Dionne G.F. Magnetic Oxides

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432 8 Spin Transport Properties

Fig. 8.30 Crystal-ﬁeld diagram illustrating the d

low-spin

S D 0

state with the free ion level

as zero-energy reference [9]

the antibonding e

–p¢ orbital states are split into lower d

and upper d

y

where the unpaired spin is located and from which the polaronic transfer of spins

to neighboring ions will take place. Note that the t

states remain labeled as non-

bonding because of their weak  overlaps with the oxygen 2p lobes. With reference

to Fig. 8.18, the Cu sites of the superconducting perovskites are either tetragonal

(with an orthorhombic component in some cases), pyramidal, or square planar. The

relevant spin occupancies of the d states are now shown in Fig. 8.30. With d

y

as the path of transfer, with single occupancy in the Cu





member and empty

for the Cu





member in a low-spin .S D 0/ state, as illustrated in Fig. 8.31

and earlier as part of Fig. 8.9.

The source of polarons differs among these compounds. In the simplest case

of the La

2x



1x



system with maximum T

 40 K[50], Sr

ions are ﬁxed negative charges in the A sublattice, and the mixed valence occurs

as tetragonally coordinated Cu

holes that are tethered to the nearest Sr

ions,

thus making the conductivity p-type. A modiﬁcation of this system that introduces

the pyramidal coordinations La

2x



1x



increased T

to 60 K

[77]. For the YBa

system with T

 95 K[51], the situation is more com-

plex. The mixed valence occurs here as a result of oxygen vacancies which establish

polarons in both the planes of Cu .2/ –O

pyramids and Cu .1/ –O

linear chains;

chemical formulae highlighting proposed Cu valence distributions that vary linearly

with polaron concentration may be written as follows:

YBa

5=2y=4

y=43=2

113y=2

3y=210

for

6:67  y  7

and YBa

5=2y=4

y=43=2

103y=2

3y=29

,for

6  y  6:67,

with Cu(1) and Cu(2) site valences v .1/ D 1:5y  8 and v .2/ D 0:25y C 0:5,

respectively.

8.4 Superconductivity in Oxides 433

Fig. 8.31 p-Type 3d

y

 2p

 O  Cu

covalent transfer in 180

perovskite bond

geometry for d

! d

(low-spin) [9]

As suggested by Fig. 8.32, the superconduction is likely to occur in the

Cu .2/ –O

planes of the pyramidal complex, because the Cu

2C.3C/

content of

the Cu .1/ –O

planes would phase over to Cu

1C.2C/

at y D6:67 as a result of

oxygen vacancies within the plane that create the Cu .1/ –O

chains. Moreover,

these vacancies would break up the continuity of the transfer couplings neces-

sary for superconduction. The origin of positive mobile polarons, therefore, would

arise from the ﬁxed negative charges of O

2

ions ﬁlling the vacancies, as y ! 7.

In Sr-free La

CuO

4Cı

[78], the excess oxygen is more correctly described by

2x



1x



, which is brought about by La cation deﬁciencies. As de-

termined earlier, a threshold value of x

 0:08 (or ıD0:04) is all that is necessary

for the onset of superconduction.

Partial veriﬁcation of this valence model was reported by Tranquada et al. [60]

who determined experimentally that the average spin of the Cu(2) ions is 0.66 Bohr

magnetons .m

/ at y D 6, and that the Cu(1) sublattice is diamagnetic. This result

indicates that most of the Cu(2) ions are 2C (with some spin canting likely reducing

the effective spin values) and that the Cu(1) ions are 1C, which is consistent with the

model in Fig. 8.33. The occurrence of Cu

ions in the Cu .1/ –O

chains should be

434 8 Spin Transport Properties

Fig. 8.32 Ordered A-layer structure of YBa

, showing breakdown of Cu–O

complexes as

y decreases from 9 (hypothetical in this case). At y D 8, oxygen is removed from Y–O

