Inter-shell exchange interaction in CdTe/ZnTe quantum dots: magneto-photoluminescence of X, X2- and XX-
Tomasz Kazimierczuk, Tomasz Smolenski, Mateusz Goryca, Lukasz Klopotowski, Piotr Wojnar, Krzysztof Fronc, Andrzej Golnik, Michal Nawrocki, Jan Gaj, Piotr Kossacki
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Inter-shell exchange interaction in CdTe/ZnTe quantum dots:magneto-photoluminescence of X, X − and XX − T. Kazimierczuk, ∗ T. Smole´nski, M. Goryca,
1, 2
L. K lopotowski, P.Wojnar, K. Fronc, A. Golnik, M. Nawrocki, J.A. Gaj, † and P. Kossacki Institute of Experimental Physics, Faculty of Physics, University of Warsaw,Ho˙za 69, 00-681 Warsaw, Poland Laboratoire National des Champs Magntiques Intenses,Grenoble High Magnetic Field Laboratory,CNRS, 38042 Grenoble, France Institute of Physics, Polish Academy of Sciences,Al. Lotnik´ow 32/64, 02-688 Warsaw, Poland (Dated: December 7, 2018)We present a comprehensive photoluminescence study of exchange interaction in self-assembledCdTe/ZnTe quantum dots. We exploit the presence of multiple charge states in the photolumines-cence spectra of single quantum dots to analyze simultaneously fine structure of different excitonictransitions, including recombination of neutral exciton/biexciton, doubly charged negative excitonand negatively charged biexciton. We demonstrate that the fine structure results from electron-holeexchange interaction and that spin Hamiltonians with effective exchange constants δ i can providea good description of each transition in magnetic field for Faraday and Voigt field geometry. Wedetermine and discuss values of the effective exchange constants for a large statistics of quantumdots. PACS numbers: 78.55.Et, 78.67.Hc, 71.70.Gm
I. INTRODUCTION
Self-assembled quantum dots (QDs) are an ideal sys-tem when it comes to study physics of closely confinedcarriers. One of interesting aspects of such a system isthe exchange interaction between the carriers. From apractical point of view, the electron-hole interaction in asingle QD is described using two characteristic energies:an isotropic contribution responsible for bright-dark exci-ton splitting and anisotropic contribution related to finestructure splitting of bright states of a neutral exciton .The studies of exchange interaction in QDs attractedwide attention several years ago after the proposal ofentangled photon generation in biexciton-exciton (XX-X) cascade . The interest was focused mainly on re-ducing anisotropic part of the e-h exchange interaction,which hindered the entanglement between emitted pho-tons. The research effort finally led to successful demon-stration of fine structure control confirmed by observa-tion of entanglement in XX-X cascade .In our present work we describe the exchange interac-tion in CdTe/ZnTe QDs.Such dots are very convenient for spectroscopy as theygive strong photoluminescence (PL) in the visible rangeof the spectrum and exhibit well resolved excitonic lines.Another outstanding advantages of such dots is a fea-sibility of incorporation a localized 5 / . A single Mn ion in a QD wasshown to exhibit long ( t = 0 . , whichmakes it an interesting candidate for quantum informa-tion storage. Precise knowledge of the exchange inter-action in such QDs is of the essence due to its role inthe optical orientation mechanism of the Mn spin . Our findings about electron-hole interaction in undoped self-assembled CdTe/ZnTe QDs should be also valid for Mn-doped dots.Most of the studies on e-h exchange were devoted tointeractions between s -shell carriers. Such an interac-tion is sufficient to describe a fine structure of neutralexciton (X), charged excitons (X + , X − ), and neutralbiexciton (XX). On the other hand, the fine structureof doubly charged exciton (X − ) and charged biexciton(XX − ) transitions is determined by an exchange interac-tion between a p -shell electron and an s -shell hole. It wasshown that this interaction can be successfully describedusing the same approach as for the interaction between s -shell carriers, i.e., by introducing respective p - s iso- andanisotropic exchange parameters as was demonstrated inRefs. 