Excitons in van der Waals materials: From monolayer to bulk hexagonal boron nitride

We present a general picture of the exciton properties of layered materials in terms of the excitations of their single-layer building blocks. To this end, we derive a model excitonic Hamiltonian by drawing an analogy with molecular crystals, which are other prototypical van der Waals materials. We employ this simplified model to analyze in detail the excitation spectrum of hexagonal boron nitride (hBN) that we have obtained from the ab initio solution of the many-body Bethe-Salpeter equation as a function of momentum. In this way, we identify the character of the lowest-energy excitons in hBN, discuss the effects of the interlayer hopping and the electron-hole exchange interaction on the exciton dispersion, and illustrate the relation between exciton and plasmon excitations in layered materials.

Figure 1

The real and imaginary parts of the macroscopic dielectric function $\mathrm{Re}{\epsilon }_{\mathrm{M}}$ and $\mathrm{Im}{\epsilon }_{\mathrm{M}}$ and the loss function $-\mathrm{Im}{\epsilon }_{\mathrm{M}}^{-1}$ calculated at two different wave vectors $\mathbf{q}$ along the in-plane $\mathrm{\Gamma }M$ direction. For improved visibility, the loss functions have been rescaled. The vertical arrows mark the smallest independent-particle GW transition energy (which for $\mathbf{q}=0$ corresponds to the direct band gap).

Figure 2

Exciton eigenvalue spectrum $E^{\lambda }(\mathbf{q}$) in bulk hBN for in-plane $\mathbf{q}$ along $\mathrm{\Gamma }M$ for (a) the electron-hole correlation function $\overline{L}$ corresponding to $\mathrm{Im}{\epsilon }_{\mathrm{M}}(\mathbf{q},\omega )$ featuring singlet excitons, (b) the electron-hole correlation function $L$ corresponding to the loss function $-\mathrm{Im}{\epsilon }_{\mathrm{M}}^{-1}(\mathbf{q},\omega )$, displaying plasmons and e-h excitations, and (c) triplet excitons. The $\mathrm{\Gamma }M$ length is 1.45 ${\AA }^{-1}$. For the explanation of the peak labels, see the main text. In the loss-function plot (b), the solid black line is a guide for the eye in order to better track the plasmon dispersion (corresponding to the ${\mathrm{A}}^{+}$ feature).

Figure 3

Same as Fig. 2 for increased interlayer distance $d=1.5d_0$ (where $d_0$ is the experimental interlayer separation of hBN).

Figure 4

Dispersion of (a) singlet and (b) triplet excitons in the BN single layer. The color key is the same as in Fig. 2.

Figure 5

Dispersion of the visible ${\mathrm{A}}^{+}$ exciton for different interlayer distances $d$ ($d_0$ is the experimental value). For each case, the exciton energies are defined with respect to the corresponding $q=0$ value.

Figure 6

Energy difference between the plasmon and the visible ${\mathrm{A}}^{+}$ exciton as a function of momentum $\mathbf{q}$ for different interlayer distances.

Figure 7

Energy difference between the singlet and triplet ${\mathrm{A}}^{+}$ excitons as a function of momentum $\mathbf{q}$ for different interlayer distances.

Figure 8

GW electronic band structure for hBN with $1.0d_0$ and for $1.5d_0$.