Anisotropic Double Exchange in Polynuclear Mixed-Valence Clusters of Transition Metal Ions
ANISOTROPIC DOUBLE EXCHANGE
AND PSEUDODIPOLAR EXCHANGE IN DIMERIC
MIXED-VALENCE CLUSTERS
Moshe Belinsky
School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel,
In the mixed-valence (MV) clusters of non-degenerate transition metal ions, taking the spin-orbit coupling into account in the Anderson-Hasegawa double exchange (DE) model results in anisotropic double exchange interaction, which is active between the states of different localization of the extra electron. For the MV clusters, the coefficients of the DE anisotropy depend linearly on the DE parameters and of the excited and ground cluster states. The anisotropic double exchange results in the zero-field splitting (ZFS) of the high-spin DE levels of the MV cluster, which is described by the effective ZFS Hamiltonian , where is the DE operator in the S representation. The ZFS parameters and are linearly proportional to the DE parameters . In the MV clusters, the ZFS operator acts between the states of different localization of the extra electron and should be added to the standard ZFS Hamiltonian, which is active in the localized states. The anisotropic DE contributions to the ZFS have different sign for the and DE states: where the “localized” contributions include the dipole-dipole, single-ion and anisotropic (pseudodipolar) exchange, contributions. The anisotropic (pseudodipolar) exchange contributions, are proportional to. Since and t>J, the AN DE contributions, to the cluster ZFS parameters, may be larger than anisotropic (pseudodipolar) exchange contributions, and single-ion contributions to,.
Introduction
The double exchange (DE) interaction was introduced to explain the magnetism of the mixed-valence manganates. The resonance splitting of the S levels of the MV dimers due to the hopping of the extra electron is described by the Anderson-Hasegawa [1] DE model: . The double exchange or spin-dependent electron transfer (ET) is isotropic: the DE levels don’t depend on the projection M of S and are not mixed by the DE coupling [1, 2]. Strong DE was found experimentally in the MV clusters [3-15]. In the [Fe(II)Fe(III)] clusters [4-15] with the Heisenberg exchange and DE {E±0(S)=JS(S+1)± (S+1/2)t0/(2s0+1)}, strong DE interaction ( cm-1) destroys the Heisenberg antiferromagnetic ordering ( cm-1, J<B) and results in the delocalized ground state with the maximal total spin . Zero-field splittings (ZFS) of the delocalized cluster ground state with is described by the standard effective ZFS Hamiltonian [16]:
(1)
with the axial and rhombic cluster ZFS parameters. The delocalized ground state of the [Fe(II)Fe(III)] clusters of the high-spin iron ions are characterized by large ZFS: [4, 7, 8, 12, 14, 15]. The ZFS contributions of individual ions to the cluster ZFS were considered the origin of the ZFS of the cluster state of the delocalized [Fe(II)Fe(III)] MV clusters [3]. The ZFS contributions connected with the anisotropic double exchange inter-ion coupling were not considered for the MV clusters.
In the mononuclear exchange clusters, the anisotropic (pseudodipolar) exchange [17, 18, 19] and Dzyaloshinsky-Moriya antisymmetric exchange [19-22], which are induced by the spin-orbit coupling (SOC) in the exchange model, strongly contribute to the ZFS and mixing of the S levels.
The effect of the spin-orbit coupling in the double exchange model is considered in the paper which results in anisotropic double exchange interaction and new form of the zero-filed splitting, specific for the MV clusters with the double exchange coupling. The anisotropic DE is compared with the anisotropic (pseudodipolar) exchange for the cluster.
Anisotropic Double Exchange in a Mixed-Valence Pair
An anisotropic double exchange interaction originates from the combined effect of the isotropic double exchange and spin-orbit interaction on the centers a and b. The two-center third-order perturbation anisotropic DE terms, including the SOC admixture of the excited d-ions states and the double exchange in the excited states, have the form
(2)
, (3)
where. The ket represents the ground S states of the cluster in the case of localization of the extra electron (hole) on the center () or . {} is the ground state wave function of the {} ion. The ket [] represents the cluster excited states coupled to the cluster ground state [] by the spin-orbit interaction [] on the -center [-center ] {dn-center }. The cluster excited states are formed by the {} ion in the excited states {} and {} ion in the ground state {}. and are the energies of the ground state and excited states of the center * {dn-center }, respectively.
The anisotropic double exchange is considered on an example of the cluster ([Cu3+-Cu2+]), using the results for the double exchange in the ground and excited states [25] and the treatment [17, 18] to the SOC admixture. The DE splittings () of the ground state set of this cluster (S=1/2, 3/2;) is described by the Anderson-Hasegawa [1] equation
, 4)
where is the ground state DE (ET) parameter for the d-functions. is the isotropic DE operator in the representation of the total spin, .
