EE5342 – Semiconductor Device Modeling and Characterization Lecture 23 - Spring 2004
Semiconductor Device Modeling and Characterization EE5342, Lecture 6-Spring 2010
description
Transcript of Semiconductor Device Modeling and Characterization EE5342, Lecture 6-Spring 2010
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L6 February 03 1
Semiconductor Device Modeling and CharacterizationEE5342, Lecture 6-Spring 2010
Professor Ronald L. [email protected]
http://www.uta.edu/ronc/
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Project 1A – Diode parameters to use
L6 February 03 2
Param Value UnitsIS 3.608E-16 AN 1IKF 1.716E-08 ARS 10 OhmISR 2.422E-12 ANR 2M 0.5VJ 755 mVCJ0 3.316E-15 FdTMOM 300 KRTH 500
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Tasks• Using PSpice or any simulator, plot the i-v curve for
this diode, assuming Rth = 0, for several temperatures in the range 300 K < TEMP = TAMB < 304 K.
• Using this data, determine what the i-v plot would be for Rth = 500 K/W.
• Using this data, determine the maximum operating temperature for which the diode conductance is within 1% of the Rth = 0 value at 300 K.
• Do the same for a 10% tolerance.• Propose a SPICE macro which would give the Rth =
500 K/W i-v relationship.
L6 February 03 3
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Example
L6 February 03 4
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L6 February 03 5
Induced E-fieldin the D.R.
xn
x-xp-xpc xnc
O-O-O-
O+O+
O+
Depletion region (DR)
p-type CNR
Ex
Exposed Donor ions
Exposed Acceptor Ions
n-type chg neutral reg
p-contact N-contact
W
0
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L6 February 03 6
Depletion approx.charge distribution
xn
x-xp
-xpc xnc
+qNd
-qNa
+Qn’=qNdxn
Qp’=-qNaxp
Charge neutrality => Qp’ + Qn’ = 0,
=> Naxp = Ndxn
[Coul/cm2]
[Coul/cm2]
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L6 February 03 7
1-dim soln. ofGauss’ law
nx
nnax
ppax
px
ndpada
daeff
npeff
bi
xx ,0E
,xx0 ,xxNq E
,0xx ,xxNq
- E
xx ,0E
,xNxN ,NN
NNN
,xxW ,qN
VaV2W
xxn xn
c
-xpc-xp
Ex
-Emax
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L6 February 03 8
Depletion Approxi-mation (Summary)• For the step junction defined by
doping Na (p-type) for x < 0 and Nd, (n-type) for x > 0, the depletion width
W = {2(Vbi-Va)/qNeff}1/2, where Vbi = Vt ln{NaNd/ni
2}, and Neff=NaNd/(Na+Nd). Since Naxp=Ndxn,
xn = W/(1 + Nd/Na), and xp = W/(1 + Na/Nd).
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L6 February 03 9
One-sided p+n or n+p jctns• If p+n, then Na >> Nd, and
NaNd/(Na + Nd) = Neff --> Nd, and W --> xn, DR is all on lightly d. side
• If n+p, then Nd >> Na, and NaNd/(Na + Nd) = Neff --> Na, and W --> xp, DR is all on lightly d. side
• The net effect is that Neff --> N-, (- = lightly doped side) and W --> x-
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L6 February 03 10
JunctionC (cont.)
xn
x-xp
-xpc xnc
+qNd
-qNa
+Qn’=qNdxn
Qp’=-qNaxp
Charge neutrality => Qp’ + Qn’ = 0,
=> Naxp =
Ndxn
Qn’=qNdxn
Qp’=-qNaxp
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L6 February 03 11
JunctionC (cont.)• The C-V relationship simplifies to
][Fd/cm ,NNV2
NqN'C herew
equation model a ,VV
1'C'C
2
dabi
da0j
21
bi
a0jj
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L6 February 03 12
JunctionC (cont.)• If one plots [C’j]
-2 vs. Va
Slope = -[(C’j0)2Vbi]-1
vertical axis intercept = [C’j0]-2 horizontal axis intercept = Vbi
C’j-2
Vbi
Va
C’j0-2
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L6 February 03 13
Arbitrary dopingprofile• If the net donor conc, N = N(x), then at xn,
the extra charge put into the DR when Va->Va+Va is Q’=-qN(xn)xn
• The increase in field, Ex =-(qN/)xn, by Gauss’ Law (at xn, but also const).
• So Va=-(xn+xp)Ex= (W/) Q’
• Further, since N(xn)xn = N(xp)xp gives, the dC/dxn as ...
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L6 February 03 14
Arbitrary dopingprofile (cont.)
p
n
j
3j
j
j
n
j
nd
ndj
p
n2j
n
p2
n
j
xNxN
1
dV
'dCq
'C
'CdVd
q
'C
xd
'Cd N with
, dV
'CddC'xd
qNdVxd
qNdVdQ'
'C further
,xN
xN1
'C
dx
dx1
Wdx
'dC
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L6 February 03 15
Arbitrary dopingprofile (cont.)
