Interplay between multipole expansion of exchange interaction and Coulomb correlation of exciton in colloidal II-VI quantum dots

Electron. Struct. 4(2022) 015006 https://doi.org/10.1088/2516-1075/ac5b7e OPEN ACCESS RECEIVED 30 November 2021 REVISED 22 February 2022 ACCEPTED FOR PUBLICATION 8 March 2022 PUBLISHED 29 March 2022 Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. PAPER Interplay between multipole expansion of exchange interaction and Coulomb correlation of exciton in colloidal II–VI quantum dots Petr Klenovsk´ y1,2,∗, Jakub Valdhans1,LucieKrej ˇ c´ ı1, Miroslav Valtr2,3, Petr Klapetek2,3and Olga Fedotova4 1Department of Condensed Matter Physics, Faculty of Science, Masaryk University, Kotl´ aˇ rsk´ a 267/2, 61137 Brno, Czech Republic 2Czech Metrology Institute, Okruˇ zn´ ı 31, 63800 Brno, Czech Republic 3CEITEC, Brno University of Technology, Purkyˇ nova 123, 612 00 Brno, Czech Republic 4Scientific and Practical Materials Research Center, National Academy of Sciences of Belarus, P. Brovki str. 19, 220072, Minsk, Belarus ∗Author to whom any correspondence should be addressed. E-mail: [email protected] Keywords: quantum dot, colloidal, multiparticle states, configuration interaction, correlation, AFM, SNOM Abstract We study the effect of Coulomb correlation on the emission properties of the ground state exciton in zincblende CdSe/ZnS core–shell and in wurtzite ZnO quantum dots (QDs). We validate our theory model by comparing results of computed exciton energies of CdSe/ZnS QDs to photoluminescence and scanning near-field optical microscopy measurements. We use that to estimate the diameter of the QDs using a simple model based on infinitely deep quantum well and compare the results with the statistics of the atomic force microscopy scans of CdSe/ZnS dots, obtaining excellent agreement. Thereafter, we compute the energy fine structure of exciton, finding striking difference between properties of zincblende CdSe/ZnS and wurtzite ZnO dots. While in the former the fine structure is dominated by the dipole terms of the multipole expansion of the exchange interaction, in the latter system that is mostly influenced by Coulomb correlation. Furthermore, the correlation sizeably influences also the exciton binding energy and emission radiative rate in ZnO dots. 1. Introduction Semiconductor quantumdots (QDs)grown by epitaxy from materials belonging to group III and V of the periodic table are one of the most promising candidates for quantum light source in quantum technology, as they combine excellent optical properties with the compatibility to semiconductor processing and the potential for scalability [1–10]. Meanwhile, they provide also a platform for photon-to-spin conversion [11,12], building up bridges between photonic and spin qubits [13–15]. Moreover, they can be used as building blocks for quantum information devices, particularly for quantum repeaters [16–18], as efficient single and entangled photon sources [2,19–28], including highly-entangled states for quantum computing [29–33], or as nanomemories [10,34–39]. One of the drawbacks of the aforementioned QD technologies is their elevated cost caused by the epitaxial growth techniques used for their fabrication, i.e., metal-organic vapor phase epitaxy (MOVPE) or molecular beam epitaxy. Compared to the epitaxially grown QDs there exist another class of QDs based on solution processed semiconducting nanocrystals with dimensions smaller than ∼20 nm, so called colloidal QDs [40,41]. Since their emergence more than 20 years ago [42,43] the ease of their manufacture and relatively low cost enabled their use in many optoelectronic applications such as lasers sources [41], light-emitting diodes (LEDs) [44,45], photodetectors [46], and solar cells [47,48]. Moreover, similarly to epitaxially grown QDs, they were utilized in integrated-photonic circuits [49–52], lab-on-chip platforms [53], optical interconnects [54–56], or advanced © 2022 The Author(s). Published by IOP Publishing Ltd Electron. Struct. 