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Chapter 7 Dopant Diffusion
1. Introduction and application.
2. Dopant solid solubility and sheet resistance.
3. Microscopic view point: diffusion equations.
4. Physical basis for diffusion.
5. Non-ideal and extrinsic diffusion.
6. Dopant segregation and effect of oxidation.
7. Manufacturing and measurement methods.
NE 343: Microfabrication and thin film technology
Instructor: Bo Cui, ECE, University of Waterloo; http://ece.uwaterloo.ca/~bcui/
Textbook: Silicon VLSI Technology by Plummer, Deal and Griffin
1
Doping in MOS and bipolar junction transistors
NMOS
BJT
p+
p well
Base Emitter
p
n+
Collector
n+
nn+
p+
p
Doping is realized by:
• Diffusion from a gas, liquid or solid source, on or above surface. (no longer popular)
• Ion implantation. (choice for today’s IC)
• Nowadays diffusion often takes place unintentionally during damage annealing…
• “Thermal budget” thus needs to be controlled to minimize this unwanted diffusion.
In this chapter, diffusion means two very different concepts: one is to dope the substrate from
source on or above surface – the purpose is doping; one is diffusion inside the substrate – the
2
purpose is re-distribute the dopant.
Application of diffusion
Diffusion is the redistribution of atoms from regions of high concentration of mobile
species to regions of low concentration.
It occurs at all temperatures, but the diffusivity has an exponential dependence on T.
In the beginning of semiconductor processing, diffusion (from gas/solid phase
above surface) was the only doping process except growing doped epitaxial layers.
Now, diffusion is performed to:
• Obtain steep profiles after ion implantation due to concentration dependent
diffusion.
• Drive-in dopant for wells (alternative: high-energy implantation), for deep p-n
junctions in power semiconductors, or to redistribute dopants homogeneously
in polysilicon layers.
• Denude near-surface layer from oxygen, to nucleate and to grow oxygen
precipitates.
• Getter undesired impurities.
3
Doping profile for a p-n junction
4
Diffusion from gas, liquid or solid source
Pre-deposition (dose control)
Drive-in (profile control)
• Silicon dioxide is used as a mask against impurity diffusion in Silicon.
• The mixture of dopant species, oxygen and inert gas like nitrogen, is passed over the
wafers at order of 1000oC (900oC to 1100oC) in the diffusion furnace.
• The dopant concentration in the gas stream is sufficient to reach the solid solubility
limit for the dopant species in silicon at that temperature.
• The impurities can be introduced into the carrier gas from solid (evaporate), liquid
(vapor) or gas source.
5
Comparison of ion implantation with solid/gas phase diffusion
Pre-deposition
Drive-in
6
Chapter 7 Dopant Diffusion
1. Introduction and application.
2. Dopant solid solubility and sheet resistance.
3. Microscopic view point: diffusion equations.
4. Physical basis for diffusion.
5. Non-ideal and extrinsic diffusion.
6. Dopant segregation and effect of oxidation.
7. Manufacturing and measurement methods.
NE 343 Microfabrication and thin film technology
Instructor: Bo Cui, ECE, University of Waterloo
Textbook: Silicon VLSI Technology by Plummer, Deal and Griffin
7
Dopant solid solubility
Solid solubility: at equilibrium, the maximum concentration for an impurity
before precipitation to form a separate phase.
Figure 7-4
8
Solid solubility of common impurities in Silicon
9
Solubility vs. electrically active dopant concentration
As in
substitutional
site, active
Inactive
Figure 7-5
V: vacancy
Not all impurities are electrically active.
As has solid solubility of 21021 cm-3.
But its maximum electrically active dopant concentration is only 21020 cm-3 .
10
Resistance in a MOS
Figure 7-1
For thin doping layers, it is convenient to find the resistance from sheet resistance.
11
Sheet resistance RS
RS 

: (bulk) resistivity
xj: junction depth, or film thickness…
xj
A
xj
w
l
l
R
A
Figure 7-2
R  RS 

xj
l
l
 l
l
R 

 RS
A
wx j x j w
w
R=Rs when l=w (square)
Important formulas
Ohm’s law:
Mobility :
By definition:
Therefore:
Finally:
Where:
: conductivity;
v: velocity;




E  J
J  E


v  E



J  q pvh  nvn 


 vhx
vn 
vnx 
 vh
  q p   n    q p  n 
E
Ex 
 E
 Ex
  q n n  p p 
vnx
n 
Ex
: resistivity;
q: charge;
vhx
p 
Ex
J: current density; E: electrical field
n, p: carrier concentration.
13
Sheet resistance

1
1
1
RS  


x j x j qNx j qQ
N is carrier density, Q is total carrier per unit area, xj is junction depth
For non-uniform doping:

