Review of Basic Semiconductor Physics

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Semiconductor Physics - 1Copyright © by John Wiley & Sons 2003
Review of Basic Semiconductor Physics
Semiconductor Physics - 2Copyright © by John Wiley & Sons 2003
Current Flow and Conductivity
Area = A
x = v t
Current
Density
= J
Electrons moving
with velocity v
• Charge in volume Ax = Q
= q n A  x = q n A vt
• Current density J = (Q/t)A
-1
= q n v
• Metals - gold, platinum, silver, copper, etc.
• n = 10
23
cm
-3
;  = 10
7
mhos-cm
• Insulators - silicon dioxide, silicon nitride, aluminum oxide
• n < 10
3
cm
-3
; < 10
-10
mhos-cm
• Semiconductors - silicon, gallium arsenide, diamond, etc.
• 10
8
< n <10
19
cm
-3
; 10
-10
<< 10
4
mhos-cm
Semiconductor Physics - 3Copyright © by John Wiley & Sons 2003
Thermal Ionization
• Si atoms have
thermal vibrations
about equilibrium
point.
• Small percentage of
Si atoms have large
enough vibrational
energy to break
covalent bond and
liberate an electron.
+
+
-
-
covalent bond
neutral silicon atom
ionized
silicon
atom
broken bond
free
electron
Semiconductor Physics - 4Copyright © by John Wiley & Sons 2003
Electrons and Holes
-
A
t = T
1
+
generation of B
B
t = T
2
-
A
recombination of B
t = T
3
apparent
movement
of "Hole"
A
-
• T
3
> T
2
> T
1
• Density of free electrons
= n : Density of free
holes = p
• p = n = n
i
(T) = intrinsic
carrier density.
• n
i
2
(T) = C exp(-qE
g
/(kT ))
= 10
20
cm
-6
at 300 K
• T = temp in K
• k = 1.4x10
-23
joules/ K
• E
g
= energy gap = 1.1 eV
in silicon
• q = 1.6x10
-19
coulombs
Semiconductor Physics - 5Copyright © by John Wiley & Sons 2003
Doped Semiconductors
A
-
empty
bond
D
-
extra valance
electron
• Extrinsic (doped) semiconductors:p = p
o
≠ n = n
o
≠ n
i
• Carrier density estimates:
• Law of mass action n
o
p
o
= n
i
2
(T)
• Charge neutrality N
a
+ n
o
= N
d
+ p
o
• P-type silicon with N
a
>> n
i
:
p
o
≈ N
a ,
n
o
≈ n
i
2
/ N
a
• N-type silicon with N
d
>> n
i
:
n
o
≈ N
d ,
p
o
≈ n
i
2
/ N
d
Semiconductor Physics - 6Copyright © by John Wiley & Sons 2003
Nonequilibrium and Recombination
• Thermal Equilibrium
- Carrier generation = Carrier recombination
• n = n
o
and p = p
o
• Nonequilibrium
- n > n
o
and p > p
o
• n = n
o
+ n and p = n
o
+ n ; n = excess carrier density
• Excess holes and excess electrons created in equal numbers by breaking of covalent
bonds
• Generation mechanisms -light (photoelectric effect), injection, impact ionization
• Recombination
- removal of excess holes and electrons
• Mechanisms - free electron captured by empty covalent bond (hole) or trapped by
impurity or crystal imperfection
• Rate equation: d(n)/dt = - n
• Solution n = n (0) e
-t

Semiconductor Physics - 7Copyright © by John Wiley & Sons 2003
Carrier Lifetimes
• = excess carrier lifetime
• Usually assumed to be constant. Changes in two important situations.
•  increases with temperature T
•  decreases at large excess carrier densities ;  = 
o
/[1 + (n/n
b
)
2
]
• Control of carrier lifetime values.
• Switching time-on state loss tradeoff mandates good lifetime control.
• Control via use of impurities such as gold - lifetime killers.

Control via electron irradiation - more uniform and better control.
Semiconductor Physics - 8Copyright © by John Wiley & Sons 2003
Current Flow

+
+
+
-
-
-
-
+
-
V
+
• J
drift
= q µ
n
n E + q p µ
p
E
• µ
n
= 1500 cm
2
/V-sec for silicon at
room temp. and N
d
< 10
15
cm
-3
• µ
p
= 500 cm
2
/V-sec for silicon at
room temp. and N
a
< 10
15
cm
-3
J
J
n
p
-
+
x
x
n

p

Drift
Diffusion
• J
diff
= J
n
+ J
p
= q D
n
dn/dx - q D
p
dp/dx
• D
n
/
n
= D
p
/
p
= kT/q ; Einstein relation
• D = diffusion constant,  = carrier mobility
• Total current density J = J
drift
+ J
diff
Semiconductor Physics - 9Copyright © by John Wiley & Sons 2003
PN Junction
P
N
metallurgical junction
x
x
N
A
A
N
N
D
D
N
-
-
N
A
N
D
N
A
N
D
Step (abrupt) junction
Linearly graded junction
Semiconductor Physics - 10Copyright © by John Wiley & Sons 2003
Formation of Space Charge Layer
• Diffusing electrons and holes
leave the region near
metallurgical junction depleted
of free carriers (depletion
region).
• Exposed ionized impurities
form space charge layer.
• Electric field due to space
charge opposes diffusion.
+
+
+
+
+
+
P
N
metallurgical
junction
space charge
layer width = W
x
ionized
acceptors
ionized
donors
+
-
-
-
+
+
Diffusing
holes
Diffusing
electrons
Electric
field
opposing
diffusion
Semiconductor Physics - 11Copyright © by John Wiley & Sons 2003
x
x


c


depletion layer
w

max

x
n
- x
p
x
-qN
a
qN
d
Quantitative Description of Space Charge Region
• Assume step junction.

