- Email: [email protected]
Development of a tunnelling model of charge-density-wve depinning
Synthetic Metals, 19 (1987) 1-6
DEVELOPMENT
1
OF A TUNNELING MODEL OF CHARGE-DENSITY-WAVE
DEPINNING
JOHN BARDEEN Department Urbana,
of Physics,
IL 6|801
University
of Illinois at Urbana-Champaign,
(U.S.A.)
J.W. LYDING, W.G. LYONS, J.H. MILLER, Jr., R.E. THORNE and J.R. TUCKER Department
of Electrical
Laboratory,
University
and Computer Engineering
of Illinois,
Urbana,
and Coordinated
IL 61801
Science
(U.S.A.)
ABSTRACT The tunneling model of transport by charge-density dimensional metals has been developed and experiment. condensate suggested
Deplnning
by the field dependence
initiated at Illinois successful
from a close interaction between theory
by coherent
in many parallel chains,
Zener tunneling of electrons
characteristic.
of the dc current.
to study detection,
Reagor
for a wide
et al. of a(~) in the
to account for CDW metals with high
such as (TaSe4)21 and the alloys Tal_xNbxS3.-
low-frequency
ac response
can be derived from parameters
frequencies.
The theory accounts
measured
The de and
measured at high
for the "narrow band" noise that accompanies
dc current flow and the subharmonic
DEVELOPMENT
tunneling theory accounts
by Gr~ner,
range i-I00 GHz suggested a modification
characteristic
In 1981, a program
mixing and harmonic generation was
The only inputs are a scaling parameter and the dc I-V
Recent measurements
pinning frequencies,
in the CDW
first applied to NbSe 3 and TaS3, was
in showing that photon-assisted
range of phenomena.
waves in quasi-one-
steps of constant current in the dc I-V
in the presence of an applied ac.
OF CONCEPTS
Recent observations
indicate that Fr~hlich conduction by moving charge-
density waves occurs in the organic transfer salt TTF-TCNQ below the Peierls transition [I]; Since the phenomena may occur in other organic metals, worthwhile
to review the background
of FrShllch conduction as developed
for inorganic metals.
Normally pinned to the lattice by impurities incommensurate
(DW's gradually become depinnned
0379-6779/87/$3.50
it may be
and current status of the tunneling theory
or lattice imperfections, and move through the lattice
© Elsevier Sequoia/Printed in The Netherlands
2 when a threshold
field is exceeded and contribute
to the current flow.
Following the discovery of FrShllch conduction in NbSe 3 in 1975, many other inorganic quasi-one-dlmensional
metals have been found that exhibit the
phenomena. Competing theories, systems[2]
one based on treating
and another as massive
freedom have been developed. development
CDW metals as macroscopic
classical systems[3] with internal degrees of
The quantum approach has undergone
in both theory and experiment,
particularly
considerable
during the past year, so
that it is now capable of accounting for a wide range of GDW phenomena quantitative
quantum
detail with a theory based on only a few measurable
While we will not go into the details of the derivation, results and how they compare with recent experiment.
in
parameters.
we will describe
It is believed
the
that
remaining problems will be resolved within the quantum approach. It was not until the remarkable
field- and frequency-dependent
was observed below Tp in NbSe 3 by Monceau, conduction was discovered. approximately
They found that the dc current could be expressed
in the form i = OaE + ObE exp[-Eo/E],
where E is the electric
field and the second term is the CDW contribution.
The exponential
suggested Zener tunneling through a small pinning gap. account for CDW depinnlng
conductivity
Ong and Portis that true Fr&~llch
form
Early attempts
to
on this basis, or by creation of solltons in an
electric field, were abandoned when it was found the required energy gap is ~ IO-4kBT. In 1979 it was pointed out that such theories are viable if the CDW is a macroscopic
quantum system with one thermal degree of freedom in a volume
containing perhaps
106 parallel
atomic chains of atoms.
