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Theory of the threshold field for the depinning transition of a charge density wave
Physica A 179 (1991) North-Holland
1-15
PHYSICA Iii
Theory of the threshold field for the depinning transition of a charge density wave L. Pietronero Dipartimento di Fisica, Universitci di Roma “La Sapienza”, 00185 Rome, Italy
Piazzale A. Moro 2,
M. Versteeg Solid State Physics Laboratory, The Netherlands
Received
University of Groningen,
Melkweg 1, 9718 EP Groningen,
30 May 1991
The Hamiltonian of an elastic string pinned by random potentials is often used to describe the depinning transition of a charge density wave in the presence of impurity pinning. The properties of the pinned states show close analogies to those of glassy systems, while the depinning transition resembles a dynamical critical phenomenon. Here we develop an approximate analytical approach based on the quasi-harmonic treatment of the nonlinear pinning potentials together with an expression of the correlation functions to lowest order. This method allows a nonperturbative treatment of a problem with many nonlinear degrees of freedom and quenched disorder. In this way it is possible to characterize the pinned configurations and to compute the threshold field as a function of the coupling constant of the model. The results are in very good agreement with the computer simulations for strong and intermediate pinning and provide an approximate but quantitative method for the determination of the pinned-unpinned phase diagram.
1. Introduction
A remarkable series of experiments on quasi-one-dimensional metals like NbSe, and similar systems have provided evidence for a contribution to nonlinear transport due to the coherent sliding of a charge density wave (CDW) [l, 21. Ab ove a threshold electric field the CDW seems to depin and to contribute to the electric current. The depinning phenomenon resembles a second-order phase transition because it shows critical behavior and a correlation length that diverges [3]. Below this threshold the system shows a number of interesting properties characteristic of glassy systems like hysteresis effects [2] and stretched exponential relaxation [3,4]. A model Hamiltonian that is often used to describe several properties of 037%43711911$03.50
0
1991-
Elsevier
Science
Publishers
B.V. (North-Holland)
these
systems
random duct
consists
various
stretched prcscnt
in the overdamped
[ 1. 5-S].
impurities
properties
exponential a simple
USC ;I different
of the
relaxation
analytical
and of the corresponding will
Numerical
depinning below
approach
of the C’DW pinned of this model
the threshold
to derive
as the
of the threshold paper
results
properties and the critical behavior at dcpinning. The Hamiltonian model is one-dirncnsional and it consists
[ 131 instead
for the
I~>
rcpro-
[ 12, 131. In this paper
In the following analytical
could
[7-l I ] as well
transition
for the calculation
phase diagram.
approach
dynamics
simulations
\ve field we
relaxation
of many
degrees
of freedom coupled elastically between them and interacting in a nonlinear wa> with randomly located pinning centers. From a theoretical point of vie\+ the combination of nonlinear interactions and quenched disorder gives rise to ;I rather complex problem. The corresponding single-particle problem can be easily solved [ 151 but the mean-field type arguments [ Ih] developed for the coupled problem appear to bc ovcrsiniplificd and difficult to relate to the original Hamiltonian. In this paper we present a new approach to the problem based on ;I Gaussian ansatz for the phase fluctuations whose variance provides a characterization of the pinned state. This variance can bc computed from the solutions of ;I set of nonlinear threshold
self-consistent equations that allow also the determination field as a function of the Hamiltonian parameters. This
of the method
therefore, allows, ;I rron/,crtllrhrrti~l~~treatment of ;I problem with many nonlinear interacting dcgrecs of freedom in the presence of quenched disorder. The nature of the approximation is rather accurate for the USC of strong pinning, but the results arc in good agreement with the computer simulations also for intcrmcdiate pinning. From the point of view of the analogy of the depinning phenomenon with a phase transition, the calculation prescntcd here for the thl-eshold field is analogous to the determination of the transition tcmpcraturc of ;I thermal transition. The nature of the corresponding critical cxponcnts requires ;I different approach that will be discusscd in the following paper [l-1]. In section 2 we describe the microscopic origin of the dynamical model and the meaning of its parameters. In section 3 the quasi-harmonic approximation is introduced for the characterization of the properties of a static configuration. It is shown how the statistical properties of a pinned configuration can be derived by ;I set of non linear self-consistent equations for the correlation coefficients. In section 4 WC generalize the theory to include an applied field and discuss the calculation of the threshold field. In section 5 the properties of the threshold field and the corresponding phase diagram are discussed and compared with the available computer simulations. In section 6 we summarize the results and discuss some conclusions.
L. Pietronero,
M. Vrrsteeg
3
I Threshold field for depinning transition of CDW
2. Hamiltonian for the phase dynamics of a charge density wave with random pinning The electron
density
case of a Peierls
p(x) =
for a one-dimensional
instability
po{1 +
can be written
c cos[2k,x
charge
density
wave (CDW)
in
as [I, 7,8]
- cp(X)]} ?
(2.1)
where p,, of the average electron density, k, is the Fermi momentum, C is the amplitude of the modulation and the phase q(x) denotes the position of the CDW.
Note
previous direction
that
the
phase
is defined
papers [7,8]. This is to have in x and cp space.
with
opposite
an applied
sign
field that
with
respect
to
acts in the same
The system described by eq. (2.1) has two types of elementary excitations: the modulations of the amplitude C and those of the phase q(x). Usually one is interested correspond the phase given by
in the phase modulations because they contribute to the current and to the lowest excitation energies [l]. The effective Hamiltonian for dynamics has been derived by various authors [ 1,5,6,17] and it is
(2.2) where p,,, is the effective constant. Both quantities
mass density of the CDW and K is an elastic force can be related to microscopic parameters [7,8, 171.
There are various sources of pinning for a CDW: commensurability, threedimensional coupling and charged impurities that usually give rise to the main effect [l, 81. We then consider impurity potentials located at random positions x, and acting only at these positions. V,, is the intensity of this interaction and the index j runs over the impurities whose (one-dimensional) density is yli. The pinning
potential
is then
(2.3)
The dynamics of a CDW is damped by various mechanisms. We assume here a phenomenological damping acting only at the impurity points because it gives rise to important mathematical simplifications. However, the dynamics of the CDW should not depend very much on whether the damping is assumed to act on the volume or at the impurity points. The damping force is then written as
1
L. Pirlronero.
___ ’ (2kJ
F,,=-
where
I Threshold
!?!zc 2
qx
T
applied
electric
-
x,)
field for dcpinning
trarwilior~ of c‘tl W
,
(2.4)
dt
,
T is a phenomenological
An c<
M. Ver.wxg
field
parameter. E’ gives
rise to a potential
energy
density
(for
1)
corresponding to a uniform shift of the CDW. Usually the dynamics of a CDW is overdamped
in the sense that the kinetic
term can be neglected with respect to the effect of the damping (1. 7,8]. In addition the equation of motion can be integrated exactly between two subsequent impurity sites. This leads to a difference equation for the values of the phases
4, at the impurity
sites [7,8].
