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Dielectric constant and threshold electric field of charge-density-waves with strong impurity pinning
Solid State Communications, Vol.51,No.5, pp.293-296, 1984. Printed in Great Britain.
0038-1098/84 ~3.00 + .00 Pergamon Press Ltd.
DIELECTRIC CONSTANT AND THRESHOLD ELECTRIC FIELD OF CHARGE-DENSITY-WAVES WITH STRONG IMPURITY PINNING Po Fo Tua and Jo Ruvalds
Department of Physics, University of Virginia, Charlottesville, Virginia 22901 UoSoAo (Received
April
6,
1984
by A.
Zawadowski)
The dielectric constant e end the threshold electric field ET for the onset of charge-denslty-wave conduction are investigated within the phenomenological model proposed by Tua and Zawadowski for the strong pinning regime. The static dielectric constant e(E) in a bias electric field E is found to be almost independent of E provided that E is not too close to E T. For relatlvely small values of the pinning strength, the product e E T becomes independent of the parameters of the theory° Good agreement is found with the available experimental data on NbSe 3 for the case of strong pinning obtained by radiation damage.
Inorganic linear chain compounds which exhibit a phase transition associated with the development of an inco~ensurate charge-densitywave (CDW) have recently attracted considerable attention. I Their highly unusual transport properties, such as nonlinear and frequencydependent conductivity, are due to the collective response of electrons condensed in the CDW state, and the observation of coherent current oscillations, 2 with frequency proportional to the current carried by the condensate, 3 is considered a strong evidence of translational CDW motion. At present, no microscopic theory is able to account for the broad variety of experimental observations. In the classical model of Gr~ner et al., 4 the CDW condensate is assmned to be a rigid object with a single degree of freedom and interacting with an average pinning potential due to the impurities present in the samples. The potential is periodic with period given by the wavelength ~ of the CDW. The Zener-tunnellng model proposed by Bardeen 5 involves a quantummechanical tunneling of macroscopic portions of the CDW through the pinning potential. Other models consider the local deformations of the CDW around impurities, 6-13 or assume that the narrow-band noise is generated by a Josephsontype mechanlsm. 14 A quite different approachl5,16 attributes the origin of the narrow-band noise to the contact regions. In general, the pinning potential plays a crucial role in determining the low field and low frequency properties. In the most studied compounds, NbSe 3 and TaS3, it is extremely small so that the static dielectric constant £o can be as hlgh 17-19 as 107-108 and the threshold electric field ET for the onset of CDWconductivity can be as iow2,19, 20 as 10-1-10 -2 V/cm. Nevertheless, there is experimental evldencel9, 21 that the product E o E T is almost independent of the strength of the pinning potential for both strong and weak impurity plnning~ The classical model predlcts 4
e O ET ffi 4~@nel,
(i)
where -e is the electron charge and n the electron density. The constant @ depends on the form but not the strength of the pinning potential: @=1/2 for a quadratic potential repeated over every wavelength and 8=(2~) -I for a sinusoidal potential. The tunneling model proposed by Bardeen 5 and the deformable medium model proposed by Fukuyama and Lee 6 and Lee and Rice 7 yield an expression analogous to Eq. (i) with @ independent of the pinning strength~ Reasonable estimates 21 give 8=0.05 for the tunneling model 5 and 8=0.02 for the FuKuyama-Lee model 6, roughly one order of magnitude smaller than the classical model. The product ne% may be determined experimentally from the ratio of the excess current ICDW arising from the depinned CDW and the frequency ~ of the first harmonic of the narrow-band noise:2, 3 ICDW/I = ne%~
(2)
P~eliminary experimental results 21 indicate that @=0~03 for orthorhombic TaS 3 in the weak pinning regime, but some uncertainties exist 22 and further tests are neededo In NbSe 3 with strong impurity pinning one may infer e o ET ~ 9 A/MHz cm2 from Ref. [19] and ICDW/~ ~ 25 A/MHz cm 2 from Ref~ [23] so that @~0.03o Furthermore, it has been found 21 that £(E) is unchanged for applied bias electric fields E less than E TIn the classical model with sinusoidal potential the static dielectric constant in a bias field E is given by 4 E(E) =
2ne%ET [i - (E/ET)2]-I/2
