The Effect of Temperature and Pinning Density on the Critical Current of a
Superconductor with a Square Periodic Array of Pinning Sites
I. M. Obaidat, U. Al Khawaja, and M. Benkraouda,
Physics Department, United Arab Emirates University,
P.O. Box 17551, Al-Ain, United Arab Emirates.
(Dated: May 25, 2005)
We have studied the effect of temperature and pinning density on the critical current density for
several values of pinning strength through an extensive series of molecular dynamic simulations on
driven vortex lattices interacting with square periodic arrays of pinning sites. We have solved the
over damped equation of vortex motion taking into account the vortex-vortex repulsion interaction,
the attractive vortex-pinning interaction, the thermal force, and the driving Lorentz force. We have
found that the critical current density as a function of the pinning density, may decrease, increase
or even remain constant, depending on the temperature and pinning strength values.
PACS numbers: 4.60.Ge; 64.60.-w; 74.25.DW
I.
INTRODUCTION
In high-Tc superconductors (HTSCs), a perfect superconducting state with a total expulsion of magnetic flux exists
up to a lower critical field Hc1 . At fields larger than the upper critical field Hc2 , the magnetic filed penetrate the
HTSC completely, and the material becomes normal. When the applied magnetic field is between Hc1 and Hc2 , the
magnetic field penetrates the HTSC in form of quantized magnetic flux lines or vortices. The total magnetic flux
each vortex contains is exactly one quantum of magnetic flux Φ0 = hc/2e = 2.07 × 10−7 G.cm2 . The vortices repel
each other and spread out over the entire superconductor volume forming a regular array, known as the Abrikosov
vortex lattice. When an applied current flows through a HTSC in the mixed state, there will be a Lorentz force on
the magnetic flux lines. The Lorentz force on each vortex is given by the expression FL = (1/c)J × Φ0 [1, 2], where
J is the transport current density, and Φ0 is a vector of magnitude hc/2e directed along the vortex length. This
Lorentz force tends to move the vortices transverse to both J and Φ0 . If the vortices move with velocity v they will
induce an electric field, E = (−1/c)B × v parallel to J, causing power to be dissipated (since the energy dissipated
per unit time per unit volume is given by E · J) and an effective resistance will appear in the superconductor. If the
vortices can be pinned and prevented from moving, no resistance will appear. Vortex pinning results from spatial
inhomogeneities of the material which produces local reductions of the free energy of a flux line, thus attracting and
holding vortices to these locations. The main goal, which allows for the use of HTSCs in technology, is to increase
the critical current density, Jc by pinning the vortices in these materials. The critical current density was found to
depend on the type of disorder; point, or extended defects [2, 3]. The pinning of vortices in HTSCs was found to
be fairly weak, especially at high temperatures [2, 3]. Hence there have been many efforts to enhance the pinning
properties in HTSCs by creating structural defects in them using energetic radiations. Irradiation by neutrons [4–8],
protons [9], electrons [10, 11], x-rays [12], and heavy ions [13–18] has been very successful in this respect. General
interest in lithographically created well-defined nanostructured periodic arrays of pinning arrays has increased now
that it is possible to construct samples with well defined periodic pinning structures in which the microscopic pinning
parameters, such as size, depth, periodicity, and density, can be carefully controlled [2, 19–24]. Periodic pinning
arrays are also of technological importance since the arrays can produce higher critical currents than an equal number
of randomly placed pins [25, 26]. This enhancement of critical currents using periodic arrays has recently been
demonstrated for high-Tc systems [27, 28]. Recent simulations of vortices interacting with periodic pinning arrays
[29, 30] or random pinning distributions [31] did not focus on the effect of pinning density on the behavior of the
critical current density as function of temperatures. Instead those studies have focused on the ordering states of
vortex lattice at integer matching fields [29] or at fractional submatching fields [30] and on the melting transition
in a random disorder and at a fixed temperature [31]. We believe that more attention should be paid to the study
of the behavior of the critical current as function of pinning density and temperature. In this paper we present the
results of detailed and extensive molecular dynamic simulations on the behavior of the critical current as a function
of temperature and pinning densities for several values of pinning strengths in square periodic arrays of pinning sites.
