Tailoring Superconductivity with Two-Level Systems

Abstract

We investigate the impact of two-level systems (TLSs) on superconductivity, treating them as soft modes localised in real space. We show that these defects can either enhance or suppress the superconducting critical temperature, depending on their surface density and average frequency. Using thin-film aluminium as a case study, we quantitatively describe how TLSs modify both the critical temperature and the zero-temperature superconducting gap. Our results thus highlight new opportunities for tailoring material properties through TLS engineering.

Figures (5)

• Figure 1: Two-level systems (TLSs) on (a) the oxidised top surface and (b) the oxidised top and bottom surfaces of an Al sample, raising the local superconducting gap $\Delta$.
• Figure 2: Change in $T_c/T_c^{(0)}$ versus TLS (vibron) frequency $\omega_{TLS}$ normalized by Einstein frequency $\omega_E$ and with $\lambda=0.3636$ held fixed. An Einstein boson model is used for both the bulk phonon and vibron, and $T_c$ is found via the Leavens-Carbotte formula Leavens2011Feb. The hatched cyan region corresponds to the frequency regime of amplified $T_c$, which becomes more pronounced for weaker coupling. The black arrow marks the maximum $T_c$, located at $\omega_{TLS}^*/\omega_E\approx 0.38$. Inset: Comparison of $\omega_{\ln}/\omega_{\ln}^{(0)}$ (red) and $\lambda/\lambda^{(0)}$ (blue) as $\omega_{TLS}$ changes for $n_{TLS}=0.3$.
• Figure 3: (a) Electron-phonon spectral function in bulk (top) and multi-layered Al found via first principles. In the monolayer limit, more spectral weight is pushed to lower frequencies. The green hatched region corresponds to the regime potentially populated by TLSs ($\omega<4$ meV). (b) Second derivative of the experimental I-V data taken on an epitaxially-grown thin film of Al $3.9$ monolayers thick vanWeerdenburg2023Mar (black) overlain with the Gaussian-broadened Dynes fit (red-dashed). The voltage regime characterized by localised dips in $d^2I/dV^2$ is highlighted in green hatch.
• Figure 4: TLS-induced modification of (a) the electron-phonon coupling, (b) the log average phonon frequency, and (c) the ratio of $T_c$ (with TLSs) to $T_c^{(0)}$ (without TLSs) in layered Al. Values for $T_c$ are found via the Leavens-Carbotte eqn. Leavens1974Jan with $\alpha^2F(\nu)$ found via first principles. The coloured error bars correspond to variations of $T_c$ upon changing the TLS density $n_{TLS}$ and the TLS frequency $\omega_{TLS}$, with the coloured shadow representing the spread of data. We use flat top distribution of $\omega_{TLS}$ in the range of 0.15 to 4 mev and $n_{TLS}$ in the range 0.01 to 0.3 in generating these plots. As we approach the single monolayer limit, $\lambda$ grows while $\omega_{\ln}$ shrinks. The net result is larger variation of $T_c$ in the single monolayer limit. As discussed in the main text, the suppression of $T_c/T_c^{(0)}$ in the analytic solution is an artifact of the constant gap approximation taken by Leavens and Carbotte Bergmann1973Aug.
• Figure 5: TLS-induced modification of (a) the critical temperature $T_c$ and (b) the gap edge $\Delta_1$. Both $T_c$ and the gap edge are found by numerically solving the Eliashberg equations. Error bars correspond to changes in the respective quantities upon introducing TLSs, and the dots correspond to average values. TLS parameters are those found in the previous section that led to maximised enhancement and suppression of $T_c/T_c^{(0)}$ when calculated numerically. As compared to the analytical calculation, TLSs tend to enhance (as opposed to suppress) the critical temperature. The change in the local gap edge $\Delta_1/\Delta_1^{(0)}$ follows a similar trend as the critical temperature, except with a more pronounced enhancement. Larger relative enhancement of the gap edge is expected, as the gap-to-$T_c$ ratio is expected to be locally enhanced from the amplified $\lambda$ induced by the TLSs Carbotte1987Carbotte1986May.