2 The hopping conduction can be regarded as an electron diffusion process in which an electron undergoes a Brow- nian motion from one molecule to another, and the diffu- sion constant D relates to P as D = P d $^{2}$, where d should be regarded as the average distance between two neigh- boring molecules. According to the Einstein relation, the electron mobility µ is given by µ = eD/ ( KT ) which is related to the conductivity in the conventional way[14]. Therefore, we can concentrate on how the tunneling ma- trix element depends on the magnetic field in order to study the magnetoresistance of the system. determined by the motion of polarons since the motion of excitons does not contribute to the electric current. The singlet excitons are, on the other hand, responsi- ble to the luminescence due to the spin-selection rule. A weak field should not change much of energy levels of various excitation states so that their populations at thermal equilibrium should not be sensitive to a magnetic field since they are given by the Fermi-Dirac distribution that depends only on the energy level distribution and the temperature. Any significant change in magnetore- sistance near the quasiequilibrium state must come from the mobility change. The question is whether a weak field of 100mT can change the mobility of polarons in organic semiconductors. In the usual inorganic crystals with lattice constant of order of angstroms, the answer is no. However, we are going to argue below that with molecule-molecule separation of tens nm in organic con- jugated materials, this can indeed happen. In the tight-binding approximation[15], one of the au- thors in an early publication[16] has generalized the Bardeen’s transfer matrix formalism to high dimension and in the presence of a magnetic field. In 3D, it is where φ$_{0}$ = c ℏ /e is the flux quanta. The integration is over the plane of x = d/ 2. For small ⃗ A when the mag- netic length l$_{B}$ = √ φ$_{0}$/ ( πB ) is bigger than d , magnetic confinement that is responsible for the exponential in- crease of resistance in the usual hopping conduction can be neglected and ψ$_{1}$ and ψ$_{2}$ do not depends on B to the zero order approximation. Then the magnitude of the field-independent part of t is order of while that of the field dependent part is about t = ℏ 2 m ∫ [( ψ ⋆ 1 ∂ψ$_{2}$ ∂x − ψ$_{2}$ ∂ψ ⋆ 1 ∂x ) − 2 i φ$_{0}$ ( ⃗ A · ˆ x ) ψ ⋆ $_{1}$ψ$_{2}$ ] |$_{x}$$_{=}$ d $_{2}$dydz, (2) ℏ 2 mξ ∫ ψ ⋆ $_{1}$ψ$_{2}$ |$_{x}$$_{=}$ d $_{2}$dydz ℏ 2 m d l 2 B ∫ ψ ⋆ $_{1}$ψ$_{2}$ |$_{x}$$_{=}$ d $_{2}$dydz. N O Al O O N N N O Al O O N N B 1 2 ( d ,0,0)  $_{!}$  $_{"}$ x y  $_{!}$ Intermolecular hopping  t  $_{!}$ FIG. 1: Schematical draw of two organic molecules (Alq 3 ) sep- arated by a distance d and aligned along x-direction. Molecule 1 is centered at the origin, and molecule 2 is centered at ( d, 0 , 0). The field is assumed to be along z-direction. ψ 1 and ψ 2 are two localized states with localization lengths ξ 1 and ξ 2 on molecules 1 and 2 , respectively. ξ − 1 = ξ − 1 2 − ξ − 1 1 , and ξ$_{1}$ and ξ$_{2}$ are the localization lengths of wavefunction ψ$_{1}$ and ψ$_{2}$ , respectively. The two terms are comparable if ξd/l 2 B is not too small. For an organic semiconductor with d = 20 nm and ξ ≃ d in a field of B = 10 mT , d/l$_{B}$ is about 0 . 1. Thus one shall expect an increase of t in the field by 1%, same order of experimentally observed OMFE. For B = 100 mT , d/l$_{B}$ is about 0 . 33 and t increases by 10%! In reality, ξ should be much bigger than d , especially when the hopping in- volves higher excited states as it is the case in photon- involved processes. Then the field-induced hopping coef- ficient could be even bigger than above estimated value, resulting in even bigger OMFE. In order to understand why a weak magnetic field can change charge carrier (electron and hole or polaron) mo- bility in an organic conjugated material, we consider a system with two molecules separated by a distance d as schematically shown in Fig. 1. One-electron Hamiltonian in a magnetic field can in general be described by where V$_{1}$ and V$_{2}$ are the potential created by molecules 1 and 2 , respectively. ⃗ A is the vector potential due to magnetic field ⃗ B . For the simplicity and clarity, we shall assume that the two molecules are aligned along x-direction, the field is along the z-direction (pointing out of the paper). The important quantity for electron transport is the tunneling matrix element between two molecules. When an electron tunnels from an initially occupied state, say ψ$_{1}$ of molecule 1 , to empty state ψ$_{2}$ of molecule 2 with tunneling matrix t , it will contribute to the hopping probability P (per unit time), proportional to | t | $^{2}$exp( − Δ ϵ$_{12}$/ ( KT )), where Δ ϵ$_{12}$ describes the rela- tive energy level with respect to the Fermi level[13, 14]. Due to the Van der Waals bonding, the electric and optical signals of organic semiconductors are too small to be detected without the illumination of a light or applying a high electric field. This is why the OMFE are measured under an optical injection of carriers (pho- toluminescence and photocurrent measurements) or an electric field above a threshold (electric injection current and electroluminescence measurements). When an or- ganic semiconductor is under the illumination of a light H = − 1 2 m ( ⃗ p − e c ⃗ A ) 2 + V$_{1}$ + V$_{2}$ (1)