8–11. However, most of the previous reports dis-cuss effects related either to X − or XX − transitions.The simultaneous access to both complexes allows thecross-comparison of all related fine structures, which isan important test of the applicability limit of the usedspin Hamiltonian model. To our knowledge, such a com-parison was reported only for highly symmetric QDs withnegligible anisotropic part of exchange interaction .In this report, we present a comprehensive study of s - s and p - s electron-hole exchange interaction in CdTe/ZnTeQDs. The experiments involved a large ( > − and XX − excitons. We show thatfine structures of these lines can be interpreted withina spin Hamiltonian featuring both iso- and anisotropicterm of exchange interaction. For simplicity, we assumethat single-particle orbitals are not affected by directCoulomb interaction, which would be important for cal-culation of absolute transition energies . Moreover, wetest the applicability of our model by introducing a mag-netic field either perpendicular (Faraday configuration)or parallel (Voigt configuration) to the QD plane. Fi-nally, we compare the exchange parameters obtained in-dependently from X, X − , and XX − transitions to cross-check the consistency of our description. II. SAMPLES AND EXPERIMENTAL SETUP
The samples were grown by molecular beam epitaxy(MBE) on GaAs substrate. The sample structure con-tained four layers deposited during the growth: a CdTebuffer (about 3 µ m), a ZnTe lower barrier (0.7 µ m), asingle CdTe QD plane, and a ZnTe capping layer (50-100nm). The QDs were formed following the original ideaof Tinjod et. al. , in which a 2D CdTe layer is tem-porarily capped with amorphous tellurium to induce thetransition to dots. A more detailed description of thesample growth can be found in Ref. 14. Some of thesamples were additionally post-processed by producing agold shadow-masks with 200nm apertures. We did notfind any significant differences in single dot propertiesbetween different samples apart from the distribution ofthe QD emission energies and the average charge state,which did not affect the results presented in this work.The sample was cooled to temperatures 1.5–10K. Spa-tial resolution defined by the laser spot diameter was0.5–2 µ m. The PL was excited non-resonantly, using anAr-ion laser (cw, 514nm), a Nd:YAG laser (cw, 532nm)or a frequency doubled Ti:Sapphire femtosecond laser(pulsed, 400nm). The choice of the excitation laser af-fected only intensities (both absolute and relative) of theobserved transitions and did not affect the energy spec-trum, thus the excitation details were not relevant tothe present work. The PL was analyzed using 0.3m -0.5m spectrographs equipped with CCD cameras and/oravalanche photodiode detectors. A λ/ FIG. 1: Three examples of typical emission pattern of aCdTe/ZnTe QD. Spectra were measured on different samplesusing non-resonant excitation (532nm or 400nm) at low tem-perature ( < III. TYPICAL SINGLE QDPHOTOLUMINESCENCE PATTERN
Studied dots were characterized by relatively largespan of emission energies from about 1800 meV (nearly700 nm) to 2250 meV (about 550 nm). A typical singledot spectra are shown in Fig. 1. Similar spectrum wasobserved by several groups , however only up to fourstrongest lines (X, X + , X − , XX) were recognized. Theselines tend to form a characteristic PL pattern with asingle line separated from the others. This single linein the high-energy side of the PL spectrum is relatedto the neutral exciton, which can be confirmed e.g. bythe anisotropy measurement (see Section IV). The nexttwo lines in the spectrum are related to the charged ex-citons (X + , X − ). The signs of their charge state weredistinguished basing on the charge tuning experimentson similar samples and observation of negative opti-cal orientation under quasi-resonant excitation . Suchorientation was previously found for negatively chargedexcitons in different material systems . Finally, thelast of the well established four lines is related to the re-combination of neutral biexciton. It is evidenced by itspolarization properties together with superlinear powerdependence. Here we extend the description of the singledot spectrum by our experimental results used to identifyX − and XX − transitions. These transitions are closelyrelated to p - s electron-hole exchange interaction but werenot identified previously in CdTe/ZnTe system.We start with a discussion of relative energies (i.e.transition energies with respect to transition energy ofneutral exciton) of these transitions in different dots. (b) E X - E li ne ( m e V ) E X - E XX (meV) XX - X X - X +
10 12 14 16 01020 C oun t E XX (meV) (a) E X - E XX ( m e V ) Energy of neutral exciton E X (meV) FIG. 2: (a) Correlation of XX relative energy and absoluteemission energy of X for different dots. A running averageshown as a solid line is drawn to guide the eye. (b) Corre-lation of relative energies of various transitions for differentdots. Solid lines mark linear ( y = ax ) fits with proportionalityconstants equal a = 0 .
64 for X + , a = 0 .
