ECE 875: Electronic Devices Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu

VM Ayres, ECE875, S14

Revised Lecture 06: Slide 19: Sze: VM Ayres, ECE875, S14

Revised Lecture 06: Slide 20: VM Ayres, ECE875, S14

Revised Lecture 06: Slide 19: Sze: Pierret: average effective mass definition includes MC VM Ayres, ECE875, S14

VM Ayres, ECE875, S14

VM Ayres, ECE875, S14

} Chp. 01 Lecture 07, 24 Jan 14 Energy levels: E-k Effective mass mij* vgroup Density of States Concentrations Degenerate Effect of temperature Contributed by traps } VM Ayres, ECE875, S14

Carrier concentration as a function of temperature: F(E) = a probability = a number between 0 and 1 (between 0% to 100%) Probability of what? Probability that an electron occupies an allowed energy level which has energy value E(k). The energy of the electron matches what is allowed by its physical environment. VM Ayres, ECE875, S14

1. Plot of F(E), temperature dependence shown For electrons with spin up/spin down, correct expression for probability is: 1. Plot of F(E), temperature dependence shown At E = EF, F(E) = ½ = 50% E – EF (eV) = 0 at E = EF Note for slide 12: Could also make the x-axis: E, not E - EF 2. For concentration, consider where EF is relative to EC and EV VM Ayres, ECE875, S14

Plot of DOS N(E): VM Ayres, ECE875, S14 http://www.mathplanet.com/education/algebra-1/radical-expressions/the-graph-of-a-radical-function VM Ayres, ECE875, S14

VM Ayres, ECE875, S14

- 3. To carry out the integral: Useful approximations: Then: must use the denominator: Use the “hot” limit in ECE 474 & 874 & 875: Si electrons - Use the “cold” limit in ECE 802-604. Nanoelectronics VM Ayres, ECE875, S14

- Need E > EF : nondegenerate doping E = EC or above is OK and EF is in usual position below EC + - Need EF > E Therefore: “Hot and cold” depend on ± |E – EF | / kT, not on whether T is large or small. Get there by heavy (degenerate) doping (Sze: 3D Si electronics) OR by naturally high carrier concentrations (2D, 1D nanoelectronics) VM Ayres, ECE875, S14

Approximation/limits on F(E): Hot limit: - = F(E) VM Ayres, ECE875, S14

Approximation/limits on F(E): Cold limit: 1 (step function at E = EF) -| | = F(E) VM Ayres, ECE875, S14

As the Hot limit approaches the Cold limit: “within the degenerate limit” F E i E V Use: VM Ayres, ECE875, S14

Carrier concentration as a function of temperature: Identify the limits Carry out the integration VM Ayres, ECE875, S14

“hot” approximation of Eq’n (16) Example: Concentration of conduction band electrons for a nondegenerate semiconductor: n: 3D: Eq’n (14) “hot” approximation of Eq’n (16) 1/3 (000) Si GaAs VM Ayres, ECE875, S14 Ge

“hot” approximation of Eq’n (16) Example: Concentration of conduction band electrons for a nondegenerate semiconductor: n: 3D: Eq’n (14) “hot” approximation of Eq’n (16) Three different variables (NEVER ignore this) VM Ayres, ECE875, S14

1st : Deal with E – EF versus E – EC: VM Ayres, ECE875, S14

2nd : Deal with E – EC versus dE: Let: Then: MAKE SURE you remember to change the limits of integration: VM Ayres, ECE875, S14

2nd : Deal with E – EC versus dE: Practically speaking, any upper energy level that takes about 0.4 eV to reach from the bottom of EC seems to require DE “infinite” amount of energy to an e-: Chance of occupancy = really low DE ~ 0.4 eV Therefore can simplify the math: VM Ayres, ECE875, S14

de VM Ayres, ECE875, S14

VM Ayres, ECE875, S14

Always do the easy one first: denominator: VM Ayres, ECE875, S14

Then do the numerator: VM Ayres, ECE875, S14

Done! For 3D non-degenerate semiconductors: MC The part in the bracket is called NC: the effective density of states at the conduction band edge. VM Ayres, ECE875, S14

Would get a similar result for holes: This part is called NV: the effective density of states at the valence band edge. Typically valence bands are symmetric about G: MV = 1 VM Ayres, ECE875, S14

F(E) considered as a distribution: F(E) = a distribution = a plot that show frequency of occurrence Frequency of occurrence of what? Consider as a histogram. x-axis: electron energy “bins” y-axis: “property”: probability. VM Ayres, ECE875, S14

VM Ayres, ECE875, S14

Example: Sze Pr. 1.10: VM Ayres, ECE875, S14

“hot” approximation of Eq’n (16) 3D: Eq’n (14) Average Kinetic Energy VM Ayres, ECE875, S14