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  t 2  )  D1t1  D1t 2  
– 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  D2t 2  
Successive Diffusions

The effective Dt product is given by
Dt eff   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
Example

Design a B diffusion for a CMOS tub such that
s=900/sq, xj=3m, and CB=11015/cc
– First, we calculate the average conductivity
 
1
1
1




3
.
7


cm
 S x j 900/sq  3 104 cm
– We cannot calculate n or  because both are
functions of depth
– We assume that because the tubs are of moderate
concentration and thus assume (for now) that the
distribution will be Gaussian

Therefore, we can use the P-type Gaussian Irvin
curve to deduce that
Example
Reading from the p-type Gaussian Irvin’s curve,
CS4x1017/cc
 This is well below the solid solubility limit for B
in Si so we may conclude that it will be driven in
from a fixed source provided either by ion
implantation or possibly by solid state
predeposition
followed
by an etch
x

3 10 
Dt 

 3.7 10 cm
C
 4 10 to be at the required
 In order for
junction
4 ln the 4

ln 
C
10


depth, we can compute
the
Dt value from the
Gaussian junction equation

4 2
2
j
17
S
B
15
9
2
Example
This value of Dt is the thermal budget for the
process
 If this is done in one step at (for example) 1100
3.7 10 9 cm 2
-13cm2/s, the
C wheretdriveDinfor
is
1.5
x
10
 B in Si

6
.
8
hrs
1.5 10 13 cm 2 /s
drive-in time will be


Q  C (0, t ) Dt  4 1017

  

3.7 109  4.3 1013 cm- 2
Given Dt and the final surface concentration, we
can estimate the dose
Example
Let us also look at doing it by predep from the
solid state (as is done in the VT lab course)
 The text uses a predep temperature of 950 C
 In this case, we will make a glass-like oxide on
the surface that will introduce the B at the solid
solubility limit 2C
Q
Dt

 At 950 C, the solubility
limit is 2.5x1020cm-3 and
D=4.2x10-15 cm2/s

S
 4.3 10 

t predep  
20 
2
.
5

10


13
2
2
 
1


 2  4.2 1015  5.5 s


Example
This is a very short time and hard to control in a
furnace; thus, we should do the pre-dep at
lower temperatures
 In the VT lab, we use 830 – 860 C
14
9
Dt

2
.
3

10

Dt

3
.
7

10
predep
drivein in?
 Does the
predep affect the drive


There is no affect on the thermal budget
because it is done at such a “low” temperature
DIFFUSION SYSTEMS
Use open tube furnaces of the 3-Zone
design
 Wafers are mounted in quartz boat in
center zone
 Use solid, liquid or gaseous impurities
for good reproducibility
 Use N2 or O2 as carrier gas to move
impurity downstream to crystals
 Common gases are extremely toxic
(AsH3 , PH3)

SOLID-SOURCE DIFFUSION
SYSTEMS
Exhaust
Platinum
source boat
Slices on
carrier
burn box
and/or scrubber
Valves and
flow meters
N2
O2
Quartz
Quartz
diffusion diffusion boat
tube
LIQUID-SOURCE DIFFUSION
SYSTEMS
Exhaust
Slices on
carrier
Burn box
and/or scrubber
Valves and
flow meters
O2
N2
Quartz
diffusion tube
Liquid source
Temperaturecontrolled bath
GAS-SOURCE DIFFUSION
SYSTEMS
Exhaust
Slices on carrier
Burn box
and/or scrubber
Quartz diffusion tube
Valves and flow meter
To scrubber system
N2
Dopant
gas
O2
Trap
DIFFUSION SYSTEMS
Al and Ga diffuse very rapidly in Si; B is
the only p-dopant routinely used
 Sb, P, As are all used as n-dopants

DIFFUSION SYSTEMS

Typical reactions for solid impurities are:
 2 CHO


B2O3  6CO2  9H2O
3 3 B 9O2
900o C
SiO2  4B
2B2O3  Si
 4POCl3  302 
 2PO

2 5 6Cl2
 
SiO2  4P
2PO
2 5 5Si
 2 As2O3  3Si
3SiO2  4 As
 2SbO
 
3SiO2  4Sb
2 3 3Si
PRODUCTION DIFFUSION
FURNACES

Commercial diffusion furnace showing the
furnace with wafers (left) and gas control
system (right).
(Photo courtesy of Tystar Corp.)
PRODUCTION DIFFUSION
FURNACES

Close-up of diffusion furnace with wafers.
Rapid Thermal Annealing

An alternative to the diffusion furnaces is
the RTA or RTP furnace
Rapid Thermal Anneling
In this system, we try to heat the wafer
quickly (but not so as to introduce fracture
stresses)
 RTAs usually use infrared lamps and heat
by radiation
 It is possible to ramp the wafer at 100 C
/sec
 Such devices are used to diffuse shallow
junctions and to anneal radiation damage
 In such a system, for the thermal
conductivity of Si, a 12 in wafer can be

Rapid Thermal Annealing
Concentration-Dependent
Diffusion
If the concentration of the doping exceeds
the intrinsic carrier concentration at the
diffusion temperature, another effect
occurs
 We have assumed that the diffusion
coefficient, D, is independent of
concentration
 This is not valid if the concentration of the
diffusing species is greater than the
intrinsic carrier concentration

Concentration-Dependent
Diffusion

The concentration profiles for P in Si look
more like the solid lines than the dashed
line for high concentrations (see French
et al)
Concentration-Dependent
Diffusion
If we define the diffusivity to be a function
of composition, then we can still use Fick’s
law to describe the dopant diffusion
 Usually, we cannot directly integrate/solve
the differential
C equations
  eff C when D is a
  DA

function of C t x 
x 
 We thus must solve the equation

Concentration-Dependent
Diffusion

It has been observed that the diffusion
coefficient usually depends on
concentration
by
either
of
the
2 following
D  (n / ni ) or D  (n / ni )
relations
Look, for example, at the diffusion of P in
Si observed by French et al
 How do we obtain information about the

Concentration-Dependent
Diffusion
B has two isotopes: B10 and B11
 We create a wafer with a high
concentration of one isotope (say B10) and
then we diffuse the second isotope into
this material
 We use SIMS to determine the
concentration of B11 as a function of
distance
 This gives us the diffusion of B as a
function of the concentration of B

Concentration-Dependent
Diffusion

We find that the diffusivity can usually be
written in the form
2
eff
A
D
for
n
 n 
 D  D    D  
 ni 
 ni 
0

 p
  p 
  D  
D dopants
D  D  and
n-type
 ni 
 ni 
eff
A
0

2
Concentration-Dependent
Diffusion
The superscripts are chosen because we
believe the interaction is between charged
vacancies and the charged diffusing
species
DAeff  D 0  D   D 
 For an n-type dopant in an intrinsic
material, the diffusivity is

 D.E 
D  D0 exp 

 kT 

All of the various diffusivities are of the
Concentration-Dependent
Diffusion
The values for all the pre-exponential
factors and activation energies are known
(see next Table)
 If we substitute into the expression for the

 n  
n

    
effective diffusion 1 coefficient,
we
find

n
n

2
DAeff  DA* 



i

1   
i





Concentration-Dependent
Diffusion
Concentration-Dependent
Diffusion
Expressed this way,  is the linear
variation with composition and  is the
quadratic variation
 Simulators like SUPREM include these
effects and are capable of modeling very
complex structures
