______________________________________________________________________________ _______________________________________ _______________________________________________ ________
7.16
(a)
(5 × 1016 )(1015 ) V bi = ( 0.0259 ) ln (1.5 × 1010 ) 2 = 0.6767 V 2 ∈ (V + V R ) N a + N d W = s bi e N N a d
7.17
(a)
V bi
1/ 2
(b)
=
= 0,
2(11.7) (8.85 ×10 −14 )( 0.6767) W = −19 1.6 ×10
5 ×1016 + 1015 × 16 15 (5 ×10 )(10 )
1/ 2
0.9452 µ
=
2.738 × 10
or W = (c)
2.738 µ
Ε max =
W
1/ 2
x p
2∈ = s
(V + V ) N bi
R
e
N a N a + N d d
1/ 2
1
1/ 2
= 5.97 × 10 −6 cm or x p
)
0.9452 × 10
−4
= 1.43 ×10
4
V
= 5 V,
(
2 0.6767 + 5 2.738 × 10
1/ 2
1
4 ×1016 1 × 17 17 16 × × + × 2 10 2 10 4 10
+ V R )
2 0.6767
/cm (ii)For V R
Ε max =
m
(
e
N N + N d a d a
cm
= 0,
(i)For V R
Ε max =
(
2 V bi
R
2(11.7) (8.85 ×10 −14 )( 0.8081 + 2.5) = − 1.6 ×10 19
1/ 2
−4
bi
2 ×1017 1 × 16 17 16 4 ×10 2 ×10 + 4 ×10
m
2(11.7 ) (8.85 ×10 −14 )( 0.6767 + 5) W = −19 1.6 ×10
5 ×1016 + 1015 × 16 15 (5 ×10 )(10 )
(V + V ) N
= 0.2987 × 10 −4 cm or x n = 0.2987 µ m
= 5 V,
(ii) For V R
2 ∈ = s
2(11.7) (8.85 ×10 −14 )( 0.8081 + 2.5) = − 1.6 ×10 19
= 9.452 × 10 −5 cm or W =
0.8081 V
(b)
x n (i) For V R
( 2 ×1017 )( 4 ×1016 ) = ( 0.0259) ln (1.5 ×1010 ) 2
−4
) = 4.15 ×10
4
V/
cm _______________________________________
= 0.0597 µ m
2 ∈ W = s
(V + V ) N + N bi
e
R
a
N a N d
d
1/ 2
2(11.7) (8.85 ×10 −14 )( 0.8081 + 2.5) = −19 1.6 ×10
______________________________________________________________________________________
2 ×1017 + 4 ×1016 × 17 16 ( 2 × 10 )( 4 ×10 ) =
0.3584 × 10
−4
1/ 2
x n
cm
Ε
max
=
(
2 V bi
+ V R )
W
=
(
)
2 0.8081 + 2.5 0.3584 ×10
−4
= 1.85 × 10 5 V/cm
= ( 2 ×10
(1.6 ×10 −19 )(11.7 ) (8.85 ×10 ) 2( 0.8081 + 2.5)
( 2 × 1017 )( 4 ×1016 ) × 17 16 2 ×10 + 4 ×10
1/ 2
= 5.78 × 10 −12 F or C = 5.78 !F _______________________________________
x p
2∈ = s
=
(c)
2
V = ni2 #x! bi V t
= (1.5 × 10 10 )
2
R
e
N a N a + N d
−6
1/ 2
cm
= 0.0283 µ m
Ε max =
(
2 V bi
+ V R )
W
(
)
2 0.740 + 10
( 2.262 + 0.0283) ×10 −
4
1/ 2
(1.6 ×10 −19 )(11.7 ) (8.85 ×10 −14 ) = 2( 0.740 + 10)
0.740
= 5.762 ×10 32
(b)
1/ 2
1
d
e ∈ s N a N d () C ′ = 2(V bi + V R )( N a + N d )
⇒ N d = 2.684 ×1015 cm 17 N a = 2.147 × 10 cm
1/ 2
= 9.38 × 10 4 V/cm
0.0259
#x!
