1. Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/
Semiconductor Device Modeling and Characterization EE5342, Lecture 3-Spring 2003
Professor Ronald L. Carter
L3 January 21

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L3 January 21

3. 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)
L3 January 21

4. 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
L3 January 21

5. 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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6. 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
L3 January 21

7. 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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8. 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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9. Electron Conc. in the MB approx.
Assuming the MB approx., the equilibrium electron concentration is
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10. 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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11. 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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12. 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
L3 January 21

13. 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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14. 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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15. 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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16. 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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17. 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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18. Equilibrium concentrations
Charge neutrality requires q(po + Nd+) + (-q)(no + Na-) = 0
Assuming complete ionization, so Nd+ = Nd and Na- = Na
Gives two equations to be solved simultaneously
1. Mass action, no po = ni2, and
2. Neutrality po + Nd = no + Na
L3 January 21

19. Equilibrium conc n-type
For Nd > Na
Let N = Nd-Na, and (taking the + root) no = (N)/2 + {[N/2]2+ni2}1/2
For Nd+= Nd >> ni >> Na we have
no = Nd, and
po = ni2/Nd
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20. Equilibrium conc p-type
For Na > Nd
Let N = Nd-Na, and (taking the + root) po = (-N)/2 + {[-N/2]2+ni2}1/2
For Na-= Na >> ni >> Nd we have
po = Na, and
no = ni2/Na
L3 January 21

21. Electron Conc. in the MB approx.
Assuming the MB approx., the equilibrium electron concentration is
L3 January 21

22. 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
L3 January 21

23. Position of the Fermi Level
Efi is the Fermi level when no = po
Ef shown is a Fermi level for no > po
Ef < Efi when no < po
Efi < (Ec + Ev)/2, which is the mid-band
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24. EF relative to Ec and Ev
Inverting no = Nc exp[-(Ec-EF)/kT] gives Ec - EF = kT ln(Nc/no) For n-type material: Ec - EF =kTln(Nc/Nd)=kTln[(NcPo)/ni2]
Inverting po = Nv exp[-(EF-Ev)/kT] gives EF - Ev = kT ln(Nv/po) For p-type material: EF - Ev = kT ln(Nv/Na)
L3 January 21

25. EF relative to Efi
Letting ni = no gives  Ef = Efi ni = Nc exp[-(Ec-Efi)/kT], so Ec - Efi = kT ln(Nc/ni). Thus EF - Efi = kT ln(no/ni) and for n-type EF - Efi = kT ln(Nd/ni)
Likewise Efi - EF = kT ln(po/ni) and for p-type Efi - EF = kT ln(Na/ni)
L3 January 21

26. Locating Efi in the bandgap
Since Ec - Efi = kT ln(Nc/ni), and Efi - Ev = kT ln(Nv/ni)
The sum of the two equations gives Efi = (Ec + Ev)/2 - (kT/2) ln(Nc/Nv)
Since Nc = 2.8E19cm-3 > 1.04E19cm-3 = Nv, the intrinsic Fermi level lies below the middle of the band gap
L3 January 21

