Current states in superconducting films: numerical results
We present numerical solutions of Aslamazov–Lempitskiy (AL) equations for distributions of the transport current density in thin superconducting films in the absence of external magnetic field, in both the Meissner and the vortex states. These solutions describe smooth transition between the regim...
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irk-123456789-1279602018-01-01T03:03:27Z Current states in superconducting films: numerical results Bezugly, E.V. Свеpхпpоводимость, в том числе высокотемпеpатуpная We present numerical solutions of Aslamazov–Lempitskiy (AL) equations for distributions of the transport current density in thin superconducting films in the absence of external magnetic field, in both the Meissner and the vortex states. These solutions describe smooth transition between the regimes of a wide film and a narrow channel and enable us to find critical currents and current-voltage characteristics within a wide range of the film width and temperature. The normalized critical currents and the electric field were found to be universal functions of the relation between the film width and the magnetic field penetration depth. We calculate the fitting constants of the AL theory and propose approximating formulas for the current density distributions and critical currents. 2015 Article Current states in superconducting films: numerical results / E.V. Bezuglyi // Физика низких температур. — 2015. — Т. 41, № 8. — С. 777–783. — Бібліогр.: 30 назв. — англ. 0132-6414 PACS: 74.25.Sv, 74.25.Uv, 74.78.–w http://dspace.nbuv.gov.ua/handle/123456789/127960 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная Bezugly, E.V. Current states in superconducting films: numerical results Физика низких температур |
| description |
We present numerical solutions of Aslamazov–Lempitskiy (AL) equations for distributions of the transport
current density in thin superconducting films in the absence of external magnetic field, in both the Meissner and
the vortex states. These solutions describe smooth transition between the regimes of a wide film and a narrow
channel and enable us to find critical currents and current-voltage characteristics within a wide range of the film
width and temperature. The normalized critical currents and the electric field were found to be universal functions
of the relation between the film width and the magnetic field penetration depth. We calculate the fitting
constants of the AL theory and propose approximating formulas for the current density distributions and critical
currents. |
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Article |
| author |
Bezugly, E.V. |
| author_facet |
Bezugly, E.V. |
| author_sort |
Bezugly, E.V. |
| title |
Current states in superconducting films: numerical results |
| title_short |
Current states in superconducting films: numerical results |
| title_full |
Current states in superconducting films: numerical results |
| title_fullStr |
Current states in superconducting films: numerical results |
| title_full_unstemmed |
Current states in superconducting films: numerical results |
| title_sort |
current states in superconducting films: numerical results |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2015 |
| topic_facet |
Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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http://dspace.nbuv.gov.ua/handle/123456789/127960 |
| citation_txt |
Current states in superconducting films: numerical results / E.V. Bezuglyi // Физика низких температур. — 2015. — Т. 41, № 8. — С. 777–783. — Бібліогр.: 30 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT bezuglyev currentstatesinsuperconductingfilmsnumericalresults |
| first_indexed |
2025-07-09T08:05:54Z |
| last_indexed |
2025-07-09T08:05:54Z |
| _version_ |
1837155858697420800 |
| fulltext |
© E.V. Bezuglyi, 2015
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8, pp. 777–783
Current states in superconducting films: numerical results
E.V. Bezuglyi
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: eugn.bezuglyi@gmail.com
Received March 24, 2015, published online June 23, 2015
We present numerical solutions of Aslamazov–Lempitskiy (AL) equations for distributions of the transport
current density in thin superconducting films in the absence of external magnetic field, in both the Meissner and
the vortex states. These solutions describe smooth transition between the regimes of a wide film and a narrow
channel and enable us to find critical currents and current-voltage characteristics within a wide range of the film
width and temperature. The normalized critical currents and the electric field were found to be universal func-
tions of the relation between the film width and the magnetic field penetration depth. We calculate the fitting
constants of the AL theory and propose approximating formulas for the current density distributions and critical
currents.
PACS: 74.25.Sv Critical currents;
74.25.Uv Vortex phases (includes vortex lattices, vortex liquids, and vortex glasses);
74.78.–w Superconducting films and low-dimensional structures.
Keywords: superconducting films, vortex state, critical current.
