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

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.

Figures (4)

• Figure 1: A conceptual drawing of what is being calculated. An electric field with polarization shown in Z is incident on a (001) or (111)-terminated slab of diamond. The current density in the diamond is then calculated and used to monitor SHG.
• Figure 2: Plot of coupling depth and |$\mathcal{B}$/$\mathcal{U}$| as a function of energy. The left axis is the effective coupling depth according to the formalism of Mizrahi and Sipe Mizrahi1988SurfaceSHG. The right axis is the value of the ratio of |$\mathcal{B}$/$\mathcal{U}$| Freund1969PRL. Points of interest are marked where black points represent wavelengths corresponding to important distances in diamond and the red circle corresponds to |$\mathcal{B}$/$\mathcal{U}$| being unity. The marked points use the coherence length with distances associated with the (111) layer separation, the (001) layer separation and the distance between two adjacent carbon atoms denoted with a black circle, triangle and square respectively. The yellow region corresponds to where one therefore might expect bulk contributions to become dominant.
• Figure 3: (a) Ratio of total $\chi^{(2)}_{ZZZ}$ for 8 active layers versus 2 layers in both (001)-terminated (red circles) and (111)-terminated (green squares) diamond slabs [left axis]. Ratio of $\chi^{(2)}_{XZZ}$ versus $\chi^{(2)}_{ZZZ}$ for 8 active layers in a (001)-terminated diamond slab (blue diamond) [right axis]. Lines connect adjacent points. (b) Calculated $\chi^{(2)}_{ZZZ}$ as a function of energy for 8 active layers in a (001)-terminated diamond slab (red circles), $\chi^{(2)}_{ZZZ}$ for a periodic bulk of diamond in its cubic cell illuminated from the $\langle$001$\rangle$ direction (blue squares), and $\chi^{(2)}_{XZZ}$ for 8 active layers in a (001)-terminated diamond slab (green diamonds). Note, $\chi^{(2)}_{XZZ}$ should be zero in the dipole limit for the (001)-terminated diamond surface with $4m$ symmetry.
• Figure 4: (a) $\chi^{(2)}_{ZZZ}$ for 2 active layers at variable depths as a function of the fundamental pulse's energy. The topmost layer is defined as layer 1. The top 2 layers (red circles), layers 3 & 4 (blue squares), layers 5 & 6 (green diamonds) and layers 7 & 8 (purple triangles) are shown for energies from 300--3000 eV. Active layers are defined as those with the C $1s$ state in their valence. (b) Ratio of $\chi^{(2)}_{ZZZ}$ for active layers 1 & 2, 3 & 4, 5 & 6 and 7 & 8 normalized to the deepest active layers (7 & 8) as a function of the incident pulses' fundamental energy. Energies shown are 300 eV (red circles), 350 eV (blue squares), 1000 eV (green diamonds) and 3000 eV (purple triangles). See text for details of these calculations and their meanings.