[<< wikibooks] Semiconductors/MESFET Transistors
== MESFET Operation ==
Assume an N channel MESFET with uniform doping and sharp depletion
region shown in figure 1.
The depletion region

W

n

{\displaystyle W_{n}}
is given by the depletion width for a
diode.  Where the voltage is the voltage from the gate to the
channel, where the channel voltage is given for a position x along
the channel as

V

g
c

(
x
)

{\displaystyle V_{gc}(x)}
.

W

n

(
x
)
=

2

ε

0

ε

r

(
Ψ
−

V

g
c

(
x
)
)

q

N

d

{\displaystyle W_{n}(x)={\sqrt {\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{gc}(x))}{qN_{d}}}}}

W

n

(
x

)

2

=

2

ε

0

ε

r

(
Ψ
−

V

g
c

(
x
)
)

q

N

d

{\displaystyle W_{n}(x)^{2}={\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{gc}(x))}{qN_{d}}}}

W

n

(
x

)

2

q

N

d

2

ε

0

ε

r

=
Ψ
−

V

g
c

(
x
)

{\displaystyle {\frac {W_{n}(x)^{2}qN_{d}}{2\varepsilon _{0}\varepsilon _{r}}}=\Psi -V_{gc}(x)}

V

g
c

(
x
)
=
Ψ
−

W

n

(
x

)

2

q

N

d

2

ε

0

ε

r

{\displaystyle V_{gc}(x)=\Psi -{\frac {W_{n}(x)^{2}qN_{d}}{2\varepsilon _{0}\varepsilon _{r}}}}

d

V

g
c

(
x
)

d

W

n

(
x
)

=
−

2

W

n

(
x
)
q

N

d

2

ε

0

ε

r

{\displaystyle {\frac {dV_{gc}(x)}{dW_{n}(x)}}=-{\frac {2W_{n}(x)qN_{d}}{2\varepsilon _{0}\varepsilon _{r}}}}
(1)The current density in the channel is given by:

J

n

=
σ
ξ

{\displaystyle J_{n}=\sigma \xi }

I

n

(
x
)
=
σ
ξ
⋅
W
⋅
b
(
x
)

{\displaystyle I_{n}(x)=\sigma \xi \cdot W\cdot b(x)}

I

n

(
x
)
=
−
σ

d

V

g
c

(
x
)

d
x

W
(
a
−

W

n

(
x
)
)

{\displaystyle I_{n}(x)=-\sigma {\frac {dV_{gc}(x)}{dx}}W(a-W_{n}(x))}
where:

ξ
=
−

d

V

g
c

(
x
)

d
x

{\displaystyle \xi =-{\frac {dV_{gc}(x)}{dx}}}
Therefore,

I

n

(
x
)
=
−
σ
a
W

(

1
−

W

n

(
x
)

a

)

d

V

g
c

(
x
)

d
W
n
(
x
)

d
W
n
(
x
)

d
x

{\displaystyle I_{n}(x)=-\sigma aW{\bigg (}1-{\frac {W_{n}(x)}{a}}{\bigg )}{\frac {dV_{gc}(x)}{dWn(x)}}{\frac {dWn(x)}{dx}}}

∫

0

L

I

n

(
x
)

d
x
=

∫

0

L

−
σ
a
W

(

1
−

W

n

(
x
)

a

)

d

V

g
c

(
x
)

d

W

n

(
x
)

d

W

n

(
x
)

d
x

d
x

{\displaystyle \int _{0}^{L}I_{n}(x)\,dx=\int _{0}^{L}-\sigma aW{\bigg (}1-{\frac {W_{n}(x)}{a}}{\bigg )}{\frac {dV_{gc}(x)}{dW_{n}(x)}}{\frac {dW_{n}(x)}{dx}}\,dx}

I

n

⋅
L
=
−
σ
a
W

∫

W
n
(
0
)

W

n

(
L
)

(

1
−

W

n

(
x
)

a

)

d

V

g
c

(
x
)

d

W

n

(
x
)

d

W

n

(
x
)

{\displaystyle I_{n}\cdot L=-\sigma aW\int _{Wn(0)}^{W_{n}(L)}{\bigg (}1-{\frac {W_{n}(x)}{a}}{\bigg )}{\frac {dV_{gc}(x)}{dW_{n}(x)}}\,dW_{n}(x)}
Substituting from equation 1:

I

n

=

−
σ
a
W

L

∫

W

n

(
0
)

W

n

(
L
)

(

1
−

W

n

(
x
)

a

)