-sin(&)+
Here
dimensionless
parameters
Q,,=2k,ln,, Q,=~(u,.,-u, electric field E is normalized consrunt
been
introduced
EQ,.
such
(2.6)
that
14, =
s,H,.
,) and the time has also been normalized. The by 2k,Cl/;,n, and B is the ejyective coupling
of the problem,
K
‘l,
B=
essentially
by the ratio of the elastic energy
(4). Defining
a random
the spacing
of impurities
2&Ldt
(2.7)
V, ’
ep,,C(2k$ defined
have
n,)-
pinning
phase
x, = 2k,x,
both in the elastic
(K) and the pinning
and neglecting
and external
B(4,+, - 24, + 4, ,) - sin(4, - x,) + E
energy
randomness
field terms,
in
we obtain (2.8)
This is the basic equation of motion that we are going to study. It describes the phases 4, of the CDW at the impurity positions. These phases are coupled by a Laplacian field that competes with the pinning terms characterized by the quenched random variables x,. In summary the model of eq. (2.8) corresponds to an elastic string pinned by impurity potentials with quenched random phases x, and subject to an external field as schematically shown in fig. 1. It should be
L. Pietronero,
M. Versteeg I Threshold field for depinning transition of CDW
5
Q)
b)
I 2Tr ‘p(x)
0
-2nt __A VI
Y “2
v3
vk
v5
Fig. 1. Schematic picture of the phase dynamics of a charge density wave pinned by randomly located impurities: (a) refers to the effect of a single impurity potential while (b) gives a more realistic idea of the dynamics in the presence of many impurity potentials with quenched random phases.
noted that our definitions in eq. (28) are slightly different from those of previous papers. In particular the parameter B here differs by a factor 2n from the B of refs. [7,8,12,13].
3. Quasi-harmonic approximation for zero field We start by considering the static properties corresponding to eq. (2.8) for the case E = 0. The statistical properties of the ground state of the CDW imply an average over the configurations of the quenched random variables { xj}. The resulting state will consist in a frustrated compromise between the elastic force and the quenched disorder of the pinning potentials.
The random probability
variables
(x,}
are uncorrelatcd
and characterized
by
uniform
a
distribution
Wx,) =
&
It is convenient
(.3. I )
--7is)(‘,52-_,
3
to introduce
the variable (3.2)
4, = 4, - x, that gives the deviation from the minimum of the pinning equation (2.8) with E = 0 gives then The equilibrium
potential
at the site j.
(3.3)
H( x, , , - 3,y, + ,y, , ) + B(r& , , - 21/j, + I/I,, , ) - sin (I/, + E = II The state of the system which
the average
can be characterized
is taken
over the disorder
in terms of average
quantities
in
configurations
(3.4) where
N is the total
mean-square
number
fluctuation
($f>
self-consistent equations. For the quenched random
of pinning
centers.
One
such
;I
quantity
for which we arc now going to construct variables
i\ the a set of
we have
(3.5)
The
diffculty
linearity
in treating
this randomness
due to the sinusoidal
force.
in cq.
At this point
(3.3)
comes
we introduce approximation
from
the non-
approxi(QHA) for an
mation that is inspired to the quasi-harmonic anharmonic vibrations [IX]. The analogy is. however. on1y mathematical because in that case the fluctuations are due to the temperature while in the present problem they are due to the quenched impurity disorder. The approximation consists in making the ansatz that the fluctuations of I/+ can be described by a Gaussian functional behavior
(3.6)
L. Pietronero,
M. Versteeg I Threshold field for depinning transition of CD W
I
in which the variance
oJz=t*:>
(3.7)
is the same for all sites. This allows to linearize because, for example
the equilibrium
equation
(3.8) The effect of this approximation scheme is, therefore, to replace the sine term by its argument multiplied by the factor exp(- $w’). In order to use this method in practice we have also to assume that -,G*jSrr.
(3.9)
This assumption is necessary to restrict the problem to the stability of a single CDW configuration for each realization of the disorder. At least in the limit of strong pinning this limitation should not be too severe. However, a more general approach should actually include all stable configurations. We can now define the following correlation coefficients: (X,Xj+i)
=
f6,n2
(X,X,+,>
=
(X,+iQi)
(3.10)
3
=
al
(3.11)
3
(#;(c;+i>= br .
(3.12)
In order to derive a self-consistent
equation for b,, we can proceed as follows. ($+, -x,+,), (+,+? - x,+~), . . and average, using the QHA. This gives the set of equations
Multiply eq. (3.3) with ($, -x,),
6,
-2b,,
b,,b r-,
+ $T’
fn’-2b, -2b,
+b,+,
+ b, - J(b,,
- a,,) =
+bZ-J(bl
-a,)=~,
-
J(b;
- a;)
=
0,
0,
(3.13) (3.14) i>l,
(3.15)
where the effective pinning force J is defined by J=
$ exp(-
+u”)
(3.16)
X
L.
We then
Pi~tronero.
repeat
M.
Versteeg
the operation
I
Threshold
for
depinrkg
with ,Y,, x, ~, , ,Y,+?, .
trmmitiot~
of C‘D W
to obtain
U, - 2a,, + :rr’ + (1, - Ju,, = 0 .
(3.17)
u,,+
(3.18)
f&2a,
+0-J/u,
=o.
(lI, I - 2a, + N, + , - Ja, = 0 , The general
solution
cr,,+
where
c -I” .
;J+
choosing
y > 0.
The
requirement
The general h, =
[r
is (3.20) (3.21)
constants
and
1,
(3.22)
that a, must
this and eqs. (3.17), al, = iI-
and (3.19)
i>O.
and q- are arbitrary
cash p=
(3.19)
.
+rr2=q+-q
q’
i>l.
of eqs. (3.18)
II, = CJ e’/” + q
From
,field
remain
(3.20)-(3.22)
hnitc
for large i implies
the value
that q * = 0.
of CI,,can be computed: (3.23)
+ vm]7T’. solution
of eqs. (3.14)
’ - iq’Jl(e”
and (3.15)
is then
- e -“)]e’” + [r + iq- J/(e”
~ e “)]e -” ,
i >o. (3.24)
h,,-
$&r+
+r
,
(3.25)
where r- and r arc arbitrary constants. Again, b, should remain finite for large i and since q’ = 0, also r’ = 0. From this and eqs. (3.13). (3.21)-(3.25) we can finally obtain
(3.26)
The two equations (3.16) and (3.26) form a set of self-consistent equations for the variable w’ that can be solved numerically. The solution is shown in fig. 2 and it gives the value of the fluctuation w = ((cl’) I” as a function of the
L. Pietronero,
M. Versteeg I Threshold field for depinning transition of CDW
9
Fig. 3.
