L08 Feb 08 1

Lecture 08 Semiconductor Device Modeling and CharacterizationEE5342 - Spring 2001

Professor Ronald L. Carterronc@uta.edu

http://www.uta.edu/ronc/

L08 Feb 08 2

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

L08 Feb 08 3

Diffnt’l, one-sided diode conductance

Va

IDStatic (steady-state) diode I-V characteristic

VQ

IQ QVa

DD dV

dIg

t

asD V

VdexpII

L08 Feb 08 4

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

L08 Feb 08 5

Charge distr in a (1-sided) short diode

• Assume Nd << Na

• The sinh (see L12) excess minority carrier distribution becomes linear for Wn << Lp

pn(xn)=pn0expd(Va/Vt)

• Total chg = Q’p = Q’p = qpn(xn)Wn/2x

n

x

xnc

pn(xn

)

Wn = xnc-

xn

Q’p

pn

L08 Feb 08 6

Charge distr in a 1-sided short diode

• Assume Quasi-static charge distributions

• Q’p = Q’p = qpn(xn)Wn/2

• dpn(xn) = (W/2)*

{pn(xn,Va+V) - pn(xn,Va)}

x

n

xxnc

pn(xn,Va)

Q’p

pn pn(xn,Va+V)

Q’p

L08 Feb 08 7

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

L08 Feb 08 8

General time-constant

np

a

nnnn

a

pppp

pnVa

pn

Va

DQd

CCC ecapacitanc diode total

the and ,dVdQ

Cg and ,dV

dQCg

that so time sticcharacteri a always is There

ggdV

JJdA

dVdI

Vg

econductanc the short, or long diodes, all For

QQ

L08 Feb 08 9

General time-constant (cont.)

times.-life carr. min. respective the

, and side, diode long

the For times. transit charge physical

the ,D2

W and ,

D2W

side, diode short the For

n0np0p

n

2p

transn,np

2n

transp,p

L08 Feb 08 10

General time-constant (cont.)

Fdd

transitminF

gC

and 111

by given average

the is time transition effective The

sided-one usually are diodes Practical

L08 Feb 08 11

Effect of non-zero E in the CNR• This is usually not a factor in a short

diode, but when E is finite -> resistor• In a long diode, there is an additional

ohmic resistance (usually called the parasitic diode series resistance, Rs)

• Rs = L/(nqnA) for a p+n long diode.

• L=Wn-Lp (so the current is diode-like for Lp and the resistive otherwise).

L08 Feb 08 12

)pn( ,ppp and ,nnn where

kTEfiE

coshn2np

npnU

dtpd

dtnd

GRU

oo

oT

i

2i

Effect of carrierrecombination in DR• The S-R-H rate (no = po = o) is

L08 Feb 08 13

Effect of carrierrec. in DR (cont.)• For low Va ~ 10 Vt

• In DR, n and p are still > ni

• The net recombination rate, U, is still finite so there is net carrier recomb.– reduces the carriers available for the

ideal diode current– adds an additional current component

L08 Feb 08 14

eff,o

taieffavgrec

o

taimaxfpfna

fnfii

fifni

x

xeffavgrec

2V2/Vexpn

qWxqUJ

2V2/Vexpn

U ,EEqV w/

,kT/EEexpnp

and ,kT/EEexpnn cesin

xqUqUdxJ curr, ecRn

p

Effect of carrierrec. in DR (cont.)

L08 Feb 08 15

High level injection effects• Law of the junction remains in the same

form, [pnnn]xn=ni

2exp(Va/Vt), etc.

• However, now pn = nn become >> nno = Nd, etc.

• Consequently, the l.o.t.j. reaches the limiting form pnnn = ni

2exp(Va/Vt)

• Giving, pn(xn) = niexp(Va/(2Vt)), or np(-xp) = niexp(Va/(2Vt)),

L08 Feb 08 16

High level injeffects (cont.)

KFKFKFsinj lh,s

i

at

i

dtKFa

appdnn

a

tainj lh,sinj lh

VJJ ,JJJ :Note

nN

lnV2 or ,n

NlnV2VV Thus

Nx-n or ,Nxp giving

V of range the for important is This

V2/VexpJJ

:is density current injection level-High

L08 Feb 08 17

Summary of Va > 0 current density eqns.• Ideal diode, Jsexpd(Va/(Vt))

– ideality factor,

• Recombination, Js,recexp(Va/(2Vt))– appears in parallel with ideal term

• High-level injection, (Js*JKF)

1/2exp(Va/(2Vt))

– SPICE model by modulating ideal Js term

• Va = Vext - J*A*Rs = Vext - Idiode*Rs

L08 Feb 08 18

Plot of typical Va > 0 current density eqns.

