6.012  Electronic Devices and Circuits
Solving the 5 basic equations  9/17/09 Version
The 5 basic equations of semiconductor device physics:
We will in general be faced with finding 5 quantities:
n
(x,t)
, p
(x,t)
, J
e
(x,t)
, J
h
(x,t)
, and E
(x,t)
,
and we have five independent equations that relate them:
(1)
J
e
(x,t)
= q µ
e
n
(x,t)
E
(x,t)
+ q D
e
∂
n
∂
(
x
x,t)
(2)
J
h
(x,t)
= q µ
h
p
(x,t)
E
(x,t)
 q D
h
∂
p
∂
(
x
x,t)
(3)
∂
n
∂
(
t
x,t)

q
1
∂
J
∂
e
(
x
x,t)
= g
L
(x,t)
 [n
(x,t)
p
(x,t)
 n
i
2
] r
(4)
∂
p
∂
(
t
x,t)
+
q
1
∂
J
∂
h
(
x
x,t)
= g
L
(x,t)
 [n
(x,t)
p
(x,t)
 n
i
2
] r
(5)
ε
∂
E
∂
(
x
x,t)
= q [p
(x,t)
 n
(x,t)
+ N
d
(x)
 N
a
(x)
]
where the assumptions we made getting these equations
are:
N
d
+
(x)
≈
N
d
(x)
, N
a

(x)
≈
N
a
(x),
R
(x,t)
≈
n
(x,t)
p
(x,t)
r
r,
ε
, µ
e
, µ
h
, D
e
, and D
h
are assumed to be independent
of position.
Temperature is assumed to be constant (isothermal).
(Note: n
i
, r, µ
e
, µ
h
, D
e
, and D
h
all depend on temperature).
1