Pair breaking of multigap superconductivity under parallel magnetic fields in the electric-field-induced surface metallic state

The roles of paramagnetic and diamagnetic pair-breaking effects in superconductivity in the electric-field-induced surface metallic state are studied using the Bogoliubov–de Gennes equation when magnetic fields are applied parallel to the surface. The multigap states of the subbands are related to the depth dependence and the magnetic field dependence of the superconductivity. In the Fermi-energy density of states and the spin density, subband contributions successively appear from higher-level subbands with increasing magnetic fields. The characteristic magnetic field dependence may be a key feature to identify the multigap structure of the surface superconductivity.

Figure 1

(a) $H$ dependence of the average $\langle \mathrm{\Delta }\rangle$ in cases (i) and (ii). Dashed line for (ii) indicates a possible first-order transition of $H_{\mathrm{c}}$. (b) Depth $z$ dependence of the pair potential $\mathrm{\Delta }\left(z\right)$ (thick solid line) and the subband decompositions (dashed lines) to $i_z=1,\cdots ,5$. $H=0$. (c) The same as (b), but $H=10$ in case (i) for only the paramagnetic pair-breaking effect. (d) The same as (b), but $H=10$ in case (ii) for both paramagnetic and diamagnetic pair-breaking effects. In (c) and (d), thin solid lines are for $H=0$ for comparison.

Figure 2

Depth $z$ dependence of the current. (a) Spin-dependent currents $J_{\uparrow }\left(z\right)$ and $J_{\downarrow }\left(z\right)$ at $H=6$. (b) $J_{\uparrow }\left(z\right)$ and $J_{\downarrow }\left(z\right)$ at $H=10$. (c) Total current $J\left(z\right)$ at $H=6$ and 10. (d) Spin current $J_{\mathrm{s}}\left(z\right)$ at $H=6$ and 10. The vertical axis is in units of $e\hslash /m^{*}$.

Figure 3

(a) Variation of internal field. Depth $z$ dependence of $\delta M\left(z\right)$ is presented at $H=6$ and 10. (b) $H$ dependence of $z_0\equiv q{\varphi }_0/\pi H$ to satisfy the current conservation. (c) $H$ dependence of magnetization $M$. The vertical axis is in units of $e\hslash /m^{*}$.

Figure 4

(a) Eigenenergy $E_{\epsilon }$ as a function of $E_{\parallel }$ for particle (up-spin) and hole (down-spin) branches of subbands $i_z=1,2,\cdots$ at $H=10$ in case (i). Fermi energy $E_{\mathrm{F}}$ corresponds to $E=0$. (b) Hole branches of down spin are converted to particle branches in order to show dispersion of up- and down-spin electrons for subbands $i_z$.

Figure 5

Electronic state at $H=10$. Left and right columns are, respectively, for cases (i) and (ii). (a) DOS $N\left(E\right)$ with the spin-resolved DOS $N_{\uparrow }\left(E\right)$ and $N_{\downarrow }\left(E\right)$. (b) Subband decompositions ($i_z=1,\cdots ,5$) of $N_{\uparrow }\left(E\right)$. (c) $H$ dependence of Fermi-energy DOS $N\left(E_{\mathrm{F}}\right)$ with the spin decompositions $N_{\uparrow }\left(E_{\mathrm{F}}\right)$ and $N_{\downarrow }\left(E_{\mathrm{F}}\right)$. $N\left(E_{\mathrm{F}}\right)$ in the normal state is presented by a straight line. (d) Subband decompositions of $N_{\uparrow }\left(E_{\mathrm{F}}\right)$ in (c).

Figure 6

(a) Depth $z$ dependence of local spin density $m\left(z\right)=n_{\uparrow }\left(z\right)-n_{\downarrow }\left(z\right)$ at $H=10$, 11, and 12 for case (i). (b) The same as (a), but for $H=8$, 10, and 11 for case (ii).

Figure 7

(a) $H$ dependence of total paramagnetic moment $M_s$ in cases (i) and (ii). The straight line is for $M_s$ in the normal state. (b) Subband decompositions ($i_z=1,\cdots ,5$) of $M_s$ in case (i) as a function of $H$. (c) The same as (b), but for case (ii).