1. Mobility 2
The average momentum is proportional to the applied force, which is qE. The electrons, on an average, collide in time n (called momentum relaxation time), so the momentum they achieve before reaching steady state is given as qnE
The average drift velocity of electrons is then given as
Note: Velocity in different directions can be different even though the field acting is the same
is the conductivity effective mass for electrons, which is the harmonic mean of the band structure effective masses. Note that this is different from the density-of-states effective mass, which is the geometric mean. n is the electron mobility
in Si (since there are 6 equivalent minima, and effective masses in three directions are ml, mt, mt)
Example:

2. Conductivity and DOS effective masses
J is given as:
m* is the conductivity effective mass
For semiconductors with a single minimum (direct bandgap materials), the current density will be different in different directions if the effective masses are different. However, for indirect bandgap materials, the current density can be isotropic even if the effective masses are not same.
For semiconductors with elliptical or cylindrical symmetry, the effective mass is same along the shorter axes
As a reminder, the density of states effective mass is given as:

3. Conductivity effective mass
The average velocity of the electrons is
The current density is given as
The mobility is given as
When there are gc equivalent conduction band minima, and total electron density, n, the electron density at each minimum is n/gc
For Si, with 6 equivalent minima, the current density in any direction is:

4. Equivalent energy minima in Si

5. Mobility 3
(1) Current caused due to motion of only electrons in applied electric field: q S
vt
(only due to electrons)
From Ohm’s Law:
(2) Total current due to both electrons and holes:
( is the hole mobility)
Note for holes,

6. Electron and hole mobility vs. bandgap
The electron and hole mobilities vary inversely with the bandgaps of the semiconductors

7. Types of mobilities Conductivity mobility
This mobility relates current density to the electric field and is given as:
Hall mobility: Measured from Hall measurement by application of magnetic field
where r is called the Hall scattering factor, and given as
Depending on the scattering mechanism, r can be significantly more than one.

8. Lorentz force and Hall effect