Nonlinear screening of negative point charges in diamond, silicon, and germanium

In the present paper we formulate a variational principle for obtaining approximate analytical solutions of a nonlinear differential equation established by Cornolti and Resta for the potentials of negative point charges embedded in pure diamond, silicon,and germanium. We consider the case of charges $Z=-1, -2, -3, \mathrm{and} -4\mathrm{}$ (in atomic units) in these semiconductors, while Cornolti and Resta considered the cases of $Z=-1\mathrm{} \mathrm{and} -4\mathrm{}$. We find that our approximate analytical results for the spatial dielectric functions of diamond, silicon, and germanium, depending on $Z$, are in excellent agreement with the numerical results of Cornolti and Resta, who have presented their results in graphical form.