Transcript THERMAL PROCESSES

Fick’s Laws

Combining the continuity equation with the first
law, we obtain Fick’s second law:
 c
 c
  D 2
 t
x
2

Solutions to Fick’s Laws depend on the
boundary conditions.

Assumptions
– D is independent of concentration
– Semiconductor is a semi-infinite slab with either
 Continuous supply of impurities that can move into
wafer
 Fixed supply of impurities that can be depleted
Solutions To Fick’s Second Law

The simplest solution is
at steady state and there
is no variation of the
concentration with time
– Concentration of diffusing
impurities is linear over
distance

This was the solution for
the flow of oxygen from
the surface to the Si/SiO2
interface in the last
chapter
c
D 2 0
x
2
c( x)  a  bx
Solutions To Fick’s Second Law

For a semi-infinite slab with a constant
(infinite) supply of atoms at the surface
 x 
c( x, t )  co erfc

 2 Dt 

The dose is

Q   cx, t dx  2c0 Dt 
0
Solutions To Fick’s Second Law
Complimentary error function (erfc) is
defined as erfc(x) = 1 - erf(x)
 The error function is defined as

erf ( z ) 
2

 exp d
z
2
0
– This is a tabulated function. There are
several approximations. It can be found as a
built-in function in MatLab, MathCad, and
Mathematica
Solutions To Fick’s Second Law
This solution models short diffusions from
a gas-phase or liquid phase source
 Typical solutions have the following shape
Impurity concentration, c(x)

c0
c ( x, t )
D3t3 > D2t2 > D1t1
1
2
cB
Distance from surface, x
3
Solutions To Fick’s Second Law

Constant source diffusion
has a solution of the form

Here, Q is the does or the
total number of dopant
atoms diffused into the Si
Q
c( x, t ) 
e
Dt

Q   c ( x, t )dx
0

The surface concentration
is given by:
Q
c(0, t ) 
Dt
 x2
4 Dt
Solutions To Fick’s Second Law
Limited source diffusion looks like
Impurity concentration, c(x)

c01
c ( x, t )
c02
D3t3 > D2t2 > D1t1
c03
1
2
cB
Distance from surface, x
3
Comparison of limited source
and constant source models
1
10-1
_
exp(-x 2 )
Value of functions
10-2
_
erfc( x)
10-3
10-4
10-5
10-6
0
0.5
1
_ 1.5_
Normalized distance from surface, x x 
2 x
2 Dt
2.5
3
3.5
Predep and Drive

Predeposition
– Usually a short diffusion using a constant
source

Drive
– A limited source diffusion
 The diffusion dose is generally the dopants
introduced into the semiconductor during the
predep

A Dteff is not used in this case.
Diffusion Coefficient

Probability of a jump is
Pj  Pv  Pm
e
 E f kT
 e  Em
kT
Diffusion coefficient is proportional to jump
probability
D  D0e
 E D kT
Diffusion Coefficient

Typical diffusion coefficients in silicon
Element Do (cm2/s) ED (eV)
B
10.5
3.69
Al
8.00
3.47
Ga
3.60
3.51
In
16.5
3.90
P
10.5
3.69
As
0.32
3.56
Sb
5.60
3.95
Diffusion Of Impurities In Silicon

Arrhenius plots of diffusion in silicon
Temperature (o C)
10-9
1400 1300 1200 1100
Temperature (o C)
1000
1200 1100 1000 900
10-4
800
700
10-10
Diffusion coefficient, D (cm2/sec)
Diffusion coefficient, D (cm2/sec)
10-5
10-11
10-12
Al
10-13
In
0.7
10-7
10-8
0.6
As
0.65
Cu
Ga
Sb
0.6
Fe
Au
B,P
10-14
Li
10-6
0.75
0.7
0.8
0.9
Temperature, 1000/T
0.8
Temperature, 1000/T (K-1)
0.85
1.0
(K-1)
1.1
Diffusion Of Impurities In Silicon

The intrinsic carrier concentration in Si is
about 7 x 1018/cm3 at 1000 oC
– If NA and ND are <ni, the material will behave
as if it were intrinsic; there are many practical
situations where this is a good assumption
Diffusion Of Impurities In Silicon

Dopants cluster into “fast” diffusers (P, B,
In) and “slow” diffusers (As, Sb)
– As we develop shallow junction devices, slow
diffusers are becoming very important
– B is the only p-type dopant that has a high
solubility; therefore, it is very hard to make
shallow p-type junctions with this fast diffuser
Limitations of Theory

Theories given here break down at high
concentrations of dopants
– ND or NA >> ni at diffusion temperature

If there are different species of the same atom
diffuse into the semiconductor
– Multiple diffusion fronts
 Example: P in Si
– Diffusion mechanism are different
 Example: Zn in GaAs
– Surface pile-up vs. segregation
 B and P in Si
Successive Diffusions

To create devices, successive diffusions of nand p-type dopants
– Impurities will move as succeeding dopant or
oxidation steps are performed

The effective Dt product is
( Dt)eff  D1 (t1  t2  )  D1t1  D1t2  
– No difference between diffusion in one step or in
several steps at the same temperature

If diffusions are done at different
temperatures
( Dt)eff  D1t1  D2t2  
Successive Diffusions

The effective Dt product is given by
Dteff   Di ti
i
Di and ti are the diffusion coefficient and time
for ith step
– Assuming that the diffusion constant is only a
function of temperature.
– The same type of diffusion is conducted (constant
or limited source)
Junction Formation

When diffuse n- and p-type materials, we
create a pn junction
– When ND = NA , the semiconductor material is
compensated and we create a metallurgical
junction
– At metallurgical junction the material behaves
intrinsic
– Calculate the position of the metallurgical
junction for those systems for which our
analytical model is a good fit
Junction Formation
Formation of a pn junction by diffusion
Impurity
Net impurity
concentration
|N(x) - NB |
concentration
N(x)
N0
(log scale)
N0 - NB
p-type Gaussian diffusion
(boron)
n-type silicon
(log scale)

p-type
region
background
NB
n-type region
xj
Distance from surface, x
xj
Distance from surface, x
Junction Formation

The position of the junction for a limited
source diffused impurity in a constant
background is given by
x j  2 Dt ln N 0 N
B

The position of the junction for a
continuous source diffused impurity is
given by
1 N
x j  2 Dt erfc
B
N0
Junction Formation
Junction Depth
Lateral Diffusion
Design and Evaluation

There are three parameters that define a
diffused region
– The surface concentration
– The junction depth
– The sheet resistance
 These parameters are not independent

Irvin developed a relationship that describes
1
these parameters  S  1 
x
 xj
q  n( x)  N B  n( x)dx
j
0
Irvin’s Curves

In designing processes, we need to use all
available data
– We need to determine if one of the analytic
solutions applies
 For example,
– If the surface concentration is near the solubility limit,
the continuous (erf) solution may be applied
– If we have a low surface concentration, the limited
source (Gaussian) solution may be applied
Irvin’s Curves

If we describe the dopant profile by either the
Gaussian or the erf model
– The surface concentration becomes a parameter in
this integration
– By rearranging the variables, we find that the surface
concentration and the product of sheet resistance and
the junction depth are related by the definite integral
of the profile

There are four separate curves to be evaluated
– one pair using either the Gaussian or the erf
function, and the other pair for n- or p-type materials
because the mobility is different for electrons and
holes
Irvin’s Curves
Irvin’s Curves

An alternative way of presenting the data
may be found if we set eff=1/sxj