Characterization of electrons in solids by structure

Mat. Res. Bull. Vol. 2, pp. 165-184, 1967. Pergamon Pres s , Inc. Printed in the United States.

CHARACTERIZATION OF d ELECTRONS IN SOLIDS BY STRUCTURE II. Spontaneous Crystallographic Distortions

John B. Goodenough Lincoln Laboratory,* Massachusetts Institute of Technology

Lexington, Massachusetts

ABSTRACT

(Received December 5, 1966; Communicated by author)

Spontaneous crystallographic distortions to a lower symmetry phase at lower temperatures may be induced either by localized _d electrons or by narrow-band electrons. Band electrons induce changes in trans- lational symmetry that introduce planes of energy discontinuity at the Fermi surface; localized electrons induce changes in local symmetry that leave the transition-metal cations in the centers of symmetry of their anionic interstices. Ferroelectric and semiconducting = metallic transitions illustrate distortions induced by collective electrons. It is also shown how the shear structures found in nonstoichiometric, homologous series of transition-metal oxides may be induced by ferroelectric-type distortions. Non. stoichiometric compounds having localized d electrons exhibit ordering of point charges and point defects. Ferroelectric distortions induced by heavy atoms in a lower valence state are also discussed.

Theory

Many transition-metal oxides exhibit spontaneous crystallo-

graphic distortions to a lower symmetry phase at lower tempera-

tures (or have a low symmetry that can be related to a higher-

symmetry phase via a diffusionless distortion). Others exhibit

*Operated with support from the U. S. Air Force.

This paper was presented at the International Conference on the Characterization of Materials, State College. Pa . , U.S.A., November 16- 18, 1966.

165

166 d ELECTRONS IN SOLIDS Vol. 2, No. 2

order-disorder transitions that may involve defect diffusion as

well as electron ordering. Since the character of these

structural changes depends upon the presence or absence of a

Fermi surface, the syn~etry of the low-temperature phase provides

a sensitive indicator of the nature of the d electrons, localized

vs collective, in the higher-symmetry phase.

A. Localized Electrons

Crystallographic distortions due to localized electrons are

characteristically local deformations, but if there is a large

enough density of these local deformations to couple elastically

with one another via VA of Eq. (3), Part I, these deformations

give a cooperative distortion of the crystal as a whole.

There are two types of local deformations: (a) ordering of

electrons among localized orbitals that produces a less symmetric

charge distribution, thus deforming the site (point symmetry) to

stabilize the order, and (b) trapping of charge carriers at

specific cationic sites via atomic deformations that create a

unique potential at those sites. The first type is illustrated

by Jahn-Teller distortions and by magnetostrictive distortions

due to spin-orbit coupling. The second is illustrated by the

formation of small polarons (Landau trapping) in compounds having

the same atom in two different valence states. For the first

type, cooperative distortions are achieved by long-range ordering

of the unique axes of the asymmetric charge distributions at

individual cations so as to minimize the elastic energy. VI of

Eq. (3), Part I, represents the elastic coupling energy between

individually distorted sites. For the second type, cooperative

distortions are achieved by ordering the small polarons - and any

Vol. 2, No. 9. d ELECTRONS IN SOLIDS 167

associated defects - so as to optimize the sum of the elastic

and electrostatic energies.

Although these local distortions may be cooperative~ thus

inducing a macroscopic distortion of the entire crystal, never-

theless the fact that localized orbit~Is are always centro-

syn~netric with respect to their cationic nucleus wherever the

higher-symmetry phase has a centrosymmetric arrangement of near-

neighbor ions imposes a necessary (but not sufficient I condition

on the symmetry of the low-temperature phase: The cations must

remain in the centers of symmetr~ of their anionic interstices (I)

Further~ these distortipns do not require an increase in the

number of molecules per primitive unit cell.

Exchange striction below a magnetic-ordering temperature is a

many-body effect that might be expected to provide an exception to

this general conclusion. However, the most general spin con-

figuration is a spiral, which gives centrosymmetric exchange

forces if the cation is centrosymmetric to the near-neighbor

cations with which it interacts. Therefore any distortions due to

interatomic exchange forces also leave the cations in the centers

of symmetry of their anionic interstices.

