L8 06Feb03 1
Semiconductor Device Modeling and CharacterizationEE5342, Lecture 8-Spring 2003
Professor Ronald L. [email protected]
http://www.uta.edu/ronc/
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)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
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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
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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.)
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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)),
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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
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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
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1N ,
V2NV
t
aexp~
1N ,
VNV
t
aexp~
Vext
ln(J)
data Effect of Rs
2NR ,
VNRV
t
aexp~
VKF
Plot of typical Va > 0 current density equations
Sexta RAJ-VV
KFS JJln
recsJln ,
SJln
KFJln
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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
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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
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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
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Ecrit for reverse breakdown (M&K**)
Taken from p. 198, M&K**
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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
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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
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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
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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 = ?
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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)
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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 = ?
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Hole lifetimes, taken from Shur***, p. 101.
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Example
• Js,long, = ?
• If xnc, = 2 micron, Js,short, = ?
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Example(cont.)• Estimate VKF
• Estimate IKF
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Example(cont.)• Estimate Js,rec
• Estimate Rs if xnc is 100 micron
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Example(cont.)• Estimate Jgen for 10 V reverse bias
• Estimate BV
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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
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Small-signal eqcircuit
CdiffCdep
l
rdiff
Cdiff and
Cdepl are both charged by
Va = VQQa
2/1
bi
ajojdepl VV ,
VV
1CCC
Va
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Diode Switching
• Consider the charging and discharging of a Pn diode – (Na > Nd)
– Wd << Lp
– For t < 0, apply the Thevenin pair VF and RF, so that in steady state • IF = (VF - Va)/RF, VF >> Va , so current source
– For t > 0, apply VR and RR
• IR = (VR + Va)/RR, VR >> Va, so current source
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Diode switching(cont.)
+
+ VF
VR
DRR
RF
Sw
R: t > 0
F: t < 0
ItI s
F
FF R
VI0tI
VF,VR >>
Va
F
F
F
aFQ R
VR
VVI
0,t for
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Diode chargefor t < 0
xn xncx
pn
pno
Dp2W
,IWV,xqp'Q
2N
TR
TRFnFnndiff,p
D
2i
noV/V
noFn Nn
p ,epV,xp tF
dxdp
qDJ since ,qAD
Idxdp
ppp
F
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Diode charge fort >>> 0 (long times)
xn xncx
pn
pno
tF V/Vnon ep0t,xp
t,xp
sppp
S Jdxdp
qDJ since ,qADI
dxdp
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Equationsummary
Q discharge to flows
R/VI current, a 0, but small, t For
RV
I ,qAD
Idxdp
AJI ,AqD
I
JqD1
dxdp
RRR
F
FF
p
F
0t,F
ssp
s
,ppt,R
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Snapshot for tbarely > 0
xn xncx
pn
pno
p
F
qADI
dxdp
p
RqAD
Idxdp
tF V/Vnon ep0t,xp
0t,xp Total charge removed, Qdis=IRt
st,xp
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I(t) for diodeswitching
ID
t
IF
-IR
ts ts+trr
- 0.1 IR
sRdischarge
p
Rs
tIQ
constant, a is qAD
Idxdp
,tt 0 For
pnp
p2is L/WtanhL
DqnI
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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.