planes and

Cu(2) sites are square-pyramids (i.e., Cu–O

), but retain C

symmetry axis. At y D 7, Cu(1) ions

become linearly coordinated in x–y plane (orthorhombic phase), with uniaxial superconduction

expected; Cu(2) ions retain square-planar coordination in x–y plane, with planar superconduction

possible. At y D 6, Cu(1) planes are fully depleted of oxygen and Cu(2) ions lose mixed-valence

with only 2C species present (see Fig. 8.33)[9]

expected, since its large radius



0:96



would preclude its occupancy of the Cu(2)

pyramidal sites; furthermore, there is already ample evidence for d

conﬁgurations

to favor linear coordinations [79].

An even more intriguing conﬁrmation of this originally proposed linear Cu va-

lence distribution has come from the “bond valence sum” analysis of Brown [80].

The results plotted in Fig. 8.34 indicate that the Cu valence distribution is basically

linear, but with an oscillation about the relevant portion of the linear curve from

Fig. 8.33, added here for comparison.

Together with the compounds discussed above, the parameters for more-

complicated “layered” structures are summarized in Table 8.1. In cases where

the Cu resides principally in sites with O

coordinations, which may provide

hop

>4meV, T

can reach 120 K. For the Bi



; Ca



2C.3C/

8Cı

system [63], the optimum Cu

concentration xD0:33 occurs because of a com-

bination of excess O

2

(i.e., ı  0:17) or the occurrence of monovalent calcium

[81]. The Tl

x1

2C.3C/

4C2xCı

compounds [82] derive their po-

laron sources from either ﬁxed-valence cation deﬁciencies (i.e., excess O

2

)orthe

8.4 Superconductivity in Oxides 435

Fig. 8.33 Proposed linear

valence model of Cu(1) and

Cu(2) as a function of the

oxygen content variation and

distribution depicted in

Fig. 8.32 [9]

Fig. 8.34 Nominal Cu valence as determined from linear model of Fig. 8.33 compared with

valence-bond-sum analysis of Brown [80]forYBa

[9]

436 8 Spin Transport Properties

Table 8.1 Layered cuprate superconductors

hop

Compound O

2

coord. T

(K) x (meV) ˇ

Ref.

p-type





2x

CuO

Cubic O

40 0.2 2.5 0.7 [50]

2x

CaCu

Pyramidal O

60 0.2 4 0.7 [77]

YBa

Pyramidal O

95 0.25 4 0.57 [51]

Sr; Ca

8Cı

Planar O

120 0.33 (>4)(<0:5)[63]

CuO

6Cı

Cubic O

80 – (2.5) (<0:5)[82]

CaCu

8Cı

Pyramidal O

110 – (4) (<0:5)[82]

CaCu

10Cı

Planar O

120 – (>4)(<0:5)[82]

n1

2nC4Cı

Pyramidal O

135 – (>4)(<0:5)[112]

n-type







2x

CuO

Cubic O

24 0.23 (2.5) (>0:7)[83]

1x

CuO

Planar O

40 0.14 (4) (>0:7)[85]

Some entries of this table are taken from Table 8.1 of reference [86]

Brackets indicate suggested values

This compound was prepared under high pressure which, together with Hg, probably increased

the covalent exchange energy b

between the Cu ions

mixed valence of Tl, which can appear as 1C or 3C to suit the ionic size or charge

requirements of its locale. With such a dual mixed-valence condition present, the

likelihood of higher polaron ordering, i.e., smaller ˇ, is also increased. Since any

defects would not be oxygen, these compounds should have chemical stability that

is superior to the YBa

system.

Another source of the enhanced T

values could be larger b values that result

from a covalent coupling between the d

y

–2p¢ antibonding state and the 6s

orbital of Bi

or Tl

. An increased exchange integral would give rise to a smaller

. A measurement of the W parameter for the compositions with maximum T

could help to sort out these possibilities.

