Visible Spectrum Quantum Light Sources Based on In x Ga 1 − x N / GaN Quantum Dots

We present a method for designing quantum light sources, emitting in the visible band, using wurtzite InxGa1−xN quantum dots (QDs) in a GaN matrix. This system is significantly more versatile than previously proposed arsenideand phosphide-based QDs, having a tuning range exceeding 1 eV. The quantum mechanical configuration interaction method, capturing the fermionic nature of electrons and associated quantum effects explicitly, is used to find shapes and compositions of dots to maximize the excitonic dipole matrix element and optimize the biexciton binding energy. These results provide QD morphologies tailored for either bright single-photon emission or entangledphoton-pair emission at any given wavelength in the visible spectrum.

T he breakthrough in the development of solid-state lighting was recently recognized by the Nobel Prize in Physics 2014 awarded to Isamu Akasaki, Hiroshi Amano, and Shuji Nakamura. 1 It is important to build upon this work and develop practical and versatile sources of quantum light because they are key components required for the advancement of quantum photonic devices. Proof-of-principle demonstrations with InAs/ GaAs, 2,3 GaN/AlN, 4 and phosphide-compatible 6,7 material systems have shown that solid-state quantum dots (QDs) can be near-ideal sources of quantum light, albeit covering narrow wavelength ranges. There are many applications of quantum light beyond quantum communications at telecoms wavelengths, such as beating the classical diffraction limit in quantum imaging 8 or reducing cell damage in the microscopy of biological systems. 9 QD light sources have already been used to exceed classical optical interferometry limits, 10 but the long wavelength of emission from the InAs QDs used in this demonstration severely limited its appeal.
Radiative recombination from excitons confined in InGaN/ GaN wurtzite QDs could provide versatility not afforded by other material systems, helping to target new applications of quantum photonics. Excitons in nitride quantum dots have large emission energies, unlocking the blue and ultraviolet spectral regions for quantum light sources (QLS) that have been inaccessible to common III−V zinc-blende compatible technologies. Stronger quantum-confinement effects, resulting from their large band gaps, band offsets, and effective masses also make wurtzite nitride-based QLS suitable for operating at higher temperatures. 4,5 Recent experiments have also reported the bound biexcitons in small III-N-based QDs. 11−13 In comparison to GaN/AlN QDs, 4,14,15 the use of In x Ga 1−x N in GaN host lattice results in longer emission wavelengths, with the modulation of the indium content providing an additional degree of freedom required to tune the emission wavelength of photon sources throughout the visible spectral band. Also, previous reports of InGaN single-QD-based spectroscopy experiments 16−18 show improved photon correlations and extraction efficiency. 19−22 This is particularly important in applications requiring cheap, high-performance single-photon detectors such as visible and free-space high-bandwidth quantum communications because InGaN/GaN QDs can offer compatibility with silicon-based avalanche photodiodes. 23 To design the nanofabrication of QDs to meet the demands of specific applications, a relationship between morphology and the confined excitonic states needs to be found. This is challenging because of the effects of strain, band mixing, quantum confinement, Coulomb interactions, and correlations among particles present in the QD system. Furthermore, particle confinement in wurtzite QDs is strongly affected by large built-in electric fields induced by both spontaneous, linear and nonlinear piezoelectric polarizations. 24 Here, we establish such a relationship using the configuration interaction (CI) method, showing that emission from InGaN/GaN QDs can be optimized for both single-and entangled-photon emission throughout the visible spectrum.
