Semiconductor Device Physics

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2 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

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Semiconductor Device Physics

Lecture 5

Dr.
Gaurav

Trivedi
,

EEE Department,

IIT Guwahati


Example: Energy
-
Band Diagram


For Silicon at 300 K, where is
E
F

if
n
= 10
17

cm

3
?

Silicon at 300 K,
n
i

= 10
10

cm

3


Consider a Si sample at 300 K doped with 10
16
/cm
3

Boron. What is its resistivity?

N
A

= 10
16
/cm
3

,
N
D
= 0

(
N
A

>>
N
D



p
-
type)

p



10
16
/cm
3
,
n



10
4
/cm
3


Consider a Si sample
doped with 10
17
cm

3
As.

How will its resistivity change when the
temperature is increased from

T
= 300 K to
T
= 400 K?

The temperature dependent factor in


(and therefore

) is

n
.

From the mobility vs. temperature curve for 10
17
cm

3
, we
find that

n
decreases from 770 at 300 K to 400 at 400 K.

As a result,


increases
by a factor of
: 770/400 =1.93





1.a.


(4.2)


Calculate the equilibrium hole concentration in silicon at
T

= 400 K if the Fermi
energy level is 0.27
eV

above the valence band energy.


1.b.



(E4.3)


Find the intrinsic carrier concentration in silicon at:

(i)
T

= 200 K and (ii)
T

= 400 K.


1.c.



(4.13)


Silicon at
T

= 300 K contains an acceptor impurity concentration of

N
A

= 10
16

cm

3
. Determine the concentration of donor impurity atoms that must
be added so that the silicon is
n
-
type and the Fermi energy level is 0.20 eV below
the conduction band edge.


What is the hole diffusion coefficient in a sample of silicon at 300 K with

p
= 410 cm
2

/ V
.
s ?


Remark:

kT/q

= 25.86 mV

at
room temperature


Consider a sample of Si doped with 10
16
cm

3

Boron, with recombination lifetime 1
μ
s
. It is
exposed continuously to light, such that electron
-
hole pairs are generated throughout the
sample at the rate of 10
20

per cm
3

per second,
i.e.

the
generation rate
G
L

= 10
20
/cm
3
/s



a)

What are
p
0

and
n
0
?

b)

What are
Δ
n

and
Δ
p
?


Hint
: In steady
-
state,
generation rate equals
recombination rate


Consider a sample of Si at 300 K doped with 10
16
cm

3

Boron, with recombination lifetime 1
μ
s
. It is exposed continuously to light, such that electron
-
hole pairs are generated
throughout the sample at the rate of 10
20

per cm
3

per second,
i.e.

the
generation rate
G
L

=
10
20
/cm
3
/s.



c)

What are
p

and
n
?

d)

What are
np
product?


Note
: The
np

product can be very
different from
n
i
2

in case of
perturbed/agitated semiconductor


Photoconductor


Photoconductivity

is an optical and electrical phenomenon in
which a material becomes more electrically conductive due to
the absorption of electro
-
magnetic radiation such as visible
light, ultraviolet light, infrared light, or gamma radiation.


When light is absorbed by a material like semiconductor, the
number of free electrons and holes changes and raises the
electrical conductivity of the semiconductor.


To cause excitation, the light that strikes the semiconductor
must have enough energy to raise electrons across the band
gap.

Photoconductor

Net Recombination Rate (General Case
)

Net
Recombination Rate (General Case)


E
T

: energy level of R

G cente
r

Chapter 3


For arbitrary injection levels and both carrier types in a non
-
degenerate semiconductor, the net rate of carrier
recombination is:

where

Net Recombination Rate (General Case)

Continuity Equation

J
N
(
x
)

J
N
(
x
+
dx
)

dx

Area
A
, volume
A.dx

Flow of current

Flow of electron


Taylor’s Series Expansion


The Continuity Equations

Continuity Equation

Minority Carrier Diffusion Equation


The minority carrier diffusion equations are derived from the
general continuity equations, and are applicable only for
minority carriers.


Simplifying assumptions:


The electric field is small, such that:



For
p
-
type material


For
n
-
type material


Equilibrium minority carrier concentration
n
0

and
p
0

are
independent of
x
(uniform doping).


Low
-
level injection conditions prevail.


Minority Carrier Diffusion Equation


Starting with the continuity equation for electrons:


Therefore


Similarly

Carrier Concentration Notation


The subscript “
n
” or “
p
” is now used to explicitly denote
n
-
type
or
p
-
type material
.


p
n

is the hole concentration in
n
-
type material


n
p

is the electron concentration in
p
-
type material


Thus, the minority carrier diffusion equations are:

Simplifications (Special Cases)


Steady state:


No diffusion current:


No thermal R

G:


No other processes:

Minority Carrier Diffusion Length

Similarly,


Consider the special case:


Constant minority
-
carrier (hole) injection at
x

= 0


Steady state, no light absorption for
x
> 0


The
hole diffusion length
L
P

is defined to be:

Minority Carrier Diffusion Length


The general solution to the equation is:


A
and

B

are constants determined by boundary conditions:


Therefore, the solution is:


Physically,
L
P

and
L
N

represent the average
distance that a minority carrier can diffuse
before it recombines with majority a
carrier.

Quasi
-
Fermi Levels


Whenever
Δ
n
=

Δ
p


0

then
np

n
i
2

and we are at non
-
equilibrium conditions.


In this situation, now we would like to preserve and use the
relations:


On the other hand, both equations imply
np

=
n
i
2
, which does
not apply anymore.


The solution is to introduce to quasi
-
Fermi levels
F
N

and
F
P

such that:


The quasi
-
Fermi levels is useful to describe the carrier
concentrations under non
-
equilibrium conditions

Example: Minority Carrier Diffusion
Length


Given
N
D
=10
16

cm

3
,
τ
p

= 10

6

s. Calculate
L
P
.


From the plot,

Example: Quasi
-
Fermi Levels


Consider a Si sample at 300 K with
N
D
= 10
17
cm

3

and

Δ
n
=
Δ
p
= 10
14
cm

3
.


The sample is an
n
-
type

a)

What are
p

and
n
?

b)

What is the
np

product?

Example: Quasi
-
Fermi Levels


Consider a Si sample at 300 K with
N
D
= 10
17
cm

3

and

Δ
n
=
Δ
p
= 10
14
cm

3
.

c)

Find
F
N

and
F
P
?

F
P

0.238 eV

F
N

0.417 eV

E
c

E
v

E
i