Physica C 332 Ž2000. 285–288
www.elsevier.nlrlocaterphysc
Superconductivity in a mesoscopic double loop:
effect of imperfections q
V.M. Fomin a,),1, J.T. Devreese a,2 , V. Bruyndoncx b, V.V. Moshchalkov b
a
b
Laboratorium Theoretische Fysica Õan de Vaste Stof, UniÕersiteit Antwerpen (UIA), UniÕersiteitsplein 1, B-2610 Antwerpen, Belgium
Laboratorium Õoor Vaste-Stoffysica en Magnetisme, Katholieke UniÕersiteit LeuÕen, Celestijnenlaan 200 D, B-3001 LeuÕen, Belgium
Abstract
Recently determined experimental normalrsuperconducting phase boundaries of mesoscopic superconducting double
loops of Al are analyzed using the network approach generalized to include short-range imperfections. The presence of
weakly scattering imperfections results in a dependence Tc ŽFrF 0 . Žof the critical temperature as a function of the relative
magnetic flux through the opening of one loop. characterized by a smooth transition between symmetric and antisymmetric
states. The best coincidence between the calculated phase boundary and the experimental data is achieved for a configuration
with imperfections in all three branches of the double loop. q 2000 Elsevier Science B.V. All rights reserved.
Keywords: Mesoscopic double loop; Imperfections; Phase boundary
Recently, topology-dependent normalrsuperconducting phase boundaries have been studied in different mesoscopic structures of Al: lines, dots, loops
w1x, double loops w2x, microladders, etc. w3x, with
sizes smaller than the superconducting coherence
length. The experimental phase boundaries for square
q
Expanded version of a talk presented at the VORTEX ESF
Conference ŽCrete, September 18–24, 1999..
)
Corresponding author. Tel.: q32-3-8202460; fax: q32-38202245.
E-mail address: fomin@uia.ua.ac.be ŽV.M. Fomin..
1
Also at: Technische Universiteit Eindhoven, P.O. Box 513,
5600 MB Eindhoven, The Netherlands. Permanent address: Department of Theoretical Physics, State University of Moldova, Str.
A. Mateevici 60, MD-2009 Kishinev, Republic of Moldova.
2
Also at: Universiteit Antwerpen ŽRUCA., Groenenborgerlaan
171, B-2020 Antwerpen, Belgium and Technische Universiteit
Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
loops w1x are in excellent agreement with calculations
w4x based on the Ginzburg–Landau ŽGL. theory. The
phase boundary of a double superconducting loop
Tc ŽFrF 0 . Žwhere F is the magnetic flux through
the opening of a loop, F 0 ' hcr2 e is the magnetic
flux quantum., according to the micronetwork approach w6x, consists of two sets of crossing parabolas
with maxima at integer and half-integer values of
FrF 0 , respectively. In order to investigate the experimentally observed w2x smooth variation of the
minima of critical temperature as a function of magnetic field shown in Fig. 1a, we consider here a
double superconducting mesoscopic square loop with
imperfections as represented in Fig. 1b. These imperfections may be brought in during fabrication of
mesoscopic structures.
In the framework of the GL approach, imperfections in a superconducting structure may be modeled
0921-4534r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 6 8 7 - 5
V.M. Fomin et al.r Physica C 332 (2000) 285–288
286
Fig. 1. Ža. Experimental phase boundaries for a double loop of Al ŽRef. w2x.. A parabolic background due to the finite width w s 130 nm of
the stripes of a double loop is plotted with a dashed line. When transforming the phase boundary to the plane TrTc – FrF 0 , the value
Q s 1302 nm is taken in order to ensure that the maxima of TrTc are at integer values of FrF 0 . Žb. A model of imperfections Žshown as
open circles. in the double loop. The positions of the imperfections represented by the values of Qs can vary within the left-hand Žs s L.,
middle Žs s M., and right-hand Žs s R. branches of the double loop, correspondingly.
by an inhomogeneity of the parameters aŽ r . and
bŽ r . in the GL equations:
1
2m
ž
2e
yi"= y
c
2
AŽ r .
/
c Ž r.
q a Ž r . c Ž r . q b Ž r . < c Ž r . < 2c Ž r . s 0,
D As
4p ie"
mc
Ž c ) =c y c =c ) . q
16p e
mc 2
Ž 1.
2
2
A< c < .
the imperfections. In our work, imperfections are
supposed to be present in all three branches of the
loop, at points characterized by ordinates Qs , s s L,
M, R Žsee Fig. 1b.. In what follows, we choose the
coherence length j ŽT . as the unit length. Eq. Ž3.
now takes on the form:
½ž
i=x q
2p
F0
2
Ax
/ ž
q i=y q
2p
F0
2
Ay
/
Ž 2.
Further, we consider the superconducting states in
the vicinity of the phase boundary. In that region, Ži.
the linearized GL equation provides an adequate
description:
1
2m
ž
2e
yi"= y
c
2
AŽ r .
/
c Ž r . q a Ž r . c Ž r . s 0,
Ž 3.
and Žii. the magnetic field in the structure is supposed to equal the applied magnetic field H.
The ‘‘short-range’’ imperfections localized in the
loop at several points rs are modeled here by choosing:
a Ž r . s a q Ý Vs d Ž r y rs . ;
Ž 4.
s
a is the GL parameter of the substance, and the
values of Vs are determined by the characteristics of
q
5
V˜s d Ž y y Qs . y 1 C s 0.
Ý
ssL ,M ,R
Ž 5.
The resulting dimensionless scattering amplitude:
V˜s s Ž 2 mVsr" 2 . j Ž T .
Ž 6.
is a function of temperature.
