3 or under a high electric field, the field dependent t will also lead to a field dependence of polaron density. Take optical injection of carriers as an example, under the il- lumination of a light, an electron in a highest occupied molecular orbit (HOMO) absorbs a photon and jumps to a higher empty molecular orbit of the same molecule. As schematically illustrated in Fig. 2, the excited elec- tron can either dump its excessive kinetic energy to the other degrees of freedom of the system and form an ex- citon with the hole left behind or jumps to neighboring molecules and become polarons. Depending on the rel- ative probabilities of excited electrons (holes) staying in the same molecules and jumping to different molecules, the polaron density shall vary with the illumination in- tensity. Let us denote the probability (per unit time) of a pair of electron and hole forming an exciton in the same molecule by P$_{0}$ ∼ ℏ /τ , where τ is the typical time for a pair of electron and hole to form an exciton. P$_{0}$ is not sen- sitive to a weak field since the field cannot change much molecule orbits that determine P$_{0}$ . Then the polaron gen- eration rate per unit volume is JP/ ( P$_{0}$ + P ) where J is the photon absorption rate per unit volume and P ∝ | t | 2 is the intermolecular hopping probability. Without the illumination of a light, polaron density shall reach its equilibrium density n ′ 0 at a rate of γ ( n − n ′ $_{0}$), where γ is polaron decay rate. At balance, JP/ ( P$_{0}$ + P ) = γ ( n − n ′ $_{0}$), thus the photon-generated polaron density n should be n ′ $_{0}$+ JP/ [ γ ( P$_{0}$ + P )]. Clearly, B -dependence of P results in a B − dependence of polaron density. molecule orbits involved in hopping are localized or not degenerated[15]. In this case, P ∝ | t | 2 = a ( B $^{2}$+ B 2 $_{0}$) and the polaron density shall depend on the magnetic field as P γ ( P 0 + P $_{)}$J + n ′ 0 = n$_{0}$ + αB $^{2}$/ ( B 2 + B 2 $_{0}$), where n$_{0}$ , n ′ $_{0}$, α and B$_{0}$ are B-independent parameters that depend on the molecule orbits involved. Thus, B $^{2}$/ ( B 2 + B 2 $_{0}$) is a natural OMFE function for t = B$_{0}$ + iaB . Interestingly, this function also appears in the magnetization expecta- tion value involving the hyperfine interactions when only quantum spin precession is considered and all other pro- cesses are neglected [17]. However, it is puzzle to us how one relates z-component spin to the resistance and how the spin polarization can survive under the huge thermal interaction. Surprisingly, the second type of empirical function can also naturally appear in when t takes a form of i ( B$_{0}$ + aB ). According to Eq. (2), this can happen when the spatial derivatives of ψ$_{1}$ or ψ$_{2}$ are the func- tions multiplied by pure imaginary numbers. Of course, this must correspond to degenerated states. In this case, the leading term in the polaron density takes a form of [ B/ ( B + B$_{0}$ )] 2 in a similar argument when P ≫ P$_{0}$ . In reality, electron (polaron) hopping between two or- ganic molecules should involve many molecule orbits, es- pecially in photophysical processes and in a high electric field. One then needs to add contributions from all hop- ping events. Thus, it is likely that both B $^{2}$/ ( B 2 + B 2 $_{0}$) and [ B/ ( B + B$_{0}$ )] 2 processes are presented, and OMFE should then be fitted by the linear combinations of these two functions, consistent with experimental findings. The novel mechanism is very robust against the tem- perature and other variations. At the room temperature, the transport of charge carriers will involve many differ- ent molecule orbits. Each hopping event will subject to the influence of this mechanism as long as magnetic con- finement is negligible ( l$_{B}$ > d ) and dξ/l 2 B is not too small (order of 1). Of course, one needs to take thermal aver- age over all hopping events. In real experiments, organic semiconductors are highly unlikely to have well defined crystal structures due to the nature of organic molecules. It is then reasonable to assume that molecule-molecule orientation of samples with a large number of molecules is quasi-random, meaning isotropic at large length scale, and the magnetic field can be along any direction with respect to the molecule-molecule bond instead of perpen- dicular direction as assumed in above discussion. This explains why OMFE is not sensitive to the field direction in devices. According to Eq. (2), different angle between the field and molecule-molecule bond leads to different hopping coefficient. It should also be emphasized that the mechanism present here does not depend on elec- tron spins, and it does not require large energy splits of different spin configurations. Obviously, the picture is equally applicable to both bipolar and hole-only (or electron-only) devices. Differ from the previous theories that try to relate the OMFE with the electronic struc- ture (either charge or spin state) changes, the present It is an experimental fact that the OMFE can often be fitted by two empirical functions B $^{2}$/ ( B 2 + B 2 $_{0}$) and [ B/ ( B + B$_{0}$ )] 2 [4]. A correct theory should be able to ex- plain why this is the case. A natural question is whether the present picture can provide a base for these func- tions. According to Eq. (2), t take a form of B$_{0}$ + iaB with B$_{0}$ and a real and field-independent parameters if ψ$_{1}$ and ψ$_{2}$ are real functions. This is the case when the Molecule 1 Molecule  2 Intramolecular exciton Intermolecular polarons ex c iton h +po laron -polaron P$_{0}$ P FIG. 2: Schematic illustration of polaron and exciton forma- tion after a pair of electron and hole is created by a photon absorption. The excited electron-hole pair has probability P jumping to the neighboring molecules to form positive- charged and negative-charged polarons, and probability P 0 to form an exciton.