Exciton spectroscopy of hexagonal boron nitride using nonresonant x-ray Raman scattering

We report nonresonant x-ray Raman scattering (XRS) measurements on the boron $K$ edge of hexagonal boron nitride for transferred momentum $\left(\mathbf{q}\right)$ from 2 to $9{\text{Å}}^{-1}$ along directions both in and out of the basal plane. A symmetry-based argument, together with real-space full multiple scattering calculations of the projected density of states in the spherical harmonics basis, reveals that a strong pre-edge feature is a dominantly $Y_{10}$-type exciton with no other $s$, $p$, or $d$ components. This conclusion is supported by a second, independent calculation of the $\mathbf{q}$-dependent XRS cross section based on the Bethe-Salpeter equation. This study demonstrates methods which should be applicable to the determination of final-state symmetries for localized resonances in other $\mathbf{q}$-dependent XRS studies of anisotropic single-crystal systems.

Figure 1

(Color online) Inset: The crystalline structure of $h$-BN has two-dimensional hexagonal sheets stacked directly on top of each other, with the boron (black) and nitrogen (gray) always being nearest neighbors (Ref. 21). Main figure: The normalized XRS spectra from $h$-BN for $\mathbf{q}\parallel c$ (top panel) and $\mathbf{q}\perp c$ (bottom panel).

Figure 2

(Color online) Ab initio FEFF8 calculation of the $l\text{-DOS}$ onto different spherical harmonics with odd (top panel) and even (bottom panel) reflection parity about the basal plane for an $h$-BN cluster of 329 atoms.

Figure 3

(Color online) First-principles calculation of the absolute values of $\text{Im}\left[ϵ\left(\mathbf{q},\omega \right)\right]$ over the experimentally measured $q$ and $\omega$ ranges along both $\mathbf{q}\perp c$ and $\mathbf{q}\parallel c$ directions.

Figure 4

(Color online) Normalized $q$ dependence of the exciton intensity $S\left(q\right)/q^2$. Thick solid lines: Simple calculations of $S\left(q\right)/q^2$ [Eq. (7)] using the boron $1s$ core state and a hydrogenlike $2p_z$ final state with a $Z_{\text{boron}}+1$ core hole, and $ϵ=7$ and 1, respectively (Ref. 6). Thin lines: Theoretically calculated $S\left(q\right)/q^2$ for the exciton state, with either a delocalized final state $1s^{1}2s^{2}2p^1$ (dashed thin line) or a bound final state $1s^{1}2s^{2}2p^2$ (solid thin line). Solid squares: Integrated exciton intensity from BSE calculation (Fig. 3) from 190–194 eV. (Filled circles with error bars): Integrated exciton intensities from the experimental XRS data (Fig. 1, for energies below 194 eV), weighted by the calculated absolute values of $\text{Im}\left[ϵ\left(\mathbf{q},\omega \right)\right]$ (Fig. 3) integrated over the 210–245 eV range. All curves are normalized to unity at $q=0$.