4(2022) 015006 PKlenovsk ´ yet al medical devices [57–60]. Moreover, colloidal QDs were realized for different material systems from the periodic table groups II–VI, III–V, or IV–VI. One of their advantages is that the properties of their quantum confinement depend predominantly on their diameter which can be controlled by the growth. In this work we investigate the electronic structure of CdSe/ZnS [61,62]core–shellandZnO[63–66]QDs. The choice of the aforementioned systems is motivated by the fact that those are currently easily commercially available for purchase. We consider the crystal structure of the former system to be that of the zincblende crystal while the latter is considered in the wurtzite phase. We focus here on the properties of the ground state exciton (X0), in particular investigating energy, fine-structure splitting (FSS), energy difference between dark and bright (BD) states and emission rate of X0. We compute the uncorrelated single-particle (SP) states of electrons and holes using the envelope function approximation based on eight-band k·ptheory [67], the results of which are used thereafter as basis states for the configuration interaction (CI) computations which elucidate the Coulomb interaction energies and correlation. We first test our theory model by comparing the results for CdSe/ZnS QDs with photoluminescence (PL) spectroscopy, scanning near-field optical microscopy (SNOM), and time-resolved PL (TRPL). Thereafter, we compare the aforementioned theory results to that obtained on ZnO QDs. 2. Methods 2.1. Single-particle states & configuration interaction We first give a description of our method of computation [27]. We start by obtaining SP states using the envelope function method based on eight-band k·papproximation using the Nextnano++ simulation suite [68]. SP states obtained that way read Ψai(r)= ν∈{s,x,y,z}⊗{↑,↓} χai,ν(r)uΓ ν,(1) where uΓ νis the Bloch wave-function of an s-like conduction band or a p-like valence band at the center of the Brillouin zone (Γpoint), ↑/↓mark the spin, and χai,νis the envelope function, where ai∈{ei,hi},whereei and himark ith state of electrons and holes, respectively. Those states are then used as basis states for the CI method [69,70] using the code which we have previously developed [27,71,72]. Let us assume that the excitonic complex |Mconsists of Neelectrons and Nhholes. The CI method uses as a basis the Slater determinants (SDs) consisting of neSP electron and nhSP hole states. The trial function of the excitonic complex then reads |M= nSD  m=1 ηm|DM m,(2) where nSD is the number of SDs in |DM m,andηmis the constant that is solved for using the variational method. The mth SD can be found as |DM m=1 √N! τ∈SN sgn(τ)φτ{i1}(r1)φτ{i2}(r2)...φ τ{iN}(rN).(3) Here, we sum over all permutations of N:=Ne+Nhelements over the symmetric group SN.For the sake of notational convenience, we joined the electron and hole wave functions of which the SD is composed of, in a unique set {φ1,...,φN}m:={Ψej,...,Ψej+Ne−1;Ψ hk,...,Ψhk+Nh−1},wherej∈ {1, ...,ne}and k∈{1, ...,nh}. Accordingly, we join the positional vectors of electrons and holes {r1,...,rN}:={re1,...,reNe;rh1,...,rhNh}. Thereafter, we solve within our CI the Schr¨ odinger equation ˆ HM|M=EM|M,(4) where EMis the eigenenergy of excitonic state |M,and ˆ HMis the CI Hamiltonian which reads ˆ HM=ˆ HM 0+ ˆ VM,where ˆ HM 0represents the SP Hamiltonian and ˆ VMis the Coulomb interaction between SP states. The matrix element of ˆ VMreads [27,39,71] DM n|ˆ VM|DM m=1 4π0 ijkl  drdrqiqj (r,r)|r−r| ×{Ψ∗ i(r)Ψ∗ j(r)Ψk(r)Ψl(r)−Ψ∗ i(r)Ψ∗ j(r)Ψl(r)Ψk(r)}.(5) In equation (5)qiand qjlabel the elementary charge |e|of either electron (−e), or hole (e), and (r,r)isthe spatially dependent dielectric function. The first (second) term in curly brackets in equation (5)represents 2 Electron. Struct. 4(2022) 015006 PKlenovsk ´ yet al the direct (exchange) Coulomb interaction. Moreover, the summation in equation (5)runsoverallSPstates considered in the CI basis and, thus, that expansion provides the effect of correlation. Note, that the Coulomb interaction is treated as a perturbation. The evaluation of the sixfold integral in equation (5) is performed using the Green’s function method [70,71,73] ∇(r)∇ˆ Uajl(r)=4πe2 0 Ψ∗ aj(r)Ψal(r), Vij,kl =drˆ Uajl(r)Ψ∗ bi(r)Ψbk(r), (6) where a,b∈{e,h}and ∇:=∂ ∂x,∂ ∂y,∂ ∂zT.Notethat(r,r) was set to bulk values [27,39]fortheCI calculations presented here. We note that in this work we consider only excitons, i.e., Ne=1andNh=1, respectively. However, in order to capture the effect of Coulomb correlation we vary the number of SP basis states neand nhfrom 2to12. Finally, we note that the multipole expansion of the exchange interaction was included in our CI for SP basis of ne=2andnh=2 following the theory outlined in references [74,75]. 