AC response of the vortex liquid in the high-T-c superconducting cylinder

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ELSEVIER Physica C 260 (1996) 8-18

PHYSICA

AC response of the vortex liquid in the high-Tc

superconducting cylinder

Chien-Jang Wu, Tseung-Yuen Tseng

Department of Electronics Engineering and Institute of Electronics, National Chiao-Tung University, Hsinchu, Taiwan Received 19 January 1996

Abstract

The ac response of the vortex liquid of a high-Te superconducting cylinder in the parallel field configuration is examined theoretically. The ac properties are analyzed from the associated complex ac permeability calculated on the basis of hydrodynamics theory. The permeability is primarily dominated by two frequency-dependent penetration lengths which are also shown to be independent of the geometries considered. In the simple flux-flow regime, the second peak in ac loss is observed to be enhanced while the first peak is depressed for the superconducting cylinder. In the viscous flux liquid, the ac response is found to rely on the viscosity, ratio of shear modulus to nonlocal compressional modulus, and the geometries discussed. The results indicate some basic difference in ae permeabilities between superconducting cylinder and slab. Keywords: Mixed state; Penetration depth,; Type II superconductors; Flux lattice

1. I n t r o d u c t i o n

The discovery of high-Tc superconductors (HTSC) has revived the study of vortex response to the ac field in these high-K superconductors. The investigation of the ac response in the mixed state can be alternatively used to understand the vortex dynamics. At present, the measurements of vortex response are performed by ac techniques such as the vibrating-reed mechan- ical resonator [ 1 ], surface impedance [ 2 - 4 ] , and ac magnetic permeability [ 5 ]. All these techniques have the common feature that a small alternating magnetic field is superimposed to a steady field. In the linear response regime, the ac response is independent of the amplitude of the alternating field. The ac field is applied parallel to the surface of the superconductor which in turn generates screening currents near the surface. The currents then produce a Lorentz force act-

ing on the vortices and cause an oscillation of vortices. The oscillating vortices propagate into the interior of the superconductor due to the vortex interaction. The moving vortices are, however, impeded by some friction or pinning. Hence vortex dynamics strongly influ- ences the ac properties of type-l/superconductors, for instance, the ac permeability or the surface impedance, which are widely considered both experimentally and theoretically.

Much effort has been made to investigate theoretically the electromagnetic response o f a vortex system in high-Te superconductors. A unified theory of the effects of vortex pinning, flux creep and flov¢ on the surface impedance and rf magnetic permeability of isotropic type-II superconductors has been developed by Coffey and Clem [6-8]. The vortex dynamics is treated self-consistently and a complex ac penetration depth is calculated to study the high-frequency vortex

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C.-J. Wu, T.-Y. Tseng / Physica C 260(1996) 8-18 response. The dissipation peak occurs when the com-

plex penetration depth is of the order of the sample size and is usually ascribed to the skin size effect [9,10] based on the model of thermally activated depinning of vortex lines. For a pinned vortex solid, Koshelev and Vinour [ 11 ] investigated the frequency response within the framework of the collective pinning theory. They have incorporated the thermal fluctuation into the Campbell theory. Based on the thermally activated flux flow (TAFF) model [12,13], Yeh [14] studied the high-frequency vortex response near the depinning threshold. A discussion of linear ac response in the mixed state from the unique macroscopic viewpoint has also been given by van der Beek et al. [ 15].

The ac techniques are also very useful in the deter- mination of the irreversibility line in the (H, T) phase diagram of the disordered HTSC [16]. The ( H , T ) phase diagram, however, exhibits a large region where the vortex lattice is in a liquid phase [ 17], i.e., the vortex-liquid. Therefore the ac response of the vortex liquid on the TAFF basis appears to be no longer useful, since the TAFF theory does not completely incorporate the vortex-vortex interaction. The ac response of flux-line liquid has recently been considered within the framework of hydrodynamics by Chen and Marchetti [18]. The hydrodynamic treatment has incorporated the viscoelastic nonlocality effect arising from intervortex interaction and entanglement. The ac permeability of a slab and surface impedance of an semi-infinite sample have been given in Ref. [18].

