Quantitative microscopic theory of superconductivity in one-atomic-layer Pb and In films on Si(111): mode-resolved electron-phonon coupling, phase-dependent $T_{\mathrm{C}}$, and disorder/fluctuation limits

Scanning tunnelling spectroscopy established that single ordered atomic layers of Pb and In epitaxially grown on Si(111) are superconducting, with transition temperatures of 1.83 K in the dense striped-incommensurate (SIC) Pb phase, 1.52 K in the $\sqrt{7}\times\sqrt{3}$-Pb phase and 3.18 K in the $\sqrt{7}\times\sqrt{3}$-In phase, and with an electron-phonon coupling constant $\lambda = 1.07 \pm 0.13$ measured by ARPES and attributed, by inference, to metal-Si interface phonon modes. The open problem is to give a quantitative, mode-resolved microscopic account of this superconductivity: identify which phonon branches (intralayer metal-metal versus interface/substrate modes) and coupling matrix elements produce the measured $\lambda$, gap and $T_{\mathrm{C}}$ in each phase; predict $T_{\mathrm{C}}$, gap and their variation with reconstruction and coverage from first principles (including the metal-Si interface); explain why the denser-packed SIC phase superconducts more strongly than the $\sqrt{7}\times\sqrt{3}$-Pb phase; and determine how disorder, finite size and superconducting phase fluctuations limit $T_{\mathrm{C}}$ and phase coherence in the single-atom layer. Later work has partially addressed pieces of this, with mutually conflicting outcomes: density-functional-perturbation-theory calculations for SIC-Pb give $\lambda$ = 0.72 and an Allen-Dynes $T_{\mathrm{C}}$ of about 1.5 K with the Si substrate frozen and spin-orbit coupling neglected; a parameter-free full-phonon superconducting density functional theory calculation yields a $T_{\mathrm{C}}$ well above experiment and attributes the suppression to order-parameter fluctuations; and a mode-resolved spin-orbit-coupled calculation for the dense (4/3)-monolayer Pb phase finds that spin-orbit interaction reduces the momentum-averaged $\lambda$ to about 0.26, that Pb-localized rather than substrate modes dominate $\alpha^2F(\omega)$, and yet recovers $T_{\mathrm{C}}$ = 1.84 K when the experimentally inferred Fermi-level position is used. Mode-resolved electron-phonon data exist for related Pb nanofilms and dense Pb monolayers; and disorder/step studies have mapped how atomic-scale defects and steps degrade or segment monolayer superconductivity. A reconciling, branch-resolved, quantitative theory consistent with all measured quantities for all superconducting monolayer phases does not yet exist.

