Cooperative Effects in the Photoluminescence of (In,Ga)As/GaAs Quantum Dot Chain Structures
Multilayer In0.4Ga0.6As/GaAs quantum dot (QD) chain samples are investigated by means of cw and time-resolved photoluminescence (PL) spectroscopy in order to study the peculiarities of interdot coupling in such nanostructures. The temperature dependence of the PL has revealed details of the confinement. Non-thermal carrier distribution through in-chain, interdot wave function coupling is found. The peculiar dependences of the PL decay time on the excitation and detection energies are ascribed to the electronic interdot coupling and the long-range coupling through the radiation field. It is shown that the dependence of the PL decay time on the excitation wavelength is a result of the superradiance effect.
KeywordsTime-resolved photoluminescence Quantum dot chain Interdot coupling
Self-assembled (In,Ga)As/GaAs quantum dots (QDs) demonstrate many favorable physical properties that makes them suitable for numerous device applications [1–4]. They possess high radiative efficiency and easily controllable areal densities; allow engineering of the energy levels permitting efficient carrier injection from a semiconductor matrix. Typically, (In,Ga)As/GaAs QDs have ~2–5 nm height and ~20–50 nm base width. Due to small QD sizes and high confinement potential, the electron and hole states are fully quantized and their wavefunctions are strictly localized within the QD area with a δ-like density of states. Therefore, the self-assembled QDs can be treated to some extent as nearly ideal zero-dimensional (0D) systems behaving themselves like a “frozen ideal gas”.
Each individual QD offers several advantages as a source for single photons. It has high oscillator strength and narrow spectral linewidth (~0.01 meV) and does not suffer from photobleaching or shelving. However, these idealizations have to be substantially modified in case of QD densities lying in the range of ~1010–1011 cm−2. These densities result in mean dot separation of the order of tens to hundreds of nanometers which is comparable with the sizes of the individual QDs. Unavoidable size fluctuations within the QD ensemble lead to an additional inhomogeneous broadening of the optical spectra, frequently making the observation of the intrinsic 0D behavior difficult. Another characteristic of QD ensemble the interdot coupling is usually grouped into two categories with regard to whether it occurs via overlapping wave functions (electronic coupling) of the spatially separated QDs or via long-range electromagnetic interactions.
Depending on the strength of electronic coupling between neighboring QDs, their individual electronic states and the relaxation of the photo-excited carriers through those states can be significantly altered. In the case of strong coupling, they can form QD molecules , resulting in new physics and a number of applications such as an excitonic qubit system with potential scalability . In the case of intermediate or weak coupling, the energy and carrier transfer between QDs occurs through quantum–mechanical tunneling that substantially affects recombination, carrier injection, and lasing in the QD system [7–11].
The long-range radiative coupling between QDs in an ensemble can be interpreted in terms of successive emission and reabsorption of photons resulting in collective modes of several QDs. In this case, the exciton state of a single QD cannot be treated as a stationary state since its excitation in an individual QD will be transferred to other QDs . In addition, energy transfer between dots can be in the form of an electrostatic dipole–dipole interaction, frequently cited as Förster energy transfer . In the case of InP QDs with interdot distance of 7 nm, the Förster time for the excitation transfer between two QDs has been estimated  to be in the range of 102–103 ps. This Förster transfer rate is proportional to the fourth power of the dipole moment matrix element and decays as the sixth power of the interdot distance . Complementary to the Förster energy transfer, the polariton coupling is also expected to have a long spatial range, of the order of a few photon wavelengths. This mechanism results in the transfer of electron–hole excitation between distant QDs mediated by the emission and reabsorption of the transverse electromagnetic field . Typical transfer rates attributed to polariton coupling are in the range of 10−3–10−4 ps−1. It is proportional to the square of the dipole moment matrix element and decays as the inverse of the interdot distance, if the transfer is mediated by a propagating field in two dimensions. Thus, in presence of radiative interaction, in fact, even very remote QDs cannot be considered as isolated systems, and the QD ensemble is expected to develop signatures of cooperative radiation or superradiation . In order to strengthen the QD coupling through their radiation field, the use of semiconductor microcavities has been proposed and strong interaction between single QDs and the cavity mode has been already demonstrated .
