Magnetic flux line in a disordered medium

PHYSICA Physica C 233 (1994) 195-202

ELSEVIER

Magnetic flux line in a disordered medium Yoshihisa Enomoto a,., Sadamichi Maekawa

b

a Department of Physics, Nagoya Institute of Technology, Gokiso, Nagoya 466, Japan b Department ofApplied Physics, Nagoya University, Nagoya 464-01, Japan

Received 27 June 1994; revised manuscript received26 August 1994

Abstract

We numerically study a single flux-line motion through a disordered medium, on the basis of the Langevin-type stochastic equation of motion. The flux line is driven by a transport current and also fluctuates by the disorder and the thermal noise. Performing large-scale simulations of the model for parameters associated with the YBCO sample, we examine some scaling exponents characterizing the thermal and disorder effects on the flux-line motion with and without the transport current.

1. Introduction

Thermodynamical and transport properties of flux states in high-To superconductors (HTSC's) have been found to exhibit fascinating behaviors, compared with conventional ones, such as a strong nonlinearity of the I - V curve and a sign reversal of the Hall resistivity [ 1 ]. It has been conjectured that these behaviors result from characteristic material properties of HTSC's, such as the small coherence length, large characteristic temperature, anisotropic structure, and high concentration of impurities. Since the discovery of HTSC's, various concepts have been proposed to explain these peculiar behaviors, such as melting of the flux-line lattice and vortex glass [ 1 ]. However, consistency among them is still inconclusive at present. This may be due to the complicated coupling of the above causes. Under these circumstances, to examine the dynamical properties of flux lines (FL's) as well as the static ones, we have recently proposed the Langevintype stochastic equation of motion for the FL posi* Corresponding author.

tion [2,3 ]. Performing computer simulations of the model for some typical cases, we have found that the model is potentially rich, and reproduces nontrivial static and dynamical properties of FL's. In the present paper, we extend our previous work on the two-dimensional interacting point vortex system [ 3 ], which is driven by the transport current in the presence of random impurities and the thermal noise, to the full three-dimensional FL system. Due to the complexity of the problem considered here, we would study rather a simple situation, that is, a single FL motion in HTSC's. The present study should therefore be considered mainly as the first step towards the study of the FL dynamics in HTSC's for a realistic situation. Using a computer simulation technique, we discuss effects of the thermal noise and disorder on the FL dynamics with and without the transport current. Similar subjects, especially the latter aspect, can be widely seen for a number of physical systems [4]. Examples include the magnetic domain wall in a random field, interface of two fluids in porous media, spreading on heterogeneous surfaces, and sliding of charge density waves in random impurities. Thus, the

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Y. Enomoto, S. Maekawa / Physica C 233 (1994) 195-202

present model can be regarded as a representative example of a general class of problems of the collective nonlinear transport of extended objects or manifold, such as domain wall and vortex, in random media driven by an external driving force [4] (transport current in this case). Moreover, it has been found that the driven manifold through a disordered medium exhibits the threshold behavior, called the pinningdepinning transition, at a certain threshold value of the driving force [ 4 ]. Above the threshold value the manifold moves on the average through random media, while below it the manifold is stuck in one of many metastable configurations. Recently, it has been pointed out [4] that such a threshold behavior can be regarded as a dynamical critical phenomenon and thus can be characterized by scaling laws and critical exponents. Some critical exponents near the threshold have been obtained at zero temperature by using a dynamic renormalization-group method [ 4 ]. However, the theoretical treatment of the FL dynamics considered here has not been solved yet. Thus, one of our present purposes is to study numerically the threedimensional FL motion, especially the I - V curve, from this point of view. Comparison of the present result with those of the above analogous systems will be discussed separately. Th e paper is organized as follows. In Section 2 we present the model. In this model equation, effects of the line tension, the thermal noise, the random impurities, and anisotropic properties are taken into account, as well as the transport current. Random impurities are modeled as the time-independent or quenched random potential, and the effect of the transport current is realized by the Lorentz force. In Section 3 we comment on the computational method to solve the model equation numerically. Values of various parameters are chosen to be those in YBCO samples. In Section 4 we discuss several simulation results. In particular, we examine some scaling exponents characterizing the FL dynamics in random impurities with and without the transport current, and also study their temperature dependence. Section 5 concludes the paper.

