According to the Kibble–Zurek model, flux lines are spontaneously created during a fast conductor–superconductor phase transition. The model predicts both the spatial density and the correlations of the flux array. We present the design of a magneto-optical system with a projected single-flux-line resolution. Such a system can allow detailed measurements of the distribution of flux created spontaneously during a conductor–superconductor phase transition.
The Kibble–Zurek model predicts the creation of topological defects following a fast order–disorder phase transition (Kibble 1976; Zurek 1985). One physical system in which this scenario can be tested is the normal–superconductor phase transition. In the superconducting state, the topological defects are quantized flux lines (vortices). A significant advantage of superconductors is that at low enough temperatures the vortices are pinned. Therefore, if topological defects are created during the transition, they will be frozen in space and can be measured afterwards. According to this scenario, in the non-equilibrium temperature interval, the state of the system is made of uncorrelated regions of the ordered phase. The average size of such a region is (the value of correlation length at the freeze-out temperature). Each region has a definite phase, uncorrelated with the phases in other regions. If the total phase difference around a vertex (the winding number) between three such regions is non-trivial (non-zero), a topological defect is created. The specific choice of winding number around each vertex is made by using the geodesic rule. This rule states that the physical system will evolve into the state with the lowest kinetic energy. Minimization of the energy requires the minimization of the square of the phase gradient (F∝|∇θ|2). Under this assumption, the probability of creating a defect can be estimated as follows. Consider a vertex (v1) in between three uncorrelated regions (I, II and III), as shown in figure 1. Since only phase differences are significant, the phase of the first region is set to zero: θ1=0. The phases of the other regions are random, meaning that they have equal probability of being anywhere between −π and π. In this case, we can calculate the winding number,(1.1)
These integrals are sensitive to the phase connection between the different domains. In our case,(1.2)where k is an integer that minimizes the gradient of the phase between regions II and III. If θ2>0 and θ3<θ2−π, minimization of the free energy requires k=1, and therefore n=1. In this case, a topological defect is created in the vertex. It is easy to calculate the probability of this process,(1.3)In the same way, it is possible to calculate the probability of creating a vortex pair in two nearby vertices (v1 and v2) having either equal or opposite signs,(1.4)(1.5)The model therefore predicts strong short-range correlations between topological defects of opposite polarity. There is no correlation between topological defects in vertices that are not nearest neighbours (P(nv1=1;nv3=1)=P(nv1=1;nv3=−1)=P(nv1=1)P(nv3=1)=1/64), and therefore there are no long-range correlations. Another mechanism of vortex creation in superconductor is a flux-trapping mechanism proposed by Hindmarsh & Rajantie (2000); see also Kibble & Rajantie (2003). In this mechanism, the flux is generated by fluctuations of the magnetic field, which become frozen out during the quench. The two mechanisms (Kibble 1976; Hindmarsh & Rajantie 2000) predict a distribution of defects with characteristically different correlation properties. In the gauge theory, vortex–vortex correlations at short distances are expected. The vortices should be formed in clusters of equal sign (Hindmarsh & Rajantie 2000; Kibble & Rajantie 2003). After the transition, topological defects with opposite polarity that are close to one another will annihilate rapidly. To overcome this problem, one should use a physical system with strong pinning forces, which can prevent migration and mutual annihilation of the topological defects after the transition.
Previous experiments on superconducting thin films detected a net flux consistent with the Kibble–Zurek prediction (Maniv et al. 2003). There are as yet no measurements of the total vortex density or of the correlation function. In order to investigate these predictions, experiments capable of resolving a single flux quantum are needed. There are several techniques having such resolution, for example scanning Hall microscopy or superconducting quantum interference device microscopy. With these techniques, one can achieve a very good spatial resolution, at the expense of speed and image size. In order to visualize an array of flux lines, a very high resolution is not required (the maximal predicted density at the cooling rates that we can achieve is one vortex/μm2). However, a fast image acquisition and a large field of view are needed. An experimental technique satisfying these requirements is magneto-optical imaging.
2. Magneto-optical imaging
Our magneto-optical system is based on the usage of the magneto-optical Kerr effect in europium selenide (EuSe). Linearly polarized light reflected from the film experiences a rotation of its polarization plane vector, i.e. if the magnetic field B is parallel to the direction of propagation (Goa et al. 2003). The angle of rotation θk is linearly proportional to the magnetic field. The outgoing light passes through a second polarizer rotated at an angle 2π−ϕ to the original polarization. The intensity of the light after the second polarizer is given by(2.1)where I0 is the incoming light intensity and e is the extinction ratio (caused by depolarization of the light). The approximation in equation (2.1) is valid for small values of θk and ϕ.