The operator V1 (2) includes the SOC {} admixture of the excited k ligand field (LF) state [23,24 ] to the ground state of the {} ion on the center a {b} in the {} localization, and isotropic double exchange interaction between the excited cluster states and formed by the excited state of the d8-ion and the ground state of the d9-ion. Using the correlations [17, 23, 24] for the d8-ion and collecting the DE terms in the excited cluster states [25], one can represent the operator in the form of the effective symmetric Hamiltonian of anisotropic DE interaction or anisotropic spin-dependent electron transfer
(5)
The parameters in the coefficients of the DE anisotropy have the form:
, (6)
where and are, respectively, the effective SOC constant and the LF intervals of the ion at the center [25]. The coefficients of anisotropy in (5) are proportional to the double exchange parameters for the excited states (; [24]). The parameters describe the transfer of the extra d-hole from the excited () state of the ion to the ground (=) state of the ion. The effective AN double exchange Hamiltonian (5) acts between the cluster ground states and of different localizations,. The one-center spin operators {} act on the ground state spin functions {} of the {} ion in the <a*b| (|ab*>) localization [25]. This action is represented by an example
. (7)
The correlation takes place for the MV cluster since the double exchange is forbidden between the corresponding excited states and .
The one-electron DE operator in (5), (7) is determined by the relation for the one-electron d-functions and connects the states with the same M:. The Hamiltonian (5) is the operator of the anisotropic spin-transfer coupling induced by the SOC.
The Hamiltonian (5) of the AN double exchange may be represented in the form
, (8)
.
In the V2 term (3), the SOC admixes the excited state to the ground state of the d8-ion at the center in the localization. In the localization, the SOC at the center admixes the excited (,) states to the ground () state of the -ion. The DE interaction in V2 (3) takes place between the different excited cluster states: and. Using the admixture of the excited states for the ion in the localization and for the ion in the localization and collecting the corresponding matrix elements of the DE interaction between excited states and [25], one can represent the V2 operator in the form of an effective Hamiltonian of the anisotropic double exchange
, (9)
where the DE parameter for the ground states S is (eq. (4)), are the parameters of anisotropy [25]. In the matrix elements, the one-center spin operators {} act on the ground state spin functions {} of the () ion on the -center in localization () [25]. For the MV cluster, the parameters of the DE anisotropy have the form:
, (10)
and are the SOC constant and LF intervals of the d9-ion, respectively.
The Hamiltonian of an anisotropic double exchange (9) may be represented in the form
(11)
The resulting Hamiltonian of the new anisotropic double exchange interaction has the form
(12)
where corresponds to and corresponds to. The coefficients of the DE anisotropy and in (12) may be represented [25], using the g-shifts and of the localized d8- and d9-ions,
(13)
Zero-field splittings induced by anisotropic double exchange
The ZFS parameters, for the S levels of the exchange monovalent clusters include the single-ion, dipole-dipole and anisotropic (pseudodipolar) exchange contributions [18].
For the double exchange AH levels of the MV cluster, the anisotropic double exchange interaction (12) results in the zero-field splittings [25, 26]. For the levels of the cluster. This zero-field splitting is described by the effective ZFS Hamiltonian
. (14)
The ZFS Hamiltonian (14) includes the DE operator which acts in the representation of the total spin S (eq. (4)). The ZFS parameters, of the ZFS Hamiltonian (14) are the combination of the contributions of (5) and (9):
(15)
The new ZFS Hamiltonian (14) acts between the states of different localizations. For example, for the S=3/2 AH states with, the axial ZFS term of (14) results in the following splitting:
(16)
The standard ZFS Hamiltonian (1) [16, 18] operates in the localized states and is not active between the and states of different localization:
Using the g-shifts, we obtain the axial and rhombic parameters (15) of the AN DE contribution to the ZFS of the AH term () in the form:
(17)
For the lowest Anderson-Hasegawa level () of the ground DE set, the anisotropic DE contributions and to the cluster ZFS parameters have opposite signs in comparison with and (17), respectively, in according with eq. (16). The value and sign of the ZFS parameters and of (14) depend on the relation between the double exchange parameters of the excited states and of the ground states and AN coefficients (g-shifts). The estimation [25] shows that the anisotropic DE contributions to the ZFS parameter may exceed (or be of the same order as) the ZFS parameters of the single d8-ion: D0(Ni2+) = -1.3 cm-1 and D0(Cu3+) = -0.19 cm-1.
The resulting ZFS parameters for the Anderson-Hasegawa DE levels have the form (18)
where the “localized”, ZFS parameters include the single-ion, dipole-dipole and anisotropic (pseudodipolar) exchange contributions.