B. Collective Electrons

The energy of a free electron of momentum ~ ~ that is moving

in a periodic potential is

E k = E o + (~2k2/2m*)

where m* is the effective mass of the electron.

is characteristic of wave propagation in periodic structures.

(1)

Bragg reflection

The

vectors that satisfy the Bragg reflection conditions define

168 d ELECTRONS IN SOLrDS Vol. 2, No. 2

energy surfaces in momentum space across which there is an energy

discontinuity Eg z in Ek vs ~. The magnitude of E z depends upon

the periodic potential. This potential at any point r in crystal

space may be expanded as a Fourier series

U(r) = 7~ k U k exp(i k.r) (2)

For an energy discontinuity that is small relative to the

bandwidth, Brillouin has shown that

s

Eg z = -2elUkl = -2elX Aktexp(i k-nt) l (3) t=l

where ~t gives the positions in real space of the ~ atoms of the

unit cell. If all the atoms are identical, then

s

U k = A k ? exp(i k.n ) = AkS k

where S k is the usual structure factor of x-ray intensities.

(4)

i. Semiconducting = metallic transitions

The resultant E k vs k curve for a linear chain of N atoms

e v e n l y s p a c e d a d i s t a n c e ~ i s shown i n F i g . 1. I f t h e c o l l e c t i v e -

¢ -_

~, a _1 r I

B A N D W I D T H

(a)

Ek

/ /

T ~ 0 ~ T - o" - 2-~ 2"~ o"

0

(b)

FIG. 1

Energy vs wave vector

for a linear chain of

atoms having one over-

lapping orbital per atom.

Fermi energy Ef corres-

ponds to a half-filled

k band, or one electron

per atom.

Vol. 2, No. 2 d E L E C T R O N S IN SOLIDS 169

electron orbitals are formed from one-each atomic orbitals and if

there is one electron per atom~ the Fermi energy is at k = ±~/2~

and the band is half-filled. Clearly any change in the trans-

lational symmetry that introduces a new energy discontinuity at

the Fermi surface will stabilize occupied states at the expense

of unoccupied states~ thereby reducing the energy Zk(E_k - Eo ) .

The l i n e a r c h a i n o f F i g . 1 may b e r e p r e s e n t e d b y two~

interpenetrating subchains each consisting of alternate ions. A

displacement of each subchain toward the other a distance 6

results in atomic pairs spaced a-26 within pairs and a+26 between

pairs. Such a displacement creates an energy discontinuity at

the Fermi surface (~ = ±~/2~) having a structure factor

S k = exp(i~6/a) + exp(i~)exp(-in6/a) = 2i sin(~6/a)

which means that

(5)

E = A6+... (6) gz

Since the elastic restoring forces are proportional to ~2 a

spontaneous static distortion requires a stabilization of the

occupied states that is linear in the displacements. Initially~

where Eg z is small compared to the bandwidth ~, the number of

states n k that are stabilized is proportional to Egz, so that the

total stabilization energy is

= -A' 62+ .... (7) = -½f Eg z

and no spontaneous distortion is anticipated. This is why semi-

conducting = metallic transitions are not a general phenomenon in

broad-band metals. However~ if the bands are so narrow that

Eqz > ~¢b for relatively small ~ a 6c~ then A' is large and n k -

saturates for 6 > 6 . In this case -- c

~E =-A'62 - -B6 for 6 > 6 c (8)


and the conditions for a finite, spontaneous distortion are met.

Further, note that for Eg z > ~¢b' the widths of the two half

bands have been reduced to nearly zero: This corresponds to

trapping the conducting electrons into homopolar, metal-metal

bonds in the linkages of length a-26 along the chain.

This type of distortion represents a semiconducting

metallic transition, since the low-symmetry phase contains a

filled band (or level) separated by a finite energy gap from an

empty band (or level) whereas the high-symmetry phase has a

partially filled band. It is also possible to show that such a

transition is first-order. It has been found in transition-metal

compounds having partially filled, cation-sublattice d bands

(Class (1) metallic oxides). The characteristic feature of these

distortions is a displacement of the cations from the centers of

s3nmnetry of their anionic interstices toward one or more near-

neighbor cations. This is accompanied by an increase in the

number of molecules per primitive unit cell. The displacements

commonly result in the formation of cationic clusters - pairs,

triangles, squares, ... - the particular cluster formation de-

pending upon the higher-symmetry structure and the number of

electrons per cation in the partially filled bands.

