Semiconductor Conductivity Ch. 1, S It is well-known that in semiconductors, there are Two charge carriers! Electrons  e - & Holes  e + What is a hole? We’ll use a qualitative definition for now! A quantitative definition will come later! Holes are often treated as “positively charged electrons”. How is this possible? Are holes really particles? We’ll eventually answer both of these questions as the course proceeds.

A Qualitative Picture of Holes (from Seeger’s book) An idealized, 2 dimensional, “diamond” lattice for e - & e + conduction

“Thought Experiment”# 1 Add an extra e - (“conduction electron”) & apply an electric field E (the material is n-type: negative charge carriers) E Field Direction   e - Motion Direction (“almost free”) e-e-

Remove an e - leaving, a “hole” e + & apply an electric field E. (the material is p-type: positive charge carriers)  e - Motion Direction e + Motion Direction  e+e+ “Thought Experiment”# 2 E Field Direction 

Crude Analogy: CO 2 Bubbles in Beer! g (gravity) Glass Beer Bubbles Bubble Motion We could develop a formal theory of bubble motion in the earth’s gravitational field. Since the bubbles move vertically upward, in this theory, the Bubbles would need “negative mass”!

Thermal Pair Generation & Annihilation Now: A classical Treatment. Simple, classical, statistical analysis. Later: Quantum Treatment Define: E g  Binding energy of a valence electron. (In the Band Picture: This is the band gap energy). Apply an energy E g to an atom (from thermal or other excitation). An e - is promoted out of a valence level (band) into a conduction level (band). Leaves a hole (e + ) behind. Later: e - - e + pair recombine, releasing energy E g (in terms of heat, lattice vibrations, …)

Schematically: e - + e +  E g This chemical “reaction” can go both ways. As the temperature T increases, more e - - e + pairs are generated & the electrical conductivity increases & the conductivity σ increases with increasing T.  e -, e + Pair Generation Recombination 

T Dependences of e - & e + Concentrations Define: n  concentration (cm -3 ) of e - p  concentration (cm -3 ) of e + Can show (& we will): np = CT 3 exp[- E g /(k B T)] (C = material dependent constant) From the “Law of mass action” from statistical physics In a pure material: n = p  n i (np = n i 2 ) n i  “Intrinsic carrier concentration” n i = C 1/2 T 3/2 exp[- E g /(2k B T)] At T = 300K Si : E g = 1.2 eV, n i =~ 1.5 x cm -3 Ge : E g = 0.67 eV, n i =~ 3.0 x cm -3

Also: Band Gaps are (slightly) T dependent! It can be shown that: E g (T) = E g (0) - αT Si : α = 2.8 x eV/K Ge : α = 3.9 x eV/K But this doesn’t affect the T dependence of n i ! n i 2 = CT 3 exp[- E g (T)/(k B T)] = Cexp(α/k B )T 3 exp[- E g (0)/(k B T)] = BT 3 exp[- E g (0)/(k B T)] where B = Cexp(α/k B ) is a new constant prefactor

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

Doped Materials: Materials with Impurities! 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 : (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

Doped Materials The 5th e - is really not free, but is loosely bound with energy ΔE D << E g The 5th e - moves when an E field is applied! It becomes a conduction e - Let: D  any donor, D X  neutral donor D +  ionized donor (e - to the conduction band) Consider the chemical “reaction”: e - + D +  D X + ΔE D As T increases, this “reaction” goes to the left. But, it works both directions We’ll show later how to calculate this!

Consider very high T  All donors are ionized  n = N D  concentration of donor atoms (constant, independent of T) It is still true that np = n i 2 = CT 3 exp[- E g /(k B T)]  p = (CT 3 /N D )exp[- E g /(k B T)]  “Minority carrier concentration” All donors are ionized  The minority carrier concentration is T dependent. At still higher T, n >>> N D, n ~ n i The range of T where n = N D  the “Extrinsic” Conduction region.

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

n vs. T

Again, consider an idealized C (diamond) lattice. (or any Group IV material). C : (Group IV) valence = 4 Replace one C with a boron. 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

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

Terminology “Compensated Material”  N D = N A “n-Type Material”  N D > N A (n dominates p: n > p) “p-Type Material”  N A > N D (p dominates n: p > n)

Doping in Compound Semiconductors This is MUCH more complicated! Semiconductor compound constituents can act as donors and / or acceptors! Example: CdS, with a S vacancy (One S -2 “ion” is missing) The excess Cd +2 “ion” will be neutralized by 2 conduction e -. So, Cd +2 acts as a double acceptor, even though it is not an impurity!  CdS with S vacancies is a p-type material, even with no doping with impurities!