39
Review of Basic Device Physics
where:
the subscript n represents the parameters for electrons
As the electrons move (diffuse) away, they leave behind positively charged
donor ions (N d
+ ), which try to pull electrons back causing drift flux of
electrons from the low- to high-concentration region. This drift of electrons from low- to high-concentration regions sets up an electric field, E x
from the high-concentration to the low-concentration regions as shown in
Figure 2.9. Then from Equation 2.30, the flux due to the drift of electrons
is given by
F
n x v
n E
n drift
d
n x
,
=
=
( )
µ
(2.44)
An equilibrium is established when diffusion = drift. Here n(x) is the number of electrons in the diffusion flux at any point x in the distribution and
≠ N d (x). Therefore, a built-in electric field is established that prevents diffusion of electrons. Then from Equations 2.43 and 2.44, we get the expression
for the built-in electric field for electrons in an n-type nonuniformly doped
substrate as
E
D
n
dn x
dx
v n
dn x
dx
x
n
n
kT
= −
= −
µ
1
1
( )
( )
(2.45)
Similarly, the built-in electric field for holes in a nonuniform p-type substrate
is given by
E
D
p
dp x
dx
v p
dp x
dx
x
p
p
kT
=
=
µ
1
1
( )
( )
(2.46)
In Equations 2.45 and 2.46 we have used Einstein’s relation given in Equation
2.42. This built-in electric field favors the transport of the minority carriers if
created by an external source.
2.2.6 Generation–Recombination
In a semiconductor under thermal equilibrium, carriers possess an average
thermal energy corresponding to the ambient temperature. This thermal
energy excites some valence electrons to reach the CB. This upward transition of an electron from the VB to CB leaves behind a hole in the VB and an
electron–hole pair is created. This process is called the carrier generation (G).
On the other hand, when an electron makes a transition from the CB to the
VB, an electron–hole pair is annihilated. This reverse process is called carrier recombination (R). Under thermal equilibrium, G = R so that the carrier
concentration remains the same and the condition pn n i
=
2 is maintained.
The thermal G–R process is shown in Figure 2.10.
Review of Basic Device Physics
where:
the subscript n represents the parameters for electrons
As the electrons move (diffuse) away, they leave behind positively charged
donor ions (N d
+ ), which try to pull electrons back causing drift flux of
electrons from the low- to high-concentration region. This drift of electrons from low- to high-concentration regions sets up an electric field, E x
from the high-concentration to the low-concentration regions as shown in
Figure 2.9. Then from Equation 2.30, the flux due to the drift of electrons
is given by
F
n x v
n E
n drift
d
n x
,
=
=
( )
µ
(2.44)
An equilibrium is established when diffusion = drift. Here n(x) is the number of electrons in the diffusion flux at any point x in the distribution and
≠ N d (x). Therefore, a built-in electric field is established that prevents diffusion of electrons. Then from Equations 2.43 and 2.44, we get the expression
for the built-in electric field for electrons in an n-type nonuniformly doped
substrate as
E
D
n
dn x
dx
v n
dn x
dx
x
n
n
kT
= −
= −
µ
1
1
( )
( )
(2.45)
Similarly, the built-in electric field for holes in a nonuniform p-type substrate
is given by
E
D
p
dp x
dx
v p
dp x
dx
x
p
p
kT
=
=
µ
1
1
( )
( )
(2.46)
In Equations 2.45 and 2.46 we have used Einstein’s relation given in Equation
2.42. This built-in electric field favors the transport of the minority carriers if
created by an external source.
2.2.6 Generation–Recombination
In a semiconductor under thermal equilibrium, carriers possess an average
thermal energy corresponding to the ambient temperature. This thermal
energy excites some valence electrons to reach the CB. This upward transition of an electron from the VB to CB leaves behind a hole in the VB and an
electron–hole pair is created. This process is called the carrier generation (G).
On the other hand, when an electron makes a transition from the CB to the
VB, an electron–hole pair is annihilated. This reverse process is called carrier recombination (R). Under thermal equilibrium, G = R so that the carrier
concentration remains the same and the condition pn n i
=
2 is maintained.
The thermal G–R process is shown in Figure 2.10.