User`s guide

EE PRO for TI-89, 92 Plus
Equations - Solid State Devices
81
Dn
kTT
q
n=
⋅
⋅
µ
Eq. 27.1.3
Dp
kTT
q
p=
⋅
⋅
µ
Eq. 27.1.4
The Fermi level EF is a measure of the chemical potential in silicon and is used to estimate the doping density. For
instance in a n-type semiconductor, equation 27.1.5 is used to determine the location of the Fermi level, while in a p-
type material equation 27.1.6 is used to establish the Fermi level. In some cases, when both donor and acceptor levels
are specified, one has to choose either equation 27.1.5 or 27.1.6 whether Nd>Na. At normal temperatures, if Nd>Na,
the material is defined as n-type and 27.1.6 would be used provided Nd in the equation is replaced by Nd-Na. The
equation 27.1.5 would be used if Nd<Na. The intrinsic Fermi level is always defined by the equation 21.1.7. The
Fermi level equations define the relative location of EF, with respect to the intrinsic Fermi Level Ei, in terms of
temperature TT, Na or Nd, and the intrinsic carrier density function ni(TT). Ei is calculated in the seventh equation in
terms of the conduction and valence band levels Ec and Ev, temperature TT, the effective mass of electrons mn and
holes mp.
Ei EF k TT
Na
ni TT
=+⋅⋅
F
H
G
I
K
J
ln
bg
Eq. 27.1.5
EF Ei k TT
Nd
ni TT
=+⋅⋅
F
H
G
I
K
J
ln
bg
Eq. 27.1.6
Ei
Ec Ev
kTT
mp
mn
=
+
+⋅⋅ ⋅
F
H
G
I
K
J
bg
2
3
4
ln
Eq. 27.1.7
The final two equations are diffusion properties of dopants in silicon based on two distinctly different conditions. The
equation 27.1.8 covers the diffusion from an infinite source while the equation 27.1.9 covers diffusion from a finite
source. The diffusion of impurities in a semiconductor subject to an infinite source with a surface concentration
N0
at a
depth
x
below the surface after a time
t
, given the diffusion coefficient
D
is shown in the next equation. The final
equation details the diffusion from a finite impurity source
Q
over a surface area
A
with a classic Gaussian
distribution.
N
N
erfc
x
Dt
0
2
=
⋅⋅
F
H
G
I
K
J
Eq. 27.1.8
N
Qtot
ADt
e
x
Dt
=
⋅⋅⋅
⋅
−
⋅⋅
π
2
4
Eq. 27.1.9
Example 27.1.1 -
Find the intrinsic and actual Fermi levels for silicon at 300_°K if the conduction band is 1.12 eV
above the valence band. The donor density is 8 x 10
-17
cm
-3
. The effective masses for electrons and holes are 0.5 and
0.85.