Microscopic theory of type-1.5 superconductivity in multiband systems

We report a self-consistent microscopic theory of characteristic length scales, vortex structure, and type-1.5 superconducting state in two-band systems using two-band Eilenberger formalism.

Figure 1

Masses ${\mu }_{L,H}$ of the composite gap function fields for (a) ${\gamma }_F=1$ and ${\lambda }_J=0.005$, (b) ${\gamma }_F=0.5$ and ${\lambda }_J=0.0025$. The position of branch cut is shown by black dashed line. The mixing angles ${\theta }_{L,H}$ are shown by blue dashed and dashed-dotted lines correspondingly. (c) Temperature dependence of the mass ${\mu }_L\left(T\right)$ (black thick curves) and the corresponding mixing angle ${\theta }_L$ determined by Eq. (11) (red thin curves) for ${\gamma }_F=1,2,5$ (solid, dashed, and dashed-dotted curves). The coupling parameters are ${\lambda }_{11}=0.25$, ${\lambda }_{22}=0.213$, and ${\lambda }_J=0.005$. (d) Temperature dependence of the mass ${\mu }_L\left(T\right)$ for different values of coupling constant ${\lambda }_{22}$.

Figure 2

Masses ${\mu }_L$ and ${\mu }_H$ (red solid lines) of the composite gap function fields for the different values of interband Josephson coupling ${\lambda }_J$ and ${\gamma }_F=1$. In the sequence of plots (a)–(d), the transformation of masses is shown for ${\lambda }_J$increasing from the small values ${\lambda }_J\ll {\lambda }_{11},{\lambda }_{22}$ to the values comparableto intraband coupling ${\lambda }_J\sim {\lambda }_{11},{\lambda }_{22}$. The particular values of coupling constants are ${\lambda }_{11}=0.25$, ${\lambda }_{22}=0.213$, and ${\lambda }_J=0.0005,0.0025,0.025;{\lambda }_{22}$ for plots (a)–(d) correspondingly. The branch cuts are shown by black dashed lines. In (a), with blue dashed-dotted lines, the masses of modes are shown for the case of ${\lambda }_J=0$. Note that, at ${\lambda }_J=0,$ the two masses go to zero at two different temperatures. Because $1/{\mu }_{L,H}$ are related to the coherence lengths, this reflects the fact that, for $\text{U}\left(1\right)\times \text{U}\left(1\right)$ theory, there are two independently diverging coherence lengths. Note that, for finite values of interband coupling, only one mass ${\mu }_L$ goes to zero at one $T_c$.

Figure 3

(a) Masses ${\mu }_L$ and ${\mu }_H$ (red solid and dotted lines) of the composite gap function fields and inverse London penetration (blue dashed lines) for the different values of $\Lambda {\mu }_L\left(T_c\right)/\sqrt{2}=$1, 2, 3, 5. The position of branch cut is shown by black dashed-dotted line. (b) The temperature dependence of the quantity $\Lambda {\mu }_L$ for $\Lambda {\mu }_L\left(T_c\right)/\sqrt{2}=$1, 2, 3, 5 (red solid, blue dashed, and black dashed-dotted lines). (c) Distributions of magnetic field $H\left(r\right)/H(r=0)$, gap functions $|{\Delta }_1|\left(r\right)/{\Delta }_{10}$ (dashed lines), and $|{\Delta }_2|\left(r\right)/{\Delta }_{20}$ (solid lines) in a single vortex for the coupling parameters ${\lambda }_{11}=0.25$, ${\lambda }_{22}=0.213$, and ${\lambda }_{21}=0.0025$ and different values of the band parameter ${\gamma }_F=$1, 2, 3, 4, 5. (d) The energy of interaction between two vortices normalized to the single vortex energy as function of the intervortex distance $d$. In panels (c) and (d) the temperature is $T=0.6$.

Figure 4

Gap-function profiles around vortex core for ${\lambda }_{11}=0.25$, ${\lambda }_{22}=0.1$, ${\lambda }_{12}={\lambda }_{21}=0.05$. (a) and (c): Variation of the gap functions $|{\Delta }_j|\left(r\right)/{\Delta }_{j0}$ ($j=1,2$) near the core. (b) and (d): The behavior of gap-function deviations from the vacuum state $\delta {\Delta }_j\left(r\right)=1-\left|{\Delta }_j\right|\left(r\right)/{\Delta }_{j0}$ at longer range. Note that, in this temperature span, the higher the temperature, the faster the long distance decay of $\delta {\Delta }_j\left(r\right)$, which reflects the found fact in the two-band system that the field mass can increase with raising temperature [see also Fig. 1d.]