d
dx
= - E(x)
E(x) =
qN
a
(x+x
p
)

; - x
p
< x < 0
E(x) =
qN
d
(x- x
n
)

; 0< x < x
n

d
2

dx
2
= -



 = - qN
a
; x < 0
 = qN
d
; x > 0

c
= -


- x
p
x
n
E(x)dx

c
= -
qN
a
x
p
2
￿+￿qN
d
x
n
2
2
Semiconductor Physics - 12Copyright © by John Wiley & Sons 2003
Contact (Built-in, Junction) Potential
• In thermal equilibrium J
n
= q µ
n
n
d
dx
+ q D
n

dn
dx
= 0
• Separate variables and integrate ;


(x
p
)
(x
n
)
d = -
D
n
µ
n




n(x
p
)
n(x
n
)
dn
n

• (x
n
) - (x
p
) = 
c
=
kT
q
ln






N
a
N
d
n
i
2
; 
c
= contact potential
• Example
• Room temperature kT/q = 0.025 eV
• N
a
= N
d
= 10
16
cm
-3
; n
i
2
= 10
20
cm
-6
• F
c
= 0.72 eV
Semiconductor Physics - 13Copyright © by John Wiley & Sons 2003
Reverse-Biased Step Junction
+
+
P
N
+
V
W(V)
x
W
o
+
+


c

+ V
c
x (V)
n
- x (V)
p
+
• Starting equations
• W(V) = x
n
(V) + x
p
(V)
• V + 
c
= -
qN
a
x
p
2
￿+￿qN
d
x
n
2
2
• Charge neutrality qN
a
x
p
= qN
d
x
n
• Solve equations simultaneously
• W(V) = W
o
1+V/
c

• W
o
=
2
c
(N
a
+N
d
)
qN
a
N
d

• E
max
=
2
c
W
o

1￿+￿V/
c

Semiconductor Physics - 14Copyright © by John Wiley & Sons 2003
Forward-Biased PN Junction
+
+
+
P
N
V
W(V)
n
po
p
no
x
W
o
p
n
(x) = p
n
(0) exp(
x
L
p
)

p
n
(0) =
n
i
2
N
d
exp(
qV
kT
)
n
p
(0)
n
i
2
N
a
exp(
qV
kT
)
=
Q
n
=

0
￿-￿∞
￿n
p
(x)dx = q








n
p
(0)￿-￿
n
i
2
N
a
￿
Q
p
=

0
￿￿∞
￿p
n
(x)dx = q








p
n
(0)￿-￿
n
i
2
N
d
￿
n
p
(x) = n
p
(0) exp(
x
L
n
)
-
p-side drift
region
n-side drift
region


• Forward bias favors
diffusion over drift.
• Excess minority
carrier injection into
both p and n drift
regions.
• Minority carrier
diffusion lengths.
• L
n
= [D
n

n
]
0.5
• L
p
= [D
p

p
]
0.5
Semiconductor Physics - 15Copyright © by John Wiley & Sons 2003
Ideal PN Junction I-V Characteristics
• Excess carriers in drift regions recombined and thus more must be constantly injected if
the distributions np(x) and pn(x) are to be maintained.
• Constant injection of electrons and holes results in a current density J given by
J =
Q
n

n
+
Q
p

p
= q n
i
2




L
n
N
a

n
+



L
p
N
d

p









exp(
qV
kT
)￿-￿1
J = J
s









exp(
qV
kT
)￿-￿1 ; J
s
= q n
i
2




L
n
N
a

n
+



L
p
N
d

p

v
J
v
J
v
- J
s
i
forward bias
reverse
bias
combined
characteristic
Semiconductor Physics - 16Copyright © by John Wiley & Sons 2003
+
+
P
N
+
V
W(V)
W
o
+
+
n (x)
p p (x)
n
n
po
p
no
+
Electric field,
J
s
x
Reverse Saturation Current
• Carrier density gradient
immediately adjacent to
depletion region causes
reverse saturation current to
flow via diffusion.
• J
s
independent of reverse
voltage V because carrier
density gradient unaffected by
applied voltage.
• J
s
extremely temperature
sensitivity because of
dependence on n
i
2
(T.)
Semiconductor Physics - 17Copyright © by John Wiley & Sons 2003
Impact Ionization
-
Si
Si
Si
-
-
-
-
-
-
Electric field E
• E ≥ E
BD
; free electron can
acquire sufficient from the field
between lattice collisions (t
c

10
-12
sec) to break covalent bond.
• Energy = 0.5mv
2
= q E
g
; v = q E
BD
t
c
• Solving for E
BD
gives
E
BD
=
2￿ E
g
￿ m
q￿ t
c
2

• Numerical evaluation
• m = 10
-27
grams, E
g
= 1.1 eV, t
c
= 10
-12
sec.
• E
BD
=
(2)￿ (1.1)￿ (10
27
)
(1.6x10
-19
)￿ (10
-24
)
= 3x10
5
V/cm
• Experimental estimates are 2-3.5x10
5
V/cm