HAL Id: jpa-00218079
https://hal.archives-ouvertes.fr/jpa-00218079
Submitted on 1 Jan 1978
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
THE METALLIC STATE IN ORGANIC MATERIALS AT HELIUM TEMPERATURES
M. Weger
To cite this version:
M. Weger. THE METALLIC STATE IN ORGANIC MATERIALS AT HELIUM TEMPERATURES.
Journal de Physique Colloques, 1978, 39 (C6), pp.C6-1456-C6-1465. �10.1051/jphyscol:19786587�.
�jpa-00218079�
JOURNAL DE PHYSIQUE
Colloque
C6,suppIPmenr au no 8, Tome 39, aozit
1978,page
C6- 1456THE M E T A L L I C STATE I N ORGANIC M A T E R I A L S A T H E L I U M TEMPERATURES
M. Weger
The Racah I n s t i t u t e
of Physics
The Hebrew University of JemsaZem, I s r d Z
Rdsum6.- L ' d t a t m d t a l l i q u e e x i s t e dans HMTSF-TCNQ sous p r e s s i o n (au-dessus de 2 , 5 kbar) jusqu'au plus b a s s e s t e m p e r a t u r e s mesurdes (2. 20 mK). I1 e s t c a r a c t d r i s d p a r une grande magndtordsistance anisotrc- pique, une h a u t e m o b i l i t d (% 40000 cm2/v s ) , un diamagnetisme de l a n d a u - P e i e r l s , des o s c i l l a t i o n s de Shubnikov-De Haas e t une s t r u c t u r e de bande p d c u l i a i r e . I1 n d c e s s i t e que l'dldment de m a t r i c e d ' e f f e t t u n n e l e n t r e l e s chafnes tl s o i t p l u s grand que l e temps de c o l l i s i o n i n v e r s e l e long d'une charne
g / ~ ~ , .
Alors l e temps d ' e f f e t t u n n e l e n t r e l e s c h a k e s ,T d e v i e n t p l u s p e t i t que T , ,.
Par c o n t r a s t e , dans 1 ' 6 t a t P une dimension, -rL > T , , e t e s t donnd p a rT L ~
= 2iT t i(%)
e t p e u t S t r e mesurd d i r e c t e - ment par NMR. Les d t a t s m e t a l l i q u e s 1 une e t B t r o i s d i m e n s i K s ( a n i s o t r o p i q u e s ) s o n t c a r a c t d r i s d s e t compards. Le taux de c o l l i s i o n T , ' e s t princip,alement di? P l a d i f f u s i o n au second o r d r e p a r li- brons b i e n que quelques a u t r e s mdcanismes a i e n t 6 t 6 proposds. Alors que l ' d t a t P une dimension s u i t une t r a n s i t i o n de P e i e r l s , 1 ' 6 t a t a n i s o t r o p i q u e 2 t r o i s dimensions e s t frdquemment superconducteur dans l e s composds i n o r g a n i q u e s , mais l a s u p r a c o n d u c t i v i t d n ' a pas encore d t 6 observde dans l ' d t a t m d t a l l i q u e organique.Abstract.- The m e t a l l i c s t a t e e x i s t s i n HMTSF-TCNQ under p r e s s u r e ( i n e x c e s s of 2.5 kbar) down t o t h e lowest measured temperatures (2. 20 mK). It i s c h a r a c t e r i z e d by a l a r g e , a n i s o t r o p i c magnetore- s i s ta n c e , a high m o b i l i t y (% 40000 cm2/v s )
,
Landau-Peierls diamagnetism, Shubnikov-De Kaas o s c i l - l a t i o n s , and a p e c u l i a r band s t r u c t u r e . It r e q u i r e s t h a t t h e t u n n e l i n g m a t r i x element between chains, tL, be b i g g e r t h a n t h e i n v e r s e c o l l i s i o n time along a c h a i n ,H / - c l , .
Then t h e t u n n e l i n g time between c h a i n s , T*, becomes s h o r t e r than T ,.
I n c o n t r a s t , i n t h e t r u e one-dimensional s t a t e , T~ > T and i s given b y TI' = ( 2 ~ / g ) t i (T / h ) , and can be measured d i r e c t l y by NMR. The anisotropic thkke dimensional, and t h e t r u e one h'imensional, m e t a l l i c s t a t e s a r e c h a r a c t e r i z e d and compared. The col- l i s i o n r a t e T i s mainly due t o second-order s c a t t e r i n g by l i b r o n s , though s e v e r a l o t h e r p o s s i b l e mechanisms ha$e been proposed. While t h e t r u e one-dimensional s t a t e undergoes a P e i e r l s t r a n s i t i o n ,t h e a n i s o t r o p i c three-dimensional s t a t e i s f r e q u e n t l y superconducting i n i n o r g a n i c compounds, b u t s u p e r c o n d u c t i v i t y has n o t y e t been observed i n t h e o r g a n i c m e t a l l i c s t a t e .
I.WHAT I S A ONE DIMENSIONAL METAL ?.- Organic mate- r i a l s w i t h a r a t h e r high e l e c t r i c a l c o n d u c t i v i t y were d i s c o v e r e d a few y e a r s ago / 1 , 2 , 3 / . The b e s t known of t h e s e compounds i s TTF-TCNQ ( t e t r a t h i o f u l v a l e n e
-
t e t r a cyano quino dimethane) which i s a c h a r g e - t r a n s f e r compound i n which t h e TTF molecu- l e s d o n a t e s 0.6 e l e c t r o n s t o t h e TCNQ molecule, and the h o l a (on t h e TTF molecule) and e l e c t r o n (on t h e TCNQ molecule) a r e o u t s i d e closed s h e l l s , and t h u s a b l e t o conduct e l e c t r i c i t y . The donor and a c c e p t o r molecules a r e s t a c k e d i n l i n e a r c h a i n s , and t h e c o n d u c t i v i t y i s mainly along t h e s e c h a i n s . By now, s e v e r a l f a m i l i e s of such c h a r g e - t r a n s f e r compounds a r e known.The c o n d u c t i v i t y of t h e s e m a t e r i a l s , a t ambient temperatures, i s about 500-2000 (a-cm)-' i n t h e c h a i n d i r e c t i o n . This c o n d u c t i v i t y corresponds t o a mean f r e e p a t h of s e v e r a l l a t t i c e c o n s t a n t s . Because of t h i s small mean f r e e p a t h /4/,R kF = 1 , ( a l t e r n a - t i v e l y phrased, ki/rll
=
EP, where T , ~ i s t h e s c a t t e -r i n g t i m e ) , and t h e Fermi s u r f a c e i s c o n s i d e r a b l y smeared o u t . Thus, t h e s e m a t e r i a l s a t ambient tempe- r a t u r e a r e on t h e border between m e t a l s , where t h e e l e c t r o n i c motion i s wavelike, and semiconductors, where the e l e c t r o n i c motion i s of a hopping type.
