EE 60556: Fundamentals of Semiconductors Lecture Note

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EE 60556 Fundamentals of SemiconductorsLecture
Note 17 (10/29/09) Carrier concentration when
given Ef, nondegenerate semiconductor, heavy
doping effects

• Outline
• Last class intrinsic and extrinsic
  semiconductors, dopants, defects,
• Carrier concentration at equilibrium for a given
  Fermi level
• Nondegenerately doped semiconductor (Boltzmann
  approximation applies)
• Heavy doping effects
• Reading materials Pierret Ch.4

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Fermi-Dirac Statistics and n p at equilibrium
(quasi-Fermi level at non-equilibrium more
later)
Ef Fermi level
Here S(E) is DOS or g(E)
n0 or P0 or
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EC,top ? 8 EC
Step 1 It will be nice to reduce limits of
integrals to 0 or 8!
Conduction Band
EV EV,bottom ? 8
Step 2 It will be nice to get rid of the
integral! This means it will be nice to consider
two effective energy levels instead of energy
bands.
Valence Band
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Fermi-Dirac Statistics and n p at equilibrium
With a reasonable approximation treating the
C.B. or V.B to be infinitely wide in energy.
Recall, for 3D,
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• The effective density of C.B. or V.B. states.
• It simplifies the entire conduction (or valence)
  band to one single energy level.
• Note do not confuse this with the general
  definition of DOS.

Conduction Band
Constant (T, m) distribution function (Ec-Ef,
Ec-Ef)
The effective mass of density of states
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When Fermi-Dirac integral Boltzmann, we
call the semiconductor is non-degenerately doped
EC EV
3kT gt small error in Boltzmann approx.
EF in this range
Useful when Ef is known, solve for n
Useful when n is known, solve for Ef
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Fermi-Dirac Statistics and n p
Streetman and Banerjee Figure 316 Schematic band
diagram, density of states, FermiDirac
distribution, and the carrier concentrations for
(a) intrinsic, (b) n-type, and (c) p-type
semiconductors at thermal equilibrium.
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Equations for n0 and p0 (under thermal
equilibrium) in non-degenerate semiconductors
Boltzmann distribution approximates closely
Fermi-Dirac distribution and ex Fermi-Dirac
integral.
gt 3kT
gt 3kT
relative to intrinsic level
relative to band edges
e2.3 10, therefore, if n gt NC/10, it is
degenerate and Fermi-Dirac integral has to be
used to be accurate. Fermi level is close to the
band edge! Often, to the first order
approximation, we assume Ef is at the band edge
for degenerately doped semiconductors.
n0 p0?
Mass action law n0 p0ni2
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Intrinsic carrier concentration

How to determine Ei?
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Under high doping concentrations, the formerly
discrete donor levels smear into a band,
effectively narrowing the band gap by an amount
?Eg. Figure 2.21
Plot of energy band diagram of a degenerate
semiconductor. The intrinsic energy gap Eg0, the
actual energy band gap Eg, and the apparent band
gap Eg are indicated. Figure 2.24
Anderson 2005
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The states for the higher donor levels can
overlap if the doping concentration is high
enough (dopant atoms close enough
together). Figure 2.22
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Reduction of room-temperature band gap ?Eg as a
function of donor density in phosphorus-doped
silicon. Figure 2.23
Apparent band-gap narrowing as a function of
impurity concentration for uncompensated n-type
and p-type Si. Figure 2.25
Anderson 2005