78 for X − , a = 0 .
88 forX − , and a = 1 .
15 for XX − . The inset presents distributionof XX relative energies. Such relative energies vary between dots with large ran-dom scatter on top of systematic changes with emissionenergy, as shown in Fig. 2(a). On average, the values ofrelative energy of XX transition in our dots are spreadaround 13.2 meV with standard deviation 1.3 meV (insetin Fig. 2(b)). We found that relative energies of varioustransitions for a single QD are strongly correlated. Datacollected in Fig. 2(b) shows a clear linear dependence be-tween relative energies of XX and charged excitons (X + ,X − , X − , and XX − ). This results in the same transi-tion sequence for all dots but scattered energetic spreadof the lines. Namely, the lines related to X − transitionare present between X − and XX lines in the typical PLspectrum while the ones related to XX − transition arebelow XX line.A crucial point of our work is a correct identifica-tion of X − and XX − transitions. A conclusive argu-ment for XX − identity was obtained by means of singlephoton correlation measurement. Such a measurement FIG. 3: (a) Photon correlation histogram demonstrat-ing XX − –X − emission cascade. (b) Anti-bunching cross-correlation between X − and X. Negative peak number cor-responds to detection of X − after X. Both correlation ex-periments were done with pulsed ps excitation of 400nm withrepetition 76MHz. (c) PL intensities of charged excitons rel-atively to the intensity of the neutral exciton. The estimateof the QD charge state is calculated as P i q i I i / P i I i , where I i is the intensity of the transition related to charge state of q i . performed with pulsed excitation gives relative probabil-ity of the emission of two photons related to differenttransitions in a single excitation event. In particular,strong correlation peak is related to observation of twotransitions from a single recombination cascade (such as | XX i → | X i → |∅i ). A histogram evidencing cascade re-combination of XX − and X − complexes is shown in Fig.3(a).The photon cascade argument is not applicable to thecase of X − transition. Single photon measurementsbetween previously identified transitions and supposedX − transition show clear anti-bunching , which is ex-pected for excitons of different charge states. Also theasymmetric shape of the histogram is characteristic forthe correlation between charged and neutral excitons ex-cited non-resonantly . The observed fine structure andmagnetic field behavior (discussed further in Sections IVand V A) indicated doubly charged exciton — X − orX . We determined the sign of the charge state bycharge tuning experiment. We exploited here the factthat the average charge state of studied QDs dependson details of optical excitation, e.g., it can be modifiedby additional weak illumination with high-energy light .In the experiment, we measured intensities of all exci-tonic transitions while exciting a single QD simultane-ously with two laser beams: 532nm and (much weaker)400nm. By varying the intensity of 400nm laser beamwe gradually changed the average charge state estimatedas P i q i I i / P i I i , where I i is an intensity of a transi-tion related to charge state of q i . The results of suchan experiment were presented in Fig. 3(c). As expectedX + and X − exhibit monotonic (increasing and decreas-ing respectively) dependence on the average QD charge δδ δδ FIG. 4: (a) A schematic energy diagram illustrating the finestructure of X and XX transitions. (b-d) Correlations be-tween various parameters showing random character of in-plane anisotropy in CdTe/ZnTe dots: (b) orientation and (c)value of the anisotropic splitting of X transition versus emis-sion energy and (d) value of anisotropic splitting and ori-entation of the in-plane anisotropy. (e) Histogram of theanisotropic splitting δ of X transition. (f) Histogram ofbright–dark exciton splitting δ . state. Data for X − follows the behavior of X − , whichprove that these two complexes share the same sign ofthe charge state. IV. FINE STRUCTURE