In the case of QD ensembles with well-separated QDs, the fingerprint of cooperative effects in radiation is the change of the photoluminescence (PL) decay time with a change in the number of interacting QDs and their respective separation [12, 17]. The number and interdot separation of the dots can be varied using postprocessing of as-grown samples. For example, the dots could be covered by a mask containing small apertures through which the optical excitation as well as the collection of the signal is done. A lateral patterning of the sample surface offers another possibility. In this case, small mesa structures are fabricated which contain only a single QD or a few QDs. In case of CdSe/ZnSe QDs studied in Ref. , mesas of the same size were arranged in a grid-like pattern over an area larger than the size of the laser spot on the sample. Mesas of sizes down to a few tens of nanometers were etched into the QD sample removing the QDs from between the mesas. It has been shown  that the radiative interaction between QDs disappears with decreasing mesa size.
Recently, the new laterally ordered self-assembled (In,Ga)As/GaAs QD stack has been grown by molecular-beam epitaxy (MBE) . QDs align in long rows forming QD chains. It has been found that the dot chains can be substantially increased in length by the introduction of growth interruptions during the initial stages of growth of the GaAs spacer layer. The growth procedure has been also used to create a template to arrange InAs dots into chains with a predictable dot density. The resulting dot chains offer the possibility to engineer interdot coupling for novel physical phenomena and potential devices. Here, the distance between neighboring QD chains can be made sufficiently large as to prevent the immediate electron–hole wavefunction tunneling as well as the Förster energy transfer between QDs belonging to different chains. This looks like as a natural mesa between chains. The objective of our research is the study of the peculiarities of interdot coupling in these new laterally ordered self-assembled (In,Ga)As/GaAs QD structures. In this case, we can expect the superradiance effect from the dot-chain ensemble. It is important that in-chain interdot coupling is easily changeable also thus permitting to study the cooperative effects caused by immediate interdot carrier tunneling.
We use the multilayer (In,Ga)As/GaAs dot-chain structure to reveal various cooperative effects arising both through overlapping wave functions of the spatially separated QDs and through long-range radiative interactions between QDs. “Experiment” includes the details of sample growth together with results of AFM analysis of the grown structure. The details of the PL are given. “Temperature cw PL measurements” includes the results of the temperature investigation of the cw PL. This results in information about the energy scheme of the system and provides evidence of interdot coupling and carrier transfer in QD chains. “Time-resolved PL measurements and mechanisms of interdot coupling” represents the results of the time-resolved PL measurements and contains the theoretical model used for the interpretations of the time-resolved PL data. Last section contains concluding remarks.
The PL was performed using the 532-nm line from a frequency doubled Nd:YAG (neodymium-doped yttrium aluminum garnet) laser for continuous-wave PL excitation and the 2-ps pulses at λexc = 750 nm from a mode-locked Ti:sapphire laser for transient PL measurements. The optical pulse train at 76 MHz and an excitation density that was varied between 109 and 2 × 1014 photons/(pulse × cm2) was used. The steady-state PL signal from the sample was dispersed by a 0.5-m single-grating monochromator and detected by a LN-cooled OMA V: InGaAs photodiode detector array. The PL transients were detected with a Hamamatsu synchroscan streak camera C5680 with an infrared enhanced S1 cathode. The overall time resolution of the system used in the time-resolved PL measurements was ~15 ps.
Temperature cw PL Measurements
with Tth and Δ being the only two fit parameters of the model. Tth is the temperature when β = 0.5 and Δ corresponds to the temperature interval over which the transition from a “glassy” to a quasi-equilibrium state occurs.
Thus, the existence of various discrete and continuous states in the multilayer In0.4Ga0.6As/GaAs QD chain sample substantially complicates the real picture of relaxation and coupling of the QDs. Based on the temperature dependence of the PL spectra, we can distinguish the low temperature effects caused by the in-chain interdot coupling and high temperature effects related to the QD coupling through the WL and GaAs barrier states. To go further, we focus on the low-temperature interdot coupling. We have demonstrated the cooperative effects caused by in-chain, interdot electronic coupling that becomes transformed into a change of the low temperature PL band and its line-shape. The features related to the interdot-chain, long-range radiative coupling are explored by means of time-resolved PL spectroscopy.
