Semiconductor Conductivity

2. Semiconductor Conductivity Twocharge carriers! Electrons  e- & Holes  e+ What is a hole? Qualitativedefinition for now! Quantitative definition later! Holes are usually treated as “positively charged electrons”. How is this possible? Are holes really particles?

3. Qualitative Picture of Holes Idealized, 2D, “diamond” lattice for e-& e+conduction

4. “Thought Experiment”# 1 • Add an extra e-(“conduction electron”)& apply electric field E(material is n-type: negative charge carriers) e- E field direction   e-motion direction (“almost free”)

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

6. Analogy: CO2 bubbles in beer! glass beer bubbles g (gravity) • Could develop a formal theory of bubble motion in the earth’s gravitational field! Bubbles move vertically upward Bubbles would need “negative mass”! bubble motion

7. Thermal Pair Generation & Annihilation • Now:Classical Treatment. Simple, classical, statistical analysis. Later: Quantum Treatment • Define:EgBinding energy of a valence electron. (In the Band Picture: This is the band gap energy). • Apply an energy Eg 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 Eg (in terms of heat, lattice vibrations, …)

8.  e-, e+pair generation Recombination  • Schematically: e- + e+ Eg • 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!

9. T Dependences of e- & e+ Concentrations • Define:n  concentration (cm-3) of e- p concentration (cm-3) of e+ • Can show (& will):np = CT3 exp[- Eg /(kBT)] (C = material dependent constant) From the “Law of mass action” from statistical physics • In a pure material:n = p  ni(np = ni2) • ni “Intrinsic carrier concentration” 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

10. Also: Band Gaps are (slightly) T dependent! • Can be shown that: Eg(T) = Eg(0) - αT Si : α = 2.8 x 10-4 eV/K Ge : α = 3.9 x 10-4 eV/K But this doesn’t affect the T dependence of ni! ni2 = CT3exp[- Eg(T)/(kBT)] = Cexp(α/kB)T3exp[- Eg(0)/(kBT)] = BT3exp[- Eg(0)/(kBT)] where B = Cexp(α/kB) is a new constant prefactor

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

12. Doped Materials: Materials with Impurities!More interesting & useful! • Consider idealized carbon(diamond)lattice (could be 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

13. Doped Materials We’ll show later how to calculate this! • 5th e- is really not free, but is loosely bound by energy ΔED<< Eg 5th e- moves when an E field is applied! (conduction e-) • 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

14. Consider very high T All donors are ionized  n = NDconcentration of donor atoms (constant, independent of T) • STILLhave np = ni2 = CT3 exp[- Eg /(kBT)]  p = (CT3/ND)exp[- Eg /(kBT)] “Minority carrier concentration” • All donors are ionized  The minority carrier concentration is Tdependent. • At still higher T, n >>> ND, n ~ ni The range of T where n = ND The “Extrinsic” Conduction region.

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

16. n vs. T

17. Again, consider an idealized C(diamond) lattice. (could be 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

18. Hole e+ is really not free. Is loosely bound by energy ΔEA<< Eg Δ EA = Energy released when B captures e-  e+ moves when E field is applied! • NA Acceptor Concentration • Let A  any acceptor, AX neutral acceptor A- ionized acceptor (e+ in 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 previous discussion!

19. 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 )

20. Doping in Compound Semiconductors • MUCHmore complicated! • Semiconductor compound constituents can act as donors and / or acceptors! • Example: CdS, with a S vacancy • (One S-2 “ion” is missing) Excess Cd+2 “ion” will be neutralized by 2 conduction e-  Cd+2acts 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!