The excited spin-triplet state of a charged exciton in quantum dots

Emission spectra collected from the investigated QDs show a number of sharp lines even when assuring the lowest possible level of the excitation power. More than 20 single QDs from the investigated sample were tested. The emission from each dot appears in slightly different spectral range and the relative intensity of emission lines changes from dot to dot. However the characteristic energy pattern of the emission lines can be always well recognized. With extensive analysis of linearly polarized emission from the investigated dots and photon correlation measurements, it was possible to distinguish three groups of lines: those related to (i) neutral, (ii) negatively-charged, and (iii) positively-charged excitons [9, 17, 22]. Here we focus our attention on a selected set of three lines which are assigned, at this point hypothetically, to three different states of a positively charged exciton: its singlet ground state (X⁺) and two excited triplet states (X^+*, and $\text{X}_{p}^{+\ast}$). This assumption is verified in the course of the paper. In figure 1(a) schematic configurations of the complexes related to X⁺ , X^+*, and $\text{X}_{p}^{+\ast}$ are shown together with a simplified picture of the respective recombination processes (green arrows).

The PL spectra of a single QD detected for different directions of linear polarization are illustrated in figures 1(b) and (c). It was found that both the X and the 2X emission lines split into two linearly-polarized components (the magnitude of splitting was equal to approx. 16 μeV). The excitation-power dependent measurements confirmed that the X line corresponds to recombination of a neutral exciton (${{e}_{s}}{{h}_{s}}$) while the 2X results from the optical recombination of a neutral biexciton ($2{{e}_{s}}2{{h}_{s}}$). The fine structure splitting (FSS) of both lines is a result of the eh exchange interaction in a dot characterized by anisotropic potential and it was intensively studied in the literature [23–27]. Additionally to the neutral excitonic complexes two emission lines (X⁺ and X⁻) appear, which have a single component. These lines are attributed to the recombination of the spin-singlet states of the positively and negatively charged excitons [26]. These charge complexes consist of an s-shell electron (e_s) and two s-shell holes (2h_s)—${{e}_{s}}2{{h}_{s}}$ or two s-shell electrons (2e_s) and an s-shell hole (h_s)—$2{{e}_{s}}{{h}_{s}}$. The eh exchange interaction influences neither the initial (in which two holes or two electrons form a closed shell with the total spin equals 0) nor the final state (only single hole or single electron left). As a result no FSS appears for the X⁺ and X⁻ lines. The identification of the sign of these charged excitons was described in details in [9]. Other lines of our interest, the X^+* and $\text{X}_{p}^{+\ast}$ display two perpendicularly partially polarized components. The fine structures of the X^+* and $\text{X}_{p}^{+\ast}$ lines are due to the splitting of the initial states. In both cases the final states consist of a single hole. The initial state of the charged exciton which corresponds to the X^+* line is in the spin-triplet configuration (${{e}_{s}}{{h}_{s}}{{h}_{p}}$). The state comprises an s-shell electron (e_s), an s-shell hole (h_s), and a p-shell hole (h_p). The two holes form a spin-singlet (J_h  =  0) and a spin-triplet (J_h  =  3) states split by the hh exchange energy (J_h is the total spin of holes). In addition, the eh exchange interaction between the respective holes and the s-shell electron e_s splits the triplet states into three doublets [7, 8]. The two optically allowed transitions from the triplet states are partially linearly polarized, due to the mixing between the ${{J}_{h}}=\pm 3$ and J_h  =  0 states [28, 29]. As a result the X^+* line comprises two partially linearly polarized components oriented along two perpendicular crystallographic directions $\left[1\,1\,0\right]$ and $\left[1\,\bar{1}\,0\right]$ [9]. The magnitude of the splitting equals approx. 130 μeV. In the case of the $\text{X}_{p}^{+\ast}$ line (${{e}_{p}}{{h}_{s}}{{h}_{p}}$), assigned to the spin-triplet configuration of the excited charged exciton, the origin of the FSS is analogous to previously discussed for the X^+* line. The initial state of the complex is composed of a p-shell electron (e_p), an s-shell hole (h_s), and a p-shell hole (h_p). Similarly, the hh and eh exchange interactions lead to two partially linearly polarized components of $\text{X}_{p}^{+\ast}$ with the splitting magnitude of approximately 60 μeV.

We note, additionally that the $\text{X}_{p}^{+\ast}$ line appears in the emission spectra at lowest excitation power densities simultaneously with the X⁺ and X^+* lines.

