Materials Considerations in Semiconductor Detectors–II S W McKnight and C A DiMarzio

Band Filling Concept Electron bands determined by lattice and ion core potentials Bands are filled by available conduction and valence electrons Pauli principle → only one electron of each spin in each (2π)3 volume in “k-space” Bands filled up to “Fermi energy” Fermi energy in band → metal Fermi energy in gap → insulator/semiconductor For T≠0, electron thermal energy distribution = “Fermi function”

k E Eg

Metal Band Structure Eg Ef 2π

Metal Under Electric Field Eg Ef

Insulator/Semiconductor Band Structure Eg Ef 2π

Temperature Effects “Occupancy” of state = PE,T = Probability that state at energy E will be occupied at temperature T = f(E) (“Fermi function) k=Boltzmann constant = 8.62 × 10-5 eV/K

Fermi Function Limits f(E)=0.5 for E=Ef For kT<<Ef: E<Ef → f(E) = 1 For kT<<Ef: E>Ef → f(E) = 0 For (E-Ef)/kT >> 1: Boltzmann distribution

Fermi Function vs. T Ef=1 eV kT=26 meV (300K) kT=52 meV (600K)

“Density of States”: N(E) N(E) dE = number of electron states between E and E+dE n = number of electrons per unit volume

Isotropic Parabolic Band

Density of States: Isotropic, Parabolic Bands NT(E) = number of states/unit volume with energy<E = “k-space” volume/(2π)3 (per spin direction)

Density of States: N(E)

Isotropic Band Density of States (2 spin states)

Electron-Hole Picture Conduction Band Ec Eg k Ef Ev Electron Vacancy = “Hole” Valence Band Unoccupied state

Electron-Hole Picture n=number of electrons/(unit volume) in conduction band p=number of vacancies (“holes”)/(unit volume) in valence band For intrinsic (undoped) material: n=p=ni

Integration of f(E)N(E) over Band Assume Ec-Ef >> kT → f(E) ≈ e -(E-Ef)/kT (Boltzmann distribution approximation)

Integration of f(E)N(E) over Band Use definite integral:

Intrinsic Carrier Concentration

Semiconductor Band Structures

Semiconductor Band Structures

Real Band Effects Eg

Real Band Effects Thermal Eg ≠ Optical Eg Electron effective mass ≠ Hole effective mass More than one electron/hole band Multiple “pockets” Overlapping bands Anisotropic electron/hole pockets Non-parabolic bands

Effect of Real Band Effects N(E)=sum of all bands and all pockets md*=“density of states” mass md*(electrons) ≠ md* (holes) Fermi level for T≠0 moves toward band with smaller density of states (smaller md*) ni=pi fixes position of Ef