SPIN WAVES IN LANTHANUM MANGANITES
L.D Falkovskaya, B.V. Karpenko,
Institute of Metal Physics, Ural Branch of Russian Academy of Sciences,
Ekaterinburg, 620041 Russia
A.V. Kuznetsov,
Ural State University, Ekaterinburg, 620083 Russia
Abstract.
The theoretical expressions are obtained for spin waves energy in the layered
antiferromagnetic structure (A-type) which is realized in perovskite-like compound
LaMnO3. Interactions of Heisenberg exchange type of the central spin moment of
three-valence manganese ion (spin S=2) with eighteen neighbors from five coordinate
spheres are taken into account together with one-ion magnetic anisotropy energy.
There are six parameters in our problem: five exchange integrals and one anisotropy
constant. At that three integrals determine interactions in 2D-ferromagnetic planes and
two ones – inter-planar interaction. Formulas for magnons energy are valid for any
direction and value of quasi-moment in the first Brillouin zone. The comparison of
theory with experimental data for dispersion curves obtained from investigations of
inelastic neutron dispersion in LaMnO3 was carried out for three main
crystallographic directions. The analysis has shown that ferromagnetic interaction in
plane can be more than four times over than antiferromagnetic interplane interaction.
These conclusions are compared with other authors’results.
Introduction
During the last years the interest to perovskite-like compounds with mixed
valence of manganese ions such as for example La1-xCaxMnO3 where instead of La
can be some other rare-earth element and instead of Ca some other bivalent metal has
essentially increased. The presence in some compound of three- and four-valence
manganese ions having four (spin S=2) and three (S=3/2) 3d-electrons leads to
interesting effects. In particular effects of double exchange, magnetic, charge and
orbital ordering, Wigner crystallization and colossal magnetoresistance can arise in
the system. These compounds are called strongly correlated electron systems due to
non-trivial interaction of magnetic, charge, orbital and lattice degrees of freedom.
Many theoretical and experimental data can be found in the fundamental review of
Dagotto (1).
The magnetic properties of perovskite-like manganites are rather various. For
example in La1-xCaxMnO3 depending on doping degree x one spin magnetic
configuration changes another. When x increases the antiferromagnetic layer structure
(A type) turns into three-dimensional ferromagnetic structure (B type), then new
antiferromagnetic configuration appears (C type) and at last one more
antiferromagnetic structure comes (G-type). The regularities of these phase transitions
are theoretically investigated in paper (2). Besides these main collinear magnetic
structures some more complicated ones are also possible. Moreover there is evidence
that non-collinear antiferromagnetism also exists.
In the present report we shall consider one aspect of magnetism namely the
energy spectrum of spin waves in the initial compound LaMnO3. The knowledge of
this spectrum can be in principle very important because the comparison with
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experiment allows to determine the value and signs of exchange integrals. Then some
judgements can be done about wave functions overlap and mechanisms of exchange
interactions. So far we should say that the double exchange is absent here because the
stoichiometric LaMnO3 has only three-valence manganese ions while the four-valence
ones are absent that is there are no conduction electrons. Most likely the exchange
interaction in this dielectric is realized due to direct overlap of wave 3d-functions of
manganese ions or the superexchange via oxygen or indirect interaction via phonons
take place.
Below the information about crystalline and magnetic structures will be
presented, the interaction parameters and form of Hamiltonian will be chosen, the
spin-wave spectrum will be obtained, the comparison with experimental dispersion
curves will be done and the numerical values of the parameters will be determined.
Structure, the Choice of Interaction Parameters
The compound LaMnO3 is crystallized in the orthorhombic syngony with
space group Pnma(N62), the structure type GdFeO3. The elementary cell is presented
on Fig.1 where only manganese ions sites are indicated. The numbers 1,2,3,4
enumerate four Bravais lattices.
Fig.1 The elementary cell of crystal LaMnO3. Only manganese sites are denoted.
The interesting for us antiferromagnetic phase A presented together with other
