Influence of surface electronic structure on quantum friction between Ag(111) slabs

The theoretical formulation developed in Phys. Rev. B 83, 205424 (2011) is applied to explore how surface electronic structure modifies the noncontact friction force between two silver slabs. We find that the friction shows three distinct regimes for different thicknesses. For very thin slabs $N=1–7$ monolayers (ML) quantum friction vs velocity shows oscillations which are due to the quantum size effect. At about $10ML$ friction rapidly increases because the surface state energies fall below $E_F$, which opens a new intraband electron-hole excitation channel. And for $N>15$ the friction is strongly enhanced due to the excitation of acoustic surface plasmons.

Figure 1

Process in which charge density fluctuation is created in the slab 1 and induces a potential in slab 2.

Figure 2

Quantized electronic bands in a $7ML$-thick silver slab. Left (right) side presents bands obtained in the jellium (Chulkov) model. Red (gray) lines highlight the even and odd energy bands evolving, upon the slab thickness increase, into a partially occupied $s-p$ surface state of the semi-infinite Ag(111) crystal.

Figure 3

Energies at $K=0$ of even surface state (black dots) and odd surface state (red [gray] squares) in the Chulkov model and topmost occupied state in the jellium model (blue [gray] diamonds) as functions of metallic slab thickness.

Figure 4

Spectral function overlap for different relative velocities; (a) $v=0$, (b) $v=0.2v_F$, (c) $v=0.4v_F$, and (d) $v=0.6v_F$. Film thicknesses, in silver atomic monolayer units, are $L_1=L_2=8ML$, which corresponds to $1.5$ nm. Parallel wave vector is $Q=0.02$ a.u. Series of peaks correspond to intraband electron-hole transitions within discrete bands.

Figure 5

Intensities of the low-lying electronic excitations in silver slabs of different thicknesses, in the jellium model. The color intensity is in arbitrary units and scales linearly as 0, black; 0.2, light blue; 0.4, light green; 0.6, red; 0.8, yellow; and 1.0, white.

Figure 6

Intensities of the low-lying electronic excitations for different Ag(111) slab thicknesses (8, 9 and $10ML$) in the (a) jellium and (b) Chulkov models. The color intensity is in arbitrary units and scales linearly as 0, black; 0.2, light blue; 0.4, light green; 0.6, red; 0.8, yellow; and 1.0, white.

Figure 7

Friction force between jellium silver slabs in relative motion as function of relative velocity $v$ with parameters given in the text. Red arrows show the position of corresponding Fermi velocities sums $v_{Fn}+v_{Fm}$.

Figure 8

The positions of the first (black squares) and second (brown [dark gray] diamonds) maxima in the friction force as functions of metallic film thickness. Red (gray) dots show the positions of $2v_{F1}$, and blue (gray) triangles the position of $v_{F1}+v_{F2}$ as functions of film thickness.

Figure 9

The friction force as function of silver film thickness in jellium model (black squares) and Chulkov model potential (red [gray] dots).Relative velocity is $v=0.1v_F$. Insert presents in more details the data in the $1–10ML$ range.

Figure 10

Friction force between two semi-infinite silver slabs in the jellium model (black dashed line) and in the Chulkov model potential (black solid line). Vertical red (gray) dashed line denotes the position of $v=2v_{ASP}$.