2.2. Atomic force microscopy Atomic force microscopy [77] (AFM) of CdSe QDs on Si substrate was measured by the Bruker Dimension ICON AFM with Bruker RTESPA-300 probe. We used the program Gwyddion [76]toanalyzetheAFMtopographic images. Since the microscope probe has typically larger radius than the size of the QD, the lateral dimensions of measured QDs are highly affected by tip convolution, which leads to their effective broadening. Therefore the size distribution was calculated on basis of QDs maxima, using the flat sample substrate as a reference. 2.3. Photoluminescence spectroscopy Standard PL setup was used in PL and TRPL measurements. In both PL and TRPL the excitation of the sample was done by a pulsed laser (510 nm, 30 MHz) with emission intensity varied by a neutral density (ND) filter from 0 to 400 μW. The samples were positioned in a cryostat, cooled to 70 K and were heated up to room temperature. The PL spectrum was detected by an Andor CCD camera in a visible part of a spectrum and the TRPL signal was measured by an avalanche diode connected to the time triggered photon counting device. 2.4. Scanning near-field optical microscopy High spatial resolution PL spectra were obtained using SNOM setup. PL spectra were measured using commercial thermomicroscopes Aurora II aperture SNOM enhanced for spectroscopic measurements. Standard SNOM probes with 100 nm aperture manufactured by thermomicroscopes were used. The CdSe/ZnS QDs on Si substrate were excited by an Ar ion laser (488 nm) and PL spectra were acquired using Avantes AvaSpec HS-TEC spectrometer at room temperature. The integration time was 60 s. In order to increase signal-to-noise ratio, we measured two spectra, one with excitation laser on, and another spectrum with excitation laser off. The SNOM PL spectrum shown in figure 1(d) was obtained as a ratio of these two datasets. 3. Results 3.1. CdSe/ZnS zincblende quantum dots We start our investigation of II–VI QDs with the discussion of QDs consisting of CdSe core and ZnS shell. We have modeled CdSe/ZnS QDs as spheres and defined the structure using Nextnano++ simulation suite [68]. Since one of the most important properties of those dots is their diameter, we modulated that in our calculations discussed in the following. Furthermore, motivated by typical properties of commercially available CdSe/ZnS dots, we have set the thickness of the ZnS shell to 1 nm in all our calculations. Since our QDs are not embedded in any bulk material nor grown by epitaxy, we do not minimize the elastic energy in the whole simulated structure, i.e., we consider our dots to be strain free. That means, we assume that the ZnS shell around the CdSe dot does not lead to any induced strain in the dot’s CdSe core. That assumption is reasonable, since capping with a 1 nm layer of material seems to be unlikely to influence the much more bulky CdSe core of the dot. Finally the eigenenergies and wavefunctions resulting from SP calculations are fed into the CI solver and we obtain states of correlated excitons as described above. However, prior to discussing our theory results, we test the setting of our simulated structure and theory tools described above by comparing that with the results of measurements using AFM, PL spectroscopy, and 3 Electron. Struct. 