Calculated potential for water enhanced crack growth in silicon

COMPUTATIONAL MATERIALS SCIENCE ComputationalMaterials Science 6 (1996) 63-70

Calculated potential for water enhanced crack growth in silicon W. Wong-Ng

*, G.S. White,

S.W. Freiman, C.G. Lindsay

Ceramics Division, National Institute of Standards and Technology, Gaithersburg,

MD 20899, LISA

Received 1 I October 1995; accepted 10 December 1995

Abstract Ab initio molecular orbital calculations have been performed on the model molecules S&H,,OH (taking into consideration of the crack wall species -OH and -H) for the crack-tip strain simulation study of silicon and its reaction with water. The calculated equilibrium structure of silicon using a 6-31G’ * basis set agreed well with experimental data. It was found that, regardless of the manner strain was applied, the values of net charge on the two crack-tip Si atoms are similar and that both become slightly more positive as strain increases. In addition to this absence of polarization of the crack-tip Si-Si bond under strain, steric hindrance also inhibited the approach of the water molecule to the Si-Si crack tip region. These calculations explain why moisture does not enhance crack-growthvia chemical reaction in Si, which is different from that observed in silica.

1. Introduction

It is known that water enhances crack growth rates in silicate glasses [l-41. Other non-aqueous molecules (i.e., ammonia, hydrazine, etc.), which possess common features such as having at least one lone electron pair (which acts as Lewis base) and a labile proton (which functions as the Brprnsted acid), also enhanced crack growth rates in silicate glasses as well as in sapphire [l-4]. The material that deviates most markedly from the environmentally enhanced fracture criteria mentioned above is silicon (Si) in both single crystal and polycrystalline forms. There has been a report of environmentally enhanced fracture of Si observed in situ in an electron microscope [5], but several studies of crack growth in Si, using a variety of macroscopic

Corresponding author. Tel.: + l-301-9755791; fax: + 1-3019752125.; e-mail: [email protected]. l

test configurations and environments [6], have not demonstrated reproducible behavior that could be attributed to environmentally enhanced crack growth. Specifically, no reproducible crack growth is observed below the crack tip stress at which catastrophic failure occurs. An understanding of why water appears to be ineffective in rupturing strained Si-Si bonds, which is different from its well-known effect of enhancing the rupture of the strained Si-0 bonds in silica, is important both because of the technological issues in the manufacture of silicon chips and because of the insight it would provide into the role of atomic structure on crack growth processes. Molecular orbital calculations (MO) [7-121, which provide basic information on atomic interactions, and which have been shown to agree with crack growth data obtained on silica [ 13-151 were employed for this study. The goals of the current study were to investigate changes in the electronic character of the

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W. Wang-Ng et al./ Computational

64

crack-tip atoms in Si as they are subjected to a tensile stress, both alone and in the presence of water, and to compare the results of the present calculations with those for silica [12-161.

2. Computational

procedure

2.1. Computer sofrware

We performed ab initio molecular orbital calculations using the Restricted Hartree-Fock (RI-IF) technique [7,1 l] on a Silicon Graphics ’ work station as well as on a Cray Y-MP computer I. The computer software suite GAUSSIAN ’ (versions 90 and 92) [ 121was used to compute the equilibrium and strained geometries. Equilibrium geometries were obtained with an optimization procedure which finds the geometry with minimum total energy. The GAUSSIAN programs provide Mulliken population analyses [ 17,181 commonly used to allocate electrons among various parts of a molecule. Mulliken population analysis partitions the overlap population into orbital contributions, which is particularly useful for the comparison of trends in charge distributions and electron donating/accepting tendencies in a series of related geometries. The net Mulliken charges (Q) on an atom, defined as the difference between the atomic number of an individual atom and the gross electron population on that atom, provides an useful estimate of the overall distribution of charge and will be used in this study to compare the crack tip Si-Si bond in different strain environments. Although the absolute values of Mulliken net charge are sensitive to the basis sets used, the trend is reliable.

