- Email: [email protected]
Dynamical scaling behavior of current-electric field characteristics in two-dimensional disordered superconductors
PHYSICA ELSEVIER
Physica C 274 (1997) 351-356
Dynamical scaling behavior of current-electric field characteristics in two-dimensional disordered superconductors Y o s h i h i s a E n o m o t o a,,, S a d a m i c h i
Maekawa b
a Department of Physics, Nagoya Institute of Technology, Gokiso, Nagoya 466, Japan b Department of Applied Physics, Nagoya University, Nagoya 464-01, Japan
Received 25 November 1996
Abstract
We study the dynamics of magnetic vortices driven by transport current in two-dimensional disordered type-II superconductors, using the Langevin dynamics simulation of the vortex model developed. Exploring the electric field induced by the vortex motion as a function of the transport current ( E - J characteristics), we find that the combined effect of random impurities and the current induced vortex-antivortex pair creation yields the dynamical scaling behavior of E - J curves near a certain critical temperature. Critical scaling exponents are found to be z --~ 2, u ~_ 2 and /z _~ 0.5, for the dynamical exponent, the static exponent and the resistivity decay exponent, respectively, which are almost independent of the magnetic induction. On the other hand, the critical temperature, Tcr, depends on the magnetic induction, B, as Tcr/Tco = 1 - B 0"68 with superconducting transition temperature Tc0 at zero field.
1. I n t r o d u c t i o n
A m o n g many striking properties of high-To superconductors, the dynamics of magnetic vortices in the mixed state is extraordinarily important from the viewpoint o f purely scientific interest as well as practical needs. Considerable work and intensive study have already been devoted to this subject, yielding many competing models and concepts for detailed description of the vortex dynamics [ 1 ]. A m o n g them, the dynamical scaling is a fascinating concept to point out an important effect of random impurities on transport properties in type-II superconductors, especially the induced electric field due to the vortex motion as a function of the transport current ( E - J characteristics), which has been theoreti* Corresponding author. E-mail: [email protected].
cally introduced by using an analogy with critical phenomena [ 2,3 ]. For three-dimensional disordered systems, the dynamical scaling behavior o f E - J curves (called the vortex glass scaling) has been supported by many experiments [ 4 - 8 ] and computer simulations for the gauge glass models [9-11 ]. On the other hand, in two-dimensional experiments, only the KosterlitzThouless (KT) type behavior o f E - J curves has been discussed, such as the universal jump o f power law exponent a at KT transition temperature defined by E oc ja [ 12]. Then, the dynamical scaling law has been less studied in two dimension. Moreover, in two dimension there is a discrepancy even among a few numerical results of Monte Carlo simulation study for the gauge-glass model [ 11,13,14] and for the classical lattice Coulomb gas model [ 15]. The purpose of the present work is to examine whether the dynamical scaling law exists or not for
0921-4534/97/$17.00 Copyright (g) 1997 Elsevier Science B.V. All rights reserved. PII S0921-4534(96) 00709-5
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Y Enomoto, S. Maekawa/Physica C 274 (1997) 351-356
two-dimensional vortex systems, by using the different numerical approach, that is, the Langevin dynamics simulation of the vortex model [ 16]. The vortex model is based on the vortex picture and is described by the stochastic equation of motion for vortex positions. Unlike other models, this model has an advantage to visualize the realistic vortex motion and thus to obtain direct information on the dynamical properties of vortex systems. In our previous works for two-dimensional vortex systems, we have found two typical behavior of E - J characteristics. One is the Kosterlitz-Thouless (KT) type behavior as E e( ja with sudden change of exponent a at KT transition temperature for the clean system including the effect of current induced vortex-antivortex pair creation [ 17]. The other is the nonlinear threshold behavior as E e( ( J - Jth)b with exponent b and threshold current Jth for the disordered system [ 18]. However, in these two-dimensional simulations we cannot observe the dynamical scaling behavior of E - J curves. Here, as one of our serial works, we show that the combined effect of random impurities and vortex-antivortex pair creation can produce the dynamical scaling behavior of E - J curves in two dimension. The paper is organized as follows. In Section 2, we review the two-dimensional vortex model, which is described by the Langevin type stochastic equation of motion for vortex positions. In this model, effects of vortex-vortex interaction, random impurities, thermal noise, and the Lorentz force due to the transport current are included, as well as the effect of current induced vortex-antivortex pair creation. We also comment on the numerical procedure to solve the stochastic equation of motion. In Section 3, we present numerical results. In particular, we show that E - J curves exhibit the dynamical scaling behavior with relevant critical exponents and the critical temperature extracted from the simulation data. We also study the magnetic induction dependence of the scaling properties. Section 4 concludes the paper.
