Lecture 4. Density of States and Fermi Energy Concepts. Reading: (Cont d) Pierret
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1 C 3040 Dr. Alan Doolittle Lecture 4 Density of States and Fermi nergy Concepts Reading: (Cont d Pierret
2 C 3040 Dr. Alan Doolittle Density of States Concept Quantum Mechanics tells us that the number of available states in a cubic cm per unit of energy, the density of states, is given by: = = ev cm States of Number Joule m States of Number unit m m g m m g v v p p v c c n n c * * 3 2 * *, ( 2 (, ( 2 ( π π
3 C 3040 Dr. Alan Doolittle Density of States Concept Thus, the number of states per cubic centimeter between energy and +d is g g c v ( d if ( d if 0 otherwise c v and, and,
4 C 3040 Dr. Alan Doolittle Probability of Occupation (Fermi Function Concept Now that we know the number of available states at each energy, how do the electrons occupy these states? We need to know how the electrons are distributed in energy. Again, Quantum Mechanics tells us that the electrons follow the Fermi-distribution function. f ( and F 1 where k Boltzman constant, T Temperature in ( F kt 1+ e Fermi energy (~ average energy of our electrons in the crystal = Kelvin f( is the probability that a state at energy is occupied 1-f( is the probability that a state at energy is unoccupied
5 C 3040 Dr. Alan Doolittle An Aside about the fermi energy: If the material had an imbalance of average electron energy from one side to another, electrons would flow from the region of high energy toward the region of low energy to balance out the average energy. Thus, gradients in the average energy would result in a current (flow of electrons until the energy balance was satisfied. If an outside force maintained such an imbalance in energy from one region to another, a continuous current would result (see quasi-fermi energy concept introduced later. When assembling our crystal, if we brought extra electrons into the crystal such as in the case of donor doping, we have to do more work to introduce the extra valence electrons. Thus, the average electron energy would increase compared to the intrinsic doped case. When assembling our crystal, if we brought fewer electrons into the crystal such as in the case of acceptor doping, we have to do less work to introduce the fewer valence electrons. Thus, the average electron energy would decrease compared to the intrinsic doped case.
6 C 3040 Dr. Alan Doolittle Probability of Occupation (Fermi Function Concept At T=0K, occupancy is digital : No occupation of states above F and complete occupation of states below F At T>0K, occupation probability is reduced with increasing energy. f(= F = 1/2 regardless of temperature.
7 C 3040 Dr. Alan Doolittle Probability of Occupation (Fermi Function Concept At T=0K, occupancy is digital : No occupation of states above F and complete occupation of states below F At T>0K, occupation probability is reduced with increasing energy. f(= F = 1/2 regardless of temperature. At higher temperatures, higher energy states can be occupied, leaving more lower energy states unoccupied (1-f(.
8 Probability of Occupation (Fermi Function Concept Si xample Valence Band States f( f ( = ( F kt 1+ e 1 For < ( f -3kT: f( ~ 1-e -(-f/kt ~1 3 kt +/-3 kt 3 kt 3 kt 3 kt f =0.55 ev [ev] No States in the bandgap T=10 K, kt= ev For > ( f +3kT: f( ~ e -(-f/kt ~0 T=300K, kt= T=450K, kt=0.039 Conduction Band States C 3040 Dr. Alan Doolittle
9 Probability of Occupation (Fermi Function Concept Si xample Valence Band States f( f ( = ( F kt 1+ e 1 Insignificant emptying of valance band states Intrinsic Case [ev] No States in the bandgap f =0.55 ev T=450K, kt=0.039 Insignificant occupation of conduction band states Conduction Band States C 3040 Dr. Alan Doolittle
10 Probability of Occupation (Fermi Function Concept Si xample Valence Band States f( f ( = ( F kt 1+ e 1 f =1.05 ev Donor Doped Donor Case [ev] Significant occupation of conduction band states No States in the bandgap T=450K, kt=0.039 Conduction Band States C 3040 Dr. Alan Doolittle
11 Probability of Occupation (Fermi Function Concept Si xample Valence Band States f( f ( = ( F kt 1+ e 1 Acceptor Case Significant emptying of valance band states [ev] No States in the bandgap f =0.05 ev Donor Doped T=450K, kt=0.039 Conduction Band States C 3040 Dr. Alan Doolittle
12 C 3040 Dr. Alan Doolittle Probability of Occupation Thus, the density of electrons (or holes occupying the states in energy between and +d is: lectrons/cm 3 in the conduction band between nergy and +d g c (f(d if c and, Holes/cm 3 in the valence band between nergy and +d g v ([1- f(]d if v and, 0 otherwise
13 C 3040 Dr. Alan Doolittle Band Occupation
14 C 3040 Dr. Alan Doolittle Intrinsic nergy (or Intrinsic Level f is said to equal i (intrinsic energy when qual numbers of electrons and holes
15 C 3040 Dr. Alan Doolittle Correction for the Presence of Additional Dopant States Intrinsic: qual number of electrons and holes n-type: more electrons than holes p-type: more holes than electrons
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Solid State Device Fundamentals ES 345 Lecture Course by Alexander M. Zaitsev alexander.zaitsev@csi.cuny.edu Tel: 718 982 2812 Oice 4101b 1 The ree electron model o metals The ree electron model o metals