Diffusion in Solids: Point sources, infinite sinks

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Diffusion in SolidsPoint sources, infinite
sinks
27-216 Spring 2004 A. D. Rollett
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Objectives

• Use case studies to illustrate different types of
  solutions.
• Study the case of a point source coupled with an
  infinite sink for the diffusant.
• Study the case of a finite source, coupled with
  an infinite sink.
• Later lectures finite sources, finite sinks.

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Overview

• It is useful to keep in mind that there are
  various characteristic classes of problem. Each
  problem class has a generic set of boundary
  conditions and a corresponding type of equation
  as its solution (generally to Ficks 2nd Law).
• 1D steady state linear for 1D Cartesian
  logarithmic for 1D radial.
• 1D finite point source, infinite sink Gaussian
  (exp-x2)
• 1D infinite source, infinite sink(s) error
  function, erf or complementary error function,
  erfc
• 1D, finite sink series with both exponential and
  cosine functions (in each term).

Examinable
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So far
Examinable
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Transient Case 1 Finite thin film source
continued
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Case Study doping of semiconductor silicon
In order to make a electronic component, Boron
has to be implanted into Si. In order to do this,
a thin film of Boron is deposited on the surface
of the 2 mm thick silicon wafer as shown below.
The thickness of the film is 5 ?m and Boron
concentration is 105 atoms/m3. Diffusion
coefficient of B in Si is 10-10 m2/s.   What
concentration of Boron would there be at a depth
of 1 mm in the Si wafer after 1 hr?              
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How to solve the problem?

• In general, one simply looks up the appropriate
  solution. However, in this case, we will examine
  the solution procedure because it is useful to
  know about it.
• Then we examine the boundary conditions.
• Finally we can apply the solution to the
  particular problem.
• As always, beware of units in diffusion problems
  here we deal with atom densities.

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Concentration profiles
t0
C(x,t)
x
Boundary Conditions
Examinable
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BCs
Solve through Laplace transformation
Examinable
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At a later time

• Does this solution fulfill our boundary and
  initial conditions?
• How does the flux change with time and position?
• How does the rate of Accumulation change with
  time and position?
• Is this solution valid forever?

Examinable
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Boron doping of Si

• First lets define the junction depth as the
  diffusion penetration depth for a given amount of
  time at which the concentration from diffusion
  alone is equivalent to the initial concentration
  of the diffusant.
• Calculating the junction depth requires an
  approximation to the thin film equation in which
  we ignore the initial concentration.

• Choosing particular conditions of B in Si PG,
  p483 T1150C (1423K) anneal time
  2hr7200sfrom Table 13.3, DB0.16
  µm2.hr-14.44.10-19m2.s-1Cinitial5.1021atoms.m-
  3 C04.5.1026atoms.m-3.

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Boron doping of Si, contd.

• Invert the equation

Therefore the junction depth, xjunction, 3.19
µm To obtain the actual concentration at this
depth, we have to use the following more exact
equation
Naturally, the surface concentration at any
given time can also be calculated with the same
equation, setting x 0.
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Deriving one solution from another
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Instantaneous planar diffusion source in an
infinite medium
Initial state
These diffusion problems concern placing a finite
amount of diffusant that spreads into the
adjacent semi-infinite solid.
Final state
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Instantaneous planar diffusion source in an
infinite medium
These diffusion problems concern placing a finite
amount of diffusant that spreads into the
adjacent semi-infinite solid.
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Transform Methods
where
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Laplace Transform of the Diffusion Equation
Linear Diffusion Equation
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Laplace Transforms
The Laplace transform of Ficks second law when
C(x,0)0
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Instantaneous planar diffusion source
Application of the Laplace transform to Ficks
second law gives
The diffusion process is subject to the mass
constraint for a unit area
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Instantaneous planar source
Reduction of the Laplace transform
The general solution for which is
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Instantaneous planar source
Mass constraint for the field
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Instantaneous planar source solution
The transform solution
M units must be compatible with units of C,
taking note of the vDt in the denominator. For
example, M can be mass per unit area (e.g.
Cdepth).
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Normalized plot of the planar source solution
C (x) M length -1
x distance
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Transient Case 2 Large source large
destination
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Follow this outline

• Choose a coordinate system
• Find boundary and initial conditions
• Decide if the problem is transient or steady
  state
• If transient, solve PDE (Ficks 2nd law)
• Solve analytically
• Find solution in diffusion/heat transfer/math
  textbooks
• Solve numerically
• In this case, we treat the problem as a series of
  thin film sources, all of equal strength, that
  are spaced out along the bar. We then superpose
  the solutions for each source.

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Superposition of solutions
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DIFFUSION COUPLE IN AN AIRCRAFT
I need to predict how much Ni I am losing from
the superalloy. T 500oC. Assume that properties
of the steel and superalloy are quite similar
with DNi 10-5 cm2/s at 500oC.
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How do we measure a diffusion coefficient?
Answer perform a thin-film diffusion penetration
experiment, measure the concentration profile,
and derive the coefficient by assuming the
solution.
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Thin-film configuration
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Procedure for Analysis of Thin-Film Data
Take logs of both sides of
A plot of lnC versus x2 yields a slope-1/4Dt
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Thin-Film Experiment
Geiger counter data after microtoning 25 slices
from a thin-film specimen in which a radioactive
species was diffused in to the specimen.
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Log Concentration versus x2
Slope-1/4Dt
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Infinite source, infinite sink

• What if we have a material that is exposed to a
  source of diffusant in a situation where the
  concentration is constant at one surface?
• This corresponds to systems where a chemical
  reaction at the surface maintains a certain
  composition, e.g. carburization of steels, rapid
  diffusion into a grain boundary (and then
  diffusion into the bulk from the boundary).
• This is known as the infinite source- infinite
  sink problem, for cases where the penetration
  distance into the material is small compared to
  the dimensions of the material.

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Initial state
Linear diffusion into a semi-infinite medium
Final state
Examinable
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Boundary Conditions
The diffusion equation is a 2nd-order PDE and
requires two boundary or initial conditions to
obtain a unique solution.
1) Initial state C 0, for x gt 0, t 0.
2) Left-hand boundary At x 0, C0 is
maintained for all t gt 0.
Examinable
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Laplace Transforms
Transform of the boundary condition at the left
hand edge
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Laplace Transforms
The concentration field associated with the image
field is found by inverting the transform either
by formal means, a look-up table, or using a
computer-based mathematics package.
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Estimation of the Error Function
For small arguments
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Estimation of the Error Function
Piecewise approximations for restricted ranges
of the argument
Useful for spreadsheet calculations.
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Error Function
Examinable
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Complementary Error Function
Examinable
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Initial state
Linear diffusion into a semi-infinite medium
Final state
Examinable
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Concentration versus the similarity variable
DimensionlessVariable
Examinable
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Concentration field versus distance
?  2(Dt)1/2 is the time tag
(Note ? has the units of distance!)
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Diffusion Penetration X
X K t1/2
Relative Conc. C/C0
Relative Conc. C(x)/C0
Distance, x units of ?1
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Penetration versus square-root of time
0.4
0.6
Penetration Distance, X(C/C0)
0.8
0.9
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Summary
Examinable