energies > 0.05 eV is t o a good approximation given by the Langevin formula. For the cases c c$^{e}$, the theoretical derivation is less straightforward. However, it will be shown in Sec. Ill that for realistic device simulations, in which c c e is common, the derived 7 in this section also works rather well. The Langevin formula can be written as: e c 2 ¾MC = 2/ bip (c)$_{-r,}$ (8) Q where ( ) ( ) ( ) is the so-called “bipolar” mobility. The analysis thus takes into account that in bipolar systems the electron-hole Coulomb interaction leads to mobilities that are slightly enhanced, by a factor , as compared t o the unipolar values. When expressing the effective (site-occupation-weighted) difference between the rates of the hnal recombination step with and without the effects of the exciton binding energy in terms of a factor ? , a comparision with Eq. (8) yields For notational simplicity, the dependence of , , and r$_{v}$ on c is not explictly shown. In the absence of energetic disorder, the occupation probabilities are at all sites equal t o c, and the unipolar mobility is (for small helds) given by = a / (k T . Furthermore, r = 1, because all hnal hops leading t o recombination are barrierless, even when E$_{exc}$ = 0. From Eq. (9), the recombination prefactor is then (no disorder). 10 For materials with Gaussian disorder, the relationship between the unipolar mobility and the nearest neighbor hop rates is less straightforward. The unipolar mobility is from the Extended Gaussian Disorder Model (EGDM) given by 0.42 2 , (11) with / ( , C)E$_{G}$DM the carrier concentration enhancement of the mobility with r e spect to the mobility for small concentrations, in the independent-particle Boltzmann regime, and with / ( ¾ ). The exponential factor expresses the temperature dependence of the mobility in the low-concentration limit, and includes the effects of