Chapter 5
UEEP2613
Microelectronic Fabrication
Diffusion
Prepared by
Dr. Lim Soo King
24 Jun 2012
Chapter 5 Diffusion .........................................................................131
5.0 Introduction ............................................................................................ 131
5.1 Model of Diffusion in Solid .................................................................... 133
5.2 Fick’s Diffusion Equation ...................................................................... 134
5.2.1 Constant Diffusivity ........................................................................................ 135
5.2.1.1 Constant Surface Concentration ............................................................................... 135
5.2.1.2 Constant Total Dopant ............................................................................................... 136
5.2.1.3 Sheet Resistance of a Diffused Layer ........................................................................ 138
5.2.1.4 Effect of Successive Diffusion Steps ........................................................................... 139
5.2.2 Concentration Dependent Diffusivity ........................................................... 140
5.2.3 Temperature Dependent Diffusivity ............................................................. 142
Exercises ........................................................................................................ 144
Bibliography ................................................................................................. 146
-i-
Figure 5.1:
Figure 5.2:
Figure 5.3:
Figure 5.4:
Figure 5.5:
Schematic of a diffusion system using liquid source ..................................... 132
Mechanism of diffusion in solid .................................................................... 133
Plot of complementary error function ............................................................ 136
A surface Gaussian diffusion with total dopant QT at the center of silicon ... 138
A typical Irvin curve for p-type Gaussian profile in an n-type background
concentration .................................................................................................. 139
Figure 5.6: Diffusivity dependent on doping concentration ............................................. 141
Figure 5.7: Concentration dependent diffusivity of common dopant in single crystal
silicon ............................................................................................................. 142
Figure 5.8: Arrhenius plot of diffusivity of the common dopants in silicon .................... 143
Figure 5.9: Temperature dependence of the diffusivity coefficient of common dopant in
silicon ............................................................................................................. 143
Figure 5.10: Intrinsic diffusivity for silicon self diffusion of common dopants ................ 144
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Chapter 5
Diffusion
_____________________________________________
5.0 Introduction
Diffusion of impurity atom or dapant in silicon is an important process in
silicon integrated circuit. Using diffusion techniques, altering the conductivity
in silicon or germanium was disclosed in a patent by William Gardner Pfann in
1952. Since then, various ways of introducing dopant into silicon by diffusion
have been studied with the goal of controlling the distribution of dopant, the
concentration of total dopant, its uniformity, and reproducibility, and for
processing large number of device wafer in a batch to reduce the manufacturing
cost.
Diffusion is used to form emitter, base, and resistor for the bipolar device
technology. It is also used to form drain and source regions and to dope
polysilicon in MOS device technology. Dopant that spans a wide range of
concentration can be introduced by a number of ways. The most common way
of diffusion is from chemical source in vapor form at high temperature. The
other ways are diffusion from a doped oxide source and diffusion and annealing
from ion implanted layer. Ion implantation can provide 1011cm-2 to greater than
1016cm-2. It is used to replace the chemical or doped oxide source wherever
possible and is extensively used in VLSI/ULSI device fabrication.
Diffusion of impurities is typically done by placing semiconductor wafers
in a carefully controlled, high temperature quartz-tube furnace and passing a gas
mixture that contains the desired dopant through it. Its purpose is to introduce
dopant into silicon crystal. Mixture of oxygen and dopants such as diborane and
phosphine are introduced in the furnace with the exposed wafer surface at
temperature ranges between 8000C and 1,2000C for silicon and 6000C and
1,0000C for gallium arsenide. The number of dopant atoms that diffused into the
semiconductor is related to the partial pressure of the dopant impurity in the gas
mixture.
The schematic of a diffusion system using liquid source is shown in Fig.
5.1.
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05 Diffusion
Figure 5.1: Schematic of a diffusion system using liquid source
Dopant can be introduced by solid source such as BN for boron, As 2O3 for
