Theoretical study of quantum size effects in thin Al(100), Al(110), and Al(111) films

We have carried out first-principles calculations for Al(100), Al(110), and Al(111) films up to 30 monolayers to study the oscillatory quantum size effects (QSEs) exhibited in the surface energy, work function, electron-phonon coupling constants, and superconductivity transition temperature $T_c$. Significant oscillatory QSEs on these physical properties are reported. These oscillations are correlated with the thickness dependence of the energies of confined electrons, which can be properly modeled by an energy-dependent phase shift of the electronic wave function upon reflection at the interface. The oscillations in the surface energy and work function for Al(111) and Al(110) films as a function of film thickness can be well fitted by a damped sinusoidal function with the periodicity determined by one Fermi wave vector along the [111] direction or a combination of three Fermi wave vectors along the [110] direction. A quantitative description of these QSE requires the full consideration of the crystal band structure.

Figure 1

Calculated energies of a quantum-well state at $\overline{\mathrm{\Gamma }}$ in (a) Al(100), (b) Al(110), and (c) Al(111) thin films as a function of thickness with the energy zero set at the Fermi level. The results are obtained at the theoretical lattice constant with fully relaxed layer separation. The right panel shows the bulk energy dispersion in the [100], [110], and [111] directions. The values of the Fermi vectors (refer to the $p$-band minimum at $\mathrm{\Gamma },X$, or $L$) in the right panels are in units of $\pi /d$, where $d$ is the ideal interlayer distance.

Figure 2

One-dimensional partial charge density of some quantum-well states at $\overline{\mathrm{\Gamma }}$ of Al(100) thin films (a) 8 monolayers and (b) 30 monolayers. The dashed vertical lines indicate the positions of Al planes. The index $n$ denotes the quantum number.

Figure 3

One-dimensional partial charge densities of the first 40 quantum-well states at $\overline{\mathrm{\Gamma }}$ for the Al(111) thin film of 25 monolayers. The dash vertical lines indicate the positions of Al planes. The index $n$ denotes the quantum number.

Figure 4

One-dimensional partial charge densities of the first 20 quantum-well states at $\overline{\mathrm{\Gamma }}$ for the Al(110) thin film of ten monolayers. The dashed vertical lines indicate the positions of Al planes. The indices $n,n^{\prime }$, and $n^{\prime \prime }$ denote the quantum numbers of three sets of QW states.

Figure 5

(a) Surface energy per $1\times 1$ unit cell and (b) work function of Al (100) thin films as a function of thickness. The calculated results (solid circles) are obtained at the theoretical lattice constant with layer spacings fully relaxed.

Figure 6

(a) Surface energy per $1\times 1$ unit cell and (b) work function of Al (110) thin films as a function of thickness. The open circles and dashed lines represent the fitted results using Eq. (3) and with a decay exponent of 2 and 1 for the surface energy and work function, respectively.

Figure 7

(a) Surface energy per $1\times 1$ unit cell and (b) work function of Al(111) thin films as a function of thickness. The open circles and dashed lines represent the fitted values using Eq. (2) and with a decay exponent of 2 and 1 for the surface energy and work function, respectively.

Figure 8

(a) EPC and superconducting critical temperature as a function of film thickness in the Al(111) films with ${\mu }^{★}=0.12$. The inset shows the thickness dependence of estimated $T_c$. The band structure of the four-monolayer and six-monolayer films is shown in (b) and (c), respectively.

Figure 9

(a) EPC and superconducting critical temperature as a function of film thickness in the Al(110) films with ${\mu }^{★}=0.12$. The inset shows the thickness dependence of estimated $T_c$. The band structure of the six-monolayer and eight-monolayer films is shown in (b) and (c), respectively.

Figure 10

The phase shift of the electronic wave function at the film-vacuum interface (see the text) as a function of the energy of quantum-well states in (a) Al(100), (b) Al(110), and (c) Al(111) thin films.

Figure 11

The inverse of energy spacing of quantum-well states near the Fermi level in (a) Al(100) and (b) Al(111) thin films as a function of thickness. The dashed line is a linear fit to the calculated values.