Nonequilibrium effects in driven superconducting rings

Figure 1.A current can be induced in a superconducting wire by subjecting it to a time dependent magnetic flux. For practical purposes one can imagine the magnetic flux to be due to an infinitely long solenoid passing through the center of the loop. That generates a magnetic flux as shown in the figure.

A quasi-one-dimensional superconducting ring provides, at least in some sense, a prototype system to study various aspects involving driven (accelerated) systems in general. In addition to the more statistical mechanics related questions, non-equilibrium superconductivity is of great interest per se (for a details, please see the the book by R. Tidecks). Indeed, the current-induced transitions in superconducting  filaments have been a subject of intense experimental and theoretical study for almost three decades. We have concentrated on current-induced phenomena, and in particular, on the emergence of the dissipative nbsp; phase-slip state in mesoscopic systems. When a superconductor (below T_c) is driven (by voltage or current source) to the critical current, several interesting phenomena may occur: the system will enter the dissipative phase-slip state, Joule heating can take place, mode locking, as well as other phenomena, may occur. In our study, we concentrated on the onset of dissipation, and its effect on the dynamics of the superconducting state.

The transitions between the current-carrying states can take place via two fundamentally different routes (see Figure 2.): 1) by a nucleation process involving thermal fluctuations and an Arrhenius activation energy barrier, or 2) the system may be driven to the point of instability by an external driving force. In the context of nucleation and metastability, the decay of persistent currents in thin superconductors is an old and extensively studied Problem (see e.g. the discussion + references in Tinkham's book). However, the latter involves a decay from a point of instability, and even in the general context of nonequilibrium dynamics it is a relatively poorly understood problem. One of the major difficulties is this: whereas in the case of nucleation the decay is from a metastable state involving thermal activation and a saddle point, in the latter case the external force drives the system to a point of instability where there is no energy barrier left, i.e., the energy landscape looks locally flat. Then, the decay and the final state depend on various factors, such as how fast the system was driven, the relative strength of fluctuations, internal excitations, and so on. This makes precise theoretical formulation of the problem difficult; it is not possible to use the free energy formulation as in the case of metastability. In our study we have used both analytical and computational methods to investigate the abovementioned problems.

Figure 2. A schematic illustration of the free energy landscape in a superconducting ring. There are many metastable states available for a mesoscopic superconductor. If at some moment of time the system resides is one of the metastable states, it can make a transition to new state provided it can jump over the energy barrier. This can occur via thermal fluctuations (inset), or due to an application of an external driving force leading to vanishing barriers.

It is important to notice that many of the phenomena observed here are by no means limited to superconducting rings, but appear in many other physical systems ranging from pattern forming systems to lasers. The hope is that the simplicity of our system makes it possible to obtain information about some of the general questions in driven non-linear systems such as state selection and the effect of dissipation on the state selection process itself.

The transitions between different current-carrying states occur via a process called phase slip. The applied driving force accelerates the supercurrent leading to an instability, and to a subsequent recovery of the perfectly superconducting state. This cycle is repeated periodically. A phase slip is a point where the system locally loses superconductivity and becomes a normal Ohmic conductor, i.e., the process involves generation of a topological defect. Since we are working below below T_c the system retains the fully superconducting state after making a transition to a state of lower current. The effect of the generation of normal current on the dynamics is an important question. At the phase slip, the system becomes locally metallic and has Ohmic resistance. Due to the resistance, heat is generated. If the heat generated is not large enough to destroy the superconducting state, the system will make a jump to a state of lower current carrying state; this is possible when T<T_c . The name "phase slip" is related to the behavior of the order parameter; the superconducting order parameter is a complex valued quantity characterized by a phase and the amplitude. "Phase slip" refers to a process where the phase can locally make a jump of 2*pi. Similar phenomena occur in Josephson junctions.

Movies from our simulations:

• A movie showing a phase slip. The z-axis is the length of the superconducting wire and the x- and y-axes represent the real and the imaginary parts of the order parameter (see Fig. 3 below): Phase slip
• The same as seen from the top (i.e., showing the real and the imaginary parts of the order parameter) : Phase slip in the order parameter plane.
• The time development of the modulus of the order parameter along the superconducting wire when multiple phase slips appear: Moduls of the order parameter.
• A movie showing a phase slip when there is a strong competition between single and double phase slips . Phase slip.
• The same in order parameter plane
• The time development of the order parameter along the wire. The wiggling is due to thermal fluctuations: Modulus of the order parameter,

Figure 3. The system is mapped to a helix. The z-axis represents the length of the superconducting wire, and the xy-plane represents the real and the imaginary parts of the order parameter. The superconducting wire can be considered as running through the center of the helix.	Figure 4. The supercurrent (y-axis) as a function of time (x-axis). When the systems reaches the critical current, it makes a transition to a state of lower current.	Figure 5. The winding number (y-axis) as a function of time. The winding number describes how many loops the helix in Fig. 3 has. At a phase slip the helix loses a loop and the winding number drops by an integer number.

References:

[1] W. Eckhaus, Studies in Non-Linear Stability Theory, Springer-Verlag (New York), 1965.
[2] P.G. de Gennes, Superconductivity of Metals and Alloys, Addision-Wesley (Reading, MA), 1966.
[3] L. Kramer and R. Rangel, Structure and Properties of the Dissipative Phase-Slip State
in Narrow Superconducting Filaments with and without Inhomogeneities,
J. Low. Temp. Phys., vol 57, p. 391 (1984).
[4] R. Tidecks, Current Induced Nonequilibrium Phenomena in Quasi-One-Dimensional Superconductors,
Springer-Verlag (Berlin), 1990.
[5] M. Tinkham, Introduction to Superconductivity, McGraw-Hill (New York), 1996.

Our publications related to superconducting rings

• Instabilities and resistance fluctuations in thin accelerated superconducting rings,
  Mikko Karttunen, K.R. Elder, Martin B. Tarlie, Martin Grant,
  Phys. Rev. E 66, 026115 (2002)
  [online] [preprint]

More:

• See our publication list.