In section II, we briefly introduce our numerical method and its parameters, and in section III we discuss our results,
then we conclude with section IV.
2
II.
SIMULATIONS
We consider a 2D transverse slice (in the x − y plane) of an infinite 3D slab containing rigid vortices and columnar
defects, all parallel to both the sample edge and the applied field H = H ẑ. These vortices attain a uniform density nv ,
allowing us to define the external field H = nv φ0 , where φ0 = hc/2e. This model is most relevant to superconductors
with periodic arrays of columnar defects or thin-film superconductors where the vortices can be approximated as
2D objects. We model the vortex-vortex force by a modified Bessel function of the first kind, K1 (r/λ), where λ is
the penetration depth [30]. For the vortex-pin interactions, we assume the pinning potential well to be parabolic
[30]. The pinning range (i.e., the radius of the parabolic well) is rp . For computational efficiency the vortex-vortex
interaction can be safely cut off at 6λ since the Bessel function decays exponentially for r greater than λ. We use finite
temperature overdamped molecular dynamics simulations in two dimensions. The overdamped equation of motion
for each vortex is given by:
fi = fivv + fivp + fiT + fd = ηvi ,
(1)
where fi is the total force on vortex i, fivv is the vortex-vortex interaction, fivp is the vortex-pin force, fd is the
driving force in the x- direction representing the Lorentz force, and fiT is the thermal force which has the properties
< fiT >= 0 and < fiT (t)fjT (t0 ) >= 2ηkB T δ(t − t0 )[30]. The total force on each vortex, due to all other vortices and
pinning sites can be expressed as follows:
fi =
Nv
X
j=1
µ
f 0 K1
|ri − rj |
λ
¶
r̂ij +
Np
X
fp
k=1
rp
µ
|ri − rk |Θ
rp − |ri − rpk |
λ
¶
(2)
Here Θ is the Heaviside step function, ri ( v i ) is the location (velocity) of the ith vortex, rkp is the location of the
k pinning site, r̂ij = (ri − rj )/|ri − rj |, r̂ik = (ri − rpk )/|ri − rpk |, fp is the pinning strength, Nv is the total number
of vortices, and Np is the total number of pinning sites. We measure all forces in units of f0 = φ20 /8π 2 λ3 , fields
in units of φ0 /λ2 , lengths in units of λ, temperature in units of λf0 /kB , and the velocity in units of f0 /η. We take
f0 = kB = η = 1.
The pinning sites are placed in square arrays at densities between np = 0.5/λ2 and 2.0/λ2 . Our system has a size
of 36λ × 36λ. The pinning radius is fixed at rp = 0.2λ, which is less than the distance between pins. Pinning sites of
this size and smaller can trap only one vortex per pinning site, which is similar to the experimental situation [9, 10].
The density of vortices is taken to be fixed, nv = 0.2/λ2 . We consider pinning strength fp varying from 0.5f0 to
3.0f0 .
The vortex phase is defined by measuring the average velocity over all Nv vortices:
th
Nv
1 X
v̄x =
vi · x̂.
Nv i=1
(3)
The transition from the pinned to moving phases are marked by a jump in the v̄x at a well defined fd .
First we obtained the static vortex configurations and then began applying an increasing driving force to obtain the
critical depinning force. We found that the time needed for the vortices to reach a steady state is 2×104 with time step
∆t = 0.02. The critical depinning force, Fdc , is determined by applying a slowly increasing, spatially uniform driving
force, fd , which would correspond to a Lorentz force from an applied current. For each drive increment we measure
the average vortex velocity in the direction of drive, v̄x . The average velocity v̄x versus the force fd curve corresponds
experimentally to a voltage-current, V (I), curve and the critical depinning force corresponds to the critical current
density. The critical depinning force Fdc is defined to be the force value at which v̄x reaches a value of 0.03 times the
ohmic (linear) response [30].
