Numerical Simulation Support for Diffusion Coefficient Measurements in Polycrystalline Thin Films

Abstract

The measurement of diffusion coefficients in today’s materials is complicated by the down scaling of the studied structures (nanometric effects in thin films, nano-crystalline layers, etc.) and by the complex production process conditions of industrial samples or structures (temperature variations, complex solute and point defect distributions, stress gradients, etc.). Often diffusion measurements have to be performed in samples for which initial experimental conditions do not offer the possibility of using conventional diffusion analytical solutions. Furthermore, phenomena involved with diffusion are sometimes so numerous and complex (stress, matrix composition inhomogeneities, time dependence of point defect generation sources, electrical effects, clustering effects, etc…) that the use of analytical solutions to solve the observed diffusion behavior is difficult. However, simulations can be of use in these cases. They are time consuming compared to the use of analytical solutions, but are more flexible regarding initial conditions and problem complexity. The use of simulations in order to model physical phenomena is quite common nowadays, and highly complex models have been developed. However, two types of simulations have to be considered: i) simulations aiming to understand and predict phenomena, and ii) simulations for measurement purposes, aiming to extract the (average) value of a physical parameter from experimental data. These two cases have different constrains. In the second case, that is the subject of this article, one of the most important stress is that the simulation has to precisely scale the experiment (sample size, experiment duration, etc.), sometimes preventing the measurement due to computational time consumption. Furthermore, the simpler the model (small number of parameters) used in the simulation, the more relevant the measurement (minimum error). In this paper, examples of recent works using two- and three-dimensional finite element simulations for diffusion coefficient measurements in thin polycrystalline films and nano-crystalline layers are presented. The possible use of simulations for diffusion coefficient measurements considering GB migration, GB segregation, or triple junctions is also discussed.

Atomic Transport in Nano-Сrystalline Thin Films

“…The SIMS profiles measured in nc-Si (Fig. 4a) could not be reproduced by the usual B diffusion model, even if B GB segregation was taken into account [14,26]. The study of the grain size distribution by atomic force microscopy (AFM) showed that the average size of the nano-grains did not change after the different annealing.…”

Anomalous B diffusion profiles in nanocrystalline Si

“…As can be seen in Figure 2, this model cannot reproduce the experimental profiles shown in Figure 1. Even considering B cluster dissolution or taking into account B GB segregation [19,20], the end of the B profiles cannot be reproduced before B atoms have reached the nc-Si/SiO 2 interface. The B diffusion coefficient variation with B concentration in Si grains cannot explain the B profiles measured in nc-Si at these temperatures (6800°C).…”

“…In this model, the B atoms are considered to be incorporated in the Si grain after the lateral migration of the GB. From the literature, D gb was set to 2.74 Â 10 À15 cm 2 s À1 in GBs at 700°C [15] (experiments support that GB diffusion is identical in immobile and mobile GBs [21]) and D g was adjusted according to the simulation results [8,9,19,20]. In order to fit the experimental profiles, we set D g = 4.33 Â 10 À19 cm 2 s À1 (Eq.…”

“…At point defect equilibrium, GB diffusion coefficients (D gb ) were found to be constant at given temperature, and for B diffusion in Si GBs D gb = 0.8 exp(À2.6 eV/k B T) cm 2 s À1 [15]. Figure 2 shows the results of two-dimensional (2-D) finite-element simulations of B diffusion in the nc-Si layer considering the Fisher geometry [18] with a grain width of 50 nm and a GB size of 0.5 nm [8,9,19,20], as well as considering the B diffusion coefficients D B and D gb and the electrical effects in grains given in the literature and described above (Eqs. (1)-(4)).…”

“…The B diffusion coefficient variation with B concentration in Si grains cannot explain the B profiles measured in nc-Si at these temperatures (6800°C). The Fisher geometry we used does not take into account the fact that GBs were mobile during annealing despite a constant average grain size [18], though mobile GBs were shown to have a significant impact on diffusion profiles in polycrystalline layers [19,20]. Figure 3 presents simulations of B diffusion in the nc-Si layer considering the same assumptions as for the simulations presented in Figure 2, but also considering GB migration [19,20] and a constant diffusion coefficient in grains (D g ).…”