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Vortex dynamics in high temperature superconductors

One of my current research topics (in the Physics Department of the Technion Israel Institute of Technology and in collaboration with Daniel Podolsky and Ronen Abravanel) is the study of the influence of temperature on the dynamics of vortices in high temperature thin films superconductors (such as YBCO for instance) where a lattice of columnar defects has been introduced artificially.

Introduction and motivation

High temperature superconductors and vortices

Superconductivity is a hundred years old; the superconducting state has been discovered in 1911 by Kamerlingh Onnes and is a state in which the material has a zero electric resistance. Furthermore, a superconducting material completely ejects a magnetic field: this is the Meissner effect that allows magnetic levitation.

Magnetic levitation
Magnetic levitation of a superconductor below its transition temperature.
The superconducting state is achieved if one lowers the temperature below a critical temperature, that is a characteristic of the material. For a long time, people believed that this temperature could not be higher than 30 K. In 1986, Bednorz and Müller (who got the Nobel prize a year later) discovered the high temperature superconductivity; this discovery opened a new research direction in both theoretical and experimental physics that remains today very dynamic. High temperature superconductivity means that the material remains superconducting within a range of temperatures above 77 K, the nitrogen boiling point. This considerably simplifies the experiments and the possible industrial practical applications of superconductivity. Indeed, in order to bring the high temperature superconductors in their superconducting state, one can use liquid nitrogen (cheap, easy to use, and available in abundance), while before 1986, the superconducting materials had to be cooled down using liquid helium or hydrogen, that are far more expensive to produce (about five times more expensive for hydrogen, and fifty times for helium, whose irreversible shortage becomes more and more worrying with time).

Since then, a very large number of high temperature superconductors have been identifies, many of them being cuprates (they contain anionic copper) among which the yttrium barium copper oxide, or YBCO (YBa\(_2\)Cu\(_3\)O\(_{7-\delta}\)) with a critical temperature of 93 K. Other cuprates like BSCCO or TBCCO have a little higher critical temperatures.

Although we have many high temperature superconductors at hand to study, this type of superconductivity is one of the unsolved problems of modern physics, has and still is the object of numerous debates and controversies. Experimental and theoretical research, fundamental and applied, has thus been extremely active in this domain for the last thirty years.

Vortices

Cuprates are type II superconducting materials: besides the superconducting Meissner phase where a magnetic field is expelled from the material, they is a mixed phase that appears when the magnetic field is high enough. In this phase, the materials can be penetrated by the magnetic field, which is transmitted through the material by flux lines called vortices (Abrikosov vortices). Inside the vortices, the superconductivity is destroyed and the material is in its "normal" state (non superconducting). Outside the vortices, the material remains fully superconducting.

Each vortex contains a quantum of magnetic flux and the higher the magnetic field, the higher the number of vortices inside the material. These vortices repel each other so that they prefer to spontaneously form a ordered lattice: the Abrikosov lattice (see figure 1). In order to fix the ideas, in YBCO, the vortices' diameter is about 2-3 nm and the average distance between two vortices is 10-30 nm for large magnetic fields (of the order of 1 T) (see Matsumoto and Mele, 2010).

Abrikosov lattice
Figure 1: Observation of an Abrikosov lattice in NbSe\(_2\) par scanning tunneling microscopy [Hess et al. 1989]

Fixing the vortices: the columnar defects

Vortices dynamic is strongly influenced by the thermal fluctuations. Above a certain temperature, the vortices become mobile inside the material and constitute a vortex liquid (as opposed to the ordered solid phase of the Abrikosov lattice; the material containing the vortices is itself still is its solid phase of course). The vortices also are weakly mobile when the material is subject to an electric current. The vortices motion is dissipative; one thus wants to enhance the characteristics and performances of the material by finding a way to pin the vortices as strongly as possible.

In this aspect, YBCO are among the most promising materials for future practical applications. In thin films YBCO, it is possible to artificially produce columnar defects (see the review by Civale, 1997) by heavy ion irradiation or by electron beam lithography: the bombing partially destroys the material leaving a set of columnar holes going through the sample. Energetically speaking, it is more favorable for a vortex to sit on such a defect: one thus spares the cost in energy to destroy superconductivity inside the vortex. The presence of these defects allows to pin the vortices very strongly. Furthermore, it is possible to artificially create true nano-lattices of columnar defects (see for instance Lin, 1996), with different geometries (square lattice, hexagonal, etc.). Figure 2 shows a square lattice of columnar defects created in a thin film of niobium (Nb) by gallium (Ga) ions irradiation.