for simplicity. We consider a thin slab of type-II superconductors, lying on a certain region in the x - y plane, with its thickness, w, less than the magnetic penetration depth, A, and its cross section area, A. An external magnetic field is applied along the positive z direction (which coincides with the c-axis in high-To superconductors), and a transport current flows along the positive y direction. Hereafter we assign a vortex to a quantized magnetic flux having its circulation along the positive z direction and an antivortex to one having negative z circulation, and also we often use the word "vortex" as a generic term including both vortex and antivortex if confusion does not occur. The position of the ith vortex at time t is described by two-dimensional position vector ri (t). Under these situations, the free energy for N-vortices system reads
i qboWj ~ c
l~i eix " ri,
(1)
i
where Ec is the core energy of a single vortex, j denotes the transport current density, and ei = 1 for a vortex and ei = - - 1 for an antivortex with :? being the unit vector along the x axis, ¢0 the flux quantum and c the light velocity. The intervortex potential, V(r), is assumed to be given by V(r I -- ri)
~
2we,eie(T) In (\ [rl~ -- ri[~ j ,
(2)
where e(T) =-- [¢0/4qr,,~(T)] 2 with the magnetic penetration depth A(T) at temperature T. Here we comment on the validity to accept the above logarithmic form (2) as the intervortex interaction. In twodimensional superconductors, the demagnetization effect is important and then the intervortex interaction V ( r ) is given by
v(r)
= -ff-~[Ho(Irl/A) ¢'~
- Yo(Ir[/A) ],
(3)
2. The model
Following our previous works [ 16-18], we here specify the dynamical model for two-dimensional magnetic vortices in random media. In the present model, the Magnus force effect has been neglected
where A denotes the effective penetration depth including the demagnetization effect defined by A = 2A2/w, and Ho(x),and Yo(x) are the zeroth order Struve function and the zeroth order Neumann function, respectively [ 19]. As will be mentioned later
Y Enomoto, S. Maekawa/Physica C 274 (1997) 351-356
in Section 3, we consider the situation with the system area [103((0)]2, its thickness w = ( ( 0 ) and the Ginzburg-Landau parameter t< = 50 where ( ( 0 ) denotes the coherence length at zero temperature. Under these conditions, the system linear size 103((0) is thus much less than the effective penetration depth A = 2 K 2 ( 1 - T / T c o ) - I ( ( o ) = 2.5x 104((0) atT/Tco = 0.8 with superconducting transition temperature To0 at zero field. Then, expanding two functions Ho(x) and Yo(x) for small x in Eq. (3) yields the logarithmic interaction form given by Eq. (2). Assuming the Langevin dynamics, the equation of motion for the ith vortex position is given by
ari(t dt
_
F({rl}) + ~ri
Vp(r ) l=l
+fi(t)
(4)
where y - 1 = o'Hc2 (T) ~bo/c 2 is the Bardeen-Stephan friction coefficient with the normal state conductivity o-, the upper critical field Hc2(r) =-- q%/2~((r) 2 and the coherence length ( ( T ) [201. The random potential, Vp(r), represents effects of quenched random impurities, assumed to be characterized by Vp(r) = 0 and
(5)
where the overline denotes the average over the impurity disorder, and np is the impurity strength and ap the area of the unit cell of the underlying crystal lattice. The last term in Eq. (4) describes the thermal noise, assumed to be characterized by zero mean and
( ( f i ~ ( t ) f t ~ ( t r ) ) ) = 2kBTy6ilt3~l~8(t - t ' ) / w ,
In the rest of this section, we comment on the numerical procedure for solving the stochastic equation (4). First of all, we rewrite Eq. (4) in terms of dimensionless variables. To do so, we take the unit of length to be ( ( 0 ) , the unit of time to be to, the unit of magnetic induction to be Hc2(0), and the unit of current density to be jo = cHc2(O)/6v~r:A(O). Then, we transform the dimensionless equation into the spatial discretized version by using a second order central difference representation for the spatial derivatives with dimensionless spacing 1. Final procedure is to integrate numerically the resulting equation of motion by using a forward Euler difference scheme with dimensionless time step AT. As boundary conditions, we use periodic boundary conditions in the x - y plane. On the other hand, as the initial condition, for the fixed area A and a given magnetic induction B, we determine the initial numbers of vortices and of antivortices, N+ and N_, respectively, so as to be satisfied with B A = N+q~0 and N_ = 0. Current induced vortex-antivortex pair creation process is treated in a probabilistic manner with a Metropolis algorithm [ 17]. For more detailed discussion on the numerical technique, see our previous works [ 16-18 ].
3. Numerical results
Vp ( r) Vp ( r') = npapE(T)2( 1 -- T/Teo) -2 × 6 ( r -- r ' ) ,
353
(6)
where f i , is the ath component of f i and ((...)) denotes the average over the thermal fluctuation. Moreover, we assume the temperature dependence of above two length scales as A(T) ---- A(0)(1 T/Teo)-1/2 and ( ( T ) --= ( ( 0 ) ( 1 - T/Teo) -a/2 at all temperature. Thus, in this case the Ginzburg-Landau parameter, K, becomes independent of temperature and is given by K -- A(0)/~:(0). We also use the relationship obtained from the microscopic theory as 47rA(T)2cr/c 2 = ( ( T ) Z / ( 1 2 D ) = t 0 ( 1 - T/Tco) -1 with to = ~rl~/ ( 48kBTco ) [20].
