Building intuition of iron evolution during solar cell processing through analysis of different process models

Abstract

An important aspect of Process Simulators for photovoltaics is prediction of defect evolution during device fabrication. Over the last twenty years, these tools have accelerated process optimization, and several Process Simulators for iron, a ubiquitous and deleterious impurity in silicon, have been developed. The diversity of these tools can make it difficult to build intuition about the physics governing iron behavior during processing. Thus, in one unified software environment and using self-consistent terminology, we combine and describe three of these Simulators. We vary structural defect distribution and iron precipitation equations to create eight distinct Models, which we then use to simulate different stages of processing. We find that the structural defect distribution influences the final interstitial iron concentration ([$\hbox {Fe}_i$]) more strongly than the iron precipitation equations. We identify two regimes of iron behavior: (1) diffusivity-limited, in which iron evolution is kinetically limited and bulk [$\hbox {Fe}_i$] predictions can vary by an order of magnitude or more, and (2) solubility-limited, in which iron evolution is near thermodynamic equilibrium and the Models yield similar results. This rigorous analysis provides new intuition that can inform Process Simulation, material, and process development, and it enables scientists and engineers to choose an appropriate level of Model complexity based on wafer type and quality, processing conditions, and available computation time.

Appendices

Appendix 1: Simulating iron precipitate nucleation sites

In all eight Models, including those with grain boundaries, iron-silicide precipitates are assumed to nucleate at precipitation sites along dislocations. In the 2D Models, the heterogeneously distributed intra-grain dislocations are in clusters, each of which has a dislocation density $N_{\rm{DL}}(x,y) = C \exp (-\frac{1}{2}(\frac{(y-y_0)^2}{L})^3-\frac{1}{2} (\frac{(x-x_0)^2}{L})^3)$ where C is the peak precipitation site density in the dislocation, parameter $L = 15$ μm adjusts how fast the dislocation density is reduced from the center of the cluster, and $x_{0}$ and $y_{0}$ are randomly chosen coordinates that determine the location of the centrum of the dislocation cluster. C is scaled so that the average dislocation density per area is $N_{\rm{avg}} = 8\times 10^{3}\ \hbox {cm}^{-2}$ within the grain. Then, the dislocation density of grid points with dislocation density $<\!10$ cm⁻² is set to zero. The grain boundary is modeled as a dense band of dislocations with an areal density of $2\times 10^{8}\ \hbox {cm}^{-2}$. The simulated grain boundary width is 10 μm, which is unrealistically wide, but it is still less than 1 % of the grain width, and for computational reasons we use this value. Most importantly, this grain boundary width paired with the dislocation density in the grain boundary preserves an accurate number of total dislocations and therefore precipitation sites at the grain boundary [95]. The precipitation site density, $N_{\rm{site}}$, is proportional to the dislocation density, $N_{\rm{DL}}$, as in $N_{\rm{site}} = 3.3\times 10^{5}\ \hbox {cm}^{-1}\times \ N_{\rm{DL}}$ [26].

Appendix 2: Detailed description of precipitation equations Model Element

Ham’s law [39] describes all the precipitates as spheres with a single average number of atoms/precipitate, $n_{\rm{avg}}$. The input parameter is the precipitate density, $N_{\rm{p}}$. The time evolution of the precipitated iron concentration, [Fe _p ], depends on g($n_{\rm{avg}}$) and d($n_{\rm{avg}}$), the precipitate size-dependent precipitate growth and dissolution rates, respectively. $C_{\mathrm{Fe}}$ is the interstitial iron concentration, and $D_{\mathrm{Fe}}$ is the iron diffusivity. $r_{\mathrm{c}}$ is the size-dependent capture radius of the precipitates. The capture radius determines how close to the center of the precipitates the dissolved iron atoms need to be in order to attach to the iron precipitate. The capture radius and local equilibrium iron concentration are defined differently in the two precipitation approaches. For the Ham’s law Model, the equilibrium iron concentration, $C_{\mathrm{Eq}}$, is the solid solubility of iron, $C_{\mathrm{S}}$, as defined in [40]. The precipitates are modeled as spheres with the volume of a unit cell containing a single iron atom in a $\beta$-FeSi$_2$ precipitate, $V_{\mathrm{p}} = 3.91\times 10^{23}\ \hbox {cm}^{3}$. These equations are summarized in the left-hand column of Table 1.

Table 1 Equations for precipitation behavior Model Element

The Fokker–Planck equation-based precipitation Model analyzes precipitates with a distribution of sizes and assigns a different spatial density for each size [25, 96]. The input parameter is the density of precipitation sites, $N_{\mathrm{prec}}$. The density of precipitates with n atoms is f(n), and the total density of precipitates is $N_{\mathrm{p}} = \int _{1}^{n_{\mathrm{max}}=10^{10}} f(n) dn$, where $n_{\mathrm{max}}$ is the maximum precipitate size. The time evolution of the precipitate distribution, f(n), is described by the FPE [25], and it is numerically solved with Cooper and Chang’s method [97]. The factor $A(n,t)=g(n,t)-d(n,t)$ is the net growth rate of the precipitates, and the factor $B(n,t)=\frac{1}{2} [g(n,t)+d(n,t)]$ describes random fluctuations in the precipitate size. The boundary conditions, $f(n=n_{\mathrm{max}}, t)$ and $f(n=1,t)$, are defined in Table 1, $p_1 = 1\times 10^4$ is a fitting parameter, and $f(n=0,t)$ is the density of empty precipitation sites. $f(n=1,t)$ describes which fraction of these sites contains an iron atom, i.e., where nucleation occurs. The Gibbs free energy of a precipitate with n atoms is $\Delta G(n)$ [98], where $E_{\mathrm{a}}$ is an energy parameter that accounts for all changes in surface energy and strain caused by the growth and dissolution of precipitates. It has been assumed to be independent of n and has been estimated in [99]. Assuming that precipitation is diffusivity-limited, the equilibrium concentration in the proximity of a precipitate is the dissolved iron concentration when $\frac{\partial \Delta G}{\partial n} = 0$. The precipitate size-dependent equilibrium iron concentration, $C_{\mathrm{Eq}}$, depends on the solid solubility of iron, $C_{\mathrm{S}}$, and the factor in the exponential captures the fact that iron has a higher chemical potential in a small cluster than in a large cluster [25]. Precipitates are modeled as flat disks [98] with thickness $a=20$ nm, and the capture radius of the precipitation site is explicitly accounted for [100]. Due to the inclusion of the size of the precipitation site, the FPE Model predicts higher capture radii at small precipitate sizes, and due to the faster expansion of 2D disks compared to 3D spheres, the growth of the capture radius remains faster at large precipitate sizes. These equations are summarized in the right-hand column of Table 1.

Note that for large precipitate sizes (n $\gg 1$), $C_{\mathrm{Eq}}\approx C_{\mathrm{S}}$ and the two precipitation models predict similar equilibrium concentrations. However, when modeling small precipitates, the models differ. The expression for the Gibbs free energy predicts a temperature- and dissolved iron concentration-dependent critical size $n_{\mathrm{crit}}$, defined as the size that maximizes $\Delta G(n)$. Thermodynamics dictates that precipitates smaller than $n_{\mathrm{crit}}$ tend to dissolve, whereas precipitates larger than $n_{\mathrm{crit}}$ tend to grow. The energy needed for the precipitates to cross from the dissolution-favoring regime into the growth regime is defined as the nucleation barrier. In the FPE precipitation Model, a certain level of local supersaturation is needed for nucleation to occur; however, in the Ham’s law Model, there is no nucleation barrier.