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Critical currents in superconducting networks and magnetic field effects
Y.Y. Wang, B. Pannetier, R. Rammal
To cite this version:
Y.Y. Wang, B. Pannetier, R. Rammal. Critical currents in superconducting networks and magnetic field effects. Journal de Physique, 1988, 49 (12), pp.2045-2057. �10.1051/jphys:0198800490120204500�.
�jpa-00210885�
Critical currents in superconducting networks and magnetic field effects
Y. Y. Wang, B. Pannetier and R. Rammal
Centre de Recherches sur les Très Basses Températures, C.N.R.S., B.P. 166X, 38042 Genoble Cedex, France (Reçu le 31 aoat 1987, accepté sous forme définitive le 18 juillet 1988)
Résumé. 2014 On a étudié les solutions des équations des réseaux supraconducteurs en présence d’un courant
extérieur. On calcule les courants critiques pour les géométries typiques. En présence d’un champ magnétique,
le courant critique est très sensible à la topologie du réseau. Ceci est illustré de façon explicite sur un réseau
carré infini. On compare nos calculs aux résultats de mesures préliminaires du courant critique effectuées sur un réseau carré de filaments d’indium submicroniques.
Abstract. 2014 Network equations for finite and extended superconducting arrays are studied in the presence of external currents. Critical currents are obtained for typical network geometries. In the presence of a magnetic field, the critical currents are shown to be very sensitive to the underlying topology of the network. This is illustrated through explicit calculations on an extended square network. Our results are compared with some preliminary critical current measurements performed on a square array made of submicronic indium wires.
Classification
Physics Abstracts
74.10 - 74.60EC - 74.60J
1. Introduction.
Recent technical and theoretical advances have led to a revival of interest in the magnetic properties of superconducting networks [1]. One line of thought
sees these structures as an appropriate tool for studying some specific frustration effects. Another
one sees the diamagnetic superconducting currents
as a sensitive tool for studying the topology of regular or irregular network structures [2-4]. Up to
now the considered networks fall into three
categories : simple finite geometries (single ring, double-loop, etc.) ; infinite regular arrays (square or honeycomb lattices, Sierpinski gasket or carpet,
Penrose quasi-periodic patterns, etc.) and disordered
structures (percolation clusters). In all these cases, attention has been focused on the following physical properties : i) the critical field for the onset of
superconductivity He, ii) magnetization M(T, H)
and derivatives aM/aT, aM/aH ant zero or finite
field, iii) mixed state (current distribution, order parameter configurations, etc.). In this respect, the important problem of the critical current Ic has not
been worked out in details. To our knowledge the only available calculation of 7p is that of reference [5]
where the zero field limit of Ic has been considered,
as well as the case a single ring in a magnetic field.
The main purpose of this paper is to report on
some new results relative to the critical currents in a
superconducting network at finite magnetic field. In
addition to theoretical considerations, we present
some preliminary critical current measurements per- formed on a square array made of submicronic in wires. The paper is organized as follows. Section 2 is
devoted to a brief summary of the mixed state
properties and the associated supercurrent densities.
The network equations with external currents are
considered in section 3, with two illustrative exam-
ples. The precise calculation of the critical currents is addressed in section 4, where the variation of
7p as function of the magnetic field is studied in details for two geometries : single loop and infinite square array. In section 5, our results are compared
with the experimental data of critical current
measurements.
2. Supercurrents in the mixed state in absence of external currents
The properties of the mixed state (magnetization,
order parameter configuration, etc.) of a supercon-
ducting network have been worked out recently in
reference [6] (see Appendix A). In this section we
summarize the main results so obtained for the supercurrents in the equilibrium state. This allows in
particular a coherent set of notations.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198800490120204500
The approach to understanding the superconduct- ing properties of wire networks below the critical temperature To is based upon the Ginzburg-Landau (GL) theory and follows the original treatment of
the mixed state of type II superconductors [4, 6]. As
a main approximation it is assumed that the super-
conducting phase is described by a complex order parameter (with neglecting fluctuations) and that the
actual order parameter is, by continuity, very close to the solution of the linearized GL equation in the
considered geometry.
The main step consists therefore in searching the
solutions of the linear equation and treating by perturbation the higher order term in the GL free energy expansion.
Since the linear solutions will be used away from the phase transition line it is convenient to write the GL equation as a eigenvalue equation for the order
parameter :
where A is the vector potential, CPo = hc/2 e the flux
quantum and l/g; (noted £(H) in the Appendix A)
is the eigenvalue which corresponds to the eigenfunc-
tion W. Using this definition g e is a function of the
magnetic field and the boundary condition, i.e. local minima of the linear GL free energy. Since this section is devoted to the equilibrium state, we will
restrict here to the solution for which 1/62 is
minimum.
