2
2.1
Diffusion and Current Flow
The Diffusion Process
The
second
principal
mechanism
of
current
flow
in
semiconductors is diffusion. If the concentration of free charge
carriers at a particular location in a semiconductor is higher than at
other surrounding locations, then the atomic forces which prevail
cause the carriers to migrate away from the region of higher
concentration towards those of lower concentration. This process is
known as diffusion and gives rise to a current known as diffusion
current. This is illustrated in Figs. 2.1, 2.2 and 2.3, which shows the
changing concentration of carriers with time after a quantity of
carriers has been released at a location.
Note that the movement of carriers is always in the direction of
decreasing concentration. The diffusion of carriers will continue as
long as the concentration gradient is maintained. The resulting current
will flow in a direction, which depends on the direction of the
concentration gradient and the charge on the carrier.
2.2 Diffusion Current
The rate at which diffusion takes place determines the charge flux
density due to diffusion. This depends on:
dn dp
,
dx dx
the carrier concentration gradients,
− q, + q
the charge on the carriers,
Dn , D p
the diffusion coefficients
The diffusion coefficients are a measure of the relative ease with
which the carriers can move within the concentration gradients. The
dimensions of these coefficients are cm2s-1 (area/unit time).
1
Fig. 2.1
The Process of Diffusion
t = t0
t1 > t0
t2 > t1
t3 > t2
t4 > t3
0
Fig. 2.2
distance
Changes in Carrier Concentration with Time During Diffusion
2
n
n
t = t2 > t1
t = t1
Carrier flow
Carrier flow
x
Fig. 2.3 Change in Concentration Gradient with Diffusion
x
The charge flux density for diffusion is then:
J n diff = -q .D n . -
dn
dn
= qD n
dx
dx
for electrons
J p diff = + q .D p . -
dp
dp
= − qD p
dx
dx
for holes
The negative signs in front of the concentration gradients above
account for the fact that the carriers move in the direction of
decreasing concentration. Note that if the concentration gradients for
both holes and electrons are in the same direction, then both types of
carrier will be travelling in the same direction and the charge flows
will tend to cancel each other. If, on the other hand, the gradients are
in opposite directions, then the flows of charge of opposite sign will
tend to add together.
For a piece of homogeneous semiconductor of uniform crosssectional area, A, the total diffusion current due to both electrons and
holes in the positive x direction is given as:
I diff = J n diff A + J p diff A = Aq(D
n
dn
dp
− Dp
)
dx
dx
The polarity of the gradients will determine the direction of flow of
individual carriers and, hence, the overall diffusion current.
3
2.6 The Einstein Relation
The diffusion coefficient is related to the mobility of the carriers by the
Einstein Relation:
Dn
kT
=
µn
q
Dp
µp
=
∴ Dn = µ n
kT
q
where VT
∴
=
kT
q
kT
= µ n VT
q
Dp = µ p
for electrons
kT
= µ p VT
q
for holes
= 26mV @ 300°°K is known as the thermal voltage.
The Thermal Voltage is the potential difference in an electric field
through which an electron with a charge of magnitude q must
accelerate to gain the same amount of energy as it has due to
temperature kT.
2.7 Carrier Injection, Diffusion and Charge Continuity
Consider a length of homogeneous p-type semiconductor of uniform
cross-sectional area, A, as shown in Fig. 2.4. Under equilibrium
conditions, the concentration of majority carriers is determined by the
doping concentration. If excess minority charge carriers are injected
into the semiconductor from an external source, this equilibrium is
disturbed as the excess minority carriers injected begin to recombine
with the majority carriers present. This gives rise to changes in the
charge concentration within the material.
The rate of change of carrier concentration depends on the rate
of injection of excess carriers and the rate of recombination in the
volume.
Notation: for electrons
Total carrier concentration: n
Equilibrium carrier concentration: n0
Excess Carrier Concentration: n’
so that:
Excess carrier concentration is:
4
n'= n− n0
excess minority
carriers recombine
with majority
carriers inside
volume of
semiconductor
A
uniform
cross
section
Jn (x)
charge flux density
becomes a function
of distance into
material
excess minority
carriers injected
from external
source
Fig 2.4
Excess Charge Carriers Injected into a Semiconductor
5
If there is no electric field across the semiconductor and the material
is homogeneous and uniform, then the injected minority carriers will
move only by diffusion in the x direction. The law governing this
process is known as the Diffusion Equation which for one dimension is
given as:
∂n' (x)
∂ 2n' (x) n' (x)
= Dn
−
∂t
τn
∂x 2
where τn is the average carrier lifetime before recombination. If the
supply of minority carriers into the material can be maintained by
some external means and the majority carriers which recombine can
also be replaced, then this will support the recombination process and
eventually a steady-state condition will be reached and the
concentration of charge within the volume will reach a stable value so
that
∂n' (x)
=0
∂t
Under these conditions…
∂ 2n' (x) n' (x)
=
2
D n τn
∂x
This is a second-order differential equation which has a general
solution of the form:
-x
n' (x) = Ae
Ln
+x
+ Be
Ln
Since recombination can only reduce the free carrier concentration in
the semiconductor there is no growth term so that B = 0. Hence the
solution is:
n' (x) = n' (x = 0)e
− Lx
n
where n’(x = 0) is the quantity of excess minority carriers injected into
the material at x = 0.
6
The minority carriers will penetrate a distance, Ln, on average into the
volume before recombining. This distance is equivalent to the depth of
penetration which would take place if the concentrations at the
boundary of the material were maintained. It is related to the average
carrier lifetime through the diffusion coefficient so that:
D n τn = L n
2
The profile of the minority carrier concentration across the material
into which they are injected therefore has an exponential form as
shown in Fig. 2.5. Note also that if a thin slice of the semiconductor is
taken, most of the injected carriers pass through the slice and emerge
at the other side but some have recombined. The quantity of minority
carriers which pass through the material depends on the thickness of
the slice and the doping concentration of the material.
n(x)
n(x = 0)
Thin slice
n’(x = 0)
n0
Ln
distance x
Fig. 2.5
Profile of Minority Carrier Concentration through the
Semiconductor
7