4(2022) 015006 PKlenovsk ´ yet al Figure 1. Comparison of the experiment and theory of CdSe/ZnS QDs. In panel (a) we show an AFM scan of commercially available CdSe/ZnS QDs emitting in yellow part of the spectrum. Panel (b) gives the statistics of QD diameters from (a) obtained using Gwyddion [76] program as yellow bars. In (c) are the computed emission energies (symbols) of X0as a function of dot diameter for two sample temperatures. The calculations were performed by eight-band k.p [68] and CI, the latter with full inclusion of the Coulomb correlation [27,71,72]. The full curves in (c) represent the fits of the theory data by our simplified model given in equation (7) (see also text). The PL and SNOM measurements of CdSe/ZnS QDs are shown in panel (d). Note that the broken lines in (b) correspond to energy positions of PL and SNOM maxima in (d) recomputed to QD diameter using fitted data from (c). SNOM. The corresponding results are given in figure 1. A commercially available sample of CdSe/ZnS QDs emitting in yellow part of the spectra was first measured using AFM {figure 1(a)} and the results of that were statistically analyzed using the Gwyddion [76] program {figure 1(b)}. We focused particularly on the distribution of QD’s diameters on the sample in order to compare to our calculations shown by symbols in figure 1(c) for sample temperatures of 70 and 300 K. The calculations were performed by eight-band k.p [68]andCI, the latter with full inclusion of the Coulomb correlation [27,71,72]. For better transferability of the theory model and to ease the comparison with experiment, we have fitted that data using a simple model motivated by results for ground state of the infinitely deep quantum well and given by E=a D2+E∞,(7) where Dmarks the QD diameter, E∞the energy for D→∞, and an additional fitting parameter a.Note that the latter two parameters (E∞and a) neatly lump the material properties of our QDs. Finally, we have measured emission from our CdSe/ZnS QDs using PL and SNOM {figure 1(d)}. Thereafter, we have picked the energies EPL corresponding to maxima in figure 1(d) and recomputed that to QD diameter using equation (7) as D=a/(EPL −E∞), where aand E∞were obtained from fitting in figure 1(c). The resulting QD diameters are shown by broken curves in figure 1(b) and are satisfactorily close to mean QD diameter of 5.5 nm observed by AFM. Thus, having validated the correct settings of our simulation toolbox, we proceed with discussion of the theory results. The SP ground state electron–hole transition energy as well as that for X0for temperature of 70 K for CdSe/ZnS QDs is given in figure 2.Wefirstnoticeinfigure2(a) the sizeable difference between the results obtained using SP and CI methods, clearly showing that SP computations only give quite imprecise results 4 Electron. Struct. 4(2022) 015006 PKlenovsk ´ yet al Figure 2. Energy difference between ground state electron and hole single particle (SP) states and the corresponding energy of the exciton computed by the configuration interaction [27,39,71] (CI) method as a function of CdSe QD diameter is shown in panel (a). Notice the difference between results for SP and CI computations, being due to direct Coulomb interaction. Binding energy of the exciton computed by CI with the SP basis of 2 electron and 2 hole SP ground states (w/o corr.) and that for 12 electron and 12 hole SP states (with corr.) as a function of ZnO QD diameter is given in panel (b). Notice that the effect of the Coulomb correlation is found to be negligible for binding energy of X0in CdSe QDs. The calculations were performed for temperature of 70 K. for these dots. The difference between SP and CI results originates from the attractive direct electron–hole Coulomb integral Jeh considered in CI. The impact of Jeh canbeviewedinfigure2(b) in the form of X0binding energy with respect to the corresponding SP result (Esp). Moreover, in that panel we compare also the results of the binding energy obtained using CI with basis of two electron and two hole SP basis states (2 ×2) and that obtained using 12 electron and 12 hole SP states (12 ×12), respectively. While the CI result with 2 ×2 basis includes a minimum of Coulomb correlation, that for 12 ×12 contains almost the full effect of the Coulomb correlation. Clearly, correlation has negligible effect on binding energy of X0in CdSe/ZnS QDs. Notice that compared to similar results in III–V QDs [71] the magnitude of Jeh in CdSe/ZnS QDs is further influenced by the dielectric constant of CdSe of =9.7[78] being almost two times smaller compared to, e.g., InAs (=15.15 [78]) or GaAs (=12.93 [68]), what effectively increases the magnitude of the Coulomb integrals in CI since is in the denominator