Effect of external voltage on collision time/ Mobility constant

Question

While studying velocity of a electron in a semiconductor under external bias we derived it something like this, $$v_d= a\ \tau + v_t,$$ where \$a\$ is the acceleration from external bias, \$\tau\$ is the mean time between two collisions and \$v_t\$ is the average thermal velocity of the electron (this is taken as zero).

The \$a\$ is further replaced as \$a=F/m^\star = (qE)/m^\star\$, where \$E\$ is the electric field due to external bias applied and \$m^\star\$ is the effective mass of a electron, giving us finally $$v_d=(q\tau/m^\star)E.$$

The quantity in the brackets is now taken as mobility constant \$\mu\$.

My question is, is the \$\mu\$ truly bias independent? Charge is constant, \$m^\star\$ is (in my knowledge) derived from the curvature of \$\text{E-K}\$ diagram (which will, presumably, simply shift in presence of an external bias).

The only quantity I am unsure about is the collision time. I think it should decrease with an increase in potential as all the electrons now tend to move to a globally common point (kind of like a crowd of people just standing as compared with a crowd of people rushing to get somewhere). I tried a surface level search but whatever came up was either too advanced or it didn't answer my question.

Any help/feedback is appreciated!

Answer 1

Of course, it's not truly bias independent, but for low bias levels it is practically bias independent. Of course there is no single definition for 'low boas levels', but it generally means up to a point where significant non-linear effects become important.

For most semiconductor devices, this is for practical, real-world bias levels. In some devices, (zener diodes, HEMTs, Gunn diodes etc.), non-linear effects are important or critical to the operation.

For the low-level bias, generally the energy imparted to each device between collisions is much less than the kT/2 energy each carrier has -- basically, the thermal energy and velocity is much higher than the E.q.t/m value. A good overview is in Carrier Transport

Answer 2

The only quantity I am unsure about is the collision time. I think it should decrease with an increase in potential

It does decrease, but the decrease is usually small enough to be considered negligible.

The average speed (i.e. absolute magnitude of velocity) of a free electron at room temperature in copper due to thermal motion is somewhere in the neighborhood of 100 km/s. Drift velocity due to an applied \$\vec{E}\$ field in copper in a typical circuit, on the other hand, might be somewhere in the neighborhood 1 cm/s or less, or about 1 part in 10,000,000. So, the drift velocity does not alter very much the total speed of free electonns, nor alter very much time between one collision of an electron and the next.

Answer 3

The model you describe is the simplest one that leads to the concept of a mobility for carriers (aknown as the Drude model). It works well for small electric fields.

For larger electric fields things are much more complicated, if interested look up high-field effects in semiconductors. There is an enormous literature.

The question seems to focus in part on voltage (maybe confusing electric potential and electric field?) when the entire discussion should be in terms of electric field.

Answer 4

mobility does depend on electric field applied?

The correct way to think about this is: the carrier velocity depends on the electric field. For small fields the relation is approximately linear and the slope is known as the mobility. For higher fields the velocity increases more slowly than linearly and eventually reaches a saturation velocity.

We must be sure to talk about electric field not bias (one of the answers again talks about bias).

The reasons for this get into the details of the scattering processes, which depend on all sorts of things.