principles of semiconductor devices-l33
TRANSCRIPT
-
8/8/2019 Principles of Semiconductor Devices-L33
1/22
Alam ECE-606 S09 1
www.nanohub.org
NCN
ECE606: Solid State DevicesLecture 33: MOSCAP Electrostatics (II)
Muhammad Ashraful [email protected]
-
8/8/2019 Principles of Semiconductor Devices-L33
2/22
Outline
Alam ECE-606 S09 2
1. Review
2. Induced charges in depletion and inversion
3. Exact solution of electrostatic problem
4. Conclusion
REF: Chapters 15-18 from SDF
-
8/8/2019 Principles of Semiconductor Devices-L33
3/22
Topic Map
Alam ECE-606 S09 3
Equilibrium DC Small
signal
Large
Signal
Circuits
Diode
Schottky
BJT/HBT
MOSCAP
-
8/8/2019 Principles of Semiconductor Devices-L33
4/22
Threshold for Inversion
4
s(V)
l o g1 0QS S( )C / c m2
~ S
~ eqS / 2kBT
~ eqS / 2kBT
0 0
0
2 A oxG s s
ox A
qN xV qN
= +
-Q
+QExposed
Acceptors
VG
VG >VT
2F
0 0
0
22 2 A ox
th
ox
F
A
F
qN xV
qN
= +
Electrons
-
8/8/2019 Principles of Semiconductor Devices-L33
5/22
What happens when surface potential is 2F ?
5
-Q
+Q
Electrons
Exposed
Acceptors
Electron concentration equals
background acceptor concentration
0 0
0
2 2 2 A oxthox
F
A
FqN xVqN
= +
( )F isE E
in e=
1sn
) ( )(( ) ( )F i bulk i bulk isE E
i
E En e e
=
( )( )i bulk isFi
E En e e
=
2F Fi
n e e =
1sn
+qs EC
EV
Ei
Ei,s
EF
F
i An e N
= =
-
8/8/2019 Principles of Semiconductor Devices-L33
6/22
A little bit about scaling .
Alam ECE-606 S09
0 0
0
22 2 A oxth
ox
F
A
F
qN xV
qN
= +
-Q
+Q
Electrons
Exposed
Acceptors
Reduce Vth by
Reducing oxide thickness
(from 1000 A in 1970s
to 10 A now)
Increase dielectric constant
(SiO2 historically, HfO2 now in
Intel Penryn)
+qs EC
EV
Ei
Ei,s
EF
-
8/8/2019 Principles of Semiconductor Devices-L33
7/22
Outline
Alam ECE-606 S09 7
1. Review
2. Induced charges in depletion and inversion
3. Exact solution of electrostatic problem
4. Conclusion
-
8/8/2019 Principles of Semiconductor Devices-L33
8/22
Induced charges below Threshold
Alam ECE-606 S09 8
( )( )i bulk isF
s
E E
i
q
n e e
Be
=
1sn
0 0
0
2 A ox
G s s
ox A
qN xV
qN
= +
-Q
+Q
Electrons
Exposed
Acceptors
s(V)
l o g1 0QS S( )
C / c m2
~ S
/
~ S Bq k T
e
~ eqS / 2kBT
+qs
EC
EV
Ei
Ei,sEF
-
8/8/2019 Principles of Semiconductor Devices-L33
9/22
Integrated charges below Threshold
9
2( )
0
2( )
0
1
1
( )
q xi
B
q xi
B
ne d
dN
dxn
e dN x
=
=
E
2( )
0 0
( )q xi
B
i nn xQ
dx e dxq N
= =
2( )
0
1
( )
q xi
B
ne d
x N
E Winv
Qs
QiQB
+qs
EC
EV
Ei
Ei,s
EF
2
( )
sqis
B
B
inv
ne
k T
qW
xn
N
=
E
-
8/8/2019 Principles of Semiconductor Devices-L33
10/22
Charges above Threshold
Alam ECE-606 S09 10
-Q
+Q
Electrons
Exposed
Acceptors
VG
VG >VT
0
( )G s o s o
ox
i s Fox
Q
x
Q
V x
+
= + = E
0
(2 )2 2
th F o F o
ox
i Fox
FV x xQQ
+= + =
E
( )0
2() 2 )(
G th s F i s F
o
x
i
o
V VQQ
x
= +
( )Gi ox thC VQ V=
+Q
VG
-Q
-
8/8/2019 Principles of Semiconductor Devices-L33
11/22
Linear Charge Build-up Above Threshold?
11
~ 2 F
s(V)
( )10log S SQ
~ S
-Q
+
Q
Electrons
Exposed
Acceptors
VG
VG >VT
Small changes s in changes Qia lot .. Change in Qichanges Eox, because Eox=Qi/0e0 Vox is large because Vox=Eox x0, i.e. most of the
drop above 2F goes to Vox.
Acts like a parallel plate capacitor, hence the inversion equation.
