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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?

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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.

If you want a physical picture, suppose that we are considering an initially uniform electron gas (you may add a neutralizing, fixed, positive charge distribution too, giving the jellium model, if you wish: the essential point is that it is the electrons that move in response to a perturbation). If we add a positive point charge, we expect the electrons to respond by increasing their density in its neighbourhood, giving the familiar electron cloud or atmosphere. If, instead, we add a negative point charge, the electrons respond by reducing their density very close to it. You could achieve the same effect by superimposing (on the initial uniform distribution of negative charge) a positively charged atmosphere, having the same form as the electron cloud that we think of surrounding a positive point charge. The effect is due to the way the mobile charges respond to the point charge, whatever its sign, and it is always a screening effect.

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