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Cluster molecular orbital description of the electronic structures of mixed-valence iron oxides and silicates
Solid State Communications, Printed in Great Britain.
Vol. 58, No. 10, pp. 719-723,
0038-1098/86 $3.00 + .OO Pergamon Journals Ltd.
1986.
CLUSTER MOLECULAR ORBITAL DESCRIPTION OF THE ELECTRONIC MIXED-VALENCE IRON OXIDES AND SILICATES
STRUCTURES
OF
D.M. Sherman U.S. Geological Survey 959, National Center, Reston, VA 22092, USA (Received
27 November
1985, itz revised
form 12
February
1986 by E.F. Bet-taut)
A molecular orbital description, based on spin-unrestricted Xo-scattered wave calculations, is given for the electronic structures of mixed valence iron oxides and silicates. The cluster calculations show that electron hopping and optical intervalence charge-transfer result from weak Fe-Fe bonding across shared edges of Fe06 coordination polyhedra. In agreement with Zener’s double exchange mode\+Fe-Fe bonding is found to stabilize ferromagnetic coupling between Fe and Fe3+ cations.
INTRODUCTION MIXED VALENCE IRON OXIDES and silicates often exhibit thermally activated electron hopping between adjacent Fe’+ and Fe3+ cations and optically induced Fe? + Fe: + Fe: + Feg charge transfer transitions. The most well-known example of a mixed valence iron oxide is magnetite (Fe30,). Examples of mixed valence iron silicates that show thermally activated electron hopping include ilvaite [ 11, deerite [2] , and cronstedite 131. A quantitative picture of the electronic structures of such mixed-valence phases would give insight on the nature and mechanisms of electron hopping and optically induced intervalence charge transfer transitions. A natural approach to describing the electronic structure of a solid is to use one-electron wavefunctions which have the translational symmetry of the crystal structure. Such are the Bloch wavefunctions of band theory. Bloch wavefunctions, however, cannot easily describe the localized electronic states found in transition metal oxides and silicates. An alternative approach is to use a cluster molecular orbital description. For example, much of the electronic spectra and even magnetic properties of iron oxides can be described in terms of the electronic structures of FeOd and Fe04 coordination polyhedra [4]. This is because the electronic states of interest are localized on the metal atom and its immediate coordination environment. Even in mixed valence phases which are said to exhibit “electron delocalization” (i.e., fast electron hopping), we are still dealing with localized electronic states. (A possible exception is magnetite over which there is some debate as to whether the electrons are itinerant or polarons [5,6] .) In this paper, the electronic structures of mixed valence Fe oxides are investigated using the cluster molecular orbital approach. The electronic structure 719
calculations were done using the Self-Consistent Field-Xo Scattered Wave (SCF-Xo-SW) method [7] . The simplest cluster that can be used to describe intervalence chargetransfer consists of two Fe06 coordination polyhedra which share a common edge (i.e., an (FezOi,,)ls-cluster). This corresponds, for example, to a pair of adjacent Fe’+ and Fe3+B-site cations in magnetite. In order to relate the electronic structure to the phenomena of electron hopping and polaron formation we need to show how the electronic states are coupled to the nuclear coordinates of the cluster. Following Austin and Mott [8] and others (e.g., [9] )the vibrational coordinate of interest is given by 4 = RA--RR, where RA and RB are the Fe-O bond lengths at sites A and B, respectively. The electronic structure of the cluster was calculated at q = 0 and at q = h. At q = h, RA and R, correspond to the equilibrium Fe3+-0 and Fe2’-0 bond lengths (i.e., RA = 2.OOA and RB = 2.16 A). At q = 0, both Fe sites are identical and correspond to the average of those at q = X (i.e., RA = RB = 2.08 A). The geometry of the cluster is shown in Fig. 1. Note that, at q = 0, the cluster has DZh symmetry: however, broken symmetry wavefunctions that transform under Czv symmetry were used so that electron delocalization is not imposed upon the cluster. This is akin to the “Xo valence bond” formalism used by Noodleman [lo] to describe the electronic structure of ferrodoxin. For the SCF-Xcr-SW method, the most important computational parameters are the atomic sphere radii. These were (in Angstroms) 1.208, 1.143 and 1.159 for the iron, bridging oxygen, and remaining oxygen atoms, respectively. The same atomic sphere radii were used at both the q = 0 and q = X cluster geometries.