1. Introduction
The main distinguishing property of current states in
wide superconducting films is an inhomogeneous distri-
bution of the current density j across the film as a result
of the Meissner screening of the current-induced magnet-
ic field. It should be noted that the current state of a film
is qualitatively different from the Meissner state of a
bulk superconductor. Whereas the transport current I in
the latter case flows basically within a surface layer with
the thickness of the order of the London penetration
depth , the current in a thin film with the thickness
d is distributed over its width w according to the
approximate power-like law [1,2], 2 2 1/2[( /2) ]j w x ,
where x is the transversal coordinate with the origin in
the middle of the film. Thus, the characteristic length
2( ) = 2 ( )/T T d [3], which is commonly referred to as
the penetration depth of the perpendicular magnetic
field, is actually not the scale of the current decay, but
rather plays the role of a “cutoff factor” in the above-
mentioned law of the current distribution at the distances
from the film edges and thereby determines the
magnitude of the edge current density. The latter was
estimated in [1] as / ,ej I d w assuming w to be
larger than and the coherence length .
In such an inhomogeneous situation, the resistive transi-
tion of a wide film occurs [1–4] when ej reaches the value
close to the critical current density GL
cj in the Ginzburg–
Landau (GL) theory. This leads to the expression
GL
c cI j d w for the critical current [1] which is
widely used in analysis of experimental data (see, e.g.,
[5,6]) and imposes a linear temperature dependence of the
critical current ( ) 1 /c cI T T T near the critical tempera-
ture .cT The quantitative theory by Aslamazov and
Lempitskiy (AL) [2] also predicts the linear dependence
( )cI T but gives its magnitude numerically larger than the
above estimate. This result has been confirmed in recent
experiments [7–9].
The instability of the current state at = cI I results in
the entry of vortices into the film which leads to formation
of the vortex part of its I–V characteristic (IVC). The mo-
tion and annihilation of the vortices of opposite signs form
a peak in the current density along the middle axis of the
film. For certain current value ,mI the magnitude of this
peak reaches ,GL
cj which causes instability of the station-
ary vortex flow [2]. Further behavior of the film depends
on the conditions of the heat removal [10] and the quality
of the films. In early experiments, an abrupt transition to
the normal state has been usually observed at = ,mI I
whereas in later researches, in which optimal heat compli-
ance was provided, a step-like structure of the IVC, associ-
ated with the appearance of phase-slip lines, was observed
at > mI I [7,11–14].
mailto:eugn.bezuglyi@gmail.com
E.V. Bezuglyi
778 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8
In the immediate vicinity of cT where ( )T unlimited-
ly grows, any film reveals the features of a narrow channel at
:w the critical current in this case is due to uniform
pair-breaking thus showing the temperature dependence of
the GL pair-breaking current 3/2( ) (1 / ) .GL
c cI T T T As
the temperature decreases, the film is expected to exhibit a
crossover to the wide film regime at ,w when the vor-
tex part of the IVC occurs and the temperature dependence
( )cI T should be linear [1,2]. However, it was found [8,9]
that in moderately wide films, the nonlinear temperature de-
pendence of cI holds down to low enough temperatures and
transforms to the linear one only when ( )T becomes
smaller than the film width by the factor of 10–20, although
the vortex state already occurs at much larger values of
(0.2–0.25) .w Similar difficulties were met in the ex-
periment [15] when trying to interpret the IVC measurements
by using asymptotical results of the AL theory, because the
strong condition c mI I used in [2] can be fulfilled only in
extremely wide films whose width exceeds ( )T by sever-
al orders of magnitude. Thus, there exist a considerable in-
termediate region of the film widths and temperatures, where
the assumptions and initial equations of the AL theory re-
main valid, but the asymptotic results cannot provide satis-
factory agreement with the experimental data.
In order to obtain a quantitative theoretical description
of the current states within a wide region of the ratio
/ ,w we perform in this paper a numerical solution of
the AL equations which describes smooth transition be-
tween the regimes of a wide film and a narrow channel.
The critical currents cI and mI normalized on the GL
critical current ,GL
cI as well as the specifically normalized
IVC, were found to be universal functions of the ratio
/ .w We calculate the fitting constants in the asymptotic
formulas of the theory [2] and propose approximating ex-
pressions for the current density distributions which are in
rather good agreement with the results of the numerical
computations.
We note that the AL model assumes rather weak pin-
ning, taking the presence of defects into consideration only
through the viscosity of the vortex fluid. The opposite case
of strong pinning corresponds to the model of critical state
with unmovable vortices, which results in quite different
distributions of the current and magnetic field (see, e.g.,
the reviews [16,17]).
