Electron-phonon interaction effects in semiconductor quantum dots: A nonperturabative approach

Multiphonon processes in a model quantum dot (QD) containing two electronic states and several optical phonon modes are considered by taking into account both intra- and interlevel terms. The Hamiltonian is exactly diagonalized, including a finite number of multiphonon processes large enough to guarantee that the result can be considered exact in the physically important energy region. The physical properties are studied by calculating the electronic Green’s function and the QD dielectric function. When both the intra- and interlevel interactions are included, the calculated spectra allow several previously published experimental results obtained for spherical and self-assembled QD’s, such as enhanced two-LO-phonon replica in absorption spectra and up-converted photoluminescence to be explained. An explicit calculation of the spectral line shape due to intralevel interaction with a continuum of acoustic phonons is presented, where the multiphonon processes also are shown to be important. It is pointed out that such an interaction, under certain conditions, can lead to relaxation in the otherwise stationary polaron system.

Figure 1

Calculated partial spectral densities of states corresponding to two electronic levels separated by $\mathrm{\Delta }=30\mathrm{meV}$ interacting with one optical phonon mode with $\hslash {\omega }_0=30\mathrm{meV}$ $({\mathrm{\Delta }}^{\prime }=0)$. Only nondiagonal coupling was included $({\alpha }_{12}=0.3)$. $T=300K$. Note the Rabi splitting in the polaron states and their shift to the lower energies.

Figure 2

Calculated room temperature absorption spectra for different values of level spacing as indicated. All the other parameters are the same as for Fig. 1.

Figure 3

Absorption (left) and emission (right) spectra for a system of two exciton levels separated by $\mathrm{\Delta }=30\mathrm{meV}$ calculated for different temperatures. The excitons interact with two optical phonon modes of the energies $\hslash {\omega }_1=25\mathrm{meV}$ and $\hslash {\omega }_2=25.2\mathrm{meV}$. The electron-phonon coupling constants are indicated in the figure. The origin of the energy axis is chosen at the energy of the lowest exciton state $\left(E_0\right)$.

Figure 4

Absorption (left) and emission (right) spectra calculated for different (compared to Fig. 3) values of the diagonal coupling constants.

Figure 5

Absorption (left) and emission (right) spectra calculated for different values of the nondiagonal coupling constant (see the explanation to Fig. 3).

Figure 6

Imaginary (left) and real (right) parts of the self-energy for a single electronic level interacting with acoustic phonons, calculated for different temperatures using the Monte Carlo technique explained in the text. The dashed curve was plotted using the one-electron approximation [Eq. (22)]. The coupling constants were calculated as explained in the Appendix. The necessary material parameters were taken as follows: ${\rho }_0=5.8\mathrm{g}∕{\mathrm{cm}}^3$, $c_l=3.8\mathrm{km}∕\mathrm{s}$ and $a_c=3.5\mathrm{eV}$. The QD radius $R=2\mathrm{nm}$.

Figure 7

Imaginary part of the self-energy calculated for a $R=4\mathrm{nm}$ QD as explained in Fig. 6. The dashed curve was plotted using one-electron approximation [Eq. (22)].

Figure 8

Absorption spectra calculated for an $R=2\mathrm{nm}$ CdSe QD including both optical and acoustic phonons for three different temperatures. Two optical phonon modes of the energies $\hslash {\omega }_1=25\mathrm{meV}$ and $\hslash {\omega }_2=25.8\mathrm{meV}$ and dimensionless interaction constants (divided by the corresponding phonon energies) ${\alpha }_{11}^1=-0.015$, ${\alpha }_{22}^1=0.027$, ${\alpha }_{12}^1=0.295$; ${\alpha }_{11}^2=0.063$, ${\alpha }_{22}^2=0.053$, ${\alpha }_{12}^2=0.095$ calculated in Ref. 39 were used. The acoustic phonon parameters were taken as for Fig. 6. The level spacing of $\mathrm{\Delta }=80\mathrm{meV}$ was assumed. The inset shows the $T=300K$ spectrum without the acoustic phonon contribution (at $T=10K$ only the main peak is seen).

Figure 9

Low-temperature absorption spectrum calculated for an InAs QD with the level spacing of $\mathrm{\Delta }=55\mathrm{meV}$ assuming two optical phonon modes (the energies are given on the figure) and neglecting coupling to acoustic phonons. The interaction constants used in this calculation are ${\alpha }_{11}^1=0.005$, ${\alpha }_{22}^1=0.005$, ${\alpha }_{12}^1=0.2$; ${\alpha }_{11}^2=0.005$, ${\alpha }_{22}^2=0.005$, ${\alpha }_{12}^2=0.15$. Notice that three features above the zero-phonon line are separated by nearly the same energy, approximately equal to $31\mathrm{meV}$. The inset shows the corresponding emission spectrum.

Figure 10

(a) Low-energy part of the absorption and emission spectra calculated for a hypothetical CdTe QD considering three optical phonon modes with parameters given on the figure. The acoustic phonon parameters were taken as for Fig. 6. The level spacing $\mathrm{\Delta }=100\mathrm{meV}$, temperature $300K$. (b) Temperature dependence of the integrated intensity of two subgap bands in the absorption spectrum (a) (lines) and experimental data of Ref. 55 (points) showing the temperature dependence of the ASPL peak amplitude.