Now we carry out two-dimensional simulations for the vortex dynamics in random media driven by transport current with effects of vortex pair creation by changing values of temperature T and magnetic induction B. In order to examine the dynamical properties of vortices, we measure the mean vortex velocity along the x direction, denoted by v, as a function of transport current j. Here, this mean velocity is calculated in dimensionless units as
v--
d r i ~ to ~c. dt l ( ( O ) '
(7)
where (...) denotes the average over both all vortices and 20 independent simulation runs. Moreover, to realize a steady state in the presence of the transport current, we typically discard 103 initial time steps and then compute the mean velocity averaged over discrete time steps ranging from 103 to 104. Note that the obtained mean velocity is proportional to the electric field along the y direction induced by the vortex
X Enomoto, S. Maekawa/Physica C 274 (1997) 351-356
354
100
........
i
......
•
,~1
....... @ +zxO
: . @ + +Ao° "+" ,% o+ o • + • @+/xO
>
•
10-1 •
"+"
~ +2° . / o• +
10-2
,
Zx
•
I
.......
T/Tc 0 © 0.715 A 0.745 + 0.775 00.805 •0.835
........
10 2
g uJ
I
........
I
T/Tco @ 0.850 [ ] 0.835 • 0.820 A 0.805 • 0.790
I
~,,;2
10 o """"¢
<>
<~
V
g
+ O A
10 .3
. . . . . . . .
l
10 -2
,
,
,,,,,,I
. . . . . . . .
J/Jo
1
. . . . . .
10 °
oa
10-2
o
g
motion and thus the following u versus j relationship corresponds to the steady state E - J curve. In actual simulations, we choose parameter values as K = 50, n p = 0.01, and the energy scale ratio kBTco/(e(O)d) = 0.02, which are of almost the same order to those values for typical high-Tc materials, respectively. We also take w = ( ( 0 ) , A = 103gr(0) x 1 0 3 ~ ( 0 ) and ap = ~ : ( 0 ) 2. The initial vortex number for B~ He2 (0) = 0.1 is 16000. The dimensionless time increment Ar is chosen to be 0.01 so as to be small enough to ensure numerical stability of the algorithm. In Fig. 1 we show the mean velocity v as a function o f j in a l o g - l o g scale at B/Ho2(O) = 0.1 for several values o f temperature. With decreasing temperature, the curvature o f these curves changes from positive to negative at a certain critical temperature denoted by Tcr ( = 0.775Tc0 in this case). To explore the dynamical scaling behavior of E - J curves, we replot our detailed simulation data by using the functional form predicted from the vortex glass model [2]. In Fig. 2 the scaled electric field Esc = ull - T / T ~ r l ~ ( d - 2 - z ) / j is plotted against the scaled current Jsc = J l 1 - T/Tor[-~(a-,)/T with the system dimensionality d = 2 and proper values of the static exponent v, the dynamical exponent z and the critical temperature Tcr. In this figure, we can see that the data collapse onto two universal curves /~4- with ~ = 2.0, z = 2.3 and Tcr/Tco = 0.775, where /~+ represent scaling functions above ( + ) and below ( - ) Tcr, respectively. Here, following the arguments in the vortex glass model [2], v, z and Tcr are esti-
0.760 0.745~
/k ~
0.730 ~:
[]
0.715:
O
0.700?
o ,
Fig. 1. M e a n velocity v as a function of the transport current J/Jo in a l o g - l o g scale at B/Hc2(O) = 0.1 for several values of temperature. Note that the mean velocity is proportional to the induced electric field due to the vortex motion.
.......
.~# '~ ";x~ ,i,* ~'
O
........
.......
I
,
, ,,,,,,I
10 °
,
.......
i
dsc
,
, ,
10 2
Fig. 2. Scaled electric field Esc as a function of scaled current Jsc for various temperature with p = 2.0, z = 2.3 and Tcr/Tco = 0.775.
,
,
,
,
,
,
,
,
N
~3 o
o
•
o
•
•
•
o
o
v:@ z:@
q
I
I
I
I
I
I
0.1
I
I
I
0.2
B/Hc2(0) Fig. 3. Static exponent p and dynamical exponent z as a function of the magnetic induction B.
mated from the slope o f v = jz+l at T = Tcr and the fitting of the linear resistivity with small j for T > Tcr to v/j ~ (T/Tcr- 1) ~z. The same type of E - J curves as shown in Figs. 1 and 2 are obtained for different values of the magnetic induction B. In Figs. 3 and 4, the critical exponents v and z, and the critical temperature Ter are plotted as a function of B. These figures show no evident B-dependence of ~, and z (v = 2.0 4- 0.1 and z = 2.1 + 0 . 2 ) , but Tcr/Teo = 1 - ( B / H c 2 ( O ) ) 0"68. In order to obtain more information on the low temperature state, we analyze the E - J curves below Tcr
Y Enomoto, S. Maekawa/Physica C 274 (1997) 351-356
355
1
1.5 O
F-b t0.5 0.5
06
. . . .
