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,

ˆ S -S
the AS DE coupling H ASDE  2iK ab T
ab
b
a

has a form of the
vector type spin-transfer interaction induced by SOC. The AS DE
coupling mixes the Anderson-Hasegawa DE states E+0(S) and E-0(S)
with the same S of the different parity. The AS DE coupling and



Dzialoshinsky-Moriya (DM) AS exchange H DM  Dab[Sb  Sa ] mix
the DE states with different S (S  1) of the same parity. AS DE
and DM AS exchange contribute in the second order to the cluster ZFS
parameters DS ( DS  [(n1Kab n1Dab )2 / n3t  n4 J )] ), where ni are

the numbers ( K ab  Dab ). DS are different for the DE states E+(S)
and E-(S), S>1/2. The AS DE vector is directed perpendicularly to the
ab-axis of dimer. The AS DE coupling results in anisotropy of gfactors. 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 AS DE coupling results in the
linear AS DE fine splittings Δ of the trigonal 2S+1E DE terms. Δ is
proportional to the trimer AS DE parameter
K Z  ( K abZ  K bcZ  K caZ ) / 3 .
The cluster AS DE vector is directed along the trigonal Z-axis:
K Z  0, K X  KY  0 . The AS DE coupling mixes the trigonal
isotropic
2S+1
A1 and
2S´+1
A2, 2S+1E and 2S´+1E DE terms, S  0, 1 . In
5
the MV trimers, the AS DE and DM AS exchange mixing of the DE
levels 2 S 1  ( S  0, 1 ) contributes in the second order to the ZFS
parameters DS of the axial anisotropy. For the delocalized
[d 9  d 10  d 10 ] cluster, the AS DE fine splitting   2 K Z 3 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
n 1
the mixed-valence (MV) rare earth manganates. In the [ d  d ] MV
dimers, the migration of the extra electron between paramagnetic
ions results in the DE splitting of the S levels [2]:
  S   (S  1/ 2)t0 /  2s0  1 , where t0 is the one-electron transfer
n
integral,
s0  s (d n ) .
The
AH
double
exchange
interaction
ˆ t (or spin-dependent electron transfer (ET) T
ˆ ) is isotropic;
H AH  T
ab 0
ab
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, singlemolecular 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+1  i [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
6
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
S Γ
i
Sj
ij
.
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 nondegenerate 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 D12 S1  S2  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
H ab 

 
,  a ,b
0
z


{t
[  + +  ]  C
[  + +  ]  C
+   C
+  }. (1)
0
The first term (with tab
 a b ) 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
0
ET parameter: tab  tab
 Cab , where Cab is the purely imaginary
vector ET parameter [19]. In the DM AS exchange interaction
H DM  ΣDij [Si  S j ],
Dij  D ji are antisymmetric vector coefficients. The Cab parameters
7
determine the vector DM AS exchange constants Dij ( 2iCij J / t ) and
~
anisotropic (AN) exchange contribution ij in the pseudodipolar term
S i Γ ijS j in the perturbation theory of AN superexchange. The DM AS
exchange leads to a canted arrangement of spins Si and S j , which
were oriented in antiferromagnetic order by the isotropic Heisenberg
interaction
H 0   J ij Si S j .
The microscopic theory of DM AS exchange was further developed
(see [21-23, 18]).
3. Double Exchange in the [ d  d ] and [ d  d ] Clusters
We will start the consideration of SOC in DE (ET) in the MV
1
dimers
Ti
3
on
an
example
0
of
 Ti 4 and V 4  V 5 pair  [24].
9
the
10
1
0
[d d ]
For the [ d  d
n
dimer
n 1
] MV
clusters, the Hamiltonian of the pair Hˆ 12  Hˆ a*b  Hˆ ab*  Vˆ12 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 d 1 ion includes the crystal field (CF) HCF

and spin-orbit interaction H LS
  L S [25-27].
Local distorted
octahedral CF forms the orbitally non-degenerate ground  a0  d xya
 