(3a)
As E approaches E T from below, e(E) diverges~ This feature is common to any pinning potential with a smooth m a x l m u m h u t is not observed experimentally~21, 22 On the other hand, the 293
DIELECTRIC CONSTANT AND THRESHOLD ELECTRIC FIELD OF CHARGE-DENSITY-WAVES
294
quadratic potential repeated over every wavelength implies 4
where =
2~neA £(E) = ET '
= ~ + X<¢>,
(4a)
+ (l+e)$i = ~ - ¢i - psin(~+~i+8i)'
(4b)
( i = l , . . , N s)
<¢>
= %1 ~ *i"
(4c)
where ¢i is the phase of the i-th segment relative to the frame and B I is random between -~ and and represents the preferred position of the l-th segment due to the particular impurity distribution; ~ is the ratio between the "external" and "internal" relaxation times and X = NsVs/VF; and p are the electric field and the Finning strength in dimensionless units. Eqs. (4) can be easily integrated by computer for flnlte N S. The 81's can be chosen equally spaced between -~ and ~, so as to avoid statistical fluctuations due to a relatively small value of N S. This model is appropriate for treating the transport properties of a moving CDW in the strong coupling case, but it is not applicable to the weak coupling region where fluctuations in the impurity distribution insld~ a Lee-Rice domain play a crucial role. 7 Choosing the initial conditions (without electric field) such that -z<¢i
p = ne~, 2~
~ +
(3b)
i~e. g is independent of Eo Even though this behavior is in agreement with the experimental data, 21, 22 the quadratic potential is certainly unphysical. In order to overcome this contradiction, we have analyzed the phenomenologlcal model proposed by Tua and Zawadowski 13 which is a modification of the classical model 4 and includes internal degrees of freedom of the CDW condensate. A detailed description of the theory can be found in Ref. [13]. A single Lee-Rlce 7 domain is considered. The long-range order of the classical model 4 is preserved by postulating the existence of a rigid and macroscopic CDW frame whose phase is @ and whose volume V F is proportional to the volume of the domain~ The frame does not interact directly with the impurities and is subject to a frictional force~ The shortrange deformations of the CDW condensate are represented by internally rigid segments which interact with a sinusoidal pinning potential and are coupled elastically to the frame. The number N S of segments is proportional to the volume of the domain. Each segment has volume V S and is subject to an "internal" frictional force in addition to the same frictional force of the frame~ Under the assumption of overdamped motion, the dynamics of the system is described by the following equations:
and
Vol. 51, No~ 5
(5a)
-/- <~>. l+×
(Sb)
The static dielectric constant e(E) in a blas electric field E is e(E) = (ne)2y -I ~(~),
(6a)
~(~)
(6b)
where
4w dP/dE,
=
and y is the elastic coupling constant (per unit volume) of the interaction between frame and segments. 13 We have calculated ~(E) for different values of N s. The results are shown in Fig. 1 for the case p = 4.0 and ~ = 0.3. It is interesting to notice that i) e(E) is not divergent as ~ + ~ and ii) the dependence of ~ on ~ decreases as the number of segments N S is increased. These features are very similar to the classical model with quadratic potential. To further investigate the problem, we have calculated the dielectric constant ~ as a function of the frequency without bias electric field. The results are shown in Flg. 2 for the case N S = ii, p = 4°0, X = 0.3 and ~ = 0.i. The curve is very similar to the one for the overdamped harmonic oscillator, i.e. it can be fitted very well by the function Re E/Co = [i + (~/~)2]-i with ~ = 0.238. The results are independent of the strength of the electric field ~ as long~as E is not too close to E T. For E close to ET, hysteresis effects appear. Experimental data 19 on N b S e 3 w i t h strong impurity pinning obtained by radiation damage are in agreement with the present theoretical results, but further tests are needed~ It is now clear that the introduction of internal degrees of freedom to the classical model with sinusoldal potential yields features analogous to the classical model with quadratic potential. This explains why the latter model, although unphysl-
1.8
'
'
'
'
I
'
'
'
II SEGMENTS. . . . . 21
16
SEGMENTS _ _ -
31 S E G M E N T S _ _
'
I
'
'
'
'
I
'
l
i
I
x=O.3
~=4o
/
o~.~1.4
~w
Fig. 1
,; /
Normalized static dielectric constant ~(~) in a bias electric field ~ vs. for ~ = 4~0, X = 0~B, and N s = ii, 21, 31.