Fig. 2
Fig. 2. Behavior of the average fluctuation of the CDW phase from the bottom of the impurity potentials 0 = ( I+$:)I” computed analytically as a function of the coupling constant B. These results refer to the static configurations of the CDW without applied field (E = 0). Fig. 3. Behavior of the elastic (H,), pinning pinned configurations discussed in fig. 2.
(HP,,) and total energy
(H,,,)
as functions
of B for the
coupling constant B. Clearly o is small for strong pinning (B < 1) and it grows monotonically for large values of B up to the asymptotic limit (3.27) which corresponds to a rigid string that is not modified by the pinning potentials. Given the value of w we can also compute the various contributions to the energy of the system as a function of B. In fig. 3 we show that the elastic energy (H,), the potential energy (H+,,) and the total energy (H,,,). The asymptotic behavior of the total energy is lim H,,, = -exp(B+=
in’)
(3.28)
and in this limit the elastic energy is zero.
4. Theory of the threshold
field
In this section we generalize the approach to include the effect of an applied field E in eq. (2.8). This induces an average shift p =
($i> .
(4.1)
In order
to retain
WC
to
have
By taking
of cq. (3.8)
P+ tJ -- (sin(/)
= E ~ cxp(--
i(o’)
t I/I,)) = fi - (sin Pcos $, + cos /‘sin
(q.
(3.8).
the term
in ccl. (2.X) can now he written
terms
sin $, becomes
E’IB
air
I//,
up to first order ~‘12). Therefore, +!J,exp(p
E - sin( P + I,//,) = - B./I& .
~
I/I,))
(1.J)
the term cos +, is replaced
.I = j/\/cxp(+fJ’)
I/),
From this follows
ti: -~ (sin I’ t $, ) = I< ~ sin I’ cos t/j, - cos I’sin
and
and
(42)
J(o’) .
The field and pinning
In the QHA.
x,
for the static case WC obtain
sin I’.
where we have used the QHA sin P = E exp(-
of the varinhlcs
4’/,as follows:
the average
0= I’-?/‘+
and the distributions
the meaning
redefine
in II/) by cxp(- w’i3) ccl. (1.6) reduces to (4.7) (4.S)
This means that if WC‘simply replace .I in ccl. (3.26) hy the present value (ccl. (4.8)) WC can gcneralizc the whole calculation of section 3 for the case in which an applied field E is present. The numerical solution of this self-consistent \et of equation therefore gives the value of w for any value of E and R. For large values of’ E one can observe that for a certain characteristic value Et the expression of J (cq. (4.X)) b ecomcs imaginary and it is not possible to obtain real solutions for CO?. This value is identified with the threshold field above which the CDW cannot stabilize in a static configuration, so it becomcs unpinned and it begins to slide and carry a finite current. The values of OJ as a function of E for different values of B are reported in fig. 4. One can see that the applied field E enhances the value of w at ;I given
L. Pietronero,
M. Versteeg I Threshold field for depinning transition of CDW
2
II
"'El
co i 1
as a function of the applied field for different Fig. 4. Fluctuations of the phase (o = ($s)“‘) values of the parameter R. The curves stop when there are no more real solutions for o. This implies the absence of self-consistent static solutions for this problem and it defines the threshold field EL. One may note that there is evidence for a divergence of dw/dE at the end points of these curves.
B. This is not surprising where the sine pinning
since the field pushes the phase towards force vanishes, This gives rise therefore
the point to larger
fluctuations of &. It can also be remarked that w remains about constant for a large range of E and then it suddenly disappears. Here we may note that at the end points of the curves there is evidence that do/dE diverges. For small values of B an analytical expression can be derived w = B7rj/2/(1This method pinned-unpinned section.
5. Properties In the maximum
E’) .
therefore phase
(4.9)
allows to derive the threshold field and to define the diagram that will be discussed in detail in the next
of the threshold
field and phase diagram
previous section we have found that, for a given B, there is a value of E above which no real solutions for w can be obtained by
the self-consistent equations. This is a sign of an instability of our starting state that is the static equilibrium configuration of the CDW. We can interpret this critical value E, as the threshold field above which there is no more static equilibrium and the CDW begins to slide. Mathematically such an instability condition is analogous to the one by which we discussed the instability of the surface of a solid [lg] also treating anharmonicity by the QHA with respect to thermal fluctuations. The computed values of EC as a function of the coupling constant B are shown in fig. 5. These values define the pinned-unpinned phase diagram of the
12
L.
I’irtrortero.
M.
Vervtreg
I
7%reshold
field
for
depinning
trumitiort
of C‘D W
expb&6) 0 Fig.
i.
Behavior
approach. good
of the threshold
field
EL as a function
This curve detincx the pinned-unpinned
agreement
intermediate
-B
with
pinning.
as B in previous
the
computer
We also indicate
papers
simulations
of U a5 computed
phase diagram (bars)
tor
the
at the top the coupling
from
for this problem
our analytrcal and it is in very
ca5c
constant
(small
W) and
R = 7~13 that was detined
17. X].
system and are compared with the values of E, previously derived by direct computer simulations (bars) [7,8]. The very good agreement with these simulations confirms that our approximation scheme is quite good for the calculation of Et at least up to B = 2. In fact the approximation scheme is certainly rather accurate for strong pinning (small B), which corresponds to small fluctuations of the phase from the minimum of the pinning potential. Instead, for weak pinning (large B) the Gaussian ansatz for the fluctuations is less accurate
and the asymptotic
behavior
(fig. S),
lim E, = exp( - i nTr2), /3j-1 for large B is inaccurate,
[81. It is field.
(5.1)
because
actually rather simple In fact, by differentiating
in this limit the value of E, should
go to zero
to find analytical expressions for the threshold eqs. (4.8) and (3.26) one obtains
(5.2) and
L. Pietronero,
M. Versteeg
I Threshold field for depinning transition of CDW
-20.1 f& exp(-W2)
.
- 2E
13
(5.3)
Combining eqs. (5.2) and (5.3) leads to the result do o dE [B2J3’*(J
+ 4) 5’2 - 2~~‘exp(-w”)]
As discussed in a previous section, dwldE that at this point B2J3’2(J
+ 4)5’2 = 27r2 exp(-w”)
= 2~r’E.