We here take a coordinate system whose z-axis coincides with the c-axis of anisotropic HTSC materials, i.e. the normal to the CuO2 planes or the ab-planes, with basic unit vectors ~, ~ and ~ along the respective direction. An external magnetic field is applied along the z-axis, and the transport current flows in the ydirection, denoted asia. The position of the FL at time t is described by r(z, t) = (x(z, t), y(z, t) ). For simplicity, we use a small curvature approximation of the FL, which is justified for the small transport current and the weak pinning considered here [ 5 ]. Since the ratio of the in-plane penetration depth 2ab to the inplane coherence length ~b is typically quite large, the Ginzburg-Landau parameter being l~ab~'~ab/ ~ab= 100, we shall often work in the London limit. In addition, we model random impurities as the timeindependent or quenched random potential Vp(r, z), characterized by zero mean and the correlation defined by [ 1 ]

Vp(r, z) Vp(r', z') =np%E~(T)(l--~c)J(r-r')J(z-z'),

where the overline denotes the average over different configuration of impurities. Here, np is the impurity strength, vp the volume of the unit cell of the underlying crystal lattice, and %(T) _= (~o/(4n2ab(T) ) )2 with flux quantum q~oat temperature T. In these situations, the free energy of a single FL, r(z, t), reads [ 1 ]

F[r(z, t) ] =

0

with the line tension ~ (T) --g% (T)ln Kab and the anisotropic parameter g ~ mab/mc, where rnab is the inplane effective mass and mc the effective mass describing the weak Josephson coupling between the CuOz planes. The dynamics of the FL can be described by [2,3 ] Or

2. The model

Now we specify the present model. We consider a single FL in a bulk HTSC sample with thickness L.

(1)

8F

F - ' - ~ = - ~r +f(z, t ) ,

(3)

where F - 1 is the friction coefficient given in the highKab limit as

Y. Enomoto,S. Maekawa/PhysicaC 233 (1994)195-202

197

wr(z-z')2nC2~ab(T) 2 ,

(4)

1 x/<<(r(z,t)_r(z,,t))2>>~lz_z,l~, with the normal-state conductivity a and the light velocity c [6]. The last term in Eq. (3) describes the thermal noise, characterized by zero mean and the correlation [ 7 ]

(fl(z, t)fm(z', t') ) =2kBTFfilm~(Z-Z')~(t-t')

,

(5)

where (...) denotes the average over thermal fluctuations,ft is the/th component off, and kBTthe usual thermal energy. The present model is relevant for the FL motion in HTSC's, as long as flux-flux interactions, the Magnus force effect, and the FL cutting can be neglected. We simply assume that the temperature dependence of the two length scales is given as _

~ab(T) =~bO(1

T ~ -1/2

-~-~]

and

(8)

where << ... >> denotes the average over randomness due to the random potential and the thermal noise as well as the sample average. In addition, to discuss the FL dynamics in random impurities driven by the transport current, especially the pinning-depinning behavior of the FL near a certain threshold value of the transport current, we investigate the velocity exponent 0 defined as follows. When the transport current j approaches its threshold value JT from above, one finds that the mean velocity v scales as [4 ]

,o ,///(0,),// Ott

v=- ¢a~

~ (j--jT) ° .

(9)

Note that v behaves as v ~ j for large values of j, as is discussed below. Note also that hereafter the current density j is measured in units of jo=-Cq)o/ ( 12x/~n2~bO~.~bo ), which denotes the critical current density at zero temperature associated with thin-film superconductors [ 6 ].

2ab=~abO(1-- ~-~c)-1/2 3. Simulation method

with the zero-field critical temperature To. Under these assumptions, the GL parameter ~¢aband the correlation of the random potential given in Eq. ( 1 ) are temperature-independent. We also obtain the normal-state conductivity tr as [ 6 ]

toc2 a = a-----~ 4n2 '

(6)

with t o - n h / ( 96kBT¢ ). Finally, we comment on the scaling exponents characterizing the present system. In order to study the thermal and disorder effects on the FL dynamics, we calculate the following exponents [4]: the dynamic exponent fl and the roughness exponent ( defined, respectively, by

wd(t-t')1 x/(((r(z,t)_r(z,t,))2))~lt_t,i

~bo

p

(7)

Now we discuss the simulation method to solve the above differential equation (3) in the system defined in the previous section. Ordinarily, the integration of a nonlinear diffusion equation with simple boundary conditions is straightforward, but here two fluctuating random terms require special care [ 8 ]. Simplifying the above model but maintaining its physical essence, we propose a computationally efficient method to simulate the dynamical behavior of the FL. First, according to Marchetti and Nelson [ 9 ], the FL may be viewed as a collection of beads connected by a spring with a spring constant g(T), and each bead is confined to move in a Cu02 plane with an interlayer spacing a2. Next, the system is divided into cells whose volume is vp. The random potential is assumed to be defined on the cells of this cubic grid. Taking the short-range nature of the random potential into account, we regard the resulting force acting on the bead as a sum of the difference between potentials at the bead's position and its nearest grid cells in