Therefore, i n o r d e r t o expose and i n v e s t i g a t e t h e m e t a l l i c p r o p e r t i e s , we u s u a l l y go t o low temperatu- r e s , where t h e c o n d u c t i v i t y i s c o n s i d e r a b l y h i g h e r
(of o r d e r 10000 ( 3
-
cm)-') and R k ~ >> 1.As w i t h o r d i n a r y m e t a l s , i t would be advantageous t o go t o helium temperatures. However, t h e s e mate- r i a l s u s u a l l y undergo a metal-to-semiconductor t r a n - s i t i o n ; TTF-TCNQ a t 5 3 K, and t h e o t h e r m a t e r i a l s , g e n e r a l l y about 20-60 K. The t r a n s i t i o n t o t h e semi- conducting s t a t e i s a P e i e r l s t r a n s i t i o n 151 which i s c h a r a c t e r i s t i c of one-dimensional m e t a l s .
I n o r d e r t o i n v e s t i g a t e o r g a n i c m e t a l s a t low tempe- r a t u r e s , some e f f o r t was made i n Orsay t o f i n d a m a t e r i a l t h a t does n o t undergo a P e i e r l s t r a n s i t i o n , by applying h y d r o s t a t i c p r e s s u r e t o t h e v a r i o u s
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786587
materials. The effort succeeded on HMTSF-TCNQ, /6,2, 31, which stays metallic down to the lowest measured temperatures ( 20 mK) at pressures in excess of
.
2.5 kbar. Up to now, the exact reason for this ef- fect of the pressure is not known, since the change of the electronic band structure parameters at these pressures is rather small. Perhaps the pressure har- dens the lattice and thus reduces electron-phonon coupling parameters A =J~
n(sF) /M u2 where J is the electronic matrix element, and M u2 the force- constant. The force-constant increases appreciably with pressure (about 10% per kbar) in these soft molecular crystals 171.A
most striking property displayed by the low-tempe- rature metallic state, is a large anisotropic magne- toresistance /6/ (figure 1). This magnetoresistance corresponds to a mobility of order 10' cm2/v s.Fig. 1 : The anisotropic magnetoresistance of HMTSF- TCNQ. Below 30 K, the magnetoresistance is large, indicating an anisotropic 3-dimensional state. Above 80 K, there is no magnetoresistance, indicating a true 1-dimensional metal. (ref /6/)
Hore detailed measurements, using the Hall effect /8,2,3/ yield a mobility of 40000 cm2/v s for the electrons, and about 12000 cm2/v s for the holes.
Such a large magnetoresistance is not uncommon in ordinary pure metals; however, a particular property of the magnetoresistance here, is that it disappears (i.e. falls by more than 4 orders) as we raise the temperature, although the conductivity itself does not change significantly in this range (a change of about 30% up to about 120 K). Thus the magnetoresis- tance cannot be related in a simple way to
a
tempe-rature-dependent collision time with a constant ef- fective mass (or a constant cyclotron frequency uc = e~/m*c).
The disappearing magnetoresistance is a peculiar pro- perty of one-dimensional metals. We can understand it as follows /9,10/.Consider two chains, with wavefunc tions and $ 2 , degenerate in energy, and a tunne- ling matrix element t
.
Then, if the electron is atI
t = 0 on chain 1, the wavefunction is given as func- tion of time by :
+(t) = @ I cos t tlK +
4,
sin t t/$.L I
At time t = T,,
,
a scattering event takes place, and the phase relationship between the two components ofI
) is destroyed. The probability to find the electron on chain 2 is given by : sin2 (tLTI
,
/d). If t z lId
>> 27~, the electron oscillates between the chains several times before it is scattered, and a phase- relationship between and @, is well defined, and for an array of chains, we can define the transverse wavevector kL. Thus, we may describe the electron by an ordinary Bloch state with quantum numbers (kx, ky, k,). This is the low-temperature (T < 50 K) sta-
te of HMTSF-TCNQ. If, on the other hand, t T /fi < 1,
I I 1
the electron is not able to oscillate between the chains before it is scattered, and the motion bet- ween the chains is a diffusive one. For motion in
the y-direction, on a chain located at (x,z), the quantum numbers are now (x,ky, 2). Since sin2( tLT
,, /M)
(t T the probability to tunnel per unit Ill
time is given by :
1 /= ~(27T ~/$) ltLI2 T I , /h (1) and TL is the hopping time between chains.
Notice the similarity to the Golden rule; here T,,/h replaces n(~~). This derivation assumes that and
$2 are degenerate in energy. Generally, states on two different chains (i.e. donor and acceptor) with the same value of k are not degenerate in energy (Figu-
Y
re 2). Clearly, only states within kgT of sF contri- bute to the interchaintunneling; thus, if K/-cll the non-degeneracy in energy can be ignored; if d/T,, <kBT, the tunneling is impeded due to the ener- gy non-conservation. This situation occurs at low temperatures, since $/rll falls faster with decrea- sing temperature than T. (It falls like T', roughly).
In analogy with the case investigated by Lyo et al.
1111
we get- = - I 2 r I t 1 2
w-c,
Id
L a(kB~)' + (H/T,~I L
(2)1.
where
a
is a number of order unity, characterizing the degree of non-degeneracy in energies of and$
'C6-
1458 JOURNAL DE PHYSIQUEdonor
I€
- a n t i bondi ng bonding
Fig. 2 : The Diffusive-to- ( a ) The energy l e v e l s f o r i s o l a t e d donor and accept- t o r chains.
(b) The e f f e c t of t h e i n t e r c h a i n m a t r i x element, i n opening up a covalency gap, and c r e a t i n g s t a t e s where t h e energy depends on (k,, ky, k Z ) .