720
ELECTRONIC
STRUCTURES
OF MIXED-VALENCE
IRON OXIDES
Vol. 58, No. 10
P-spin 6
---=55= eg _---
4
_--==== =I==
I-
s: 3 k 6
i Fe(3c-i) t 29 I
a-spin 2
eg
-m
5
\
l O(2P)
Fig. 1. Geometry coordinate 4.
of the (FezOio)rs-
L
cluster along the
The (Fe20i0)r5- cluster is surrounded by an outer sphere of radius 14.64 A at the 4 = 0 geometry and 15.408 A at the 4 = h geometry. The (Y parameters were taken from Schwarz [ 1 l] . The electrostatic stabilization energy of the crystal is simulated by surrounding the cluster by a Watson sphere with charge + 15.0. Partial waves with maximum angular momentum quantum numbers 1= 1, 2 and 3 were used for the oxygen, iron and outer spheres, respectively. RESULTS AND DISCUSSION General
The electronic structure of the (Fez0ro)i5- cluster in the ferromagnetic state (S = 912) at 4 = 0 is shown in Fig. 2. In an isolated FeOe cluster [4], the Fe(3d) orbitals are split into a set of tzg orbitals (Fe-O rrantibonding) and a set of eg orbitals (Fe-O uantibonding). The energy separation between the tzg and eg orbitals corresponds to the 1ODq parameter of ligand field theory. All of the orbitals are split by the exchange energy into a-spin (majority spin) and &spin (minority spin) manifolds. The effect of the ligand field and exchange splitting of the Fe(3d) type orbitals is also seen in the (Fez010)15- cluster. This cluster, however, also allows for Fe-Fe bonding to occur; overlap of the Fe,(tsJ and Fe,(t,,) orbitals results in the cluster molecular orbital analogue of a narrow Fe(3d) band. The p-spin Fe(3d) orbitals are shown in more detail in Fig. 3. The Fe(tzg)-Fe(t2& u-bonding overlap is strong enough to completely delocalize the Fe2+ &spin electron over the two metal atoms so that each Fe atom has a formal valence of + 2.5. As will be shown below, this is not the case if the Fe atoms are antiferromagnetically coupled. The energy levels of the cluster at 4 = X are given in Fig. 4. At this cluster geometry, the Fe’+ b-spin
Fig. 2. Calculated energy level diagram for (Fe20,0)‘5at the coordinate 4 = 0 (all Fe-O bond lengths are 2.08 A) when the two Fe atoms are ferromagnetically coupled. Dashed levels correspond to unoccupied orbitals. electron tends to be localized on the larger Fe site. This gives distinguishible Fe2+ and Fe3+ cations in the cluster. The effect of localizing the Fe’+ P-spin electron is to split the tzn and eg Fe(3d) bands into Fe2+(t2&, Fe3+(t2J, Fe”(e,) and Fe3+(e,) sub-bands. Cluster model of electron hopping
Given two Fe atoms located at sites A and B, the electronic states of the pair can be described in terms of the zeroth-order wavefunctions +i and \ks, which correspond to the ionic configurations FeyFeg and FeTFeg:, respectively. Associated with each wavefunction 1s a potential energy surface as shown in Fig. 5. If there exists an off-diagonal matrix element J which can coupled \kr and \ks, then the true states of the mixed valence pair will be: \k+ = (Y\ki + (1 -f_YZ)“*Qk2, and 9_ = (1 - a2)“2\ki From perturbation
-c&2. theory,
(Y’ = (1 - AE/(AE’
+ 4J2)“2)/2,
where AE is the energy difference between \ki and \k2. In terms of the states \ki and \k2, the ground state electronic structure of the (Fe20,0)‘5- cluster at 4 = 0 corresponds to the wavefunction: \k+ = l/d/2(*,
+ **).
The excited state \k_ corresponds to the configuration in which the &spin electron occupies the 17ar (Fe-Fe a-antibonding) orbital rather than the 16aI (Fe-Fe
ELECTRONIC
.Vol. 58, No. 10
STRUCTURES
OF MIXED-VALENCE
721
IRON OXIDES P-spin
13b:! ---_.
Fe3* IIII
::::
eV 5.5 -
Fe3+ x=1= ---_II_ Fe*+
z 19a, ____A
eg
I \ f29
a-spin
6 5 2
Fe3+ x X
s
12bp ---_I
Fez+
Fe*+
.-9
3
18a, ----.
o-
B O(2P) -2 -
\
17a,
&do 0 49
____/
CT’
Fig. 4. Electronic structure of the cluster at 4 = h. The Fe(3d) bands split into Fe2’ and Fe3+ orbitals.
4.0 llb 1 ---_ / 7a,
____-
6a,
----\
lob, ----
\
3.5 16a, -
\
u
\ Fig. 3. Details of the &spin Fe(3d) energy levels at 4 = 0 showing the different Fe(3d)-Fe(3d) bond types.
u-bonding) orbital (Fig. 3). At 4 = X, the ground state of the cluster corresponds to the wavefunction \k+ with 0 < c? < l/2. From the orbital compositions, a is found to be 0.12. The physical meaning of the coupling integral J is that it describes the weak Fe-Fe bonding interaction. At 4 = 0, the energy difference between the states \k+ and \k_ is 2J and is given by the energy of the one-electron transition 16ar -+ 17ar (Fig. 3). Note, in passing, that this quantity is also the cluster molecular orbital estimate of the Fe(f& 3d bandwidth. Using the transition state formalism [ 121 , the 16ar -+ 17ar transition energy is found to be 0.84 eV; hence, J = 0.42 eV. This is comparable to values estimated for other mixed valence pairs (e.g., Ti3+--Ti4+, Ni’+-Ni3+) [8] and is close to that estimated [ 131 (0.3 eV) for Fe304. At 4 = h, the energy of the 16ar -+ 17ar (9+(h)--+ \k_(X)) transition corresponds to Eop in Fig. 5. This is calculated to be 1.3 1 eV. Optical absorption spectra of mixed valence Fe oxides and silicates [14] typically
show a strong absorption band near 1.65 eV that is assigned to optical intervalence charge transfer (sometimes referred to as “photon induced hopping”). Since, at 4 = h, the 16ar -+ 17ar transition is an Fe’+ + Fe3+ + Fe3+ + Fe’+ charge-transfer process, the electronic structure calculations support the band assignment. Moreover, like intervalence charge-transfer bands observed in the optical absorption spectra of mixed valence Fe silicates [14], the 16ar + 17ar transition will be polarized along the Fe-Fe internuclear vector in accordance with the Laporte selection rule. Note that transitions from the 16ar Fe-Fe u-bonding orbital to the 1 lbr rr- and 7a2 b-antibonding orbitals are not expected to be important given the negligible Fe-Fe rr- and b-bonding overlap. The activation energy for electron hopping (E,) can be estimated from the calculated values for E,, and J: Ea = (2&p - AE(h))/4 From perturbation
-J(O).
