In a magnetic field B perpendicular

In a magnetic field B perpendicular to the 2DEG, the current is no longer in the direction of the electric field due to the Lorentz force. Consequently, the conductivity is no longer a scalar but a tensor _, related via the Einstein relation _ = e2ρ(EF)D to the diffusion tensor Equation (1.23) follows from a straightforward generalization of the argument leading to the scalar relation (1.19) [but now the ordering of v(t) and v(0) matters]. Between scattering events the electrons at the Fermi level execute circular orbits, with cyclotron frequency ωc = eB/m and cyclotron radius lcycl = mvF/eB. Taking the 2DEG in the x−y plane, and the magnetic field in the positive z-direction, one can write in complex number notation