For t u n n e l i n g between c-axis neighbours which a r e of t h e same kind ( i . e . donorMonor, and a c c e p t o r t a c c e p - t o r ) , t h e energy l e v e l s a r e d e g e n e r a t e and consequen:
t l y a=0. This accounts f o r t h e experimental observa- t i o n ( F i g u r e 2 ) / 1 2 / t h a t ucc i n c r e a s e s a t low tempe- r a t u r e s more t h a n
a,,,
f o r which o n l y d o n o r t a c c e p t o r t u n n e l i n g c o n t r i b u t e s .The t r a n s v e r s e c o n d u c t i v i t y i s g i v e n by
o
1 = ne pA2
[n(cF)kB~] e ( e ~ l / k ~ ~ ) = n ( ~ ~ ) e ' d ' / r ~ (3) where d i s t h e i n t e r c h a i n d i s t a n c e . Since t h e p a r a l -l e l c o n d u c t i v i t y i s given by
o I I
= n e 2r I I
/m*, we havea l l / o - T , ; TI ,
and a s long as T
I
I =
I / T , ,u
/U i s independent of T , ~,
and a s a r e s u l t tempe- I 1I
r a t u r e and p r e s s u r e independent. However, when
i
i s L g i v e n by (2),u , ,
/ U i n c r e a s e s . T h e r e f o r e , paradoxy-I
c a l l y , t h e c o n d u c t i v i t y i n t h e low-temperature 3D s t a t e (T /T < 1 ) i s more a n i s o t r o p i c t h a n i n t h e
I
J I-Coherent T r a n s i t i o n
(c) When
HIT,,
> t,. t h e covalency gap i s smeared o u t , and t h e motion between c h a i n s i s d i f f u s i v e ; t h e s t a t e s a r e c h a r a c t e r i z e d by ( x , k y , 2).(d) A t low temperatures, t u n n e l i n g between c h a i n s i s f u r t h e r reduced because s t a t e s w i t h t h e same v a l u e of ky a r e no l o n g e r d e g e n e r a t e i n energy. This in- c r e a s e s t h e a n i s o t r o p y .
high-temperature I D s t a t e ( T * / T ~ ~ > 1 ) . Experimen- t a l l y , T , , i n c r e a s e s roughly l i k e T-', and T decrea-
I s e s roughly a s T, i n TTF-TCNQ. T h e r e f o r e , t h e aniso- t r o p y i n c r e a s e s ( v e r y roughly) l i k e T-' 1121. T , , i s about 3 x 1 0 - ' ~ s a t ambient, and
r
i s about 10-"s.L
Thus, an e l e c t r o n s c a t t e r s a few thousand times i n a c h a i n b e f o r e d i f f u s i n g away t o a n o t h e r chain. I n HMTSF-TCNQ, a t low temperature, T , becomes s h o r t e r
-L
t h a n
r , ,
( t h e y a r e of - o r d e r 10-13s t h e r e ) . The d e r i - v a t i o n of (2) r e q u i r e s t h a t t <H/rII
< kBT. I n TTF-I
TCNQ, t h e t u n n e l i n g m a t r i x element bztween two neiglr bouring c h a i n s i s about 1 meV /13/ which i s s u f f i - c i e n t l y small t o s a t i s f y t h i s r e l a t i o n s h i p . Each c h a i n has s e v e r a l (about 6) neighbouring c h a i n s , t o which t h e e l e c t r o n may t u n n e l , t h e r e f o r e t h e e f f e c -
t i v e v a l u e of t i s about 6 meV.
L
A v e r y convenient and r e l i a b l e way t o measure t h e
escape time
Tis by Nuclear Magnetic Resonance
I/14,9/. The relaxation time TI is field dependent, increasing like v'k at moderate fields
(%50 kOe). At low fields, it is field independent. This follows from the basic property of random walk in one dimen- sion, to return an infinite number of times to the initial position. This would cause a divergence of the relaxation rate at zero magnetic field, were it not for the escape of the electron from the chain, which renders the motion somewhat three dimensional.
Thus the field Hc at which the divergent behaviour of the relaxation rate stops, yields the escape time by the relation
T =(2.1 H )-I, where ye is the elec-
I e c
tronic gyromagnetic ratio.
T*
also dominates the Electron Spin Resonance beha- viour 1151; in a purely 1D system, relaxation by the Elliott mechanism is suppressed; r / T is a mea-
1 1
I
sure of the deviation from the
IDstate. Since the relaxation by the Elliott mechanism is proportional to (Ag)
'T,, -I, the relaxation rate in the quasi- 1D state is proportional to [(ng2)
-c -I] LT,,ITL]
I I
=
( ~ g ) ~
T,--I.The width of the ESR line AH is domi- nated by this relaxation rate, therefore
AH= (Ag)'
T1-
(4)
AH,
Tand a are plotted as function of tempera-
I 1.
ture in figure 3, and are seen to behave similarly.
Fig.3
:The increase of the ESR linewidth 11 51 AH; the escape rate from the chain 191 r -I; and the trans- verse conductivity 1121 Ua, oc, ht low temperatures.
The increase in the collision time T I , is also shown.
T
is also plotted. The drop of
'CL-'below the
I I
value predicted by
( 1 )at low temperatures, due to the impeded tunneling [equation ( 2 ) ] is seen clearly.
Thus, let us summarize
:Organic metals may possess a one-dimensional metal- lic state, in which r I
>r I I
;the ratio
TIT,, is a
L
measure of the one-dimensionality. In this state, the transverse motion is diffusive,
DL =d2/TL, and the quantum numbers are (x, ky, z). It possesses the following properties
:1)
The resistivity is given by I161
p = p o
+ B(P)T~, n = 2.3.
p odepends on impurities and defects and is essentially zero in a high-quali- ty material. B(P) is sample independent (for reaso- nable quality samples) but is strongly pressure de- pendent; 1171 3 Rn B(P) IaRn b = 40.
2) The resistivity is probably strongly frequency dependent, increasing significantly in the far in infra-red 1181.
3) There is no magnetoresistance 1191 (within lo-'
at 50 kOe); the Hall constant is given 1201 by the classical value RH
=I/no ec, where no is the densi- ty of conduction electrons
(%0.6 per molecule).
4) The NMR relaxation time T, is strongly field de- pendent.
5) The ESR line is strongly narrowed.
6)
The magnetic susceptibility exceeds the Pauli value, and increases with temperature 12
11.
7) There is a Peierls transition, around 30-60 K.
T increases (slightly) with pressure 117,221.
P
Some organic metals, such as HMTSF-TCNQ at helium temperatures, possess a three-dimensional metallic state, in which
TI
<-ell . In this state, the trans- verse motion of the electrons is coherent, as well as the longitudinal one, and the quantum numbers are (kx, ky, k , ) . Thus, a transverse effective mass mx can be defined. This state possesses the follo- wing properties I
:1) It may be metallic at OK; the
resistance there is mainly residual. 2) The magne- toresistance is enormous, and transverse. 3) The Hall constant is enormous, RH 1000/noec, indica-
ting a huge mobility,
p= 40000 cm2/v s. 4) Quan- tum (Shubnikov-de Haas oscillations can be observed 1231.
