Determination of electric field distributions in superconductors via magneto-optical imaging and the Faraday law

The determination of electric field distributions within superconductors is of fundamental importance for both basic research and applications. The analysis of electric fields allows insights into different modes of vortex dynamics in superconductors as well as the determination of local maxima of dissipation which could destabilize superconducting transport. We present an analysis of electric field distributions for thin films in magnetization experiments. We show that electric field distributions can be determined experimentally by time-resolved magneto-optical measurements of flux density distributions via the Faraday law. We give a detailed analysis of the inductive and the potential contributions to the electric fields ${\mathbf{E}}_i$ and ${\mathbf{E}}_p$, respectively. Experimental results are demonstrated for a magnetized square-shaped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.95}$ thin film.

Figure 1

Sketch of an electrically polarizable sample in an inductive electric field ${\mathbf{E}}_i$, which is induced by a local temporal evolution of $B_z$. According to the Faraday law, electric field components $E_{i,x}$ and $E_{i,y}$ are induced along any contour containing ${\dot{B}}_z\ne 0$ (black arrows). This inductive field ${\mathbf{E}}_i$ polarizes the medium, and consequently a polarization field $\mathbf{P}$ (white arrows) and a charge density $n^{in}=-{ϵ}_0\mathbf{\nabla }\bullet \mathbf{P}$ are induced in the sample (here only depicted at the sample surfaces). The induced charge density is the source of a potential electric field ${\mathbf{E}}_p=-\mathbf{P}$ in the sample.

Figure 2

Sketch of the induced surface charge densities $n_s^{in}$ due to electric field components perpendicular to a relaxing current density. (a) Contribution of the relaxation of the normal flux density component ${\dot{B}}_z$ to the electric field in the $(x,y)$ plane of a square-shaped film with a relaxing Bean-like current density (b) (bold lines represent current flow lines). According to the Faraday law, in each current domain, there are inductive electric field components parallel and perpendicular to the current density. Whereas the ${\mathbf{E}}_i$ component $\mid \mid \mathbf{j}$ reflects the occurrence of a finite resistivity, the ${\mathbf{E}}_i$ component perpendicular to $\mathbf{j}$ induces a charge density and a polarization $\mathbf{P}$ which is related to inhomogeneous vortex motion. Note that the total electric field ${\mathbf{E}}_i+{\mathbf{E}}_p$ (with ${\mathbf{E}}_p=-\mathbf{P}$) is parallel to the current density. (c) Possible occurrence of an $E_z$ component of the inductive electric field due to the relaxation of in-plane flux densities. Shown is a $j_x$ domain of a square-shaped film with a ${\partial }_t{B}_y$. Since the total electric field must be parallel to $\mathbf{j}$, $E_z$ is totally canceled by $E_{p,z}$ due to surface charge densities at the $z=\pm d∕2$ [see (d)] and the $z=0$ planes.

Figure 3

(a) Scheme for the integration of the Faraday law according to the Stokes theorem. If the integration area is divided into cells, each containing a time change of the flux density component $B_z$, the electric field components can be calculated along the contours of each cell. However, the neighboring cells have to be taken into account. (b) Strip geometry for the one-dimensional current distribution.

Figure 4

Calculated temporal change of the normal magnetic flux component $B_z\left(x\right)$ of a thin strip of thickness $d=200\mathrm{nm}$, width $W=8\mathrm{\mu }\mathrm{m}$ in an external magnetic field of ${\mu }_{0}H=42\mathrm{mT}$. The strip is in a nearly fully penetrated state and carries a constant magnetization current density with a value of $j_c\left(t_1\right)=\pm {10}^{11}\mathrm{A}∕{\mathrm{m}}^2$ which relaxes to $j_c\left(t_2\right)=\pm 0.9\times {10}^{11}\mathrm{A}∕{\mathrm{m}}^2$. $\mathrm{\Delta }{B}_z=B_z\left(t_2\right)-B_z\left(t_1\right)$ denotes the corresponding change of the flux density.

