Electron & Hole Statistics in Semiconductors A “Short Course”. BW, Ch. 6 & S. Ch 3

1. Electron & Hole Statistics in SemiconductorsA “Short Course”. BW, Ch. 6 & S. Ch 3

2. NOTE:The following discussion assumes a knowledge of elementary statistical physics. • We know that the electronic energy levels in the bands (which are solutions to the Schrödinger Equation in the periodic crystal) are actually NOT continuous, but are really discrete. We have always treated them as continuous, because there are so many levels & they are so very closely spaced. • For the next discussion, lets treat them as discrete for a while. • Assume that there are Nenergy levels (N >>>1): ε1, ε2, ε3, … εN-1, εN with degeneracies: g1, g2,…,gN

3. A result fromquantum statistical physics: Electrons have the followingFundamental Properties: • They are indistinguishable particles • For statistical purposes, they areFermions, withSpin s = ½ • So, they obey thePauli Exclusion Principle: That is, when doing statistics (counting) for the occupied states: There can be at most, one e- occupying a given quantum state (including spin) • Consider the band state labeled nk(with energy Enk, & wavefunction nk). It can hold, at most, 2 e- : 1 e- with spin “up” ( ) + 1 e- with spin “down” (  ). So, energy level Enk can have 2 e- :  or 1 e- : or1 e- : , or 0 e- : __

4. Fermi-Dirac Distribution • Statistical Mechanics Results for Electrons: consider a system of n e-, with N single e- levels (ε1, ε2, ε3, … εN-1, εN ) with degeneracies (g1, g2,…, gN) at absolute temperature T: The probability that energy level εj, with degeneracy gj, is occupied is: (<nj/gj ) ≡ (exp[(εj - εF)/kBT] +1)-1 (< ≡ ensemble average, kB ≡ Boltzmann’s constant) Physical Interpretation <nj = average number of e- in energy level εjat temperature T εF≡ Fermi Energy (or Fermi Level, discussed next)

5. Define: The Fermi-Dirac Distribution Function (or Fermi distribution) f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1 • Clearly, the probability of occupation of level j is (<nj/gj ) ≡ f(ε)

6. Now, lets look at theFermi Functionin more detail. f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1 • Physical InterpretationofεF≡ Fermi Energy: εF≡The energy of the highest occupied level atT = 0. • Consider the limitT  0: f(ε)  1, ε < εF f(ε)  0, ε > εF and, for all T f(ε) = ½, ε = εF

7. Fermi Function: f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1 • LimitT  0: f(ε)  1, ε < εF f(ε)  0, ε > εF for all T f(ε) = ½, ε = εF • What is the order of magnitude of εF? Any solid state physics text discusses a simple calculation of εF. • Typically, it is found, in temperature units that εF  104 K. • Compare with room temperature (T  300K): kBT  (1/40) eV  0.025 eV So, obviouslywe always haveεF >> kBT

8. Fermi-Dirac Distribution • NOTE!Levels within ~ 2 kBT of εF (in the “tail”, where it differs from a step function) are the ONLY ones which enter conduction (transport) processes! Within that tail, f(ε) ≡ exp[-(ε - εF)/kBT]= Maxwell-Boltzmann distribution

9. Free Electrons in Metals at 0 K Metal Fermi Energy – highest occupied energy state: Vacuum Level F: Work Function Fermi Velocity: EF Energy Fermi Temperature: Band Edge

10. f k T B 1 T = 0 K Vacuum Occupation Probability, Level Increasing T 0 E F F Work Function, Electron Energy, E Effect of Temperature Fermi-Dirac Equilibrium Distribution = The probability of electron occupying energy level Eat temperatureT

11. Number and Energy Densities Number density: Energy density: Density of States Number of electron states available between energy E and E+dE 3D spherical bands only!