Polarization resolved photoluminescence measure-ments represent an important and very time-effectivecharacterization tool. In general, polarization depen-dence of the QD emission spectrum is related to in-planeanisotropy of the confining potential. Possible physicalorigins of this anisotropy include non-cylindrical sym-metry of the QD shape, in-plane strain or electric fieldor the symmetry of the interfaces between barriers andthe QD . The in-plane anisotropy was thoroughly stud-ied for many years because of its destructive role in thescheme of entangled photon pair generation . It wasshown that a satisfactory description of the anisotropy-induced exciton splitting can be achieved by introducingan anisotropy of exchange interaction between electronand heavy hole. This interaction can be parametrized by three quantities:2 h↓⇑| H exch |↓⇑i = − h↓⇓| H exch |↓⇓i = δ (1)2 h↓⇑| H exch |↑⇓i = δ (2)2 h↓⇓| H exch |↑⇑i = δ (3)where H exch is the effective exchange interaction and ↓ and ⇓ represent z-component spin projection of the elec-tron and the hole respectively . These parameters areusually related to the fine structure of a neutral exciton(Fig. 4(a)): splitting between dark and bright branch( δ ), splitting between two bright configurations ( δ ), andsplitting between two dark excitons ( δ ). In the presentwork we neglect the presence of dark exciton splitting(i.e., δ = 0). This assumption is justified by very smallvalues of δ (of order of a few µ eV ), never resolved inour experiments.We performed systematic measurements of PL polar-ization properties to determine the influence of the in-plane anisotropy on the typical emission pattern of singleQDs in our samples. In the experiment we recorded thePL spectra for a number of linear polarization directionsfor each studied dot. Such a procedure was necessary dueto a large scatter of the anisotropy axis between differentQDs, usually observed in II-VI systems . The collecteddata allowed us to determine actual principal axis of eachQD and analyze the corresponding PL spectra. For dotswith small anisotropic splitting (smaller than our experi-mental resolution), we determined the value of the split-ting by fitting a gaussian profile to the whole collecteddataset as described in Ref. 29.We start our discussion with analysis of the spectrallines related to the recombination of the neutral excitoniccomplexes: X and XX. In the former case, the zero-fieldemission consist of two closely spaced lines related to spinconfigurations build from ↑⇓ and ↓⇑ states . These twolines are separated by the energy δ and are visible intwo perpendicular linear polarizations. The same under-lying splitting of neutral exciton state affects also XXtransition which also features polarization-resolved dou-blet split by δ . Different ordering of the components ofXX and X transitions results from different role of neu-tral exciton state in both cases: as final and initial stateof the transition respectively. Figure 4(b-d) shows cor-relations of various anisotropy-related parameters for aset of measured dots. Coherently with previous reportson the anisotropy in a similar system , we observe nopreferential direction of the in-plane anisotropy. Neitherthe splitting value nor the anisotropy direction exhibitsignificant correlation with the transition energy or thebiexciton relative energy.Single charged excitons (X + , X − ) do not exhibit no-ticeable fingerprints of in-plane anisotropy. This is ex-pected since the electron-hole exchange interaction influ-ences neither the initial (where two majority carriers areforming closed shell with S = 0) nor the final state (onlyone carrier left) of the transition. Although some de-gree of linear polarization of charged exciton transitions FIG. 5: Polar plots presenting orientation of anisotropy-related linear polarization of various lines in the spectrum ofa typical single QD. Empty symbols are related to the higher-energy component of each transition. may arise due to valence band mixing , we did not con-centrate on this effect in our study due to its negligibleintensity in the studied samples.The remaining two transitions — X − and XX − —exhibit more complex fine structure . Interestingly, weobserved that the orientation of the linear polarizationof X − and XX − lines can be different than the orien-tation of the linear polarization of X and XX lines (Fig.5). The mismatch between these orientation is relativelysmall and varies from dot to dot. Nevertheless, this differ-ence clearly indicates a presence of additional anisotropicinteraction in X − and XX − complexes. We identify itas an exchange interaction between a hole and p -shellelectron.In further considerations we tentatively assume thatthe in-plane anisotropy significantly lifts the degeneracybetween p x and p y orbitals. We also assume that thenon-radiative relaxation of the excitonic complex