In the case of partially filled, crystalline d bands (Class

(2) metallic oxides), such a semiconducting = metallic transition

could be manifested as a disproportionation into two valence

states for the ions. On the other hand, there may simply be a

change in the number of molecules per primitive unit cell. In

either case the cations are not displaced from the center of

symmetry of their anionic interstice, but a change in the number

Vol. 2, No. 2 d ELECTRONS IN SOLE)S 171

of molecules per primitive unit_cell is required.

Thus distortions due to semiconducting = metallic transitions

can almost always be distinguished from those due to localized-

electron ordering~ and they represent a Fermi-surface-dependent

property that is most prominant where the bands are narrowest. A

similar statement holds for the ferroelectric-type distortions.

2. Ferroelectric transitions

A linear chain of N atoms consisting of alternating

cations and anions has distinguishable subchains without any

atomic displacements~ so that the ~k vs ~ curve has a discon-

tinuity at ~ = e~/2~ as shown in Fig. 2. To make contact with

FIG. 2

Energy vs wave vector for

a linear chain of alter-

o- - '0 o- . -0 o- . -0 o-- ( a )

E k

• ~ , / Y

-27r/0 - -,r/o 0 ~/o 2 ~/a

(b)

nating anions and cations.

Fermi energy Ef corres-

ponds to a half-filled

band. Given two over-

lapping orbitals per atom~

cationic d ~ id and _y~ zx

anionic P xx ~ ipy~ there

are two degenerate bands

and the two states per

k atom (four states per

anion) below Ef form a

bonding ~ band~ the two

states per atom (four per

cation) above Ef form an

antibonding n* band.

1"t2 d ELECTRONS IN SOL.'~S Vol. 2, No. 2

the perovskite BaTiO3, consider only the two anion Pn orbitals Px

and py (z-axis along chain) and the two cationic d orbitals dz__~ x

and dy z. Because of the difference in potential at the cations

and anions, the ~ bands represented by Fig. 2 consist of a bonding

band for -n/2~ < k < ~/2~ and an antibonding Z* band for

[~/2~l<k<In/~ [ . The bonding orbitals are primarily anionic in

character, the antibonding orbitals are crystalline d orbitals.

With two electrons per atom (four electrons per cation-anion

molecule), the ~ band is filled and the ~* band is empty. From

Eq. (3), the energy gap between ~ and ~* bands is

= -2elAM-AoJ (9) Egz

where A M and A 0 are the A k for the cation and anion, respectively.

If the two sublattices are displaced toward one another a distance

~, then this energy increases:

Eg z = -2e{ IAM-AoI + (AM+Ao)q6+...] (10)

Therefore if the two bands are so narrow that the number of states

that are stabilized by the distortion is n k - 2N for small ~,

then a finite, spontaneous distortion occurs at lower T. Note

also that if ~ additional electrons are introduced, the

stabilization energy is reduced because ~ occupied ~* states are

destabilized by the transition. Therefore additional electrons

either reduce the transition temperature or suppress the

transition entirely.

This type of distortion represents a ferroelectric or anti-

ferroelectric transition, since it is characterized by a dis-

placement of the cations from the centers of symmetry of their

anionic interstices toward one or more near-neighb_or anions. This

need not be accompanied by an increase in the number of molecules

Vol. 2, No. 2 d ELECTRONS IN SOT.TDS 173

per primitive unit cell.

