EUROPHYSICS LETTERS
1 February 1996
Europhys. Lett., 33 (4), pp. 297-302 (1996)
Avalanches in the Bean critical state:
a characteristic of the random pinning potential
O. Pla1 , N. K. Wilkin2 and H. Jeldtoft Jensen2
1
Instituto de Ciencia de Materiales, CSIC
Universidad Autónoma de Madrid C-III - E-28049 Madrid, Spain
2
Department of Mathematics, Imperial College - 180 Queen’s Gate, London
SW7 2BZ, United Kingdom
(received 8 September 1995; accepted in final form 12 December 1995)
PACS. 74.60Ge – Flux pinning, flux creep, and flux-line lattice dynamics.
PACS. 05.40+j – Fluctuation phenomena, random processes, and Brownian motion.
Abstract. – We report on two-dimensional simulations of the avalanche response of vortices
in the Bean critical state of type-II superconductors. Two different kinds of repulsive-particle
models are considered in order to make the findings generic. We use repulsive two-body
potentials to represent the vortex-vortex interaction and an attractive two-body potential to
represent the interaction between the static pinning wells and the vortices. We find that a
collection of very short-ranged dense pinning centres leads to a broad distribution of response
avalanches, whereas broader and less dense pinning centres produce a narrow distribution of
responses.
The magnetic vortex system in superconductors is a prominent example of the interaction
between a flexible elastic system and a static random background potential. The nature of the
deformations of the vortex system induced by the random pinning potential depends on the
characteristics of the random potential, as well as the manner in which the vortex system is
driven. In the presence of an applied force plastic deformations will always be present in the
depinning regime [1], [2], [3]. In thermal equilibrium, without any applied force, it appears
that the vortex lattice may or may not be plastically deformed depending on the strength of
the pinning potential and the dimensionality of the system [4].
In this paper we study the response of the vortex system in the spatially inhomogeneous
configuration of the Bean state. We confine our study to zero temperature and two spatial
dimensions [5]. We show that the response distribution depends on the properties of the
random pinning potential. Sharp and dense pins lead to a broad distribution of vortex
avalanches. Pinning centres of longer range and lower density produce a narrow distribution
of events.
Model. – In our simulations we treat the vortices as classical particles with a repulsive
interaction and approximate the pinning centres by attractive potentials. Our understanding
c Les Editions de Physique
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EUROPHYSICS LETTERS
vortices
added
vortices
removed
vortices
removed
–L
L
– x 1 x1
Fig. 1. – The configuration we use for our simulations (• vortex, × pin). There are periodic boundary
conditions in the y-direction.
of the importance of the pinning landscape is achieved by using two different models. In the
Gaussian model (GM) we use Gaussian potentials for both the vortex-vortex, Uvv , and vortexpin, Uvp , interactions which enables us to simulate the low-density long-range pin landscape,
a situation relevant to columnar pinning [6]. In the parabolic model (PM) we use parabolic
potentials for both interactions and we are then able via relaxation techniques to simulate
the sharp dense pin landscape. This situation is of relevance to pinning by microscopic point
defects [6].
Explicitly the different potentials are given by
X
X
U =
Uvv (|rvi − rvj |) +
Uvp (|rvi − rpj |),
i,j
i,j6=i
(
Uvv (r) =
(GM),
Av (r − ξv ) Θ(ξv − r)
(PM),
−Ap exp[−r2 /ξp2 ]
¡
¢
(fc /2ξp ) r2 − ξp2 Θ(ξp − r)
(GM),
2
(
Uvp (r) =
Av exp[−r2 /ξv2 ]
(1)
(PM),
where U is the total potential, riv are the vortex positions, rip the pin positions, Av and Ap
are the prefactors of the potentials, and ξv and ξp the vortex and pin ranges, respectively.
The force fc is the strength of the pinning well and is the force needed to detach a vortex
from a pinning centre in the one-body problem. The Heaviside step function Θ(x) has values
Θ(x) = 1 for x ≥ 0 and Θ(x) = 0 for x < 0. The free parameters in the simulation are Av or
Ap , the number of pins, Np , number of vortices Nv and ξp , with all other parameters being
set equal to one. Due to numerical accuracy the GM needs NpGM ≤ Nvcrit , whereas the PM
can make NpGM > Nvcrit , where Nvcrit is the average number of vortices in the critical state.