with anelectron at the higher-energy orbital is much faster thanits optical recombination. In other words, we interpretthe splitting patterns taking into account only single ex-cited ( p -shell) level without any contribution from orbitalangular momentum. The spin part of the wavefunctionis sufficient to describe all observed effects.Recombination of X − was previously studied in GaAs-based dots by several groups with results qualitativelydifferent in terms of light polarization . In our sam-ples, we found that the X − transition consists mainly of FIG. 6: (color online) (a) Typical photoluminescence signa-ture of X − recombination in investigated dots. Black andred (gray) curves correspond to two orthogonal linear polar-izations. (b) Structure of the X − emission spectrum. Curvefill denotes line polarization (plain fill corresponds to full lin-ear polarization, striped pattern corresponds to unpolarizedline). (c) A set of X − exchange parameters for 15 differ-ent dots. The line corresponds to relation 2 δ ps = − δ ps . (d)A schematic illustration of the X − fine structure in analogyto the structure of the neutral exciton shown in Fig. 4(a).The final state in the figure corresponds to the spin tripletconfiguration. two emission lines separated by approximately 0 . × ) weaker than the lower-energy one and iscompletely linearly polarized, similarly to the case de-scribed in Ref. 11. The lower energy line exhibits a par-tial linear polarization in the orthogonal direction. Thissplitting pattern arises from the electron-hole exchangeinteraction in the initial state of the transition. In theinitial state, only two out of three electrons can accom-modate in the lowest s orbital forming a closed shell. Theremaining electron on the p shell interacts with the hole,similarly to the interaction between carriers forming aneutral exciton. This analogy allows us to understandthe arising splitting pattern, however one has to notedifferences in the exchange energies (denoted by δ ps and δ ps for isotropic and anisotropic part respectively) andselection rules (each state of X − is bright).The final state of the considered transition consists oftwo electrons which form either spin singlet or tripletconfiguration, typically separated by several meV . Theoptical transitions involve recombination of an electronand a hole of opposite spin orientations, therefore the“dark” configurations ( (cid:12)(cid:12)(cid:12) ↑↑↓ ⇑ E , (cid:12)(cid:12)(cid:12) ↓↑↓ ⇓ E ) recombine to thetriplet states ( (cid:12)(cid:12)(cid:12) ↑↑ E , (cid:12)(cid:12)(cid:12) ↓↓ E ). The “bright” states can re-combine either to singlet or triplet state. Therefore, forthe spin singlet final state we expect only a pair of linesof equal intensity in the PL spectrum. Experimentallymeasured spectrum is more complex, thus we supposethat the observed transitions are related to the tripletconfiguration in the final state. Indeed, in such a case,one expect three emission lines (Fig. 6(d)): unpolarizedemission from the “dark” state and two linearly polarizedlines from anisotropy-split “bright” doublet. One shouldnote that “bright”/“dark” labels are used here only inanalogy to the case of neutral exciton. Actual intensityof “dark” exciton recombination is two times strongerthan “bright” exciton recombination in case of X − , asthe oscillator strength of the “bright” state recombina-tion is divided into spin singlet and triplet configurationsof the final state.A typical PL spectrum of X − and corresponding de-composition into elementary transitions discussed aboveare presented in Figs. 6(a-b). Experimental data indicatea close coincidence of energies of the unpolarized line andlower component of the anisotropy-split doublet. Such acoincidence can be expressed in terms of exchange pa-rameters as a relation: 2 | δ ps | ≈ | δ ps | . This property wasobserved for all our dots for which X − transition wasseen. Surprisingly, a similar coincidence was observedalso for anisotropic InAs/GaAs QD . No underlyingreason for this coincidence has been proposed so far.In order to analyze quantitatively the relation between δ ps and δ ps we performed simultaneous fitting of spectrain both polarizations. This procedure allowed us to sep-arate isotropic and anisotropic contributions to the ex-change interaction. The results of the fitting procedureare presented in Fig. 6(c). By averaging the extractedvalues we established typical values of exchange parame-ters as δ ps = (0 . ± .