bi
2.83 × 10
=
"# $in
80 N d
(V + V ) N
1 1 × 17 15 80 2.147 ×10 + 2.684 ×10 or x p
N N = V t ln a 2 d ni 80 N d 2 = V t ln 2 ni
e
2(11.7) (8.85 ×10 −14 )( 0.740 + 10) = −19 1.6 ×10
7.18
(a) V bi
R
1/ 2
= 2.262 ×10 −4 cm or x n = 2.262 µ m
( )( ) 2 V + V N + N bi R a d
−4
bi
a N d N a + N d 1
80 1 × 1 2.147 ×1017 + 2.684 ×1015
1/ 2
e ∈ s N a N d
() C = A
(V + V ) N
2(11.7) (8.85 ×10 −14 )( 0.740 + 10) = − 1.6 ×10 19
or W = 0.3584 µ m Also W = x n + x p = 0.3584 µ m (c)
2 ∈ = s
3
−
3
−
( 2.147 × 1017 )( 2.684 ×1015 ) × 17 15 2.147 ×10 + 2.684 ×10
1/ 2
C ′ = 4.52 × 10 −9 F/cm 2 _______________________________________ 7.19
______________________________________________________________________________________
(
(a) V bi 3 N a
) − V ( N ) bi
(b)
a
V bi
N ( 3 N ) N N = V t ln d 2 a − V t ln d 2 a ni ni
or
V bi
N N N N = V t ln( 3) + ln d 2 a − V t ln d 2 a ni ni
= V t ln( 3) = ( 0.0259 ) ln ( 3 ) = 0.02845 V e ∈ s N a C ′ ≅ 2(V bi + V R )
(3 ×10 ) 5
(
( )
C ′ N a
) = 3 N
a
N a
3
2(1.6 × 10 −19 )(V bi + V R ) = (11.7 ) (8.85 × 10 −14 )
( 4 ×10 )( 4 ×10 ) 4 × 10 16
(V
17
+ 4 × 1017
so 'a'
+ V R ) = 8.008 V
bi
ic i#ls V R = 7.18 V (c)
1/ 2
=
2
16
%o
C ′ 3 N a
= 0.826 V
"# a-#
×
1/ 2
(b)
( 4 ×1016 )( 4 ×1017 ) = ( 0.0259) ln (1.5 ×1010 ) 2
= 1.732
(c) For a lar&#r o!in&, '# s!ac# car&# i' narros ic r#s*l's in a lar&#r ca!aci'anc#. _______________________________________
V bi
( 4 ×1017 )( 4 ×1017 ) = ( 0.0259) ln (1.5 ×1010 ) 2
or
V bi
7.20
= 0.886 V
"# a-#
(a)
V bi
( 4 ×1015 )( 4 ×1017 ) = ( 0.0259) ln (1.5 ×1010 ) 2
(3 ×10 ) 5
2
or
V bi
= 0.766 V ×
+o
2e(V bi + V R ) N a N d Ε max = ∈ s N + N d a
1/ 2
or
(3×10 ) 5
2(1.6 ×10 −19 )(V bi + V R ) = (11.7) (8.85 ×10 −14 ) ( 4 × 1015 )( 4 × 10 17 ) × 4 × 1015 + 4 × 10 17
2
or
9 ×10
10
so 'a'
(V
bi
= 1.224 ×10 9 (V bi + V R )
( 4 ×10 )( 4 ×10 ) 17
4 × 10 17 so 'a'
(V
bi
17
+ 4 × 1017
+ V R ) = 1.456 V
ic i#ls V R = 0.570 V _______________________________________ 7.21
(a)
( ) W ( B ) W A
+ V R ) = 73.53 V
ic i#ls V R = 72.8 V
2(1.6 ×10 −19 )(V bi + V R ) = (11.7) (8.85 ×10 −14 )
or
2 ∈ s (V biA + V R ) N a + N dA e N N a dA = 2 ∈ s (V biB + V R ) N a + N dB e N N a dB
1/ 2
1/ 2
______________________________________________________________________________________
( ) = (V + V ) ⋅ ( N + N ) ⋅ N W ( B ) (V + V ) ( N + N ) N W A
biA
R
a
dA
biB
R
a
dB
dB dA
1/ 2
N V + V N + N dB = dA biB R a N V V N N + + dA dB biA R a
1/ 2
1015 5.8139 1018 + 1016 = 16 18 15 5 . 7543 + 10 10 10
1/ 2
"# $in
(1018 )(1015 ) = 0.7543 = ( 0.0259) ln (1.5 ×1010 ) 2
V biA
or
( ) = 0.319 C ′ ( B ) C j′ A
V
(1018 )(1016 ) = 0.8139 = ( 0.0259) ln (1.5 × 1010 ) 2
V biB
j
_______________________________________ 7.22
(a) "# a-# 1/ 2
V
