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Density of States and Fermi Energy Concepts How do Electrons and Holes Populate the Bands? q Density of States Concept The number of conduction band states/cm3 lying in the energy range between E and E + dE (if E ³ Ec). The number of valence band states/cm3 lying in the energy range between E and E + dE (if E £ Ev). General energy dependence of gc (E) and gv (E) near the band edges. How do Electrons and Holes Populate the Bands? q Density of States Concept Quantum Mechanics tells us that the number of available states in a cm3 per unit of energy, the density of states, is given by: Density of States in Conduction Band Density of States in Valence Band How do electrons and holes populate the bands? q Probability of Occupation (Fermi Function) Concept Ø Now that we know the number of available states at each energy, then how do the electrons occupy these states? Ø We need to know how the electrons are “distributed in energy”. Ø Again, Quantum Mechanics tells us that the electrons follow the “Fermi-distribution function”. Ef ≡ Fermi energy (average energy in the crystal) k ≡ Boltzmann constant (k=8.617´10-5eV/K) T ≡Temperature in Kelvin (K) v f(E) is the probability that a state at energy E is occupied. v 1-f(E) is the probability that a state at energy E is unoccupied. Ø Fermi function applies only under equilibrium conditions, however, is universal in the sense that it applies with all materials-insulators, semiconductors, and metals. How do electrons and holes populate the bands? q Fermi-Dirac Distribution Ef How do electrons and holes populate the bands? q Probability of Occupation (Fermi function) Concept kT = 0.0259eV @300K v At T=0K, occupancy is “digital”: No occupation of states above Ef and complete occupation of states below Ef . v At T>0K, occupation probability is reduced with increasing energy. f(E=Ef ) = 1/2 regardless of temperature. How do electrons and holes populate the bands? q Probability of Occupation (Fermi function) Concept kT = 0.0259eV @300K v At T=0K, occupancy is “digital”: No occupation of states above Ef and complete occupation of states below Ef . v At T>0K, occupation probability is reduced with increasing energy. f(E=Ef ) = 1/2 regardless of temperature. v At higher temperatures, higher energy states can be occupied, leaving more lower energy states unoccupied [1 - f(Ef )]. How do electrons and holes populate the bands? q Probability of Occupation (Fermi function) Concept Ø If E ³ Ef +3kT Ú Ø Consequently, above Ef +3kT the Fermi function or filled-state probability decays exponentially to zero with increasing energy. How do electrons and holes populate the bands? Example 2.2 The probability that a state is filled at the conduction band edge (Ec) is precisely equal to the probability that a state is empty at the valence band edge (Ev). Where is the Fermi energy locate? Solution The Fermi function, f(E), specifies the probability of electron occupying states at a given energy E. The probability that a state is empty (not filled) at a given energy E is equal to 1- f(E). How do electrons and holes populate the bands? q Probability of Occupation Concept The density of electrons (or holes) occupying the states in energy between E and E + dE is: Electrons/cm3 in the conduction band between E and E + dE (if E ³ Ec). Holes/cm3 in the conduction band between E and E + dE (if E £ Ev). 0 Otherwise How do electrons and holes populate the bands? q Fermi function and Carrier Concentration How do electrons and holes populate the bands? q Probability of Occupation Concept How do electrons and holes populate the bands? Fermi-Dirac distribution function describing the probability that an allowed state at energy E is occupied by an electron. The density of allowed states for a semiconductor as a function of energy; note that g(E) is zero in the forbidden gap between Ev and Ec. The product of the distribution function and the density-of-states function How do electrons and holes populate the bands? q Typical band structures of Semiconductor g (E) µ (E–Ec)1/2 E E E Ec+c [1–f(E)] CB For Area electrons Ec nE(E) Ec number of number of electrons per unit states per unit probability of energy per unit volume EF energy per unit EF occupancy of The area under nE(E) vs. E is the volume a state electron concentration. Ev Ev pE(E) Area = p For holes VB 0 g(E) fE) nE(E) or pE(E) Energy band Density of states Fermi-Dirac g(E) X f(E) diagram probability Energy density of electrons in function the CB Metals vs. Semiconductors Ef Ef Metal Semiconductor Ø Allowed electronic-energy-state systems for metal and semiconductors. Ø States marked with an X are filled; those unmarked are empty. Metals vs. Semiconductors q Allowed electronic-energy states g(E) Fermi level Ef immersed in the The Fermi level Ef is at an intermediate continuum of allowed states. energy between that of the conduction band edge and that of the valence band edge. Ef Ef Metal Semiconductor How do electrons and holes populate the bands? q Fermi function and Carrier Concentration Ø Note that although the Fermi function has a finite value in the gap, there is no electron population at those energies. (that's what you mean by a gap) Ø The population depends upon the product of the Fermi function and the electron density of states. So in the gap there are no electrons because the density of states is zero. Ø In the conduction band at 0K, there are no electrons even though there are plenty of available states, but the Fermi function is zero. Ø At high temperatures, both the density of states and the Fermi function have finite values in the conduction band, so there is a finite conducting population. How do electrons and holes populate the bands? q Energy Band Occupation How do electrons and holes populate the bands? q Intrinsic Energy (or Intrinsic Level) Ef is said to equal equal number of Ei (intrinsic energy) electrons and holes. when… How do electrons and holes populate the bands? q Additional Dopant States Intrinsic Equal number of electrons and holes n-type More electrons than Holes p-type More holes than electrons How do electrons and holes populate the bands? q Pure-crystal energy-band diagram How do electrons and holes populate the bands? q n-type material How do electrons and holes populate the bands? q p-type material Intrinsic, n-Type, p-Type Semiconductors q Energy band diagrams CB Ec Ec Ec Ef n Ef i Ef p Ev Ev Ev VB (a) intrinsic (b) n-type (c) p-type np = ni2 Note that donor and acceptor energy levels are not shown. How do electrons and holes populate the bands? q Heavily Doped Dopant States E CB CB EFn Impurities forming Ec Ec bands g(E) Ev Ev EFp VB Degenerated n-type semiconductor Degenerated p-type Large number of donors form a semiconductor band that overlaps the CB

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posted: | 7/1/2013 |

language: | English |

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