magnetic structures on Fig.2 consists of ferromagnetic ab-planes whose total magnetic
moments are directed in turn along and against the crystal b-axis. One sub-lattice
includes the first and the second Bravais lattices, while the other one - the third and
the fourth.
115
Fig.2 The fragment of antiferromagnetic A structure in LaMnO3. The other possible
magnetic phases of compound La1-xCaxMnO3 are also presented. The arrows show the
directions of magnetic moments.
Most often at the analysis of exchange interactions the interactions between
nearest neighbors are only taken into consideration though generally speaking this can
be insufficient. On the other hand it is impossible to say beforehand how many
neighbors should be taken into account for an adequate description of the situation.
We shall choose fairly at will interactions with eighteen neighbors from four
coordinate spheres directing our attention only on interstitial distances though it's
known that the direction of the neighbors vector can play the main role. So the
vectors-coordinates of the neighbors are the following:
1   2  (0, a / 2, b / 2);  3   4  (0, a / 2,b / 2);
(2.1)
(2.2)
(2.3)
(2.4)
 5   6  (0, a,0);
 7   8  (0,0, b) ;
 9  10  (с / 2,0,0);
11  12  (c / 2, a / 2, b / 2); 13  14  (c / 2, a / 2,b / 2);
15  16  (с / 2,a / 2, b / 2); 17  18  (c / 2,a / 2,b / 2) .
;
(2.5)
In these formulas a, b, c present lattice parameters. According to paper (3) a=5.533 Ǻ,
b=5.7461 Ǻ, c=7.6637 Ǻ. Then the numerical values of vectors length from (2.1)(2.5) are as following
1   2   3   4  3.9886 Ǻ,
(2.6)
 5   6  5.5333 Ǻ
(2.7)
 7   8  5.7451 Ǻ,
(2.8)
 9  10  3.8319 Ǻ,
(2.9)
11  12  13  14  15  16  17  18  5.531 Ǻ.
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(2.10)
The next neighbors are on distance 6.7306 Ǻ, but we shall not take them into
account. From the symmetry consideration we have five different parameters of
exchange interaction:
I1 for vectors 1   4 ,
I 2 for vectors  5   6 ,
I 3 for vectors  7   8 ,
I 4 for vectors  9  10 ,
(2.11)
(2.12)
(2.13)
(2.14)
I5 for vectors 11  18 .
(2.15)
Besides these five parameters we shall introduce also anisotropy constant D
which orients the antiferromagnetism axis along crystallographic b-axis.
The Hamiltonian, the Energy Spectrum
Let's consider that the exchange interaction is described by Heisenberg
Hamiltonian while the anisotropy energy has one-ion character:
2 18
2
Hˆ      I (mi , mi   j ) S (mi ) S (mi   j )  D   ( S mz ) 2 ,
i
i 1 j 1mi
i 1 mi
(3.1)
where i numbers sub-lattices while mi runs N values of radius-vectors of each sublattice sites. It is supposed that D is positive. Then passing in a usual manner from
spin operators to second quantization operators and making diagonalization with the
help of Fourier transformation we obtain the expression for magnons energy w as
function of the reduced quasi-moment q :
w(q)  t12  t 22 ,
t1  2S I1 ( 1  4)  I 2 ( 2  2)  I 3 ( 3  2)  2I 4  8I 5  D,
t 2  2S I 4 4  I 5 5 ,
 1  4 cos q y cos q z ,
(3.3)
(3.4)
(3.5)
 2  2 cos 2q y ,
(3.6)
 3  2 cos 2q z ,
 4  2 cos q x ,
 5  8сosq x cos q y cos q z .
(3.7)
(3.8)
(3.9)
(3.2)
The reduced quasi-moment q is connected with the “real” quasi-moment k by
relations:
ak y
ck
bk
qx  x , q y 
, qz  z .
(3.10)
2
4
4
The expression for w(q ) is valid in the whole Brillouine zone.
Comparison with Experiment
Moussa with collaborators (3) investigated the inelastic dispersion of neutrons
and obtained the dispersion curves for main crystallographic directions in LaMnO3.
The experimental points are presented on Figs.3, 4 and 5.
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w,THz
9
8
7
6
5
4
3
2
1
0
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
qy
w,THz
Fig.3 The dispersion curve for direction [100]. Small squares denote the
experimental points (3), the solid curve is obtained with the help of equations (4.1)
and (4.4)-(4.8).
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
2.2 2.4 2.6 2.8
3
qy
Fig.4 The dispersion curve for direction [110]. Small squares denote the experimental
points (3), the solid curve is obtained with the help of equations (4.2) and (4.9)-(4.11).
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1.4
1.2
w,THz
1
0.8
0.6
0.4
0.2
0
-3
-2.8
-2.6
-2.4
-2.2
-2
-1.8
-1.6
-1.4
-1.2
-1
qx
Fig.5 The dispersion curve for direction [001]. Small squares denote the experimental
points (3), the solid curve is obtained with the help of equations (4.3) and (4.12)(4.13).
Now we shall try to compare our results with these experimental data for the
directions [100], [110] and [001]. From relations (3.2)-(3.9) we have obtained