1
RS  

x j x j
1
xj
q  nx   N B  nx dx
0
This relation is calculated to generate the so-called Irvin’s curves.
See near the end of this slide set.
14
Chapter 7 Dopant Diffusion
1. Introduction and application.
2. Dopant solid solubility and sheet resistance.
3. Microscopic view point: diffusion equations.
4. Physical basis for diffusion.
5. Non-ideal and extrinsic diffusion.
6. Dopant segregation and effect of oxidation.
7. Manufacturing and measurement methods.
NE 343 Microfabrication and thin film technology
Instructor: Bo Cui, ECE, University of Waterloo
Textbook: Silicon VLSI Technology by Plummer, Deal and Griffin
15
Diffusion from a macroscopic viewpoint
Fick’s first law of diffusion
F is net flux.
C  x, t 
F  x, t    D
x
This is similar to other laws
where cause is proportional to
effect (Fourier’s law of heat flow,
Ohm’s law for current flow).
Figure 7-6
C is impurity concentration (number/cm3), D is diffusivity (cm2/sec).
D is related to atomic hops over an energy barrier (formation and migration of mobile
species) and is exponentially activated.
Negative sign indicates that the flow is down the concentration gradient.
16
Intrinsic diffusivity Di
Intrinsic: impurity concentration NA, ND < ni (intrinsic carrier density).
Note that ni is quite high at typical diffusion temperatures, so "intrinsic" actually
applies under many conditions. E.g. at 1000oC, ni =7.141018/cm3.
Ea
Di  D exp( 
)
kT
0
Ea: activation energy
B
In
P
As
Sb
D0(cm2/s)
1.0
1.2
4.70
9.17
4.58
Ea(eV)
3.46
3.50
3.68
3.99
3.88
Figure 7-15, page 387
17
Fick’s second law
The change in concentration in a volume element
is determined by the change in fluxes in and out
of the volume.
Within time t, impurity number change by:
A
Cx, t  t   Cx, t  A  x
During the same period, impurity diffuses in and
out of the volume by:
Figure 7-7
F x, t   F x  x, t  A  t  F x  x, t   F x, t  A  t
Therefore:
Cx, t  t   Cx, t  A  x  F x  x, t   F x, t  A  t
Or,
Since:
We have:
C ( x, t )
F ( x, t )

t
x
C  x, t 
F  x, t    D
x
C  x, t 
F  x, t    C  x, t  


D

t
x
x 
x 
If D is constant:
C x, t 
 2 C  x, t 
D
2
t
x
18
Solution to diffusion equation
C x, t 
 2 C  x, t 
D
2
t
x
At equilibrium state, C doesn’t change with time.
C
 2C
D 2 0
t
x
C  a  bx
x
Cg
Cs
Diffusion of oxidant (O2 or H2O)
through SiO2 during thermal
oxidation.
C*
CI=0
SiO2
Si
19
Gaussian solution in an infinite medium
as t 0
for
x>0
C  as t 0
for
x=0
C0
At t=0, delta
function dopant
distribution.
C(x,t)dx=Q (limited source)
At t>0
 x2 
 x2 
Q
  C 0, t  exp  

C x, t  
exp  
2 Dt
 4 Dt 
 4 Dt 
This corresponds to, e.g. implant a very
narrow peak of dopant at a particular depth,
which approximates a delta function.
Important consequences:
• Dose Q remains constant
• Peak concentration (at x=0) decreases as 1/ t
• Diffusion distance from origin increases as 2 Dt
Figure 7-9
20
Gaussian solution near a surface
 x2 
Q

C x, t  
exp  
Dt
 4 Dt 
1. Pre-deposition
for dose control
Figure 7-10
A surface Gaussian diffusion can be
treated as a Gaussian diffusion with
dose 2Q in an infinite bulk medium.
21
2. Drive in for
profile control
Note: Pre-deposition by diffusion can also
be replaced by a shallow implantation step.
Gaussian solution near a surface
Surface concentration
decreases with time
Concentration gradient
C x, t 
  x C x, t 
x
2Dt
QT
CS  C 0, t  
Dt
Junction depth
At p-n junction
2CB  Cs 
C x, t 


ln 
x x j
xj
 CB 
 Cs 

x j  2 Dt ln 
 CB 
 QT
 2 Dt ln 
 C B Dt
22




Error function solution in an infinite medium
This corresponds to, e.g. putting a
thick heavily doped epitaxial layer on a
lightly doped wafer.
At t=0
C=0 for x>0
C=C for x<0.
An infinite source of material in the halfplane can be considered to be made up of
a sum of Gaussians. The diffused solution
is also given by a sum of Gaussians,
known as the error-function solution.
Figure 7-11
  x  xi  

