Theory of vortices in hybridized ballistic/diffusive-band superconductors

We study the electronic structure in the vicinity of a vortex in a two-band superconductor in which the quasiparticle motion is ballistic in one band and diffusive in the other. This study is based on a model appropriate for such a case, that we have introduced recently [Tanaka et al., Phys. Rev. B 73, 220501(R) (2006)]. We argue that in the two-band superconductor ${\mathrm{MgB}}_2$, such a case is realized. Motivated by the experimental findings on ${\mathrm{MgB}}_2$, we assume that superconductivity in the diffusive band is “weak,” i.e., mostly induced. We examine intriguing features of the order parameter, the current density, and the vortex core spectrum in the “strong” ballistic band under the influence of hybridization with the “weak” diffusive band. Although the order parameter in the diffusive band is induced, the characteristic length scales in the two bands differ due to Coulomb interactions. The current density in the vortex core is dominated by the contribution from the ballistic band, while outside the core the contribution from the diffusive band can be substantial, or even dominating. The current density in the diffusive band has strong temperature dependence, exhibiting the Kramer-Pesch effect when hybridization is strong. A particularly interesting feature of our model is the possibility of additional bound states near the gap edge in the ballistic band, that are prominent in the vortex center spectra. This contrasts with the single band case, where there is no gap-edge bound state in the vortex center. We find the above-mentioned unique features for parameter values relevant for ${\mathrm{MgB}}_2$.

Figure 1

(Color online) The bulk $\mid {\Delta }_{\sigma }\mid ∕T_c$ in the zero-temperature limit as given by Eqs. (25, 26): (a) as a function of ${\rho }_0={\Delta }_{\pi }^0∕{\Delta }_{\sigma }^0$ for $n_{\pi }=n_{\sigma }$ and various values of $\Lambda$, and (b) as a function of $n_{\pi }∕n_{\sigma }$ for various sets of ${\rho }_0$ and $\Lambda$. For a fixed $n_{\pi }∕n_{\sigma }$, $\mid {\Delta }_{\sigma }\mid ∕T_c$ shows nonlinear behavior as a function of ${\rho }_0$, where the BCS value is recovered at ${\rho }_0=0$ and 1. The bulk gap is enhanced as $n_{\pi }∕n_{\sigma }$ increases, and this effect is larger for larger ${\rho }_0$ and for smaller $\Lambda$.

Figure 2

(Color online) Energy gaps ${\Delta }_{\pi }\left(T\right)$ and ${\Delta }_{\sigma }\left(T\right)$ (a) and the ratio ${\Delta }_{\pi }\left(T\right)∕{\Delta }_{\sigma }\left(T\right)$ (b) in the homogeneous case as a function of temperature $T$ for various values of $\Lambda$, for fixed ${\rho }_0=0.3$ and $n=\sqrt{n_{\sigma }∕n_{\pi }}=1$. There is a second transition temperature associated with the $\pi$ band for $\Lambda \ge {\Lambda }_c$ (see text).

Figure 3

(Color online) The ${\Delta }_{\sigma }^0∕T_c$ [(a) and (b)] and ${\Delta }_{\pi }^0∕T_c$ [(c) and (d)] as a function of the gap ratio at $T_c$, $\rho ={\lim }_{T\to T_c}{\Delta }_{\pi }∕{\Delta }_{\sigma }$, obtained from Eqs. (25, 26), for $\Lambda <0$ [(a) and (c)] and $\Lambda >0$ [(b) and (d)]. While for $-1\le \Lambda \le 0$ there is a unique solution, for $\Lambda >0$, for sufficiently small $\rho$ there are three solutions. This indicates possible first-order transitions as a function of temperature, if $\Lambda$ for a given $\rho$ is larger than a critical value.

Figure 4

(Color online) (a) Order parameters $\mid {\Delta }_{\pi }\left(x\right)\mid$ and $\mid {\Delta }_{\sigma }\left(x\right)\mid$ and (b) the ratio $\mid {\Delta }_{\pi }\left(x\right)∕{\Delta }_{\sigma }\left(x\right)\mid$ as a function of coordinate $x$ along the path $y=0$ for various values of $\Lambda$, for ${\xi }_{\pi }∕{\xi }_{\sigma }=3$, ${\rho }_0=0.3$, $T∕T_c=0.1$, $n=1$. The Coulomb repulsion (which reduces $\Lambda$) renormalizes the recovery lengths of the order parameters in the two bands differently.

