The Franz–Keldysh effect is a change in
optical absorption
In physics, absorption of electromagnetic radiation is how matter (typically electrons bound in atoms) takes up a photon's energy — and so transforms electromagnetic energy into internal energy of the absorber (for example, thermal energy). A ...
by a
semiconductor
A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...
when an
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
is applied. The effect is named after the German physicist
Walter Franz
Walter Franz (8 April 1911, in Munich – 16 February 1992, in Münster) was a German theoretical physicist who independently discovered the Franz–Keldysh effect.
Franz was a student of Arnold Sommerfeld at the University of Munich. He wa ...
and Russian physicist
Leonid Keldysh.
Karl W. Böer observed first the shift of the optical
absorption edge An absorption edge, absorption discontinuity or absorption limit is a sharp discontinuity in the absorption spectrum of a substance. These discontinuities occur at wavelengths where the energy of an absorbed photon corresponds to an electronic tran ...
with electric fields during the discovery of high-field domains and named this the Franz-effect. A few months later, when the English translation of the Keldysh paper became available, he corrected this to the Franz–Keldysh effect.
As originally conceived, the Franz–Keldysh effect is the result of
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
s "leaking" into the band gap. When an electric field is applied, the
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
and
hole
A hole is an opening in or through a particular medium, usually a solid body. Holes occur through natural and artificial processes, and may be useful for various purposes, or may represent a problem needing to be addressed in many fields of en ...
wavefunctions become
Airy functions rather than plane waves. The Airy function includes a "tail" which extends into the classically forbidden band gap. According to
Fermi's golden rule, the more overlap there is between the wavefunctions of a free electron and a hole, the stronger the optical absorption will be. The Airy tails slightly overlap even if the electron and hole are at slightly different potentials (slightly different physical locations along the field). The absorption spectrum now includes a tail at energies below the band gap and some oscillations above it. This explanation does, however, omit the effects of
exciton
An exciton is a bound state of an electron and an electron 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. The ...
s, which may dominate optical properties near the band gap.
The Franz–Keldysh effect occurs in uniform, bulk semiconductors, unlike the
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 th ...
, which requires a quantum well. Both are used for
electro-absorption modulator An electro-absorption modulator (EAM) is a semiconductor device which can be used for modulating the intensity of a laser beam via an electric voltage. Its principle of operation is based on the Franz–Keldysh effect, i.e., a change in the absorpti ...
s. The Franz–Keldysh effect usually requires hundreds of
volt
The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827).
Defi ...
s, limiting its usefulness with conventional electronics – although this is not the case for commercially available Franz–Keldysh-effect electro-absorption modulators that use a waveguide geometry to guide the optical carrier.
Effect on modulation spectroscopy
The
absorption coefficient is related to the
dielectric constant
The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insulat ...
(especially the complex part
2). From Maxwell's equation, we can easily find out the relation,
:
''n''
0 and ''k''
0 are the real and complex parts of the refractive index of the material.
We will consider the direct transition of an electron from the valence band to the
conduction band
In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in w ...
induced by the
incident light
In optics a ray is an idealized geometrical model of light, obtained by choosing a curve that is perpendicular to the ''wavefronts'' of the actual light, and that points in the direction of energy flow. Rays are used to model the propagation o ...
in a
perfect crystal
Crystalline materials (mainly metals and alloys, but also stoichiometric salts and other materials) are made up of solid regions of ordered matter (atoms placed in one of a number of ordered formations called Bravais lattices). These regions are kn ...
and try to take into account of the change of absorption coefficient for each Hamiltonian with a probable interaction like electron-photon, electron-hole, external field. These approach follows from.
[C. Hamaguchi, "Basic Semiconductor Physics", Springer (2001)] We put the 1st purpose on the theoretical background of Franz–Keldysh effect and third-derivative modulation spectroscopy.
One electron Hamiltonian in an electro-magnetic field
:
where A is the
vector potential and ''V''(r) is a periodic potential.
: