Dionne G.F. Magnetic Oxides

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392 8 Spin Transport Properties

Incoherent electrical conduction by thermal hopping is signiﬁcant where E

hop

small or temperature is large. Because spin alignment determines U

in a magnetic

material, the term magnetic polaron can be added to the lexicon. In simple terms,

when neighboring spins of the trapped carrier are aligned parallel, the local envi-

ronment of the carrier is ferromagnetic and cos  D 1, thereby removing the U

part of the polaron trap. The carrier would then be constrained only by its dielectric

(elastic) trap. The ferromagnetic polaron therefore has the transport properties of a

conventional dielectric polaron.

8.1.3 Transfer by Covalent Tunneling

At low temperatures, a more efﬁcient transfer mechanism can produce remarkable

effects when the right conditions of chemical bonding and spin ordering are present.

In the context of molecular-orbital states charge transfer can also be viewed as a

quantum mechanical probability stemming directly from the covalent bond, i.e.,

through a “covalent-transfer” mechanism. This concept can be readily appreciated

by imagining the concentric ring diagram in Fig.8.2 as analogous to a hydrogen

atom (with reversed polarity in this case), where the rings would represent a se-

ries of higher energy states. The wavefunction of the mobile polaron charge could

then be viewed as an s orbital wavefunction and its probability of occupancy at

a distance r from the source would then be determined by its contribution to the

molecular-orbital state as the carrier moves toward the outer rings. Such an exercise

can be approximated from the eigenfunctions of (8.6) after being modiﬁed to antic-

ipate the variable polaron trap energy U

.r/ of (8.10) and is sketched in the energy

level diagram of Fig. 8.3.

For the small polaron limit b

 .E

 E

D U

, application of the proce-

dureusedtoderive(8.6) yields

small



1 









(8.11)

and the transfer probability of the polaron carrier from site a to site b can be deﬁned

as the square of the coefﬁcient c

of '

. If we deﬁne a transfer efﬁciency as the

ratio of the two probabilities

.r/

small





=2U



1 



=2U









; (8.12)

which tends toward zero in this approximation, thereby conﬁrming the ineffective-

ness of quantum tunneling to extend the range of the carrier beyond the small-

polaron ring.

8.1 Polarons and Charge Transfer 393

For the large polaron limit b

 U

,the'

eigenfunction for a bonding state

with the coefﬁcients deﬁned in (8.6), would be

large





1 C





1 



; (8.13)

and

.r/

large



1  U

1 C U

 1 

: (8.14)

From the electrostatic attraction between two oppositely charged particles in a

medium of dielectric constant K and M

to M

dimension a,ther-dependent part

of U

becomes

.r/ 









1 



: (8.15)

Therefore, the charge-transfer probabilities can be related to quantum mechanical

tunneling to a distance r from the polaron source by substituting (8.15) (or another

appropriate function) into (8.12)and(8.14). Figure 8.4 compares the  efﬁciency pa-

rameter variation with r=a for large polarons. A somewhat more rigorous approach

would consider consecutive discrete transfers of the polaron carrier from its source.

To this point, the only reference to temperature has been in relation to the Boltzmann

Fig. 8.4 Polaron transfer efﬁciency as a function of reduced lattice length for two ratios of elec-

trostatic energy to polaron half-bandwidth

394 8 Spin Transport Properties

activation probability. Although not stated explicitly, there is a T dependence of the

spin alignment that is characterized by the Curie temperature through the Brillouin–

Weiss theory. The temperature will also affect the value of E

hop

and the extent to

which tunneling is allowed to control the lifetime of the polaron in its trap. However,

even if perfect spin alignment were possible, lattice vibrations would still create a

temperature dependence of b

, which will now be related to a polaron bandwidth.