V61. 5], No. 5 I
295
DIELECTRIC CONSTANT AND THRESHOLD ELECTRIC FIELD OF CHARGE-DENSITY-WAVES I
I
l
tor,
)
I
1.0
=OJ =03
0.8
From Eqs. (4)-(6) one easily obtains: £im
~_~
~
4~
x(z+x)
o
(8)
"
=4.0
~0.6 3 x,~ 0.4 ~D
n,0,2 O0 0.0 Fig. 2
I
]
I
I
I
I
O.I
02
03
0,4
0.5
0.6
0.7
Real part of the dielectric constant vs. frequency ~ for p = 4.0, X = 0.3, and N s = ii. The frequency ~ is In dimensionless units (see text).
cal, is in agreement with the experimental datao19-22 We have also investigated the combined dependence of ~o and ~T upon the pinning strength p for different values of the parameter Xo Fig. 3 shows ~ -i vs. ET with N s = ii. For small values o~ p (i.eo small values of ~T) the product ~o ~T is independent of p and XThe constant 8 appearing in Eq. (i) can be expressed in terms of the dimensionless quantities ~o and ~T through the relation
Using the radiation damage technique, Fuller et al. 19 were able to vary the defect concentration in NbSe 3 samples. They found that the product e o E T was roughly independent of the defect concentration while both e O and E T varied about one order of magnitude. It is tempting to assume that the pinning strength p of the present model can be related to the defect concentration in the NbSe B samples. Within this a s s ~ p t l o n , the present theory for relatively small values of p is in agreement with the experimental data of Fuller et al. 19 Finally, we have considered the dependence of ~o and ~T on the parameter X which is the ratio between the total v o l ~ e of the segments and the volume of the frame. Fig. 4 shows the case with N s = ii and p = 4.0. As X-~° both T O and ~T approach a constant value. In this limit the volume of the frame vanishes and the present model becomes equivalent to the mean field theory proposed by Fisher. II Viceveraa, as x+O the total pinning of the domain vanishes so that ET also vanishes and ~o diverges.
II
segmenls
/
x = 0.7
id' @ = Eo~T/(8~2).
(7)
///
In the present case with N s = ii, we obtain @=0.015 in good agreement with the previously extrapolated value @=0.03 for NbSe 3 with strong impurity plnnlng. 19 From Fig° 3 it appears that as ~T -~= (and ~-~o) the dielectric constant go approaches a finite value° This behaviour can be easily explained° As ~-~o the segments become "localized" and the frame, which interacts with the segments hut not with the Dinning potential, behaves as a simple harmonic oscilla-
2x,O' l . .
........
'
.........
'
......
6.
~
x=O.I
id'
Iz~'°'
id'
,o'
101 I/
Id'
,
,
,
~
I
,
I~'
Io~
,d'
16'
,o °
Id
rcf
E, zx~d
zxso'
,d f'"
,d
Zx~O°
Zx~O°
i 0 O, l, l,,, , ( ~
,,IK:f . . . . . . .
Idl
......
I0a
i0 o
x
Fig. 3
Inverse static dielectric constant ~-i vs. threshold electric field ~T as t~e pinning strength p is varied°
Fig. 4
Static dielectric constant go and threshold electric field ~T vs. X for N s = ii and p = 4.0.
In conclusion, we have found that the static properties of the present theory are analogous to those of the classical mode! 4 with quadratic potential and are in agreement with the available experimental data 19 on NhSe 3 for the strong pinning regime. One of us (P.F.T.) is very grateful to G. Gr~ner and G. Mozurkewich for stimulating discussions. This work was sunnorted in part by the DOE Grant DE-AS05-81-ERI0959.