Lattice of columnar defects
Figure 2 : Lorentz micrographs allow to observe a square lattice of columnar defects (the pink dots) for different values of the magnetic field. (a) Strong magnetic field: it induces a number of vortices (in blue) four times higher than the number of defects ; (b) the magnetic field induces a number of vortices equal to the number of defects (matching field); (c) weak magnetic field: the lattice of defects is at quarter filling. [After Tonomura 1999]

In the followong, we attempt to study the influence of temperature on the vortices in materials where such a lattice of columnar defects has been produced.

Bipartite lattice at half-filling

One considers a square bipartite lattice (that possesses two equivalent sublattices A and B, as shown on figure 3) of columnar defects in a type-II superconductor. We subject the material to a magnetic field such that exactly half of the defects are occupied by vortices. The inter-vortex interactions are repulsive, so that the energy is minimized when one every two site is occupied; there are two degenerate ground states: one where the sublattice A is occupied, and B is empty; the other state has B occupied and A empty.

Bipartite lattice
Figure 3: Bipartite lattice.

The experiment starts at low temperature in one of the two ground states; then one increases the temperature up to above the transition from the "vortex solid" state (Abrikosov lattice) to the vortex liquid with mobile vortices. The vortices are not strongly pinned to the defects anymore and start to explore the system as the temperature increases. Let us imagine that after some time, we decrease again the temperature and go back to the initial temperature where the vortices are all frozen on a sublattice.

Now the question is: what is the probability that the system changed its ground state? Or stated otherwise: to which point the vortices have been able to explore the system? Have they been able to wander far enough away from the defects to which they were attached, in order to "jump" to the neighboring defects and stay there as the transition was crossed down again? To which point this depends on how high the temperature has been raised above the transition temperature before going back down again?

Model for an "itinerant" vortex

In order to address this question, one must take two important effects into account: the columnar defect tend to attract the vortices, and the inter-vortex interactions are repulsive.

Attractive potential created by a columnar defect

Let us consider a defect at the origin. A vortex experiences the attractive potential created by this defect and one can model this with a cylindric well: \(V_{\text{defect}}(\br)=-V_0<0\) for \(\br \leq a\) and \(V_{\text{defect}}(\br)=0\) for \(\br > a\), i.e. : $$\begin{equation} V_{\text{defect}}(\br)=-V_0 \theta(a-|\br|) \end{equation}$$ where \(\theta(x)\) is the Heaviside function.

However, defining the defect potential with a discontinuous function is problematic, especially if one want to solve the equations numerically. We can smoothen the potential by choosing a continuous function that approaches asymptotically the Heaviside function. There are several choices and we choose here to use the Fermi-Dirac distribution (with a purely technical and non physical motivation) shown on figure 4.

Fermi-Dirac distribution
Figure 4: the Fermi-Dirac distribution \(\theta_F\) asymptotically approaching the Heaviside function \(\theta(0.5-|\br|)\) for \(d \to 0\).
The potential created at point \(\br\) by a defect at the origin is thus: $$\begin{equation} V_{\text{defect}}(\br)=-V_0\theta_F^a(\br) = -V_0\frac{1}{1+\exp\left(\frac{|\br|-a}{d}\right)} \end{equation}$$ where \(d\) must be small enough to correctly model the potential well; \(a\) is the cylindric well radius.

If we now consider the entire system, an itinerant vortex at point \(\br\) feels the sum of the potentials created by all the defects (at points \(\br_j\)) : $$\begin{equation} V_{\text{lattice}}(\br)=-V_0\sum_{j \in \text{lattice}}\frac{1}{1+\exp\left(\frac{|\br-\br_j|-a}{d}\right)} \end{equation}$$