Materids

Science 6 (1996) 63-70

as models of local environments in silicate crystals. These investigators employed basis sets ranging from the minimal-basis STO-3G* * to the more extensive split-valence 6-31G * * set. The agreement between calculated and observed bond lengths and bond angles was rather good, expecially when the 6-31G* * basis sets were used. Accordingly we used 6-31G’ * basis sets for all calculations in the present study. These basis sets include six d-type polarization functions on each Si and 0 atom, and the three p-type polarization functions on each H atom, that are necessary for accurate description of interatomic bonding and charge density distributions [16,19-231. 2.3. Molecular model 2.3.1. Basic structure of Si The general approach to choosing a model for MO calculations is to use a fragment of the unit cell to represent the solid. Since isolated molecules (i.e., H6Si,0, and Si(OH),) have been shown to be valid models for calculation of bonding parameters in solid silicates [19,20], we adopted this procedure for silicon. MO calculations for a system which contains a large number of Si atoms would require extensive disk space and computer time; the size of our model is therefore limited by computational constraints. Silicon has a diamond-type cubic structure in which all the atoms have tetrahedral symme,_y [24], as shown in Fig. la. Currently no literature model is available which is appropriate for the present study. Most reported quantum chemical studies of Si clusters (as summarized in the review article by Raghavachari [25]) are concerned with the under-

2.2. Basis set In studies of silicate chemistry, Gibbs and coworkers [ 19,201, using MO methods, calculated minimum-energy geometries of silicate molecules used

’ These commercial products are identified in this article to specify the calculation or experimental procedure. In no instance does such identification imply recommendation or endorsement by the National Institute of Standards and Technology.

(a)

(b)

Fig. I. (a) Crystal structure of Si [l]. (b) Two neighboring Si, units used for this study.

W. Wang-Ng et al./Computational

standing of the nature of their structures, stabilities and the fragmentation behavior. It has been shown that the structure of small clusters (Si,-Si,,) are more compact and different from any structure which may be derived from the diamond lattice. A molecular model of a crack tip in an Si crystal should be constructed with a central Si-Si bond that models the crack tip site. The Si atom at either end of this bond should be tetrahedrally coordinated to mimic the Si crystal structure near the crack tip. To create such a molecular model, we choose two Si, units, joined by an Si-Si bond, from the Si crystal structure. Fig. la portrays the Si crystal structure; our molecular model of that structure is shown in Fig. lb. This model uses H atoms to mimic the second coordination shell around the Si atoms joined by the central bond. These H atoms serve to close the valence orbitals of the outer Si atoms in a way that maintains the tetrahedral coordination geometry found in the Si crystal structure. The resulting molecular model has the formula S&H,, (Si,H,Si,H,). It is analogous to the H$i,O, model used in our study of environmentally enhanced fracture in silica [ 121. 2.3.2. Crack wall consideration After a Si-Si bond is broken during crack propagation, the newly separated Si atoms on opposite sides of the crack have the potential to react with

Materials Science 6 (195%) 63-70

65

Table 1 Summary of molecular distortion for strain. Simulation for Si,,H,,OH rl

r2

a

Angles fied

(9 (ii) (iii) (iv) (v) (vi)

E e

l

0.2c 0.46 0.66 0.8~ l.Or

E c E

0.2P 0.4e 0.6~

l l

E

Angles strained

(vii) (viii) (ix)

0.2c 0.46 0.66

r 1 is the distance between Si,-Si,, r2 is the distance between Si , , and Si, to the other 3 connected Si’s. a is the value of the six tetrahedral angles Si,-Si,-Si and Si,-Si,-Si.

species in the environment. Various spectroscopic studies and photostimulated desorption experiments [26-301 have shown that after exposure of the surface of a silicon crystal to water, H atoms and OH groups are found on that surface, but H,O groups are not. This is consistent with simple thermodynamic arguments that favor chemisorption of H and OH over chemisorption of H,O on an Si surface [3 1,321. Accordingly, the molecular model we finally adopted for our calculations was derived from the molecular model of the basic Si structure by replacing one of the H atoms with an OH group. Fig. 2 shows the configuration of this molecule and also a schematic drawing of the crack wall. In this model, all the Si-Si bond lengths and Si-Si-Si bond angles are constrained to be the same. Si ,-Si, is considered to be the crack-tip bond that will ultimately break when sufficient strain is applied. 2.4. Strain simulation

Fig. 2. Model molecules of SisH ,70H for silicon in the presence of water, a sketch of the sharp crack wall along with the -OH and -H groups on the wall is also shown.