Our system is configured as shown in fig. 1 with periodic boundary conditions in the ydirection. The vortices are randomly added to the system in the pin-free region −x1 < xv < x1
and, at sufficiently high vortex density, the mutual repulsion forces them into the pinned
regions. The choice of a symmetric configuration obviates the need to define an additional
force to push the vortices into the system. The vortices are removed from the system when
they reach |xv | = L. This configuration is similar to that of a recent experiment of Field et
al. [7] in which they measure avalanche distributions of vortices in a type-II superconductor
O. PLA
et al.:
AVALANCHES IN THE BEAN CRITICAL STATE: ETC.
299
with a hollow-cylinder geometry. The pin-free region corresponds to the exterior of the
superconductor with the density of vortices being proportional to the applied magnetic field.
The region with |xv | > L corresponds to the centre of the sample where the external field is
kept equal to zero.
We drive the systems in the following way. In the GM we add Nin vortices and we then let
the system relax by use of overdamped molecular dynamics, i.e. the velocity of the vortices are
put equal to the total force on that particle, v = ftot (1 ). When the critical state of the system
has been reached, Nout vortices leave the system as a consequence of adding Nin vortices. A
similar procedure is followed for the PM but we relax the system by solving the force balance
equation [5].
The number of vortices in the region x1 ≤ |xv | ≤ L increases until vortices begin to enter
the region |xv | > L where they are removed. We have then reached the Bean critical state [8].
This state is characterized by a balance between the net vortex forces produced by the density
gradient and the pinning forces.
n
∂n
= Fp (n, ∇n, r),
∂r
(2)
where n is the vortex density and Fp the total pinning force. In order to maintain this critical
state, subsequent addition of vortices to the system results in other vortices leaving the system.
Results. – We have studied the distribution D(Nout ) of avalanches of vortices leaving the
sample when Nin new vortices are added to the vortex system in the Bean critical state. We
find that D(Nout ) depends strongly on the nature of the pinning potential. See fig. 2. Sharp
dense pinning wells lead to a broad distribution of avalanche sizes. Whereas smooth and more
sparse pinning wells produce a narrow distribution peaked around Nin .
This observation can be understood as follows. The sharp dense pins produce a large number
of metastable configurations which the vortex density relaxes abruptly between. The reason
for this is that the effective pinning force Fp in eq. (2) produced by the sharp pins will exhibit
threshold behaviour locally, [9] something like
Fp (n, ∇n, r) = n
∂n
Θ(δc (r) − |∇n|).
∂r
(3)
When thresholds exist many different microscopic configurations will exist for a given set of
boundary conditions n(x = x1 ) = n0 and n(x = L) = 0. Although all these configurations
will be stable because none of them locally exceeds the thresholds δc (r) their degree of
stability may be very different. Some configurations will become stable again after only minor
rearrangements whereas other configurations once made unstable will need a repositioning of
all the vortices.
The smooth potentials, on the other hand, are not likely to produce a very sharp threshold [9]. If no thresholds are present the pinning force Fp (n, ∇n, r) will be a simple function
of n. For instance Fp = nα [10]. In this case eq. (2) will have a unique solution for a given
set of boundary conditions. The response of this system will be smooth. A perturbation
anywhere will always be carried through and felt by the entire system. Hence a one-to-one
correspondence is expected between perturbation and response. Let us recall that the original
argument for the existence of broad (power law) behaviour characteristic of self-organized
criticality focused on the existence of threshold dynamics and a large number of metastable
states [11]. This has recently been made very explicit in the paper by Cafiero et al. [12].
(1 ) Here we have set the friction coefficient equal to 1.