of that, see equation (5). That, e.g., means that Jeh should be twice as large in CdSe QDs than in InAs dots, given they are of the same size, shape, and alloy distribution. Next, we study the energy splitting of the bright exciton Kramers doublet of X0of CdSe/ZnS QDs, i.e., FSS, and the energy difference between the optically bright and dark doublet of X0, i.e., BD. The results are given in figure 3. Here, we study the effect of various physical phenomena influencing the exchange interaction and, thus, FSS and BD. Namely, those are (i) the exchange Coulomb interaction between ground state electron and hole wavefunction (marked as ‘2 ×2’ in figure 3), (ii) the multipole expansion of exchange interaction [75](markedas‘2×2 multipole’ in figure 3), and (iii) the effect of Coulomb correlation [71,79](markedas ‘12 ×12’ in figure 3). Clearly, the dominant contribution to both FSS and BD in CdSe/ZnS QDs comes from the multipole expansion of the exchange interaction, similarly as in references [75,80] for III–V QDs. We further note, that the dominant contribution comes from the dipole–dipole term of that expansion for CdSe QDs. We finally note that our results quantitatively agree with previous experiments [81]aswellasatomistic theoretical results [82]. Furthermore, alongside the CI calculations we have computed also the emission radiative rate [71]ofthe recombination of X0for CdSe/ZnS QDs using the Fermi’s golden rule [83], see figure 4.Ascanbeseen,the 5 Electron. Struct. 4(2022) 015006 PKlenovsk ´ yet al Figure 3. Panel (a) gives the CdSe QD diameter dependence of the fine structure splitting (X0FSS) of the bright exciton computed with the SP basis of 2 electron and 2 hole SP ground states (2 ×2) [71], that including the multipole expansion of the exchange interaction [75](2×2 multipole), and computation for CI basis of 12 electron and 12 hole SP states (12 ×12). In panel (b) the energy difference between bright and dark excitonic doublet (X0BD) as a function of ZnO QD diameter is shown. Notice that both FSS and BD are dominated by multipole expansion terms of the exchange interaction in CdSe QDs. The calculations were performed for temperature of 70 K. computations of the rate with and without the inclusion of Coulomb correlation lead to similar results, i.e., the dominant contribution to that comes already from the recombination of the electron and hole ground states. Furthermore, we notice that the rate reduces considerably with QD size. However, since we compute the rate using the Fermi’s golden rule between the envelope functions only, we admit that our computed results might provide smaller values than those seen in experiment. Nevertheless, the obtained values are close to those experimentally obtained from time-resolved photoluminescence measurements (TRPL). We have tested that by measuring TRPL of an ensemble of our CdSe/ZnS QDs. The reduction of the single photon rate after each laser pulse was fitted by a double mono-exponential (2ME) decay model, see also references [84,85] I(t)=A1exp{−(t−t0)/τ1}+A2exp{−(t−t0)/τ2},(8) where the amplitude of ME A1(A2) and the decay time τ1(τ2) characterize the slow (fast) decay process and t0is the time of the QD emission triggered by the pulse laser and the start of the decay of intensity. Note that A1+A2is the maximum intensity emitted from QDs. We can see in figure 5(a) that the next pulse of the laser comes sooner than the slow ME completely decays which we call repumping, the effect which we included in the fitting as well. In order to compare with our theory we introduce τPL that represents the weighted arithmetic mean of two decay times τ1and τ2for 2ME model and is given by τPL = 2  i=1 wiτi;wi=Aiτi 2 i=1Aiτi ,(9) where wiis the weight defined by parameters of 2ME model. From figure 5(b) we can infer that the emission radiative rate of our QDs, which is found almost constant for all studied temperatures, is ∼0.071 ns, a value close to computations of CdSe QDs in figure 4. 6 Electron. Struct. 