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Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/
Semiconductor Device Modeling and Characterization EE5342, Lecture 6-Spring 2010 Professor Ronald L. Carter L6 February 03
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Project 1A – Diode parameters to use
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Tasks Using PSpice or any simulator, plot the i-v curve for this diode, assuming Rth = 0, for several temperatures in the range 300 K < TEMP = TAMB < 304 K. Using this data, determine what the i-v plot would be for Rth = 500 K/W. Using this data, determine the maximum operating temperature for which the diode conductance is within 1% of the Rth = 0 value at 300 K. Do the same for a 10% tolerance. Propose a SPICE macro which would give the Rth = 500 K/W i-v relationship. L6 February 03
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Example L6 February 03
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Induced E-field in the D.R.
Ex p-contact N-contact O - O + p-type CNR n-type chg neutral reg O - O + O - O + Exposed Acceptor Ions Depletion region (DR) Exposed Donor ions W x -xpc -xp xn xnc L6 February 03
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Depletion approx. charge distribution
+Qn’=qNdxn +qNd [Coul/cm2] -xp x -xpc xn xnc -qNa Charge neutrality => Qp’ + Qn’ = 0, => Naxp = Ndxn Qp’=-qNaxp [Coul/cm2] L6 February 03
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1-dim soln. of Gauss’ law Ex -xpc -xp xn xnc x -Emax L6 February 03
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Depletion Approxi- mation (Summary)
For the step junction defined by doping Na (p-type) for x < 0 and Nd, (n-type) for x > 0, the depletion width W = {2e(Vbi-Va)/qNeff}1/2, where Vbi = Vt ln{NaNd/ni2}, and Neff=NaNd/(Na+Nd). Since Naxp=Ndxn, xn = W/(1 + Nd/Na), and xp = W/(1 + Na/Nd). L6 February 03
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One-sided p+n or n+p jctns
If p+n, then Na >> Nd, and NaNd/(Na + Nd) = Neff --> Nd, and W --> xn, DR is all on lightly d. side If n+p, then Nd >> Na, and NaNd/(Na + Nd) = Neff --> Na, and W --> xp, DR is all on lightly d. side The net effect is that Neff --> N-, (- = lightly doped side) and W --> x- L6 February 03
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Charge neutrality => Qp’ + Qn’ = 0, => Naxp = Ndxn
Junction C (cont.) r +Qn’=qNdxn +qNd dQn’=qNddxn -xp x -xpc xn xnc -qNa Charge neutrality => Qp’ + Qn’ = 0, => Naxp = Ndxn dQp’=-qNadxp Qp’=-qNaxp L6 February 03
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Junction C (cont.) The C-V relationship simplifies to L6 February 03
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Junction C (cont.) If one plots [C’j]-2 vs. Va Slope = -[(C’j0)2Vbi]-1 vertical axis intercept = [C’j0]-2 horizontal axis intercept = Vbi C’j-2 C’j0-2 Va Vbi L6 February 03
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Arbitrary doping profile
If the net donor conc, N = N(x), then at xn, the extra charge put into the DR when Va->Va+dVa is dQ’=-qN(xn)dxn The increase in field, dEx =-(qN/e)dxn, by Gauss’ Law (at xn, but also const). So dVa=-(xn+xp)dEx= (W/e) dQ’ Further, since N(xn)dxn = N(xp)dxp gives, the dC/dxn as ... L6 February 03
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Arbitrary doping profile (cont.)
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Arbitrary doping profile (cont.)
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Arbitrary doping profile (cont.)
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Arbitrary doping profile (cont.)
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n x xn Nd Debye length The DA assumes n changes from Nd to 0 discontinuously at xn, likewise, p changes from Na to 0 discontinuously at -xp. In the region of xn, the 1-dim Poisson equation is dEx/dx = q(Nd - n), and since Ex = -df/dx, the potential is the solution to -d2f/dx2 = q(Nd - n)/e L6 February 03
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Debye length (cont) Since the level EFi is a reference for equil, we set f = Vt ln(n/ni) In the region of xn, n = ni exp(f/Vt), so d2f/dx2 = -q(Nd - ni ef/Vt), let f = fo + f’, where fo = Vt ln(Nd/ni) so Nd - ni ef/Vt = Nd[1 - ef/Vt-fo/Vt], for f - fo = f’ << fo, the DE becomes d2f’/dx2 = (q2Nd/ekT)f’, f’ << fo L6 February 03
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Debye length (cont) So f’ = f’(xn) exp[+(x-xn)/LD]+con. and n = Nd ef’/Vt, x ~ xn, where LD is the “Debye length” L6 February 03
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13% < d < 28% => DA is OK
Debye length (cont) LD estimates the transition length of a step-junction DR (concentrations Na and Nd with Neff = NaNd/(Na +Nd)). Thus, For Va=0, & 1E13 < Na,Nd < 1E19 cm-3 13% < d < 28% => DA is OK L6 February 03
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Example An assymetrical p+ n junction has a lightly doped concentration of 1E16 and with p+ = 1E18. What is W(V=0)? Vbi=0.816 V, Neff=9.9E15, W=0.33mm What is C’j? = 31.9 nFd/cm2 What is LD? = 0.04 mm L6 February 03
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Ideal Junction Theory Assumptions Ex = 0 in the chg neutral reg. (CNR)
MB statistics are applicable Neglect gen/rec in depl reg (DR) Low level injections apply so that dnp < ppo for -xpc < x < -xp, and dpn < nno for xn < x < xnc Steady State conditions L6 February 03
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Forward Bias Energy Bands
Ev Ec EFi xn xnc -xpc -xp q(Vbi-Va) EFP EFN qVa x Imref, EFn Imref, EFp L6 February 03
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Law of the junction (follow the min. carr.)
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Law of the junction (cont.)
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Law of the junction (cont.)
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Injection Conditions L6 February 03
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Apply the Continuity Eqn in CNR
Ideal Junction Theory (cont.) Apply the Continuity Eqn in CNR L6 February 03
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Ideal Junction Theory (cont.)
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Ideal Junction Theory (cont.)
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Excess minority carrier distr fctn
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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 L6 February 03
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Minority carrier currents
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Evaluating the diode current
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Special cases for the diode current
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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] L6 February 03
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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 L6 February 03
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Diffnt’l, one-sided diode conductance
Static (steady-state) diode I-V characteristic IQ Va VQ L6 February 03
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Diffnt’l, one-sided diode cond. (cont.)
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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 L6 February 03
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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 L6 February 03
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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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