Descriptive analysis of the crystal structure of the 1-D semiconducting TCNQ salt : TEA(TCNQ)2, as a function of temperature. - II. Charge distribution on the conducting TCNQ columns

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Descriptive analysis of the crystal structure of the 1-D semiconducting TCNQ salt : TEA(TCNQ)2, as a

function of temperature. - II. Charge distribution on the conducting TCNQ columns

J.P. Farges

To cite this version:

J.P. Farges. Descriptive analysis of the crystal structure of the 1-D semiconducting TCNQ salt : TEA(TCNQ)2, as a function of temperature. - II. Charge distribution on the conducting TCNQ columns. Journal de Physique, 1985, 46 (7), pp.1249-1254. �10.1051/jphys:019850046070124900�.

�jpa-00210068�

Descriptive analysis ^{of the} crystal

^structure

^{of the 1-D}

semiconducting TCNQ ^{salt :} TEA(TCNQ)2,

^{as a}

function of temperature.

II. Charge distribution

^on

the conducting TCNQ ^columns

J. P. Farges

Laboratoire de Biophysique, Université de Nice-Valrose, ^{06034 Nice} Cedex, ^France

(Reçu le 10 décembre 1984, révisé le 28 fevrier ^1985, accepte ^{Ie 5 mars} 1985)

Résumé. ²⁰¹⁴ La reconsidération de résultats récents de rayons X permet ^{de mettre en évidence une} dépendance

notable avec la température ^{de la} distribution de charge ^sur les deux sites non équivalents TCNQ ^{A et} TCNQ ^B dans une colonne conductrice de TEA(TCNQ)2. ^D’une façon plus particulière, ^une simple ^{loi de la forme :}

qA ⁼ (1/2 ⁺ T0/T) e ct qB = (1/2 - To/T) e,avec To ^{= 24 K, rend bien compte de cette} dépendance ^{et conduit à}

la limite remarquable : qA = qB ⁼ e/2 lorsque ^{T~ oo. Ces nouveaux} aspects ^sont pris ^en considération dans ^une analyse théorique succincte de l’unité dimère dans la colonne de TCNQ, ^{et leurs} implications possibles ^{sur les} propriétés électriques ^du semiconducteur sont également ^discutées.

Abstract. ²⁰¹⁴ By reconsidering ^recent X-ray results, evidence is presented ^{for a} noticeable temperature dependence

of the charge distribution on the two non-equivalent ^sites TCNQ ^{A and} TCNQ ^{B in a} conducting ^{column of} TEA(TCNQ)2. ^In particular, ^a simple ^{law of} the form : qA ⁼ (1/2 ⁺ To/T) e and qB ⁼ (1/2 - To/T) e, ^with To ^{= 24} K, ^{well accounts for this} dependence, giving the remarkable limit : qA ⁼ qB = e/2 ^{when T ~ oo. These} findings ^are considered in a brief theoretical analysis ^{of the dimer unit in the} TCNQ column, ^{and their} possible implications ^to the electronic properties ^{of the} semiconductor are also discussed.

Classification

Physics ^Abstracts

61.50L - 72 . 80L

Introduction.

A recent paper

by

Filhol and Thomas

[1] reported

the results of an

impressive study

^{of the} crystal

structure of the organic semiconductor

triethyl-

ammonium

(tetracyano-7,7,8,8,p-quinodimethane)2

^or

TEA(TCNQ)2. They

^{have measured the X-ray struc-}

ture at 110, 173, 234 and 345 K, ^and

they

^{have com-}

pleted

^{this work}

by

^also

reprocessing

^the X-ray ^{data of}

Jaud et al. at 295 K.

The results of Filhol and Thomas have

already

^been

used in Paper ^I

[2]

^to

develop

^a

descriptive analysis

^of

the temperature

dependence

of the intermolecular distortions of a

conducting

TCNQ ^{column in}

TEA(TCNQ)2.

^To

complement

^this

analysis,

^the

present paper ^now considers the temperature

depen-

dence of the charge distribution on the

TCNQ

^mono-

mers, ^a result obtainable, ⁱⁿ

principle,

^{from the intra-}

molecular distortions of the columns.

It is

appropriate,

first, ^{to recall}

briefly

^{the main}

features of the structural arrangement

of TEA(TCNQ)2 [1-3].

