Driving force on flowing quantum vortices in type-II superconductors with finite Ginzburg-Landau parameter

The origin of the driving force on quantum vortices in superconductors has long been discussed. We investigate the origin of this force using the momentum flux tensor $\mathcal{P}$, Maxwell stress tensor $\mathcal{T}$, and numerical solutions for a flowing rectilinear vortex in the time-dependent Ginzburg-Landau (TDGL) theory for three-dimensional superconductors with finite Ginzburg-Landau parameter $\kappa$ and the Maxwell equations. We calculate the hydrodynamic force ${\mathit{F}}_{\mathrm{hydro}}\left(C\right)$ and magnetic Lorentz force ${\mathit{F}}_{\mathrm{mag}}\left(C\right)$ respectively using the contour integral of $\mathcal{P}$ and $\mathcal{T}$ along a closed path that winds around the vortex line. The calculations show that neither ${\mathit{F}}_{\mathrm{hydro}}\left(C\right)$ nor ${\mathit{F}}_{\mathrm{mag}}\left(C\right)$ reaches the full magnitude of the driving force. However, when the path $C$ is farther than the penetration depth from the vortex line and hence the energy dissipation is negligible on $C$, the sum of the two forces becomes independent of the choice of $C$ and accounts for the full magnitude of the driving force on the vortex. We demonstrate the applicability of this result to a flowing vortex described in the generalized or modified version of the TDGL equation and to a pinned vortex. We then discuss the driving force on the Pearl vortex in two-dimensional superconductors and a curved vortex line in three-dimensional superconductors. We propose an experiment that locally probes the magnetic field with a pinned vortex to verify our results that the contribution of the magnetic pressure (Lorentz force) to the total driving force on the vortex is less than half.

Figure 1

Flow of ideal fluid around an object. Circulation is positive and thus this object is subjected to the Magnus force.

Figure 2

(a) Schematic diagram of the order parameter, the current density, and magnetic field around a single vortex located at $\mathit{r}=\mathit{0}$. Green solid line denotes the current density on the plane $x=0$. Pink dotted line denotes the current density in the absence of a vortex. These lines have common asymptotes. Blue-violet and blown lines, respectively, represent the modulus of order parameter and magnetic field. (b) Order parameter and current density in the absence of a vortex. Pink dotted line is the same as that in (a).

Figure 3

Current densities and magnetic fields $\kappa =10$ and $\zeta =1/3$. (a) ${\mathit{j}}_0\left(\mathit{r}\right)$ (arrows) with the unit $\frac{\sqrt{2}{B}_{\mathrm{c}}}{{\mu }_0\lambda }$ and ${\mathit{h}}_0\left(\mathit{r}\right)$ (density) with the unit $\sqrt{2}{B}_{\mathrm{c}}$. (b) ${\mathit{j}}_1\left(\mathit{r}\right)$ (arrows) with the unit $\frac{\sqrt{2}v{B}_{\mathrm{c}}}{{\mu }_0\lambda v_0}$ and ${\mathit{h}}_1\left(\mathit{r}\right)$ (density) with the unit $\frac{\sqrt{2}v{B}_{\mathrm{c}}}{v_0}$. (c) Normal component of the current density ${\mathit{j}}_{\mathrm{n}}\left(\mathit{r}\right)$ with the unit $\frac{\sqrt{2}v{B}_{\mathrm{c}}}{{\mu }_0\lambda v_0}$. This figure can be seen as that for $\mathit{\varepsilon }\left(\mathit{r}\right)$ with the unit $2v{B}_{\mathrm{c}}$. (d) Supercurrent density ${\mathit{j}}_{\mathrm{s}1}\left(\mathit{r}\right)$ with the unit $\frac{\sqrt{2}v{B}_{\mathrm{c}}}{{\mu }_0\lambda v_0}$.

Figure 4

Backflow ${\mathit{j}}_1\left(\mathit{r}\right)-{\mathit{j}}_{\mathrm{tr}}\left(\mathit{r}\right)$ (arrows) and magnetic field ${\mathit{h}}_1\left(\mathit{r}\right)-{\mathit{h}}_{\mathrm{tr}}\left(\mathit{r}\right)$ (density plot) for $\kappa =10$ and $\zeta =1/3$. Left and right color bars show values for the magnetic field and current density, respectively, in the units $\frac{\sqrt{2}v{B}_{\mathrm{c}}}{v_0}$ and $\frac{\sqrt{2}v{B}_{\mathrm{c}}}{{\mu }_0\lambda v_0}$.

Figure 5

${\mathit{j}}_0\left(\mathit{r}\right)+{\mathit{j}}_1\left(\mathit{r}\right)$ (arrow) and ${\mathit{h}}_0\left(r\right)+{\mathit{h}}_1\left(\mathit{r}\right)$ (density plot) for $\kappa =10, \zeta =1/3$, and $v/v_0=0.05$. Left and right color bars show values for the magnetic field and current density, respectively, in the units $\sqrt{2}{B}_{\mathrm{c}}$ and $\frac{\sqrt{2}{B}_{\mathrm{c}}}{{\mu }_0\lambda }$.

