of this long-range order can be regarded as a Bose-Einstein con-
densation of magnons. The phase boundaries are predicted to obey power laws B
c
( T ) -B
c
(0) T
. The
determined phase boundaries could be fitted to power laws yielding
1
=2.630.10 for B
c 1
( T ) and
2
=2.550.10 for B
c 2
( T ).
DOI: 10.1103/PhysRevB.69.020405 PACS numbers: 75.10.Jm, 75.30.Kz
Magnetic systems having a singlet ground state, and with
an excitation gap that can be overcome by magnetic fields
obtainable in laboratory, have received considerable attention
lately,
1–9
due to the interesting quantum effects exhibited.
Antiferromagnetic spin dimers,
1–6
S =1 antiferromagnetic
chains,
7
S =1/2 alternating chains,
8
and even-leg spin
ladders
9
are examples. In these systems, an external mag-
netic field can lower one of the upper energy levels, which
eventually get close to the ground level creating the neces-
sary degeneracy for the appearance of magnetic order. The
presence of a small exchange interaction can lead to long-
range order LRO at a critical field B
c 1
. Further increasing
the field should lead to a second transition at B
c 2
after which
the system saturates. As pointed out many years ago,
10
the
onset of this LRO can be viewed as a Bose-Einstein conden-
sation BEC of magnons and recent papers have addressed
specifically to this point.
1–6
The first attempt to interpret this field-induced order in the
context of BEC was done by Nikuni et al.
2
Their Hartree-
Fock-Popov calculation HFP-BEC was able to explain
qualitative aspects of the experimental data for the com-
pound TlCuCl
3
a coupled spin-dimer system which were in
clear disagreement with the predictions of mean-field theory
MFT.
11,12
One of such aspects is the behavior of the mag-
netization M as a function of temperature T near B
c 1
where a
cusplike dip was observed. Another aspect is the power-law
behavior of the observed phase boundary
B
c
T -B
c
0 T
. 1
Although the BEC picture does predict a power law, in quali-
tative agreement with the data, the value of
1
calculated by
Nikuni et al.
2
for B
c 1
( T ),
1
=1.5, differed somewhat from
the best fit to the TlCuCl
3
data which yielded
1
=2.2.
2,5
Subsequent neutron-diffraction studies, confirmed the BEC
character of the transition.
1,6
To further investigate this point, Wessel et al.
3
have ob-
tained the entire field-induced phase diagram applying quan-
tum Monte Carlo technique to the three-dimensional 3D
antiferromagnetic spin-1/2 Heisenberg model with spatially
anisotropic exchange couplings. Magnetization simulations
with Hamiltonians appropriate for weakly coupled dimers
such as TlCuCl
3
) and weakly coupled Heisenberg ladders
were made. The general characteristics of the M ( T ) curve
were found to agree qualitatively with the isotropic HFP-
BEC. Both phase boundaries, B
c 1
( T ) and B
c 2
( T ), deter-
mined from the extrema in M ( T ), could be fitted to power
laws up to temperatures close to the highest-ordering tem-
perature, for both Hamiltonians. The best exponents for the
dimers case were
1
=2.70.2 and
2
=2.30.2 for B
c 1
( T )
and B
c 2
( T ), respectively, and
1
=3.10.2 and
2
=1.8
0.2 for the ladder case. The values of for B
c 1
, however,
are quite different from the isotropic HFP-BEC prediction.
So, although at least part of the disagreement could be attrib-
uted to the inadequacy of the Hartree-Fock approach in the
critical region, it was concluded that the values of the expo-
nents should depend strongly on the dimensionality and
quantum dynamics of the particular system.
So far, the focus of the attention has been on low-
dimensional systems such as dimers and chains. Field-
induced order, however, can also exist in three-dimensional
compounds with large single-ion anisotropy, and they also
should exhibit BEC.
2
In an S =1 spin system, such as in a
Ni
++
compound, for instance, a cubic crystalline field leaves
a spin triplet as the ground state and distortions of the cubic
symmetry can lift the degeneracy. An axially symmetric dis-
tortion of the proper sign can split the triplet into a lower
singlet separated from an excited doublet by an energy | D|
usually D is chosen to be negative in this case. With a
singlet ground state, small antiferromagnetic exchange inter-
actions (2 z | J | | D| , where z is the number of neighbors
cannot produce LRO. The presence of an external magnetic
field parallel to the distortion axis, however, induces LRO
between B
c 1
( T ) and B
c 2
( T ). MFT Refs. 12 and 13 gives
for isotropic J and T =0) g
B
B
c 1
(0) =| D| -2 z | J | = and
g
B
B
c 2
(0) =| D| +4 z | J | , where g
is the parallel spectro-
scopic factor and
B
is the Bohr magneton. One example of
such a system is the compound NiCl
2
.4SC(NH
2
)
2
, dichloro-
tetrakisthiourea-nickel II, hereon referred to as DTN, for
short.
14
In this paper we report magnetization measurements
on DTN in fields and temperatures covering the entire or-
dered phase. B
c 1
( T ) and B
c 2
( T ) were determined and could
be fitted to Eq. 1 yielding values for
1
and
2
.
DTN is a tetragonal crystal, space group I 4, with two
molecules in the unit cell. Its magnetic susceptibility up to an
external field of 7 K has already been investigated.
14
The
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