EE 5340Semiconductor Device TheoryLecture 15 - Fall 2009

Professor Ronald L. Carterronc@uta.edu

http://www.uta.edu/ronc

Minority hole lifetimesMark 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.

2DAorefD

op NCNN1 τ

ττ

L 15 Oct 13 2

Minority electron lifetimesMark 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.

2DAorefD

on NCNN1 τ

ττ

L 15 Oct 13 3

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, 2003.

2. 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.

3. 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, 1990. Technical Digest., International 9-12 Dec. 1990 Page(s):357 – 360.

4. 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.

5. 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.

6. D.B.M. Klaassen, “A Unified Mobility Model for Device Simulation – I. Model Equations and Concentration Dependence”, Solid-State Electr. Vol. 35, pp. 953-959, 1992. 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. 961-967, 1992. Download at DbmK.pdf.

L 15 Oct 13 4

L 15 Oct 13 5

Taken from Synopsys [1] manual

Taken from Synopsys [1] Table 3-6. Default … parameters –

L 15 Oct 13 6

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(vD/NVt) – 1) term in the diffusion current.

2. The multiplier of the exp(vD/(NRVt)) 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 13 7

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=.5369Vj=.75 Isr=1.707n Nr=2 BV = 6

L 15 Oct 13 8

L 15 Oct 13 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

pt

apop

nt

anon

V

V-

pononoV

V-V

pon

t

biaponno

xx at ,1VV

expnn sim.

xx at ,1VV

exppp so

,epp ,pepp

giving V

V-Vexpppp

t

bi

t

bia

InjectionConditions

10

L 15 Oct 13

Ideal JunctionTheory

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

np < ppo for -xpc < x < -xp, and pn < nno for xn < x < xnc

• Steady State conditions

11

L 15 Oct 13

Ideal Junction Theory (cont.)

ppcn

ncnp

xxx- ,Jq1

dtdn

tn

0

and , xxx ,Jq1

dtdp

tp

0

CNR the to Equation Continuity the applying

and , 0tn

tp

case, (static) state steady the In

12

L 15 Oct 13

Ideal JunctionTheory (cont.)

ppc

nn

p2p

2

ncnpp

n2n

2

ppx

nnxx

xxx- for ,0D

n

dx

nd

and ,xxx for ,0D

p

dx

pd

giving dxdp

qDJ and

dxdn

qDJ CNR, the in 0E Since

13

L 15 Oct 13

Ideal JunctionTheory (cont.)

)contacts( ,0xnxp and

,1en

xn

pxp

B.C. with

.xxx- ,DeCexn

xxx ,BeAexp

So .D L and D L Define

pcpncn

VV

po

pp

no

nn

ppcL

xL

x

p

ncnL

xL

x

n

pp2pnn

2n

ta

nn

pp

14

L 15 Oct 13

0.1

1.0

10.0

100.0

1000.0

1.E+13 1.E+14 1.E+15 1.E+16 1.E+17 1.E+18 1.E+19 1.E+20

Doping Concentration (cm̂ - 3)

Diff

usio

n Le

ngth

, L

(mic

rons

)electrons holes

Diffusion Length model

2imim

minN36E5.4N18E7.71

sec45

L = (D)1/2 Diffusion Coeff. is Pierret* model

15

L 15 Oct 13

Excess minoritycarrier distr fctn

1eLWsinh

Lxxsinhnxn

,xxW ,xxx- for and

1eLWsinh

Lxxsinhpxp

,xxW ,xxx For

ta

ta

VV

np

npcpop

ppcpppc

VV

pn

pncnon

nncnncn

16

L 15 Oct 13

Forward Bias Energy Bands

1eppkT/EEexpnp ta VV0nnFpFiiequilnon

1/exp 0 ta VV

ppFiFniequilnon ennkTEEnn

Ev

Ec

EFi

xn xnc-xpc -xp 0

q(Vbi-Va)

EFPEFNqVa

x

Imref, EFn

Imref, EFp

17

L 15 Oct 13

CarrierInjection

xn-xpc 0

ln(carrier conc)ln Naln Nd

ln ni

ln ni2/Nd

ln ni2/Na

xnc-xp

x

~Va/Vt~Va/Vt

1enxn t

aV

V

popp

1epxp t

aV

V

nonn

18

L 15 Oct 13

Minority carriercurrents

1eLWsinh

Lxxcosh

LNDqn

xxx- for ,qDxJ

1eLWsinh

Lxxcosh

LN

Dqn

xxx for ,qDxJ

ta

p

ta

n

VV

np

npc

na

n2i

ppcdx

ndnn

VV

pn

pnc

pd

p2i

ncndxpd

pp

19

L 15 Oct 13

Evaluating thediode current

p/nn/pp/nd/a

p/n2isp/sn

spsns

VV

spnnp

LWcothLN

DqnJ

sdefinition with JJJ where

1eJxJxJJ

then DR, in gen/rec no gminAssu

ta

20

L 15 Oct 13

Special cases forthe diode current

nd

p2isp

pa

n2isn

nppn

pd

p2isp

na

n2isn

nppn

WN

DqnJ and ,

WND

qnJ

LW or ,LW :diode Short

LN

DqnJ and ,

LND

qnJ

LW or ,LW :diode Long

21

L 15 Oct 13

Ideal diodeequation• Assumptions:

– 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] 22

L 15 Oct 13

Ideal diodeequation (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

23

L 15 Oct 13

Diffnt’l, one-sided diode conductance

Va

IDStatic (steady-state) diode I-V characteristic

VQ

IQ QVa

Dd dV

dIg

t

asD V

VdexpII

24

L 15 Oct 13

Diffnt’l, one-sided diode cond. (cont.)

DQ

t

dQd

QDDQt

DQQd

tat

tQs

Va

DQd

tastasD

IV

g1

Vr ,resistance diode The

. VII where ,V

IVg then

, VV If . V

VVexpI

dV

dIVg

VVdexpIVVdexpAJJAI

Q

25

L 15 Oct 13

Charge distr in a (1-sided) short diode

• Assume Nd << Na

• The sinh (see L10) excess minority carrier distribution becomes linear for Wn << Lp

pn(xn)=pn0expd(Va/

Vt)

• Total chg = Q’p =

Q’p = qpn(xn)Wn/2

x

n

x

xnc

pn(xn

)

Wn = xnc-

xn

Q’p

pn

26

L 15 Oct 13

Charge distr in a 1-sided short diode

• Assume Quasi-static charge distributions

• Q’p = +qpn(xn,Va)Wn/2

• Q’p =q(W/2) x {pn(xn,Va+V)

- pn(xn,Va)}

• Wn = xnc - xn (Va)

xn

xxnc

pn(xn,Va)

Q’p

pn pn(xn,Va+V)

Q’p

27

L 15 Oct 13 28

Cap. of a (1-sided) short diode (cont.)

p

x

x p

ntransitQQ

transitt

DQ

pt

DQQ

taaa

a

Ddx

Jp

qVV

V

I

DV

IV

VVddVdV

dVA

nc

n2W

Cr So,

. 2W

C ,V V When

exp2

WqApd2

)W(xpqAd

dQC Define area. diode A ,Q'Q

2n

dd

2n

dta

nn0nnn

pdpp

L 15 Oct 13

References

[1] Taurus Medici Medici User Guide Version A-2008.09, September 2008, ©SYNOPSYS Inc pg 3-306 – 3-315. 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. 284-291, Feb. 1982.

[3] Semiconductor Device Fundamentals , 2nd edition, by Robert F. Pierret, Addison Wesley, New York, 1996.

29