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Semiconductor Device Modeling and Characterization EE5342, Lecture 7-Spring 2004. Professor Ronald L. Carter [email protected] http://www.uta.edu/ronc/. MidTerm and Project Tests. MidTerm on Thursday 2/12 Cover sheet to be posted at http://www.uta.edu/ronc/5342/tests/ - PowerPoint PPT Presentation
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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/
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
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]
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
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
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
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
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
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
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).
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
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
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.)
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)),
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
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
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
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
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
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
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
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
L7 February 10 23
Ecrit for reverse breakdown (M&K**)
Taken from p. 198, M&K**
Casey Model for Ecrit
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
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
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
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,
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
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
.
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.
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
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 = ?
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)
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 = ?
L7 February 10 35
Hole lifetimes, taken from Shur***, p. 101.
L7 February 10 36
Example
• Js,long, = ?
• If xnc, = 2 micron, Js,short, = ?
L7 February 10 37
Example(cont.)• Estimate VKF
• Estimate IKF
L7 February 10 38
Example(cont.)• Estimate Js,rec
• Estimate Rs if xnc is 100 micron
L7 February 10 39
Example(cont.)• Estimate Jgen for 10 V reverse bias
• Estimate BV
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
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
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
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
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
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
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
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
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
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.