EE 5340 Semiconductor Device Theory Lecture 14 - Fall 2009 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc

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/tp, (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/tn, (prop to exc min carr) L 14 Oct 08

Minority hole lifetimes Mark E. Law, E. Solley, M. Liang, and Dorothea E. Burk, “Self-Consistent Model of Minority-Carrier Lifetime, Diffusion Length, and Mobility, IEEE ELECTRON DEVICE LETTERS, VOL. 12, NO. 8, AUGUST 1991 The parameters used in the fit are τo = 10 μs, Nref = 1×1017/cm2, and CA = 1.8×10-31cm6/s. L 14 Oct 08

Minority electron lifetimes Mark E. Law, E. Solley, M. Liang, and Dorothea E. Burk, “Self-Consistent Model of Minority-Carrier Lifetime, Diffusion Length, and Mobility, IEEE ELECTRON DEVICE LETTERS, VOL. 12, NO. 8, AUGUST 1991 The parameters used in the fit are τo = 30 μs, Nref = 1×1017/cm2, and CA = 8.3×10-32 cm6/s. L 14 Oct 08

References for Part A Device Electronics for Integrated Circuits, 3rd ed., by Richard S. Muller, Theodore I. Kamins, and Mansun Chan, John Wiley and Sons, New York, 2003. Mark E. Law, E. Solley, M. Liang, and Dorothea E. Burk, “Self-Consistent Model of Minority-Carrier Lifetime, Diffusion Length, and Mobility, IEEE ELECTRON DEVICE LETTERS, VOL. 12, NO. 8, AUGUST 1991. D.B.M. Klaassen; “A UNIFIED MOBILITY MODEL FOR DEVICE SIMULATION”, Electron Devices Meeting, 1990. Technical Digest., International 9-12 Dec. 1990 Page(s):357 – 360. David Roulston, Narain D. Arora, and Savvas G. Chamberlain “Modeling and Measurement of Minority-Carrier Lifetime versus Doping in Diffused Layers of n+-p Silicon Diodes”, IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-29, NO. 2, FEBRUARY 1982, pages 284-291. M. S. Tyagi and R. Van Overstraeten, “Minority Carrier Recombination in Heavily Doped Silicon”, Solid-State Electr. Vol. 26, pp. 577-597, 1983. Download a copy at Tyagi.pdf. L 14 Oct 08

S-R-H rec for deficient 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 L 14 Oct 08

The Continuity Equation The chain rule for the total time derivative dn/dt (the net generation rate of electrons) gives L 14 Oct 08

The Continuity Equation (cont.) L 14 Oct 08

The Continuity Equation (cont.) L 14 Oct 08

The Continuity Equation (cont.) L 14 Oct 08

The Continuity Equation (cont.) L 14 Oct 08

The Continuity Equation (cont.) L 14 Oct 08

The Continuity Equation (cont.) L 14 Oct 08

Review of depletion approximation pp << ppo, -xp < x < 0 nn << nno, 0 < x < xn 0 > Ex > -2Vbi/W, in DR (-xp < x < xn) pp=ppo=Na & np=npo= ni2/Na, -xpc< x < -xp nn=nno=Nd & pn=pno= ni2/Nd, xn < x < xnc x xn xnc -xpc -xp Ev Ec qVbi EFi EFn EFp L 14 Oct 9

Review of D. A. (cont.) Ex -xpc -xp xn xnc x -Emax L 14 Oct 9

Forward Bias Energy Bands Ev Ec EFi xn xnc -xpc -xp q(Vbi-Va) EFP EFN qVa x Imref, EFn Imref, EFp L 14 Oct 9

Law of the junction: “Remember to follow the minority carriers” L 14 Oct 9

Law of the junction (cont.) L 14 Oct 9

Law of the junction (cont.) L 14 Oct 9

Injection Conditions L 14 Oct 9

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 injection applies so that dnp < ppo for -xpc < x < -xp, and dpn < nno for xn < x < xnc Steady State conditions L 14 Oct 9

Ideal Junction Theory (cont.) L 14 Oct 9

Ideal Junction Theory (cont.) L 14 Oct 9

Ideal Junction Theory (cont.) L 14 Oct 9

Diffusion Length model L = (Dt)1/2 Diffusion Coeff. is Pierret* model L 14 Oct 9

Excess minority carrier distr fctn L 14 Oct 9

Forward Bias Energy Bands Ev Ec EFi xn xnc -xpc -xp q(Vbi-Va) EFP EFN qVa x Imref, EFn Imref, EFp L 14 Oct 9

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 L 14 Oct 9

Minority carrier currents L 15 Oct 14

Evaluating the diode current L 15 Oct 14

Special cases for the diode current L 15 Oct 14

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] L 15 Oct 14

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 L 15 Oct 14

Diffnt’l, one-sided diode conductance Static (steady-state) diode I-V characteristic IQ Va VQ L 15 Oct 14

Diffnt’l, one-sided diode cond. (cont.) L 15 Oct 14

Charge distr in a (1- sided) short diode dpn Assume Nd << Na The sinh (see L10) 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 L 15 Oct 14

Charge distr in a 1- sided short diode dpn Assume Quasi-static charge distributions Q’p = +qdpn(xn,Va)Wn/2 dQ’p =q(W/2) x {dpn(xn,Va+dV) - dpn(xn,Va)} Wn = xnc - xn (Va) dpn(xn,Va+dV) dpn(xn,Va) dQ’p Q’p x xn xnc L 15 Oct 14

References * Semiconductor Device Fundamentals, by Pierret, Addison-Wesley, 1996 ** Physics of Semiconductor Devices, M. Shur, Wiley. L 14 Oct 9