011. . . . .
i
).2
B/Hc2(O) Fig. 4. Critical temperature Tcr as a function magnetic induction B. Solid line indicates the
i
i
I
0.1
i
I
i
i
0.2
B/Hc2(O) of the relation
TcrlTco = 1 -- ( BIHc2(O) ) 0"68.
Fig. 5. Critical exponent/~ as a function of the magnetic induction B.
nents and the critical temperature. by using the functional form/7;_ (x) ~ e x p ( - 1 / x ~ ) , predicted by several models [ 1 ]. This means that the linear resistivity E / J vanishes as J ~ 0 and thus the low temperature state is a purely superconducting phase, since the vortices are frozen into some disordered patterns [2]. The procedure to estimate the exponent /x is as follows: we replot the v ( j ) data as I n ( v / j ) versus j - ~ for T < Tcr with various /z and determine the appropriate/z with which the replotted data form a straight line. In Fig. 5, we show the exponent/x as a function o f the magnetic induction B, This result indicates that/z is almost constant a s / z = 0.49 4- 0.02. Finally, we comment on the related competing simulation results in two-dimensional vortex systems. Some equilibrium and dynamical simulations for the gauge glass model [ 11,13] have shown to be u = 2 b u t Tcr = 0. However, some dynamical simulations asserted the non-zero critical temperature with z = 2 for the lattice Coulomb gas model [ 15], and with z = 2.2, l, = 2 . 2 , / z = 0.5 for the gauge glass model [ 14], which are consistent with our present data. At present, physical reasons and/or physical meaning for such a discrepancy on the critical temperature are not clear. We need carefully check the consistency among situations considered in various works. Unlike the present study, there is no previous simulations to examine the detailed B-dependence of critical expo-
4. Conclusion To conclude, we have carried out the twodimensional Langevin dynamics simulations of magnetic vortices interacting logarithmically each other, including effects of current induced creation of vortex-antivortex pairs and random impurities. By changing values of temperature and magnetic induction, we have measured numerically steady state E - J curves. Then, we have demonstrated that in two dimension the combined effect of random impurities and of current induced vortex pair creation can yield the dynamical scaling behavior for E - J curves with proper values of critical exponent v, z, /z and the critical temperature Tcr. We have also found that u, z and /x are constants (u "~ 2, z ,.o 2 and /x _~ 0.5), but Tot/Too = 1 - (B/Hc2(O))0.68 in the range of the magnetic induction considered here. This is the first dynamical simulation to demonstrate the existence of the dynamical scaling behavior of E J curves from the point of view of the vortex picture. The present result for the dynamical exponent z --~ 2 is consistent with that of the Kosterlitz-Thouless model and experimental results for two-dimensional vortex systems at KT transition temperature [ 12]. However, in these experiments, the dynamical scaling behavior
356
Y Enomoto, S. Maekawa/Physica C 274 (1997) 351-356
o f E - J curves has been less discussed. To i m p r o v e the present study, it is h i g h l y desirable to e x a m i n e E J curves near the K T transition temperature in t w o d i m e n s i o n a l d i s o r d e r e d systems f r o m the v i e w p o i n t o f the d y n a m i c a l scaling. H e r e w e have restricted the present simulation to study the vortex d y n a m i c s in disordered twod i m e n s i o n a l superconductors. However, the qualitative p r e d i c t i o n s o b t a i n e d in this paper w o u l d still give a useful g u i d e to study the d y n a m i c a l scaling behavior even for high-Tc s u p e r c o n d u c t i n g bulk systems, at least for h i g h l y anisotropic B i - b a s e d materials.
Acknowledgements This w o r k was supported by a P r i o r i t y - A r e a Grant f r o m the M i n i s t r y o f Education, S c i e n c e and C u l t u r e o f Japan.
References [ 1] G. Blatter, M.V. Feigelman, V.B. Geshkenbein, A.I. Larkin and V.M. Vinokur, Rev. Mod. Phys. 66 (1994) 1125. [2] M.P. Fisher, Phys. Rev. Lett. 62 (1989) 1415. [3] D.S. Fisher, M.RA. Fisher and D.A. Huse, Phys. Rev. B 43 (1991) 130.
