L07 04Feb02 1
Semiconductor Device Modeling and CharacterizationEE5342, Lecture 7-Spring 2002
Professor Ronald L. [email protected]
http://www.uta.edu/ronc/
L07 04Feb02 2
Ideal JunctionTheory
Assumptions
• Ex = 0 in the chg neutral reg. (CNR)
• MB statistics are applicable• Neglect gen/rec in depl reg (DR)• Low level injections apply so that
np < ppo for -xpc < x < -xp, and pn
< nno for xn < x < xnc
• Steady State conditions
L07 04Feb02 3
Ideal JunctionTheory (cont.)
Apply the Continuity Eqn in CNR
ncnn
ppcp
xxx ,Jq1
dtdn
tn
0
and
xxx- ,Jq1
dtdp
tp
0
L07 04Feb02 4
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
L07 04Feb02 5
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
L07 04Feb02 6
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
L07 04Feb02 7
Forward Bias Energy Bands
xn xnc-xpc -xp 0
Ev
Ec
q(Vbi-Va)
EF
EF
EFi
qVa
x
Imref, EFn
Imref, EFp
1ennkT/EEexpnn ta VV0ppFiFniequilnon
1eppkT/EEexpnp ta VV0nnFpFiiequilnon
L07 04Feb02 8
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
L07 04Feb02 9
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
L07 04Feb02 10
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
L07 04Feb02 11
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
L07 04Feb02 12
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]
L07 04Feb02 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
L07 04Feb02 14
Diffnt’l, one-sided diode conductance
Va
IDStatic (steady-state) diode I-V characteristic
VQ
IQ QVa
DD dV
dIg
t
asD V
VdexpII
L07 04Feb02 15
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
L07 04Feb02 16
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/2
x
n
x
xnc
pn(xn
)
Wn = xnc-
xn
Q’p
pn
L07 04Feb02 17
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
L07 04Feb02 18
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
L07 04Feb02 19
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
L07 04Feb02 20
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
L07 04Feb02 21
General time-constant (cont.)
Fdd
transitminF
gC
and 111
by given average
the is time transition effective The
sided-one usually are diodes Practical
L07 04Feb02 22
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).
L07 04Feb02 23
)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
L07 04Feb02 24
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
L07 04Feb02 25
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.