2

72
EXTENDED FIVE

STREAM MODEL FOR DIF
FUSION MASS
TRANSFER OF IMPLANTE
D DOPANT ATOMS IN
SEMICONDUCTOR SILICO
N
Boris B. Khina
1
, Valeriy A. Tsurko
2
and Galina M. Zayats
2
1
Physico

Technical Institute, National Academy of Sciences, Minsk, Belarus
2
Institute of Ma
thematics, National Academy of Sciences, Minsk, Belarus
Ion implantation of dopants (donors and acceptors) into monocrystalline
silicon
with subsequent short

tem thermal annealing at a high temperature) is used for the
formation of ultra

shallow p

n junct
ions in modern VLSI technology. The
experimentally observed phenomenon of transient enhanced diffusion (TED), which
hinders further downscaling of the transistor thickness in VLSI circuits, is typically
ascribed to the interaction of diffusing species with
non

equilibrium point defects.
However, mathematical models of dopant diffusion, which are based on the “five

stream” approach, and software packages (e.g., SUPREM4 by Silvaco Data Systems)
encounter severe difficulties in describing TED. In this work, an
extended five

stream
model for diffusion in silicon is developed taking into account all the possible charge
states of both point defects (vacancies and silicon self

interstitials/interstitialcies) and
diffusing pairs “dopant atom

vacancy” and “dopant ato
m

silicon self

interstitial”. The
model includes drift terms for diffusing species in the internal electric field and the
kinetics of interaction between unlike species. The equations for determining initial
conditions are derived. For implanted dopant ato
ms, the experimental results obtained
by SIMS are used. The profiles of point defect at the annealing temperature at t=0 are
determined from a set of non

linear equations.
1.
INTRODUCTION
The mainstream in modern VLSI technology is further miniaturizatio
n. Of
particular importance is decreasing the depth of
p

n
junctions in transistors down to
nanometric size, which permits minimizing the leakage from drain to source when the
transistor is off (the so

called short channel effect). Currently, u
l
trashallow
p

n
junctions (USJ) in modern VLSI
technology
are produced by low

energy
(~1

10 keV)
high

dose ion implantation of donor (As, P, Sb) or acceptor (e.g., boron) dopants into
a silicon waver with subsequent rapid thermal annealing (RTA). The goal of the latte
r
is healing the lattice defects generated during implantation and
performing
electrical
activation of the dopant
atom
s. However, during RTA, as well as during other kinds
of post

implantation thermal treatment
(e.g., spike annealing)
,
the phenomenon of
tr
ansient enhanced diffusion (TED) is observed
: the apparent diffusion coefficient of
the impurity atoms increases by several orders of magnitude, and near the outer
surface uphill diffusion takes place [
1

3
]
. This
complex
phenomenon is currently a
subject o
f extensive experimental investigation because it hampers obtaining the
optimal concentratio
n profile of the dopants and hence
hinders
attaining
the required
current

voltage characteristics of
the
transistor
in a VLSI circuit [
1
,2]
.
TED is typically ascrib
ed to the interaction of diffusing species with non

equilibrium point defects (vacancies and silicon self

interstitials), which are
accumulated in silicon due t
o ion damage, and with small clu
sters that form and
dissolve in the course of diffusion. Solving
the intricate problem of TED suppression
2

73
is impossible without mathematical modeling of this complex phenomenon. However,
modern technology computer

aided design (TCAD) s
oftware packages such as
SUPREM
4 (Silvaco Data Systems) encounter severe difficulties
in predicting TED of
implanted dopants. Therefore, development of novel models that are supposed to give
a correct physical description of TED is an urgent problem in this area. Most of the
models used in this area, including the model implem
ented in popu
lar package
SUPREM
4, employ the so

called “five

stream” approach
[
4

6
]
, which was
first put
forward
in Ref.
[7
].
Unlike metals where solid

state diffusion occurs via simple direct
mechanisms (by exchange with vacancies or by jumps over interstitial position
s), in
crystalline silicon, which possesses a
diamond

type lattice, diffusion of impurity
atoms can proceed only by
indirect mechanisms
[8]
: diffusion of pairs “dopant atom

vacancy” (AV) and “dopant atom

silicon self

interstitial” (AI)
. The impurity atoms
A
in lattice sites are considered immovable
. Also, the diffusion of point defects X
V,I
(vacancies V and silicon self

interstitials/interstitialcies I), which can exist in five
charge states X
(
=0,
1,
2) is considered
[
4