Time-Resolved PL Measurements and Mechanisms of Interdot Coupling
Electronic Wave Function Interdot Coupling
Here, N(E) is the number of QDs with the ground state emission energy E; D(E) is the density of states function; τ T (E, E′) is the tunneling time from the state E to the state E′, and G(E, Eexc) is the carrier generation rate in the state E by light with energy Eexc. For the QD ensemble, the D(E) is taken as a Gaussian, (see “Temperature cw PL measurements”). In fact Eq. (3) with the relaxation terms (5) and (6) describes carrier dynamics with a cascade-like carrier transfer from upper states to the lower QD states defined by the tunneling rate This equation allows for consideration of non-linear effects as well because the inequality (7) must hold strictly.
For the sake of simplicity, we define a single tunneling time for all QDs, τ T (E, E′) = τ T , and independent of E and Eexc set the optical recombination time as τop(E, Eexc) = τop = 1 ns. Then, considering the case of a weak electromagnetic field and comparatively slow interdot transfer, τ T = 1.4 ns [10, 27, 28], we also assume that N(E) > >n(E t). This allows us to linearize Eqs. (3), (5), and (6) and solve them numerically. The result of such simulation is shown in Fig. 8 by solid line. This demonstrates that the mechanism of interdot carrier tunneling allows us to qualitatively reproduce the observed rise of τd(Edet) with decreasing Edet through the range of the QD PL spectrum. The difference between experimental and calculated dependences can be reduced significantly if the saturation of the QD ground states is taken into account. This would allow us to shift the energies at which the calculated τd(λdet) function saturates closer to the PL maximum as it is observed for the measured τd(λdet) dependence. Thus, this mechanism of in-chain interdot tunneling is consistent with two items mentioned earlier: rapid increasing the τd(Edet) value within the QD PL spectrum and reaching the maximal τd(Edet) values at the energy corresponding to the energy of PL maximum. However, this simplified model cannot reproduce in principle two experimental facts: decreasing the maximal τd(Edet) values with decreasing the excitation energy Eexc and appearance of the dip on the τd(Edet) dependences for the detection energies lower than E0. These two features are formed by complementary mechanisms: interdot coupling through radiation field and QD coupling with the host materials and inherent defects.
Radiation Field Interdot Coupling
The appearance of the dip on the dependences for the detection energies lower than E0 (see Fig. 7) can be attributed to the coupling of longitudinal optical phonons with confined electrons and holes in QDs and to the presence of deep levels allowing the holes tunnel to these states even at low temperatures . The states of deep levels are closer to the states of larger QDs. Therefore, the hole tunneling is more effective for larger QDs that form the low energy side of the PL spectrum thus reducing the value observed experimentally in the spectral range below the QD PL maximum.
Finally, peculiar dependences of the PL decay time on the excitation and detection energies are revealed in the multilayer In0.4Ga0.6As/GaAs QD chain sample and ascribed to the peculiarities of the carrier and energy relaxation caused by both immediate electronic interdot coupling and long-range coupling through the radiation field. Summarizing, we have established: i) appearance of low temperature asymmetry of the PL band caused by the non-thermal carrier distribution through in-chain interdot wave function coupling. The energy scheme of the In0.4Ga0.6As/GaAs QD chain sample is specified and 1D WL, 2D InGaAs WL, and the heavy hole excited states are identified from the temperature cw PL measurements: ii) the in-chain interdot carrier transfer forms the swing of the decay time τd versus detection energy dependence and can lead to the state saturation effect shifting the maximal τd value close to QD PL maximum; iii) superradiant mode of the emission field couples the QDs of different dot chains and leads to a significant change of the decay time value just in the vicinity of the QD PL maximum. This radiant interdot coupling defines the dependence of the τd value on the excitation wavelength. Coexistence of various mechanisms of interdot coupling is determined here by special design of the QD samples providing effective interdot in-chain electronic coupling and effective radiative interchain coupling due to superradiance. This latter type of coupling could play a significant role in ultrafast applications.
Marega Jr: On leave from Departamento de Física e Ciência dos Materiais, Instituto de Física de São Carlos-USP São Carlos SP 13560-970, Brazil.
The authors acknowledge the financial support of the National Science Foundation of the U.S. through Grant # DMR-0520550.
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