Политематический сетевой электронный научный журнал Кубанского государственного аграрного университета, 2013, №90

Научный журнал КубГАУ, №90(06), 2013 года

http://ej.kubagro.ru/2013/06/pdf/51.pdf

УДК 531

ВЛИЯНИЕ ЗАВИСИМОЙ ОТ СТРЕССА
ПОВЕРХНОСТНОЙ ГЕНЕРАЦИИ
МЕЖДОУЗЛИЙ НА РОСТ ДЕФЕКТОВ
УПАКОВКИ И ДИФФУЗИЮ ЛЕГИРУЮЩИХ
ПРИМЕСЕЙ В КРЕМНИИ

Лойко Константин Валерьевич
Ph.D. 
Freescale Semiconductor, Остин, Техас, США
 
Разработана модель зависимой от стресса
поверхностной генерации и рекомбинации
точечных дефектов в кремнии. С ее помощью
смоделированы такие явления, как рост дефектов
упаковки и диффузия легирующих примесей в
кремнии
 
Ключевые слова: КРЕМНИЙ, МЕЖДОУЗЛИЯ,
ТОЧЕЧНЫЕ ДЕФЕКТЫ, ДЕФЕКТЫ УПАКОВКИ,
ДИФФУЗИЯ ЛЕГИРУЮЩИХ ПРИМЕСЕЙ

UDC 531
 
INFLUENCE OF STRESS-DEPENDENT
SURFACE GENERATION OF INTERSTITIALS
ON  STACKING FAULT GROWTH AND
DOPANT DIFFUSION IN SILICON
 

Loiko Konstantin Valeryevich
Ph.D. 
 Freescale Semiconductor, Austin, Texas, U.S.A. 
 
A model is developed for stress-dependent surface
generation and recombination of point defects in
silicon. Using the model, such phenomena as stacking
fault growth and stress-mediated dopant diffusion in
silicon are simulated

 
Keywords: SILICON, INTERSTITIALS, POINT
DEFECTS, STACKING FAULTS, DOPANT
DIFFUSION

Introduction

Silicon self-interstitials play an important role in the growth of stacking faults and

contribute to dislocation nucleation. The effects of stress on dopant diffusion in silicon are

also attributed to the behavior of intrinsic point defects in stress fields [1-5]. 

Equilibrium  concentrations  of  silicon  self-interstitials  and  vacancies  are  known  to

depend on stress as follows [6,7]:

)
exp(
*
0
*

kT
V
P
C
C
i
i
i





,                                                       (1)

)
exp(
*
0
*

kT
V
P
C
C
v
v
v




,                                                         (2)

where  
 and  
 are  the  equilibrium  concentrations  of  interstitials  and  vacancies,

respectively, in the presence of the stress field, 
 and 
 are the stress-free equilibrium

concentrations,  
is  pressure,  
 and  
 are  the  activation  volumes,  k  is  Boltzmann's

constant, T is absolute temperature. The terms 
 and 
 describe a reversible work

process of the point defect formation under stress. They show a change in the Gibbs free

energy of the formation of an interstitial or vacancy due the presence of the stress field [7].

Научный журнал КубГАУ, №90(06), 2013 года

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Therefore, the activation volumes in Equations 1 and 2, being the coefficients of pressure in

the reversible work process, are indeed thermodynamic formation volumes. These activation

volumes have been calculated based on the assumption of sphericity of the interstitial and

vacancy with radii of 1.11 Å and 2.47 Å, respectively [1-3], or using ab initio calculations

[4,5,8]. Nevertheless, there is no agreement in literature on the values for these parameters.

From Equations 1 and 2, the equilibrium concentration of self-interstitials decreases

and the equilibrium concentration of vacancies increases under compressive stress. Tensile

stress has an opposite effect on the equilibrium point defect concentrations. Since boron and

phosphorus  are  known  to  diffuse  mainly  in  pairs  with  interstitials  [9],  retardation  of  the

diffusion of these dopants is expected under compressive stress. There is some experimental

evidence of such retarded diffusion [3,10]. At the same time, boron diffusion enhancement

under compressive stress has been reported as well [11,12]. This discrepancy could possibly

be  explained  by  differences  in  point  defect  interaction  with  the  free  surface  [4].  Most

models  of  stress-dependent  point  defect  and  dopant  diffusion  consider  only  bulk

interactions.  However,  stress-dependent  surface  generation  and  recombination  of  point

defects may cause significant changes in their distributions. These processes are not well

understood due to a lack of experimental results.  