∈ s N a N d 2(V )( N + N ) C j′ ( 0 ) = bi a d C j′ (10 ) ∈ s N a N d 2(V + V )( N + N ) bi R a d
"# $in
( ) = 5.7543 10 W ( B ) 5.8139 10
1/ 2
+ 1015 18 + 1016 18
W A
1016 × 15 10
or
( ) = = V + V 3.13 C ′ (10) V C j′ 0
or
( ) = 3.13 W ( B ) W A
Ε( A) = Ε( B )
(
+ V R )
( )
W A 2 V biB + V R
(
( )
bi
R
j
(b) 2 V biA
biA
R
biB
R
= 10 V, # $in ( 3.13) 2 V bi = V bi + 10
For V R
V bi x p
#n x p
x n
or
Ε ( A) = 0.316 Ε ( B )
= 0.2W = 0.2 x p + x n
= 0.25 =
N d N a
+o
V bi
(c)
∈ s N a N dA C j′ ( A) 2(V biA + V R )( N a + N dA ) = C j′ ( B ) ∈ s N a N dB 2(V + V )( N + N ) biB R a dB
= 1.137 V
(b)
W B
1 5.7543 = 3.13 5.8139
bi
1/ 2
or
( ) ⋅ V + V = ) W ( A) V + V W B
1/ 2
1/ 2
1/ 2
N N = V t ln a 2 d ni
so
0.25 N a2 1.137 = ( 0.0259 ) ln (1.8 ×10 6 ) 2 "# can '#n ri'#
N a
=
1.8 ×10 6 0.25
1.137 2( 0.0259)
#x!
ic i#ls
N a
= 1.23 × 1016 cm
3
−
______________________________________________________________________________________
an
(b)
N d = 3.07 ×10 cm _______________________________________ 15
3
−
V bi
7.23
V bi
( 2 ×1016 )(5 ×1015 ) = ( 0.0259) ln (1.8 ×10 6 ) 2
= 1.162 V C ′ ∝
%o
1
V bi
( )= C ′(V )
1.162 + V R 2
1.50 =
C =
1.662 = 2.58 V
7.24
( 2 ×1015 )( 4 ×1016 ) = ( 0.0259) ln (1.5 ×1010 ) 2
2(V bi + V R )( N a + N d ) e ∈ s N a N d
= (5 × 10
−4
(1.6 ×10 −19 )(11.7) (8.85 ×10 ) 2( 0.6889 + V R )
( 2 ×10 )( 4 ×10 ) × ( 2 ×10 + 4 ×10 ) 15
16
15
C =
1/ 2
(i) For V R
0.6889 + V R
= 0,
C = 7.567 !F (ii) For V R = 5 V, C = 2.633 !F
16
1.157 + V R
= 0,
C = 6.178 !F (ii) For V R = 5 V,
C = 2.678 !F _______________________________________ 7.25
V bi
( 2 ×1017 )(5 ×1015 ) = ( 0.0259) ln (1.5 ×1010 ) 2
= 0.7543 V
(a) 1/ 2
e ∈ s N a N d C = AC ′ = A 2( V bi + V R )( N a + N d )
= (8 ×10
−4
16
6.2806 × 10 −12
1/ 2
6.6457 × 10 −12
(i) For V R
= 0.6889 V
1/ 2
) )
16
15
1.162 + V R 2
C = AC ′ = A
(1.6 × 10−19 )(13.1) (8.85 × 10−14 ) ) 2(1.157 + V R )
15
1.162 + 0.5
=
2
−4
( 2 ×10 )( 4 ×10 × ( 2 ×10 + 4 ×10
ic i#ls V R 2 _______________________________________
V bi
1/ 2
e ∈ s N a N d C = AC ′ = A 2( V bi + V R )( N a + N d )
= (5 × 10
+ V R2 V bi + V R1
V bi
R 2
(a)
= 1.157 V
+ V R
C ′ V R1
(1.50)
( 2 ×1015 )( 4 ×1016 ) = ( 0.0259) ln (1.8 ×10 6 ) 2
(1.6 ×10 −19 )(11.7) (8.85 ×10 −14 ) ) 2( 0.7543 + 10)
( 2 ×10 )(5 ×10 × ( 2 ×10 + 5 ×10 17
15
17
1/ 2
) )
15
C = 4.904 × 10 −12 F f =
1 2π LC
⇒ L =
1
(
)
C 2π f
2
______________________________________________________________________________________
1
=
L
4 N d 2 = ( 0.0259) ln (1.8 × 10 6 ) 2
( 4.904 ×10 )[2π (1.25 ×10 )] −12
6
2
= 3.306 × 10 −3 = 3.306 m (b) (i) For V R
f =
1/ 2
e ∈ s N a N d C = AC ′ = A 2( V bi + V R )( N a + N d )
= 1 V, C = 12.14 !F 1
[(
)(
2π 3.306 × 10 − 3 12.14 × 10 −12
)]
1/ 2
= 0.794
