 |
If you are not very familiar with
diffusion in general, it would be wise to consult some other Hyperscripts: |
|
 |
Basic diffusion in "Introduction to Materials
Science I" (at present in
German) |
|
|
Point defects and diffusion in "Defects in
Crystals" |
|
The diffusion of dopants is of course
one of the major topics in all process and device considerations. For any
modern Si technology you must be able to have exactly the right
concentration of the right dopant at the right place - with tolerances as small
as 1% in critical cases. |
|
|
And it is not good enough to assure the proper
doping right after the doping process -
what counts is only the dopant distribution in the finished device. |
|
|
Annoyingly, every time a high temperature process
is executed after one of the doping steps, all dopants already put in place will diffuse again,
and this must be taken into consideration. |
|
|
Even more annoying, the diffusion of the dopants
may depend on the process - it may, e.g., be different if other dopants are
present. |
|
|
A well-known example is the so-called emitter-dip
or emitter-push effect which makes it difficult to achieve very thin base
regions in bipolar transistors. The effect is due to a changed diffusion
coefficient of B in the presence of P; it is
covered in more detail in
an advanced module. |
|
The only way to master diffusion in
making devices is an extensive simulation of the concentration profiles as a
function of all parameters involved - always in conjunction with feed-back from
measurements. This requires a mathematical framework that can be based on three
qualitatively different approaches: |
|
|
Use equations that describe typical solutions to
diffusion problems and determine a sufficient number of free parameters
experimentally. Observed but poorly understood phenomena may simply be included
by adding higher order terms with properly adjusted parameters. This will
always work for problems within a certain range of the experimental parameters
for which the fit has been made - but not necessarily for other regions in
parameter space. |
|
|
Solve macroscopic diffusion equations matched to
the problem; i.e. equations of the type expressed in Ficks 1st and
2nd law. The input are the diffusion coefficients together with the
relevant boundary conditions. This works fine if you know the the dependence of
the diffusion coefficients on everything else (which you usually don't). |
|
|
Base the math on the proper atomic mechanisms. If
all mechanisms and interactions are fully known, they will contain all
informations and the results will be correct by necessity. Unfortunately, all
mechanisms and interactions are not fully known - neither in Si, nor in
all the other semiconductors. |
|
So none of theses approaches works
satisfactorily by itself - what is needed is a combination. |
|
|
In the eighties, e.g., it proved necessary to
include diffusion mechanisms mediated by Si self-interstitials; a
diffusion mechanisms not observed in most other materials. |
|
|
This would be not necessary for
"simple" diffusion as expressed in Ficks laws with a constant
diffusion coefficients - regular vacancy or interstitial mechanisms are not
distinguishable at this level. |
|
|
Special effects, however, may occur and it is far
easier to include these effects if the additional mathematical terms reflect
the atomic mechanisms - the alternative is to add correction terms with
adjustable parameters. |
|
In any case, diffusion in Si
(and the other semiconductors) is complicated and an issue of much research and
debate. It has become extremely important to include all possible
"classical" effects usually neglected because very high precision is
needed for very short diffusion times (or penetration depth), but the atomic
mechanisms of diffusion in Si are still not entirely clear. |
|
|
In what follows a few basic facts and data will
be given; in due time some advanced modules with more specific items may
follow. |
|
Basic equations are the two
phenomenological laws known as "Ficks laws"
which connect the (vector) flux j of diffusion particles to the
driving force and describe the local change in particle density, r(x,y,z,t) and the
Einstein-Smoluchowski
relations which connect Ficks laws with the atomic mechanisms of
diffusion. Ficks first and second law are |
|
First
law: |
|
|
|
|
|
|
|
|
|
|
|
With c = concentration of the
diffusing particles, D = diffusion constant and Ñ = Napla operator. We have |
|
|
|
|
|
| Ñc |
= vector = |
æ
ç
è |
¶c
¶x |
, |
¶c
¶y |
, |
¶c
¶y |
ö
÷
ø |
|
|
|
|
|
|
Second
law:
|
|
|
|
|
|
|
|
|
|
|
With D = Delta
operator (= Ñ2), and
Dc given by |
|
|
|
|
|
| Dc |
= |
¶2c