In this paper, a model is developed for the stress-induced redistribution of intrinsic

point defects in silicon. It incorporates equilibrium conditions different for defects at the

surface and in the bulk of silicon, taking into account stress-dependent surface generation

and  recombination  of  point  defects.  Using  the  model,  such  phenomena  as  stacking  fault

growth and stress-mediated dopant diffusion are simulated.

Calculation of Activation Volumes for Interstitials and Vacancies

In  the  derivation  of  the  activation  volumes  for  intrinsic  point  defects,  silicon  is

considered as an elastic continuum. A Si-interstitial is modeled as a non-compressible ball.

The radius of the ball equals to the radius of a silicon atom, Ra = 1.32 Å [13]. The ball is

inserted into a spherical hole having the radius Ri of an interstitial site. The radius of an

interstitial site is the maximum radius of a ball that can be inserted into the interstitial site

Научный журнал КубГАУ, №90(06), 2013 года

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without deforming it. Two interstitial sites in silicon are considered. First, the octahedral

site, such as the one which center has coordinates    in a unit cell of the silicon

lattice. The coordinates of the six nearest lattice sites are  ½, ½, 1;  ½, 1, ½;  1, ½, ½;  3

3;  33;  33. The distance between the center of the octahedral site and these

lattice sites is 
11 a0/8. Therefore, Ri = 11 a0/8 - Ra ≈ 0.93 Å, where a0 = 5.4307 Å is the

silicon lattice constant. The second interstitial site is the tetrahedral one, for example the site

with coordinates The four nearest lattice sites are  ;  33;  3

3;  33. Therefore, Ri =
3 a0/4 - Ra ≈ 1.03 Å. The preferential interstitial site (the

bigger one) has Ri  ≈ 1.03 Å. The difference of volumes of the Si atom and interstitial site

ΔV = 4/3π(Ra3 - Ri3) ≈ 5 Å3 might be regarded as the activation volume of self-interstitials.

However, this approach does not take into account such factors as "compressibility" of the

silicon atom, chemical-valency effects, and local relaxations [7]. Thus, the derived value is

overestimated. A more reasonable activation volume of Si-interstitials seems to be ΔVi =

V

2
1

 ≈ 2.5 Å.3

A vacancy is considered to be formed by removing a spherical ball from the elastic

continuum of silicon resulting in the formation of a hole. The radius of the ball is Ra. The

size of the hole becomes smaller than the size of the ball because of surface tension. The

activation volume of a vacancy is the difference in volumes of the ball Va and the hole Vh.

The surface energy of the hole before contraction E0 = 4πRa2γ, where γ ≈ 1.5 J/m2 is the

silicon surface energy [13]. After the contraction by ΔR, the energy of the hole E = 4π(Ra 
ΔR)2γ. Thus, the total force of surface tension F ≈ (E0 – E)/ΔR  ≈ 8πRaγ.  This gives the

contracting pressure P = F/(4πRa2) = 2γ/Ra. This pressure causes the reduction in the radius

of the hole by u = 0.5PRa(1+ )/Y, where 
 ≈ 0.28 is Poisson’s ratio and Y ≈ 170 GPa is

Young’s modulus of silicon [14]. Then, u  =  γ(1+ )/Y  ≈ 0.113  Å. Finally,  the  activation

volume of a vacancy is found as ΔVv = 4/3π{Ra3 – (Ra-u)3} ≈ 2.3 Å.3

The  obtained  values  for  the  activation  volumes  of  point  defects  are  close  to  those

Научный журнал КубГАУ, №90(06), 2013 года

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suggested in some reports [2,3,8].

Diffusion Equations for Point Defects

The following diffusion equations are considered for interstitials and vacancies in the

bulk of silicon:

b
R
J
t
C






,                                                 (3)

)
(
*
*

C
C
DC
J




,                                                (4)

)
(
*
*

v
i
v
i
b
b
C
C
C
C
K
R


,                                           (5)

where C , 

*
C , D , and J are the concentration, equilibrium concentration, diffusivity, and

flux  of  interstitials  or  vacancies,  
b
R  and  
b
K are  the  bulk  recombination  rate  and  its  rate

factor, i  and v  denote parameters for interstitials and vacancies, respectively.

