RAPID COMMUNICATIONSPHYSICAL REVIEW B
CONDENSED MATTER AND MATERIALS PHYSICSTHIRD SERIES, VOLUME 57, NUMBER 4

15 JANUARY 1998-II

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Rapid Communications are intended for the accelerated publication of important new results and are therefore given priority treatment both in the editorial office and in production. A Rapid Communication in Physical Review B may be no longer than four printed
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Dimensional crossover of quantum nucleation processes in charge-density-wave phase slips
Noriyuki Hatakenaka
NTT Basic Research Laboratories, Atsugi, Kanagawa 243-0198, Japan

Masato Shiobara, Ken-ichi Matsuda, and Satoshi Tanda
Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan~Received 10 July 1997; revised manuscript received 21 November 1997!
A phenomenological model for quantum phase slips of charge-density waves that takes into account thesystem-size effect is presented. The process of quantum nucleation leading to the phase slip changes fromvortex-pair to vortex-ring creations as the external electric field increases, which is analogous to the evolutionof a ripple in a rectangular water tank. The crossover field is determined by the system size. The present model
describes a number of features observed in the nonohmic conductivity in TaS3 at low temperature.
@S0163-1829~98!52004-8#

Since the beginning of 1980, the applicability of quantum
mechanics on a macroscopic scale has been tested throughtypical quantum effects. One such effect is macroscopic
quantum tunneling ~MQT!, which has been established via
quantum tunneling of the phase difference of Josephson
junctions with small capacitance.1–3 It has been clarified thattunneling on a macroscopic scale is strongly affected by themacroscopic degrees of freedom. Later, MQT was verified invarious macroscopic systems, including the domain wall in
ferromagnets4 and the spin-density wave in ~TMTSF!2 PF6 .5
Although the charge-density waves ~CDW’s! are anothertype of macroscopic quantum phenomena, the issue of quantum tunneling in CDW’s ~Refs. 6 and 7! remains unsettled
for two main reasons: at high temperatures quantum processes are probably masked by classical processes, and there
is a lack of the evidence associated with macroscopic numbers of electrons. Zaitsev-Zotov recently made abreakthrough.8 He observed temperature-independent nonlinear conduction at low temperatures and that the electric
current is transferred by portions of charge corresponding todisplacement of the CDW’s by a distance of the order of
their wavelength. Although he tried to explain his experimental results in terms of quantum creep, they might be possibly explained by MQT of CDW phase slips. The interpretation of the results are still an open question. At least heclarifies previously uncertain points and provides a physicalpicture of nonohmic conductivity at low temperatures, thatis, the phase slip model.9,10
Zaitsev-Zotov’s most important discovery was to find ascaling form of nonlinear conductivity expressed asexp(2const/ e 2 ), where e is an applied electric field. This
form was a problem for the existing tunneling models6,7 withexp(2const/ e ). To overcome this problem, Duan has proposed a homogeneous quantum phase-slippage model basedon the vortex-ring creation in bulk CDW systems.11 This
successfully explains the scaling form. However, he implicitly assumes that the low-temperature conduction of thicksamples proceeds by the same mechanism as in thin ones.
Thus his model cannot explain the system-size effect. The
low-temperature conduction in thick and thin samples ofTaS3 is similar, but there are important differences:12 ~a!
The lnI(E,T) as a function of temperature does not level out
for T>4.2 K, whereas in thin samples it levels out at T
<10 K. ~b! At T,10 K, the slope of the Arrhenius plot of
the current, ] ln@I(E,T)#/](1/T), increases with a decrease in
the electric field, E, whereas it decreases in thin samples.
In this paper, we consider the effect of system size onquantum nucleations mechanisms leading to phase slips inthe CDW of TaS3 . In particular, we make a possible conjecture to explain the experimental results in terms of the di-0163-1829/98/57~4!/2003~3!/$15.00

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HATAKENAKA, SHIOBARA, MATSUDA, AND TANDA

