Order 1252715: Condensed Matter physics
tutorthammy
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Semiconductors 2
Statistical Mechanics of Semiconductors
Recall from Lecture 3 – density of states for free electrons
per unit volume
Electrons in conduction band like
free electrons but with mass m*, can write:
Similarly the density of states for holes
near the top of the valence band are:
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The Fermi-Dirac distribution for a semiconductor
• For a metal, the Fermi energy is the highest occupied energy at 0 K. The chemical potential is temperature- dependent (but not much) and so the two are essentially the same.
• For a semiconductor, the definition of the Fermi energy is not so clear. We better use the chemical potential.
• Some (many) people also use the term “Fermi energy” for semiconductors but then it is temperature-dependent.
Earlier (Lecture 3) we wrote it as:
In this Section, we call it the Fermi-Dirac function to reflect relates to
electrons – recall it gives the probability that an available energy state E will
be occupied by an electron at absolute temperature T.
Statistical Mechanics of Semiconductors
For a given chemical potential, the total number of electrons in
the conduction band as a function of temperature is:
where
For
We have
And then,
Boltzman distribution
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Statistical Mechanics of Semiconductors
Want to solve integral: multiply Eq (1) by
(1)
Standard Equation for density of electrons
Statistical Mechanics of Semiconductors Similarly can get the number of holes in the valence band p as:
When substantially above the top of the valence
band we have:
and
Standard Equation for density of holes
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Number of electrons excited into conduction band must equal number
of holes left behind in valance band so
Law of Mass Action
Intrinsic Semiconductors
Forming product of density of electrons in conduction band, and holes in the
valence band we obtain important relation:
Depends only on band gap
Dividing the density of electrons in conduction band n(T), and holes in the valence
band p(T) we obtain:
Intrinsic Semiconductors
Taking the log of both sides of the below
and solving for gives:
That is, an expression for the chemical potential – at T=0 gives exactly in
the middle of the band-gap
Using the law of Mass Action above with n=p we obtain:
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Extrinsic/Doped Semiconductors
Law of Mass Action also holds for doping when we have
Concentrations n and p
From the law of mass action we have
Consider intrinsic case:
np =
Example
gap size (eV) n in m -3
at 150 K
n in m -3
at 300 K
InSb 0.18 2x10 22
6x10 23
Si 1.11 4x10 6
2x10 16
diamond 5.5 6x10 -68
1x10 -21
Using prefactor
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Dopants, n- and p-type
Majority and minority carriers
equal number of
electrons and holes
majority: electrons
minority: holes
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Band diagram, density of states, Fermi-Dirac distribution, and the carrier concentrations at thermal equilibrium
Intrinsic
semiconductor
n-type
semiconductor
p-type
semiconductor
Consider a Si sample maintained at T = 300K under
equilibrium conditions, doped with Boron to a
concentration 2×1016 cm-3 : Given the intrinsic
concentration n i = p i is 1x10 10 cm-3
• What are the electron and hole concentrations (n
and p) in this sample? Is it n-type or p-type?
Example
Suppose the sample is doped additionally with Phosphorus
to a concentration 6×1016 cm-3.
• Is the material now n-type or p-type?
What are the n and p concentrations now?