[4] R.H. Koch, V. Foglietti, W.J. Gallagher, G. Koren, A. Gupta and M.P.A. Fisher, Phys. Rev. Lett. 63 (1989) 1511. [5] RL. Gammel, L.F. Schneemener and D.J. Bishop, Phys. Rev. Lett. 66 (1991) 953. [6] C. Dekker, W. Eideloth and R.H. Koch, Phys. Rev. Lett. 68 (1992) 3347. [7] J. Deak, M. McElfresh, R. Muenchausen, S. Foltyn and R. Dye, Phys. Rev. B 48 (1993) 1337. [8] RJ.M. Woltgen, C. Dekker, J. Swuste and H.W. de Wijn, Phys. Rev. B 49 (1994) 6890. [9] J.D. Reger, T.A. Tokuyasu, A.P. Young and M.P.A. Fisher, Phys. Rev. B 44 (1991) 7147. [10] K.H. Lee and D. Stroud, Phys. Rev. B 44 (1991) 9780. [11] MJ. Gingras, Phys. Rev. B 45 (1992) 7547. [12] R Minnhagen, Rev. Mod. Phys. 59 (1987) 1001; K.H. Fischer, Physica C 210 (1993) 179. [13] M.RA. Fisher, T.A. Tokuyasu and A.P. Young, Phys. Rev. Lett. 66 (1991) 2931. [14] Y.H. Li, Phys. Rev. Lett. 69 (1992) 1819. [15] J.R. Lee and S. Teitel, Phys. Rev. B 50 (1994) 3149. [16] Y. Enomoto, R. Kato and S. Maekawa, Studies of High Temperature Superconductors, ed., A.V. Narlikar (Nova Science, New York, 1993), Vol. 11, p. 309. [171 Y. Enomoto and S. Maekawa, Physica C 247 (1995) 156. [18] Y. Enomoto, K. Katsumi and S. Maekawa, Physica C 215 (1993) 51. [19] J. Pearl, Appl. Phys. Lett. 5 (1964) 65; A.L. Fetter and P.C. Hohenberg, Phys. Rev. 159 (1967) 330. [20] M. Tinkham, Introduction to Superconductivity (MacGrawHill, New York, 1975).
Physica C 274 (1997) 351-356
Dynamical scaling behavior of current-electric field characteristics in two-dimensional disordered superconductors Y o s h i h i s a E n o m o t o a,,, S a d a m i c h i
Maekawa b
a Department of Physics, Nagoya Institute of Technology, Gokiso, Nagoya 466, Japan b Department of Applied Physics, Nagoya University, Nagoya 464-01, Japan
Received 25 November 1996
Abstract
We study the dynamics of magnetic vortices driven by transport current in two-dimensional disordered type-II superconductors, using the Langevin dynamics simulation of the vortex model developed. Exploring the electric field induced by the vortex motion as a function of the transport current ( E - J characteristics), we find that the combined effect of random impurities and the current induced vortex-antivortex pair creation yields the dynamical scaling behavior of E - J curves near a certain critical temperature. Critical scaling exponents are found to be z --~ 2, u ~_ 2 and /z _~ 0.5, for the dynamical exponent, the static exponent and the resistivity decay exponent, respectively, which are almost independent of the magnetic induction. On the other hand, the critical temperature, Tcr, depends on the magnetic induction, B, as Tcr/Tco = 1 - B 0"68 with superconducting transition temperature Tc0 at zero field.
1. I n t r o d u c t i o n
A m o n g many striking properties of high-To superconductors, the dynamics of magnetic vortices in the mixed state is extraordinarily important from the viewpoint o f purely scientific interest as well as practical needs. Considerable work and intensive study have already been devoted to this subject, yielding many competing models and concepts for detailed description of the vortex dynamics [ 1 ]. A m o n g them, the dynamical scaling is a fascinating concept to point out an important effect of random impurities on transport properties in type-II superconductors, especially the induced electric field due to the vortex motion as a function of the transport current ( E - J characteristics), which has been theoreti* Corresponding author. E-mail: [email protected].