theory,
E op = (AJY(X)~ + 4J(h)2)“2. Assuming that J(h) ==z J(O), E, is calculated to be - 0.016 eV which implies that there is no thermal barrier for electron hopping (i.e., the potential energy surface corresponds to the dashed lines in Fig. 5). The lack of an activation barrier implies that small polarons (i.e., distinguishable Fe2+ cations) are unstable when the Fe-Fe separation is less than 2.95 A. This, in turn, suggests that small polarons are not the charge-carriers in magnetite [5, 61. Increasing the Fe-Fe distance will decrease J so that, eventually, there will be an activation barrier for hopping. Such a barrier, on the order of
722
ELECTRONIC E
STRUCTURES
OF MIXED-VALENCE
l!!LL v,
IRON OXIDES P-spin
: 1: 1 Fe’+le,)
(z-spin
6
VZ!
l2,31
Vol. 58, No. 10
IIII
Fe2+(t2,1
13,21
-A
0
h
q
-A
0
h
q
Fig. 5. Potential energy surfaces for the \kr and \kZ states of an Fe*“-Fe3+ pair: (a) in the absence of coupling (J = 0), and (b) when coupling is present. The dashed curves correspond to the case where the coupling integral J is so large that only a single minimum occurs in the potential energy surface of the ground electronic state.
Fig. 6. Electromc structure of the (Fe20re)‘5- cluster when the two Fe atoms are antiferromagnetically coupled. The absence of Fe-Fe bondin in this configuration results in distinct Fe2+ and Fe3$ cations even at 4 = 0. SUMMARY
0.1-0.35 eV, is found in ilvaite [ 11, deerite [2] , and cronstedite [3] where the Fe-Fe distances are somewhat larger than that in the cluster described here. Magnetism and Mixed Valency
Adjacent Fe” and Fe3+ cations will be magnetically coupled either by the antiferromagnetic Fe-O-Fe superexchange interaction or by the ferromagnetic double exchange mechanism. Consistent with the notion of double exchange, the Fe’+ &spin electron is delocalized over the two Fe sites in the ferromagnetic configuration. The electronic structure of the (FezO,,)‘scluster in the antiferromagnetic state (S = l/2) is shown in Fig. 6. In this configuration, there is no Fe-Fe bonding and, consequently, the Fe*’ P-spin electron is completely localized to one Fe site giving distinguishible Fe” and Fe3+ cations even at 4 = 0. Using the “magnetic transition state” formalism [ 12, 151 the ferromagnetic configuration is found to be more stable than the antiferromagnetic configuration by 0.94 eV. Ferromagnetic coupling between Fe*+ and Fe3+ cations occupying edge-sharing coordination sites is predicted by the double exchange model and is found in all mixed valence Fe silicates investigated thus far [16]. The relative stability of the ferromagnetic state, however, is greater than the stabilization energy resulting from electron delocalization. Hence, ferromagnetic coupling by “direct exchange” [S, 61 must also be important.
The results presented here give a molecular orbital description of polarons in mixed-valence iron oxides and silicates. Electron hopping in such phases results from the formation of chemical bonds between next nearest neighbor Fe*+ and Fe3’ cations; if such bonds are sufficiently strong, thermally induced electron hopping will give way to collective electron behavior. The idea that electron hopping results from metal-metal bonding is consistent with the observation that such hopping is not observed between cation sites that do not share a common edge or face [14]. In agreement with the observed magnetic structures of mixed valence iron oxide and silicates, Fe2+ and Fe3+ cations are predicted to be ferromagnetically coupled. Finally, electronic transitions between Fe-Fe bonding and antibonding orbitals correspond to optical intervalence charge transfer and are responsible for the polarized absorption bands observed in the optical spectra of mixed-valence Fe silicates. REFERENCES 1. 2.
J. Phys. Chem. Solids
B.J. Evans & G. Amthauer, 41,985
(1980).
H. Pollack,
R. Quartier
& W. Bruyneel,
Phys.
Chem. Mineral. 7, 10 (1981). 3.
J.M.D. Coey, A. Mourkarika
4.
Solid State Commun. 41, 797 (1982). D.M. Sherman, Phys. Chem. Mineral. (1985).
