Finite-momentum excitons in rubrene single crystals

Excitons with nonzero momenta and their energy dependence are important for time dependent phenomena such as transport properties or the coupling to external or internal fields, for example in electron energy loss spectroscopy. In this paper we calculate the momentum dependent energy landscape of excitons in rubrene single crystals. We show that singlet excitons exhibit a dispersion that is qualitatively similar to the electronic valence and conduction bands, namely a relatively large bandwidth along $\mathrm{\Gamma }$–Y and much flatter bands along $\mathrm{\Gamma }$–X and $\mathrm{\Gamma }$–Z. However, the absolute value of the bandwidths is significantly weaker than for both electronic bands. Triplet excitons, on the other hand, show much less dispersion and the exciton bands are much flatter than their singlet counterparts.

Figure 1

Atomic structure of rubrene single crystals. (a) isolated molecule, (b) view of the crystal along the $c$ axis showing the herringbonelike arrangement, and (c) view of the crystal along the $a$ axis showing the two separated $ab$ planes. For better visibility the second $ab$ plane is not shown in (b). Carbon atoms are shown in cyan and hydrogen atoms in gray. The unit cell is drawn in blue.

Figure 2

Electronic band structure within LDA and GWA along Z–$\mathrm{\Gamma }$–Y–X–$\mathrm{\Gamma }$. The two band structures are aligned at the valence band maximum at $\mathrm{\Gamma }$ which is set to zero energy. The band structure has a large bandwidth along $\mathrm{\Gamma }$–Y for the highest valence and lowest conduction bands. The bandwidths along $\mathrm{\Gamma }$–X and especially $\mathrm{\Gamma }$–Z are much smaller.

Figure 3

Imaginary parts of the macroscopic dielectric function of bulk rubrene single crystals for different exciton momenta. The momenta are in $Q_y$ direction and range from zero momentum ($\mathbf{Q}=\mathrm{\Gamma }$) to momenta spanning half the Brillouin zone ($\mathbf{Q}=\text{Y}$). For these calculations the full BSE without the TDA was used. The lowest exciton energy increases with increasing exciton momentum.

Figure 4

Plot of the band structure of singlet and triplet excitons of bulk rubrene single crystals along X–$\mathrm{\Gamma }$–Y. The results of the full BSE are shown with solid lines and those for the TDA-BSE with dotted lines. The dispersion of the triplet energies is much lower than that of the singlet energies.

Figure 5

Left: Probability density $|\chi (\mathbf{x},{\mathbf{x}}_h){|}^2$ for a singlet exciton with a total momentum near Y, calculated within the Tamm-Dancoff approximation. The hole was fixed at ${\mathbf{x}}_h$ on the center molecule as indicated by the red dots (these correspond to density maxima of the valence band). The volume enclosed by the indicated isosurface contains 64 % of the probability. Right: Contributions of the different $\mathbf{k}$ points of the $3\times 7\times 1$ grid to the same exciton wave function. The contributions are symmetric around $k_x=0$ due to symmetry. With respect to $k_y$ the symmetry is broken due to the nonzero exciton momentum.

Figure 6

Expectation values of the different elements of the BSE for the investigated singlet and triplet excitons within the TDA as functions of the exciton momentum $\mathbf{Q}$. (a) Expectation value of the quasiparticle gap $E_{c,\mathbf{k}+\mathbf{Q}}^{\text{QP}}-E_{v,\mathbf{k}}^{\text{QP}}$, (b) expectation value of the exchange electron-hole interaction (only singlet) and (c) of the direct electron-hole interaction. Note the different scale of the energy axis in the different subfigures and the break in the energy axis of the direct interaction in (c).

Figure 7

Left: Probability density $|\chi (\mathbf{x},{\mathbf{x}}_h){|}^2$ for a triplet exciton with a total momentum near Y, calculated within the Tamm-Dancoff approximation. The hole was fixed at ${\mathbf{x}}_h$ on the center molecule as indicated by the red dots (these correspond to density maxima of the valence band). The volume enclosed by the indicated isosurface contains 92 % of the probability. Right: Contributions of the different $\mathbf{k}$ points of the $3\times 7\times 1$ grid to the same exciton wave function. The contributions are symmetric around $k_x=0$ due to symmetry. With respect to $k_y$ the symmetry is broken due to the nonzero exciton momentum.

Figure 8

Imaginary parts of the macroscopic dielectric function of rubrene thin films for different exciton momenta. The momenta are in the $Q_y$ direction and range from no momentum ($\mathbf{Q}=\mathrm{\Gamma }$) to momenta spanning half the Brillouin zone ($\mathbf{Q}=\text{Y}$). For these calculations the full BSE without the TDA was used. The lowest exciton energy again increases with increasing exciton momentum.

Figure 9

Plot of the band structure of singlet and triplet excitons of rubrene thin films along X–$\mathrm{\Gamma }$–Y. The results of the full BSE are shown with solid lines and those for the TDA-BSE with dotted lines. Again, the dispersion of the triplet energies is much lower than that of the singlet energies. The overall energies are similar to those of the bulk crystal.