Nonlocal and quantum-tunneling contributions to harmonic generation in nanostructures: Electron-cloud-screening effects

Our theoretical examination of second- and third-harmonic generation from metal-based nanostructures predicts that nonlocal and quantum-tunneling phenomena can significantly exceed expectations based solely on local, classical electromagnetism. Mindful that the diameter of typical transition-metal atoms is approximately 3 Å, we adopt a theoretical model that treats nanometer-size features and/or subnanometer-size gaps or spacers by taking into account (i) the limits imposed by atomic size to fulfill the requirements of continuum electrodynamics, (ii) spillage of the nearly free electron cloud into the surrounding vacuum, and (iii) the increased probability of quantum tunneling as objects are placed in close proximity. Our approach also includes the treatment of bound charges, which add crucial, dynamical components to the dielectric constant that are neglected in the conventional hydrodynamic model, especially in the visible and UV ranges, where interband transitions are important. The model attempts to inject into the classical electrodynamic picture a simple, perhaps more realistic description of the metal surface by incorporating a thin patina of free electrons that screens an internal, polarizable medium.

Figure 1

Top left: Scale illustration of $d$ and $s$ orbitals within a single metal atom obtained via Hartree-Fock theory. The radii of the $d$ and $s$ orbitals, $r_d$ and $r_s$, respectively, correspond to the maxima of each calculated electronic wave function. Parameters for Au: $r_d$ = 0.64 Å and $r_s$ = 1.56 Å; Ag: $r_d$ = 0.55 Å and $r_s$ = 1.53 Å; Cu: $r_d$ = 0.33 Å and $r_s$ = 1.37 Å [61, 62], and $t$ ∼ 1 Å. Bottom right: The electron cloud composed of outer $s$-shell electrons permeates the entire volume. At the same time, free electrons that belong to atoms near the surface spill outside and screen the hard, ionic background.

Figure 2

(a) Metal array composed of a mixture of free and bound charges. $d$ and $s$ orbitals overlap, so that bound electron orbits graze the surface. (b) The illustration of Fig. 1 yields a picture where an outer, free-electron shell approximately 2 Å thick covers each nanowire. The green particles inside the gap region of width $g$ represent tunneling electrons.

Figure 3

Thin Curves: Reflection, transmission, and absorption vs. wavelength for the gold-silver nanowire array in Fig. 2, where free and bound charges extend all the way to the surface. Thick Curves: Reflection, transmission, and absorption for the gold nanowire array in Fig. 2, where bound charges are covered by a shallow, free-electron layer.

Figure 4

Differential, free charge density distribution $\delta n$ for (a) local and (b) nonlocal cases. In (b) the metal surface is more elastic compared to (a), and can accommodate a smoother, but lower-amplitude charge distribution around the 2-Å-thick nanoshell. Peak values of $\delta n$ are larger (by nearly a factor of 4) and sharper in (a) compared to (b); this leads to larger SHG conversion efficiencies in the local case (a) compared to the nonlocal case (b). The distribution in (a) is numerically noisier compared to (b) because the field derivatives are very close to zero, giving rise to unphysical fluctuations inside the volume that have been averaged out.

Figure 5

Electric field intensity distributions between two adjacent Au-Ag nanowires without (a), (c), (e) and with (b), (d), (f) quantum-induced linear and nonlinear currents.