Figure 5

Different contributions to the total inductive electric field in the $(x,z)$ plane of a thin-film strip being extended infinitely in the $y$ direction. The strip is in a fully penetrated state and the time evolution of the flux and current distribution are shown in Fig. 4. (a) Total inductive electric field $E_{i,y}$ according to Eq. (28) for $\mid {\partial }_t{j}_y\mid ={10}^{10}\mathrm{A}∕\left({\mathrm{m}}^2\sec \right)$. In (b) and (c) the two contributions to the inductive electric field distribution according to Eqs. (31, 32) are depicted. Note that the size of the cross section of the strip $(W∕d=40)$ can be seen in the grey-scale pattern of (b).

Figure 6

Distribution of the total inductive electric field $E_{i,y}$ in the $(x,z)$ plane (fully penetrated state) for different widths $W$ of a thin strip. The width of the strip is increased from $W=2\mathrm{\mu }\mathrm{m}$ (a), $W=8\mathrm{\mu }\mathrm{m}$, (b) to $W=400\mathrm{\mu }\mathrm{m}$ (c). All other parameters are the same as in Figs. 4, 5.

Figure 7

Calculated temporal change of the normal magnetic flux component $B_z\left(x\right)$ of a thin strip in a partly penetrated state. The thickness of the strip is $d=200\mathrm{nm}$, the width $W=4\mathrm{\mu }\mathrm{m}$ and ${\mu }_{0}H=10\mathrm{mT}$. The current density in the flux penetrated area is relaxed from $j_c\left(t_1\right)=\pm 2\times {10}^{11}\mathrm{A}∕{\mathrm{m}}^2$ to $j_c\left(t_2\right)=\pm 1.8\times {10}^{11}\mathrm{A}∕{\mathrm{m}}^2$. $\mathrm{\Delta }{B}_z=B_z\left(t_2\right)-B_z\left(t_1\right)$ denotes the corresponding change of the flux density. For a clearer representation, $\Delta B_z$ was shifted by a value of $-0.005\mathrm{T}$.

Figure 8

Electric field distribution in a $(x,z)$ plane of a thin strip due to current relaxation in a partly penetrated state for the same parameters as in Fig. 7. The total inductive electric field $E_{i,y}(x,z)$ [Eq. (28)] is shown in (a), whereas (b) and (c) depict the contributions $E_{i,y}^{\left(1\right)}$ and $E_{i,y}^{\left(2\right)}$ according to Eqs. (31, 32), respectively.

Figure 9

(a) Calibrated magneto-optical image showing the normal magnetic flux density distribution of a square-shaped $\mathrm{YBaCuO}$ film as a grey-scale image. The thin film has a thickness of $500\mathrm{nm}$ and a width of $1.2\mathrm{mm}$. The image is taken $1.2\sec$ after ramping up the external field from a zero-field-cooled state at $T=8\mathrm{K}$ to ${\mu }_0{H}_{ex}=86\mathrm{mT}$. (b) Flux density distribution with superimposed current flow lines and current density profiles for the same state as in (a). The current distribution is obtained by 2D inversion of the Biot-Savart law.

Figure 10

(a) Time change of the flux density $\mathrm{\Delta }{B}_z(x,y)=B_z(x,y,t=1.2$ $\sec )-B_z(x,y,t=0.7$ $\sec )$ due to thermally activated flux creep. (b) Grey-scale map of the absolute value of inductive electric field distribution in the $z=0$ plane at $t=1.2\sec$ of the same state of the sample shown in Fig. 9. The profiles of the electric field are taken along the black lines.

Figure 11

Grey-scale map of the $E_{i,x}(x,y)$ (a) and $E_{i,y}(x,y)$ (b) components of the inductive electric field in the $z=0$ plane for the same conditions as in Fig. 10.

Figure 12

Grey-scale map of the $E_{p,x}(x,y)$ (a) and $E_{p,y}(x,y)$ (b) components of the potential electric field ${\mathbf{E}}_p$ in the $z=0$ plane for the same conditions as in Fig. 10. Some small spots in the image are experimental artifacts due to defects in the mirror of the magneto-optical active sensor film.

Figure 13

Grey-scale representation of the induced surface charge density $n_s^{in}=-{ϵ}_0\mathbf{\nabla }\bullet \mathbf{P}$ for the same conditions as in Fig. 10.