When specialized to a wire network, the above equation reduces to the following set of finite
difference equation [2, 4] :
The considered network is supposed made of
strands (i,j) of lengths f i j connecting the nodes i where the order parameter assumes the value
1Jt i. In equation (2.1b), yij denotes the phase factor
and s is the curvilinear coordinate along the strand (i,j ) (see Fig. 1 for notation).
Equation (2.1b) has been successfully used to
determine the phase transition line of superconduct- ing networks in various geometries (see Ref. [1] and
reference therein). Here the critical temperature
Tc(H) is determined by the minimum eigenvalue :
Fig. 1. - Used notations at the junction i of different strands.
where g (T) = g (0)/ 1- T/To is temperature de-
pendent coherence length, and To = Tc(H)IH=o.
The determination of the supercurrent induced by
the magnetic field requires an explicit knowledge of
the order parameter over the network. For a given
solution of equation (2.1b), the order parameter
value and the supercurrent between nodes i and j are given as a function of the W values at the nodes by
the following expression :
where m * refers to the mass of an electron pair.
1’is is the phase factor
For a finite network with N nodes, it is convenient to write 1/1’ as 1/1’ = APO Ii where Ii is the normalized
order parameter at node i and ip 0 2 is the mean square of the amplitude over the whole network. In regular
networks (fij = f ), the normalization condition sim-
ply writes L 1 Ii ,2 = N (see Appendix B).
Similarly we define the normalized current jij by lij = Jo jij where Jo is given by :
In equation (2.4) we have assumed that ~ij = f is
the same for all strands. This is appropriate for regular structures such as the square network. The normalized current jij becomes :
Because of the linearity of equation (2.1), 1/1’ 0 and
therefore Jo have to be formed by considering the
non linear terms in the GL equations. This pertur- bative approach to the mixed state is outlined in
appendix A. Briefly it consists in minimizing the
fourth order term in the GL free energy expansion
as function of both the order parameter and the
magnetic induction. At equilibrium (no external current) 1/e2 is minimum and 410 and F, can be expressed in terms of the phase transition line
Tc(H) which is defined by equation (2.2). In this
situation only we have :
a) The mean amplitude of the order parameter
W2 is linear in Tc(H) - T :
b) The free energy difference Fs - Fn is given by
Here W§§ (0 ) = ’/I’ (T) I T = 0 and W§§ ( T) is the usual
expression W§§ (T) = - a (T)/p of the uniform bulk solution at which the free energy AF =
a [ W [ 2
+ ’
[ W [ 4 assumes its minimum value and /3 A is the generalizedAbrikosov parameter [8] which is generally field dependent.
c) The configuration of the order parameter and the distribution of supercurrent are obtained from
equation (2.3) and reflect the specific vortex struc- ture of the mixed state. For example [6] in a square network the equilibrium configuration consists in a periodic arrangement of basic supercells.
From equations (2.6) and (2.7) it is clear that the
specific features of the network topology appears
through the field dependence of the critical tempera-
ture Tc(H) and the numerical value of f3 A. In a type II superconductor, the minima of the free energy are
fixed by those of 13 A [7] and the reader is directed to
reference [6] for further details.
It is important to notice that the above formulation is also valid in a bulk superconductor. For instance at H = 0, 1JI’ becomes uniform and 8 A = 1, the
order parameter being
Using equation (2.6), the average supercurrent amplitude Jo reads
In this expression g e is taken at the band edge
(1/03BE2e minimum) and the prefactor J,, is given by
(rc = GL parameter). As it should be, Jo vanishes as Tc(H) is approached.
The above expressions, valid in the equilibrium
state (zero external current) are no longer valid in
presence of an applied current since 6, and therefore
the critical temperature are expected to be modified.
In the following Tc (H) will be kept as the actual
transition temperature in zero current. The approxi-
mate expression for Jo, valid without restriction on
the transport current is obtained in terms of 03BEe using equations (2.4) and (A. 16) :
The influence of an external current on 03BEe is
discussed in the next section.
3. Networks with external currents
When external currents are injected at some nodes
of the network, one must solve the Ginzburg-Land-
au equation with taking into account this additional
boundary condition. Within our approximation of
the mixed state one has to calculate the new
eigenstates 03BEe of the linearized Ginzburg-Landau equation. e now becomes dependent on both the magnetic field and the external current. In this situation equation (2.1) and therefore equation (2.6)
must be modified. A convenient procedure for including external currents effects in equation (2.1)
has been described in reference [9]. Here we recall briefly this procedure and then describe two simple examples which illustrate this method and its limi- tations. Critical current calculations are based on the results of this section.