The optimal QLS parameters differ depending on the type of light source required, as illustrated in Figure 1a,b. For singlephoton sources it is desirable to maximize the biexciton (XX) shift so that in the two-photon cascade through the intermediate exciton (X) state the two photons have a large spectral separation 4 as shown in Figure 1a. Successful spectral separation of the two photons always requires an optical component, but in practice, its ease of implementation is dependent on the energetic separation of the exciton and biexciton photons. At the limit of large separation an efficient, low-cost cold mirror can be employed, whereas differentiation below <1 meV requires expensive impractical filtering. 3 The former minimizes the resources required to select emission from a single excitonic state because the required precision in the wavelength-dependent optics reduces with increasing energy separation between the biexciton and exciton photons. In contrast, entangled photons can be generated through time reordering when the biexciton shift is zero, 25 as depicted in Figure 1b. We want to point out that we are not considering here an entangled-photon sources scheme that is based on the vanishing fine structure splitting, FSS = 0, already realized and elaborated in the literature. 3,26,27 We focus our analysis to the time-reordering scheme to generate entangled-photon pairs that requires B XX = 0. 28,29 The biexciton shift can be expressed as B XX = (E XX − E X ) − E X , where E XX is the energy of the biexciton in its ground state and E X is the average energy of the two intermediate bright exciton states. In this material system, it has been shown that the internal electric field localizes the electrons at the top and the holes at the bottom of the dot. 30 As a consequence of this charge separation, the repulsive e − e and h − h Coulomb interactions rapidly dominate as the dot height increases, leading to an increase in the biexciton shift. 11,15 As the electron and hole separation increases, however, the optical transition matrix element for exciton recombination, p X , decreases, undesirably reducing the maximum repetition rate of the photon source. 31 Thus, the two requirements for an ideal single-photon source, i.e., maximum intensity and spectral purity, have competing demands on the dot's height. We express this conflict in an optimization function Ξ that can be used to find the best balance between the requirements; it is defined as follows: 15,32 where p X (x) is the value of the x component of the dipole matrix element of the exciton transition, p X (0) is equal to 10 −4 p X (x),max , and p X (x),max is the maximal value of p X (x) for all quantum dots considered. Although the choice of p X (0) is somewhat arbitrary, we find that the position of the maxima of the optimization function are weakly dependent on its value when it is changed within reasonable limits. Within the range of energies in which the optimization function, eq 1, is studied, the maximum biexciton shift remains within the small limit, i.e., within an order of magnitude of the typical line width of emission. 3 In this limit the assumption discussed earlier, i.e., that larger shifts are advantageous for efficient cheap filtering, holds true. In our theoretical model, the single-particle electron and hole states of wurtzite In x Ga 1−x N/GaN QDs were modeled using the eight-band k·p Hamiltonian, 15,33,34 consisting of the kinetic part that includes spin−orbit interaction, the crystal-field splitting, the strain part, and additional terms arising from the presence of spontaneous and piezoelectric polarization. The strain, spontaneous, and piezo field were relaxed on augmented embedding box until convergency in several lowest electron (e) and highest hole (h) states was achieved. 15 After the singleparticle states were found, the (bi)exciton states were obtained using the CI method, 15,35 i.e., by direct diagonalization of the CI Hamiltonian. The many-body CI Hamiltonian contains only particle-conserving terms and is given by To get excitonic states, we write the Hamiltonian, eq 2, in a two-particle basis (one electron and one hole) |i, j⟩ = |e i ⟩ |h j ⟩, and for biexcitons, we use a four-particle basis |i, j,k,l⟩ = |e i ⟩ |h j ⟩ | e k ⟩ |h l ⟩. 36 Coulomb integrals, V ijkl , required for the diagonalization of the CI Hamiltonian were evaluated in reciprocal space and then corrected using the Makov-Payne method 37 adapted for a hexagonal lattice 15 by adding the first few terms (monopole, dipole, and quadrupole) in the multipole expansion to compensate for the effect of the mirror charges induced by periodic boundary conditions. In this way, the model faithfully represents electronic and excitonic structure of a single QD. 15 All calculations presented here are performed with kppw parallel code. 38 In our analysis of single InGaN/GaN QDs, we have assumed a hexagonal truncated-pyramidal shape, most often used in the analysis of other wurtzite QD, e.g., GaN/AlN QD. Although InGaN QDs grown on the top of quantum wire (QWR) exhibit quite regular form, it is worth pointing out that the actual shape of InGaN QDs is still unclear and could differ from the perfectly symmetric one; the choice of shape here should be considered as an approximation to the real one. A model QD, in the shape of a hexagonal truncated pyramid is illustrated in Figure 1c. It is assumed that the QD is positioned on a wetting layer (WL) that is 0.5185 nm thick. The dimensions of the QD are controlled by two independent parameters: the diameter, D, of the circumscribed circle around the hexagon at the WL−QD interface at z = 0, and the angle between pyramid base and pyramid side edges, which is assumed to be α = 30°. The size of the QDs was controlled by D/h aspect ratio, where h is the QD height. The calculations are performed over a wide range of QDs with In content ranging from 10 to 80%. All the CI calculations involved single-particle electron (N e = 12) and hole (N h = 18) states, including both spins.
