semiconductor device modeling and characterization ee5342, lecture 5-spring 2003
DESCRIPTION
Semiconductor Device Modeling and Characterization EE5342, Lecture 5-Spring 2003. Professor Ronald L. Carter [email protected] http://www.uta.edu/ronc/. First Assignment. Send e-mail to [email protected] On the subject line, put “5342 e-mail” In the body of message include - PowerPoint PPT PresentationTRANSCRIPT
L5 28Jan03 1
Semiconductor Device Modeling and CharacterizationEE5342, Lecture 5-Spring 2003
Professor Ronald L. [email protected]
http://www.uta.edu/ronc/
L5 28Jan03 2
First Assignment
• Send e-mail to [email protected]– On the subject line, put “5342 e-mail”– In the body of message include
• email address: ______________________• Last Name*: _______________________• First Name*: _______________________• Last four digits of your Student ID: _____
* As it appears in the UTA Record - no more, no less
L5 28Jan03 3
S-R-H net recom-bination rate, U• In the special case where no = po
= o = (Ntvtho)-1 the net rec. rate, U is
)pn( ,ppp and ,nnn where
kTEfiE
coshn2np
npnU
dtpd
dtnd
GRU
oo
oT
i
2i
L5 28Jan03 4
S-R-H rec forexcess min carr• For n-type low-level injection and net
excess minority carriers, (i.e., no > n = p > po = ni
2/no),
U = p/o, (prop to exc min carr)
• For p-type low-level injection and net excess minority carriers, (i.e., po > n = p > no = ni
2/po),
U = n/o, (prop to exc min carr)
L5 28Jan03 5
Minority hole lifetimes. Taken from Shur3, (p.101).
L5 28Jan03 6
Minority electron lifetimes. Taken from Shur3, (p.101).
L5 28Jan03 7
Parameter example
• min = (45 sec) 1+(7.7E-18cm3Ni+(4.5E-36cm6Ni
2
• For Nd = 1E17cm3, p = 25 sec
– Why Nd and p ?
L5 28Jan03 8
Direct rec forexcess min carr• Define low-level injection as
n = p < no, for n-type, andn = p < po, for p-type
• The recombination rates then areR’n = R’p = n(t)/n0, for p-type,
and R’n = R’p = p(t)/p0, for n-type
• Where n0 and p0 are the minority-carrier lifetimes
L5 28Jan03 9
S-R-H rec fordeficient min carr• If n < ni and p < pi, then the S-R-H net
recomb rate becomes (p < po, n < no):
U = R - G = - ni/(20cosh[(ET-Efi)/kT])
• And with the substitution that the gen lifetime, g = 20cosh[(ET-Efi)/kT], and net gen rate U = R - G = - ni/g
• The intrinsic concentration drives the return to equilibrium
L5 28Jan03 10
The ContinuityEquation• The chain rule for the total time
derivative dn/dt (the net generation rate of electrons) gives
n,kz
jy
ix
n
is gradient the of definition The
.dtdz
zn
dtdy
yn
dtdx
xn
tn
dtdn
L5 28Jan03 11
The ContinuityEquation (cont.)
vntn
dtdn then
,BABABABA Since
.kdtdz
jdtdy
idtdx
v
is velocity vector the of definition The
zzyyxx
L5 28Jan03 12
The ContinuityEquation (cont.)
etc. ,0xx
dtd
dtdx
x
since ,0dtdz
zdtdy
ydtdx
xv
RHS, the on term second the gConsiderin
.vnvnvn as
ddistribute be can operator gradient The
L5 28Jan03 13
The ContinuityEquation (cont.)
.Equations" Continuity" the are
Jq1
tp
dtdp and ,J
q1
tn
dtdn
So .Jq1
tn
vntn
dtdn
have we ,vqnJ since ly,Consequent
pn
n
n
L5 28Jan03 14
The ContinuityEquation (cont.)
z).y,(x, at p
or n of Change of Rate Local explicit"" the
is ,tp
or tn
RHS, the on term first The
z).y,(x, space in point particular a at p or
n of Rate Generation Net the represents
Eq. Continuity the of -V,dtdp or
dtdn LHS, The
L5 28Jan03 15
The ContinuityEquation (cont.)
