Summary. — We report an enhancement, up to a doubling, of the maximum resistance-less transport current passing through a superconducting thin film in the mixed state. These enhancements occur when an external magnetic field is applied near parallel to the broad planar surface. A description is presented in which entropy in the form of flux disks flows into the film creating an enhanced inhomogeneous current distribution corresponding to a state of driven broken symmetry, permitting the passage of an enhanced maximum resistance-less transport current.
PACS 74.50 – Proximity effects, weak links, tunneling phenomena, and Josephson effects.
1. – Introduction
Josephson [1] considered the existence of flux spots (originally proposed by Pearl [2]) in superconducting thin films in the mixed state. Unless the externally applied field is exactly in the plane of the film, flux spots, disks, or pancake [3] vortices where the superconducting order parameter is zero, occur. These flux disks are short flux line elements, practically normal to the film surface, surrounded by extended circulating supercurrents [2].
Dissipation in the mixed state is associated with the motion of these magnetic flux disks across the width of the film. This motion is achieved when the Lorentz-like force
J 3B from the action of the transport current upon the flux disks exceeds the pinning
force coming from film inhomogeneities, and is evidenced through a steady voltage (V) between the ends of the superconducting track for a particular transport current density (J), greater than the threshold (Jc) which defines the critical transport
current.
2. – Experimental
Amorphous MoSi thin films were prepared on a silicon substrate by sputter deposition, and have Tc A 6 .3 K, and Jc( 4 .2 K , H 40) D1Q109 A Om2. All films were
G. STEPHENS 88
Fig. 1. – Schematic of field m0H(u , f) incident upon superconducting track carrying transport
current I.
0.3 mm in thickness, with a composition of 30% silicon. The films were patterned chemically using a wet etching technique, the track and contact pads being defined lithographically. Palladium contacts were thereafter deposited upon the pads to aid adhesion for the silver paint used to connect voltage and current leads. An electric-field criterion of 15 mV Ocm has been adopted for the Jc measurements. A
superconducting solenoid together with a set of superconducting Helmholtz coils were used to direct the externally applied field H. Figure 1 shows a schematic of this field,
m0H(u , f), incident upon the superconducting track carrying transport current, I. All
measurements were performed at 4.2 K. An estimate of the penetration depth was taken from dHc2O dTNTc, r0, rrt, using Gor8kov theory [4] and found to be about l4
0 .98 mm (Hc2is the upper critical field, rrtis the room temperature resistivity and r0is
the normal state resistivity extrapolated to T 40 K). This is in agreement with amorphous MoSi films of a similar composition and thickness in the literature [5]. The penetration depth is greater than the film thickness l Dd. In such situations, one expects a 2D vortex lattice (instead of 3D flux line) in which the flux line elements are directed perpendicularly to the film surface [6]. Also, the perpendicular vortices are practically straight, even when the magnetic-field lines are strongly curved [7].
Figures 2a)-d) show the critical transport current density Jcvs. external field m0H ,
for different angles f between the transport current and the applied field measured in the plane of the MoSi thin film, with m0H incident near parallel to the broad planar
surface, or with some offset u . Figures 2a), b) and d) show data for only the transport current direction, fig. 2c) shows the effect of reversing the transport current and is denoted by the dashed line. Track widths are 75 mm, 70 mm, 70 mm, 105 mm for figs. 2a)-c) and 2d), respectively. All tracks are 240 mm in length.
We see the trend is from a symmetric enhancement about H 40 T for the
f 407 configuration to an asymmetric m0H vs. Jc characteristic for the f 4907
configuration. Hart and Swartz [8, 9] likewise found an asymmetric transport critical current about the zero-field mark in the f 4907 configuration for their 1 mm thick PbTl films, although they did not observe an enhancement above the self-field transport critical current as we do because they did not make measurements at low enough fields. They developed a critical flux disk state model in order to describe their results. This type of asymmetry has recently been observed
Fig. 2. – Dependence of the enhanced Jcon the angle f between the current and the applied field.
a) f 407, track width 70 mm, with u offset indicated; b) f4227, track width 70 mm, with u offset indicated; c) f 4457, track width 70 mm, with u offset indicated; d) f4907, track width 105 mm, with u offset indicated.
in some High Temperature Superconducting (HTS) films by Roas et al. [10]. The origin of this asymmetry is unclear.
