Non-London electrodynamics in a multiband London model: Anisotropy-induced nonlocalities and multiple magnetic field penetration lengths

The London model describes strongly type-2 superconductors as massive vector field theories, where the magnetic field decays exponentially at the length scale of the London penetration length. This also holds for isotropic multiband extensions, where the presence of multiple bands merely renormalizes the London penetration length. We show that, by contrast, the magnetic properties of anisotropic multiband London models are not this simple, and the anisotropy leads to the interband phase differences becoming coupled to the magnetic field. This results in the magnetic field in such systems having $N+1$ penetration lengths, where $N$ is the number of field components or bands. That is, in a given direction, the magnetic field decay is described by $N+1$ modes with different amplitudes and different decay length scales. For certain anisotropies we obtain magnetic modes with complex masses. That means that magnetic field decay is not described by a monotonic exponential increment set by a real penetration length but instead is oscillating. Some of the penetration lengths are shown to diverge away from the superconducting phase transition when the mass of the phase-difference mode vanishes. Finally the anisotropy-driven hybridization of the London mode with the Leggett modes can provide an effectively nonlocal magnetic response in the nominally local London model. Focusing on the two-component model, we discuss the magnetic field inversion that results from the effective nonlocality, both near the surface of the superconductor and around vortices. In the regime where the magnetic field decay becomes nonmonotonic, the multiband London superconductor is shown to form weakly-bound states of vortices.

Figure 1

(Left) Two Fermi surfaces in an anisotropic multiband superconductor, qualitatively similar to that of the uniaxial compound ${\mathrm{MgB}}_2$. The crystal structure anisotropy axis is $c$. The magnetic field $\mathit{B}$ is declined with respect to the $c$ axis. (Right) Polarizations of one ordinary ${\mathit{B}}_0$ and two extraordinary ${\mathit{B}}_{e1,2}$ magnetic modes in a multiband superconductor with uniaxial anisotropy. The ordinary mode is decoupled from interband phase difference ${\theta }_{12}$ while the extraordinary modes give a coupled magnetic field and ${\theta }_{12}$.

Figure 2

(a) Orientation of the superconductor boundary with respect to the crystal axes. The anisotropy axis $z\left(c\right)$ lies in the plane perpendicular to the boundary and makes the angle $\theta$ to the normal direction. The field $\mathit{B}$ applied parallel to the boundary depends on the coordinate $r$ along the surface normal. (b) The cylinder of anisotropic superconductor subjected to the external magnetic field applied along the cylinder axis $\mathit{y}$.

Figure 3

Wave numbers of the extraordinary magnetic modes as functions of the interband Josephson coupling $k_{1,2}=k_{1,2}\left(k_0\right)$. (a) Strong anisotropy ${\lambda }_{2z}=0.1{\lambda }_{2\perp }=0.1\lambda$. (b) Weak anisotropy ${\lambda }_{2z}=0.8{\lambda }_{2\perp }=0.8\lambda$.

Figure 4

A plot of the negative magnetic field at $\theta =\pi /4$ for strong anisotropy ${\lambda }_{2x}={\lambda }_{1y}=1$ and ${\lambda }_{1x}={\lambda }_{2y}=0.1$ for various values of $k_0$.

Figure 5

A contour plot and a radial slice at various angles of the magnetic field, negative magnetic field, and phase difference for the 1d boundary problem solution with strong anisotropy in a single component $k_0=0.7, {\lambda }_{2x}={\lambda }_{2y}={\lambda }_{1y}=1$, and ${\lambda }_{1x}=0.1{\lambda }_{1y}$. Vertical and horizontal axes on upper panels correspond to $y$ and $x$ directions, respectively. A radial curve from the center of the plot represents the field orthogonal to a 1d boundary crossing the origin in the $x-y$ plane. This way every possible direction (or $\theta$) is plotted for equations (24) and (25) with the radial distance representing $r$. The plot quantities are: magnetic field $B_z$ (left), negative magnetic field $|B_z|-B_z$ (center), and phase difference ${\theta }_{12}$ (right).