Potential due to repulsive inter-vortex interactions

A vortex also feels the potential due to the interactions with all the other vortices of the system. We start by studying the motion of one vortex alone. Let us take this vortex to be at the origin at the beginning of the experiment and consider for the moment that it is the only one to move; all the other vortices are frozen and pinned onto their defects (that belong to the same sublattice as the defect at the origin). The inter-vortex potential is represented by a Bessel function (see Blatter 1994): at small distance, it can be approximated by a logarithm (in our case, it is valid since a type II superconductor has a large penetration depth); we can thus take: $$\begin{equation} V_\text{v}(r)=-\rho_S \ln\left(r\right)+C \end{equation}$$ as the interaction potential between two vortices at a distance \(r\). \(C\) is a numerical constant without much meaning here that we will discard in the following. Adding together all the logarithmic potential created by all the vortices (at the points \(\br_i\) of the sublattice), we obtain the total interaction potential seen by at \(\br\) by our itinerant vortex. To this, we must add the potential in \(|\br|^2\) created by the neutralizing background and that allows for the confinement of the vortices: $$\begin{equation} V_\text{vortex}(\br)=\rho_S\frac{\pi}{4}|\br|^2-\sum_{i \in \text{sublattice}}\rho_S \ln\left(|\br-\br_i|\right) \end{equation}$$ Figure 5 shows this potential, and also includes the potential created by the columnar defect at the origin. In each point where there is another vortex, the potential diverges.

Potential seen by a vortex close to the origin
Figure 5: The potential due to all the vortices of the system, seen by a vortex around the origin - the origin where sits the defect where it can initially pinned. (The attractive potential due to this particular defect is shown on the figure, but not the potential created by all the other defects).

The itinerant vortex as a random walker in a potential

Langevin equation - Brownian motion

In order to estimate the ability of the vortex to explore the vicinity of its initial position, we consider the vortex as a random walker in a potential. This problem in analog to the Brownian motion and can be addressed with Langevin equation: we write the equations of motion and add by hand a stochastic force - or Langevin force - that accounts for the thermal effects: the random walker is immersed in a thermal bath whence it can collect energy.

The forces applying on the vortex (to which we assign a inertia "mass" \(m\)) are a force of friction with the "background" or environment (with a friction coefficient \(c=m\gamma\) ; \(\tau=1/\gamma\) is the relaxation time of the vortex in this "background"), a force deriving from the potential modeled above and a stochastic force (the thermal fluctuations) : \(m\Gamma(t)\). The equations of motion are then: $$\begin{equation} m\ddot \br = - c \dot \br - m\nabla V (\br) + m\Gamma(t) \frac{\br}{|\br|} \end{equation}$$ We must choose an appropriate form for \(\Gamma(t)\): without the potential \(V(\br)\), we must recover the results of classical statistical mechanics, and in particular the equipartition of energy. We thus choose a white noise, of intensity \(q=2\gamma kT/m\) : $$\begin{equation} \left<\Gamma(t)\right>=0 \quad\text{et :}\quad \left<\Gamma(t)\Gamma(0)\right>=q\delta(t) \end{equation}$$ It is easy to show that one indeed recovers the expected results (see Risken).

The friction being important (\(\gamma\gg 1\)) - the relaxation time is small - one can neglect the second derivative to obtain the following Langevin equation: $$\begin{equation} \dot \br =\frac{1}{\gamma}\left( -\nabla V(\br) + \Gamma(t) \frac{\br}{|\br|}\right) \end{equation}$$

Fokker-Planck equation

Even for a simple system and a simple potential, the Langevin equation are not easy to solve analytically, nor even numerically. One often prefers to address the problem in the Fokker-Planck equation formalism, that is equivalent to the Langevin equation.

The Fokker-Planck equation that corresponds to our problem is also called the Schmoluchowski equation: $$\begin{equation} \frac{\pd W(\br,t)}{\pd t}= \left[-\frac{1}{\gamma}\vec\nabla V'(x) + \frac{q}{2\gamma^2}\nabla^2\right] W(\br,t) \label{schmol} \end{equation}$$ with \(W(\br,t)\) being the probability density to find the walker (the vortex) at point \(\br\) at time \(t\).

Ongoing research

A numerical study of the Fokker-Planck equation (\ref{schmol}) will allow us to compute the characteristics of the vortex in its potential: mean free path, escape rate, etc.

Then, in the light of the results obtained for a unique itinerant vortex, we intend to attack the dynamics of the set of all vortices, in the Fokker-Planck formalism, or with numerical simulations.

Bibliography