Unbiased system

The potential for a quantum well may be written as

${\displaystyle V(z)={\begin{cases}0;&|z|<L/2\\V_{0};&{\mbox{otherwise}}\end{cases}}}$,

where ${\displaystyle L}$ is the width of the well and ${\displaystyle V_{0}}$ is the height of the potential barriers. The bound states in the well lie at a set of discrete energies, ${\displaystyle E_{n}}$ and the associated wavefunctions can be written using the envelope function approximation as follows:

${\displaystyle \psi (\mathbf {r} )=\phi _{n}(z){\frac {1}{\sqrt {A}}}e^{i(k_{x}\cdot {x}+k_{y}\cdot {y})}u(\mathbf {r} ).}$

In this expression, ${\displaystyle A}$ is the cross-sectional area of the system, perpendicular to the quantization direction, ${\displaystyle u(\mathbf {r} )}$ is a periodic Bloch function for the energy band edge in the bulk semiconductor and ${\displaystyle \phi _{n}(z)}$ is a slowly varying envelope function for the system.

On the left: wave functions corresponding to the n=1 and n=3 levels in a quantum well with no applied electric field (${\displaystyle {\vec {F}}=0}$). On the right: the perturbative effect of the applied electric field ${\displaystyle {\vec {F}}\neq 0}$ modifies the wave functions and decreases the energy ${\displaystyle E}$ of the n=1 transition.

If the quantum well is very deep, it can be approximated by the particle in a box model, in which ${\displaystyle V_{0}\to \infty }$. Under this simplified model, analytical expressions for the bound state wavefunctions exist, with the form

${\displaystyle \phi _{n}(z)={\sqrt {\frac {2}{L}}}\times {\begin{cases}\cos \left({\frac {n\pi z}{L}}\right)&n\,{\text{odd}}\\\sin \left({\frac {n\pi z}{L}}\right)&n\,{\text{even}}\end{cases}}.}$

The energies of the bound states are

${\displaystyle E_{n}={\frac {\hbar ^{2}n^{2}\pi ^{2}}{2m^{*}L^{2}}},}$

where ${\displaystyle m^{*}}$ is the effective mass of an electron in a given semiconductor.

Biased system

Supposing the electric field is biased along the z direction,

${\displaystyle \mathbf {F} =F\mathbf {z} ,}$

the perturbing Hamiltonian term is

${\displaystyle H'=eFz.}$

The first order correction to the energy levels is zero due to symmetry.

${\displaystyle E_{n}^{(1)}=\langle n^{(0)}|eFz|n^{(0)}\rangle =0}$.

The second order correction is, for instance n=1,

${\displaystyle E_{1}^{(2)}=\sum _{k\neq 1}{\frac {|\langle k^{(0)}|eFz|1^{(0)}\rangle |^{2}}{E_{1}^{(0)}-E_{k}^{(0)}}}\approx {\frac {|\langle 2^{(0)}|eFz|1^{(0)}\rangle |^{2}}{E_{1}^{(0)}-E_{2}^{(0)}}}=-24\left({\frac {2}{3\pi }}\right)^{6}{\frac {e^{2}F^{2}m_{e}^{*}L^{4}}{\hbar ^{2}}}}$

for electron, where the additional approximation of neglecting the perturbation terms due to the bound states with k even and > 2 has been introduced. By comparison, the perturbation terms from odd-k states are zero due to symmetry.

Similar calculations can be applied to holes by replacing the electron effective mass ${\displaystyle m_{e}^{*}}$ with the hole effective mass ${\displaystyle m_{h}^{*}}$. Introducing the total effective mass ${\displaystyle m_{tot}^{*}=m_{e}^{*}+m_{h}^{*}}$, the energy shift of the first optical transition induced by QCSE can be approximated to:

${\displaystyle \Delta E\approx -24\left({\frac {2}{3\pi }}\right)^{6}{\frac {e^{2}F^{2}m_{tot}^{*}L^{4}}{\hbar ^{2}}}.}$^[4]

The downward shift in the confined energy level discussed in the above equation is referred to as the Franz-Keldysh effect.

The approximations made so far are quite crude, nonetheless the energy shift does show experimentally a square law dependence from the applied electric field,^[5] as predicted.

Absorption coefficient

Experimental demonstration of quantum-confined Stark effect in Ge/Si${\displaystyle _{0.18}}$Ge${\displaystyle _{0.82}}$ quantum wells.

