Rise and fall of shape resonances in thin films of BCS superconductors

The confinement of a superconductor in a thin film changes its Fermi-level density of states and is expected to change its critical temperature $T_c$. Previous calculations have reported large discontinuities of $T_c$ when the chemical potential coincides with a subband edge. By solving the BCS gap equation exactly, we show that such discontinuities are artifacts and that $T_c$ is a continuous function of the film thickness. We also find that $T_c$ is reduced in thin films compared with the bulk if the confinement potential is lower than a critical value, while for stronger confinement $T_c$ increases with decreasing film thickness, reaches a maximum, and eventually drops to zero. Our numerical results are supported by several exact solutions. We finally interpret experimental data for ultrathin lead thin films in terms of a thickness-dependent effective mass.

Figure 1

BCS critical temperature in a thin film, modeled as a square potential of infinite depth, as a function of film thickness $L$. The Thompson–Blatt approximate result (TB) is compared with the exact result for a dimensionless 3D density $\tilde{n}=1$ and a coupling $\overline{\lambda }=1$. Similar results are obtained for other densities and couplings. The 3D critical temperatures are $k_{\mathrm{B}}{T}_c^{3\mathrm{D}}/\hslash {\omega }_{\mathrm{D}}=0.500$ and 0.436 in the TB and exact calculations, respectively. The band mass is used as the reference mass: $\tilde{L}=2^{1/3}{(m_b{\omega }_{\mathrm{D}}/2\pi \hslash )}^{1/2}L$.

Figure 2

Relative variation of the critical temperature with varying film thickness (curve 0) and its interpretation (curves 1 to 6); see text. The vertical bars at the top mark the film thicknesses where $\mu =E_q-\hslash {\omega }_{\mathrm{D}}$ for $q=2,3,4,5$.

Figure 3

(a) Evolution of the approximate (dashed-blue) and exact (solid-red) chemical potentials as a function of film thickness. The dimensionless density is $\tilde{n}=3$, the coupling is $\overline{\lambda }=1$, and all energies are measured from the lowest subband $E_1$, and expressed in units of $\hslash {\omega }_{\mathrm{D}}$. (b)–(g) Chemical potential in different ranges of film thickness and energies $E_q$ (thin solid lines), $E_q-\hslash {\omega }_{\mathrm{D}}$ (thin dashed), and $E_q+\hslash {\omega }_{\mathrm{D}}$ (dotted) for $q=1,...,6$. In (d) and (g), the gray line shows the $1/L$ behavior at large $L$. The horizontal axis in all graphs is $\tilde{L}=2^{1/3}{(m_b{\omega }_{\mathrm{D}}/2\pi \hslash )}^{1/2}L$.

Figure 4

Critical temperature as a function of film thickness for (a) various densities at fixed coupling and (b) various couplings at fixed density. $T_c$ is expressed in units of $\hslash {\omega }_{\mathrm{D}}/k_{\mathrm{B}}$ and $n$ in units of $2{[m_b{\omega }_{\mathrm{D}}/\left(2\pi \hslash \right)]}^{3/2}$. The dashed lines show the asymptotic behavior at low $L$, Eq. (11).

Figure 5

Characteristic length of the $T_c$ oscillations as a function of density and coupling, calculated numerically at large $L$. The Thompson–Blatt (TB) expression (7) and weak-coupling expression (8) are shown for comparison. The band mass is used as the reference mass $\left(m=m_b\right),\tilde{n}=n/\left[2{(m_b{\omega }_{\mathrm{D}}/2\pi \hslash )}^{3/2}\right]$, and $\mathrm{\Delta }\tilde{L}=2^{1/3}{(m_b{\omega }_{\mathrm{D}}/2\pi \hslash )}^{1/2}\mathrm{\Delta }L$.

Figure 6

(a) Thickness $L^{max}$ leading to the maximum $T_c$ normalized by the coupling $\overline{\lambda }$ and (b) maximum $T_c$ as a function of $\overline{\lambda }\tilde{n}$ for various values of $\overline{\lambda }$. The broken thick lines show the asymptotic formulas in the various regimes.

Figure 7

Relative change of the critical temperature as a function of film thickness for various strengths of the confinement potential. The dotted lines are calculated by neglecting the contribution of scattering states. (Inset) Critical potential as a function of $\overline{\lambda }\tilde{n}$ for various couplings. The thick gray line is the function $4.65{\left(A\overline{\lambda }\tilde{n}\right)}^{1/2}$ with $A=(3\sqrt{\pi }/8){(m/m_b)}^{3/2}$. The confinement potential is measured in units of $\hslash {\omega }_{\mathrm{D}}$. The band mass is used as the reference mass.

Figure 8

Maximum $T_c$ as a function of $\overline{\lambda }\tilde{n}$ for a confinement potential $\tilde{U}=10$. The dashed line is the result for $\tilde{U}=\infty$ and $\overline{\lambda }=2$ (Fig. 6). The gray line shows the variation of ${\tilde{T}}_c^{3\mathrm{D}}$ with $\overline{\lambda }$ at the critical density.

Figure 9

Superconducting transition temperature of Pb thin films as a function of film thickness $L$. The diamonds with error bars show the measurements of Ref. [25]; different colors correspond to different data sets. (a) Prediction of the model using parameters of bulk Pb with a strong (dotted line) and weak (dashed line) confinement potential. A running average is shown by the solid lines. (b) Interpretation of the data by means of thickness-dependent coupling parameter and band mass. The $L$ dependence of these quantities is shown in the inset.