processes other
G R thermal p
processes other
G R thermal N
p p
p
N n
N
p N
t p t
J p q t p
t n t
J n q t n
dx qD dp pE q J
dx qD dn nE q J
J J J
AJ I
1 1
(3.46), From
5. Ideal diode equation
(6.6b) ,
and (6.5b) ,
0
(6.6a) ,
and (6.5a) ,
0
(3.51) (3.51) 1
1 1
and
. 0 when
, since
And system.
l dimensiona -
one for the 1
1
2 2
2 2
2 2 2
2 0
0
n n
p p
p
p p
N N
n N
L processes other
n G
R thermal
N N
N N
N
N N
n N
N N
x x x
qD p p J
x D p
-x x x
qD n n J
x D n
From t G
n
n From t
n
x D n x
D n x
qD n x q x J J q
q
x E qD n
x qD n nE q J
x n x
n x
n x
n
Δn n
x n J J q
q
(5-1) Quasineutral Region Considerations
:Under the steady state conditions with GL=0, the minority carrier diffusion equations appropriate for the p and n quasineutral regions are given by the following equations:
(5-2) Depletion Region Considerations :Within the depletion region, E 0.
).
( ) (
region, depletion
In the region.
depletion the
of
edges at the evaluated
solutions region
al quasineutr the
from determined are
) (x J and ) (-x J where
) ( )
(
) ( )
(
allows region
depletion t the
throughou currents
carrier the
of constancy The
region.
depletion the
inside position of
t independen constants
be to determined are
J and J Thus
. 0 and
0
region, depletion
he through t negligible
is generation -
ion recombinat mal
that ther Assuming
0 1 0 1
n p p
N p N
n p n
p p
p N n
p N
N p
G R thermal p
G R thermal N
p p
p
N n
N
x J x J J
x J x x -x J
x J x x -x J
x J x
J
t p x
J q
t n x
J q
dx qD dp pE q J
dx qD dn nE q J
Boundary Conditions
(a) At the ohmic contacts
The ideal diode is usually taken to be “wide-base” diode, or a diode whose contacts are several minority carrier diffusion lengths or more from the edges of the depletion region. In a wide-base diode any perturbation in the carrier concentrations created at the edges of the depletion region will decay to zero before reaching the contacts. The contacts may effectively be viewed as being
positioned at x= .
0 ) (
0 ) (
x p
x n
n p
(b) At the depletion region edges
kT i qV
kT qV A i p
kT qV i A p p
p
n p
kT qV i
A p
N A
Fp Fn
p N
kT Fp F i
kT F E i kT
E F i
A A
A A
N
p i i
N
n e x
n x
n x
n
N e x n
n
e n N
x n x
p x n
x x x e
n np
qV -F
F qV
E E
F F
e n np
e n p e
n n
/ 2
/ 2
2 / 2 /
Fn
Fp p
N /
) 2 (
/ ) / (
) (
) ( )
( )
( ) (
) ( ) ( ) (
region, depletion
the of edge - p At the
(6.12) .
,
, region,
depletion in the
assumed mostly
If
.
6.4(b) Fig.
From junction.
the of side - n on the far
E o junction t the
of side - p on the far
E from lly monotonica vary
will levels F
and F the assume to
reasonable is
It
(6.11) and
(3.72), From
levels.
Fermi -
quasi of
use he through t conditions
rium nonequilib to
extended is
diagram band
energy the
from inspection
by ions concentrat carrier
the deduce to
able being of
e convenienc The
(6.18) ),
1 (
) (
) ( )
(
) , ) (
( Since
) ( )
( )
(
) ( )
( ) (
Similarly,
(6.15) ),
1 (
) (
) ( )
(
) , ) (
( Since
) ( )
( )
(
/ 2
2 /
2 0
/ 2
2
0 2 0
0 2 0
2 0
0
/ 2
0
2 / /
2
2 /
2 0
/ 2
2
0 2 0
0 2 0
2 0
0
/ 2 0
kT qV D i n
D kT i
qV D i n
kT qV D i n
D i n
i n
i i
kT qV D i n
n n
kT qV i n
D n
n
kT qV A i p
A kT i
qV A i p
kT qV A i p
A i p
i p
i i
kT qV A i p
p p
A
A A
A A
A
A A
A
N e x n
p
N e n
N x n
p N e
x n p
N n x
n x n
n p p n
n p n
N e x n
p x
p x
p
e n x
p N x
p x n
N e x n
n
N e n
N x n
n N e
x n n
N n x
p x n
p n n n
n p n
N e x n
n x
n x
n
diode ideal
an in flowing Current
(c)
region.
depletion e
within th E
F and E
F at constant are
levels Fermi
- quasi the
ly, equivalent or
region, depletion
t the throughou qV
F - F that assumed We
:
Fp p
Fn N
A p
N
(6.23) 0
, )
1 (
) 21 . 6 ( )
(
) 1 (
) 1 (
) 0 (
(6.20b), and
(6.21) From
0 0
) (
(6.20a), and
(6.21) From
(6.20b) ),
1 (
) 0 (
(6.18), From
(6.20a) ,
0 ) (
(6.10b), From
: conditions Boundary
(6.21) ,
) (
0 ,
0
(6.5b) From
0).