state. The excited CF states   d yz

and va  d 2 2
x y

a
,a   d zx a , ua   d3 z2 r 2 
are separated by the CF intervals  ,  ,  and 
a
a
a
u
a
a
v,
a
respectively [25]. The one-site SOC admixes excited CF states  m into
the ground state  0 of the d 1 ion of the MV pair in the localizations
8
|a*b> and |ab*>. In the first order perturbation, the renormalized SOCadmixed ground state functions for the center  have a form
  (m  1/ 2)  [ 0 (1/ 2)  i v va (1/ 2)   a ( 1/ 2)  i  ( 1/ 2) ,(2)
where d-orbital functions refer to the local Cartesian frame and
 v    v ,     2 ,   ,. In the resonance representation
   S   [ a*b  S , M    ab*  S , M ]/ 2 , the non-diagonal AS DE
v
K ab
terms in the DE matrix [24] describe the SOC mixing of the AH
DE states  S  and  S  . The double exchange with the taking
into account the real vector transfer term K ab ( Cab  iK ab ) results
1
0
splitting for the [ d  d ] MV dimer has the form
2
2
.
'   tab
 K ab
(3)
1
0
2
2
is the effective DE (ET) parameter of the [ d  d ]
teff  tab
 K ab
Y 2
MV dimer, K 2ab  ( K abX ) 2  ( K ab
)  ( K abZ ) 2 .
0
The transfer integral tab has the form tab  tab
   tab ,
0
tab
    a0 | Vˆ12 |  b0 ,
  [ v2 v a vb   2  a b   2 a || b ].
tab
The tab   term is the standard isotropic ET integral for the
transfer between the neighboring ζ0 d-orbitals in the ground state
0
 term is the contribution to t ab of the ET in the
without SOC. The t ab
excited states due to the SOC admixture. In the real antisymmetric
( K ab  K ba ) vector transfer parameter




 K ab
 iK ab
 K aby  iK abx ):
K ab
( K ab

Kab
 [ btab ( a0 , b )   atab (a ,  b0 )],

Kab
 [  tab (a ,  b0 )   btab ( a0 ,b )],

(4)
K ab
 [ vbtab ( a0 , v b )   va tab (v a ,  b0 )],
t ( 0 ,  )   0 | Vˆ12 |   is the transfer integral between the
0
ground orbital   state on the center   1,2 and the excited  
9
state on the center   2,1 . 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 dorbitals  n to the ground  0 orbital. The local CF 3d-orbitals may be


written in the common xyz-coordination system [18]   
2 :
va '  va   a a  , ua  ua   3 a  a  , a  a 




 a  3ua   a , a  a    va  3ua   a ,
 a   a   a a  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  b

states:
tab  a , b   [tab    tab  ], tab  a ,b   [tab    tab  ].
b
The corresponding transfer integrals between the ground state
and excited  a
state have the opposite sign:
tab  b , a   tab  a , b  , tab  b ,a   tab  a ,b  .
The

components of AS DE vector parameter K ab
are the following:
KabX    [t  t ],
The
resonance
E   t  K  K
2
ab
t
'
eff
Y
Kab
  [t  t ],
2
x
2
y
splitting
in
the
Kabz  0.
DE
+
SOC
(5)
model
depends on the effective transfer parameter
 t K K .
2
ab
2
x
2
y
The AS DE contribution to the g-factors is described by
the eq. (6):
gα=g0{(t2+Kα2)/(t2+K2)}1/2.
(6)
In the case K x , y  0 , Kz=0, we obtain an anisotropy of g-factors
(gz<gx) induced by antisymmetric double exchange [24].
The estimations of the value of the AS DE vector parameters
10
Cab
 | K ab | were obtained by Moriya [19]
Cab tab  Kab tab ~  g g .
The AS DE vector parameter K ab may be the quantities of the order of
1-5% of the double exchange parameter t ab .


The hole-transfer in the Cu 2  Cu  d 9  d 10 pair of holedoped La2CuO4 was considered in [18]. The components of the AS

vector transfer parameter K ab
are the following [18, 24]:
KabX  [ atab (vb0 , a )   btab (v0a , b )]   [tv  t ]/ 2  ,
(7)
Y
Kab
 [atab (vb0 ,a )  btab (v0a ,b )]   [tv  t ]/ 2 , K abz  0.
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]:
H SO   Cij   ci b c j    H .c. .
(8)
ij
An anisotropic term HAN  Si ΓijS j is characterized by the
symmetric anisotropic parameter [18, 19]:
Γij  4[CijCji + CjiCij  1  (CijCji )]/ U  J [CijCji + CjiCij  1  (CijCji )]/ t 2 . (9)
This anisotropic interaction is not active in the [d 1  d 0 ]
([Ti 3  Ti 4 ]) and [d 9  d 10 ] ([Cu 2  Cu  ]) 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
11
following form

ˆ S -S
H ASDE  2iK ab T
ab
b
a


(10)

ˆ
where K ab is an antisymmetric K ab = -K ba vector coefficient, T
ab
is the transfer operator, and Sa , Sb are spins of the ions of the
d n  d n1 pair. The effective AS double exchange Hamiltonian (10) of
the second-order perturbation theory describes the vector type spintransfer interaction induced by SOC. The matrix elements of H ASDE
between the states of different localization with different spins S and S
have the form

 a*b  S , M  H ASDE  ab* S , M
2iK ab





.