3. Shear structures

Several nonstoichiometric oxide systems have been shown

to consist of a homologous series of discrete phases rather than

a broad range of solid solubility with randomly distributed de-

fects. Magn~li and his coworkers have identified, for example,

the series TinO2n_l (2), (W,M)nO3n_2 with M = Nb or Ta (3), and

(W,MO)nO3n_l (4). These homologous series are to be distinquished

from other ordered-defect phases, such as are found in PRO2_ x or

FeOl+x, by the characteristic occurrence of crystal shear at

regularly spaced intervals. The other compounds, which contain

localized ~ or d electrons, are characterized by ordered point

defects. The systems exhibiting shear structures, on the other

hand, appear to contain collective d ombitals, which suggests

that the surprising regularity of the spacings between shear

planes is due to the existence of a Fermi surface and the energy

stabilization to be achieved by either creating, or increasing the

energy discontinuity across, a Brillouin-zone surface at the

Fermi surface.

Magn~li (5) has already pointed out how condensation of the

defects into common planes minimizes the elastic energy of the

crystal, but this does not explain why these planes are

regularly spaced. However, it does show that any stabilization

due to the creation of a new periodicity via regular spacing only

has to be greater than the entropic energy at temperatures high

enough for diffusion to occur. This stabilization is greater the

narrower the bands, just as for the semiconducting = metallic

transitions and the ferroelectric transitions.


To illustrate this concept, consider TinO2n_l ~ which has

rutile slabs ~ titanium atoms thick that are connected by shear

planes across which the titanium octahedra share common faces~ as

shown in Fig. 3. These shear planes are so spaced that if the

titanium ions are given formal charges +3 and +4, all of the Ti

SHEAR PLANE (lOT)

3+

X OCTAH EDRAL- SITE VACANCl ES

• TITANIUM OF UPPER SLAB

"T • T,TAN,u. OF LOWER SLAB

O OXYGEN SLAB

n+½)a TH CK

c, a, (lOT) REFER TO AXES OF RUTILE SLABS

FIG. 3

Shear plane in TinO2n_l structure.

ions could face one another across the shared octahedral-site

faces. Howeverj with the exception of low-temperature Ti305, the

Ti-Ti spacing across these faces is relatively large, indicating

that Ti3+-Ti 3+ homopolar bonding across the shear planes does not

Vol. 2, No. 2 d ELECTRONS IN SOLIDS 175

take place at room temperature. This means that at high tempera-

tures the semiconductor = metallic transition is not competitive

with a ferroelectric-type transition, which displaces the cations

away from each other across the shear planes. Displacements of

the cations towards one another would reduce the electrostatic

repulsion across a shared face by concentrating electronic charge

in homopolar Ti3+-Ti 3+ bonds. Displacements away from one

another reduces the electrostatic repulsion by increasing both

the cation-cation separation and the anionic shielding. In

either mode, the effect is enhanced by a periodic spacing~ since

this concentrates cooperatively the influence of the distortions

to a definite group of collective-electron orbitals, stabilizing

some at the expense of others. (Localized orbitals would have

only a short-range influence.) In the "ferroelectric mode," a few

unoccupied orbitals are destabilized a great deal whereas a larger

number of occupied orbitals are each stabilized a smaller amount.

In the "semiconductor = metallic mode", a smaller number of

occupied orbitals (2/n per TiO 2 molecule) are stabilized a greater

amount. Since the energy gap between those occupied and un-

occupied orbitals influenced by the transition is larger for the

ferroelectric mode~ this mode dominates at high temperatures

where the periodic structures are initially formed. In a case

like W35Ta4Ol15, the crystalline d bands are all empty~ so that

stabilization via the ferroelectric mode is alone possible. How-

ever, in the system TinO2n_l it is necessary to entertain the

possibility of a semiconductor = metallic transition at lower

temperatures. This would be accompanied by a reversal in the

sign of the Ti-atom displacements at the shear planes.

