Impacts of Coulomb Interactions on the Magnetic Responses of Excitonic Complexes in Single Semiconductor Nanostructures
We report on the diamagnetic responses of different exciton complexes in single InAs/GaAs self-assembled quantum dots (QDs) and quantum rings (QRs). For QDs, the imbalanced magnetic responses of inter-particle Coulomb interactions play a crucial role in the diamagnetic shifts of excitons (X), biexcitons (XX), and positive trions (X − ). For negative trions (X − ) in QDs, anomalous magnetic responses are observed, which cannot be described by the conventional quadratic energy shift with the magnetic field. The anomalous behavior is attributed to the apparent change in the electron wave function extent after photon emission due to the strong Coulomb attraction by the hole in its initial state. In QRs, the diamagnetic responses of X and XX also show different behaviors. Unlike QDs, the diamagnetic shift of XX in QRs is considerably larger than that of X. The inherent structural asymmetry combined with the inter-particle Coulomb interactions makes the wave function distribution of XX very different from that of X in QRs. Our results suggest that the phase coherence of XX in QRs may survive from the wave function localization due to the structural asymmetry or imperfections.
KeywordsQuantum dots Quantum rings Magnetophotoluminescence Diamagnetic shift
Single semiconductor nanostructures, such as quantum dots (QDs) and quantum rings (QRs), have attracted much attention due not only to their fundamental interest, but also to their potential applications in the prospective quantum information technology [1–8]. In semiconductor nanostructures, excitonic effects play a central role in their optical properties [9–15]. Electrons and holes form a variety of exciton complexes in QDs and QRs due to the enhanced Coulomb interactions by the spatial confinements. The spatial extents of excitonic wave functions thus reflect the both the combined effects of spatial confinements and the inter-particle Coulomb interactions among constituent charged carriers [9, 10, 13, 14]. Applying a magnetic field B is an efficient way to probe the spatial extent of excitonic wave functions. In a magnetic field B, the exciton emission energy increases quadratically with B, i.e., the diamagnetic shift ΔE = γB2, with a diamagnetic coefficient γ proportional to the area of the excitonic wave function. Depending on the confinement regime of the nanostructures, the measured γ could have different physical meanings . In the strong confinement regime, where the single-particle energy dominates over the Coulomb energies, γ is a measure of the spatial confinement of the QDs, while the magnetic responses of inter-particle Coulomb energies only appear as correction terms to the overall diamagnetism. On the other hand, when confined excitons are in the weak confinement regime, the Coulomb energies become prominent, such that γ will be mainly determined by the magnetic response of the inter-particle Coulomb interactions. For exciton complexes confined in nanostructures [14, 15], the diamagnetic behaviors are more complicated because of the more elaborate Coulomb interactions among the constituent charged carriers. Therefore, a systematic investigation is essential into further understanding the magnetic responses of different exciton complexes in QDs or QRs.
In this work, we report on the diamagnetic response of different exciton complexes, including neutral excitons (X), biexcitons (XX), positive and negative trions (X+ and X−) confined in single InAs/GaAs self-assembled QDs and QRs. In QDs, the influences of imbalanced magnetic responses of Coulomb energies on the overall diamagnetisms of X, XX, and X+ are observed and discussed. As for the magnetic response of X− in QDs, we observe an anomalous diamagnetic behavior, which cannot be described by the conventional quadratic energy shift with the applied B. Such anomalous behaviors for X− can be attributed to the apparent change in the electron wave function extent after photon emission due to the strong Coulomb attraction by the hole in its initial state. In QRs, the diamagnetic responses of X and XX also show different behavior. The impacts of inherent structural asymmetry of the self-assembled QRs, as well as the inter-particle Coulomb interactions, on the distribution of X and XX wave functions are discussed. Our results suggest that the phase coherence of XX in QRs may survive from the wave function localization due to the structural asymmetry or imperfections.
The investigated samples were grown on GaAs (100) substrates by a molecular beam epitaxy system. For the QD sample, a layer of InAs self-assembled QDs (2.0 MLs) was grown on GaAs at 480°C without substrate rotations, yielding a gradient in area dot density ranging from 108 to 1010 cm−2. Atomic force microscopy (AFM) of uncapped samples reveals that the dots are lens shaped, with an average height and diameter of ≈2(±0.5) and ≈15 nm, respectively. The QDs were finally capped by a 100-nm undoped GaAs layer.
For the QR sample, a low-density InAs QD layer was first grown by depositing 2-ML InAs at 520°C as QR precursors. The substrate temperature was then lowered to 500°C for depositing a thin GaAs layer (1.7 nm) to cover the QD sidewalls, followed by a 50-s growth interruption for expelling the indium atoms from the QD center for the QR formation [16, 17]. Surface topography of uncapped QRs has also been investigated by AFM. The area density of surface QRs is about 107 cm−2. Typical surface QRs have a rim diameter of 35 nm, a height of ≈1.3 nm, and a center dip of about 2 nm. The realistic dimension of the embedded QRs is expected to be much smaller . We also found that the QR is anisotropic; the rim of the surface QR is higher along [ ].
Results and Discussions
where is the single-particle diamagnetic coefficient, and accounts for the magnetic response of V αβ (B), with a constant k defined as . If we take the Coulomb energies as perturbations to the single-particle energy in the strong confinement regime, the diamagnetic shift of X should be corrected as . On the other hand, the diamagnetic shift of XX will deviate from that of X by an amount of . Because varies as ∼ℓ3, a slight difference between ℓ e and ℓ h can lead to very different values for and . Since ℓ h < ℓ e and hence , we have that leads to . The same arguments can be applied to the X+ case, where is also reduced by a similar amount, in agreement with our experimental observations. The reason for can also be understood from the fact that the emission energies of XX and X+ are usually correlated even in different QDs , because their energy difference equals roughly to V ee V eh , which is value less sensitive to the magnetic field.
