1. Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/
Semiconductor Device Modeling and Characterization EE5342, Lecture 7-Spring 2005
Professor Ronald L. Carter
L7 February 08

2. Injection Conditions
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3. Apply the Continuity Eqn in CNR
Ideal Junction Theory (cont.)
Apply the Continuity Eqn in CNR
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4. Ideal Junction Theory (cont.)
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5. Ideal Junction Theory (cont.)
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6. Excess minority carrier distr fctn
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7. Carrier Injection ln(carrier conc) ln Na ln Nd ln ni ~Va/Vt ~Va/Vt
ln ni2/Nd
ln ni2/Na x
-xpc
-xp
xnc xn
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8. Minority carrier currents
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9. Evaluating the diode current
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10. Special cases for the diode current
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11. Ideal diode equation Assumptions: Current dens, Jx = Js expd(Va/Vt)
low-level injection
Maxwell Boltzman statistics
Depletion approximation
Neglect gen/rec effects in DR
Steady-state solution only
Current dens, Jx = Js expd(Va/Vt)
where expd(x) = [exp(x) -1]
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12. Ideal diode equation (cont.)
Js = Js,p + Js,n = hole curr + ele curr
Js,p = qni2Dp coth(Wn/Lp)/(NdLp) = qni2Dp/(NdWn), Wn << Lp, “short” = qni2Dp/(NdLp), Wn >> Lp, “long”
Js,n = qni2Dn coth(Wp/Ln)/(NaLn) = qni2Dn/(NaWp), Wp << Ln, “short” = qni2Dn/(NaLn), Wp >> Ln, “long”
Js,n << Js,p when Na >> Nd
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13. Diffnt’l, one-sided diode conductance
Static (steady-state) diode I-V characteristic IQ Va VQ
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14. Diffnt’l, one-sided diode cond. (cont.)
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15. Charge distr in a (1- sided) short diode
dpn
Assume Nd << Na
The sinh (see L12) excess minority carrier distribution becomes linear for Wn << Lp
dpn(xn)=pn0expd(Va/Vt)
Total chg = Q’p = Q’p = qdpn(xn)Wn/2
Wn = xnc- xn
dpn(xn)
Q’p x xn
xnc
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16. Charge distr in a 1- sided short diode
dpn
Assume Quasi-static charge distributions
Q’p = Q’p = qdpn(xn)Wn/2
ddpn(xn) = (W/2)* {dpn(xn,Va+dV) - dpn(xn,Va)}
dpn(xn,Va+dV)
dpn(xn,Va)
dQ’p
Q’p x xn
xnc
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17. Cap. of a (1-sided) short diode (cont.)
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18. Diode equivalent circuit (small sig)ID
h is the practical “ideality factor” IQ VD VQ
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19. Small-signal eq circuit
Cdiff and Cdepl are both charged by
Va = VQ Va
Cdiff
rdiff
Cdepl
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20. General time- constant
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21. General time- constant (cont.)
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22. General time- constant (cont.)
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23. Effect of carrier recombination in DR
The S-R-H rate (tno = tpo = to) is
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24. Effect of carrier rec. in DR (cont.)
For low Va ~ 10 Vt
In DR, n and p are still > ni
The net recombination rate, U, is still finite so there is net carrier recomb.
reduces the carriers available for the ideal diode current
adds an additional current component
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25. Effect of carrier rec. in DR (cont.)
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26. Effect of non- zero E in the CNR
This is usually not a factor in a short diode, but when E is finite -> resistor
In a long diode, there is an additional ohmic resistance (usually called the parasitic diode series resistance, Rs)
Rs = L/(nqmnA) for a p+n long diode.
L=Wn-Lp (so the current is diode-like for Lp and the resistive otherwise).
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27. High level injection effects
Law of the junction remains in the same form, [pnnn]xn=ni2exp(Va/Vt), etc.
However, now dpn = dnn become >> nno = Nd, etc.
Consequently, the l.o.t.j. reaches the limiting form dpndnn = ni2exp(Va/Vt)
Giving, dpn(xn) = niexp(Va/(2Vt)), or dnp(-xp) = niexp(Va/(2Vt)),
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28. High level inj effects (cont.)
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29. Summary of Va > 0 current density eqns.
Ideal diode, Jsexpd(Va/(hVt))
ideality factor, h
Recombination, Js,recexp(Va/(2hVt))
appears in parallel with ideal term
High-level injection, (Js*JKF)1/2exp(Va/(2hVt))
SPICE model by modulating ideal Js term
Va = Vext - J*A*Rs = Vext - Idiode*Rs
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30. Plot of typical Va > 0 current density equations
ln(J)
data
Effect of Rs
Vext
VKF
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31. Reverse bias (Va<0) => carrier gen in DR
Va < 0 gives the net rec rate, U = -ni/2t0, t0 = mean min carr g/r l.t.
