Transcript Semiconductor Device Modeling and Characterization – EE5342 Lecture 4 – Spring 2011 Professor Ronald L.

Semiconductor Device Modeling
and Characterization – EE5342
Lecture 4 – Spring 2011
Professor Ronald L. Carter
[email protected]
http://www.uta.edu/ronc/
First Assignment
• e-mail to [email protected]
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Second Assignment
• Submit a signed copy of the document
that is posted at
www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf
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Semiconductor Electronics concepts thus far
• Conduction and Valence states due to
symmetry of lattice
• “Free-elec.” dynamics near band edge
• Band Gap
– direct or indirect
– effective mass in curvature
• Thermal carrier generation
• Chemical carrier gen (donors/accept)
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Counting carriers - 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 = 1E10/cm3
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Calculating the
equilibrium no
• The ideal 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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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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Donor ionization
• The density of elec trapped at
donors is nd = Nd/{1+[exp((EdEF)/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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Classes of
semiconductors
• Intrinsic: no = po = ni, since Na&Nd << ni
=[NcNvexp(Eg/kT)]1/2,(not easy to get)
• n-type: no > po, since Nd > Na
• p-type: no < po, since Nd < Na
• Compensated: no=po=ni, w/ Na- = Nd+ > 0
• Note: n-type and p-type are usually
partially compensated since there are
usually some opposite- type dopants
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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.
• 1Device Electronics for Integrated Circuits, 2 ed.,
by Muller and Kamins, Wiley, New York, 1986.
• 2Physics of Semiconductor Devices, by S. M. Sze,
Wiley, New York, 1981.
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