Download Estimation of Exchange Interaction Strength of

Estimation of Exchange Interaction Strength of Nd3 Ga5 SiO14 from Mean-Field
Analysis
(Dated: September 7, 2009)
We carried out the estimation of exchange interaction using mean-field theory using the low
temperature magnetization data from Bordet et al [1]. The upper limit of the interaction energy is
found to be on the order of 0.01 meV.
INTRODUCTION
There has been the controversy in estimating the magnitude of the exchange interaction from the magnetic behavior of Nd3 Ga5 SiO14 . At first, the high temperature
Curie-Weiss behavior gives |θCW | >50 K, suggesting significant exchange interaction, although the crystal field
effect was already a concern [2, 3]. Recently the crystal
field issued was investigated by different groups and it
turned out that the model of isolated Nd3+ sites with
crystal field levels can quantitative explain the magnetization experiments, both high temperature and low temperature [4, 5]. However, so far the crystal field study
can not rule out the possibility of hidden exchange interaction that has a lower energy scale, say ∼1 K, due to
the experimental uncertainty. This possibility plus evidence of slower spin fluctuation at low temperature is
so tempting that Zorko et al [6] still chose to use the
Curie-Weiss temperature and mean field theory to estimate exchange couple strength Je ∼1.4 K. One can have
several argument against this estimation: 1) the CurieWeiss behavior has its most contribution from crystal
field levels, 2) even pure Nd has only |θCW |= 11 K and
the superexchange is normally not much higher than the
RKKY interaction. Still, the question is if this estimation
is not valid, what is the valid way? Below, we will try
to make a better estimation of exchange interaction from
low temperature magnetization using mean field theory.
REVIEW OF MEAN FIELD THEORY
The Curie-Weiss mean field theory is also called molecular field theory, for it assumes a so called molecule field
which is proportional to the magnetization M :
Bm = γM.
where {µi } and {Ei } are the set of magnetic moments
and energy levels, N is the number density of magnetic
sites.
Note that Eq. (1) is written for a general case. If
there is no crystal field splitting, one has {Ei = 0}, and
{µi = gJzi µB }, where J is the angular moment quantum
number. Next, we discuss this case. One can also find
the discussion in most text books.
Although Eq. (1) can not be solved analytically, in
principle, as long as γ is known one can solve Eq. (1)
numerically to get M as a function of T and Bext . Following is an approach that helps understand the physics
of Eq. (1).
Defining
µ = gµB J
µB
,
x =
kB T
Eq. (1) becomes
M = N µBJ (x),
where BJ (x) is the Brillouin function.
If the external field Bext is zero, one has the equations:
M = N µBJ (x)
kB T
x
M =
γµ
Both functions in Eq. (2) is plotted in Fig. 1. One
can see that at low temperature, there are two solutions
of Eq. (2) including one with M 6= 0, indicating ferromagnetism, while at high temperature, there is only one
solution with M = 0. The critical temperature θ can be
found when the slope of the two functions in Eq. (2) are
the same, giving
kB θ =
Plus the external field Bext , the microscopic field is
B = Bext + Bm = Bext + γM.
X
µi e−(σµi B+Ei )/kB T /
i,σ=−1,1
X
1
γN J(J + 1)g 2 µ2B .
3
(3)
The high temperature behavior can also be explained by
this model. At high temperature, x ≪ 1. Then
With this assumption, one has the self consistent equation:
M= N
(2)
e−(σµi B+Ei )/kB T
i,σ=−1,1
(1)
BJ (x) =
(J + 1)
x.
3J
Therefore
M=
N J(J + 1)g 2 µ2B
(B + γM )
3kB T
2
measurements [4]), which seems too low to be compatible
with the observed slower fluctuation at ∼1 K. In fact,
there is another way to estimate the exchange interaction
from the ”ferromagnetic” state of Nd3 Ga5 SiO14 at low
temperature, which is at least more valid than using Eq.
(4).