q).( holes and (-q) electrons for signs
in difference the Note z).y,(x, point
the of" out" flowing ionsconcentrat
p or n of rate local the is Jq1
or
Jq1
RHS, the on term second The
p
n
L5 28Jan03 16
The ContinuityEquation (cont.)
inflow of rate rate generation net
change of rate Local
:as dinterprete be can Which
Jq1
dtdp
tp
:as holes the for equation
continuity the write-re can we So,
p
L5 28Jan03 17
Poisson’sEquation• The electric field at (x,y,z) is
related to the charge density =q(Nd-Na-p-n) by the Poisson Equation:
silicon for 7.11
andFd/cm, ,14E85.8
with , ypermitivit the is
xE
E where, ,E
r
o
ro
x
L5 28Jan03 18
Poisson’sEquation• For n-type material, N = (Nd - Na) >
0, no = N, and (Nd-Na+p-n)=-n +p +ni
2/N
• For p-type material, N = (Nd - Na) < 0, po = -N, and (Nd-Na+p-n) = p-n-ni
2/N
• So neglecting ni2/N, [=(Nd-Na+p-n)]
carriers. excess with material type-p
and type-n for ,npq
E
L5 28Jan03 19
Quasi-FermiEnergy
used. be must level
Fermi-quasi the then ,nnn i.e.,
m,equilibriu not in ionconcentrat the If
kT
EEexp
nn and ,
nn
lnkTEE
:by given are level Energy Fermi the and
conc carrier mequilibriu the m,equilibriu In
o
fif
i
o
i
ofif
L5 28Jan03 20
Quasi-FermiEnergy (cont.)
kT
EE
nnn
nnn
kTEE
fifn
i
o
i
ofifn
exp
:is density carrier the and
, ln
:defined is (Imref) level Fermi-Quasi The
L5 28Jan03 21
Quasi-FermiEnergy (cont.)
kT
EE
npp
npp
kTEE
fpfi
i
o
i
ofpfi
exp
:is density carrier the and
, ln
:as defined is
(Imref) level Fermi-Quasi the holes, For
L5 28Jan03 22
Energy bands forp- and n-type s/c
p-type
Ec
Ev
EFi
EFp
qp= kT ln(ni/Na)
Ev
Ec
EFi
EFnqn= kT ln(Nd/ni)
n-type
L5 28Jan03 23
JunctionC (cont.)
xn
x-xp
-xpc xnc
+qNd
-qNa
+Qn’=qNdxn
Qp’=-qNaxp
Charge neutrality => Qp’ + Qn’ = 0,
=> Naxp =
Ndxn
Qn’=qNdxn
Qp’=-qNaxp
L5 28Jan03 24
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
L5 28Jan03 25
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
L5 28Jan03 26
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 ...
L5 28Jan03 27
Arbitrary dopingprofile (cont.)
p
n3j
nn
p
n2j
n
p2n
xNxN
1
dVdC
q
'C
dCVd
qC
dxCd
N with
,dVCd
dCxd
qNdVxd
qNdVdQ
C further
,xN
xN1
'C
dx
dx1
WdxdC
L5 28Jan03 28
Arbitrary dopingprofile (cont.)
,VV2
qN'C where , junctionstep
sided-one to apply Now .
dVdC
q
'C xN
profile doping the ,xN xN orF
abij
3j
n
pn
L5 28Jan03 29
Arbitrary dopingprofile (cont.)
bi0j
bi
23
bi
a0j
23
bi
a30j
V2qN
'C when ,N
V1
VV
121
'qC
VV
1'C
N so
L5 28Jan03 30
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
L5 28Jan03 31
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
L5 28Jan03 32
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
L5 28Jan03 33
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
L5 28Jan03 34
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
L5 28Jan03 35
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
L5 28Jan03 36
Ideal JunctionTheory (cont.)
Apply the Continuity Eqn in CNR
ncnn
ppcp
xxx ,Jq1
dtdn
tn
0
and
xxx- ,Jq1
dtdp
tp
0
L5 28Jan03 37
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
L5 28Jan03 38
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
L5 28Jan03 39
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
L5 28Jan03 40
References
• 1Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 1986.
• 2Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981.
• 3 Physics of Semiconductor Devices, Shur, Prentice-Hall, 1990.