1ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityHandout 14Statistics of Electrons in Energy BandsIn this lecture you will learn:ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityExample: Electron Statistics in GaAs - Conduction BandConsider the conduction band of GaAs near the band bottom at the -point:eeemmmM1000100011This implies the energy dispersion relation near the band bottom is:ecezyxccmkEmkkkEkE22222222Suppose we want to find the total number of electrons in the conduction band:We can write the following summation:FBZin2kckfN2ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityfcKTEkEcEkEfkffcexp11Where the Fermi-Dirac distribution function is:We convert the summation into an integral:KTEkEkdVkfNfckcexp11222FBZ33FBZ in Then we convert the k-space integral into an integral over energy:??FBZ33exp1122fcfcEEfEgdEKTEkEkdVNWe need to find the density of states function gc(E) for the conduction band and need to find the limits of integrationExample: Electron Statistics in GaAs - Conduction BandFBZin2kckfNAnother way of writing itEfECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityDensity of States in Energy BandsEnergyxkaasEssV4akVEkExsssxcos2sssVE2sssVE2Consider the 1D energy band that results from tight binding: We need to find the density of states function g1D(E):dEEgLdEdEdkLdkLdkLssssssssssssxVEVEDVEVExaxaaxk221220FBZ in 224222akaVdkdExssxsin2 221212sssDEEVaEgEEgD1sEsssVE2sssVE23ECE 407 – Spring 2009 – Farhan Rana – Cornell University??FBZ33exp1122fcfcEEfEgdEKTEkEkdVNExample: Electron Statistics in GaAs - Conduction BandecezyxccmkEmkkkEkE22222222Energy dispersion near the band bottom is:Electrons will only be present near the band bottom(parabolic and isotropic)Since the electrons are likely present near the band bottom, we can limit the integral over the entire FBZ to an integral in a spherical region right close to the -point:  point32FBZ3384222fccEkEfdkkVkfkdVNEfECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityExample: Electron Statistics in GaAs - Conduction Band point32842fcEkEfdkkVNSince the Fermi-Dirac distribution will be non-zero only for small values of k, one can safely extend the upper limit of the integration to infinity:032842fcEkEfdkkVNemdkkdkdE2ceEEmk 22andWe have finally:ecezyxccmkEmkkkEkE22222222cEfcfcEEfEgdEVEkEfdkkVN032842We know that:Ef4ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityWe have finally:Where the conduction band density of states function is:cecEEmEg 2322221cEfcfcEEfEgdEVEkEfdkkVN032842EEgccEExample: Electron Statistics in GaAs - Conduction BandThe density of states function looks like that of a 3D free electron gas except that the mass is the effective mass and the density of states go to zero at the band edge energy emcEEfECE 407 – Spring 2009 – Farhan Rana – Cornell UniversitycecEEmEg 2322221cEfcEEfEgdEnfEEfEfEEgcEfKTfEEeEEffIf then one may approximate the Fermi-Dirac function as an exponential:KTEEfcKTEEKTEEEEffffexpexp11KTEENEEfEgdEnfccEfccexpcEWhere:23222KTmNecMaxwell-Boltzman approximationExample: Electron Statistics in GaAs - Conduction BandEffective density of states (units: #/cm3)5ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityExample: Electron Statistics in GaAs - Valence Band and HolesEf• At zero temperature, the valence band is completely filled and the conduction band is completely empty• At any finite temperature, some electrons near the top of the valence band will get thermally excited from the valence band and occupy the conduction band and their density will be given by:• The question we ask here is how many empty states are left in the valence band as a result of the electrons being thermally excited. The answer is (assuming the heavy-hole valence band):• We call this the number of “holes” left behind in the valence band and the number of these holes is P:KTEENnfccexpFBZ in 12kfhhEkEffhhkfhhEkEfkdVEkEfP 12212FBZ33FBZ in ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityExample: Electron Statistics in GaAs - Valence Band and Holesparabolic approx.fhhEkEfkdVP 122FBZ33hhvhhzyxhhmkEmkkkEvkE22222222Energy dispersion near the top of the valence band is:Holes will only be present near the top of the valence bandSince the holes are likely present near the band maximum, we can limit the integral over the entire FBZ to an integral in a spherical region right close to the -point:  point321842fhhEkEfdkkVPEf6ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityExample: Electron Statistics in GaAs - Valence Band and Holes0321842fhhEkEfdkkVPSince the Fermi-Dirac distribution will be non-zero only for small values of k, one can safely extend the upper limit of the integration to infinity: