Franz–Keldysh effect
Franz–Keldysh effect
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).
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 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 the complex part \kappa2). 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. These approach follows from.[5] 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
![{\displaystyle A={1 \over 2}A_{0}e[e^{i(k_{p}\cdot r-\omega t)}+e^{-i(k_{p}\cdot r-\omega t)}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12f419af720d167871c258fba984e9144a5c35e9)
- and e are the wave vector of em field and
the transition probability can be obtained such that
![{\displaystyle w_{cv}={2\pi \over \hbar }|\langle ck'|{e \over m}A\cdot p|vk\rangle |^{2}\delta [\mathrm {E} _{c}(k')-\mathrm {E} _{v}(k)-\hbar \omega ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3cab3f5d9194aeb63b91b5f31f1fb65118b868c)
![{\displaystyle ={\pi e^{2} \over 2\hbar m^{2}}A_{0}^{2}|\langle ck'|exp(ik_{p}\cdot r)e\cdot p|vk\rangle |^{2}\delta [\mathrm {E} _{c}(k')-\mathrm {E} _{v}(k)-\hbar \omega ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c3df0fcec9897b3dab3425fb6930e57e2af6469)
Power dissipation of the electromagnetic waves per unit time and unit volume gives rise to following equation
![{\displaystyle \kappa ={{\pi e^{2}} \over {\epsilon _{0}m^{2}\omega ^{2}}}\sum _{k,k'}|e\cdot p_{cv}|^{2}\delta [\mathrm {E} _{c}(k')-\mathrm {E} _{v}(k)-\hbar \omega ]\delta _{kk'}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87ccfc42e3fc10344584fd327142441dce53d2fb)
2-body(electron-hole) Hamiltonian with EM field
And we can take a total wave vector K such that
If V varies slowly over the distance of the integral, the term can be treated like following.
![{\displaystyle [\mathrm {E} _{c}(k_{e})+\mathrm {E} _{h}(k_{h})+V(r_{e}-r_{h})-\epsilon ]A_{c,V}^{n,K}(k_{e},k_{h})=0(*)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f64673e7939a3f5753b56a40f4b2dbb1ba3a46)
![{\displaystyle [(-{\hbar ^{2} \over 2M}\nabla ^{2})+(-{\hbar ^{2} \over 2\mu }\nabla ^{2}-{e^{2} \over 4\pi \epsilon r})]\Phi ^{n,k}(r,R)=[\mathrm {E} -\mathrm {E} _{G}]\cdot \Phi ^{n,K}(r,R)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a3f464f91c5225cff7414159fe5ec57fb0cd10)
then the solution of eq is given by
then, the dielectric function is
detailed calculation is in.[5]
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
![{\displaystyle [-{\hbar ^{2} \over 2\mu }\nabla ^{2}-eE\cdot r]\psi (r)=\epsilon \psi (r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f355e5c51b39fc820c03ae3bc1940050f57fc8bb)
And the equation can be expressed,
![{\displaystyle [-{\hbar ^{2} \over 2\mu }{d^{2} \over {dr_{i}^{2}}}-eE_{i}r_{i}-\epsilon _{i}]\psi (r_{i})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5e860bb7e31cc51d6169e3958ea45fdb8389e11)
Using change of variables:
then the solution is
![{\displaystyle \kappa _{2}(\omega ,E_{x})={1/2}\kappa _{2}(\omega )exp[{-4 \over 3}({{\epsilon _{G}-\hbar \omega } \over {\hbar \theta _{x}}})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10145a3f886d20c9e6bc3f88c89dc70818ed99c1)
Therefore, the dielectric function for the incident photon energy below the band gap exist! These results indicate that absorption occurs for an incident photon.
See also
Quantum-confined Stark effect