cally introduced by using an analogy with critical phenomena [ 2,3 ]. For three-dimensional disordered systems, the dynamical scaling behavior o f E - J curves (called the vortex glass scaling) has been supported by many experiments [ 4 - 8 ] and computer simulations for the gauge glass models [9-11 ]. On the other hand, in two-dimensional experiments, only the KosterlitzThouless (KT) type behavior o f E - J curves has been discussed, such as the universal jump o f power law exponent a at KT transition temperature defined by E oc ja [ 12]. Then, the dynamical scaling law has been less studied in two dimension. Moreover, in two dimension there is a discrepancy even among a few numerical results of Monte Carlo simulation study for the gauge-glass model [ 11,13,14] and for the classical lattice Coulomb gas model [ 15]. The purpose of the present work is to examine whether the dynamical scaling law exists or not for
0921-4534/97/$17.00 Copyright (g) 1997 Elsevier Science B.V. All rights reserved. PII S0921-4534(96) 00709-5
352
Y Enomoto, S. Maekawa/Physica C 274 (1997) 351-356
two-dimensional vortex systems, by using the different numerical approach, that is, the Langevin dynamics simulation of the vortex model [ 16]. The vortex model is based on the vortex picture and is described by the stochastic equation of motion for vortex positions. Unlike other models, this model has an advantage to visualize the realistic vortex motion and thus to obtain direct information on the dynamical properties of vortex systems. In our previous works for two-dimensional vortex systems, we have found two typical behavior of E - J characteristics. One is the Kosterlitz-Thouless (KT) type behavior as E e( ja with sudden change of exponent a at KT transition temperature for the clean system including the effect of current induced vortex-antivortex pair creation [ 17]. The other is the nonlinear threshold behavior as E e( ( J - Jth)b with exponent b and threshold current Jth for the disordered system [ 18]. However, in these two-dimensional simulations we cannot observe the dynamical scaling behavior of E - J curves. Here, as one of our serial works, we show that the combined effect of random impurities and vortex-antivortex pair creation can produce the dynamical scaling behavior of E - J curves in two dimension. The paper is organized as follows. In Section 2, we review the two-dimensional vortex model, which is described by the Langevin type stochastic equation of motion for vortex positions. In this model, effects of vortex-vortex interaction, random impurities, thermal noise, and the Lorentz force due to the transport current are included, as well as the effect of current induced vortex-antivortex pair creation. We also comment on the numerical procedure to solve the stochastic equation of motion. In Section 3, we present numerical results. In particular, we show that E - J curves exhibit the dynamical scaling behavior with relevant critical exponents and the critical temperature extracted from the simulation data. We also study the magnetic induction dependence of the scaling properties. Section 4 concludes the paper.
for simplicity. We consider a thin slab of type-II superconductors, lying on a certain region in the x - y plane, with its thickness, w, less than the magnetic penetration depth, A, and its cross section area, A. An external magnetic field is applied along the positive z direction (which coincides with the c-axis in high-To superconductors), and a transport current flows along the positive y direction. Hereafter we assign a vortex to a quantized magnetic flux having its circulation along the positive z direction and an antivortex to one having negative z circulation, and also we often use the word "vortex" as a generic term including both vortex and antivortex if confusion does not occur. The position of the ith vortex at time t is described by two-dimensional position vector ri (t). Under these situations, the free energy for N-vortices system reads
i qboWj ~ c
l~i eix " ri,
(1)
i
where Ec is the core energy of a single vortex, j denotes the transport current density, and ei = 1 for a vortex and ei = - - 1 for an antivortex with :? being the unit vector along the x axis, ¢0 the flux quantum and c the light velocity. The intervortex potential, V(r), is assumed to be given by V(r I -- ri)
~
2we,eie(T) In (\ [rl~ -- ri[~ j ,
(2)
where e(T) =-- [¢0/4qr,,~(T)] 2 with the magnetic penetration depth A(T) at temperature T. Here we comment on the validity to accept the above logarithmic form (2) as the intervortex interaction. In twodimensional superconductors, the demagnetization effect is important and then the intervortex interaction V ( r ) is given by
v(r)
= -ff-~[Ho(Irl/A) ¢'~
- Yo(Ir[/A) ],
(3)
2. The model
Following our previous works [ 16-18], we here specify the dynamical model for two-dimensional magnetic vortices in random media. In the present model, the Magnus force effect has been neglected
where A denotes the effective penetration depth including the demagnetization effect defined by A = 2A2/w, and Ho(x),and Yo(x) are the zeroth order Struve function and the zeroth order Neumann function, respectively [ 19]. As will be mentioned later
Y Enomoto, S. Maekawa/Physica C 274 (1997) 351-356
in Section 3, we consider the situation with the system area [103((0)]2, its thickness w = ( ( 0 ) and the Ginzburg-Landau parameter t< = 50 where ( ( 0 ) denotes the coherence length at zero temperature. Under these conditions, the system linear size 103((0) is thus much less than the effective penetration depth A = 2 K 2 ( 1 - T / T c o ) - I ( ( o ) = 2.5x 104((0) atT/Tco = 0.8 with superconducting transition temperature To0 at zero field. Then, expanding two functions Ho(x) and Yo(x) for small x in Eq. (3) yields the logarithmic interaction form given by Eq. (2). Assuming the Langevin dynamics, the equation of motion for the ith vortex position is given by
ari(t dt
_
F({rl}) + ~ri
Vp(r ) l=l
+fi(t)
(4)
where y - 1 = o'Hc2 (T) ~bo/c 2 is the Bardeen-Stephan friction coefficient with the normal state conductivity o-, the upper critical field Hc2(r) =-- q%/2~((r) 2 and the coherence length ( ( T ) [201. The random potential, Vp(r), represents effects of quenched random impurities, assumed to be characterized by Vp(r) = 0 and
(5)
where the overline denotes the average over the impurity disorder, and np is the impurity strength and ap the area of the unit cell of the underlying crystal lattice. The last term in Eq. (4) describes the thermal noise, assumed to be characterized by zero mean and
( ( f i ~ ( t ) f t ~ ( t r ) ) ) = 2kBTy6ilt3~l~8(t - t ' ) / w ,
In the rest of this section, we comment on the numerical procedure for solving the stochastic equation (4). First of all, we rewrite Eq. (4) in terms of dimensionless variables. To do so, we take the unit of length to be ( ( 0 ) , the unit of time to be to, the unit of magnetic induction to be Hc2(0), and the unit of current density to be jo = cHc2(O)/6v~r:A(O). Then, we transform the dimensionless equation into the spatial discretized version by using a second order central difference representation for the spatial derivatives with dimensionless spacing 1. Final procedure is to integrate numerically the resulting equation of motion by using a forward Euler difference scheme with dimensionless time step AT. As boundary conditions, we use periodic boundary conditions in the x - y plane. On the other hand, as the initial condition, for the fixed area A and a given magnetic induction B, we determine the initial numbers of vortices and of antivortices, N+ and N_, respectively, so as to be satisfied with B A = N+q~0 and N_ = 0. Current induced vortex-antivortex pair creation process is treated in a probabilistic manner with a Metropolis algorithm [ 17]. For more detailed discussion on the numerical technique, see our previous works [ 16-18 ].