4(2022) 015006 PKlenovsk ´ yet al Figure 4. Radiative rate of bright exciton as a function of CdSe QD diameter. The data are shown for calculation using CI with basis of 2 electron and 2 hole SP ground states (w/o corr.) and that for 12 electron and 12 hole SP states (with corr.). Notice that Coulomb correlation has negligible effect on the emission of CdSe QDs. The calculations were performed for temperature of 70 K. Figure 5. (a) Deconvolved TRPL signal (black solid curve) measured at 70 K by 2ME model (blue solid curve). The slow (fast) ME is represented by red (green) solid curve and decay intensity of slow ME from a previous pulse by dashed black curve. (b) Fitted radiative rate 1/τ1,1/τ2and 1/τPL as a function of temperature [86]. 3.2. ZnO wurtzite quantum dots Next II–VI QD system which we study is ZnO wurtzite QDs. Compared to CdSe QDs, the different crystal system in ZnO QDs leads to strikingly different properties of the latter system. The hexagonal crystal structure of ZnO QDs considerably influences the topology of the hole wavefunctions, starting already with the ground state as can be seen in figure 6. While for zincblende CdSe QDs, as well as that for III–V QDs [7,8,72,79,87,88], the hole SP ground state has s-like spherical symmetry, in ZnO QDs the hole ground state has ring-like topology with symmetry axis oriented along c-axis of the wurtzite crystal. 7 Electron. Struct. 4(2022) 015006 PKlenovsk ´ yet al Figure 6. Cuts through origin of the probability densities of the ground single particle (SP) states of electrons (leftmost column) and holes (remaining three columns) of CdSe/ZnS (top row) and ZnO (bottom row) QDs. The hole densities are shown in three planes defined by the orientation of the corresponding unit vectors (yz,xz,xy). The computations were performed using the Nextnano++ computation suite [68]. Notice the difference of the hole probability densities of ZnO QDs between xy plane and that of xz and yz planes originating from the asymmetry of wurtzite crystal along aand ccrystal axes (here cis along zaxis). Note that electron ground state SP probability density has spherical symmetry in both studied systems. Figure 7. (a) Energy difference between SP and CI states and (b) binding energy of the exciton for ZnO QDs. The outline and marking is the same as that in figure 2. Moreover, in ZnO QDs the Coulomb correlation considerably increases the binding energy of X0with respect to the SP ground state electron–hole transition by ∼50 meV as compared to the uncorrelated result. Of course, the dominant contribution to binding energy seen in figure 7is due to the direct Coulomb integrals Jeh similarly as that for CdSe QDs in figure 2and discussed in the aforementioned. Intriguingly, the Coulomb correlation influences sizeably also FSS, BD, and radiative rate of X0[89,90] as can be seen in figures 8and 9, respectively. Namely, in figure 8the effect of correlation on the exchange interaction even surpasses the multipole expansion of that in ZnO QDs. On the other hand, the overall total magnitude of FSS is found smaller in ZnO QDs compared to that in CdSe/ZnS QDs what is a result of the increased charge separation [75] in the former system. Similarly, calculating the Fermi’s golden rule also for SP excited states higher in energy and including the result of that into the final emission rate, increases the magnitude of the rate by many orders of magnitude compared to that when Fermi’s rule is computed only from overlap of the electron and hole SP ground states. 8 Electron. Struct. 4(2022) 015006 PKlenovsk ´ yet al Figure 8. (a) FSS and (b) BD for ZnO QDs as a function of dot diameter. The outline of the figure is the same as that in figure 3. Notice that (i) both FSS and BD is relatively smaller and (ii) is dominated by Coulomb correlation in ZnO QDs compared to that for CdSe QDs in figure 3. Figure 9. Radiative rate of bright exciton as a function of ZnO QD diameter. The data are shown for calculation using CI with basis of 2 electron and 2 hole SP ground states (w/o corr.) and that for 12 electron and 12 hole SP states (with corr.). Notice the considerable influence Coulomb correlation on the emission rate of ZnO QDs. Naturally, the emission rate increases with ZnO QD diameter, since in particular the hole SP states are more closely spaced in energy with increasing QD size and, thus, their relative importance in the correlated state of X0is larger. 9