The Surface Sensitivity of X-ray Second Harmonic Generation as a Function of Energy

TL;DR

This study investigates how the surface sensitivity of x-ray second harmonic generation in diamond varies with energy, revealing a transition from surface to bulk dominance as energy increases, using analytical and computational methods.

Contribution

It provides a detailed analysis of the energy-dependent surface sensitivity of SHG in diamond, incorporating full multipole expansion and crystal orientation effects.

Findings

Surface sensitivity is high near the C K-edge (~285 eV).

Bulk quadrupole response dominates SHG above 1000 eV.

Crystal orientation significantly affects surface sensitivity.

Abstract

The surface sensitivity and probe depth in the x-ray regime of diamond for second harmonic generation (SHG) was investigated both analytically and computationally with velocity gauge real-time time-dependent density functional theory (VG-RT-TDDFT), which includes a full multipole expansion. This was accomplished using two different approaches, by changing the number and location of layers that can generate SHG computationally and by exploiting the symmetry of a crystal, a similar pattern emerged. We find that by 1000 eV, well above the ~285 eV of the C $K$-edge, the SHG of diamond is dominated by the bulk, quadrupole response, in agreement with our analytic calculations. The bulk response continues to grow as the energy is increased, becoming overwhelming by 7000 eV. Near the C $K$-edge the measurement is quite surface sensitive, however, this surface sensitivity reduces as the energy increases such that by 1000 eV (and certainly by 3500 eV) SHG is largely bulk sensitive. Moreover, we find that the specific details of the crystal orientation (i.e., comparing a (001)-terminated and (111)-terminated surface) appear to have significant effects on the surface sensitivity.