Point defects density influence on the silicon oxide ultrathin film mechanical stress by molecular simulation

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Journal of Non-Crystalline Solids 221 (1997) 97-102

Letter to the Editor

Point defects density influence on the silicon oxide ultrathin film mechanical stress by molecular simulation A. Tandia *, S. Aouba LAAS / CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse cedex, France Received 26 June 1996; revised 7 July 1997

Abstract We investigate the influence of the vacancy densities on the oxide ultrathin films, et the mechanical stress minimization. By means of many body potential energy function, Si (two unit lattices in each direction), (001) oriented, on which the et cristobalite is variation of the stress in the region of the interface and in the oxide upper layers with © 1997 Elsevier Science B.V.

1. Introduction Often, electronic devices and integrated circuits do not work as expected once they are produced. A critical reason is the existence of some defects in the bulk material or at the interface, which are issued from the material itself or are introduced during various steps of its processing. It is well known that defects play a main role in the semiconductors chips thanks to their interactions with free carders. Sometimes, defects in MOS chips are generated by the interfacial stress relaxation because dielectric films under tension often relieve their stresses by viscous flow during anneals. So it seems very interesting to study the influence of the defects density on the stress distribution in S i / S i O 2 based chips. To better understand the stress relaxation phe-

* Corresponding author. Tel.: +33-5 61 33 69 30; fax: +33-5 61 33 62 08; e-mail: [email protected]. Presented at the Franco-Italian Conference on Structure and Defects in SiO 2, Fundamentals and Applications, Agelonde, France, 23-25 September 1996.

cristobalite structure, for the need of we analyze a structure composed of grown. The simulation points out a the defect densities in the substrate.

nomena, we analyze the influence of the defect densities on the stress distribution. The lack of performant tools able to give atomic information of the interface without having any influence on the atoms distribution in the materials [1] and the difficulties in the microscopic images interpretations are good reasons to use simulation at the atomic scale. In this work, we first focus on the simulator description and validation. Second, we will describe the procedure to study the defect densities influence on the stress distribution. The silicon substrate crystalline orientation and the temperature of the oxidation process are the two parameters we will use to investigate the link between defect densities and stress in S i / S i O 2 based chips. The third part is devoted to the simulation results discussion.

2. Simulation description Many studies have been done in the past to analyze the influence of the defect densities on the S i / S i O 2 mechanical stress, but many of them are

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A. Tandia, S. Aouba / Journal of Non-Crystalline Solids 221 (1997) 97-102

98

essentially based on experimental tools. Since the experimental tools may interact with the atom patterns, which is in our case on ultrathin film (thickness ~ 81 nm), it seems more realistic to use atomic simulation. For the small systems that have physical processes such as oxidation, it has been shown elsewhere [2] that the many body interaction potential model is a powerful tool [3,4]. The ab initio approach and the molecular mechanics are not suitable in the case of defect densities influence analysis because the effects do not really consider the distortions and even the disorder of the oxide layer. These factors are the main reasons why the many body interaction model for the defect densities effects on the stress distribution analysis for Si/SiO 2 based chips at the atomic scale is chosen. The many body potential model we use in this simulation is limited to the two first terms. The features of these potentials are well known to be l

N

N

E = 2--~.E

E

V0~2)

a a:#/3

X(R[o,/3])

1

+ T . ' E E E V 0 '3~ o,

/3

Si-Si-Si

O-O-O

Si-O-O

Si-Si-O

5276.96

22.59

385.71

486.58

eter for two-atoms interaction and Z~t~ that for three-atoms, 0n is the angle formed by a triplet of atoms. Tables 1 and 2 give the values of the different simulation parameters. We suppose that the silicon oxidation is due to oxygen molecules. Each molecule that arrives on the substrate surface goes into two atoms and each one tries to reach a more stable position in order to minimize the structure's energy. For each type of atom in the structure (O, Si), we define a critical radius, ~, for the limitations of its interactions, which is the radius of the sphere containing the other atoms that may interact with that atom at its center. For a model using the Lennard-Jones potential, a critical radius less or equal to 2.5/' is advised [5], with

~...

