EE 5340
Semiconductor Device Theory
Lecture 04 – Spring 2011
Professor Ronald L. Carter
ronc@uta.edu
http://www.uta.edu/ronc
Review the Following
• R. L. Carter’s web page:
– www.uta.edu/ronc/
• EE 5340 web page and syllabus. (Refresh all
EE 5340 pages when downloading to assure the
latest version.) All links at:
– www.uta.edu/ronc/5340/syllabus.htm
• University and College Ethics Policies
– www.uta.edu/studentaffairs/conduct/
• Makeup lecture at noon Friday (1/28) in 108
Nedderman Hall. This will be available on the
web.
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First Assignment
• Send e-mail to ronc@uta.edu
– On the subject line, put “5340 e-mail”
– In the body of message include
• email address: ______________________
• Your Name*: _______________________
• Last four digits of your Student ID: _____
* Your name as it appears in the UTA
Record - no more, no less
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Second Assignment
• Submit a signed copy of the document
posted at
www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf
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Quantum density
of states function
• 1 dim electron wave #s range for n+1
“atoms” is 2p/L < k < 2p/a where a is
“interatomic” distance and L = na is
the length of the assembly (k = 2p/l)
• Shorter ls, would “oversample”
• if n increases by 1, dp is h/L
• Extn 3D: E = p2/2m = h2k2/2m so a vol
of p-space of 4pp2dp has h3/LxLyLz
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QM density of
states (cont.)
• So density of states, gc(E) is
(Vol in p-sp)/(Vol per state*V) =
4pp2dp/[(h3/LxLyLz)*V]
• Noting that p2 = 2mE, this becomes
gc(E) = {4p(2mn*)3/2/h3}(E-Ec)1/2
and E - Ec = h2k2/2mn*
• Similar for the hole states where
Ev - E = h2k2/2mp*
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Fermi-Dirac
distribution fctn
• The probability of an electron having
an energy, E, is given by the F-D distr
fF(E) = {1+exp[(E-EF)/kT]}-1
• Note: fF (EF) = 1/2
• EF is the equilibrium energy of the
system
• The sum of the hole probability and
the electron probability is 1
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Fermi-Dirac
DF (continued)
• So the probability of a hole having
energy E is 1 - fF(E)
• At T = 0 K, fF (E) becomes a step
function and 0 probability of E > EF
• At T >> 0 K, there is a finite
probability of E >> EF
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Maxwell-Boltzman
Approximation
• fF(E) = {1+exp[(E-EF)/kT]}-1
• For E - EF > 3 kT, the exp > 20, so
within a 5% error,
fF(E) ~ exp[-(E-EF)/kT]
• This is the MB distribution function
• MB used when E-EF>75 meV (T=300K)
• For electrons when Ec - EF > 75 meV
and for holes when EF - Ev > 75 meV
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Electron Conc. in
the MB approx.
• Assuming the MB approx., the
equilibrium electron concentration is
no 
Emax
 gc (E fF (E dE
Ec
  (Ec  EF  
no  Nc exp 

kT


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Electron and Hole
Conc in MB approx
• Similarly, the equilibrium hole
concentration is
po = Nv exp[-(EF-Ev)/kT]
• So that nopo = NcNv exp[-Eg/kT]
• ni2 = nopo, Nc,v = 2{2pm*n,pkT/h2}3/2
• Nc = 2.8E19/cm3, Nv = 1.04E19/cm3
and ni = 1.45E10/cm3
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Calculating the
equilibrium no
• The idea is to calculate the
equilibrium electron concentration no
for the FD distribution, where
fF(E) = {1+exp[(E-EF)/kT]}-1
gc(E) = [4p(2mn*)3/2(E-Ec)1/2]/h3
no 
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Emax
 gc (E fF (E dE
Ec
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Equilibrium concentration for no
• Earlier quoted the MB approximation
no = Nc exp[-(Ec - EF)/kT],(=Nc exp hF)
• The exact solution is
no = 2NcF1/2(hF)/p1/2
• Where F1/2(hF) is the Fermi integral
of order 1/2, and hF = (EF - Ec)/kT
• Error in no, e, is smaller than for the
DF: e = 31%, 12%, 5% for -hF = 0, 1, 2
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Equilibrium concentration for po
• Earlier quoted the MB approximation
po = Nv exp[-(EF - Ev)/kT],(=Nv exp h’F)
• The exact solution is
po = 2NvF1/2(h’F)/p1/2
• Note: F1/2(0) = 0.678, (p1/2/2) = 0.886
• Where F1/2(h’F) is the Fermi integral
of order 1/2, and h’F = (Ev - EF)/kT
• Errors are the same as for po
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Figure 1.10
(a) Fermi-Dirac distribution
function describing the
probability that an allowed state
at energy E is occupied by an
electron.
(b) The density of allowed states
for a semiconductor as a function
of energy; note that g(E) is zero
in the forbidden gap between Ev
and Ec.
(c) The product of the
distribution function and the
density-of-states function.
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(p. 17 - M&K)
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Degenerate and
nondegenerate cases
• Bohr-like doping model assumes no
interaction between dopant sites
• If adjacent dopant atoms are within
2 Bohr radii, then orbits overlap
• This happens when Nd ~ Nc (EF ~ Ec),
or when Na ~ Nv (EF ~ Ev)
• The degenerate semiconductor is
defined by EF ~/> Ec or EF ~/< Ev
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Figure 1.13
Energy-gap narrowing
Eg as a function of
electron concentration.
[A. Neugroschel, S. C.
Pao, and F. A. Lindhold,
IEEE Trans. Electr.
Devices, ED-29, 894
(May 1982).] taken from
p. 25 - M&K)
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Donor ionization
• The density of elec trapped at donors
is nd = Nd/{1+[exp((Ed-EF)/kT)/2]}
• Similar to FD DF except for factor
of 2 due to degeneracy (4 for holes)
• Furthermore nd = Nd - Nd+, also
• For a shallow donor, can have Ed-EF >>
kT AND Ec-EF >> kT: Typically EF-Ed ~
2kT
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Donor ionization
(continued)
• Further, if Ed - EF > 2kT, then
nd ~ 2Nd exp[-(Ed-EF)/kT], e < 5%
• If the above is true, Ec - EF > 4kT, so
no ~ Nc exp[-(Ec-EF)/kT], e < 2%
• Consequently the fraction of unionized donors is
nd/no = 2Nd exp[(Ec-Ed)/kT]/Nc
= 0.4% for Nd(P) = 1e16/cm3
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Figure 1.9
Electron
concentration vs.
temperature for
two n-type doped
semiconductors:
(a) Silicon doped
with 1.15 X 1016
arsenic atoms cm3[1],
(b) Germanium
doped with 7.5 X
1015 arsenic atoms
cm-3[2]. (p.12 in
M&K)
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References
*Fundamentals of Semiconductor Theory
and Device Physics, by Shyh Wang,
Prentice Hall, 1989.
**Semiconductor Physics & Devices, by
Donald A. Neamen, 2nd ed., Irwin, Chicago.
M&K = Device Electronics for Integrated
Circuits, 3rd ed., by Richard S. Muller,
Theodore I. Kamins, and Mansun Chan,
John Wiley and Sons, New York, 2003.
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