Charge transport in organic molecular semiconductors from first principles: The bandlike hole mobility in a naphthalene crystal

Predicting charge transport in organic molecular crystals is notoriously challenging. Carrier mobility calculations in organic semiconductors are dominated by quantum chemistry methods based on charge hopping, which are laborious and only moderately accurate. We compute from first principles the electron-phonon scattering and the phonon-limited hole mobility of naphthalene crystal in the framework of ab initio band theory. Our calculations combine GW electronic bandstructures, ab initio electron-phonon scattering, and the Boltzmann transport equation. The calculated hole mobility is in very good agreement with experiment between $100–300\mathrm{K}$, and we can predict its temperature dependence with high accuracy. We show that scattering between intermolecular phonons and holes regulates the mobility, though intramolecular phonons possess the strongest coupling with holes. We revisit the common belief that only rigid molecular motions affect carrier dynamics in organic molecular crystals. Our paper provides a quantitative and rigorous framework to compute charge transport in organic crystals and is a first step toward reconciling band theory and carrier hopping computational methods.

Figure 1

The monoclinic crystal structure of naphthalene, with two molecules in the unit cell. The molecules are arranged in a herringbone pattern in the $ab$ planes (left), which are stacked in the $c$ crystallographic direction (right). The $c^{*}$ direction normal to the $ab$ plane is also shown.

Figure 2

The hole mobility in naphthalene, given, from left to right in separate panels, in the two in-plane directions $a$, $b$ and in the plane-normal direction $c^{*}$. Circle markers are the computed mobilities and black markers the experimental data from Ref. [6]. Straight lines are best fits to the power law function $T^{-n}$ of the data points in the $100–300\mathrm{K}$ temperature range, and the exponent $n$ for each data set is also given. The error bars are obtained by assuming a $10\%$ error on both the phonon frequencies and the GW band stretching factor. These error sources are assumed to be independent and combined together.

Figure 3

Mode resolved e-ph scattering rates, ${\mathrm{\Gamma }}_{n\mathbf{k}}^{\left(\nu \right)}$, for (a) the 12 intermolecular phonon modes and (b) selected intramolecular phonon modes (note the $y$-axis log scale). Also sketched in (b) are the dominant e-ph scattering processes below and above the phonon emission threshold energy $\hslash {\omega }_0$, which is shown as a vertical dashed line for mode 90. The black dashed curve represents the integrand in Eq. (2) and shows that only hole states within a $50–100\mathrm{meV}$ energy window of the valence band maximum (VBM) contribute to the mobility. (c) Mode-resolved scattering rates averaged over the energy window contributing to the mobility. In all plots, the zero of the energy axis is the VBM, and the hole energy increases moving away from the VBM into the valence band.

Figure 4

(a) The absolute value of the local coupling constant [see Eq. (4)] between each of the phonon modes and the HOMO Wannier function. (b) The square of the HOMO Wannier function. The potential perturbation ${\mathrm{\Delta }}_{\nu \mathbf{q}}{V}^{\mathrm{KS}}$ at $\mathbf{q}=0$ is shown for (c) mode $\nu =88$ and (d) mode $\nu =89$. These modes correspond to the peak (mode 88) and sudden drop (mode 89) in e-ph coupling in (a). In panels (b)–(d), yellow is used for positive, and blue for negative isosurfaces.

Figure 5

Band structures and phonon dispersions of naphthalene crystal, for the structure used at 300 K. (a) The HOMO and $\mathrm{HOMO}-1$ electronic bands, where black is used for the DFT bands and red for the bands with the GW correction. (b) Dispersion of the 12 intermolecular phonon modes. (c) Sketch of the first Brillouin zone.

Figure 6

The mobility at 300 K, obtained using a structure relaxed with the TS-vdW correction, is shown with black crosses. The values fall within the error bars.

Figure 7

Calculated dispersion of the 12 intermolecular phonon modes for perdeuterated naphthalene, with lattice constants taken from Refs. [64],[52]. The markers are the experimental data at 6 K from Ref. [65].