Quantum-confined Stark effect
Encyclopedia
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, electron
s and holes
within the quantum well may only occupy states within a discrete
set of energy subbands. Consequently, 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 (ie. 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 exciton
s 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 modulator
s, 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 embebed in a nanowire.
.
,
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 quantisation 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.
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 electrons.
Similar calculations can be applied to holes by replacing the electron effective mass with the hole effective mass.
, which says that transition probability is proportional to the overlap, optical transition strength is weakened. Using this, light absorption of materials can be controlled by changing electric field and can be used as an optical modulator.
Stark effect
The Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to presence of an external static electric field. The amount of splitting and or shifting is called the Stark splitting or Stark shift. In general one distinguishes first- and second-order Stark effects...
(QCSE) describes the effect of an external electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
upon the light absorption spectrum or emission spectrum
Emission spectrum
The emission spectrum of a chemical element or chemical compound is the spectrum of frequencies of electromagnetic radiation emitted by the element's atoms or the compound's molecules when they are returned to a lower energy state....
of a quantum well
Quantum well
A quantum well is a potential well with only discrete energy values.One technology to create quantization is to confine particles, which were originally free to move in three dimensions, to two dimensions, forcing them to occupy a planar region...
(QW). In the absence of an external electric field, electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s and holes
Electron hole
An electron hole is the conceptual and mathematical opposite of an electron, useful in the study of physics, chemistry, and electrical engineering. The concept describes the lack of an electron at a position where one could exist in an atom or atomic lattice...
within the quantum well may only occupy states within a discrete
Discrete
Discrete in science is the opposite of continuous: something that is separate; distinct; individual.Discrete may refer to:*Discrete particle or quantum in physics, for example in quantum theory...
set of energy subbands. Consequently, 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 (ie. fluorescence quantum yield
Quantum yield
The quantum yield of a radiation-induced process is the number of times that a defined event occurs per photon absorbed by the system. The "event" may represent a chemical reaction, for example the decomposition of a reactant molecule:...
) 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 exciton
Exciton
An exciton is a bound state of an electron and hole which are attracted to each other by the electrostatic Coulomb force. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and some liquids...
s 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 modulator
Optical modulator
An optical modulator is a device which is used to modulate a beam of light. The beam may be carried over free space, or propagated through an optical waveguide. Depending on the parameter of a light beam which is manipulated, modulators may be categorized into amplitude modulators, phase...
s, 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
Band gap
In solid state physics, a band gap, also called an energy gap or bandgap, is an energy range in a solid where no electron states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference between the top of the valence band and the...
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 embebed 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 theoryPerturbation theory (quantum mechanics)
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...
.
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 quantisation direction, is a periodic Bloch functionBloch function
In mathematics, Bloch function may refer to* a periodic function which appears in the solution of the Schrödinger's equation with periodic potential; see Bloch wave.* an analytic function in the unit disc which is an element of the Bloch space....
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
Particle in a box
In quantum mechanics, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems...
model, in which
. Under this simplified model, analytical expressions for the bound state wavefunctions exist, with the formThe energies of the bound states are
where
is the effective massEffective mass
In solid state physics, a particle's effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. It can be shown that electrons and holes in a crystal respond to electric and magnetic fields almost as if they were particles with a mass dependence in their...
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 electrons.
Similar calculations can be applied to holes by replacing the electron effective mass with the hole effective mass.
Absorption coefficient
Addition to energy level shift, the DC electric field causes decrease of absoption coefficient. Because electron and hole are forced to opposite direction by the field, the overlap of relating valence and conduction band in transition is decreased. Thus, according to Fermi's golden ruleFermi's golden rule
In quantum physics, Fermi's golden rule is a way to calculate the transition rate from one energy eigenstate of a quantum system into a continuum of energy eigenstates, due to a perturbation....
, which says that transition probability is proportional to the overlap, optical transition strength is weakened. Using this, light absorption of materials can be controlled by changing electric field and can be used as an optical modulator.