x ( junction the
of side - n al quasineutr on the
holes with first work
us et (i)
/ / 2
/ 2 /
1 / 2
1
/ 2
1 0 0
2 1
0 1
2 2
0 1 /
2 / 2 /
1 2 2
x e
N e n
e A e
A x
p
N e A n
N e A n e
A e
A x
p
A e
A e
A x
p N e x n
p x p
e A e
A x
p
p x x
d p D d
L
A p
p A
A A
p
L kT x
i qV
Lp L x
x n
kT qV D i
kT qV D i n
n
kT qV D i n
n
Lp L x
x n
p n n
p
(5-3) Carrier Currents
(6.24) 0
, )
1 (
) 6 . 6 (
(6.23) 0
, )
1 (
) (
/ / 2
/ / 2
x e
N e n L q D
x b d
p qD d
J
x e
N e x n
p
A p A p
L kT x
qV D i p
p
n p
p
L kT x
qV D i n
(6.25) 0
, )
1 (
) (
) 1 (
) 1 (
) 0 (
0 0
) (
) 1 (
) 0 (
(6.18), From
0 ) (
(6.10a), From
: conditions Boundary
) (
0 ,
0
(6.5a) From
0).
x ( junction the
of side - p al quasineutr on the
electrons work with
second us
et (ii)
/ /
2
/ 2 /
1 / 2
1
/ 2
1 0 0
2 1
0 1
2 2
0 1
/ 2
/ 2 /
1 2 2
x e
N e n
e A e
A x
n
N e A n
N e A n e
A e
A x
n
A e
A e
A x
n
N e x n
n x n
e A e
A x
n
n x x
d n D d
L
N A
N N
A
A A
N N
L x kT
qV A i
L x L
x p
kT qV A i
kT qV A i p
p
kT qV A i p
p
L x L
x p
n p p
N
) 1 (
) 1 (
) (
(6.27b) ),
1 (
) 1 (
) 0 (
) (
(6.27a) ),
1 (
) 1 (
) 0 (
) (
(6.26) 0
, )
1 (
) 6 . 6 ( )
(
(6.25) 0
, )
1 (
) (
/ 2
/ 2
/ 2
0 / /
2 /
2
0 /
/ 2
/ /
2
/ / 2
kT qV D i p kT p
qV A i N
N p
N kT qV D i p
p
x L kT x
qV D i p
p p
n p
kT qV A i N
N
x L x kT
qV A i N
N N
p N
L x kT
qV A i N
N
p N
p N
N
L kT x
qV A i p
A A
A
A p A
N A
A N
N A
N e n L qAD N e
n L qAD A
J J JA I
N e n L q D
e N e
n L q D x
J x
x J
N e n L q D
e N e
n L q D x
J x
x J
x e
N e n L q D
x a d
n qD d
x d
n qD d
x J
x e
N e x n
n
factor ideality
: (6.30) ,
where
equation diode
: (6.29) ),
1 (
) 1 (
) 1 (
) 1 (
) (
2 2
0
/ 0
/ 2
2
/ 2
/ 2
D i p
p A
i N
N kT qV
kT qV D
i p
p A
i N
N
kT qV D i p kT p
qV A i N
N p
N
N n L D N
n L qA D I
ideal e
I
N e n L D N
n L qA D
N e n L qAD N e
n L qAD A
J J JA I
A
A
A A
(6.32b) diodes
p - n for
(6.32a) diodes
n - p for
current saturation
The (iii)
).
ln(I of intercept ed
extrapolat an
and q/kT of
slope region
linear a
has plot that semilog
0 VA a predicts theory
ideal The :
ln ln
, 3
kT/q few
a an greater th biasing
forward For
(ii)
, 3
few a an greater th biases
reverse For
(i)
) 1 (
) 1 (
2 0
2 0
0 0
/ 0
0
/ 0
/ 2
2
A i N
N D i p
p
A kT
qV
A A
kT qV kT
qV D
i p
p A
i N
N
N n L qAD I
N n L qAD I
kTV I q
I e I I
kT/q) (V
I I
kT/q) kT/q (V
e I N e
n L D N
n L qA D I
A
A A
A kT
qV kT
qV kT qV
kT qV
kTV I q I
e I e
I I
ii
I e
I I
e I I from
V . kT/q
A A
A
A
0 /
0 /
0
0 /
0
/ 0
ln ln
) 1 (
kT/q, few
a an greater th biasing
forward for
) (
) 1 (
), 1 (
of equation diode
ideal the from
and kT/q, few
a an greater th biasing
reverse for
and ) 026 0 (
re temperatu room
At (i)
10
(3.25b) )
/ , (
) 1 (
) 1 (
. /cm 10
N , /cm 10
N re.