= 7.94 ×10 (ii) For V R = 5 V, C = 6.704 !F 5
f =
×
1
[(
2π 3.306 ×10
−3
)(6.704 ×10 )] −12
= (10
1/ 2
−4
(1.6 ×10 −19 )(13.1) (8.85 ×10 −14 ) ) 2(V bi + 2 ) 1/ 2
( 4 N ) 2 d
( 5 N ) d
= 1.069 × 10 6 = 1.069 _______________________________________ 7.26
2e(V bi + V R ) N d Ε max ≅ ∈ s 2#' V bi ≅ 0.75 V
(
(a)
2.5 × 10
5
)
1/ 2
5
cm
3
+2
3
,
3
,
−
−
cm
= 4 x p
= N d xn = N d 4x p ⇒ N a = 4 N d N a N d (a) V bi = V t ln ni2
V bi
N d
= 2.724 ×10 − 20
V bi
+5
= 2.976 ×1015 cm = 1.19 ×1016 cm = 1.135 V
3
−
3
−
,
,
_______________________________________ 7.28
3
= ( 0.20)W = ( 0.20) x n + x p ( 0.8) x p = ( 0.2 ) x n
x p
N a
−
_______________________________________
−12
'rial an #rror,
N d
⇒ N d = 3.01×10
N a x p
0.6 ×10
2
15
x n
N a
−
2(1.6 ×10 −19 )( 0.75 + 10) N d = (11.7) (8.85 ×10 −14 ) 7.27
V bi
(b) From !ar' (a),
⇒ N d = 1.88 ×10 10
N d
= 2.724 ×10 − 20
= 1.504 ×1015 cm = 6.016 ×1015 cm = 1.10 V
N d
V bi
16
( )
−12
'rial an #rror,
2
2(1.6 ×10 −19 )( 0.75 + 10) N d = (11.7) (8.85 ×10 −14 ) (b)
0.6 ×10
(a) V bi
(5 × 10 15 )(1014 ) = ( 0.0259) ln (1.5 × 1010 ) 2
or
V bi
= 0.5574 V
(b)
x p
2 ∈ V = s bi e
2(11.7 ) =
N d 1 N N N + a a d
(
8.85 × 10
−14
1.6 ×10
−19
)(
0.5574
)
1/ 2
______________________________________________________________________________________
1 1014 + 5 × 1015
10 × 15 5 × 10 14
x p
1/ 2
x n
x p = 5.32 ×10
Also x n
2 ∈ V = s bi e
N d N a
x n
cm
N a 1 + N N N d a d
1/ 2
1014 = ( 50 ×10 − 4 ) 16 10
2(11.7 ) =
(
8.85 × 10 1.6 × 10
−19
)(
0.5574
= 0.50 × 10 −4 cm = (c)
Ε max ≅ −14
N ⇒ x n = x p a N d
so
or −6
=
)
2V R W
=
0.50
(
µ m
)
2 193.15 50.5 ×10
−4
or
Ε max = 7.65 ×10 4 V/cm _______________________________________
5 ×1015 × 14 10
1 1014 + 5 × 1015
7.30
1/ 2
(a)
V bi
or
= 2.66 × 10 −4 cm (c) For x n = 30 µ m, # a-# x n
30 × 10
−4
5 ×1015 × 14 10
2(11.7 ) (8.85 × 10 −14 )(V bi + V R ) = 1.6 × 10 −19 1 1014 + 5 × 1015
−6
= 1.269 ×10 −7 (V bi + V R )
_______________________________________ 7.29
= 1014 cm
2 ∈ V ≅ s R eN a
× (11.7 ) (8.85 × 10 −14 )( 2 × 1015 )} 1.287 × 10 − C = V bi + V R (i) For V R = 1 V, C = 9.783 ×10 − (ii) For V R = 3 V, C = 6.663 × 10 − (iii) For V R = 5 V, C = 5.376 × 10 −
,
(50 ×10 ) = 2
14
F
7.31
1/ 2
(a) V bi
(8 ×1016 ) N d = ( 0.0259) ln (1.8 ×10 6 ) 2
= 1.20
)(
2 11.7 8.85 ×10
ic i#ls V R = 193 V (b)
(
F
F _______________________________________
or −4
14
14
3
−
(a) A on#si# 4*nc'ion an ass*m# V R >> V bi . #n
x p
(1.6 ×10 −19 ) ) 2(V bi + V R ) 13
= 70.4 V
An n + p 4*nc'ion i' N a
−5
1/ 2
"# $in
V R
= 0.7305 V 1/ 2 e ∈ s N d C = AC ′ ≅ A ⋅ 2( V bi + V R )
= (10
1/ 2
ic b#com#s
9 ×10
(b)
( 2 ×1017 )( 2 ×1015 ) = ( 0.0259) ln 10 2 (1.5 ×10 )
−14
V R
(8 ×10 ) N = (1.8 ×10 ) 16
(1.6 ×10 − )(10 ) 19
6
d
2
1.20 0.0259
#x!
