Fermi Energy and Carrier Concentration
Again, we look at a "perfect" semiconductor, where doping has been achieved by replacing some lattice atoms by suitable doping atoms without  in the ideal world  changing anything else.  
We have now sharp allowed energy energy levels in the band gap, belonging to electrons of the doping atoms, or since electrons can not be distinguished, to all electrons in the semiconductor.  
These levels may or may not be occupied by an electron. If it is not occupied by an electron, it is by necessity occupied by a hole; the Fermi distribution will give the probability for occupancy as before. In analogy to the intrinsic case, we now have the following highly stylized picture.  


While the picture may look complicated, it is actually just a device to keep track of the number of charges in the semiconductor. Since there must be an overall charge neutrality, we must have  


This equation can be used to calculate the exact position of the Fermi energy as we will see. Since most everything else follows from the Fermi energy, lets look at this in some detail.  
First lets count the negative charges  always relative to the perfect semiconductor at zero Kelvin, where all electrons are in the conduction band and charge neutrality automatically prevails.  
There are, first, the electrons in the conduction band. Their concentration as spelled out before was  


More generally and more correctly, however, we have  


The Fermi energy E_{F} is now included as a variable in the Fermi function, because the density of electrons depends on its precise value which we do not yet know. In this formulation the electron concentration comes out always correct, no matter where the Fermi energy is positioned in the band gap.  
Then there are, second, the negatively charged acceptor atoms  ions in fact. Their concentration N_{A}^{–} is given by the density of acceptor states (which is just their density N_{A}) times the probability that the states are occupied, and that is given by the value of the Fermi distribution at the energy of the acceptor state, f(E_{A}, E_{F},T). We have accordingly  


Note: The Fermi distribution in this case should be slightly modified to be totally correct. The difference to the straightforward formulation chosen here is slight, however, so we will keep it in this simple way. More about this in an advanced module.  
Now lets count the positive charges.  
First, we have the holes in the valence band. Their number is given by the number of electrons that do not occupy states in the valence band; in other words we have to multiply the effective density of states with the probability that the state is not occupied.  
The probability that a state is not occupied is just 1 minus the probability that it is occupied, or simply 1 – f(E, E_{F}, T). This gives the density of holes to  


Once more, the better general formula for any value of the Fermi energy whatsoever is  


Second, we have the positively charged donors, i.e. the donor atoms that lost an electron. Their concentration N_{D}^{+} is equal to the density of states at the donor level which is again identical to the density of the donors themselves times the probability that the level is not occupied; we have  


Charge equilibrium thus demands  




If we insert the expression for the Fermi distribution  


where E_{n} stands for E_{C,V,D,A}, we have one equation for the one unknown quantity E_{F} !  
Solving this equation for any given semiconductor (which specifies E_{C,V}) and any concentration of (ideal) donors and acceptors, will not only give us the exact value of the Fermi energy E_{F} for any temperature T, i.e. E_{F}(T), but all the carrier concentrations as specified above.  
Unfortunately, this is a messy transcendental equation  it has no direct solution that we can write down.  
Before the advent of cheap computers, this was a problem  you now had to do case studies and use approximations: High or low temperatures, only donors, only acceptors and so on. This is still very useful, because it helps to understand the essentials.  
However, here we are going to use a program (written by J. Carstensen), that solves the transcendental equation and provides all functions and numbers required. It is contained as an independent module in the illustration section.  
You should work extensively with this program. The module contains a number of suggestions for exercises  use it!  
Here we show only a screen shot with the resultt for the typical case of Si with  
1. An acceptor concentration of 10^{15} cm^{3} (red line).  
2. An donor concentration of 10^{17} cm^{3} (blue line).  


One of many important points to note about carrier densities is the simple, but technologically supremely important fact that the majority carrier concentration for many semiconductors in a technically useful temperature interval is practically identical to the dopant concentration.  
For Si, this useful temperature interval is between ....^{o}C and ....^{o}C, (find it out for yourself by using the simulation module!). This is then the temperature regime were we can hope to have only a weak temperature dependence of electronic properties and thus of devices made from Si.  
For Ge, this interval is about ...^{o}C to ....^{o}C, (find it out for yourself by using the simulation module!) which makes Ge much less useful for many applications. 
The equations used for charge neutrality also allow to deduce an extremely important relation, the mass action law (for electrons and holes), as follows:  
First we consider the product  


