Dynamics of Topological Defects in Type-II Superconductors under Gradients of Temperature/Spin Density

Abstract

We theoretically investigate the motion of a domain wall and a vortex in type-II superconductors driven by inhomogeneities of temperature or spin density. The model consists of the time-dependent Ginzburg-Landau equation and the thermal/spin diffusion equation, whose transport coefficients (the thermal/spin conductivity and the spin relaxation time) depend on the order parameter, interpolating between values in the superconducting and normal states. Numerical and analytical calculations indicate that the domain wall moves toward the higher-temperature/spin-density region, where the order parameter is suppressed. We also derive an analytical expression for the vortex velocity. We can understand the dynamics of topological defects as processes that reduce the loss of condensation energy. We also analyze the driving force, the viscous force, the thermal force, and the force due to the spin accumulation gradient, on the basis of the momentum balance relations.

Figures (12)

• Figure 1: Schematic picture of the setup we discuss. A sample of the superconductor $(x_{\textrm{L}} \leq x\leq x_{\textrm{R}})$ is placed between heat baths with temperatures $T_{\textrm{L}}$ at $x=x_{\textrm{L}}$ and $T_{\textrm{R}}$ at $x=x_{\textrm{R}}$ respectively. We describe the superconductivity in terms of the dimensionless condensate wavefunction, namely the order parameter $\psi(x,t)$ represented by the vertical axis. We impose the boundary conditions with different signs (see Eqs. \ref{['boundconds for gradT third']} and \ref{['boundconds for gradT last']}), and thus we have the domain wall structure of the order parameter in our system.
• Figure 2: Snapshots of numerical solutions to the TDGL Eq. \ref{['TDGLdimensionless']} coupled with the thermal diffusion Eq. \ref{['TDdimensionless']}. The figures in the first and second rows show the solutions for the order parameter $\psi(x,t)$ and the temperature $\tau(x,t)$, respectively. The time evolves from (a) to (d). (a): An initial state satisfying the boundary conditions Eqs. \ref{['boundconds for gradT third']} and \ref{['boundconds for gradT last']}. (b): The initial profile is deformed, and the domain wall-like structure appears. (c), (d): The domain wall keeps flowing against the heat flow.
• Figure 3: Comparison between the analytic solution up to the first order, namely $\psi_0(x) + \psi_1(x)$ plotted in the blue solid curve, and the snapshot at the time $t=0$ in the time-evolution obtained from the numerical calculation we discuss in Sec. \ref{['gradtnum']} plotted in the red dashed curve.
• Figure 4: Time-dependence of the coordinate of the domain wall. The solid red line and the blue curve represent the analytical and numerical result under the parameters $k =1/20$, $\tilde{\gamma}\kappa_{\textrm{n}}/(C\xi^2)=1$, $\tau_{\textrm{L}} = 0.990$, $\tau_{\textrm{R}} = 0.990+10^{-6}$, $x_{\textrm{R}}/\xi=25$, respectively. The ratio between the analytically obtained velocity $v_{\rm ana}$ and the numerically obtained velocity $v_{\rm sim}$ is $v_{\rm sim} / v_{\rm ana}=1.0039$ (see Supplemental Material for temperature dependency of the ratio.).
• Figure 5: Illustration of the relations among the heat flow, velocity, and forces. According to Eq. (\ref{['vvsqtransport']}), the velocity of the domain wall is directed to the hotter region since the heat flows toward the colder region. The direction of the thermal force follows from Eq. \ref{['Fthq']}. The viscous force, which is against the velocity, is balanced with the thermal force. The driving force is negligible when the system size is much larger than $\xi$, i.e., $|x_{\rm L}|,x_{\rm R}\gg \xi$.