Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002
Semiconductor Device Modeling and Characterization EE5342, Lecture 7-Spring 2004
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Transcript of Semiconductor Device Modeling and Characterization EE5342, Lecture 7-Spring 2004
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L7 February 10 1
Semiconductor Device Modeling and CharacterizationEE5342, Lecture 7-Spring 2004
Professor Ronald L. [email protected]
http://www.uta.edu/ronc/
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L7 February 10 2
MidTerm andProject Tests• MidTerm on Thursday 2/12
– Cover sheet to be posted at http://www.uta.edu/ronc/5342/tests/
• Project 1 draft assignment will be posted 2/13.– Project report to be used in doing:– Project 1 Test on Thursday 3/11– Cover sheet will be posted as above
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L7 February 10 3
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]
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L7 February 10 4
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
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L7 February 10 5
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
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L7 February 10 6
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
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L7 February 10 7
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
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L7 February 10 8
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
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L7 February 10 9
General time-constant (cont.)
Fdd
transitminF
gC
and 111
by given average
the is time transition effective The
sided-one usually are diodes Practical
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L7 February 10 10
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).
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L7 February 10 11
)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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L7 February 10 12
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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L7 February 10 13
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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L7 February 10 14
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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L7 February 10 15
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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L7 February 10 16
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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L7 February 10 17
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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L7 February 10 18
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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L7 February 10 19
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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L7 February 10 20
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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L7 February 10 21
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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L7 February 10 22
Reverse biasjunction breakdown
8/3
4/3
0
4/3
2/3
20
161/
1.1/ 120 so
,161/
1.1/ 60 gives *,***
usually , 2
D.A. theand diode sided-one a Assuming
EN
EqNVE
EN
EVBVCasey
BVqN
EBV
g
Sicrit
B
g
icritSi
i
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L7 February 10 23
Ecrit for reverse breakdown (M&K**)
Taken from p. 198, M&K**
Casey Model for Ecrit
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L7 February 10 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
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L7 February 10 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
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L7 February 10 26
rpc
rprj
rnrnc
Gauss’ Law
Surface r
rErdSE0
Surfacein Enclosed2 Q)(4
2
3
amax
33a2
3
qN so
,3
4qN 4
j
pjr
Surface
pr
r
rrEE
rrErdSE
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L7 February 10 27
Spherical DiodeFields calculations
2
3
d2
2
max 3
qN
r
rr
r
rEE jj
r Setting Er = 0 at r = rn, we get
3
d
max
qN
31
jjn r
Err
Note that the equivalent of the lever law for this spherical diode is
33d
33a NN jnpj rrrr
For rj < ro ≤ rn,
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L7 February 10 28
Spherical DiodeFields calculations
Assume Na >> Nd, so rn – rj d >> rj – rp. Further, setting the usual definition for the potential difference, and evaluating the potential difference at breakdown, we havePHIi – Va = BV and Emax = Em = Ecrit = Ec. We also define = 3eEm/qNd[cm].
njj
njjnj rr
rrr
rrr11
E11E
2
E BV 2
c3c22c
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L7 February 10 29
Showing therj ∞ limit
C1. Solve for rn – rj = as a function of Emax and solve
for the value of in the limit of rj . The solution for
rn is given below.
theorem.binomial apply the limit, thegwhen takin
11 so
,qN
E3 , 1
1/3
,0d
crit
1/3
jjjn
Sirj
jn
rrrr
rrr
.
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L7 February 10 30
Solving for theBreakdown (BV)
Solve for BV = [i – Va]Emax = Ecrit,
and solve for the value of BV in the limit of rj . The solution for BV is given
below.
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L7 February 10 31
Spherical diodeBreakdown Voltage
1.0
10.0
100.0
1.00E+14 1.00E+15 1.00E+16 1.00E+17
Substrate Concentration (cm^-3)
Bre
ak
do
wn
Vo
lta
ge
(V
olt
)
rj = 0.1 micron
rj = 0.2 micron
rj = 0.5 micron
rj = 1.0 micron
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L7 February 10 32
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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L7 February 10 33
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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L7 February 10 34
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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L7 February 10 35
Hole lifetimes, taken from Shur***, p. 101.
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L7 February 10 36
Example
• Js,long, = ?
• If xnc, = 2 micron, Js,short, = ?
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L7 February 10 37
Example(cont.)• Estimate VKF
• Estimate IKF
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L7 February 10 38
Example(cont.)• Estimate Js,rec
• Estimate Rs if xnc is 100 micron
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L7 February 10 39
Example(cont.)• Estimate Jgen for 10 V reverse bias
• Estimate BV
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L7 February 10 40
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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L7 February 10 41
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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L7 February 10 42
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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L7 February 10 43
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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L7 February 10 44
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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L7 February 10 45
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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L7 February 10 46
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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L7 February 10 47
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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L7 February 10 48
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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L7 February 10 49
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.