Accumulation
We apply a gate to bulk voltage that is smaller than the flat-band voltage:
V GB V FB
NOTICE:
for an MOS structure with
n+ poly-silicon gate and
p-type substrate this is a
negative value
The gate potential is more negative
than the potential of he bulk
Since the bulk is of p-type, there is plenty of positively charged holes in it
that will be attracted toward the substrate surface, so the MOS capacitor
start to store positive charge at the substrate surface. (And as a by-product
of the neutrality principle the gate charge become negative)
Accumulation
The piling-up of holes at the surface increases
the density of holes at the surface ps above
the normal bulk level of NA.
s V T
p s n i e
ps
NA
s V T ln
V T ln
ni
ni
p
However, the logarithmic function is weak so it is a reasonable approximation
that s p in accumulation (for ps 10 NA at room temperature the s
would increases of only 60 mV)
Accumulation drop across the semiconductor
mn+
V ox
sp
pm
V GB mn+ V ox sp pm
0
V GB mn+ pmV ox s p
BUILT-IN V FB
V GB V FBV ox
In accumulation the charge density is identical to that of
a parallel plate capacitor, thus:
for V GB V FB :
ox
acc G V ox ox V ox
t ox
for V GB V FB :
G ox V ox ox V GB V FB Accumulation
M
0
O
S
Depletion
We apply a gate to bulk voltage bigger than the flat-band voltage but
smaller than a certain threshold:
V FB V GB V TH
With respect to the flat-band condition the
potential of the poly-silicon will be shifted
VGB above the potential of the substrate
The holes in the semiconductor will be
repelled down in the substrate and leave
negatively charged fixed acceptor ions
behind (i.e. a depletion region is created
near the surface).
The negatively charged depletion region
in the substrate is mirrored by
a positive charge sheet on the gate
NOTE:
we are already familiar with this case: this the same case we have analyzed
for the MOS capacitor in thermal equilibrium
Example:
VGB = 250 mV
Depletion
0
BUILT-IN V FB ( n+,0 p,0 )
V GB mn pm V MOS
V ox V B
V GB V FBV ox V B
(n+ p )
Depletion
V GB V FBV ox V B
Carefully adapting
the results found in TE
Depletion Width :
s
X d
ox
Charge of the MOS capacitor :
Surface Potential :
2 2ox
1
V GB V FB 1
qN A s
for V FB V GB V TH
Q G Q B q N A X d for V FB V GB V TH
qNA Xd
qNAXd
s n+ V GB n+,0 ox
ox
for V FB V GB V TH
Depletion
NOTE:
In reality in the substrate besides the holes being
repelled down we have also free electrons (minority
carriers) being attracted up, toward the surface.
The concentration of electrons at the surface is:
n s n i e
s V T
Example:
with s=185 mV the surface electron
concentration at room temperature is:
n s ni e
s V T
1010 e185261.21013 cm3
which is negligible compared to the
ionized acceptor concentration NA
(=1017 cm-3)
V FB V GB V TH
Depletion
• In order to keep the analysis simple we assume that in
depletion condition the surface electron concentration is
always negligible
• As VGB get closer and closer to the critical value VTH this is
not necessarily very accurate
V GB V FB
&
close to V TH
Depletion
• The electron concentration increases exponentially as the
surface potential increases, to the point where it
eventually becomes the dominant component of the
negative charge in the silicon substrate
n s n i e
s V T
• As a result the charge density in the silicon substrate must
be modified to include the electron contribution
x q N An x q N An i e
x V T
for 0x X d
Threshold
As we keep increasing the gate to bulk
voltage above the flat-band level the surface
potential continue to rise and eventually, there
will be a significant electron concentration
at the surface.
q N A X d V GB for V FB V GB V TH : s V GB n+,0
ox
At some point the surface will become so rich of electrons that it is no longer in
depletion but at the threshold of inversion. In other words, it looks like if the
surface inverted from p-type to n-type (a “material” with a lot of free mobile
electrons).