27. Sample calculations
Efi = (Ec + Ev)/2 - (kT/2) ln(Nc/Nv), so at 300K, kT = meV and Nc/Nv = 2.8/1.04, Efi is 12.8 meV or 1.1% below mid-band
For Nd = 3E17cm-3, given that Ec - EF = kT ln(Nc/Nd), we have Ec - EF = meV ln(280/3), Ec - EF = eV =117meV ~3x(Ec - ED) what Nd gives Ec-EF =Ec/3
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28. Equilibrium electron conc. and energies
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29. Equilibrium hole conc. and energies
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30. vx = axt = (qEx/m*)t, and the displ
Carrier Mobility
In an electric field, Ex, the velocity (since ax = Fx/m* = qEx/m*) is
vx = axt = (qEx/m*)t, and the displ
x = (qEx/m*)t2/2
If every tcoll, a collision occurs which “resets” the velocity to <vx(tcoll)> = 0, then <vx> = qExtcoll/m* = mEx
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31. Carrier mobility (cont.)
The response function m is the mobility.
The mean time between collisions, tcoll, may has several important causal events: Thermal vibrations, donor- or acceptor-like traps and lattice imperfections to name a few.
Hence mthermal = qtthermal/m*, etc.
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32. Carrier mobility (cont.)
If the rate of a single contribution to the scattering is 1/ti, then the total scattering rate, 1/tcoll is
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33. Drift Current
The drift current density (amp/cm2) is given by the point form of Ohm Law
J = (nqmn+pqmp)(Exi+ Eyj+ Ezk), so
J = (sn + sp)E = sE, where
s = nqmn+pqmp defines the conductivity
The net current is
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34. Drift current resistance
Given: a semiconductor resistor with length, l, and cross-section, A. What is the resistance?
As stated previously, the conductivity, s = nqmn + pqmp
So the resistivity, r = 1/s = 1/(nqmn + pqmp)
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35. Drift current resistance (cont.)
Consequently, since
R = rl/A
R = (nqmn + pqmp)-1(l/A)
For n >> p, (an n-type extrinsic s/c)
R = l/(nqmnA)
For p >> n, (a p-type extrinsic s/c) R = l/(pqmpA)
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36. Drift current resistance (cont.)
Note: for an extrinsic semiconductor and multiple scattering mechanisms, since
R = l/(nqmnA) or l/(pqmpA), and
(mn or p total)-1 = S mi-1, then
Rtotal = S Ri (series Rs)
The individual scattering mechanisms are: Lattice, ionized impurity, etc.
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37. Exp. mobility model function for Si1
Parameter As P B
mmin
mmax
Nref e e e17 a
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38. Exp. mobility model for P, As and B in Si
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39. Carrier mobility functions (cont.)
The parameter mmax models 1/tlattice the thermal collision rate
The parameters mmin, Nref and a model 1/timpur the impurity collision rate
The function is approximately of the ideal theoretical form: 1/mtotal = 1/mthermal + 1/mimpurity
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40. Carrier mobility functions (ex.)
Let Nd = 1.78E17/cm3 of phosphorous, so mmin = 68.5, mmax = 1414, Nref = 9.20e16 and a = Thus mn = 586 cm2/V-s
Let Na = 5.62E17/cm3 of boron, so mmin = 44.9, mmax = 470.5, Nref = 9.68e16 and a = Thus mn = 189 cm2/V-s
L3 January 21

41. Lattice mobility
The mlattice is the lattice scattering mobility due to thermal vibrations
Simple theory gives mlattice ~ T-3/2
Experimentally mn,lattice ~ T-n where n = 2.42 for electrons and 2.2 for holes
Consequently, the model equation is mlattice(T) = mlattice(300)(T/300)-n
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42. Ionized impurity mobility function
The mimpur is the scattering mobility due to ionized impurities
Simple theory gives mimpur ~ T3/2/Nimpur
Consequently, the model equation is mimpur(T) = mimpur(300)(T/300)3/2
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43. Net silicon (ex- trinsic) resistivity
Since r = s-1 = (nqmn + pqmp)-1
The net conductivity can be obtained by using the model equation for the mobilities as functions of doping concentrations.
The model function gives agreement with the measured s(Nimpur)
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44. Net silicon extr resistivity (cont.)
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45. Net silicon extr resistivity (cont.)
Since
r = (nqmn + pqmp)-1, and
mn > mp, (m = qt/m*) we have
rp > rn
Note that since
1.6(high conc.) < rp/rn < 3(low conc.), so
1.6(high conc.) < mn/mp < 3(low conc.)
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46. Net silicon (com- pensated) res.
For an n-type (n >> p) compensated semiconductor, r = (nqmn)-1
But now n = N = Nd - Na, and the mobility must be considered to be determined by the total ionized impurity scattering Nd + Na = NI
Consequently, a good estimate is r = (nqmn)-1 = [Nqmn(NI)]-1
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47. References
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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