SPIN-ORBIT COUPLING IN MIXED-VALENCE CLUSTERS WITH DOUBLE EXCHANGE INTER-ION INTERACTION. ANTISYMMETRIC DOUBLE EXCHANGE
M. Belinsky
School of Chemistry, Tel Aviv University,
69978 Tel Aviv, Israel
The spin-orbit coupling (SOC) effect is considered for the mixed valence (MV) dimeric and trimeric clusters with the double exchange (DE) and Heisenberg exchange inter-ion interactions (t-J model). The SOC effect in DE results in an antisymmetric (AS) double exchange coupling and anisotropic DE interaction. In the MV dimers, the ASDE coupling has a form of the vector type spin-transfer interaction induced by SOC. The ASDE coupling mixes the Anderson-Hasegawa DE states E+0(S) and E-0(S) with the same S of the different parity. The ASDE coupling and Dzialoshinsky-Moriya (DM) AS exchange mix the DE states with different S of the same parity. ASDE and DM AS exchange contribute in the second order to the cluster ZFS parameters DS (), where are the numbers (). are different for the DE states E+(S) and E-(S), S>1/2. The ASDE vector is directed perpendicularly to the ab-axis of dimer. The ASDE coupling results in anisotropy of g-factors. In the MV dimers with si>1/2, SOC and local crystal field splittings result in an anisotropic DE interaction.
In the trimeric MV clusters, the ASDE coupling results in the linear ASDE fine splittings Δ of the trigonal 2S+1E DE terms. Δ is proportional to the trimer ASDE parameter
.
The cluster ASDE vector is directed along the trigonal Z-axis: . The ASDE coupling mixes the trigonal isotropic 2S+1A1 and 2S´+1A2, 2S+1E and 2S´+1E DE terms, . In the MV trimers, the ASDE and DM AS exchange mixing of the DE levels () contributes in the second order to the ZFS parameters DS of the axial anisotropy. For the delocalized cluster, the ASDE fine splitting of the ground 2E DE term determines strong anisotropy of g-factors and magnetic moment.
1.Introduction
The double exchange (DE) model was introduced by Ziner [1] and Anderson and Hasegawa (AH) [2] for description of magnetism of the mixed-valence (MV) rare earth manganates. In the [] MV dimers, the migration of the extra electron between paramagnetic ions results in the DE splitting of the S levels [2]: , where t0 is the one-electron transfer integral, . The AH double exchange interaction (or spin-dependent electron transfer (ET)) is isotropic; the DE splittings do not depend on the projection M of S. The DE concept is widely used in magnetism of the MV compounds, particularly DE magnets, in the theory of the MV metal clusters in inorganic and bioinorganic chemistry. The MV clusters are attracting considerable attention in investigations of molecular magnetism, single-molecular magnets [3-5] and the structural elements of many biological systems.
The isotropic AH double exchange in dimeric clusters was investigated in detail (see reviews [6-8]). Strong DE interaction (t=1350 cm-1) was found experimentally in the [Fe(II)Fe(III)] MV center of the [Fe2(OH)3(tmtacu)2]2+ cluster [9,10]. The DE concept was developed for trimeric [11-14], tetrameric [15, 16] and more complicated clusters [16,17] (see also [6-8]). In polynuclear MV clusters, the DE coupling and Heisenberg exchange (t-J model) form isotropic exchange-resonance terms 2S+1i [11, 13, 15].
Usually the effects of spin-orbital coupling (SOC) are not considered in the theory of the Anderson-Hasegawa DE between orbitally non-degenerate ions. However SOC should be taken into account for description of the zero-field splittings (ZFS) of the DE states, anisotropy of the Zeeman splittings and g-factors, and anisotropy of magnetic characteristics. SOC for the transfer of holes between the neighboring sites in doped La2CuO4 was considered in [18]. An extension of the t-J model [18] includes the spin-orbit hopping term and anisotropic term . The consideration [18] is restricted by the systems of the hole transfer in the low-spin systems (si=1/2). The aim of the work is the consideration of SOC in DE for the dimeric [dn-dn+1] and trimeric [dn-dn-dn+1] MV clusters of orbitally non-degenerate ions. The taking into account SOC results in an antisymmetric DE interaction and anisotropic DE coupling, ZFS, magnetic anisotropy of levels and EPR spectra.