¶x2 |
+ |
¶2c
¶y2 |
+ |
¶2c
¶ z2 |
|
|
|
|
|
|
An atomic view of diffusion
considering the elementary jumps of diffusing atoms (or vacancies) over a
distance a (closely related to the lattice constant) yields not
only a justification of Ficks laws,
but the
relations |
|
|
|
|
| D |
= |
g · a2 · n |
|
| |
|
|
|
| n |
= |
n0 · exp – |
EM
kT |
|
|
|
|
|
|
With g = geometry factor describing
the symmetry of the situation, i.e. essentially the symmetry of the lattice,
and n = jump frequency of the diffusion
particle, EM = activation enthalpies of
migration. |
|
If the diffusion mechanisms involves
intrinsic point defects as vacancies (V) or self-interstitials
(i), their concentration is given by |
|
|
|
|
|
|
|
|
|
|
|
With EF = formation
enthalpy of the point defect under consideration. |
|
The problem may get
complicated if more than one atomic mechanism is involved. A relevant example
for Si is the so-called "kick-out" mechanism
for extrinsic point defects (= impurities): |
|
|
A foreign atom (most prominent is Au)
diffuses rather fast as interstitial impurity, but on occasion "kicks
out" a lattice atom and then becomes substitutional and diffuses very
slowly. However, the substitutional Au atom may also be kicked out by
Si interstitials and then diffuses fast again. An
animation of
this process can be seen in the link. |
|
|
The "kick-out" process is not adequately described by the simple version of
the Fick equation given above. |
|
Since even the simple Fick equations
are notoriously difficult to solve even for simple cases, not to mention
complications by more involved atomic mechanisms, only the two most simple
standard solutions shall be briefly discussed. |
|
|
|
Consider the following
situation: |
|
|
On the surface of a Si crystal the
concentration c0 of some dopant species is kept
constant - e.g. by immersing the Si in a suitable gas with constant
pressure or by depositing a thick layer of the substance on the surface. |
|
|
The dopant will then diffuse into the Si
and since the source of dopant atoms is the surface, there will be a drop in
concentration of the dopant from the value c0 at the
surface to zero deep in the crystal. |
|
|
Independent of the dopant concentrations outside
the Si, the maximum concentration in the Si next to surface
cannot be larger than the solubility of the dopant atom an the temperature
considered; we take c0 than as solubility limit. |
|
The general one-dimensional solution
of the differential equations of Ficks laws for this boundary condition of an
inexhaustible source then is given by |
|
|
|
|
|
|
|
|
|
|
|
With L = 2(D ·
t)1/2 = diffusion length, and erfc(x) =
complementary errorfunction = 1 – erf(x) and erf
(z) = errorfunction given by
|
|
|
|
|
|
| erf (z) |
= |
2
p1/2 |
· |
z
ó
õ
0 |
exp – a2 · da |
|
|
|
|
|
|
|
The errorfunction can not be written in closed
form; its values, however are tabulated. A typical solution of the diffusion
problem may look like this: |
|
|
|
|
|
|
|
|
|
|
|
The error functions and
real solutions to the
"infinite source" diffusion situation can be found in an advanced
module comprising an JAVA simulation tool where you can generate solutions to
your own problems. |
|
The interesting quantity is the
diffusion length L which is a direct measure of how far the
diffusion particles have penetrated into the Si. At a distance
L from the surface, the concentration of the dopant is about
1/2 c0 or, to be exact 0,4795 ·
c0. |
|
|
This diffusion length
for dopants or any other kind of atoms is not to be confused with
the diffusion length of
minority carriers as introduced before. Of course, the physics is exactly
the same - the diffusion length for electron and holes as introduced before
could just as well be obtained from solving the Fick equations for these
particles. |
|
|
Note in passing that while all
definitions of diffusion lengths contain the (D ·
t)1/2 term, the factor 2 (or on occasion
21/2) may or may not be there, depending on the exact
solution - but this is of little consequence for qualitative discussions. |
|
The total quantity of dopant atoms
now in the Si expressed as a concentration
ctotal can be obtained by integrating the "diffusion profile", i.e. the curve of
the concentration versus depth. This is analytically possible if the
integration runs from 0 to ¥ - a
very good approximation for slowly diffusing atoms and thick wafers. The result
is |
|
|
|
|
|
| ctotal |
= |
L · c0
p1/2 |
= |
0,56 · L · c0 |
|
|
|
|
|
|
The other standard solution for
diffusion problems deals with the case of a finite
source; i.e. only a limited amount of diffusion particles is
available. |
|
|
This is the standard case for, e.g.
