The simplest model of electrons in metals, each metal atom contributes one or more valence electrons, which spread out between the atoms and hold the metal together. To understand what happens, consider an fcc gold crystal. Gold is a good electrical conductor, and an electric current will flow if a small electric field is applied. Next, imagine increasing the lattice constant to one centimeter. If an electric field is applied when the atoms are far apart, the atoms will polarize; the center of the negative charge of the atom will shift away from the center of the positive charge of the atom, but no current will flow. Gold with a large lattice constant will be an electrical insulator. As the lattice constant is decreased and the gold atoms are brought back together, eventually, the electron clouds of the atoms will start to overlap. This results in electron screening. The outer electrons of the gold atoms are not as tightly bound to the positive nucleus because the electrons of the neighboring atoms come in between. The form of a screened Coulomb potential is, [1]

$$ V = \frac{e}{4\pi\epsilon_0|\vec{r}-\vec{r}'| } \exp(-k_s|\vec{r}-\vec{r}'|),$$

where $k_s$ is the screening parameter, which is related to the electron density $n$. Adjust the screening parameter in the plot below and notice that as the electron density increases, the potential that the electron sees becomes narrower.

$V(r)$ eV
	$r$ Å

$k_s=$0.2 Å-1
$n$ $\Large =\frac{\pi^4\hbar^6\epsilon_0^3k_s^6}{2m^3e^6}=$ m-3

The valence electrons become unbound due to electron screening, which allows them to move in a metal when an electric field is applied.