Anisotropic SYMMETRIC EXCHANGE INTERACTION in metal cluster consisting of exhange-coupled copper ions
V.V. Bannikov1,S.I. Klokishner2, V.Ya. Mitrofanov3, B. Tsukerblat4
1Institute of Solid State Chemistry, Ural Branch of Russian Academy of Sciences, 620041Ekaterinburg, Russia
2Institute of Applied Physics, Academy of Sciences of Moldova Republic, Kishinev, MD-2028 Moldova
3Institute of Metallurgy, Ural Branch of Russian Academy of Sciences, 620016 Ekaterinburg, Russia
4Ben-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.
We shall denote the two ions in the pair by a(u) and b(v). Then for the axial system the spin Hamiltoniancan be written as
(1)
where the exchange constant J = 2 cm-1 [1], the second term represents the anisotropic exchange interactions, HZ describes the Zeeman term. Ja = Jmd + Jpd, where is the dipolar part of the 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 = -28610-4 cm-1 was obtained from the measured value of Ja= (466 3)10-4 cm-1 and calculated value of Jmd = 54010-4 cm-1 [1].
It is interesting to note that in the insulating cuprates K2CuF4, 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 ionsa andb and are of the second order in spin-orbital interaction of ionsa(b). Schematically these contributions for a pair of copper ions can be presented in the following way [4, 5]:
(2)
where cc is the notation for terms that are Hermitian conjugate to the above terms and distinguished by rearrangement ab. 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]
(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 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 anisotropy Ja only due to a spin-orbit correction to Vor. This circumstance is connected by that mutual cancellation of contributions due to spindependent and spinindependent 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
(4)
where j = j, ju = ju = j1, jv = jv = 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 ju0.003 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 undesired anisotropy Jpd (Jpd > 0), since , where the index p denotes the contribution of the "potential" exchange to j.
In order to clarify an essential point, i.e. the origin of Jpd 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]
(6)
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
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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