w[100]  a1  b1 cos q y  c1 cos 2 q y  d1 cos 3 q y  f1 cos 4 q y
w[110]  (a 2  b2 sin 2 q y  c 2 sin 4 q y )
w[001]  (a 3  b3 cos 2 q x )
1
1

1
2,
(4.1)
2,
(4.2)
2,
(4.3)
where the following notations are introduced


a1  16 16( I1  I 2 ) 2  (2 I 4  8I 5  D)(2 I 4  8I 5  D  8I1`  8I 2 )  4 I 4 2 ,
(4.4)
b1  128I1 (4 I1  4 I 2  2I 4  8I 5  D)  4 I 4 I 5  ,
(4.5)


c1  128 2I 12  I 2 (4 I 1  4I 2  2I 4  8I 5  D)  8I 52 ,
d1  512I1 I 2 ,
(4.6)
(4.7)
f1  256I 22 ,
a 2  16 D4( I 4  4I 5 )  D ,
b2  128 ( I1  I 2  I 3 )( 2I 48I 5  D)  4( I 4  4I 5 ) I 5  ,
(4.8)
(4.9)
(4.10)
c2  256 ( I1  I 2  I 3 ) 2  4 I 52 ,
(4.11)


a3  162I 4  8I 5  D ,
(4.12)
b3  64I 4  4I 5 2 .
(4.13)
2
We supposed here S=2.
In order to find the numerical values of parameters I1  I 5 and D we used the
least square method. However, because of the root character of energy dependence
upon quasi-moment the corresponding system of equations turns out to be too
complicated. That’s why we decided to minimize not the form
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F   ( wtheoret  wexp erim ) 2 ,
(4.14)
but the form
2
2
2
G   ( wtheoret
 wexp
erim ) .
(4.15)
As a result the following values of parameters were obtained
I1=11.39 K, I2=-1.86 K, I3=-6.58 K, I4=6.04 K, I5=-2.28 K, D=3.25 K.
(4.16)
We have introduced two additional characteristic parameters
J1  2 I1  I 2  I 3 , J 2  I 4  4 I 5 .
(4.17)
The first one gives the planar interaction while the second one – the inter-planar.
Positive sign corresponds to ferromagnetic interaction, while negative one – to
antiferromagnetic interaction. With our values of parameters (4.16) we obtain
J1=14.31K, J2=-3.09 K.
The calculated dispersion curves for all three directions are presented on
Figs. 3-5.
Discussion
The theoretical magnon spectrum was obtained also in paper (3). The authors
of this paper considered only interactions I1 and I4 (in our notations) and D. The
formulas of paper (3) follow as a particular case from our expressions (4.1)-(4.13) if
one assigns in the last I2=I3=I5=0. Comparison with experiment gives I1=9.6K, I4=6.72K, D=1.92K. All these values differ from our parameters. The main difference is
in the fact that integrals I4 have different signs – ours is positive and that from paper
(3) is negative. This difference is the principle one. The antiferromagnetic interaction
between planes in (3) is caused by the negative sign of integral I4 while in our case by the negative sign of parameter J 2.
As a whole the results of the present work justify our model of taking into
account more distant neighbors: eight negative integrals I5 cause antiferromagnetic
bond between ferromagnetic layers and the absolute value of the plane integral I3 less
than twice differs from the integral between the nearest neighbors I1. The ratio value
J1 / J 2  4.6 allows to talk about the two-dimensional antiferromagnetism.
This work was carried out with the support of the program of Presidium of
Ural Branch of Russian Academy of Sciences “New materials and structures” and of
RFBR Project N 06-03-90893 Mol-a.
References
1. Dagotto E.: The physics of manganites and related compounds. SpringerVerlag,
Berlin-Heidelberg-New-York-London-Paris-Tokyo-Hong CongBarcelona-Budapest, 2002, 1-448
2. Karpenko B.V., Falkovskaya L.D., Kuznetsov A.V.: Magnetic structure and
double exchange in hypothetical compound La1-xCaxMnO3. cond-mat 2005.
0509507, 1-31
3. Moussa F., Hennion M., Rodriguez-Carvajal J., Mouden H., Pinsard L. and
Revcolevschi A.: Spin waves in the antiferromagnetic perovskite LaMnO3: A
neutron scattering study. Phys. Rev.B 1996 54 15 149-55
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