Chapter 6
–
part 1
Mosfet
•
Consider the layering of Metal

Oxide

Semiconductor (p

type) as shown below.
•
Here it is assumed that the metal and semi

conductor
Fermi

Levels are the same. We will consider the case
where they are not later.
•
The structure is a capacitor with one plate as a semi

conductor.
•
A negative voltage will deposit
–
charge on the metal
and the capacitor action will require + charge to be
deposited on the semiconductor side. That is holes will
accumulate next to the oxide and the bands will bend up.
Mosfet
•
MOS capacitor
–
flat band condition
=
E
c
E
Fs
E
i
E
v
E
FM
E
o
q
SiO
2
Metal
Oxide
Semiconductor
q
q
q
q
Si
Depends on carrier type
q
Si
x
Mosfet
•
Accumulation
E
c
E
i
E
F
E
v
E
FM
M
O
S
Σ
V<0
qV
Accumulation
q <0

Q
+Q
bulk
Mosfet
•
Of course a negative voltage will raise an electrons
energy.
•
Notice that the work functions
Φ
M,S
are independent of
voltage and therefore the oxide bands will also bend.
Since the oxide is thin it will be a straight line change.
•
Since hole accumulation occurs at the oxide surface on
the semiconductor side we see that E
i

E
F
must change
there.
•
As in a capacitor no current passes through the structure
so E
i
does not change deep inside the semiconductor
(just near the surface).
Mosfet
•
A positive applied voltage causes positive charge to
effectively be deposited on the metal and corresponding
•
Negative charge to be deposited on the semiconductor
surface next to the oxide.
•
This negative charge arises from depletion of holes at
the semiconductor surface. Now E
i
is closer to E
F
.
•
Continuing to increase the applied voltage will cause E
i
to cross E
F
. We call this inversion. Here the
semiconductor looks locally n

type.
•
When
Φ
S
= 2
Φ
F
we have strong inversion and an n

channel has formed. Moreover the charge in the channel
is mobile (in a direction normal to the figure).
Mosfet
•
Depletion
E
c
E
i
E
f
E
v
E
FM
M
O
S
Σ
V>0
qV

Q
+Q
bulk
W
D
Mosfet
•
In depletion the charge density is
•
Poissons’s equation becomes
•
We can integrate twice to get
φ
S
as
•
giving
Mosfet
•
Then from Gauss’s law we have
•
Where Q
2D
DEP
is the charge density in the Si
•
At strong inversion we have
Mosfet
•
Inversion
E
c
E
i
E
F
E
v
E
FM
M
O
S
Σ
V>>0
qV

Q
+Q
electrons
exposed acceptors
bulk
W
D
q >0
Mosfet
•
Note:
•
The electron and hole concentrations are related to the
potential. Here
•
We can also get the electron concentration at any value
Φ
as
•
There is a similar expression for holes as
Mosfet
•
Using
•
We have
•
Since
ρ
=0 and
φ
in the bulk we write
•
or
•
Thus we can write
ρ
(x) as
•
or with the above
Mosfet
•
from which we can find
•
This expression can be integrated from deep in the
semiconductor to the edge of the oxide. We rewrite
multiply both sides by d
φ
/dx and rewrite as
•
Multiply both sides by dx and integrate from as
Mosfet
•
giving
•
Using
•
We can rewrite the above after evaluating it at the
semiconductor surface x = 0 as
•
We can also get the amount of charge at the surface per
unit area from
Mosfet
•
The voltage between the gate and the channel drops
mostly across the oxide as the former are highly
conductive
•
A large electric field is generated in the oxide and at the
oxide Si interface we have
•
In general for the potentials in the semiconductor we
have
Mosfet
•
Inversion occurs when
•
Beyond the onset of strong inversion, electrons are
spilled into the channel without widening the depletion
layer further.
•
Relationships amongst the variables are
for charge neutrality across the device that
Mosfet
•
For the voltage across the whole device we have
•
At the onset of strong inversion
•
giving V
TH
.
•
Capacitance can now be discussed from the above
•
It is frequency dependent as described above with
Mosfet
V
G
C
2D
MOS
V
TH
C
2D
MOS,min
low f 10 hz
high f 1 Mhz
ε
OC
/d
OX
pulsed
Mosfet
•
If there is a work function difference between the metal
and semiconductor, we will have a non zero flat band
voltage V
FB
= we can add to this the fact that
the oxide can have charge trap states adding to the flat
band voltage as
•
Now the threshold voltage is written as
•
Gauss’s Law can be used to relate electric field to
charge as
Mosfet
•
Recall for charge in general we had
•
Therefore we can write, replacing expressions above
•
Where V is the channel to source voltage
E
c
E
i
E
F
E
v
E
FM
Σ
V>>0
qV
q
Q
2D
Σ
Φ
x
x
x
d
ox
W
DM
Q
m
2D
Q
n
2D
Q
D
2D
V
V
OX
Φ
S
V=V
OX
+
Φ
S
Q
m
2D
=positive charge in metal
Q
n
2D
=negative charge in channel
Q
D
2D
=negative depletion charge
Mosfet
•
Summary of Si surface
charge per unit area
vs. surface potential.
Mosfet
•
Capacitance (book)
•
END
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