2. Basic equations and asymptotic results of AL theory
A starting point of the AL theory are static GL equa-
tions for the dimensionless modulus F of the order pa-
rameter (normalized on its equilibrium value in the GL
theory) and the gauge-invariant vector potential
1
eff= ,Q A
2 2 2 2
eff (1 ) = 0,F F F Q (1)
2rot rot = ( ).F zQ Q (2)
Here the electromagnetic vector potential A is measured in
units of 0 /2 , 0 is the magnetic flux quantum, is
the order parameter phase and eff = / is the effective
GL parameter. The axis z is perpendicular to the film
whose thickness is assumed to be infinitely small, and all
distances are measured in units of .
Usually in thin films, the GL parameter is large,
eff 1. Assuming the film width much larger than
( ),T one thus can neglect the gradient term in Eq. (1) and
use the local relation 2 2= 1F Q between the order pa-
rameter and the vector potential. Inside the thin film, the
latter has only one component yQ Q and can be found
from equation
/2 2
/2
1 ( )[1 ( )]
= – , = / ,
2
w
w
dQ Q x Q x
dx w w
dx x x
(3)
with the Biot–Savard integral which relates the magnetic
field /dQ dx to the dimensionless density 2= (1 )j Q Q of
the surface current. Equations (1)–(3) determine the stabil-
ity threshold of the Meissner state, when the vortices begin
to penetrate into the film, and the edge value of the vector
potential appears to be close to its critical value
=1/ 3GL
cQ in the GL theory for narrow channels.* The
asymptotic value of the critical current at w has
been calculated in [2] and then refined in [18]:
1/2= 15/8 ( / ) .AL GL
c cI I w (4)
The resistive vortex state of a wide film is described by
including the contribution of the vortices 0n ( ( )n x is the
vortex density) to the net magnetic field induction in the
hydrodynamic approximation [1,2,19]. Using the continui-
ty equation for the flux density nv of the vortex fluid,
expressing the vortex velocity v through the linear current
density j and the viscosity coefficient as [21]
1
0= sign ,v j x (5)
and the average electric field — through the flux density as
0= ,E nv one obtains equation
1 3
01
( )
4 2 – = sign
( )
dj j x dx c E
x
w dx x x j x
. (6)
* Actually, according to the definition of Q, the correct formula for the dimensionless current density must have the opposite sign,
j = –Q(1 – Q
2
). Thus, to deal with positive values of the transport current, one should consider negative values of Q. To avoid this
inconvenience, the authors of [1,2] define j as presented in the text and use positive Q values, that does not affect the results. In
our paper, we also follow this convention.
Current states in superconducting films: numerical results
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8 779
Here and below, the coordinate x is normalized on the
film half-width /2,w and the expression sign x indicates
the opposite direction of the vortex motion in different
halves of the film.
An asymptotic analysis of Eq. (6) at w shows [2]
that the IVC is linear in the vicinity of ,cI whereas at large
currents, the voltage grows quadratically,
2
0
0 02 2 3
( )/ , ; 8
= = ,
( / ) , ,
c c c c c
c c
I I I I I I I
V E L E
C I I I I w c
(7)
until the transport current reaches the threshold of stability
of the vortex state,
1/2= ( / ).ln
GL
m cI C I w (8)
In Eqs. (7) and (8), L is the film length, GL
cI is the GL
critical current formally calculated for uniform current
distribution and C , C are fitting constants which cannot
be determined by the asymptotic analysis.
We note that the validity of the AL theory for the analy-
sis of the vortex state is confined by applicability of the
static Ginzburg–Landau equations to the description of the
vortex motion. With this approach, the order parameter
relaxation time ( / )cT ( is the energy relaxation
time) is assumed to be much smaller than other characteris-
tic times of the system. In the opposite case, finiteness of
results in a considerable deformation of the vortex core
and in occurrence of a wake with suppressed order parame-
ter behind the moving vortex. As shown by numerical sim-
ulation [20], this may anomalously enhance the vortex ve-
locity and lead to creation of rapidly moving chains of
vortices treated in [21] as nuclei of the phase-slip lines.
