We report on optical properties of coupled three-dimensional (3D) Ge quantum dot crystals (QDCs). With increasing the vertical periodic number of the QDCs, the photoluminescence (PL) spectral linewidth decreased exponentially, and so did the peak energy blueshift caused by increasing excitation power, which are attributed to the electronic coupling and thus the formation of miniband. In the PL spectra, the relative intensity of the transverse-optical (TO) phonon replica also decreases with increasing the vertical periodic number, which is attributed to the increased Brillouin-zone folding effect in vertical direction and therewith the relaxation of indirect transition nature of exciton recombination. Besides, the optical reflectivity at the interband transition energy was much more reduced for the QDCs than for the in-plane disordered QDs grown with the same parameters, indicating a higher interband absorption of the QDCs due to the miniband formation.
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Correlation and coupling of quantum dots (QDs) have been extensively studied in several material systems [1–3]. The coupling can be clearly observed and modulated when the QDs are laterally  or vertically  close enough, which allows for strong excition wave-function overlapping. Attractive physical phenomena, such as coupling and entangling quantum states , enhanced intermediate-band light absorption , enhancement of thermoelectric figure-of-merit  and negative differential conductance  have been reported in coupled QD systems. It has also been reported that the formation of extended electron or hole states due to the electronic state coupling in vertically stacked three-dimensional (3D) quantum dot superlattices (QDSLs) [10–12]. Furthermore, the extended states will evolve into mininbands when the QDs are small and homogeneous, 3D ordered and closely adjacent to each other [13, 14].
High-density, homogeneous and 3D ordered QD arrays have also been refered to as quantum dot crystals (QDCs) . In 3D artificial QDCs, the QDs play the role similar to that of atoms in real crystals. The band offsets at the interfaces between the QD and the matrix in the QDCs also play a role analogous to the periodic potential in real crystals. Minibands are formed in QDCs due to the 3D coupling. Lazarenkova et al. presented the first semi-analytical model for predicting miniband energy states in 3D QDCs and showed that density of states (DOS) spikes of electrons or holes will emerge in each miniband and will significantly change the optical and transport properties of the QDCs [16, 17]. Considerable efforts have been devoted to the fabrication of ordered QDs via different patterning or lithography techniques [15,18–21]. However, the fabrication of dense ordered QDCs is still a challenge due to difficulties in the fabrication of patterned substrates with a high areal density of nanopits on the surfaces. In addition, most works investigating the coupling effect in QD ensembles were carried on in-plane disordered QDSLs samples and by using photoluminescence (PL) spectroscopy [10–12, 22]. Few experimental works have been reported on the coupling effect of 3D ordered QDs, especially of dense ordered QDCs. Considering the compatibility with the Si integration technology and the promising applications in optoelectronic devices, extensive study of the optical properties of the Ge QDCs with the formation of minibands is in demand.
In this paper, we employ excitation power dependent PL and photoreflectance (PR) measurements to investigate optical properties of coupled 3D Ge QDCs. The QDCs are in-plane hexagonally ordered and vertically aligned with thin Si spacer layers. The periodic number in vertical direction is varied. Obvious changes with the periodic number were observed in PL spectra, including the changes in spectral linewidth, peak energy blueshift caused by increasing excitation power and weight of phonon component in entire PL spectrum. PR results reveal a much more reduced optical reflectivity at the interband transition energy for the QDCs as compared to the in-plane disordered QDs grown with the same parameters. These experimental results are interpreted in the context of a coupling model in which the localized hole ground state of each Ge QD couples and extended into delocalized hole minibands.
2. Experimental methods
The Ge QDCs samples are grown in a Riber EVA-32 molecular-beam epitaxy (MBE) system on pre-patterned Si (001) substrates via nanosphere lithography. The pre-patterned Si substrates were chemically cleaned, outgassed in vacuum at 780°C for 5 min in the MBE chamber. And then the Si buffer layer was deposited. The first Ge QD layer was grown by depositing Ge of nominal coverage of 8 MLs with increasing the substrate temperature from 450 to 550°C. After that, layer stacking of QDs started with 5.5 nm thick Si spacer layers. For further experimental details on the sample growth and the micro-structures one can refer to reference 21. To investigate coupling effect in 3D QDCs, Ge QDCs with single, 5, 10, 15 and 20 stacked dot layers were fabricated and characterized. All the QDCs samples were capped with a 100 nm thick Si layer at 400°C to suppress the surface non-radiative recombination of excitons in PL measurements. The surface morphologies of the QDCs without Si capping were characterized by atomic force microscopy (AFM) while the vertical ordering of the QDCs was characterized by cross-sectional transmission electron microscopy (TEM). For PL measurements, the samples were placed in a closed-cycle helium cryostat with a temperature range from 16 K to room temperature. An Ar+ laser (488 nm) was used as the excitation source. The PL spectra were recorded by an extended InGaAs detector using the standard lock-in technique. The PR measurements were performed at room temperature using a step-scan VERTEX-70 Fourier-transform infrared spectrometer. The incident light from a tungsten halogen lamp was unpolarized and the samples were under normal incidence.
