49
Review of Basic Device Physics
2.2.7.4 Transport Equations
In Section 2.2.5.5, we have shown that the electron diffusion current
density J n,diff due to concentration gradient in a semiconductor is given
by Equation 2.40. On the other hand, the electron current density due to
drift of electrons by an applied electric field described in Section 2.2.5.2 is
given by Equation 2.30. Thus, when an electric field is present in addition
to a concentration gradient, both the drift and diffusion current will flow
through the semiconductor. The total electron current density J n at any
point x is then simply the sum of the diffusion and drift currents, that is,
J n (=J n,drift + J n,diff ). Therefore, the total electron current in a semiconductor
is given by
J
qn E qD
dn
dx
n
n
n
=
+
µ
(2.76)
Similarly, the total hole current density J p (=J p,drift + J p,diff ) is given by
J
qp E qD
dp
dx
p
p
p
=
−
µ
(2.77)
so that the total current density J = J n + J p . The current Equations 2.76 and 2.77
are often referred to as the transport equations.
Under thermal equilibrium no current flows inside the semiconductor and
therefore, J n = J p = 0. However, under nonequilibrium conditions J n and J p can
be written in terms of quasi-Fermi potentials f n and f p for electric field, E, in
Equations 2.76 and 2.77, respectively, to get
J
qn
d
dx
J
qp
d
dx
n
n
n
p
p
p
= −
= −
µ
µ
φ
φ
(2.78)
2.2.7.5 Continuity Equations
When carriers diffuse through a certain volume of semiconductor, the current density leaving the volume may be smaller or larger depending upon
the recombination or generation taking place inside the volume. Let us consider a small length Δx of a semiconductor as shown in Figure 2.13 with
cross-sectional area A in the yz plane.
From Figure 2.13, the hole current density entering the volume A.Δx is
J p (x) whereas the density leaving is J p (x + Δx). From the conservation of
charge, the rate change of hole concentration in the volume is the sum of
(1) net holes flowing out of the volume and (2) net recombination rate. That is,
Review of Basic Device Physics
2.2.7.4 Transport Equations
In Section 2.2.5.5, we have shown that the electron diffusion current
density J n,diff due to concentration gradient in a semiconductor is given
by Equation 2.40. On the other hand, the electron current density due to
drift of electrons by an applied electric field described in Section 2.2.5.2 is
given by Equation 2.30. Thus, when an electric field is present in addition
to a concentration gradient, both the drift and diffusion current will flow
through the semiconductor. The total electron current density J n at any
point x is then simply the sum of the diffusion and drift currents, that is,
J n (=J n,drift + J n,diff ). Therefore, the total electron current in a semiconductor
is given by
J
qn E qD
dn
dx
n
n
n
=
+
µ
(2.76)
Similarly, the total hole current density J p (=J p,drift + J p,diff ) is given by
J
qp E qD
dp
dx
p
p
p
=
−
µ
(2.77)
so that the total current density J = J n + J p . The current Equations 2.76 and 2.77
are often referred to as the transport equations.
Under thermal equilibrium no current flows inside the semiconductor and
therefore, J n = J p = 0. However, under nonequilibrium conditions J n and J p can
be written in terms of quasi-Fermi potentials f n and f p for electric field, E, in
Equations 2.76 and 2.77, respectively, to get
J
qn
d
dx
J
qp
d
dx
n
n
n
p
p
p
= −
= −
µ
µ
φ
φ
(2.78)
2.2.7.5 Continuity Equations
When carriers diffuse through a certain volume of semiconductor, the current density leaving the volume may be smaller or larger depending upon
the recombination or generation taking place inside the volume. Let us consider a small length Δx of a semiconductor as shown in Figure 2.13 with
cross-sectional area A in the yz plane.
From Figure 2.13, the hole current density entering the volume A.Δx is
J p (x) whereas the density leaving is J p (x + Δx). From the conservation of
charge, the rate change of hole concentration in the volume is the sum of
(1) net holes flowing out of the volume and (2) net recombination rate. That is,