5)There is a diamagnetic (Landau-Peierls) susceptibility 1241.
Inorganic chains materials, such as (SN), and NbSe, are probably in this class 1251. These inorganic materials are superconducting (NbSe3 under pressure).
An intermediate state, with r " T I , , may also exist
1.It is characterized by an extreme sensitivity of the
resistivity to radiation damage 1261 and by a very
fast drop of the Peierls transition temperature
with pressure 1271. HMTSF-TCNQ around 30 K is in
this intermediate state. Possibly HMTSF-TNAP 1281
C6-1460
JOURNAL DE PHYSIQUEand HMTTF-TCNQ 1271 a r e a l s o i n t h i s i n t e r m e d i a t e s t a t e , because of t h e r a p i d drop of Tp w i t h p r e s s u r e i n t h e s e m a t e r i a l s .
The r e a s o n f o r t h e s e n s i t i v i t y of t h e i n t e r m e d i a t e s t a t e i s , t h a t a s m a l l p e r t u r b a t i o n can e a s i l y take i t over t o a one-dimensional s t a t e , which undergoes a P e i e r l s t r a n s i t i o n and t h u s i s i n s u l a t i n g a t he- lium temperatures ( t h i s i s t h e e f f e c t of r a d i a t i o n damage), o r t o t h e three-dimensional s t a t e , which i s m e t a l l i c a t low temperatures ( t h i s i s t h e e f f e c t of p r e s s u r e ) .
2. THE RESISTANCE I N THE METALLIC STATE.- Quite a number of t h e o r i e s a t t e m p t t o account f o r t h e r e s i s - t i v i t y of TTF-TCNQ i n t h e m e t a l l i c s t a t e , and t h i s f i e l d i s r a t h e r w i l d . I n Table I we l i s t a number of t h e s e t h e o r i e s , a s w e l l a s t h e experimental f e a - t u r e s of t h e r e s i s t i v i t y t h a t must be accounted f o r .
We do not a t t e m p t a d e t a i l e d survey ( t h i s would t a k e u s o u t s i d e t h e Framework of t h e p r e s e n t t a l k ) , b u t o n l y l i s t w i t h a
J
t h e main experimental f e a t u r e s t h a t they claim t o account f o r , and by an X t h e ex- p e r i m e n t a l f e a t u r e t h a t we f e e l t h a t t h e y have not been a b l e t o account f o r . The l i b r o n t h e o r y proposed by Gutfreund and t h e a u t h o r 1291 c o n s i d e r s t h e cou- p l i n g between t h e e l e c t r o n and t h e molecular l i b r a - t i o n , which i s a second o r d e r i n t e r a c t i o n because t h e e l e c t r o n cannot d i s t i n g u i s h between a molecular r o t a t i o n a t an a n g l e +O and a r o t a t i o n a t an angle-@.
Thus, t h e t r a n s f e r i n t e g r a l t , , i s g i v e n by t , , = ty,+
1 a2t:, and t h e r e f o r e t h e c o l l i s i o n time2
g
02i s g i v e n by he Golden r u l e
Because of t h e low l i b r o n frequency /30/(35-60 cm-l) TABLE I
THEORY -t
EXPERImNT
+
Theory e x i s t s , s t a r t i n g from Hami 1 t o n i a n Momentum t r a n s f e r t o l a t t i c e Absolute v a l u e T-dependence (P=O) P-dependence T-dependence
( b c o n s t . ) /32/
a-dependence /21/
R a t i o acceptor- donor ( r e l a t i o n w i t h TEP), RH 1451 D i f f e r e n t
m a t e r i a l s 1461 R a d i a t i o n
damage 1471
D i e l e c t r i c
c o n s t a n t 1481 R e l a t i o n w i t h
xs
R e l a t i o n w i t h /9/
R e l a t i o n w i t h K 1491 R e l a t i o n w i t h RH AP/P,H*
,xd.
/2,24/S i n g l e Ordinary phonon- t r a n s l a t i o n
J
J
X X
?
X X
E l e c t r o r magnon
IJeromek 3 /
X
J
1
J ,
4
E l e c t r o n I n t r a - molecular
Seiden/s 1 /
X J
X
/
X
PhononF l u c t u a t i n g C D W ABB
/
34 // 3 5 /
X
X
-
E l e c t r o nI n t e r - c h a i n
Seidenl42
X
/
d
X
E l e c t r o n-
C o l l e c t i v e R i g i d
C D W LRA Ileeger
X
J
J
X X X
X P a r t i c l e
Libron
GW
2 9
50 2 9 29,7
7
7,44 50
3 1 , t
46
t
5 1 52 53
9 5 1
2,24
L o c a l i z a t i o n Abrikosov/38/
Rashba
/
3 9 1 / 3 6 , 3 7 / Y . ~ . a e n / 4 0 /X
X
X
I n t r a -molecular phonon Cunwell/3 3 /
/
V '
J
X
X
Boltzmann statistics apply above
100K, and
<8'>=
kB~/1w; , where I is the appropriate moment of inertia. wL
=% (T,V) is temperature and volume de- pendent. This dependence can be calculated for the Lennard-Jones intermolecular forces 171, yielding
-3RnwL/a Rn b =
10at constant T, and w = wo
[l
+ 0.2 (T/300)] at constant b. This volume and temperature dependence yields
-3Rn 013 En b - 40,
in excellent agreement with experiment, and
~T'/w'
=T2/
[l+ 0.2(~/300)]'. This last law is quasi-linear in the temperature range 150 K -
500 K, with a negative intercept at T
=0, in excel- lent agreement with experiment 1321. At P
=0, this theory yields the law p
=po+ B T " ~ observed expe- rimentally. The resistivity of the TTF stack in TTF- TCNQ is particularly high because of the very long frequency of libration of TTF in its plane, i.e.
35 cm-' 130,311. The two tri-methylene bridges in HMTTF, HMTSF contribute to the rigidity of the stack, and thus lower the resistivity considerably. We list in Table I the references where each experimental feature of the conductivity is accounted for by the libron theory. For a critical comparison with the other theories, it is necessary to complete the ta- ble for them.