08) meV and δ ps = ( − . ± . δ ps is related to the fact, thatthe √ (cid:16)(cid:12)(cid:12)(cid:12) ↑↑↓ ⇓ E + (cid:12)(cid:12)(cid:12) ↓↑↓ ⇑ E(cid:17) state has lower energy than √ (cid:16)(cid:12)(cid:12)(cid:12) ↑↑↓ ⇓ E − (cid:12)(cid:12)(cid:12) ↓↑↓ ⇑ E(cid:17) state, while in the case of the neu-tral exciton √ (cid:16)(cid:12)(cid:12)(cid:12) ↑ ⇓ E + (cid:12)(cid:12)(cid:12) ↓ ⇑ E(cid:17) state has higher energythan √ (cid:16)(cid:12)(cid:12)(cid:12) ↑ ⇓ E − (cid:12)(cid:12)(cid:12) ↓ ⇑ E(cid:17) state. However, one has totake into account the difference in the selection rules forX and X − . Namely, the lines related to recombinationof √ (cid:16)(cid:12)(cid:12)(cid:12) ↑↑↓ ⇓ E + (cid:12)(cid:12)(cid:12) ↓↑↓ ⇑ E(cid:17) and √ (cid:16)(cid:12)(cid:12)(cid:12) ↑ ⇓ E + (cid:12)(cid:12)(cid:12) ↓ ⇑ E(cid:17) stateshave two opposite linear polarizations. This differencearises due to the fermionic nature of the electrons andis related to the sign of the P ± matrices given in the Ap-pendix A. Therefore, in spite of negative value of δ ps , thelinearly polarized emission lines of X and X − exhibit thesame order in the PL spectrum.We studied the role of the p -shell electron also in thefine structure of the XX − transition. The typical spec-trum of XX − transition is presented in Fig. 7(a). It con-sist of two lines. The intensity ratio of these lines is closeto 2:1 in favor of the lower energy line. Both lines exhibit δ δ δ FIG. 7: (color online) (a) Typical PL signature of XX − re-combination. Black and red (gray) curves correspond to twoorthogonal linear polarizations. (b) Simulated XX − emissionspectrum. Curve fill denotes polarization (c) Correlation be-tween two figures of merit for XX − : e δ and e δ . The presenteddata were obtained by fitting field dependence as discussed inSection V B. (d) Schematic energy diagram illustrating theorigin of XX − fine structure partial linear polarization. In this case the fine structureis determined mainly by the final state. In the initialstate (3 electrons + 2 holes) the electron-hole exchangeinteraction is cancelled by paired hole spins. Similarlyto the previously discussed X − transition, spectroscopicsignature of XX − transition fits to the case of triplet con-figuration of electrons remaining in the final state (seeFig. 7(d)). Its degeneracy is lifted by the exchangeinteraction with remaining hole forming three nearly-equidistant levels. In the simple picture of a symmetricdot, the splittings should be equal to e δ = ( δ + δ ps )as they result from the interaction between both s and p electron with the hole. Only two out of three (four outof six including Krammers degeneracy) configurations ofthe final state are optically active, giving rise to two emis-sion lines of XX − transition in the PL spectrum. Theforbidden configurations correspond to parallel spin ori-entations of all confined carriers ( (cid:12)(cid:12)(cid:12) ↑↑ ⇑ E and (cid:12)(cid:12)(cid:12) ↓↓ ⇓ E ).The in-plane anisotropy manifests itself as mixing ofthe abovementioned states by off-diagonal element pro-portional to e δ = ( δ + δ ps ). In general, this addi-tion should account for (possible) misorientation betweenanisotropy of s and p shells, however in most cases theexperimentally determined mismatch between the twoorientations is small. The strength of this mixing canbe evaluated by measuring the degree of linear polariza-tion of the corresponding PL lines. By diagonalization ofa Hamiltonian including both isotropic and anisotropicpart of the exchange interaction we found that the split-ting between two components of XX − transition is givenby qe δ + e δ . Linear polarization degree ( P = I ⊥ − I || I ⊥ + I || )of both lines is given by: P = 4 β ± p β (4)where β = e δ / e δ and sign ‘+’ corresponds to the strongerof the two lines (i.e. line corresponding to (cid:12)(cid:12)(cid:12) ↑↑ ⇓ E and (cid:12)(cid:12)(cid:12) ↓↓ ⇑ E configurations in case of a symmetrical dot).The presented relations enable us to determine e δ and e δ values separately. We found that in our dotsthe average values of the exchange parameters were: e δ = (0 . ± .
16) meV and e δ = ( − . ± .
08) meV.Within a spin Hamiltonian picture used here, these valuescan be independently obtained by measuring separatelyelectron-hole exchange parameters for s - and p -shell elec-tron. Using previously determined δ i and δ psi values wefound: e δ (calc)0 = 12 ( δ + δ ps0 ) = (0 . ± .
08) meV (5) e δ (calc)1 = 12 ( δ + δ ps1 ) = ( − . ± .