^{In this} material, ^a

plane

^to plane

stacking

^{of the} planar TCNQ ^monomers results in the formation of

parallel

^and

conducting

TCNQ columns. Each column may ^most

conveniently

^{be viewed as built} up from identical A-B dimer units, ^each

containing

^one TCNQ ^{A and one} TCNQ B monomers,

according

^to

the tetramerized _{sequence :}

At _any temperature, ^the intradimer distance

d(A-B)

is the shortest ^one in the _sequence while the intra- dimer

overlap

^is

optimal.

^{The two monomers A and}

B are

non-equivalent

^{in the} sequence, and this is most

clearly

^seen

by considering only

^{their nearest}

neighbours,

^{as shown in}

figure

^1, although ^the asym- metric

neighbouring

^{cations TEA+} contribute also to the

non-equivalence.

^{On the} reasonable

assumption

of complete charge transfer of one electron _per cation

TEA +,

^{and as a result} of the 1 : 2

stoichiometry

^{of the} material, ^{there is}

formally

^one

unpaired

electron per dimer unit in the TCNQ column. The _average charge

per

TCNQ

^{site is}

then qo

⁼

e/2, e

⁰ being ^the

electron charge, ^{and the}

fractional

charges qA and qB distributed ^on the two

independent

^{sites are}

related

by qB

^{= e -} qA.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019850046070124900

1250

Fig. ^{1. - Schematic} position ^{of the nearest} neighbours ^to TCNQ A, 1, ^and TCNQ B, 2, ^{in a} TCNQ ^{column of} TEA(TCNQ)2. ^{N is the normal to} the molecular plane ^of

the monomers and L is the elongation axis of the molecules.

Any ^{A or B site is} surrounded by ^one site A and one site B

(interplanar distances such that : d(A-B) d(A-A)

d(B-B)).

Table I. ^- The values, ^at

five

temperatures,

of

^the

fractional charges

^{on the two}

independent

^sites

TCNQ

^A

and

TCNQ

^{B in}

TEA(TCNQ)2 :

a) from

^{the work}

of

Filhol and Thomas

(see

^{Table 5}

of Ref. [1]) ^(esd

= estimated standard

deviations),

In reference

[1],

the values of the fractional charges qA

and qB

^{have been} estimated, ^{at each} temperature of

investigation,

from the internal bond

lengths

^{of the}

two

independent

^{monomers A and B,}

according

^{to a}

method

suggested by

^{Flandrois and Chasseau} [4]. ^In

the

opinion

of the authors of reference

[1],

these values show that the observed difference

between qA and qB

is not very

significant

^{for each structure taken indivi-}

dually,

^{while the}

corresponding

^{mean values}

clearly

show a

significant

difference :

qA

⁼ 0.60 e and

qB

⁼

0.40 e.

The charge distribution in the semiconductor

being

information of

primary importance

^to

explain

^its

electrical

properties,

the aim of the present paper is to reconsider these numerical results ^more

dosely.

^Of

course, there is an inherent limitation in the method of Flandrois and Chasseau. This method is, however, ^the

only

^{one that is}

practically

available, and the trends its suggests ^{must then be}

exploited

^{to the} fullest,

although

^{with some}

prudence.

In the present ^case, the overall

consistency

^{of the}

numerical results under consideration is

certainly

enhanced

by

the fact that

they proceed

^{from the same}

group and the same

experimental

X-ray procedure

(we

^shall

disregard

here the results at 40 K

reported by

^{Filhol et ale in a} separate paper [5], ^as

they

^are

deduced from neutron data which

required

^{an inde-}

pendent analysis).

Inspection

of the results of reference [1] ^reveals that, ⁱⁿ

spite

of their inherent limited _accuracy,

they

do not vary in an erratic manner from one temperature

to the next On the contrary,

they

^{are found to} vary with an

unexpected regularity,

^and

according

^{to a}

remarkably simple

^{law. This fact should not be}

entirely

fortuitous.

In the

following

^{sections, we shall}

analyse

^{this beha-}

viour in detail and,

speculating

^{on its}

reliability,

^we

shall also consider its most direct consequences.

1. Charge distribution.

The values of the fractional charges qA

and qB

^deduced

by

Filhol and Thomas from the X-ray structures at

110, 173, 234, 295 and 345 K [1] ^are

reported

together

with their estimated standard deviations in table I.