ˆ   S, M    S, M  S - S 
 a*b  S , M  T
ab
ab*
ab*
b
a
ab* S , M
(11)
In the case of the states with the same total spin S  S , the
ˆ
first multiplier in eq. (11) represents the Anderson-Hasegawa DE T
ab
contribution
ˆ   S , M    S  1/ 2   2 Sa  1  .
a*b  S , M  T
ab
ab*


n
n 1
For the [ d  d ] MV dimers, matrix elements of the AS DE
coupling (10) between the states of different localization with
S  S , M  M depend on projection M [24]:
 2S  1   Sb  Sb  1  Sa  Sa  1
M . (12)

S  S  1
 2Sa  1 
z
 a*b  S , M  H ASDE  ab*  S , M   iK ab

The mixing of the Anderson-Hasegawa DE states with different
total spin S ( S  S ), may be described by the effective operator of the
AS DE
mixing


ˆ ' S -S ,

H ASDE
 2iK ab T
ab
b
a
where
ˆ    S , M    2S  1 / 2.
a*b  S , M  T
ab
ab*
a
The AS DE coupling mixes the S and S states with S  1
[24].
12
5. Antisymmetric Double Exchange in the [d 9  d 8 ] Cluster
The total spin of the [d a8  d b9 ] pair is S = 3/2, ½. The AS DE
coupling (10), (12) mixes the states of different localization with the
same total spin S:
 a*b  3 / 2, 3 / 2  H ASDE  ab*  3 / 2, 3 / 2   tv  iK abz ,
 a*b  3 / 2, 1/ 2   ab*  3 / 2, 1/ 2   tv  i K abz 3,
(13)
 a*b 1/ 2, 1/ 2   ab* 1/ 2, 1/ 2   tv 2  5i K abz 6,
In the resonance representation Ψ±(S, M), AS DE mixes the
Anderson-Hasegawa DE states S=3/2 of different parity:
  3/ 2 [ E0 (3/ 2)  t  3J ] and    3/ 2  [ E0 (3 / 2)  t  3 J ] ,
and also the AH DE states
 1/ 2 [ E0 (1/ 2)  t / 2] and
 1/ 2 [ E0 (1/ 2)  t / 2] with the same S = 1/2.
AS DE 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), AS DE 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   (3/ 2) and  (1/ 2) of the
same parity depends on both AS double exchange and DM AS
exchange (ASE) parameters:

   (3 / 2, 3 / 2) ||   (1/ 2, 1/ 2)  i[ K   3DDM
/ 2] / 6,
Z
   (3 / 2,|1/ 2 |) ||   (1/ 2,|1/ 2 |)  i 2[ K Z  3DDM
/ 2] / 3, (14)

   (3 / 2, 1/ 2) ||   (1/ 2, 1/ 2)  i[ K   3DDM
/ 2] / 3 2,


where DDM
are the components of the DM AS exchange.
 Dab
Since K X ~ t (g / g ), DDM ~ J (g / g ), t  J and K X  DDM in
13
the delocalized MV cluster, the AS DE contributions to the mixing (14)
S are stronger than the DM AS exchange contributions.
In the [d a8  d b9 ] MV cluster, an initial zero-field splitting of

the d a8 -ion ( H ZFS
 D0  S2z  S  S  1 3 ) 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 AS DE and DM AS exchange mixing results in
the following second order AS exchange contributions to the ZFS
parameters of the high spin DE states:
DS ( S  3 / 2)  D0  DS ,
DS  2[ K 2 / t  K 2 /(t  6 J )]/ 9 , (15)

K 2  ( K X2  KY2 ) / 2, K 2  [( K X ) 2  ( KY ) 2 ]/ 2, K  K  3DDM
/2
in
the
case K X ,Y  0, K Z  0 .
For
the
cluster
with
X
Y
Z
|| DDM
| DX  DDM , DDM
0,
| K X || KY |, KZ  0 and | DDM