Tensile stress is assumed to be applied along the central Si,-Si, bond, which is perpendicular to the crack propagation direction. Familiar measures such as changes in bond lengths and bond angles are used to describe strain [ 131. The response of the Si structure to strain (E) is not known; i.e., whether the strain will be localized on a crack-tip Si-Si bond or whether it will affect

W. Wong-Ng et al./Computationul

66 Table 2 Calculated

equilibrium

and experimental

(Exp’tl) geometry

of SisH,,

and SisH ,,OH as defined in Fig. 2

Bond angle (“)

Bond length (A)

Si-Si Si-H O-H Si-0

Materials Science 6 (1996) 63-70

SisHrs

Si,H,,OH

Exp’tl

2.3122 1.478 1 _

2.3687

2.33 (241 1.48 [33] 0.96 [341 1.63- 1.7 1201

1.4788 0.9425 1.6531

SiEH18

the molecule to a larger extent. Consequently, a total of 9 different ways to apply strain (Table 11 were investigated. The maximum strain values that we quote are for the crack tip Si,-Si, bond. Strains on the surrounding bond lengths and angles are relative to that on the Si,-Si, bond. Some of these strain geometries involve only distortion of bonds while others involve angular distortions as well. Strain was simulated at 3 levels: 10, 15, and 20%. A 10% strain referred to the stretching of a bond, or changing of an angle of 10% from its equilibrium value. In case (i), strain, strain was only applied to the central crack-tip Si,-Si, bond (r 1) (Fig. 2). In the other models, strain was also applied to the bonds and angles around the crack-tip (r2) at various levels compared to the central crack tip strain. For the systems silicon + H,O, strain was only applied to the central crack-tip Si, -Si, bond (r 1) and geometry optimizations were conducted without constraining any bond angle or bond length to be equal.

3. Results and discussion The optimized H,,Si, and H,,SisOH structures are shown in Table 2; bond lengths and angles agree

Si-Si-Si Si-Si-0 Si-O-H Si-Si-H

110.03 1 _

SisH,,OH

Exp’tl

110.507 106.693 120.130 110.507

109.54 108-125

[20]

reasonably well with experimental values. For example, the Si-Si length was measured to be ca. 2.33 A in an X-ray diffraction study [24], which compares well with our calculated values of 2.372 and 2.369 w for SisH,, and H,,Si,OH, respectively. The angles < Si-Si-Si, 110.03” (Si,H,s) and 111.52” (H,,Si,OH) are close to the tetrahedral angle of 109.5”. The Si-H distances of 1.478 (SisH,,) and 1.479 A (H,,SisOH) bracket the measured value of ca. 1.48 A, which was determined from a variety of molecules containing Si-H bonds [33]. The calculated Si-0 distances, 1.614 and 1.653 A, do not deviate significantly from those of 1.63 to 1.70 A calculated by Gibbs and co-workers [ 19,201. The overall results indicate that the choice of the 6-3 1G * * basis set led to a valid approximation of the Si structure. Our calculations also show that, in the equilibrium structure, separation between the crack wall species -H and -OH is about 2.83 A at the tip (Table 3). In the unstrained state, this separation is too narrow to allow a water molecule to pass through to react with the crack-tip Si, and Si, atoms. Fig. 3 shows the effect of strain on the wall separation for the nine different strain modes. The modes which involved angle distortion yielded larger open spaces than those for which the angles were held rigid.

