The Wannier equation describes a quantum mechanical eigenvalue problem in solids where an electron in a conduction band and an electronic vacancy (i.e. hole) within a valence band attract each other via the Coulomb interaction.
For one electron and one hole, this problem is analogous to the Schrödinger equation of the hydrogen atom; and the bound-state solutions are called excitons.
An excited solid typically contains many electrons and holes; this modifies the Wannier equation considerably.
Since an electron and a hole have opposite charges their mutual Coulomb interaction is attractive.
) is the elementary charge related to an electron (hole),
The solutions of the hydrogen atom are described by eigenfunction
and the wavefunction size are orders of magnitude different from the hydrogen problem because the relative permittivity
As a result, the exciton radius can be large while the exciton binding energy is small, typically few to hundreds of meV, depending on material, compared to eV for the hydrogen problem.
[1][2] The Fourier transformed version of the presented Hamiltonian can be written as where
The Coulomb sums follows from the convolution theorem and the
The Wannier equation can be generalized by including the presence of many electrons and holes in the excited system.
One can start from the general theory of either optical excitations or light emission in semiconductors that can be systematically described using the semiconductor Bloch equations (SBE) or the semiconductor luminescence equations (SLE), respectively.
Therefore, the homogeneous parts of the SBE and SLE provide a physically meaningful way to identify excitons at arbitrary excitation levels.
The resulting generalized Wannier equation is where the kinetic energy becomes renormalized by the electron and hole occupations
weakens the Coulomb interaction via the so-called phase-space filling factor that stems from the Pauli exclusion principle preventing multiple excitations of fermions.
Due to the phase-space filling factor, the Coulomb attraction becomes repulsive for excitations levels
At this regime, the generalized Wannier equation produces only unbound solutions which follow from the excitonic Mott transition from bound to ionized electron–hole pairs.
Once electron–hole densities exist, the generalized Wannier equation is not Hermitian anymore.
As a result, the eigenvalue problem has both left- and right-handed eigenstates
The left- and right-handed eigenstates have the same eigen value
(that is real valued for the form shown) and they form a complete set of orthogonal solutions since The Wannier equations can also be generalized to include scattering and screening effects that appear due to two-particle correlations within the SBE.
This extension also produces left- and right-handed eigenstate, but their connection is more complicated[4] than presented above.
As main consequences, an excitation tends to weaken the Coulomb interaction and renormalize the single-particle energies in the simplest form.
Once also correlation effects are included, one additionally observes the screening of the Coulomb interaction, excitation-induced dephasing, and excitation-induced energy shifts.
All these aspects are important when semiconductor experiments are explained in detail.
Due to the analogy with the hydrogen problem, the zero-density eigenstates are known analytically for any bulk semiconductor when excitations close to the bottom of the electronic bands are studied.
strongly deviates from the ideal two- and three-dimensional systems due to finite quantum confinement of electronic states.
Hence, one cannot solve the zero-density Wannier equation analytically for those situations, but needs to resort to numerical eigenvalue solvers.
In general, only numerical solutions are possible for all semiconductor cases when exciton states are solved within an excited matter.
Further examples are shown in the context of the Elliott formula.