III.
RESULTS
Figure 1 shows the average velocity of all vortices along the direction of the driving Lorentz force with increasing
applied current for different values of temperature T , for a pinning density np = 1.5λ−2 , and pinning strengths
fp = 1.0f0 , 2.0f0 . At zero temperature and at low driving forces, the vortices remain pinned until a critical value of
the current is reached, at which a sharp increase in the average velocity occurs. This sudden jump of the average
velocity corresponds to a state where all the vortices are depinned and flow in the same direction. As the temperature
increases, the onset of depinning shifts to lower driving forces. This is due to the thermal energy which suppresses the
3
effect of pinning forces. From these dynamic phase diagrams we see two distinct phases at low temperatures; a plastic
phase at low driving forces and an elastic phase at high driving forces. The plastic phase appears when only a small
number of vortices are depinned and move under the influence of the driving force. The elastic phase appears when
all the vortices become depinned and flow collectively in one direction, giving a sharp rise in the average velocity. The
transition from plastic to elastic phase is smeared out as temperature increases and disappears at high temperatures,
where we see only an elastic phase. At high temperatures where all the vortices are already depinned, the average
velocity increases linearly with the driving force. It can be seen from figure 1 that as the driving force becomes very
large all the curves converge and a linear regime is attained independently of the temperature. This is expected as
all the vortices are all depinned and the average velocity becomes directly proportional to the applied driving force.
Similar dynamic phase diagrams have been seen for the other values of pinning strength used in this study. From
these diagrams we calculated the critical depinning force Fdc (which is directly related to the critical current density)
at each temperature for specific values of fp and np . The critical depinning force is the value of the driving force at
which an abrupt change in the average velocity of the vortices occurs. From figure 1, we see that while Fdc can be
exactly defined at low temperatures, it becomes difficult to define at higher temperatures. In addition, the appearance
of the ohmic behavior in the plastic region prohibits using a constant value of the average velocity as a criterion to
define Fdc . To overcome these difficulties, we found, as in Ref. [30], that a suitable criterion for the critical depinning
force is the value of the driving force at which v̄x reaches a value of 0.03 times the ohmic response.
Figure 2 shows the critical depinning force Fdc as function of temperature for several values of pinning density np
at fixed pinning strength fp . As it can be seen from the figure, Fdc decreases with temperature for all values of np .
The rate of decrease of Fdc with temperature is fast for small values of fp and slows down as fp increases reaching its
slowest rate for fp = 3.0f0 where Fdc decreases linearly throughout most of the temperature range. This behavior is
in a qualitative agreement with the experimental results of Ref. [32] where it was shown that fast Kr ion irradiation
on Bi2 Sr2 CaCu2 O8 single crystals produces a slower rate of decrease of the critical current density with temperature
than in the case of irradiation by fast Ar ions.
The increase in the pinning density enhances Fdc values at all temperatures for most values of fp . This effect is
more pronounced at T = 0, and diminishes as the temperature increases until it vanishes at high temperatures. These
results are in excellent agreement with the experimental results of Ref. [33] where they studied the effect of neutron
irradiation fluence on the critical current density in Y Ba2 Cu3 O7−δ single crystals. They showed that increasing the
fluence suppresses the critical current density as the temperature is increased. Figure 2 also shows that fp enhances
Fdc for all np values at a given temperature. At T = 0, as fp is increased, the difference in Fdc values for all np values
starts to shrink, converging to the same value for the largest fp = 3.0f0 . Appreciable values of Fdc persist even at high
temperatures for large fp values. Interestingly enough, we found, as figure 2c shows, that for fp = 2.0f0 , the pinning
density has no significant effect on the critical depinning force for all temperatures.