Equation 4 is different from the standard expression for the diffusion flux 
C
D
J




[15] and accounts for non-uniformity in the equilibrium concentrations of point defects due

to their interaction with external fields. Substituting Equations 1 and 2, which describe the

exponential  dependency  of  the  equilibrium  point  defect  concentrations  on  pressure,  into

Equation 4, and ignoring the drift of charged point defects in the electric field, it is possible

to express the total point defect flux as a sum of the diffusion flux and the drift flux in the

pressure field:

 

P
kT
V
C
D
C
D
J
i
i
i
i
i
i







,                                     (6)

.                                   (7)

Surface Generation and Recombination of Point Defects

At a stress-free surface, self-interstitial and vacancy fluxes due to surface generation

and recombination are given by [15]

,                                      (8)

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)
(
*
0
v
v
vs
vs
C
C
K
J



,                                             (9)

where  

*
0
i
C
 and  

*
0
v
C
 are  the  stress-free  equilibrium  concentrations  of  interstitials  and

vacancies, 
is
K  and 
vs
K  are their surface generation-recombination rate factors, and 
is
G  is the

interstitial  injection  rate  due  to  oxidation.  Terms  
i
isC
K
 and  
v
vsC
K
 describe  surface

recombination, while 

*
0
i
isC
K
 and 

*
0
v
vsC
K
 describe surface generation of point defects. 

In the presence of the stress field, these equations have to be modified to account for

the free energy change associated with the surface generation-recombination process:

is
i
s
i
i
s
i
is
is
G
kT
V
V
P
C
kT
V
V
P
C
K
J











)]
)
(
exp(
))
(
exp(
[
*
0
,        (10)

)]
)
(
exp(
))
(
exp(
[
*
0
kT
V
V
P
C
kT
V
V
P
C
K
J
v
s
v
v
s
v
vs
vs












,              (11)

where  
s
V

 is  the  silicon  volume  change  occurring  when  a  silicon  atom  is  added  to  or

removed from the silicon surface. 
s
V

 ≈ 20 Å3, which is the volume of silicon per atom. 

s
V


iV


Si

Figure 1: Self-interstitial formation in silicon by surface generation. ΔVs > ΔVi.

The  process  of  surface  generation-recombination  of  point  defects  is  illustrated  in

Figure  1  showing  the  generation  of  a  Si-interstitial.  When  an  interstitial  is  formed  by

removing  an  atom  from  the  silicon  surface  and  inserting  it  into  the  crystal,  there  is  a

thermodynamic volume expansion 
, associated with the fact that the size of the Si atom

is larger than the size of the interstitial site. On the other hand, there is a decrease in the

volume of the whole Si crystal because one atom has left its surface. This volume change is

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equal to the volume of silicon per atom 
s
V

 ≈ 20 Å3 > 
iV

. The difference of these two

volumes (
i
s
V
V



) is found in the exponents in Equation 10, which shows how interstitial

surface generation and recombination change under pressure. 

As follows from Equations 10 and 11, the surface generation of interstitials, leading

to the thermodynamic volume decrease, increases with pressure while their recombination

decreases. The effect of pressure on vacancies is opposite. Since (
V
Vs



) is positive for

both  interstitials  and  vacancies,  interstitial  generation  increases  under  compression  and

vacancy generation increases under tension. 

Under the equilibrium conditions, the surface flux of point defects is zero. Therefore,

Equations 10 and 11 effectively describe the deviation of equilibrium defect concentrations

at the surface from their values in the bulk. 

Simulation of Stacking Fault Growth

Stacking faults in silicon are oxidation-induced defects. Therefore, their preferential

growth in the field areas, where interstitial injection due to oxidation has the highest rate,

seems logical and is supported by experimental results. At the same time, the appearance of

stacking  faults  at  the  edges  of  active  areas  has  been  reported  by  Jarreau  [16].  An

enhancement  in  stacking  fault  growth  at  active  edges  can  be  explained  by  either  an

increased  density/size  of  their  nucleation  centers  due  to  some  damage  (for  example,

plasma-induced  damage)  or  by  enhanced  interstitial  generation  in  this  area.  The  later  is

consistent  with  the  proposed  model  for  stress-dependent  surface  generation  and

recombination of point defects.

Figure 2 shows simulated pressure, equilibrium concentration and  concentration  of

interstitials, as well as stacking fault radius after steam field oxidation performed at 1100

C.  The  results  were  obtained  using  a  2D  process  simulator  TSUPREM-4  [17].

Stress-dependent  surface  generation  and  recombination  of  point  defects  was  taken  into

account. As can be seen, pressure is at maximum under the nitride edge (Figure 2a). This

decreases  equilibrium  concentration  of  interstitials  in  this  area  (Figure  2b),  but  increases

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their concentration due to enhanced surface generation (Figure 2c). Consequently, stacking

fault size is also increased under the nitride edge (Figure 2d). The stacking fault radius at

the position of the pressure peak is about the same as that under the field oxide. A uniform

density  of  stacking  fault  nucleation  centers  and  their  constant  size  was  used  in  this

simulation. To obtain even larger stacking faults at the active edge, higher density or larger

size of their nucleation centers should be assumed.

