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Published byAlison Mason Modified about 1 year ago

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GOAL: The density of electrons (n o ) can be found precisely if we know 1. the number of allowed energy states in a small energy range, dE: S(E)dE “the density of states” 2. the probability that a given energy state will be occupied by an electron: f(E) “the distribution function” n o = band S(E)f(E)dE

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Energy Band Diagram conduction band valence band ECEC EVEV x E(x) S(E) E electron E hole note: increasing electron energy is ‘up’, but increasing hole energy is ‘down’.

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Reminder of our GOAL: The density of electrons (n o ) can be found precisely if we know 1. the number of allowed energy states in a small energy range, dE: S(E)dE “the density of states” 2. the probability that a given energy state will be occupied by an electron: f(E) “the distribution function” n o = band S(E)f(E)dE

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Fermi-Dirac Distribution The probability that an electron occupies an energy level, E, is f(E) = 1/{1+exp[(E-E F )/kT]} –where T is the temperature (Kelvin) –k is the Boltzmann constant (k=8.62x10 -5 eV/K) –E F is the Fermi Energy (in eV) –(Can derive this – statistical mechanics.)

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f(E) = 1/{1+exp[(E-E F )/kT]} All energy levels are filled with e - ’s below the Fermi Energy at 0 o K f(E) 1 0 EFEF E T=0 o K T 1 >0 T 2 >T 1 0.5

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Fermi-Dirac Distribution for holes Remember, a hole is an energy state that is NOT occupied by an electron. Therefore, the probability that a state is occupied by a hole is the probability that a state is NOT occupied by an electron: f p (E) = 1 – f(E) = 1 - 1/{1+exp[(E-E F )/kT]} ={1+exp[(E-E F )/kT]}/{1+exp[(E-E F )/kT]} - 1/{1+exp[(E-E F )/kT]} = {exp[(E-E F )/kT]}/{1+exp[(E-E F )/kT]} =1/{exp[(E F - E)/kT] + 1}

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The Boltzmann Approximation If (E-E F )>kT such that exp[(E-E F )/kT] >> 1 then, f(E) = {1+exp[(E-E F )/kT]} -1 {exp[(E-E F )/kT]} -1 exp[-(E-E F )/kT] …the Boltzmann approx. similarly, f p (E) is small when exp[(E F - E)/kT]>>1: f p (E) = {1+exp[(E F - E)/kT]} -1 {exp[(E F - E)/kT]} -1 exp[-(E F - E)/kT] If the Boltz. approx. is valid, we say the semiconductor is non-degenerate.

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Putting the pieces together: for electrons, n(E) f(E) 1 0 EFEF E T=0 o K T 1 >0 T 2 >T 1 0.5 EVEV ECEC S(E) E n(E)=S(E)f(E)

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Putting the pieces together: for holes, p(E) f p (E) 1 0 EFEF E T=0 o K T 1 >0 T 2 >T 1 0.5 EVEV ECEC S(E) p(E)=S(E)f(E) hole energy

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Energy Band Diagram intrinisic semiconductor: n o =p o =n i conduction band valence band ECEC EVEV x E(x) n(E) p(E) E F =E i where E i is the intrinsic Fermi level

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Energy Band Diagram n-type semiconductor: n o >p o conduction band valence band ECEC EVEV x E(x) n(E) p(E) EFEF

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Energy Band Diagram p-type semiconductor: p o >n o conduction band valence band ECEC EVEV x E(x) n(E) p(E) EFEF

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The intrinsic carrier density is sensitive to the energy bandgap, temperature, and m *

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