Figure 3 shows the critical depinning force Fdc as function of pinning density np for several temperature values at
fixed pinning strength fp . It can be seen from the figure that for fp = 0.5f0 and at low temperatures, Fdc increases
linearly with a small slope as np is increased. This increase in Fdc with np at low temperatures diminishes as fp increases
and ceases to increase for fp = 3.0f0 . Also the figure shows that at high temperatures, np has a significant effect on
Fdc only for the largest fp value, 3.0f0 , where we start to see a very small linear increase of Fdc as np is increased.
These results are in good qualitative agreement with the experimental results in Refs. [32], and [33]. In Ref. [33], it
was shown that the fluence effect of fast neutron irradiation on the critical current density in Y Ba2 Cu3 O7−δ single
crystals becomes insignificant as the temperature is increased. As in figure 2c, figure 3c shows that for fp = 2.0f0 ,
Fdc depend has very weak dependence on the pinning density for all temperatures. It is also seen in figure 3, that Fdc
may decrease as np increases. This is also in agreement with the experimental results of Ref. [34] where it was shown
that for N dBa2 Cu3 O7−δ and Y2 Ba4 Cu8 O16 single crystals irradiated with fast neutrons, the critical current density
might even decrease with increasing irradiation fluence at high temperatures.
IV.
CONCLUSION
We have conducted extensive simulation study on the effect of pinning density and temperature on the critical
current density in a superconductor with square periodic arrays of pinning sites. We have found that the effect of
the pinning density np on the critical depinning force is significant at low temperatures for small values of pinning
strength fp , whereas at high temperatures, the effect of np on the critical depinning force is noticeable only at large
values of fp . However, for fp = 2.0f0 , we found no significant dependence of the critical depinning force on the
pinning density. Furthermore, we found that for fp = 1.0f0 and at high temperatures the critical depinning force
might decrease with the pinning density. Our results agree qualitatively with numerous experimental results. While
this study was done at a fixed vortex density, future work will focus on the role that the vortex density plays on the
critical current density dependence on temperature.
4
V.
ACKNOWLEDGEMENT
This work was financially supported by the Research Affairs at the UAE University under a contract no. 04-02-211/05.
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5
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1.5
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T=0.0
T=0.5
T=2.0
T=4.5
T=8.0
T=12.5
T=18.0
T=24.5
T=32.0
T=40.5
T=50.0
fp=2.0
np=1.5Λ-2
fd
FIG. 1: The average velocity v̄x vs. the driving force fd at different temperatures. v̄x is in units of f0 /η and fd is in units of
f0 .
6
0.5
ó
ç
0.4æ
Fcd
HaL
0
1
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20
T
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20
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T
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30
HdL fp=3.0
3.5
Fcd
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20
HcL fp=2.0
2.5
Fcd
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np =0.5
np =1.0
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40
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50
np =0.5
np =1.0
np =1.5
np =2.0
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40
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50
np =0.5
np =1.0
np =1.5
np =2.0
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40
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50
np =0.5
np =1.0
np =1.5
np =2.0
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40
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50
FIG. 2: The critical depinning force Fdc vs. the temperature T at different values of the pinning density np . Fdc and fp are in
units of f0 and T is in units of λf0 /kB , and np is in units of 1/λ2 .
7
ç
æ
ƒ
0.4 à
à
0.3
0.6
0.5
Fcd
fp=0.5 HaL
T=50.0
T=12.5
T=0.5
T=0.0
à
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1
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1
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T=50.0
T=12.5
T=0.5
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T=12.5
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T=12.5
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ç
2
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fp=3.0 HdL
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1
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2
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1
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5
np
fp=2.0 HcL
3.5
3
np
æ
ç
2
np
FIG. 3: The critical depinning force Fdc vs. the pinning density np at different temperatures. Units are defined in the caption
of figure 2.