296
DIELECTRIC CONSTANT AND THRESHOLD ELECTRIC FIELD OF CHARGE-DENSITY-WAVES Vol. 51, No.' 5 References
10
2°
3.
4.
5~
6. 7. 8. 9~ i0.
See, for review, R. M. Fleming, in Physics ~n One Di~enslon, Springer Series ~n SolEd State Sciences, Volo 23, edited by Jo Bernasconl and T. Schneider (SpringerVerlag, Berlin, 1981); N. P~ Ong, Can. Jo Phys° 6_~0, 757 (1982); Go Gr~ner, Mol. Crysto Liqo Crysto 81, 735 (1982), and Co~ents Solid State Phys~ iO, 183 (1983)o R. M° Fleming and Co Co Grimes, Phys. Rev. Lett. 42, 1423 (1979); Ro Mo Fleming, Solid State Commun. 43, 167 (1982). P. Monceau, J. Richard, and Mo Renard, Phys. Rev. Lett. 4~5, 43 (1980); Jo Bardeen, E, Ben Jacob, A. Zettl, and G, Gruner, Phys. Rev~ Lett. 49, 493 (1982)o G. Grbner, A. Zawadowski, and Po M. Chaikln, Phys. Revo Letto 46, 511 (1981) o See also P. Monceau, J. Richard, and Mo Renard, Phys. Rev. B 25, 931 (1982). J. Bardeen, Phys. Rev. Lett~ 45, 1978 (1980); MOlo Crysto Liq. Cryst. 81, 1 (1982). H. Fukuyama and P. A. Lee, Phys. Rev. BI7, 545 (1978). P . A . Lee and T. M. Rice, Phys. Rev. BI9, 3970 (1979). J . B . Sokoloff, Phys. Rev. B 23, 1992 (1981). L. Sneddon, M. C. Gross, and D. S. Fisher, Phys. Rev. Lett. 49, 292 (1982). R. A. Klemm and J. R. Schrieffer, Phys. Rev. Lett. 51, 47 (1983).
Iio
Do S. Fisher, Physo Rev. Letto 50, 1486 (1983). 12o L° Peitronero and S. Strassler, Physo Rev. B28, 5863 (1983) 13. P. F. Tua and Ao Zawadowski, Solid State Comun~ 49, 19 (1984)~ 14~ So Eo Barnes and Ao Zawadowski, Physo Rev. Lett~ 51, 1003 (1983)o 15. Lo P. Gorkov, JETP Letters 38, 87 (1983) and JETP Letters, in press. 16o N. P. OnE, G. Verma and K. 1~aki, Phys. Rev. Left. 52, 663 (1984). 17. G. Gr~ner, L. C. Tipple, Jo Sanny, W. Go Clark, and N. P. Ong, Phys. Rev. Left. 45, 935 (1980). 18. A. Zettl, C. M. Jackson, and G= Gruner, Phys~ Rev. B 26, 5773 (1982). 19. W. W. Fuller, G. Gr~ner, P. M. Chaikin, and No P. Ong, Physo ReVo B 23, 6259 (1981). 20° A. Zettl, Go Gr~ner, and A. H. Thompson, Phys. Revo B 26, 5760 (1982). 21. Wei-yu Wu, A. J~nossy, and G. Gr~ner, Solid State Communo, 49, 1013 (1984). 22, G. Mih~ly, L. M ~ l y , and Ho Mutka, Solid State Cu~mun., 49, 1009 (1984). 23. P. Monceau, Mo Renard, J. P~chard, M. C. Salnt-Lager, H. Salva, and Zo Zo Wang, Phys, Rev. B 28, 1646 (1983).
0038-1098/84 ~3.00 + .00 Pergamon Press Ltd.