arsenic, and P2O5 for phosphorus, gases source such as B2H6, AsH3, and PH3,
and liguid source such as BBr3, AsCl3, and POCl3. However, liquid source is the
commonly used method. The chemical reaction for phosphorus diffusion using
liquid source POCl3 is shown as follow.
4POCl3 + 3O2 → 2P2O5 + 6Cl2 ↑
(5.1)
The P2O5 forms a glass-on-silicon wafer and then reduces to phosphorus by
silicon following the equation.
2P2O5 + 5Si → 4P + 5SiO2 ↑
(5.2)
The phosphorus is released and diffused into silicon Si and chlorine Cl2 gas is
vented.
For diffusion in gallium arsenide, the high vapor pressure of arsenic
requires special method to prevent loss of arsenic by decomposition or
evaporation. These methods include diffusion in sealed ampules with over
pressure of arsenic and diffusion in an open-tube furnace with doped oxide
capping layer such as silicon nitride. Most of the studies on p-type diffusion
have been confined to the use of zinc in the forms of Zn-Ga-As alloys and
ZnAs2 for the sealed-ampule approach or ZnO-SiO2 for the open-tube approach.
The n-type dopants in gallium arsenide include selenium and tellurium. To
complete the process, 'drive in' or re-distribution of dopant is done in nitrogen or
wet oxygen where silicon dioxide SiO2 is grown at the same time.
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05 Diffusion
5.1 Model of Diffusion in Solid
At high temperature, point defects such as vacancy and self interstitial atom are
generated in a single crystalline solid. When concentration gradient of the host
or impurity atom exists, such point defect affects the atom movement namely as
diffusion. Diffusion in solid can be treated as the atomic movement of diffusant
either impurity atom or host atom in crystal lattice by vacancy and self
intersitial.
There are several types of diffusion mechanisms. They are vacancy
diffusion, intersitial diffusion, divacancy or diffusion assisted by a double
vacancies, and interstitialcy diffusion. Figure 5.2 illustrates the diffusion
mechanisms by vacancy, interstitial, and interstitialcy. At elevated temperature,
the atom in crystal lattice vibrates in its equilibrium site. Occasionally, the atom
acquires sufficient energy to leave its equilibrium site and becomes a self
interstitial atom. If there is an impurity atom (red color) around or a neigboring
host atom, it can occupy this vacant site and this type of diffusion is termed as
diffusion by a vacancy. This type of diffusion is illustrated by diffusion
mechanism 1 shown in Fig. 5.2(a). If the the migrating atom is a host atom, it is
called self diffusion. If it is a impurity atom, then it is called impurity diffusion.
(a) Vacancy diffusion and interstitial diffusion
(b) Interstitialcy diffusion
Figure 5.2: Mechanism of diffusion in solid
If the movement of impurity atom is in between equilibrium site of the crystal
lattice that does not involve occupying lattice site, it is called interstitial
diffusion as illustrated by diffusion mechanism 2 shown in Fig. 5.2(a).
Diffusion assisted by a double vacancy or divacancy is a diffusion mechanism
involving impurity atom has to move to a second vacancy that is at the nearest
neighbor of the original vacancy site.
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05 Diffusion
Interstitialcy diffusion is shown in Fig. 5.2(b). The mechanisms are shown
by four steps. In step 1, a self interstitial host atom displaces an impurity atom
(read color) from the lattice site and makes this impurity atom as interstitial
atom (step 2). This interstitial impurity atom then displaces a host atom (step 3)
from its equilibrium site and makes this host atom to become interstitial atom
(step 4).
Vacancy and interstitialcy diffusions are commonly happened for
phosphorus P, boron B, arsenic As, antimony Sb impurity diffusion in silicon.
However, for phosphorus P and boron B diffusion, interstitialcy diffusion is
more dominant than vacancy diffusion. Vacancy diffusion is more dominant
than interstitialcy diffusion for arsenic As and antimony Sb diffusion. Group 1
and VIII elements have small ionic radii and are fast diffuser in silicon. The
diffusion is normally involved interstitial diffusion.
5.2 Fick’s Diffusion Equation
In 1855, Adolf Fick published the theory of diffusion. His theory was based on
the analogy between material transfer in a solution and heat transfer by
conduction. Fick’s assumed that in dilute liquid or gaseous solution, in the
absence of convection, the transfer solute atom per unit area in one direction
flow can be described by Fick’s first law of diffusion shown in equation (5.3).
J  D
C( x , t )
x
(5.3)
where J is the local rate of transfer of solute per unit area or the diffusion flux, C
is the concentration of solute is a function of x and t, x is the coordinate axis in
the direction of solute flow, t is the diffusion time, and D is the diffusivity or at
time it is called diffusion coefficient or diffusion constant. The negative sign of
the equation denotes that the solute flows to the direction of lower concentration.
From the law of conservation of matter, the change of solute concentration
with time must be the same as the local decrease of the diffusion flux in the
absence of a source or sink. Thus,
C( x , t )
J ( x , t )