(5.4)
is infinite at the threshold field, so
.
(5.5)
This relation, together with eqs. (3.26) and (4.8) defines the threshold field E, and the values of J and o as functions of B. From this it is possible to compute the asymptotic behaviors for small and large values of B: E,=I-TV?B
(small B) ,
E, = exp(-
an*) + ~IT~‘~~-‘“‘~B-*‘” exp(-
= 0.1930 + 0.05078 B-2’3
(5.6) AT*)
(large B).
(5.7)
From the definition of the coupling constant B (eq. (2.7)) and the definition of the dimensionless field E (section 2) we can define the dependence of the real threshold field Ei on the impurity concentration n,. We obtain EL = an, - bnf
(small ni) ,
EL 2: O.l93an, + ~n;‘~
(5.8)
(large ni) ,
(5.9)
where a, b and c are constant, independent of ni. This behavior is schematically reported in fig. 6. Given the nature of our approximations, these results should apply basically to the case of strong pinning (small B). In fact for weak pinning (large B) a
-+ni Fig. 6. Behavior of the threshold field as a function of the impurity our analytical approach for the one-dimensional Hamiltonian.
concentration
as derived
from
variational suggest
argument
gives
~5:.= H: ’ [S, X]. The
experimental
results
instead
E{ = tl:’ with cy between I and 2 [ 191. It is possible. this observed behavior cannot be explained by ;I purely onu-
a dependence
however,
that
dimensional
model.
because
it may be affected
to two- and three-dimensional
pinned
impurities
couplings.
6. Summary
and conclusions
The properties
of ;I CDW
by random
can
he
dcscribcd
by
an elastic string pinned by random sinusoidal impurity potentials. If an external field is applied, one observes that above a critical value, the threshold field, the CDW depins and it begins to move coherently. Below this tield the CDW configurations are characterized by many metastable states and show relaxation properties and memory effects analogous to those of glassy systems. In this paper we have presented ;I simple analytical approach in which
;I
pinned contiguration is characterized by the average of the fuctuations of the phase from the bottom of the impurity potential. The analysis is restricted to a single CDW configuration for each realization of the random variables. By making the ansatz of Gaussian distribution for the fluctuations. it is possible to compute the properties of these tluctuations from the solution of :I set ot self-consistent equations for the correlation coefticicnts. The approach can be generalized to the cast of an applied field and the breakdown of self-consistency for the static solutions allows to characterize the threshold ficld. The behavior of the threshold held as a function of a coupling constant of the model defines the pinned-unpinned phase diagram. The approximation scheme is rather appropriate for the cast the results are in good agreement with the computer intermediate
of strong pinning but simulations also for
pinning.
The calculation of the threshold field for this problem is analogous to that of the transition temperature for ;I thermal phase transition. In analogy with thermal critical phenomena also the dcpinning transition gives rise to :I diverging correlation length and to critical exponents. Since the nature of the present theoretical approach is mean field (even though it includes some correlations). it allows to compute the threshold field rather accurately. but it is not suitable for the analysis of the exponents that characterize the critical behavior. In the following paper [ 141 we will present a scaling theory that allows to determine these exponents and the stretched cxponcntial relaxation below threshold. These exponents arc in very good agreement with various computer simulations in various dimensions and they allow to clarify the nature of the critical properties of the depinning transition.
L. Pietronero,
M. Versteeg I Threshold field for depinning transition of CDW
15
References (11 H. Fukuyama and H. Takayama, in: Electronic Properties of Inorganic Quasi One Dimensional Materials, P. Monceau. ed. (Reidel, Dordrecht, 1985) p. 41. P. Monceau, in: Electronic Properties of Inorganic Quasi One Dimensional Materials, P. Monceau, ed. (Reidel, Dordrecht. 1985) p. 139. [2] G. Gruner, Rev. Mod. Phys. 60 (1988) 1129. P. Monceau, in: Application of Statistical and Field Theory Methods to Condensed Matter, D. Baeriswyl et al., eds., NATO AS1 Series (Plenum, New York. 1990) p. 357. [3] Z.Z. Wang and N.P. Ong. Phys. Rev. Lett. 58 (1987) 2375. [4] G. Kriza and G. Mihaly, Phys. Rev. Lett. 56 (1986) 2529. [5] H. Fukuyama and P.A. Lee, Phys. Rev. B 17 (1978) 535. P.A. Lee and T.M. Rice, Phys. Rev. B 19 (1979) 3970. (61 M.V. Feigel’man and V.M. Vinokur, in: Charge Density Waves in Solids, L.P. Gorkov and G. Gruner, eds. (North-Holland, Amsterdam 1989) p. 293. R.A. Klemm and J.R. Schrieffer, Phys. Rev. Lett. 51 (1983) 47. [7] L. Pietronero and S. Strassler, Phys. Rev. B 28 (1983) 5863. (81 L. Pietronero. in: Highlights of Condensed-matter Theory, F. Bassani, F. Fermi and M.P. Tosi. eds. (North-Holland, Amsterdam, 1985) p. 442. [9] J.B. Sokoloff, Phys. Rev. B 23 (1981) 1992; 31 (1985) 2270. [lo] P. Sibani and P.B. Littlewood, Phys. Rev. Lett. 64 (1990) 1305. P.B. Littlewood, in: Charge Density Waves in Solids, L.P. Gorkov and G. Gruner. eds. (North-Holland, Amsterdam, 1989) p. 321. [ll] A.A. Middleton and D.S. Fisher, Phys. Rev. Lett. 66 (1991) 92. [12] A. Erzan, A. Veermans, R. Heijungs and L. Pietronero, Phys. Rev. B 41 (1990) 11522. [13] A. Erzan, E. Veermans, R. Heijungs and L. Pietronero, Physica A 166 (1990) 447. [14] G. Parisi and L. Pietronero, Physica A 179 (1991) 16, following paper, this volume. [15] G. Gruner, A. Zawadowski and P. M. Chaikin, Phys. Rev. Lett. 46 (1981) 511. [16] D.S. Fisher, Phys. Rev. B 31 (1985) 1396. R.K. Pitala and J.A. Hertz, Phys. Ser. 34 (1986) 264. [17] L. Pietronero, S. Strassler and G.A. Tombs, Phys. Rev. B 12 (1975) 5213. [18] L. Pietronero and E. Tosatti, Solid State Commun. 32 (1979) 255. [19] N.P. Ong., J.W. Brill, J.C. Eckert, J.W. Savage, S.K. Khanna and R.B. Somoano, Phys. Rev. Lett. 42 (1979) 811; Phys. Rev. B 23 (1981) 1517.