Y. Enomoto, S. Maekawa/Physica C 233 (1994) 195-202

198

the same plane. Finally, we take the unit of length to be ~ab0 and time to be to, respectively. In these units, we use the dimensionless variables such as z - t/to and s~az/~abO. We also use R(n, z)--r(z=naz, t)/~ab0, which represents the position vector of a bead of the FL on the nth plane where 0 < n < N w i t h N=L/a=. Using this modelling, we approximately obtain the discretized dimensionless equation of motion for R (n, z) with time increment Az:

R(n, z+ Az) =R(n, z) + Az{ (.~. 5R(n, z) )So + (fJ.~R(n, z)))3} +7(n, z) ,

(10)

with

which values are of almost the same order as those for the YBCO sample. The impurity strength is changed from 0.01 to 0.05, since it is estimated as np_~0.01 for the YBCO sample [ 10]. Moreover, we take a large simulation system size, that is, L = 2048~ab0 with Vp = ~3b0 and a= = ~abO,which values are chosen only for a computational reason and do not directly describe a specific real system. The dimensionless time interval Az is empirically chosen to be satisfied with the condition Az [6R(n, z)[ < 0.01. As initial condition, we take the straight-line configuration. We also impose the boundary conditions periodic in the x - y plane and free in the z-direction.

8R(n, z ) = g l n tGb{R(n+ 1, z) 4. Simulation results

+ R ( n - 1 , z ) - 2 R ( n , z)}/s 2

+ 4 ~ (1- ~-)-' JA:

Now we carry out several computer simulations for a single FL motion in r a n d o m impurities by changing the values of the transport current j, the impurity strength np, and the temperature T.

NN

- ~ {V( [R(n, z)] +n~, n) - ~'( [R(n, z)], n)}n,~, Ol

(11) where the argument [R(n, z)] denotes the nearest grid cell point to R ( n, z) at the same height n on which the random potential is defined, and the symbol Y~ N means that n~'s are s u m m e d over +:~ and +~. The dimensionless r a n d o m potential due to impurities, P, is the r a n d o m n u m b e r with a normal distribution [2,3] with mean zero and variance

and is taken independently on each grid cell point at r = 0 . On the other hand, each component of the dimensionless thermal noise, ],, is the r a n d o m number with a normal distribution [ 8 ] with zero mean and variance

4.1. Thermal fluctuation of a single FL We first discuss the thermal behavior of a single FL in the absence of both r a n d o m impurities and transport current. In this situation the FL fluctuates only by the thermal noise. We study this simple case to check the reliability of the above simulation method, because both dynamic and roughness exponents have been obtained analytically in this case [ 11 ]. In Figs.

103 Wd(t) 0.25

~

!02

o

•

T

with energy-scale ratio E=- kBT¢/(%(0)2abO), and is taken independently on each position of the FL at every time step. In actual simulations, we choose the values of physical quantities as •ab = 50, g = 0.04, and E = 10- 4,

o °°°°°°

1(]

i

,

,

o o°°

•

o oe

• e ee=e

i~l,,I

. . . . . . .

10 a 103 t Fig. 1. Logarithmic plot of wd(t) against t with np=0 and j=0 for T~Tc= 0.1 ( • ) and 0.4 ( O ). A straight line is also shown with its slope indicated.

199

Y. Enomoto, S. Maekawa I Physica C 233 (1994) 195-202 10 2

i

o

Wr(Z)

0.5

o

0.4

o o

o

a•

o o o o

o

o

•

,

i

'0.4

o

B •

••o

.0.2

0.2

•••

,

° >

o

10.

o

o

o

o

o• o°

101

o

o

.....

i

102 z

. . . . . . .

103

Fig. 2. Logarithmic plot of wr(z) against z with np=0 and j = 0 for T/Tc=0.1 ( O ) and 0.4 (O). A straight line is also shown with its slope indicated.