Tunneling
electrons would then add coherently in this volume, as do electrons through an insulating
layer in the Josephson
relate the depinning field,
effect.
were made of o(~) in the megahertz
that if tunneling is involved, tunneling
tunneling
An attempt was made to
Eo, with the pinning frequency.
The expression
given for E o is close to that in the present version of the theory. after measurements
of individual
In 1980,
region, it was suggested
it should be possible to apply photon-asslsted
(PAT) theory to derive the ac response from the de.
One consequence
is that o(~) should scale with idc/E , so that the CDW response is o(~) = abexp(-~s/m),
where u s is the scaling frequency.
A program was initiated in 1981 to study detection, and harmonic generation
to test predictions
found to be in good quantitative
mixing,
of the PAT theory.
or semi-quantltative
harmonic mixing Results were
agreement when biased
above threshold or if the frequency to be detected
is above a critical value.
In some cases results were qualitatively
from those expected from
classical models.
A phenomenological
range of phenomena when the frequency
different
theory was developed
to account for a
is lower than the critical value, called
the ac-dc decoupling
frequency.
In this region the expected
compensated
at least in part by polarization
THEORETICAL
BACKGROUND
When a CDW moves with a drift velocity, to -kF+q, kF+q, where Nq = my d. the current
response
is
fields.
Vd, the ID Fermi surface, ~kF, moves
The Peierls gaps move with the Fermi sea and
flow is the same as if the gaps were not present.
The total kinetic
energy is (I/2)(m + MF)V~, where M F >> m is that associated with the moving ions that accompany
the wave.
Changes in electron density ~nd in v d can be described by a phase ~(x,t) such that the electron density
is
(2)
P = Po + 01 c°s[2kFX + ~(x,t)]
For uniform flow, ~ = -~d t, where ~d = 2kFVd is the frequency of the macroscopically difference
occupied phonon that accompanies
the wave.
between opposite sides of the FS is ~ d "
density are proportional
~o(X),
More generally,
changes in
to ~#/Sx and drift velocity to - ~ / ~ t .
In the Fukuyama-Lee-Rice deformations,
The energy
theory of weak impurity pinning,
change the phase to minimize
fluctuations.
The phase is adjusted
deformations
with an average wavelength
static
the energy from impurity
in regions of average length, Ld, by lp = 2L d = 2~Co/~p, where
c o = m/~7~F v F is the phason velocity and ~p/2~ the pinning frequency.
More than
half the pinning energy can be maintained with current flow if the phase oscillates
between two space variations
in the sign of the space variations.
~A(X) and ~B(X) = ~ - ~A(X) that differ
In an idealized model,
#A =
-(~/2)cos(~X/Ld) , giving the minimum energy when the space-average 8 = <#> = 0 (mod 2~). (mod 2w).
When a current
flows and 8 changes in time, the system oscillates
between 8 A and 8 B for each change w in 8. maintained
phase
The state 8 B gives the minimum energy when =
In the model (Fig.
i) the phase is
at -~/2 for x = n2L d as 8 goes from 0 to -w and at -(3w/2)
x = (2n + I)L d as e goes from -~ to -2~.
at
The variation of the pinning energy
with phase would then be V(8) = -V o - Vlcos8 j (see Fig. lb). The phase variations Eg = (2/~)4MF/m ~ p
8 A and 8 B give rise to a pinning gap
at the Fermi surface.
The reason for the gap is that the
charge added in going from 8 A to 8 B must be localized within 2L d. is that required to add 2k F to the wave-vector acceleration antlsoliton
step.
In the idealized model,
of the electrons
The energy E g
in an
this is the energy of a ~-soliton-
pair, where a ~-soliton corresponds
to half a state, or, when doubly
occupied,
to a charge e per chain.
The Zener expression
for the probability
tunneling
through the pinning gap is P(E) = exp(-Eo/E) , where
of
A
2~'-
/i-'--'--,,~ B
",., t
-@(x)
~. .
.
.
.