176 d ELECTRONS IN SOI/DS Vol. 2, No. 2

4. Heavy-atom ferroelectric distortions

In a free atom or ion, splitting between ~ and ~ orbitals

of the same pricipal quantum number is due to different screening

by the core electrons. Therefore the splitting is small for

light atoms, but increases with atomic number until, in the

heavier atoms, it may become comparable to or greater than the

width of the collective-electron ~ and ~ bands. This is why the

heavier atoms may occur in a low formal-valence state. The ions

pb 2+ and Bi 3+, for example, have core electrons (6~ 2 in the free

ions) of large radial extension and of energy intermediate between

bonding and antibonding ~ and ~ orbitals. Because the core

electrons have the same principal quantum number as the cationic

orbitals primarily responsible for binding, they present a core-

repulsion potential of such relatively large radial extension

that the overlap integrals of the bonding and antibonding orbitals

are relatively small. This, coupled with a large electro-

negativity difference between ions as in oxides, can mean narrow

and ~ bands, which provides the possibility of a ferroelectric

transition in which cooperative cationic displacements relative

to the anionic subarray increase the energy gap between bonding

and antibonding states. This possibility is further enhanced by

the large polarizability of the outer core electrons, which

strongly reduces the elastic restoring forces. This

polarizability is due to the relatively small separation between

6~ and 6~ orbitals, which allows for the formation of hybridized

2 6(~sin2~ + ~ cos ~) core orbitals that concentrate the core

electrons on the side of the cation opposite to a nearest-neighbor

anion. This is the reason that spontaneous ferroelectric dis-

tortions may be induced by the heavy atoms Pb 2+ and Bi 3+ even


though the more obvious source of narrow-band orbitals~ viz.

orbitals~ is absent. Although we are rarely in doubt about the

collective-electron character of outer ~ and ~ electrons~ it is

impDi-tant to appreciate the generality of our concept of the

origin of the ferroelectric-type distortion.

C. Magnetic Order

Although this paper is not concerned with characterization or

the d electrons from magnetic properties~ it is relevant to point

out that the periodicity seen by an electron includes the magnetic

structure. Therefore it is the magnetic space group given by

neutrons~ not the crystallographic space group given by x-rays~

that determines the scattering factors S k and hence the surfaces

of energy discontinuity. In the linear-chain situation of Fig. i~

for example~ an energy discontinuity can be introduced at k =

±~/2~ by the introduction of an antiferromagnetic spin-density

wave (6) ~ since this doubles the number of atoms per unit cell.

The atomic moment created by a spin-density wave is usually small

(<0.5UB) compared to localized-electron moments. Because screen-

ing effects damp out spin-density waves (7)~ this effect has been

unambiguously identified only in a few materials with specialized

conditions. However~ onset of a semiconductor = metallic transi-

tion may stabilize a spin-density wave in the low-temperature

phase~ the magnetic periodicity simply enhancing the energy dis-

continuity introduced by the new lattice periodicity (8). The

significant point is that the appearance of long-range magnetic

order below a semiconducting = metallic transition does not

distinguish a magnetic vs crystallographic origin for the

transition. Such transitions may have a purely magnetic origin~

178 d ELECTRONS Eq SOT,TnS Vol. 2, No. 2

a purely crystallographic origin, or may be due to both mechanisms

operating cooperatively. For our purposes, the point to be

emphasized is that such transitions signal the presence of

collective electrons. In particular~ although the electrons may

be trapped in molecular-cluster orbitals at low temperatures~

these transitions do not reflect localized-electron = collective-

electron transitions in the sense of crystal-field-orbital =

molecular-orbital transitions.

Application

Table I lists some relevant crystallographic properties of

several transition-metal oxides. These have been chosen to

illustrate the several kinds of spontaneous crystallographic dis-

tortions that can be encountered and the type of information about

the d electrons that they provide.

Distortions induced by localized d electrons are illustrated

by several types of Jahn-Teller distortions (9),spin-orbit dis-

tortions (i0), Jahn-Teller vs spin-orbit distortions (ii), small-

polaron ordering, exchange striction, dipole-dipole interactionsj

and high-spin = low-spin transitions (12). The last three items,

however, are not exclusive to localized electrons~ so that the

conclusions that MnO~ CrO 2 and NiO contain localized d electrons

follow from supplementary magnetic data. The compound LaNiO3, on

the other hand, is included because octahedral-site, low-spin Ni 3+

ions are Jahn-Teller ions if the single e electron per ion is q

localized. It was the absence of a Jahn-Teller distortion that led

to the prediction and subsequent discovery (13) that LaNiO 3 is a

metallic oxide.


Distortions induced by collective d electrons are illustrated

by semiconductor = metallic transitions~ with and without coopera-

tive spin-density-wave formation~ and various ferroelectric-type

distortions. VO 2 represents cooperative semiconductor = metallic

and ferroelectric-type transitions.

