= Wannier-Mott exciton
{c}
{title2=$a_B^*,R^*$}
= Mott-Wannier exciton
{c}
{synonym}
= Wannier exciton
{c}
{synonym}
A <Wannier-Mott exciton> has an internal <wave function> extending over many <unit cells>. Its simplest model assumes a three-dimensional bulk crystal, approximately parabolic <valence band> and <conduction band>, positive isotropic scalar <effective masses> $m_e^*,m_h^*$, a constant scalar <relative permittivity> $\epsilon_r$, and a dilute pair population. Replace microscopic crystal forces by those <effective masses> and <dielectric screening>, and neglect exchange corrections and coupling to <phonons>. These assumptions must hold on the spatial and <energy> scales of the particular <exciton>.
With relative coordinate $\mathbf r=\mathbf r_e-\mathbf r_h$, the internal <Hamiltonian operator> is the three-dimensional <hydrogen atom> problem with a <reduced mass> and a screened <Coulomb potential>:
$$
H_{\rm rel}=-\frac{\hbar^2}{2\mu}\nabla_{\mathbf r}^2-\frac{e^2}{4\pi\epsilon_0\epsilon_r r},\qquad
\mu=\frac{m_e^*m_h^*}{m_e^*+m_h^*},\qquad M=m_e^*+m_h^*.
$$
Here $e$ is the <elementary charge>, $\epsilon_0$ the <vacuum permittivity>, and $\hbar$ the reduced <Planck constant>. The internal <energy eigenvalues>, radius scale, and <exciton binding energies> are
$$
a_B^*=\frac{4\pi\epsilon_0\epsilon_r\hbar^2}{\mu e^2},\qquad
R^*=\frac{\mu e^4}{2(4\pi\epsilon_0\epsilon_r)^2\hbar^2},\qquad
E_{{\rm rel},n}=-\frac{R^*}{n^2},\qquad E_{B,n}=\frac{R^*}{n^2}.
$$
The <principal quantum number> is $n=1,2,\ldots$. This reduction and these scales are derived in <Excitons in Semiconductor Quantum Dots by Matjaž Humar> (§3, printed pp. 5–6, equations (13)–(26)).
To check the scales by <dimensional analysis>, write $C=e^2/(4\pi\epsilon_0\epsilon_r)$, which has units of <energy> times <length>. Then $a_B^*=\hbar^2/(\mu C)$ is a <length> and $R^*=\mu C^2/(2\hbar^2)$ is an <energy>. Rescaling $r=a_B^*\rho$ makes $H_{\rm rel}/R^*=-\nabla_\rho^2-2/\rho$, the dimensionless <hydrogen atom> operator. The <Bohr radius> $a_B^*$ is a <wave function> scale, not the radius of a classical orbit; for this model's lowest internal <bound state>, the mean separation is $3a_B^*/2$.
For the diagrams' hypothetical inputs $m_e^*=0.10m_0$, $m_h^*=0.40m_0$, $\epsilon_r=10$ and $E_g=1.500\,\mathrm{eV}$, where $m_0$ is the free <Electron> mass, independent substitution gives $\mu=0.080m_0$, $a_B^*\simeq6.615\,\mathrm{nm}$ and $R^*\simeq10.885\,\mathrm{meV}$. Here <electronvolts> express <energy>. The <exciton binding energies> for $n=1,2,3$ are approximately $10.885$, $2.721$ and $1.209\,\mathrm{meV}$. These are model calculations, not measured material parameters. The large-radius approximation additionally requires $a_B^*$ to be much larger than the crystal's <unit cell> length scale. In a thin layer, with anisotropic <effective masses>, or with spatially varying <dielectric screening>, this three-dimensional formula is not generally valid.
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