In this paper, we extend the work of Chen and Marchett [ 18 ] and calculate the ac response of the vortex liquid in the superconducing cylinder. As cus- tomary, the most commonly used geometries for ac or microwave response in HTSC are the slab and cylinder. The calculated complex ac permeabilities for slab and cylinder in various approaches are now available [ 12,19]. It is therefore of interest to calculate the corresponding permeability for a cylinder in the vortex liquid state which has not been seen thus far. We also illustrate some basic difference in both real and imaginary parts of permeabilities for the slab and cylinder. The problem considered here is applicable in the microwave cavity perturbation experiment especially in the parallel field configuration [20].

2. Basic equations

We consider a uniaxial type-II superconducting cylinder (assumed to be infinitely long) with axis in the z-direction and cross section parallel to x - y plane. For high-Te cuprate, the c-axis is chosen as the z-direction and the a - b plane as the x - y plane. A static magnetic field H = ~H is applied parallel to the surface of cylinder to generate the vortex array. The density of vortices no and the average flux density B0 = ~B0 are related by no = Bo/qbo and ao "~ 1/ v ' ~ , where ~b0 is the flux quantum defined by ~b0 = h c / 2 e and a0 is the intervortex spacing.

In this paper, we are interested in the regime Hcl << B << Hc2, namely, ~:ab << a0 <<

,~ab,

where

Gab

and Aab are the coherence and penetration lengths in the a - b plane, respectively. In the parallel configuration, the demagnetization factor is ignored so that B0 ,~ H. We assume the vortex array is in a liquid phase and the vortex dynamics is described by some hydrodynamic fields coarse-grained over several lattice spacings.

The electrodynamics of a superconductor is well described by Maxwell's equations

1 ab V x e . . . . (1) c at 4¢r 1 0e V x b = - - j + (2) c c at V . e = 4¢rp (3) V . b = 0 , (4) two-fluid equation J = Jn + Js, (5)

and London equation in the presence of vortices, [ 18 ] V x A . Js = ~ ( - b + ~b0T), (6) where e and b are the local fields, Js the supercurrent, Jn the normal current specified by Jn = One, A is a diagonal tensor with elements Axx = Ayy = A~b and Azz = A2c , and T is defined by T = ~:n + 7" with n and ~" the coarse-grained hydrodynamic density and tilt fields of the vortex array, respectively. By making use of the above equations, one easily has

4~o',,A 2 a b

(3)

10

In obtaining the Eq. (7), we have dropped the term arising from displacement current density because the ac response considered in this context is restricted to frequencies no higher than microwave.

In addition to the above-described equation, to investigate the response of the vortex liquid to the ac field, we include some hydrodynamic equations to describe the vortex dynamics. For the linear response regime, the linearized hydrodynamic equations are given by [ 18] O 8n/Ot + n o V ± - v = O, (8) - y v + rI,V2v + r / b V ± ( V ± • v) 02V 1B0 x j = 0. ( 9 ) +~Tz OZ 2 c

Eq. (8) in fact is the equation of continuity for the areal density of vortices with 8n = n - no, the devia- tion from the equilibrium value, no. Eq. (9) behaves as an equation of motion for the moving vortex with velocity v, and friction coefficient y. The coefficients, rl~, r/b, and r/z are the shear, bulk, and tilt viscosity constants, respectively. A detailed description of the viscosity coefficients can be found in Ref. [ 18]. Eqs. ( 7 ) - ( 9 ) serve as the foundation for investigating the ac response o f the flux-line liquid in type-II superconductors.