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32. Reverse bias (Va< 0), carr gen in DR (cont.)
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33. Reverse bias junction breakdown
Avalanche breakdown
Electric field accelerates electrons to sufficient energy to initiate multiplication of impact ionization of valence bonding electrons
field dependence shown on next slide
Heavily doped narrow junction will allow tunneling - see Neamen*, p. 274
Zener breakdown
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34. Reverse bias junction breakdown
Assume -Va = VR >> Vbi, so Vbi-Va-->VR
Since Emax~ 2VR/W = (2qN-VR/(e))1/2, and VR = BV when Emax = Ecrit (N- is doping of lightly doped side ~ Neff)
BV = e (Ecrit )2/(2qN-)
Remember, this is a 1-dim calculation
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35. Reverse bias junction breakdown
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36. Ecrit for reverse breakdown (M&K**)
Taken from p. 198, M&K**
Casey Model for Ecrit
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37. Junction curvature effect on breakdown
The field due to a sphere, R, with charge, Q is Er = Q/(4per2) for (r > R)
V(R) = Q/(4peR), (V at the surface)
So, for constant potential, V, the field, Er(R) = V/R (E field at surface increases for smaller spheres)
Note: corners of a jctn of depth xj are like 1/8 spheres of radius ~ xj
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38. BV for reverse breakdown (M&K**)
Taken from Figure 4.13, p. 198, M&K**
Breakdown voltage of a one-sided, plan, silicon step junction showing the effect of junction curvature.4,5
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39. Gauss’ Law
rpc rp rj rn
rnc
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40. Spherical Diode Fields calculations
For rj < ro ≤ rn,
Setting Er = 0 at r = rn, we get
Note that the equivalent of the lever law for this spherical diode is
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41. Spherical Diode Fields calculations
Assume Na >> Nd, so rn – rj  d >> rj – rp.
Further, setting the usual definition for the potential difference, and evaluating the potential difference at breakdown, we have
PHIi – Va = BV and
Emax = Em = Ecrit = Ec.
We also define a = 3eEm/qNd[cm].
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42. Showing the rj  ∞ limit
C1. Solve for rn – rj = d as a function of Emax and solve for the value of d in the limit of rj  . The solution for rn is given below. .
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43. Solving for the Breakdown (BV)
Solve for BV = [fi – Va]Emax = Ecrit,
and solve for the value of BV in the limit of rj  . The solution for BV is given below .
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44. Spherical diode Breakdown Voltage
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45. Example calculations
Assume throughout that p+n jctn with Na = 3e19cm-3 and Nd = 1e17cm-3
From graph of Pierret mobility model, mp = 331 cm2/V-sec and Dp = Vtmp = ?
Why mp and Dp?
Neff = ?
Vbi = ?
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46. L7 February 08

47. Parameters for examples
Get tmin from the model used in Project tmin = (45 msec) (7.7E-18cm3)Ni+(4.5E-36cm6)Ni2
For Nd = 1E17cm3, tp = 25 msec
Why Nd and tp ?
Lp = ?
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48. Hole lifetimes, taken from Shur***, p. 101.
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49. Example
Js,long, = ?
If xnc, = 2 micron, Js,short, = ?
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50. Example (cont.)
Estimate VKF
Estimate IKF
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51. Example (cont.) Estimate Js,rec Estimate Rs if xnc is 100 micron
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52. Example (cont.) Estimate Jgen for 10 V reverse bias Estimate BV
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53. Diode Switching Consider the charging and discharging of a Pn diode
(Na > Nd)
Wd << Lp
For t < 0, apply the Thevenin pair VF and RF, so that in steady state
IF = (VF - Va)/RF, VF >> Va , so current source
For t > 0, apply VR and RR
IR = (VR + Va)/RR, VR >> Va, so current source
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54. Diode switching (cont.)
VF,VR >> Va
F: t < 0 Sw RF
R: t > 0 VF + RR D VR +
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55. Diode charge for t < 0 pn
pno x xn
xnc
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56. Diode charge for t >>> 0 (long times)pn
pno x xn
xnc
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57. Equation summary
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58. Snapshot for t barely > 0pn
Total charge removed, Qdis=IRt
pno x xn
xnc
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59. I(t) for diode switchingID IF ts
ts+trr t
- 0.1 IR
-IR
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