8.1.4 The Holstein Polaron Theory

In his quantum mechanical analysis of polaron motion [6]. Holstein deﬁned three

stages of conductivity that were determined by the relative magnitudes of two en-

ergy parameters (1) the polaronic charge exchange energy b

, which establishes the

lifetime 

of the polaron carrier in its site according to 

„=b

, and thereby

presents b

as the width of the polaron energy band that is the product of the full

electronic exchange integral b and a lattice vibronic coupling factor, and (2) the en-

ergy U

of the electrostatic potential well in which the carrier resides and which is

shaped by the charge of the carrier and a neighboring charge of opposite sign that

created it, i.e., the other half of an extendible dipole. The temperature dependence of

is determined by the number of vibrational modes (phonons) available to interact

with the carriers. The vibrational overlap integral is a decreasing function of T and

its contribution to b

is maximum at absolute zero, diminishing rapidly with rising

temperatures.

Based on these concepts, Holstein deﬁned the criterion for a large polaron (one in

which the carrier extends beyond its immediate environment by spontaneous charge

transfer) if b

 E

hop

, and that of a small polaron (which is limited to its imme-

diate neighboring sites) if b

hop

. These transfer mechanisms are often referred

to as coherent or adiabatic because of their energy conserving nature. In the ﬁnal

stage which sets in at higher temperatures, the metallic conductivity begins to break

down and semiconduction by incoherent nonadiabatic electron hopping becomes

dominant. With increasing temperatures, the large polaron condition begins to fail

because higher-frequency lattice vibrations allow the carriers to stabilize in their

traps by permitting elastic adjustments to occur more quickly, thereby lengthening

the polaron lifetime of 

. This action leads to a decrease in b

which begins to set

in as the temperature approaches the Debye temperature 

The relation between the Debye energy k

and U

in determining the actual activation energy

hop

can be appreciated if one recognizes that the electrostatic potential of the polaron dipole ﬁeld

has an intimate tie to the elastic distortion that dresses the polaron charge site. The distortion of the

lattice can be viewed as a reaction to the dipolar ﬁeld and both energies involve the polarizability of

the lattice through the dielectric constant K. As origins of the trap energy, they could be considered

to have some equivalence.

8.1 Polarons and Charge Transfer 395

The polaron bandwidth b

is reduced by vibronic (orbit–lattice) coupling that

produces a temperature-dependent narrowing given by [12].

.T / D b exp



 coth







(8.16)

that ranges from b

.T / D b exp .–/ at T D 0 to the simpliﬁcation

.T / D b exp











; (8.17)

when T  

=2.

Although not readily discernible in this brief summary, another facet of

Holstein’s model is that the threshold where thermal activation becomes a signiﬁcant

competitor to tunneling is T  

=2. As seen in the complete function (8.16) plot-

ted in Fig. 8.5, the polaron bandwidth is still a sizable fraction of b exp ./ at this

point. The parameter   50=K

opt

would be 0.2 to 2.0 for transition-metal oxides

with an optical dielectric constant K

opt

in the 15 to 5 range. As a consequence,

the decrease in b

will eventually serve to reduce the carrier population according

to a probability approximated by the Boltzmann relation exp



E

hop

=kT



in the

usual diffusion relation for electron hopping by thermal activation. Therefore, the

transport of a polaronic carrier requires that an increase in energy E

hop

be supplied

to remove it from its trap. A qualitative summary of the polaron stages is presented

in Fig. 8.6.

Fig. 8.5 Reduced polaron bandwidth as a function of reduced temperature for different values of 

396 8 Spin Transport Properties

Fig. 8.6 Diagram of polaron stages based on the relative magnitudes of b

and E

hop

8.2 Metallic Oxides with Polarized Spins

For electronic conduction in solids, two conditions must be satisﬁed (1) there must

be charge carriers and (2) there must be a mechanism to enable their transport.

Carriers can be “free,” as in collective-electron metals, excited as in band-model

semiconductors, or activated from polaron traps in ionic compounds. The second

requirement involves the environment of the destination sites and what lies in be-

tween, i.e., the mobility. The temperature dependence of these variables is what

generally deﬁnes whether the material is a metal (carrier density decreasing with T )

or insulator (both carrier density and mobility increasing with T ). There is, how-

ever, another signiﬁcant distinction that applies particularly to magnetically ordered

materials.