The

following

^{features are} revealed from an

inspec-

tion of these values :

- The

T-dependence

of qA

and qB

^is

quite regular

for the five temperatures ^studied

- These values

fit

accurately ^{a T -1} law, ^{as is} shown in

figure

^2.

according

^{to this law are also}

reported

^{in table I.}

Fig. ^{2. - Linear} dependence ^{of the} fractional charges q, and qB ^on the reciprocal temperature. ^{Black circles are the} data of reference [1], ^{and the} vertical bars reproduce ^their

e.s.d. The two straight ^{lines are the functions} qA ⁼ (1/2 ⁺

- A linear

extrapolation

of qA

and qB

^{versus T-1}

to

high

temperature

gives

the remarkable limit

(Fig. 2) :

In conclusion,

figure

² suggests ^{that the} charge difference Aq ⁼ qA - qB between the two dimer sites in

TEA(TCNQ)2

^is

significantly

temperature

depen-

dent : it is rather large ^{at 100} K, Aq ⁼ 0.48 e, ^and

rather small at 300 K, Aq ^{= 0.16 e}

(note

^{that the dis-}

tance between the two

charged

^{sites in a} tetrad unit is the shortest interdimer distance

d(A-A) [1]).

^More-

over, the fit of

figure

^{2 also} suggests, ^{but not}

definitely

proves, that the

charges

qA

and qB

vary

according

^{to a}

simple

^{T -1 law. The} remarkable

extrapolated

^limit

of this law for T-> oo : qA ⁼ qB ⁼

e/2, greatly

reinforces such a

possibility.

2.

Origin

^{of the site} charge difference.

It was

pointed

^{out to us}

by

^J. Kommandeur [6] ^that

TEA(TCNQ)2

^had

interesting

^common features with the simplest ^model system ^{that can show 2}

kF

^and

4 kF

distortions. Such a system,

consisting

^{of two}

electrons and four sites

arranged

^{on a} circle, ^was

investigated by

^the

Groningen

group

[7] by

^{means of}

an intersite

dependent

Hubbard Hamiltonian with an

intrasite Coulomb

repulsion

^{U. For}

large

enough U,

a

figure applicable

^to

TEA(TCNQ)2 [3],

^the system first is tetramerized at low temperature, ^{with both}

2

kF

^{and 4}

kF

distortions, ^{then dimerized at}

higher

temperatures

(with only

^at

4 kF distortion).

^For

U = 6 t, for instance, t

being

the scale of the transfer

integrals,

^the

4 kF

transition is found to occur at T = 2 t in this system. ^{In the} tetramerized

phase,

^the

four sites form two dimers with identical

charge

^distri-

butions and intradimer

separations,

^{but two}

larger

^and

unequal

interdimer distances. As in the

TEA(TCNQ)2

tetrad unit

[1],

^{the two electrons are not}

evenly

shared on the four sites, ^but

they

^are concentrated on

the two sites with the shortest interdimer spacing. ^The

charge

difference decreases ^as T increases and it vanishes above the 4

kF

transition temperature.

It is however unclear whether a transition to a true

dimerized

phase

may or may ^not be observed in

TEA(TCNQ)2.

^From

crystallographic

considerations

[1,

2], the tetramerization of the columns

clearly

persists ^{at the}

highest

temperatures,

probably

up ^to the

crystal

decomposition.

On the other hand, ^recent

transfer

integral

calculations

[8]

suggest that, whereas dimerization is

always

present, tetramerization could become

insignificant

above 300 K, ^{from a}

purely

electronical

point

^{of view.}

In fact, ^{there is a}

major difficulty

ⁱⁿ

applying

^the

oversimplified

^{model of reference}

[7]

^{to a} real material such as

TEA(TCNQ)2.

^{The model accounts for the}

electronic interactions, whereas it

ignores

^the strong ionic interactions also present ⁱⁿ the material. In

TEA(TCNQ)2,

^both dimerization and tetramerization

are

probably

^dominated

by

the latter interactions. This

results from general considerations on the symmetry and temperature

dependence

^{of the 3-D} crystal

packing [1, 2].

^More

precisely,

dimerization can be

explained by

^the

particular

^{1 : 2}

stoichiometry,

^i.e.

one cation TEA+ _per two

TCNQ

sites, whereas the

subsequent

tetramerization can be

explained by

^the

alternating

arrangement

of

^the asymmetric ^{cations. We}

shall

develop

^this

point

^{of view} here,

considering

^that

the

charge

distribution is also dominated

by

^{the ionic}

interactions.