2.Dzyaloshinsky-Moriya Antisymmetric Exchange and Symmetric Anisotropic Exchange
The importance of the SOC effect in the theory of the Anderson superexchange between two identical d-ions was shown first by Moriya [19]. He showed that an antisymmetric (AS) Dzyaloshinsky-Moriya (DM) spin-spin interaction introduced by Dzyaloshinsky [20] for explanation of weak ferromagnetism is a result of the taking into account SOC in the theory of superexchange. The Moriya [19] one electron Hamiltonian of the inter-ion coupling between Cu(II) ions was written in conventional terms of annihilation and creation operators
(1)
The first term (with ) in eq. (1) is the convenient transfer (DE) term for the t-J models. The taking into accounts SOC in the virtual ET process between Cu(II) ions results in modification of the ET parameter: , where is the purely imaginary vector ET parameter [19]. In the DM AS exchange interaction
are antisymmetric vector coefficients. The parameters determine the vector DM AS exchange constants and anisotropic (AN) exchange contribution in the pseudodipolar term in the perturbation theory of AN superexchange. The DM AS exchange leads to a canted arrangement of spins and , which were oriented in antiferromagnetic order by the isotropic Heisenberg interaction
.
The microscopic theory of DM AS exchange was further developed (see [21-23, 18]).
3. Double Exchange in the [] and [] Clusters
We will start the consideration of SOC in DE (ET) in the MV dimers on an example of the [] dimer [24]. For the [] MV clusters, the Hamiltonian of the pair is the sum of the single-site terms for the extra electron localizations |a*b>, |ab*> and inter-ion interaction of the direct type (Coulomb interaction) or indirect interaction through the ligand bridge.
We consider that each d-ion has octahedral coordination with tetragonal distortion and octahedra are titled relatively the joint axes. The single-site term for the ion includes the crystal field (CF) HCF and spin-orbit interaction [25-27]. Local distorted octahedral CF forms the orbitally non-degenerate ground state. The excited CF states and are separated by the CF intervals, respectively [25]. The one-site SOC admixes excited CF states into the ground state of the ion of the MV pair in the localizations |a*b> and |ab*>. In the first order perturbation, the renormalized SOC-admixed ground state functions for the center have a form
,(2)
where d-orbital functions refer to the local Cartesian frame and In the resonance representation , the non-diagonal ASDE terms in the DE matrix [24] describe the SOC mixing of the AH DE states and . The double exchange with the taking into account the real vector transfer term () results splitting for the [] MV dimer has the form
. (3)
is the effective DE (ET) parameter of the [] MV dimer, .
The transfer integral has the form,
The term is the standard isotropic ET integral for the transfer between the neighboring ζ0 d-orbitals in the ground state without SOC. The term is the contribution to of the ET in the excited states due to the SOC admixture. In the real antisymmetric () vector transfer parameter
():
(4)
is the transfer integral between the ground orbital state on the center and the excited state on the center . The transfer integrals are different from zero due to the tilt of the distorted octahedra [19, 18]. The tilting of the MO6 octahedra results in a small (~θ) admixture of the excited d-orbitals to the ground orbital. The local CF 3d-orbitals may be written in the common xyz-coordination system [18]:
in the |a*b> localization. In the case of the |ab*> localization, should be changed to -. Because of the tilt, there is a non-zero transfer (~) between the ground state and excited states:
The corresponding transfer integrals between the ground state and excited state have the opposite sign: . The components of ASDE vector parameter are the following:
(5)
The resonance splitting in the DE + SOC model depends on the effective transfer parameter .
The ASDE contribution to the g-factors is described by theeq.(6):
gα=g0{(t2+Kα2)/(t2+K2)}1/2. (6)
In the case , Kz=0, we obtain an anisotropy of g-factors (gz<gx) induced by antisymmetric double exchange [24].
The estimations of the value of the ASDE vector parameters were obtained by Moriya [19]
The ASDE vector parameter may be the quantities of the order of 1-5% of the double exchange parameter .
The hole-transfer in the pair of hole-doped La2CuO4 was considered in [18]. The components of the AS vector transfer parameter are the following [18, 24]:
(7)
In the second-order perturbation theory [18], the effective Hamiltonian of hole transfer between the neighboring i and j Cu sites has the form [18]:
. (8)
An anisotropic term is characterized by the symmetric anisotropic parameter [18, 19]:
(9)
This anisotropic interaction is not active in the and MV pairs with the total spin S=1/2 and J=0.
