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First some numbers from the literature. According to "Semiconductor
Materials", the intrinsic electrical conductivity of Si at 300 K is
3.16 µS/cm. The NSM
archive has rather similar numbers. |
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[S] = "Siemens" is a quaint German measure of conductivity, it is simply
1/W |
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This translates into a room temperature resistivity r of
r = 1/s = 316 000 Wcm. |
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Alternatively , numbers for the intrinsic carrier density found in the sources given above
or in arbitrary books are somewhere in between 1.00 · 1010 cm– 3 or
1.38 · 1010 cm– 3. |
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Lets see if we can get numbers like this by calculation: |
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The carrier density is given by
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| ne = Neeff · exp – |
EC – EF
kT |
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Neeff can
be estimated from the free electron gas model in a fair approximation to |
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| Neeff | = 2 · |
æ
ç
è |
2 p m kT
h2
| ö
÷
ø |
3/2 |
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The dimension of this Neeff is |
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| [Neff] | = |
kg3/2 · eV3/2 · eV– 3 · s– 3 |
= kg3/2 · eV– 3/2 · s– 3 |
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That is a bit strange. Nevertheless it is right - try to do something about the kg! If you have
problems of figuring out how to get the proper dimension m– 3, use
the link. |
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Inserting numbers (me = 9,109 · 10– 31 kg; k·T
= 1/40 eV, h2 = (4,1356 · 10– 18)2 eV2s2 = 1,71 ·
10– 35 eV2s2), we obtain |
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| Neff | = |
4.59 · 1015 · T3/2 cm–3 |
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= |
2.384 · 1019 cm–3 | |
T = 300 K | | |
= |
2.384 · 1025 m–3 |
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The intrinsic carrier density thus is |
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| ne = 3.22 · 1019 cm– 3 · exp –
| EC – EF
kT |
= 3.22 · 1019 cm– 3 · exp – |
0.55 eV
0.025 eV |
= 9 · 109cm– 3 |
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This is just a little bit smaller than than the values given above; rather
amazing, considering that the free electron gas model is just a very simple approximation. |
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Now we can see what kind of mobility m
we would get with ni = 1 · 1010 cm– 3 and a
conductivity s = 3.16 µS/cm = 3.16 · 10– 6
W – 1cm– 1 |
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We had the simple law
s = 2eµni (the factor two takes into account that
we have holes and electrons), and thus µ = s/2e ni.
This gives us |
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| µ | = |
3.16 · 10– 6
2 · 1,602 · 10– 19
· 1 · 1010 |
W– 1 · cm– 1 · C–
1 · cm3 |
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With [W]= [V/A] = [V · s/C] we have |
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| µ | = |
986 cm2 · s– 1 · V– 1 |
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as an expected result. The unit [cm2 · V– 1 · s–
1] comes from the original definition of µ,
which was (drift) velocity divided by field strength. |
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Looking around a bit we find tabulated values of, e.g., 1400 cm2/Vs, which is just off by a factor of two - and that
we do not take seriously. So, what have we learned so far? |
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1. It is not so easy to really calculate the intrinsic
properties. Getting the right order of magnitudes is already pretty good. This is due, of course, to the fact that we have
approximations a plenty, coupled with lots of exponentials which are quite sensitive to small changes in the argument. |
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2. If we accept an intrinsic carrier concentration for one kind of carrier at room
temperature around ni = 1 · 1010 cm–3, we would need a dopant
concentration that is at least an order of magnitude smaller if we want to claim truly intrinsic properties. That means we demand
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| Ndop |
£ |
1 · 10– 9 cm– 3 £ 20 ppqt
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Find out what ppqt means yourself. |
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The minimum doping concentration Nmin achievable (corresponding to
the maximum resistivity rmax
of 1000 Wcm or the minimum conductivity
smin of 1 · 10– 3
W– 1 · cm– 1) must be about |
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| Nmin | = |
316 000
1000 |
» 300 ni = 3 · 1012
cm– 3 |
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In the "master" curve
for resistivity vs. doping, we find a value between 5 · 1012 cm– 3 and 1
· 1013 cm– 3, so again we are close enough. |
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The final answer thus is: We are still at least a factor of 100 away from "perfection"
with respect to unwanted doping. |
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And of course, we can not make any statement about the perfection achieved with respect to impurities that
do not influence the carrier concentrations |
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© H. Föll (Semiconductors - Script)
Ohm's Law and Materials Properties 
With frame