DS (3 / 2)
the
terms
in the
cluster
ZFS parameters
DS (3 / 2)  DS  DS are different for the Anderson-Hasegawa DE
states E0 (3 / 2) and E0 (3 / 2) of the different parity:
DS (3 / 2)  4[ JK X2 / t  0.5DDM ( K X  0.75 DDM )]/ 3(t  6 J ). (16)
The AS DE [~ JK X2 / t (t  6 J )] contribution to the cluster ZFS
parameter is stronger than the pure Dzialoshinsky-Moriya AS exchange
2
contribution [~ DDM
/(t  6 J )] since K X  DDM , t  J . 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
1
T2 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
14
ˆ [ S 2  S ( S  1) / 3] ,
H ANDE  (3 / 2)T
(17)
ab
Z
where Tˆ ab 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  '(3 / 2) AN DE
parameter of the easy-axis two-ion anisotropy
 '(3/ 2)  {t ( 1   2 )2  2(t  tv )( 12   22 ) 
(18)
 [t' ( 1    2 )2  2(t'  tv )( 12  22 )  3 2tv ( 1    2 )  ]}/ 6
  ,
is proportional to the t ( 1   2 ) 2 terms [24],  1 ,  2 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 [d n  d n  d n 1 ] MV clusters, the AS DE interaction
results in the new effects, which can not be obtained in the MV dimers.
Strong double exchange and Heisenberg exchange interactions in the
[d n  d n  d n1 ] clusters form the isotropic exchange-resonance (DE)
eigenstates
2 S 1
i characterized by the total spin S and irreducible
representations ( i  A1 , A2 , E ) of the trigonal D3 group [13].
7.1. The MV Cluster [d 9-d 10-d 10] ([d 1-d 0-d 0])
2
2
The DE levels are the A1 and E trigonal terms. The group
theoretical consideration and effective Hamiltonian method show that
2
the E DE term must be linearly split by the spin-orbit coupling into
2
two Kramers sublevels. This fine splitting of the ground E terms
(19)
  2 E  t  K Z 3
 
is produced by the AS DE coupling [28, 29]. The splittings of the E
terms are proportional to the cluster AS DE parameter
15

K Z  K abZ  K acZ  K caZ

(20)
3,
Z
where K is the pair AS DE contribution. The microscopic
consideration of SOC (AS DE) in the DE trimer [28] shows that the
vector K of the AS DE coupling is directed along the trigonal Z-axis of
the MV cluster:
K Z    3(tv  2t ), K X  KY  0 ,
(21)
 is the small angle between the local z-axis and the C2 -axis of the
2
trimer. The AS DE splitting of the E DE term   2K Z 3 is linear
with respect to the SOC parameter λ and ET parameter t. The AS DE
2
2
coupling does not mix the E and A1 terms.
For the delocalized [Cu(II)Cu2(I)] cluster, the linear AS DE fine
splitting   2K Z 3 of the ground 2E DE term determines strong
anisotropy of the Zeeman splitting. In the external magnetic field
   Z parallel to the trigonal Z-axis, the Zeeman splitting of the
2
AS DE sublevels E  1 2 and E  1 2 of the DE E term is
linear [28, 29] ( hZ  g 0  Z 2 ):
1,2  t  K Z 3  hZ ,  3,4  t  K Z 3  hZ .
(22)
In the case    X , the Zeeman sublevels depend non-linearly on
magnetic field
1, 2  t  3K Z2  hX2 ,  3, 4  t  3K Z2  hX2
.
(23)
Along with the fine splitting , AS DE 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    Z , we obtain from eq.
(22) the cluster g-values g ||  g 0 for the E  1 2  and E  1 2 
Kramers sublevels. In the case    X , we obtain the cluster g-value
g   0 . The strong anisotropy of the Zeeman splitting determines
anisotropy of the magnetic susceptibility (  ||    ) of the MV trimer.
16
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
2S 1
 DE terms of the trigonal [d 10  d 9  d 9 ] MV cluster:
 ( 3E )  t  2 J ,  ( 3 A2 )  2t  2 J ,
 ( 1E )  t ,  ( 1 A1 )  2t . The
AS DE 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


Z
Z ˆ
H ASDE
 2i K
T SZ  SZ ,

Z
Z
where K
,
  K
(24)
Tˆ is the isotropic transfer operator for the 
Z
pair,  = ab, bc, ca. The operator H ASDE
(24) describes the vectortype spin-transfer interaction in the MV trimer induced by SOC. The
matrix elements of the AS DE interaction depend on the projection M
of total spin S. The AS DE describes the non-collinear orientation of
spins.
3
The AS DE coupling determines the linear splitting of the E
DE term
Z
   3 E, M  H t  J  H ASDE
   3 E , M   t  2 J  3K Z M
3
1
(25)
of the [d9-d9-d10] MV cluster. The A2 and A1 , E and E DE terms
with different S are mixed by the AS double exchange (24) and
Dzialoshinsky-Moriya AS exchange, which has the form
3
1
Z
Z
S  Sb  in the delocalized MV trimer. The AS
H DM
  D