4. Operator of Antisymmetric Double Exchange
For dimeric MV dn-dn+1 clusters in the t-J model, the Moriya [19] vector transfer operator (1) and the “spin-hopping” Hamiltonian HSO (8) [18] describe the SOC action in the form of the creation and annihilation operators. The SOC action in DE may be represented as the effective Hamiltonian in the form of the transfer and spin operators. This effective Hamiltonian of antisymmetric double-exchange has the following form
(10)
where is an antisymmetric vector coefficient, is the transfer operator, and are spins of the ions of the pair. The effective AS double exchange Hamiltonian (10) of the second-order perturbation theory describes the vector type spin-transfer interaction induced by SOC. The matrix elements of HASDE between the states of different localization with different spins S and have the form
(11)
In the case of the states with the same total spin , the first multiplier in eq. (11) represents the Anderson-Hasegawa DE contribution
.
For the [] MV dimers, matrix elements of the ASDE coupling (10) between the states of different localization with depend on projection M [24]:
(12)
The mixing of the Anderson-Hasegawa DE states with different total spin S (), may be described by the effective operator of the ASDE mixing , where
The ASDE coupling mixes the S and states with [24].
5.Antisymmetric Double Exchange in the Cluster
The total spin of the pair is S = 3/2, ½. The ASDE coupling (10), (12) mixes the states of different localization with the same total spin S:
(13)
In the resonance representation Ψ±(S, M), ASDE mixes the Anderson-Hasegawa DE states S=3/2 of different parity: and, and also the AH DE states and with the same S = 1/2.
ASDE mixes also the states of different localization with different total spin S: Φa*b(S=3/2) with Φab* (1/2), and Φa*b(1/2) with Φab*(3/2). In the resonance representation Ψ±(S, M), ASDE mixes the Anderson-Hasegawa DE states E+(3/2) [E-(3/2)] with E+(1/2) [E-(1/2)] of the same parity with different S. In comparison with the DM AS exchange, which mixes the localized states with different S, an antisymmetric DE in the delocalized system mixes the AH states E+0(S) and E-0(S) of the different parity with the same S and also the AH states of the same parity with different total spin S.
The mixing of the DE states and of the same parity depends on both AS double exchange and DMAS exchange (ASE) parameters:
(14)
where are the components of the DM AS exchange. Since and in the delocalized MV cluster, the ASDE contributions to the mixing (14) S are stronger than the DM AS exchange contributions.
In the MV cluster, an initial zero-field splitting of the -ion () essentially contributes to ZFS ΔZFS0(S=3/2)=2DS of the S=3/2 cluster states of the [d8-d9] pair, where DS=D0+. The ASDE and DM AS exchange mixing results in the following second order AS exchange contributions to the ZFS parameters of the high spin DE states:
, (15)
in the case. For the cluster with and , the terms in the cluster ZFS parameters are different for the Anderson-Hasegawa DE states and of the different parity:
(16)
The ASDE contribution to the cluster ZFS parameter is stronger than the pure Dzialoshinsky-Moriya AS exchange contribution since . Since KX~(Δg/g)t and DDM~(Δg/g)J, one can suppose, for example, that the strength of the AS double exchange may be more than 10 cm–1 and KX>DDM in the MV [Fe(III)Fe(II)] cluster where strong double exchange (t=1350, J~70 cm-1) was found experimentally [9, 10].
6. Anisotropic Double Exchange
In the [d8-d9] MV clusters, the SOC admixture of the 3T2 and 1T2 excited terms to the ground 3A2 state of the d8-ion results in the AN splitting of the high-spin DE levels with S=3/2 which is described by the effective anisotropic DE Hamiltonian
, (17)
where is the AH transfer operator (<Φ±(3/2)|Tab|Φ±(3/2>=±1). The axial anisotropic DE contributions to the zero-field splittings are different for the E±(3/2) levels: 2[DS± Γ΄(3/2)]. The AN DE parameter of the easy-axis two-ion anisotropy
(18)
is proportional to the terms [24], coefficients determine the SOC admixture of the components of the 3T2 and 1T2 terms, respectively.
7.Antisymmetric Double Exchange in Trimeric MV Clusters
In trimeric MV clusters, the ASDE interaction results in the new effects, which can not be obtained in the MV dimers. Strong double exchange and Heisenberg exchange interactions in the clusters form the isotropic exchange-resonance (DE) eigenstates characterized by the total spin S and irreducible representations () of the trigonal D3 group [13].
