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

VM Ayres, ECE875, S14 Chp. 01 Energy levels: E-k Effective mass m ij * v group Density of States Concentrations Effective DOS Lecture 06, 22 Jan 14

VM Ayres, ECE875, S14 Electronics: Transport: e-’s moving in an environment Correct e- wave function in a crystal environment: Bloch function: Sze:  r,k) = exp(jk.r)U b (r,k) =  (r + R,k) Correct E-k energy levels versus direction of the environment: minimum = E gap Correct concentrations of carriers n and p: contributions of (1) degeneracy and (2)traps Correct current and current density J: moving carriers I-V measurement J: V ext direction versus internal E-k: E gap direction Fixed e-’s and holes: C-V measurement Motivation: x Probability f 0 that energy level is occupied q n, p velocity Area (KE + PE)  (r,k) = E  (r,k)

VM Ayres, ECE875, S14 Carrier concentration: example: electron concentration n:

VM Ayres, ECE875, S14 Review: DOS:

VM Ayres, ECE875, S14 Typically you don’t know N T (E) or dE. To deal with this, note that E and k are related. Simple example: 1. Find N T (k): 2. Find/use the connector E = f(k) to convert N T (k) to N T (E) 3. Finish expression for DOS N(E)

VM Ayres, ECE875, S14 1. Find N T (k): An electron described by a wave fits into Volume (or Area, Line) in a a way that is bounded: a b c Macroscopic Volume in real space Bounded wave in reciprocal space OR

VM Ayres, ECE875, S14

GaAsGeSi Pierret

VM Ayres, ECE875, S14 GaAsGeSi Sze Pr. 1.07

VM Ayres, ECE875, S14

Lecture 04, Pr. 1.07: The parabola approximation and the equivalent constant energy surface ellipsoid (“cigar shaped minima”) description are the same: Parabola: Ellipsoid: Wikipedia: ellipsoid. Set b = c

VM Ayres, ECE875, S14 GaAsGeSi Vol-ellipsoid =Vol-sphere =

VM Ayres, ECE875, S14 GaAsGeSi M C = 8/2 = 4 Conduction band M C = 6M C = 1 Ge: lowest valley at the zone boundary along [111]

VM Ayres, ECE875, S14

GaAsGeSi Effective k-space volumeActual k-space volumes:

VM Ayres, ECE875, S14 Sze equation 14:

VM Ayres, ECE875, S14

Carry out the integration:

VM Ayres, ECE875, S14 This part is called N C : the effective density of states at the conduction band edge. MCMC Result: Use in Pr Note: m de ’s are given in this problem