C. -J. Wu, T.- Y. Tseng / Physica C 260 (1996) 8-18

skin depth given by An? = 4¢riwtrn/c 2. On the other hand, Eqs. (8) and (9) can be expressed as

l d

iwtn - noo--~o(pvp) = O, (11)

Top - ~Te--~pp (PV°) + --c = O, (12)

where we have used the velocity of moving vortex as

v = - ~ v o ( p ) , and ~le = ~ls + ~Tb is the wavevector and

frequency-dependent longitudinal viscosity given by Ref. [ 181,

He(q, w) = r/e (13)

1 + iw~'t(q)"

with static viscosity r/t = He, (q = 0, w -- 0) and char- acteristic relaxation time ft. In the case of w >> 1/re, the hydrodynamic theory reduces to the continuum elasticity for the vortex lattice system, correspondingly

[181,

3. C a l c u l a t i o n a n d d i s c u s s i o n o f ac p e r m e a b i l i t y for a c y l i n d e r

We now consider the above configuration for a cylinder with radius a in cross sectional area. An ac field Ha = ~ n a e iwt is applied parallel to the surface

of the cylinder (also parallel to H ) , with 8Ha << H. The local field b in Eq. (7) is thus assumed to be

b = ~SBz ( p ) e iwt and ~- = 0 since the vortex lines are

on average aligned along the z-direction. Eq. (7) in this case becomes

1 d f d t B z , ( 1 1 )

-iT- 6Bz

p .

p \

p

) +

--: +

:

= ~ t n ( p ) , (10)

where we have denoted the London penetration length Aab as A for simplicity, and Any is the normal-fluid

7"t = r l d [ c H ( q ) + G - cL(q)], (14)

zs = n , / C , (15)

where cll and G are the compressional and shear elastic moduli of a vortex line elastic medium, respectively. Also the longitudinal compressional modulus

CL(q) and Cll are related to Cll (q) = cL(q) -{- C66 with

C66 "~ G << CL(q). Here C66 i s the dispersive shear modulus. The expression for bulk modulus cL(q) can be found in Ref. [21].

The response to the ac field can be obtained by solving 8Bz in Eq. (10). Eq. (10) is solvable with the help of Eqs. (11) and (12) together with the boundary condition 8Bz ( p = a) = 6Ha. In this article, we consider two conditions in solving Eq. (10). One is the sim- plest case with f/e = 0, i.e., the flux-flow regime. The other is the viscous liquid with ~)e =/= 0 in which the effect of viscosity on the response can be elucidated.

Let us begin to consider the simple flux flow condition at Ht = 0. By combining Eqs. (2), (11) and (12), we have

l d

f

d S B z ' ~

8 n ( p ) =l~.f-~'~#____ ~ p - - ~ p ) , (16) where a f is the ac penetration length of a vortex liquid given by Af 2 = iWy/CL(O) with cL(O) = B~/4zr. A

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C-J. Wu, T.-E Tseng/Physica C 260 (1996) 8-18

//'l+A.~'~ 1 d f d S B 2 ~

t,P---Z p )

(,,)

Tg-- 8B2 = 0. (17)

+ ~7 + ,~,f

The solution of Eq. (17) is easily obtained from the boundary condition; the result is

8Bz ( p ) = 8Halo(P/Aac) l / o ( a l A a c ) , (18)

where I0 is the modified Bessel function of the first kind of order zero, and the complex ac penetration length Aac is defined by

( w ) : 1-¢ a- a ) ' (19)

which is exactly the same as that given in Ref. [ 18]. It alternatively means that the ac penetration length relies only on the model considered and has nothing to do with the geometry used. By the way, the ac penetration length/~ac given in Eq. (19) is very similar to that developed by Coffey and Clem [7,19] in the absence of creep. The associated complex ac is directly evaluated from the integral

2 ~ r a I x ( w ) - ~a2~$H a ¢ $ B z ( p ) p d p d O , (20) o o which is given by I x ( w ) = 2 Aac I1 (a/Aac) (21) a l o ( a / a a c ) '

where ll is again the modified Bessel function of the first kind of order one. Here the argument of the modified Bessel function is complex-valued and can be numerically evaluated through the integrals

I o ( z )

= --¢rl

f cosh(z cosO)dO,

(22)

0 ~ r = _ e z cos0 cosOdO. (23) I I ( Z ) 7r 0

The penetration length in Eq. (19) strongly depends on the frequency of the applied ac field and classified into three regimes by two frequency scales. One is the

11 g == o 10 0 10 ~ . . . ' \ . . . l ' , . . . I , , , , ' " ' 1 . . . ,,,i ' ' " ' " ' i . . . ' "

\

C % f / % = 5 0 q=0 . . . I 10-2 10-1 10 0 101 10 2 10 a 10 4

reduced frequency co/%

Fig. 1. The normalized complex ac penetration length IAac(W)I/~ in Eq. (19) as a function of reduced frequency w/w[ for the nonviscous vortex liquid with Wnf/Wf = 50. Three regimes are identified as described in the text.

frequency related to flux diffusion, w f = cz.(O)/'yA 2.

