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EE 5340

Semiconductor Device Theory

Lecture 14


Spring 2011

Professor Ronald L. Carter

ronc@uta.edu

http://www.uta.edu/ronc

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2

S
-
R
-
H net recom
-

bination rate, U


In the special case where
t
no

=
t
po

=
t
o
= (N
t
v
th
s

)
-
1

the net rec. rate, U is

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S
-
R
-
H “U” function

characteristics


The numerator, (np
-
n
i
2
) simplifies in
the case of extrinsic material at low
level injection (for equil., n
o
p
o
= n
i
2
)


For n
-
type (n
o

>

n㴠

瀠pp
o
= n
i
2
/n
o
):

(np
-
n
i
2
) = (n
o
+

n)(p
o
+


-
n
i
2





= n
o
p
o

-

n
i
2

+ n
o

p +


o

+

n




~

n
o

p (largest term)


Similarly, for p
-
type, (np
-
n
i
2
)
~

p
o

n

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S
-
R
-
H rec for

excess min carr


For n
-
type low
-
level injection and net
excess minority carriers, (i.e., n
o

>




瀠>⁰
o
= n
i
2
/n
o
),

U =


t
p
, (prop to exc min carr)


For p
-
type low
-
level injection and net
excess minority carriers, (i.e., p
o

>




瀠>
o
= n
i
2
/p
o
),

U =


t
n
, (prop to exc min carr)

Minority hole lifetimes

Mark E. Law, E.
Solley
,
M. Liang, and Dorothea
E. Burk, “Self
-
Consistent Model of
Minority
-
Carrier
Lifetime, Diffusion
Length, and Mobility,
IEEE ELECTRON
DEVICE LETTERS,
VOL. 12, NO. 8,
AUGUST 1991


The parameters used in
the fit are

τ
o

= 10
μ
s,

N
ref

= 1
×
10
17
/cm
2
, and

C
A

= 1.8
×
10
-
31
cm
6
/s.

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Minority electron lifetimes

Mark E. Law, E.
Solley
,
M. Liang, and Dorothea
E. Burk, “Self
-
Consistent Model of
Minority
-
Carrier
Lifetime, Diffusion
Length, and Mobility,
IEEE ELECTRON
DEVICE LETTERS,
VOL. 12, NO. 8,
AUGUST 1991


The parameters used in
the fit are

τ
o

= 30
μ
s,

N
ref

= 1
×
10
17
/cm
2
, and

C
A

= 8.3
×
10
-
32

cm
6
/s.

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6

Minority Carrier Lifetime, Diffusion Length and
Mobility Models in Silicon


A. [40%] Write a review of the model equations for minority
carrier (both electrons in p
-
type and holes in n
-
type
material) lifetime, mobility and diffusion length in silicon.
Any references may be used. At a minimum the material
given in the following references should be used.

Based on the information in these resources, decide which
model formulae and parameters are the most accurate for
D
n

and
L
n

for electrons in p
-
type material, and
D
p

and
L
p

holes in n
-
type material.



B. [60%] This part of the assignment will be given by 10/12/09.
Current
-
voltage data will be given for a diode, and the
project will be to determine the material parameters (
Nd
,
Na, charge
-
neutral region width, etc.) of the diode.


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References for Part A

Device Electronics for Integrated Circuits
, 3
rd

ed., by Richard S.
Muller, Theodore I. Kamins, and Mansun Chan, John Wiley and Sons,
New York, 2003.

Mark E. Law, E. Solley, M. Liang, and Dorothea E. Burk, “Self
-
Consistent Model of Minority
-
Carrier Lifetime, Diffusion Length,
and Mobility, IEEE ELECTRON DEVICE LETTERS, VOL. 12, NO. 8,
AUGUST 1991.

D.B.M. Klaassen; “A UNIFIED MOBILITY MODEL FOR DEVICE
SIMULATION”, Electron Devices Meeting, 1990. Technical Digest.,
International 9
-
12 Dec. 1990 Page(s):357


360.

David Roulston, Narain D. Arora, and Savvas G. Chamberlain “Modeling
and Measurement of Minority
-
Carrier Lifetime versus Doping in
Diffused Layers of n
+
-
p Silicon Diodes”, IEEE TRANSACTIONS ON
ELECTRON DEVICES, VOL. ED
-
29, NO. 2, FEBRUARY 1982, pages
284
-
291.

M. S. Tyagi and R. Van Overstraeten, “Minority Carrier Recombination
in Heavily Doped Silicon”, Solid
-
State Electr. Vol. 26, pp. 577
-
597,
1983. Download a copy at
Tyagi.pdf
.

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S
-
R
-
H rec for

deficient min carr


If n < n
i

and p

< p
i
, then the S
-
R
-
H net
recomb rate becomes (p < p
o
, n < n
o
):

U = R
-

G =
-

n
i
/(2
t
0
cosh[(E
T
-
E
fi
)/kT])


And with the substitution that the
gen lifetime,
t
g

= 2
t
0
cosh[(E
T
-
E
fi
)/kT],
and net gen rate U = R
-

G =
-

n
i
/
t
g


The intrinsic concentration drives the
return to equilibrium

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The Continuity

Equation


The chain rule for the total time
derivative dn/dt (the net generation
rate of electrons) gives

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The Continuity

Equation (cont.)

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The Continuity

Equation (cont.)

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The Continuity

Equation (cont.)

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The Continuity

Equation (cont.)

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The Continuity

Equation (cont.)

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The Continuity

Equation (cont.)

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Review of depletion

approximation

Depletion Approx.


p
p

<< p
po
,
-
x
p
< x < 0


n
n

<< n
no
, 0

< x < x
n


0 > E
x

>
-
2V
bi
/W,
in DR (
-
x
p
< x < x
n
)


p
p
=p
po
=N
a

& n
p
=n
po
=
n
i
2
/N
a
,
-
x
pc
< x <
-
x
p


n
n
=n
no
=N
d

& p
n
=p
no
=
n
i
2
/N
d
, x
n

< x < x
nc

x

x
n

x
nc

-
x
pc

-
x
p

0

E
v

E
c

qV
bi

E
Fi

E
Fn

E
Fp

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Review of

D. A. (cont.)

x

x
n

x
nc

-
x
pc

-
x
p

E
x

-
E
max

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Forward
Bias Energy
Bands

E
v

E
c

E
Fi

x
n

x
nc

-
x
pc

-
x
p

0

q(V
bi
-
V
a
)

E
FP

E
FN

qV
a

x

Imref, E
Fn

Imref, E
Fp

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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
, Addison
-
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.