It was recalled in the introduction that ^monomers A and monomers B occupy ^two

non-equivalent

positions

in the

crystal

lattice of

TEA(TCNQ)2.

^{There exists,}

consequently,

^a steric energy

difference

^{between the A}

and B sites

(primarily

^an effect of the

adjacent

cations),

which contributes to a charge redistribution into the

TCNQ columns. An estimation of this site _energy difference will be

provided

^{in the next section.}

3. Dimer model and site energy estimation.

The _energy difference between the two

independent

TCNQ ^{sites of a}

conducting

^{column in}

TEA(TCNQ)2

may be

reasonably

^well estimated, within the

approxi-

mation of

non-interacting

^{dimers, from a}

microscopic

theoretical

analysis

^{of the mean} distribution of the

unpaired

electron in the dimeric

(A-B)

^{unit of the}

column.

In this

simplified

one-electron

approach

^which

closely

^{follows that of} Rice et al.

[9],

^Coulomb

repul-

sion is thus

ignored,

^and

only

^the largest, intradimer,

charge

^transfer

integral : t(A-B)

^{= t, is retained.}

The

energies

^of the n-molecular orbitals of the two monomers

TCNQ

^{A and} TCNQ ^{B are} then, respec-

tively :

Eo

^{is an}

arbitrary

^reference energy and the additional

energy

^{E is a} consequence of the site

non.equivalence.

The

experimental fact : I qA I > I qB

I

implies

^{here :}

E > 0

(note

^{that, as in reference}

^[9],

^{the net value of s}

could result from the enhancement,

by

^{a vibronic}

coupling,

^{of an}

initially slight

^lattice energy difference 80 : ^{E =} 80 ⁺ As , ^{the new term As}

being

^attribu-

table to the internal molecular

distortions).

^The

secular

equation

^{is then :}

from which one obtains the _energy

Ei

^{of the}

bonding

two-site n-molecular orbital :

One

subsequently

^{obtains the}

expression

^{of the}

orbital itself, ^which accommodates the

unpaired

1252

electron of the dimer, ^{and then :}

The latter

equation

^{can also be} arranged ^{in the} form :

or,

equivalently :

This is the relation between charge difference and site energy difference for ^an isolated dimer with ^an

unpaired

^electron.

According

^{to section 1, a}

plausible assumption

^for

TEA(TCNQ)2

^{is :}

The ratio

61t

^is

plotted,

^versus

Aqle

^{and versus}

1fT,

in

figure

3, ^{curve a. It is seen in this}

figure

^{that a linear}

behaviour is

approached

^{to a rather}

good approxima-

tion above 100 K. For the present estimate, ^the

following

substitution _appears, then, ^as

quite

accep- table :

This

simplified

linear function is shown in

figure

3, ^as line b. A further step ^{in the} present

analysis proceeds

from the results

by

^the

Groningen

group,

already

mentioned in section 1, of a transfer

integral

calculation

applied

^{to- the case of}

TEA(TCNQ)2 ^[8]. According

to these results, ^the intradimer transfer

integral t

^{is a}

decreasing

^{function of} temperature,

closely reproduced

above 100 K

(and

^{below 350}

K) by

^{the linear} equa- tion :

When this result is combined with the

preceding

^one,

the site energy difference ⁸ may

finally

^{be evaluated at}

any temperature between 100 and 350 K from the linear function :

E/to = elt- 2 TOIT’ 0

^{with 2}

To/To

^{= 0.05 .}

The

corresponding

^variation is also shown in

figure

^{3, as line c.}

Although

inaccurate, ^{the calculated} absolute value

of to : to

^{0.2 eV}

[6],

may be taken as a

reasonable basis for numerical estimates. One obtains

Fig. ^{3. - Curve a :} dependence ^of E/t ^on Aqle ^{for an isolated}

dimer with an unpaired electron, ^{line b : linear} approxima-

tion of curve a : e/ t ⁼ Aqle, ^{line c :} dependence ^of s/to ^on Aqle in the latter approximation (e/t- alto ⁼² Tol To’= 0.05).