t
x
(5.4)
Substitute equation (5.3) into equation (5.4) yields equation of Fick’s second
law in one dimensional form, which is
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05 Diffusion
C( x, t )   J( x, t ) 

D
t
x 
x 
(5.5)
When the concentration of solute is low, the diffusivity at a given temperature
can be considered as a constant then equation (5.5) shall become
C( x, t )
 2 J( x, t )
D
t
x 2
(5.6)
Equation (5.6) is another form of Fick’s second law of diffusion. In equation
(5.6), D is given in unit of cm2/s or m2/h and C(x, t) is in unit of atom/cm3. The
solution for equation (5.6) for various initial condition and boundary condition
shall be dealt in next sub-section.
5.2.1 Constant Diffusivity
The solution of diffusion equation shown in equation (5.6) has constant
diffusivity or diffusion coefficient for constant surface concentration and
constant total dopant will be discussed in this sub-section. The sheet resistance
of a diffused layer of constant diffusivity will be discussed too.
5.2.1.1 Constant Surface Concentration
For the case of constant surface concentration, the initial condition at time t = 0
is C(x, 0) = 0 and the boundary conditions are C(0, t) = C s and C(, t) = 0. The
solution of equation (5.6) is equal to
 x 
C( x , t )  CSerfc

 2 Dt 
(5.7)
where Cs is the surface concentration, D is the constant diffusivity, x is the
distance, t is the diffusion time, and erfc is the complementary error function.
The plot of complementary error function erfc of equation (5.7) is shown in
Fig. 5.3.
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05 Diffusion
Figure 5.3: Plot of complementary error function
Since erfc(x) = 1 – erf(x), equation (5.7) is also equal to

 x 
C( x, t )  CS 1  erf 

 2 Dt 

(5.8)
From the result shown in Fig. 5.2, the error function solution is approximately a
triangular function, so that the total amount of dopant per unit area introduced
can be approximated by QT  CS Dt . A more accurate answer for the total
amount of dopant introduced per unit area is


2C
 x 
QT   CS 1  erf 
dx  S Dt

 2 Dt 

0
(5.9)
5.2.1.2 Constant Total Dopant
If a thin layer of dopant is deposited onto the silicon surface with a fixed or
constant total amount of dopant QT per unit area. This dopant diffuses only into
the silicon and all the dopants remain in the silicon. The initial and boundary
conditions are initial condition C(x, 0) = 0 and boundary condition
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05 Diffusion

 C(x, t)dx  Q
T
and C(, t) = 0. The solution of the diffusion equation shown in
0
equation (5.6) shall be
C(x, t ) 
 x2 
QT

exp 
Dt
 4Dt 
(5.10)
If x = 0, equation (5.10) is equal to surface concentration CS, which is
CS  C(0, t ) 
QT
Dt
(5.11)
Combining equation (5.10) and (5.11) yields equation (5.12).
 x2 