3. Results of numerical calculations
In our calculations, we perform numerical solution of
Eq. (6) with a certain modification. As is obvious, the left-
hand side of Eq. (6) is the approximate form of Eq. (3) , in
which the vector potential Q in the gradient term is re-
placed by the current density j. Such an approximation
corresponds to the linear London relation j Q between
the current and the vector potential which assumes inde-
pendence of of the vector potential. For this reason,
Eq. (6) is usually referred to as a generalized London equa-
tion [1,2]. This does not essentially affect the asymptotic
results [2] because the gradient term is small at ;w
however, in our calculations, we will use the full
nonlinearized version of Eq. (6) in a dimensionless form
(see also [18]):
1
1
1 ( ) sign
– = ,
4 ( )
dQ i x dx E x
dx x x i x
(9)
where the following definitions are introduced,
2
0
3 3
= (1 ), = , ( ) = ( ),
2
GL
cIi Q Q E E E j x i x
w
20
0 2 3
54
= ( ) , =
2
GL
cE I
ww c
. (10)
The distribution of the vector potential is obviously
symmetric, ( ) = ( ),Q x Q x which enables us to consider
Eq. (9) only in the region > 0x and to reduce the integral
in (9) to the region > 0x . After integration of Eq. (9)
from the film edge to a given point x , we finally get*
1 2 2
2
0 1
1
[ ( ) ] = ( ) ln ,
4 ( )1
x
e
x x dx
Q x Q i x dx E
i xx
(11)
where (1)eQ Q is the edge value of the vector potential.
In these notations, the net current I is given by equation
1 1
0 0
3 3
= ( ) = ( )
2
GL
cI w j x dx I i x dx . (12)
At < cI I , the quantity eQ increases with the current
and has to be determined self-consistently from Eqs. (11)
and (12) at zero electric field; this procedure simultaneous-
ly gives the solution for the current distribution across the
film. As noted above, the resistive state of a wide film at
= cI I occurs when eQ approaches the critical value
= 1/ 3.cQ Such a relation is also obviously valid for nar-
row channels, that makes it reasonable to extend it over the
films of arbitrary width. In the resistive vortex state,
> ,cI I the quantity eQ holds its critical value ,cQ and
Eqs. (11) and (12) determine the dependence ( ),E I i.e.,
the IVC, 0( ) = ( ) .V I E I E L
A specific property of these equations is that their solu-
tions, i.e., the normalized current density distribution ( )i x
and the electric field ,E are universal functions of the
parameters /w and / .GL
cI I This implies that the nor-
malized critical current / GL
c cI I and the maximum current
of existence of the vortex state, / ,GL
m cI I as well as the
normalized maximum electric field 0= ( )/ ,m mE E I E are
universal functions of the parameter / .w Thus, the tem-
perature dependencies of these quantities, being expressed
through the variable / ( ),w T must coincide for the films
with different widths and thicknesses, which has been
demonstrated in experiments [8,9].
3.1. Solution in subcritical regime cI I
Solution of Eqs. (11) and (12) can be found by iteration
method, using eQ as initial approximation for the function
( ).Q x Although the iteration parameter 1 /w is
large for a wide film, convergence of the iterations can be
* Similar method has been applied to the problem of critical magnetic field of a wide film at I = 0 [22].
E.V. Bezuglyi
780 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8
nevertheless provided by introducing certain weight factors
for contributions of previous and current iterations. The
result of numerical calculation of the reduced critical cur-
rent, shown in Fig. 1, describes transition from the uni-
formly distributed GL depairing current 3/2( )GL
c cI T T
in a narrow channel to the critical current AL
c cI T T for
a wide film (4). As seen from Fig. 1, the asymptotic de-
pendence (4) can be achieved with appropriate accuracy
only at rather large ratio / >w 10–20.
It should be noted that in some experiments [9,7,23],
the behavior of ( )cI T at the beginning of transition to the
wide film regime was found to be different from the
smooth dependence following from the AL theory. Name-
ly, when the temperature decreases and the ratio /w
exceeds 4–5, the critical current sharply falls to the value
( ) 0.8 ( )GL
c cI T I T and holds this level until / 10.w
Within this temperature interval, the film enters the vortex
state at > ,cI I although the temperature dependence of
cI is similar to the case of a vortex-free narrow channel.
Analogous behavior of the critical current in wide films
has been registered in early experiments [24,25]. To ex-
plain such a specific dependence of ( ),cI T it was sup-
posed [9] that the Pearl vortices [26,27] in moderately
wide films may overcome the edge barrier at the edge cur-
rent density 2(1 / )cT T much smaller than the GL criti-
cal current density 3/2(1 / ) ,cT T possibly due to interac-
tion with the opposite film edge.
In Fig. 2(a) we present the current density distribution
across the film at the resistive transition point = cI I and
different ratios / .w Interestingly, these distributions are
well approximated by function
1
2 2
( ) = .
