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EE5342 – Semiconductor Device Modeling and Characterization Lecture 05-Spring 2010

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## EE5342 – Semiconductor Device Modeling and Characterization Lecture 05-Spring 2010

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**EE5342 – Semiconductor Device Modeling and**CharacterizationLecture 05-Spring 2010 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/**First Assignment**• Send e-mail to ronc@uta.edu • On the subject line, put “5342 e-mail” • In the body of message include • Your email address • Your Name as it appears in the UTA Record - no more, no less • Last four digits of your Student ID: _____ • The name you would like me to use when speaking to you.**Second Assignment**• e-mail to listserv@listserv.uta.edu • In the body of the message include subscribe EE5342 • This will subscribe you to the EE5342 list. Will receive all EE5342 messages • If you have any questions, send to ronc@uta.edu, with EE5342 in subject line.**E**- - Ec Ec Ef Efi gen rec Ev Ev + + k Direct carriergen/recomb (Excitation can be by light)**Direct gen/recof 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**Direct rec forlow-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**Shockley-Read-Hall Recomb**E Indirect, like Si, so intermediate state Ec Ec ET Ef Efi Ev Ev k**S-R-H trapcharacteristics1**• 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”**S-R-H trapchar. (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**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**S-R-Hrecomb. (cont.)**• In the special case where tno = tpo = to the net recombination rate, U is**S-R-H “U” functioncharacteristics**• 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**S-R-H “U” functioncharacteristics (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**S-R-H net recom-bination rate, U**• In the special case where tno = tpo = to = (Ntvthso)-1 the net rec. rate, U is**S-R-H rec forexcess 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)**Parameter example**• tmin = (45 msec) 1+(7.7E-18cm3)Ni+(4.5E-36cm6)Ni2 • For Nd = 1E17cm3, tp = 25 msec • Why Nd and tp ?**S-R-H rec fordeficient min carr**• If n < ni and p< pi, then the S-R-H net recomb rate becomes (p < po, n < no): U = R - G = - ni/(2t0cosh[(ET-Efi)/kT]) • And with the substitution that the gen lifetime, tg = 2t0cosh[(ET-Efi)/kT], and net gen rate U = R - G = - ni/tg • The intrinsic concentration drives the return to equilibrium**The ContinuityEquation**• The chain rule for the total time derivative dn/dt (the net generation rate of electrons) gives**p-type**Ec Ec Ev EFn qfn= kT ln(Nd/ni) EFi Ev Energy bands forp- and n-type s/c n-type EFi qfp= kT ln(ni/Na) EFp**Eo**Making contactin a p-n junction • Equate the EF in the p- and n-type materials far from the junction • Eo(the free level), Ec, Efi and Ev must be continuous N.B.: qc = 4.05 eV (Si), and qf = qc + Ec - EF qc(electron affinity) qf (work function) Ec Ef Efi qfF Ev**EfN**Band diagram forp+-n jctn* at Va = 0 Ec qVbi = q(fn -fp) qfp < 0 Ec Efi EfP Ev Efi qfn > 0 *Na > Nd -> |fp|> fn Ev p-type for x<0 n-type for x>0 x -xpc xn 0 -xp xnc**Band diagram forp+-n at Va=0 (cont.)**• A total band bending of qVbi = q(fn-fp) = kT ln(NdNa/ni2) is necessary to set EfP = EfN • For -xp < x < 0, Efi - EfP < -qfp, = |qfp| so p < Na = po, (depleted of maj. carr.) • For 0 < x < xn, EfN - Efi < qfn, so n < Nd = no, (depleted of maj. carr.) -xp < x < xn is the Depletion Region**DepletionApproximation**• Assume p << po = Nafor -xp < x < 0, so r = q(Nd-Na+p-n) = -qNa, -xp < x < 0, and p = po = Nafor -xpc < x < -xp, so r = q(Nd-Na+p-n) = 0, -xpc < x < -xp • Assume n << no = Ndfor 0 < x < xn, so r = q(Nd-Na+p-n) = qNd, 0 < x < xn, and n = no = Ndfor xn < x < xnc, so r = q(Nd-Na+p-n) = 0, xn < x < xnc**Poisson’sEquation**• The electric field at (x,y,z) is related to the charge density r=q(Nd-Na-p-n) by the Poisson Equation:**Poisson’sEquation**• For n-type material, N = (Nd - Na) > 0, no = N, and (Nd-Na+p-n)=-dn +dp +ni2/N • For p-type material, N = (Nd - Na) < 0, po = -N, and (Nd-Na+p-n) = dp-dn-ni2/N • So neglecting ni2/N, [r=(Nd-Na+p-n)]**Depletion approx.charge distribution**r +Qn’=qNdxn +qNd [Coul/cm2] -xp x -xpc xn xnc Charge neutrality => Qp’ + Qn’ = 0, => Naxp = Ndxn -qNa Qp’=-qNaxp [Coul/cm2]**Induced E-fieldin the D.R.**• The sheet dipole of charge, due to Qp’ and Qn’ induces an electric field which must satisfy the conditions • Charge neutrality and Gauss’ Law* require thatEx = 0 for -xpc < x < -xp and Ex = 0 for -xn < x < xnc h0**O**O O O O O + + + - - - Induced E-fieldin the D.R. Ex N-contact p-contact p-type CNR n-type chg neutral reg Depletion region (DR) Exposed Donor ions Exposed Acceptor Ions W x -xpc -xp xn xnc 0**1-dim soln. ofGauss’ law**Ex -xp xn xnc -xpc x -Emax**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).**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. • 3 Physics of Semiconductor Devices, Shur, Prentice-Hall, 1990.