Different kinds of electric current conductions
Preface
Why do charge carriers flow and form electric current?
In principle, the charges flow because the system is in a inequilibrium state. This inequilibrium can be due to inhomogeneous electric potential, chemical potential, concentration, and probability distribution.
1) Drifting.
Electric carriers move because of the electric field gradient. Here the carrier move from the high potential positions to low potential places. The master equation is basically Newton’s second law:
Apparently, this speed of a carrier will diverge in this case if there is no damping mechanism. Normally, the damping mechanism is the collision of carriers to other particles, e.g. phonons, impurities etc.
2) Diffusion.
Electric carriers move because of the concentration gradient. Her the carrier diffuse from the high concentration place to the low concentration places. The master equation is:
,
where j is the number density current and D is called diffusion coefficient.
How will the charge carrier flow stop?
1) Collision
In this case, the carriers collide with other carriers or other quasi particles, e.g. phonons, impurities. The detailed behavior (e.g. the temperature dependence) depends on the kind of collisions. On the other hand, this kind of mechanism can be described by the time interval between the collision, or the relaxation time 
, which is related to the probability of collision.
2) Combination between positive and negative carriers
The good example is the combination of electron and holes. This process can also be described by a lifetime , which is related to the probability of the recombination.
Note that we often look at the combination of effect of different mechanism, there one needs to be careful.
There are different kinds of conductions that may involve finite distance or just an interface. Here we try to enumerate them as many as possible.
1. Conduction in a homogeneous media
Most of the conduction is energy dispersive, in other words, resistive. Then the question is what’s the dependence of resistivity 
(or conductivity) on applied voltages. There are actually many types of conduction mechanism. We will try to list as many as possible and hopefully to unify them.
Conduction in a metal
This is the most common case. The idea is that the electrons get accelerated by a electric field and return to equilibrium through collision.
The acceleration can be described as:
,
where m is the effective mass in the quasi-classical motion of the carriers, v is the speed, q is the charge and E is the electric field. This is basically Newton’s second law.
Assuming as the mean time span between collisions, the maximum velocity is
.
Now we look at the current density
,
where n is the charge carrier density.
Then
.
Hence we get the conductivity
.
This actually makes a lot of sense.
The conductivity is proportional to the carrier density, charge squared, mean collision interval and inversely proportional to the mass.
Another interesting concept that we can introduce here is the mobility
.
Then the current density and conductivity become:
.
We can see that the mobility
or 
which describes how fast the carrier can get in an electric field.
Some important message here:
1) The linear dependence of current on field (or voltage) is quite “normal” because it obeys Ohm’s law.
2) The electric field and charge carriers are supposed to be constant across the conducting wire.
3) Some numbers for typical metal copper:
n=8e22 /m3
q=e=1.6e-19 columb
m=9.1e-31 kg
sigma=6e7 S/m (300 K)
tau=2.6e-8 s
Conduction in an insulator (space charge limited conduction)
Here we are concerned with the conduction in an insulator or an intrinsic semiconductor. In this case, there is basically no intrinsic charge carrier in the media we are dealing with, any conduction needs to rely on the carrier injected from the electrodes.
The master equations in this case are (we only consider one dimensional case for simplicity):
.
The first equation describes the current density, the second is basically the Poisson equation.
Since j must be constant to conserve charge, the variables in those equations are actually n and E.
Combining the two equations to cancel n, one has:
or
.
One can integrate this equation and get
,
which is the relation between E and x, assuming E=0 when x=0.
Rewrite the above equation, one gets:
,
One can find the voltage in this case
or
,
which is called the Mott-Gurney Law.
Let’s look at the special features here:
1) The relation between current j and voltage V is not linear, but rather quadratic.
2) Neither the electric field nor the carrier density is a constant across the media. In fact:
3) The media is not charge neutral due to the carrier injection. The charge carrier density drops gradually with distance.
4) For the same voltage the current drops dramatically with a 
, as opposed to the 
in the conductor’s case.
5) since , one can estimate the current density:
q=e=1.6e-19 columb
mu=0.1 m2/s/V
epsilon=10
j=1.3e-30 V2/x3
Obviously, it is very sensitive to the thickness x. If we apply one volt, j=1.3e-9 A/m2 for x=100nm, j=1.3e-6 A/m2 for x=10 nm.
=8.9e-6 V/x /m3. If we apply one volt, at 100 nm away, n=890/m3.
In any case, these are very small currents. After all, the media here is an insulator. Note that we did not consider any effect of contact, only fixed the voltages at the boundary of the insulator.