Figure 6

A contour plot and a radial slice at various angles for the 1d boundary problem solution with strong anisotropy in opposite directions in each component $k_0=0.7, {\lambda }_{2x}={\lambda }_{1y}=1$, and ${\lambda }_{1x}={\lambda }_{2y}=0.1$. Vertical and horizontal axes on upper panels correspond to $y$ and $x$ directions, respectively. A radial curve from the center of the plot represents the field orthogonal to a 1d boundary crossing the origin in the $x-y$ plane. This way every possible direction (or $\theta$) is plotted for equations (24) and (25) with the radial distance representing $r$. The plot quantities are magnetic field $B_z$ (left), negative magnetic field $|B_z|-B_z$ (center), and phase difference ${\theta }_{12}$ (right).

Figure 7

Meissner state numerical solution for strong anisotropy in opposite directions on a disk of radius 5 ${\lambda }_{x1}^{-1}={\lambda }_{y2}^{-1}=1, {\lambda }_{x2}^{-1}={\lambda }_{y1}^{-1}=0.1, {\eta }_{12}=0.5$, and ${\gamma }_1={\gamma }_2=10$. The full 2d plots are accompanied by 1d slices at $\theta =0,\frac{\pi }{4},\frac{\pi }{2}$ below. The quantities plotted are (a) $B_z$ magnetic field, (b) $B_z-\left|B_z\right|$ negative magnetic field, (c) $\mathcal{E}$ energy density, (d) ${\left|{\varphi }_1\right|}^2$, (e) ${\left|{\varphi }_2\right|}^2$, and (f) ${\theta }_{12}$ phase difference.

Figure 8

Meissner state numerical solution for strong anisotropy in opposite directions on a disk of radius 12 ${\lambda }_{x1}^{-1}={\lambda }_{y2}^{-1}=1, {\lambda }_{x2}^{-1}={\lambda }_{y1}^{-1}=0.1, {\eta }_{12}=0.5$, and ${\gamma }_1={\gamma }_2=10$. The full 2d plots are accompanied by 1d slices at $\theta =0,\frac{\pi }{4},\frac{\pi }{2}$ below. The quantities plotted are (a) $B_z$ magnetic field, (b) $B_z-\left|B_z\right|$ negative magnetic field, (c) $\mathcal{E}$ energy density, (d) ${\left|{\varphi }_1\right|}^2$, (e) ${\left|{\varphi }_2\right|}^2$, and (f) ${\theta }_{12}$ phase difference.

Figure 9

Meissner state numerical solution for strong anisotropy in similar directions on a disk of radius 12 ${\lambda }_{x1}^{-1}={\lambda }_{y1}^{-1}={\lambda }_{y2}^{-1}=1, {\lambda }_{x2}^{-1}=0.1, {\eta }_{12}=0.5$, and ${\gamma }_1={\gamma }_2=10$. The full 2d plots are accompanied by 1d slices at $\theta =0,\frac{\pi }{4},\frac{\pi }{2}$ below. The quantities plotted are (a) $B_z$ magnetic field, (b) $B_z-\left|B_z\right|$ negative magnetic field, (c) $\mathcal{E}$ energy density, (d) ${\left|{\varphi }_1\right|}^2$, (e) ${\left|{\varphi }_2\right|}^2$, and (f) ${\theta }_{12}$ phase difference.

Figure 10

Negative part of the magnetic field ${\tilde{B}}_z=B_z-\left|B_z\right|$ distribution around a single vortex, normalized by ${\mathrm{\Phi }}_0/{\lambda }^2$. The parameters correspond to either (a) strong anisotropy or (b) weak anisotropy.

