Effects of electron-electron and electron-phonon scatterings on the linewidths of intersubband transitions in a quantum well

We study the effects of electron-electron and electron-longitudinal optical (LO) phonon scatterings on the spectral lineshape and linewidth of intersubband transitions (ISBTs) within the second Born approximation. In particular, we show that there is a strong cancellation of the contributions from the so-called diagonal and nondiagonal scattering terms in the ISBT polarization equation to the linewidth. The omission of the nondiagonal scattering leads to a significant overestimate of the ISBT linewidth by several times for GaAs quantum wells. Such cancellation effect is less significant for semiconductors with strong nonparabolicity because of the dominance of the inhomogeneous broadening due to $k$-dependent transitions. In addition, there is a cancellation effect between the electron-electron and electron-phonon scatterings, leading to a property of nonadditiveness of contributions from individual scattering processes to the linewidth. Finally, we compare our theoretical linewidth calculation with existing experimental measurements and very good agreement is achieved.

Figure 1

Top panel: Absorption spectra for a $10\mathrm{nm}$ GaAs quantum well at $300\mathrm{K}$. Solid lines represent cases with both in- (or nondiagonal) and out- (diagonal) scatterings included, while dashed lines represent the cases with out-scattering only. Bottom panel: The same curves as in the top panel, but plotted on log-scale. The curves marked by 1(I), 2(II), and 3(III) represent cases with e-ph scattering, e-e scattering, and both e-ph, and e-e scatterings, respectively.

Figure 2

Normalized absorption spectra with e-ph scattering (top), with e-e scattering (middle), and with both e-ph and e-e scatterings (bottom). Parameters are the same as in Fig. 1. In each case, the solid and dashed lines represent the case with and without in-scattering, respectively. The linewidth [full-width at half maximum (FWHM)] is indicated in the figure where (D) and (F) stand for “diagonal” only and full scattering terms (both in- and out-scatterings included), respectively.

Figure 3

Normalized absorption spectra for a $10\mathrm{nm}$ InAs quantum well at $77\mathrm{K}$ and density of $1\times {10}^{12}{\mathrm{cm}}^{-2}$ with e-e scatterings. The effective mass model is used for this calculation with masses being 0.027 and 0.039, respectively, in the unit of free electron mass. The solid and dot-dashed lines are for the cases with and without in-scattering term, while the dashed curve is obtained by assuming a constant dephasing rate of $1\mathrm{meV}$. The inset shows the same results on a log-scale to highlight the spectral features away from the line center.

Figure 4

Absorption spectra for a $10\mathrm{nm}$ InAs quantum well at $77\mathrm{K}$ and density of $1\times {10}^{12}{\mathrm{cm}}^{-2}$ with only e-e scattering included. The “1-band” stands for results obtained using parabolic bands with effective masses, while “8-band” for results obtained using band structures of an 8 band $k\bullet p$.

Figure 5

Linewidth (FWHM) as a function of temperature for a $10\mathrm{nm}$ GaAs quantum well at a density level of $1.0\times {10}^{12}{\mathrm{cm}}^{-2}$. Lines with circles and triangles represent linewidth when the e-ph and e-e scattering are separately considered, respectively. The curve with squares is the linewidth when both e-e and e-ph scatterings are included, while the curve with diamonds is simply the addition of curve with circles and that with triangles.

Figure 6

Comparison of theory (solid line) and experiment (circles) (Refs. 25, 33) on linewidth as function of temperature for a $9\mathrm{nm}$ GaAs quantum well. Density is at $9.8\times {10}^{11}{\mathrm{cm}}^{-2}$.