1. Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/
Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2004
Professor Ronald L. Carter
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Submit a signed copy to Dr. Carter
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4. 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
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5. Equilibrium electron conc. and energies
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6. Equilibrium hole conc. and energies
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7. Mobility Summary
The concept of mobility introduced as a response function to the electric field in establishing a drift current
Resistivity and conductivity defined
Model equation def for m(Nd,Na,T)
Resistivity models developed for extrinsic and compensated materials
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8. 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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9. Exp. mobility model function for Si1
Parameter As P B
mmin
mmax
Nref e e e17 a
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10. Exp. mobility model for P, As and B in Si
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11. 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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12. 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 mp = 189 cm2/V-s
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13. 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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14. 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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15. 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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16. Net silicon extr resistivity (cont.)
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17. 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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18. 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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19. Equipartition theorem
The thermodynamic energy per degree of freedom is kT/2
Consequently,
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20. Carrier velocity saturation1
The mobility relationship v = mE is limited to “low” fields
v < vth = (3kT/m*)1/2 defines “low”
v = moE[1+(E/Ec)b]-1/b, mo = v1/Ec for Si
parameter electrons holes
v1 (cm/s) E9 T E8 T-0.52
Ec (V/cm) T T1.68
b E-2 T T0.17
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21. vdrift [cm/s] vs. E [V/cm] (Sze2, fig. 29a)
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22. Carrier velocity saturation (cont.)
At 300K, for electrons, mo = v1/Ec = 1.53E9(300)-0.87/1.01(300) = 1504 cm2/V-s, the low-field mobility
The maximum velocity (300K) is vsat = moEc = v1 = 1.53E9 (300) = 1.07E7 cm/s
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23. Diffusion of carriers
In a gradient of electrons or holes, =p and =n are not zero
Diffusion current,`J =`Jp +`Jn (note Dp and Dn are diffusion coefficients)
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24. Diffusion of carriers (cont.)
Note (=p)x has the magnitude of dp/dx and points in the direction of increasing p (uphill)
The diffusion current points in the direction of decreasing p or n (downhill) and hence the - sign in the definition of`Jp and the + sign in the definition of`Jn
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25. Diffusion of Carriers (cont.)
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26. Current density components
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27. Total current density
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28. Doping gradient induced E-field
If N = Nd-Na = N(x), then so is Ef-Efi
Define f = (Ef-Efi)/q = (kT/q)ln(no/ni)
For equilibrium, Efi = constant, but
for dN/dx not equal to zero,
Ex = -df/dx =- [d(Ef-Efi)/dx](kT/q) = -(kT/q) d[ln(no/ni)]/dx = -(kT/q) (1/no)[dno/dx] = -(kT/q) (1/N)[dN/dx], N > 0
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29. Induced E-field (continued)
Let Vt = kT/q, then since
nopo = ni2 gives no/ni = ni/po
Ex = - Vt d[ln(no/ni)]/dx = - Vt d[ln(ni/po)]/dx = - Vt d[ln(ni/|N|)]/dx, N = -Na < 0
Ex = - Vt (-1/po)dpo/dx = Vt(1/po)dpo/dx = Vt(1/Na)dNa/dx
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30. The Einstein relationship
For Ex = - Vt (1/no)dno/dx, and
Jn,x = nqmnEx + qDn(dn/dx) = 0
This requires that nqmn[Vt (1/n)dn/dx] = qDn(dn/dx)
Which is satisfied if
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31. Direct carrier gen/recombk Ec Ev
(Excitation can be by light)
gen
rec - + Ev Ec Ef
Efi
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32. Direct gen/rec of excess carriers
Generation rates, Gn0 = Gp0
Recombination rates, Rn0 = Rp0
In equilibrium: Gn0 = Gp0 = Rn0 = Rp0
In non-equilibrium condition:
n = no + dn and p = po + dp, where nopo=ni2
and for dn and dp > 0, the recombination rates increase to R’n and R’p
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33. Direct rec for low-level injection
Define low-level injection as dn = dp < no, for n-type, and dn = dp < po, for p-type
The recombination rates then are R’n = R’p = dn(t)/tn0, for p-type, and R’n = R’p = dp(t)/tp0, for n-type
Where tn0 and tp0 are the minority-carrier lifetimes
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34. Shockley-Read- Hall Recomb
Indirect, like Si, so intermediate state Ec Ec ET Ef
Efi Ev Ev k
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35. S-R-H trap characteristics1
The Shockley-Read-Hall Theory requires an intermediate “trap” site in order to conserve both E and p
If trap neutral when orbited (filled) by an excess electron - “donor-like”
Gives up electron with energy Ec - ET
“Donor-like” trap which has given up the extra electron is +q and “empty”
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36. S-R-H trap char. (cont.)
If trap neutral when orbited (filled) by an excess hole - “acceptor-like”
Gives up hole with energy ET - Ev
“Acceptor-like” trap which has given up the extra hole is -q and “empty”
Balance of 4 processes of electron capture/emission and hole capture/ emission gives the recomb rates
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37. tpo = (Ntvthsn)-1, where sn~p(rBohr)2
S-R-H recombination
Recombination rate determined by:
Nt (trap conc.),
vth (thermal vel of the carriers),
sn (capture cross sect for electrons),
sp (capture cross sect for holes), with
tno = (Ntvthsn)-1, and
tpo = (Ntvthsn)-1, where sn~p(rBohr)2
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38. S-R-H recomb. (cont.)
In the special case where tno = tpo = to the net recombination rate, U is
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39. S-R-H “U” function characteristics
The numerator, (np-ni2) simplifies in the case of extrinsic material at low level injection (for equil., nopo = ni2)
For n-type (no > dn = dp > po = ni2/no):
(np-ni2) = (no+dn)(po+dp)-ni2 = nopo - ni2 + nodp + dnpo + dndp ~ nodp (largest term)
Similarly, for p-type, (np-ni2) ~ podn
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40. S-R-H “U” function characteristics (cont)
For n-type, as above, the denominator = to{no+dn+po+dp+2nicosh[(Et-Ei)kT]}, simplifies to the smallest value for Et~Ei, where the denom is tono, giving U = dp/to as the largest (fastest)
For p-type, the same argument gives U = dn/to
Rec rate, U, fixed by minority carrier
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41. S-R-H net recom- bination rate, U
In the special case where tno = tpo = to = (Ntvthso)-1 the net rec. rate, U is
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42. S-R-H rec for excess min carr
For n-type low-level injection and net excess minority carriers, (i.e., no > dn = dp > po = ni2/no),
U = dp/to, (prop to exc min carr)
For p-type low-level injection and net excess minority carriers, (i.e., po > dn = dp > no = ni2/po),
U = dn/to, (prop to exc min carr)
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43. Minority hole lifetimes. Taken from Shur3, (p.101).
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44. Minority electron lifetimes. Taken from Shur3, (p.101).
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45. Parameter example tmin = (45 msec) 1+(7.7E-18cm3)Ni+(4.5E-36cm6)Ni2
For Nd = 1E17cm3, tp = 25 msec
Why Nd and tp ?
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46. 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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