semiconductor device modeling and characterization ee5342, lecture 6-spring 2010
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Semiconductor Device Modeling and Characterization EE5342, Lecture 6-Spring 2010. Professor Ronald L. Carter [email protected] http://www.uta.edu/ronc/. Project 1A – Diode parameters to use. Tasks. - PowerPoint PPT PresentationTRANSCRIPT
L6 February 03 1
Semiconductor Device Modeling and CharacterizationEE5342, Lecture 6-Spring 2010
Professor Ronald L. [email protected]
http://www.uta.edu/ronc/
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
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
Example
L6 February 03 4
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
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]
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
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).
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-
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
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
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
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 ...
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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]
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
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
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
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
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
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
L6 February 03 44
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
L6 February 03 45
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
L6 February 03 46
General time-constant (cont.)
Fdd
transitminF
gC
and 111
by given average
the is time transition effective The
sided-one usually are diodes Practical