# Screening due to electron-electron interaction

In the random phase approximation we obtain that the Coulomb interaction between charges is replaced by an effective interaction $$V(r)=\frac{1}{r}e^{-k_{D}r}$$ In other words, the electric field only extends over a distance $$\frac{1}{k_D}$$. If it is a positive charge it is clear, the electron cloud leads to screening, but what does occur when the charge is negative? It seems that in this case there isn't any screening; is it correct that screening occurs only for positive charge?

Actually I believe that the screening occurs whether the point charge has the same sign, or the opposite sign, to the mobile charge carriers. The derivation of the screened Coulomb potential involves solving Poisson's equation in the presence of a perturbing point charge (or charge distribution); either the Debye-Huckel theory or the Thomas-Fermi approximation is used. The result is basically the same, and the screening occurs independently of the sign of the point charge. The Poisson equation with screening, for a point charge $$Q$$ at the origin, is $$\left[\nabla^2 - k_D^2 \right] V(\mathbf{r}) = -\frac{Q}{\varepsilon_0}\delta(\mathbf{r})$$ and the solution is $$V(r) = \frac{Q}{4\pi\varepsilon_0 r} \exp(-k_D r)$$ irrespective of whether $$Q$$ is positive or negative.