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In all semiconductors, lattice defects change the electronic properties of the
material locally, and this may result in electronic energy states in the band gap of the semiconductor and this is true
for all kinds of lattice defects |
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Semiconductor technology actually depends completely on this fact. Doping a semiconductor, after all, mostly means the incorporation of (usually) substitutional extrinsic point defects in defined concentrations in defined regions
of the crystal - we have B, As and P for Si. |
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Our extrinsic point defects now exist in two states: we have some concentration [P]0
of e.g. a neutral donor like P and some concentration [P]+ of ionized donors; and [P]0
+ [P]+ = [P0], the total concentration of P holds at all conditions. |
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The concentration [P]+ is simply given by the the total concentration times
the probability that the electronic state associated with the P impurity atom is not
occupied by an electron. |
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If this electrons state is at an energy ED in the band gap, basic semiconductor physics tells us that for a given
EF and temperature T the concentration of ionized impurity atoms is given by
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| [P+] |
= [P0] · |
{1 – f(ED, EF; T)} | | | |
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» [P0] · |
exp |
EF – ED
kT |
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There is no reason whatsoever that a vacancy (or any other point defect you care
to come up with) should not have a energy level (or even more than one) in the band gap of its host semiconductor. This
level then will be occupied or not occupied by electrons exactly like the extrinsic point defect. |
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If the vacancy is mobile at the temperature considered, it will diffuse around - exactly like
an extrinsic mobile defect. |
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If the temperature changes, the intrinsic point defects concentration changes to the extent
that it can establish equilibrium - in pronounced contrast to the extrinsic point defects. |
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It should be clear form this, that intrinsic point defects in semiconductors are
not all that simple. Charge states must be considered that depend on primary doping with extrinsic point defects and temperature.
If things get really messy, the intrinsic point defects change the actual doping and their mobility (or diffusion coefficient)
depends on their charge state. |
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Looking at jus at few topics in the case of Si, we obtain a bunch of complex
relations, which shall only be touched upon: |
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Once again, the equilibrium concentration of charged point defects depends on the Fermi energy
EF (which is the chemical potential of the electrons). As an example, for a negatively charged vacancy we obtain
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| c(V – ) |
= c(V) · exp |
EF – EA
kT |
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With EF = Fermi energy, and EA = acceptor level of the vacancy in the band gap. |
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This tells us that besides the formation energies and entropies, we now also must know the
energy levels of the defects in the band gap! |
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The dependence of the concentration of arbitrarily charged point defects on the carrier concentration
(i.e. on doping) is given by |
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cVx(n)
cVx(ni) | = |
æ
ç
è |
n
ni |
ö
÷
ø |
–x |
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With ni, n = (intrinsic) carrier density, x
= charge state of point defect. |
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As a Si special, we also must consider
self-interstitials (which, if you remember, we always can safely neglect
for just about any other elemental crystal) |
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Local equilibrium between vacancies and interstitials follows this relation: |
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| cV(loc) · ci(loc) |
» |
cV(equ) · ci(equ) |
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Considering that carrier densities and the Fermi energy depend on the temperature,
too, things obviously get complicated! |
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It thus should not be a big surprise that the scientific community still has not
come up with reliable, or least undisputed numbers for the basic properties of intrinsic point defects in Si, not
to mention the more complicated semiconductors. |
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But do not let yourself be deceived by this: While you
might have problems coming up with numbers for e.g. the vacancy concentration in Si at some temperature and so on,
the Si crystal has no problems whatsoever to "produce" the concentration
that is just right for this condition. |
© H. Föll (Defects - Script)