BiFeO 3 is included because it represents a ferroelectric

compound having localized d electrons~ the ferroelectric dis-

tortion being due to the existence of a heavy ion Bi 3+ in a low-

valence state. The compound LaCoO 3 illustrates a first-order

semiconductor = metallic transition~ and the key to this

identification was a structural study (12). The metallic compound

PbRu03~ which has a defect pyrochlore structure~ illustrates trap-

mediated Pb-Pb bonding~ which has again been identified by mainly

structural considerations (14).

Although this summary of illustrations is brief~ it does

reveal the power of structural analysis as an aid in the

characterization of the thermodynamic state of the outer

electrons in transition-metal compounds.

TABLE I. CRYSTALLOGRAPHIC PROPERTIES OF SEVERAL OXIDES AND THEIR INTERPRETATIONS

Nomenclature: Mono~ 0 or 0'~ R~ Tet~ C = monoclinc~ orthorhombic

(~<c_//2<b or c_//2<a<b) ~ rhombohedral~ tetragonal~ and cubic. Tt~

TN~ T c = transition~ N~el~ and Curie temperature. J-T = Jahn-

Teller distortion~ L'S = cooperative spin-orbit-coupling ordering

via magnetic ordering with a collinear-spin configuration; B-site

and A-site = octahedral and tetrahedral cations; polarons = small

polarons; s.c. m metal = semiconductor m metallic transition; Co 3+

vs Co III= high-spin vs low-spin trivalent cobalt; ferroel. =

ferroelectric-type transition;ior~cation or anion vacancies; SDW

= spin-density wave. Class (i) = cation-sublattice d bands; Class

(2) = crystalline d bands.

