Lecture 8: Capacitors and PN Junctionsee105/fa03/handouts/lectures/Lecture8.pdf · Lecture 8:...
24
Department of EECS University of California, Berkeley EECS 105 Fall 2003, Lecture 8 Lecture 8: Capacitors and PN Junctions Prof. Niknejad
Transcript of Lecture 8: Capacitors and PN Junctionsee105/fa03/handouts/lectures/Lecture8.pdf · Lecture 8:...
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8
Lect
ure
8:
Cap
acito
rs a
nd P
N J
unct
ions
Prof
. Nik
neja
d
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Lect
ure
Out
line
Rev
iew
of E
lect
rost
atic
sIC
MIM
Cap
acito
rsN
on-L
inea
r Cap
acito
rsPN
Junc
tions
The
rmal
Equ
ilibr
ium
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Elec
tros
tatic
s R
evie
w (1
)El
ectri
c fie
ld g
o fr
om p
ositi
ve c
harg
e to
neg
ativ
e ch
arge
(by
conv
entio
n)
Elec
tric
field
line
s diverge
on c
harg
e
In w
ords
, if t
he e
lect
ric fi
eld
chan
ges m
agni
tude
, th
ere
has t
o be
cha
rge
invo
lved
!R
esul
t: In
a c
harg
e fr
ee re
gion
, the
ele
ctric
fiel
d m
ust b
e co
nsta
nt!
++++
++++
++++
++++
++++
+
−−−−−−−−−−−−−−−
ερ=
⋅∇E
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Elec
tros
tatic
s R
evie
w (2
)
Gau
ss’L
aw e
quiv
alen
tly sa
ys th
at if
ther
e is
a net
elec
tric
field
leav
ing
a re
gion
, the
re h
as to
be
posi
tive
char
ge in
that
regi
on:
++++
++++
++++
++++
++++
+
−−−−−−−−−−−−−−−
Elec
tric
Fie
lds
are
Leav
ing
This
Box
!
∫=
⋅εQ
dSE
∫∫
==
⋅∇
VV
QdV
dVE
εερ
/εQ
dSE
dVE
SV
∫∫
=⋅
=⋅
∇
Rec
all:
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Elec
tros
tatic
s in
1D
Ever
ythi
ng si
mpl
ifies
in 1
-D
Con
side
r a u
nifo
rm c
harg
e di
strib
utio
n
ερ=
=⋅
∇dxdE
Edx
dEερ
=
')'
()
()
(0
0dx
xx
Ex
Ex x∫
+=
ερ
)(xρ
xx
dxx
xE
x
ερε
ρ0
0
')'
()
(=
=∫
Zero
fiel
dbo
unda
ryco
nditi
on
1x
0ρ
1x
)(xE
10x
ερ
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Elec
tros
tatic
Pot
entia
l
The
elec
tric
field
(for
ce) i
s rel
ated
to th
e po
tent
ial
(ene
rgy)
:
Neg
ativ
e si
gn sa
ys th
at fi
eld
lines
go
from
hig
h po
tent
ial p
oint
s to
low
er p
oten
tial p
oint
s (ne
gativ
e sl
ope)
Not
e: A
n el
ectro
n sh
ould
“flo
at”
to a
hig
h po
tent
ial
poin
t:
dxdE
φ−
= dxde
qEF e
φ−
==
1φ
2φ
dxde
F eφ
−= e
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Mor
e Po
tent
ial
Inte
grat
ing
this
bas
ic re
latio
n, w
e ha
ve th
at th
e po
tent
ial i
s the
inte
gral
of t
he fi
eld:
In 1
D, t
his i
s a si
mpl
e in
tegr
al:
Goi
ng th
e ot
her w
ay, w
e ha
ve P
oiss
on’s
equ
atio
n in
1D
:
∫⋅
−=
−C
ldE
xx
)(
)(
0φ
φ
)(
0xφ
)(xφ
Eld
∫−=
−x x
dxx
Ex
x0
')'
()
()
(0
φφ
ερφ
)(
)( 2
2x
dxx
d−
=
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Bou
ndar
y C
ondi
tions
Pote
ntia
l mus
t be
a co
ntin
uous
func
tion.
If n
ot, t
he fi
elds
(f
orce
s) w
ould
be
infin
ite
Elec
tric
field
s nee
d no
t be
cont
inuo
us.
We
have
alre
ady
seen
that
the
elec
tric
field
s div
erge
on
char
ges.
In fa
ct,
acro
ss a
n in
terf
ace
we
have
:
Fiel
d di
scon
tiuity
impl
ies c
harg
e de
nsity
at s
urfa
ce!