The other w , f =

c2/(4~ro-.a 2)

is dominated by normal fluid. For w << w f , the field penetration is con- trolled by simple flux flow, and Aac(W) ~ Af(w) ,-,

w -1/2. In the higher frequency regime, w f << w <<

w , f , one has Aac(W) ,,~ A which indicates the static

Messner response with a real penetration length. As the frequency becomes much higher, say w >> Wnf,

t~ac( W ) ~ ~.nf( W) is again proportional to w -1/2. T h e

response is thus dominated by the normal fluid. The overall frequency-dependent feature of Aac is given in Fig. 1 of Ref. [18], and replotted here in Fig. 1 for convenience of discussion. In Fig. 2, we illustrate the ac permeability of the cylinder in Eq. (21) where comparsion with the slab has been made, too. The permeability of a slab with thickness 2a was calculated by Chen et al. [ 18 ] as

bt(w) = ( aacla) tanh(a/Aac). (24)

As can be seen in Figs. 2a,b, the behavior of br =

IX# _ iixtt for the cylinder is somewhat distinc from

that of the slab. In Fig. 2a, a / A = 10, the peak height in IX" at lower frequency is depressed from 0.417 for the slab to 0.367 for the cylinder, whereas, the second peak height at higher frequency is enhanced. Fur- thermore, the peak frequencies for the cylinder are in-

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12 C-J. Wu, T.-Y. Tseng/Physica C 260 (1996)8-18 =~. ,,= =~ 8 g 1.0 .8 .6 .4 .2 1.0 (a) ... ... ' ... 'c'L"er ... ":. \ ... slab :\ g, ! ~ ~nf/O)f=50 :~i \ a/L=lO 10-2 10 4 10 0 101 10 2 10 3 .8 ":::L _II .6 ::L =~ .E .4 .2 0.0 0.0 10 .3 10 4 reduced frequency m/% (b) slab i

o~.,m=5o

~.,..

\

~i

10-2 10-1 10 0 101 10 2 10 a

reduced frequency (o/%

Fig. 2. (a) The real and imaginary parts of ac permeabilities for both cylinder (radius = a) in Eq. (21) and slab (half width = a) in Eq.

(24) at f / = 0, wnf/wf = 50 and a = 10A. (b) The real and imaginary parts of ac permeabilities for both cylinder (radius = a) and slab

(half width = a) at ~ = 0, wnf/wy = 50 and a = 2A.

creased, especially for the peak located at the lower frequency regime. Another significant feature shown in the cylinder is the disappearance of the coincidence in # ' a n d / z " which occurs around

w/ff

~ 0.1 for the slab. For the smaller ratio of

a/A,

depicted in Fig. 2b, the above features remain observable. Furthermore, the depression in the first peak and the enhancement in second peak o f / x " for the cylinder appears to be more pronounced.

The results shown in Fig. 2 elucidate the main difference arising from the different geometries considered. Concerning the nature of this two-peak behavior in /z", one is refered to Ref. [ 18]. Here we only demon- strate the difference due to the geometric effect. The real part,/z' of the cylinder is found to be larger than that of the slab for all frequencies. This signals that the resonance frequency is strongly geometry-dependent, when the superconductor is considered to be a part of the resonant circuit.

We now go on to study the ac response of viscous vortex liquid for a superconducting cylinder. For a viscous liquid, the response due to vortex-vortex interaction and entanglement becomes nonlocal. The nonlocality is manifested by a new viscous length scale

( w ) = (~ (w)/y

) 1/2. The electric field generated by

the vortex motion is

Eo(p) = (1/c) Bovp + ( 4¢ra2iw/c) js.

