Download EXTENDED FIVE-STREAM MODEL FOR DIFFUSION MASS

EXTENDED FIVE-STREAM MODEL FOR DIFFUSION MASS
TRANSFER OF IMPLANTED DOPANT ATOMS IN
SEMICONDUCTOR SILICON
Boris B. Khina1, Valeriy A. Tsurko2 and Galina M. Zayats2
1
Physico-Technical Institute, National Academy of Sciences, Minsk, Belarus
2
Institute of Mathematics, 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 junctions 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 “fivestream” 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 atom-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 atoms, 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 miniaturization. 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, ultrashallow 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 latter
is healing the lattice defects generated during implantation and performing electrical
activation of the dopant atoms. However, during RTA, as well as during other kinds
of post-implantation thermal treatment (e.g., spike annealing), the phenomenon of
transient 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 of extensive experimental investigation because it hampers obtaining the
optimal concentration 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 ascribed to the interaction of diffusing species with nonequilibrium point defects (vacancies and silicon self-interstitials), which are
accumulated in silicon due to ion damage, and with small clusters that form and
dissolve in the course of diffusion. Solving the intricate problem of TED suppression
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is impossible without mathematical modeling of this complex phenomenon. However,
modern technology computer-aided design (TCAD) software packages such as
SUPREM4 (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 implemented in popular package
SUPREM4, 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 positions), in
crystalline silicon, which possesses a diamond-type lattice, diffusion of impurity
atoms can proceed only by indirect mechanisms [8]: diffusion of pairs “dopant atomvacancy” (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 reactiondiffusion 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 this
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 MODEL
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:
CI/t = div JI  RIV + RAI + RAVI ,
(1)
CV/t = div JV  RIV + RAV + RAIV ,
(2)
CAV/t = div JAV + RAV  RAVI  RAVAI ,
(3)
CAI/t = div JAI + RAI  RAIV  RAVAI .
(4)
Since the dopant atoms located in the lattice sites are considered immovable,4-7
the balance equation is written
CA/t = RAIV + RAVI + 2RAVAI  RAI  RA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:
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C A 
 C (AI)   C (AV)  
 1
 1
C
 1, 2
I



 1, 2
C
V

pn0
(6)
Here CY, 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, RYZ are the reaction terms describing the rate of interaction between
different species (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 atoms in the lattice sites,  = +1 for donor atoms (A  A+ = As+, P+,
Sb+) and  = –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
A1A and A2, the system of Eqs.(1)-(4) should be supplemented with similar
equations for diffusion of pairs A2I and A2V, an equation similar to Eq.(5) should be
written for dopant A2, the reaction terms accounting for interaction of pairs A1X and
A2X, XV,I, should be added to the right-hand side of all the reaction-diffusion
equations, and the terms describing the charges of pairs A2X and atoms A2 in
substitutional positions are to be added to Eq.(6).
2.2. Diffusion fluxes
To formulate expressions for diffusion fluxes JY, 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:
J
X

 D
X

Here D
C
X

X


X
qC
X

.
(7)
is the diffusion coefficient of point defects in charge state , 
X

is
their mobility, q is the charge of electron,  =  is the vector of the electric field
strength,  is the electric potential. Equation (7) is simplified by applying the
Einstein’s formula for mobility    D  /( k BT) , where kB is the Boltzmann
X
X
constant, and using the Boltzmann’s distribution of charged particles in a potential
field: n = ni exp[q/(kBT)]. Here ni is the intrinsic concentration of charge carriers;
2
from the Boltzmann’s distribution it follows that n i  np , where n and p are the
density of electrons and holes, correspondingly. Assuming, similarly to Refs.[5-8],
that the diffusion coefficient of point defects is independent of their charge
D X  D  , from Eq.(7) we obtain the following expression for the diffusion flux of
X
point defects

p 
J X   D X  C X  C X  X  ln ,
ni 

X
1

X
2
 p
 K X  n
 i
  2


 ,


X
2
 p
  K  
X
 ni
  2
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

 ,


(8)
where K
X

are the equilibrium constants for ionization reactions X0 + e–  X,
which are known in literature in the Arrhenius form (it is obvious that K
X
0
 1 ).
Similarly 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
J AX
X
1

X
1
 p
 D (AX) K AX K X  n
 i
  1
 2
 p
  K   K  
A X
X
 ni
 0
where K




 


p 
 C AX  C AX X  ln  ,

X
n i 



 p
 ,  X  K  0  K  2 K 2 

n
A X
A X
X

 i




2
,
(9)
are the equilibrium constants for pairing reactions A + X 
 
A X
(AX), =0,1, XV,I, i.e. K
AX
C
*
( AX)
C
*
*
C
A X 
; 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 X0 + e–  X and summarizing over all
the charge state of point defects, we obtain
C
X

K
X

CX  p

 X  n i


 , =0,12, XV,I.