3. Numerical results
Vp ( r) Vp ( r') = npapE(T)2( 1 -- T/Teo) -2 × 6 ( r -- r ' ) ,
353
(6)
where f i , is the ath component of f i and ((...)) denotes the average over the thermal fluctuation. Moreover, we assume the temperature dependence of above two length scales as A(T) ---- A(0)(1 T/Teo)-1/2 and ( ( T ) --= ( ( 0 ) ( 1 - T/Teo) -a/2 at all temperature. Thus, in this case the Ginzburg-Landau parameter, K, becomes independent of temperature and is given by K -- A(0)/~:(0). We also use the relationship obtained from the microscopic theory as 47rA(T)2cr/c 2 = ( ( T ) Z / ( 1 2 D ) = t 0 ( 1 - T/Tco) -1 with to = ~rl~/ ( 48kBTco ) [20].
Now we carry out two-dimensional simulations for the vortex dynamics in random media driven by transport current with effects of vortex pair creation by changing values of temperature T and magnetic induction B. In order to examine the dynamical properties of vortices, we measure the mean vortex velocity along the x direction, denoted by v, as a function of transport current j. Here, this mean velocity is calculated in dimensionless units as
v--
d r i ~ to ~c. dt l ( ( O ) '
(7)
where (...) denotes the average over both all vortices and 20 independent simulation runs. Moreover, to realize a steady state in the presence of the transport current, we typically discard 103 initial time steps and then compute the mean velocity averaged over discrete time steps ranging from 103 to 104. Note that the obtained mean velocity is proportional to the electric field along the y direction induced by the vortex
X Enomoto, S. Maekawa/Physica C 274 (1997) 351-356
354
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motion and thus the following u versus j relationship corresponds to the steady state E - J curve. In actual simulations, we choose parameter values as K = 50, n p = 0.01, and the energy scale ratio kBTco/(e(O)d) = 0.02, which are of almost the same order to those values for typical high-Tc materials, respectively. We also take w = ( ( 0 ) , A = 103gr(0) x 1 0 3 ~ ( 0 ) and ap = ~ : ( 0 ) 2. The initial vortex number for B~ He2 (0) = 0.1 is 16000. The dimensionless time increment Ar is chosen to be 0.01 so as to be small enough to ensure numerical stability of the algorithm. In Fig. 1 we show the mean velocity v as a function o f j in a l o g - l o g scale at B/Ho2(O) = 0.1 for several values o f temperature. With decreasing temperature, the curvature o f these curves changes from positive to negative at a certain critical temperature denoted by Tcr ( = 0.775Tc0 in this case). To explore the dynamical scaling behavior of E - J curves, we replot our detailed simulation data by using the functional form predicted from the vortex glass model [2]. In Fig. 2 the scaled electric field Esc = ull - T / T ~ r l ~ ( d - 2 - z ) / j is plotted against the scaled current Jsc = J l 1 - T/Tor[-~(a-,)/T with the system dimensionality d = 2 and proper values of the static exponent v, the dynamical exponent z and the critical temperature Tcr. In this figure, we can see that the data collapse onto two universal curves /~4- with ~ = 2.0, z = 2.3 and Tcr/Tco = 0.775, where /~+ represent scaling functions above ( + ) and below ( - ) Tcr, respectively. Here, following the arguments in the vortex glass model [2], v, z and Tcr are esti-
0.760 0.745~
/k ~
0.730 ~:
[]
0.715:
O
0.700?
o ,
Fig. 1. M e a n velocity v as a function of the transport current J/Jo in a l o g - l o g scale at B/Hc2(O) = 0.1 for several values of temperature. Note that the mean velocity is proportional to the induced electric field due to the vortex motion.
.......
.~# '~ ";x~ ,i,* ~'
O
........
.......
I
,
, ,,,,,,I
10 °
,
.......
i
dsc
,
, ,
10 2
Fig. 2. Scaled electric field Esc as a function of scaled current Jsc for various temperature with p = 2.0, z = 2.3 and Tcr/Tco = 0.775.