_L--
-_!
_~_- -
o
(a)
X/Ld----,-
t v(8) 0
7r/2
3~r/2
5rr/2
777/2
__.
(b) Figure 1.
w -w. (a) Phase variations for e = 0, - ~,
(b) Pinning energy as
function of B ffi <@(x,t)>.
wE 2 M~u2 E ° =----g--= P 4~vFe wmvFe
(2)
The equation for the acceleration is dq ~-=
dVd m dt
m eEF(E) mF
(3)
where m/M F is the probability that the added momentum goes to the electrons rather than the macroscopic phonons.
When integrated over a relaxation time 3*,
m
~u d ffi 2~kFV d = 2VF~ q ~ F
(4)
eE2VFT*P(E)
where VFT* ~ Co/U p and o b = ne2~*/M.
The scaling frequency is the drift
frequency for free acceleration in a field Eo, or [~u
m
s ffi ~
(5)
eEo2VF T*
These expressions for Eo and u s are in good agreement with experiment for "pure" NbSe 3 and TaS3, but require modification when the pinning frequency is very high, as in the alloy Tal_xNbxS 3 and (TaSe4)21.
For the latter, it is
necessary to take into account the relaxation time between electrons and macroscoplcally occupied phonons.
If T = Tel_p h + T* is the total relaxation
time for momentum added to the electrons,
the revised expressions
for E o and u s
are
E' = o
m' = T4~W/~ m S s
(T/T*)Eo
EXPERIMENTAL
RESULTS
Careful experiments excellent
(6)
agreement[4]
on well-prepared
specimens
with the predictions
Zener form and the magnitudes
of E o and m s.
form holds for fields as large as I00 E T. agreement
of NbSe 3 and TaS 3 are in
of the tunneling theory for both the In NbSe 3 below TI, the exponential The revised expressions
for the alloy and the iodide as long as temperatures
thermally activated
region.
The expressions
are in good
are above the
are expected to be valid up to
fields as high as m d ~ ~MF/m m , or to l/T, the upper frequency cut-off on the P CDW response. The expression
(4) for the ~m d is similar to that for the overdamped
oscillator model, but with a field dependent
relaxation time, T*P(E).
E is the total field, E = Eappl + Epo I (O) where Epo I is proportional -~V(O)/~0.
The potential
to
V(O) has a rich harmonic content that differs markedly
from that of the sinusoidal
variation of the sine-Gordon model generally
in the overdamped oscillator model. oscillating
The field
At high fields,
taken
Eappl >> E o, the
current density from Epol, with O = mdt , is
ipo I = ObEpol(mdt)
(7)
A Fourier expansion of -~V(8)/~e decreases
indicates
that the amplitude of the harmonics
as I/n, consistent with experiment.
Because of rounding at the breaks
between 8 A and 8B, there is some decrease in harmonic content at high fields. Further confirmation
of the form for V(8) comes from the subharmonic
constant current observed when the dc I-V characteristic presence of an applied ac current. rational fractions
Measurements
p/q occur for integers
at UCLA have shown that all
less than about 12.
steps can be estimated by a simple argument.
The width of the
If it is assumed that the only
currents present are the applied dc and ac currents,
8 = -mdt
steps of
is measured in the
the phase is
mI m cosmt + 8 °
(s)
where e o is an adjustable current flow.
phase that may depend on the magnitude
With this expression
have dc components
corresponding
rational fraction p/q.
of the dc
for 8, the Fourier expansion of Epol(t) will
to a drift frequency m d = (p/q)m for every
Thus the total dc field E = Eappl + Epol(8o)dc
can
remain constant at the value required for the p/q step for a range of values of
Eappl extending from E - Em to E + Em, where Em is the maximum value of Epol(eb)dc as eo is varied.
This method has been used[5] to calculate the step
width for the rational fractions p/q.