09) meV (6)The values determined by combining the X and X − finestructure parameters are close to the values obtainedfrom analysis of XX − emission. A small but not negligi-ble difference between them gives a measure of applica-bility of spin Hamiltonian approach to the fine structureof the studied transitions. V. MAGNETOPHOTOLUMINESCENCE
Our study was completed by PL measurements in themagnetic field. The experiment was performed for twofield configurations — in-plane (Voigt configuration) oralong the growth axis (Faraday configuration). In bothcases, the influence of the magnetic field can be well de-scribed by linear Zeeman term related to the spin andquadratic diamagnetic shift related to the extension ofthe exciton wavefunction. In the present work we wereinterested mainly in the influence of the magnetic fieldthrough the Zeeman term on the fine structure of theexcitonic states.The magnitude of the magnetic interaction is governedby values of electron and hole g-factors. For simplic-ity, in further considerations we abstract from valenceband effects and introduce pseudospin for the s -shellhole with effective anisotropic g-factor encapsulating e.g.heavy-light hole mixing. In this convention the Zeemansplitting of the neutral exciton in Faraday configurationis given by ( g z h − g z e ) µ B B . For the same purpose, we as-sume that s - and p -shell electrons are characterized bythe same g-factor. We also assume no contribution of FIG. 8: (color online) (a) Experimental and (b) simulated PLspectrum of X − in magnetic field in Faraday configuration.Color saturation denotes degree of circular polarization (blue— σ +, red — σ − ) and brightness denotes the PL intensity.(c) Schematic of the corresponding energy diagram. the orbital angular momentum of p -shell electron, possi-bly due to anisotropy-related degeneracy lifting of two p orbitals. A. Magnetophotoluminescence of X − We start the discussion with the case of X − transitionin the Faraday configuration. A typical data obtained insuch an experiment is shown in Fig. 8(a). The magneticfield splits the lower energy component of X − transitioninto three lines (denoted B, C, D in Fig. 8(b)). Thehigher energy component (denoted A) does not split. Allfour lines are naturally organised in two pairs, each con-sisting of σ + and σ − polarized (fully or partially) com-ponents of equal intensities. The stronger pair (B andD) is characterized by splitting linear with field and itoriginates from the lower energy component at B = 0.Conversely, the other pair (A and C) is already split at B = 0 and the magnetic field induces only small increaseof the splitting.The observed behavior can be analyzed in analogy toneutral exciton, invoked in the previous Section. In sucha picture, the two pairs of lines correspond to the twobranches of the initial state: “bright states” and “darkstates”. The magnetic field acts on each branch inde-pendly. The splitting of each branch is increasing ac-cording to q δ + ( gµ B B ) where δ corresponds to split-ting at B = 0 and g is a respective excitonic Land`e fac-tor. In such approach, the main difference between thetwo branches is their zero-field splitting. Perfectly linearsplitting of the “dark branch” originates from negligi-ble δ ps value while substantial zero-field splitting of the“bright branch” ( δ ps ) dominates over the field-dependentcontribution in the latter case.The presented analogy is instructive, however it is notperfect. Particularly, it ignores the structure of the fi-nal state which is also affected by the magnetic field.It is important especially for recombination of “darkstates”, which according to selection rules lead to S = ± FIG. 9: (color online) (a) Experimental and (b) simulatedPL spectrum of X − in magnetic field in Voigt configuration.Color saturation denotes degree of linear polarization (red— ~E k ~B ext , blue — ~E ⊥ ~B ext ) and brightness denotes thePL intensity. A mismatch between orientation of magneticfield and the QD anisotropy was 16 ◦ . (c) Schematic of thecorresponding energy diagram. Transitions related to char-acteristic splitting of higher-energy component were markedwith arrows. sult, the optically observed splitting of a “dark states”is governed by the same excitonic Land`e factor as the“bright states” (i.e., g z e − g z h ) and not the Land`e factor ofthe real dark neutral exciton (i.e., g z e + g z h ). This effectcan be also seen as a result of the fact that the unpairedelectron from the initial state is not the same electronthat is recombining with the hole.PL measurements in Voigt configuration revealed qual-itatively different behavior of X − transition (Fig. 9(a)).This was confirmed by repetition of the measurementsfor a number of dots with different relative orientationsof the in-plane anisotropy axis and the magnetic field.In each case we observed splitting of the higher-energycomponent into two lines and difficult to resolve multiplesplitting of lower-energy component.This spectral features