The question now is: how is it possible to pass an enhanced resistance-less transport current for the cases in which the enhancements were found?
3. – Background to the model
The Bean critical state model [11] describes a configuration in which the flux vortices are distributed such that the whole sample carries a single «critical current
G. STEPHENS 90
density». For the case in hand, of a superconducting thin film carrying its maximum resistance-less transport current, the field induction profile and corresponding flux disk distribution, as sketched in fig. 3i) is brought into existence by the self-field of the transport current. The «transport critical current» defines the departure from an immutable field state to one in which there is continuous movement, causing dissipation. This movement takes the form of flux disks of opposite circulatory sense entering at each edge, a corresponding redistribution, and annihilation of vortex–anti-vortex pairs in the central region of the film. The field induction profile shown in fig. 3i), fig. 4 (top diagram), is the simplest, i.e. the linear one. Recent numerical calculations suggest that, for an infinitely thin film, the field induction (and current density) may be peaked very close to the film edges as a consequence of the
Fig. 3. – Open circles: vortex. Crosses: anti-vortex. Closed circles: vortex nucleated from the externally applied field m0H .
Fig. 4.
thin film geometry [12]. A linear field profile is used in the description of the model for the purpose of simplicity, cf. fig. 4.
In this films (d Gl), the forces on the vortices are given by the current. This current is comprised of the Meissner current and the current from vortex curvature and a tiny contribution from vortex gradient [13]. This may be seen by writing the curl B as (˜B) 3B×1 B(˜ 3 B×) with B×4 B O B . The first term is the current from the density gradient and the second is the current from the curvature. It is the second term which dominates in thin films. This means that the standard grad B driven critical state model is an oversimplification. It should be noted, though, that Bean’s main
G. STEPHENS 92
assumption that the flux lines start to move only when J(r) reaches Jc(r) is still
valid [12], and we may have a critical state model driven predominantly by the degree of vortex curvature rather than the vortex density gradient.
Starting from an initial state in which the film is carrying its maximum resistence-less current, fig. 3i)-iii) shows how a state in which one side of the film is effectively devoid of flux disks, and the other is at the critical state, may be achieved under the application of a small external field directed near parallel to the broad planar surface.
4. – The model
At a certain externally applied field intensity m0Hd, dependent on the thickness of
the film, d, flux disks will nucleate and try to distribute themselves uniformly across the film. Externally introduced flux disks appearing on the left hand half of the film cause crowding and subsequent reorganisation of the flux disk induction already present associated with the self field of the transport current, cf. fig. 3ii), iii). These disturbances are propagated towards the axis of the film, mediated by the strong vortex-vortex repulsion resulting in flux disks being displaced towards the flux disks of opposite sense existing on the other side of the film. These displaced flux disks will then be within a range d Edann at which the pinning forces can no longer restrain the
migration and annihilation of such vortex–anti-vortex pairs. Flux disks appearing on the right hand half of the film will have only a transient existence within the film. Because upon their formation, they will immediately migrate and annihilate with the flux disks of opposite circulatory sense already existing due to the self field of the transport current.
The extreme nature of this effect is brought forward when one considers the large penetration depth l A 0.98 mm of a flux spot within this superconducting system. Thus we are left with a state (at an appropriate external field strength) in which one half of the film is at the critical state and the other half devoid of flux spots, cf. fig. 3iii).
The characteristic dissipation process for superconductors (and all superfluid flow) is phase slippage. Phase slip occurs when vortex–anti-vortex pairs annihilate. The p.d. generate by one annihilating pair may be cancelled by the p.d. generated by the other elsewhere in the film. Hence, vortex motion without dissipation is possible provided the net phase slip between the ends of the superconducting track is zero.
