14.1 Introduction

The purpose of this work is to find a calibrated model for the formation and dissolution of silicon self-interstitial clusters of {113} or {311} defects. A good calibrated model for self-interstitial clustering is important for accurately simulating the TED (transient enhanced diffusion) of impurities. TED is the fast displacement of impurities in the first thermal step just after implantation and the simulation of its evolution and magnitude is important in the manufacturing processes of submicron devices [99].

The source of the silicon self-interstitials was shown to be the $\{113\}$ defects which are rod like clusters of interstitials [30]. Counting the amount of self-interstitials is a non-trivial task: from transmission electron micrographs the number of interstitials in each defect and thus the total number has to be measured. In [95] one can find measurements giving the number of interstitials as a function of time for annealing at four temperatures ( $670\,\text{\textcelsius}$, $705\,\text{\textcelsius}$, $738\,\text{\textcelsius}$, and $815\,\text{\textcelsius}$) and $5\cdot10^{13}\,\mathrm{cm}^{-2}$, $40\,\mathrm{keV}$ implants. These measurements are shown in detail in Figure 14.1 and provided the basis for this inverse modeling problem. For the computations we used TSUPREM-4 [8] and the optimization framework SIESTA [45,139].

We were interested in finding solutions for two different technologies corresponding to different values of several TSUPREM-4 variables. In the following we will call these parameter sets the high and the low parameter set (the latter being the TSUPREM-4 default values). The parameters and their values are shown in Table 14.1.

Since the rate of formation and dissolution is not yet fully understood, the model used contains several proposed models (e.g., [99]) as special cases [8]. After describing the model and the details of the inverse modeling problem we present the results and the calibrated model.

Figure 14.1: The silicon self-interstitial density (in cm**(-3)) as a function of time (in s) for different annealing temperatures (interstitials stored in {113} or {311} defects after 5x10**(13)cm**(-2), 40keV implants).

Table 14.1: The differences between the high and low (defaults) parameter set. Variables ending in 0 are pre-exponential constants and those ending in E are energies in Arrhenius laws. The first and last two parameters are used in modeling the equilibrium concentrations of interstitials and vacancies in silicon, respectively. The third and fourth parameter determine the bulk recombination factor of interstitials. The next two parameters determine part of the surface recombination rate of point defects. The seventh and eighth parameter occur in the Arrhenius law for the diffusivity of vacancies in silicon[8].

TSUPREM-4 statement	High Parameter Set	Low Parameter Set	Unit
interstitial mat=silicon cequil.0	$2.9\cdot{10}^{24}$	$1.25\cdot{10}^{29}$	$\mathrm{cm}^{-3}$
interstitial mat=silicon cequil.E	$3.18$	$3.26$	$\mathrm{eV}$
interstitial mat=silicon kb.0	$1.2\cdot{10}^{-5}$	$1.0\cdot{10}^{-21}$	$\mathrm{cm}^3 \mathrm{s}^{-1}$
interstitial mat=silicon kb.E	$1.77$	$-1.0$	$\mathrm{eV}$
interstitial silicon /oxide ksurf.0	$5.1\cdot{10}^{7}$	$1.4\cdot{10}^{-6}$	$\mathrm{cm} \mathrm{s}^{-1}$
interstitial silicon /oxide ksurf.E	$1.77$	$-1.75$	$\mathrm{eV}$
vacancy mat=silicon d.0	$3.0\cdot{10}^{-2}$	$3.65\cdot{10}^{-4}$	$\mathrm{cm}^2 \mathrm{s}^{-1}$
vacancy mat=silicon d.e	$1.8$	$1.58$	$\mathrm{eV}$
vacancy mat=silicon cequil.0	$1.4\cdot{10}^{23}$	$1.25\cdot{10}^{29}$	$\mathrm{cm}^{-3}$
vacancy mat=silicon cequil.E	$2.0$	$3.26$	$\mathrm{eV}$

Table 14.2: Variables, their intervals, and their units.

Variable	Interval	Unit
${\mathrm{d0}}$	$[25, 1000]$	$\mathrm{cm}^2 \mathrm{s}^{-1}$
${\mathrm{dE}}$	$[1.4, 1.85]$	$\mathrm{eV}$
${\mathrm{kfi0}}$	$[10^{20},10^{28}]$	$\mathrm{cm}^{-3(1+{\mathrm{isfi}}-{\mathrm{ifi}})}\mathrm{s}^{-1}$
${\mathrm{kfiE}}$	$[3.4, 6.0]$	$\mathrm{eV}$
${\mathrm{ifi}}$	constant $=2$	$1$
${\mathrm{isfi}}$	constant $=2$	$1$
${\mathrm{kfc0}}$	$[10^{17}, 7\cdot 10^{19}]$	$\mathrm{cm}^{-3(1+{\mathrm{isfc}}-{\mathrm{ifc}}-{\mathrm{cf}})}\mathrm{s}^{-1}$
${\mathrm{kfcE}}$	$[4.9, 5.2]$	$\mathrm{eV}$
${\mathrm{ifc}}$	constant $=1$	$1$
${\mathrm{isfc}}$	constant $=1$	$1$
$\alpha$	$[0, 5000]$	$1$
${\mathrm{cf}}$	constant $=1$	$1$
${\mathrm{kr0}}$	$[1.5\cdot 10^{16}, 10^{18}]$	$\mathrm{cm}^{-3(1-{\mathrm{cr}})}\mathrm{s}^{-1}$
${\mathrm{krE}}$	$[3.0, 3.62]$	$\mathrm{eV}$
${\mathrm{cr}}$	constant $=1$	$1$

Table 14.3: Results for the high and low parameter sets with the above free variables. The mean relative error is $0.389666$ for the high and $0.504462$ for the low parameter set.

High Parameter Set (mean relative error $0.389666$):
Variable	SIESTA variable	Best point found
$\mathrm{d0}$	d-0	$51.7282$
$\mathrm{dE}$	d-e	$1.76996$
$\mathrm{kfi0}$	kfi-0	$4.97576\cdot10^{24}$
$\mathrm{kfiE}$	kfi-e	$3.77408$
$\mathrm{kfc0}$	kfc-0	$4.36789\cdot10^{19}$
$\mathrm{kfcE}$	kfc-e	$4.95$
$\alpha$	kfci	$1099.63$
$\mathrm{kr0}$	kr-0	$2.77935\cdot10^{16}$
$\mathrm{krE}$	kr-e	$3.56997$
Low Parameter Set (mean relative error $0.504462$):
Variable	SIESTA variable	Best point found
$\mathrm{kfi0}$	kfi-0	$1.14156 \cdot 10^{25}$
$\mathrm{kfiE}$	kfi-e	$3.94079$
$\mathrm{kfc0}$	kfc-0	$1.5051 \cdot 10^{19}$
$\mathrm{kfcE}$	kfc-e	$5.81858$
$\alpha$	kfci	$1563.1$
$\mathrm{cf}$	cf	$1.01287$
$\mathrm{kr0}$	kr-0	$1.06467 \cdot 10^{17}$
$\mathrm{krE}$	kr-e	$3.84503$
$\mathrm{cr}$	cr	$0.9639$