C( x, t )  CS exp 
 4Dt 
(5.12)
Equation (5.10) is often called the Gaussian distribution and the diffusion
concentration is referred to dopant concentration of the pre-deposited thin layer
source or drive-in diffusion from a fixed total dopant concentration. Impurity
atom distribution from ion implantation into amorphous material can be
approximated by Gaussian function.
For the case whereby a thin layer of dopant is deposited in the center the
silicon surface, the diffusion profile looks as what is shown in Fig. 5.4. The
diffusion flux will be equal to
 x2 
QT

C( x, t ) 
exp 
2 Dt
 4Dt 
whereby the assumption is that half of the QT will diffuse virtually.
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(5.13)
05 Diffusion
Figure 5.4: A surface Gaussian diffusion with total dopant QT at the center of silicon
5.2.1.3 Sheet Resistance of a Diffused Layer
For a diffused layer that form pn junction, an average sheet resistance RS is
defined and related to junction depth xj, the carrier mobility , and the impurity
distribution C(xi) by the equation (5.14).
RS 
1
xj
q  C( x )dx
0

1
(5.14)
xj
q eff  C( x )dx
0
Empirical expression of mobility  versus impurity distribution C has been
determined for concentration above 1016cm-3 in silicon. The donor dopant
mobility n is
n 
1360  90
 92.0cm2 / Vs
17 0.91
1  (C / 1.3x10 )
(5.15)
For acceptor concentration in silicon for acceptor dopant, the mobility p is
p 
468  49.7
 49.7cm2 / Vs
1  (C / 1.6x1017 ) 0.7
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(5.16)
05 Diffusion
The sheet resistance RS is also equal to
RS 

xj
(5.17)
where  is the resistivity. Thus, the effective conductivity  is equal to

1
R Sx j
(5.18)
Once the surface resistance, surface concentration, and junction depth are
known, one can design a diffused layer. There is a useful design curve called
Irvin curve that can be used to determine the surface concentration CS versus the
effective conductivity on background concentration CB, which shown in Fig. 5.5.
Figure 5.5: A typical Irvin curve for p-type Gaussian profile in an n-type background
concentration
5.2.1.4 Effect of Successive Diffusion Steps
Since there are often multiple diffusion steps in a fully integrated circuit
process, they must be added in some ways before the final profile can be
predicted. It is clear that if all the diffusion steps occurred at a constant
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05 Diffusion
temperature where the diffusivity is the same then the effective Dt product is
given by
(Dt )eff  D1 ( t1  t 2  ...)  D1t1  D1t 2  .....
(5.19)
In other words doing a single step in a furnace for a total time of t 1 + t2 is the
same as doing two separate steps, one for time t1 and one for time t2.
Mathematically, one could increase the time t2 by a numerical factor D2/D1
and re-write equation (5.19) as
D 
(Dt )eff  D1t1  D1 2 t 2  D1t1  D2 t 2
 D1 
(5.20)
Thus, the derived formula for the total effective Dt for a dopant that is diffused
at a temperature T1 with diffusivity D1 for time t1 and then diffused at
temperature T2 with diffusivity D2 for time t2. The total effective Dt is given by
the sum of all the individual Dt products.
5.2.2 Concentration Dependent Diffusivity
At high concentration, when the diffusion conditions are closed to the constant
surface concentration case or the constant total dopant case, the measured
impurity profiles are not the same as the constant diffusivity cases. For high
concentration case, it can be represented by concentration dependent diffusivity.
Anderson and Lisak obtained the concentration dependent diffusivity equation
by changing the diffusivity of equation (5.5) with equation (5.19), which is
C
D  2Di  
 ni 
r
(5.21)
where Di is the constant diffusivity at low concentration or intrinsic diffusivity;
C is doping concentration; and ni is the intrinsic concentration, and r is a
constant. We shall further discuss this equation.
Based on many experiment results, the diffusivity of common dopants in
silicon has been characterized and found to depend linearly or sometimes
quadratically on the carrier concentration as shown in Fig. 5.6.
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05 Diffusion
Figure 5.6: Diffusivity dependent on doping concentration
The effective diffusivity under extrinsic condition based on equation (5.6) can
be written as
2
D
eff
A
n
n
 D  D    D 2    for n-type dopant
 ni 
 ni 
0