Table 3 O... H (A> Separation on the opposite sides of the crack wall of H ,,Si,OH, Table 1 using the 6-3 1G * basis set

calculated

for the 9 different modes of straining

as shown in

l

% strain

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

10 20 25

3.300 3.416 3.531

3.343 3.494 3.615

3.385 3.543 3.701

3.428 3.608 3.787

3.472 3.673 3.875

3.515 3.739 3.%4

3.598 3.881 4.137

3.906 4.337 4.776

4.222 4.823 5.437

W. Wang-Ng

0

10

20

Strain

et al./ Computational Materials Science 6 (1996) 63-70

40

30

00

%

Fig. 3. A plot of the crack wall H. . .O distance in H,,SisOH (see Fig. 2) as a function of strain, (a) assuming only the stretching of the crack-tip Si-Si bond, and (b) with the widening of angle a as well.

The 6-31G* * set yielded a Mulliken net charge value of -0.35 and -0.38 for Si, and Si,. Since in the silicon crystal there is only one type of element, we expect Q to be very close to zero. We presume that the deviations from neutrality are artifacts resulting from our use of hydrogen to terminate the Si dangling bonds. However, since we are interested in determining trends rather than absolute values, we will use the net charge of the unstrained molecule as a reference point for comparing changes of Q (SQ) as a function of simulated crack-tip strain. Fig. 4a and 4b show SQ of the crack tip Si,-Si, atoms as a function of different modes of strain application (IO, 15, and 20%), respectively. At each strain level, Si, and Si, atoms have similar Q values. The charges on the two atoms are not identical because of the presence of the -H and -OH groups on opposite sides of the wall. Also, in all cases, SQ values for Si, and Si, are small but positive and monotonically increase as the strain level increases. This increase becomes more substantial when the strain is applied in a relatively more delocalized way. There is no tendency for the crack tip bond to polarize with strain despite the lack of symmetry at the crack wall. Since it is expected that an increase in crack tip polarization would assist in orienting the environmental molecule with respect to the crack tip atoms, the lack of polarization in the Si-Si bond implies that there is no driving force for attracting or orienting polar environments.

67

The picture of the net charge on the Si crack tip as a function of strain is quite different from that obtained for silica [ 111. Regardless of the way the molecule H,Si,O, (the model for silica) is strained, the net charge on the OS- becomes more negative (less neutral), while that on the Si’+, becomes more positive as the strain increases. This bond polarization resulted in increasing ionic character of the crack-tip Si-0 bond. In addition to the lack of polarization with strain of the Si-Si bond, steric hindrance in Si prevents access to the crack tip bonds even for large str?ns. We estimated that an opening of ca. 4-4.2 A is needed for a water molecule to pass thorough to the track-tip. This value was obtained by assuming a 1.6 A hydrogen bonding between the 0 (water) and OH (silica) on one side of the wall and a contact distance of 2.6 A (sum of van der Waals radii of 0 (1.4 A> and H (1.2 A)) between the 0 (water) and H (silica) 0.1

x (a)

0.08

x

5 0.06 CY 3 0.04

: x

2 0.02

0 . .

Or 0

5

10

j 15 % Strain

’ 20

25

x

(b) -

ii7 @_0.06 (3 B 0.04 $

x

x

0.02 0.00

f I

a I .

0.10 0.08

3

.

. . c:

6

f

10 15 % Strain

20

2 . l

0

5

25

Fig. 4. Effect of s@ain on the net charge, Q, of (a) crack-tip Si(2) and(b) crack-tip Si(5) on Si,H,,OH using the basis set C31G . The nine modes of distortion according to Table I are:(i) filled hourglass, (ii) v , (iii) 0, (iv) A, (v) + , (vi) open hourglass, (vii) 0, (viii) v , and (ix) n . l