DIELECTRIC CONSTANT AND THRESHOLD ELECTRIC FIELD OF CHARGE-DENSITY-WAVES WITH STRONG IMPURITY PINNING Po Fo Tua and Jo Ruvalds
Department of Physics, University of Virginia, Charlottesville, Virginia 22901 UoSoAo (Received
April
6,
1984
by A.
Zawadowski)
The dielectric constant e end the threshold electric field ET for the onset of charge-denslty-wave conduction are investigated within the phenomenological model proposed by Tua and Zawadowski for the strong pinning regime. The static dielectric constant e(E) in a bias electric field E is found to be almost independent of E provided that E is not too close to E T. For relatlvely small values of the pinning strength, the product e E T becomes independent of the parameters of the theory° Good agreement is found with the available experimental data on NbSe 3 for the case of strong pinning obtained by radiation damage.
Inorganic linear chain compounds which exhibit a phase transition associated with the development of an inco~ensurate charge-densitywave (CDW) have recently attracted considerable attention. I Their highly unusual transport properties, such as nonlinear and frequencydependent conductivity, are due to the collective response of electrons condensed in the CDW state, and the observation of coherent current oscillations, 2 with frequency proportional to the current carried by the condensate, 3 is considered a strong evidence of translational CDW motion. At present, no microscopic theory is able to account for the broad variety of experimental observations. In the classical model of Gr~ner et al., 4 the CDW condensate is assmned to be a rigid object with a single degree of freedom and interacting with an average pinning potential due to the impurities present in the samples. The potential is periodic with period given by the wavelength ~ of the CDW. The Zener-tunnellng model proposed by Bardeen 5 involves a quantummechanical tunneling of macroscopic portions of the CDW through the pinning potential. Other models consider the local deformations of the CDW around impurities, 6-13 or assume that the narrow-band noise is generated by a Josephsontype mechanlsm. 14 A quite different approachl5,16 attributes the origin of the narrow-band noise to the contact regions. In general, the pinning potential plays a crucial role in determining the low field and low frequency properties. In the most studied compounds, NbSe 3 and TaS3, it is extremely small so that the static dielectric constant £o can be as hlgh 17-19 as 107-108 and the threshold electric field ET for the onset of CDWconductivity can be as iow2,19, 20 as 10-1-10 -2 V/cm. Nevertheless, there is experimental evldencel9, 21 that the product E o E T is almost independent of the strength of the pinning potential for both strong and weak impurity plnning~ The classical model predlcts 4
e O ET ffi 4~@nel,
(i)
where -e is the electron charge and n the electron density. The constant @ depends on the form but not the strength of the pinning potential: @=1/2 for a quadratic potential repeated over every wavelength and 8=(2~) -I for a sinusoidal potential. The tunneling model proposed by Bardeen 5 and the deformable medium model proposed by Fukuyama and Lee 6 and Lee and Rice 7 yield an expression analogous to Eq. (i) with @ independent of the pinning strength~ Reasonable estimates 21 give 8=0.05 for the tunneling model 5 and 8=0.02 for the FuKuyama-Lee model 6, roughly one order of magnitude smaller than the classical model. The product ne% may be determined experimentally from the ratio of the excess current ICDW arising from the depinned CDW and the frequency ~ of the first harmonic of the narrow-band noise:2, 3 ICDW/I = ne%~
(2)
P~eliminary experimental results 21 indicate that @=0~03 for orthorhombic TaS 3 in the weak pinning regime, but some uncertainties exist 22 and further tests are neededo In NbSe 3 with strong impurity pinning one may infer e o ET ~ 9 A/MHz cm2 from Ref. [19] and ICDW/~ ~ 25 A/MHz cm 2 from Ref~ [23] so that @~0.03o Furthermore, it has been found 21 that £(E) is unchanged for applied bias electric fields E less than E TIn the classical model with sinusoidal potential the static dielectric constant in a bias field E is given by 4 E(E) =
2ne%ET [i - (E/ET)2]-I/2
(3a)
As E approaches E T from below, e(E) diverges~ This feature is common to any pinning potential with a smooth m a x l m u m h u t is not observed experimentally~21, 22 On the other hand, the 293