7.1. The MV Cluster [d 9-d 10-d 10] ([d 1-d 0-d 0])
The DE levels are the and trigonal terms. The group theoretical consideration and effective Hamiltonian method show that the DE term must be linearly split by the spin-orbit coupling into two Kramers sublevels. This fine splitting of the ground terms
(19)
is produced by the ASDE coupling [28, 29]. The splittings of the E terms are proportional to the cluster ASDE parameter
(20)
where is the pair ASDE contribution. The microscopic consideration of SOC (ASDE) in the DE trimer [28] shows that the vector of the ASDE coupling is directed along the trigonal Z-axis of the MV cluster:
, (21)
is the small angle between the local z-axis and the -axis of the trimer. The ASDE splitting of the DE term is linear with respect to the SOC parameter λ and ET parameter t. The ASDE coupling does not mix the and terms.
For the delocalized [Cu(II)Cu2(I)] cluster, the linear ASDE fine splitting of the ground 2E DE term determines strong anisotropy of the Zeeman splitting. In the external magnetic field parallel to the trigonal Z-axis, the Zeeman splitting of the ASDE sublevels and of the DE term is linear [28, 29] ():
. (22)
In the case , the Zeeman sublevels depend non-linearly on magnetic field
. (23)
Along with the fine splitting , ASDE results in the axial anisotropy of the Zeeman splitting. The g-factors of the MV cluster [d9-d10-d10] are strongly anisotropic: in the magnetic field , we obtain from eq. (22) the cluster g-values for the and Kramers sublevels. In the case , we obtain the cluster g-value . The strong anisotropy of the Zeeman splitting determines anisotropy of the magnetic susceptibility of the MV trimer.
7.2 The MV Cluster [d 9-d 9-d 10] ([d 1-d 1-d 0])
The isotropic DE and Heisenberg exchange form the following DE terms of the trigonal MV cluster: . The ASDE coupling in the delocalized [d9-d10-d10] and [d9-d9-d10] DE clusters (si=1/2) may be represented in the form of the effective antisymmetric DE Hamiltonian
(24)
where , is the isotropic transfer operator for the pair, = ab, bc, ca. The operator (24) describes the vector-type spin-transfer interaction in the MV trimer induced by SOC. The matrix elements of the ASDE interaction depend on the projection M of total spin S. The ASDE describes the non-collinear orientation of spins.
The ASDE coupling determines the linear splitting of the DE term
(25)
of the [d9-d9-d10] MV cluster. The and, and DE terms with different S are mixed by the AS double exchange (24) and Dzialoshinsky-Moriya AS exchange, which has the form in the delocalized MV trimer. The AS mixing results in the second-order ASDE and DM ASE contributions to the cluster ZFS parameters of the axial anisotropy
(26)
for the DE trigonal term and
,
for the DE term The double exchange (the migration of the extra electron or hole among all three ions in the MV cluster) forms the cluster Dzialoshinsky-Moriya AS parameter
(27)
The vector of the DM ASE interaction is directed along the trigonal Z-axis of the MV trimer:=,.
7.3 The MV Trimers with High Individual Spins and
Spin Frustration
In all MV trimers, only the ASDE coupling determines the linear splittings of the DE terms with maximal total spin S:
(28)
For the DE terms with non-maximal total spin S, characterized by the DE mixture of intermediate spins (spin frustration), both ASDE and DM ASE interactions determine the linear splittings i(n1KZ+n2GZ),since these AS couplings mix the frustrated states with the same S and different intermediate spins , Thus, the parameters of the linear splittings of the and states of the [d1-d1-d2] cluster have the form , [28]. The ASDE contribution (KZ)to the linear splittings is stronger than the DM ASE contribution (GZ) since .
The ASDE coupling (and DM ASE) mixes only and , and trigonal DE terms, In the trimeric clusters with high individual spin Si, both AS double exchange and DM ASE form the second order ZFS parameters: the DM ASE mixes the localized Heisenberg levels with different S () and also the frustrated states with the same S (). The AS double exchange mixes the DE terms of the delocalized trimer. The ASDE plus DM ASE mixing of the DE levels with different S and also the AS mixing of the terms (with the same S) with different intermediate spins Sij determine the second order AS contributions to the cluster ZFS parameters of the axial anisotropy. In general, the both ASDE and DM ASE (KZ and GZ parameters), t and J determine the second order ZFS parameters of the axial anisotropy
, (30)