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. When an exciton's radius extends over several unit cells, it is referred to as a Wannier exciton in contrast to Frenkel excitons whose size is comparable with the unit cell. An excited solid typically contains many electrons and holes; this modifies the Wannier equation considerably. The resulting generalized Wannier equation can be determined from the homogeneous part of the semiconductor Bloch equations or the semiconductor luminescence equations.
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Background
Since an electron and a hole have opposite charges their mutual Coulomb interaction is attractive. The corresponding Schrödinger equation, in relative coordinate
with the potential given by
Here,
In a solid, the scaling of
The Fourier transformed version of the presented Hamiltonian can be written as
where
Generalized Wannier equation
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. The homogeneous parts of these equations produce the Wannier equation at the low-density limit. 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
where
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 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 than presented above. Additionally,
Physically, the generalized Wannier equation describes how the presence of other electron–hole pairs modifies the binding of one effective pair. 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.
Applications
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. In nanostructured materials, such as quantum wells, quantum wires, and quantum dots, the Coulomb-matrix element