The Franz–Keldysh effect is a change in optical absorption by a semiconductor when an electric field is applied.
Karl W. Böer observed first the shift of the optical absorption edge with electric fields [1] during the discovery of high-field domains[2] and named this the Franz-effect.
[3] A few months later, when the English translation of the Keldysh paper became available, he corrected this to the Franz–Keldysh effect.
[4] As originally conceived, the Franz–Keldysh effect is the result of wavefunctions "leaking" into the band gap.
When an electric field is applied, the electron and hole 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 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 excitons, which may dominate optical properties near the band gap.
The Franz–Keldysh effect usually requires hundreds of volts, 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.
The absorption coefficient is related to the dielectric constant (especially the complex part
From Maxwell's equation, we can easily find out the relation, n0 and k0 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 induced by the incident light in a perfect crystal 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.
[5] We put the 1st purpose on the theoretical background of Franz–Keldysh effect and third-derivative modulation spectroscopy.
(j = v, c that mean valence band, conduction band) the transition probability can be obtained such that Power dissipation of the electromagnetic waves per unit time and unit volume gives rise to following equation
From the relation between the electric field and the vector potential,
And finally we can get the imaginary part of the dielectric constant and surely the absorption coefficient.
But Assume the slight difference of the momentum due to the photon absorption is not ignored and the bound state- electron-hole pair is very weak and the effective mass approximation is valid for the treatment.
(i, j are the band indices, and re, rh, ke, kh are the coordinates and wave vectors of the electron and hole respectively) And we can take the center of mass momentum Q such that
Then, Bloch functions of the electron and hole can be constructed with the phase term
If V varies slowly over the distance of the integral, the term can be treated like following.
(1), the effective mass equation for the exciton may be written as
The ground state of the exciton is given in analogy to the hydrogen atom.
Now we're thinking about the effective mass equation for the relative motion of electron hole pair when the external field is applied to a crystal.
When the Coulomb interaction is neglected, the effective mass equation is
is the value in the direction of the principal axis of the reduced effective mass tensor) Using change of variables:
The dielectric constant can be obtained inserting this expression into Eq.
(the Joint density of states is nothing but the meaning of DOS of both electron and hole at the same time.)
Therefore, the dielectric function for the incident photon energy below the band gap exist!
These results indicate that absorption occurs for an incident photon.