Quantum-confined Stark effect
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.
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.
[1] 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.
[2] 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.
[3] 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.
is a periodic Bloch function for the energy band edge in the bulk semiconductor and
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.
Similar calculations can be applied to holes by replacing the electron effective mass
, the energy shift of the first optical transition induced by QCSE can be approximated to: The approximations made so far are quite crude, nonetheless the energy shift does show experimentally a square law dependence from the applied electric field,[5] as predicted.
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 description of quantum-confined Stark effect given by second order perturbation theory is extremely simple and intuitive.
Excitons are quasiparticles consisting of a bound state of an electron-hole pair, whose binding energy in a bulk material can be modelled as that of an hydrogenic atom where
If an electric field is applied to a bulk semiconductor, a further redshift in the absorption spectrum is observed due to Franz–Keldysh effect.
This somewhat limits the applicability of Franz-Keldysh for modulation purposes, as the redshift induced by the applied electric field is countered by shift towards higher energies due to the absence of exciton generations.
This problem does not exist in QCSE, as electrons and holes are confined in the quantum wells.
Furthermore, quantum wells behave as two dimensional systems, which strongly enhance excitonic effects with respect to bulk material.
[6] Quantum-confined Stark effect's most promising application lies in its ability to perform optical modulation in the near infrared spectral range, which is of great interest for silicon photonics and down-scaling of optical interconnects.
[2][7] A QCSE based electro-absorption modulator consists of a PIN structure where the instrinsic region contains multiple quantum wells and acts as a waveguide for the carrier signal.
An electric field can be induced perpendicularly to the quantum wells by applying an external, reverse bias to the PIN diode, causing QCSE.
This mechanism can be employed to modulate wavelengths below the band gap of the unbiased system and within the reach of the QCSE induced redshift.
[8] Differently from III/V semiconductors, Ge/SiGe quantum well stacks can be epitaxially grown on top of a silicon substrate, provided the presence of some buffer layer in between the two.
This is a decisive advantage as it allows Ge/SiGe QCSE to be integrated with CMOS technology[9] and silicon photonics systems.
is the optical fiber`s transparency window and the most extensively employed wavelength for telecommunications.
Electro-optic modulation by QCSE using Ge/SiGe quantum wells has been demonstrated up to 23 GHz with energies per bit as low as 108 fJ.
