Exciton-exciton interaction: A quantitative comparison between complimentary phenomenological models

Many-body interactions such as exciton-exciton interactions significantly affect the optical response of semiconductor nanostructures. These interactions can be rigorously modeled through microscopic calculations. However, these calculations can be computationally intensive and often lack physical insights. An alternative is to use phenomenological many-body-interaction models such as the modified optical Bloch equations and the anharmonic oscillator model. While both these models have separately been used to interpret experimental data, to the best of our knowledge, an explicit and direct correspondence between these models has not been established. Here, we show the empirical equivalence between these two complimentary models through two-dimensional coherent spectroscopy simulations. A quantitative correspondence between the parameters used to incorporate the exciton-exciton interactions in these two models are obtained. We also perform a quantitative comparison of these phenomenological models with experiments, which highlights their usefulness in interpreting experimental results.

Figure 1

(a) Exciton as a two-level system used for the MOBE calculations with excitation-density-dependent dephasing rate $\gamma$ and resonance frequency $\omega$. $|0\rangle$ and $|1\rangle$ represent the ground and excited states, respectively. (b) Energy level scheme for anharmonic exciton ladder is illustrated. $|0\rangle$ : ground state, $|1\rangle$ : single-exciton state, $|2\rangle$: two-exciton state, ${\gamma }_{01}$: dephasing rate of the $|0\rangle \leftrightarrow |1\rangle$ transition, $\xi$: excitation-induced dephasing (EID), $\mathrm{\Delta }$: excitation-induced shift (EIS).

Figure 2

The (a) absolute and (b) real part of the simulated 2D spectra using MOBEs for a pulse area of $3\times {10}^{-3}\pi$, $N{\gamma }^{\prime }$ = 100 meV and $N{\omega }^{\prime }$ = 50 meV. The (c) absolute and (d) real part of the simulated 2D spectra using the best-fit parameters for the AO model. Dashed diagonal line indicates equal excitation and emission energy amplitudes. The cross-diagonal (X-Diag) and Diagonal (Diag) directions are indicated by arrows in (a). The dashed contours in (b) and (d) show the zero level.

Figure 3

Pulse-area-dependent cross-diagonal slices of absolute value 2D spectra obtained from MOBE model are shown. Excitation-induced dephasing (EID) causes a broadening of the cross-diagonal slices with increasing pulse area. A slight blue shift in the peak of the cross-diagonal slices and asymmetric lineshape indicates excitation-induced shift (EIS). The black arrows indicate increasing pulse area. The dashed line shows the cross-diagonal slice in the absence of MBIs.

Figure 4

Slices obtained from MOBE simulations (green dashed line) and the corresponding fits using the AO model (solid red line) are shown. Slices for the MOBE simulation are taken from the absolute value [Fig. 2] and real part [Fig. 2] 2D spectra, respectively. The particular slice is indicated by the label above each plot.

Figure 5

The symmetry-breaking fit parameters in the AO model—EID ($\xi$) in (a) and EIS ($\mathrm{\Delta }$) in (b) measured as a function of excitation densities for different values of $N{\gamma }^{\prime }$ and $N{\omega }^{\prime }$. The error bars are standard deviation in parameters value obtained by fitting 2D spectra in multiple energy ranges.

Figure 6

Linear variation of homogeneous linewidth with excitation density for different combinations of $N{\gamma }^{\prime }$ and $N{\omega }^{\prime }$. Linear fits to data are shown by the dashed lines.

Figure 7

Excitation-density dependence of the homogeneous linewidth ${\gamma }_{01}$ obtained by fitting the experimental data (red circles) and MOBE simulations (blue diamonds) to the AO model. MOBE simulations were performed for $N{\gamma }^{\prime }$ = 100 meV, $N{\omega }^{\prime }$ = 150 meV. The linear fits are shown by the blue and red dashed lines and show an almost perfect overlap with each other.

Figure 8

Excitation-density dependence of the shift in the resonance frequency ${\omega }_{01}$ in experiments (red circles) and MOBE simulations (blue diamonds). The data and simulations are the same as those used in Fig. 7.