Now, we determine the occupation of the electronic levels.

The first part, we studied the band structure,

and then these are possible solution of the Schrödinger equation,

that is to say, the possible values of energy.

Now, we deal with the occupations in order to determine

the number of charges to finally calculate the conductivity.

The conductivity will therefore depend on the number of free charges.

The number of charges will be the function of the possible energy states

affected by the filling probability in accordance with the Pauli principle.

This electron is a particle or spin-off,

it can be shown in statistical physics that is follow the Fermi-Dirac law,

which provide the probability of occupancy of an energy state E.

The chemical potential mu in the denominator correspond to the change

in free energy of the system when one introduces an additional particle.

If two systems can exchange particle,

the system with highest chemical potential releases particle to

the system with the lowest chemical potential until equalization of chemical potentials.

This is a quasi-static of the thermodynamic equilibrium.

At zero degree K,

the filling probability is displayed in the figure on the left.

It shows that all those states are occupied up to

the value of the energy that correspond to the chemical potential.

The chemical potential corresponds to the value of your last occupied state.

It is a state of higher energy because of the filling process by increasing energy.

Now, at T different from zero,

we see that the Fermi-Dirac on the right figure

slightly departs from zero to one behavior.

In a temperature gradient,

depending on kT, the filling probability is about a half.

As a function of the distance between E_c,

the minimum of the conduction band on mu as compare to kT,

the Fermi-Dirac law can be approximated by Maxwell-Boltzman expression,

which means that f so probability is very small as compared to one.

In this case, if E_c minus mu is large as compared to kT,

from a statistical point of view,

the electron behaves like a classical particle.

Now, the probability of occupancy for a hole,

is a probability that this state is not occupied by an electron.

Therefore, it is one minus f(E).

Since the electrons in the conduction bands are almost free,

they contribute to the conductivity.

We have to assess the contribution in the first step.

So, a number of such carriers is obtained by integrating of the density of

state weighted by the probability of occupancy of these energy states.

It's correspond to the integer two, one here.

We can assume that the gap is very large as compared to kT.

As an example, silicon meets this from our hypothesis.

Let us recall that close to the minimum of the conduction band,

the electron behaves as a free particle.

So, we can deduce the square root dependance of

D(E) on those who can integrate the equation two, one.

Going faster, for the calculation,

we can obtain the carrier density is a conduction band N,

which is found to exponentially vary as

a function of the distance between the Fermi level on E_c,

the minimum energy of the condition band.

We can't proceed likewise for the holes.

Consider as a lack of electrons in valence band.

In the same way, we obtain the expression for B,

which is also found exponentially depending on

the distance between the maximum of valence band on the Fermi level.

Therefore, we obtain the mass action law for a product np

equal n_i squared that depends on the temperature for beta on the band here.

Likewise, we obtain the location of the chemical potential of the Fermi level,

within the band gap,

the Fermi level or the chemical potential is found very close to the middle of

the band gap only affected by logarithmic dependence of the effective mass ratio.

We display here the variation of Carrier concentrtation as a function

of temperature for the free crystalline semi-conductor previously mentioned.

The slope of the ratio is correlated the band gap value.

The carrier density is strongly affected by

the temperature of the many orders of magnitude.

In particular, in case of silicon,

the intrinsic ionization at room temperature,

which is related to thermal transition

between valence and conduction band is extremely low,

10 to the minus 12.

On the other hand,

it is impossible to achieve such a weak impurity level in practical conditions.

As a matter of fact,

the best purity in crystalline silicon used in

microelectronics is 10 to the minus eight called 8N,

for order of magnitude more than a thermal ionization at room temperatrure.

In photovoltaics application, as we shall see later,

because of economic issues,

so silicon purity is relaxing to 10 to the minus 6.

It shows that the impurities will actually

control the electrical properties of semiconductor since it

is impossible to process semiconductor with level impurity as low as 10 to the minus 12.

In particular, impurities related to levels

corresponding to energy levels close to the band edges,

the so-called shallow levels,

which are therefore easy to ionize,

will play major influence on the electrical properties of semiconductors.

Then, let's finish the seconds.

So I remind you that the effective masses

on the whole concept are detailed in the next two. Thank you.