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EUROPHYSICS LETTERS
600
500
a)
b)
30000 pins
700 pins
5000 pins
400
400
D(s)
D(s)
300
200
200
100
0
0
20
s
40
0
0
5
10
15
s
Fig. 2. – Avalanche distributions, D(s), where ‘s’ is the size of the avalanche. In a) the PM is used
and 10 particles are added at a time. • Many and narrow pinning centres; ◦ few and wide ones. In
b) the GM is used. Note that the wider pins of the PM are tending towards a distribution similar to
that generated by the GM. See table I for the other parameters used.
This threshold dynamics could arise just because it is microscopically put into the system
by the pinning force in the PM, which jumps discontinuously from 0 to fc at ξp . To investigate
this possibility we have simulated the PM with the modified potential
©
¡
¢ ¡
¢ ¡
¢ª
Uvp (r) = (fc /ξp ) −(r − ξp )2 Θ(ξp − r) 1 − Θ(ξp /2 − r) + r2 − ξp2 /2 Θ ξp /2 − r , (4)
which is continuous but still gives a linear force (necessary for the relaxation techniques). The
results agree well qualitatively with the original PM demonstrating that our threshold effects
are of macroscopic origin.
The different nature of the Bean profile produced by the two different types of pinning
potentials is also seen directly from the spatial dependence of the density profile n(x) =
hn(x, y)iy averaged over the y-direction. The set of longer-ranged sparse pinning centres used
in the GM model leads to a density profile which behaves like
n(x) = n(0)(1 − x/L)α ,
(5)
with α = 2/5. This finding is in agreement with the prediction obtained from eq. (2) by
combining collective pinning theory with elasticity theory [10]. The density profile obtained
in the case of the sharp pins in the PM model cannot be fitted to the functional form shown
in eq. (5). This is consistent with having large fluctuations in the system.
In order to compare the pinning in the two models, we define the individual pinning
strength as γ = |∂r Uvp |max /∂r Uvv |max |, and the average pinning strength in the system as
β = Np πξp2 γ/Ap , where Ap is the area of the system that contains pinning centres. The
values of γ and β relevant to fig. 2 are given in table I. It can be seen that the two models
have similar overall pinning strength such that the difference in behaviour is due to the actual
pinning landscape.
O. PLA
et al.:
AVALANCHES IN THE BEAN CRITICAL STATE: ETC.
301
Table I. – Parameters for the simulations shown in fig. 2.
Model
Av
Ap
Np
ξp
γ
Ap
β
Ncrit
PM
PM
GM
1.2
1.2
1.0
1.0
1.0
0.04
30 000
700
5 000
0.005
0.5
0.125
0.42
0.42
0.41
180
180
480
0.005
1.27
1.64
650
800
2000
Conclusion. – We have demonstrated that the avalanche response in the Bean critical
state depends on the nature of the pinning potential. Sharp pins lead to broad avalanche
distributions whereas broader and less dense pins produce a narrow distribution of avalanches.
We understand this difference in terms of the number of metastable states produced by the
two different types of pinning potentials.
The observed connection between the shape of the pinning potential and the response
allows for a new method of experimentally investigating the nature of the pinning potential.
By accurately measuring the magnetic avalanche distribution as done by Field et al. [7] and
Zieve et al. [13], one should be able to distinguish between the various morphologies of the
potential landscape.
In fact our simulations appear to be in accordance with experimental findings. Field et
al. [7] used a conventional superconductor Nb47%wt Ti47%wt and found that their magnetic
avalanche distribution tended towards a power law distribution. However, Zieve et al. [13] find
that in an untwinned YBaCuO7−δ crystal the avalanche distribution is strongly peaked. The
Nb47%wt Ti47%wt is known to strongly pin the vortices whereas YBaCuO7−δ in its untwinned
state only weakly pins the vortices.
***
We would like to thank S. Spenser and D. O’Kane for many useful discussions and the
British Council-Ministerio de Educación y Ciencia for providing funding for this collaboration.
NKW and HJJ were supported by the EPSRC grant no. Gr/J 36952 and OP by CYCIT
MAT94-0982-C02-02.
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on the shape of the pinning potential.
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EUROPHYSICS LETTERS
[10] Jensen H. J., J. Phys. C, 6 (1994) 149.
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