1 and 2 we show logarithmic plots of wd (t) and wr (z) against t and z, respectively, for T/To= 0.1 and 0.4. Note that these and the following results have been obtained by averaging over 20 independent simulation runs, as well as time average ranging from 5 × 102to to 103to. From these results we find the dynamic exponent fl= 0.25 _+0.01 and the roughness exponent ( = 0.51 +_0.02, regardless of the temperature, which are in good agreement with the analytical result fl= ~ and ( = ½ [ 11 ]. Note that the temperature affects only the amplitude of wd(t) and Wr(Z), but not the scaling exponents. These results confirm the reliability of the present simulation method. Next, we study the thermal fluctuation of the FL in the presence of random impurities without transport current. In Fig. 3 we show the dynamic exponent fl and the roughness exponent ( as a function of the temperature at np=0.01. Note that since at zero temperature the FL is pinned by random impurities on the initial configuration, fl and ( are to be zero in the present case. We comment that with increasing impurity strength, the depinning temperature, below which the FL is pinned, increases. However, the results of the simulation are not enough to give a numerical estimate of the temperature. Thus, we do not discuss it anymore. Except for low temperatures (for T~ Tc < 0.1 in this case), scaling exponents are found to be independent of temperature (at least for T~ To<0.4) with fl=0.21_+0.02 and (=0.41_+0.02. Note that these values offl and (are smaller than those for the case in the absence of random impurities ob-

(

I

O0

i

0.2

'

TFFc

0

0.4

Fig. 3. The dynamic exponent fl ( • ) and the roughness exponent ( ( O ) as a function of the reduced temperature T/Tc with np=0.01 and j = 0 .

(c)

(d)

Fig. 4. Typical motion of a single FL at j=0.16; (a) (T/Tc, np) = (0, 0.01 ), (b) (0, 0.03), (c) (0, 0.05), and (d) (0.2, 0.05). FL is plottedafter every 50totime intervalfrom the initial straightline configuration. tained above. Random impurities reduce the thermal fluctuation of the FL, as is expected. No systematic deviation from the above values can be found for other values of the impurity strength (np_< 0.05 ). 4.2. Driven FL dynamics

Here we study the driven FL motion by the transport current through random impurities and then obtain the velocity exponent 0 as well as fl and (. In Fig.

Y. Enomoto, S. Maekawa /Physica C 233 (1994) 195-202

200

4 we visualize some cases of the FL motion for the small system size L = 50~abO. A typical dynamical behavior of the FL can be seen in these figures, such as the disturbed FL motion, pinning-depinning behavior, and the pinned FL state. First, we study the zero-temperature case. In Fig. 5 the typical example of the j - v curve is shown for the case with T = 0 and np=0.05. In this figure, we can see the pinning-depinning transition at a certain value of the transport current. To estimate the velocity exponent 0, we have plotted v 1/° againstj and the value of 0 was changed until the best straight line was obtained. By extrapolation of that line to v = 0 , we simultaneously obtain the threshold value of the driving force JT. In Fig. 6 we plot loglo(v) against 1OglO(/'--jx) withjx=0.18 for T = 0 and np=0.05. The

linear regime and the nonlinear part are seen as straight lines of different slopes, the slope of the nonlinear part being 0 and that of the linear part being close to 1. The value of 0 for the nonlinear regime and the corresponding value ofjT obtained in this way are shown in Fig. 7 as a function of the disorder strength np at T-- 0. The dynamic exponent fl and the roughness exponent ( for the same region are shown in Fig. 8 as a function of np at T = 0. N o systematic np dependence (np < 0.05 ) has been found for these exponents. Within the present data we find 0=0.31 _+0.03, fl= 1.52_+0.03, and (=0.75_+ 0.02 at zero temperature. On the other hand, the threshold value of the transport current, JT, is found to be an increasing function of n v Finally we briefly study the temperature effect on

, o

0.,~

0.4

0.2

o 'C @

0

v

•

JT

0 0

0.1

0

0.,~

0.2

0

)-

0 0

I

, ::=

0

I

,

0.2

I

q

J

0.4

l

Fig. 5. Mean velocity v as a function of the transport current j with np=0.05 and T = 0 .

101

. . . . . . . .

=

. . . . . . . .

l

'

'0.02 '

'

' 0.04 '

np

Fig. 7. The velocity exponent 0 ( • ) and the threshold value of the transport current Jr ( 0 ) as a function of the disorder strength np at T = 0.

. . . . . . . .

2

,

,

,

V

0 0 0 0

I0 °

o

o

o

>o

o

D.8

0.31 .....J"~=

10 -1

Q

[1.4

,

1012 ,-3

1.5

o •°••~

,

,,,,,,I

,

,

10 -2

,,....i

10 -1

. . . . . .

10 °

J-JT

Fig. 6. Logarithmic plot of v against J--JT with jx--0.18 for np= 0.05 and T = 0. Straight lines are also shown with their slopes indicated.