mensional crossover of the quantum nucleation from a vortex pair to a vortex ring as a function of an applied electricfield.
The slope of the Arrhenius plot of the current is thought
to be created by the competition between classical and quantum processes. The first difference ~a! might be understoodby considering the electric field to be fixed. The tunnelingprocess is temperature-independent, while the thermal activation process depends exponentially on temperature. The
crossover from a classical to a quantum process occurs as thetemperature is reduced past a certain point, and finally thetunneling process becomes dominant at low temperatures.
The crossover temperature shifts to a higher temperature as
the tunneling rate increases. One can easily imagine that thetemperature-dependent current regions will shrink when thetunneling rate is high. We will show later that the tunnelingrate in a thick sample is lower than that in a thin sample. Infact, Zaitsev-Zotov argued this in his experiment.8
Next, we discuss the electric field dependence of theslope. In general, both tunneling and thermal activation rates
increase with an increasing external electric field. The slope
decreases as e increases if ] G thermal / ] e , ] G tunneling / ] e ,where G thermal and G tunneling are thermal activation and tunneling rates, respectively. In the thick sample, this situation
is supposed to be attained. On the other hand, in the thinsamples, temperature-independent current has already been
observed at lower electric fields. Then, the angle of the slope
increases as the external electric field increases. This meansthat the above relation changes to ] G thermal / ] e. ] G tunneling / ] e . This cannot be explained by monoscaling
theory. An important clue to solving case ~b! is also contained in Zaitsev-Zotov’s experiments; the form of scaling
seems to turn into 1/e 2 from 1/e in an external electric fieldat 2 K ~see the inset of Fig. 3 of Zaitsev-Zotov!. Until now,
all previous tunneling theories were considered not to bechanged in their scaling form regardless of the electric fieldmagnitudes. If we accept the change of scaling form as afunction of an external electric field in one sample, it may bepossible to attain ] G thermal / ] e . ] G tunneling / ] e . In what follows, we investigate the sample-size dependence of the tunneling rate and the change of scaling form as a function of anexternal electric field.
First let us consider how the scaling form changes as afunction of an external electric field. The scaling form isdetermined by the type of nucleation process of a bubble
leading to phase slips. Maki pointed out that the process isrelated to the anisotropy of the system; a phase vortex-pairprocess gives rise to a 1/e scaling form in a strongly anisotropic system, while a phase vortex-ring process leads to a1/e 2 scaling form in a more isotropic system. Unfortunately,
the vortex pair and ring creations have been discussed separately, since anisotropy was previously thought to be a peculiar characteristic of the sample matter. In this case, the scaling form cannot be changed in one sample. Here we insist
that this anisotropy depends on sample-size effect.
TaS3 is nearly isotropic within the xy plane perpendicular
to the conduction direction because j x ; j y ;10 Å. Thus wediscuss how anisotropy results from such an isotropic situation. Suppose that you dropped one drop of water in thecenter of a rectangular water tank. It will create a ripple on
the water surface that will spread through the whole water

57

FIG. 1. The time evolution of a ripple created by one drop ofwater. A topological dislocation will grow in the same way.

tank, with time. At first, the ripple will spread in concentric
circles (R5finite!. After it reaches a long edge of the tank,the water wave front will be parallel to the edge
(R5infinity! and will move toward another edge. Thissimple analogy demonstrates the continuous changes fromring to pair creations; that is the appearance of anisotropy
from an isotropic situation. This is a key idea for the mechanism of change of nucleation processes in one sample. ~SeeFig. 1.!
Now let us apply this idea to the CDW system. The phaseslips in CDW behave like water drops: a phase slip proceedsfrom a vortex ring to a vortex pair. In the imaginary-time
path-integral formalism, the rate of quantum phase slips isexpressed as
G5Aexp@ 2DS Eucl /\ # ,~1!

where DS Eucl is the difference between saddle-point action
and metastable-state action. The action is basically described
asDS Eucl52 a ~ D ! R D11 e 1 b ~ D ! R D ,~2!

where D is a system dimension and R is the bubble radius.
The first term of the right-hand side of Eq. ~2! is volume
energy of a bubble gained from an external electric field andthe second is the surface tension energy of the bubble. TheM, determines the tunneling rate
extremal of the action, DS Eucl
since the prefactor A, originating from small fluctuations
around the tunneling path, is less important. The action isestimated by using the radius expressed as
RM5b ~ D !Da~D!

1,~ D11 ! e~3!

which is derived from ] DS Eucl / ] R50. R M is inversely proportional to an external electric field. The larger e is, thesmaller R M . Therefore, it is important to know whether R M
is larger than the system size. It determines the nucleation
process: ~i! L x ,L y ,R M ; phase slips occur coherently atwhole cross sections. It is not necessary to create the phaseslip center at the xy plane. ~ii! L x ,R M ,L y ; the phase slip
proceeds through the vortex-pair creations. ~iii! R M ,L x,L y ; the vortex-ring creation is dominant. In this way, thenucleation process changes from a vortex pair to a ring. The
maximum action is given as
D11 !
DS ~Eucl~RM!

\5F

1 b~D! b~D! D\ D11 a ~ D ! D11GD

1e ~0D11 !

eeD

5D.
~4!

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DIMENSIONAL CROSSOVER OF QUANTUM NUCLEATION . . .

The crossover field e cr can be estimated by setting R M.L x /2 as2b ~ D !De cr 5.
L x a ~ D ! ~ D11 !

~5!

