 |
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. |
| 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 |
| |
| Ñ c | = vector =
| æ
ç
è
| ¶c
¶x | , | ¶c
¶
y | , | ¶
c
¶y |
ö
÷
ø | |
|
|
Second law:
|
| | |
| | With D = Delta operator (= Ñ2 ), and Dc
given by |
| |
| D c | =
| ¶ 2c
¶x2 | + |
¶ 2c
¶y2 | + |
¶2 c
¶ 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. |
| |
| Diffusion from an Unlimited Surface Source |
| | |
|
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 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. |
|
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 (a key word is: "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 (Semiconductors - Script)