1 (1 )
e
a
j x j
a x
(13)
Equation (13) represents a modification of the asymptotic
function [1,2,19] 2 1/2( ) = (0)(1 )j x j x with a regulariza-
tion parameter 1= (0)/ ea j j which provides finiteness of
the approximated current density (13) at the film edges. As
follows from its definition, this parameter characterizes
suppression of the current in the middle of the film due to
the Meissner screening. Substituting Eq. (13) with
= /GL GL
e c cj j I w into Eq. (12), we obtain equation for its
value = cosca at the critical current,
/ = / tan .GL
c cI I (14)
In the case of a wide film, ,w the coefficient ca is
small, 1,ca and it can be estimated by using the asymp-
totic value (4) of the critical current as
1/2= 2.74( / ) .ca w (15)
Within the framework of the generalized London's
equation (6) in which the effect of the current on the order
parameter is neglected, the current distribution is deter-
mined only by geometric factors, therefore the coefficient
a is independent of the current and holds a constant value
ac. The edge current density in this approximation varies
linearly with the transport current, that reproduces the re-
sult of [1],
= ( / ) GL
e c cj I I j (16)
(see dashed lines in Fig. 2(b))*. In the general case de-
scribed by Eq. (9), the dependence ( )ej I appears to be
nonlinear, and the coefficient a increases with the current
and approaches a maximum value ca at = cI I (solid lines
in Fig. 2(b)). Physically, this is due to suppression of the
* Numerical solutions of (6) have been obtained in [28] and used for determination of the penetration depth.
Fig. 1. Numerically calculated critical current (1) compared to the
asymptotical estimate (4) (2).
Fig. 2. Current distributions over the film width at the resistive
transition point, I – Ic, numerically calculated for different values
w/ = 1, 5, and 20 (solid lines). Dashed lines show the approxi-
mating dependence (13) with the critical value of the coefficient
a = ac found from Eq. (14) (a); numerically calculated dependen-
cies of the edge current density je and the current suppression
coefficient in the middle of the film, a = j(0)/je, on the transport
current I for the wide film w/ = 20 (solid lines). For compari-
son, the values (15) and (16) found from the generalized Lon-
don's equation (6) are shown by the dashed lines (b).
Current states in superconducting films: numerical results
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8 781
Fig. 3. Solid lines — distributions of the net current density (a)
and the vortex contribution (b) in the vortex state at w/ = 20
numerically calculated at different values of the transport current:
I = Ic (1), I = 0.5(Ic + Im) (2), I = Im (3). The approximating dis-
tributions (17) are shown by dashed lines.
order parameter by the transport current, that weakens the
screening effect while the current increases.
3.2. Solution in the vortex state, <c mI I I
In the region of the vortex resistivity, the distribution of
the screening current is superimposed by the distribution
associated with the vortex motion and having a peak at the
middle of the film, as shown in Fig. 3. The logarithmic
feature 1/2( / )ln w of this peak predicted in [2] appears
to be rather weak and remains visible only for a certain
intermediate current value; at ,mI I this feature practi-
cally vanishes. Such an inhomogeneous current distribu-
tion with three maxima in the vortex state of wide films
has been visualized experimentally [29] by using the laser
scanning microscope.
For moderately wide films, in which the above-men-
tioned logarithmic factor is of the order of unity, the vortex
contribution can be approximated by a piecewise-linear
function
2( ) = (1 | |)GL
cj x j b x (17)
depicted in Fig. 3(b) by dashed lines. As follows from (17),
the parameter 2= (0)/ GL
cb j j represents the relative (in
units of )GL
cj current density created by vortices in the
middle of the film. Within such an approximation, this
parameter linearly depends on the transport current,
( ) = 2( )/ .GL
c cb I I I I (18)
According to [2], the vortex state becomes unstable
when the height of the central peak of the current distribu-
tion approaches .GL
cj Using this condition and solving
Eqs. (9) and (11) at the critical edge value of the vector
potential, = ,e cQ Q we determine the maximum current of
existence of the vortex state mI and the normalized maxi-
mum electric field = ( ).m mE E I The results of numerical
calculation compared to the asymptotic results of the AL
theory are presented in Fig. 4 by solid and dashed lines,
respectively. At large enough values of /w 20–30,
the asymptotic dependencies [2]
21/2
1 2 3/ = ( / ), = ( / ) ,ln
GL GL
m c m m cI I C C w E C I I (19)
can be fitted to the numerical results by an appropriate
choice of the fitting constants of the AL theory (shown in
the caption of Fig. 4) which cannot be evaluated within the
framework of the asymptotical analysis. In order to obtain
a satisfactory agreement, one has to introduce an additional
constant 2C into the argument of the logarithm, since the
formulas (19) were derived in [2] with logarithmic accura-
cy. At smaller / 20,w the asymptotic results (19)
considerably overestimate the numerically obtained values
of mI and .mE
Another useful expression for mI suitable for a rather
wide range of film widths can be obtained from the ap-
proximating current distributions (13) and (17). At the sta-
bility threshold of the vortex state, where (0) =j
1 2(0) (0) = ,GL
cj j j the relation =1 cb a is fulfilled
which leads to equation
= 0.5 (1 ).GL
m c c cI I I a (20)
As seen from Fig. 4, this approximation (dotted line) rather
well reproduces the result of numerical calculations of ,mI
up to the point of nucleation of the vortex resistivity at
/ 4.w For extremely wide films, in which m cI I
and the logarithmic peak of the current is well pronounced,
the AL asymptotic expression (19) for mI with numerical-
ly calculated fitting constants is more preferable.
