1Density of StatesandFermi Energy ConceptsHow do Electrons and Holes Populate the Bands?dEEgc)(The number of conduction band states/cm3lying in the energy range between E and E + dE (if E ≥ Ec).The number of valence band states/cm3lying in the energy range between E and E + dE(if E ≤ Ev).dEEgv)( Density of States ConceptGeneral energy dependence of gc(E) and gv(E) near the band edges.How do Electrons and Holes Populate the Bands? Density of States ConceptQuantum Mechanics tells us that the number of available statesin a cm3per unit of energy, the density of states, is given by:Density of States in Conduction BandDensity of States in Valence BandHow do electrons and holes populate the bands? Probability of Occupation (Fermi Function) Concept¾ Now that we know the number of available states at each energy,then how do the electrons occupy these states?¾ We need to know how the electrons are “distributed in energy”.¾ 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) f(E) is the probability that a state at energy E is occupied. 1-f(E) is the probability that a state at energy E is unoccupied.()kTEEfeEf/)(11−+=2 Fermi-Dirac DistributionHow do electrons and holes populate the bands? How do electrons and holes populate the bands? Probability of Occupation (Fermi function) Concept At T=0K, occupancy is “digital”: No occupation of states above Efand complete occupation of states below Ef. At T>0K, occupation probability is reduced with increasing energy.f(E=Ef) = 1/2 regardless of temperature.()kTEEfeEf/)(11−+=kT = 0.0259eV @300KHow do electrons and holes populate the bands? Probability of Occupation (Fermi function) Concept At T=0K, occupancy is “digital”: No occupation of states above Efand complete occupation of states below Ef. At T>0K, occupation probability is reduced with increasing energy.f(E=Ef) = 1/2 regardless of temperature. At higher temperatures, higher energy states can be occupied, leaving more lower energy states unoccupied [1 - f(Ef)].()()kTEEfeEf/11−+=kT = 0.0259eV @300KHow do electrons and holes populate the bands? Probability of Occupation (Fermi function) Concept3How do electrons and holes populate the bands?Example 2.2The 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?SolutionThe 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).() ()VCEfEf −= 1()()kTEECFCeEf/11−+=()() ()kTEEkTEEVVFFVeeEf//111111−−+=+−=−kTEEkTEEFVFC−=−2VCFEEE+=The density of electrons (or holes) occupying the statesin energy between E and E + dE is:How do electrons and holes populate the bands? Probability of Occupation Concept0 Otherwise dEEfEgc)()(Electrons/cm3in the conduction band between E and E + dE (if E ≥ Ec).Holes/cm3in the conduction band between E and E + dE (if E ≤ Ev).dEEfEgv)()( Fermi function and Carrier ConcentrationHow do electrons and holes populate the bands? How do electrons and holes populate the bands? Probability of Occupation Concept4 Typical band structures of SemiconductorEvEc0Ec+χEFVBCBEg(E)g(E)∝(E–Ec)1/2fE)EFEForelectronsFor holes[1–f(E)]Energy band diagramDensity of states Fermi-Diracprobability functionprobability of occupancy of a statenE(E)orpE(E)EnE(E)pE(E)Area = pAreaEcEvndEEnE==∫)(g(E) X f(E)Energy density of electrons in the CBnumber of electrons per unit energy per unit volumeThe area under nE(E) vs. E is the electron concentration.number of states per unit energy per unit volumeHow do electrons and holes populate the bands?¾ 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). ¾ The population depends upon the product ofthe Fermi function and the electron density of states. So in the gap there are no electrons because the density of states is zero.¾ In the conduction band at 0K, there are no electrons even though there are plenty of available states, but the Fermi function is zero.¾ 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. Fermi function and Carrier ConcentrationHow do electrons and holes populate the bands?Intrinsic Semiconductor.movHow do electrons and holes populate the bands? Current Flow of Intrinsic SemiconductorHow do electrons and holes populate the bands? Energy Band Occupation5How do electrons and holes populate the bands? Intrinsic Energy (or Intrinsic Level)Ef is said to equal Ei(intrinsic energy) when…equal number of electrons and holes.IntrinsicEqual numberof electronsand holesn-typeMore electrons than Holesp-typeMore holes than electronsHow do electrons and holes populate the bands? Additional Dopant States Energy band diagramsnp = ni2Note that donor and acceptor energy levels are not shown.EcEvEFiCBEFpEFnEcEvEcEvVB(a) intrinsic(c) p-type(b) n-typeIntrinsic, n-Type, p-Type SemiconductorsCBg(E)EImpuritiesforming a bandEFpEvEcEFnEvEcCBVBDegenerated n-type semiconductorLarge number of donors form a band that overlaps the CBDegenerated p-typesemiconductorHow do electrons and holes populate the bands? Heavily Doped Dopant