Intervalley Excitonic Hybridization, Optical Selection Rules, and Imperfect Circular Dichroism in Monolayer $h\text{−}\mathrm{BN}$

We perform first-principles $GW$ plus Bethe-Salpeter equation calculations to investigate the photophysics of monolayer hexagonal boron nitride ($h\text{−}\mathrm{BN}$), revealing excitons with novel $\mathbf{k}$-space characteristics. The excitonic states forming the first and third peaks in its absorption spectrum are $s$-like, but those of the second peak are notably $p$-like, a first finding of strong co-occurrence of bright $s$-like and bright $p$-like states in an intrinsic 2D material. Moreover, even though the $\mathbf{k}$-space wave function of these excitonic states are centered at the $K$ and $K^{\prime }$ valleys as in monolayer transition metal dichalcogenides, the $\mathbf{k}$-space envelope functions of the basis excitons at one valley have significant extents to the basin of the other valley. As a consequence, the optical response of monolayer $h\text{−}\mathrm{BN}$ exhibits a lack of circular dichroism, as well as a coupling that induces an intervalley mixing between $s$- and $p$-like states.

Figure 1

(a) DFT-PBE (dash black) and full-frequency eigenvalue-self-consistent $GW$ (solid red) band structures. The top of the valence band is set at 0 eV. (b) Calculated absorption spectrum with ($GW$-BSE, solid pink) and without ($GW$-RPA, dash blue) electron-hole interactions for linear-polarized light. The first three peaks are labeled as $A$, $B$, and $C$, respectively. A Gaussian broadening factor of 100 meV is included.

Figure 2

Calculated amplitude and phase in $\mathbf{k}$ space of the envelope function $A_{\mathit{k}}^S$ for the first four lowest energy singlet excitons, from a calculation restricting coupling only of interband transitions within the $K$-valley region. The amplitude of $A_{\mathit{k}}^S$ of the state corresponding to (a) peak $A$, (b) peak $B$, (c) a dark exciton near peak $B$, and (d) peak $C$, as labeled in Fig. 1. The amplitude indicates the magnitude of the free electron-hole pair excitation at each $\mathbf{k}$ point. The plotted value of each point has been normalized to its overall largest value in the BZ. The color scale bar is shown under panels (a)–(d). The phase of $A_{\mathit{k}}^S$ is presented for states in (e) peak $A$, (f) peak $B$, (g) a dark exciton near peak $B$, and (h) peak $C$. The angle of the arrows with respect to the $k_x$ direction gives the phase of $A_{\mathit{k}}^S$. Length of arrows’ tail represents the relative amplitude of $A_{\mathit{k}}^S$. The winding number about the $K$ point, $m$, denotes the angular quantum number of $A_{\mathit{k}}^S$. All plots use Cartesian coordinates where the origin is set at $K$ point. The blue dashed circle in (a) has radius $\frac{1}{3}(2\pi /a)$. The black dashed circle in (h) highlights the nodal structure as seen in (d). Note that the $\mathbf{k}$-space regions in (a)–(d) are different from those in (e)–(h).

Figure 3

Calculated amplitude of the envelope function $A_{\mathit{k}}^S$ of bright excitons contributing to the first three peaks in the optical absorption spectrum of monolayer $h\text{−}\mathrm{BN}$, in the presence of $K$- and $K^{\prime }$-valley interaction. The square root of the sum of amplitude square of excitons corresponding to (a) peak $A$, (b) peak $B$, and (c) peak $C$ in the full BZ region, normalized to its overall largest value with the color scale bar shown below the panels. The summation index $S$ represents degenerate bright excitonic states.