3.1 NETWORK EQUATIONS WITH EXTERNAL CUR- RENTS. - In order to take into account the presence of an injected current at node i, we separate in equation(2.1) the term associated with strand
(t, n) :
A simple calculation shows that for vanishing
fin’ the expression
becomes a pure imaginary number :
where we have defined Qi by (’Pi =- I ’Pi I e’v’) :
Clearly, Qi has the meaning of a gauge-invariant velocity field of pairs, related to the external current
Jext at node i by
Accordingly the network equation at node i becomes
As we have mentioned, in presence of external current, 03BEe depends on H and Je, according to equations (3.5) and (3.6). For a fixed external cur-
rent, 1/03BE2e reaches its local minima at a new equilib-
rium state, which are the solutions of the new
boundary conditions (Eqs. (3.5) and (3.6)), and give
the critical temperature in presence of external current.
3.2 Two EXAMPLES. - The physical meaning and
the limitations of equation (3.6) become clear when simple cases are considered. This is illustrated here
on two examples : single loop and infinite networks.
a) Single loop case (Fig. 2). - The secular
equation of the linear system (Eq. (3.6)) correspond- ing to this geometry reads simply (Q = Q1= - Q2)
Fig. 2. - A square single loop with injected currents at nodes 1 and 2.
where a refers to the side of the square loop and
y = 2,7rO/Oo is the reduced flux (tP = Ha 2). It is
found from equations (3.6) and (3.7) that the order
parameters at nodes 1 and 2 have the same ampli-
tude : 11,1 1 2 =1 "2 12 , as expected by symmetry.
Solving this equation for Q, one obtains the follow-
ing expression for the current Jext :
Note that the factors I fl 12 = I f2l2 =1 have been
ignored.
Equation (3.8) can also be solved for 03BEe under the given external current Jext. This point of view has
been adopted in reference [9]. Here we just point
out the relation between equation (3.8) and the
usual definition of supercurrents [7]. Let a be the
phase difference between the two nodes : W2 = W, eia. We have trivial solutions of equation (3.6) :
and
from which one deduces :
therefore
Comparing this equation and the « derivative of free energy F (see Eq. (A. 17)) one has the simple result
for the mean current of one single strand J/2 :
Clearly equation (3.11) reproduces the usual defi-
nition of J as derivative of the free energy F with respect to the phase of the order parameter. Actu- ally, equation (3.11) is a very general relation which
holds for an arbitrary geometry and can then be used
as an alternative to equation (3.5).
b) Square network (Fig. 3). - For an infinite network, it is not convenient to use equation (3.6)
and this particularly for massive contacts as shown
Fig. 3. - An infinite square network with horizontal current injected through massive contacts.
on figure 3. For this we adopt a slightly different point of view, by considering first the solutions of
equation (2.1) corresponding to the lowest free
energy. With obvious notations, the network
equations reduce to [2, 4]
where y = 2 7T4> /4>0’ e = 4 cos (a/03BEe) and f m n =
e’k’ f m is a translationally invariant solution. For
rational flux y = 2 irplq, the general solutions are q-periodic : f m + q = f m ei a with a Floquet factor
a. This leads in particular to a determinental
equation [10, 11]
where Pq(’-) = Eq + ...is a polynomial of degree q in B.
In what follows we consider mainly the normalized
average current per elementary strand = J_ /Jo (horizontal) and jt = Jt /Jo (vertical). Due to the q periodicity, we have simply
These expressions can be simplified further by using
the following identities [11]
and
where P’(B)= dPq(B)/dB. Therefore one obtains
finally
where the normalization has been used. Note that equation (3.16) have been derived here for particular wave functions f m, n. However it
is not difficult to check the validity of this result for
an arbitrary solution
Therefore, the final expression for the current
densities become
Clearly, J_ and Jt involve in addition to T and H, the energy e of the corresponding solution as well
as the phase factors a and qk. For example at the spectrum edge of equation (3.12) (ground state) one
has P (c) = 4, a = qk = 0 and then J_ = Jt = 0. In general equation (3.17) will be used to calculate the allowed solutions (e) giving a set of external currents
J_ and Jt. In cases where more than one solution is
allowed, the lowest in energy will be considered as
corresponding to the real state. Making further
restriction qk = 0 or 7r, only the horizontal current
remains and a is fixed by a =
Arc cos [P (e )/2 =+ 1 ]. In this case, one gets
and
- 1-1
This complicated expression for the horizontal current can actually be written in terms of other
properties of the spectrum of equation (3.12). In- deed, in the appendix C, the following relations are
derived :
where 9 (e) is the density of states of equation (3.12),
E =1/03BE2e and F refers to the free energy (Eq.
(A.17)).
More generally, the obtained expression for the
currents can be cast in a more transparent form.
Using equations (3.15) and (A.17), one gets
with Jo = 4 ealh. Note that equation (3.22) is ident-
ical to equation (3.11) and this is a non surprising
common feature to finite and infinite networks.