DIELECTRIC CONSTANT AND THRESHOLD ELECTRIC FIELD OF CHARGE-DENSITY-WAVES
294
quadratic potential repeated over every wavelength implies 4
where =
2~neA £(E) = ET '
= ~ + X<¢>,
(4a)
+ (l+e)$i = ~ - ¢i - psin(~+~i+8i)'
(4b)
( i = l , . . , N s)
<¢>
= %1 ~ *i"
(4c)
where ¢i is the phase of the i-th segment relative to the frame and B I is random between -~ and and represents the preferred position of the l-th segment due to the particular impurity distribution; ~ is the ratio between the "external" and "internal" relaxation times and X = NsVs/VF; and p are the electric field and the Finning strength in dimensionless units. Eqs. (4) can be easily integrated by computer for flnlte N S. The 81's can be chosen equally spaced between -~ and ~, so as to avoid statistical fluctuations due to a relatively small value of N S. This model is appropriate for treating the transport properties of a moving CDW in the strong coupling case, but it is not applicable to the weak coupling region where fluctuations in the impurity distribution insld~ a Lee-Rice domain play a crucial role. 7 Choosing the initial conditions (without electric field) such that -z<¢i
p = ne~, 2~
~ +
(3b)
i~e. g is independent of Eo Even though this behavior is in agreement with the experimental data, 21, 22 the quadratic potential is certainly unphysical. In order to overcome this contradiction, we have analyzed the phenomenologlcal model proposed by Tua and Zawadowski 13 which is a modification of the classical model 4 and includes internal degrees of freedom of the CDW condensate. A detailed description of the theory can be found in Ref. [13]. A single Lee-Rlce 7 domain is considered. The long-range order of the classical model 4 is preserved by postulating the existence of a rigid and macroscopic CDW frame whose phase is @ and whose volume V F is proportional to the volume of the domain~ The frame does not interact directly with the impurities and is subject to a frictional force~ The shortrange deformations of the CDW condensate are represented by internally rigid segments which interact with a sinusoidal pinning potential and are coupled elastically to the frame. The number N S of segments is proportional to the volume of the domain. Each segment has volume V S and is subject to an "internal" frictional force in addition to the same frictional force of the frame~ Under the assumption of overdamped motion, the dynamics of the system is described by the following equations:
and
Vol. 51, No~ 5
(5a)
-/- <~>. l+×
(Sb)
The static dielectric constant e(E) in a blas electric field E is e(E) = (ne)2y -I ~(~),
(6a)
~(~)
(6b)
where
4w dP/dE,
=
and y is the elastic coupling constant (per unit volume) of the interaction between frame and segments. 13 We have calculated ~(E) for different values of N s. The results are shown in Fig. 1 for the case p = 4.0 and ~ = 0.3. It is interesting to notice that i) e(E) is not divergent as ~ + ~ and ii) the dependence of ~ on ~ decreases as the number of segments N S is increased. These features are very similar to the classical model with quadratic potential. To further investigate the problem, we have calculated the dielectric constant ~ as a function of the frequency without bias electric field. The results are shown in Flg. 2 for the case N S = ii, p = 4°0, X = 0.3 and ~ = 0.i. The curve is very similar to the one for the overdamped harmonic oscillator, i.e. it can be fitted very well by the function Re E/Co = [i + (~/~)2]-i with ~ = 0.238. The results are independent of the strength of the electric field ~ as long~as E is not too close to E T. For E close to ET, hysteresis effects appear. Experimental data 19 on N b S e 3 w i t h strong impurity pinning obtained by radiation damage are in agreement with the present theoretical results, but further tests are needed~ It is now clear that the introduction of internal degrees of freedom to the classical model with sinusoldal potential yields features analogous to the classical model with quadratic potential. This explains why the latter model, although unphysl-
1.8
'
'
'
'
I
'
'
'
II SEGMENTS. . . . . 21
16
SEGMENTS _ _ -
31 S E G M E N T S _ _
'
I
'
'
'
'
I
'
l
i
I
x=O.3
~=4o
/
o~.~1.4
~w
Fig. 1
,; /
Normalized static dielectric constant ~(~) in a bias electric field ~ vs. for ~ = 4~0, X = 0~B, and N s = ii, 21, 31.
