Finite-size nanowire at a surface: Unconventional power laws of the van der Waals interaction

We study the van der Waals interaction of a metallic or narrow-gap semiconducting nanowire with a surface, in the regime of intermediate wire-surface distances $(v_F/c)L\ll d\ll L$ or $L\ll d\ll (c/v_F)L$, where $L$ is the nanowire length, $d$ is the distance to the surface, and $v_F$ is the characteristic velocity of nanowire electrons (for a metallic wire, it is the Fermi velocity). Our approach, based on the Luttinger liquid framework, allows one to analyze the dependence of the interaction on the interplay between the nanowire length, wire-surface distance, and characteristic length scales related to the spectral gap and temperature. We show that this interplay leads to nontrivial modifications of the power law that governs van der Waals forces, in particular to a nonmonotonic dependence of the power-law exponent on the wire-surface separation.

Figure 1

Schematic view of the model considered: a finite-length nanowire immersed in the medium with the dielectric constant ${\varepsilon }_m$, interacting with the substrate with the dielectric constant ${\varepsilon }_s$.

Figure 2

Typical zero-temperature vdW interaction energy calculated via the full diagonalization of the matrix $M$, Eq. (14) (symbols), compared to the results of using the diagonal approximation (15). The parameters taken are $b_1=b_2=1$.

Figure 3

Zero-temperature vdW interaction energy at $b_1=b_2=1$, expressed in units of $\hslash v_F/L$ as a function of the dimensionless distance $d/L$, for several values of the nanowire length $L$ and radius $r$, with the elongation factor $L/r$ kept fixed.

Figure 4

Plots of scaling functions $f\left(b_2\right)$ and $g\left(b_2\right)$, defined by Eqs. (24) and (29), respectively, for several values of the elongation factor $L/r$.

Figure 5

Behavior of the vdW power-law exponent defined by Eq. (33) as a function of the wire-surface separation $d$, for different values of the temperature $T$ and the spectral gap $\mathrm{\Delta }$ expressed in units of $\hslash v_F/L$. Nonmonotonic behavior of $\alpha$ is suppressed with the increase of the gap.