Drift of Carriers in electric and magnetic fields: conductivity and mobility
In addition to the values of n and p, we must be able to take into account the collisions of the charge carriers with the lattice and with the impurities. These processes will affect the ease with which electrons and holes can flow through the crystal, that is, their mobility within the solid.
Conductivity and Mobility:
If an electric field ξx is applied in the jc-direction, each electron experiences a net force -qξx from the field. This force may be insufficient to alter appreciably the random path of an individual electron, the effect when averaged over all the electrons, however, is a net motion of the group in the-x-direction. If px is the x component of the total momentum of the group, the force of the field on the n electrons/cm3 is
To find the total rate of momentum change from collisions, we investigate the collision probabilities more closely. If the collisions are truly random, there will be a constant probability of collision at any time for each electron. Let us consider a group of N0 electrons at time t = 0 and define N(t) as the number of electrons that have not undergone a collision by time t. The rate of decrease in N(t) at any time t is proportional to the number left unscattered at t,
where t is a constant of proportionality. The solution of the just above equ is an exponential function
and t represents the mean time between scattering events, called the mean free time. The probability that any electron has a collision in the time interval dt is . Thus the differential change in px due to collisions in time dt is
The rate of change of px due to the decelerating effect of collisions is
The sum of acceleration and deceleration effects must be zero for steady state. we have
The average momentum per electron is
where the angular brackets indicate an average over the entire group of electrons. As expected for steady state, Eq. indicates that the electrons have on the average a constant net velocity in the negative x-direction
The current density resulting from this net drift is just the number of electrons crossing a unit area per unit time n(vx) multiplied by the charge on the electron {—q):