The Franz–Keldysh effect is a change in optical absorption by a semiconductor when an electric field is applied. The effect is named after the German physicist Walter Franz and Russian physicist Leonid Keldysh (nephew of Mstislav Keldysh).
Contents
- Effect on modulation spectroscopy
- One electron Hamiltonian with EM electro magnetic field
- 2 bodyelectron hole Hamiltonian with EM field
- References
Karl W. Böer observed first the shift of the optical absorption edge 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 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 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 excitons, which may dominate optical properties near the band gap.
The Franz–Keldysh effect occurs in uniform, bulk semiconductors, unlike the quantum-confined Stark effect, which requires a quantum well. Both are used for Electro-absorption modulators. 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.
Effect on modulation spectroscopy
The absorption coefficient is related to the dielectric constant (especially complex term). From Maxwell's equation, we can easily find out the relation,
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. These approach follows from. We put the 1st purpose on the theoretical background of Franz–Keldysh effect and third-derivative modulation spectroscopy.
One electron Hamiltonian with EM (electro magnetic) field
Neglecting the square term
Then using the Bloch function
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.
2-body(electron-hole) Hamiltonian with EM field
An electron in the valence band(wave vector k) is excited by photon absorption into the conduction band(the wave vector at the band is
Thinking about the direct transition,
And we can take a total wave vector K such that
Then, Bloch functions of the electron and hole can be constructed with the phase term
If V slowly over the distance of the integral, the term can be treated like following.
here we assume that the conduction and valence bands are parabolic with scalar masses and that at the top of the valence band
Now, The Fourier transform of
then the solution of eq is given by
then, the dielectric function is
detailed calculation is in.
Franz–Keldysh effect means an electron in a valence band can be allowed to be excited into a conduction band by absorbing a photon with its energy below the band gap. 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. But we are not to take a mutual potential of electron-hole pair into the Hamiltonian.
When the Coulomb interaction is neglected, the effective mass equation is
And the equation can be expressed,
Using change of variables:
then the solution is
where
For example,
The dielectric constant can be obtained inserting this equation to the (**) (above block), and changing the summation with respect to λ to
The integral with respect to
where
Then we put
And think about the case we find
Finally,
Therefore, the dielectric function for the incident photon energy below the band gap exist! These results indicate that absorption occurs for an incident photon.