Figure 5

(Color online) (a) Order parameters $\mid {\Delta }_{\pi }\left(x\right)\mid$ and $\mid {\Delta }_{\sigma }\left(x\right)\mid$ and (b) the ratio $\mid {\Delta }_{\pi }\left(x\right)∕{\Delta }_{\sigma }\left(x\right)\mid$ as a function of coordinate $x$ along the path $y=0$ for various density of states ratios $n_{\pi }∕n_{\sigma }$, and fixed ${\xi }_{\pi }∕{\xi }_{\sigma }=3$, ${\rho }_0=0.3$, $T∕T_c=0.1$, $\Lambda =-0.1$. The $\pi$-band density of states renormalizes the recovery lengths of the order parameters, and the bulk magnitudes of the order parameters are larger for larger $n_{\pi }∕n_{\sigma }$.

Figure 6

(Color online) (a) Order parameter in the $\pi$ band $\mid {\Delta }_{\pi }\left(x\right)\mid$ as a function of coordinate $x$ along the path $y=0$ for various values of ${\rho }_0$ and ${\xi }_{\pi }∕{\xi }_{\sigma }$, for $T∕T_c=0.1$ and $n_{\pi }∕n_{\sigma }=1$; for (a) $\Lambda =-0.1$ and (b) $\Lambda =0.1$. For $\Lambda >0$, $\mid {\Delta }_{\pi }\left(x\right)\mid$ is depleted around the vortex core for larger ${\xi }_{\pi }∕{\xi }_{\sigma }$, and this effect is larger for larger ${\rho }_0$. When the Coulomb repulsion dominates $(\Lambda <0)$, for small ${\rho }_0$ (e.g., 0.1), $\mid {\Delta }_{\pi }\left(x\right)\mid$ is enhanced near the vortex center for larger ${\xi }_{\pi }∕{\xi }_{\sigma }$.

Figure 7

(Color online) (a) The ratio $\mid {\Delta }_{\pi }\left(x\right)∕{\Delta }_{\sigma }\left(x\right)\mid$ as a function of coordinate $x$ along the path $y=0$ for various values of ${\rho }_0$ and ${\xi }_{\pi }∕{\xi }_{\sigma }$, for $T∕T_c=0.1$ and $n_{\pi }∕n_{\sigma }=1$; for (a) $\Lambda =-0.1$ and (b) $\Lambda =0.1$. For negative (positive) $\Lambda$, the ratio is larger (smaller) than ${\rho }_0$ around the vortex core, and the area of the enhancement (depletion) is larger for larger ${\xi }_{\pi }∕{\xi }_{\sigma }$. As ${\rho }_0$ increases, the ratio changes more drastically as $x$ approaches zero.

Figure 8

(Color online) Order parameters $\mid {\Delta }_{\pi }\left(x\right)\mid$ and $\mid {\Delta }_{\sigma }\left(x\right)\mid$ as a function of coordinate $x$ along the path $y=0$ for $T∕T_c=0.1$, $\Lambda =-0.1$, $n_{\pi }∕n_{\sigma }=1$; for (a) various values of ${\rho }_0$ for ${\xi }_{\pi }∕{\xi }_{\sigma }=3$ and (b) different values of ${\xi }_{\pi }∕{\xi }_{\sigma }$ for ${\rho }_0=0.3$. For a fixed ${\xi }_{\pi }∕{\xi }_{\sigma }$, as ${\rho }_0$ increases, the core area is enhanced in both bands. For a fixed ${\rho }_0$, for larger ${\xi }_{\pi }∕{\xi }_{\sigma }$ the core area is enlarged in the $\sigma$ band.

Figure 9

(Color online) The ratio $\mid {\Delta }_{\pi }\left(x\right)∕{\Delta }_{\sigma }\left(x\right)\mid$ as a function of position $x$ along the path $y=0$ for ${\xi }_{\pi }∕{\xi }_{\sigma }=3$ and $n_{\pi }∕n_{\sigma }=1$, for various values of ${\rho }_0$ and $T∕T_c$; for (a) $\Lambda =-0.1$ and (b) $\Lambda =0.1$. The difference between the length scales in the two bands and its temperature dependence are enhanced as ${\rho }_0$ increases, especially for $\Lambda <0$. As temperature is raised, the difference between the recovery lengths of the two order parameters becomes smaller.