,VV2
qN'C where , junctionstep
sided-one to apply Now .
dV'dC
q
'C xN
profile doping the ,xN xN orF
abij
3j
n
pn
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L6 February 03 16
Arbitrary dopingprofile (cont.)
bi0j
bi
23
bi
a0j
23
bi
a30j
V2qN
'C when ,N
V1
VV
121
'qC
VV
1'C
N so
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L6 February 03 17
Arbitrary dopingprofile (cont.)
)( and ,
12
and
when area),(A and V, , '
,quantities measured of in terms So,
22
0
VCxN
dV
CdqA
NxNxNN
CAC
jnd
j
rapnd
jj
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L6 February 03 18
Debye length• The DA assumes n changes from Nd to
0 discontinuously at xn, likewise, p changes from Na to 0 discontinuously at -xp.
• In the region of xn, the 1-dim Poisson equation is dEx/dx = q(Nd - n), and since Ex = -d/dx, the potential is the solution to -d2/dx2 = q(Nd - n)/
n
xxn
Nd
0
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L6 February 03 19
Debye length (cont)• Since the level EFi is a reference for
equil, we set = Vt ln(n/ni)
• In the region of xn, n = ni exp(/Vt), so d2/dx2 = -q(Nd - ni e
/Vt), let = o + ’, where o = Vt ln(Nd/ni) so Nd - ni e
/Vt = Nd[1 - e/Vt-o/Vt], for - o = ’ << o, the DE becomes d2’/dx2
= (q2Nd/kT)’, ’ << o
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L6 February 03 20
Debye length (cont)• So ’ = ’(xn) exp[+(x-xn)/LD]+con.
and n = Nd e’/Vt, x ~ xn, where LD is the “Debye length”
material. intrinsic for 2n and type-p
for N type,-n for N pn :Note
length. transition a ,q
kTV ,
pnqV
L
i
ad
tt
D
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L6 February 03 21
Debye length (cont)• LD estimates the transition length of a step-
junction DR (concentrations Na and Nd with Neff =
NaNd/(Na +Nd)). Thus,
bi
efft
da0V
dDaDV2
NV
N1
N1
W
NLNL
a
• For Va=0, & 1E13 < Na,Nd < 1E19
cm-3
13% < < 28% => DA is OK
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L6 February 03 22
Example
• An assymetrical p+ n junction has a lightly doped concentration of 1E16 and with p+ = 1E18. What is W(V=0)?
Vbi=0.816 V, Neff=9.9E15, W=0.33m
• What is C’j? = 31.9 nFd/cm2
• What is LD? = 0.04 m
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L6 February 03 23
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
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L6 February 03 24
Forward Bias Energy Bands
1eppkT/EEexpnp ta VV0nnFpFiiequilnon
1/exp 0 ta VV
ppFiFniequilnon ennkTEEnn
Ev
Ec
EFi
xn xnc-xpc -xp 0
q(Vbi-Va)
EFPEFNqVa
x
Imref, EFn
Imref, EFp
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L6 February 03 25
Law of the junction(follow the min. carr.)
t
bia
n
p
p
na
t
bi
no
po
po
no
po
not
no
pot2
i
datbi
V
V-Vexp
n
n
pp
,0V when and
,V
V-exp
n
n
pp
get to Invert
.nn
lnVp
plnV
n
NNlnVV
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L6 February 03 26
Law of the junction (cont.)
t
a
pt
a
n
t
a
t
a
t
bi
t
bia
VV
2ixpp
VV
2ixnn
VV
no
2iV
V
pono
pon
VV
nopoVV-V
pn
ennp also ,ennp
Junction the of Law the
enn
epn
np have We
enn nda epp for So
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L6 February 03 27
Law of the junction (cont.)
dnonapop
ppnn
ppopppop
nnonnnon
a
Nnn and Npp
injection level- low Assume
.pn and pn Assume
.ppp ,nnn and
,nnn ,ppp So
. 0V for nnot' eq.-non to Switched
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L6 February 03 28
pt
apop
nt
anon
V
V-
pononoV
V-V
pon
t
biaponno
xx at ,1VV
expnn sim.
xx at ,1VV
exppp so
,epp ,pepp
giving V
V-Vexpppp
t
bi
t
bia
InjectionConditions
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L6 February 03 29
Ideal JunctionTheory (cont.)
Apply the Continuity Eqn in CNR
ncnn
ppcp
xxx ,Jq1
dtdn
tn
0
and
xxx- ,Jq1
dtdp
tp
0
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L6 February 03 30
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
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L6 February 03 31
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
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L6 February 03 32
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
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L6 February 03 33
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
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L6 February 03 34
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
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L6 February 03 35
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
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L6 February 03 36
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
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L6 February 03 37
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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L6 February 03 38
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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L6 February 03 39
Diffnt’l, one-sided diode conductance
Va
IDStatic (steady-state) diode I-V characteristic
VQ
IQ QVa
DD dV
dIg
t
asD V
VdexpII
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L6 February 03 40
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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L6 February 03 41
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
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L6 February 03 42
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
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L6 February 03 43
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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L6 February 03 44
General time-constant
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L6 February 03 45
General time-constant (cont.)
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L6 February 03 46
General time-constant (cont.)
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