It is worth to emphasize that the systematic differences among γ X ,γ XX , and could be observed only when ℓ e and ℓ h exhibit a large difference. For small InAs QDs, electron gradually lose confinement as the dot size reduces, which in turn push the electron level toward the wetting-layer continuum, resulting in a more extended electron wave function penetrating into the barrier material . On the other hand, the hole wave function remains well localized in the QDs, leading to a large difference between ℓ e and ℓ h . For larger In(Ga) As QDs, since ℓ e and ℓ h are expected to be similar, the diamagnetic response of all exciton complexes will be nearly identical , so that the magnetic response of interparticle Coulomb energies of such strongly confined few-particle systems becomes unable to resolve experimentally.
where and . Because of ℓ h < ℓ e and , and only have minor influences on the overall diamagnetism. Accordingly, we obtain and . Equation (3) makes clear how the difference in ℓe,i and ℓe,f can lead to anomalous diamagnetic shifts for X−. For a normal case of ℓe,i ≈ ℓe,f = ℓ e , we have and . That is, the X− diamagnetic shift behaves as the usual B2 dependence with a coefficient similar to that of X. A very interesting case occurs when , which leads to 2γe,i = γe,f and cancels out the B2 term, resulting in a dominant B4-dependent energy shift. The energy shift shown in Fig. 3(b) exhibits both quadratic and quartic B dependence. Therefore, the measured curvature can also be a sensitive probe to the change in wave function extents between the initial and the final-state electrons of X−.
The lack of rotational symmetry in the potential of embedded QRs has significant impacts on the diamagnetic responses of X and XX. For X confined in QRs, the height variation strongly localizes the hole in one of the potential valleys due to the large effective mass. Consequently, the electron will be bound to the same valley by the electron-hole Coulomb attraction. This means that the wave function extent of X is determined mainly by the confinement of the potential valley and the Coulomb interaction. Therefore, the diamagnetic response of X is similar to an elliptic QD.
For XX confined in QRs, the two holes may be separately localized in different valleys due to the strong hole–hole Coulomb repulsion. Due to the Coulomb attractions of the two separately localized holes, the electron wave functions of XX are more likely to spread over the two valleys and become more extended than that of X. Unlike X, the wave function extent of XX is determined mainly by the diameter of the embedded QRs. This explains why γ XX is significantly larger than γ X for QRs.
Before conclusions, we would like to comment on the implications of the larger diamagnetic shift of XX in QRs. Charged particles confined to a QR are expected to show the well-known Aharanov–Bohm (AB) effect due to the quantum interference of the carrier’s wave function in the ring-shaped geometry [6–8]. However, for an exciton confined in ring-like nanostructures, the AB effect is not expected to occur for such a charge-neutral composite unless the electron and hole can propagate coherently in different trajectories with a non-zero electric dipole moment . This can occur naturally in type-II QD systems, where the electron and hole are spatially separated [28, 29]. However, for InGaAs self-assembled QRs where both the electron and hole are confined, it is still an open question whether the excitonic AB effect can be observed. In fact, this issue is further complicated by the inherent structural asymmetry and/or imperfections presented inevitably in self-assembled QRs [18, 19]. Our results suggest that the inherent structural asymmetry combined with the inter-particle Coulomb interactions tends to localize the X wave function and hence may smear out its phase coherence. On the other hand, due to hole–hole repulsion, the XX wave function is more extended and spread over the entire ring. Therefore, the phase coherence of XX may be more likely to survive from such wave function localizations at lower magnetic field.
In conclusion, the diamagnetic response of different exciton complexes, including neutral excitons (X), biexcitons (XX), positive and negative trions (X+ and X−) confined in single InAs/GaAs self-assembled QDs and QRs has been investigated. In QDs, we found a systematic trend of caused by the imbalanced magnetic responses of inter-particle Coulomb interactions, which could be observed only when the confined electron and hole wave functions exhibit a large difference in their lateral extents. Furthermore, the difference was found to scale as the cube of its single-particle wave function extent and therefore can be a sensitive probe to the electron-hole wave function asymmetry. On the other hand, the magnetic response of X− in QDs fell into a special regime where the conventional quadratic diamagnetic shift failed to describe its magnetic response. Our measurements show that the X− diamagnetic shifts in most QDs investigated are even smaller and non-quadratic. Such anomalous behaviors for X− were explained by the apparent change in the electron wave function extent after photon emission due to the strong Coulomb attraction induced by the hole in its initial state. In QRs, the diamagnetic responses of X and XX also show different behavior. Unlike single QDs, XX in QRs shows a considerably larger diamagnetic coefficient than the X. Numerical model calculations indicate that the inherent structural asymmetry and imperfection of the self-assembled QRs, combined with the inter-particle Coulomb interactions, play a crucial role in the distribution of X and XX wave functions. Our results suggest that the phase coherence of XX in QRs may survive from the wave function localization due to the structural asymmetry or imperfections.
This work was supported in part by the program of MOE-ATU and the National Science Council of Taiwan under Grant Nos.: NSC-97-2112-M-009-015-MY2, NSC-97-2120-M009-004 and NSC-97-2221-E009-161. We also acknowledge the Center for Nanoscience and Technology (CNST) at National Chiao Tung University.
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