Figure 2: Contour plots of simulated (a) pressure (Pa), (b) equilibrium concentration and (c) concentration
of  interstitials  (cm-3),  (d)  stacking  fault  radius  (Å)  after  steam  field  oxidation  performed  at  1100  C.
Stress-dependent surface generation and recombination of point defects was taken into account.

Simulation of Stress-Mediated Dopant Diffusion

The effect of stress on dopant diffusion in silicon is usually attributed to the behavior

of  intrinsic  point  defects  and  dopant-defect  pairs  in  stress  fields  [1-5].  Since  boron  and

phosphorus  are  known  to  diffuse  mainly  in  pairs  with  interstitials  [9],  retardation  of  the

Научный журнал КубГАУ, №90(06), 2013 года

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diffusion of these dopants is expected under compressive stress reducing the equilibrium

concentration  of  interstitials.  However,  the  existing  models  for  stress-mediated  dopant

diffusion  fail  to  explain  conflicting  experimental  results.  Both  retardation  [10]  and

enhancement of boron diffusion under compressive stress have been reported [11,12]. 

This discrepancy could possibly be explained by considering point defect interaction

with the silicon surface. In different experiments, surface generation and recombination of

point defects could play more or less pronounced role in the defect and therefore dopant

distribution depending upon the proximity of the surface to the studied region.

Using the developed model for the stress-induced point defect redistribution and their

surface  generation  and  recombination,  the  impact  of  stress  on  boron  diffusion  was

simulated. In TSUPREM-4, the five-stream diffusion model was used for boron [18]. This

model assumes that dopant diffusion occurs through the formation of dopant/defect pairs

that diffuse as a unit. The concentration of such pairs depends upon dopant concentration

and point defect supersaturation. Therefore, a change in point defect concentrations due to

their redistribution in the stress fields is expected to directly affect boron diffusion.

In  the  simulation,  an  experimental  setup  used  by Osada  et  al.  [10]  was  replicated.

Boron  with  a  dose  of  7.5    1013  cm-3  was  implanted  at  70  keV  into  a  p-type  (100)  Si

substrate through a 50 nm thick screen oxide.  After 30 min annealing at 900 C in N2 and

removing SiO2, a 500 nm thick Si3N4 film was deposited and patterned. Then an anneal in

N2 at 1014 C for 120 min was performed.

Simulation  results  on  the  pressure  distribution  and  equilibrium  concentration  of

self-interstitials, as well as concentrations of interstitials, boron-interstitial pairs, and boron

are  shown  in  Figures  3  -  6.  Two  groups  of  the  results  are  presented.  In  one  simulation,

stress-dependent  surface  generation  and  recombination  of  point  defects  was  taken  into

account. In the other simulation, it was neglected.

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Figure 3: Simulated (a) pressure (Pa) and  (b) equilibrium concentration of interstitials (cm-3) after 1014 C
anneal in N2 for 120 min. 

Figure 4: Simulated interstitial concentration (cm-3) after 1014 C anneal in N2 for 120 min (a) with and (b)
without taking into account stress-dependent surface generation and recombination of point defects.

As  can  be  seen  in  Figure  3,  the  equilibrium  concentration  of  self-interstitials

decreases under the nitride, in the region of compression. It increases at the nitride edge

under tensile stress. The concentration of interstitials increases under the nitride in Figure 4a

due to their enhanced surface generation. In Figure 4b, the interstitial concentration almost

exactly follows the contours of the equilibrium concentration. 

In Figure 5a, the concentration of boron-interstitial pairs decreases under the nitride

edge due to a lower interstitial supersaturation and slightly increases under the nitride. On

the contrary, in Figure 5b the concentration of boron-interstitial pairs is almost independent

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of a lateral position. Same relative lateral uniformity of the boron concentration can be seen

in Figure 6b.

Figure 5: Simulated concentration of boron-interstitial pairs (cm-3) after 1014 C anneal in N2 for 120 min
(a)  with  and  (b)  without  accounting  for  stress-dependent  surface  generation  and  recombination  of  point
defects.

Figure 6: Contour plots of simulated boron concentration (cm-3) after 1014 C anneal in N2 for 120 min (a)
with and (b) without accounting for stress-dependent surface generation and recombination of point defects.

Figure 6a shows that taking into account the stress-dependent surface generation and

recombination  of  point  defects  results  in  an  increase  in  boron  concentration  under  the

nitride  edge  at  the  peak  of  tensile  stress.  At  the  same  time,  the  boron  concentration  is

decreased in the bulk. This is opposite to what can be seen under the nitride, where stress is

compressive. Near the surface, the boron concentration is lower than that in the open area,

and the contours lie closer to the interface, which might be interpreted as a slower diffusion.