Figure 10

(Color online) Order parameters (a) $\mid {\Delta }_{\sigma }\left(x\right)\mid$ and (b) $\mid {\Delta }_{\pi }\left(x\right)\mid$ as a function of $x$ along the path $y=0$ for ${\rho }_0=0.3$, ${\xi }_{\pi }∕{\xi }_{\sigma }=3$, $n_{\pi }∕n_{\sigma }=1$, $\Lambda =-0.1$. The weak $\pi$ band makes the temperature dependence of the $\mid {\Delta }_{\sigma }\mid$ profile more drastic than in the single-band case.

Figure 11

(Color online) The core size ${\xi }_c$ defined by Eq. (34) in the two bands for ${\rho }_0=0.3$, ${\xi }_{\pi }∕{\xi }_{\sigma }=3$, and $n_{\pi }∕n_{\sigma }=1$, for (a) $\Lambda =0.1$ and (b) $\Lambda =-0.1$. The Kramer-Pesch effect is induced in the $\pi$ band, although absent in a single diffusive band (dashed-dotted curves). As seen from (a), the core radius ${\xi }_c$ is larger in the $\pi$ band than that in the $\sigma$ band when $\Lambda >0$. However, the opposite is true for $\Lambda <0$, as shown in (b).

Figure 12

(Color online) The $\pi$-band LDOS $N_{\pi }(ϵ,r)$ as a function of energy $ϵ$ at various distances $r$ from the vortex center ($r∕{\xi }_{\sigma }$ from 0 to 40 with an increment of 5). The parameter values are ${\xi }_{\pi }∕{\xi }_{\sigma }=1$, ${\rho }_0=0.3$, $T∕T_c=0.1$, $n_{\pi }∕n_{\sigma }=1$, and $\Lambda =0$. The $N_{\pi }(ϵ,r)$ shows no sign of localized states in the vortex core, consistent with the experiment on ${\mathrm{MgB}}_2$ (Ref. 17).

Figure 13

(Color online) The zero-bias LDOS $N_{\pi }(ϵ=0,r)$ as a function of distance $r$ from the vortex center for $n_{\pi }∕n_{\sigma }=1$; (a) ${\rho }_0=0.3$, $T∕T_c=0.1$, $\Lambda =-0.1$ for various values of ${\xi }_{\pi }∕{\xi }_{\sigma }$; (b) ${\xi }_{\pi }∕{\xi }_{\sigma }=3$, $T∕T_c=0.1$, $\Lambda =-0.1$ for various values of ${\rho }_0$; (c) ${\rho }_0=0.3$, ${\xi }_{\pi }∕{\xi }_{\sigma }=3$, $T∕T_c=0.1$ for various values of $\Lambda$; (d) ${\rho }_0=0.3$, ${\xi }_{\pi }∕{\xi }_{\sigma }=3$, $\Lambda =-0.1$ for various temperatures. By comparing with the experimental data (Ref. 17) we estimate ${\xi }_{\pi }∕{\xi }_{\sigma }\sim 2$ for ${\mathrm{MgB}}_2$, taking ${\rho }_0=0.3$.

Figure 14

(Color online) The $\sigma$-band LDOS $N_{\sigma }(ϵ,r)$ as a function of energy $ϵ$ at various distances $r$ from the vortex center ($r∕{\xi }_{\sigma }$ from 0 to 10 with an increment of 1). The parameter values are ${\xi }_{\pi }∕{\xi }_{\sigma }=3$, ${\rho }_0=0.3$, $T∕T_c=0.1$, $n_{\pi }∕n_{\sigma }=1$, and $\Lambda =-0.1$. Coupling to a diffusive band results in bound states at the gap edge in the clean band, in addition to the well-known Caroli–de Gennes–Matricon bound states.