(25) After some manipulation, we have

dp

(pv~)

Bo dp

-~p (pEe)

4¢rA2iwd(lff~

)

Bo d p ( P L ) •

(26)

With the help of Eqs. (25) and (26), Eq. (12) further becomes

=-pfjo+a~A:iw [js-t32J~(1J-~(pjs))].

(27) Here the flux flow resistivity py --

B2o/cZy

has been introduced. On the other hand, Eq. (12) supplemented by Eq. (2) gives

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1 d (pd(3Bz~

p dp \

dp ]

47"ry 1 d

- Bo pdp (pc°)

4~Y(321d--d- [P ff'; (1J-p(pV°))]

p dp

( 2 8 )

By making use of Eqs. (11) and (28), Eq. (10) can finally be, after some calculation, expressed as

[ d ( 1 d f ~,

,],]]

1 + A 2

An: p dp

1 d { d(31)z'X

-(:

+

ko

) +(38z

= 0. (29)

The solution for

6B z

in Eq. (29) can be assumed to be of the form

(3Bz (p) = AIo(p/Al) + BIo(p/A2),

(30) where we have introduced two ac penetration lengths A1 and A2, and the coefficients A and B will be determined later. Feeding Eq. (30) into Eq. (29), we find A2 1 [ 2 (32

1,2 = ~ ,a,,~ +

4- (((32 + A2ac)2

4A2(32

(31) which is again identical to the ac penetration lengths when the slab is considered [ 18]. Based on Eqs. (19) and (31 ) we conclude that the complex ac penetration length seems to be an intrinsic property related solely to the model for vortex dynamics being considered and does not depend on the geometry of the sample and radius of the cylinder or width of the slab. This important feature is also observed in the theory of self- consistent treatment of vortex dynamics as considered by Coffey and Clem [ 19]. The complex penetration lengths may be dependent on the orientation of the static field generated inside the supercondutor as can be seen in Ref. [8]. Nevertheless, it turns out to be also geometry-independent. Eq. (31) reveals that the penetration lengths AI,2 are closely related to the flux

C.-J. Wu, T.- Y. Tseng / Physica C 260 (1996) 8-18

13

flow penetration length

Aac

in Eq. (19), viscous length (3, normal-fluid skin depth

(3n f,

and the static London penetration depth A.

Next, the coefficients A and B in Eq. (30) can be evaluated as follows. By substitutingEq. (30) into Eq. (10) together with

(3n(p

= a) = 0 and the continuity condition

(3Bz (p = a)

=

(3Ha,

we obtain the complete expression for

(3B z (p),

(3H a (3nz ( p ) = ,,~2 2 -- ,,~12 / I o ( P / A 1 ) x

~,(a~ -2

-

a - :

-

a2f)

_ ( A / 2 _ 2_ 2 _ An?)

IO(p/A2)

/0(a/a2) )

(32)

Hence, the corresponding electric field

Eo

is given, through Eq. (1), by

E o ( p ) =

-iw

_{c A -2 - ~-2}

8 H a

x [(A~ -2 - A -2 - A~-:) "AI-[o(a/A1)II(P/A1----~)

Ii

(p/a2)

- ( a ? 2 - a -2 - a 2 ~ ) '

^2-fo(p/ a2---

5 ]"

(33)

a~-2 _ ai-2

x

A 2 An~f a lo(a/Al)

1 1 ) A2 2Ii(a/A2) ]

(34) It is worthwhile to compare the ac permeability of the slab in the viscous flux liquid, the result is given as [ 18],

z ( w ) =

[ ( 1 2 1 1 ) A l t a n h ( a )

~(w) =

The associated complex ac permeability is thus directly calculated as

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14 C.-J. Wu, T.-Y. Tseng / Physica C 260 (1996) 8-18 (a) 4o 8 15 o)1/%= 0.001 G/CL(0)=I 0 .3 8JZ=l . . . I . . . I . . . I = ~ H d . . . , , d , , , ~ , , , d 10 .2 10 -1 10 ° 11 "E 1.0 0 . . . I . . . 0 . 0 10-3 101 1 0 2 10-3

reduced frequency (o/%

(b)