Zero-spin polarons can also occur in cuprates with n-type conduction prop-

erties. In A

perovskites, Cu

–O

2

–Cu

combinations can produce su-

perconductivity with Cu



;S D 0



ions as the minority carrier instead of



;S D 0



. Through Ce

substitutions in A sites and the creation of O

2

vacancies by a reducing atmosphere anneal, Tokura et al. [83] reported n-type su-

perconductivity with T

 24 KinNd

2x



1x2y

xC2y



4y

,for

z D 0:15 and y D 0:04. As a consequence, stoichiometry is maintained with a

concentration of x D 0:23.

Although the critical temperatures of this system are not particularly large,

these results are very signiﬁcant for providing insight about the microscopic

mechanism of superconductivity. The substitution of tetravalent cations into the

Since Nd

is a magnetic rare-earth ion with S D 3=2, superexchange involving the A sublattice

may result in Ce

clustering that would lead to larger ˇ parameters.

8.4 Superconductivity in Oxides 437

Fig. 8.35 n-Type 3d

y

 2p

 O  Cu

covalent transfer in 180

perovskite bond

geometry for d

! d

[9]

A sublattice creates negative polarons (electrons), and the appearance of super-

conduction veriﬁes the prediction of n-type Cu

! Cu

C e



orbital transfer

predicted in the initial report on the CET model [84]. As shown in the orbital trans-

fer diagram of Fig. 8.35,thed

y

orbital is again the transfer path. Here the Cu

ions provide the S D 0 states required for spin transfer and the onset on magnetic

frustration in the Cu

host lattice with S

D 1=2. In addition, the existence of

n-type superconduction signiﬁcantly weakens the argument that high-T

phenom-

ena are based on hole transport through the oxygen sublattice, i.e., local peroxide

1

formation. For the oxygen ligands to provide conduits for electrons, O

3

ions

would have to be postulated – a situation even more unlikely than the peroxide case.

Another interesting feature of these compounds is the necessity to create oxygen

defects as part of their preparation. In this case, the Cu

ions occupy square pla-

nar sites .Cu–O

/, with an occasional missing oxygen, As shown in Fig. 8.32 for

YBa

, the Cu(1) sites that are proposed to harbor Cu

ions are also square

planar to linear, as y falls below 7. Since Cu

ions are usually not accepted in sites

of higher coordination, e.g., octahedral O

, because of their large radii



0:96



this result is entirely consistent with traditional metal-oxide chemistry.

A primitive layered n-type family Sr

1x



1x



was reported by

Smith et al. [85]. In this case, the Cu ions reside in planar coordinations, with the A

438 8 Spin Transport Properties

sites forming oxygen-free layers that are interleaved between Cu–O

planes. The n-

type compositions are noteworthy because of the increased T

D 40 Kandthelower

value of x

max

D 0:14. In the context of the foregoing discussion, these results may

be explained by an E

hop

higher than that of the compound containing Ce

, but with

alargerˇ parameter, as compared in Table 8.1. This interpretation remains in accord

with the general conclusions that T

through E

hop

has a crystal-ﬁeld dependence re-

lated to oxygen coordination, and that cation spatial ordering is essential for high T

Mixed-valence combinations involving zero-spin transition ions were examined

further in the context of HTS materials design [86]. Figure 8.21 suggests how var-

ious zero-spin polaron possibilities could be developed from low-spin states to

participate in coherent spin tunneling. In certain lattice structures, the t

shell can

be the source of superconductivity, as conﬁrmed by the critical temperature of 4.7K

reported for low-spin Co



;S D 0



and Co



;S D 1=2



combinations in

1x

that becomes n-type when hydrated, with likely composition

0:35

0:65

 1:3H

O[87].