(

−

2

W

n

(
x
)
q

N

d

2

ε

0

ε

r

)

d
W
n
(
x
)

{\displaystyle I_{n}={\frac {-\sigma aW}{L}}\int _{W_{n}(0)}^{W_{n}(L)}{\bigg (}1-{\frac {W_{n}(x)}{a}}{\bigg )}{\bigg (}-{\frac {2W_{n}(x)qN_{d}}{2\varepsilon _{0}\varepsilon _{r}}}{\bigg )}\,dWn(x)}

I

n

=

σ
a
W
2
q

N

d

2

ε

0

ε

r

L

∫

W

n

(
0
)

W

n

(
L
)

(

W

n

(
x
)
−

W

n

(
x

)

2

a

)

d
W
n
(
x
)

{\displaystyle I_{n}={\frac {\sigma aW2qN_{d}}{2\varepsilon _{0}\varepsilon _{r}L}}\int _{W_{n}(0)}^{W_{n}(L)}{\bigg (}W_{n}(x)-{\frac {W_{n}(x)^{2}}{a}}{\bigg )}\,dWn(x)}

I

n

=

2
σ
a
W
q

N

d

2

ε

0

ε

r

L

[

W

n

2

(
x
)

2

−

W

n

3

(
x
)

3
a

]

W

n

(
0
)

W

n

(
L
)

{\displaystyle I_{n}={\frac {2\sigma aWqN_{d}}{2\varepsilon _{0}\varepsilon _{r}L}}{\bigg [}{\frac {W_{n}^{2}(x)}{2}}-{\frac {W_{n}^{3}(x)}{3a}}{\bigg ]}_{W_{n}(0)}^{W_{n}(L)}}

I

n

=

2
σ
a
W
q

N

d

2

ε

0

ε

r

L

[

W

n

2

(
L
)
−

W

n

2

(
0
)

2

−

W

n

3

(
L
)
−

W

n

3

(
0
)

3
a

]

{\displaystyle I_{n}={\frac {2\sigma aWqN_{d}}{2\varepsilon _{0}\varepsilon _{r}L}}{\bigg [}{\frac {W_{n}^{2}(L)-W_{n}^{2}(0)}{2}}-{\frac {W_{n}^{3}(L)-W_{n}^{3}(0)}{3a}}{\bigg ]}}

I

n

=

2
σ
a
W
q

N

d

a

2

6
L
⋅
2

ε

0

ε

r

[

3
(

W

n

2

(
L
)
−

W

n

2

(
0
)
)

a

2

−

2
(

W

n

3

(
L
)
−

W

n

3

(
0
)
)

a

3

]

{\displaystyle I_{n}={\frac {2\sigma aWqN_{d}a^{2}}{6L\cdot 2\varepsilon _{0}\varepsilon _{r}}}{\bigg [}{\frac {3(W_{n}^{2}(L)-W_{n}^{2}(0))}{a^{2}}}-{\frac {2(W_{n}^{3}(L)-W_{n}^{3}(0))}{a^{3}}}{\bigg ]}}
One defines constant Β as the channel conductance with no
depletion.  And the work function to deplete the channel
W00 [1]:

W

00

=
Ψ
−

V

t
o

=

q

N

d

a

2

2

ε

0

ε

r

{\displaystyle W_{00}=\Psi -V_{to}={\frac {qN_{d}a^{2}}{2\varepsilon _{0}\varepsilon _{r}}}}

β
=

σ
a

3
L

W

00

{\displaystyle \beta ={\frac {\sigma a}{3LW_{00}}}}
We now define Vto, the voltage such that the channel is pinched off.  d is the ratio of channel depletion to maximum depletion for the drain.  s the ratio of channel depletion to
maximum depletion for the source.

d
=

W

n

(
L
)

a

=

2

ε

0

ε

r

(
Ψ
−

V

g
d

)

q

N

d

2

ε

0

ε

r

(
Ψ
−

V

t
o

)

q

N

d

=

Ψ
−

V

g
d

W

00

{\displaystyle d={\frac {W_{n}(L)}{a}}={\frac {\sqrt {\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{gd})}{qN_{d}}}}{\sqrt {\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{to})}{qN_{d}}}}}={\sqrt {\frac {\Psi -V_{gd}}{W_{00}}}}}

s
=

W

n

(
0
)

a

=

2

ε

0

ε

r

(
Ψ
−

V

g
s

)

q

N

d

2

ε

0

ε

r

(
Ψ
−

V

t
o

)

q

N

d

=

Ψ
−

V

g
s

W

00

{\displaystyle s={\frac {W_{n}(0)}{a}}={\frac {\sqrt {\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{gs})}{qN_{d}}}}{\sqrt {\frac {2\varepsilon _{0}\varepsilon _{r}(\Psi -V_{to})}{qN_{d}}}}}={\sqrt {\frac {\Psi -V_{gs}}{W_{00}}}}}
Substituting:

