The quantum-confined Stark effect (QCSE) describes the effect of an external electric field upon the light absorption spectrum or emission spectrum of a quantum well (QW). In the absence of an external electric field, electrons and holes within the quantum well may only occupy states within a discrete set of energy subbands. Only a discrete set of frequencies of light may be absorbed or emitted by the system. When an external electric field is applied, the electron states shift to lower energies, while the hole states shift to higher energies. This reduces the permitted light absorption or emission frequencies. Additionally, the external electric field shifts electrons and holes to opposite sides of the well, decreasing the overlap integral, which in turn reduces the recombination efficiency (i.e. fluorescence quantum yield) of the system.
The spatial separation between the electrons and holes is limited by the presence of the potential barriers around the quantum well, meaning that excitons are able to exist in the system even under the influence of an electric field. The quantum-confined Stark effect is used in QCSE optical modulators, which allow optical communications signals to be switched on and off rapidly.
Even if Quantum Objects (Wells, Dots or Discs, for instance) emit and absorb light generally with higher energies than the band gap of the material, the QCSE may shift the energy to values lower than the gap. This was evidenced recently in the study of quantum discs embedded in a nanowire.
Theoretical description
The shift in absorption lines can be calculated by comparing the energy levels in unbiased and biased quantum wells. It is a simpler task to find the energy levels in the unbiased system, due to its symmetry. If the external electric field is small, it can be treated as a perturbation to the unbiased system and its approximate effect can be found using perturbation theory.
Unbiased system
The potential for a quantum well may be written as
, where is the width of the well and is the height of the potential barriers. The bound states in the well lie at a set of discrete energies, and the associated wavefunctions can be written using the envelope function approximation as follows:
In this expression, is the cross-sectional area of the system, perpendicular to the quantization direction, is a periodic Bloch function for the energy band edge in the bulk semiconductor and is a slowly varying envelope function for the system.
If the quantum well is very deep, it can be approximated by the particle in a box model, in which . Under this simplified model, analytical expressions for the bound state wavefunctions exist, with the form
The energies of the bound states are
where is the effective mass of an electron in a given semiconductor.
Biased system
Supposing the electric field is biased along the z direction,
the perturbing Hamiltonian term is
The first order correction to the energy levels is zero due to symmetry. . The second order correction is, for instance n=1,
for electron, where the additional approximation of neglecting the perturbation terms due to the bound states with k even and > 2 has been introduced. By comparison, the perturbation terms from odd-k states are zero due to symmetry.
Similar calculations can be applied to holes by replacing the electron effective mass with the hole effective mass . Introducing the total effective mass , the energy shift of the first optical transition induced by QCSE can be approximated to:
The downward shift in the confined energy level discussed in the above equation is referred to as the Franz-Keldysh effect. The approximations made so far are quite crude, nonetheless the energy shift does show experimentally a square law dependence from the applied electric field, as predicted.
Absorption coefficient
Additionally to the redshift towards lower energies of the optical transitions, the DC electric field also induces a decrease in magnitude of the absorption coefficient, as it decreases the overlapping integrals of relating valence and conduction band wave functions. Given the approximations made so far and the absence of any applied electric field along z, the overlapping integral for transitions will be: . To calculate how this integral is modified by the quantum-confined Stark effect we once again employ time independent perturbation theory. The first order correction for the wave function is . Once again we look at the energy level and consider only the perturbation from the level (notice that the perturbation from would be due to symmetry). We obtain