1-15
PHYSICA Iii
Theory of the threshold field for the depinning transition of a charge density wave L. Pietronero Dipartimento di Fisica, Universitci di Roma “La Sapienza”, 00185 Rome, Italy
Piazzale A. Moro 2,
M. Versteeg Solid State Physics Laboratory, The Netherlands
Received
University of Groningen,
Melkweg 1, 9718 EP Groningen,
30 May 1991
The Hamiltonian of an elastic string pinned by random potentials is often used to describe the depinning transition of a charge density wave in the presence of impurity pinning. The properties of the pinned states show close analogies to those of glassy systems, while the depinning transition resembles a dynamical critical phenomenon. Here we develop an approximate analytical approach based on the quasi-harmonic treatment of the nonlinear pinning potentials together with an expression of the correlation functions to lowest order. This method allows a nonperturbative treatment of a problem with many nonlinear degrees of freedom and quenched disorder. In this way it is possible to characterize the pinned configurations and to compute the threshold field as a function of the coupling constant of the model. The results are in very good agreement with the computer simulations for strong and intermediate pinning and provide an approximate but quantitative method for the determination of the pinned-unpinned phase diagram.
1. Introduction
A remarkable series of experiments on quasi-one-dimensional metals like NbSe, and similar systems have provided evidence for a contribution to nonlinear transport due to the coherent sliding of a charge density wave (CDW) [l, 21. Ab ove a threshold electric field the CDW seems to depin and to contribute to the electric current. The depinning phenomenon resembles a second-order phase transition because it shows critical behavior and a correlation length that diverges [3]. Below this threshold the system shows a number of interesting properties characteristic of glassy systems like hysteresis effects [2] and stretched exponential relaxation [3,4]. A model Hamiltonian that is often used to describe several properties of 037%43711911$03.50
0
1991-
Elsevier
Science
Publishers
B.V. (North-Holland)
these
systems
random duct
consists
various
stretched prcscnt
in the overdamped
[ 1. 5-S].
impurities
properties
exponential a simple
USC ;I different
of the
relaxation
analytical
and of the corresponding will
Numerical
depinning below
approach
of the C’DW pinned of this model
the threshold
to derive
as the
of the threshold paper
results
properties and the critical behavior at dcpinning. The Hamiltonian model is one-dirncnsional and it consists
[ 131 instead
for the
I~>
rcpro-
[ 12, 131. In this paper
In the following analytical
could
[7-l I ] as well
transition
for the calculation
phase diagram.
approach
dynamics
simulations
\ve field we
relaxation
of many
degrees
of freedom coupled elastically between them and interacting in a nonlinear wa> with randomly located pinning centers. From a theoretical point of vie\+ the combination of nonlinear interactions and quenched disorder gives rise to ;I rather complex problem. The corresponding single-particle problem can be easily solved [ 151 but the mean-field type arguments [ Ih] developed for the coupled problem appear to bc ovcrsiniplificd and difficult to relate to the original Hamiltonian. In this paper we present a new approach to the problem based on ;I Gaussian ansatz for the phase fluctuations whose variance provides a characterization of the pinned state. This variance can bc computed from the solutions of ;I set of nonlinear threshold
self-consistent equations that allow also the determination field as a function of the Hamiltonian parameters. This
of the method
therefore, allows, ;I rron/,crtllrhrrti~l~~treatment of ;I problem with many nonlinear interacting dcgrecs of freedom in the presence of quenched disorder. The nature of the approximation is rather accurate for the USC of strong pinning, but the results arc in good agreement with the computer simulations also for intcrmcdiate pinning. From the point of view of the analogy of the depinning phenomenon with a phase transition, the calculation prescntcd here for the thl-eshold field is analogous to the determination of the transition tcmpcraturc of ;I thermal transition. The nature of the corresponding critical cxponcnts requires ;I different approach that will be discusscd in the following paper [l-1]. In section 2 we describe the microscopic origin of the dynamical model and the meaning of its parameters. In section 3 the quasi-harmonic approximation is introduced for the characterization of the properties of a static configuration. It is shown how the statistical properties of a pinned configuration can be derived by ;I set of non linear self-consistent equations for the correlation coefficients. In section 4 WC generalize the theory to include an applied field and discuss the calculation of the threshold field. In section 5 the properties of the threshold field and the corresponding phase diagram are discussed and compared with the available computer simulations. In section 6 we summarize the results and discuss some conclusions.
L. Pietronero,
M. Vrrsteeg
3
I Threshold field for depinning transition of CDW
2. Hamiltonian for the phase dynamics of a charge density wave with random pinning The electron
density
case of a Peierls
p(x) =
for a one-dimensional
instability
po{1 +
can be written
c cos[2k,x
charge
density
wave (CDW)
in
as [I, 7,8]
- cp(X)]} ?
(2.1)
where p,, of the average electron density, k, is the Fermi momentum, C is the amplitude of the modulation and the phase q(x) denotes the position of the CDW.
Note
previous direction
that
the
phase
is defined
papers [7,8]. This is to have in x and cp space.
with
opposite
an applied
sign
field that
with
respect
to
acts in the same
The system described by eq. (2.1) has two types of elementary excitations: the modulations of the amplitude C and those of the phase q(x). Usually one is interested correspond the phase given by
in the phase modulations because they contribute to the current and to the lowest excitation energies [l]. The effective Hamiltonian for dynamics has been derived by various authors [ 1,5,6,17] and it is
(2.2) where p,,, is the effective constant. Both quantities
mass density of the CDW and K is an elastic force can be related to microscopic parameters [7,8, 171.
There are various sources of pinning for a CDW: commensurability, threedimensional coupling and charged impurities that usually give rise to the main effect [l, 81. We then consider impurity potentials located at random positions x, and acting only at these positions. V,, is the intensity of this interaction and the index j runs over the impurities whose (one-dimensional) density is yli. The pinning
potential
is then
(2.3)
The dynamics of a CDW is damped by various mechanisms. We assume here a phenomenological damping acting only at the impurity points because it gives rise to important mathematical simplifications. However, the dynamics of the CDW should not depend very much on whether the damping is assumed to act on the volume or at the impurity points. The damping force is then written as
1
L. Pirlronero.
___ ’ (2kJ
F,,=-
where
I Threshold
!?!zc 2
qx
T
applied
electric
-
x,)
field for dcpinning
trarwilior~ of c‘tl W
,
(2.4)
dt
,
T is a phenomenological
An c<
M. Ver.wxg
field
parameter. E’ gives
rise to a potential
energy
density
(for
1)
corresponding to a uniform shift of the CDW. Usually the dynamics of a CDW is overdamped
in the sense that the kinetic
term can be neglected with respect to the effect of the damping (1. 7,8]. In addition the equation of motion can be integrated exactly between two subsequent impurity sites. This leads to a difference equation for the values of the phases
4, at the impurity
sites [7,8].