In the three cases, the temperature dependence ^{is also}

deduced from the relation : Aqle ^{= 2} To/ T (the ^vertical dashed line indicates _{the upper} T-limit, ³⁵⁰ K, ^of the expe- rimental data).

in this _{way :}

Independent

estimations, ^from

optical

^{data at}

300 K, ^for

MEM(TCNQ)2 (MEM

⁼

methyl-ethyl- morpholinium) [9]

gave :

then :

Aq’

^{= 0.24 e.}

In conclusion, ^{the dimer} model indicates that, ^{as T} increases, there is also ^a

significant

^{reduction of the}

energy difference E

between

^{the two} sites of the dimer unit in

TEA(TCNQ)2,

which results from a

quasi-

linear

dependence of Aq

^{on e,} of the form :

slt

^#

Aqle.

4. Charge distribution and electrical properties.

4.1. A first comment,

again

^{in reference}

[1],

^concerns

the observation

by

^{Belousov et al.}

[10]

^{of a doublet}

structure of the vibrational bands in the IR spectrum

of TEA(TCNQ)2,

attributed to

single

modes from both

TCNQ-

^and

TCNQO species.

^{The doublet structure}

is observed at 100 K but not at 300 K, and the authors of reference

[1]

admitted that this could be the conse-

quence of a

change

^{in the}

charge

distribution. The present

analysis explains

^{this as follows : at 100} K, the

charge

distribution : _qA ⁼ 0.74 e

and qB

^{= 0.26 e,}

is close to the ground ^state distribution : (TCNQ A)-

and

(TCNQ B)°,

^{so that two} distinct modes are in fact observed in the spectrum. ^{On the} contrary, ^the charge distribution at 300

K : qA

^{= 0.58 e}

and qB

^{= 0.42 e,}

is close to the distribution in the

high-T

^{limit :}

(TCNQ A)-1/2

^and

(TCNQ B)-1/2,

^{so that}

only

^one

(averaged)

mode is observed.

4.2. Three

orthogonal directions,

1, ^{2 and} 3, ^were defined in the past ^{as the}

principal

directions of the electrical

conductivity

^{Q-tensor in} crystals ^of

TEA(TCNQ)2 [3],

^the

a-anisotropy,

^{measured from}

60 to 400 K,

being

^{such that :} a 1 >> a2 >> a 3.

Direction 1 is

strictly

parallel ^{to the}

crystallographic

c axis, the axis of the TCNQ columns, whereas the two transverse directions, ^{2 and} 3, ^are

roughly

parallel (to

an

angle

^{of 20° or} better) ^{to the}

crystallographic

^{b and}

a axes,

respectively.

It must be

pointed

^{out that the}

TCNQ

^{sites are}

regularly spaced

^{in the two a} and b directions

[1],

^but

the shortest intermolecular links ^occur

only

^between

TCNQ

^A sites, ^{in direction} b, ^and

only

^between

TCNQ

^B sites, in direction a

(see

^{Table 7 of}

[1]).

When a

simple (narrow-)

^band

picture

^is

adopted,

different

band-fillings

^{are then}

suggested

^{from the}

results of section 1 for the two transverse directions,

according

^{to the} charge densities :

In large-U systems, ^carriers may be taken as elec- trons or as holes,

depending

whether the site

density

is less or more than

1/2 [11,

12].

Application

^{of this}

picture

^to

TEA(TCNQ)2 provides

^a

rough

^but

plausible

argument to account for the distinctive signs ^{of the two} transverse components of the thermo- power Q. ^{From 100 to 300 K}

[3, 13], Q2

^is

positive,

^as

for holes, ^and

Q3

^is

negative,

^as for electrons

(the

same argument would also

explain

^the positive sign

reported

for the Hall constant

[3,14]).

When, ^{on the other} hand, band effects ^are

ignored,

or in the

high- T

limit, ^the

thermopower

^{for a} large- U

system ^is

simply given by [ 15] :

where k is the Boltzmann constant From this for- mula,

Q2,

^like

qA/e, is

^a

decreasing

function of T, and

Q3,

^like

qB/e,

^{is an} increasing function of T, ^as observed

[3,

13]. Furthermore, ^as pi ⁼

qole

⁼

1 j2,

the latter formula also gives for direction 1 :

Q,

⁼

(k/e)

^{Ln 2 = - 60}

JlV/K,

^{which is}

precisely

the observ- ed saturation value of

Q1 ^[3,13].

^This distinctive satura- tion of

Q,

^has

already

been used in the past

[3,11,16],

as one of the strongest arguments ^for

considering TEA(TCNQ)2

^{as a} large-U system.