The threshold is defined as the condition when the surface electron
concentration ns is equal to the bulk doping concentration NA
Threshold
V GB V TH
• At the threshold the surface is as much n-type as the bulk
is p-type:
n
threshold s
n i e
threshold s
V T
N A
• And since at T.E. the bulk concentration is:
NA
e ni
p,0
V T
• It follows that at the threshold the surface potential
becomes equal and opposite to the fermi potential of the
bulk:
threshold
p,0F-bulk
s
Threshold Voltage
• Let's write KVL at the onset of inversion (threshold):
(threshold)
(threshold)
V (threshold)
V
V
V
GB
FB
ox
B
V TH
All we have to do is to find
Vox and VB
(threshold)
V TH V FBV (threshold)
V
ox
B
• The potential drop across the depletion region is a known
quantity:
(threshold)
(threshold)
V (threshold)
2
2
2
B
s
p0
p0
F-bulk
s
Threshold Voltage
• At the onset of inversion the depletion width has increased to its
maximum value Xd,max
• The charge density in the depletion region is (x)=qNA
X d,max
Gauss at the Boundary:
0+
X d,max
dE x 0+
q N A X d,max
E X d,max E 0+
s
0
q N A X d,max
x
dx
s
s
q N A X d,max
E 0+
s
Threshold Voltage
q N A x
x
dE x dx 0+
0+
s
s
x
Gauss:
x
q N A x
E x E 0+
s
q N A x qN A
E x E 0+
X d,max x
s
s
q N A X d,max
s
X d,max
Potential:
0+
X d,max
E x dx
d x
0+
V (threshold)
B
Threshold Voltage
X d,max
(threshold)
VB
0+
X d,max
E x dx
0+
2
q NA
q N A X d,max
X d,max x dx
2
s
s
2
X
q
N
(threshold)
A
d,max
V (threshold)
2
2
B
F-bulk
s
2
s
(threshold)
4
s s
X d,max q NA
Threshold Voltage
• The charge in the depletion region at the onset of inversion is:
(threshold)
(threshold)
q
N
X
q
N
4
B
B,max
A
d,max
A s s
• Let's know find the voltage drop across the oxide:
t ox
0-
t ox
t ox
d x E x dx E ox
(threshold)
0-
0-
(threshold)
V ox
(threshold)
V (threshold)
E
t ox
ox
ox
(threshold)
dx E ox
t ox
Threshold Voltage
- t ox
t ox
0-
0-
dE x • Gauss:
0
t ox
E t ox E 0- 0-
t
ox
t ox
x
dx
ox
(threshold)
G
x
ox
t ox
dx t ox
(threshold)
G(threshold) x
G
dx
ox
ox
(threshold)
E ox
(threshold)
(threshold)
G
B
E (threshold)
ox
ox
ox
(threshold)
4q
N
t
(threshold)
(threshold)
ox
s
A s
4q
V (threshold)
E
t
N
ox
ox
ox
s
A s
ox
ox
Threshold Voltage
• Finally putting everything together:
NOTE:
I got tired of dragging the
superscript threshold on
all the equations so I simply
put a prime at its place
V TH V FB V 'ox V 'B
'
2
q
N
2
'
A s
s
V TH V FB 2 s ox
• Or equivalently:
V TH V FB 2 F-bulk
For an n-type material
this is a positive value:
= - Q B'
F-bulk
ND
KT
ln
q
ni
For a p-type material
this is a negative value
2 q N A s 2 F-bulk ox
with :
NA
KT
F-bulk ln
q
ni
Threshold
V GB V TH
Inversion
• We apply a gate to bulk voltage bigger than the threshold
voltage
Q N Q inv
Q GQ invQ dep Q B
electrons at the
oxide-substrate
surface constitute
a sheet of charge
Inversion
• After inversion occurs, further increases
in VGB only slightly increase the surface
potential s. Any further increase in the
surface potential would induce a much
larger increases in the surface electron
density and we cannot expect the surface
electron density to increase indefinitely
• For sake of simplification we assume that for VGB> VTH the
surface potential stop increasing and it remain pinned to 's
• If s does not increases neither will the depletion region width,
that approximately will reach its maximum value:
2 s 2 's X d,max qNA
Inversion
V GB V FB V B V ox
• Since we assumed the surface potential remains pinned at s'
and the bulk is at a fixed reference potential, the drop across
the substrate is the same as in threshold condition: V B 2 's
• So any increase of VGB above VTH is all picked up by Vox
G B
dep,max inv 2 q N A s 2 's inv
V ox ox ox
ox
ox
ox
ox
'
2
q
N
2
inv
'
A s
s
V GB V FB2 s ox
ox
V TH
inv
V GB V TH ox
Example: VGB = 1.0V
Inversion
inv
V GB V TH ox
Beside the voltage
offset of VTH it
behaves just as a
linear capacitor
inv ox V GB V TH G V GB dep,max inv dep,max ox V GB V TH Inversion
G Q dep,max Q inv
E ox ox
ox
E 0 E ox
ox G Q dep,max Q inv
s
s
s
ox 1
s 3
Qinv is negative
Q dep,max
Q inv
E 0+
E 0
s
s
Vox is the area
of the rectangle
of height Eox and
base -tox, 0
V ox E oxt ox
Inversion
• So far, we assumed that in inversion the electrons appears at
the surface instantaneously.
• However, since there are few electrons in the p-type body, it can
take minutes for thermal generation to generate the
necessary electrons to form the inversion layer.
• The MOS transistor structure solve this problem. The inversion
electrons are supplied by the n+ junctions.
Gate Charge of MOS capacitor
Accumulation:
Depletion:
Inversion:
The depletion charge reach its maximum value for VGB = VTH
dep,max VGB = VTH
Gate Charge of MOS capacitor
V GB B V GB NOT a very “good” name
dep,max
Substrate Charge of MOS Cap.
V gb