-sin(&)+
Here
dimensionless
parameters
Q,,=2k,ln,, Q,=~(u,.,-u, electric field E is normalized consrunt
been
introduced
EQ,.
such
(2.6)
that
14, =
s,H,.
,) and the time has also been normalized. The by 2k,Cl/;,n, and B is the ejyective coupling
of the problem,
K
‘l,
B=
essentially
by the ratio of the elastic energy
(4). Defining
a random
the spacing
of impurities
2&Ldt
(2.7)
V, ’
ep,,C(2k$ defined
have
n,)-
pinning
phase
x, = 2k,x,
both in the elastic
(K) and the pinning
and neglecting
and external
B(4,+, - 24, + 4, ,) - sin(4, - x,) + E
energy
randomness
field terms,
in
we obtain (2.8)
This is the basic equation of motion that we are going to study. It describes the phases 4, of the CDW at the impurity positions. These phases are coupled by a Laplacian field that competes with the pinning terms characterized by the quenched random variables x,. In summary the model of eq. (2.8) corresponds to an elastic string pinned by impurity potentials with quenched random phases x, and subject to an external field as schematically shown in fig. 1. It should be
L. Pietronero,
M. Versteeg I Threshold field for depinning transition of CDW
5
Q)
b)
I 2Tr ‘p(x)
0
-2nt __A VI
Y “2
v3
vk
v5
Fig. 1. Schematic picture of the phase dynamics of a charge density wave pinned by randomly located impurities: (a) refers to the effect of a single impurity potential while (b) gives a more realistic idea of the dynamics in the presence of many impurity potentials with quenched random phases.
noted that our definitions in eq. (28) are slightly different from those of previous papers. In particular the parameter B here differs by a factor 2n from the B of refs. [7,8,12,13].
3. Quasi-harmonic approximation for zero field We start by considering the static properties corresponding to eq. (2.8) for the case E = 0. The statistical properties of the ground state of the CDW imply an average over the configurations of the quenched random variables { xj}. The resulting state will consist in a frustrated compromise between the elastic force and the quenched disorder of the pinning potentials.
The random probability
variables
(x,}
are uncorrelatcd
and characterized
by
uniform
a
distribution
Wx,) =
&
It is convenient
(.3. I )
--7is)(‘,52-_,
3
to introduce
the variable (3.2)
4, = 4, - x, that gives the deviation from the minimum of the pinning equation (2.8) with E = 0 gives then The equilibrium
potential
at the site j.
(3.3)
H( x, , , - 3,y, + ,y, , ) + B(r& , , - 21/j, + I/I,, , ) - sin (I/, + E = II The state of the system which
the average
can be characterized
is taken
over the disorder
in terms of average
quantities
in
configurations
(3.4) where
N is the total
mean-square
number
fluctuation
($f>
self-consistent equations. For the quenched random
of pinning
centers.
One
such
;I
quantity
for which we arc now going to construct variables
i\ the a set of
we have
(3.5)
The
diffculty
linearity
in treating
this randomness
due to the sinusoidal
force.
in cq.
At this point
(3.3)
comes
we introduce approximation
from
the non-
approxi(QHA) for an
mation that is inspired to the quasi-harmonic anharmonic vibrations [IX]. The analogy is. however. on1y mathematical because in that case the fluctuations are due to the temperature while in the present problem they are due to the quenched impurity disorder. The approximation consists in making the ansatz that the fluctuations of I/+ can be described by a Gaussian functional behavior
(3.6)
L. Pietronero,
M. Versteeg I Threshold field for depinning transition of CD W
I
in which the variance
oJz=t*:>
(3.7)
is the same for all sites. This allows to linearize because, for example
the equilibrium
equation
(3.8) The effect of this approximation scheme is, therefore, to replace the sine term by its argument multiplied by the factor exp(- $w’). In order to use this method in practice we have also to assume that -,G*jSrr.
(3.9)
This assumption is necessary to restrict the problem to the stability of a single CDW configuration for each realization of the disorder. At least in the limit of strong pinning this limitation should not be too severe. However, a more general approach should actually include all stable configurations. We can now define the following correlation coefficients: (X,Xj+i)
=
f6,n2
(X,X,+,>
=
(X,+iQi)
(3.10)
3
=
al
(3.11)
3
(#;(c;+i>= br .
(3.12)
In order to derive a self-consistent
equation for b,, we can proceed as follows. ($+, -x,+,), (+,+? - x,+~), . . and average, using the QHA. This gives the set of equations
Multiply eq. (3.3) with ($, -x,),
6,
-2b,,
b,,b r-,
+ $T’
fn’-2b, -2b,
+b,+,
+ b, - J(b,,
- a,,) =
+bZ-J(bl
-a,)=~,
-
J(b;
- a;)
=
0,
0,
(3.13) (3.14) i>l,
(3.15)
where the effective pinning force J is defined by J=
$ exp(-
+u”)
(3.16)
X
L.
We then
Pi~tronero.
repeat
M.
Versteeg
the operation
I
Threshold
for
depinrkg
with ,Y,, x, ~, , ,Y,+?, .
trmmitiot~
of C‘D W
to obtain
U, - 2a,, + :rr’ + (1, - Ju,, = 0 .
(3.17)
u,,+
(3.18)
f&2a,
+0-J/u,
=o.
(lI, I - 2a, + N, + , - Ja, = 0 , The general
solution
cr,,+
where
c -I” .
;J+
choosing
y > 0.
The
requirement
The general h, =
[r
is (3.20) (3.21)
constants
and
1,
(3.22)
that a, must
this and eqs. (3.17), al, = iI-
and (3.19)
i>O.
and q- are arbitrary
cash p=
(3.19)
.
+rr2=q+-q
q’
i>l.
of eqs. (3.18)
II, = CJ e’/” + q
From
,field
remain
(3.20)-(3.22)
hnitc
for large i implies
the value
that q * = 0.
of CI,,can be computed: (3.23)
+ vm]7T’. solution
of eqs. (3.14)
’ - iq’Jl(e”
and (3.15)
is then
- e -“)]e’” + [r + iq- J/(e”
~ e “)]e -” ,
i >o. (3.24)
h,,-
$&r+
+r
,
(3.25)
where r- and r arc arbitrary constants. Again, b, should remain finite for large i and since q’ = 0, also r’ = 0. From this and eqs. (3.13). (3.21)-(3.25) we can finally obtain
(3.26)
The two equations (3.16) and (3.26) form a set of self-consistent equations for the variable w’ that can be solved numerically. The solution is shown in fig. 2 and it gives the value of the fluctuation w = ((cl’) I” as a function of the
L. Pietronero,
M. Versteeg I Threshold field for depinning transition of CDW
9
Fig. 3.