5. Conclusion.

In

spite

^{of its}

speculative

character, ^the present ^ana-

lysis provides

^{the first}

quantitative approach

^{of the} charge distribution in ^an

organic

TCNQ salt,

namely

the semiconductor

TEA(TCNQ)2.

^{In this material,}

it is shown that the mean charges ^{on the two non-}

equivalent

sites of the TCNQ ^{columns are}

signifi- cantly

^{different and} temperature

dependent.

^The charge difference decreases in a

regular

^{manner as T}

increases and is

roughly proportional

^{to the site}

energy difference. From these

findings,

^{several unex-}

plained

aspects of the electrical

properties

^{of this}

important

^{material can be} clarified, and, ⁱⁿ

parti-

cular, ^the

sign anomaly

^{of the}

thermopower.

The

analysis

suggests ^{to consider}

TEA(TCNQ)2

as a large- U system in which intermolecular as well

as intramolecular distortions ^{of the} TCNQ ^columns

are dominated

by

^ionic interactions, rather than

by

electronic interactions.

Acknowledgments.

The author wishes to thank Prof. J. Kommandeur

(University

^of

Groningen,

^the Netherlands), ^{Dr. M. J.}

Rice (Xerox, ^{Webster N.} Y., U.S.A.), ^{Dr. K. Cameiro}

(University

^of Copenhagen,

Denmark),

^{Dr. M. Almeida}

(University

^{of Lisbon,}

Portugal),

^{Dr. A. Filhol}

(Laue- Langevin

Institute, Grenoble, France) ^{and Dr. A. Brau}

(University

^{of Nice,}

France),

for their critical

reading

of ^a first version of the

manuscript,

^{and for numerous}

clarifying

^{comments. Dr. ten Bosch}

helped

^{with the}

English

translation.

1254

References

[1] ^{FILHOL, A. and} THOMAS, M., ^Acta Cryst. ^{B 40} (1984)

44.

[2] ^{FARGES, J.} P., J. Physique ⁴⁶ (1985) ^465.

[3] ^{For a review on} TEA(TCNQ)2, ^{see the article} by FARGES, ^J. P., ⁱⁿ Physics ^and Chemistry of ^Low-

Dimensional Solids, proceedings of the NATO ASI in TOMAR, Portugal, ²⁶ August-7 Sep- tember 1979, ^L. Alcacer, ^Ed. (Dordrecht : Reidel)

1980, p. 223-232.

[4] FLANDROIS, ^{S. and} CHASSEAU, D., ^Acta Cryst. ^{B 33} (1977) ^2744.

[5] FILHOL, A., ZEYEN, ^{C. M.} E., CHENAVAS, P., GAULTIER,

J. and DELHAES, P., ^Acta Cryst. ^{B 36} (1980) ^2719.

[6] KOMMANDEUR, J., University ^of Groningen, the Nether-

lands, private communication.

[7] ^HUIZINGA, S., KOMMANDEUR, J., JONKMAN, ^{H. T.}

and HAAS, C., Phys. ^{Rev. B 25} (1982) ^1717.

[8] JANSSEN, G., VISSER, R., JONKMAN, ^H. T., ^DE BOER, ^J.

and KOMMANDEUR, J., ^J. Physique Colloq. ⁴⁴ (1983) ^C3-1587.

[9] ^{RICE, M. J.,} YARTSEV, ^{V. M. and} JACOBSEN, ^C. S., Phys. ^{Rev. B 21} (1980) ^3437.

[10] ^{BELOUSOV, M. V.,} VAINRUB, A. M. and VLASOVA, ^R. M.,

Fiz. Tverd. Tela (Leningrad) ¹⁸ (1976) ^2637.

[11] CONWELL, ^{E. M.,} Phys. ^{Rev. B 18} (1978) ^1818.

[12] CONWELL, E. M. and HOWARD, ^I. A., ^J. Physique Colloq.

44 (1983) ^C3-1487.

[13] FARGES, ^{J. P. and} BRAU, A., Phys. ^{Status Solidi b 24} (1974) ^269.

[14] FARGES, J. P. and BRAU, A., Phys. ^{Status Solidi b 92} (1979) ^K131.

[15] ^BENI, G., KWAK, ^{J. F. and} CHAIKIN, ^P. M., ^{Solid State} Commun. 17 (1975) ^1549.

[16] CHAIKIN, ^P. M., KWAK, ^{J. F. and} EPSTEIN, ^A. J., Phys. Rev. Lett. 42 (1979) ^1178.