14
⇒ N d = 5.36 ×1015 cm
3
−
(b) 1/ 2
e ∈ s N a N d C = AC ′ = A 2( V bi + V R )( N a + N d )
______________________________________________________________________________________
(1.6 ×10 −19 ) −12 × = A 1.10 10 2(1.20 + 1.0)
or
13.1) (8.85 ×10 − )( 8 ×10 )( 5.36 ×10 ( × 14
(8 ×10
16
1
+ 5.36 ×10
16
15
)
−12
(1.6 ×10 ) = ( 7.56 ×10 −5 ) 2(V bi + V R )
13.1) (8.85 ×10 − )( 8 ×10 )( 5.36 ×10 ( ×
(8 ×10
1
=
2.1585 × 10
V bi
( x n − x) ∈ s "# also a-# Ε 2 = Ε 1 a' x = x O
#n
eN dO x O
+ V R
dx r
1
∈ s
∈ s
Ε
1
dx
eN aO x
∈ s
+ C 1
eN aO
∈ s
∈ s
∈
eN dO x
∈
d Ε 2 dx
=
x x − ∈ 2
eN dO
O
n
s
s
=
2
∈
s
2
( ) = eN
∈ s
−2 <
=
dO
∈ s
x
dx
< −1 µ m, ρ ( x) = +eN d
eN d
∈ s
⇒ Ε =
eN d x
%o
C 1
=
eN d x O
∈ s
#n
Ε =
eN d
∈ s
( x + x ) O
= 0 , Ε ( 0 ) = Ε ( x = −1 ) , so
Ε ( 0) =
< x < xn
ρ x
For
A' x
+ C
∈ s
+ C 1 ∈ s A' x = −2 µ m ≡ − x O , Ε = 0
eN dO
s
nr#&ion, xO
∈
dx 2
dx
p
ρ ( x )
2
2
−
( ) = − ρ ( x ) = − d Ε ( x )
%o d Ε
( x + x )
s
1
eN aO x p
− x p < x < 0
or
Ε =
∈ s
x xn − O 2
d 2φ x
(a)
=
)
7.34
nr#&ion, 0 < x < x O
d
− x O
aO
Ε = 0 a' x = − x p ⇒ C 1 = −
n
_______________________________________
"# a-#
Ε = −
eN dO
eN dO x
Ε =
( ) = − eN
#n $or
( x
∈ s
#n $or 0 < x < x O # a-#
N N = V t ln aO 2 dO ni
Ε = −
=−
C 2
7.33
=
eN dO
ic &i-#s
_______________________________________
(c) !r#&ion d Ε ρ x
+ C 2 = −
2 ∈ s
−8
⇒ V bi + V R = 4.161 = 1.20 + V R V R = 2.96 V
(a) V bi
eN dO
Ε 2 = −
+ 5.36 ×1015 )
16
1.0582 × 10 −8
16
∈ s < x < xn , # a-#
so 'a' $or xO
−19
14
eN dO x n
⇒ C 3 = −
(c)
0.80 ×10
+ C 3 ∈ s Ε 2 = 0 a' x = x n
"# a-#
A = 7.56 × 10 −5 cm 2
⇒
eN dO x
Ε 2 =
eN d
∈ s
( − 1 + 2) ×10 −
4
______________________________________________________________________________________
(b)
(1.6 ×10 )(5 ×10 ) (1×10 ) = (11.7) (8.85×10 ) −19
15
N B
−14
or Ε ( 0 ) = 7.726 × 10 4 V/cm (c) a&ni'*# o$ !o'#n'ial i$$#r#nc# is eN d φ = Ε dx = x + x O dx ∫
∈ s
∫ (
0
= N a = 2.59 ×1016 cm
3
−
7.36
)
N a
=
2 ∈ s Ε crit
2eV B
− 11.7) (8.85 × 10 )( 4 ×10 ) ( = − 2(1.6 ×10 )( 80)
5 2
14
19
= 6.47 × 1015 cm
= − x O , '#n
a' x
2
_______________________________________
x 2 + xO ⋅ x = + C 2 ∈ s 2 2#' φ =
5
19
r N B
