Книги2 / 1993 P._Lloyd,__C._C._McAndrew,__M._J._McLennan,__S._N

annealing time (min)

than this, significant implant-induced nonequilibrium defect populations are present for most of the anneal time and influence dopant diffusion. Defects generated by ion implantation are in general represented either as an initial amount of vacancies and interstitials located in the bulk at the same place as the dopants, or as dislocations at the boundary of the amorphised silicon, or as amounts of defects which decrease with time independently from diffusion and recombination phenomena. The algorithm implemented in STORM is robust enough for all these cases. The influence of defects on dopant diffusion was shown even at temperatures as low as 850°C.

5.1.3Dopant-Dopant Interactions

The precipitation kinetics of boron and antimony in silicon at high concentrations are accurately modelled by the LAMEL SOLMI model incorporated into the STORM code by CNETj the model also simulates very high concentration As precipitation. In STORM, the model has been combined with the MMG model described above. In order to take the precipitation into account, an additional term P(x,t) which acts as a source or sink for the dopant has been added to the diffusion equation. P(x,t) is the amount of dopant going to or leaving the precipitates in the interval dt:

This source term depends on the dopant solubility Cso!, and the number Np and radius rp of precipitates. Csol is a function of temperature and co-diffusing dopant

6.1Diffusion and Reaction of Oxidizing Species

For the diffusion and reaction of the oxidizing species a straight-forward generalization of the well known Deal & Grove model [37] is used, which assumes stationary diffusion with an effective diffusivity Deff in the oxide

Here, e· is the surface concentration resulting from Henry's law, and k, and k. are the transfer coefficient at the free surface and the reaction rate at the Si/Si02 interface, respectively. Diffusion and reaction coefficient depend on temperature, Hel concentration and stress:

Here, P is the hydrostatic pressure, and 0""" is the normal stress along the Si/Si02 interface. The model parameters used in eqs. (14-18) have been taken from literature.

The diffusion-reaction equations are solved numerically by using a standard finiteelement method with piecewise linear shape functions and numerical quadrature rules.

6.2Oxide growth near Si/Si02 interfaces

During a time step 6..t, an infinitesimal new oxide layer with thickness 6 is grown:

For the calculation of the movement of the interface between Si and Si02 , two different approaches are being used. In the first one, the interface between Si and Si02 is displaced by the incremental additional oxide thickness 6. Afterwards, suitable equations of motion are solved to express the mechanical reaction of the silicon. In the second approach, the interface is displaced by 0.44 ·6 to express the consumption of the silicon, and the dilatation of the layers is considered afterwards. In this latter case, a mesh is built in the newly formed oxide in each time step. This also decreases the numerical noise otherwise resulting from unstructured triangular meshes. In contrast to the first approach this one also correctly includes that part of the stress which results from ID oxidation, whereas the first approach includes 2D extra stress only. Both principles are shown in Fig. 11.

6.3Displacement resulting from surface tension and dilatation forces

Following the approaches described above, the oxidation kinetics either leads to Dirichlet boundary conditions on velocities at the interface between rigid Si and Si02 ,

and the normal stress along any boundary or interface depends on the normal component Vn and the tangential component VT of the velocity via

The boundary conditions are that if is zero in rigid materials, O"nn is zero on the free surface, and Vx is zero at the left and right sides of the structure. For the interfaces between Si02 and silicon,

Here, N1 is the number of oxidizing molecules incorporated into a unit volume of the oxide layer.

In STORM, the elastic model outlined here is used as the default one for silicon nitride, and also recommended for oxide layers at temperatures below 1000°C.

The link between viscosity and elasticity coefficients in silicon dioxide is achieved by introducing a relaxation time T, which is applied to cumulative stresses in the expressions of diffusivity and reaction rate. The Maxwell viscoelastic model is assumed for the shear modulus G and the viscosity f.L:

In STORM, thermal dilatation is accounted for during thermal ramps (under oxidizing or inert ambients) by simulating the different expansion between bulk silicon and the other materials. Thermal stresses are computed in all materials defined as elastic or visco-elastic. The internal forces which result from temperature variations lead to an additional term in the elasticity or visco-elastic equations; from first principles, the internal pressure is given as CithK 6T, where Cith is the volume expansion coefficient, K is the bulk modulus, and 8T denotes the temperature variation.

6.3.2Visco-elastic Model

Alternatively, a viscoelastic model may be used in STORM. In this model, the components of the strain tensor e depend on the components of the stress tensor 0' via the stress deviator 0", the hydrodynamic pressure p, the shear stress modulus G, the viscosity J-l and the components Dx and Dy of the dilatation force:

In addition to the boundary condition (30) of the elastic model, the surface tension at the free surface is inversely proportional to the local curvature radius R:

This condition has negligible influence in case of thermal oxidation, but is the driving force for glass reflow.

In this model, the pressure is computed as a piecewise constant solution and is used a a Lagrange multiplier for expressing the uncompressibility of the materials at a fixed temperature. A modified version of the well-known Uzawa algorithm is used to combine this uncompressibility, the elastic term in eqs. (32-34), and the different dilatation forces.

6.3.3Stress-reduced Viscosity

In STORM, stress reduced viscosity has been expressed according to the work by Suturdja and Oldham [40]:

with

Here, P is the local viscosity, (J'.heo.r is the shear stress, and the parameters VI" Po, and EI' are taken from literature [41).

6.3.4 Viscosity of BPSG Glasses

The viscosity parameters referred to above are valid for thermal oxidation. For planarization, materials with much lower viscosity are used, such as phospho-silicate and boro-phospho-silicate glasses. For BPSG, the preexponential term Po has been expressed versus boron concentration for different deposition processes, whereas EI' shows no clear dependence on process conditions.

6.3.5Finite Element Simulation of Dilatation Forces

Dilatation forces result either from the transformation of Si into Si02 or from thermal dilatation. They can be expressed as gradient of a scalar function d. This fits well to the Finite Element approach to solve the oxidation problem. If some silicon is not treated as a rigid body, the full stress tensor needs to be evaluated, including 1D stress occuring during full wafer oxidation. In this case, the scalar function d in the new oxide layer is

Otherwise, only extra stress resulting from local 2D effects is simulated.

Furthermore, at each time step the thermal dilatation is simulated, depending on the dilatation coefficient 0th of each material:

Figure 12: Comparison between experiment and STORM/TITAN simulatiom of local oxidation at 920°C for 300 min