Self-consistent calculation of dephasing in quantum cascade structures within a density matrix method

Dephasing in terahertz quantum cascade structures is studied within a density matrix formalism. We self-consistently calculate the pure dephasing time from the intrasubband interactions within the upper and lower lasing states. Interface roughness and ionized impurity scattering interactions are included in the calculation. Dephasing times are shown to be consistent with measured spontaneous emission spectra, and the lattice temperature dependence of the device output power is consistent with experiment. The importance of including multiple optical transitions when a lower miniband continuum is present and the resulting multi-longitudinal modes within the waveguide resonant cavity are also shown.

Figure 1

The FWHM linewidth as a function of the pure dephasing time calculated for a generalized LO phonon depopulated three-level QC structure where $\mathrm{\Delta }{E}_{3-2}=16\mathrm{meV}$ and $\mathrm{\Delta }{E}_{2-1}=\hslash {\omega }_{\mathrm{LO}}$. The effect of pure dephasing on spectral broadening is illustrated by the increasing FWHM for shorter pure dephasing times. Also shown is the corresponding decrease in the unsaturated peak optical gain for shorter pure dephasing times. The representative lifetimes used in these calculations are ${\tau }_3=2\mathrm{ps}, {\tau }_{3-1}=6\mathrm{ps}$, and ${\tau }_2=0.5\mathrm{ps}$, with $x_{ul}=3.5\mathrm{nm}$ and the average doping concentration per QC period is $n_{3\mathrm{D}}=6\times {10}^{15}\mathrm{c}{\mathrm{m}}^{-3}$. The anticrossing ${\mathrm{\Delta }}_{1-3^{\prime }}=1\mathrm{meV}$ and unity injection efficiency are assumed.

Figure 2

(a) The peak optical power as a function of lattice temperature calculated using the density matrix Monte Carlo method for the QC structure in Ref. [14]. The experimentally measured peak optical power is also shown. A $\mathrm{\Delta }T=15\mathrm{K}$ between the lattice and the heat sink temperature scales is set corresponding to the approximated temperature rise typically measured in QC devices. The inset: the saturated optical gain spectra at 25 K are shown for the combined $|5\rangle \to |4\rangle ,|3\rangle$ optical transition as well as the spectra for the $|5\rangle \to |4\rangle$ and $|5\rangle \to |3\rangle$ optical transitions. The optical gain peak for the combined $|5\rangle \to |4\rangle ,|3\rangle$ bound-to-continuum (miniband) transition is clamped at the finite element method calculated threshold $({\alpha }_{\mathit{w}}+{\alpha }_m)/\mathrm{\Gamma }=55.1\mathrm{c}{\mathrm{m}}^{-1}$ at $\sim 3.4\mathrm{THz}$ which agrees with experiment, whereas the peaks for the $|5\rangle \to |4\rangle$ and $|5\rangle \to |3\rangle$ transitions are red (blue) shifted. (b) The calculated electron distributions of two cascaded four-well QC periods at 25 and 122 K are shown. At 122 K, the electrons are distributed higher in energy from the levels band edges.

Figure 3

The pure dephasing time as a function of lattice temperature self-consistently calculated using the density matrix Monte Carlo method for the QC structure in Ref. [14]. Impurity and interface roughness scattering interactions are included in the calculations. Both the unsaturated and the saturated cases are included for an interface roughness of $\mathrm{\Delta }=1\mathrm{ML}$ with $\mathrm{\Lambda }=5\mathrm{nm}$. The unsaturated pure dephasing times are shorter than the saturated times at lower lattice temperatures. As expected the saturated case approaches the unsaturated values at higher lattice temperatures and corresponding lower output powers. Also shown are the pure dephasing times for the unsaturated case of 2 ML interface roughness with $\mathrm{\Lambda }=5\mathrm{nm}$, which results in shorter pure dephasing times due to the increased interface roughness interaction. The contribution to the pure dephasing time from the intrasubband interface roughness interaction can be seen from the inset interface roughness pure dephasing rates as a function of in-plane kinetic energy for the two main optical transitions within the QC structure.