& C.M. McDonagh, 12,
167
Vol. 58, No. 10 5.
6. 7. 8. 9. 10.
ELECTRONIC
STRUCTURES
OF MIXED-VALENCE
J.B. Goodenough, Mixed-Valence Compounds, (Edited by D.B. Brown), D. Reidel: Dordrecht, Netherlands, 4 13 -425 (1980). J.B. Goodenough, Prog. Solid State Chem. 5, 145 (1971). K.H. Johnson, Adv. Quantum Chem. 7, 143 (1973). LG. Austin & N.F. Mott, Adv. Phys. 18,41(1969). K.Y. Wong &‘P.N. Schatz, Prog. Inorg. Chem. 28, 369 (1981). Noodleman et al. J. Amer. Chem. Sot. 107, 3418 (1985).
11. 12.
13. 14. 15. 16.
IRON OXIDES
723
K.P. Schwarz, Phys. Rev. B5, 2466 (1972). J.C. Slater, The self-consistent field method for atoms, molecules and solids, McGraw Hill, New York (1974). U. Buchenau & I. Muller, SoZid State Commun. 11, 1291 (1972). G. Amthauer & G.R. Rossman, Phys. Chem. Mineral. 11, 37 (1984). V.A. Gubanov & D.E. Ellis, Phys. Rev. Lett. 44, 1633 (1980). J.M.D. Coey, A. Moukarika & 0. Ballet, J. Appl. Phys. 53,832O (1982).
Vol. 58, No. 10, pp. 719-723,
0038-1098/86 $3.00 + .OO Pergamon Journals Ltd.
1986.
CLUSTER MOLECULAR ORBITAL DESCRIPTION OF THE ELECTRONIC MIXED-VALENCE IRON OXIDES AND SILICATES
STRUCTURES
OF
D.M. Sherman U.S. Geological Survey 959, National Center, Reston, VA 22092, USA (Received
27 November
1985, itz revised
form 12
February
1986 by E.F. Bet-taut)
A molecular orbital description, based on spin-unrestricted Xo-scattered wave calculations, is given for the electronic structures of mixed valence iron oxides and silicates. The cluster calculations show that electron hopping and optical intervalence charge-transfer result from weak Fe-Fe bonding across shared edges of Fe06 coordination polyhedra. In agreement with Zener’s double exchange mode\+Fe-Fe bonding is found to stabilize ferromagnetic coupling between Fe and Fe3+ cations.
INTRODUCTION MIXED VALENCE IRON OXIDES and silicates often exhibit thermally activated electron hopping between adjacent Fe’+ and Fe3+ cations and optically induced Fe? + Fe: + Fe: + Feg charge transfer transitions. The most well-known example of a mixed valence iron oxide is magnetite (Fe30,). Examples of mixed valence iron silicates that show thermally activated electron hopping include ilvaite [ 11, deerite [2] , and cronstedite 131. A quantitative picture of the electronic structures of such mixed-valence phases would give insight on the nature and mechanisms of electron hopping and optically induced intervalence charge transfer transitions. A natural approach to describing the electronic structure of a solid is to use one-electron wavefunctions which have the translational symmetry of the crystal structure. Such are the Bloch wavefunctions of band theory. Bloch wavefunctions, however, cannot easily describe the localized electronic states found in transition metal oxides and silicates. An alternative approach is to use a cluster molecular orbital description. For example, much of the electronic spectra and even magnetic properties of iron oxides can be described in terms of the electronic structures of FeOd and Fe04 coordination polyhedra [4]. This is because the electronic states of interest are localized on the metal atom and its immediate coordination environment. Even in mixed valence phases which are said to exhibit “electron delocalization” (i.e., fast electron hopping), we are still dealing with localized electronic states. (A possible exception is magnetite over which there is some debate as to whether the electrons are itinerant or polarons [5,6] .) In this paper, the electronic structures of mixed valence Fe oxides are investigated using the cluster molecular orbital approach. The electronic structure 719