, ~ . ... slab -

:~, \ o I/o)~0.001

"~ \ G/ct(0)=l 0 ~ -

10 -2 10 "1 10 o 101 10 2

reduced frequency o/%

Fig. 3. (a) The normalized complex penetration lengths [A1,2 (w) l / a given in Eq. (31 ) as a function of the reduced frequency w/wf at

we/wf = 0.001, G/cc(O) = 10 - 3 and t$0/A = 1. The la2(w)l/a is negligibly small and is not observable. The normal-fluid contribution

has been neglected in Figs.3-6. (b) The frequency dependence o f complex permeabilities in Eqs. (34) and (35), at a/A = 10 and same

conditions as in (a).

( ~

1 1 ) A~tanh(~

) _

A 2 An~f ' a

(35)

The coefficients, aside from terms including the dependence on size, are both identical for slab and cylinder (which is reflected in the coefficients A, B described previously). The change in the vortex areal density due to ac field penetration is readily obtained from Eq. (10), we have

S n ( p ) = qbo A2a - A 2 \ I o ( a / A 1 )

Io(p/az)

lo(a/A2) ) "

(36)

The location P0 where 6n reaches a maximum is easily determined, it satisfies the equation

A1 l o ( a / A l ) I

Ii (po/AI) = A2 ~ 1 (p0/A2), (37) which can be numerically evaluated for P0 using the fact that JAil >> [A21 as described below. The condition for P0 is clearly more complicated than the slab where P0 is approximated at

la21 [ 18].

The complex ac permeability and surface impedance of the superconductors are the main quantities used extensively to study the ac response. For a superconductor in the geometry of occupying semi-infinite space, the most relevant element is surface impedance

Zs = Rs - iXs, where Rs is the surface resistance and

Xs the surface reactance. For a superconductor in the shape of a cylinder, slab or prism, one usually defers the ac permeability for a theoretical investigation. The role of the imaginary part of the ac permeability is equivalent to the surface resistance, both of them represent the ac dissipation of the superconductor. Similarly, the behavior in Xs is similar to the real part of permeability. From the above results, together with the results reported in the literature [9,12,19,20,22], one concludes that the permeability of a slab or prism varies as a function of the hyper-tangent with complex argument, whereas the cylinder depends on the modified Bessel function, also with complex argument. All the arguments are essentially related with the model-dependent complex penetration length along the sample dimension, as illustrated in Eqs. (34) and (35). These two special functional dependences of the permeabilities for slab and cylinder appeared

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C.-J. Wu, T.-Y. Tseng/Physica C260 (1996) 8-18 15 15 " ~ lO Z (0t/(0f=0.01 G/CL(0)= 10 -2 a0/Z=l

(a)

1~2(m)l/~. 1 0 - 2 1 0 - 1 1 0 0 1.0 0.8 u 0.6 .E o.4 0.2 0 , . . . i ' ' ' ' ' " ' 0 , 0 10 -3 101 102 10 -3

reduced frequency o)/%

(b) , ",. \ ... slab - o) t/%=0.01 i

!

\

G/c~(o)=lO-~

\

5JZ=1

i

10 "2 10 -1 10 ° 101 102 reduced frequency 0>/%

Fig. 4. (a) The normalized complex penetration lengths IA1,2(w)[/A in Eq. (31) at we~w~ = 0.01, G/cL(O) = 10 - 2 and 6o/A = 1. The

IA2(w) I/a again is very small. (b) The frequency dependence of complex permeabilities for slab and cylinder at a/A = 10 and same conditions as in (a). There is no observable change in the response compared with Fig. 3b.

to show universal behavior. As regards the surface impedance, it is usually described not only in terms of the ac penetration length or the ac resistivity [ 18]. The primary issue is the related model for vortex dynamics in determining the key result, the complex ac penetration length.