The relative step widths are in good
agreement with experiment, although the magnitudes are smaller than predicted. Experiments on field dependence of current, harmonic content of narrow band noise and the subharmonlc steps all indicate that pinning energy is maintained to very high fields.
This is strong evidence for the quantum tunneling model of
deplnnlng, but is contrary to the basic assumption of the classical model of Sneddon, Cross and Fisher[3]. The latter assume that at high fields the CDW is essentially free and treat interaction with impurities by perturbation theory. At low temperatures, particularly in TaS3, (TaSe4)21 and K0.3MoO 3, the response is thermally activated.
While threshold fields are not far from
predicted values, the magnitudes of the currents are much smaller than predicted and scaling of the PAT theory no longer applies.
It is believed[13] that what is
involved is time to screen the COW fluctuations in the eA ÷ e B deplnnlng process by the normal conductivity from electrons and holes thermally excited across the Pelerls gap.
The theory as given above omits Coulomb forces from density
fluctuations and is valid only at high temperatures when the screening is complete. ACKNOWLEDGEMENTS The authors express gratitude for the close interaction with George Gr~ner and his students, S. E. Brown, M. Makl, G. Mozurkewlch, D. Reagor and S. Srldhar, that has been essential for the work described here to be carried out.
RET wishes to acknowledge support provided by an NSERC (Canada)
Postgraduate Scholarship and JHM for an IBM Fellowship. Funds were provided by the U.S. Joint Services Electronics Program under contract No. N00014-84-C-0149.
REFERENCES 1
R. C. Lacoe, H. J. Schulz, D. J~r~me, K. Bechgaard and L. Johannsen./ PhTs. Rev. Lett., 55
2
(1985) 2351.
R. E. Thorne, J. H. Miller, Jr., W. G. Lyons, J. W. Lydlng and J. R. Tucker, Phys. Rev. Lett., 5 5
(1985) 1006; J. Bardeen, Phys. Rev. Lett., 55 (1985) 1010
and references therein. 3
L. Sneddon, M. C. Cross and D. S. Fisher, Phys. Rev. Lett., 49 (1982) 292. D. S. Fisher, Phys. Rev. B,31 (1985) 1396.
4
R. E. Thorne, W. E. Lyons, J. H. Miller, Jr., J. W. Lydlng and J. R. Tucker, to be published.
5
R. E. Thorne, J. R. Tucker, J. Bardeen, S. E. Brown and G. Gr~ner, PhTs. Rev. B, 33 (May 15) (1986).
6
J.R. Tucker, Proc. Yamada Conference, to be published in Physica B, December 1986
DEVELOPMENT
1
OF A TUNNELING MODEL OF CHARGE-DENSITY-WAVE
DEPINNING
JOHN BARDEEN Department Urbana,
of Physics,
IL 6|801
University
of Illinois at Urbana-Champaign,
(U.S.A.)
J.W. LYDING, W.G. LYONS, J.H. MILLER, Jr., R.E. THORNE and J.R. TUCKER Department
of Electrical
Laboratory,
University
and Computer Engineering
of Illinois,
Urbana,
and Coordinated
IL 61801
Science
(U.S.A.)
ABSTRACT The tunneling model of transport by charge-density dimensional metals has been developed and experiment. condensate suggested
Deplnning
by the field dependence
initiated at Illinois successful
from a close interaction between theory
by coherent
in many parallel chains,
Zener tunneling of electrons
characteristic.
of the dc current.
to study detection,
Reagor
for a wide
et al. of a(~) in the
to account for CDW metals with high
such as (TaSe4)21 and the alloys Tal_xNbxS3.-
low-frequency
ac response
can be derived from parameters
frequencies.
The theory accounts
measured
The de and
measured at high
for the "narrow band" noise that accompanies
dc current flow and the subharmonic
DEVELOPMENT
tunneling theory accounts
by Gr~ner,
range i-I00 GHz suggested a modification
characteristic
In 1981, a program
mixing and harmonic generation was
The only inputs are a scaling parameter and the dc I-V
Recent measurements
pinning frequencies,
in the CDW
first applied to NbSe 3 and TaS3, was
in showing that photon-assisted
range of phenomena.
waves in quasi-one-
steps of constant current in the dc I-V
in the presence of an applied ac.
