Symmetry effects on nonlocal electron-phonon coupling in organic semiconductors

The electronic and electrical properties of crystalline organic semiconductors, such as the dispersions of the electronic bands and the dependence of charge-carrier mobility on temperature, are greatly impacted by the nonlocal electron-phonon interactions associated with intermolecular lattice vibrations. Here, we present a theoretical description that underlines that these properties vary differently as a function of the symmetry of the nonlocal electron-phonon coupling mechanism. The electron-phonon coupling patterns in real space are seen to have a direct and significant impact on the interactions in reciprocal space. Our findings demonstrate the importance of aspects that are usually missing in current transport models. Importantly, an adequate description of the electronic and charge-transport properties of organic semiconductors requires that the models take into account both antisymmetric and symmetric contributions to the nonlocal electron-phonon coupling mechanism.

Figure 1

Schematic diagrams of examples of lattice vibrations (top part) and e-ph coupling constants $|{\upsilon }_{kq}{|}^2$ in reciprocal space (bottom part): (a) symmetric coupling; (b) antisymmetric coupling. Here, $\delta t_{mn}=t_{mn}-t^{\left(0\right)}$; $k$ and $q$ are the electron and phonon wave vectors, respectively.

Figure 2

Sketch of the crystal structure of pentacene and illustration of the molecular pairs presenting the largest transfer integrals and considered in the calculations.

Figure 3

Energy-resolved electronic density of states (in units of $N/t^{\left(0\right)}$) and localization length for different e-ph coupling strengths: (a) and (b) symmetric coupling only ($c=1$); (c) and (d) antisymmetric coupling only ($c=0$); (e) and (f) both couplings treated on an equal footing ($c=0.5$). Here, $\hslash \omega =50$ ${\mathrm{cm}}^{-1}$, $t^{\left(0\right)}/\hslash \omega =10$, $T=300$ $\mathrm{K}$, $N=500$, and the values of $L$ are shown in (a).

Figure 4

Spectral function $A(k,E)$ for different e-ph coupling strengths. All parameters are the same as in Fig. 3. The values of $L$ are shown on the top of each column sharing the same e-ph coupling strength: (a)–(c) $c=1$; (d)–(f) $c=0$; and (g)–(i) $c=0.5$.

Figure 5

Full width at half maximum (FWHM) of the spectral function for different $k$ values as a function of temperature. Here, $\hslash \omega =50$ ${\mathrm{cm}}^{-1}$, $t^{\left(0\right)}/\hslash \omega =10$, $L/\hslash \omega =1$, $T=300$ K, and $N=500$. (a) $c=1$; (b) $c=0$; and (c) $c=0.5$.

Figure 6

Band dispersion obtained from the maximum position of the spectral functions. All parameters are the same as in Fig. 5. The black short-dashed line is the dispersion in a perfect lattice without e-ph coupling. The inset is an enlarged view of one of the band edges.

Figure 7

Energy-resolved electronic relaxation time (a) and electron mobility calculated by the Boltzmann theory (b) and the Kubo formula (c) as a function of the reduced temperature. All parameters are the same as in Fig. 5. In the case of Boltzmann calculations for the symmetric coupling, a value of $c=0.99$ instead of $c=1$ is used in order to avoid the singularity shown by $\tau$ at the band center. The inset in (c) is an enlarged view near room temperature showing the power-law behavior of the mobility.