Because these situations require strong cation–anion interactions, the possibility

of utilizing cations with the longer reaching radial components of the 4d

and 5d

shells might be attractive candidates for future investigation. Larger U

values to-

gether with greater  and b

integrals could result in increased T

values through

favorable E

hop

=W ratios in (8.39). Low-spin combinations of the strongly covalent



;S D 0



and Ru



;S D 1=2



ions in highly reduced stoichiom-

etry, for example, would satisfy the U

D 0 condition that allows b

 U

and charge ordering could then parameter optimize the polaron dispersal for spin

transport. In the octahedral sites of a spinel lattice, a T

 12 K derived from t

shell d

–d

¢ delocalization exchange



$ 3d



was observed from n-type

! Ti

C e



transfers [88].

In summation, the mixed-valence manganites and cuprates are metallic if the po-

laron trap energy from antiferromagnetic exchange is eliminated. La

1x

MnO

is metallic for x<0:5because of the occurrence of ferromagnetism that results

from vibronic Jahn–Teller effects and delocalization exchange between partially

ﬁlled e

orbital states. Above x D 0:5, the crystallographic structures are inﬂuenced

by a complexity of factors that include J–T distortions that in turn control the split-

ting of the e

states. Magnetic order can then be established in a variety of antifer-

romagnetic conﬁgurations, often tenuous and subject to insulator–metal transitions

when high magnetic ﬁelds reverse the sign of the resultant exchange ﬁeld to restore

the ferromagnetism and create and produce a signiﬁcant magnetoresistance effect.

An internal spontaneous magnetic moment acting as a magnetic source, however,

would oppose the ﬂux exclusion (Meissner effect) that is a fundamental property of

superconductivity. p-Type La

2x

2C.3C/

is metallic because zero-spin

“hole” carriers (or Cu

carriers in n-type La

2x

2C.1C/

) can

transport as large polarons in the charge-ordered B sublattice. These conditions

remove the possibility of magnetic exchange trapping and frustrate any antifer-

romagnetic order above a concentration threshold of only a few percent. In both

materials systems, large polarons can become itinerant, in one case contributing to a

8.5 Supercurrents and Magnetic Fields 439

condition from which a magnetoresistance anomaly may occur at the Curie temper-

ature, and in the other, making possible the coherent transfer among S D 1=2 and

S D 0 charge-ordered and spin-frustrated ions that is necessary for a superconduct-

ing state. Other compounds where the absence of a magnetic exchange trap allows

polaronic conductivity to condense into superconductivity are the mixed-valence

Ti, V, and Bi oxides [89].

8.5 Supercurrents and Magnetic Fields

From this model, two-ﬂuid functions can be developed to analyze penetration

depths, critical magnetic ﬁelds, critical current densities, and other properties as a

function of temperature [23]. The end result in each case is determined by the super-

carrier density n

. One central result of this approach to the superconducting state

that arises from the Boltzmann-type partitioning of the carriers, however, is that the

supercarrier population will decrease accordingly with temperature until the density

of supercarriers falls below the threshold value of n

and the superconducting state

is extinguished. As a consequence, high supercurrent densities would be expected

only at the lower temperatures regardless of the critical temperature value.

Although the basic requirement for zero resistance has been deﬁned as n

 n

this condition alone is not sufﬁcient for superconduction. Superconductivity is a

thermodynamic state of energy lower than the normal state, and condensation to

the superconducting state occurs when this stabilization energy is transferred to a

supercurrent ﬂow. In the following sections, the role of magnetic ﬁeld in limiting this

current is described, and the factors that determine critical magnetic ﬁeld, critical

current density, and related phenomena are examined [9].

8.5.1 Supercurrent Formation

It is important to remember that a superconductor is not a “perfect” conductor.