)(
11
εE
)(
22
εE
∫=
+−
=⋅
inside
QS
ES
EdS
E2
21
1ε
εε
x∆0
0
→
→∆x
inside
Q
02
21
1=
+−
SE
SE
εε
12
21
εε=
EES
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
IC M
IM C
apac
itor
Bot
tom
Pla
teB
otto
m P
late
Top
Pla
te
CV
Q=
Thin
Oxi
de
Con
tact
s
By
form
ing
a th
in o
xide
and
met
al (o
r pol
ysili
con)
pla
tes,
a ca
paci
tor i
s for
med
Con
tact
s are
mad
e to
top
and
botto
m p
late
Para
sitic
cap
acita
nce
exis
ts b
etw
een
botto
m p
late
and
su
bstra
te
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Rev
iew
of C
apac
itors
∫=
⋅εQ
dSE
++++
++++
++++
++++
++++
+
−−−−−−−−−−−−−−−
+ −V
s
sox
Vt
Edl
E=
=⋅
∫0
oxs tVE
=0
∫=
=⋅
εQA
EdS
E0
εQA
tV oxs=
∫−
=⋅
εQdS
E
sCV
Q=
oxtAC
ε=
For a
n id
eal m
etal
, all
char
ge m
ust b
e at
surf
ace
Gau
ss’l
aw:
Surf
ace
inte
gral
of e
lect
ric fi
eld
over
cl
osed
surf
ace
equa
ls c
harg
e in
side
vol
ume
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Cap
acito
r Q-V
Rel
atio
n
Totalc
harg
e is
line
arly
rela
ted
to v
olta
geC
harg
e de
nsity
is a
del
ta fu
nctio
n at
surf
ace
(for
pe
rfec
t met
als)
sCV
Q=
sV
Q
y
)(yQ
y++
++++
++++
++++
++++
+++
−−−−−−−−−−−−−−−
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
A N
on-L
inea
r Cap
acito
r
We’
ll so
on m
eet c
apac
itors
that
hav
e a
non-
linea
r Q-V
re
latio
nshi
pIf
pla
tes a
re n
ot id
eal m
etal
, the
cha
rge
dens
ity c
an p
enet
rate
in
to su
rfac
e
)(sV
fQ=
sV
Q
y
)(yQ
y++
++++
++++
++++
++++
+++
−−−−−−−−−−−−−−−
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Wha
t’s th
e C
apac
itanc
e?
For a
non
-line
ar c
apac
itor,
we
have
We
can’
t ide
ntify
a c
apac
itanc
eIm
agin
e w
e ap
ply
a sm
all s
igna
l on
top
of a
bia
s vo
ltage
:
The
incr
emen
tal c
harg
e is
ther
efor
e:
ss
CV
Vf
Q≠
=)
(
sV
Vs
ss
vdVV
dfV
fv
Vf
Qs
=
+≈
+=
)(
)(
)(
Con
stan
t cha
rge
sV
Vs
vdVV
dfV
fq
s=
+≈
+=
)(
)(
0
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Smal
l Sig
nal C
apac
itanc
e
Bre
ak th
e eq
uatio
n fo
r tot
al c
harg
e in
to tw
o te
rms:
sV
Vs
vdVV
dfV
fq
s=
+≈
+=
)(
)(
0
Incr
emen
tal
Cha
rge
Con
stan
tC
harg
e
ss
VV
vC
vdVV
dfq
s
==
=
)(
sVV
dVV
dfC
=
≡)
(
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Exam
ple
of N
on-L
inea
r Cap
acito
r
Nex
t lec
ture
we’
ll se
e th
at fo
r a P
N ju
nctio
n, th
e ch
arge
is a
func
tion
of th
e re
vers
e bi
as:
Smal
l sig
nal c
apac
itanc
e:
bp
aj
Vx
qNV
Qφ
−−
=1
)(
Con
stan
tsC
harg
e A
t N S
ide
of J
unct
ion
Volta
ge A
cros
s N
PJu
nctio
n
b
j
b
b
pa
jj
VC
Vx
qNdVdQ
VC
φφ
φ−
=−
==
111
2)
(0
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Car
rier C
once
ntra
tion
and
Pote
ntia
l
In th
erm
al e
quili
briu
m, t
here
are
no
exte
rnal
fiel
ds
and
we
thus
exp
ect t
he e
lect
ron
and
hole
cur
rent
de
nsiti
es to
be
zero
:
dxdnqD
Eqn
Jo
nn
n+
==
00
0µ
dxdn
kTqE
nD
dxdno
onn
o0
0φ
µ
=
−
=
00
00
ndnV
ndnqkT
dth
o=
=φ
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Car
rier C
once
ntra
tion
and
Pote
ntia
l (2)
We
have
an
equa
tion
rela
ting
the
pote
ntia
l to
the
carr
ier c
once
ntra
tion
If w
e in
tegr
ate
the
abov
e eq
uatio
n w
e ha
ve
We
defin
e th
e po
tent
ial r
efer
ence
to b
e in
trins
ic S
i:
)(
)(
ln)
()
(0
000
00
xnx
nV
xx
th=
−φ
φ
inx
nx
==
)(
0)
(0
00
0φ
00
00
ndnV
ndnqkT
dth
o=
=φ
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Car
rier C
once
ntra
tion
Vers
us P
oten
tial
The
carr
ier c
once
ntra
tion
is th
us a
func
tion
of
pote
ntia
l
Che
ck th
at fo
r zer
o po
tent
ial,
we
have
intri
nsic
ca
rrie
r con
cent
ratio
n (r
efer
ence
).