calculations were done using the Self-Consistent Field-Xo Scattered Wave (SCF-Xo-SW) method [7] . The simplest cluster that can be used to describe intervalence chargetransfer consists of two Fe06 coordination polyhedra which share a common edge (i.e., an (FezOi,,)ls-cluster). This corresponds, for example, to a pair of adjacent Fe’+ and Fe3+B-site cations in magnetite. In order to relate the electronic structure to the phenomena of electron hopping and polaron formation we need to show how the electronic states are coupled to the nuclear coordinates of the cluster. Following Austin and Mott [8] and others (e.g., [9] )the vibrational coordinate of interest is given by 4 = RA--RR, where RA and RB are the Fe-O bond lengths at sites A and B, respectively. The electronic structure of the cluster was calculated at q = 0 and at q = h. At q = h, RA and R, correspond to the equilibrium Fe3+-0 and Fe2’-0 bond lengths (i.e., RA = 2.OOA and RB = 2.16 A). At q = 0, both Fe sites are identical and correspond to the average of those at q = X (i.e., RA = RB = 2.08 A). The geometry of the cluster is shown in Fig. 1. Note that, at q = 0, the cluster has DZh symmetry: however, broken symmetry wavefunctions that transform under Czv symmetry were used so that electron delocalization is not imposed upon the cluster. This is akin to the “Xo valence bond” formalism used by Noodleman [lo] to describe the electronic structure of ferrodoxin. For the SCF-Xcr-SW method, the most important computational parameters are the atomic sphere radii. These were (in Angstroms) 1.208, 1.143 and 1.159 for the iron, bridging oxygen, and remaining oxygen atoms, respectively. The same atomic sphere radii were used at both the q = 0 and q = X cluster geometries.
720
ELECTRONIC
STRUCTURES
OF MIXED-VALENCE
IRON OXIDES
Vol. 58, No. 10
P-spin 6
---=55= eg _---
4
_--==== =I==
I-
s: 3 k 6
i Fe(3c-i) t 29 I
a-spin 2
eg
-m
5
\
l O(2P)
Fig. 1. Geometry coordinate 4.
of the (FezOio)rs-
L
cluster along the
The (Fe20i0)r5- cluster is surrounded by an outer sphere of radius 14.64 A at the 4 = 0 geometry and 15.408 A at the 4 = h geometry. The (Y parameters were taken from Schwarz [ 1 l] . The electrostatic stabilization energy of the crystal is simulated by surrounding the cluster by a Watson sphere with charge + 15.0. Partial waves with maximum angular momentum quantum numbers 1= 1, 2 and 3 were used for the oxygen, iron and outer spheres, respectively. RESULTS AND DISCUSSION General
The electronic structure of the (Fez0ro)i5- cluster in the ferromagnetic state (S = 912) at 4 = 0 is shown in Fig. 2. In an isolated FeOe cluster [4], the Fe(3d) orbitals are split into a set of tzg orbitals (Fe-O rrantibonding) and a set of eg orbitals (Fe-O uantibonding). The energy separation between the tzg and eg orbitals corresponds to the 1ODq parameter of ligand field theory. All of the orbitals are split by the exchange energy into a-spin (majority spin) and &spin (minority spin) manifolds. The effect of the ligand field and exchange splitting of the Fe(3d) type orbitals is also seen in the (Fez010)15- cluster. This cluster, however, also allows for Fe-Fe bonding to occur; overlap of the Fe,(tsJ and Fe,(t,,) orbitals results in the cluster molecular orbital analogue of a narrow Fe(3d) band. The p-spin Fe(3d) orbitals are shown in more detail in Fig. 3. The Fe(tzg)-Fe(t2& u-bonding overlap is strong enough to completely delocalize the Fe2+ &spin electron over the two metal atoms so that each Fe atom has a formal valence of + 2.5. As will be shown below, this is not the case if the Fe atoms are antiferromagnetically coupled. The energy levels of the cluster at 4 = X are given in Fig. 4. At this cluster geometry, the Fe’+ b-spin
