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Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/
Semiconductor Device Modeling and Characterization EE5342, Lecture 7-Spring 2004 Professor Ronald L. Carter L7 February 10
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MidTerm and Project Tests
MidTerm on Thursday 2/12 Cover sheet to be posted at Project 1 draft assignment will be posted 2/13. Project report to be used in doing: Project 1 Test on Thursday 3/11 Cover sheet will be posted as above L7 February 10
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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] L7 February 10
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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 L7 February 10
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Diffnt’l, one-sided diode cond. (cont.)
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Cap. of a (1-sided) short diode (cont.)
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General time- constant
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General time- constant (cont.)
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General time- constant (cont.)
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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). L7 February 10
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Effect of carrier recombination in DR
The S-R-H rate (tno = tpo = to) is L7 February 10
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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 L7 February 10
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Effect of carrier rec. in DR (cont.)
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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)), L7 February 10
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High level inj effects (cont.)
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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 L7 February 10
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Plot of typical Va > 0 current density equations
ln(J) data Effect of Rs Vext VKF L7 February 10
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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. L7 February 10
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Reverse bias (Va< 0), carr gen in DR (cont.)
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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 L7 February 10
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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 L7 February 10
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Reverse bias junction breakdown
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Ecrit for reverse breakdown (M&K**)
Taken from p. 198, M&K** Casey Model for Ecrit L7 February 10
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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 L7 February 10
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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 L7 February 10
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Gauss’ Law rpc rp rj rn rnc L7 February 10
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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 L7 February 10
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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]. L7 February 10
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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. . L7 February 10
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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 . L7 February 10
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Spherical diode Breakdown Voltage
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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 = ? L7 February 10
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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 = ? L7 February 10
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Hole lifetimes, taken from Shur***, p. 101.
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Example Js,long, = ? If xnc, = 2 micron, Js,short, = ? L7 February 10
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Example (cont.) Estimate VKF Estimate IKF L7 February 10
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Example (cont.) Estimate Js,rec Estimate Rs if xnc is 100 micron
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Example (cont.) Estimate Jgen for 10 V reverse bias Estimate BV
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Diode equivalent circuit (small sig)
ID h is the practical “ideality factor” IQ VD VQ L7 February 10
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Small-signal eq circuit
Cdiff and Cdepl are both charged by Va = VQ Va Cdiff rdiff Cdepl L7 February 10
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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 L7 February 10
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Diode switching (cont.)
VF,VR >> Va F: t < 0 Sw RF R: t > 0 VF + RR D VR + L7 February 10
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Diode charge for t < 0 pn pno x xn xnc L7 February 10
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Diode charge for t >>> 0 (long times)
pn pno x xn xnc L7 February 10
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Equation summary L7 February 10
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Snapshot for t barely > 0
pn Total charge removed, Qdis=IRt pno x xn xnc L7 February 10
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I(t) for diode switching
ID IF ts ts+trr t - 0.1 IR -IR L7 February 10
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References * Semiconductor Physics and Devices, 2nd ed., by Neamen, Irwin, Boston, 1997. **Device Electronics for Integrated Circuits, 2nd ed., by Muller and Kamins, John Wiley, New York, 1986. ***Physics of Semiconductor Devices, Shur, Prentice-Hall, 1990. L7 February 10
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