3. THE PEIERLS TRANSITION.- A basic property of the 1-D metallic state, is to undergo a Metal-to-Semi- conductor i.e., a Peierls transition. The temperatu- re of this transition is given in the mean field approximation approximately by 151
:T 1 2
E~exp(-111)
5'n(cF)/Y a ' . P
When no coupling exists between the chains, the sys- tem is purely one-dimensional, and does not possess a phase transition; i.e. the Peierls transition is smeared out by the fluctuations. This is the situa- tion in the organic system KCP. When some coupling exists between the chains, either due to Coulomb coupling, or due to mechanical interaction, or due to electron tunneling, the fluctuations are reduced and the Peierls transition can take place 1541. In TTF-TCNQ-like systems, the coupling is strong enou*
so that Tp is close to the mean field value 1551.
Coupling between the chains also affects the mean- field value of.Tp. Coulomb coupling increases ~ ~ 1 5 6 1 . This is because the field produced by the neighbou- ring chains, enforces the field inducing the transi- tion. The phase of the charge-density-wave on neiglr bouring chains reverses, to minimize the Coulomb interaction. Naive estimates of this Coulomb cou- plingindicate that it is extremely strong 1561 and would tend to make both chains undergo a Peierls
transition together, at a very high temperature
(a few hundred degrees, or higher). These naive estima- tes use the continuum model, in which the charge is assumed to be distributed continuously along the chain according to a cosine law, q
L-q, cos(2kFb), and the potential due to such a charge distribution is calculated classically. Actually, because of the atomic distribution of charge, and the relative tilt of the molecules on neighbouring stacks, this Cou- lomb coupling is greatly attenuated - probably by more than an order of magnitude 1571. Coupling via
lattice strain (involving translation, or rotation, of the molecules) similarly increases Tp, however, the interaction via neighbouring chains may cause a phase reversal between next nearest neighbour chains 1581. Coupling via electron tunneling, on the other hand, reduces Tp, since the Fermi surface is made less planar 1591. In a one-chain-family system, the Fermi surface, though not planar, is nested, and as a result Tp of a CDW with a transverse period of 2a (i.e. again a phase reversal between neighbouring chains) is not appreciably suppressed 1551 unless the 3D coupling is extremely strong. In a 2 chain- family system, this perfect nesting is not present I601 and as a result Tp is strongly depressed by the interchain tunneling, accounting for the metallic state of systems like HMTSF-TCNQ, HMTTF-TCNQ.
The interchain tunneling in IMTSF-TCNQ-like systems is particularly strong, because molecules of neigh- bouring chains are nearly parallel, giving rise to a large overlap of the pZ orbitals, and thus to a large value of tL (of order 20 meV) 1241. In TTF- TCNQ-like systems, there is an angle of approximate ly 55' between molecules on neighbouring chains, significantly reducing the overlap and the value of tl (of order 6 meV, including the coupling to all neighbours 191. The reduced distance between the atoms, mainly the donor seleniums (or sulphurs) and acceptor nitrogens, also plays some role in enhan- cing the value of
t,of HMTSF-TCNQ 1611.
By now, the behaviour of the Peierls transition in quite a number or systems belonging to this family is known 128,461. An interesting different system, recently discovered 1631 is TTT213. This system maintains a relatively high conductivity down to helium temperature, and apparently possesses a ra- ther low value of Tp. We still do not know the cause for this.
Summarizing, a metallic state at helium temperature
and the suppression of the Peierls transition, re-
quires
:I) A high conductivity, i.e. a well-ordered
JOURNAL DE PHYSIQUE
c r y s t a l . T h i s a s s u r e s a long c o l l i s i o n time T I ; . of p o s s i b l e s u p e r c o n d u c t i v i t y . By s u p e r c o n d u c t i v i t y 2 ) A s t r o n g i n t e r c h a i n t u n n e l i n g m a t r i x element tL. we mean BCS type s u p e r c o n d u c t i v i t y , t o b e d i s t i n - This i s a i d e d by n e a r l y p a r a l l e l s t a c k i n g of molecu- guished from e l e c t r i c f i e l d induced movement of l e s on neighbouring c h a i n s . 3) Conducting donor and charge d e n s i t y waves, which may, p e r h a p s , g i v e r i s e a c c e p t o r c h a i n s , t o p r e v e n t t h e p e r f e c t n e s t i n g t o some d i m u n i t i o n i n t h e r e s i s t i v i t y , a s suggested c h a r a c t e r i s t i c of one-chain-family systems. by F r o h l i c h /5/ and perhaps observed r e c e n t l y i n
TTF-TCNQ a t helium temperatures 1641 above Tp 1651, and i n NbSeB 1661. I n o r g a n i c quasi-one-dimensional m e t a l s , f r e q u e n t l y a r e superconducting. Examples a r e
t h e A-IS'S, such a s Nb3Sn, Nb3Ge, e t c . which a r e superconducting up t o 23 K 1671 (SN),, which i s superconducting 1251 below 1 K, NbSe3, which i s superconducting under p r e s s u r e 1661. A t h e o r i c a l a r - gument i n favour of s u p e r c o n d u c t i v i t y of I-D m e t a l s , i s a s f o l l o w s 1681. The electron-phonon coupling c o n s t a n t i s given by A = J~ n(zF)/hw2. Due t o t h e Kohn anomaly of I-D m e t a l s ( i . e . t h e approach t o t h e P e i e r l s t r a n s i t i o n ) , w d e c r e a s e s , t h u s i n c r e a - s i n g A . Since Tc i s given by t h e BCS formula, Tc
"
w exp (- 1 /A), t h e d e c r e a s e i n h should i n c r e a s e Tc. C l e a r l y , Tc cannot exceed w and q u a n t i t a t i v e c a l c u l a t i o n s 1691 show t h a t (Tc) max"
1 WO,
where w o i s t h e b a r e phonon frequency. S t i l l , f o r o r g a n i c m e t a l s w i t h a,, 2 60 K, we may e x p e c t super- c o n d u c t i v i t y around I K, under f a v o r a b l e circumstan- ces. S t i l l , we should be aware of f a c t o r s unfavora- b l e f o r s u p e r c o n d u c t i v i t y .
-
1) F l u c t u a t i o n s . - I n a s t r i c t I-D m a t e r i a l , f l u c t u a - t i o n s p r e v e n t any phase t r a n s i t i o n ( f o r f i n i t e r a n g f o r c e s ) . I n quasi-I-D m a t e r i a l s , f l u c t u a t i o n s tend t o lower t h e temperature of t h e phase t r a n s i t i o n (whether t o a superconducting s t a t e , i n s u l a t i n g s t a - t e 1541, magnetic s t a t e 1701, e t c . ) . C o u p l i n g between t h e chains reduces t h i s e f f e c t , and i t probably i s n o t impor'ant i n TTF-TCNQ and s i m i l a r m a t e r i a l s . 2) The Covalency Gap.- I n two c h a i n f a m i l y system, t h e donor-acceptor t u n n e l i n g m a t r i x element causes a covalency gap. This gap may r e n d e r t h e m a t e r i a l i n s u l a t i n g o r a t l e a s t d e c r e a s e s t h e d e n s i t y of s t a t e s a t t h e Fermi l e v e l 1711. This reduces t h e e f f e c t i v e A, and thus Tc. Low-lying molecular e x c i - t e d s t a t e s may reduce t h i s e f f e c t / 6 0 / .