OF CONCEPTS
Recent observations
indicate that Fr~hlich conduction by moving charge-
density waves occurs in the organic transfer salt TTF-TCNQ below the Peierls transition [I]; Since the phenomena may occur in other organic metals, worthwhile
to review the background
of FrShllch conduction as developed
for inorganic metals.
Normally pinned to the lattice by impurities incommensurate
(DW's gradually become depinnned
0379-6779/87/$3.50
it may be
and current status of the tunneling theory
or lattice imperfections, and move through the lattice
© Elsevier Sequoia/Printed in The Netherlands
2 when a threshold
field is exceeded and contribute
to the current flow.
Following the discovery of FrShllch conduction in NbSe 3 in 1975, many other inorganic quasi-one-dlmensional
metals have been found that exhibit the
phenomena. Competing theories, systems[2]
one based on treating
and another as massive
freedom have been developed. development
CDW metals as macroscopic
classical systems[3] with internal degrees of
The quantum approach has undergone
in both theory and experiment,
particularly
considerable
during the past year, so
that it is now capable of accounting for a wide range of GDW phenomena quantitative
quantum
detail with a theory based on only a few measurable
While we will not go into the details of the derivation, results and how they compare with recent experiment.
in
parameters.
we will describe
It is believed
the
that
remaining problems will be resolved within the quantum approach. It was not until the remarkable
field- and frequency-dependent
was observed below Tp in NbSe 3 by Monceau, conduction was discovered. approximately
They found that the dc current could be expressed
in the form i = OaE + ObE exp[-Eo/E],
where E is the electric
field and the second term is the CDW contribution.
The exponential
suggested Zener tunneling through a small pinning gap. account for CDW depinnlng
conductivity
Ong and Portis that true Fr&~llch
form
Early attempts
to
on this basis, or by creation of solltons in an
electric field, were abandoned when it was found the required energy gap is ~ IO-4kBT. In 1979 it was pointed out that such theories are viable if the CDW is a macroscopic
quantum system with one thermal degree of freedom in a volume
containing perhaps
106 parallel
atomic chains of atoms.
Tunneling
electrons would then add coherently in this volume, as do electrons through an insulating
layer in the Josephson
relate the depinning field,
effect.
were made of o(~) in the megahertz
that if tunneling is involved, tunneling
tunneling
An attempt was made to
Eo, with the pinning frequency.
The expression
given for E o is close to that in the present version of the theory. after measurements
of individual
In 1980,
region, it was suggested
it should be possible to apply photon-asslsted
(PAT) theory to derive the ac response from the de.
One consequence
is that o(~) should scale with idc/E , so that the CDW response is o(~) = abexp(-~s/m),
where u s is the scaling frequency.
A program was initiated in 1981 to study detection, and harmonic generation
to test predictions
found to be in good quantitative
mixing,
of the PAT theory.
or semi-quantltative
harmonic mixing Results were
agreement when biased
above threshold or if the frequency to be detected
is above a critical value.
In some cases results were qualitatively
from those expected from
classical models.
A phenomenological
range of phenomena when the frequency
different
theory was developed
to account for a
is lower than the critical value, called
the ac-dc decoupling
frequency.
In this region the expected
compensated
at least in part by polarization
THEORETICAL
BACKGROUND
When a CDW moves with a drift velocity, to -kF+q, kF+q, where Nq = my d. the current
response
is
fields.
Vd, the ID Fermi surface, ~kF, moves
The Peierls gaps move with the Fermi sea and
flow is the same as if the gaps were not present.
The total kinetic
energy is (I/2)(m + MF)V~, where M F >> m is that associated with the moving ions that accompany
the wave.