180 d ELECTRONS IN SOI3"DS Vol. 2, No. 2

O IM

I-I

I-I r~

<

r~

I-4

D~

r~ O

I-I

0

8 <

C O

-,-I a}

,M

o C o O

C O

-,-I

r0 o

-,-I tW .,~

'O I-..I

,.-I ,..4

O

.,-I

.M

.,-I

o~

4a

O

a~ O

C

0

R 0

'O 'O 'O 'O 'O • • • • • N N N N N

• M -,-~ .M .M -M

O O O D O O O O O O

C -M

o

I I 0

C • • • • O

4a o 4J .~ • ~ .,"l .,-I -,.-I t~

I I I I 0

O

O ~ O O on m O ~ ~ •

X X X X

T T T T T

~1' m ~ ~ ,-4 ,--4 A V

v ^ v

-IJ ~ 4-; ~ T

~ ~ ~ 0

" " " " 0"*

O O O

+ + • +

+ + ~ +

t r

,10 , o • • • a~

' o ' o ' o - - . .--. ---. • • • e~ ~ ,--4 N N N v ..... • ,-I -M -M

O O O • r0 tO o o ,...4 ,~ ,~

O O~ -,-I izl

.,-4

• ,~ • •

4.J 1,4 0 4-s 4.J ~1

-,4 • • •

O~ ~ I

O o m~o~ d 6 d x - .~ • • •

o o o o ~

o o x x x ~ ~

,-I ,-4 ,-I O 11~ x T T T

x x T • )¢ .N 'I~ -,--I

,-t ,-4 u~ 4.a 4a I I II Z I I • •

O~

T

0 0 o

~ ~ ~ ~ 0

X +

o 0 0 0 0 0 ~>

C,,l J~

A A

,M ,-4

U} tn tO

,-I ,-I O O

.%.-% ,-.-I , - I n~ to

t~ t~

i

d o

t '~ O m

r~ O • M {~I

X X {",I

T T

o o ,.-4 U~ u~ ,-4 ~b I I

.r'~ O

t ~ O m

e~ O

Vol. 2, No. 2 _d ELECTRONS IN SOLIDS 181

C O

.,-I

O c o O

c o .,~ . o

o -,-I

.H 4~

o ~O

tu r-4

o o

o >

-,-I

-,-I

~h

O

O

O

O

C

o

O

C -H 0 -H 0

4~ 4~ 4J

m m m tu a~ t~ O

,-.I ,--I ,--I O r.9 ro ro .-3

O

-W O

4-1 m

• • /

• 1'1 • c O O a~

o $4 o • • :~

tl-I

04 O . ~ c~ O

O > O

x x X

O4 ~

(~ c~

o m

o

0J

@ @

,-4 I

txl O

C e~ O -,~ O

> o

O~ ~n O~ c~ e~ O~ (]; 013

c~ • 0) • C • • c~

.H -H .H ,Q .H -H

m tU ~ ~ 0 ~ • ,-4 m m o o o I o o O ,-4 0 0 0 -.4 0 0

E-I I I-~

i-~ • ,, .-~

@ 0 ,'0 1.-i i.-i .iJ O H O C

• ~ rO m • o • ~ II , 0 0 I-I 0 • "~

• - - I 0 • fl/ 0 I 0 0 I

0'3 t ~ ¢ q 0 3 t 'q t n O O O O O

O C • • O O l> ~ t n r~ [~ o o

o • a~ o -,~ . ~ • a~

E~ x x < x x x x ~ ~-4 ~ ¢xl txl O4 T I' x T T T T

X

O3 I ~ t ~ 0 3

ro cq ,-4 O ,-~ ,--4 k0 O

O II o II II O ,--4 o ~ o O ~ u'~

A ,--4

0 V o T O ~n t

C@ O

O ,--4 ~ v • I~

ml 0 ~ .- t,.

I -~ ,I~ o

~ , .a O

~o O

+

+

O m O ~ O O O • ,~ O c ~ • O .,-~

UI IN

C

t~ C

-,-t

o 0

D

O

X

t~ o

u~

o @ -,-I

o 0 -,-4

o @ 0

-,-I

o

0

ix]

O C

o


TABLE I (CONT'D)

a. Number of molecules per primitive unit cell. Asterisk refers

to magnetic cell, where different from crystalline cell.

b. T = 43°K. c

c. The system Fe3_xCrxO 4 exhibits Tet(c/a<l)~ orthorhombic, and

Tet(c/a>l) symmetries for different values of ~, and these all

correspond to different Jahn-Teller modes for A-site Fe 2+

ions (15).

d. T = 90°K. _c3+..2+ ~+ 3+

e. re x ~ll_ x [Ni Cr2_x]O 4. (a) x < 0.2, (b) x > 0.25.

f. T = 858OK c

g. Two ordered FeOl+ x phases have been identified (16). D and

Fe 3+ ions order so as to minimize the Madelung energy.

h. Although structure alone cannot distinguish antiferromagnetic~

localized e electrons from collective e electrons with a g g

spin-density wave~ the contrast with the PdO structure plus

the nearly spin-only atomic moment indicate these electrons

are localized. Nevertheless~ the high N~el temperature indi-

cates Aca c ~ Ac~ which may account for the ambiguity of the

> A c has been confirmed; transport data. In La2Ni04~ a Aca c

it is Pauli paramagnetic and metallic (17). The Ni-O-Ni

distances in Ni0 and La2NiO 4 are 4.168~ and 3.865A,

respectively.

i. The assumption that triangular clusters are formed in (iii)

planes of V 3+ ions has not yet been established by x-rays.

j. A decrease in c/a ratio with decreasing temperature indicates

stabilization of t o orbitals relative to te orbitals. In this

compound a s.c. = metal transition does not require a change in

symmetry since there is no inversion symmetry about cations

(these are in pairs along c-axis), and it can therefore be

second-order. There is nc antiferromagnetic order below

T t (18).

k. Here s.c. = metal transition causes formation of cation-cation

pairs in basal planes, so that each V 3+ ion has two nearest-

neighbor V 3+ ions and two at a larger separation. T t = 180°K

for rising temperature.

i. At low temperatures a s.c. = metal transition might compete

with the ferroelectric distortion~ the conducting d electrons

Vol. 2, No. 2 d ELECTRONS IN SOLIDS 183 m

becoming trapped in Ti-Ti bonds across the shear planes.

m. At low temperatures, c-axis V-V chains break up into 2.65~ V-V

bonds separated by 3.12~ V-V distances. Hence, there is a

s.c. = metal transition like that of Fig. 1 for the t o orbitals

directed along the c-axis. There is also a tilting of the

pairs to make one shortest Ti-O bond, which indicates a

simultaneous ferroelectric-type transition for the t± orbitals,

the two remaining t2g orbitals that overlap with a free,

anionic p~ orbital.

n. Below the ferromagnetic Curie temperature, there is an

anomalous increase in the c-axis with decreasing temperature.

