Band Structure and Oscillatory Electron-Phonon Coupling of Pb Thin Films Determined by Atomic-Layer-Resolved Quantum-Well States

Using a low temperature growth method, we have prepared atomically flat Pb thin films over a wide range of film thickness on a Si-(111)-$7\times 7$ surface. The Pb film morphology and electronic structure are investigated in situ by scanning tunneling microscopy and angle-resolved photoemission spectroscopy. Well-defined and atomic-layer-resolved quantum-well states of the Pb films are used to determine the band structure and the electron-phonon coupling constant ($\lambda$) of the films. We found an oscillatory behavior of $\lambda$ with an oscillation periodicity of two atomic layers. Almost all essential features in the $\mathrm{Pb}/\mathrm{Si}(111)$ system, such as the growth mode, the oscillatory film stability, and the 9 monolayer envelope beating pattern, can be explained by our results in terms of the electron confinement in Pb films.

Figure 1

Room temperature STM images ($2000\mathrm{nm}\times 2000\mathrm{nm}$) of the Pb films at the completion of 13 ML (a) and 14 ML (b). All stable films exhibit essentially the same morphology as that shown in (a), and are (111) oriented. In (b), the letter “$\mathrm{A}$” indicates the 2 ML height islands on top of the 13 ML film labeled with the letter “$\mathrm{B}$.” (c) Normal emission spectra measured at 110 K. The successive spectra have been graphically offset for clarity. Binding energy at 0.0 eV corresponds to the Fermi level.

Figure 2

(a) Binding energy of the QWS as a function of Pb film thickness (the shadowed squares and diamonds correspond to the experimental and the fitting results, respectively). For each measured “branch” of QWS in the photoemission data, the reduced quantum member $P$ is indicated at the right panel of the figure. The inserted panel shows the experimentally determined energy band $E(k)$ along the $\Gamma L$ direction in the first Brillouin zone. (b) Calculated relative surface energy as a function of the film thickness.

Figure 3

Temperature-dependent photoemission spectra (75–300 K) obtained from both even (22 ML) (a) and odd (23 ML) (b) layered films.

Figure 4

(a) Lorentzian peak widths of the QWS of the 22 (square) and 23 ML (triangle) films as a function of the sample temperature. (b) Measured $\lambda$ (triangles) as a function of the film thickness, and the calculated superconductivity transition temperature (diamonds) by the formula $T_c=\frac{{\Theta }_D}{1.45}\exp [-\frac{1.04(1+\lambda )}{\lambda -{\mu }^{*}-0.62\lambda {\mu }^{*}}]$ [24]. In the calculation, typical values of 0.1 for ${\mu }^{*}$ (the effective Coulomb interaction) and 105 K for ${\Theta }_D$ (the Debye temperature) were used. As a comparison, the experimental transition temperature of the Pb films covered with 4 ML Au is also displayed (balls).