3) P o s s i b l e E l e c t r o n - E l e c t r o n i n t e r a c t i o n s . - Perhaps Fig. : The nearly parallel tilt of molecules of e l e c t r o n - e l e c t r o n Coulomb i n t e r a c t i o n s ("Big U") neighbouring c h a i n s i n HMTSF-TCNQ and HMTTF-TCNQ p l a y a r o l e i n o r g a n i c m e t a l s 1721. Such i n t e r a c -
(bottom) i s an e s s e n t i a l f e a t u r e f o r t h e l a r g e over-
lap, and hence large tl- giving rise to the 3-D state. t i o n s a r e known i n o r d i n a r y m e t a l s t o s u p p r e s s t h e I n TTF-TCNQ, TSF-TCNQ, t h e l a r g e a n g l e between t h e superconducting t r a n s i t i o n temperature 1731.
molecules ( t o p ) reduces tL s i g n i f i c a n t l y . P o s s i b l e , t h i s e f f e c t a l r e a d y r e n o r m a l i z e s t h e P e i e r l s t r a n s i t i o n temperature, and t h u s i s a l r e a d y 4. POSSIBLE SUPERCONDUCTIVITY?.- A most i n t r i g u i n g taken into account when we use the value of deri- q u e s t i o n concerning o r g a n i c m e t a l s , i s t h e q u e s t i o n ved from t h e P e i e r l s t r a n s i t i o n temperature. I n t h a t
C6-1463
case, we don't have to worry about "Big U" ince DDIS /74/.
4) The Peierls transition.- 1-D materials tend to undergo a Peierls transition to an insulating state.
Clearly, an insulating state cannot be superconduc- ting. Therefore, to achieve a superconducting state, we must suppress the Peierls transition. Such a sup- pression can be achieved in several ways,
(i) Geology .- The electron-phonon coupling cons- tant g(q) depends on the magnitude of the momentum transfer q. In 1-D materials, only momentum trans- fers of q=0 and q=2kF conserve energy. Thus, only gj = g(2kp) and g = g(0) contribute effectively, g tends to give rise to the Peierls state, but g2
gives rise to superconductivity /75/. Thus, a large g / g is favourable for superconductivity. In figure 5, we show a phase diagram, in the g!~g2 pi2"
ne, of the many diverse phases of 1-D materials, for an infinite phonon frequency (instantaneous in- teraction), in the mean field approximation /76/.
The essence of this phase diagram is that the effec- tive coupling constant 'g'=-g2/}ko for phonon-mediated interactions for the various phases is given by a linear combination of g and g appropriate for the given phase, namely. For the ordinary (singlet) superconducting phase S : geff = gj + g2. For the Peierls phase (Charge Density Wave) P: g ,,. =2g -g . For the antiferromagnetic (Spin Density Wave) phase AF: gef£ = g2. For the triplet superconducting phase T: g£ f f = -gj + g2. The phase with the largest ge f f
dominates. Umklapp processes (when 2kF is close to a reciprocal lattice vector) modify these coupling pa- rameters.
(ii) High Phonon Frequency.- A high phonon frequency is favorable to superconductivity vis-a-vis the Peierls transition. In figure 5 we show the phase diagram, in the relevant quadrant of the g,-g2 plane, when the finite phonon frequency is taken into ac- count /69/. It is seen that the importance of the
Ii A . . . . ratio g2/gj is diminished.
I Fig. 5 : A The phase diagram of I-D metals for an instantaneous (w •*• <*>)
interaction /76/.
AF : antiferromagnetic P : Peierls state
S : singlet superconductivity T : triplet superconductivity
F : ferromagnetic M : martensitic (i.e. lattice distortion).
L : localized paired electrons (bi-polarons)
Q The effect of the retarded interaction, for several phonon frequencies /69/.
c6-1464 JOURNAL DE PHYSIQUE ( i i i ) I n t e r c h a i n Coupling.- I n t e r c h a i n c o u p l i n g , v i a
e l e c t r o n t u n n e l i n g , g i v e s r i s e t o a 3-D s t a t e , i n which t h e P e i e r l s t r a n s i t i o n may b e s u p p r e s s e d . This
i s t h e c a s e i n t h e i n o r g a n i c quasi-1-D systems, l i k e t h e A-15's, (SN),, NbSea under p r e s s u r e , e t c . which a r e indeed s u p e r c o n d u c t i n g w i t h o u t a P e i e r l s t r a n s i - t i o n . T h e o r e t i c a l c o n s i d e r a t i o n s / 7 7 / a l s o i n d i c a t e t h a t t h i s c o u p l i n g f a v o r s s u p e r c o n d u c t i v i t y , w h i l e Coulomb c o u p l i n g f a v o r s t h e P e i e r l s s t a t e .
( i v ) Gorkov 1781 s u g g e s t e d t h e p o s s i b l e e x i s t e n c e of a s t a t e which i s a h y b r i d of a P e i e r l s s t a t e and a BCS s u p e r c o n d u c t o r . A l l t h e s e f a c t o r s concern s u p e r c o n d u c t i v i t y due t o alectron-phonon coupling;
an e n t i r e l y d i f f e r e n t mechanism, i s s u p e r c o n d u c t i - v i t y due t o e l e c t r o n - e l e c t r o n c o u p l i n g ( e x c i t o n i c i n t e r a c t i o n ) / 7 9 / . A t h e o r e t i c a l c a l c u l a t i o n 1801 i n d i c a t e s t h a t t h i s mechanism can s e r v e a s a s o u r c e f o r s u p e r c o n d u c t i v i t y i n c h a i n systems, i n which t h e conducting s t r a n d i s surrounded from a l l s i d e s by e x c i t o n i c dye molecules. However, systems r e s e m b l i n g
t h i s proposed one have n o t y e t been s y n t h e s i z e d . Thus, t h i s f i e l d i s s t i l l wide open, and s e a r c h f o r p o s s i b l e s u p e r c o n d u c t i v i t y i n o r g a n i c m e t a l s a t t e m p e r a t u r e s below 1 K may p r o v e t o b e an i n t e r e s - t i n g s u b j e c t i n t h e f u t u r e .
