Modeling excitonic line shapes in weakly disordered semiconductor nanostructures

Excitonic spectra of weakly disordered semiconductor heterostructures are simulated on the basis of a one-dimensional tight-binding model. The influence of the length scale of weak disorder in quantum wells on the redshift of the excitonic peak and its linewidth is studied. By calculating two-dimensional Fourier-transform spectra we are able to determine the contribution of disorder to inhomogeneous and also to homogeneous broadenings separately. This disorder-induced dephasing is related to a Fano-type coupling and leads to contributions to the homogeneous linewidth that depends on energy within the inhomogeneously broadened line. The model includes heavy- and light-hole excitons and yields smaller inhomogeneous broadening for the light-hole exciton if compared to the heavy-hole exciton, which agrees qualitatively with the experiment.

Figure 1

Dependence of the standard deviation of the disorder potential on the length scale $L/a$ before renormalization. The horizontal line indicates the standard deviation of the potential resulting from the original box distribution. Inset: representation of the renormalized disorder potentials for different length scale.

Figure 2

Upper row: FWHM of the absorption spectra of the exciton and lower row: shift of the maximum of the excitonic absorption line. Left-hand column: plotted vs $L/a$ and right-hand column: plotted vs $L/a_B$. Various curves refer to $a_B/a$ as given in (b). Disorder parameter is $W/J=0.2$.

Figure 3

FWHM extracted from the linear spectra of the excitonic line for $N=100$ and the number of realizations $M=100$ and $W/J=0.4$. Disorder length scale is $L=a$. Crosses depict values for $J^v$ of 1.8 and 2.25 meV, which we will discuss in Sec. 6.

Figure 4

(Color online) (a) From top to bottom: level scheme showing transition into a single state $|a⟩$ with dipole matrix element ${\mu }_{ag}$ and phenomenological dephasing rate $\gamma$ and into a continuum of states with matrix element ${\mu }_0$; analytical results for the linear spectrum and amplitude 2DFTS, rephasing mode, corresponding to the level scheme above. (b) From top to bottom: level scheme showing transition into a single state $|a⟩$ and into a continuum of states with matrix element ${\mu }_0$, $\Gamma$ is the coupling rate between $|a⟩$ and the continuum; analytical results for the linear spectrum and amplitude 2DFTS, rephasing mode, corresponding to the level scheme above.

Figure 5

(Color online) Upper figure: linear spectra for the ordered (solid line) and disordered (dot-dashed line) situation. Normalized imaginary-part 2DFTS, rephasing mode, (a) for the ordered and (b) for the disordered cases in the Pauli-blocking limit. Disorder amplitude of the heavy-hole exciton is 0.334 meV, corresponding to the disorder parameter $W/J=0.1$. The number of realizations is 40. Normalized amplitude for (c) ordered and (d) disordered cases, respectively. The energies at the vertical axis follow from reflection of the horizontal axis at the diagonal.

Figure 6

(Color online) Upper figure: normalized linear spectra for the ordered (solid line) and disordered (dot-dashed line) situation at the heavy-hole excitonic peak. Normalized imaginary-part 2DFTS rephasing mode: (a) experimental data are taken from Ref. 14 and (b) modeled disorder with the amplitude of 0.72 meV, $W/J=0.4$, and the number of realizations is 13. Disorder length scale $L=a$. The energies at the vertical axis follow from reflection of the horizontal axis at the diagonal.

Figure 7

Inhomogeneous broadening of the excitons (black identifies heavy-hole exciton and gray the light-hole exciton) vs ${\Omega }^{hom}$, extracted from the 2DFTS, rephasing mode. The squares show experimental data for GaAs/AlGaAs taken from (Ref. 14), diamonds are the data of the ordered 2DFTS, which have been broadened by Gaussian with the FWHM of 0.72 meV, and triangles are the data extracted from the present calculation. We used the following disordered parameters: $W/J=0.4$, $W^h=0.72\text{meV}$, $W^l=0.9\text{meV}$, the number of realizations is 13, and disorder length scale $L=a$. Estimated error bars are shown.

Figure 8

(Color online) Upper figure: normalized linear spectra for the ordered (solid line) and disordered (dot-dashed line) situation at the light-hole excitonic peak. Normalized imaginary-part 2DFTS rephasing mode: (a) experimental data are taken from Ref. 14 and (b) modeled disorder with the amplitude of 0.9 meV, $W/J=0.4$, and number of realizations is 13. Disorder length scale $L=a$. The energies at the vertical axis follow from reflection of the horizontal axis at the diagonal.