Charge carriers in metals are drawn mainly from s and p states, despite the for-

mation of hybrids with d states in the collective intermingling. As a result, the

d -electron magnetic carriers represent only a fraction of the population that pro-

duces current. In transition-metal oxides, the s and p electrons of the cations are

transferred to the oxygen to establish the anion lattice, and have only negligible hy-

brid mixing with the d shell. Therefore, in the rare situations where the d electrons

become itinerant, the current has the spin polarization of the transfer states, which

is usually ferromagnetic in contrast to the more random situation in a ferromagnetic

metal. Since the vast majority of the carriers are of only one spin orientation, metal-

lic compounds of the transition-metal series are frequently called “half-metals.” The

possible origins of polarized spin transport in select metallic oxides can be reviewed

in the context of molecular-orbital theory.

In systems where only the t

orbital states are occupied, i.e., the lighter mem-

bers of a 3d

transition series, metallic conduction can occur if the t

levels retain

a degeneracy in a lower symmetry crystal ﬁeld. The most common occurrences are

found with d

and d

conﬁgurations in simple monocation-site compounds. Be-

8.2 Metallic Oxides with Polarized Spins 397

cause the cation sites are principally octahedral, the t

electrons usually form weak

 bonds to the O

2

anions and therefore feature small exchange stabilization that

would normally be expected to provide low antiferromagnetic N´eel temperatures

. At room temperature, even with dynamic Jahn–Teller (J–T) or spin–orbit (S–O)

splittings



10

2



of the triplet, the t

states would then form partially ﬁlled

–p–t

states resulting from the localized metal–ligand–metal superexchange

[8,9].

8.2.1 Simple Oxides

In cases such as TiO and TiO

, single d electron transfer can occur as a result

of hopping electron between 3C and 4C mixed-valence states of the Ti ions when

the compound is off stoichiometry, e.g., from O

2

vacancies. Spontaneous magnetic

ordering is usually not involved here because the spin density is too small to produce

anything but paramagnetism. A more interesting situation occurs with stoichiomet-

ric CrO





. The crystal structure is rutile (tetragonal cation site symmetry D

with c=a <1), which splits the t

triplet into a lower d

singlet and an upper d

xz:yz

doublet, as sketched in Fig. 8.7a. The molecular-orbital hybrid states populated ac-

cording to the Aufbau qualitative guidelines would then have a single electron in

each level, but with competing spin-polarization tendencies. In the lower state, the

half-ﬁlled d

–p–d

hybrid would be expected to follow the rules of antifer-

romagnetic superexchange [13]. In the upper d

xz:yz

–p–d

xz:yz

doublet, however,

Hund’s rule high-spin alignment can exist within the two orthogonal states because

the degeneracy allows the two electrons to select different orbitals, thereby avoiding

the Pauli principle while providing ferromagnetic exchange stabilization through

itinerant charge-transfer characteristic of metallic conduction [8, 9]. The net result

would be determined by the spin conﬁguration that produces the lowest energy.

Consequently, the antibonding half of d

–p–d

is stabilized to accommodate

the electrons with the lower energy SD1 parallel spin alignment. The ferromag-

netic order gives the full 2m

magnetic moment per cation (100% polarization) at

low temperatures and coexists with metallic conductivity up to a Curie temperature

of 400 K [14,15]. A more in-depth discussion of the various orbital interactions that

lead to the magnetic order and electrical conductivity of these compounds is given

by Goodenough [16].

8.2.2 Complex Oxides

Another uncommon situation occurs in mixed (or double) perovskites (Fig. 8.1)

in which charge-ordered dissimilar octahedral-site cations can produce metal-

lic conduction and long-range spin order that reach above room temperature.

