ECE 340

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1 Νοε 2013 (πριν από 3 χρόνια και 5 μήνες)

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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