7].
The dopant atoms, which have
charge +1 (donors) or
–
1 (acceptors), form diffusing pairs (AX)
with point defects of
an opposite charge and neutral ones, thus the pairs
can
exist in three charge states
=0,
1. During diffusion, the generation and annihilation of pairs and point defects
takes place
, thus the corresponding kinetic terms are to be included in the reaction

diffusion equations
. In the models used for studying TED, typically only few of the
possible charge states of pairs and point defects are taken into account
[
4

7].
In thi
s
work, as the first step, an extended “five

stream” model is developed that takes into
account all the possible charge states of both point defects and pairs (AI and AV).
2.
FORMULATION OF THE M
ODEL
2.1.
Reaction

diffusion equations
The model consists
of four reaction

diffusion equations for pairs AV and AI and point
defects I and V, which include sink/source terms describing the interaction of
diffusing species of different kinds:
C
I
/
t =
div
J
I
R
I
V
+ R
A
I
+ R
AV
I
,
(1)
C
V
/
t =
div
J
V
R
I
V
+
R
A
V
+ R
AI
V
,
(2)
C
AV
/
t =
div
J
AV
+ R
A
V
R
AV
I
R
AV
AI
,
(3)
C
AI
/
t =
div
J
AI
+ R
A
I
R
AI
V
R
AV
AI
.
(4)
Since the dopant atoms located in the lattice sites are considered immovable,
4

7
the balance equation is written
C
A
/
t = R
AI
V
+ R
AV
I
+ 2
R
AV
AI
R
A
I
R
A
V
.
(5)
Eqs.
(1)

(5) are supplemented with the condition of local electroneutrality
because the mobility of free charge carriers (electrons and holes) is much higher than
that of charged diffusing species:
2

74
0
n
p
C
C
C
C
C
2
,
1
V
2
,
1
I
1
)
AV
(
1
)
AI
(
A
(
6
)
He
re C
Y
, Y
I,V,AI,AV is the volumetric concentration of diffusing species
(point defects and pairs) as a sum over all charge states, J
Y
is the corresponding
diffusion flux, R
Y
Z
are the reaction terms describing the rate of interaction between
different spec
ies (e.g., generation and decomposition of pairs and annihilation of point
defects),
and
are the charges of pairs and point defects, correspondingly,
is the
charge of dopant
atom
s in the lattice sites,
=
+1 for donor atoms
(A
A
+
= As
+
, P
+
,
Sb
+
)
a
nd
=
–
1 for acceptors
(A
A
–
= B
–
, Al
–
), p and n are the concentrations of
holes and free electrones
. To study diffusion mass transfer of two different dopants
A
1
A and A
2
,
the
system
of Eqs.
(1)

(4) should be supplemented with similar
equations for diff
usion of pairs A
2
I and A
2
V, an equation similar to Eq.(5) should be
written for dopant A
2
, the reaction terms accounting for interaction of pairs A
1
X and
A
2
X, X
V,I, should be added to the right

hand side of all the reaction

diffusion
equations, and the te
rms describing the charges of pairs A
2
X and atoms A
2
in
substitutional positions
are to
be added to Eq.(6).
2.2. D
iffusion
fluxe
s
To formulate expressions for diffusion fluxes J
Y
, the first Fick’s law together
with a drift term accounting for the effect
of built

in electric field on the diffusion of
differently charged species is used
:
X
X
X
X
X
C
q
C
D
J
.
(7)
Here
X
D
is the diffusion coefficient of point defects in charge state
,
X
is
their mobility, q i
s the charge of electron,
㴠
is the vector of the electric field
strength,
is the electric potential. Equation (
7
) is simplified by applying the
Einstein’s formula for mobility
)
T
k
/(
D
B
X
X
, where k
B
is the Boltzmann
constant, and using t
he Boltzmann
’s
distribution of charged particles in a potential
field
:
n
=
n
i
exp[q
/(k
B
T)]
. Here
n
i
is
the intrinsic concentration of charge carriers;
from the
Boltzmann
’s
distribution
it follows that
np
n
2
i
, where n and p are the
density
of electrons and holes, correspondingly
. Assuming, similarly to Refs.[5

8],
that the diffusi
on coefficient
of point defects is independent of their charge
X
X
D
D
,
from Eq.(7)
we
obtain
the following expression for the diffusion flux of
po
int defects
,
n
p
ln
C
C
D
J
i
X
X
X
X
X
,
n
p
K
,
n
p
K
1
2
2
i
X
X
2
2
i
X
X
X
(8)
2