(4)

(1)

where Vo{2) and Vo(3) are respectively the potentials of Lennard-Jones and Axilrod-Teller:

/3) = ~""t[, R[,~/3] )

i-j-k Zijk (eV 9)

F = d,~

y

x(R[~/3], R[~,/], R[/3~,1),

vg,(o~,

Table 2 Parameters of the Axilrod-Teller potential

Rio,/3] (2)

The Metropolis test [6] is the main point of our simulator because it manages the energy minimization. Each atom in the limiting sphere moves in a random direction in the space, for energy minimization. For the equilibrium state, each atom moves in a random way. The initial step (S) for the considered atom is defined as follows:

1 + 3 cos 0~ cos 0a cos 0~

v0~3~(~,/3, ~) = z~, (R[~/3IR[~]R [ / 3 r ] )

Zd~

3' S=

(3) R~¢ is the distance between atoms c~ and /3, d ~ is their distance when in equilibrium, e ~ is the paramTable 1 Parameters of the Lennard-Jones potential

NP

'

where N is the number of atomic species in the structure, P [7] is the sampling of the mean interatomic distance at the equilibrium state. We calculate the difference between the final and the initial energies for the considered atom A E = E l - E i.

i-j

Si-Si

O-O

Si-O

Eij (eV) dlj (,~)

3.24 2.25

5.15 1.20

4.06 1.62

(5)

(6)

If A E < 0, the displacement is accepted and we take another atom. If A E > 0, the displacement can be accepted with a certain probability.

A. Tandia, S. Aouba / Journal of Non-Crystalline Solids 221 (1997) 97-102

To realize that choice, we generate a random number, n, in the range [0,1 ]. If the condition on the Boltzmann factor e -AE/kT > n

on the calculation of the oxidation rate compared to the experimental data of Morita et al. [8] (Fig. 1). It is important to notice that the initial distance between the oxide and the substrate must be less than the smallest interaction critical radius of the different species in the structure (Si, O), otherwise some interactions in the interface will not be considered. After deposition, we choose an ambient temperature T = 300 K and relax the structure to minimize the final energy of the system. At the equilibrium state, the interface is already in its stable configuration. For analyzing the defect densities affects, we consider silicon SiO 2 ultra thin layers in the aim of a fundamental understanding. The grown silicon dioxide small layer ( a cristobalite) [9] is composed of 208 SiO 2 molecules (thickness about 7 nm, Fig. 2). We introduce, for fixed temperature and crystalline orientation, increasing defect rates in a (001) silicon substrate with two silicon unit lattices in each direction (2 × 2 × 2). The defects we consider here are the vacancies with their densities ranging from 0 to 4 per silicon unit lattice. For a defect chosen rate, n defects per

(7)

is satisfied for an atom then its displacement is accepted otherwise it is refused and that atom remains in its initial position. The step S is changed during the simulation in order to balance the ratio (R) of accepted and refused atomic displacements (balance in the range [40% to 60%]). The step modification law is R < 40% ~ S = 1.1S,

(8)

R > 60% ~ S = 0.9S;

(9)

elsewhere the step is kept constant. If the ratio R is small (less than 40%) then there are more accepted atomic displacements and the Boltzmann factor, e aE/kr, is great in the range [0,1]. To increase the ratio, we have to decrease the previous factor; that means we increase A E or the step S. The opposite reasoning is substantial for great ratio. The sensitivity of the simulator to the temperature is expressed by the Boltzmann factor in the Metropolis test. The validation of the simulator lies mainly

J

+

5.5

0 +

4.5

99

Morita data simulation

== 4 3.5 0 3

2.5

I

20

5~

1~0

15 2000 O~dationTime(mn)

2500

3000

Fig. 1. Comparison between Morita experimental data and our Si oxidation simulation. Lines are drawn as guides for the eye.

100

A. Tandia, S. Aouba / Journal of Non-Crystalline Solids 221 (1997) 97-102 z

A ' -

/

silicon unit lattice ( n / u l ) , we generate in a random way n defects in each one of the eight silicon unit lattices. We divide the structure (substrate and oxide) into several layers and calculate in each one the mean stress per atom which is defined as follows [10,11]:

70 ,~s

/

~,.i[ ~

]

= } E {v,,=~[,n3 ]R[ o,t~ 1,~,[,~

],~j[,~ ])

/34= ot

Oxide

+-'3 E {E

{R[.t3 ]v('~[ ~# ]¢,[ ~# ]

× ~[ ,~# ] + R[ ,~./] v(3~[ ,~, ] ~,[ ,~, ] ~j[ ,~, ] Substrate 10.8/~s

+e[vt3 ]Vl~[ v~ ] ~,[vt3 ] ~[vt~ 1}}

/ i 0

/

' "

/ 10.8 ~s

i

/~,

Y

with

X

q~i[ ol/3 ] =

x;[. ] - xi[ # ] R[ oz/3 ]

(11)

Fig. 2. Geometry of the simulated structure.

Stress at the interface

10~s 16

i

I

14

.=0/ul

x=l/ul +=2/ul

12

(lO1

tO.8 ,~,s

*=3/ul o=4/ul

~8 g m 6 4 2

05= =

"

=1o

~v

"~" " 15"" Distance (A)

Fig. 3. The Si/SiO 2 interface stress dependence on the vacancies densities. Lines are drawn as guides for the eye.