eN d
(11.7) (8.85 ×10 − )( 4 ×10 ) = 2(1.6 × 10 − )( 20) 14
−4
3
−
_______________________________________ 7.37
− x + C ⇒ C = eN x 0= 2∈ ∈ 2 2
eN d x O
d
2
O
2
O
s
(b)
#n # can ri'#
φ =
eN d 2
( x + x )
2
=
(1.6 ×10 )(5 ×10 ) [( − 1 + 2) ×10 ] 2(11.7 ) (8.85 × 10 ) 15
−4
2
−14
or
Ε max %#'
3.863
V o'#n'ial i$$#r#nc# across '# in'rinsic r#&ion
V bi φ 2
= Ε ( 0) ⋅ d = ( 7.726 ×10 4 )( 2 ×10 −4 )
smm#'r, '# !o'#n'ial i$$#r#nc# across '# !r#&ion s!ac#car&# r#&ion is also 3.863 V. # 'o'al r#-#rs#bias -ol'a&# is '#n
∈ s Ε crit 2
2eN B
or
=
#n
∈ s Ε
2eV B N B
=
(11.7) (8.85 ×10 )( 4 ×10 − 2(1.6 ×10 )( 40) −14
2(1.6 ×10 −19 )(V bi + V R ) = (11.7) (8.85×10 −14 )
(b) 5
19
= N a = 1.294 ×1016 cm
5
1/ 2
7.35
N B
crit
( 2 ×1016 )( 2 ×1016 ) × 16 16 2 ×10 + 2 ×10 ⇒ V bi + V B = 51.77 V %o V B = 51.04 V
_______________________________________
2 crit
max
1/ 2
( 2 ×1016 )( 2 ×1016 ) = ( 0.0259) ln (1.5 ×1010 ) 2
4 ×10
= 2( 3.863) + 15.45 = 23.2 V
=
2e( V bi + V R ) N a N d = ∈ s + N N d a Ε = Ε an V R = V B
#n
φ 2 = 15.45 V
(a) V B
,
= 0.7305 V
or
V R
3
(a) From 9*a'ion (7.36),
−19
φ 1 =
−
, $rom Fi&*r#
7.38
s
A' x = −1 µ m
φ 1
≅ 75 V For N d = 1015 cm V B ≅ 450 V
3
_______________________________________
O
∈
−
7.15, V B
2
s
= 1016 cm
(a) For N d
2
3
−
V bi
(5 ×1015 )(5 ×1015 ) = ( 0.0259) ln (1.5 ×1010 ) 2
= 0.6587 V
______________________________________________________________________________________
#n
4 ×10
5
2(1.6 ×10 −19 )( V bi + V R ) = (11.7) (8.85 ×10 −14 ) 15
7.39
3
−
x n
(
x n min
_______________________________________ Ass*m# silicon: For an n + p 4*nc'ion
x p
1/ 2
4
cm
1/ 2
=
V R
5.09 µ
V +o
eN d x n
(
)(
2 11.7 8.85 ×10
(1.6 ×10 )(10 −19
R
15
= 150 µ m 2
(
)(
2 11.7 8.85 ×10
−14
(1.6 ×10 )(10 −19
)V )
R
15
ic i#ls
V R
= 1.74 × 10 4 V
+o'#: From Fi&*r# 7.15, '# br#a;on -ol'a&# is a!!roxima'#l 300 V. %o, in #ac cas#, br#a;on is r#ac# $irs'. _______________________________________
18
a
n
14
=
2 × 1018 2 × 10
−4
= 10 22 cm
4
−
From Fi&*r# 7.15, V B ≅ 15 V _______________________________________
ic i#ls
= 6.47 × 10 −6 cm
7.43
(a) For '# lin#arl &ra# 4*nc'ion ρ x = eax
+o
( )
(V + V ) N e