Perfect conduction implies simply zero electrical resistivity – the unrealizable case

of universally unimpeded charge transport. For a material to be a perfect conductor,

there can be neither scattering among carriers nor energy loss in the form of thermal

dissipation through electron–phonon interaction. In reality, electrons in a solid can

never be completely uncoupled from the lattice, and where so-called “free” electrons

are involved, the state occupation limitations imposed by the Pauli exclusion princi-

ple create a repulsive action that also limits current ﬂow. Superconduction, however,

requires spatial rigidity of the supercurrent in order to fulﬁll the E D 0; B D 0 con-

ditions. As pointed out in the discussion of (8.26), supercurrent rigidity .ri

D 0/

requires both rA D 0 (the London gauge) and rn

D 0, where the latter is

also a sufﬁcient condition that implies real-space ordering of supercurrent carriers

440 8 Spin Transport Properties

Fig. 8.36 Two-dimensional model of polaron condensation to superconducting state: (a)be-

low percolation threshold and (b) after ferroelectric alignment with ordered supercurrent ﬂow

at R D R

[9]

that may also involve some form of dynamic spin ordering. The spatial ordering of

charges .rn

! 0/ suggests a MO scheme as a source of rigid conduits for super-

current ﬂow.

If mixed valence can provide conduction among bonding electrons delocalized

within the directed lobes of their covalent bonds, electrostatic ordering of these car-

riers must exist; lattice periodicity alone would suggest a regularity for any carriers

participating in the bonding. The charge balance requirements for optimizing the

Madelung energy also dictate such a spatial distribution of valence states. Since part

of the trapping or stabilization energy of the carrier is due to the electrostatic attrac-

tion to its polaron source, i.e., the other half of its electric dipole, the carrier ordering

is dictated by the dispersal of the ﬁxed polaron sources, as depicted in Fig. 8.36a. For

vanishingly small concentrations, the carriers are isolated, with radially symmetric

cell proﬁles in an x–y plane (in reality, only the fourfold symmetry of a d

x2y2

orbital in the CuO

square-planar case). As the spatial density of polaron “cells”

increases, the shapes of the overlapping regions converge into a chain to assume the

minimum energy orientation of aligned electric dipoles. At a threshold density of

ﬁxed polaron sources, condensation of ordered chains would occur spontaneously

in a manner depicted by Fig. 8.36b.

In the preceding sections, the necessary condition that  D 0 was discussed

in terms of a series of parallel, independent chains that could be represented by

a one-dimensional model, since only one completed chain would be required for

zero resistance. For phenomena related to the buildup of supercurrent and its as-

sociated magnetic effects, however, orbital transfer can no longer be treated as an

isolated linear chain, but rather as series of interconnected chains distributed across

a macroscopic area.

8.5 Supercurrents and Magnetic Fields 441

The completion of a single path is contingent on a threshold supercarrier density

, occurring at T D T

. Since a fraction of the electrons is in thermal activation at

all times, the total n

cannot be expected to complete every possible superconduc-

tion linkage simultaneously. Even if the chemical ordering were ideal .P D 1/,the

incidence of electron–phonon encounters is still random. An analog to this effect

would be the electrical breakdown of a gaseous medium, which begins with a single

irregular striation that moves about as dictated by random molecular collisions. As

the ionizing collisions increase with density, multiple striations percolate and the

gas is eventually transformed into a plasma continuum with a large fraction of the

ionized gas participating in the current.

At this point, we may reasonably assume that the current increases continuously

from zero, with only the excess of n

over n

initially contributing to the super-

current. An effective carrier density at T D T

is therefore deﬁned as n

/ D

/  n

/ D 0.AtT D 0, however, there are no hopping electrons and we

must assume that all of n

is contributing to the supercurrent, i.e., n

.0/ D n

.0/.To

satisfy these boundary conditions in the most direct manner, the above relation may

be generalized to all temperatures, according to

.T / D n

.T /  n

.T /; (8.40)

where n

.T / D n



1  n

.T / = n

.0/



Thus, the fractional contribution of the

threshold carrier density n

/ to the supercurrent increases in direct proportion

to the buildup of the effective carrier density n

.T / as T ! 0. The logic of

(8.40) is displayed in Fig. 8.37,wherethen

.T / population declines because of the

Fig. 8.37 Pictorial representation of the simultaneous decrease of n

.T / and the growth of

.T / as T ! T

,wheren

and n

converge to establish the threshold density for the onset of

superconductivity [9]