First-principles calculations of anisotropic charge-carrier mobilities in organic semiconductor crystals

The orientational dependence of charge-carrier mobilities in organic semiconductor crystals and the correlation with the crystal structure are investigated by means of quantum chemical first-principles calculations combined with a model using hopping rates from Marcus theory. A master equation approach is presented which is numerically more efficient than the Monte Carlo method frequently applied in this context. Furthermore, it is shown that the widely used approach to calculate the mobility via the diffusion constant along with rate equations is not appropriate in many important cases. The calculations are compared with experimental data, showing good qualitative agreement for pentacene and rubrene. In addition, charge-transport properties of core-fluorinated perylene bisimides are investigated.

Figure 1

The charge carrier is “trapped” between two lattice sites. a) The surrounding of the monomer causes an energetic “pit.” b) Strongly differing jump rates lead to a capturing.

Figure 2

The potential energy surfaces of the neutral and the charged monomer. The dashed arrows indicate the vertical transitions from one state to the other. ${\lambda }_0$ and ${\lambda }_c$ are the two contributions to the reorganization energy, see Eq. (19).

Figure 3

Mobility, $\mu$, (left) diffusion coefficient, $D$, (middle), and the ratio $D/\mu$ (right) as a function of the energetic disorder, calculated with the rate equations (10) and (15) respectively, and via Monte Carlo simulation. The calculations were conducted at $T=300$ K and $F={10}^5$ V/m.

Figure 4

The molecules investigated in this work: a) pentacene, b) rubrene, c) PBI-F${}_2$, d) PBI-(C${}_4$F${}_9$)${}_2$.

Figure 5

The mobility for holes (top) and electrons (bottom) in the pentacene crystal for $F={10}^7$ V/m and $T=300$ K.

Figure 6

The mobility for holes and electrons in the pentacene crystal in the $\mathit{ab}$ plane (left), $\mathit{ac}$ plane (middle) and $\mathit{bc}$ plane (right). The parameters are the same as in Fig. 5. For comparison some experimental values[53] are plotted. Note that in the experiment the crystal orientation could not be determined[53] and therefore the experimental data are rotated to fit best.

Figure 7

The most important hopping paths in the pentacene crystal. Direction of view is parallel to the $c^{*}$ axis.

Figure 8

The most important hopping paths in the rubrene crystal. Direction of view is parallel to the $a$ axis (left) and the $b$ axis (right), respectively. The black and the gray monomers have a different position in $b$ direction.

Figure 9

The mobility for holes and electrons in the rubrene crystal in the $\mathit{ba}$ plane (left), $\mathit{ac}$ plane (middle), and $\mathit{bc}$ plane (right). The parameters are $F={10}^7$ V/m, $T=300$ K. For comparison some experimental values for hole mobilities[54, 59, 60] are plotted for the $\mathit{ba}$ plane.

Figure 10

The mobility for holes in the rubrene crystal in all three dimensions. The parameters are the same as in Fig. 9.

Figure 11

The most important hopping paths in the PBI-F${}_2$ crystal.

Figure 12

The mobility for holes (top) and electrons (bottom) in the PBI-F${}_2$ crystal in all three dimensions. The parameters are $F={10}^7$ V/m, $T=300$ K.

Figure 13

The mobility for electrons and holes in the PBI-F${}_2$ crystal in the $\mathit{ab}$ plane (left), $\mathit{ac}$ plane (middle), and $\mathit{bc}$ plane (right). The parameters are $F={10}^7$ V/m, $T=300$ K.

Figure 14

The PBI-F${}_2$ HOMO (left) and the LUMO (right) orbital for the dimer which is built by a $b$ shift and leads to the coupling $V_1$, compare Table  and Fig. 11.

Figure 15

Comparison of master equation and Monte Carlo results for the electron mobility in PBI-F${}_2$ in the $\mathit{ab}$ plane. The parameters are $F={10}^7$ V/m, $T=300$ K. The two methods show very good agreement.

Figure 16

The most important hopping paths in the PBI-(C${}_4$F${}_9$)${}_2$ crystal.

Figure 17

Projection of the charge trajectory onto the respective direction with the highest mobility for PBI-(C${}_4$F${}_9$)${}_2$ ($a$ direction, top) and PBI-F${}_2$ ($b$ direction, bottom). The parameters are $F={10}^7$ V/m, $T=300$ K.

Figure 18

Comparison of the mobility in the $\mathit{ab}$ plane of PBI-(C${}_4$F${}_9$)${}_2$ calculated via the diffusion coefficient [Eq. (15)] and the Einstein relation [Eq. (11)] for $F=0$ (green, dotted) and $F={10}^7$ V/m (black, dashed), calculated directly [Eq. (10), red, solid], and calculated with Monte Carlo [Eq. (17), blue points] for $F={10}^7$ V/m ($T=300$ K in all cases).