The SOC for the transfer of hole in doped La2CuO4 results in the SOC creation-annihilation hopping term [22]. Taking SOC into account in the MV clusters results in antisymmetric double exchange [26- 27]
where is the real antisymmetric vector coefficient, is the scalar DE operator (4), and are spins of the ions of the pair [26]. For the MV pair, the components have the form, [22, 26], for; is small angle of the tilting of the CuL6 octahedra [22]. The estimation of the antisymmetric DE parameter is. The antisymmetric DE vector in the MV dimer is oriented perpendicularly to the ab axis [22, 26] and results in the canting of the spin vectors of the MV pair. The AS double exchange mixes the Anderson-Hasegawa double exchange states E+0(S) and E-0(S) with the same S of the different parity and DE states with different total spin S of the same parity,. The mixing of the DE levels by the antisymmetric double exchange results in the second order contributions to ZFS. These contributions to the ZFS parameters are smaller than the anisotropic DE contributions.
Anisotropic (pseudodipolar) exchange in the localized cluster
The origin of the magnetic anisotropy and anisotropic (pseudodipolar (pd)) exchange
(19)
in the monovalent exchange pairs (Cu-Cu, Ni-Ni) was considered by Kanamori [17] and Owen and Harris [18]. For the [d9-d9] pair, for example, the coefficients in the parameters of the exchange anisotropy in eq. (19) are described by equation [49]:
. (20)
The parameters in (19) describes the Heisenberg exchange between the ground state and excited states , of the identical ions in the monovalent pair [49]. For the [Cu2+-Cu2+] pair, the ZFS parameters in the ZFS Hamiltonian (1) are determined by the pseudodipolar contributions, [49]:
, . (21)
The anisotropic (pseudodipolar) exchange Hamiltonian for the localized [d8-d9] pair, which was obtained [25] in the third order perturbation, has the form
. (22)
The parameters of the exchange anisotropy in (22) have the form
. (23)
The parameters of the exchange anisotropy in the AN exchange Hamiltonian (22) for the localized [d8-d9] pair
depend on the anisotropy coefficients (20) of the d9 ion and of the d8 ion. The coefficients of the exchange anisotropy are proportional to. The exchange integrals {} in (22), (23) describe the Heisenberg exchange between the ground state {} of the d9 {d8} ion and the excited state {} of the d8 {d9} ion [25]. The anisotropic (pd) contributions to the ZFS parameters of (1) for the localized [d8-d9] pair have the form
, . (24)
Comparison of the anisotropic double exchange operator (12) with the anisotropic (pseudodipolar) exchange Hamiltonian Hpd (22) demonstrates the following essential differences: 1) The Hamiltonian of the AN double exchange (12) acts between the different localized states. The anisotropic (pseudodipolar) exchange operator Hpd (22) acts in the localized states and is not active between the states of different localization. 2) The coefficients of anisotropy, in the AN double exchange Hamiltonian (12) are proportional to the double exchange (ET) parameters and tv. In comparison, the coefficients of anisotropy (23) of the anisotropic (pseudodipolar) exchange Hpd(19) are proportional to the exchange parameters 3) The estimate of the AN double exchange is, the estimate of the AN (pseudodipolar) exchange is. Since usually the DE interaction is stronger than the Heisenberg exchange coupling in the MV dimers, the contributions of the anisotropic DE interaction (12) to the resulting anisotropy may be larger than the contributions of the anisotropic (pseudodipolar) exchange (22).
Conclusion
The third-order perturbation terms, including the spin-orbit coupling admixture of the excited states and DE in the excited states, results in symmetric anisotropic double exchange interaction (12). The AN double exchange is active between the ground states ( and, ) of different localization of the extra electron. The Hamiltonian of anisotropic double exchange or anisotropic spin-dependent electron transfer has the form of anisotropic spin-transfer interaction (12). The coefficients of the DE anisotropy ~ , are linearly proportional to the DE parameters and. The anisotropic double exchange terms (~) differ from the exchange anisotropy term [19, 22] with.
For the double exchange Anderson-Hasegawa levels, the anisotropic DE interactions (12) results in the zero-field splittings. For the DE levels of the cluster, this ZFS induced by the AN double exchange is described by the effective ZFS Hamiltonian (16) which includes the DE operator. The ZFS Hamiltonian (16) is active between the states and () of different localization and differs from the standard ZFS Hamiltonian (1) which acts in the localized states. In the DE clusters, the ZFS operator should be added to the standard “localized” ZFS Hamiltonian (1). The AN double exchange contributions to the cluster ZFS parameters have different sign for the Anderson-Hasegawa DE levels and:. The axial and rhombic ZFS parameters of the ZFS Hamiltonian (16) are linearly proportional to the DE parameters (, ~1-10). Since t>J, the anisotropic DE contributions, to the cluster ZFS parameters, of the MV dimer may be larger than the single-ion, dipole-dipole and anisotropic (pseudodipolar) exchange contributions to the ZFS parameters of the localized cluster.
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