)V )
(1.6 ×10 )(10 ) x = (11.7) (8.85 ×10 − )
bi
−14
7.42
∈ s −19
x n
= 75 µ m
2
(b) For x p
so
2∈ = s
<< V R
(150 ×10 ) =
(1018 )(1018 ) = 0.933 = ( 0.0259) ln (1.5 ×1010 ) 2
x n
eN a
1/ 2
= 4.35 × 10 3 V
−4
Ε max =
R
ic i#ls
"# $in
10
bi
( 75 ×10 ) =
7.40
6
Ass*m# V bi
−4
m _______________________________________
V bi
(V + V )
2∈ = s
(a) For x p
) = 5.09 ×10 −
= 5.54 V
V R
≅ 100 V,
an V B
2(11.7 ) (8.85 ×10 −14 )(100) = 15 −19 (1.6 ×10 )(5 ×10 ) or
1 1018 + 1018
or
'#n, n#&l#c'in& V bi # a-#
2 ∈ V ≅ s B eN d
2(11.7 ) (8.85 × 10 −14 ) = 1.6 × 10 −19
2
7.41
For a silicon p + n 4*nc'ion i'
= 5 ×1015 cm
−6
ic i#ls V bi + V R = 6.468 V
%o V R ≅ 206 V _______________________________________
N d
( 6.47 ×10 )
1018 × (V bi + V R ) 18 10
1/ 2
(5 ×10 )(5 ×10 ) × 15 15 5 ×10 + 5 × 10 ⇒ V bi + V R = 207.1 15
#n
R
a N d N a + N d 1
1/ 2
#n
______________________________________________________________________________________
d Ε dx
( ) = eax
7.45
ρ x
=
∈ s
∈ s
(a) C j
+o
Ε =
eax
∫ ∈
=
dx
s
A' x %o
⋅
∈ s
2
2
+ C 1
ea xO
0=
∈ s 2
2 + C 1 ⇒ C 1 = − ea xO ∈ s 2
V bi
ea 2
( x − x ) 2
∈ s
C j
2
O
= 0.45 ×10
−12
(1.6 ×10 −19 ) = A 2( 0.7648 + 5)
× (11.7 ) (8.85 ×10 −14 )( 5 ×1015 )}
( ) = −∫ Εdx = − %#' φ =
0 a'
ea x 2
3
∈ s 3
− xO ⋅ x + C 2
2
= − x O , '#n
x
3
O
⇒
A
2
2
s
#n
φ ( x ) = − − x 2 ∈ 3 3
2 O
s
eax ⋅ x + 3 ∈
3
O
s
_______________________________________ 7.44
"# a-# 'a'
ea ∈ C ′ = 12(V + V ) 2
1/ 3
s
bi
#n 7.2 × 10
−9
R
)
3
a (1.6 ×10 −19 )[(11.7 ) (8.85 × 10 −14 )] 2 = 12( 0.7 + 3.5) ic i#ls a = 1.1× 10 20 cm 4 _______________________________________ −
(
= (5.309 ×10
−5
=
)
2
(1.6 ×10 −19 ) ) 2(V bi + V R )
× (11.7 ) (8.85 ×10 −14 )( 5 ×1015 )} C j
1/ 2
= A 8.476 × 10 −9
= 5.31× 10 −5 cm
s
ea x
−12
3
O
3
O
0.45 × 10
(b) C j
− x eax 0=− x C C + + ⇒ = 2 ∈ 3 3∈ ea
(
(3 ×1017 )(5 ×1015 ) = ( 0.0259) ln (1.5 ×1010 ) 2
+o
φ x
== N d
= 0.7648 V
(b)
3
−
1/ 2
#n
#n
Ε=
= 5 × 1015 cm
2#' N a
= + x O an x = − x O , Ε = 0 2
ea x
e ∈ N = AC ′ ≅ A ⋅ s a 2(V bi + V R )
1/ 2
1.0805 × 10 −12
(i) For V R
V bi
+ V R
= 2.5 V, = 0,
C j
= 0.598 !F = 1.24 !F
C j (ii) For V R _______________________________________