4cknowledgement.- T h i s work i s a r e s u l t o f extended c o l l a b o r a t i o n w i t h J . F r i e d e l , D. JQrome, J.R.
Cooper, K. Bechgaard, H. Gutfreund and M. Kaveh.
References
/ I / "LOW Dimensional Co-operative PhenomenaH,Keller H . J .
,
Ed. Nato A s i S e r i e s (Plenum P r e s s , New York) 1975./2/ "Chemistry and P h y s i c s o f One-Dimensional We- t a l s " , K e l l e r , H . J . , Ed. Nato A s i S e r i e s
=,
(Plenum P r e s s . New York) 1977.
/3/ "Organic Conductors and Semi-conductors", Con- f e r e n c e P r o c e e d i n g s , S i o f o k , Hungary, 1976, L e c t u r e Notes i n P h y s i c s , 65, ( S p r i n g e r Verlag)
1977.
141 Thomas, G.A. e t a l . , Phys. Rev.= (1976) 5105.
/5/ P e i e r l s , R.E., "Quantum Theory of S o l i d s " , (Clarendon P r e s s , Oxford), 1955; F r o h l i c h , H., Proc. Roy. Soc.
A223
(1954) 296.161 Cooper, J.R., Weger, M., Jerome, D . , L e f u r , D . , Bechgaard, K., Cowan, P.O., Bloch, A.N., S o l i d S t a t e Conmn.
2
(1976) 749; Jerome, D. and Weger, M., i n r e f . / 2 / ; r e f . 131, pp. 363, 371./ 7 / Gutfreund, H., Weger, M. and Kaveh, M., S o l i d S t a t e Commun.
2
(1978).181
Cooper, J . R . , Weger, M., Delplanque, G., Jerome, D., Bechgaard, K., J. Physique L e t t .37
(1976) L-349./ 9 / Soda, G., Jerome, D., Weger, M., A l i z o n , J . , Gallice, J . , R o b e r t , H., F a b r e , J.M. and G i r a l , L., J. Physique
3
(1977) 931./ l o / Weger, M . , B u l l e t i n of t h e EPS, J u l y , 1978.
G l i c k . A. and Gutfreund, H. ( u n p u b l i s h e d ) . 1 1 1 / H o l s t e i n , T., Lyo, S.K., Orbach, R., Phys. Rev.
Bl5 (1977) 4693.
-
/12/ Coleman, L., T h e s i s , U n i v e r s i t y of P e n n s y l v a n i a 1131 B e r l i n s k y , A.J., C a r o l a n , J . F . , W e i l e r , L.,
S o l i d S t a t e Commun.
15
(1974) 795./14/ Soda, G . , Jerome, D., Weger, M., F a b r e , J.M.
and G i r a l , L . , S o l i d S t a t e Commun.
2
(1976) 1417.1151 Tomkievicz, Y . , S c o t t , B.A., Tao, L.J. and T i t l e R.S., Phys. Rev. L e t t .
2
(1974) 1363.1161 G r o f f , R.P., Suna, A. and M e r r i f i e l d , R.E., Phys. Rev. L e t t .
2
(1974) 418; F e r r a r i s , J . P . , Finnegan, T.F., S o l i d S t a t e Commun.18
(1976) 1169./17/ Chu, C.W., Harper, J . M . , G e b a l l e , T.H., Greene, R.L., Phys. Rev. L e t t .
2
(1973) 1431; Jerome, D . , M u l l e r , W. and Weger, M., J. Physique L e t t . 35 (1974) L-77; Cooper, J.R., Jerome, D. Weger,-
M. and Etemad, S . , J . Phys. L e t t .36
(1975)L219./18/ Tanner, D.B., Jacobsen, C.S., G a r i t o , A.F. and Heeger, A. J . , Phys. Rev.
B13
(1976) 3381.1191 T i e d j e , T., Carolan, J . F . , B e r l i n s k y , A.J. and W e i l e r , C., Can. J . Phys.
53
(1975) 1593./20/ Cooper, J.R., M i l j a k , M., Delplanque, G. ,Jerome D., Weger, M., Fabre, J . M . and G i r a l , L., J.
Physique
38
(1977) 1097./21/ S c o t t , J . C . , G a r i t o , A.F. and Heeger, A.J.,
Phys. Rev. (1974) 3131.
1221 F r i e n d , R.H., M i l j a k , M. and Jerome, D . , Phys.
Rev. L e t t .
40
(1978) 1048.1231 M i l j a k , M . , Andrieux, A , , F r i e n d , R.H., Manlfait G . , Jerome, D. and Bechgaard, K . , S o l i d S t a t e Commun.
1241 Soda, G . , Jerome, D . , Weger, M., Bechgaard, K.
and P e d e r s e n , E., S o l i d S t a t e Commun.
20
(1976) 107.1251 Greene, R.L., G i l l , W.D., Azavedo, L.J., C l a r k , W.G. and D e u t s c h e r , G., p. 603 i n r e f . / 3 / ; G i l l , W.D., B e r g e r , W . , S t r e e t , G.B., p. 619 i n r e f . / 3 / .
1261 Z u p p i r o l i , L . , Arionceau, J . , Weger, W., Bech- g a a r d , K. and Weyl, C . , J. Phys. L e t t . ,
2
(1978) L-170.
1.271 F r i e n d , R.H., Jerome, D., F a b r e , J . M . , G i r a l , L. and Bechgaard, K., J . Phys. (1978) 263.
1281 Mogensen, B., F r i e n d , R.H., Jerome, D . , Bech- g a a r d , K. and C a r n e i r o , K.
1291 Gutfreund, H. and Weger, M., Phys. Rev.
(1977) 1753.
1301 Kuzmany, H. and S t o l t z , H.J., J . Phys.
(1977) 2241.
/31/ S c h u l t z , A.J., Stucky, G.D., B l e s s i n g , R.H. an3 Coppens, P . , J. Am. Chem. Soc.
98
(1976) 3194.13.21 Cooper, J.R., Phys. Rev. B Comments.; F r i e n d , R.H., M i l j a k , M., Jerome, D . , Decker, D.L. and Debray, D., J. Phys. L e t t .
2
(1978) L-134./33/ Conwell, E.M., Phys. Rev. L e t t .
2
(1977) 777.1341 A l l e n d e r , D . , Bray, J . W . , Bardeen, J . , Phys.
Rev.
2
(1974) 119.1351 Weger, M., H o r o v i t z , B. and Gutfreund, H., Phys.
Rev.
