Quantitative mobility evaluation of organic semiconductors using quantum dynamics based on density functional theory

We present an order-$N$ methodology to evaluate mobilities of charge carriers coupled with molecular vibrations using quantum dynamics based on first-principles calculations that can be applied to micron-scale soft materials. As a demonstration, we apply it to several organic semiconductors and show that the calculated intrinsic hole mobilities and their temperature dependences are quantitatively in good agreement with those obtained in experiments. We also clarified which vibrational modes dominate the transport properties. The methodology paves the way for quantitative prediction of the transport properties of various soft materials.

Figure 1

Time-dependent diffusion coefficients $D^{\left(v\right)}\left(t\right)$ (red bold line) and $D^{\left(x\right)}\left(t\right)$ (blue bold line) at 300 K for a simple one-dimensional model Hamiltonian [22, 33]. The vertical line represents the characteristic time of vibrations $1/{\omega }_0$. To compare the effects of dynamical disorder and static disorder (${\omega }_0=0$), the diffusion coefficients in the presence of static disorder are shown by the broken lines. Inset: Time-dependent diffusion coefficients $D^{\left(v\right)}\left(t\right)$ (red) and $D^{\left(x\right)}\left(t\right)$ (blue) at 300 K for two-dimensional square lattice.

Figure 2

(a)–(d) Calculated HOMO band structure with symmetry points of $\mathrm{\Gamma }(0,0,0)$, $X(1/2,0,0)$, $Y(0,1/2,0)$, $S(1/2,1/2,0)$ for ${\mathrm{C}}_{10}$-DNBDT (stand), ${\mathrm{C}}_8$-BTBT, pentacene, and ${\mathrm{C}}_{10}$-DNBDT (sleep), respectively. The origin of energy axis is set to the on-site energy (HOMO energy level). (e) Thermally activated ${\mu }_{\mathrm{TDWPD}}$ of ${\mathrm{C}}_{10}$-DNBDT (sleep) and power-law temperature dependence of ${\mu }_{\mathrm{TDWPD}}$ of ${\mathrm{C}}_{10}$-DNBDT (stand) and ${\mathrm{C}}_8$-BTBT, shown by solid circles. The different crystal structures of the ${\mathrm{C}}_{10}$-DNBDT polymorphs are shown in the insets. The alkyl side chains are omitted for visibility. The mobilities are calculated along the column direction shown by the arrow. The activation energy of the mobility for ${\mathrm{C}}_{10}$-DNBDT (sleep) is estimated to be 9.5 meV. The experimental data points for ${\mathrm{C}}_{10}$-DNBDT (stand) and ${\mathrm{C}}_8$-BTBT are shown by circles [77] and squares [52], respectively. (f) Temperature-independent ${\mu }_{\mathrm{TDPWD}}$ of pentacene shown by solid circles. The ${\mu }_{\mathrm{band}}$ [49, 58] and ${\mu }_{\mathrm{hop}}$ [49] are represented by gray curves. The experimental data are plotted by diamonds [61] and triangles [66]. (g) Calculated time-dependent diffusion coefficients $D^{\left(v\right)}\left(t\right)$ at 300 K for pentacene, ${\mathrm{C}}_8$-BTBT, and the stand and sleep phases of ${\mathrm{C}}_{10}$-DNBDT.

Figure 3

Amplitudes of vibrating transfer integrals of ${\mathrm{C}}_{10}$-DNBDT (stand) induced by the molecular vibrations with the frequencty $h{\omega }_{l\mathbf{q}=0}^{\text{inter}}$ at 300 K. Red arrows in the schematic molecular structures represent the normal-mode coordinates with its frequencies $h{\omega }_{l\mathbf{q}=0}^{\text{inter}}$.