Band gap renormalization, carrier mobilities, and the electron-phonon self-energy in crystalline naphthalene

Organic molecular crystals are expected to feature appreciable electron-phonon interactions that influence their electronic properties at zero and finite temperature. In this work, we report first-principles calculations and an analysis of the electron-phonon self-energy in naphthalene crystals. We compute the zero-point renormalization and temperature dependence of the fundamental band gap, and the resulting scattering lifetimes of electronic states near the valence- and conduction-band edges employing density functional theory. Further, our calculated phonon renormalization of the $GW$-corrected quasiparticle band structure predicts a fundamental band gap of 5 eV for naphthalene at room temperature, in good agreement with experiments. From our calculated phonon-induced electron lifetimes, we obtain the temperature-dependent mobilities of electrons and holes in good agreement with experimental measurements at room temperature. Finally, we show that an approximate energy self-consistent computational scheme for the electron-phonon self-energy leads to the prediction of strong satellite bands in the electronic band structure. We find that a single calculation of the self-energy can reproduce the self-consistent results of the band gap renormalization and electrical mobilities for naphthalene, provided that the on-the-mass-shell approximation is used, i.e., if the self-energy is evaluated at the bare eigenvalues.

Figure 1

Naphthalene is the smallest acene that crystallizes in a herringbone structure. There are two molecules in the monoclinic unit cell, each situated at inversion centers.

Figure 2

Electronic band structure of naphthalene calculated with DFT. The locations of the conduction-band minimum (CBM) and valence-band maximum (VBM) are indicated with black dots.

Figure 3

(a) Renormalization and temperature dependence of the band edge states at $\mathrm{\Gamma }$ and $\mathrm{A}$, with ${\mathrm{\Omega }}_{\mathrm{DFT}}$. The dotted lines indicate the ZPR, connecting the bare eigenvalues calculated with PBD-D3 (circles) with the renormalized energies at $0\mathrm{K}$. The renormalized energies for ${\mathrm{\Omega }}_{295\mathrm{K}}$ (squares) at $300\mathrm{K}$ are plotted for comparison. (b) ZPR (dotted) and temperature dependence (solid) of the indirect band gap of naphthalene for ${\mathrm{\Omega }}_{\mathrm{DFT}}$. The red square shows the renormalization at $300\mathrm{K}$ using ${\mathrm{\Omega }}_{295\mathrm{K}}$.

Figure 4

Individual contributions of the phonon modes to the renormalization of the CBM and VBM plotted against frequency, with a Lorentzian smearing of $1\mathrm{meV}$ (red solid line, left axis). The gray dotted line at $19\mathrm{meV}$ indicates the separation of inter- from intramolecular modes. The blue dashed line (right axis) shows the cumulative integral of the individual contributions.

Figure 5

The energy-resolved decomposition of the mobility according to Eq. (12) of holes (top) and electrons (bottom) at $300\mathrm{K}$ and the experimental room-temperature structure with ${\mathrm{\Omega }}_{295\mathrm{K}}$. The velocity (orange solid) and the lifetime (blue dash-dot) are associated with the left and right $y$-axes, respectively. The density of states $D\left(\varepsilon \right)$ (green dashed), the derivative of the Fermi-Dirac distribution (red dotted), and the mobility integrand (gray filled) are in arbitrary units, but share the same scale across all plots.

Figure 6

The real (solid) and imaginary part (dashed) of the electron-phonon self-energy of naphthalene, evaluated for the VBM at $\mathrm{A}$ (left) and CBM at $\mathrm{\Gamma }$ (right). The features of the self-energy correlate with the electronic DOS (filled).

Figure 7

(a) The DFT-PBE-D3 band structure of naphthalene of the two highest valence and two lowest conduction bands. (b) and (c) The spectral function of the full band structure calculated using the (b) one-shot and (c) self-consistent method. To highlight the renormalized band structure, the highest peak for each state $n\mathbf{k}$, i.e., the solution to Eq. (17) with the smallest imaginary part, is marked with a dot. While the one-shot spectral function displays a continuous quasiparticle band structure, the self-consistent result shows discontinuities.