# Semiconductor Device Theory

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EE 5340

Semiconductor Device Theory

Lecture 11

Spring 2011

Professor Ronald L. Carter

ronc@uta.edu

http://www.uta.edu/ronc

-
24Feb2011

2

Metal/semiconductor

system types

n
-
type semiconductor

Schottky diode
-

blocking for
f
m

>
f
s

contact
-

conducting for
f
m

<
f
s

p
-
type semiconductor

contact
-

conducting for
f
m

>
f
s

Schottky diode
-

blocking for
f
m

<
f
s

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24Feb2011

3

Real Schottky

band structure
1

Barrier transistion
region,
d

Interface states

above
f
o

acc, p neutrl

below
f
o

dnr, n neutrl

D
it
d

-
> oo, q
f
Bn

=

E
g
-

f
o

Fermi level “pinned”

D
it
d

-
> 0, q
f
Bn

=

f
m
-

c

Goes to “ideal” case

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24Feb2011

4

Fig 8.4
1

(a) Image charge and electric field

at a metal
-
dielectric interface (b) Distortion

of potential barrier at E=0 and (c) E

0

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24Feb2011

5

Poisson’s Equation

The electric field at (x,y,z) is related
to the charge density

=q(Nd
-
Na
-
p
-
n)
by the Poisson Equation:

-
24Feb2011

6

Poisson’s Equation

n = n
o

+
d
nⰠ慮d瀠㴠p
o

+
d

-
equ楬

For n
-
type material, N = (N
d
-

N
a
) > 0,
n
o

= N, and (N
d
-
N
a
+p
-
n)=
-
d
n+
d

i
2
/N

For p
-
type material, N = (N
d
-

N
a
) < 0,
p
o

=
-
N, and (N
d
-
N
a
+p
-
n) =
d
p
-
d
n
-
n
i
2
/N

So neglecting n
i
2
/N

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24Feb2011

7

Ideal metal to n
-
type

barrier diode
(
f
m
>
f
s
,V
a
=0)

E
Fn

E
o

E
c

E
v

E
Fi

q
f
s,n

q
c
s

n
-
type s/c

q
f
m

E
Fm

metal

q
f
Bn

q
f
bi

q
f

n

No disc in E
o

E
x
=0 in metal
==> E
o
flat

f
Bn
=
f
m
-

c
s
=
elec mtl to
s/c barr

f
bi
=
f
Bn
-
f
n
=
f
m
-
f
s

elect
s/c to mtl
barr

Depl reg

0 x
n

x
nc

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24Feb2011

8

Depletion

Approximation

For 0 < x < x
n
,
assume n << n
o

= N
d
, so

㴠q(N
d
-
N
a
+p
-
n) = qN
d

For x
n
< x < x
nc
,
assume n = n
o

= N
d
, so

㴠q(N
d
-
N
a
+p
-
n) = 0

For x = 0
-
, there is a pulse of charge
balancing the qN
d
x
n
in 0 < x < x
n

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24Feb2011

9

Ideal n
-
type Schottky

depletion width (V
a
=0)

x
n

x

qN
d

d

Q’
d
=
qN
d
x
n

x

E
x

-
E
m

x
n

(Sheet of negative charge on metal)=
-
Q’
d

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24Feb2011

10

Debye

length

The DA assumes n changes from N
d

to 0 discontinuously at x
n
.

In the region of x
n
, Poisson’s eq is


E

=

/

dE
x
/dx = q(N
d

-

n),
and since E
x

=
-
d
f
/dⰠe桡he

-
d
2
f
/dx
2
= q(N
d

-

n)/

t⁢e獯l癥d

n

x

x
n

N
d

0

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24Feb2011

11

Debye length

(cont)

Since the level E
Fi

is a reference for
equil, we set
f

㴠V
t

ln(n/n
i
)

In the region of x
n
, n = n
i

exp(
f

t
),
so

d
2
f
/dx
2
=
-
q(N
d

-

n
i

e
f
/Vt
), let

f

f
o

+
f
’, where
f
o

= V
t

ln(N
d
/n
i
)
so

N
d

-

n
i

e
f
/Vt

= N
d
[1
-

e
f
/Vt
-
f
/Vt
],
for
f

-

f
o

=
f
’ <<
f
o
, the DE becomes

d
2
f
’/dx
2
= (q
2
N
d
/

kT)
f
’,
f
 
f
o

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24Feb2011

12

Debye length

(cont)

So

f

=
f
’(x
n
) exp[
+
(x
-
x
n
)/L
D
]+con.

and n = N
d

e
f
’/Vt
, x ~ x
n
, where

L
D

is the “Debye length”

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24Feb2011

13

Debye length

(cont)

