States and state filling So far, we saw how to calculate bands for solids Kronig-Penny was a simple example Real bandstructures more complex Often look like free electrons with effective mass m* Given E-k, we can calculate ‘density of states’ High density of conducting states would imply metallicity ECE 663

Carrier Statistics Carrier populations depend on number of available energy states (density of states) statistical distribution of energies (Fermi-Dirac function) Assume electrons act ‘free’ with a parametrized effective mass m* ECE 663

Labeling states allows us to count them! Si GaAs ECE 663

Where are the states? E dE dE dk k x k For 1D parabolic bands, DOS peaks at edges ECE 663

Where are the states? E dE dE dk k x k (# states) = 2(dk/Dk) = Ldk/p Dk = 2p/L (# states) = 2(dk/Dk) = Ldk/p DOS = g = # states/dE = (L/p)(dk/dE) ECE 663

Analytical results for simple bands dE dE dk k x k Dk = 2p/L E = h2k2/2m* + Ec  dk/dE = m/2ħ2(E-Ec) DOS = Lm*/2p2ħ2(E-Ec) ~ 1/(E-Ec) ECE 663

Increasing Dimensions # k points increases due to angular integral along circumference, as (E-Ec) dNs = 2 x 2pkdk/(2p/L)2 g ~ Sq(E-Ec), step fn E dE dE dE dk 2pkdk . k k In higher dimensions, DOS has complex shapes ECE 663

From E-k to Density of States Σ1 dNs = g(E)dE = 2  2 (dk/[2p/L]) for each dimension k Use E = Ec + ħ2k2/2mc to convert kddk into dE ECE 663

From E-k to DOS for free els Ec DOS 3-D 2-D 1-D (Wmc/2p2ħ3)[2mc(E-Ec)]1/2 (Smc/2pħ2)q(E-Ec) (mcL/pħ)/√2mc[E-Ec] ECE 663

Real DOS needs computation VB CB E (eV) ECE 663

Keep in mind 3-D Density of States In the interest of simplicity, we’ll try to reduce all g‘s to this form… ECE 663

(Ev-E) (Ev-E) lh ECE 663

What if ellipsoids? Such that number of states is preserved E – EC = ħ2k12/2ml* + ħ2k22/2mt* + ħ2k32/2mt* b a 1 = k12/a2 + k22/b2 + k32/b2 a = 2ml*(E-EC)/ħ2 b = 2mt*(E-EC)/ħ2 ECE 663

What if ellipsoids? mn* = (ml*mt*2)1/3(Nel)2/3 a = 2ml*(E-EC)/ħ2 b = 2mt*(E-EC)/ħ2 Total k-space volume of Nel ellipsoids = (4pab2/3)Nel where k-space volume of equivalent sphere = (4pk3/3) k = 2mn*(E-EC)/ħ2 where mn* = (ml*mt*2)1/3(Nel)2/3 Equating K-space Volumes ECE 663

Valence bands more complex where ECE 663

Valence bands more complex hh lh But can try to fit two paraboloids for heavy and light holes and sum ECE 663

Valence bands more complex hh lh 4pk3/3 = 4pk13/3 + 4pk23/3 k = 2mp*(EV-E)/ħ2 k1 = 2mhh(EV-E)/ħ2 k2 = 2mlh(EV-E)/ħ2 ECE 663

So map onto 3D isotropic free-electron DOS gC(E) = mn*[2mn*(E-EC)]/p2ħ3 gV(E) = mp*[2mp*(EV-E)]/p2ħ3 with the right masses mn* = (ml*mt*2)1/3(Nel)2/3 mp* = (mhh3/2 + mlh3/2)2/3 ECE 663

Density of states effective mass for various solids mn* = (ml*mt*2)1/3(Nel)2/3 mp* = (mhh3/2 + mlh3/2)2/3 ml* mt* Nel mhh mlh mn* mp* GaAs Si Ge 1 6 8 0.98 0.19 0.49 0.16 1.64 0.082 0.067 0.45 0.28 0.042 0.067 0.473 1.084 0.5492 0.89 0.29 ECE 663

But how do we fill these states?

Fermi-Dirac Function Find number of carriers in CB/VB - need to know Number of available energy states (g(E)) Probability that a given state is occupied (f(E)) Fermi-Dirac function derived from statistical mechanics of “free” particles with three assumptions: Pauli Exclusion Principle – each allowed state can accommodate only one electron The total number of electrons is fixed N=Ni The total energy is fixed ETOT =  EiNi ECE 663

Fermi-Dirac Function ECE 663

Carrier concentrations Can figure out # of electrons in conduction band And # of holes in valence band gc(E) State Density Occupancy per state El. Density f(E) ECE 663

Full F-D statistics ECE 663

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Carrier Concentrations If EC-EF >> kT the integral simplifies – nondegenerate semiconductors Fermi level more than ~3kT away from bottom/top of band We then have h << -1, so drop +1 in denominator of F1/2 function For electrons in conduction band: Or equivalently CB lumped density of states ECE 663

Carrier Concentrations Can do same thing for holes (nondegenerate approximation) VB lumped density of states ECE 663

Intrinsic Semiconductors For every electron in the CB there is a hole in the VB Fermi level is in middle of bandgap if effective masses not too different for e and h ECE 663

Intrinsic Semiconductors Can plug in Fermi energy to find intrinsic carrier concentrations Electrical conductivity proportional to n so intrinsic semiconductors have resistance change with temperature (thermistor) but not useful for much else. ECE 663

Doped Semiconductors Boron has 1 less el than P has 1 more el than tetrahedral Extra el loosely tied (why?) It n-dopes Si (1016/cm3 means 1 in 5 million) Boron has 1 less el than tetrahedral It steals an el from a nb. Si to form a tetrahedron. The deficit ‘hole’ p-dopes Si ECE 663

Carrier Concentrations - nondegenerate Independent of Doping (This is at Equilibrium) ECE 663

A more useful form for n and p For intrinsic semiconductors n = nie(EF-Ei)/kT p = nie(Ei-EF)/kT EF Ei EF Ei ECE 663

Charge Neutrality Poisson’s Equation In equilibrium, E=0 and =0 - - Relationship Number of ionized donors: (gD = 2 for e’s, 4 for light holes, 1 for deep traps) ECE 663

Charge Neutrality ED “1” PN  e-(EN-EFN)/kT “0” ED “1” PN  e-(EN-EFN)/kT “0” <N> = 0.P0 + 1.P1 = f(ED-EF) = 1/[1 + e-(EF-ED)/kT] (gD = 2 for e’s, 4 for light holes, 1 for deep traps) ECE 663

Charge Neutrality   2ED + U0 PN  e-(EN-EFN)/kT   ED <N> = 0.P0 + 1.(P + P) + 2.P (gD = 2 for e’s, 4 for light holes, 1 for deep traps) ECE 663

Charge Neutrality Can equivalently alter ED to account for degeneracy n ECE 663

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Nondegenerate Fully Ionized Extrinsic Semiconductors n-type (donor) ND >> NA, ND >> ni n  ND p = ni2/ND p-type (acceptor) NA >> ND, NA >> ni p  NA n = ni2/NA ECE 663

Altering Fermi Level with doping (… and later with fields) Recall: Take ln n-type p-type ECE 663

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In summary Labeling states with ‘k’ index allows us to count and get a DOS In simple limits, we can get this analytically The Fermi-Dirac distribution helps us fill these states For non-degenerate semiconductors, we get simple formulae for n and p at equilibrium in terms of Ei and EF, with EF determined by doping Let’s now go away from equilibrium and see what happens