HOMO band dispersion of crystalline rubrene: Effects of self-energy corrections within the $GW$ approximation

We investigate the band dispersion and relevant electronic properties of rubrene single crystals within the $GW$ approximation. Due to the self-energy correction, the dispersion of the highest occupied molecular orbital (HOMO) band increases by 0.10 eV compared to the dispersion of the Kohn-Sham eigenvalues within the generalized gradient approximation, and the effective hole mass consequently decreases. The resulting value of 0.90 times the electron rest mass along the $\Gamma$-$\mathrm{Y}$ direction in the Brillouin zone is closer to experimental measurements than that obtained from density-functional theory. The enhanced bandwidth is explained in terms of the intermolecular hybridization of the HOMO($\mathrm{Y}$) wave function along the stacking direction of the molecules. Overall, our results support the bandlike interpretation of charge-carrier transport in rubrene.

Figure 1

Atomic structure of (a) isolated rubrene molecules and (b) rubrene single crystals. The C and H atoms are indicated by yellow and light blue spheres, respectively. In the single crystal, the tilted long axes of the molecules are stacked along the $b$ direction.

Figure 2

PBE and $G_0{W}_0$ band structure of the eight bands (HOMO$-$3 to LUMO$+$3) along $\mathrm{Z}-\Gamma \text{-Y}\text{-X}-\Gamma$ in the Brillouin zone. The HOMO energy at $\Gamma$ is set to zero. There are four pairs of degenerate bands that originate from the two inequivalent molecule pairs in adjacent $ab$ planes separated by a distance $c/2$. The calculated data points, indicated by symbols, are connected by lines to guide the eye.

Figure 3

Isosurfaces of the HOMO wave function (real part) of (a) an isolated rubrene molecule as well as (b) HOMO($\mathrm{Y}$) and (c) HOMO($\Gamma$) in a single crystal. The atomic configuration and orientation in (a) are identical to the molecular units in the single crystal, indicated by the black dashed circle in (b) and (c). The amplitude of the isosurfaces is $\pm 7.14$ Å${}^{-3/2}$, and different colors indicate the sign. In addition, contour plots in the $ab$ plane cutting through the orange dashed lines in (b) and (c), along which the largest concentration of the isosurfaces is located, are displayed. The black solid lines mark the unit cell. Note that HOMO($\mathrm{Y}$), which is centered on one of the four molecules in the unit cell, is degenerate with each of the HOMO$-3$($\mathrm{Y}$) to HOMO$-1$($\mathrm{Y}$) states, which are centered on the other three molecules and exhibit a similar spatial distribution. Some of the isosurfaces in the left panels of (b) and (c) are truncated in order to avoid excessive overlap and improve the visual clarity of the plots.

Figure 4

Isosurfaces of the HOMO$-1$ wave function (real part) of (a) an isolated rubrene molecule as well as (b) HOMO$-1$($\mathrm{Y}$) in a single crystal. See the explanation of Fig. 3 for details.