L
D

estimates the transition length of
a step
-
junction DR. Thus,

For V
a
= 0,
f
i

~ 1V, V
t

~ 25 mV

d

<

11%

DA

assumption OK

-
24Feb2011

14

Effect of V

0

Define an external voltage source, V
a
,
with the +term at the metal contact
and the
-
term at the n
-
type contact

For V
a

> 0, the V
a

induced field tends
to oppose E
x

caused by the DR

For V
a

< 0, the V
a

induced field tends
to aid E
x
due to DR

Will consider V
a

< 0 now

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24Feb2011

15

Effect of V

0

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24Feb2011

16

Ideal metal to n
-
type

Schottky (V
a
> 0)

qV
a
= E
fn
-

E
fm

Barrier for
electrons
from sc to
m reduced
to

q(
f
bi
-
V
a
)

q
f
Bn
the same

DR decr

E
Fn

E
o

E
c

E
v

E
Fi

q
f
s,n

q
c
s

n
-
type s/c

q
f
m

E
Fm

metal

q
f
Bn

q(
f
i
-
V
a
)

q
f

n

Depl reg

-
24Feb2011

17

Schottky diode

capacitance

x
n

x

qN
d

-
Q
-
d
Q

Q’
d
=
qN
d
x
n

x

E
x

-
E
m

x
n

d
Q’

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24Feb2011

18

Schottky Capacitance

(continued)

The junction has +Q’
n
=qN
d
x
n

(exposed
donors), and Q’
n
=
-

Q’
metal

(Coul/cm
2
),
forming a parallel sheet charge
capacitor.

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24Feb2011

19

Schottky Capacitance

(continued)

This Q ~ (
f
i
-
V
a
)
1/2

is clearly non
-
linear, and Q is not zero at V
a

= 0.

Redefining the capacitance,

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24Feb2011

20

Schottky Capacitance

(continued)

So this definition of the capacitance
gives a parallel plate capacitor with
charges
d
Q’
n

and
d
Q’
p
(=
-
d
Q’
n
),
separated by, L (=x
n
), with an area A
and the capacitance is then the ideal
parallel plate capacitance.

Still non
-
linear and Q is not zero at
V
a
=0.

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24Feb2011

21

Schottky Capacitance

(continued)

The C
-
V relationship simplifies to

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24Feb2011

22

Schottky Capacitance

(continued)

If one plots [C
j
]
-
2

vs.
V
a

Slope =
-
[(C
j0
)
2
V
bi
]
-
1

vertical axis intercept = [C
j0
]
-
2

horizontal axis intercept =
f
i

C
j
-
2

f
i

V
a

C
j0
-
2

Diagrams for ideal metal
-
semiconductor Schottky diodes. Fig. 3.21 in Ref 4.

-
24Feb2011

23

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24Feb2011

Energy bands for

p
-

and n
-
type s/c

p
-
type

E
c

E
v

E
Fi

E
FP

q
f
P
= kT ln(n
i
/N
a
)

E
v

E
c

E
Fi

E
FN

q
f
n
= kT ln(N
d
/n
i
)

n
-
type

24

-
24Feb2011

Making contact

in a p
-
n junction

Equate the E
F

in
the p
-

and n
-
type
materials far from
the junction

E
o
(the free level),
E
c
, E
fi
and E
v

must
be continuous

N.B.: q
c

=4⸰5敖e(Si),

and q
f

=ⁱ
c

E
c

-

E
F

E
o

E
c

E
F

E
Fi

E
v

q
c

(electron
affinity)

q
f
F

q
f

(work function)

25

-
24Feb2011

Band diagram for

p
+
-
n jctn* at V
a

= 0

E
c

E
FN

E
Fi

E
v

E
c

E
FP

E
Fi

E
v

0

x
n

x

-
x
p

-
x
pc

x
nc

q
f
p
< 0

q
f
n
> 0

qV
bi

= q(
f
n
-

f
p
)

*N
a

> N
d

-
> |
f
p
|

>
f
n

p
-
type for x<0

n
-
type for x>0

26

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24Feb2011

A total band bending of

qV
bi

= q(
f
n
-
f
p
) = kT ln(N
d
N
a
/n
i
2
)

is necessary to set E
Fp

= E
fn

For
-
x
p
< x < 0, E
fi
-

E
FP

<
-
q
f
p
, = |q
f
p
|
so p < N
a
= p
o
, (depleted of maj. carr.)

For 0 < x < x
n
, E
FN

-

E
Fi

< q
f
n
,

so n < N
d
= n
o
, (depleted of maj. carr.)

-
x
p
< x < x
n

is the Depletion Region

Band diagram for

p
+
-
n at V
a
=0 (cont.)