The analysis of the FSS magnitude of the X^+* and $\text{X}_{p}^{+\ast}$ lines was performed on many single QDs present in the sample (23 in total). Results of this study are summarized in table 1. In average, the FSS of the X^+* line is found to be almost two times bigger than for the $\text{X}_{p}^{+\ast}$ line. This difference is very likely associated to the variation in the strength of eh exchange interaction in one case implying the electron from the s-shell and in another case from the p-shell, each shell having different spatial extent and symmetry [30].

Table 1. The amplitude of the FSS splitting measured for dot 1 as compared to its mean value deduced from experiments performed on 23 different QDs.

	Dot 1	Ensemble
X+*	130 μeV	($118.8\pm 2.3$) μeV
$\text{X}_{p}^{+\ast}$	60 μeV	($68.2\pm 4.1$) μeV

Our assignment of the X⁺ , X^+*, and $\text{X}_{p}^{+\ast}$ states as due to different states (ground and excited) of the same complex (positively charged exciton) is strongly supported by results of photon correlation experiments. Histograms, which correspond to cross-correlation between the emission of investigated lines are presented in figure 2. All histograms display the long time scale processes with bunching like character, of the order of 10–20 ns, on the top of which short scale (<1 ns) processes are seen (antibunching dip). The long-scale processes reflect the charge fluctuation effects (random capture of single carriers) in the QD under a quasi-resonant excitation (below the energy of the barrier) [16, 31] The cross-correlation histograms between emission lines, which originate from the recombination of different charge states of excitonic complexes (neutral, positively or negatively charged), show an antibunching character with the same time scale of the order of tens of ns, analogous to the bunching-like peaks described above (see [16, 17] for details). Sharp antibunching dips around zero delay time (characteristic of processes on the scale of the radiative recombination time) occur only between the lines belonging to the same charge family and indicate that at the same time photons are emitted either within the X⁺ , or the X^+*, or the $\text{X}_{p}^{+\ast}$ recombination process. This is in agreement with our assumption of three different recombination channels each leading to "disappearance" of the same (three-particle) object.

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The relationship between X⁺ , X^+*, and $\text{X}_{p}^{+\ast}$ can be also seen when comparing the PL and PLE spectra of the investigated dots. This is demonstrated in figure 3 with pairs of PL and PLE spectra measured for three different dots. The PLE spectrum of the X^+* emission shows a clear (absorption) resonance which perfectly coincides with the $\text{X}_{p}^{+\ast}$ emission peak, and similar behaviour is observed for the PLE spectrum of the X⁺ emission peak (data not shown, for details see [17]). The observation of the $\text{X}_{p}^{+\ast}$ resonance in the PLE spectra of both the conventional spin-triplet state (detected at X^+*) and the ground state (X⁺) of the charged exciton confirms the role of the excited triplet-state of the charged exciton in feeding the complexes. The relaxation of an electron from the p-shell to the s-shell (${{e}_{p}}\to {{e}_{s}}$) must proceed the X^+* emission. The same relaxation combined the hole relaxation from the p-shell to the s-shell must proceed the X⁺ emission. Additionally, the both relaxation processes must occur in the timescale comparable to the radiative lifetime of the excited triplet-state of the charged exciton as the $\text{X}_{p}^{+\ast}$ line can also be observed in the PL spectrum of the dot. In other words, the appearance of the $\text{X}_{p}^{+\ast}$ resonance in the PL spectra implies that the relaxation from the $\text{X}_{p}^{+\ast}$ is quite suppressed but on the other hand it remains efficient enough to give raise to the resonance in the PLE of X^+* (and X⁺) emission lines. Notably, the $\text{X}_{p}^{+\ast}$ resonance shows the characteristic FSS doublet structure both in PL and PLE (absorption) spectra. This is not surprising as exchange effects are active in both the recombination and the absorption processes, in their initial and final states, respectively.

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More can be learned on the properties of the excited spin-triplet state from PL and PLE measurements performed in magnetic fields. A collection of data which demonstrate the evolution of the corresponding $\text{X}_{p}^{+\ast}$ resonance with magnetic field in both types of experiments is displayed in figure 4 (for dot 3). It is important to emphasize that the magnetic-field dependence of the $\text{X}_{p}^{+\ast}$ resonance obtained from PLE spectra follows exactly the same evolution as that of the $\text{X}_{p}^{+\ast}$ line in PL spectra (see figure 4(b)).