temperatu room
at diodes junction
step n - p ideal Two 6.1) Ex
1 2 2
1 02
01
2 1
2 0
/ 0
/ 2
2
3 16 D2
3 15 D1
D D p
p p p
p p p
p p
p p p
p D i p
p
kT qV kT
qV D
i p
p A
i N
N
N N I
I Also
q from kT
D D
D L
D N
n L qAD I
e I N e
n L D N
n L qA D
I A A
) 1 (
) 1 (
) (
(6.27b) ),
1 (
) 0 (
) 1 (
) (
(6.27a) ),
1 (
) (
(6.26) 0
, )
1 (
) (
/ 2
/ 2
/ 2
/ / 2
/ 2
/ /
2
kT qV D i p kT p
qV A i N
N p
N
kT qV D i p
p p
L kT x
qV D i p
p n
p
kT qV A i N
N p
N
L x kT
qV A i N
N N
A A
A A p
A N A
N e n L qAD N e
n L qAD A
J J JA I
N e n L q D x
J
e N e
n L q D x
x J
N e n L q D x
x J
x e
N e n L q D x
J
Carrier Currents
:As in Fig. 6.7, whereas electrons and holes both contribute to the current through the depletion region, the hole current dominates far from the junction on the p-side of the device and the electron current dominates far from the junction on the n-side of the device.
(5-4) Carrier Concentrations
: (a) Forward biasing increases the carrier
concentrations over their respective equilibrium values, while reverse biasing lowers the
concentrations below the equilibrium values. In either case, the perturbations decay
exponentially as one proceeds away from the edges of the depletion region. Moreover, after several diffusion lengths the perturbations effectively die out and the minority carrier
concentrations approach their equilibrium values;
i L i
kT i qV
i
D i n
D L i
kT qV
D i
D i n
A
A i p
kT qV
A L i
kT qV
A i
A i p
A
D i n
kT qV
D L i
kT qV
D i
D i n
L x kT
qV
A i
A i p p
p
L kT x
qV
D i
D i n n
n
L x kT
qV
A i p
L kT x
qV
D i n
A
D i n n
A i p p
n n
n n
N or n
x N p
e n N e
n N p n
kT qV N
or n x
n N e
e n N e
n N n n
kT qV N
or n x
p N e
e n N e
n N p n At
x e
N e n N n n n
n
x e
N e n N p n p
p
x e
N e x n
n
x e
N e x n
p V b
x as N n p p
x as N n n n
a
A p
A N
A
p A A
N A
A p N A
A p
2 2
/ /
2 2
2 2
0 / /
2 2
2
p /
2
1 / 0 /
2 2
2
n /
2
1 / / 0
2 2
/ /
2 2
0
/ / 2
2
0
/ /
2
/ / 2
2 0 2
0
ln ) x (
ln )
1 (
, x and x
At
ln ) x x 0 (
ln )
1 (
ln ) x x 0 ( ln )
1 (
0, x
0 ,
) 1 (
0 ,
) 1 (
) 25 . 6 ( 0 ,
) 1 (
) (
) 23 . 6 ( 0 ,
) 1 (
) (
: 0 )
(
bias.
applied the
of t independen /
and /
) (
L x kT qV N
x n n e
N e n N e n
N e n N n n
L x kT qV N
x n p e
N e n N e n
N e n N p n For
x e
N e n N n n n
n
x e
N e n N p n p
p
N A
A i p
L x kT qV
A i
A L i
x kT
qV
A i
A i p
p A
D i n
L kT x qV
D i
D L i
kT x qV
D i
D i n
L x kT
qV
A i
A i p p
p
L kT x
qV
D i
D i n n
n
N A
N A
A p A p
N A
A p
ln 1 ) ( ln )
1 (
ln 1 ) ( ln )
1 (
x and x
0 ,
) 1 (
0 ,
) 1 (
2 /
/ 2 2
/ /
2 2
2 / /
2 2
/ / 2
2
/ /
2 2
0
/ / 2
2
0
A L i
kT V q
A i
A i p
D L i
kT V q
D i
D i n
A L i
kT V q
A i
A i p
D L i
kT V q
D i
D i n
L x kT
qV
A i
A i p p
p
L kT x
qV
D i
D i n n
n A
N e n
N e n N n n
N e n
N e n N p n
N e n
N e n N n n
N e n
N e n N p n At
x e
N e n N n n n
n
x e
N e n N p n p
p V c
A N A p A N
A p
N A
A p
2
0 / /
2 2
2
0 / /
2 2
2
1 / 0
0 / 2
2
2
1 / 0
0 / 2
2
p n
/ /
2 2
0
/ / 2
2
0
) 1 (
) 1 (
, x and x
At
) 1 (
) 1 (
), x - or x (or x 0 x
0 ,
) 1 (
0 ,
) 1 (
: 0 )
(