1. Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/
Semiconductor Device Modeling and Characterization – EE5342 Lecture 4 – Spring 2011
Professor Ronald L. Carter

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3. Second Assignment
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4. 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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5. 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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6. 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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7. 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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8. 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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9. 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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10. Electron Conc. in the MB approx.
Assuming the MB approx., the equilibrium electron concentration is
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11. 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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12. 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
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13. Equilibrium con- centration 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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14. Equilibrium con- centration 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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15. 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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16. 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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17. 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 un-ionized donors is nd/no = 2Nd exp[(Ec-Ed)/kT]/Nc = 0.4% for Nd(P) = 1e16/cm3
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18. 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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19. 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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