x
exp



i
4
Dt
i 1


  x   2 
0
C
C  x, t  
exp 
d
4
Dt
2 Dt 


x     
2 Dt
C 
2
C  x, t  
exp


d

C
C  x, t  
2 Dt

2
n
x 2 Dt
 
erf  z  
2

z


2
exp


d

0
C
 x
C  x, t   1  erf 
2
 2 Dt
erfc x   1  erf  x 
C
 x
C  x, t   erfc
2
 2 Dt






erfc: complementary error function
23
Error function solution in an infinite medium
Evolution of erfc diffused profile
Figure 7-12
Important consequences of error function solution:
• Symmetry about mid-point allows solution for constant surface concentration to be derived.
• Error function solution is made up of a sum of Gaussian delta function solutions.
• Dose beyond x=0 continues to increase with annealing time.
24
Error function solution in an infinite medium
Properties of Error Function erf(z) and Complementary Error Function erfc(z)
erf x  
2

x
 
2
exp
u
du

erfc x   1  erf x  

exp -u du


2
2
x
0
erf 0  0
erfc 0  1
erf   1
erfc   0
d erf x  2

exp  x 2
dx

 

 erfc ( x)dx 
0
d 2 erf x 
4
2


x
exp

x
dx 2

2
erf ( x) 
x
For x << 1
1

 

For x >> 1

1 exp  x 2
erfc x  
x


25
Error function solution near a surface
Constant surface concentration at all times, corresponding to, e.g., the situation of
diffusion from a gas ambient, where dopants “saturate” at the surface (solid solubility).
Boundary condition: C(x,0)=0, x0; C(0,t)=Cs; C(,t)=0
 x
C x, t   Cs erfc 
 2 Dt


x 2 Dt
e
u 2
Pre-deposition dose
du

2C
 x 
Q   Cs erfc 
dx  s

 2 Dt 
0
Dt
Cs is surface concentration,
limited by solid solubility,
which doesn’t change too
fast with temperature.
½
Constant 1/2
 2Cs



26
Successive diffusions
• Successive diffusions using different times and temperatures
• Final result depends upon the total Dt product
Dt tot   Di t i
For example, the profile is a Gaussian i
function at time t=t0, then after further
diffusion for another 3t0, the final profile is
still a Gaussian with t=4t0=t0+3t0.

(The Gaussian solution holds only if the Dt used to
introduce the dopant is small compared with the
final Dt for the drive-in i.e. if an initial /delta
function approximation is reasonable)
When D is the same (same temperature)
Dt eff
When diffused at different temperatures Dt eff
 Dt1  t 2  ...  t n 
 D2 
 D1t1  D2t2  ...  D1t1  D1t2    ...
 D1 
As D increases exponentially with temperature, total diffusion (thermal
budget) is mainly determined by the higher temperature processes.
27
Irvin’s curves
Motivation to generate Irvin’s curves: both NB (background carrier concentration), Rs
(sheet resistance) and xj can be conveniently measured experimentally but not N0 (surface
concentration). However, these four parameters are related by:

1
1
xj
RS  
 xj
1
 
 x dx
x j x j
xj 0
q  nx   N B  nx dx

0
Irvin’s curves are plots of N0 versus (Rs, xj) for various NB, assuming erfc or half-Gaussian
profile. There are four sets of curves for (n-type and p-type) and (Gaussian and erfc).
1-
Irvin’s curves
Figure 7-17
Four sets of curves: p-type erfc, n-type erfc, p-type half-Gaussian, n-type half-Gaussian
Explicit relationship between: N0, xj, NB and RS.
Once any three parameters are know, the fourth one can be determined.
1-
Example
Design a boron diffusion process (say for the well or tub of a CMOS process) such that
s=900/square, xj=3m, with CB=11015/cm3.
The average conductivity of the layer is
From (half-Gaussian) Irvin’s curve, we find
Cs << solubility of B in Si, so it is correct to assume pre-deposition (here by ion
implantation) plus drive-in, which indeed gives a Gaussian profile.
30
Example (cont.)
 x 
Q

C x, t  
exp  
Dt
 4 Dt 
2
Dt 
x 2j
 Cs 

4 ln 
 CB 

3 10 
4 2
 4 1017 

4 ln 
15
 10 
 x 2j 

C B  Cs exp  
 4 Dt 


 3.7 109 cm 2
Assume drive-in at 1100oC, then D=1.5×10-13cm2/s.
tdrivein
3.7 10 9 cm 2

 6.8 hours
13
2
1.5 10 cm sec
Pre-deposition dose
Q  Cs Dt  4 1017    3.7 10 9  4.3 1013 cm 2
31
Example (cont.)
Now if we assume pre-deposition by diffusion from a gas or solid phase at 950oC, solid
solubility of B in Si is Cs=2.5×1020/cm3, and D=4.2×10-15cm2/s.
The profile of this pre-deposition is erfc function.
Dt
Q  2Cs

2
 4.3  10


 
 
 5.5 sec
20 
15
 2  2.5  10  4.2  10
13
tpre dep



Dt predep 2.3 1014  Dt drivein 3.7 109

However, the pre-deposition time is too short for real processing,
so ion-implantation is more realistic for pre-deposition.
32