 Solution

Take the screened Coulomb potential of a singly ionized donor as $V(r)=-\kappa /r$, with $\kappa =e^2/(4\pi {ϵ}_0{ϵ}_r)$ and $r=|\mathbf{r}-{\mathbf{r}}_0|$. The electron's motion is two-dimensional, but its electrostatic interaction is the ordinary three-dimensional $1/r$ interaction restricted to the layer; it is not a logarithmic two-dimensional electrostatic potential.

For the rotationally symmetric bound state, the two-dimensional radial Schrödinger equation isFor $\psi =A{e}^{-br}$, the radial Laplacian is $(b^2-b/r)\psi$. Cancel the $1/r$ term to obtainThus the positive two-dimensional hydrogenic donor binding energy is four times the three-dimensional effective Rydberg:The decay length is $1/b=a_B^{\ast }/2\simeq 15.3\mathrm{nm}$. Normalization, if desired, follows from $1=2\pi |A{|}^2{\int }_0^{\infty }r{e}^{-2br}dr=\pi |A{|}^2/(2b^2)$, giving $A=b\sqrt{2/\pi }$ up to a constant phase. This ideal result neglects finite well thickness, dielectric mismatch at the interfaces and short-range impurity corrections.