We have solved Eq. Ž5. using the micronetwork
approach. A new conceptual ingredient of the present
work, as compared to Refs. w5,6x, is the appearance
of additional nodal points of the micronetwork at the
positions of the imperfections. By integrating Eq. Ž5.
over y from Qs y ´ to Qs q ´ Ž ´ q0., we obtain
an additional condition that is imposed on the derivatives to the left and to the right of the new nodal
point y s Qs :
™
dC
dC
s V˜sC Ž Qs . .
y
dy
ys Q sq e
dy
ysQ sy e
Ž 7.
V.M. Fomin et al.r Physica C 332 (2000) 285–288
The approach based on Eq. Ž7., when applied to a
single loop of length L with one imperfection,
leads to the following secular equation:
cos Ž 2pFrF 0 . s cos Ž L . q 12 V˜s sin Ž L . .
Ž 8.
For a superconducting ring with a lateral arm of
length L, the onset of superconductivity is described
w5x by an equation, which differs from Eq. Ž8. by a
substitution V˜s s ytanŽ L.r2. After some algebra,
we obtain the secular equation, which establishes a
relation between the magnetic flux F and the temperature T along the phase boundary for a double
loop with imperfections in all three branches Žsee
Fig. 1b.. The values of the scattering amplitudes ŽEq.
Ž6.. for s s L,M,R are chosen, as explained below,
from the fitting of the calculated phase boundary to
the experimental data.
The presence of imperfections with small scattering amplitudes results in a continuous change from a
symmetric superconducting order parameter at integer values of FrF 0 to an antisymmetric state at
half-integer values of the relative magnetic flux. This
is illustrated by Fig. 2. When the scattering amplitudes are increased above a certain threshold, gaps
appear for values of the magnetic flux around halfinteger values of FrF 0 .
The best coincidence of the calculated phase
boundaries with the experimental ones w2x in the
vicinity of integer values of FrF 0 occurs for the
zero-temperature coherence length j 0 s 128 nm. This
287
value is in good agreement with previous estimates
Žsee, e.g., Ref. w1x.. The resulting phase boundaries
calculated for the ratio of the temperature T Žalong
the phase boundary. to the critical temperature in the
bulk are shown in Fig. 2a. Within our model, the
presence of imperfections diminishes the critical
temperature of a double loop at zero magnetic field.
Note that experimental data are represented for a
ratio of the temperature along the phase boundary to
the critical temperature measured in a double loop
without magnetic field. Therefore, for a comparison
between theory and experiment, the temperature scale
for the calculated phase boundaries should be renormalized, taking the value of the critical temperature
characteristic of a mesoscopic double loop with imperfections at zero magnetic field Žinstead of the
critical temperature in the bulk. as the unit temperature. The resulting set of phase boundaries is shown
in Fig. 2b.
It follows from the comparison of the phase
boundaries for different configurations of imperfections that the critical temperature at the half-integer
values of FrF 0 is mainly determined by the imperfections in the middle branch. As shown in Fig. 3,
the best coincidence between the calculated phase
boundary and the experimental data is achieved for a
configuration with imperfections in all three branches
of the double loop.
For a perfect double loop, the phase boundary
consists of segments of two types of parabolas w2x.
Fig. 2. Theoretical phase boundaries Žsolid lines. for a double loop with an imperfection in one side branch at Q L s 0.5Q for Q s 1302 nm
and j 0 s 128 nm. The dimensionless scattering amplitude is V˜L s 0.02r 1 y TrTc , where Tc is the critical temperature in the bulk. The
calculated results are represented for two different kinds of normalization of the temperature along the phase boundary: Ža. by the critical
temperature in the bulk; Žb. by the critical temperature characteristic of a mesoscopic double loop with imperfections at zero magnetic field.
Experimental data of Ref. w2x are shown with squares.
'
288
V.M. Fomin et al.r Physica C 332 (2000) 285–288
Fig. 3. Experimental phase boundaries of Ref. w2x for an aluminum mesoscopic double loop Žshown with squares., compared with theoretical
phase boundaries obtained within the micronetwork approach Žsolid lines.: Ža. for a perfect double loop and Žb. for a double loop with
imperfections. The parameters are: Q L s Q R s Q M s 0.5Q for Q s 1302 nm and j 0 s 128 nm. The dimensionless scattering amplitudes,
which provide the best fitting to the experimental data, are: V˜L s yV˜R s 0.017r 1 y TrTc and V˜M s 0.090r 1 y TrTc .
'
The parabolas with maxima at integer values of the
relative magnetic flux refer to different numbers of
magnetic flux quanta penetrating each loop: L s 0,
1, 2, . . . . The parabolas with maxima at half-integer
values of FrF 0 correspond to different odd numbers of magnetic flux quanta penetrating the double
loop as a whole, so that one counts L s 1r2, 3r2,
. . . per loop. These two groups of parabolas are
crossing each other at certain points Žsee Fig. 3a.. In
the crossing points, the left and right derivatives of
the lowest Landau level ELLLŽFrF 0 ., which correlates to the relative condensation temperature w3x, are
different and, hence, the persistent current should
discontinuously change. This unphysical behaviour is
removed when imperfections are present in the
mesoscopic double loop Žsee Fig. 3b..
In summary, the critical temperature at half-integer values of the relative magnetic flux is shown to
be determined mainly by the imperfections in the
central branch of a mesoscopic double loop. The
phase boundary for a mesoscopic double square loop
with imperfections calculated within the micronetwork approach using a fitting of the scattering amplitudes allows an excellent agreement with the experiment.
'
Acknowledgements
The authors thank V.N. Gladilin for fruitful interactions. This work has been supported by the IUAP;
the F.W.O.-V. projects Nos. G.0287.95, G.0232.96,
W.O.G. WO.025.99N ŽBelgium., the PHANTOMS
Research Network, and the ESF Programme VORTEX.
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