This indicates half-filled t O orbitals, which would be

stabilized by antiferromagnetic coupling, are forced to have

parallel spins and hence to be localized. The metallic con-

ductivity in this compound occurs because of Class (2) t±

orbitals. (One-electron energies for CrO 2 are given in Ref.

19,

o. One-electron energies for the perovskite structure may be

found in Ref. 20.

p T N 100°K. Although • = Aca c ~ A c and the low-temperature phase

has an enlarged primitive unit cell, the magnetic order and

the distortions are both well interpreted on a crystal-field

model with Jahn-Teller distortions to an orthorhombic (rather

than tetragonal) local-site symmetry [21).

REFERENCES

i. This is clearly not applicable where there are displacements from the centers of symmetry of the anionic interstices in the high-symmetry phase, as in A1203, as a result of a non-centrosymmetric ionic arrangement about the cations.

2. S. Andersson and A• Magn~li, Naturwis 43, 495 (1956); Acta Chem. Scand. i i~ 164 (1950).

3. P. Gado~ B. Holmberg, and A. Magn~li~ Acta Chem. Scand 19~ 2010 (1965).

4. A. Magn~li~ Acta Cryst. ~, 495 (1953)•

5. A. Magn~li~ Nature I15~ 356 (1950); Arkiv Kemi ~, 513 (1950).

6. A. W. Overhauser, Phys. Rev. 128, 1437 (1962)•

7. P. A. Fedders and P. C. Martin, Phys. Rev. 143, 245 (1966).

184 d ELECTRONS ~ SOLIDS Vol. 2 No. 2

8. Alternately~ the magnetic coupling may be treated as strong coupling within cationic clusters and weak coupling between clusters. In the linear-chain case of Fig. 3~ the electrons form ideal homopolar (singlet state) bonds only if there is no interaction between pairs. Interaction between pairs introduces a finite atomic spin density and long-range magnetic order. The magnitude of the atomic moments depends upon the ratio of the coupling within a pair to that between pairs.

9. J. B. Goodenough~ J. Phys. Chem. Solids 25~ 151 (1964).

i0. J. Kanamori~ Progr. Theoret. Phys. (Kyoto) 17~ 177~ 197 (1957)

ii. J. B. Goodenough~ J. Phys. Soc. Japan 17~ Suppl. B-I~ 185 (1962).

12. P. M. Raccah and J. B. Goodenough~ Phys. Rev. (to be published).

13. J. B. Goodenough and P. M. Raccah~ J. Appl. Phys. Suppl. 36~ 1031 (1965).

14. J. M. Longo~ P. M. Raccah~ and J. B. Goodenough~ J. Chem. Phys. (to be published).

15. M. H. Francombe~ J. Phys. Chem. Solids ~ 37 (1957).

16. P. Vallet and P. M. Raccah~ C.R. Acad. Sci. Paris 258~ 3679 (1964); C. Carel~ D. Weigel and P. Vallet~ C.R. Acad. Sci. Paris 260~ 4325 (1965).

17. J. M. Longo and P. M. Raccah (unpublished research).

18. A. Arrott (personal communication) has shown that the report of antiferromagnetic order by S. C. Abrahams~ Phys. Rev. 130, 2230 (1963)~ is incorrect. J. M. Honig and L. VanZandt also report (unpublished research) that the transport data is incompatible with antiferromagnetic order.

19. J. B. Goodenough~ Bull. Soc. Chim. France~ 1200 (1965).

20. J. B. Goodenough~ J. Appl. Phys. 37~ 1415 (1966).

21. J. B. Goodenoughj Phys. Rev. i00~ 564 (1965); Magnetism and the Chemical Bond~ (Interscience, John Wiley and Sons~ New York-London~ 1963).

Documents

Characterization of electrons in solids by structure