ion
implantation, where a precisely measured number of dopant atoms is
implanted into a surface near area. |
|
|
For simplifying the math, we may assume that
these dopants are all contained in one atomic layer - a delta function type
distribution at the surface. |
|
|
This is of course not true for a real ion
implantation, where there is some depth distribution of the concentration below
the implanted surface, but as long as the diffusion length obtained in this
case is much larger than the distribution width after implantation, this is a
good approximation. |
|
It is more convenient to resort from
volume concentrations cof atoms to
areal densities C because that is
what an ion implantation measures: the total number of P-, As- or
B-atoms shot into the wafer per cm2 called the
dose = atoms/cm2. With
C0 = implanted dose and C(x, t)
the area density in the Si, the following solution is obtained: |
|
|
|
|
|
| C(x,t) |
= |
æ
ç
è |
C0
p·D·t |
ö
÷
ø |
1/2 |
· exp – |
x2
4D t |
|
|
|
|
|
|
|
This is simply one half of a
Gaussian distribution (the "–" sign in front of
x2 takes care of this) with a "half-width"
of (Dt)1/2; what it looks like is shown in the
picture. |
|
|
|
|
|
|
|
|
|
|
|
The curves can be characterized by a
(Dt)1/2 product, which again gives a typical diffusion length. |
|
|
This looks reasonable;
simulation module allows to
practice with real cases. |
|
The quantity of prime importance is
always the diffusion coefficient of the diffusing particle. Only for
"simple" mechanisms it is a simple function of the prime parameters
of the point defect involved as implicitly stated
above. |
|
|
D(T) then follows a simple
Arrhenius kind of behavior; examples for the common dopants are shown in the
figure: |
|
|
|
|
|
|
|
|
The lines shown are perfect straight lines over
more than 8 orders of magnitude - provided there are no complications. |
|
The example of an ion-implanted layer
as the source for diffusion, however, provides a good example for some of the
complications that may be encountered in real Si diffusion: |
|
|
First of all,
if the distribution of implanted dopant atoms cannot be treated as a delta
function, but must be taken into account as it is. Solutions then can only be
obtained numerically - with some effort. |
|
|
Second, if
only a small area has been implanted through a mask, at least a two-dimensional
problem must be solved - which is much more complicated. |
|
|
Third, some
dopant atoms will reach the surface after some random walk. The idealized
solution assumes that they will go back into the bulk, i.e. the surface does
not act as sink for diffusion atoms. This is, however, not always true and will
lead to complications. |
|
|
Fourth, while
all of the above still only amounts to a mathematical exercise in solving Ficks
differential equations, there are physical problems, too: Ion implantation
produces lots of surplus vacancies and interstitials which will become mobile
during the diffusion procedure. The point defect concentration at the diffusion
temperature thus is not identical to the
equilibrium concentration (at least for some time), and the diffusion
coefficient which always reflects the underlying atomic mechanism for
equilibrium conditions, will be changed and become time dependent - a very
messy situation! |
|
In fact, the usual goal after ion
implantation is to keep the implanted profile in place as much as possible - no
diffusion would just be great. But you must get rid of the crystal lattice
defects produced by the implantation and for that you must anneal at elevated
temperatures for some time - and diffusion will take place! |
|
|
What is better: Long anneals at low temperatures
or short anneals at high temperatures to remove the defects but keep your
dopants in place. Not an easy question; the answer must depend on the kinetics
of the defect annealing and the diffusion peculiarities of the atom under
consideration. |
|
|
However, the second case is usually preferred,
and a whole industry has developed around this point under the catch phrase
"rapid thermal annealing or
rapid thermal processing (RTA or
RTP, respectively). |
|
But there are more complications
yet: |
|
|
The diffusion of an atom may be changed if there
are noticeable concentrations of other foreign atoms around - and this includes
the own species. P, as an example, diffuses faster in large
concentrations and also enhances the diffusivity of B (see
emitter push effect). |
|
|
Some processes (notably thermal oxidation)
produces non-equilibrium point defects (oxidation produces Si
interstitials) which will be felt by atoms diffusing via these point defects -
their diffusivity will be different if the Si is oxidized compared to an
inert surface. |
|
|
Some atoms, as already mentioned above, diffuse
by more complicated mechanisms, e.g. the kick-out
mechanism. In a treatment with Ficks equations, this calls for two
superimposed mechanisms, each with its own diffusion constant and some boundary
conditions to assure particle conservation etc. |
|
|
A review from 1988 (which almost certainly
will have been contested in the meantime in some points) covering just
fast
diffusing elements in Si and discussing some of the complications mentioned
above, is provided in the link. |
|
|
Some
prominent cases of deviations
from simple diffusion behavior can be found in an advanced module |
|
It should come as no surprise than
that diffusion in Si, as far as the application to devices is concerned,
is an active area of research and development, and that no process engineer
will ever believe the results of a simulation for diffusion under a new set of
conditions without experimental verification. |
© H. Föll