As a first step in our optimization, we calculate the biexciton shift of a variety of InGaN/GaN QD structures, therefore assessing its qualification as a QLS. One of the major shortcomings of the mean-field theories, such as Hartree or Hartree−Fock approaches, is the lack of electron correlation effects, whereas the CI approach is capable of capturing such quantum effects explicitly. This is of paramount importance in small QDs, where the direct Coulomb integrals can be comparable in energy to the correlation effect. For a series of smaller In x Ga 1−x N/GaN QDs, we have calculated the excitonic and biexcitonic states with In content varying from 10 to 80% and changing the QD heights in the range of h = 1−2.1 nm with 0.1 nm steps, as shown in Figure 2. The crossover between unbound to bound biexcitonic states is observed for all In compositions greater than or equal to 20% and for the aspect ratios D/h = 4, 5, 6, and 7. As the In content reduces, the QD confinement potential for electrons and holes diminishes, leading in turn to a reduction in the number of confined states. This limits the effect of correlations, which eventually (In x < 20%) prevents the formation of bound biexciton.
By analyzing the results presented in Figure 2, several observations regarding the excitonic structure of In x Ga 1−x N/ GaN single QDs can be made: (1) By varying the aspect ratio (D/h) and height h, i.e., QD dimensions, for each In composition greater than or equal to 20%, it was possible to identify the range of excitonic energies at which transition from unbound to bound biexciton occurs, i.e., B XX = 0. This is important because such morphologies are suitable to fulfill the criteria for the operation of entangled-photon sources. (2) By moving from smaller (right side of the traces) to larger QDs (left side of the traces), it can be observed that smaller QDs support bound biexcitons, whereas large ones do not. Small QDs produce less strain because of smaller volume and, consequently, the smaller piezoelectric field around them, with less ability to reduce the overlap between electron and hole wave function. Good overlap between wave functions produces strong correlation that in turn makes the biexciton bound. With increasing QD size, more states can be confined, contributing more toward correlation, and we observe characteristic minima in B XX for certain QD sizes and for In content between 30 and 60%. For large QDs, the piezoelectric field becomes so strong that the overlap of wave functions is strongly reduced, and in this way, the correlation effect is reduced as well. In such QDs, the correlation effect is no longer strong enough to overcome the e − e and h − h repulsive interactions in a biexciton, leading to its unbound character. In Figure 4, the top line shows the exciton emission energy and indium contents that satisfy the B XX = 0 condition. (3) For certain aspect ratios from the D/h = 5, 6, and 7 range and In contents of 40−50%, we observe a sudden drop in B XX and strongly bound biexcitons. This phenomenon arises from the confinement of increased number of states (particularly in VB where h states exhibit much larger density of states because of larger effective masses) with a small variation in QD dimensions that lead to increased correlations. However, for QDs with large In content (70%), the effect of internal electric field dominates and destroys the conditions for strong correlations as explained in item 2. In QDs with smaller In content of 20 and 30%, such effect is not observed because of two reasons: (a) Such QDs exhibit shallower confinement with much less bound states capable of contributing to correlations. (b) A smaller amount of In in a QD results in a smaller effect of an internal electric field on wave function overlap within QD structures. (4) Also, we want to highlight that for the case of 50% In content and aspect ratio D/h = 5 we observe five crossings between bound and unbound states over the range of 30 meV of excitonic energies from E X = 2.60−2.62 eV (inset in Figure 2). This is the region with the most suitable morphology for entangled-photon source devices.
In the Supporting Information, all morphologies that exhibit B XX = 0 in Figure 3 are listed, together with excitonic energy deviations from the excitonic energy at which B XX = 0.