Numerical simulation of the absorption coefficient of Ge/Si${\displaystyle _{0.18}}$Ge${\displaystyle _{0.82}}$ quantum wells

Additionally to the redshift towards lower energies of the optical transitions, the DC electric field also induces a decrease in magnitude of the absorption coefficient, as it decreases the overlapping integrals of relating valence and conduction band wave functions. Given the approximations made so far and the absence of any applied electric field along z, the overlapping integral for ${\displaystyle n_{valence}=n_{conduction}}$ transitions will be:

${\displaystyle \langle \phi _{c,n}|\phi _{v,n}\rangle =1}$.

To calculate how this integral is modified by the quantum-confined Stark effect we once again employ time independent perturbation theory. The first order correction for the wave function is

${\displaystyle \phi _{n}^{'}=\sum _{k\neq n}{\frac {\langle \phi _{n}|H'|\phi _{k}\rangle }{E_{n}-E_{k}}}|\phi _{k}\rangle }$.

Once again we look at the ${\displaystyle n=1}$ energy level and consider only the perturbation from the level ${\displaystyle n=2}$ (notice that the perturbation from ${\displaystyle n=3}$ would be ${\displaystyle =0}$ due to symmetry). We obtain

${\displaystyle \phi _{c,1}=\phi _{c,1}^{0}+\phi _{c,1}^{'}={\frac {1}{A}}\left(\cos \left({\frac {\pi z}{L}}\right)-\left({\frac {2}{3\pi }}\right)^{4}{\frac {2m_{e}^{*}eFL^{3}}{\hbar ^{2}}}\sin \left({\frac {\pi z}{L}}\right)\right)}$

${\displaystyle \phi _{v,1}=\phi _{v,1}^{0}+\phi _{v,1}^{'}={\frac {1}{A}}\left(\cos \left({\frac {\pi z}{L}}\right)+\left({\frac {2}{3\pi }}\right)^{4}{\frac {2m_{h}^{*}eFL^{3}}{\hbar ^{2}}}\sin \left({\frac {\pi z}{L}}\right)\right)}$

for the conduction and valence band respectively, where ${\displaystyle A}$ has been introduced as a normalization constant. For any applied electric field ${\displaystyle {\vec {F}}\cdot {\hat {z}}\neq 0}$ we obtain

${\displaystyle \langle \phi _{c,1}|\phi _{v,1}\rangle <1}$.

Thus, according to Fermi's golden rule, which says that transition probability depends on the above overlapping integral, optical transition strength is weakened.

Excitons

The description of quantum-confined Stark effect given by second order perturbation theory is extremely simple and intuitive. However to correctly depict QCSE the role of excitons has to be taken into account. Excitons are quasiparticles consisting of a bound state of an electron-hole pair, whose binding energy in a bulk material can be modelled as that of an hydrogenic atom

${\displaystyle E_{X,n}={\frac {\mu }{m_{e}\varepsilon _{r}^{2}}}{\frac {R_{H}}{n^{2}}}}$

where ${\displaystyle R_{H}}$ is the Rydberg constant, ${\displaystyle \mu }$ is the reduced mass of the electron-hole pair and ${\displaystyle \varepsilon _{r}}$ is the relative electric permittivity. The exciton binding energy has to be included in the energy balance of photon absorption processes:

${\displaystyle h\nu >E_{g}-E_{X}}$.

Exciton generation therefore redshift the optical band gap towards lower energies. If an electric field is applied to a bulk semiconductor, a further redshift in the absorption spectrum is observed due to Franz–Keldysh effect. Due to their opposite electric charges, the electron and the hole constituting the exciton will be pulled apart under the influence of the external electric field. If the field is strong enough

${\displaystyle -e{\vec {F}}\cdot {\vec {r_{X}}}>|E_{X}|}$

then excitons cease to exist in the bulk material. This somewhat limits the applicability of Franz-Keldysh for modulation purposes, as the redshift induced by the applied electric field is countered by shift towards higher energies due to the absence of exciton generations.

This problem does not exist in QCSE, as electrons and holes are confined in the quantum wells. As long as the quantum well depth is comparable to the excitonic Bohr radius, strong excitonic effects will be present no matter the magnitude of the applied electric field. Furthermore, quantum wells behave as two dimensional systems, which strongly enhance excitonic effects with respect to bulk material. In fact, solving the Schrödinger equation for a Coulomb potential in a two dimensional system yields an excitonic binding energy of

${\displaystyle E_{X,n}={\frac {\mu }{m_{e}\varepsilon _{r}^{2}}}{\frac {R_{H}}{n^{2}-1/2}}}$

which is four times as high as the three dimensional case for the ${\displaystyle 1s}$ solution.^[6]