Fig. 2
Fig. 2. Behavior of the average fluctuation of the CDW phase from the bottom of the impurity potentials 0 = ( I+$:)I” computed analytically as a function of the coupling constant B. These results refer to the static configurations of the CDW without applied field (E = 0). Fig. 3. Behavior of the elastic (H,), pinning pinned configurations discussed in fig. 2.
(HP,,) and total energy
(H,,,)
as functions
of B for the
coupling constant B. Clearly o is small for strong pinning (B < 1) and it grows monotonically for large values of B up to the asymptotic limit (3.27) which corresponds to a rigid string that is not modified by the pinning potentials. Given the value of w we can also compute the various contributions to the energy of the system as a function of B. In fig. 3 we show that the elastic energy (H,), the potential energy (H+,,) and the total energy (H,,,). The asymptotic behavior of the total energy is lim H,,, = -exp(B+=
in’)
(3.28)
and in this limit the elastic energy is zero.
4. Theory of the threshold
field
In this section we generalize the approach to include the effect of an applied field E in eq. (2.8). This induces an average shift p =
($i> .
(4.1)
In order
to retain
WC
to
have
By taking
of cq. (3.8)
P+ tJ -- (sin(/)
= E ~ cxp(--
i(o’)
t I/I,)) = fi - (sin Pcos $, + cos /‘sin
(q.
(3.8).
the term
in ccl. (2.X) can now he written
terms
sin $, becomes
E’IB
air
I//,
up to first order ~‘12). Therefore, +!J,exp(p
E - sin( P + I,//,) = - B./I& .
~
I/I,))
(1.J)
the term cos +, is replaced
.I = j/\/cxp(+fJ’)
I/),
From this follows
ti: -~ (sin I’ t $, ) = I< ~ sin I’ cos t/j, - cos I’sin
and
and
(42)
J(o’) .
The field and pinning
In the QHA.
x,
for the static case WC obtain
sin I’.
where we have used the QHA sin P = E exp(-
of the varinhlcs
4’/,as follows:
the average
0= I’-?/‘+
and the distributions
the meaning
redefine
in II/) by cxp(- w’i3) ccl. (1.6) reduces to (4.7) (4.S)
This means that if WC‘simply replace .I in ccl. (3.26) hy the present value (ccl. (4.8)) WC can gcneralizc the whole calculation of section 3 for the case in which an applied field E is present. The numerical solution of this self-consistent \et of equation therefore gives the value of w for any value of E and R. For large values of’ E one can observe that for a certain characteristic value Et the expression of J (cq. (4.X)) b ecomcs imaginary and it is not possible to obtain real solutions for CO?. This value is identified with the threshold field above which the CDW cannot stabilize in a static configuration, so it becomcs unpinned and it begins to slide and carry a finite current. The values of OJ as a function of E for different values of B are reported in fig. 4. One can see that the applied field E enhances the value of w at ;I given
L. Pietronero,
M. Versteeg I Threshold field for depinning transition of CDW
2
II
"'El
co i 1
as a function of the applied field for different Fig. 4. Fluctuations of the phase (o = ($s)“‘) values of the parameter R. The curves stop when there are no more real solutions for o. This implies the absence of self-consistent static solutions for this problem and it defines the threshold field EL. One may note that there is evidence for a divergence of dw/dE at the end points of these curves.
B. This is not surprising where the sine pinning
since the field pushes the phase towards force vanishes, This gives rise therefore
the point to larger
fluctuations of &. It can also be remarked that w remains about constant for a large range of E and then it suddenly disappears. Here we may note that at the end points of the curves there is evidence that do/dE diverges. For small values of B an analytical expression can be derived w = B7rj/2/(1This method pinned-unpinned section.
5. Properties In the maximum
E’) .
therefore phase
(4.9)
allows to derive the threshold field and to define the diagram that will be discussed in detail in the next
of the threshold
field and phase diagram
previous section we have found that, for a given B, there is a value of E above which no real solutions for w can be obtained by
the self-consistent equations. This is a sign of an instability of our starting state that is the static equilibrium configuration of the CDW. We can interpret this critical value E, as the threshold field above which there is no more static equilibrium and the CDW begins to slide. Mathematically such an instability condition is analogous to the one by which we discussed the instability of the surface of a solid [lg] also treating anharmonicity by the QHA with respect to thermal fluctuations. The computed values of EC as a function of the coupling constant B are shown in fig. 5. These values define the pinned-unpinned phase diagram of the
12
L.
I’irtrortero.
M.
Vervtreg
I
7%reshold
field
for
depinning
trumitiort
of C‘D W
expb&6) 0 Fig.
i.
Behavior
approach. good
of the threshold
field
EL as a function
This curve detincx the pinned-unpinned
agreement
intermediate
-B
with
pinning.
as B in previous
the
computer
We also indicate
papers
simulations
of U a5 computed
phase diagram (bars)
tor
the
at the top the coupling
from
for this problem
our analytrcal and it is in very
ca5c
constant
(small
W) and
R = 7~13 that was detined
17. X].
system and are compared with the values of E, previously derived by direct computer simulations (bars) [7,8]. The very good agreement with these simulations confirms that our approximation scheme is quite good for the calculation of Et at least up to B = 2. In fact the approximation scheme is certainly rather accurate for strong pinning (small B), which corresponds to small fluctuations of the phase from the minimum of the pinning potential. Instead, for weak pinning (large B) the Gaussian ansatz for the fluctuations is less accurate
and the asymptotic
behavior
(fig. S),
lim E, = exp( - i nTr2), /3j-1 for large B is inaccurate,
[81. It is field.
(5.1)
because
actually rather simple In fact, by differentiating
in this limit the value of E, should
go to zero
to find analytical expressions for the threshold eqs. (4.8) and (3.26) one obtains
(5.2) and
L. Pietronero,
M. Versteeg
I Threshold field for depinning transition of CDW
-20.1 f& exp(-W2)
.
- 2E
13
(5.3)
Combining eqs. (5.2) and (5.3) leads to the result do o dE [B2J3’*(J
+ 4) 5’2 - 2~~‘exp(-w”)]
As discussed in a previous section, dwldE that at this point B2J3’2(J
+ 4)5’2 = 27r2 exp(-w”)
= 2~r’E.