-
8/8/2019 Principles of Semiconductor Devices-L33
12/22
Tunneling Current
Alam ECE-606 S09 12
( ) ( )GqV
T i G m th J Q V qn e T E
=
EC
EV
EF
EG
T( )
( ) ox C
C oxC
T s g g s
m th
qV E
i G m t
i G
h
qVE E
J J J
e qn e e
Q V q
Q
e T
V
n T e
= =
=
-
8/8/2019 Principles of Semiconductor Devices-L33
13/22
Outline
Alam ECE-606 S09 13
1. Review
2. Induced charges in depletion and inversion
3. Exact solution of electrostatic problem
4. Conclusion
-
8/8/2019 Principles of Semiconductor Devices-L33
14/22
A step back: Exact Solution ofQS(
S)
Alam ECE-606 S09 14
EC
EV
EFEG
qS
S> 0
D=
Jn q( )= GR( )
Jp q
( )= GR( )
0 0
0
2 A ox
s sG
ox A
qN xV
qN
= +Approximate
2
0 02
0
( ) ( )D A
si
d qp x n x N N
dx
+ = +
-
8/8/2019 Principles of Semiconductor Devices-L33
15/22
Normalized Variable (to save some writing)
Alam ECE-606 S09 15
q x( )qS x( )= 0
EC(x) = constant q(x)
qVG > 0
x
, ( )( )
C bulk C E E x
xq
=
( ) ( )( )/
i bulk i x
B B
E Exu
k T q k T
= =
( ) ( )
/
i bulk i surfaceS
B B
SuE E
k T q k T
= =
( )
/
i bu
F
lk FF
B B
E E
k T q k T u
= =
-
8/8/2019 Principles of Semiconductor Devices-L33
16/22
Normalized Variable (to save some writing!)
Alam ECE-606 S09 16
q x( )qS x( )= 0
qVG > 0
x
[ ( ) ] ( )( ) F Fi
E x E U
i
U
i p x n e n e + = =
[ ( ) ] ( )( ) Fi F
E x E U
i
U
in x n e n e = =
,[ ] ( )F i bulk FE E
D i
U
i N n e n e + = =
,[ ] ( )F i bulk FE EA i
Ui N n e n e
= =
-
8/8/2019 Principles of Semiconductor Devices-L33
17/22
Poisson-Boltzmann Equation
Alam ECE-606 S09 17
2
20
( ) ( )D A
s
d q
p x n x N N dx
+ = +
( ) ( )2
2
0
( , )F F F FU U U U U U
F
i
i i
B s
qnq d Ue e n e n e
k dg U
xU
T
+ = +
2
2
0
(2 , ) 2i B
s
F
n k Td U
d
dU dU
dx dxxg U U
=
2
2 2( , )12 D
Fd d g U UU dUdx dxdx dx L dx =
2( ) ( )
2
0 0
( , )1
q x kT U
F
x
D
dUd dU
LxUg
dU
=
E
Can be evaluated
at any U
-
8/8/2019 Principles of Semiconductor Devices-L33
18/22
Exact Solution (continued)
Alam ECE-606 S09 18
2 ( ) 2
2 2
0
1 ( , )( ,
( ))
U x
D
F
D
FF U Ug U Uq x
dU
L LkT
=
E
2( ) ( )
2
0 0
( , )1
q x kT U
D
F
x
d dUL
dUdx
g U U
=
E
( , )B
D
Ss FF U
k T
qLU=E
0 0( , )s s B
G s s s
ox ox
s
D
F
k TV
LU x
qUx F
= + = +
E
0 0
0
2 A ox
s sG
ox A
qN xV
qN
= +
Compare
Vox
-
8/8/2019 Principles of Semiconductor Devices-L33
19/22
How does the calculation go
19
2 ( ) 2
2 2
0
1 ( , )
( ,
( )
)
U x
D
F
D
FF U U
g U U
xq
dUkT L L
=
E
0 0( , )s s B
G s s s
ox ox D
s F
k TV x x
qLF U U
= + = +E
Begin with a surface potential
Calculate Us and then divide Us by N points.
Calculate g(U, UF) at those points
and integrate to find F(Us,UF)
Find VG.
-
8/8/2019 Principles of Semiconductor Devices-L33
20/22
Exact Solution
Alam ECE-606 S09 20
Accumulation Depletion Inversion (weak)
Inversion (strong)
-
8/8/2019 Principles of Semiconductor Devices-L33
21/22
Exact solution is not really exact
Alam ECE-606 S09 21
EC
EV
EF
EG
qS
S> 0
2
2
2( ) ( ) ( )
D A
xd q
p x n x N N dx
+ = +
Wave function should be accounted for
Bandgap widening near the interface
must also should be accounted for.
Assumption of nondegeneracy may not
always be valid
wavefunction, not potential !
-
8/8/2019 Principles of Semiconductor Devices-L33
22/22
Conclusion
Alam ECE-606 S09 22
Our discussion today was focused on calculating the
induced charge in the depletion and inversion region as a
function of gate bias.
We found that we could calculate the tunneling currentfrom the inversion changes by using the thermionic
emission theory.
We also discussed the exact solution of the MOS-capacitor electrostatics. The exact solution is
mathematically exact, but not necessarily physically exact
solution of the electrostatic problem.