(10)
Similarly, for pairing reaction A + X  (AX) we have
C
( AX)

C AX
X
 p
K    K   
A X
X
 ni




 
, =0,1, XV,I.
(11)
The concentration of equilibrium point defects in charge state , which will be used
further, is determined as
*
C
*
X
K
X
CX
*
X
,  *X 
2

  2
K
X
, XV,I,
*
(12)
where CX is the equilibrium concentration of point defects (XV,I) in all the charge
states, which is known in literature.
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2.4. Reaction terms
The next step is the formulation of sink/source terms RYZ, Y,ZI,V,AI,AV, YZ,
which enter the right-hand side of Eqs.(1)-(5). For this purpose, small deviation from
the local equilibrium for a corresponding bimolecular reaction is assumed. For one
charge state (e.g., =0) the recombination rate of vacancies and silicon selfinterstitials is expressed as R I V  4r (DI  DV )(CICV  C*IC*V ) , r is the capture
radius; typically it is considered that r=a0 where a0 is the crystal lattice period of
silicon. Summarizing over all charge states, we obtain
C C
*
*
R I  V  4a 0 (D I  D V ) I  V  I V  C 0 C 0
I
V
 I V

,


 IV   I   V  1 .
(13)
Let us consider terms RAI and RAV that describe the kinetics of pairing
reactions using acceptor dopant 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 K     k f / k r , where k f and k r are the rates
A X
of
forward

R A X
and

 k f C  C  
A
X
reverse

 kr C
,
( AX)
reactions,

kf
we
can
write:
 4rD X . Then, summarizing over all
the charge states and assuming r=a0, we obtain
R A X
2
C C
 p
C 
 4 a 0 D X  X  A X  AX  ,  X   K  
X
X 
 X
 ni
 0


 .


(14)
Using a similar approach for bimolecular recombination reactions of pairs (AI)
and (AV) with vacancies and silicon self-interstitials, correspondingly, we obtain the
following expressions:
*
*
I0
* *
4 a 0 (D I  D AV )( C AV C I  C A C I C 0
V
R AI V  4 a 0 (D V  D AI )( C AI C V  C A C V C
I) ,
R AV I 
V) .
(15)
, XV,I.
(16)
Here terms DAX are determined as follows:
D AX
1

X
1
 p
 D (AX) K AX K X  n
 i
 1




 
To derive an expression for bimolecular recombination reaction AV + AI 
it is assumed that its kinetics is independent of the charge state of pairs AX.
Then, using the method described above for Eq.(14), we receive
2A,
2
*
I
R AVAI  4 a 0 (D AV  D AI )( C AV C AI  C A C 0 C
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*
V0
IV) .
(17)
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 closed 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:
CX(x=0) = CX(x=L) = CX(y=0) = CX(y=W) = CX*, XI,V.
(18)
This is because the outer surface of a crystalline solid is typically considered as a sink
for point defects of infinite capacity, and outside the implanted domain the
concentration of point defects corresponds to thermal equilibrium.
For diffusion of pairs AX, the second-kind, or Neumann boundary conditions to
Eqs.(3),(4) are formulated:
JAX(x=0) = JAX(x=L) = JAX (y=0) = JAX(y=W) = 0.
(19)
Hence, different-type boundary conditions are posed to the conjugate system of
diffusion equations (1)-(4).
Formulating the initial conditions , CY(t=0), YI,V,AI,AV, is a more difficult
problem, which is typically passed over in silence in literature on modeling TED [47]. In experiments, 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 CAX,
XI,V. Therefore at t=0 we have to write
(SIM S)
CA
 C A  C AI  C AV ,
(20)
where C (ASIMS) 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 temperature. 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]:
( net )
 C I  C AI ,
(21)
( net )
 C V  C AV ,
(22)
CI
CV
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where CX(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]:
C AI  C A
C AV
CI
 ,
I I
C
 CA V V .
V
(23)
(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)-(24)
together with the condition of local electroneutrality (6) and expressions (10), (11)
which link together the concentrations of species in different 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 the 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 regimes 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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