Figure 15

(Color online) The $N_{\sigma }(ϵ,r)$ as a function of energy $ϵ$ at various distances $r$ from the vortex center ($r∕{\xi }_{\sigma }$ from 0 to 10 with an increment of 1) for $T∕T_c=0.1$; (a) the single-band case, (b) ${\xi }_{\pi }∕{\xi }_{\sigma }=5$, ${\rho }_0=0.2$, (c) ${\xi }_{\pi }∕{\xi }_{\sigma }=2$, ${\rho }_0=0.3$, and (d) ${\xi }_{\pi }∕{\xi }_{\sigma }=5$, ${\rho }_0=0.3$. In the two-band cases, $n_{\pi }∕n_{\sigma }=1$ and $\Lambda =-0.1$. The gap-edge bound states, which are absent in the single-band case, are enhanced (and the number of branches increases) for larger ${\xi }_{\pi }∕{\xi }_{\sigma }$ and ${\rho }_0$.

Figure 16

(Color online) The total LDOS in the $\sigma$-band at the vortex center, $N_{\sigma }(ϵ,r=0)$, for energy $ϵ$ close to the energy gap, for ${\xi }_{\pi }∕{\xi }_{\sigma }=5$ and $T∕T_c=0.1$ for different values of $\Lambda$; for ${\rho }_0=0.2$ [(a),(b)] and 0.5 [(c),(d)] and for $n_{\pi }∕n_{\sigma }=1$ [(a),(c)] and 2 [(b),(d)]. Bound states exist near the gap edge for $\Lambda$ smaller than a certain value.

Figure 17

(Color online) Same as Fig. 16, but for fixed $\Lambda =0$, and for [(a),(b)] various values of ${\xi }_{\pi }∕{\xi }_{\sigma }$ for ${\rho }_0=0.5$ and [(c),(d)] different values of ${\rho }_0$ for ${\xi }_{\pi }∕{\xi }_{\sigma }=5$. Gap-edge bound states develop as ${\rho }_0$ or ${\xi }_{\pi }∕{\xi }_{\sigma }$ is increased.

Figure 18

(Color online) Same as Fig. 16, but for $n_{\pi }∕n_{\sigma }=2$ and $\Lambda =0$, for four sets of other parameters (see text) and various temperatures. The energy gap is reduced significantly as temperature increases and the energies of the gap-edge bound states are shifted together with the energy gap.

Figure 19

(Color online) The angle-resolved LDOS $N_{\sigma }(ϵ,p_y=0,y)$ as a function of energy $ϵ$, for quasiparticles moving in the $x$ direction at various positions along the $y$ axis (from $y=-18{\xi }_{\sigma }$ to $y=18{\xi }_{\sigma }$ with an increment of 3${\xi }_{\sigma }$). The parameter values are ${\xi }_{\pi }∕{\xi }_{\sigma }=5$, $T∕T_c=0.5$, $n_{\pi }∕n_{\sigma }=1$, $\Lambda =-0.1$, with (a) ${\rho }_0=0.3$ and (b) ${\rho }_0=0.5$. An increase in ${\rho }_0$ results in more gap-edge bound states and enhancement of their dispersion.

Figure 20

(Color online) The energy of the bound states as a function of $y$ obtained from Fig. 19. There are two branches of gap-edge bound states for ${\rho }_0=0.3$. For ${\rho }_0=0.5$ there is a third branch in the vortex core: the lowest branch is extended over large distances and its dispersion is enhanced considerably compared with the case for ${\rho }_0=0.3$.

Figure 21

(Color online) Magnitude of the supercurrent density in the two bands, $j_{\pi }$ and $j_{\sigma }$, as a function of distance $r$ from the vortex center, for $n_{\pi }∕n_{\sigma }=1$ and $\Lambda =0$, for various temperatures. In the left-hand panels, we have ${\rho }_0=0.1$, for (a) ${\xi }_{\pi }∕{\xi }_{\sigma }=1$ and (b) 3. The right-hand panels have ${\rho }_0=0.5$, for (c) ${\xi }_{\pi }∕{\xi }_{\sigma }=1$ and (d) 3. The $\pi$-band contribution can be substantial, or even dominating, and have strong temperature dependence far outside the vortex core for large enough ${\rho }_0$ and ${\xi }_{\pi }∕{\xi }_{\sigma }$.