Fig. 2. Calculated energy level diagram for (Fe20,0)‘5at the coordinate 4 = 0 (all Fe-O bond lengths are 2.08 A) when the two Fe atoms are ferromagnetically coupled. Dashed levels correspond to unoccupied orbitals. electron tends to be localized on the larger Fe site. This gives distinguishible Fe2+ and Fe3+ cations in the cluster. The effect of localizing the Fe’+ P-spin electron is to split the tzn and eg Fe(3d) bands into Fe2+(t2&, Fe3+(t2J, Fe”(e,) and Fe3+(e,) sub-bands. Cluster model of electron hopping
Given two Fe atoms located at sites A and B, the electronic states of the pair can be described in terms of the zeroth-order wavefunctions +i and \ks, which correspond to the ionic configurations FeyFeg and FeTFeg:, respectively. Associated with each wavefunction 1s a potential energy surface as shown in Fig. 5. If there exists an off-diagonal matrix element J which can coupled \kr and \ks, then the true states of the mixed valence pair will be: \k+ = (Y\ki + (1 -f_YZ)“*Qk2, and 9_ = (1 - a2)“2\ki From perturbation
-c&2. theory,
(Y’ = (1 - AE/(AE’
+ 4J2)“2)/2,
where AE is the energy difference between \ki and \k2. In terms of the states \ki and \k2, the ground state electronic structure of the (Fe20,0)‘5- cluster at 4 = 0 corresponds to the wavefunction: \k+ = l/d/2(*,
+ **).
The excited state \k_ corresponds to the configuration in which the &spin electron occupies the 17ar (Fe-Fe a-antibonding) orbital rather than the 16aI (Fe-Fe
ELECTRONIC
.Vol. 58, No. 10
STRUCTURES
OF MIXED-VALENCE
721
IRON OXIDES P-spin
13b:! ---_.
Fe3* IIII
::::
eV 5.5 -
Fe3+ x=1= ---_II_ Fe*+
z 19a, ____A
eg
I \ f29
a-spin
6 5 2
Fe3+ x X
s
12bp ---_I
Fez+
Fe*+
.-9
3
18a, ----.
o-
B O(2P) -2 -
\
17a,
&do 0 49
____/
CT’
Fig. 4. Electronic structure of the cluster at 4 = h. The Fe(3d) bands split into Fe2’ and Fe3+ orbitals.
4.0 llb 1 ---_ / 7a,
____-
6a,
----\
lob, ----
\
3.5 16a, -
\
u
\ Fig. 3. Details of the &spin Fe(3d) energy levels at 4 = 0 showing the different Fe(3d)-Fe(3d) bond types.
u-bonding) orbital (Fig. 3). At 4 = X, the ground state of the cluster corresponds to the wavefunction \k+ with 0 < c? < l/2. From the orbital compositions, a is found to be 0.12. The physical meaning of the coupling integral J is that it describes the weak Fe-Fe bonding interaction. At 4 = 0, the energy difference between the states \k+ and \k_ is 2J and is given by the energy of the one-electron transition 16ar -+ 17ar (Fig. 3). Note, in passing, that this quantity is also the cluster molecular orbital estimate of the Fe(f& 3d bandwidth. Using the transition state formalism [ 121 , the 16ar -+ 17ar transition energy is found to be 0.84 eV; hence, J = 0.42 eV. This is comparable to values estimated for other mixed valence pairs (e.g., Ti3+--Ti4+, Ni’+-Ni3+) [8] and is close to that estimated [ 131 (0.3 eV) for Fe304. At 4 = h, the energy of the 16ar -+ 17ar (9+(h)--+ \k_(X)) transition corresponds to Eop in Fig. 5. This is calculated to be 1.3 1 eV. Optical absorption spectra of mixed valence Fe oxides and silicates [14] typically
show a strong absorption band near 1.65 eV that is assigned to optical intervalence charge transfer (sometimes referred to as “photon induced hopping”). Since, at 4 = h, the 16ar -+ 17ar transition is an Fe’+ + Fe3+ + Fe3+ + Fe’+ charge-transfer process, the electronic structure calculations support the band assignment. Moreover, like intervalence charge-transfer bands observed in the optical absorption spectra of mixed valence Fe silicates [14], the 16ar + 17ar transition will be polarized along the Fe-Fe internuclear vector in accordance with the Laporte selection rule. Note that transitions from the 16ar Fe-Fe u-bonding orbital to the 1 lbr rr- and 7a2 b-antibonding orbitals are not expected to be important given the negligible Fe-Fe rr- and b-bonding overlap. The activation energy for electron hopping (E,) can be estimated from the calculated values for E,, and J: Ea = (2&p - AE(h))/4 From perturbation
-J(O).