We proceed to examine the effects of viscosity and geomety on the ac permeability of the cylinder in Eq. (34). From Eq. (13), the viscous length 8(w) is expressed by S(w) = 60(1 + iwT"e) 1/2 with static vis- cous length 60 = r ~ - / y . In order to analyze the viscous ac response, two frequency scales are introduced [18], namely, we = 1/~'e = G/rle and w, 7 = cL(O)/rle where we describes the relaxation of shear stresses and dominates the length 6 ( w ) , whereas the w, 7 is the fre- quency such that I,~f(w) 1 ~ 6o. Clearly, both of them decrease with increasing viscosity Be. Also, the ratio G/cL(O) = w J w n will be much smaller than unity for HSTC at the temperature of interest [ 23 ].

The frequency we significantly indicates the crossover frequency from liquid-like to solid-like response of the vortex lattice. At low frequency w << we << w, 7 the dynamics is simply due to flux flow of a Vortex liquid. For frequencies w >> we, the model then describes the flux flow of a vortex solid. As

discussed by Chen and Marchetti [ 18], the only difference between liquid-like and solid-like responses arises from the small difference in the compressional moduli. In Fig. 3a, we plot the ac penetration lengths

lall/a

and

IA21/A

as a function of reduced frequency w / w y in conditions of neglecting the normal-fluid contribution we/wy = 0.001, G/ct.(O) = 10 -3, and

80/a --1. la21/a is

negligibly small and obscure for all frequencies. That is, al (w) ~ Aac, and A2(w) ~ O, consequently, the effect of viscosity is very unobserv- able. The response shown in Fig. 3b is very similar to the Fig. 2a at w / w f < 10.

The disappearance of the second small peak in Fig. 3b is natural, since we have now disregarded the influence of the normal-fluid. The same situation is also observed in Figs. 4a,b, where we have taken G/cL(O) = 10 -2 and small viscosity 6o/A = 1. From the results displayed in Figs. 3 and 4, one is tempted to conclude that the viscous effect may not be observable at very small values of G/cL(O) and low static viscosity. The difference in response between cylinder and slab is again clearly illustrated in Figs. 3b, 4b. To observe the viscous effect on the permeability, we take G/ct.(O) = 1 and 8o/A = 10 and the results are shown in Figs. 5a,b where the normal-fluid contribution is again ne-

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16 C.-£ Wu, T.-Y. Tseng/Physica C 260 (1996)8-18 15 "-~ 10 1.0 (a) . . . '1 . . . " 1 , , , ' ' " q ' ' . . . I ' ' ' . . . %/%=0.01 G/CL(0)=I 5o/2= 10 i;q(o,)i/x -T.~-,,,,,I ... L ... , , ,?;-:~,.r--~- ... 10 -2 10 -I 10 ° 101 102 0.8 ,,= 0.6 = L 0.4 0.2 0 0.0 10 "3 10 "z

reduced frequency co/%

(b) . . . I . . . I ' ' ' ' ' " ' 1 . . . I ' ' ' ' " ~ ' ~ cylinder ' " . ~ ... slab ''". /1, C ' " : ~ o t/%=0.01 '"~. ~ G/CL(0)=I "',., ~ ao/Z.=10 10 -2 10 -1 10 ° 10 ~ 102

reduced frequency co/COl

Fig. 5. (a) The complex penetration lengths la,,2(w)/al at

wdwf

= 0.01,

a/cL(O)

= 1 and 80/A = 10. The Ia2(w)I/A in all frequencies

is not observable. (b) The frequency dependence of complex permeabilities for slab and cylinder at a/A = 10, w , / w y = 0.01, G/cL(O) = 1

and ~0/A = 10. A new second peak at very low frequency region is observed because of the large viscosity.

glected. As can be seen, the viscous effect makes A2 no longer negligible and is of the order of the static penetration length A which in turn will impede the flux penetration. Fig. 5a shows that at

w/wf

<< 10 -2 or w ( ( w,/, ~1 ~ /~f and A2 is negligible. For

w/w/>

10 -2, we have, A1 ~ 80 and A2 ~ A. Therefore, a new second peak occurs at w ,-~ w, 1 = 10-2wf, as demon- strated in Fig. 5b. The new peak height is larger than the one to the right and has lower peak frequency. Furthermore the peak height has been depressed considerably due to the large viscosity. For much higher frequencies,

