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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
rof 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 pm kT
h2
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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 |
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T = 300 K |
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= |
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/2eni. 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, 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 (Semiconductor - Script)
Ohms Law and Materials Properties 
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