After identifying suitable In contents, we proceed now with the optimization of single QDs for single photon source (SPS) devices. For QDs with the In content in the range from 20 to 70%, the aspect ratio in the range D/h = 6−11, and the QD height in the range h = 1−6 nm, we have calculated the excitonic ground state energy, E X , its dipole, p X , the biexcitonic energy, E XX , and the optimization function, Ξ, eq 1. The results of the optimization procedure are given in Figure 3. We have observed that for each In concentration it is possible to identify a maximum in function Ξ. QD morphologies that support these maxima will provide the best SPS devices at particular wavelengths. Optimization function Ξ, eq 1, assumes only the unbound biexcitons. However, by replacing E XX − 2E X with |E XX − 2E X |, the search can be extended to bound biexcitons as well. We have found local maxima of Ξ in that region to be negligible. In Figure 4, we plot the exciton energy (wavelength) versus In concentration for the most optimal SPS QDs. The dimensions of those QDs, i.e., aspect ratio, height in nanometers, and In concentrations (%) together with the emitting wavelengths in micrometers, are (6, 2.6, 20%, 0.401), (9, 2.1, 30%, 0.439), (7, 2.2, 40%, 0.466), (7, 2.4, 50%, 0.539), (7, 2.2, 60%, 0.603), and (7, 2.2, 70%, 0.743). In contrast, for very small QDs with D/h = 4 and 5, B XX is small so that local maxima for these aspect ratios are lower than the one for D/h = 6; in addition, biexcitons become bound before Ξ can reach local maxima for all In concentrations considered. In the Supporting Information, all morphologies that exhibit maxima in Figure 3 are listed, together with their emission wavelengths. From Figure 4, it is clearly seen that with suitable modulation of the In content and careful design of QD dimensions it is possible to achieve the In x Ga 1−x N/GaN QD-based SPS emission in the range from λ min = 0.403 μm to λ max = 0.754 μm that covers the entire visible spectrum.
Recently, Deshpande et al. 20 realized good quality electrically injected single-photon sources that are based on InGaN quantum dots in GaN nanowires. The dots in ref 20 have a height of 2 nm and a diameter of about 25 nm. Our results indicate that optimal dot dimensions are typically slightly larger than 2 nm and therefore agree with the fact that good quality single-photon sources were realized in ref 20 with dots of similar dimensions.
In addition to the emission wavelength other factors, such as the repetition rate, the output coupling efficiency, and coherence of the emission, need to be taken into consideration in the design of SPS sources. We have estimated from the excitonic lifetime in InGaN QDs that the repetition rate of our SPS sources will be in the gigahertz range. Purcell enhancement in an optical cavity could increase this even further. 41 As-grown QDs typically have poor output coupling efficiency, which however could be rectified with structural engineering. 42 The emission from QDs can be highly coherent, 43 although many applications of quantum light do not require coherency of light. Also, it is very likely that the addition of the Al in the barrier material, i.e., InGaN/AlGaN QD system, could provide for better thermal characteristics because of larger confinement potentials for electrons and holes, leading further toward higher temperature operations.
We have shown that with suitable variation of the In/Ga ratio in single In x Ga 1−x N/GaN QDs it is possible to tune both SPS and entangled-photon emission covering the whole visible spectrum. Using the CI method, including the effect of electron exchange and correlations interactions, we have shown that in small QDs the exchange-correlation effect is sufficient enough to compensate for the increase of the direct Coulomb energy of two excitons in a QD. Because InGaN materials have smaller piezoelectric and spontaneous polarization effects than those in GaN, the exchange-correlation effect is not capable of simultaneously reducing the electron−hole attraction (and exchange-correlation) and increasing the pairwise Coulomb repulsion. It is currently thought that InGaN QDs are formed as local composition fluctuations in quantum wells. Although it is unlikely that such structures provide the level of control over QD morphology required to optimize SPS and entangledphoton devices for planar structures, InGaN QDs formed in GaN nanowires present a promising alternative system where good SPSs have already been realized. 20,22,39,40 Our designs can serve as a guidance of further development of this technology. In combination with recent work optimizing the brightness, 41 efficiency, 42 and coherence of light from quantum dots, 43 the results presented here can be used to design light sources for a number of applications including quantum imaging and communications.

Notes
The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS
S.T. thanks the Royal Society, London (grant "High Performance Computing in Modelling of Innovative Photo-Voltaic Devices"), EU-COST (project MP1406), and the STFC, United Kingdom, for providing the computational resources. J.P. thanks EPSRC for a Doctoral Prize Fellowship. R.J.Y. acknowledges support by the Royal Society through a University Research Fellowship (UF110555). N.V. was supported by European Community FP7 Marie Curie Career Integration Grant (ELECTROMAT) and Serbian Ministry of Education, Science and Technological Development (project ON171017). We thank Mark Holmes for valuable suggestions and discussions.