As a crude approximation, an impurity pentavalent atom in a (tetravalent) silicon lattice can be treated as a one-electron atom, in which the extra electron orbits a net positive charge of 1. Because this "atom" is not in free space, however, the permitivity of free space, ${\mathit{\epsilon }}_{\mathbf{0}}$. must be replaced by $\mathit{\kappa }{\mathit{\epsilon }}_{\mathbf{0}}$, where$\mathit{\kappa }$ is the dielectric constant of the surrounding material. The hydrogen atom ground-state energies would thus become

$\mathit{E}\mathbf{=}\mathbf{-}\frac{\mathbf{m}{\mathbf{e}}^{\mathbf{4}}}{\mathbf{2}{\mathbf{\left(}\mathbf{4}\mathbf{\pi }\mathbf{\kappa }{\mathbf{\epsilon }}_{\mathbf{0}}\mathbf{\right)}}^{\mathbf{2}}{\mathbf{\hslash }}^{\mathbf{2}}}\frac{\mathbf{1}}{{\mathbf{n}}^{\mathbf{2}}}\mathbf{=}\frac{\mathbf{-}\mathbf{13}\mathbf{.6}\mathbf{\text{eV}}}{{\mathbf{\kappa }}^{\mathbf{2}}{\mathbf{n}}^{\mathbf{2}}}$

Given $\mathit{\kappa }\mathbf{=}\mathbf{12}$for silicon, how much energy is needed to free a donor electron in its ground state? (Actually. the effective mass of the donor electron is less than , so this prediction is somewhat high.)

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