l

68

W. Wang-Ng ei al./ Computational Materials Science 6 (19%) 63-70

on the other [34]. The lack of a driving force at the crack tip to draw the water molecule past the steric barriers along the crack wall means that, even as the strain increases, there is no driving force for environmental interaction with the crack tip. The equilibrium geometry of the system (Si + H,O) indicates that in the absence of strain the distance between 0 (water) and the crack-tip Si, and Si, are at a relatively large separation distance of 5.32 and 5.19 A, respectively. Even with the application of a 15% strain on the Si-Si crack-tip bon!, the above 0 . . . Si distances remain 5.37 and 4.97 A. In both cases, the closest crack-wall separation remained smal$ at ca. 3.4 A and weak hydrogen bonds (2.09 Al were formed between H (water) and the crack-wall oxygen (Fig. 5). It seems possible that a wider opening of the crack would render the crack tip more accessible to a water molecule. To assess this possibility, we performed calculations using a model in which the crack tip spans of three Si-Si bonds, as shown in Fig. 6a. Again, the crack wall consists of the -OH and - H units. Optimization of the system (H ,,Si,O + H,O) shows that the water molecule forms a weak hydrogen bond with the crack wall (2.06 A>, but there is no tendency for the water molecule to be

(a)

\

I

z Fig. 6. (a) Model molecule of H,,Si,O showing 3 Si-Si crack tip bonds. (b) Reaction of H ,4Si60 + H,O, showing the equilibrium distance (ca. 4 A) between water and the crack tip Si-Si bond at a strain level of 15%.

drawn to the crack-tip. Fig. 6b shows the condition of a 15% strain applied to the crack-tip Si-Si bond. The distance between Si, and Si, with O@O), remains in the neighborhood of 3.74 and 3.95 A. It is apparent that although the effect of straining in silicon favors the approach of the incoming environmental molecule by reducing steric hindrance, the resulting electronic character does not assist chemical reaction. These calculations explain why moisture does not enhance crack-growth via chemical reaction in Si. In other words, the crack-tips bonds are broken due entirely to strain, which is different from that observed in silica.

4. Summary Fig. 5. Reaction of H ,,Si,OH + H,O, showing the large equilibrium distance (ca. 5 8) between water and the crack tip Si-Si bond at a strain level of 15%.

Molecular orbital calculations using an ab initio, self-consistent field technique were used to rational-

W. Wang-Ng et al. / Computational Materials Science 6 (19%) 63-70

ize the absence of experimental evidence for environmentally enhanced crack growth in Si. The effect of strain on the charge redistribution on the crack tip Si atoms was investigated using the model molecule (H,,SisOH). Strain effects were simulated using bond length and bond angle distortions. Although we have not determined the actual changes of bond lengths and angles around the Si-Si crack tip region in response to strain on an Si crystal, we have found that the geometric arrangement of Si-Si bonds apparently has no systematic effect on silicon-water interactions. In examining charge polarizations estimated from Mulliken Populations, we found that the Si-Si bond showed no tendency toward charge polarization regardless of the geometry variations we imposed on the molecular model. This is in contrast to the results we obtained in our modeling of silica 1161,in which the 0 became increasingly negatively charged, and Si became increasingly positively charged, as the Si-0 bond was stretched. In this way, straining of the Si-0 bond can be envisioned as leading to an increasingly attractive electrostatic force between the Si-0 bond and a water molecule, which would be a relatively long-range force. Our present results suggest that straining an Si-Si bond does not lead to an increasingly attractive long-range force between an Si-Si bond and a water molecule. Consequently water molecules may be less likely to be drawn toward a crack tip in silicon than in silica. Similarly, if this electrostatic force is not significant in a crack tip region in silicon, we should expect the probability of a water molecule being drawn into a favorable orientation for reaction to be lower for silicon than for silica. In addition, steric hindrances, which are much more severe in Si than in SiO, remain a serious obstacle to the approach of water molecules. These results are consistent with the absence of incontrovertible experimental observations demonstrating environmental enhancement of crack growth in this material.

Acknowledgements Support from the Office of Naval Research for this work is gratefully acknowledged. Drs. R. Thom-

69

son and E.R. Fuller from NIST and Dr. M. Peterson from University of Toronto are acknowledged for their discussions. The , critical review of this manuscript by Drs. K. Irikura, T.J. Chuang and S. Dapkunas is also appreciated.

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