W/Wf

> 20, i.e.

w/w, 1

> 200, t h e / z " becomes negligibly small a n d / z t approaches a con- stant. The position

W/Wy ~

20 in Fig. 5b reveals AI A2 indicating that the field penetration is considerably small. In other words, the dissipation in/.," can be disregarded in the high frequency regime. Concerning the influence of sample geometry, Fig. 5b clearly demon- strates the main features in/z where the two peaks in /z" are more obvious in the cylinder than in the slab, Also, the variation in /z" in between the two peaks o f / z " seems to be more pronounced for the cylinder. This again suggests the response depends strongly not only on the complex penetration length but the sample geometry. In Fig. 6a, we consider the case of very

large viscosity 80/A = 100 and

G/cL(O)

= 102. In this case the crossover occurs around

w/wf

= 10 -4 or w = wn = 10-4w f, where/z" attains a maximum. The very high viscosity has effectively reenhanced the dissipation peak and makes the peak frequency smaller than that in Fig. 4b by two orders of magnitude. The most important observation is that the peak frequency in this case is almost the same for both cylinder and slab. This can be understood from Fig. 6a where

lall

>> a at the crossover frequency and [A2I ~ 0 indicating that the ac field cannot penetrate into the sample due to the viscous screening [ 18]. Therefore the crossover frequency naturally does not vary with the geometry considered. Nevertheless, the overall magnitudes of the response actually depend on the shape of the sample. As displayed in Fig. 6b, there exists a crossing between/.d and/*" in the slab at 10 -4 < w/wf < 10 -3, while the cylinder does not exhibit this crossing and is fully separated i n / ~ a n d / ~ ' .

4. Summary

The linear ac response analysis of the flux-line liquid of the high-Te superconductor in the shape of a cylinder has been carded out. The complex ac perme-

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C.-J. Wu, T.-Y. Tseng/Physica C 260(1996) 8-18 17

(a)

15 ... ,,,~ ... v

% 1%=0.01 ,-e G/CL(0)=102 10 ~5°/~'=100 .c _.e .10-5 104 . . . I . . .

IZ~(~)l/I.

i . . . I , , ,,,,.,I , , ... I , , ,,,,, 10 -a 10 -2 10 "t 10 °

reduced frequency o)/%

"3 =L :=. =., (b) I.o ... I ... I ' , . . . i ... .

(o~/0)~0.01

- D6 ~ " \ G/CL(0)=I 0 ~ -_

\~

Vz=1oo

"

.4

.. 7.

a/;~=10

- .2 . ... , - i ' 1 0.0 10-5 104 10-5 10-2 10-1

reduced frequency ~o/%

Fig. 6. (a) The complex penetration lengths IA1,2(w)I/A at we/wf = 0.01, G/cc(O) = 102 and 8o/A = 100. The crossover occurs at

w / w f ,~ 10 -4. (b) The frequency dependence of complex permeabilifies of slab and cylinder at a/A = 10 and conditions given in (a).

ability for the c y l i n d e r is given and comparsion with the slab is also made. In the nonviscous flux liquid, the two peaks i n / z " d e p e n d on both the sample g e o m - etry and size. The p e a k located in the lower frequency region indicates the crossover o f response from flux liquid to flux solid and the height is decreased for the cylinder. The higher frequency p e a k arising from the contribution o f the normal-fluid is however increased in the cylinder. In the case o f the viscous flux liquid, c o m b i n e d with the neglect o f the normal-fluid, the response o f low viscosity shows some increase in p e a k height and a shift in p e a k frequency for the cylinder. For very large viscosity, w e f i n d that the peak frequency essentially remains unchanged for both cylinder and slab. The analysis presented here provides s o m e fundamental information about the p e r m e a b i l i t y which is o f p a r a m o u n t i m p o r t a n c e in the arrangement o f s a m p l e d i m e n s i o n and geometry when experiments are performed.

A c k n o w l e d g e m e n t s

The w o r k is supported b y National Scientific Coun- cil through N o . N S C 85-2112-M009-037 PH.

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