theory,
E op = (AJY(X)~ + 4J(h)2)“2. Assuming that J(h) ==z J(O), E, is calculated to be - 0.016 eV which implies that there is no thermal barrier for electron hopping (i.e., the potential energy surface corresponds to the dashed lines in Fig. 5). The lack of an activation barrier implies that small polarons (i.e., distinguishable Fe2+ cations) are unstable when the Fe-Fe separation is less than 2.95 A. This, in turn, suggests that small polarons are not the charge-carriers in magnetite [5, 61. Increasing the Fe-Fe distance will decrease J so that, eventually, there will be an activation barrier for hopping. Such a barrier, on the order of
722
ELECTRONIC E
STRUCTURES
OF MIXED-VALENCE
l!!LL v,
IRON OXIDES P-spin
: 1: 1 Fe’+le,)
(z-spin
6
VZ!
l2,31
Vol. 58, No. 10
IIII
Fe2+(t2,1
13,21
-A
0
h
q
-A
0
h
q
Fig. 5. Potential energy surfaces for the \kr and \kZ states of an Fe*“-Fe3+ pair: (a) in the absence of coupling (J = 0), and (b) when coupling is present. The dashed curves correspond to the case where the coupling integral J is so large that only a single minimum occurs in the potential energy surface of the ground electronic state.
Fig. 6. Electromc structure of the (Fe20re)‘5- cluster when the two Fe atoms are antiferromagnetically coupled. The absence of Fe-Fe bondin in this configuration results in distinct Fe2+ and Fe3$ cations even at 4 = 0. SUMMARY
0.1-0.35 eV, is found in ilvaite [ 11, deerite [2] , and cronstedite [3] where the Fe-Fe distances are somewhat larger than that in the cluster described here. Magnetism and Mixed Valency
Adjacent Fe” and Fe3+ cations will be magnetically coupled either by the antiferromagnetic Fe-O-Fe superexchange interaction or by the ferromagnetic double exchange mechanism. Consistent with the notion of double exchange, the Fe’+ &spin electron is delocalized over the two Fe sites in the ferromagnetic configuration. The electronic structure of the (FezO,,)‘scluster in the antiferromagnetic state (S = l/2) is shown in Fig. 6. In this configuration, there is no Fe-Fe bonding and, consequently, the Fe*’ P-spin electron is completely localized to one Fe site giving distinguishible Fe” and Fe3+ cations even at 4 = 0. Using the “magnetic transition state” formalism [ 12, 151 the ferromagnetic configuration is found to be more stable than the antiferromagnetic configuration by 0.94 eV. Ferromagnetic coupling between Fe*+ and Fe3+ cations occupying edge-sharing coordination sites is predicted by the double exchange model and is found in all mixed valence Fe silicates investigated thus far [16]. The relative stability of the ferromagnetic state, however, is greater than the stabilization energy resulting from electron delocalization. Hence, ferromagnetic coupling by “direct exchange” [S, 61 must also be important.
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