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EE 5340 Semiconductor Device Theory Lecture 13 - Fall 2009 Professor Ronald L. Carter [email protected] http://www.uta.edu/ronc Reverse bias junction breakdown E crit for reverse breakdown (M&K**) Taken from p. 198, M&K** Casey Model for E crit Reverse bias junction breakdown

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EE 5340Semiconductor Device TheoryLecture 13 - Fall 2009

Professor Ronald L. Carter

[email protected]

http://www.uta.edu/ronc

Reverse biasjunction breakdown

Ecrit for reverse breakdown (M&K**)

Taken from p. 198, M&K**

Casey Model for Ecrit

Reverse biasjunction 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

Junction curvatureeffect 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

Table 4.1 (M&K* p. 186) Nomograph for silicon uniformly doped, one-sided, step junctions (300 K).(See Figure 4.15 to correct for junction curvature.) (Courtesy Bell Laboratories).

E doped, one-sided, step junctions (300 K).

-

-

Ec

Ec

Ef

Efi

gen

rec

Ev

Ev

+

+

k

Direct carriergen/recomb

(Excitation can be by light)

Direct gen/rec doped, one-sided, step junctions (300 K).of 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 for doped, one-sided, step junctions (300 K).low-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- doped, one-sided, step junctions (300 K).Hall Recomb

E

Indirect, like Si, so intermediate state

Ec

Ec

ET

Ef

Efi

Ev

Ev

k

S-R-H trap doped, one-sided, step junctions (300 K).characteristics*

• 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 trap doped, one-sided, step junctions (300 K).char. (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 doped, one-sided, step junctions (300 K).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 = (Ntvthsp)-1, where sn,p~p(rBohr,n.p)2

S-R-H net recom- doped, one-sided, step junctions (300 K).bination rate, U

• In the special case where tno = tpo = to = (Ntvthso)-1 the net rec. rate, U is

S-R-H “U” function doped, one-sided, step junctions (300 K).characteristics

• 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 rec for doped, one-sided, step junctions (300 K).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)

Minority hole lifetimes doped, one-sided, step junctions (300 K).

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.

Minority electron lifetimes doped, one-sided, step junctions (300 K).

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.

Minority Carrier Lifetime, Diffusion Length and Mobility Models in Silicon

A. [40%] Write a review of the model equations for minority carrier (both electrons in p-type and holes in n-type material) lifetime, mobility and diffusion length in silicon. Any references may be used. At a minimum the material given in the following references should be used.

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.

B. [60%] This part of the assignment will be given by 10/12/09. Current-voltage data will be given for a diode, and the project will be to determine the material parameters (Nd, Na, charge-neutral region width, etc.) of the diode.

References for Part A Models in Silicon

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.

S-R-H rec for Models in Silicondeficient 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 Continuity Models in SiliconEquation

• The chain rule for the total time derivative dn/dt (the net generation rate of electrons) gives

The Continuity Models in SiliconEquation (cont.)

The Continuity Models in SiliconEquation (cont.)

The Continuity Models in SiliconEquation (cont.)

The Continuity Models in SiliconEquation (cont.)

The Continuity Models in SiliconEquation (cont.)

The Continuity Models in SiliconEquation (cont.)

References Models in Silicon

[M&K] Device Electronics for Integrated Circuits, 2nd ed., by Muller and Kamins, Wiley, New York, 1986.

[2] Devices for Integrated Circuits: Silicon and III-V Compound Semiconductors, by H. Craig Casey, Jr., John Wiley & Sons, New York, 1999.

Bipolar Semiconductor Devices, by David J. Roulston, McGraw-Hill, Inc., New York, 1990.