Langevin recombination

It is interesting to note that in the Langevin recombination mechanism with the yeh expressed by the sum of the individual carrier mobilities (4), the interrelation between z/e,h and po switches to the relations between electron and hole... 

Equation (319) expresses that the field dependence of the ratio xrec/xt is governed by the field dependence of the mobilities, ne, Hh, recombination coefficient, y, and injection efficiency, j/jscL- For the Langevin recombination mechanism, the y is governed by the carrier motion [see Eq. (4)] so that Eq. (319) can be simplified to... 

Figure 9 The effect of Langevin recombination on the photoinduced discharge.

Van der Holst JJM, van Cost FWA, Coehoom R, Bobbert PA (2009) Electron-hole recombination in disordered organic semiconductors validity of the Langevin formula. Phys Rev B 80 235202... 

The velocity relaxation time is again f/rn and the mean square velocity (up = k T/m. Schell et al. [272] have used the Langevin equation to model recombination of reactants in solutions. Finally, from the properties of the fluctuating force (see above)... 

There have been several studies of the iodine-atom recombination reaction which have used numerical techniques, normally based on the Langevin equation. Bunker and Jacobson [534] made a Monte Carlo trajectory study to two iodine atoms in a cubical box of dimension 1.6 nm containing 26 carbon tetrachloride molecules (approximated as spheres). The iodine atom and carbon tetrachloride molecules interact with a Lennard—Jones potential and the iodine atoms can recombine on a Morse potential energy surface. The trajectives were followed for several picoseconds. When the atoms were formed about 0.5—0.7 nm apart initially, they took only a few picoseconds to migrate together and react. They noted that the motion of both iodine atoms never had time to develop a characteristic diffusive form before reaction occurred. The dominance of the cage effect over such short times was considerable. 

Hynes et al. [298] and later Schell et al. [272] have developed a numerical simulation method for the recombination of iodine atoms in solution. The motions of iodine atoms was governed by a Langevin equation, though spatially dependent friction coefficients could be introduced to increase solvent structure. The force acting on iodine atoms was obtained from the mutual potential energy of interaction, represented by a Morse potential and the solvent static potential of mean force. The solvent and iodine atoms were regarded as hard spheres. The probability of reaction was calculated by following many trajectories until reaction had occurred or was most improbable. The importance of the potential of... 

Further applications to new fields can be found in the work of P. Langevin On the Recombination of Electrically Dissociated Gases (ThSse, Paris 1902) and On the Magnetic Permeability of Gases (/. d. Phye., 4 [1905], 678). 

The classic treatment of carrier recombination can be related to the notion of the recombination time. The recombination time represents a combination of the carrier motion time (im), i.e. the time to get the carriers within capture radius (it is often assumed to be the Coulombic radius rc = e2/An o kT), and the elementary capture time (tc) for the ultimate recombination event (actual annihilation of charge carriers), tree1 = m1 -I Tc 1 (cf. Fig. 3). Following the traditional description of recombination processes in ionized gases, a Langevin-like [22] and Thomson-like [23] recombination can be defined if Tcsolid-states physics, these two cases have been distinguished... 

The long-time balance between recombination and drift of carriers as expressed by the y/n ratio has been analyzed using a Monte Carlo simulation technique and shown to be independent of disorder [40]. Consequently, the Langevin formalism would be expected to obey recombination in disordered molecular systems as well. However, the time evolution of y is of crucial importance if the ultimate recombination event proceeds on the time scale comparable with that of carrier pair dissociation (Tc/Td l). The recombination rate constant becomes then capture—rather than diffusion-con-trolled, so that Thomson-like model would be more adequate than Langevin-type formalism for the description of the recombination process (cf. Sec. 5.4). 

The question arises what is the reason for the high field decrease of (jOg(F). One of them could be a transition from the Langevin to Thomson description of the volume recombination process (see Sec. 1.3). The recombination coefficient y in Eq. (319) cannot be longer expressed by the mobility of charge carriers [see Eq. (4)] and TrecAt follows a field increasing function of the mobility in the numerator of Eq. (319) or/and field-decreasing y. The Thomson-like recombination occurs whenever the capture time (tc) in the ultimate step of the recombination process becomes comparable with the dissociation time (tj) of an initial (Coulombically correlated) charge pair (CP). Such a recombination scheme, depicted in Fig. 172, allows PR to be expressed by Eq. (3). However, to complete this picture, the overall recombination probability should also... 

The probability of bimolecular recombination is proportional to the product of the electron and hole concentrations and may become important for high-intensity exposures. In low-mobility materials, bimolecular recombination is usually described by a theoiy due to Langevin (1903). LangeviiTs expression for the recombination coefficient is... 

Reciprocity failure in aggregate photoreceptors has been described by Mey et al. (1979), as shown in Fig. 3. The exposure wavelength was at the absorption maximum, 680 nm. A loss in sensitivity of 0.14 log z was observed for positive surface potentials and 0.12 log z for negative surface potentials. The reciprocity failure was explained by Langevin (1903) recombination in conjunction with a field-dependent photogeneration process. 

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