27

-
24Feb2011

Depletion

Approximation

Assume p << p
o

= N
a

for
-
x
p
< x < 0, so

㴠q(N
d
-
N
a
+p
-
n) =
-
qN
a
,
-
x
p
< x < 0,
and p = p
o

= N
a

for
-
x
pc
< x <
-
x
p
, so

㴠q(N
d
-
N
a
+p
-
n) = 0,
-
x
pc
< x <
-
x
p

Assume n << n
o

= N
d

for 0 < x < x
n
, so

㴠q(N
d
-
N
a
+p
-
n) = qN
d
, 0 < x < x
n
,
and n = n
o

= N
d

for x
n
< x < x
nc
, so

㴠q(N
d
-
N
a
+p
-
n) = 0, x
n
< x < x
nc

28

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24Feb2011

Depletion approx.

charge distribution

x
n

x

-
x
p

-
x
pc

x
nc

+qN
d

-
qN
a

+Q
n
’=qN
d
x
n

Q
p
’=
-
qN
a
x
p

Due to Charge
neutrality
Q
p
’ + Q
n
’ = 0,
=> N
a
x
p =
N
d
x
n

[Coul/cm
2
]

[Coul/cm
2
]

29

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24Feb2011

Induced E
-
field

in the D.R.

The sheet dipole of charge, due to
Q
p
’ and Q
n
’ induces an electric field
which must satisfy the conditions

Charge neutrality and Gauss’ Law*
require that

E
x

= 0 for
-
x
pc
< x <
-
x
p

and E
x

= 0 for
-
x
n
< x < x
nc

30

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24Feb2011

Induced E
-
field

in the D.R.

x
n

x

-
x
p

-
x
pc

x
nc

O

-

O

-

O

-

O

+

O

+

O

+

Depletion
region (DR)

p
-
type
CNR

E
x

Exposed
Donor ions

Exposed
Acceptor Ions

n
-
type chg
neutral reg

p
-
contact

N
-
contact

W

0

31

-
24Feb2011

Induced E
-
field

in the D.R. (cont.)

Poisson’s Equation

E

=

/

Ⱐ桡猠t桥
ne
-
d業en獩sn慬景rmⰠ

x
/dx =

/

which must be satisfied for

-

a
,
-
x
p
< x < 0, and

㴠+qN
d
, 0 < x < x
n
, with

E
x

= 0 for the remaining range

32

-
24Feb2011

Soln to Poisson’s

Eq in the D.R.

x
n

x

-
x
p

-
x
pc

x
nc

E
x

-
E
max

33

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24Feb2011

Soln to Poisson’s

Eq in the D.R. (cont.)

34

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24Feb2011

Soln to Poisson’s

Eq in the D.R. (cont.)

35

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24Feb2011

E
x

and V
bi

V
bi

is not measurable externally since
E
x

is zero at both contacts

The effect of E
x

does not extend
beyond the depletion region

The lever rule [N
a
x
p
=N
d
x
n
] was
obtained assuming charge neutrality.
It could also be obtained by requiring

E
x
(x=0

d

 
E
x
(x=0
d
)

E
max

36

-
24Feb2011

Sample

calculations

V
t

25.86 mV at 300K

r

= 11.7*8.85E
-
14 Fd/cm

= 1.035E
-
12 Fd/cm

If N
a

5E17/cm
3
, and N
d

2E15 /cm
3
,
then for n
i

1.4E10/cm
3
, then what is
V
bi
= 757 mV

37

-
24Feb2011

Sample

calculations

What are N
eff
, W ?

N
eff
, = 1.97E15/cm
3

W = 0.707 micron

What is x
n
?

= 0.704 micron

What is E
max
? 2.14E4 V/cm

38

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24Feb2011

39

References

1
Device Electronics for Integrated Circuits
, 2 ed., by
Muller and Kamins, Wiley, New York, 1986.
See
Semiconductor Device Fundamentals
-
Wesley, 1996, for another treatment of the
m

model.

2
Physics of Semiconductor Devices
, by S. M. Sze,
Wiley, New York, 1981.

3
Semiconductor Physics & Devices
, 2nd ed., by
Neamen, Irwin, Chicago, 1997.

4
Device Electronics for Integrated Circuits, 3/E
by
Richard S. Muller and Theodore I. Kamins. © 2003
John Wiley & Sons. Inc., New York.

-
24Feb2011

40

References

1 and
M&K
Device

Electronics for Integrated
Circuits
, 2 ed., by Muller and
Kamins
, Wiley,
New York, 1986. See
Semiconductor Device
Fundamentals
, by
Pierret
-
Wesley,
1996, for another treatment of the
m

model.

2
Physics of Semiconductor Devices
, by S. M.
Sze
,
Wiley, New York, 1981.

3 and **
Semiconductor Physics & Devices
, 2nd ed.,
by
Neamen
, Irwin, Chicago, 1997.

Fundamentals of Semiconductor Theory and
Device Physics
, by
Shyh

Wang, Prentice Hall,
1989.