AprimeexampleisSr





[17]. The charge-transfer and mag-

netic properties are believed to originate from the half-ﬁlled/half-ﬁlled correlation

398 8 Spin Transport Properties

Fig. 8.7 Molecular-orbital approximations for two special cases of spin-polarized metallic charge

transfer: (a)Cr





where a ferromagnetic moment occurs because of the survival of a

spin–orbit d

xz;yz

doublet that forms a “band” with partially ﬁlled orthogonal states allowing Hund’s

rule to apply and (b)Sr





, which is a paramagnet/unbalanced antiferromag-

net (quasi-ferrite) that occurs by delocalization superexchange involving the sharing of the single

electron. Oxygen ligand states are not shown

superexchange [13] between the two lowest states of Fe





and





that form the 3d

.Fe/ –2p–5d

(Mo) hybrid in the B sub-

lattice. Antiferromagnetism is expected in this case, but with a net “ferrimagnetic”

moment associated with the remaining four unpaired spins of the Fe

ion. As a

consequence, the compound is a quasi-ferrimagnet with only one crystallographic

sublattice and only one molecular-ﬁeld coefﬁcient. If this material were an ideal un-

canted ferrimagnet, the magnetic moment per formula unit would be 4m

. Although

measurements indicate that the actual value at low temperatures is closer to 3m

the conductivity is still metallic. One possible explanation can be gleaned from

the molecular-orbital model in Fig. 8.7b. Beginning with the conductivity issue, we

can reason that the single spin resident in the t

shell of the Mo





ion is

shared with the Fe





ion by delocalization exchange. Since the outermost

8.2 Metallic Oxides with Polarized Spins 399

electron of the Fe





ion is loosely bound in a ﬁlled d

orbital, i.e., the ﬁrst

one added to the half-ﬁlled t

shell, we can argue that two ionic conﬁgurations can

exist simultaneously through spin transfer reactions

C Mo

, Fe

C Mo

C U

eff

C t

, t

C t

C U

eff

The degree of sharing and hence the mobility of the single Mo

spin will de-

pend on the relative magnitudes of U

eff

and b,whereU

eff

is the difference in the

binding energy between the initial and ﬁnal states of the transfer and b is the ex-

change integral between the Fe and Mo ions mediated by the O

2

ligand (not

shown in Fig. 7b). If the single Mo

spin becomes itinerant in this sense, spin-

polarized transport could take place in the narrow half-ﬁlled band constructed from

the 3d

.Fe/ –2p–5d

(Mo) hybrid. Inspection of the above charge-transferreac-

tions reveals that the left-hand side represents an antiferromagnet (or a quasi-ferrite)

and the right-hand side a paramagnet. Therefore, the probability of obtaining the

maximum 4m

is reduced by the probability of paramagnetism that occurs when

is missing its t

spin. If a paramagnetic contribution is present, experiments

at very high magnetic ﬁelds might help to clarify the situation.

Another point of interest is that the 3m

value of the net magnetic moment per

molecule lends credence to the suggestion that the Goodenough–Kanamori (G–K)

rules [9] for predicting the most probable type of spin ordering might not apply in

this case. If only the e

states are considered because of the conventional wisdom

that the t

–p hybrids are effectively nonbonding and can be ignored, the logical

conclusion for 180

¢ bonding of the perovskites is that ferromagnetic delocaliza-

tion exchange will result between the empty Mo

and half-ﬁlled Fe

shells,

producing magnetic moment limit of 6m

. However, the experimental result of 3m

suggests that the d

–p bond of the t

shell is strong enough to stabilize the type

of ferrimagnetic ordering proposed.

In a series of compositions with Cr, Mn, and Co used in place of Fe, [18] ferro-

magnetism was found only with Cr. Since it assumes its 3C valence with a highly

stable electronic conﬁguration t

, 3d

-d

antiferromagnetic (ferrimagnetic)

superexchange is again the likely result. No evidence of metallic conduction was

reported, however, which suggests that the value of U

eff

is greater than that of the

Fe case. The other ions also showed insulating properties, but no magnetic order-

ing. The prospects of antiferromagnetism are reduced because the most stable forms

of Mn and Co are divalent, which would force the Mo valence to be raised to the

diamagnetic 6C state, thereby lowering the chance of spontaneous magnetism. Sim-

ilar analysis can be applied to other combinations of transition metals, such as Ti,

W, and Re, but the prospects for strong superexchange with spin-polarized metallic

conductivity between the alternating B-site cations probably remains highest with a

–d

combination.