A. Tandia, S. Aouba / Journal of Non-C~stalline Solids 221 (1997)97-102

q~i[ cx/3 ] is the ith Cartesian coordinate axis,

a V(on) V(")[ a ] - a0R[ -~--] "

(12)

In the case of small stress relaxation propagation, due to the small size of the structures and the small variation of the displacement of the atoms for energy minimization, we can admit the following equality

Vy )

s]

R[ s---l"

(13)

With l atoms in the considered layer m, the mean stress per atom in this layer is a=/

1

Si~= E -f%i[a] •

(14)

o~=1

We are interested here on the stress across the layer

101

Fig. 3 deals with the defects rate effect on the substrate and the oxide first layer stresses. In that figure, we observe that the substrate stress is not affected by the defect densities. The defect densities in the substrate seems to have no effects on the stress distribution because it is well known that the defects are inclined to sink into clusters for dislocation formation. Fig. 3 also shows a stress variation in the first oxide layers for a substrate without defects (0/ul). This variation lies in a thickness of about 1 nm in the oxide. The stress in the Si/SiO 2 interface region is maximal when the defect density is one defect per silicon unit lattice ( l / u l ) . Large stress magnitude in the region of the interface can be explained by the adjustment of the crystalline structure [9]. So the stress classification at the Si/SiO 2 interface is as follows:

m.

1/ul > 3 / u l > 2 / u l > 4 / u l > 0 / u l 3. Discussion

Figs. 3 and 4 represent the plots of the square stress per atom versus the distance of the containing layer from the back of the substrate. x 15

1027

in the range [0, 3.74 × 1014 N/m2]. Fig. 4 deals with the influence of defects rate on the oxide buried and upper layers. In the oxide buried layers, in the thickness range [2.5 to 6.8 nm], we observe a weak stress dependence on the defects

Stress in the oxide i

.=0/ul x:l/ul +=2/ul *=3/ul <~10

o=4/ul

Cq

E

co 5

:~' 60 " ~" , , 46'2

' ' "~4

, 66 -

(15)

~ .... 76

72

74

I 76

~ F ....80

Distance (A)

Fig. 4. The SiO 2 stress dependence on the vacancies densities. Lines are drawn as guides for the eye.

102

A. Tandia, S. Aouba / Journal of Non-Crystalline Solids 221 (1997) 97-102

rate, Thus, at 6.6 nm from the substrate back, there is a maximum stress for a perfect substrate with a magnitude of 3.16 × 1013 N / m 2. But as we approach the top of the oxide, the stress is affected by defect density. We observe in Fig. 4 that the stress in the oxide upper layers (about 7.0 nm from the substrate base) increases with a smaller magnitude than that at the interface region. The stress distribution in the oxide upper layers can be explained by the surface reconstruction of dangling bonds. It is generally known that atoms of the surface are inclined to sink into dimers for the surface energy minimization [3,4]. The stress magnitude classification is as follows: 0 / u l > 2 / u l > 1/ul = 3 / u l > 4 / u l

(16)

in the range [0, 1.09 X 1014 N/m2].

4. Conclusion In this work, the influence of vacancy densities on the silicon dioxide ot cristobalite has been calculated. We observe a variation of the Si/SiO 2 interface stress with the defect densities and a weak dependence of the oxide upper layer stress on that density. It appears that there is no significant classification of

the stress distribution in the oxide layers with increasing or decreasing defect densities. Nevertheless, we observe that the stress is non-existant in the oxide when the substrate is perfect, except in the upper layers where that density produces a stress. The study of the interface stress points out that the density of 1/ul gives the greatest stress.

References [1] J.S. Johannessen, W.E. Spicer, J. Appl. Phys. 47 (1976) 7. [2] T. Halicioglu, P.J. White, J. Vac. Sci. Technol. 17 (5) 0980) 1213. [3] A. Tandia, Microelectron. Eng. 33 (1997) 423. [4] A. Tandia, G. Sarrabayrousse, A. Martinez, Solid Thin Films 296 (1997) 122. [5] D. Levesque, L. Verlet, J. Kurkijarvi, Phys. Rev. A7 (1973) 1690. [6] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, J. Chem. Phys. 21 (6) (1953) 1087. [7] K. Binder, Monte Carlo Methods in Statistical Physics (Springer, Berlin, 1979). [8] M. Morita, T. Ohmi, E. Hasegawa, M. Kawa-kami, M. Ohwada, J. Appl. Phys. 68 (3) (1990) 1272. [9] W.G. Wyckoff Ralph, Crystal structures, 2nd ed., vol. 1, (Interscience, Berlin, 1956). [10] A. Tandia, D. Esteve, submitted for publication. [ll] D.C. Wallace, Thermodynamics of Crystals (Wiley, New York, 1972).