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ANISOTROPIC SYMMETRIC EXCHANGE INTERACTION IN
METAL CLUSTER CONSISTING OF EXHANGE-COUPLED
COPPER IONS
V.V. Bannikov1, S.I. Klokishner2, V.Ya. Mitrofanov3, B. Tsukerblat4
1
Institute of Solid State Chemistry, Ural Branch of Russian Academy of
Sciences, 620041 Ekaterinburg, Russia
2
Institute of Applied Physics, Academy of Sciences of Moldova Republic,
Kishinev, MD-2028 Moldova
3
Institute of Metallurgy, Ural Branch of Russian Academy of Sciences,
620016 Ekaterinburg, Russia
4
Ben-Gurion University of the Negev, Beer-Sheva, 84105 Israel
Molecule-based materials with usable physical properties, in
particular, magnetic and optical, are in the focus of contemporary
materials science research. The building blocks for these materials are
the coordinated metal ions or metal clusters consisting of several metal
ions coordinated by inorganic or/and organic ligands. In the magnetic
clusters the constituent metal ions are coupled by the isotropic and
anisotropic exchange interactions that give rise to the magnetic ordering
in solids and specific magnetic behavior of the molecular clusters. The
main goal of this communication is the elaboration of the microscopic
model approach to the anisotropic symmetric exchange interactions in
metal clusters consisting of copper ions in frames of a conventional
approach peculiar for a weakly covalent system.
Examination of electron paramagnetic resonance (EPR) spectra of
the isolated pairs of ions gives a valuable information about exchange
interactions in the ground state. Interesting EPR spectra of Cu2+ pairs in
CaO were observed in [1]. The spectra are typical for S = 1 in a
tetragonal symmetry with the pair axes parallel to the [100], [0110] or
[001] directions. The g factor of the pair is isotropic, while the hyperfine
parameters are highly anisotropic. This behavior was explained by
assigning the spectra to linear bonds in the Cu2+—O2—Cu2+ fragment
where the two Cu ions of a pair are coupled by an antiferrodistortive
ferromagnetic interaction. Thus the ground state of the pair is composed
of a u state (3z2 – r2) of one Cu ion and v state (x2 - y2) on the other with
an exchange constant J = -2 cm-1.
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We shall denote the two ions in the pair by a(u) and b(v). Then for
the axial system the spin Hamiltonian can be written as
H  J (S a S b )  J a [S az Sbz 1 / 3(S a S b )]  H z ,
(1)
where the exchange constant J = 2 cm-1 [1], the second term represents
the anisotropic exchange interactions, HZ describes the Zeeman term.
2 / r 3 is the dipolar part of the
Ja = Jmd + Jpd, where J md   3g a g b  B
ab
spin-spin interaction with rab the distance between the two Cu2+ ions, Jpd
is a pseudo-dipolar term which might originate from exchange
anisotropy [2]. For copper pairs in CaO a surprisingly small value of Jpd
= -28610-4 cm-1 was obtained from the measured value of Ja = (466 
3)10-4 cm-1 and calculated value of Jmd = 54010-4 cm-1 [1].
It is interesting to note that in the insulating cuprates K 2CuF4,
KCuF3, La2CuO4 there exists two-ion anisotropy which confines the
spins in the CuX2 (X = O, F) besides a small canting from a plane (weak
ferromagnetism) for KCuF3 and La2CuO4. Since the classical work by
Kanamori [2] on the anisotropic exchange, in a rather large number of
publications it have pointed out that the anisotropic direct (potential)
exchange is the origin interaction in describing of the origin of the
“easy-the magnetism of cuprates. In particularly, Hanzawa [3] has
shown that only the spin-orbit corrections to the direct (potential)
exchange give rise to the desired anisotropy Ja with correct order of
magnitude and sign in K2CuF4 and YBa2Cu3O6.3. In paper [4] and
especially in [5] it has been shown that rigorous analysis of anisotropic
exchange leads to a reconsideration of many established concepts
regarding the spin-orbit structure.
Two-ion anisotropy of the exchange-relativistic origin appears as
the third order terms within the perturbation theory. These terms are
linear in exchange interaction of ions a and b and are of the second
order in spin-orbital interaction of ions a (b). Schematically these
contributions for a pair of copper ions can be presented in the following
way [4, 5]:
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Van 
Vso ( a ) Vex ( ab) V or( ab) Vso ( a )
V ( a )Vso (b) Vex ( ab) V or( ab)
 so