,
,
,
,
,
,
,
,
N
~3 o
o
•
o
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•
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o
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q
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I
I
I
I
I
0.1
I
I
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0.2
B/Hc2(0) Fig. 3. Static exponent p and dynamical exponent z as a function of the magnetic induction B.
mated from the slope o f v = jz+l at T = Tcr and the fitting of the linear resistivity with small j for T > Tcr to v/j ~ (T/Tcr- 1) ~z. The same type of E - J curves as shown in Figs. 1 and 2 are obtained for different values of the magnetic induction B. In Figs. 3 and 4, the critical exponents v and z, and the critical temperature Ter are plotted as a function of B. These figures show no evident B-dependence of ~, and z (v = 2.0 4- 0.1 and z = 2.1 + 0 . 2 ) , but Tcr/Teo = 1 - ( B / H c 2 ( O ) ) 0"68. In order to obtain more information on the low temperature state, we analyze the E - J curves below Tcr
Y Enomoto, S. Maekawa/Physica C 274 (1997) 351-356
355
1
1.5 O
F-b t0.5 0.5
06
. . . .
011. . . . .
i
).2
B/Hc2(O) Fig. 4. Critical temperature Tcr as a function magnetic induction B. Solid line indicates the
i
i
I
0.1
i
I
i
i
0.2
B/Hc2(O) of the relation
TcrlTco = 1 -- ( BIHc2(O) ) 0"68.
Fig. 5. Critical exponent/~ as a function of the magnetic induction B.
nents and the critical temperature. by using the functional form/7;_ (x) ~ e x p ( - 1 / x ~ ) , predicted by several models [ 1 ]. This means that the linear resistivity E / J vanishes as J ~ 0 and thus the low temperature state is a purely superconducting phase, since the vortices are frozen into some disordered patterns [2]. The procedure to estimate the exponent /x is as follows: we replot the v ( j ) data as I n ( v / j ) versus j - ~ for T < Tcr with various /z and determine the appropriate/z with which the replotted data form a straight line. In Fig. 5, we show the exponent/x as a function o f the magnetic induction B, This result indicates that/z is almost constant a s / z = 0.49 4- 0.02. Finally, we comment on the related competing simulation results in two-dimensional vortex systems. Some equilibrium and dynamical simulations for the gauge glass model [ 11,13] have shown to be u = 2 b u t Tcr = 0. However, some dynamical simulations asserted the non-zero critical temperature with z = 2 for the lattice Coulomb gas model [ 15], and with z = 2.2, l, = 2 . 2 , / z = 0.5 for the gauge glass model [ 14], which are consistent with our present data. At present, physical reasons and/or physical meaning for such a discrepancy on the critical temperature are not clear. We need carefully check the consistency among situations considered in various works. Unlike the present study, there is no previous simulations to examine the detailed B-dependence of critical expo-
4. Conclusion To conclude, we have carried out the twodimensional Langevin dynamics simulations of magnetic vortices interacting logarithmically each other, including effects of current induced creation of vortex-antivortex pairs and random impurities. By changing values of temperature and magnetic induction, we have measured numerically steady state E - J curves. Then, we have demonstrated that in two dimension the combined effect of random impurities and of current induced vortex pair creation can yield the dynamical scaling behavior for E - J curves with proper values of critical exponent v, z, /z and the critical temperature Tcr. We have also found that u, z and /x are constants (u "~ 2, z ,.o 2 and /x _~ 0.5), but Tot/Too = 1 - (B/Hc2(O))0.68 in the range of the magnetic induction considered here. This is the first dynamical simulation to demonstrate the existence of the dynamical scaling behavior of E J curves from the point of view of the vortex picture. The present result for the dynamical exponent z --~ 2 is consistent with that of the Kosterlitz-Thouless model and experimental results for two-dimensional vortex systems at KT transition temperature [ 12]. However, in these experiments, the dynamical scaling behavior
356
Y Enomoto, S. Maekawa/Physica C 274 (1997) 351-356
o f E - J curves has been less discussed. To i m p r o v e the present study, it is h i g h l y desirable to e x a m i n e E J curves near the K T transition temperature in t w o d i m e n s i o n a l d i s o r d e r e d systems f r o m the v i e w p o i n t o f the d y n a m i c a l scaling. H e r e w e have restricted the present simulation to study the vortex d y n a m i c s in disordered twod i m e n s i o n a l superconductors. However, the qualitative p r e d i c t i o n s o b t a i n e d in this paper w o u l d still give a useful g u i d e to study the d y n a m i c a l scaling behavior even for high-Tc s u p e r c o n d u c t i n g bulk systems, at least for h i g h l y anisotropic B i - b a s e d materials.
Acknowledgements This w o r k was supported by a P r i o r i t y - A r e a Grant f r o m the M i n i s t r y o f Education, S c i e n c e and C u l t u r e o f Japan.
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