Changes in electron density ~nd in v d can be described by a phase ~(x,t) such that the electron density
is
(2)
P = Po + 01 c°s[2kFX + ~(x,t)]
For uniform flow, ~ = -~d t, where ~d = 2kFVd is the frequency of the macroscopically difference
occupied phonon that accompanies
the wave.
between opposite sides of the FS is ~ d "
density are proportional
~o(X),
More generally,
changes in
to ~#/Sx and drift velocity to - ~ / ~ t .
In the Fukuyama-Lee-Rice deformations,
The energy
theory of weak impurity pinning,
change the phase to minimize
fluctuations.
The phase is adjusted
deformations
with an average wavelength
static
the energy from impurity
in regions of average length, Ld, by lp = 2L d = 2~Co/~p, where
c o = m/~7~F v F is the phason velocity and ~p/2~ the pinning frequency.
More than
half the pinning energy can be maintained with current flow if the phase oscillates
between two space variations
in the sign of the space variations.
~A(X) and ~B(X) = ~ - ~A(X) that differ
In an idealized model,
#A =
-(~/2)cos(~X/Ld) , giving the minimum energy when the space-average 8 = <#> = 0 (mod 2~). (mod 2w).
When a current
flows and 8 changes in time, the system oscillates
between 8 A and 8 B for each change w in 8. maintained
phase
The state 8 B gives the minimum energy when
In the model (Fig.
i) the phase is
at -~/2 for x = n2L d as 8 goes from 0 to -w and at -(3w/2)
x = (2n + I)L d as e goes from -~ to -2~.
at
The variation of the pinning energy
with phase would then be V(8) = -V o - Vlcos8 j (see Fig. lb). The phase variations Eg = (2/~)4MF/m ~ p
8 A and 8 B give rise to a pinning gap
at the Fermi surface.
The reason for the gap is that the
charge added in going from 8 A to 8 B must be localized within 2L d. is that required to add 2k F to the wave-vector acceleration antlsoliton
step.
In the idealized model,
of the electrons
The energy E g
in an
this is the energy of a ~-soliton-
pair, where a ~-soliton corresponds
to half a state, or, when doubly
occupied,
to a charge e per chain.
The Zener expression
for the probability
tunneling
through the pinning gap is P(E) = exp(-Eo/E) , where
of
A
2~'-
/i-'--'--,,~ B
",., t
-@(x)
~. .
.
.
.
_L--
-_!
_~_- -
o
(a)
X/Ld----,-
t v(8) 0
7r/2
3~r/2
5rr/2
777/2
__.
(b) Figure 1.
w -w. (a) Phase variations for e = 0, - ~,
(b) Pinning energy as
function of B ffi <@(x,t)>.
wE 2 M~u2 E ° =----g--= P 4~vFe wmvFe
(2)
The equation for the acceleration is dq ~-=
dVd m dt
m eEF(E) mF
(3)
where m/M F is the probability that the added momentum goes to the electrons rather than the macroscopic phonons.
When integrated over a relaxation time 3*,
m
~u d ffi 2~kFV d = 2VF~ q ~ F
(4)
eE2VFT*P(E)
where VFT* ~ Co/U p and o b = ne2~*/M.
The scaling frequency is the drift
frequency for free acceleration in a field Eo, or [~u
m
s ffi ~
(5)
eEo2VF T*
These expressions for Eo and u s are in good agreement with experiment for "pure" NbSe 3 and TaS3, but require modification when the pinning frequency is very high, as in the alloy Tal_xNbxS 3 and (TaSe4)21.
For the latter, it is
necessary to take into account the relaxation time between electrons and macroscoplcally occupied phonons.
If T = Tel_p h + T* is the total relaxation
time for momentum added to the electrons,
the revised expressions
for E o and u s
are
E' = o
m' = T4~W/~ m S s
(T/T*)Eo
EXPERIMENTAL
RESULTS
Careful experiments excellent
(6)
agreement[4]
on well-prepared
specimens
with the predictions
Zener form and the magnitudes
of E o and m s.
form holds for fields as large as I00 E T. agreement
of NbSe 3 and TaS 3 are in
of the tunneling theory for both the In NbSe 3 below TI, the exponential The revised expressions
for the alloy and the iodide as long as temperatures
thermally activated
region.
