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

Electronic Energy Levels in the Bands The following assumes basic knowledge of elementary statistical physics. We know that Electronic Energy Levels in the Bands (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.

treat them as discrete for a while Though we normally treat these levels as if they were continuous, in the next discussion, lets treat them as discrete for a while Assume that there are N energy levels (N >>>1): ε1, ε2, ε3, … εN-1, εN with degeneracies: g1, g2,…,gN

Fermions, Spin s = ½ Electrons have the following Results from quantum statistical physics: Electrons have the following Fundamental Properties: They are indistinguishable For statistical purposes, they are Fermions, Spin s = ½

Pauli Exclusion Principle: Indistinguishable Fermions, with Spin s = ½. This means that they must obey the Pauli 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)

Energy level Enk can have Electrons obey the Pauli Exclusion Principle: So, when doing statistics for the occupied states: There can be at most, one e- occupying a given quantum state (including spin) Consider the band state (Bloch Function) labeled nk (energy Enk, & wavefunction nk): Energy level Enk can have 2 e- , or 1 e- , or 1 e- , or 0 e- _

Fermi-Dirac Distribution Statistical Mechanics Results for Electrons: Consider a system of n e-, with N Single e- energy Levels (ε1, ε2, ε3, … εN-1, εN ) with degeneracies (g1, g2,…, gN) at absolute temperature T: See any statistical physics book for the proof that 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 εj at temperature T εF ≡ Fermi Energy (or Fermi Level, discussed next)

f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1 Physical Interpretation: Define: The Fermi-Dirac Distribution Function (or Fermi distribution) f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1 Physical Interpretation: The occupation probability for level j is (<nj/gj ) ≡ f(ε)

Physical Interpretation: Look at the Fermi Function in more detail. f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1 Physical Interpretation: εF ≡ Fermi Energy ≡ Energy of the highest occupied level at T = 0. Consider the limit T  0. It’s easily shown that: f(ε)  1, ε < εF f(ε)  0, ε > εF and, for all T f(ε) = ½, ε = εF

f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1 The Fermi Function: f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1 Limit T  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, obviously we always have εF >> kBT

Fermi-Dirac Distribution ≡ Maxwell-Boltzmann Distribution NOTE! Levels within ~  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 - ε F)/kBT] ≡ Maxwell-Boltzmann Distribution

“Free Electrons” in Metals at 0 K Properties of the Free Electron Gas: The Fermi Energy EF & related properties Fermi Energy EF  Energy of the highest occupied state. Related Properties Fermi Velocity vF  Velocity of an electron with energy EF Fermi Temperature TF  Effective temperature of an electron with energy EF Fermi Wavenumber kF  Wave number of an electron with energy EF Fermi Wavelength λF  de Broglie wavelength of an electron with energy EF

ηe  Electron Density in the material Free Electron Gas: Fermi Energy EF  Energy of highest occupied state. Fermi Velocity vF  Velocity of electron with energy EF Fermi Temperature TF  Effective temperature of an electron with energy EF ηe  Electron Density in the material Fermi Wavenumber kF  Wave number of an electron with energy EF: EF = [ħ2(kF)2]/(2m)  kF  (3π2ηe)⅓ Fermi Wavelength λF  Wavelength of an electron with energy EF : λF  (2π/kF)  λF  [2π/(3π2ηe)⅓]

Sketch of a typical experiment Sketch of a typical experiment. A sample of metal is “sandwiched” between two larger sized samples of an insulator or semiconductor. Vacuum Level  Metal Band Edge  F EF  EF  Fermi Energy F  Work Function Energy

Using typical numbers in the formulas for several metals & calculating gives the table below:

Fermi-Dirac Distribution 1 EF Electron Energy Occupation Probability Work Function F Increasing T T = 0 K kBT

Number and Energy Densities Number Density: Energy Density: Density of States De(E)  Number of electron states available between energies E & E+dE. For 3D spherical bands only, it’s easily shown that:

T Dependences of e- & e+ Concentrations n  concentration (cm-3) of e- p  concentration (cm-3) of e+ Using earlier results & making the Maxwell-Boltzmann approximation to the Fermi Function for energies near EF, it can be shown that np = CT3 exp[- Eg /(kBT)] (C = material dependent constant)

ni = C1/2T3/2exp[- Eg /(2kBT)] For all temperatures, it is always true that np = CT3 exp[- Eg /(kBT)] (C = material dependent constant) In a pure material: n = p  ni (np = ni2) ni  “Intrinsic carrier concentration”. So, ni = C1/2T3/2exp[- Eg /(2kBT)] At T = 300K Si : Eg= 1.2 eV, ni =~ 1.5 x 1010 cm-3 Ge : Eg = 0.67 eV, ni =~ 3.0 x 1013 cm-3

Intrinsic Concentration vs. T Measurements/Predictions Note the different scales on the right & left figures!

 P is a DONOR (D) impurity Doped Materials: Materials with Impurities! As already discussed, these are more interesting & useful! Consider an idealized carbon (diamond) lattice (we could do the following for any Group IV material). C : (Group IV) valence = 4 Replace one C with a phosphorous P. P : (Group V) valence = 5 4 e-  go to the 4 bonds 5th e- ~ is “almost free” to move in the lattice (goes to the conduction band; is weakly bound). P donates 1 e- to the material  P is a DONOR (D) impurity

It becomes a conduction e- Doped Materials The 5th e- isn’t really free, but is loosely bound with energy ΔED << Eg (Earlier, we outlined how to calculate ΔED!) The 5th e- moves when an E field is applied! It becomes a conduction e- If there are enough of these, a current is created

But, it works both directions Doped Materials Let: D  any donor, DX  neutral donor D+ ionized donor (e- to conduction band) Consider the chemical “reaction”: e- + D+  DX + ΔED As T increases, this “reaction” goes to the left. But, it works both directions

 The “Extrinsic” Conduction Region. Consider very high T  All donors are ionized  n = ND  concentration of donor atoms (a constant, independent of T) It is still true that np = ni2 = CT3 exp[- Eg /(kBT)]  p = (CT3/ND)exp[- Eg /(kBT)]  “Minority Carrier Concentration” All donors are ionized  The minority carrier concentration is T dependent. At still higher T, n >>> ND, n ~ ni The range of T where n = ND  The “Extrinsic” Conduction Region.

Almost no ionized donors & no intrinsic carriers n vs. 1/T Almost no ionized donors & no intrinsic carriers lllll   High T Low T  

n vs. T   Low T High T  

 B is an ACCEPTOR (A) impurity Again, consider an idealized C (diamond) lattice. (or any Group IV material). C : (Group IV) valence = 4 Replace one C with a boron B. B : (Group III) valence = 3 B needs one e- to bond to 4 neighbors. B can capture e- from a C  e+ moves to C (a mobile hole is created) B accepts 1 e- from the material  B is an ACCEPTOR (A) impurity

e++A-  AX + ΔEA NA  Acceptor Concentration The hole e+ is really not free. It is loosely bound by energy ΔEA << Eg Δ EA = Energy released when B captures e-  e+ moves when an E field is applied! NA  Acceptor Concentration Let A  any acceptor, AX  neutral acceptor A-  ionized acceptor (e+ in the valence band) Chemical “reaction”: e++A-  AX + ΔEA As T increases, this “reaction” goes to the left. But, it works both directions Just switch n & p in the previous discussion!

“n-Type Material”  ND > NA Terminology “Compensated Material”  ND = NA “n-Type Material”  ND > NA (n dominates p: n > p) “p-Type Material”  NA > ND (p dominates n: p > n)