Sharp transitions in small exciton spectra for multi-orbital lattice systems

TL;DR

The paper investigates strongly bound excitons with radii on the order of the lattice constant and shows that continuum approximations fail qualitatively in multi-orbital lattice systems. By developing an efficient lattice two-particle Green's-function method, the authors study 1D and 2D models with multi-orbital valence bands and a single conduction band, revealing sharp transitions in the momentum and orbital character of the lowest-energy exciton that cannot be captured by continuum theories. They demonstrate both 1D and 2D examples where the ground-state exciton changes its momentum (and sometimes orbital composition) as the interaction strength $U$ is varied, including cases where the lowest-energy exciton resides away from the gap minimum despite a direct gap. The work highlights the necessity of lattice models to accurately describe small excitons in organic and low-dimensional semiconductors and provides a practical framework for exploring complex exciton physics in realistic multi-orbital systems.

Abstract

We demonstrate that strongly bound excitons, whose radii approach the lattice constant, display physics that eludes continuum descriptions not just quantitatively but qualitatively. We investigate such phenomenology by calculating the exciton spectra for several one- and two-dimensional lattice models, whose valence band has a multi-orbital character. We identify sharp transitions in the character and momentum of the lowest-energy exciton, driven directly by the multi-orbital nature of the lattice models. Such transitions cannot occur in simple continuum descriptions.

Figures (12)

• Figure 1: 1D exciton with ${K}=0$ binding energy (left panel) and its radius in units $a=1$ (right panel) from the continuum approximation (blue line) and a 1D lattice model (red circles) with a one-orbital conduction band with $t_c=1$, and a $s,p_x$ two-orbital valence band with $t_{ss}=1.1, t_{pp}=-1, t_{sp}=0.5$. The corresponding valence band structure is shown in Fig. \ref{['fig:2']}. Here, we assume $U_s=U_p=U$. The vertical dashed line marks the value of $U$ above which the continuum approximation starts to fail. See text for more details.
• Figure 2: The two valence bands $E_{v}({k,\pm})$ (red lines) corresponding to the parameters used in Fig. \ref{['fig:dE_U2']}. The top of the valence band is at $k=0$, as is the bottom of the conduction band (not shown), indicating a direct gap $K=0$. The dashed blue line shows the continuum approximation, which matches the exact dispersion to within $1\%$ for $|k|\lesssim \pi/6$ (dashed vertical lines).
• Figure 3: Gap energy $E_{\rm gap}(K)$ (black solid line) for the 1D lattice model with parameters as in Fig. \ref{['fig:dE_U2']}. Red circles show the exciton dispersion $E_{\rm exc}(K)$ for $U_p=U_s=0.5$, while blue crosses show $E_{\rm exc}(K)$ for $U_p=3U_s=1.5$. The latter case stabilizes an exciton with momentum $K=\pi$ even though this is a direct gap ($K=0$) semiconductor, because the top-valence band states near $k=\pi$ have dominant $p$-character.
• Figure 4: Heat maps showing the spectral weight $A_{\alpha \alpha}({\bf K}, \omega) = - {1\over \pi} \text{Im}G_{\alpha \alpha} ({\bf K}, \bm{0}, z)$ for ${\bf K}$ along high-symmetry lines in the Brillouin zone, when $\alpha = d_{x^2-y^2}$ (top panel), $\alpha = d_{xy}$ (middle panel) and $\alpha = d_{3z^2-r^2}$ (bottom panel). Here $U=12$ and all other parameters are as listed in Appendix A. The superimposed thick black line marks the lower edge of the electron-hole continuum, $E_{\rm gap}({\bf K})$. See text for more details.
• Figure 5: (top) Dispersion of the lowest-energy exciton $E_{\rm exc}({\bf K})$ for different values of $U$. The blue dashed line is the exciton energy calculated with a cut-off at $\delta =2$ (see Appendix \ref{['App:1shell']} for details), and the dot marks its predicted lowest-energy exciton; (bottom) $E_{\rm exc}({\bf K})$ when ${\bf K}=M$ (blue triangles) and when ${\bf K}$ is the indirect gap, i.e. local minimum for $E_{\rm gap}({\bf K})$ along the $K-\Gamma$ line (red circles).