The expressions
are in good
are above the
are expected to be valid up to
fields as high as m d ~ ~MF/m m , or to l/T, the upper frequency cut-off on the P CDW response. The expression
(4) for the ~m d is similar to that for the overdamped
oscillator model, but with a field dependent
relaxation time, T*P(E).
E is the total field, E = Eappl + Epo I (O) where Epo I is proportional -~V(O)/~0.
The potential
to
V(O) has a rich harmonic content that differs markedly
from that of the sinusoidal
variation of the sine-Gordon model generally
in the overdamped oscillator model. oscillating
The field
At high fields,
taken
Eappl >> E o, the
current density from Epol, with O = mdt , is
ipo I = ObEpol(mdt)
(7)
A Fourier expansion of -~V(8)/~e decreases
indicates
that the amplitude of the harmonics
as I/n, consistent with experiment.
Because of rounding at the breaks
between 8 A and 8B, there is some decrease in harmonic content at high fields. Further confirmation
of the form for V(8) comes from the subharmonic
constant current observed when the dc I-V characteristic presence of an applied ac current. rational fractions
Measurements
p/q occur for integers
at UCLA have shown that all
less than about 12.
steps can be estimated by a simple argument.
The width of the
If it is assumed that the only
currents present are the applied dc and ac currents,
8 = -mdt
steps of
is measured in the
the phase is
mI m cosmt + 8 °
(s)
where e o is an adjustable current flow.
phase that may depend on the magnitude
With this expression
have dc components
corresponding
rational fraction p/q.
of the dc
for 8, the Fourier expansion of Epol(t) will
to a drift frequency m d = (p/q)m for every
Thus the total dc field E = Eappl + Epol(8o)dc
can
remain constant at the value required for the p/q step for a range of values of
Eappl extending from E - Em to E + Em, where Em is the maximum value of Epol(eb)dc as eo is varied.
This method has been used[5] to calculate the step
width for the rational fractions p/q.
The relative step widths are in good
agreement with experiment, although the magnitudes are smaller than predicted. Experiments on field dependence of current, harmonic content of narrow band noise and the subharmonlc steps all indicate that pinning energy is maintained to very high fields.
This is strong evidence for the quantum tunneling model of
deplnnlng, but is contrary to the basic assumption of the classical model of Sneddon, Cross and Fisher[3]. The latter assume that at high fields the CDW is essentially free and treat interaction with impurities by perturbation theory. At low temperatures, particularly in TaS3, (TaSe4)21 and K0.3MoO 3, the response is thermally activated.
While threshold fields are not far from
predicted values, the magnitudes of the currents are much smaller than predicted and scaling of the PAT theory no longer applies.
It is believed[13] that what is
involved is time to screen the COW fluctuations in the eA ÷ e B deplnnlng process by the normal conductivity from electrons and holes thermally excited across the Pelerls gap.
The theory as given above omits Coulomb forces from density
fluctuations and is valid only at high temperatures when the screening is complete. ACKNOWLEDGEMENTS The authors express gratitude for the close interaction with George Gr~ner and his students, S. E. Brown, M. Makl, G. Mozurkewlch, D. Reagor and S. Srldhar, that has been essential for the work described here to be carried out.
RET wishes to acknowledge support provided by an NSERC (Canada)
Postgraduate Scholarship and JHM for an IBM Fellowship. Funds were provided by the U.S. Joint Services Electronics Program under contract No. N00014-84-C-0149.
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L. Sneddon, M. C. Cross and D. S. Fisher, Phys. Rev. Lett., 49 (1982) 292. D. S. Fisher, Phys. Rev. B,31 (1985) 1396.
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