V61. 5], No. 5 I
295
DIELECTRIC CONSTANT AND THRESHOLD ELECTRIC FIELD OF CHARGE-DENSITY-WAVES I
I
l
tor,
)
I
1.0
=OJ =03
0.8
From Eqs. (4)-(6) one easily obtains: £im
~_~
~
4~
x(z+x)
o
(8)
"
=4.0
~0.6 3 x,~ 0.4 ~D
n,0,2 O0 0.0 Fig. 2
I
]
I
I
I
I
O.I
02
03
0,4
0.5
0.6
0.7
Real part of the dielectric constant vs. frequency ~ for p = 4.0, X = 0.3, and N s = ii. The frequency ~ is In dimensionless units (see text).
cal, is in agreement with the experimental datao19-22 We have also investigated the combined dependence of ~o and ~T upon the pinning strength p for different values of the parameter Xo Fig. 3 shows ~ -i vs. ET with N s = ii. For small values o~ p (i.eo small values of ~T) the product ~o ~T is independent of p and XThe constant 8 appearing in Eq. (i) can be expressed in terms of the dimensionless quantities ~o and ~T through the relation
Using the radiation damage technique, Fuller et al. 19 were able to vary the defect concentration in NbSe 3 samples. They found that the product e o E T was roughly independent of the defect concentration while both e O and E T varied about one order of magnitude. It is tempting to assume that the pinning strength p of the present model can be related to the defect concentration in the NbSe B samples. Within this a s s ~ p t l o n , the present theory for relatively small values of p is in agreement with the experimental data of Fuller et al. 19 Finally, we have considered the dependence of ~o and ~T on the parameter X which is the ratio between the total v o l ~ e of the segments and the volume of the frame. Fig. 4 shows the case with N s = ii and p = 4.0. As X-~° both T O and ~T approach a constant value. In this limit the volume of the frame vanishes and the present model becomes equivalent to the mean field theory proposed by Fisher. II Viceveraa, as x+O the total pinning of the domain vanishes so that ET also vanishes and ~o diverges.
II
segmenls
/
x = 0.7
id' @ = Eo~T/(8~2).
(7)
///
In the present case with N s = ii, we obtain @=0.015 in good agreement with the previously extrapolated value @=0.03 for NbSe 3 with strong impurity plnnlng. 19 From Fig° 3 it appears that as ~T -~= (and ~-~o) the dielectric constant go approaches a finite value° This behaviour can be easily explained° As ~-~o the segments become "localized" and the frame, which interacts with the segments hut not with the Dinning potential, behaves as a simple harmonic oscilla-
2x,O' l . .
........
'
.........
'
......
6.
~
x=O.I
id'
Iz~'°'
id'
,o'
101 I/
Id'
,
,
,
~
I
,
I~'
Io~
,d'
16'
,o °
Id
rcf
E, zx~d
zxso'
,d f'"
,d
Zx~O°
Zx~O°
i 0 O, l, l,,, , ( ~
,,IK:f . . . . . . .
Idl
......
I0a
i0 o
x
Fig. 3
Inverse static dielectric constant ~-i vs. threshold electric field ~T as t~e pinning strength p is varied°
Fig. 4
Static dielectric constant go and threshold electric field ~T vs. X for N s = ii and p = 4.0.
In conclusion, we have found that the static properties of the present theory are analogous to those of the classical mode! 4 with quadratic potential and are in agreement with the available experimental data 19 on NhSe 3 for the strong pinning regime. One of us (P.F.T.) is very grateful to G. Gr~ner and G. Mozurkewich for stimulating discussions. This work was sunnorted in part by the DOE Grant DE-AS05-81-ERI0959.
296
DIELECTRIC CONSTANT AND THRESHOLD ELECTRIC FIELD OF CHARGE-DENSITY-WAVES Vol. 51, No.' 5 References
10
2°
3.
4.
5~
6. 7. 8. 9~ i0.
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Iio
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