Locking of length scales in two-band superconductors

A model of a clean two-band $s$-wave superconductor with cylindrical Fermi surfaces, different Fermi velocities $v_{1,2}$, and a general $2\times 2$ coupling matrix $V_{\alpha \beta }$ is used to study the order parameter distribution in vortex lattices. The Eilenberger weak coupling formalism is used to calculate numerically the spatial distributions of the pairing amplitudes ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ of the two bands for vortices parallel to the Fermi cylinders. For generic values of the interband coupling $V_{12}$, it is shown that, independently of the couplings $V_{\alpha \beta }$, of the ratio $v_1/v_2$, of the temperature, and the applied field, the length scales of spatial variation of ${\mathrm{\Delta }}_1$ and of ${\mathrm{\Delta }}_2$ are the same within the accuracy of our calculations. The only exception from this single length-scale behavior is found for $V_{12}\ll V_{11}$, i.e., for nearly decoupled bands.

Figure 1

(a) Pairing amplitudes $|{\mathrm{\Delta }}_1\left(\mathit{r}\right)|$ and $|{\mathrm{\Delta }}_2\left(\mathit{r}\right)|$ (in units $\pi T_{c,1})$ vs distance $r$ (in units of $R_1=\hslash v_1/2\pi T_{c,1})$ from the vortex center to the midpoint between nearest neighbor vortices. In this calculation, $V_{12}/V_{11}=0.32,t=T/T_{c,1}=0.5$, and $B=0.1$ (in units ${\varphi }_0/2\pi R_1^2)$. Solid lines are for $V_{22}=0$, dashed lines are for $V_{22}/V_{11}=0.32$. (b) Nearly constant ratios $|{\mathrm{\Delta }}_2\left(\mathit{r}\right)|/|{\mathrm{\Delta }}_1\left(\mathit{r}\right)|$ imply the same length scales for both pairing amplitudes.

Figure 2

(a) Temperature dependence of maximum values ${\mathrm{\Delta }}_{\mathrm{m},\alpha }$ of pairing amplitudes $|{\mathrm{\Delta }}_{\alpha }\left(\mathit{r}\right)|$ at $B=0.1$ for $V_{22}=0$ and $V_{12}=0.32V_{11}$. Zero-field $|{\mathrm{\Delta }}_{\alpha }|$ are shown by dotted lines. (b) The same as (a) for $V_{22}=0.32V_{11}$. (c),(d) $T$ dependences of core sizes ${\xi }_{\alpha }^{\left(c\right)}$, and (e) of ${\xi }_2^{\left(c\right)}/{\xi }_1^{\left(c\right)}$ for $B=0.1$. Temperature $T$ is in units of $T_{c,1}$.

Figure 3

(a) Magnetic field dependence of ${\mathrm{\Delta }}_{\mathrm{m},\alpha },\alpha =1,2$. (b),(c) $B$ dependence of the core sizes ${\xi }_{\alpha }^{\left(c\right)}$ and (d) the ratio ${\xi }_2^{\left(c\right)}/{\xi }_1^{\left(c\right)}$. Inputs: $t=0.5,V_{12}=0.32V_{11}$, solid lines are for $V_{22}=0$, dashed lines for $V_{22}=0.32V_{11}$.

Figure 4

(a) $|{\mathrm{\Delta }}_1\left(\mathit{r}\right)|$ and $|{\mathrm{\Delta }}_2\left(\mathit{r}\right)|$ vs distance $r$ from the vortex center to the midpoint between nearest neighbor vortices. (b) $|{\mathrm{\Delta }}_2\left(\mathit{r}\right)|/|{\mathrm{\Delta }}_1\left(\mathit{r}\right)|$. Input parameters are $V_{12}=0.01V_{11},V_{22}=0.85V_{11}$, and $t=0.5$; solid lines are for $B=0.1$, dashed lines are for $B=0.01$.

Figure 5

(a) Temperature dependence of ${\mathrm{\Delta }}_{\mathrm{m},\alpha }$ at $B=0$ (dotted lines), $B=0.03$ (dashed lines), and $B=0.1$ (solid lines). (b) $T$ dependence of the vortex core radii ${\xi }_1^{\left(c\right)}$ and ${\xi }_2^{\left(c\right)}$, and (c) the ratio ${\xi }_2^{\left(c\right)}/{\xi }_1^{\left(c\right)}$. $V_{12}=0.01V_{11}$ and $V_{22}=0.85V_{11}$.

Figure 6

(a) The field dependence of ${\mathrm{\Delta }}_{\mathrm{m},\alpha }$. (b) $B$ dependence of core radii ${\xi }_1^{\left(c\right)}$ and ${\xi }_2^{\left(c\right)}$, and (c) the ratio ${\xi }_2^{\left(c\right)}/{\xi }_1^{\left(c\right)}$. Input parameters: $t=0.5,V_{12}=0.01V_{11}$ and $V_{22}=0.85V_{11}$.

Figure 7

(a) ${\xi }_{\alpha }^{\left(c\right)}$ vs interband coupling $V_{12}/V_{11}$ for $V_{22}=0.83V_{11}$ at $t=0.2$ and $B=0.03$. (b) Ratios ${\xi }_2^{\left(c\right)}/{\xi }_1^{\left(c\right)}$ and ${\mathrm{\Delta }}_{\mathrm{m},2}/{\mathrm{\Delta }}_{\mathrm{m},1}$ vs $V_{12}/V_{11}$ for the same parameters as (a).

Figure 8

Solving the first-order ordinary differential Eq. (A2) along the trajectory ${\mathit{r}}^{\prime }=\mathit{r}+s{\widehat{\mathit{v}}}_{\alpha }$ for $a$ at $s=0$. Real and imaginary parts of $a$ are shown for start positions $s_0=-8.2$ (A), $-16.4$ (B), and $-32.7$ (C). It is seen that $a$ converges to the same solution at $s=0$. Input parameters are $\varphi =1.{25}^{\circ }$ for $\mathit{k},\alpha =1,\omega =\pi T$, and $V_{22}=0$ in the case of Fig. 1. $\mathit{r}$ is near the midpoint $(-a_0/2,0)$ between nearest neighbor vortices.