Percolation model for a selective response of the resistance of composite semiconducting $np$ systems with respect to reducing gases

It is shown that a two-component percolation model on a simple cubic lattice can explain an experimentally observed behavior [Savage et al., Sens. Actuators B 79, 17 (2001); Sens. Actuators B 72, 239 (2001).], namely, that a network built up by a mixture of sintered nanocrystalline semiconducting $n$ and $p$ grains can exhibit selective behavior, i.e., respond with a resistance increase when exposed to a reducing gas A and with a resistance decrease in response to another reducing gas B. To this end, a simple model is developed, where the $n$ and $p$ grains are simulated by overlapping spheres, based on realistic assumptions about the gas reactions on the grain surfaces. The resistance is calculated by random walk simulations with $nn$, $pp$, and $np$ bonds between the grains, and the results are found in very good agreement with the experiments. Contrary to former assumptions, the $np$ bonds are crucial to obtain this accordance.

Figure 1

(a) and (b) Schematized models of (a) an $n$ grain and (b) a $p$ grain, covered by adsorbed oxygen that traps electrons on the surface (small black dots), thereby becoming ionized and creating depletion shells or enrichement shells, respectively. The depletion shell of (a) with thickness ${\lambda }_n\left(0\right)$ contains no electrons, whereas the enrichment shell of (b) with thickness ${\lambda }_p\left(0\right)$ has an increased hole density as compared to the bulk.

Figure 2

Illustration of the random walk method: (a) Sketch of a 2D random walk on a square lattice (lattice sites not shown) starting at the lattice site labeled 0 and ending at the site labeled 50 where the numbers represent the time steps. At some selected lattice sites, all step numbers are displayed, i.e., “20,36” indicates that the walker passes at this site at time steps 20 and 36. The vector $\vec{r}(t=50)$ of length $r$ is indicated by the arrow. (b) The length $r\left(t\right)=\sqrt{r^2\left(t\right)}$ is shown for the walk of (a) as a function of $t$ for the first 50 time steps. (c) $r^2\left(t\right)$ is shown as a function of $t$ for the first 50 time steps as an average over $10,{10}^2$, and ${10}^4$ random walks (circles, squares, and triangles, respectively).

Figure 3

Two-dimensional sketches of networks of (a) $n$ grains (gray core with a white depletion shell), (b) $p$ grains (gray core with a dark gray enrichment shell), and (c) heterogeneous networks of both types of grains.

Figure 4

Schematized models of overlapping grains with a geometrical neck of diameter ${\Delta }_0$ between them: (a) $nn$ bond, (b) $pp$ bond, and (c) $np$ bond.

Figure 5

Under the reaction factors ${\alpha }_n={\alpha }_p=1,R\left(N_r\right)/R\left(0\right)$ is plotted versus $N_r$ for $N_z=9,\langle D\rangle =100\mathrm{nm},T=500\mathrm{K},p=0.8$, and $\epsilon =8$. For $n$ grains we have $N_D=3.5\times {10}^{-3}{\mathrm{nm}}^{-3}$ and ${\lambda }_n\left(0\right)=10\mathrm{nm}$, whereas for $p$ grains, we chose $q{V}_s=0.3\mathrm{eV}$ and different values of $N_A$: (a) and (b) $N_A=1.4\times {10}^{-4}{\mathrm{nm}}^{-3}$, (c) and (d) $N_A=2.9\times {10}^{-4}{\mathrm{nm}}^{-3}$, and (e) and (f) $N_A=8\times {10}^{-4}{\mathrm{nm}}^{-3}$ referring to ${\lambda }_p\left(0\right)>{\lambda }_n\left(0\right),{\lambda }_p\left(0\right)={\lambda }_n\left(0\right)$, and ${\lambda }_p\left(0\right)<{\lambda }_n\left(0\right)$, respectively. The different colors and symbols indicate (from bottom to top) $p_n=1$, i.e., only $n$ sites (black circles), $p_n=0.75$ (red squares), $p_n=0.5$ (green diamonds), $p_n=0.25$ (blue triangles up), and $p_n=0$, i.e., only $p$ sites (cyan triangles down). Whereas in the top row (a), (c), and (e), the $np$ bonds have been included into the calculations, they have been set to zero in the bottom row (b), (d), and (f). The insets (g)–(i) show the corresponding values for ${\lambda }_n\left(N_r\right)$ (black $+$) and ${\lambda }_p\left(N_r\right)$ (red stars).

Figure 6

Under the reaction factors ${\alpha }_n={\alpha }_p=1,R\left(N_r\right)/R\left(0\right)$ is plotted versus $N_r$ for $\langle D\rangle =200\mathrm{nm}$. All other parameters are the same as in Fig. 5 as well as the meaning of the different colors, symbols, and insets (g)–(i).

Figure 7

Some data of Fig. 5 of Ref. [12] are redrawn: (a) $R/R_0$ under the influence of ${\mathrm{CH}}_4$ for five films of anatase-rutile ($n\text{−}p$) composites with different mass ratios $n$-anatase:$p$-rutile: 1:0 (circles), 3:1 (boxes), 1:1 (diamonds), 1:3 (triangles up), and 0:1 (triangles down). (b) $R/R_0$ for the sample with the 1:3 ratio of $n$-anatase:$p$-rutile under the influence of ${\mathrm{CH}}_4$ [open triangles up, see also (a)] and CO (filled triangles up). One can see that $R/R_0$ stays (nearly) constant under ${\mathrm{CH}}_4$, whereas it decreases considerably under CO.

Figure 8

$R\left(N_r\right)/R\left(0\right)$ is plotted versus $N_r$ for different reaction factors (a) ${\alpha }_n=0.75,{\alpha }_p=1$, (b) ${\alpha }_n=0.5,{\alpha }_p=1$, (c) ${\alpha }_n=0.3,{\alpha }_p=1$, (d) ${\alpha }_n=1,{\alpha }_p=0.75$, (e) ${\alpha }_n=1,{\alpha }_p=0.5$, and (f) ${\alpha }_n=1,{\alpha }_p=0.3$. All other parameters are the same as in Fig. 5 as well as the meaning of the different colors, symbols, and of the insets (g)–(l), i.e., the data represent the material of Fig. 5 under the influence of different gases.

Figure 9

$R\left(N_r\right)/R\left(0\right)$ is plotted versus $N_r$ for different reaction factors (a) ${\alpha }_n=0.75,{\alpha }_p=1$, (b) ${\alpha }_n=0.5,{\alpha }_p=1$, (c) ${\alpha }_n=0.3,{\alpha }_p=1$, (d) ${\alpha }_n=1,{\alpha }_p=0.75$, (e) ${\alpha }_n=1,{\alpha }_p=0.5$, and (f) ${\alpha }_n=1,{\alpha }_p=0.3$. All other parameters are the same as in Fig. 5 as well as the meaning of the different colors, symbols, and of the insets (g)–(l).

Figure 10

$R\left(N_r\right)/R\left(0\right)$ is plotted versus $N_r$ for the occupation probability $p=1$. All other parameters and the reaction factors are the same as in Fig. 9.

Figure 11

Some numerical data from Fig. 9 are shown for comparison with the experimental data from Fig. 7. The same system (the same value of $p_n$) may show an increase, a decrease, or (nearly) constant behavior of $R$ with $N_r$. From top to bottom: (a) Curves for $p_n=0.25$ from Fig. 9 (open triangles up), from Fig. 9 (small triangles up) and from Fig. 9 (big filled triangles up). (b) Curves for $p_n=0.5$ from Figs. 9 and 9.