I

n

=
W
⋅

σ
a
⋅

W

00

3
L

[

3
(

d

2

−

s

2

)
−
2
(

d

3

−

s

3

)

]

{\displaystyle I_{n}=W\cdot {\frac {\sigma a\cdot W_{00}}{3L}}{\big [}3(d^{2}-s^{2})-2(d^{3}-s^{3}){\big ]}}

I

n

=
W
⋅
β

W

00

2

[

3
(

d

2

−

s

2

)
−
2
(

d

3

−

s

3

)

]

{\displaystyle I_{n}=W\cdot \beta W_{00}^{2}{\big [}3(d^{2}-s^{2})-2(d^{3}-s^{3}){\big ]}}
(2)Equation 2 is Shockley's expression [2] for drain current in the linear region.  When the device enters saturation, one end is pinched off(normally the drain). Thus $d=1$ and one may derive the equation for the saturation region:

I

s
a
t

=
β

W

00

2

(
1
−
3

s

2

+
2

s

3

)

{\displaystyle I_{sat}=\beta W_{00}^{2}(1-3s^{2}+2s^{3})}

g

m

=
3
β

W

00

(
s
−
1
)

{\displaystyle g_{m}=3\beta W_{00}(s-1)}

G

D
S

=
3
β

W

00

(
1
−
d
)

{\displaystyle G_{DS}=3\beta W_{00}(1-d)}

== Simpler Model ==

I

d
s

=

3
2

β

W

00

2

[

(

V

g
s

−

v

t
o

)

2

W

00

2

−

(

V

g
d

−

v

t
o

)

2

W

00

2

]

{\displaystyle I_{ds}={\frac {3}{2}}\beta W_{00}^{2}{\bigg [}{\frac {(V_{gs}-v_{to})^{2}}{W_{00}^{2}}}-{\frac {(V_{gd}-v_{to})^{2}}{W_{00}^{2}}}{\bigg ]}}

g

m

=
3
β

W

00

(

V

g
s

−

V

t
o

)

{\displaystyle g_{m}=3\beta W_{00}(V_{gs}-V_{to})}

G

d
s

=
3
β

W

00

(

V

g
d

−

V

t
o

)

{\displaystyle G_{ds}=3\beta W_{00}(V_{gd}-V_{to})}

=== General power law: ===
It was found that a general power law provided a better fit for real devices [3].

I

d
s

=
β

[

(

V

g
s

−

V

t
o

)

Q

−
(

V

g
d

−

V

t
o

)

Q

]

{\displaystyle I_{ds}=\beta {\big [}(V_{gs}-V_{to})^{Q}-(V_{gd}-V_{to})^{Q}{\big ]}}
Where Q is dependent on the doping profile and a good fit is usually obtained for Q between 1.5 and 3. A general power law is approximately equal to Shockley's equation for Q = 2.4. Β is also empirically chosen and is proportion to the previous Β

β

proportial to

σ
a
W

3
L

W

00

{\displaystyle \beta {\mbox{ proportial to }}{\frac {\sigma aW}{3LW_{00}}}}
Modelling the various regions is done though model binning.  This however infers that a sharp transition exists from one region to another, which may not be accurate.

I

d
s

=

{

0

V

g
s

<

V

t
o

β

[

(

V

g
s

−

V

t
o

)

Q

−
(

V

g
d

−

V

t
o

)

Q

]

V

g
s

≤

V

g
d

β
(

V

g
s

−

V

t
o

)

Q

V

g
s

>

V

g
d

{\displaystyle I_{ds}=\left\{{\begin{matrix}0&V_{gs}V_{gd}\end{matrix}}\right.}

== References ==
[1] A. E. Parker. Design System for Locally Fabricated Gallium Arsenide Digital
Integrated Circuits. PhD thesis, Sydney University, 1990.
[2] W. Shockley. A unipolar field-effect transistor. IEEE Trans/ Electron Devices, 20(11):1365–1376, November 1952.
[3] I. Richer and R.D. Middlebrook. Power-law nature of field-effect transistor experimental characteristics. Proc. IEEE, 51(8):1145–1146, August 1963.