15.1 Introduction and Measurements

The inverse modeling problem is to find the parameter values of the diffusion model of arsenic in silicon, where a large number of 19 measurements ought to be matched. Table 15.1 shows the process conditions of the experiments for which measurements were obtained. In the table the temperature of the wafer surface, the time period where the wafer was exposed to arsenic, and the arsenic concentration at the wafer surface are given. The last column indicates if the measurement refers to the electrically active or total concentration.

Table 15.1: The 19 process conditions for which measurements of the arsenic profile were performed.

Measurement	Temperature	Time	As surface	Active/total
Number			concentration	concentration
	$[\text{\textcelsius}]$	$[\mathrm{s}]$	$[1/\mathrm{cm}^3]$
1	850	12575	$3 \cdot 10^{20}$	total
2	900	2020	$3 \cdot 10^{20}$	total
3	950	425	$4 \cdot 10^{20}$	total
4	950	1680	$1 \cdot 10^{21}$	total
6	1000	100	$4 \cdot 10^{20}$	total
7	1000	505	$1 \cdot 10^{21}$	total
8	1050	20	$6 \cdot 10^{20}$	total
10	1050	60	$9.45\cdot 10^{20}$	total
11	1050	1140	$1 \cdot 10^{21}$	total
12	1100	350	$2 \cdot 10^{21}$	total
13	1108	120	$3 \cdot 10^{19}$	total
14	1108	200	$5 \cdot 10^{18}$	total
15	1150	100	$2 \cdot 10^{21}$	total
16	1200	60	$4 \cdot 10^{18}$	total
1a	850	12575	$3 \cdot 10^{20}$	active
2a	900	2020	$3 \cdot 10^{20}$	active
3a	950	425	$4 \cdot 10^{20}$	active
6a	1000	100	$4 \cdot 10^{20}$	active
8a	1050	20	$6 \cdot 10^{20}$	active

Table 15.2: Variable of the model to be determined, their intervals, and their units.

Variable	Interval	Unit
$\mathrm{f}$	$[0,1]$	$1$
$\mathrm{dx0}$	$[1,10^{10}]$	$\mathrm{cm}^2/\mathrm{s}$
$\mathrm{dxE}$	$[3.43,3.6]$	$\mathrm{eV}$
$\mathrm{dm0}$	$[10^9,10^{12}]$	$\mathrm{cm}^2/\mathrm{s}$
$\mathrm{dmE}$	$[4.05,4.6]$	$\mathrm{eV}$
$\mathrm{ctn0}$	$[10^{-20},10^{-12}]$	$\mathrm{cm}^{-3/(\mathrm{ctnF}-1)}$
$\mathrm{ctnE}$	$[-0.9,-0.39]$	$\mathrm{eV}$
$\mathrm{ctnF}$	$[3.7,4.4]$ or constant $=4$	$1$

Table 15.3: Result using optimizer DONOPT (cf. Section 9.3.5) with the above free variables yielding the respective values. The value of the objective function at this point is $0.0265418$.

Variable	Best point found
$\mathrm{f}$	$0.2$
$\mathrm{dx0}$	$8.5\cdot 10^8$
$\mathrm{dxE}$	$3.44$
$\mathrm{dm0}$	$2.7\cdot 10^{11}$
$\mathrm{dmE}$	$4.11028$
$\mathrm{ctn0}$	$4.7\cdot 10^{-18}$
$\mathrm{ctnE}$	$-0.47$

Figure 15.1: Measurement and resulting simulation corresponding to parameter values shown in Table 15.3. The measurements used are numbered 1, 2, 3, 6, 7, and 8 in Table 15.1.

Table 15.4: Result using optimizer DONOPT (cf. Section 9.3.5) with the above free variables yielding the respective values. The value of the objective function at this point is $0.0219828$.

Variable	Best point found
$\mathrm{f}$	$0.200121$
$\mathrm{dx0}$	$7.92893\cdot 10^8$
$\mathrm{dxE}$	$3.44045$
$\mathrm{dm0}$	$2.52394\cdot 10^{11}$
$\mathrm{dmE}$	$4.12253$
$\mathrm{ctn0}$	$5.66261\cdot 10^{-18}$
$\mathrm{ctnE}$	$-0.483025$
$\mathrm{ctnF}$	$4.00856$

Figure 15.2: Measurement and resulting simulation corresponding to parameter values shown in Table 15.4. All 19 measurements from Table 15.1 were used.

The simulations were performed using TSUPREM-4 [8]. The model of diffusion of arsenic and of point defects, i.e., interstitials and vacancies, in silicon depends on several variables. The variables of the model to be determined are shown in Table 15.2. Their meaning is as follows. $\mathrm{f}$ is a factor used in the pre-exponential constants of the diffusivities. The diffusivity of arsenic with neutral interstitials $D_{i,0}$ is given by

$$D_{i,0} = \mathrm{f}\cdot\mathrm{dx0} \cdot \mathrm{e}^{\mathrm{dxE} /kT},$$

where $\mathrm{f}\cdot\mathrm{dx0}$ is the pre-exponential constant and $\mathrm{dxE}$ the activation energy. The diffusivity of arsenic with neutral vacancies $D_{v,0}$ is given by

$$D_{v,0} = (1-\mathrm{f})\cdot\mathrm{dx0} \cdot \mathrm{e}^{\mathrm{dxE} /kT}.$$

The diffusivity of arsenic with singly-negative interstitials $D_{i,-1}$ is given by

$$D_{i,-1} = \mathrm{f}\cdot\mathrm{dm0} \cdot \mathrm{e}^{\mathrm{dmE} /kT},$$

and the diffusivity of arsenic with singly-negative vacancies $D_{v,-1}$ is given by

$$D_{v,-1} = (1-\mathrm{f})\cdot\mathrm{dm0} \cdot \mathrm{e}^{\mathrm{dmE} /kT}.$$

Thus the pre-exponential constants of the interstitials and the vacancies add to unity in the two cases of neutral and charged particles.

Finally $\mathrm{ctn0}$ is the pre-exponential constant for the clustering of arsenic, $\mathrm{ctnE}$ its activation energy, and $\mathrm{ctnF}$ its exponent of concentration.