This second way of estimation has to do again with
the solution of Eqs. (2). In principle, one can predict the
dependence of M on T and Bext , if γ is known by solving
the Eqs. (2) at least numerically. If the experimental
dependence of M on T and Bext is known, one can find
γ. Next, we will use the magnetization data reported by
Bordet et al [1] to estimate γ.
2.0
M=N
B (x)
J
M=xk T/
, (T> )
M=xk T/
, (T< )
B
1.5
M
B
1.0
0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
FIG. 1: Relation between M and x.
and
N J(J + 1)g 2 µ2B
.
3kB (T − θ)
Now we can see that the parameter γ is really the key.
It is a microscopic parameter that can be estimated from
the macroscopic parameter, e.g. θ.
One can related the molecular field to the exchange
interaction by comparing the interaction energy:
1 2
γµ N ≈ zJe J(J + 1),
2
where Je is the exchange interaction strength, z is the
number of nearest neighbors. One can also relate macroscopic parameter θ with Je using Eq. (3) :
kB θ ≈
2z
Je J(J + 1).
3
(4)
Maagnetization per Nd (
x
M=
1.6
B
)
0.0
Experiment
1.2
Fit, U=2.7e-4 meV
U=0.1 meV
U=0.01 meV
0.8
0.4
0.0
0
1
2
4
Magnetic Field (T)
FIG. 2: Fit of low temperature 1.6 K experimental magnetization data using Mean-Field theory. Two curves with of U =
0.01 meV and U = 0.1 meV are also plotted for comparison.
Figure 2 shows the estimation, where the experimental data digitized from Bordet et al [1] is plotted as dots.
Here we assume only the ground state doublet is participating at T ∼1 K. Then the magnetization follows:
APPLICATION OF MEAN FIELD THEORY
M = N µg
For Nd3 Ga5 SiO14 , it is more or less a consensus that
the materials does not exhibit long range order at even
T ≈50 mK. The open question is what the magnitude of
the exchange interaction is. Although the interaction is
believed to be antiferromagnetic type (otherwise there is
no frustration), the observation of ”exotic” field induced
ferromagnetism at low temperature is intriguing. This
may be the reason that Zorko et al [6] used the Eq. (4)
to estimate the exchange interaction. Given the crystal
field splitting information we know [4], this estimation is
unlikely to be valid. On the other hand, all the other
estimations show a much lower energy scale, say ∼0.01
meV (from possible doublet splitting [5] and PL lifetime
3
eµg B/kB T − e−µg B/kB T
,
eµg B/kB T + e−µg B/kB T
where µg is the ground state magnetic moment. Note
that here B = Bext + γM is the microscopic magnetic
field. Fitting the experimental curve, one get
µg = 1.63µB
γN = 1.7 × 10−3 T/µB ,
which is pretty small. One can estimate the energy
scale of the exchange interaction as
U ≈ γN µg µg = 2.7 × 10−4 meV
3
This is even smaller than any other estimation so far.
To see how much uncertainty this fit has, two curves corresponding to U =0.01 and 0.1 meV are plotted too. One
can see that U =0.1 meV is too much off from the experimental observation and U =0.01 meV is much closer.
Therefore, we can conclude pretty conservatively that
from this estimation, the upper limit of the exchange
interaction energy should be on the order of 0.01 meV.
CONCLUSION
We carried out the estimation of exchange interaction
with mean-field theory using the low temperature magnetization data from Bordet et al. The upper limit of
the interaction energy is found to be on the order of 0.01
meV, which is consistent with the result from the sudies
on photophysical properties.
[1] P. Bordet et al., J. Phys.: Condens. Matter 18, 5147
(2006).
[2] J. Robert et al., Phys. Rev. Lett. 96, 197205 (2006).
[3] H. D. Zhou et al., Phys. Rev. Lett. 99, 236401 (2007).
[4] X. S. Xu et al., submitted to Phys. Rev. Lett.
[5] V. Simonet et al., Phys. Rev. Lett. 100, 237204 (2008).
[6] A. Zorko et al., Phys. Rev. Lett. 101, 026405 (2008).