Influence of stress-dependent surface generation of interstitials on stacking fault growth and dopant diffusion in silicon

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Influence of stress-dependent surface generation of interstitials on stacking fault growth and dopant diffusion in silicon

Loiko Konstantin Valeryevich

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]:

, (1)

, (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]. 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.

1. 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 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. The distance between the center of the octahedral site and these lattice sites is a0/8. Therefore, Ri =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. Therefore, Ri =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 ÄV = 4/3ð(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 ÄVi = ? 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 = 4ðRa2ã, where ã ? 1.5 J/m2 is the silicon surface energy [13]. After the contraction by ÄR, the energy of the hole E = 4ð(Ra - ÄR)2ã. Thus, the total force of surface tension F ? (E0 - E)/ÄR ? 8ðRaã. This gives the contracting pressure P = F/(4ðRa2) = 2ã/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 = ã(1+)/Y ˜ 0.113 Å. Finally, the activation volume of a vacancy is found as ÄVv = 4/3ð{Ra3 - (Ra-u)3} ˜ 2.3 Å.3

The obtained values for the activation volumes of point defects are close to those suggested in some reports [2,3,8].

2. Diffusion Equations for Point Defects

dopant diffusion equation defect

The following diffusion equations are considered for interstitials and vacancies in the bulk of silicon:

, (3), , (4), , (5)

where , , , and are the concentration, equilibrium concentration, diffusivity, and flux of interstitials or vacancies, and are the bulk recombination rate and its rate factor, and denote parameters for interstitials and vacancies, respectively.

Equation 4 is different from the standard expression for the diffusion flux [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:

, (6), . (7)

3. 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)

, (9)

where and are the stress-free equilibrium concentrations of interstitials and vacancies, and are their surface generation-recombination rate factors, and is the interstitial injection rate due to oxidation. Terms and describe surface recombination, while and 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:

, (10)

, (11)

where is the silicon volume change occurring when a silicon atom is added to or removed from the silicon surface. ˜ 20 Å3, which is the volume of silicon per atom.

Figure 1 - Self-interstitial formation in silicon by surface generation. ÄVs > ÄVi

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 equal to the volume of silicon per atom ˜ 20 Å3 > . The difference of these two volumes () 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 () 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.

4. 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 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

5. 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 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.

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 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. However, deeper in the substrate, the boron concentration under the nitride is slightly higher than that in the open region. This shows an increase in boron diffusion and agrees with the results of Zhao at al. [11,12], whose measurements suggest enhanced boron diffusivity under compressive stress. Therefore, our results demonstrate the importance of surface proximity in analyzing the effect of stress on dopant diffusion.

It is necessary to point out that the stress-dependencies of dopant-defect pairing coefficients and equilibrium pair concentrations were not taken into account in this simulation. Since these effects may play a more or less significant role in stress-mediated dopant diffusion, a detailed study of them is required in the future.

Conclusions

In this paper, the model was developed for the stress-induced redistribution of point defects in silicon device structures. It incorporates equilibrium conditions different for defects at the surface and in the bulk of silicon and takes into account stress-dependent surface generation and recombination of point defects. In the model, equilibrium concentrations of intrinsic point defects depend exponentially on hydrostatic pressure in silicon. For bulk interactions, the calculations yielded the activation volumes of about 2.5 Å3 and 2.3 Å3 for interstitials and vacancies, respectively. The activation volumes for surface generation and recombination of point defects close to 18 Å3 were used.

Computer simulations showed that the enhancement in surface generation of interstitials under compressive stress may explain the growth of larger stacking faults at the edges of active areas.

The importance of taking into account the proximity of the silicon surface in analyzing the effect of stress on dopant diffusion was demonstrated using simulations. It is shown that surface generation and recombination of point defects play a pronounced role in the point defect and therefore dopant distribution.

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