3. Results and discussion
After multi-layer stacking growth, 3D ordered Ge QDs are formed in a hexagonal crystal structure. A typical surface morphology of the uncapped QDCs after 15 layer stacking was shown in Fig. 1(a). Highly uniformed and well ordered QDs with hexagonal close-packed structure is clearly seen. The inset shows the Fast Fourier Transform (FFT) image of Fig. 1(a). The 4-th order component spots are clearly seen, indicating a nearly perfect in-plane ordering of QDs. The in-plane periodicity of the QDCs is about 100 nm. Figure 1(b) showed a typical high resolution TEM (HRTEM) image of the 10 layer QDCs. Vertical alignment of the dots is demonstrated and the vertical periodicity is found to be ∼7 nm. A small interdot spacing of ∼2.5 nm was found in the growth direction, which corresponds to a strong electronic coupling . A lateral interdot spacing of ∼10 nm was also found by TEM in large area . A schematic structure of the 3D Ge QDCs was illustrated in Fig. 1(c).
Figure 2 shows the normalized excitation power dependent PL spectra of the Ge QDCs samples with various numbers of stacked QD layers. The measurements were carried out at 16 K. The PL spectrum changed significantly with the vertical periodic number, the linewidth becoming narrower and the peak energy blueshift caused by increasing excitation power (indicated by dash lines) becoming less. The linewidth and the peak energy blueshift (at the excitation powers of 0.07 and 1.2 W respectively) as a function of the periodic number are plotted in Fig. 3. Both can be modeled by an exponential fit based on tight-binding formalism, which was proposed by Solomon et al. . In Solomon’s work, a similar behavior of the PL spectral linewidth with increasing vertical periodic number was reported for the in-plane disordered InGaAs QD-SLs. They explained such behavior by the electron state coupling of InGaAs islands, and the associated ground state miniband filling. For Si/Ge QDs, the band offsets mainly exist at the valence band and thus the formed minibands in the QDCs are hole minibands. In our QDCs, the interdot spacings in both the vertical and the lateral direction are small, which will lead to strong electronic state coupling of QDs and hence the formation of hole minibands. With increasing the vertical periodic number of the Ge QDCs, the delocalization of the hole state in each QD layer will further extend and the miniband related physical effects will also become more prominent. Considering the existing of a DOS maximum peak in the lowest energy of the ground state miniband  and the relatively small excitation power density in our experiments, all the photon-generated carriers will fill the lowest energy states in the ground state hole mini-band. Such filling will give rise to a reduced PL spectra linewidth and a reduced peak energy blueshift. In addition, it is worthy to note that the peak energy blueshift of the QDCs is close to zero when the periodic number is over 15, as shown in Fig. 3 (also see Fig. 2(d)). This PL characteristic is evidently different from the large blueshifts observed usually in the QDs with type-II band alignment due to the induced band bending and band filling with increasing excitation power, which have been reported in both laterally disordered [23,24] and laterally ordered but weakly coupled [25, 26] QDSLs. Such non-shift characteristic with increasing excitation power may also reflect the formation of minibands in QDCs.
In regard to theory, minbands are formed only in an ideal QDC in which the QDs are identical and are positioned periodically with large wavefunction overlapping between neighboring QDs. As a matter of fact, although the 3D ordering of QDs can be achieved in artificial QDCs via patterning and multilayer stacking techniques, the number of coupled QDs along the growth direction is limited by the number of stacked layers realized in practical fabrication. In our QDCs, the very large in-plane periodic number as well as the small in-plane interdot spacing (∼10 nm) surely give rise to in-plane extended states for holes (mainly valence band offset). In the growth direction, although the coupling between neighboring QDs is stronger due to the smaller interdot spacing (∼2.5 nm), however, miniband formation along the growth direction still needs the vertical periodic number to exceed a certain value. From the nearly zero-blueshfit shown in Fig. 3, it can be concluded that the hole ground state miniband is essentially formed in our QDCs samples when the vertical periodic number is over 15.
