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DEE4521 Semiconductor Device Physics Lecture 3A: Density-of-States (DOS), Fermi-Dirac Statistics, and Fermi Level Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University October 1, 2013 1 What are States? • Pauli exclusion principle: No two electrons in a system can have the same set of quantum numbers. • Here, Quantum Numbers represent States. 2 DOS • We have defined the effective masses (ml* and mt*) in a valley minimum in Brillouin zone. • We now want to define another type of effective mass in the whole Brillouin zone to account for all valley minima: DOS Effective Mass m*ds • Here DOS denotes Density of States. • States (defined by Pauli exclusion principle) can be thought of as available seats for electrons in conduction band as well as for holes in valence band. 3 DOS Ways to derive DOS and hence its DOS effective mass: •Solve Schrodinger equation in x-y-z space to find corresponding k solutions •Again apply the Pauli exclusion principle to these k solutions – spin up and spin down •Mathematically Transform an ellipsoidal energy surface to a sphere energy surface, particularly for Si and Ge Regarding this point, textbooks would be helpful. 4 3-D Carriers S(E): DOS function, the number of states per unit energy per unit volume. mdse*: electron DOS effective mass, which carries the information about the DOS in conduction band mdsh*: hole DOS effective mass, which carries the information about the DOS in valence band * dse 3 / 2 2 1 2m S(E) 2 ( ) 2 * dsh 3 / 2 2 1 2m S(E) 2 ( ) 2 E EC EV E 5 3-D Case 1. Conduction Band • GaAs: mdse* = me* • Silicon and Germanium: mdse* = g2/3(ml*mt*2)1/3 where the degeneracy factor g is the number of ellipsoidal constant-energy surfaces lying within the Brillouin zone. For Si, g = 6; For Ge, g = 8/2 = 4. 2. Valence Band – Ge, Si, GaAs mdsh* = ((mhh*)3/2 + (mlh*)3/2)2/3 (Here for simplicity, we do not consider the Split-off band) 6 Fermi-Dirac Statistics Fermi-Dirac distribution function gives the probability of occupancy of an energy state E if the state exists. f (E) 1 1 e ( E E f )/ kBT 1 - f(E): the probability of unfilled state E Ef: Fermi Level 7 Fermi level is related to one of laws of Nature: Conservation of Charge Extrinsic case 8 2-13 Case of EV < Ef < EC (Non-degenerate) C = (Ef – EC)/kBT Electron concentration n S ( E ) f ( E )dE EC n NC exp(C) EV p S ( E )(1 f ( E ))dE Hole concentration Effective density of states in the conduction band p NV exp(V) V= (EV – Ef)/kBT NC = 2(mdse*kBT/2ħ2)3/2 NV = 2(mdsh*kBT/2ħ2)3/2 Effective density of states in the valence band Note: for EV < Ef < EC, Fermi-Dirac distribution reduces to Boltzmann distribution. 9 Case of EV < Ef < EC (Non-degenerate) n NC exp(C) C = (Ef – EC)/kBT p NV exp(V) V= (EV – Ef)/kBT NC = 2(mdse*kBT/2ħ2)3/2 NV = 2(mdsh*kBT/2ħ2)3/2 For intrinsic case where n = p, at least four statements can be drawn: •Ef is the intrinsic Fermi level Efi •Efi is a function of the temperature T and the ratio of mdse* to mdsh* •Corresponding ni (= n = p) is the intrinsic concentration •ni is a function of the band gap (= Ec- Ev) 10 11 12 (Continued from Lecture 2) Conduction-Band Electrons and Valence-Band Holes and Electrons Hole: Vacancy of Valence-Band Electron 13 No Electrons in Conduction Bands All Valence Bands are filled up. 14 15