Figure 11

Magnetic field $B_z$ for a fractional vortex in $U\left(1\right)\times U\left(1\right)\mathrm{model}$ (i.e., $k_0=0$), with various types of anisotropy (a) ${\lambda }_{1x}=0.8{\lambda }_{1y}, {\lambda }_{2x}={\lambda }_{2y}={\lambda }_{1y}$, (b) ${\lambda }_{1x}=0.4{\lambda }_{1y}, {\lambda }_{2x}={\lambda }_{2y}={\lambda }_{1y}$, (c) ${\lambda }_{1x}={\lambda }_{2y}=0.8{\lambda }_{1y}$, (d) ${\lambda }_{1x}={\lambda }_{2y}=0.4{\lambda }_{1y}, {\lambda }_{2x}={\lambda }_{1y}$. The field is in units of ${\mathrm{\Phi }}_0/{\lambda }_{1y}^2$.

Figure 12

$N=1$ single quanta numerical solution for weak anisotropy in one band ${\lambda }_{x1}={\lambda }_{y1}={\lambda }_{x2}=1, {\lambda }_{y2}=0.5, {\eta }_{12}=0.5$, and ${\gamma }_1={\gamma }_2=2$. The contour plots are (a) $B_z$ magnetic field, (b) $\left|B_z\right|-B_z$ negative magnetic field, (c) $\mathcal{E}$ energy density, (d) ${\left|{\varphi }_1\right|}^2$, (e) ${\left|{\varphi }_2\right|}^2$, and (f) ${\theta }_{12}$ phase difference.

Figure 13

$N=1$ single quanta numerical solution for strong anisotropy in one band ${\lambda }_{x1}={\lambda }_{y1}={\lambda }_{x2}=1, {\lambda }_{y2}=0.1, {\eta }_{12}=0.5$, and ${\gamma }_1={\gamma }_2=2$. The contour plots are (a) $B_z$ magnetic field, (b) $\left|B_z\right|-B_z$ negative magnetic field, (c) $\mathcal{E}$ energy density, (d) ${\left|{\varphi }_1\right|}^2$, (e) ${\left|{\varphi }_2\right|}^2$, and (f) ${\theta }_{12}$ phase difference.

Figure 14

$N=1$ single quanta numerical solution for anisotropy in equivalent directions in both bands ${\lambda }_{1x}^{-1}=0.7, {\lambda }_{1y}^{-1}=0.4, {\lambda }_{2x}=1$ and ${\lambda }_{2y}^{-1}=0.1, {\eta }_{12}=0.5$, and ${\gamma }_1={\gamma }_2=2$. The contour plots are (a) $B_z$ magnetic field (b) $\left|B_z\right|-B_z$ negative magnetic field, (c) $\mathcal{E}$ energy density, (d) ${\left|{\varphi }_1\right|}^2$, (e) ${\left|{\varphi }_2\right|}^2$, and (f) ${\theta }_{12}$ phase difference.

Figure 15

$N=1$ single quanta numerical solution for strong anisotropy in opposite directions ${\lambda }_{x1}={\lambda }_{y2}=1, {\lambda }_{y1}^{-1}={\lambda }_{x2}^{-1}=0.1, {\eta }_{12}=0.5$, and ${\gamma }_1={\gamma }_2=2$. The contour plots are (a) $B_z$ magnetic field, (b) $\left|B_z\right|-B_z$ negative magnetic field, (c) $\mathcal{E}$ energy density, (d) ${\left|{\varphi }_1\right|}^2$, (e) ${\left|{\varphi }_2\right|}^2$, and (f) ${\theta }_{12}$ phase difference.

Figure 16

$N=1$ single quanta numerical solution for strong anisotropy in opposite directions but with no Josephson coupling ${\gamma }_1={\gamma }_2=2, {\eta }_{12}=0.0, {\lambda }_{x1}={\lambda }_{y2}=1$, and ${\lambda }_{y1}^{-1}={\lambda }_{x2}^{-1}=0.1$. Note that the grid size of $50\times 50$ is due to the long range nature of the negative magnetic field. (a) $B_z$ magnetic field, (b) $\left|B_z\right|-B_z$ negative magnetic field, (c) $\mathcal{E}$ energy density, (d) ${\left|{\varphi }_1\right|}^2$, (e) ${\left|{\varphi }_2\right|}^2$, and (f) ${\theta }_{12}$ phase difference.