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The $\text{X}_{p}^{+\ast}$ resonance splits in magnetic field. In a first approximation two well resolved components can be observed. The average energy of those components red shift with the magnetic field. The red shift is due to Zeeman interaction of the exciton orbital momentum with magnetic field and it is a fingerprint of resonances which involve p-shell states. This is in contrast to the magnetic field evolution of emission lines related to the excitons involving carriers from the s-shell. The blue shift of their energy in magnetic field results from the diamagnetic shift (the orbital momentum of carriers at the s-shell equals zero) [32, 33]. The observation stays in agreement with our assignment of the $\text{X}_{p}^{+\ast}$ line to the excited spin-triplet state of the charged exciton. The energy separation between magnetically split components of the $\text{X}_{p}^{+\ast}$ resonance is a linear function of magnetic field, similarly as it was shown for neutral and charged excitons in InGaAs QDs [26]. The splitting is due to the Zeeman interaction of the $\text{X}_{p}^{+\ast}$ spin with magnetic field. The X⁺ and X^+* lines also split in magnetic field. The amplitude of the X⁺ and X^+* splittings is about three times lower then the corresponding splitting of the $\text{X}_{p}^{+\ast}$ line. This can be seen in figure 5, which depicts a similar magnetically-induced splitting of the X^+* and $\text{X}_{p}^{+\ast}$ emission lines at B  =  3 T and at B  =  1 T, respectively. In our opinion, this effect may be related to the spin–orbit Zeeman effect for the p-shell states. Two circularly polarized components in magnetic field are separated by $g{{\mu}_{B}}B$, where g is the excitonic g-factor. We found that for the X⁺ and X^+* lines the g factors are 1.46 and 1.54, respectively, while g-factor of the $\text{X}_{p}^{+\ast}$ resonance equals 4.19. The data presented in figure 5 demonstrate that each magnetically split component consists in fact of a doublet, in both cases of the X^+* and $\text{X}_{p}^{+\ast}$ lines. The fine, doublet structure of the magnetically split components of the X^+* line is a characteristic feature observed for the transitions involving the spin-triplet state of the charged exciton, e.g. ${{\text{X}}^{\pm \ast}}$ and 2${{\text{X}}^{\pm}}$ [34]. The analogous effect observed for the $\text{X}_{p}^{+\ast}$ line confirms our hypothesis on the origin of this resonance.

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We believe that altogether our experimental data provide convincing arguments for the identification of three characteristic spectral lines in the investigated QDs as due to three different X⁺ , X^+*, and $\text{X}_{p}^{+\ast}$ states of a positively charged exciton. Most interesting are the properties of the $\text{X}_{p}^{+\ast}$ resonance that involves p-shell electrons. This line appears up to 20 meV above the X⁺ and X^+* states (both comprising s-shell electrons). Its observation in the PL spectra implies the partial blocking of the relaxation of the $\text{X}_{p}^{+\ast}$ state towards lower energy X⁺ and X^+* states. The suppression of the relaxation processes between QD states is most often discussed in reference to spin flip transitions which are often speculated to be inefficient. The suppressed efficiency of the spin relaxation invokes the explanation of the appearance of the PL signal from the X^+* resonance (the X^+* to X⁺ relaxation implies the change of the spin of the exciton and/or of a hole in a simplified view) [7–9]. Spin-blocking arguments could also account for the suppressed relaxation between $\text{X}_{p}^{+\ast}$ and X⁺ but in the case of the $\text{X}_{p}^{+\ast}$ to X^+* transition, in a simplified picture, does not imply the flip of a spin. On the other hand the magnetic field splitting of the $\text{X}_{p}^{+\ast}$ resonance is much larger than that of the X^+* (and X⁺) states, which may suggest much complex picture of spin physics (coupling of spin and orbital momenta) associated with those states. Optionally, the observed reduced efficiency of relaxation from the p- to s-shell states is due to the fact that the energy separation between these states is exceptionally small in the investigated dots, being in the range of 10 meV–20 meV, depending on the studied QD [22]. This separation is much smaller than any characteristic energy of optical phonons of the host lattice and the relaxation process with optical phonons is forbidden. Therefore, the discussed relaxation process must involve acoustic phonons, the longitudinal-acoustic (LA) predominantly [35]. It was demonstrated that in the system of reduced dimensionality, like in QDs, the relaxation with LA phonons is reduced due to the decreased transition probability (form factor) and electrons tend to stay at the excited levels [36, 37]. The acoustic phonons are therefore not well coupled to the electronic states which prevents the relaxation process.

Although the overall properties of $\text{X}_{p}^{+\ast}$ state are quite similar to those exhibited by the X^+* state, the amplitude of the effects such as fine structure and magnetic field splitting are very different for both states. For the $\text{X}_{p}^{+\ast}$, the fine structure splitting is of about two-times smaller whereas the magnetic splitting is three times larger as compared to the case of the X^+* state. We believe that this is due to a different symmetry of s- and p-shell electrons involved in these two states, but more theoretical studies are needed to quantify those effects.