To evaluate the sample-size-dependent tunneling rate andthe crossover field, we recall the previous theories.13 Since
the amplitude coherence length of the CDW is orders ofmagnitude smaller than typical sample dimensions, the CDW
phase slip should be three dimensional ~3D!. Starting from a311D phase Hamiltonian, one can obtain the action
DS Eucl5A 0E )m 50,1,2,3

dx m @~ ] m D f ! 2 12 ~ ] m D f • ] m f !# ,~6!

mwhere x is one component of four-dimensional space-timecoordinates: x 0 5 t , x 1 5x, x 2 5y, and x 3 5z, D f is thephase difference from the metastable state ( f m) defined by
D f 5 f 2 f m , and A 0 is constant.14 We ignore the impurity
potential because the temperature-independent behavior of
the current at fixed e is observed only in a pure sample.15 In
the 111D case, the action is given by

S DE )111 !
DS ~Eucl
5A 0 L xvFva

m 50,2,312 ~ ] m D f • ] m f !# ,dx m @~ ] m D f ! 2~7!

where v F and v a are the Fermi velocities for z and x directions, respectively. The factor L x v F / v a means physically thelength of the vortex lines and shows the sample-size effect~the thicker the sample, the bigger the action!. Thus the tunneling rate decreases as the sample size increases. This elucidates case ~a!.

R. F. Voss and R. A. Webb, Phys. Rev. Lett. 47, 265 ~1981!.
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M. Uehara, B. Barhara, B. Dieny, and P. C. E. Stamp, J. Phys.
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G. Mihály, Y. Kim, and G. Grüner, Phys. Rev. Lett. 67, 2713~1991!.
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J. Bardeen, Phys. Rev. Lett. 42, 1498 ~1979!.
8
S. V. Zaitsev-Zotov, Phys. Rev. Lett. 71, 605 ~1993!.
12R2005

Next, we discuss the crossover field to explain case ~b!.
To do this, it is necessary to compare the action in 111Dwith that in 211D. Unfortunately, the parameters e (111)and0e (211)maydependonmanyfactors
andaredifficulttoesti0
mate using the existing theories. Instead of the theoreticalvalues, we have used the experimental data. Zaitsev-Zotov
exp@2e(111)/e# and I (211)
fitted his data as I (111) 5I (111)00(211)(211) 25I 0
exp@2e0 /e #. The former is a good fit for lower
fields, while the latter is good for higher ones. Using the50.015 Å, e (111)5900 V/cm, I (211)51.5
values I (111)000(211)27
Å, and e 0517 433 V2 /cm2 , we obtained 43.8
310
V/cm as the crossover field. This is almost consistent withthe field where the slope appears in Fig. 2 of Zaitsev-Zotov.
The current I (211) is indeed smaller than I (111) . This meansthat at low temperatures where the thermal activation is really excluded the quantum phase slip rate is slowing down as
the field increases. This may cause the situation that makes aslope in I-~1/T! curves: ] G thermal / ] e . ] G tunneling / ] e .
In summary, we have discussed the quantum phase slipsof charge-density waves in TaS3 , taking into account thesystem-size effect. The phase slip process has been changed
by increasing the external electric field from a phase vortex
pair to a phase vortex ring. The present model describes anumber of features observed in the nonohmic conductivity inTaS3 at low temperatures. Our model will be tested by observing the crossover field predicted by Eq. ~5! throughsamples with various dimensions.
We would like to thank Kazumi Maki, Sergei V. ZaitsevZotov, Takashi Sambongi, Hiroshi Matsukawa, Hiroyuki
Miyake, Nobuo Matsumoto, and Kazuhiko Yamaya for valuable discussions. We also thank Mariko Uchitani for her kindassistance. One of us ~S.T.! is grateful for his financial support through a Grant-in-Aid for Scientific Research from theMinistry of Education, Science, and Culture of Japan.

9N. P. Ong, G. Verma, and K. Maki, Phys. Rev. Lett. 52, 663~1984!.
10For review papers, see, e.g., S. N. Artemenko and A. F. Volkov,in Charge Density Waves in Solids, edited by L. P. Gor’kov andG. Grüner ~Elsevier, Amsterdam, 1989!, Vol. 6; also, L. P.
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11J.-M. Duan, Phys. Rev. B 48, 4860 ~1993!.
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S. V. Zaitsev-Zotov, Phys. Rev. Lett. 72, 587 ~1994!.
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K. Maki, Phys. Lett. A 202, 313 ~1995!.
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K. Maki, Ferroelectrics 176, 353 ~1996!.
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S. V. Zaitsev-Zotov, N. I. Baklanov, G. Remenyi, and P. Monceau, Czech. J. Phys. , 46, 2615 ~1996!.