In Fig. 5 a normalized IVC per unit length of a wide
film ( / = 20)w is shown by the curve 1 within the re-
Fig. 4. Dependencies of the maximum current of existence of the
vortex state Im and the normalized maximum electric field E m on
the parameter w/ (solid lines). Dashed lines show their asymptot-
ic behavior (19) in the AL theory with the fitting constants C1 =
= 1.2, C2 = 0.4, C3 = 0.062. Dotted line depicts the approximating
dependence (20) of Im, in which the result of numerical calculation
of Ic (see Fig. 1) and the formula (15) for the parameter ac were
used.
E.V. Bezuglyi
782 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 8
gion of the stable vortex state < < 1.7 .c m cI I I I Its ini-
tial part coincides with the linear AL asymptotic (line 2) at
c cI I I ,
2
4
( ) = 1 0.873 1 ,
27
c
GL
c cc
I I I
E I
I w II
(21)
obtained by using Eq. (4) for cI . At > 1.4 ,cI I the IVC is
well described by the modified AL asymptotic for :cI I
2 2 2
1 2 1 2
4
( ) = 0.873
27
c
GL
c cc
I I I
E I C C C C
I w II
(22)
with the fitting constants 1 = 0.97C and 2 = 0.7.C Intro-
duction of an additional constant 2 ,C which shifts the
original AL parabola, enables us to generalize the result
obtained in [2] for the case of large supercriticality,
,cI I to the region of currents comparable with .cI
Such a modification of the AL asymptotic formulas has
been successfully used for fitting of the parabolic part of
the IVC [15]. In experiments with films of relatively small
width (in which the vortex state nevertheless exists), the
region of vortex resistivity is rather narrow, ,m c cI I I
and only a linear part of the IVC is observed [7,15,23].
Similar current distributions and IVCs were obtained by
numerical simulation of the vortex motion in an infinitely
long and thick superconducting slab [20]. Although these
results cannot be quantitatively applied to the thin film
because of essential difference between Abrikosov vortices
in a bulk slab and Pearl vortices in a thin film [26,27], they
give an additional theoretical evidence of intrinsic nonline-
arity of the IVCs in the vortex state, which is often as-
cribed to the flux creep or to the nonequilibrium state of
quasiparticles in the vortex core [30].
4. Summary
We studied distributions of the transport current density
in thin superconducting films in zero external magnetic
field within a wide range of the film widths w and tem-
peratures, using numerical solutions of the integro-
differential equations for the gauge-invariant vector poten-
tial. We found that these solutions can be approximated by
rather simple analytical formulas, the parameters of which
have a clear physical meaning and can be relatively easily
calculated.
We found that the reduced critical current cI and the
reduced maximum current of existence of the vortex state
mI (both normalized on the Ginzburg–Landau critical
current in a uniform current state), as well as the reduced
maximum electric field in the vortex state are universal
functions of the parameter / ;w this has been confirmed
in the experiment. We calculated numerically the current-
voltage characteristic of a wide film in the vortex state and
propose a modification of the asymptotic results [2] which
provides much better fitting with the experimental data.
For wide enough films, /w 20–30, our results coin-
cide with the asymptotic dependencies [2,18] with properly
chosen fitting constants.
The author is grateful to I.V. Zolochevskii for helpful
discussions and advices.
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