Figure 17

Minimal energy multiquanta solution to the London model with weak anisotropy in both bands, in opposite directions, found using simulated annealing for composite vortices. The parameters are ${\lambda }_{1x}={\lambda }_{2y}=\lambda , {\lambda }_{2x}={\lambda }_{1y}=0.5\lambda$, and $k_0=0.7/\lambda$.

Figure 18

Minimal energy multiquanta solution to the London model with stronger anisotropy in both bands, in opposite directions, found using simulated annealing for composite vortices. The parameters are ${\lambda }_{1x}={\lambda }_{2y}=\lambda , {\lambda }_{2x}={\lambda }_{1y}=0.3\lambda$, and $k_0=0.7/\lambda$.

Figure 19

Minimal energy multiquanta solution to the London model with stronger anisotropy in both bands, in opposite directions, found using simulated annealing for composite vortices. The parameters are ${\lambda }_{1x}={\lambda }_{2y}=\lambda , {\lambda }_{2x}={\lambda }_{1y}=0.1\lambda$, and $k_0=0.7/\lambda$.

Figure 20

Minimal energy multiquanta solution to the London model with stronger anisotropy in one band, found using simulated annealing for composite vortices. The parameters are ${\lambda }_{1x}={\lambda }_{2y}={\lambda }_{1y}=\lambda , {\lambda }_{2x}=0.3\lambda$, and $k_0=0.7/\lambda$.

Figure 21

$N=2$ two quanta numerical solution for strong anisotropy in a single component ${\gamma }_1={\gamma }_2=2, {\eta }_{12}=0.5, {\lambda }_{x1}={\lambda }_{y1}={\lambda }_{x2}=1$, and ${\lambda }_{y2}^{-1}=0.1$. (a) $B_z$ magnetic field, (b) $\left|B_z\right|-B_z$ negative magnetic field, (c) $\mathcal{E}$ energy density, (d) ${\left|{\varphi }_1\right|}^2$, (e) ${\left|{\varphi }_2\right|}^2$, and (f) ${\theta }_{12}$ phase difference.

Figure 22

$N=2$ two quanta numerical solution for strong anisotropy in both bands in opposite directions ${\gamma }_1={\gamma }_2=2, {\eta }_{12}=0.5, {\lambda }_{y1}={\lambda }_{x2}=1$, and ${\lambda }_{x1}^{-1}={\lambda }_{y2}^{-1}=0.1$. (a) $B_z$ magnetic field, (b) $\left|B_z\right|-B_z$ negative magnetic field, (c) $\mathcal{E}$ energy density, (d) ${\left|{\varphi }_1\right|}^2$, (e) ${\left|{\varphi }_2\right|}^2$, and (f) ${\theta }_{12}$ phase difference.

Figure 23

$N=3$ three quanta local solutions for anisotropy in both bands in opposite directions ${\gamma }_1={\gamma }_2=2, {\eta }_{12}=0.5, {\lambda }_{x2}={\lambda }_{y1}=1$, and ${\lambda }_{y2}^{-1}={\lambda }_{x1}^{-1}=0.5$. (a) $B_z$ magnetic field, (b) $\left|B_z\right|-B_z$ negative magnetic field, (c) ${\left|{\varphi }_1\right|}^2$, (d) ${\left|{\varphi }_2\right|}^2$, and (e) ${\theta }_{12}$ phase difference.

Figure 24

$N=3$ three quanta local solutions for strong anisotropy in both bands in opposite directions ${\gamma }_1={\gamma }_2=2, {\eta }_{12}=0.5, {\lambda }_{x2}={\lambda }_{y1}=1$, and ${\lambda }_{y2}^{-1}={\lambda }_{x1}^{-1}=0.1$. (a) $B_z$ magnetic field, (b) $\left|B_z\right|-B_z$ negative magnetic field, (c) ${\left|{\varphi }_1\right|}^2$, (d) ${\left|{\varphi }_2\right|}^2$, and (e) ${\theta }_{12}$ phase difference.