With an appropriate external field m0H , it would now be possible to pass more
transport current through one side of the superconducting track without causing dissipation, cf. fig. 4. The density of which would once again be determined by the pinning strength coming from film inhomogeneities. Thus an additional ( 1 O2) Ic would
be allowed to pass along the strip without causing dissipation. A further increase m0h1
in the external field would wipe out flux disks on the right hand 1 O4 of the film, thereby leaving it available for the passage of more transport current, up to a maximum once again determined by the critical state (state on brink of motion) below which no net continuous motion of flux takes place across the width of the film.
This process could, in principle, be repeated up to a doubling of the maximum resistance-less transport current as indicated in the schematic figure 4.
and annihilation will occur is denoted dann. This property of spontaneous annihilation
below dann was first suggested by Pearl [2]. This point leads to the conclusion that a straightforward superposition of the external field and the self field is not valid.
The density of flux disks introduced by the external field m0H must be dilute in
order to observe any enhancement effect. Their number ndisks(m0H», w) per unit
length, must be of the order of those introduced by the self-field (w 4width of film,
m0H»4perpendicular introduction).
This explains why above a small angle u between the externally applied field and the broad planar surface, there is no enhancement above that of the self field Jc and
why the effect is only observed at small applied fields.
If the external field is applied perpendicular to the broad planar surface, diamagnetic screening occurs, proceeded by rapid flux entry in which the excluded flux rushes in from the edges at once forming a relatively dense flux disk lattice. Two recent papers describe theoretically and show experimentally this particular type of geometrical barrier [14, 15]. In the case in which the externally applied field is directed near parallel to the broad planar surface, flux disks enter at a rate determined by the degree of suppression of the order parameter across the whole surface which is dependent on the magnitude of the external field. For this reason the enhancement is not readily observed in the perpendicular geometry.
We are dealing with a complex many-bodied system containing irreversible processes. In such situations, we may better describe the system in a global way. Prigogine [16] has de-veloped a thermodynamics of irreversible processes in which the system under considera-tion is separated into «internal» and «external» environment. Entropy (matter, energy) flows in from the external environment and though a series of concurrent irreversible proc-esses, the system evolves to a state of increased order or lower entropy.
For the system considered here, this would correspond to the continual flow of flux disks from the external field m0H into the superconducting film carrying its maximum
resistance-less transport current. The irreversible processes would be the vortex–anti-vortex annihilation, the vortex–anti-vortex-vortex–anti-vortex repulsion, forces from film inhomogeneities, (image attraction to the edge) etc. The irreversible processes would have corresponding rates Ji, and forces Xi.
A result claimed to be «so general that it can really be called a universal evolution criterion, valid throughout the whole range of macroscopic physics» [p. 99, ibid.] is the inequality
dF 4
dn!
J 8kd X 8kG 0 ,G. STEPHENS 94
where X 8k are the forces and J 8k the flows, for inhomogeneous systems or for
homogeneous systems, simply
dX[G0 ,
(2)
where [4diS Odt and S is the entropy for the internal system, which expresses the
result that the internal system is robust in going to the more ordered state when the ordering or disordering forces are changed, corresponding for instance to small variations in the thickness of the film, changing the effective penetration depth and so the flux disk interaction forces. The dashes in the J 8k, X 8k indicate that different sets of
generalised forces and flows may be used, but no physical consequences should depend on the particular choice. The step-like variation in the current distribution across the width of the film corresponds to this more ordered state. If an order parameter h is defined as the extra resistance-less transport current density over and above that of the self-field case J0such that h 4NJ2J0N O NJ0N and 0 G h G 1, then the extra J may
the thought of as an «emergent property» of a driven broken symmetry system. The phenomena described here may have particular relevance to the strong anisotropy in Jc of some high-temperature superconductors. Recently, extremely high
Jc values have been observed when the applied field is aligned virtually parallel to the
CuO2 planes and have been conjectured to be due to a kind of lock-in effect in which
flux line segments are strongly pinned between the superconducting planes which contain the flux disks [17]. What might actually be happening, in this case, is that the perpendicular component of the applied field is comparable to self-field induction existing in the CuO2 planes. Thus, the conditions for a reorganising behaviour leading
to higher Jc’s outlined in this paper would be realised.
* * *
I should like to thank H. H. STøLUM, R. J. LEVETTfor discussion of the manuscript. R E F E R E N C E S
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