Classical and quantum Hall effects

We consider next what happens if in addition to the perpendicular magnetic field we apply also an electric field in the x direction. Ex, as shown in Fig. 7.12. The applied electric field will induce a current jx = —e)nvx in the same direction, where Vx is the velocity and n is the density of electrons in the plane. Due to the presence of 

Using the expression for the current, and the fact that the motion in the y direction can be associated with an effective electric field Ey, we obtain 

The expression derived in Eq. (7.38) is defined as the Hall conductivity an it is the ratio of the current in the x direction to the effective electric field Ey in the perpendicular direction, as usually defined in the classical Hall effect in electrodynamics. In more general terms, this ratio can be viewed as one of the off-diagonal components of the conductivity tensor cr, (f, j = x, y), which relates the current to the electric field  

The electric field can be expressed in terms of the current using the resistivity tensor 

We assume next that the conditions in the system of confined electrons are such that cTxx = 0 Pxx = 0 the meaning of this assumption will become evident shortly. We then hnd for the Hall conductivity 

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