Figure 25

$N=3,4$ three and four quanta local solutions for strong anisotropy in one direction ${\gamma }_1={\gamma }_2=2, {\eta }_{12}=0.5, {\lambda }_{x2}={\lambda }_{x1}={\lambda }_{y1}=1$, and ${\lambda }_{y2}^{-1}=0.1$. (a) $B_z$ magnetic field, (b) $\left|B_z\right|-B_z$ negative magnetic field, (c) ${\left|{\varphi }_1\right|}^2$, (d) ${\left|{\varphi }_2\right|}^2$, and (e) ${\theta }_{12}$ phase difference.

Figure 26

$N=4$ four quanta local solutions for strong anisotropy in both bands in opposite directions ${\gamma }_1={\gamma }_2=2, {\eta }_{12}=0.5, {\lambda }_{x2}={\lambda }_{y1}=1$, and ${\lambda }_{y2}^{-1}={\lambda }_{x1}^{-1}=0.1$. (a) $B_z$ magnetic field, (b) $\left|B_z\right|-B_z$ negative magnetic field, (c) ${\left|{\varphi }_1\right|}^2$, (d) ${\left|{\varphi }_2\right|}^2$, and (e) ${\theta }_{12}$ phase difference.

Figure 27

$N=4$ our quanta local solutions for anisotropy in both bands in opposite directions ${\gamma }_1={\gamma }_2=2, {\eta }_{12}=0.5, {\lambda }_{x2}^{-1}={\lambda }_{y1}^{-1}=1$, and ${\lambda }_{y2}^{-1}={\lambda }_{x1}^{-1}=0.5$. (a) $B_z$ magnetic field, (b) $\left|B_z\right|-B_z$ negative magnetic field, (c) ${\left|{\varphi }_1\right|}^2$, (d) ${\left|{\varphi }_2\right|}^2$, and (e) ${\theta }_{12}$ phase difference.

Figure 28

A contour plot and a radial slice at various angles of the magnetic field, negative magnetic field, and phase difference for the 1d boundary problem solution with anisotropy in a single component (${\lambda }_{2x}={\lambda }_{2y}={\lambda }_{1y}=1$). A radial curve from the center of the plot represents the field orthogonal to a 1d boundary crossing the origin in the $x-y$ plane. This way every possible direction (or $\theta$) is plotted for equations (24) and (25) with the radial distance representing $r$. The plot quantities are magnetic field $B_z$ (left), negative magnetic field $|B_z|-B_z$ (center), and phase difference ${\theta }_{12}$ (right) for various strengths of anisotropy from weak to strong (a) ${\lambda }_{1x}=0.8{\lambda }_{1y}$, (b) ${\lambda }_{1x}=0.5{\lambda }_{1y}$, and (c) ${\lambda }_{1x}=0.1{\lambda }_{1y}$.

Figure 29

A contour plot and a radial slice at various angles for the 1d boundary problem solution with anisotropy in opposite directions in each component (${\lambda }_{2x}={\lambda }_{1y}=1$ and ${\lambda }_{1x}={\lambda }_{2y}$). A radial curve from the center of the plot represents the field orthogonal to a 1d boundary crossing the origin in the $x-y$ plane. This way every possible direction (or $\theta$) is plotted for equations (24) and (25) with the radial distance representing $r$. The plot quantities are magnetic field $B_z$ (left), negative magnetic field $|B_z|-B_z$ (center), and phase difference ${\theta }_{12}$ (right) for various strengths of anisotropy from weak to strong (a) ${\lambda }_{1x}={\lambda }_{2y}=0.8{\lambda }_{1y}$, (b) ${\lambda }_{1x}={\lambda }_{2y}=0.5{\lambda }_{1y}$, (c) ${\lambda }_{1x}={\lambda }_{2y}=0.1{\lambda }_{1y}$.