Taking the derivative with respect to
x
x of Fick's first law
ddxFlux=−(D∂2Nxt∂x22)
x
Flux
D
x2
Nxt
(1) and then substituting the continuity equation into it,
we have
Fick's second law of diffusion:
∂Nxt∂t=D∂2Nxt∂x22
t
Nxt
D
x2
Nxt
(2) This is a standard diffusion equation, and one which
shows up over and over again when one is dealing with such phenomena.
In order to get a solution to the diffusion equation, we must
first assume some boundary conditions. We will deal with a
semi-infinite wafer, and assume that
limit
x
→
∞
Nxt=0
x
Nxt
0
(3)
This is a reasonable assumption, since at most our diffusion will only
penetrate a micron or so into the wafer, and the whole wafer itself is
several hundred microns thick.
We also have to decide something about initial conditions. We
will make the assumption that we have at time
t=0
t0 and
x=0
x0
some surface concentration of impurities which we will call
Q0
Q0(
impuritiescm2
impurities
cm
2
). This is the situation we would have if we introduce
the impurities using a relatively shallow implant step. An
alternative surface boundary condition would be one where the
concentration of impurities remains at some fixed value. This is
what happens when there are impurities in the gas flow over the
wafer during the time that they are in the diffusion oven. This is
called an infinite source diffusion.
The first condition is called a limited source
diffusion and that is what we shall consider further
here. It is not too hard to show that with this initial condition, the
solution to the diffusion equation is:
Nxt=
Q
0
πDte−x24Dt
N
x
t
Q
0
D
t
x
2
4
D
t
(4) Note that
Nxt
Nxt is a function of distance into the wafer, and time
tt. The time is, of course, the
time of the diffusion process.
DD, the diffusion constant, depends on
the temperature at which the diffusion takes place. Figure 1 is a plot of DD for three of the most
commonly used dopants in silicon. Phosphorus and boron are the
most common acceptor and donor respectively. Arsenic is
sometimes used because it is significantly bigger in diameter
than either P or B and thus, moves around less after an
implant.
Suppose we do a relatively shallow implant of boron
into our p-type wafer, and deposit a
Q0
Q0 of 5×1013
51013
phosphorus
atomscm2
atoms
cm
2
. We then perform an anneal diffusion at 1100 °C
for 60 minutes. At 1150°C, DD for
phosphorus seems to be about
2×10-13
cm2sec
2
10-13
cm
2
sec
. We will make a plot of
Nx
N x
for various times. If you do this at home, be sure to
put time in seconds, not minutes, hours, or fortnights. Looking
at Figure 1, is pretty easy to see
how the impurities move into the semiconductor, and how the
concentration at the surface,
N0t
N0t, decreases as more and more of the impurities moves
deeper into the wafer.
If the substrate had been doped at
1016
acceptorscm3
1016
acceptors
cm3
where would be the location of the p-n junction
between the implanted phosphorus layer, and the background
boron?
About 1.2 μm after 1 hour of
diffusion time. You know this because for
x<1.2
μm x1.2μm the phosphorus concentration is
greater than that of boron, and so the material is
n-type. For
x>1.2
μm
x1.2μm, the boron concentration exceeds
that of the phosphorous, and so the material is now p-type.)
