Resonant and crossover phenomena in a multiband superconductor: Tuning the chemical potential near a band edge

Resonances in the superconducting properties, in a regime of crossover from BCS to mixed Bose-Fermi superconductivity, are investigated in a two-band superconductor where the chemical potential is tuned near the band edge of the second miniband generated by quantum confinement effects. The shape resonances at $T=0$ in the superconducting gaps (belonging to the class of Feshbach-like resonances) is manifested by interference effects in the superconducting gap at the first large Fermi surface when the chemical potential is in the proximity of the band edge of the second miniband. The case of a superlattice of quantum wells is considered and the amplification of the superconducting gaps at the Lifshitz transition of the type neck-collapsing of Fermi surface topology is clearly shown. The results are found to be in good agreement with available experimental data on a superlattice of honeycomb boron layers intercalated by Al and Mg spacer layers.

Figure 1

(Color online) Pictorial view of the evolution of the Fermi surfaces (FS) for the superconductor made of nonhybridized two-bands by moving the chemical potential from panel (a) to panel (d) so that it crosses two Lifshitz electronic topological transitions (ETT). The first ETT occurs moving $E_F$ across the band edge $E_{edge}$ of the second band so that the superconductivity goes from the single FS in panel (a) with a single condensate to the two FS in panel (b) with two condensates: the first (1) has a 2D topology and the second (2) has a 3D topology. Here, $\xi$ is the transversal band dispersion. Changing $E_F$ the system crosses the critical energy $E_{3\text{D-}2\text{D}}$ where the second FS undergoes a 3D-2D ETT shown in panel (b) changing its topology: the second closed 3D FS [panel (b)] becomes the tubular 2D FS in panel (d). The first large 2D FS (1) remains nearly constant when the chemical potential is moved.

Figure 2

(Color online) The crossover regime reached by decreasing the charge density in the superlattice from the two-band BCS superconductor to the mixed Bose-Fermi regime at the lover band edge $E_{2,L}$, as a function of the electron charge density $\left(\rho -{\rho }_{edge}\right)/{\rho }_c$, where ${\rho }_{edge}$ $\left({\rho }_{edge}+{\rho }_c\right)$ is the charge density where $E_F$ is tuned at $E_{edge}$ $\left(E_{3\text{D-}2\text{D}}\right)$. Panel (a). The ratio of the DOS of the second miniband $N_2$ and the DOS of the first miniband $N_1$. Upper part of panel (b): the relative variation of the Lifshitz parameter $\left({\mu }_n-E_{2,L}\right)/\xi$ as a function of electron charge; lower part panel (b): the variation of the chemical potential, $\left({\mu }_s-{\mu }_n\right)/{\mu }_n$, as a function of charge density $\rho$, where ${\mu }_n$ and ${\mu }_s$ are the chemical potentials in the normal and superconducting phase, $E_{2,L}$ is the lower band-edge energy of the second miniband.

Figure 3

(Color online) Density histograms of the pairing interaction matrix elements $I_{\mathbf{k},{\mathbf{k}}^{\prime }}^{\ell ,{\ell }^{\prime }}$ between the first and the second minibands, for the case with miniband dispersion $\xi ={\omega }_0$. Histograms 11, 22, 21, and 12, shows, respectively, the pairing interaction matrix elements $I_{\mathbf{k},{\mathbf{k}}^{\prime }}^{1,1}$, $I_{\mathbf{k},{\mathbf{k}}^{\prime }}^{2,2}$, $I_{\mathbf{k},{\mathbf{k}}^{\prime }}^{2,1}$, and $I_{\mathbf{k},{\mathbf{k}}^{\prime }}^{1,2}$.

Figure 4

(Color online) Calculation of the gap anisotropy, i.e., the dependence of the gap in the first miniband, with $c_{22}/c_{11}=2.17$, $c_{11}=0.22$ and $c_{12}/c_{11}=0.43$ as function of the wave vector $k_z$ in the transversal direction. We show in the figure two characteristics cases with different values of the averaged gap obtained solving the BCS-like equations at finite temperatures $T=5.7\text{K}$ (solid line) and $T=27.2\text{K}$ (dashed line) in the first miniband.

Figure 5

(Color online) The evolution of the gap, averaged in the $k_z$ direction, in the first miniband [panel (a)] and in the second one [panel (b)] as a function of the Lifshitz parameter, with $c_{22}/c_{11}=0.45$ and $c_{11}=0.22$. The ratio of the gaps as a function of the Lifshitz parameter is reported in panel (c). Several interband coupling ratio cases are reported: curve 1 (solid line) shows the case for $c_{12}/c_{11}=-2.73$, curve 2 (dashed line) shows the case for $c_{12}/c_{11}=-1.59$, curve 3 (dotted line) shows the case for $c_{12}/c_{11}=-1.04$, curve 4 (dot-dashed line) shows the case for $c_{12}/c_{11}=-0.91$ and curve 5 (solid line with open squares) shows the case for $c_{12}/c_{11}=-0.68$. The curve 3 (dotted line) shows the critical case of interband pairing $-c_{12}^{critical}=\sqrt{c_{11}{c}_{22}}/0.656$ where the gap ratio remains constant going from the two-gap BCS Fermi case to the mixed Bose-Fermi regime at the band edge.

