Percolation and the electron-electron interaction in an array of antidots

A square lattice of microcontacts with a period of 1 $\mu$m in a dense low-mobility two-dimensional electron gas is studied experimentally and numerically. At the variation of the gate voltage $V_g$, the conductivity of the array varies by five orders of magnitude in the temperature range $T$ from $1.4$ to $77$ K in good agreement with the formula $\sigma(V_g)=A(V_g-V_g^*(T))^{\beta}$ with $\beta=4$. The saturation of $\sigma(T)$ at low temperatures is absent because of the electron-electron interaction. A random-lattice model with a phenomenological potential in microcontacts reproduces the dependence $\sigma(T, V_g)$ and makes it possible to determine the fraction of microcontacts $x(V_g, T)$ with conductances higher than $\sigma$. It is found that the dependence $x(V_g)$ is nonlinear and the critical exponent in the formula $\sigma\propto(x-1/2)^t$ in the range $1.3<t(T,V_g)<\beta$.