Z

mixing results in the second-order AS DE and DM ASE contributions
to the cluster ZFS parameters of the axial anisotropy
DS
 A   K
3
2
Z
Z
 DDM
4
 t  J 2
2
3
for the A2 DE trigonal term and
DS
 E  K
3
Z
Z
 DDM
2

2
17
2 t  J  ,
(26)
for the E DE term {H AN  DS ( 2 S 1 i )[ S z2  S ( S  1) / 3]}. 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
Z
(27)
DDM
 DabZ  DbcZ  DcaZ 3.
3


The vector of the DM ASE interaction is directed along the
Z
X
Y
 D DM
0.
trigonal Z-axis of the MV trimer: D DM
= 4 3  J 0 , DDM
7.3 The MV Trimers with High Individual Spins and
Spin Frustration
In all MV trimers, only the AS DE coupling determines the
2 S 1
E DE terms with maximal total spin S:
(28)
 ASDE  2 Smax 1E , M    K Z M 3 S max
linear splittings of the
2 S 1
E DE terms with non-maximal total spin S,
For the
characterized by the DE mixture of intermediate spins S ij (spin
frustration), both AS DE 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 S ij , Sij  1.
Thus, the parameters 1 of the linear splittings
3
1M of the E and
3
E ' states of the [d1-d1-d2] cluster have the form
1  9 3[ KZ  2GZ / 3]/10 , 1'  3( KZ  6GZ ) /10 [28]. The
AS DE contribution (KZ) to the linear splittings  i is stronger than
Z
the DM ASE contribution (GZ) since K ab  Gab , GZ  DDM
.
The AS DE coupling (and DM ASE) mixes only
2 S 1
A2 ,
2 S 1
E and
2 S 1
2 S 1
E trigonal DE terms, S  0, 1.
A1 and
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 ( S '  S  1 ) and
18
also the frustrated states with the same S ( Sij'  Sij  1 ). The AS double
exchange mixes the DE terms
2 S 1
i of the delocalized trimer. The
AS DE plus DM ASE mixing of the DE levels
2 S 1
i with different S
2 S 1
i terms (with the same S) with
and also the AS mixing of the
different intermediate spins Sij determine the second order AS
contributions to the cluster ZFS parameters DS ( 2 S 1 i ) of the axial
anisotropy. In general, the both AS DE and DM ASE (KZ and GZ
parameters), t and J determine the second order ZFS parameters of the
axial anisotropy
DS  2 S 1 i   [(nK Z  pGZ )2 /(qt  rJ )] ,

(30)

which contribute to the cluster ZFS DS 2 S 1 i parameters along with
the individual (local) ZFS contributions D0. For the DE terms of the
MV trimers, which have the same DE energy, the strongest second
2
order contribution to ZFS (the K Z / J terms) are determined by the
AS DE admixture of the levels with the same DE energy separated only
by the Heisenberg exchange intervals ~J. The second order AS DE
contributions to the ZFS parameters DS are stronger than the pure DM
ASE contributions since KZ  GZ , KZ / GZ  t / J . The AS DE
contributions are different for different 2S+1A1, 2S+1A2 and 2S+1E trigonal
DE terms of the MV trimers.
8. Conclusion
An antisymmetric double exchange interaction and anisotropic
double exchange describe the spin-orbit coupling in the MV clusters
with the double exchange and Heisenberg exchange inter-ion coupling,
individual crystal field and ZFS splittings.
The results of the AS DE coupling in the MV trimers are
essentially different from the AS DE results for the MV dimers. In the
MV dimers, the AS DE coupling results in the mixing of the AndersonHasegawa E ( S ) and E_ ( S ) levels of the different parity with the
same S, the mixing of the E ( S ) and E ( S ' ) DE levels of the same
19
parity with different total spin ( S '  S  1 ), and also in the second
order zero-field splittings of the Anderson-Hasegawa DE states and
small anisotropy of g-factors. In the MV trimers, the AS DE exchange
2 S 1
E DE terms, strong trigonal
results in the linear splittings of the
anisotropy of the Zeeman splittings, g-factors and magnetic moment,
mixing of the
2 S 1
A1 and
2 S ' 1
A2 ,
2 S 1
E and
2 S ' 1
E ' DE levels
( (S  0, 1) , in the second-order contributions to the trigonal ZFS
2 S 1
E terms.
parameters of the axial anisotropy for the 2 S 1 Ai and
In the mixed-valence clusters of orbitally non-degenerate d-ions
with the double exchange and Heisenberg exchange inter-ion
interaction, an antisymmetric double exchange essentially contributes
to the zero-field splittings of the cluster states, mixing of the DE levels,
magnetic anisotropy and anisotropy of g-factors.
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21