This course requires that you must be familiar with some solid state physics including a working knowledge of thermodynamics and quantum theory.  
The free electron gas model is a paradigm for the behavior of electrons in a crystal, you should be thoroughly familiar with it.  
In case of doubt, refer to the Hyperscript "MaWi II"  which, however, is in German.  
In the following, the essentials of the model are repeated  briefly, without much text. If you have serious problems with the topic already here, you do indeed have a problem with this course!  
The Energy Levels of Electrons in a Constant Potential
The free electron gas model works with a constant potential. This is, of course, a doubtful approximation; essentially only justified because it works  up to a point.  
Approximations: Constant potential U = U_{0} = 0 within a crystal with length L in all directions; U = ¥ outside; only one electron is considered.  


The major formulas and interpretations needed are:  
Time independent onedimensional
Schrödinger equation : 



= h
"dash" = h/2p = Plancks constant/2p. m = electron mass. y = wave function. E = total energy = kinetic energy + potential energy. Here it is identical to the kinetic energy because the potential energy is zero. 

Potential U(x) as defined above; i.e. U(x) = U_{0} = const = 0 for 0 £ x £ L, or ¥ otherwise.  
For the (potential) energy, there is always a free choice of zero point; here it is convenient to put the bottom of the potential well at zero potential energy. We will, however, change that later on.  
Boundary conditions: Several choices are possible, here we use periodic conditions (also called Born  von Karmann conditions).  


Solution (for 3dimensional case)





A somewhat more general form for crystals with unequal sides can be found in the link.  
There are infinitely many solutions, and every individual solution is selected or described by a set of the three quantum numbers n_{x}, n_{y}, n_{z}. The solution y describes a plane wave with amplitude (1/L)^{3/2} moving in the direction of the wave vector k.  
Next, we extract related quantities of interest in connection with moving particles or waves:  
The wavelength l of the "electron wave" is given by  


The momentum p of the electron is given by  


From this and with m = electron mass we obtain the velocity v of the electron to be  


The numbers n_{x}, n_{y}, n_{z} are quantum numbers; their values (together with the value of the spin) are characteristic for one particular solution of the Schrödinger equation of the system. A unique set of quantum numbers (alway plus one of the two possibilities for the spin) describes a state of the electron  
Since these quantum numbers only appear in the wave vector k, one often denotes a particular wave function by indexing it with k instead of n_{x, y, z} because a given k vector denotes a particular solution or state just as well as the set of the three quantum numbers.  


In other words, in a formal sense we can regard the wave vector more abstract as a kind of vector quantum number designating a special solution of the Schrödinger equation for the given problem.  
Since the total energy E is identical to the kinetic energy E_{kin} = ½mv^{2} = p^{2}/2m, we have  


We now have expressed the total energy as a function of the wave vector. Any relation of this kind is called a dispersion function. Spelt out we have  


This is the first important result: There are only discrete energy levels for the electron in a box with constant potential that represents the crystal.  
This result (as you simply must believe at this point) will still be true if we use the correct potentials, and if we consider many electrons. The formula, however, i.e. the relation between energy and wave vectors may become much more complicated.  
The boundary conditions chosen and the length L of the box are somewhat arbitrary. We will see, however, that they do not matter for the relevant quantities to be derived from this model.  
Density of States  
Knowing the energy levels, we can count how many energy levels are contained in an interval DE at the energy E. This is best done in k  space or phase space.  

In phase space a surface of constant energy is a sphere, is a as schematically shown in the picture.  
Any "state", i.e. solution of the Schroedinger equation with a specific k, occupies the volume given by one of the little cubes in phase space.  
The number of cubes fitting inside the sphere at energy E thus is the number of all energy levels up to E. 
Counting the number of cells (each containing one possible state of y) in an energy interval E, E + DE thus correspond to taking the difference of the numbers of cubes contained in a sphere with "radius" E + DE and E. We thus obtain the density of states D as  


With N(E) = Number of states between E = 0 and E per volume unit, and V = L^{3 } = volume of the crystal.  
Taking into account that every state (characterized by its set of quantum numbers n_{x}, n_{y}, n_{z}) can accommodate 2 electrons (one with spin up; one with spin down), the final formula is  


The derivation of this formula and more to densities of states (including generating some numbers) can be found in the link.  
Some important points are:  


If we fill the available states with the available electrons at a temperature of 0 K (since we consider the free electrons of a material this number will be about 1 (or a few) per atom and thus is principially known), we find a special energy called Fermi energy E_{F} at the value where the last electron finds its place.  


In order to get (re)acquainted with the formalism, we do two simple exercises  


Carrier Statistics  
We have the number of energy states for a given energy interval and want to know how many (charge) carriers we will find in the same energy interval in thermal equilibrium. Since we want to look at particles other than electrons too, but only at charged particles, we use the term "carrier" here.  
In other words, we want the distribution of carriers on the available energy levels satisfying three conditions:  
The Pauli exclusion principle: There may be at most 2 carries per energy state (one with spin "up", one with spin "down"), not more.  
The equilibrium condition: Minimum of the appropriate thermodynamic potential, here always the free Enthalpy G (also called Gibbs energy).  
The conservation of particles (or charge) condition; i.e.constant number of carriers regardless of the distribution.  
The mathematical procedure involves a variation principle of G. The result is the wellknown FermiDirac distribution f(E,T):  
f(E,T) = probability for occupation of (one!) state at E for the temperature T  


If you are not very familiar with the distributions in general or the Fermi distribution in particular, read up on it in the (German) link.  
This is the "popular" version with the Fermi energy E_{F} as a parameter. In the "correct" version, we would have the chemical potential m instead of E_{F}.  
Since the Fermi energy is a quantity defined independently of the equilibrium considerations above, equating E_{F} with m is only correct at T = 0 K. Most textbooks emphasize that small differences may occur at larger temperatures, but do not explain what those differences are. We, like everybody else, will ignore these fine points and use the term "Fermi Energy" without reservations.  
In all experience, many students (and faculty) of physics or materials science have problems with the concept of the "chemical potential". This is in part psychological (we want to do semiconductor physics and not chemistry) but mostly due to little acquaintance with the subject. The link provides some explanations and examples which might help.  
The FermiDirac distribution has some general properties which are best explained in a graphic representation.  


It contains a convenient definition of the Fermi energy: The energy where exactly half of the available levels are occupied (or would be occupied if there would be any!) is the Fermi energy:  


The width of "soft zone" is » 4 kT = 1 meV at 3 K, and 103 meV at 300K.  
For E – E_{F} >> kT the Boltzmann approximation can be used:  


In this case the probability that a given energy state is occupied by more than 1 electron is negligible, and the exclusion principle is not important because there are always plenty of free states around the electrons behave akin to classical particles.  
This leads to the final formula for the incremental number or density of electrons, dN, in the energy interval E, E + DE (and, of course, in thermodynamic equilibrium).  


This is an extremely important formula, that is easily generalized for most everything. The number (or density) of something is given by the density of available places times the probability of occupation.  
This applies to the number of people found in a given church or stadium, the number of photons inside a "black box", the number of phonons in a crystal, and so on.  
The tricky part, of course, is to know the probabilities or the distribution function in each case. However, if we do not consider church goers or soccer fans, but only physical particles (including electrons and holes, but also "quasiparticles" like phonons, excitons, ...), there are only two distribution functions (and the Boltzmann distribution as an approximation): The FermiDirac distribution for Fermions, and the BoseEinstein distribution for Bosons. Mother nature here made life real easy for physicists.  
Since all available electrons N must be somewhere on the energy scale, we always have a normalization condition.  



© H. Föll (Semiconductor  Script)