The contrast of the magneto-optical signal is then defined as(2.2)To obtain maximal contrast, the angle between the polarizers should be . In this case, we find(2.3)Typical magneto-optical imaging systems used for research on superconductivity use ferromagnetic garnet film (FGF) sensors of 0.7–5 μm thickness with an in-plane magnetization. If one uses the Kerr effect in EuSe, then the thickness of the indicator layer can be reduced to 40 nm. EuSe has another major advantage, being diamagnetic down to 4 K. Consequently, EuSe does not introduce an uncontrolled magnetic field as an FGF does. On the other hand, the magneto-optical sensitivity of EuSe is high within a narrow region of temperatures, between 4 and 20 K (Schoenes & Wachter 1976). Our samples consist of a niobium superconducting film (Tc=9±0.1 K) on a sapphire substrate, capped by an aluminium thin film mirror and a magneto-optically active layer. The signal that we want to detect is a modulation of the magnetic field above the surface of the superconductor. Carneiro & Brandt (2000) have shown that the scalar potential describing the magnetic field of a vortex above the surface of the superconductor can be approximated by(2.4)where Φ0 is the quantum of magnetic flux; λ is the penetration depth (0.1 μm for Nb); r is the in-plane distance from the location of the flux line; and z is the distance from the surface. As can be seen from figure 2, the magnetic field generated by a vortex decays rapidly with the distance from the surface of the film on a micrometre scale and so the indicator layer should be as close to the surface as possible. Figure 3 shows a sketch of our setup. The sample stage is connected by a flexible thermal link to a cold finger cooled by a liquid helium cryostat. We use a helium bath cryostat instead of a flow cryostat in order to avoid vibrations. The pump that evacuates the vacuum chamber is disconnected during the measurements. The entire system is positioned on a floating optical table. Therefore, our sample is in a very quiet environment. We detected no vibrations on a 0.1 μm scale or larger. After cooling, the sample stage is focused under the microscope objective using an XYZ manipulator. The objective is mounted inside the vacuum chamber and therefore the vacuum window is not in a converging part of the beam. The vacuum window itself is tilted by a small angle to eliminate reflections. To illuminate the sample, we use a monochromatic light beam (546 nm) from a 100 W mercury lamp. The light intensity is detected by a Hamamatzu Peltier-cooled charge-coupled device (CCD) camera. At the maximal magnification, each pixel covers 0.13 μm of the sample. At this magnification, the field of view is 0.01 μm2. Spatial resolution is limited by optical diffraction, which ‘smears’ out the image on the length scale of a wavelength (approx. 0.5 μm).
The source of heat in the present experiment is a pulsed Nd:YAG laser. Single pulses, 10 ns long, are used to heat the film. After passing through a vacuum window, the laser pulse passes through the substrate and illuminates the film homogeneously. The sapphire substrate is transparent at the wavelength we use (1.06 μm), but the Nb layer absorbs approximately 50 per cent of the light. Hence, only the film heats up, while the substrate remains near the base temperature of 5 K. After the heating pulse is over, the heat from the film escapes into the substrate, which acts as a heat sink while the film is cooling. With this method, we can potentially achieve cooling rates up to 1011 K s−1 (Leiderer et al. 1993).
At temperatures significantly lower than Tc, the magnetic field enters the superconducting film in dendrite-like fractal shape due to a thermo-magnetic instability (Rudnev et al. 2005). A low-magnification measurement of the magnetic flux inside the sample is shown in figure 4. The flux was introduced by applying a uniform external magnetic field of 80 G. An image of the film without magnetic field was subtracted to reduce the background. The line in the middle of the sample is a crack in the film, which forms an edge of the superconductor. The intensity of the image is proportional to the magnetic field trapped inside the superconducting film. Bright regions represent positive magnetic field, and dark regions negative magnetic field. At higher magnification (figure 5), only two branches of such a structure are seen. If the external magnetic field reduced to zero, then a magnetic field with opposite polarity is produced by screening currents. In figure 6, magnetic flux lines with both positive and negative polarities are present. The distance between the regions with opposite magnetic flux in the image shown here is less than 5 μm. This distance, at which flux with opposite polarity can coexist without annihilation, may depend on the cooling rate. When achieved, single-flux-line resolution will give us a unique opportunity to measure topological defect formation frozen in time.
We thank S. Hoida, L. Iomin and O. Shtempluk for their technical assistance. This work was supported by Israel Science Foundation and Technion Fund for Research.
One contribution of 16 to a Discussion Meeting Issue ‘Cosmology meets condensed matter’.
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