Figure 6

(Color online) The evolution of the gap, averaged in the $k_z$ direction, in the first miniband [panel (a)] and in the second one [panel (b)] as a function of the Lifshitz parameter, with $c_{22}/c_{11}=2.17$ and $c_{11}=0.22$. The ratio of the gaps as a function of the Lifshitz parameter is reported in panel (c). Several interband coupling ratio cases are reported: curve 1 (solid line) shows the case for $c_{12}/c_{11}=-6.09$, curve 2 (dashed line) shows the case for $c_{12}/c_{11}=-4.34$, curve 3 (dotted line) shows the case for $c_{12}/c_{11}=-2.24$, curve 4 (dot-dashed line) shows the case for $c_{12}/c_{11}=-1.08$ and curve 5 (solid line with open squares) shows the case for $c_{12}/c_{11}=-0.43$. The curve 3 (dotted line) shows the critical case of interband pairing $-c_{12}^{critical}=\sqrt{c_{11}{c}_{22}}/0.656$ where the gap ratio remains constant from the two-gap BCS Fermi case to the mixed Bose-Fermi regime at the band edge.

Figure 7

(Color online) Panel (a): the variation of the critical chemical potential (solid line with filled circles) as a function of intraband coupling ratio $c_{22}/c_{11}$. We observe that for the intraband coupling ratio approaching unity the critical chemical potential goes to infinity. Panel (b): the variation of the critical interband pairing as a function of $\sqrt{c_{11}{c}_{22}}$. We observe a linear fit $\sqrt{c_{11}{c}_{22}}=-0.656c_{12}^{critical}$ (solid line) of calculated $c_{12}^{critical}$ points for the cases of $\sqrt{c_{11}{c}_{22}}$ (open circles) that divide the plane space in two regions: the upper one, for $\sqrt{c_{11}{c}_{22}}>-c_{12}^{critical}$, in weak interband regime, where $\partial \left(-{\Delta }_2/{\Delta }_1\right)/\partial \mu >0$ and the lower one, for $\sqrt{c_{11}{c}_{22}}<-c_{12}^{critical}$, in strong interband regime, where $\partial \left(-{\Delta }_2/{\Delta }_1\right)/\partial \mu <0$.

Figure 8

(Color online) The case of strong coupling in the second band (the so-called diboride case) with $c_{22}/c_{11}=2.17$ and $c_{11}=0.22$. The variable ratio $u=N_2/N_1$ (solid black line) is shown in panel (a). The critical temperature $T_c$ [panels (a) and (b)], the ratio $2{\Delta }_1/T_c$ [panel (c)], the ratio $-2{\Delta }_2/T_c$ [panel (d)], as functions of the reduced Lifshitz parameter $\left(\mu -E_{2,L}\right)/{\omega }_0$. Several values of the interband coupling ratio are reported: curve 1 (solid line) shows the case for $c_{12}/c_{11}=-6.09$, curve 2 (dashed line) shows the case for $c_{12}/c_{11}=-4.34$, curve 3 (dotted line) shows the case for $c_{12}/c_{11}=-2.24$, curve 4 (dot-dashed line) shows the case for $c_{12}/c_{11}=-1.08$ and curve 5 (solid line with open squares) shows the case for $c_{12}/c_{11}=-0.43$. For the largest value of $c_{12}$ considered here, the critical temperature reaches 300 K, in the range $1<\left(\mu -E_{2,L}\right)/{\omega }_0<2$. At the antiresonance for $\left(\mu -E_{2,L}\right)/{\omega }_0=0$, $T_c$ and $2{\Delta }_1/T_c$ reach the minimum value for weak interband interaction. On the contrary, for large interband interaction, $2{\Delta }_1/T_c$ and $T_c$ reach the minimum value at $\left(\mu -E_{2,L}\right)/{\omega }_0=-1$, where $T_c$ reaches room temperature for the chosen set of parameters.

Figure 9

(Color online) Panel (a): The superconducting critical temperature in Al for Mg substitution $\left({\text{Mg}}_{1-x}{\text{Al}}_x\right){\text{B}}_2$, (open circles),[53] in Sc for Mg substitution $\left({\text{Mg}}_{1-x}{\text{Sc}}_x\right){\text{B}}_2$,(open triangles)[56] and for the C for B substitution in the $\text{Mg}{\left({\text{C}}_x{\text{B}}_{1-x}\right)}_2$ system (open squares)[58] as function of the Lifshitz parameter $“z”=\left(E_{\Gamma }-E_F\right)/\left(E_A-E_{\Gamma }\right)$ for diborides.[53, 56, 58] Panel (b). The gaps ${\Delta }_1={\Delta }_{\sigma }$ (open symbols) and ${\Delta }_2={\Delta }_{\pi }$ (filled symbols) in Al for Mg substitution $\left({\text{Mg}}_{1-x}{\text{Al}}_x\right){\text{B}}_2$,[59] and for the C for B substitution in the$\text{Mg}{\left({\text{C}}_x{\text{B}}_{1-x}\right)}_2$ system[64] as function of the Lifshitz parameter $“z”=\left(E_{\Gamma }-E_F\right)/\left(E_A-E_{\Gamma }\right)$ calculated for aluminum[53, 54] and carbon[57] substitutions based on theoretical calculations and structural data. Panel (c). The BCS ratios curves $2{\Delta }_1/T_c$ (solid line) and $-2{\Delta }_2/T_c$ (dashed line), as functions of $T_c$, extracted from the experiments measuring in the same run the two gaps well below the critical temperature and $T_c$.[59, 60, 61, 62, 63] Carbon doped samples,[64] (open circles). Al doped samples from Daghero et al.[59] (open squares), and from Samuely et al.[61] (triangles). The experimental data are compared with the calculated curves in the case $c_{22}/c_{11}=2.17$, $c_{12}/c_{11}=-0.43$, where $-c_{12}=0.10$ is lower than the value $\left(-c_{12}^{critical}\right)=0.52$, where the critical temperature, and the gaps are calculated as a function of $\left(\mu -E_{2,L}\right)/{\xi }_0$.