Finally, on figure 4 is shown a diagonal configur-
ation of the same square network. For such a
diagonal injection of currents, we have Jt = J_ and
this gives kq = a and Pq( e) = 4 cos a respectively.
Similar calculation leads in this case to the following expressions of the current :
which are the counterparts of equations (3.18) and (3.19). This configuration is relevant for the exper- imental results of section 5.
Fig. 4. - Same as figure 3 with a diagonal injection of the
currents.
3.3 DISCUSSION. - Let us conclude this section with some remarks. Whereas the formalism of reference [9] is appropriate for finite networks with
point contacts, infinite networks call for a slightly
modified formulation. Indeed in that situation, and
far away from the massive contacts, there is a
spontaneous current pattern required by the minimi-
zation of the free energy. Such a pattern has to be
matched to the boundary conditions in order to fit the current injection. Fortunately, in a trans- lationally invariant network, the conjugate variable
to J, i.e. the phase a (or qk) is introduced in a
natural way and this allows for the simple result
given above. This valuable simplification is at the
basis of the critical currents calculation of section 4.
4. Critical current of a superconducting network
Besides the thermodynamical properties, critical
currents Jc in superconducting networks are of basic
interest. Already in bulk superconductors, Jc is a
very important topic both because of its origin (flux jump, thermal instabilities, pair-breaking, etc.) and
for its dependence on various parameters (tempera-
ture, magnetic field, voltage, ...). In the case worked here, we have a slight simplification, due to the fact
that magnetic fluxes are fixed and the inductive fields are smaller than H, ,2 around wires.
4.1 DEFINITION AND SIMPLE EXAMPLE. - The criti- cal current Jc is defined usually [7] as the value of the
external current beyond which the normal state is recovered. Let us illustrate this definition on the
example of a single wire in zero magnetic field as
described in reference [7]. Assuming a small width d f (T), one can write W (s) =- Wo eitp(s) for the
order parameter along the wire, Wo being indepen-
dent of the coordinate s. In this limit, the free energy
assumes the following form :
where vs denotes the velocity of pairs
The conjugate variables J and v s are related by
and the explicit expression of Jc is obtained in two
steps. First, the free energy is minimized with respect to 11,12. In the present case this yields
Next, J as given by equation (4.3) is maximized under the constraint (Eq. (4.4)). Eliminating I T 1 1,
J can be expressed as a cubic function of vs :
J, is given by the maximum of equation (4.5), and
this leads to the 3/2 law
In order to make contact with the procedure used below, we give another derivation of this result. For this we notice that (m * vsllt)2 corresponds to the eigenvalue 1162 of the linearized GL equation:
which for a single wire reads simply
The eigenvalue is trivially
and this provides a one to one correspondence between and v,. With this change of variables, equations (4.4) and (4.5) become
and
Maximizing equation (4.10) with respect to 6, repro- duces the above expression (Eq. (4.6)) of Jc.
4.2 SINGLE LOOP GEOMETRY. - Consider the con-
figuration depicted on figure 2 and let calculate the critical current Jc. The first step of minimization of the free energy leads to the network equation
considered in section 2. The resulting expression of
the current J (Eq. (3.8)) is
The critical current Jc is obtained, as for the single wire, by taking the maximum of J with respect to ç e. In general J,, is a decreasing function of tempera-
ture T. At zero field, one gets :
As expected, Ie is twice that of the single wire equation (4.6). For finite magnetic fields, J,, as
obtained numerically from equation (4.11) is still given by a similar expression
where C ((P / 0 0) is a numerical constant shown in
figure 5. At half quantum flux 4> /4>0 = 1/2,
C (0 / 0 0) vanishes and this is due to the symmetric
location of nodes 1 and 2 which leads to 2 a /03BEe =
?r/2, i.e. Q = 0 at this value of the magnetic field.
Such an accidental degeneracy is no longer present for a generic disposition of nodes.
The result shown in figure 5 has to be compared
with the recent calculation reported by Fink et al.
(Ref. [5]). Agreement is found for 0/00 = 0, 1/2
and 1. However, our approach is valid close to the critical temperature and this allows us to consider various geometries. In this respect, the work re-
ported in reference [5] is an exact treatment for the single ring case with arms and then the difference with our results appears far away from T,.
Fig. 5. - Field dependence of the critical current (Eq.
(4.13)) for the square single loop.
4.3 INFINITE SQUARE NETWORK. - As for the
single loop case the maximum of J -t> with respect to
e (or ge) will be identified as the critical current. The
simple limit H = 0 and T To can be worked out explicitly since J_ (Eq. (3.19)) takes a simple form (p = 0, q = 1 and P1(03B5)=E):
and 03BE2(0)/03BE2e 1 - T/To is vanishing when T - To.
One gets the 3/2 law for Jc :
When compared with the single wire result, the only difference between equations (4.15) and (4.6) is