75
where
X
K
are the equilibrium constants for ionization reactions X
0
+
e
–
X
,
which are known in literature
in the Arrhenius form (it is obvious that
1
K
0
X
)
.
Similar
ly
to Eq.(
8
), the expression for diffusion flux of pairs AX, X
V,I, can be
formulated accounting for a difference in diffusion coefficients for differently charged
pairs (AX)
,
=0,
1
. Formulating diffusion equations similarly to Eq.(8) for
differently charged pairs and summarizing over all charge states, we arrive at
,
n
p
ln
C
C
n
p
K
K
D
1
J
i
X
X
AX
AX
1
1
i
X
X
A
)
AX
(
X
AX
2
0
i
X
X
A
X
n
p
K
K
,
2
i
X
X
A
X
A
X
n
p
K
K
K
2
2
0
,
(9)
where
X
A
K
are the equilibrium constants for pairing reactions A
+ X
(
AX)
,
=0,
1, X
V,I, i.e.
*
X
*
A
*
)
AX
(
X
A
C
C
C
K
; here superscript *
denotes the equilibrium concentration. For some dopants these values can be found in
literature.
2.3.
Connection between concentrations of differently charged species
To determine the
concentrations of diffusing species (point defects and pairs) in
different charge states, which appear in the local electroneutrality condition (6), the
quasi

chemical approach is used implying that the deviation from equilibrium is
small. Considering the
ionization reaction X
0
+
e
–
X
and summarizing over all
the charge state of point defects, we obtain
i
X
X
X
X
n
p
C
K
C
,
=0,
1
2
, X
V,I
.
(10)
Similarly, for
pairing reaction A
+ X
(AX)
we have
i
X
X
A
X
AX
)
AX
(
n
p
K
K
C
C
,
=0,
1, X
V,I
.
(
11)
The concentration of equilibrium point defects in charge state
, which will be used
further, is determined as
*
X
*
X
X
*
X
C
K
C
,
2
2
X
*
X
K
, X
V,I
,
(12)
where
*
X
C
is the equilibrium concentration of point d
efects (X
V,I)
in all the charge
states, which is known in literature.
2

76
2.4.
Reaction
terms
The next step is the formulation of sink/source terms R
Y
Z
, Y,Z
I,V,AI,AV, Y
Z
,
which enter the right

hand side of Eqs.(1)

(5)
. For this purpose, small deviation f
rom
the local equilibrium for a corresponding bimolecular reaction is assumed. For one
charge state (e.g.,
=0) the recombination rate of vacancies and silicon self

interstitials is expressed as
)
C
C
C
C
)(
D
D
(
r
4
R
*
V
*
I
V
I
V
I
V
I
, r is the capture
radius
; typically i
t is
considered that
r
=
a
0
where a
0
is the crystal lattice period of
silicon. Summarizing over all charge states, we obtain
1
,
C
C
C
C
)
D
D
(
a
4
R
V
I
V
I
*
V
*
I
V
V
I
I
V
I
V
I
0
V
I
0
0
.
(13)
Let us consider terms R
A
I
and R
A
V
that describe the kinetics of pairing
reactions using acceptor do
pant as an example. For one charge state
, assuming
small deviation from equilibrium and bearing in mind that for reaction
A
+ X
(AX)
,
the equilibrium constant is
r
f
X
A
k
/
k
K
, where
f
k
and
r
k
a
re the rates
of forward and reverse reactions, we can write:
X
f
)
AX
(
r
X
A
f
X
A
rD
4
k
,
C
k
C
C
k
R
. Then, summarizing over all
the charge states and assuming r=a
0
, we obtain
X
AX
X
X
A
X
X
0
X
A
C
C
C
D
a
4
R
,
2
0
i
X
X
n
p
K
.
(14)
Using a similar approach for bimolecu
lar recombination reactions of pairs (AI)
and (AV)
with vacancies and silicon self

interstitials, correspondingly, we obtain the
following expressions:
)
C
C
C
C
C
(
)
D
(D
a
4
R
I
*
I
*
V
A
V
AI
AI
V
0
V
AI
0
,
)
C
C
C
C
C
(
)
D
(D
a
4
R
V
*
V
*
I
A
I
AV
AV
I
0
I
AV
0
.
(15)
Here terms D
AX
are determined as follows:
1
1
i
X
X
A
)
AX
(
X
AX
n
p
K
K
D
1
D
, X
V,I
.
(16)
To derive an expression for bimolecular recombination reaction AV + AI
2A
, it is assumed that its kinetics is independent of the charge state of pairs AX.
Then, using the method described above for Eq.(1
4
), we rece
ive
)
C
C
C
C
C
(
)
D
(D
a
4
R
V
I
*
V
*
I
2
A
AI
AV
AI
AV
0
AI
AV
0
0
.
(17)
2

