Role of the reorganization energy for charge transport in disordered organic semiconductors

While it is commonly accepted that the activation energy of the thermally activated polaron hopping transport in disordered organic semiconductors can be decoupled into a disorder and a polaron contribution, their relative weight is still controversial. This feature is quantified in terms of the so-called $C$ factor in the expression for the effective polaron mobility: ${\mu }_e\propto \exp \left[-E_a/k_{B}T-C{\left(\sigma /k_{B}T\right)}^2\right]$, where $E_a$ and $\sigma$ are the polaron activation energy and the energy width of a Gaussian density of states (DOS), respectively. A key issue is whether the universal scaling relation (implying a constant $C$ factor) regarding the polaron formation energy is really obeyed, as recently claimed in the literature [Seki and Wojcik, J. Chem. Phys. 145, 034106 (2016)]. In the present work, we reinvestigate this issue on the basis of the Marcus transition rate model using extensive kinetic Monte Carlo simulations as a benchmark tool. We compare the polaron-transport simulation data with results of analytical calculations by the effective medium approximation and multiple trapping and release approaches. The key result of this study is that the $C$ factor for Marcus polaron hopping depends on first the degree of carrier localization, i.e., the coupling between the sites, further whether quasiequilibrium has indeed been reached, and finally the $\sigma /E_a$ ratio. This implies that there is no universal scaling with respect to the relative contribution of polaron and disorder effect. Finally, we demonstrate that virtually the same values of the disorder parameter $\sigma$ are determined from available experimental data using the $C$ factors obtained here irrespective of whether the data are interpreted in terms of Marcus or Miller-Abrahams rates. This implies that molecular reorganization contributes only weakly to charge transport, and it justifies the use of the zero-order Miller-Abrahams rate model for evaluating the DOS width from temperature-dependent charge transport measurements regardless of whether or not polaron effects are accounted for.

Figure 1

Kinetic Monte Carlo simulations (symbols) of the diffusion coefficient as a function of disorder-normalized temperature for the Miller-Abrahams rate and the nearest-neighbor hopping regime in an isotropic 3D disordered organic system with different energetic disorder values ($\sigma =50,70$, and 100 meV, indicated by red squares, green circles, and blue triangles, respectively). The KMC simulations are performed (a) for equilibrium transport, using the ODOS and (b) for nonequilibrium transport, using a conventional DOS as starting distribution. Dotted lines in both figures represent a linear fit to $\ln \left(D\right)\propto -C{\left(\sigma /k_{B}T\right)}^2$. Dashed and solid curves in (a) are the results obtained by MTR and EMA theories, respectively. These calculated curves are shifted vertically relative to each other for clarity. The arrow in (b) depicts the crossover from nondispersive to dispersive transport.

Figure 2

Temperature dependencies of the diffusion coefficient obtained for the Marcus rate by kinetic Monte Carlo simulations (symbols) for the nearest-neighbor hopping in an isotropic 3D disordered organic system at different number of hops during simulations (ranging from ${10}^3$ to ${10}^5$) performed using (a), (c) conventional DOS and (b), (d) equilibrated ODOS-simulation approaches. Simulations are done at $\sigma =100$ meV and $E_a=30$ meV. Straight lines in (c), (d) represent the linear fit of the above simulated super-Arrhenius plot (symbols) made in the temperature range where the $\ln \left[D/\exp \left(-E_a/k_{B}T\right)\right]\propto {\left(\sigma /k_{B}T\right)}^2$ law is obeyed.

Figure 3

$C$ factor vs $\sigma /E_a$ derived from kinetic Monte Carlo simulations of the nearest-neighbor hopping diffusivity $D\left(T\right)$ using a Marcus rate in an isotropic 3D system and $2\gamma a=10$. Also shown are results from analytic MTR and EMA calculations. The KMC simulations marked “DOS, full” refer to the case of nonequilibrated transport; all other data are obtained for transport under equilibrium. For comparison, values obtained using a Marcus-type hopping rate with a constant preexponential factor is also shown (“ODOS, simpl.”)

Figure 4

The $C$ factor as a function of localization derived from kinetic Monte Carlo simulations of hopping diffusivity $D$($T$) in both the nearest-neighbor hopping (empty symbols) and variable-range hopping (filled symbols) regimes using (a) a Miller-Abrahams or (b) a Marcus-type hopping rate. Results of analytic MTR calculations in the VRH regime for MA and Marcus rates are shown by red stars in (a), (b), respectively.

Figure 5

(a) The temperature dependence of the mobility plotted as $\mu$ vs $1/T^2$ on a semilog scale, along with the zero-field TOF mobility for copolymers 1 (red squares), 3 (green circles), 7 (pink triangles), and 9 (blue diamonds) as reported in Ref. [35]. Solid lines represent the fitting results using Eq. (1) with $C=0.40$ and $E_a$ values presented in Table 1. (b) Comparison of the fits done by using Eq. (1) with $C=0.35,0.40$, and $0.44$, and by using Eq. (12b) (MA rate).