EE 5340 Semiconductor Device Theory Lecture 15 - Fall 2009 Professor Ronald L. Carter

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 τ o = 10 μs, N ref = 1×10 17 /cm 2, and C A = 1.8× cm 6 /s. L 15 Oct 132

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 τ o = 30 μs, N ref = 1×10 17 /cm 2, and C A = 8.3× cm 6 /s. L 15 Oct 133

References for Part A: Based on the information in these resources, decide which model formulae and parameters are the most accurate for Dn and Ln for electrons in p-type material, and Dp and Lp holes in n-type material. 1. Device Electronics for Integrated Circuits, 3rd ed., by Richard S. Muller, Theodore I. Kamins, and Mansun Chan, John Wiley and Sons, New York, 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 Note: This article is removed from the list and items 6 and 7 are added. D.B.M. Klaassen; “A UNIFIED MOBILITY MODEL FOR DEVICE SIMULATION”, Electron Devices Meeting, Technical Digest., International 9-12 Dec Page(s):357 – 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 M. S. Tyagi and R. Van Overstraeten, “Minority Carrier Recombination in Heavily Doped Silicon”, Solid-State Electr. Vol. 26, pp , Download a copy at Tyagi.pdf. 6. D.B.M. Klaassen, “A Unified Mobility Model for Device Simulation – I. Model Equations and Concentration Dependence”, Solid-State Electr. Vol. 35, pp , See below. 7. D.B.M. Klaassen, “A Unified Mobility Model for Device Simulation – II. Temperature Dependence of Carrier Mobility and Lifetime”, Solid-State Electr. Vol. 35, pp , Download at DbmK.pdf. L 15 Oct 134

5 Taken from Synopsys [1] manual

Taken from Synopsys [1] Table 3-6. Default … parameters – L 15 Oct 136

Part of a SPICE model for the Motorola 1N5233 Zener diode is shown in Table 1. For purposes of this assignment, this means that 1. IS may be interpreted as the multiplier of the (exp(v D /NV t ) – 1) term in the diffusion current. 2. The multiplier of the exp(v D /(NRV t )) term in the recombination current may be interpreted as ISR. 3. The M value implies that this is essential a step diode. L 15 Oct 137

Table 1. A SPICE model for the Motorola 1N5233 diode.model D1N5233 Is=629E-18 Rs=1.176 N=1 Xti=3 Eg=1.11 Cjo=140p M=.5369 Vj=.75 Isr=1.707n Nr=2 BV = 6 L 15 Oct 138

9 Use the information given to make the best estimate of the following: 1. Diode area. 2. Concentration of donors or acceptors on the lightly doped side. Support your conclusion as to the type of Si on the lightly doped side. 3. Concentration and type of the heavily doped side. 4. Estimate the value IKF might have. The multiplier of the exp(vD/(2NVt)) term in the high level injection current may be interpreted as √(IS×IKF). 5. Length of the charge neutral region on the lightly doped side. 6. Show that the estimates are self-consistent for all regions of diode operation – especially capacitance, BV, recombination, and diffusion ranges.

L 15 Oct 13 Injection Conditions 10

L 15 Oct 13 Ideal Junction Theory Assumptions E x = 0 in the chg neutral reg. (CNR) MB statistics are applicable Neglect gen/rec in depl reg (DR) Low level injection applies so that  n p < p po for -x pc < x < -x p, and  p n < n no for x n < x < x nc Steady State conditions 11

L 15 Oct 13 Ideal Junction Theory (cont.) 12

L 15 Oct 13 Ideal Junction Theory (cont.) 13

L 15 Oct 13 Ideal Junction Theory (cont.) 14

L 15 Oct 13 Diffusion Length model L = (D  ) 1/2 Diffusion Coeff. is Pierret* model 15

L 15 Oct 13 Excess minority carrier distr fctn 16

L 15 Oct 13 Forward Bias Energy Bands EvEv EcEc E Fi xnxn x nc -x pc -x p 0 q(V bi -V a ) E FP E FN qV a x Imref, E Fn Imref, E Fp 17

L 15 Oct 13 Carrier Injection xnxn -x pc 0 ln(carrier conc) ln N a ln N d ln n i ln n i 2 /N d ln n i 2 /N a x nc -x p x ~V a /V t 18

L 15 Oct 13 Minority carrier currents 19

L 15 Oct 13 Evaluating the diode current 20

L 15 Oct 13 Special cases for the diode current 21

L 15 Oct 13 Ideal diode equation Assumptions: –low-level injection –Maxwell Boltzman statistics –Depletion approximation –Neglect gen/rec effects in DR –Steady-state solution only Current dens, J x = J s expd(V a /V t ) –where expd(x) = [exp(x) -1] 22

L 15 Oct 13 Ideal diode equation (cont.) J s = J s,p + J s,n = hole curr + ele curr J s,p = qn i 2 D p coth(W n /L p )/(N d L p ) = qn i 2 D p /(N d W n ), W n > L p, “long” J s,n = qn i 2 D n coth(W p /L n )/(N a L n ) = qn i 2 D n /(N a W p ), W p > L n, “long” J s,n > N d 23

L 15 Oct 13 Diffnt’l, one-sided diode conductance VaVa IDID Static (steady- state) diode I-V characteristic VQVQ IQIQ 24

L 15 Oct 13 Diffnt’l, one-sided diode cond. (cont.) 25

L 15 Oct 13 Charge distr in a (1- sided) short diode Assume N d << N a The sinh (see L10) excess minority carrier distribution becomes linear for W n << L p  p n (x n )=p n0 expd( V a /V t ) Total chg = Q’ p = Q’ p = q  p n (x n )W n /2 xnxn x x nc  p n (x n ) W n = x nc - x n Q’ p pnpn 26

L 15 Oct 13 Charge distr in a 1- sided short diode Assume Quasi- static charge distributions Q’ p = +q  p n (x n,V a )W n /2  Q’ p =q(W/2) x {  p n (x n,V a +  V) -  p n (x n,V a )} W n = x nc - x n (V a ) xnxn x x nc  p n (x n,V a ) Q’ p pnpn  p n (x n,V a +  V)  Q’ p 27

L 15 Oct 1328 Cap. of a (1-sided) short diode (cont.)

L 15 Oct 13 References [1] Taurus Medici Medici User Guide Version A , September 2008, ©SYNOPSYS Inc pg – This reference also quotes [2] below. [2] D.J Roulston, N.D. Arora and S. G Chamberlain, “Modeling and Measurement of Minority-Carrier Versus Doping in Diffused Layers of n + -p Silicon Diodes,” IEEE Trans, Electron Devices, Vol. ED-29, pp , Feb [3] Semiconductor Device Fundamentals, 2 nd edition, by Robert F. Pierret, Addison Wesley, New York,