Background

Superconductivity in a single atomic layer is the ultimate two-dimensional limit of crystalline superconductors: in two dimensions the superconducting condensate is fragile against quantum and thermal phase fluctuations, and prior experiments had only demonstrated superconductivity in films of several atomic layers or more. Zhang, Cheng, Li, Sun, Wang, Zhu, He, Wang, Ma, Chen, Wang, Liu, Lin, Jia and Xue (Nature Physics 6, 104-108, 2010) demonstrated by low-temperature scanning tunnelling microscopy and spectroscopy (STM/STS, in which a sharp tip probes the local density of states of a surface) that ordered one-atomic-layer films of Pb and In grown epitaxially on the Si(111) substrate are genuinely superconducting, confirming the phase-coherent state by direct imaging of superconducting vortices under magnetic field. The relevant atomic structures are coverage-driven surface reconstructions (ordered arrangements of adatoms that minimize the interface energy): the striped incommensurate (SIC) Pb phase, a dense, nearly close-packed 4/3-monolayer phase of areal density 10.44 Pb atoms per nm^2, and the commensurate $\sqrt{7}\times\sqrt{3}$ phases of Pb and In at 9.40 metal atoms per nm^2. Measured values were: SIC-Pb gap 0.35 meV with $T_{\mathrm{C}}$ = 1.83 K, $\sqrt{7}\times\sqrt{3}$-Pb gap 0.27 meV with $T_{\mathrm{C}}$ = 1.52 K, and $\sqrt{7}\times\sqrt{3}$-In gap 0.57 meV with $T_{\mathrm{C}}$ = 3.18 K, each with a Bardeen-Cooper-Schrieffer (BCS)-like gap ratio near 4.1-4.4, an upper critical field of about 1450 G and a Ginzburg-Landau coherence length (the characteristic size of the superconducting order-parameter variation) of about 49 nm. In-situ angle-resolved photoemission spectroscopy (ARPES, which maps the electronic band dispersion) showed two-dimensional, free-electron-like metallic overlayer bands, and the temperature dependence of the ARPES quasiparticle linewidths - analysed with a Debye-model phonon spectrum - yielded the dimensionless electron-phonon coupling constant $\lambda = 1.07 \pm 0.13$ for SIC-Pb, enhanced relative to values below 0.9 in much thicker Pb films; a similar $\lambda$ of order 1 was found for the In monolayer. Here $\lambda$ is the standard mass-renormalization electron-phonon coupling constant, which in the Eliashberg theory of strong-coupled superconductivity is the Fermi-surface average of the electron-phonon matrix elements weighted by inverse phonon frequency, usually obtained by integrating the Eliashberg spectral function $\alpha^2F(\omega)$ (phonon density of states weighted by the squared coupling matrix elements); the ARPES value above is instead the state-dependent coupling of the probed occupied overlayer band, not this Fermi-surface average, so the two must not be conflated. The Fermi-surface-averaged $\lambda$, together with the logarithmically averaged phonon frequency $\omega_{\log}$, enters the McMillan/Allen-Dynes formula that estimates $T_{\mathrm{C}}$ given a Coulomb pseudopotential $\mu^*$. The source paper attributed the enhancement of $\lambda$ to phonon modes associated with the metal-Si covalent interface bonds, but this attribution was an inference: no mode-resolved spectroscopy, no isotope test and no first-principles calculation identifying the contributing vibrational branches was performed. Bulk Pb, for comparison, has $T_{\mathrm{C}}$ = 7.2 K and computed $\lambda$ near 1.4-1.55, and first-principles work on ultrathin Pb films had shown that electron-phonon coupling and $T_{\mathrm{C}}$ are suppressed as film thickness decreases. The same experimental study also showed that the superconductivity is insensitive to the bulk Si dopant concentration, so the pairing resides in the metallic overlayer and its interface. Two further concepts matter for what remains open. First, solving the Eliashberg equations (the strong-coupled gap equations that go beyond the McMillan/Allen-Dynes estimate), or equivalently using superconducting density functional theory (SCDFT, a parameter-free density-functional framework in which the gap equation is solved directly from computed electron-phonon and screened Coulomb interactions), is the standard route to a quantitative $T_{\mathrm{C}}$ prediction; in a quasi-two-dimensional, substrate-coupled monolayer the usual bulk approximations (isotropic coupling, bulk $\mu^*$, decoupling of substrate phonons) may fail. Second, in two dimensions, phase fluctuations (as captured by Berezinskii-Kosterlitz-Thouless physics, in which vortex-antivortex unbinding destroys quasi-long-range order at a finite stiffness-dependent temperature) and disorder can suppress or fragment superconductivity even when pairing survives, so the measured monolayer $T_{\mathrm{C}}$ may lie below the mean-field, strong-coupling value.

Problem Statement

Establish a quantitative microscopic theory of superconductivity in the one-atomic-layer Pb and In films grown on Si(111), covering the three experimentally characterized superconducting phases (the striped incommensurate 4/3-monolayer SIC-Pb phase with $T_{\mathrm{C}}$ = 1.83 K, the $\sqrt{7}\times\sqrt{3}$-Pb phase with $T_{\mathrm{C}}$ = 1.52 K, and the $\sqrt{7}\times\sqrt{3}$-In phase with $T_{\mathrm{C}}$ = 3.18 K). The jointly required results, which together support this single objective, are: (1) a mode-resolved microscopic identification of which phonon modes - intralayer Pb-Pb (or In-In) modes versus metal-Si interface and substrate modes, their frequencies and coupling matrix elements - quantitatively account for the measured state-dependent electron-phonon coupling constant $\lambda = 1.07 \pm 0.13$ (ARPES) and for the measured gaps and transition temperatures in each phase, going beyond the inference from the measured enhancement of $\lambda$; (2) a first-principles prediction of $T_{\mathrm{C}}$ and gap magnitude, and of their variation with surface reconstruction and coverage, including the metal-Si interface in the electron-phonon coupling, at a level that resolves the present discrepancy between the frozen-substrate Allen-Dynes estimate ($\lambda$ = 0.72, $T_{\mathrm{C}}$ = 1.5 +/- 0.6 K, in apparent agreement with experiment), the full-phonon ab initio result ($T_{\mathrm{C}}$ well above experiment, with fluctuations invoked as the suppressing agent), and the spin-orbit-coupled mode-resolved result (momentum-averaged $\lambda$ = 0.26 dominated by Pb-localized modes, with $T_{\mathrm{C}}$ = 1.84 K recovered only under an assumed Fermi-level position); (3) a microscopic explanation of the structural dependence of the superconductivity, namely why the denser-packed SIC-Pb phase superconducts more strongly than the $\sqrt{7}\times\sqrt{3}$-Pb phase and why the In monolayer reaches 3.18 K; and (4) a quantitative determination of how disorder, finite-size effects and superconducting phase fluctuations set the ultimate limits of $T_{\mathrm{C}}$ and phase coherence in a single-atom layer (for example the dependence on island size, controlled disorder, and substrate-induced strain). The theory must be quantitatively consistent with the scanning-tunnelling gap values, the vortex/upper-critical-field data, and the ARPES-measured $\lambda$ and band structure, with the disorder, finite-size and phase-fluctuation limits of (4) determined quantitatively rather than by fitted corrections. All four results are jointly required for this single objective.

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