(5.4)
is infinite at the threshold field, so
.
(5.5)
This relation, together with eqs. (3.26) and (4.8) defines the threshold field E, and the values of J and o as functions of B. From this it is possible to compute the asymptotic behaviors for small and large values of B: E,=I-TV?B
(small B) ,
E, = exp(-
an*) + ~IT~‘~~-‘“‘~B-*‘” exp(-
= 0.1930 + 0.05078 B-2’3
(5.6) AT*)
(large B).
(5.7)
From the definition of the coupling constant B (eq. (2.7)) and the definition of the dimensionless field E (section 2) we can define the dependence of the real threshold field Ei on the impurity concentration n,. We obtain EL = an, - bnf
(small ni) ,
EL 2: O.l93an, + ~n;‘~
(5.8)
(large ni) ,
(5.9)
where a, b and c are constant, independent of ni. This behavior is schematically reported in fig. 6. Given the nature of our approximations, these results should apply basically to the case of strong pinning (small B). In fact for weak pinning (large B) a
-+ni Fig. 6. Behavior of the threshold field as a function of the impurity our analytical approach for the one-dimensional Hamiltonian.
concentration
as derived
from
variational suggest
argument
gives
~5:.= H: ’ [S, X]. The
experimental
results
instead
E{ = tl:’ with cy between I and 2 [ 191. It is possible. this observed behavior cannot be explained by ;I purely onu-
a dependence
however,
that
dimensional
model.
because
it may be affected
to two- and three-dimensional
pinned
impurities
couplings.
6. Summary
and conclusions
The properties
of ;I CDW
by random
can
he
dcscribcd
by
an elastic string pinned by random sinusoidal impurity potentials. If an external field is applied, one observes that above a critical value, the threshold field, the CDW depins and it begins to move coherently. Below this tield the CDW configurations are characterized by many metastable states and show relaxation properties and memory effects analogous to those of glassy systems. In this paper we have presented ;I simple analytical approach in which
;I
pinned contiguration is characterized by the average of the fuctuations of the phase from the bottom of the impurity potential. The analysis is restricted to a single CDW configuration for each realization of the random variables. By making the ansatz of Gaussian distribution for the fluctuations. it is possible to compute the properties of these tluctuations from the solution of :I set ot self-consistent equations for the correlation coefticicnts. The approach can be generalized to the cast of an applied field and the breakdown of self-consistency for the static solutions allows to characterize the threshold ficld. The behavior of the threshold held as a function of a coupling constant of the model defines the pinned-unpinned phase diagram. The approximation scheme is rather appropriate for the cast the results are in good agreement with the computer intermediate
of strong pinning but simulations also for
pinning.
The calculation of the threshold field for this problem is analogous to that of the transition temperature for ;I thermal phase transition. In analogy with thermal critical phenomena also the dcpinning transition gives rise to :I diverging correlation length and to critical exponents. Since the nature of the present theoretical approach is mean field (even though it includes some correlations). it allows to compute the threshold field rather accurately. but it is not suitable for the analysis of the exponents that characterize the critical behavior. In the following paper [ 141 we will present a scaling theory that allows to determine these exponents and the stretched cxponcntial relaxation below threshold. These exponents arc in very good agreement with various computer simulations in various dimensions and they allow to clarify the nature of the critical properties of the depinning transition.
L. Pietronero,
M. Versteeg I Threshold field for depinning transition of CDW
15
References (11 H. Fukuyama and H. Takayama, in: Electronic Properties of Inorganic Quasi One Dimensional Materials, P. Monceau. ed. (Reidel, Dordrecht, 1985) p. 41. P. Monceau, in: Electronic Properties of Inorganic Quasi One Dimensional Materials, P. Monceau, ed. (Reidel, Dordrecht. 1985) p. 139. [2] G. Gruner, Rev. Mod. Phys. 60 (1988) 1129. P. Monceau, in: Application of Statistical and Field Theory Methods to Condensed Matter, D. Baeriswyl et al., eds., NATO AS1 Series (Plenum, New York. 1990) p. 357. [3] Z.Z. Wang and N.P. Ong. Phys. Rev. Lett. 58 (1987) 2375. [4] G. Kriza and G. Mihaly, Phys. Rev. Lett. 56 (1986) 2529. [5] H. Fukuyama and P.A. Lee, Phys. Rev. B 17 (1978) 535. P.A. Lee and T.M. Rice, Phys. Rev. B 19 (1979) 3970. (61 M.V. Feigel’man and V.M. Vinokur, in: Charge Density Waves in Solids, L.P. Gorkov and G. Gruner, eds. (North-Holland, Amsterdam 1989) p. 293. R.A. Klemm and J.R. Schrieffer, Phys. Rev. Lett. 51 (1983) 47. [7] L. Pietronero and S. Strassler, Phys. Rev. B 28 (1983) 5863. (81 L. Pietronero. in: Highlights of Condensed-matter Theory, F. Bassani, F. Fermi and M.P. Tosi. eds. (North-Holland, Amsterdam, 1985) p. 442. [9] J.B. Sokoloff, Phys. Rev. B 23 (1981) 1992; 31 (1985) 2270. [lo] P. Sibani and P.B. Littlewood, Phys. Rev. Lett. 64 (1990) 1305. P.B. Littlewood, in: Charge Density Waves in Solids, L.P. Gorkov and G. Gruner. eds. (North-Holland, Amsterdam, 1989) p. 321. [ll] A.A. Middleton and D.S. Fisher, Phys. Rev. Lett. 66 (1991) 92. [12] A. Erzan, A. Veermans, R. Heijungs and L. Pietronero, Phys. Rev. B 41 (1990) 11522. [13] A. Erzan, E. Veermans, R. Heijungs and L. Pietronero, Physica A 166 (1990) 447. [14] G. Parisi and L. Pietronero, Physica A 179 (1991) 16, following paper, this volume. [15] G. Gruner, A. Zawadowski and P. M. Chaikin, Phys. Rev. Lett. 46 (1981) 511. [16] D.S. Fisher, Phys. Rev. B 31 (1985) 1396. R.K. Pitala and J.A. Hertz, Phys. Ser. 34 (1986) 264. [17] L. Pietronero, S. Strassler and G.A. Tombs, Phys. Rev. B 12 (1975) 5213. [18] L. Pietronero and E. Tosatti, Solid State Commun. 32 (1979) 255. [19] N.P. Ong., J.W. Brill, J.C. Eckert, J.W. Savage, S.K. Khanna and R.B. Somoano, Phys. Rev. Lett. 42 (1979) 811; Phys. Rev. B 23 (1981) 1517.