Ferromagnetism in metallic oxides also occurs with spins in the e

shell of

the mixed-valence perovskite



1x



1x



, for temperatures

reaching above 300 K. The origins of this ferromagnetism are complex and are

400 8 Spin Transport Properties

discussed in Sect. 8.3.1. Normally when the e

shell is occupied, strong ¢ bonds are

formed that produce antiferromagnetic stabilizations that can survive to well above

room temperature. Simple oxides of the heavier 3d

elements, MnO through CuO,

all feature antiferromagnetism and usually p-type semiconduction when mixed va-

lence occurs, despite the ferromagnetic tendencies that can still exist with the t

shell partially ﬁlled.

8.2.3 Classical Resistivity–Temperature Model

Returning to the discussion that produced (8.10), we can now examine the ques-

tion of conductivity in polaronic oxides. Most transition-metal oxides are termed

mobility-activatedsemiconductors in contrast to more conventionalband model ver-

sion with the usual description of holes and electrons dictated by Fermi statistics.

The electrical resistivity of mixed-valence oxides obeys the standard relation for

mobility-activated semiconduction in which the polaron trap energy E

hop

and the

activation energy are equivalent in the temperature regime where random hopping

is dominant [19–22],

 D



eff





1

exp





; (8.18)

where the carrier concentration per chemical formula unit, x

eff

D n

eff

=V , the ratio

of the volume carrier density to the volume of a formula unit V , the factor (eD/kT )

is the Einstein diffusion mobility, and D the diffusion constant that equals the ratio

of the square of the mean-free path to the carrier lifetime d

hop

=

hop

. To account for

the probability of a transfer being completed, an effective carrier concentration is

deﬁned by x

eff

D xP ,whereP is a polaron dispersal probability equal to (1  x)for

a random distribution. Further dependencies of relevance are that 

hop

D 

1

hop

,which

is on the order of the Debye frequency and d  a=.1x/ to account for the average

hop distance as a function of concentration [23]. Upon substituting appropriately

into (8.18), a working expression for the resistivity becomes

 D

C.1 x/ kT

exp



hop



C.1 x/ kT

for kT  E

hop

; (8.19)

where C D V=e



hop

. As illustrated in Fig. 8.8,  plotted as a function of T

for E

hop

D 10 meV from (8.19) will show insulating behavior (decreasing) at low

temperatures and pass through a minimum at T D E

hop

=k.AsT increases further,

the curve will approach a metallic straight line of (increasing) slope C .1–x/ k=x.

Note that the asymptote of the curve will pass through the origin at T D 0,which

is a result that is not expected in experiment because residual resistivity 

effects

would appear. This model is discussed further in relation to superconductivity in

Sect. 8.4.

8.3 Magnetoresistance in Oxides (CMR) 401

Fig. 8.8 Generic plots of  vs. T for E

hop

D 0 and 10 meV, deﬁning relations for T

min

,and

asymptote slope @=@T j

and intercept 

. Figure reprinted from G.F. Dionne, IEEE Trans. Magn.

27, 1190 (1991) with permission.

 1991 by the IEEE

The temperature of metal–insulator transition therefore resides in the value of

hop

, which from (8.10) is controlled greatly by the magnetic state of the spin

system. There are two prominent examples where E

hop

is reduced to the basic elas-

tic polaron limit E

hop



D U



by the absence or removal of E

hop

.D U

/.Ifthe

spins are collinear, cos D 1 and E

hop

 0. In oxides, this condition is uncom-

mon because superexchange usually dominates to produce antiferromagnetism. An

important exception can occur with Mn ions in certain ferromagnetic perovskites

compounds that feature metallic properties but also display large magnetoresistance

effects at the Curie temperature.

8.3 Magnetoresistance in Oxides (CMR)

In the generic





MnO

perovskite system, the conditions for metal-

lic conduction in the Mn sublattice can occur in a ferromagnetic phase that ap-

pears at lower temperatures. When the ferromagnetism is dissipated at the Curie