77
Thus, the model is complete: all the expressions for the diffusion fluxes,
sink/source
terms and concentrations of species in different charge states are derived.
2.
5
. Boundary and initial conditions
To formulate the c
losed system, Eqs.(1)

(5) for the reaction

diffusion of dopants are
to be supplemented with relevant boundary and initial conditions.
In two

dimensional domain x
[0,L], y
[0,W],
the
first

kind
, or
Dirichlet
boundary conditions are posed to Eqs.(1),(2) for
diffusion of point defects:
C
X
(x=0) = C
X
(x=L) = C
X
(y=0) = C
X
(y=W) = C
X
*
, X
I,V.
(18)
This is be
cause the outer surface of a crystalline solid is typ
ically considered as a sink
for
point defects
of infinite capacity, and outside the implanted domain the
c
oncentration of point defects corresponds to thermal equilibrium.
For diffusion of pairs AX,
the
second

kind
, or
Neumann
boundary conditions to
Eqs.(3),(4) are
formulat
ed:
J
AX
(x=0) = J
AX
(x=L) = J
AX
(y=0) = J
AX
(y=W) = 0.
(19)
Hence, different

type boundar
y conditions are posed to the conjugate system of
diffusion equations (1)

(4).
Formulating the initial conditions
, C
Y
(t=0),
Y
I,V,AI,AV
,
is a more difficult
problem, which is typically passed over in silence in literature on modeling TED
[4

7].
In experim
ents, the depth profile of dopant atoms after implantation is measured
using the second

ion mass spectrometry (SIMS).
However, this method cannot
distinguish between the impurity atoms in the lattice sites (C
A
) and in pairs C
AX
,
X
I,V
. Therefore at t=0 we
have to write
AV
AI
A
)
SIMS
(
A
C
C
C
C
,
(20)
where
)
SIMS
(
A
C
is the experimentally determined distribution of dopants.
The
depth
profile of point defects
immediately after implantation
can be
obtained by Monte Carlo simulation
(MCS)
[9
,10
]
:
those are the so

called “net
vacancies” and “net interstitials”, which remain in silicon after fast recombination of
the Frenkel pairs
; the latter takes place after implantation and espacuially during the
initial stage of heating to the annealing temperatu
re. The concentration of net
interstitials
exceeds that of dopants throughout the implanted depth by the factor of
about 3

4 for As and about 1.1

1.2 for light atoms such as boron [10]
.
Upon
heating,
the pairs AX, X
I,V, are considered to form quickly
via
reaction
A
+ X
(AX)
,
X
I,V.
Hence the balance equations are to be written at t=0 [11]:
AI
I
)
net
(
I
C
C
C
,
(21)
AV
V
)
net
(
V
C
C
C
,
(22)
2

78
where C
X
(net)
, X
I,V, is the concentration of net point defects determined by MCS.
Assuming that a
deviation from equilibrium for the pairing reaction is small, we
can write, using the above described formalism, the following equations [11]:
I
I
I
A
AI
C
C
C
,
(23)
V
V
V
A
AV
C
C
C
.
(24)
Hence the initial conditions to reaction

diffusion
equations (1)

(
5
) can be
determined by solving numerically the set of non

linear algebraic equations (
20
)

(2
4
)
together with the condition of local electroneutrality (
6
) and
expressions (10), (11)
which link together the concentrations of species in differ
ent charge states. After that,
numerical solution of the whole problem (1)

(6) can be performed.
3.
CONCLUSION
Thus, a closed system of equation describing the diffusion of dopants during RTA
after ion implantation is formulated taking into account all t
he possible charge states
of the species. Currently, the work on numerical solution of the formulated problem is
underway. Computer simulation for particular systems using the parameter values
available in literature will permit determining the optimal reg
imes of ion implantation
and subsequent RTA for obtaining a desirable profile of dopants and hence for
producing USJ with required current

voltage characteristics.
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