(5.22)
2
D
eff
A
 p
 p
 D  D    D 2    for p-type dopant
 ni 
 ni 
0

(5.23)
D0 and D+ etc. are chosen because on the atomic level. These different terms are
thought to occur because of the interaction with neutral and charged point defect.
The diffusivity under intrinsic condition for an n-type dopant (n = p = ni) is
D*A  D0  D   D 2 
(5.24)
The concentration dependent diffusivity of common dopant in single crystal
silicon is shown in Fig. 5.7.
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05 Diffusion
By re-writing equation (5.22) and (5.24), the diffusivity measured under
extrinsic conditions can be elegantly described by equation (5.25).
Deff
A
2


 1   n   n  
n  

ni
 i
 D*A 

1   






(5.25)
where   D / D0 and   D2  / D0 .
o
D ,0
Do, E
D+, 0
D+, E
D-, 0
D-, E
D2-, 0
D2-, E
Si
560
4.76
B
0.05
3.5
0.95
3.5
In
0.6
3.5
0.6
3.5
As
0.011
3.5
Sb
0.214
3.65
P
3.85
3.66
31.0
4.15
15.0
4.08
4.44
4.0
44.2
43.7
Unit
cm2s-1
eV
cm2s-1
eV
cm2s-1
eV
cm2s-1
eV
Figure 5.7: Concentration dependent diffusivity of common dopant in single crystal silicon
5.2.3 Temperature Dependent Diffusivity
The diffusivity determined experimentally over range of diffusion temperature
can be expressed in Arrhenius form as
 E 
D  D o exp  a 
 kT 
(5.26)
where k is the Boltzmann constant and T is the temperature. The activation
energy Ea has a typical value between 3.5 to 4.5eV for impurity dopant in
silicon. Plots of the diffusivity of common dopant in silicon are shown in Fig.
5.8 and 5.9 corresponding to the intrinsic diffusivity and activation energy
shown in Fig. 5.10. Figure 5.10 also represents an Arrhenius fit to the
diffusivity under intrinsic condition.
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05 Diffusion
Figure 5.8: Arrhenius plot of diffusivity of the common dopants in silicon
Figure 5.9: Temperature dependence of the diffusivity coefficient of common dopant in
silicon
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05 Diffusion
Do
Ea
Si
560
4.76
B
1.0
3.5
In
1.2
3.5
As
9.17
3.99
Sb
4.58
3.88
P
4.70
3.68
Unit
cm2s-1
eV
Figure 5.10: Intrinsic diffusivity for silicon self diffusion of common dopants
Exercises
5.1.
Given the solution of Fick’s diffusion equation that satisfies the initial

x 
 . Prove that the total
 2 Dt 
and boundary conditions is C( x, t )  CSerfc
number of dopant atoms per unit area of the semiconductor is
QT ( t ) 
2
Cs Dt .

