Time-consistent hopping transport with vibration-mode-resolved electron-phonon couplings

Charge transport in organic semiconductors is affected by the complex interaction between charge carriers and molecular vibrations. A common way to treat the molecular vibrations in hopping approaches is by condensing them into a single analytical parameter, the reorganization energy. In contrast, here we present a nonadiabatic hopping transport approach that avoids this approximation by dividing the vibrational spectrum of organic molecules into three distinct analytical classes, namely the quasistatic, low-frequency dynamic, and high-frequency dynamic modes. The quasistatic and dynamic regimes are separated time consistently with respect to the timescale of the hopping events, which results in charge transfer events driven by a set of strongly coupling driving modes. Using these time-consistent hopping rates, we compute the charge carrier mobilities for a set of hopping transport materials and a control set of band-transport materials and compare them to experimental values. The resulting mobilities are consistent for both sets by showing similar values for the hopping transport materials and lower values for the control set of band-transport materials due to the absence of coherent transport contributions. We further study other popular hopping approaches such as the Marcus theory and the Levich-Jortner theory and observe substantial inconsistencies for them.

Figure 1

Exemplary separation of the intramolecular vibrations in the TCH approach based on the spectral distribution of the vibration-mode-resolved reorganization energy of NTMTI.

Figure 2

Schematic comparison of the impact of the intramolecular modes on the energetics of the molecular sites in the TCH approach (a) and LJH/MH approach (b) using the color scale for the different modes from Fig. 1.

Figure 3

Time-consistent hopping. (a) Schematic representation of the mutual dependencies: the hopping rate ${\nu }_{MN}$ depends on the particular separation of modes through the quantities ($\mathrm{\Delta }{E}^{\text{qs}}, {\sigma }^{\text{qs}}, {\mathrm{\Lambda }}_{\text{th}}, {\mathrm{\Lambda }}_{\text{red}}$). The separation of the modes in turn depends on the hopping rate. (b) Example of the iterative determination of the transport parameters for a single hopping event. Displayed are the preliminary transport parameters at each iteration step. Top panel: Dynamic reorganization energy (orange) and static disorder strength (green). Lower panel: Hopping rate (black) and current trial rate (blue). The last iteration determines ${\nu }_{\text{TCH}}$. Lines are guide to the eye. (c) As a consequence of the time-consistent treatment, the CT is mainly driven by a set of strongly coupled driving modes and the CT takes place on the timescale of these modes.

Figure 4

Materials used to study the TCH and other nonadiabatic hopping approaches. The materials are classified into the hopping (localized) transport regime (LOC, green) and bandlike (delocalized) transport regime (DEL, violet).

Figure 5

(a)–(d) Comparison of different simulated hopping mobilities: (a) TCH approach with separation of dynamic and quasistatic modes; (b) mLJH approach (without TC); and (c) MH approach (without TC). The simulations in (a)–(c) include environmental contributions. (d) MH approach (without TC) and neglecting environmental contributions. Material classes are indicated as colored points: hopping (green) and bandlike (violet). The gray shaded lines in (a)–(d) indicate relative deviations between simulated and experimental mobilities. (e)–(h) Distribution of the ratios between hopping rate of the considered model and the TCH rate for hopping materials (top panel) and bandlike materials (bottom panel). The average ratios are indicated in green and violet, while the ideal ratio of one is highlighted in blue. The violet dashed in line in (f) represents the averaged ratio without pentacene.

Figure 6

Degree of localization of the charge carrier. (a) Degree of localization and the coefficients $|c_{1/2}{|}^2$ calculated in the two-state model for TTF-$\beta$ (left) and rubrene (right). (b) Degree of localization ${\theta }_{\text{LOC}}$ at room temperature versus ${\theta }_{\text{HOP}}$ for all 10 materials studied in this work. Materials are classified into the regime of localized transport (LOC, green) and delocalized transport (DEL, violet) according to ${\theta }_{\text{LOC}}$

Figure 7

Collection of the rate and vibrational spectrum for NTMTI and rubrene. Left: Comparison of the exact spectrum Eq. (B1) in black and approximated spectrum Eq. (B2) for $E_{\text{sep}}=700{\text{cm}}^{-1}$ in red for (a) NTMTI and (b) rubrene. The dashed lines show the approximated spectra for $E_{\text{sep}}=500{\text{cm}}^{-1}$ (green) and $E_{\text{sep}}=1000{\text{cm}}^{-1}$ (blue). Center: Vibrational mode spectrum for (c) NTMTI and (d) rubrene. The red, green, and blue bars mark the different separation energies used to compute the spectra in (a) and (b). Right: Comparison of the Marcus spectrum Eq. (B3) in black and the approximated mLJ spectrum Eq. (B2) in red for (e) NTMTI and (f) rubrene. While in Marcus theory all modes contribute to the energy barrier imposed by the reorganization energy ${\mathrm{\Lambda }}_{\text{tot}}$, in the mLJ approach only the low-frequency dynamic vibrations contribute to the reduced reorganization energy ${\mathrm{\Lambda }}_{\text{red}}$, yielding an increased hopping rate around $E\approx 0$ (violet dashed line).