If w
e do
a si
mila
r cal
cula
tion
for h
oles
, we
arriv
e at
a
sim
ilar e
quat
ion
Not
e th
at th
e la
w o
f mas
s act
ion
is u
phel
d
thVx
ienx
n/)
(0
0)
(φ
=
thVx
ienx
p/)
(0
0)
(φ−
=
2/)
(/)
(2
00
00
)(
)(
iV
xV
xi
ne
en
xp
xn
thth
==
−φ
φ
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
The
Dop
ing
Cha
nges
Pot
entia
lD
ue to
the
log
natu
re o
f the
pot
entia
l, th
e po
tent
ial c
hang
es
linea
rly fo
r exp
onen
tial i
ncre
ase
in d
opin
g:
Qui
ck c
alcu
latio
n ai
d: F
or a
p-ty
pe c
once
ntra
tion
of 1
016
cm-3
, the
pot
entia
l is -
360
mV
N-ty
pe m
ater
ials
hav
e a
posi
tive
pote
ntia
l with
resp
ect t
o in
trins
ic S
i
100
0
0
0
00
10)
(lo
g10
lnm
V26
)(
)(
lnm
V26
)(
)(
ln)
(x
nx
nx
nx
nx
nV
xi
ith
≈=
=φ
100
010
)(
log
mV
60)
(x
nx≈
φ
100
010
)(
log
mV
60)
(x
px
−≈
φ
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
PN J
unct
ions
: O
verv
iew
The
mos
t im
porta
nt d
evic
e is
a ju
nctio
n be
twee
n a
p-ty
pe re
gion
and
an
n-ty
pe re
gion
Whe
n th
e ju
nctio
n is
firs
t for
med
, due
to th
e co
ncen
tratio
n gr
adie
nt, m
obile
cha
rges
tra
nsfe
r nea
r jun
ctio
n El
ectro
ns le
ave
n-ty
pe re
gion
and
hol
es le
ave
p-ty
pe re
gion
Thes
e m
obile
car
riers
bec
ome
min
ority
ca
rrie
rs in
new
regi
on (c
an’t
pene
trate
far d
ue
to re
com
bina
tion)
Due
to c
harg
e tra
nsfe
r, a
volta
ge d
iffer
ence
oc
curs
bet
wee
n re
gion
sTh
is c
reat
es a
fiel
d at
the
junc
tion
that
cau
ses
drift
cur
rent
s to
oppo
se th
e di
ffus
ion
curr
ent
In th
erm
al e
quili
briu
m, d
rift c
urre
nt a
nd
diff
usio
n m
ust b
alan
ce
n-ty
pe
p-ty
pe ND
NA
−−−−−−
+ +
+ +
++
+ +
+ +
+ +
+ +
+
−−−−−−
−−−−−−
− V +
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
PN J
unct
ion
Cur
rent
s
Con
side
r the
PN
junc
tion
in th
erm
al e
quili
briu
mA
gain
, the
cur
rent
s hav
e to
be
zero
, so
we
have
dxdnqD
Eqn
Jo
nn
n+
==
00
0µ
dxdnqD
Eqn
on
n−
=0
0µ
dxdnn
qkTndxdn
DE
n
on
0
00
01
−=
−=
µ
dxdpp
qkTndxdp
DE
p
op
0
00
01
−=
=µ
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
PN J
unct
ion
Fiel
ds
n-ty
pep-
type
ND
NA
)(
0x
paN
p=
0
di Nnp
2
0=
diff
J0E
ai Nnn
2
0=
Tran
sitio
n R
egio
n
diff
J
dN
n=
0
––
+ +
0E0px−
0nx
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Tota
l Cha
rge
in T
rans
ition
Reg
ion
To so
lve
for t
he e
lect
ric fi
elds
, we
need
to w
rite
dow
n th
e ch
arge
den
sity
in th
e tra
nsiti
on re
gion
:
In th
e p-
side
of t
he ju
nctio
n, th
ere
are
very
few
el
ectro
ns a
nd o
nly
acce
ptor
s:
Sinc
e th
e ho
le c
once
ntra
tion
is d
ecre
asin
g on
the
p-si
de, t
he n
et c
harg
e is
neg
ativ
e:
)(
)(
00
0a
dN
Nn
pq
x−
+−
=ρ
)(
)(
00
aN
pq
x−
≈ρ
0)
(0
<x
ρ0p
Na>
00
<<
−x
x p
Dep
artm
ent o
f EEC
SU
nive
rsity
of C
alifo
rnia
, Ber
kele
y
EE
CS
105
Fall
2003
, Lec
ture
8P
rof.
A. N
ikne
jad
Cha
rge
on N
-Sid
e
Ana
logo
us to
the
p-si
de, t
he c
harg
e on
the
n-si
de is
gi
ven
by:
The
net c
harg
e he
re is
pos
itive
sinc
e:
)(
)(
00
dN
nq
x+
−≈
ρ0
0nx
x<
<
0)
(0
>x
ρ0n
Nd> ai Nn
n2
0=
diff
J
dN
n=
0
––
+ +
0E
Tran
sitio
n R
egio
n