5.2.
Find the diffusivity and total impurity from a known impurity profile.
Assume that boron is diffused into an n-type silicon single crystal
substrate with doping concentration of 1015cm-3 and that the diffusion
profile can be described by a Gaussian function. Using diffusion time of
one hour, one obtains a measured junction depth of 2.0m and surface
concentration of 1.0x1018cm-3.
5.3.
A boron diffusion process such that the surface concentration is
4.0x1017cm-3, thickness x = 3.0m, substrate concentration CB =
1.0x1015cm-3. Calculate the drive in time if the diffusion temperature is
1,100oC.
5.4.
Boron pre-deposition is performed at 950oC for 30 minutes in a neutral
ambient. Given the activity energy of boron Ea = 3.46eV, D0 = 0.76
cm2/sec and the boron surface concentration is Cs = 1.8x1020cm-3.
(i) . Calculate the diffusion length.
(ii). Determine the total amount of dopant introduced.
5.5.
Assume the measured phosphorus profile can be represented by a
Gaussian function with a diffusivity D = 2.3x10-13cm2s-1. The measured
surface concentration is 1.0x1018cm-3, and the measured junction depth is
1.0m at a substrate concentration of 1.0x1015 atoms/cm3.
(i) . Calculate the diffusion time.
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05 Diffusion
(ii). Find the total dopant in the diffused layer.
5.6.
For boron diffusion in silicon at 1,000oC, the surface concentration is
maintained at 1.0x1019cm-3 and the diffusion time 1.0hr. If the diffusion
coefficient of boron at 1,000oC is 2.0x10-14cm2/s, calculate
(i). The total dopant QT diffused into silicon.
(ii). The location where the dopant concentration reaches 1.0x1018cm-3.
5.7.
Calculate the effective diffusion coefficient at 1,000oC for two different
box shaped arsenic profile grown by silicon epitaxy, one doped at
1.0x1018cm-3 and the other doped at 1.0x1020cm-3.
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05 Diffusion
Bibliography
1.
JD Pummer, MD Del, and Peter Griffin, “Silicon VLSI Technology”
Fundamentals, Practices, and Modeling”, Prentice Hall, 2000.
2.
Hong Xiao, “Introduction to Semiconductor Manufacturing Technology”,
Pearson Prentice Hall, 2001.
3.
SM Sze, “VLSI Technology”, second edition, McGraw-Hill, 1988.
4.
CY Chang and SM Sze, “ULSI Technology”, McGraw-Hill, 1996.
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Index
Intrinsic diffusivity .............................................. 140
Irvin curve .......................................................... 139
A
Activation energy ............................................... 142
Adolf Fick ........................................................... 134
Antimony ........................................................... 134
Arrhenius ........................................................... 142
Arsenic ....................................................... 132, 134
Arsenic trioxide .................................................. 132
As2O3 ......................................... See Arsenic trioxide
M
Mobility .............................................................. 138
P
Boltzmann constant ........................................... 142
Boron ................................................................. 134
P2O5 ............................... See Phosphorus pentoxide
Phosphine .......................................................... 131
Phosphorus ................................................ 132, 134
Phosporus pentoxide ......................................... 132
pn junction ......................................................... 138
Point defect ........................................................ 133
C
S
Chlorine ............................................................. 132
Complementary error function ......................... 135
Selenium ............................................................ 132
Self diffusion ...................................................... 133
Semiconductor
Gallium arsenide .................................... 131, 132
Germanium .................................................... 131
Sheet resistance ................................. 135, 138, 139
Silicon ................................................................. 138
Silicon dioxide .................................................... 132
B
D
Diborane ............................................................ 131
Diffusion ............................................................ 131
Diffusion coefficient .......................................... 134
Diffusion time .................................................... 135
Diffusivity ........................................................... 134
Divacancy diffusion ............................................ 133
Dopant ............................................................... 131
T
Tellurium ............................................................ 132
E
U
Effective conductivity ........................................ 139
Ultra large scale integration............................... 131
F
V
Fick’s first law of diffusion ................................. 134
Fick’s second law of diffusion ............................ 135
VLSI .................................................................... 131
W
G
William Gardner Pfann....................................... 131
Gaussian function .............................................. 137
Z
I
Zinc arsenate ...................................................... 132
ZnAs2 ............................................ See Zinc arsenate
Zn-Ga-As alloy .................................................... 132
Impurity diffusion .............................................. 133
Interstitial diffusion ........................................... 133
Interstitialcy diffusion ........................................ 134
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