Figure 4 shows the integrated PL intensity as a function of the excitation power (I∼Pm) for the QDCs with various periodic numbers. The extracted slope m is in the range of 0.5 to 0.7, as indicated in Fig. 4. This means that the band alignment for all the Ge QDCs is still type-II due to the basically unchanged spatial separation of electrons and holes. Noteworthy is that the integrated PL intensity attenuates for the QDCs with periodic number over 15 as compared to that of QDCs with periodic number of 10. Such PL attenuation maybe attributed the degraded crystalline quality of the QDCs with increasing stacked periodic number. A higher growth temperature for QDCs may improve their optical emission properties.
To further understand the PL properties of the Ge QDCs, a decomposition analysis of the non-phonon (NP) recombination and its transverse-optical (TO) phonon replica was performed. Figure 5 shows the ratios of the integrated intensity of the TO phonon component and that of the entire spectrum (ITO/IPL) as a function of the periodic number. The decomposition of the TO and NP peaks for the 5 layer QDCs sample is shown by the dash lines in the inset of Fig. 5. The ratios of ITO/IPL monotonically decrease with increasing the periodic number. Two physical reasons are suggested to account for the attenuation of TO phonon component. Firstly, as we know, for the Ge QDs coherently embedded in the Si host, the most tensely strained Si is located at the vertical interdot Si spacer region. The biaxial tensile strain in Si splits the sixfold-degenerated Δ valleys into the fourfold-degenerate in-plane Δ(4) valleys and the twofold-degenerate Δ(2) valleys along the  growth direction . The Δ(2) conduction band minimum has a lower energy which constructs an electron well with respect to the surrounding less strained or unstrained Si and Ge QDs. Due to the vertical periodicity of 7 nm of the QDCs, Brillouin-zone folding effect [28,29] along the growth direction will become important with increasing the vertical periodic number, which will result in conduction band minima to be close to the Γ point in the reciprocal space and thus relax the indirect transition nature of exciton recombination. In effect, the indirect interband transition changes to a partially direct one. The emergence of the partially direct interband transition would decrease the relative intensity of the TO phonon component. Secondly, theoretical investigations have demonstrated that the acoustic-phonon dispersion in QDCs undergoes modification and leads to emergence of quasi-optical phonon branches with much lower energies . Those quasi-optical phonons will participate the exciton recombination process and thus reduce the intensity of the TO phonon replica.
For a fully characterization of the optical properties of the Ge QDCs, PR spectrum of the 20 layer QDCs (Fig. 2(d)) was measured to determine its optical reflection characteristics. For comparison, PR spectrum of a 20 layer in-plane disordered Ge QDs reference sample grown on a flat Si substrate with the same growth parameters was also measured. Both samples were capped with 100 nm Si and showed flat surfaces after capping. The PR spectra of the 20 layer QDCs and the reference sample are shown in Fig. 6. The PL spectrum of the 20 layer QDCs measured at 16 K is also plotted in Fig. 6. Clearly two reflectivity minima were observed for both the QDCs and the reference sample, as indicated by two dash lines and denoted by P1 and P2. Considering the temperature dependent band gap of bulk Si and Ge (Varishni’s relation), the minima denoted by P1 and located at around 780 meV is ascribed to the interband absorption . In reference to the Si TO replica at 1090 meV in the PL spectrum, the minima denoted by P2 and located around 1120 meV is ascribed to the Si band edge absorption. The reflectivity at P1 position is much more reduced for the QDCs as compared to the reference sample, which indicates a stronger optical interband absorption. This result proves that the optical absorption of Ge QDs can be enhanced by formation of miniband via 3D dot ordering with small interdot spacing. It is consistent with the theoretical prediction that the formation of miniband in Ge QDCs could result in a higher optical absorption .
In summary, we investigated the optical properties of the coupled Ge QDCs with various periodic numbers along vertical direction via excitation power dependent PL and PR measurements. Obvious changes were observed in PL spectra with the vertical periodic number. The PL spectral linewidth, as well as the peak energy blueshift caused by increasing excitation power, decreases exponentially with the periodic number. PR results reveal a much more reduced optical reflectivity at the interband transition energy for the QDCs as compared to the in-plane disordered QDs grown with the same parameters. Those observed optical properties suggest that the hole ground state miniband is essentially formed in our QDCs when the vertical periodic number is over 15 and the miniband formation enhances the optical interband absorption.
This work was supported by the special funds for the Major State Basic Research Project (No. 2011CB925601 and No. 2009CB929300) of China, and the Natural Science Foundation of China (NSFC) under Project No. 61274016 and 10974031. Yingjie Ma also thanks the support from Fudan University by the Research Support Project for Outstanding Ph. D Students.
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