B12
(1975) 1086.1361 Lee, P.A., R i c e , T.M. and Anderson, P.W., Phys.
Rev. L e t t .
31
(1973) 462.1371 Heeger, A . J . , i n r e f s . / I , 2 , 31
1381 Abrikoson, A.A. and Ryshkin, I.A., S o l i d S t a t e Commun.
2
(1977) 319.1391 Rashba, E . I . , Gogolin, A.A. and Melnikov, V . I . , r e f . 131, p. 265.
1401 Cohen, M.H., r e f . 131, p. 225.
1411 S e i d e n , P.E. and Cabib, D . , Phys. Rev.
B13
(1976) 1846.
1421 S e i d e n , P.E. ( P r e p r i n t ) .
1431 Jerome, D., J. Physique L e t t .
2
(1977) L-489.1441 G u t f r e u n d , H. and Weger, M., Dubrovnik Confe- r e n c e , Aug. 1978.
1451 C h a i k i n , P.M., Greene, R.L., Etemad, S. and E n g l e r , E.M.
,
Phys. Rev.B13
(1976) 1627.1461 J a c o b s e n , C.S., Mortensen, K., Andersen, J . P . and Bechgaard, K.
,
Phys. Rev. B (1978) ; Bechgaard, K., J a c o b s e n , C.S., Hessel-Andersen, N . , S o l i d S t a t e Commun.25
(1978) 875.1471 Chiang, C.K., Cohen, M . J . , Newman, P.R. and Heeger, A.J., Phys. Rev.
B16
(1977) 3163.I 4 8 1 Gunning, W . J . , Heeger, A . J . , Shchegolev, I.F.
and Z o l o t u k i a , S.P., S o l i d S t a t e Commun.
6
(1978) 981.
1491 Salomon, M.B., Bray, J . W . , de P a s q u a l i , G . , Craven, R.A., S t u c k y , G. and S c h u l t z , A., Phys.
Rev. (1975) 619.
1501 Weger, M. and G u t f r e u n d , H . , Comments on S o l i d S t a t e P h y s i c s , S e p t . 1978.
1511 Kaveh, M., Gutfreund, H. and Weger, M., Phys.
Rev. B and P r e p r i n t .
1521 Weger, M. and Kaveh, M. ( P r e p r i n t ) . 1531 M a r i a n e r , S . , Weger, Y. and G u t f r e u n d , H.
( P r e p r i n t )
.
1541 Thielmans, M . , D e l t o u r , R., Jerome, D. and Cooper, J . R . , S o l i d S t a t e Commun.
19
( 1976) 2 1 . 1551 H o r o v i t z , B., G u t f r e u n d , H. and Weger, M., Phys.Rev.
812
(1975) 3174.1561 B j e l i s , A. and B a r i s i c , S., r e f . 131, p. 291.
1571 Weger, M. and F r i e d e l , J . , J. P h y s . 3 8 (1977)241.
1581 Weger, M. and F r i e d e l , J. i b i d ,
2
(1977) 881.1591 B e n i , B , S o l i d S t a t e Commun.
15
(1974) 269.1601 Weger, M., S o l i d S t a t e Commun.
19
(1976) 1149;S h i t z k o v s k y , S . , Weger, M. and G u t f r e u n d , H . , J . Phys.
39
(1978) 71 1 .1611 P h i l l i p s , T.E., Kistenmacher, T . J . , Bloch, A.N.
and Cowan, D.O., J.C.S. Chemical Commun.
334
( 1976)
1621 Cooper, J . R . , Jerome, D . , Etemad, S. and Engler, E.M., S o l i d S t a t e Commun.
2
(1977) 257.1631 Kaminski, V.F., Khidekel, M.L., L j u b o v s k i i , R.E Shchegolev, I . F . , S h i b a e v a , R.D., Y a g u b s k i i , E.B., Zvarykina, A.V. and Zuereva, G.L., Physica S t a t u s S o l i d i ( 1978).
1641 Cohen, M . J . and Heeger, A . J . , Phys. Rev.
B16
(1977) 688.
1651 S e e g e r , K. and Maurer, W . , S o l i d S t a t e Comun.
(1978).
1661 Monceau, P . , Ong, N.P., P o r t i s , A.M., Meerschant, A . , Rouxel, J . , Phys. Rev. L e t t .
2
(1976) 602.Monceau, P . , P e y r a r d , J . , R i c h a r d , J. and Moli- n i e , P . , Phys. Rev. L e t t .
2
(1977) 161.1671 G a v a l e r , J.R., Appl. Phys. L e t t .
12
(1973) 480;Weger, M. and Goldberg, I . , S o l i d S t a t e P h y s i c s 28 (1973) 1 ; S e i t z , T u r n b u l , E h r e n r e i c h E d t s .
-
1681 G u t f r e u n d , H . , H o r o v i t z , B., Weger, M., J. Phys.
C7 (1974) 383.
-
1691 Birnboim, A. and H o r o v i t z , B., S o l i d S t a t e Com- mun. 19 (1976) 91; H o r o v i t z , B . , Phys. Rev.
e
( 1 9 7 7 F 3 9 4 3 .
1701 Azevedo, L . J . , C l a r k , W.G., H u l i n , D. and McLean, E.O., Phys. L e t t .
58A
(1976) 255.1711 B e r n s t e i n , U . , C h a i k i n , P.M., P i n c u s , P . , Phys.
Rev. L e t t .
2
(1975) 271.1721 T o r r a n c e , J . B . , r e f . 1 2 1 , p. 137.
1731 McMillan, W.L., Phys. Rev.
167
(1968) 331.1741 Caron and Jbrome,D. ( P r e p r i n t ) .
1751 Menyhard, N. and Solyom, J . , J. Low Temp. Phys.
12 (1973) 529.
-
1761 H o r o v i t z , B . , S o l i d S t a t e Commun.
18
(1976) 445 1771 Klemm, R.A. and G u t f r e u n d , H . , Phys. Rev.2
(1976) 1086.
1781 Bychkov, Y.A., Gorkov, L.O. and D z y a l o s h i n s k i i , I . E . , JETP
3
(1966) 489.1791 L i t t l e , W.A., Phys. Rev.
134
(1964) A1416;Ginzburg, V . I . , JETP
5
(1964) 2318; A l l e n d e r , P . , Bray, J . , Bardeen, J . , Phys. Rev. (1973)1020.
1801 G u t f r e u n d , H. and L i t t l e , W.A., i n "Highly Con- d u c t i n g One-Dimensional S o l i d s " ; d e v r e e s e , J.
and E r v a r d , R., Ed.( Plenum P r e s s ) 1978