Figure 6

Amplitude of the order parameter for $\kappa =10, \zeta =1/3$, and $v/v_0=0.003$. (a) $f_1\left(\mathit{r}\right)$ in the unit $v/v_0$ and (b) $f_0\left(r\right)+f_1\left(\mathit{r}\right)$.

Figure 7

Local Lorentz force density $[{\mathit{j}}_0\left(\mathit{r}\right)+{\mathit{j}}_1\left(\mathit{r}\right)]\times [{\mathit{h}}_0\left(\mathit{r}\right)+{\mathit{h}}_1\left(\mathit{r}\right)]$ subtracted by ${\mathit{j}}_0\left(\mathit{r}\right)\times {\mathit{h}}_0\left(\mathit{r}\right)$ (arrows) and magnetic energy density ${[{\mathit{h}}_0\left(\mathit{r}\right)+{\mathit{h}}_1\left(\mathit{r}\right)]}^2/2{\mu }_0$ subtracted by ${\mathit{h}}_0{\left(\mathit{r}\right)}^2/2{\mu }_0$ (density plot) for $\kappa =10, \zeta =1/3$, and $v/v_0=0.003$. Left and right color bars show values for the magnetic energy density and local Lorentz force density, respectively, in the units $\frac{B_{\mathrm{c}}^{2}v}{{\mu }_0{v}_0}$ and $\frac{c\left(\kappa \right)j_{\mathrm{tr}}\left|{\mathit{\varphi }}_0\right|}{\lambda }$.

Figure 8

Local hydrodynamic force density $-{\partial }_{\nu }{\mathcal{P}}_{\mu \nu }$ for $\kappa =10, \zeta =1/3$, and $v/v_0=0.003$. The unit for intensity is $\frac{c\left(\kappa \right)j_{\mathrm{tr}}|{\mathit{\varphi }}_0|}{\lambda }$.

Figure 9

Dissipation function $W\left(\mathit{r}\right)$ defined in Eq. (23) (density) and $O\left(v^2\right)$ terms of the energy current density ${\mathit{j}}_E\left(\mathit{r}\right)$ defined in Eq. (22) (arrows) for $\kappa =10, \zeta =1/3$, and $v/v_0=0.003$. Left color bar shows values for $W$ in the unit $2{\sigma }_n{v}^2{B}_{\mathrm{c}}^2$ and right one those of $O\left(v^2\right)$ terms of ${\mathit{j}}_E\left(\mathit{r}\right)$ in the unit of $\frac{v^2{B}_{\mathrm{c}}^2}{v_0{\mu }_0}$.

Figure 10

Radius dependence of the hydrodynamic force ${\mathit{F}}_{\mathrm{hydro}}\left(C\right)$, Eq. (41), magnetic Lorentz force ${\mathit{F}}_{\mathrm{mag}}\left(C\right)$, Eq. (6), and total force ${\mathit{F}}_{\mathrm{tot}}\left(C\right)={\mathit{F}}_{\mathrm{hydro}}\left(C\right)+{\mathit{F}}_{\mathrm{mag}}\left(C\right)$, Eq. (8), per unit length along the vortex axis for $\zeta =1/3$ and $\kappa =3,5,7,10$ [from (a) to (d)]. Horizontal dotted lines indicate $|{\mathit{j}}_{\mathrm{tr}}\left(0\right)\left|\right|{\mathit{\varphi }}_0|$.

Figure 11

Setup for calculation of the force on the Pearl vortex in a 2D superconductor. The transport magnetic field in the $z$ direction is perpendicular to the wide surface of the slab. The transport current flows uniformly in the $x$ direction and varies along the $y$ axis. The closed surface $S$ consists of the surface $S_{+}$, bottom $S_{-}$, and side $S^{\prime }$. The normal vector $S$ points outward.

Figure 12

Force on Pearl vortex in a 2D superconductor with thickness $d$ that is much smaller than the penetration depth $\lambda$. Calculation is based on the Maxwell-London equation Eq. (52), namely the radial dependence of the hydrodynamic force Eq. (54) and the magnetic force Eq. (55) on the vortex in the presence of the transport current $j_{\mathrm{tr}}$. The characteristic scale ${\lambda }_{\mathrm{eff}}=2{\lambda }^2/d$ is the effective magnetic screening length in the 2D superconductor. The top panel (a) shows the spatial range on the order of $R\sim {\lambda }_{\mathrm{eff}}$ and the bottom panel (b) shows the results in a much wider range of $R$.

Figure 13

Local Cartesian coordinates for the calculation of the force on a segment of the vortex line. The transport current flows in the $x$ direction and varies along the $y$ axis. The transport magnetic field ${\mathit{h}}_{\mathrm{tr}}$ points along the $z$ direction and varies along the $y$ axis. The $z^{\prime }$ direction is parallel to the tangential direction along the vortex line at $\mathit{r}\left(s\right)$. Here, $s$ is the length along the vortex line. The closed surface $S=S_{+}(R;s+\mathrm{\Delta }s)\cup S_{-}(R;s+\mathrm{\Delta }s)\cup S^{\prime }(R;s,s+\mathrm{\Delta }s)$ surrounds the segment of the vortex line. The normal vector $S$ points outward.