' 0.02

'

np

'

' 0.()4

'

0

Fig. 8. The dynamic exponent fl ( • ) and the roughness exponent ( ( O ) as a function of the disorder strength np at T = 0.

Y. Enomoto, S. Maekawa / Physica C 233 (1994) 195-202

1

,

. . . .

. . . .

0,2

0

JT

0 0

0 0

0.5

0.1

I

0 )<

0 ¸ 0

*

0

i

0

0

,

I

0.2

,

i

T/-I- c

O

,

i

0

0.4

Fig. 9. The velocityexponent 0 ( • ) and the threshold valueJT (O) as a function of the reduced temperature T/Tc at nv = 0.0 5. the above result for T ! Tc < 0.4. In principle, the thermal activation from one metastable configuration to another may play an important role in the depinning dynamics of the FL. In Fig. 9 the velocity exponent 0 and the threshold value jT are shown for np=0.05 as a function of the temperature T. From this result and further simulations for different values ofnp ( < 0.05) we find that the threshold valuejT decreases with increasing temperature, while for the velocity exponent there is no essential or systematic deviation from the zero-temperature value. Note that a similar tendency has been found forfl and (.

201

fl=1.52+0.03, and the roughness exponent ¢=0.75_+0.02. Within the present simulations, no systematic np and Tdependence can be seen for these values, nor any system-size dependence. However, more work, such as the study on strong-impurity and/ or high-temperature effects is needed to properly address this point. In fact, the temperature dependence of the velocity exponent has been observed for the two-dimensional interacting-vortices system [ 3 ]. In contrast to scaling exponents, the threshold value Jx of the transport current has been found to be an increasing function of the disorder strength but a decreasing one of temperature. The present simulations have been restricted to rather simple situations. For the FL dynamics of real HTSC's we need to take realistic effects, such as fluxflux interactions, the Magnus force effect, and/or the FL cutting, into account. The present model can be easily extended so as to include these effects. On the other hand, the present model is relevant for the threshold behavior of other analogous systems mentioned in Section 1. Thus, in order to study the generic properties of the threshold dynamics it would be interesting to compare the present results with those of such systems.

Acknowledgement 5. Conclusion In conclusion, we have numerically studied various aspects of a single FL dynamics in HTSC's on the basis of the Langevin-type stochastic equation of motion. Large-scale simulations have been performed for the disorder strength np<0.05 and the temperature T < O . 4 T c with parameters corresponding to the YBCO sample. For the FL dynamics in random impurities without the transport current, we have studied the thermal behavior of the FL, and have found the dynamic exponent r = 0.21 + 0.02 and the roughness exponent ~=0.41+0.02. On the other hand, for the FL dynamics in random impurities with transport current, the pinning-depinning transition has been found at a certain threshold value of the current. Near the threshold we have obtained the velocity exponent 0=0.31 _+0.03, the dynamic exponent

This work has been supported by a Grant-in-Aid from the Ministry of Education, Science and Culture of Japan.

References [ 1] D.R. Nelson, J. Stat. Phys. 57 (1988) 511; D.S. Fisher, M.P. Fisher and D.A. Huse, Phys. Rev. B 43 (1991) 130. [2] Y. Enomoto, R. Kato and S. Maekawa, Studies of High Temperature Superconductors, vol. 1I, ed. A.V. Narlikar (Nova Science,New York, 1993). [ 3] Y. Enomoto, K. Katsumi and S. Maekawa, PhysicaC 215 (1993) 51. [4 ] T. Nattermann, S. Stepanow, L.H. Tangand H. Leschhorn, J. Phys. II (France) 2 (1992) 1483; O. Narayan and D.S. Fisher, Phys. Rev. E 49 (1994) 9469 and referencescited therein.

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[5] M. Kardar, G. Parisi and Y.C. Zhang, Phys. Rev. Lett. 56 (1986) 889; O. Narayan and D.S. Fisher, Phys. Rev. B 48 ( 1993 ) 7030. [ 6 ] M. Tinkham, Introduction to Superconductivity (MacGrawHill, New York, 1975). [7]M. Toda, R. Kubo and Y. Saito, Statistical Physics I (Springer, Berlin, 1992).

[8 ] M.P. Allen and D.J. Tildesly, Computer Simulation of Liquid (Clarendon, Oxford, 1982 ). [9] M.C. Marchetti and D.R. Nelson, Phys. Rev. B 42 (1990) 9938. [ 10 ] T. Nattermann, M. Feigelmann and I. Lyuksyutov, Z. Phys. B84 (1991) 353. [ 11 ] A.T. Dorsey, Phys. Rev. B 46 (1992) 8376.