Solving the Bethe-Salpeter equation on a subspace: Approximations and consequences for low-dimensional materials

It is well known that the ambient environment can dramatically renormalize the quasiparticle gap and exciton binding energies in low-dimensional materials, but the effect of the environment on the energy splitting of the spin-singlet and spin-triplet exciton states is less understood. A prominent effect is the renormalization of the exciton binding energy and optical strength (and hence the optical spectrum) through additional screening of the direct Coulomb term describing the attractive electron-hole interaction in the kernel of the Bethe-Salpeter equation. The repulsive exchange interaction responsible for the singlet-triplet splitting, on the other hand, is unscreened within formal many-body perturbation theory. However, Benedict argued that in practical calculations restricted to a subspace of the full Hilbert space, the exchange interaction should be appropriately screened by states outside of the subspace, the so-called $S$ approximation [L. X. Benedict, Phys. Rev. B 66, 193105 (2002)]. Here, we systematically explore the accuracy of the $S$ approximation for different confined systems, including a molecule and heterostructures of semiconducting and metallic layered materials. We show that the $S$ approximation is actually exact in the limit of small exciton binding energies (i.e., small direct term) and can be used to significantly accelerate convergence of the exciton energies with respect to the number of empty states, provided that a particular effective screening consistent with the conventional Tamm-Dancoff approximation is employed. We further find that the singlet-triplet splitting in the energy of the excitons is largely unaffected by the external dielectric environment for most quasi-two-dimensional materials.

Figure 1

(a) BSE kernel diagrams consisting of an exchange term $K^x$ and a direct term $K^d$. (b) Expansion of the electron-hole correlation function $L$ in the BSE in powers of the BSE kernel. (c) Effective screened exchange interaction kernel ${\overline{K}}_{AA}^x$, which contains electron-hole bubbles in the $B$ subspace.

Figure 2

Convergence of exciton excitation energies for a benzene molecule with respect to the number of unoccupied bands ($N_{\mathrm{unocc}}$) used to diagonalize the BSE Hamiltonian of (a) the singlet excitation energy for the lowest-energy dark exciton $E_{d,1}^{\mathrm{S}}$ (which is dark due to a separate selection rule), (b) the excitation energy for the triplet state [spin partner to the state in (a)] associated with the lowest-energy dark exciton $E_{d,1}^{\mathrm{T}}$, (c) the singlet excitation energy for the lowest-energy bright exciton $E_{b,1}^{\mathrm{S}}$, and (d) the excitation energy for the triplet state [spin partner to the state in (c)] associated with the lowest-energy bright exciton $E_{b,1}^{\mathrm{T}}$. Red circles give singlet excitation energies calculated in the $A$ subspace only. Blue triangles are singlet excitation energies calculated with the $S$ approximation. The red dashed line is the linear extrapolation of the red circles to $1/N_{\mathrm{unocc}}\to 0$. The green squares show the convergence of the triplet state, which does not include any exchange contribution.

Figure 3

Convergence of singlet excitation energy in a benzene molecule for $E_{d,1}$ (top panel) and $E_{b,1}$ (bottom panel) with respect to the number of unoccupied bands ($N_{\mathrm{unocc}}$) used to diagonalize the BSE Hamiltonian with only the exchange interaction included in the interaction kernel (i.e., $K^d=0)$. Red circles are excitation energies in the restricted subspace only. Blue triangles are excitation energies calculated with the $S$ approximation. The dashed line is the linear extrapolation of the red circles to $1/N_{\mathrm{unocc}}\to 0$.

Figure 4

(a) Illustration of supercell with strained h-BN encapsulated in ${\mathrm{MoS}}_2$. The dashed rectangle indicates the boundaries of the supercell used in the calculation, which is periodic in all directions. (b) Band structure at LDA level of the ${\mathrm{MoS}}_2$ and h-BN supercell (red lines) superimposed on the band structure of an isolated ${\mathrm{MoS}}_2$ slab (blue circles).

Figure 5

(a) Illustration of supercell with a superlattice of alternating layers of graphene and monolayer h-BN. The dashed rectangle indicates the boundaries of the periodic supercell used in the calculation, which is periodic in all directions. (b) Band structure at LDA level of the graphene and h-BN supercell (red lines) superimposed on the band structure of the isolated h-BN monolayer (blue circles).