PN Junction Diodes
Sung June Kim
kimsj@snu.ac.kr
http://helios.snu.ac.kr
Chapter 5.
Contents
q Drift
q Diffusion
q Generation-Recombination q Equations of State
2
q Preliminaries
• Junction Terminology/Idealized Profiles
Net doping profile
ü The step junction is an acceptable approximation to an ion- implantation or shallow diffusion into a lightly doped starting wafer
Step (abrupt) junction Linearly graded junction
• Poisson’s Equation
K
S 0r Ñ × E = e
S 0
d
dx K r
= e E
1-Dimension
Ks is the semiconductor dielectric constant and e0 is the permittivity of free space. r is the charge density (charge/cm3)
D A
( )
q p n N N
r = - + -
• Qualitative Solution
ü Let us assume an equilibrium conditions
dx to dE al
proportion
r is
ü It is reasonable to expect regions far removed from the metallurgical junction to be identical to an isolated semiconductor.
ü Under equilibrium conditions, the Fermi level is a constant
P-N Diode Junction Energy Band
P
Ec
Ef Ev
N
Ec Ef
Ev
Equilibrium P-N Junction
P N
Ec
Ef Ev
Ec Ef
Ev
V=0
Forward Biased P-N Junction
P N
Ec
Ef Ev
Ec Ef
Ev
V>0
Reverse Biased P-N Junction
P N
Ec
Ef Ev
Ec Ef
Ev
V<0
1 (
c ref)
V E E
= - q -
Eref
ü V versus x relationship must have the same functional form as the
“ upside-down” of Ec
dV
= - dx E
S 0
d
dx K r
= e E
ü The voltage drop across the junction under equilibrium conditions and the appearance of charge near the metallurgical boundary
ü Where does this charge come from?
ü Charge neutrality is assumed to prevail in the isolated, uniformly doped semiconductors
hole electron
Charge redistribution
Charge density Space charge
region or
depletion region
ü The build-up of charge and the associated electric field continues until the diffusion is precisely balanced by the carrier drift
ü The individual carrier diffusion and drift components must of course cancel to make JN and JP separately zero
• The Built-in Potential (V
bi)
üConsider a nondegenerately-doped junction
dV
= - dx E
n n
p p
( )
n p bi
( )
( ) ( )
x V x
x
dx
V xdV V x V x V
- -
- ò E = ò = - - =
ü Integrating
N n N
dn 0
J q n qD m dx
= E + =
ü Solving for and making use of the Einstein relationship, we obtain
E
N n
/ /
D dn dx kT dn dx
n q n
= - m = -
E
n n
p p
( )
n
bi ( )
p
ln ( )
( )
x n x
x n x
n x
kT dn kT
V dx
q n q n x
- -
é ù
= - = = ê ú
ê - ú
ë û
ò E ò
2 i
n D p
A
( ) , ( ) n
n x N n x
Q = - = N
A D
bi 2
i
ln N N V kT
q n
æ ö
= ç ÷
è ø
• The Depletion Approximation ü It is very hard to solve
(1) The carrier concentrations are negligible in
(2) The charge density outside the depletion region=0
) (
0
A D
s
N N
n k p
q dx
dE = - + -
e
n
p
x x
x £ £
-
S 0
D A
S 0
( )
d
dx K
q p n N N K
r e e
=
= - + -
E
ü Exact
ü Depletion Approximation
D A p n
S 0
p n
( ) . . .
0 . . . and
q N N x x x
d K
dx x x x x
e
ì - - £ £
@ í ï
ï £ - ³ î
E
q Quantitative Electrostatic Relationships
• Assumptions/definitions
• Step Junction with V
A=0 ü Solution for r
ü Solution for
E
A p
D n
p n
. . . 0
. . . 0
0 . . . and
qN x x
qN x x
x x x x
r
- - £ £ ì
ï £ £ í
ï £ - ³ î
A S p
D S n
p n
/ . . . 0
/ . . . 0
0 . . . and
qN K x x
d qN K x x
dx x x x x
e e
0 0
- - £ £
ì
ï £ £ í
ï £ - ³ î
E
p
( ) A
0 S 0
x x
x
d qN dx
K e
¢ = -
-¢
ò
EE ò
ü For the p-side of the depletion region
ü Similarly on the n-side
0 n
D
( ) S 0
x
x x
d qN dx
K e
¢ = - ¢
ò
EE ò
D
n n
S 0
( ) qN ( ) . . . 0
x x x x x
K e
= - - £ £ E
A p D n
N x = N x
A
p p
S 0
( ) qN ( ) . . . 0
x x x x x
K e
= - + - £ £
E
ü Solution for V ( )
E = - dV dx /
ü With the arbitrary reference potential set equal to zero at x=-xp and Vbi across the depletion region equilibrium conditions
p
bi n
0 at at
V x x
V V x x
= = -
= =
A
p p
S 0 D
n n
S 0
( ) . . . 0
( ) . . .
qN x x x x
dV K
qN
dx x x x x
K e e
ì + - £ £ ï ï
= í
ï - 0 £ £
ï î
ü For the p-side of the depletion region
p
( ) A
0 p
S 0
( )
V x x
x
dV qN x x dx
K e
¢ =
-+ ¢ ¢
ò ò
A 2
p p
S 0
( ) ( ) . . . 0
2
V x qN x x x x
K e
= + - £ £
ü Similarly on the n-side of the junction
D 2
bi n n
S 0
( ) ( ) . . . 0
2
V x V qN x x x x
K e
= - - £ £
2 2
A D
p bi n
S 0 S 0
2 2
qN qN
x V x
K e = - K e
@ x=0ü Solution for xn and xp
A p D n
N x = N x Q
1/ 2
S 0 A
n bi
D A D
2
( )
K N
x V
q N N N
e
é ù
= ê ú
ë + û
1/ 2
D n S 0 D
p bi
A A A D
2
( )
K
N x N
x V
N q N N N
é e ù
= ê ú
ë + û
1/ 2
S 0 A D
n p bi
A D
2K N N
W x x V
q N N
é e æ + ö ù
º + = ê ç ÷ ú
è ø
ë û
Depletion width
• Step Junction with V
A¹0
ü When V
A> 0, the externally imposed voltage drop lowers
the potential on the n-side relative to the p-side
ü The voltage drop across the depletion region, and hence the boundary condition at x=xn, becomes Vbi-VA
1/ 2
S 0 D
p bi A
A A D
2 ( )
( )
K N
x V V
q N N N
e
é ù
= ê - ú
ë + û
1/ 2
S 0 A
n bi A
D A D
2 ( )
( )
K N
x V V
q N N N
é e ù
= ê - ú
ë + û
1/ 2
S 0 A D
bi A
A D
2 K N N ( )
W V V
q N N
é e æ + ö ù
= ê ç ÷ - ú
è ø
ë û
• Examination/Extrapolation of Results
ü Depletion widths decrease under forward biasing and increase under reverse biasing
üA decreased depletion width when V
A> 0 means less charge around the junction and a correspondingly smaller -field.
Similarly, the potential decreases at all points when V
A>0 ü The Fermi level is omitted from the depletion region
E
A Fn
Fp
E qV
E - = -
pn junction energy band diagrams.
Summary
31
Summary
32