Vext

ln J

data

ln(JKF)

ln(Js)

ln[(Js*JKF) 1/2]

Effect

of Rs

t

aV

Vexp~

t

aV2

Vexp~

VKF

ln(Jsrec)

Effect of high level injection

low level injection

recomb. current

Vext-Vd=JARs

L08 Feb 08 19

Reverse bias (Va<0)=> carrier gen in DR• Va < 0 gives the net rec rate,

U = -ni/, = mean min carr g/r l.t.

NNN/NNN and

qN

VV2W where ,

2Wqn

J

(const.) U- G where ,qGdxJ

dadaeff

eff

abi

0

igen

x

xgen

n

p

L08 Feb 08 20

Reverse bias (Va< 0),carr gen in DR (cont.)

gens

gen

gengensrev

JJJ

JSPICE

JJJJJ

or of largest the set then ,0

V when 0 since :note model

VV where ,

current generation the plus bias negative

for current diode ideal the of value The

current the to components two are there

bias, reverse ,)0V(V for lyConsequent

a

abi

ra

L08 Feb 08 21

Reverse biasjunction breakdown• Avalanche breakdown

– Electric field accelerates electrons to sufficient energy to initiate multiplication of impact ionization of valence bonding electrons

– field dependence shown on next slide

• Heavily doped narrow junction will allow tunneling - see Neamen*, p. 274– Zener breakdown

L08 Feb 08 22

Ecrit for reverse breakdown (M&K**)

Taken from p. 198, M&K**

L08 Feb 08 23

Reverse biasjunction breakdown• Assume -Va = VR >> Vbi, so Vbi-Va-->VR

• Since Emax~ 2VR/W = (2qN-VR/())1/2, and VR = BV when Emax = Ecrit (N

- is doping of lightly doped side ~ Neff)

BV = (Ecrit )2/(2qN-)

• Remember, this is a 1-dim calculation

L08 Feb 08 24

Junction curvatureeffect on breakdown• The field due to a sphere, R, with

charge, Q is Er = Q/(4r2) for (r > R)

• V(R) = Q/(4R), (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

L08 Feb 08 25

BV for reverse breakdown (M&K**)

Taken from Figure 4.13, p. 198, M&K**

Breakdown voltage of a one-sided, plan, silicon step junction showing the effect of junction curvature.4,5

L08 Feb 08 26

Example calculations• Assume throughout that p+n jctn with Na

= 3e19cm-3 and Nd = 1e17cm-3

• From graph of Pierret mobility model, p

= 331 cm2/V-sec and Dp = Vtp = ? • Why p and Dp?

• Neff = ?

• Vbi = ?

L08 Feb 08 27

0

500

1000

1500

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)

Mob

ility

(cm̂

2/V

-se

c)P As B n(Pierret) p(Pierret)

L08 Feb 08 28

Parameters forexamples• Get min from the model used in Project

2 min = (45 sec) 1+(7.7E-18cm3Ni+(4.5E-36cm6Ni

2

• For Nd = 1E17cm3, p = 25 sec

– Why Nd and p ?

• Lp = ?

L08 Feb 08 29

Hole lifetimes, taken from Shur***, p. 101.

L08 Feb 08 30

Example

• Js,long, = ?

• If xnc, = 2 micron, Js,short, = ?

L08 Feb 08 31

Example(cont.)• Estimate VKF

• Estimate IKF

L08 Feb 08 32

Example(cont.)• Estimate Js,rec

• Estimate Rs if xnc is 100 micron

L08 Feb 08 33

Example(cont.)• Estimate Jgen for 10 V reverse bias

• Estimate BV

L08 Feb 08 34

Diode equivalentcircuit (small sig)

ID

VDVQ

IQ

t

Q

dd

VD

D

V

I

r1

gdVdI

Q

is the practical

“ideality factor”

Q

tdiff

t

Qdiffusion

mintrdd

IV

r , V

IC

long) for short, for ( , Cr

L08 Feb 08 35

Small-signal eqcircuit

CdiffCdep

l

rdiff

Cdiff and

Cdepl are both charged by

Va = VQQa

2/1

bi

ajojdepl VV ,

VV

1CCC

Va

L08 Feb 08 36

References

* Semiconductor Physics and Devices, 2nd ed., by Neamen, Irwin, Boston, 1997.

**Device Electronics for Integrated Circuits, 2nd ed., by Muller and Kamins, John Wiley, New York, 1986.

***Physics of Semiconductor Devices, Shur, Prentice-Hall, 1990.