Then we insert the formula for the Fermi distribution and note that  


We get the famous and very important mass action law (try it)  


We are now in a position to calculate the concentration of majority and minority carriers with very good precision if we use the complete formula, and with a sufficient precision for the appropriate temperature range were we can use the following very simple relations  


We know that the conductivity s of the semiconductor is given by  


With m = mobility of the carriers.  
We now have the concentrations; obviously we now have to consider the mobility of the carriers.  
Finding simple relations for the mobility of the carriers is just not possible. Calculating mobilities from basic material properties is a farfetched task, much more complicated and involved than the carrier concentration business.  
However, the carrier concentrations (and their redistribution in contacts and electrical fields) is far more important for a basic understanding of semiconductors and devices than the carrier mobility.  
At this point we will therefore only give a cursory view of the essentials relating to the mobility of carriers.  
The mobility µ of a carrier in an operational sense is defined as the proportionality constant between the average drift velocity v_{D} of a (ensemble of) carriers in the presence of an electrical field E:  


The average (absolute) velocity v of a carrier and its drift velocity v_{D} must not be confused; for a detailed discussion consult the links to two basic modules:  
The simple linear relationship between the drift velocity and the electrical field as a driving force is pretty universal  it is the requirement for ohmic behavior (look it up in the link)  but not always obeyed. In particular, the drift velocity may saturate at high field strengths, i.e. increasing the field strength does not increase v_{D} anymore. We come back to that later.  
Here we only want to get a feeling for the order of magnitudes of the mobilities and the major factors determining these numbers.  
As we (should) know, the prime
factor influencing mobility is the average time between scattering processes.
In fact, the mobility µ
may be written as 



With t_{s} = mean scattering time. We thus have to look at the major scattering processes in semiconductors.  
There are three important mechanisms:  
The first (and least important one) is scattering at crystal defects like dislocations or (unwanted) impurity atoms.  
Since we consider only "perfect" semiconductors at this point, and since most economically important semiconductors are pretty perfect in this respect, we do not have to look into this mechanism here.  
However, we have to keep an open mind because semiconductors with a high density of lattice defects are coming into their own (e.g. GaN or CuInSe_{2}) and we should be aware that the mobilities in these semiconductors might be impaired by these defects.  
Second, we have the scattering at wanted impurity atoms, in other word at the (ionized) doping atoms.  
This is a major scattering process which leads to decreasing mobilities with increasing doping concentration. The relation, however, is nonlinear and the influence is most pronounced for larger doping concentration, say beyond 10^{17} cm^{–3} for Si.  
Examples for the relation between doping and mobilities can be found in the illustration.  
As a rule of thumb for Si, increasing the doping level by 3 orders of magnitude starting at about about 10^{15} cm^{–3} will decrease the mobility by one order of magnitude, so the change in conductivity will be about only two orders of magnitude instead of three if only the carrier concentration would change.  
The scattering at dopant ions decreases with increasing temperature.  
Third we have scattering at phonons  the other important process. Phonons are an expression of the thermally stimulated lattice vibrations and such strongly dependent on temperature.  
This part must scale with the density of phonons, i.e. it must increase with increasing temperature (with about T ^{3/2}). It is thus not surprising that it dominates at high temperatures (while scattering at dopant atoms may dominate at low temperatures).  
Scattering at phonons and dopant atoms together essentially dominate the mobilities.  
The different and opposing temperature dependencies almost cancel each other to a certain extent for medium to high doping levels (see the illustration), again a very beneficial feature for technical applications where one doesn't want strongly temperature dependent device properties.  
With a relatively simple law for the carrier densities and no direct equations for the mobility, but relatively simple behavior of the mobility, conductivity data can now be compiled.  
Some data showing the relationship between doping and resisitivity (= 1/conductivity) or the temperature dependence of the conductivity are shown in the illustration.  
The deviation from a straight line for lower doping concentrations in the resistivity  temperature plot for is due to the temperature dependence of the mobility.  
The illustrations show different behavior for n and ptype Si, which, in our present simplified treatment awould not be predicted; it is due to different mobilities of the different carrieres.  
Our present assumptions of equal parameters (mass, mobility, life time, and so on) for electrons and holes, while justifiable in a 1st degree approximation, are to simple. Specific differences between holes and electrons exist and will be dealt with in subsequent chapters. They are responsible for the differences in mobility, conductivity, and so on between holes and electrons.  
© H. Föll (Semiconductor  Script)