2
E
2

Vso ( a )[Vex ( ab)  Vor ( ab)]Vso (b)
 cc
E 2
(2)
where cc is the notation for terms that are Hermitian conjugate to the
above terms and distinguished by rearrangement ab. The term Vex(ab)
in equation (2) represent essentially the isotropic exchange interaction
between the ground and excited states of neighboring magnetic ions and
can be written as [6]
H ex    [(1 / 4)na ( )nb ( )  (s a ( )s b ( ))] j ,  ,
 ,   
(3)
using the operators and notations introduced in [6]. Here , , ,  are
the symbols of summation over the d-orbitals. Up to the second order of
the perturbation theory the expression for j ,   contains two
competing contributions-antiferromagnetic “kinetic” exchange and
ferromagnetic “potential” exchange [7]
if we omit the terms
corresponding to the exchange polarization and two-electron jumps. Vor
in equation (2) describes the Coulomb energy operator.
The data of experimental studies specify that magnitude and a
sign of two-ion anisotropy Ja in cuprates essentially depend on the type
of orbital ordering and orientation of a copper pair in crystal. In the case
of ferrodistortive orderings of copper orbitals J = 1117 K, Ja = 0.41 K
for YBa2Cu3O6.3 [8], J=1530 К [9], Ja = 0.23 K [10] in La2CuO4. If the
antiferrodistortive ordering of copper orbitals takes place then J = 2
cm-1, Ja = 0.046 cm-1 in CaO:Cu [1] and J = 23.8 K, Ja = 0.18 K in
K2CuF4 [11]. In KCuF3 the corresponding parameters for Cu2+ ions
along the c axis are equal to J = 406 K, Jac = 0.08 K [12]. The analysis
presented here has shown that the mechanism I is active for copper pairs
in CaO and K2CuF4, while the mechanisms I and III contribute to the
anisotropic exchange Ja in La2CuO4 and KCuF3. The anisotropic direct
exchange interaction in contrast to [3] does not contribute to Ja in
La2CuO4 due to mutual cancellation of corresponding contributions from
the I and III mechanisms in (2). It is necessary to note also that the III
mechanism in (2) gives rise to a nonzero contribution to two-ion
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anisotropy Ja only due to a spin-orbit correction to Vor. This
circumstance is connected by that mutual cancellation of contributions
due to spindependent and spinindependent exchange in (3) takes place
in the excited pair states. Then, if we shall consider the contribution
from the first mechanism in (2) to two-ion anisotropy Ja for Cu2+ pair in
CaO we get
2
 Δg x(b)
 jξu 
 g 
J pd   
Δg
g

2
2
2
2
 Δg y(b) 
1  Δg x(a )
1  Δgy (a )
1  Δgz (b )




 g  jηu  3  g  j ξv  3  g  jηv  4  g  jζu ,








  e L g  E
e
 Eg

1
(4)
,
e
where j = j, ju = ju = j1, jv = jv = j2, g is the spin-orbit
correction to the g-factor (g = 2).
In accordance to [1] the factors g/g in equation (4) are estimated
as gx(a)/g = gy(a)/g = 0.171, gx(b)/g = gy(b)/g =
=0.057, gz(b)/g= 0.228.
Then expression (4) can be represented in the following form
Jpd = 0.013 ju0.003 j1-0.01 j2 .
(5)
It can be easily seen form equation (5) that an account of a
ferromagnetic "potential" exchange in j  (3) only gives rise to an
p
p
p
undesired anisotropy Jpd (Jpd > 0), since j  j  j , where the
ζu
ξv
ξu
index p denotes the contribution of the "potential" exchange to j.
In order to clarify an essential point, i.e. the origin of J pd in Cu2+
pairs in CaO, here we consider the traditional contribution to the
isotropic exchange interaction arising in the third order of perturbation
theory and taking into account the transfer of an electron from the closed
orbital into a half-filled orbital of the neighboring magnetic ions [7]
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4b 2 I
(III)
(III)
p
(III)
(III) (III)
jζv  j
, jξv  jηv  j
, jξu  jηu  j  j
 j
, j
 σ ,
σ
π
1
σ
π
σ
U2
4b 2 I
π ,
(6)
2
U
where I is the intraatomic exchange integral, b and b are the hopping
integrals via p and p orbitals of the ligands. It is easy to see from
equations (4-6) that the sign of pseudo-dipolar interaction Jpd strongly
depends on the details of the electron structure of Cu2+ ion in crystal,
interrelation between hopping integrals b and b as well as relative
magnitude of the contributions of the first and third order in equation
(6). Nevertheless if we accept the following approximations
(III)
j

π
b2   b2 , j1p  j( III )  j( III ) ,
we obtain the right sign of pseudo-dipolar anisotropy in the case of
reasonable value   3.
In conclusion, we have considered a problem of anisotropic
(symmetric) exchange coupling in Cu2+ pairs in CaO. It is obvious that
qualitative explanation can be reached by examining the spin-orbit
corrections to the superexchange or the magnetic dipole-dipole. In our
case the dipole interaction gives rise to the observed anisotropy Ja. The
spin-orbit corrections to the “potential” exchange (direct exchange)
cannot explain a sign of pseudo-dipolar anisotropy Jpd. In light of the
obtained results, which testify to essential compensation of the various
contributions in parameter Jpd, represents the doubtless interest to
estimate the spin-orbit correction to Vor in the equation (2).
The present work was supported by RFBR grants N 06-03-90893 and N
06-02-72021.
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