are completely reproduced bythe model based on spin Hamiltonian given in AppendixA as shown in Fig. 9(b). The simulations confirmthat the double splitting of the higher-energy componentdoes not depend on the relative orientation of in-planeanisotropy. On the other hand, such a dependence wasfound in the spectral lines split from the lower-energycomponent. Detailed analysis of the involved energy lev-els allowed us to conclude that the two characteristic linessplit from the higher-energy component are transitionsfrom the same initial state to S x = ± g x e = 0 . B. Magnetophotoluminescence of XX − As it was shown in Section IV, the zero field spectralsignature of XX − consists of two emission lines. Figure10(a) presents an evolution of this structure with exter-nal magnetic field in the Faraday configuration. Initially, FIG. 10: (color online) (a) Magnetophotoluminescence ofXX − transition in Faraday configuration (field along thegrowth axis). Blue and red represents σ + and σ − polariza-tion respectively. (b) Result of simulation using spin Hamilto-nian given in Appendix A. (c) Degree of circular polarizationfor different components of XX − as a function of magneticfield for the same QD. (d) Magnetophotoluminescence of XX − transition for a different dot measured in a resistive magnet(no polarization resolution). Fitted model is shown by dashedlines. (e) Schematic of QD energy levels. for field up to a few tesla, both lines exhibit typical Zee-man splitting into two circularly polarized components.However, the linearity of the Zeeman effect is perturbedin the stronger field and an anticrossing between two in-ner spin-split components is observed. The anticrossingis accompanied by characteristic exchange of line polar-ization, depicted in Fig. 10(c).The observed anticrossing is precisely reproduced bythe model based on the spin Hamiltonian (Fig. 10(b)).Similarly to the fine structure at B = 0, the field de-pendence is governed mainly by the final states of thetransitions. The anticrossing is due to the off-diagonalanisotropic part of the e-h exchange interaction ( e δ ). Themodel calculation reproduce also quantitatively the mea-sured polarization behavior (Fig. 10(c)). It is worth tonote that for none of the spectral lines the field corre-sponding to complete linear polarization (and thus zerocircular polarization) coincides with the actual anticross-ing point determined as a point of minimum energy sep-aration between the lines. Instead, at the anticrossingpoint both involved lines exhibit elliptical polarizationwith the same contribution of σ + polarization. Such aneffect is not related to the properties of the anticross-ing states, but rather to the difference in the values of Transition Field dependence ( B || z ) from the spin HamiltonianX (cid:18) δ ± q δ + (( g z e − g z h ) µ B B ) (cid:19) X + and X − ± ( g z e − g z h ) µ B B X − (cid:18) δ ps ± q ( δ ps ) + (( g z e − g z h ) µ B B ) (cid:19) ( − δ ps ± ( g z e − g z h ) µ B B )XX − − e δ + r e δ + (cid:16)e δ ± ( g z e − g z h ) µ B B (cid:17) ! − e δ − r e δ + (cid:16)e δ ± ( g z e − g z h ) µ B B (cid:17) ! TABLE I: Field dependence of transition energies in Faraday configuration obtained from the spin Hamiltonian (neglectingdiamagnetic shift). transition matrix elements (see Appendix A).The field dependence of XX − transition energies andparticularly the anticrossing strength is a direct measureof e δ . We verified that indeed the same pair of e δ i valuesfits transition energies and corresponding polarization de-grees with and without magnetic field. For example, themagnetic field measurements on a dot in Fig. 10(a,c) al-lowed us to obtain e δ = 0 .
69 meV and e δ = − .
29 meV.By substituting these values to Eq. 4 we predict degreeof zero field linear polarization for both PL lines of thisparticular dot as 0 .
37 and − .
68, while in the indepen-dent measurement we found them to be equal to 0 .
36 and − . − finestructure) evidence of negligible value of δ ps exchangeparameter.Finally, we studied the XX − transition in the Voigtgeometry. Both the experiment (Fig. 11(a)) and themodel (Fig. 11(b)) show only small variation of the PLspectrum of XX − . Interesting point in this configurationwould be an observation of the transition leading to op-tically forbidden branch of the final state, in analogy tothe dark neutral exciton, which is partially allowed by in-plane magnetic field. The transition would produce emis-sion lines at energy approximately e δ above the higherenergy component of the XX − in the PL spectrum. Nosuch lines were found experimentally. The reason is thatin contrast to the case of neutral exciton, different linesof XX − transition share the same initial state and a small FIG. 11: (color online) (a) Experimental and (b) simulatedPL spectrum of XX − in magnetic field in Faraday configura-tion. Color saturation denotes degree of linear polarization(red — ~E k ~B ext , blue — ~E ⊥ ~B ext ) and brightness denotesthe PL intensity. A mismatch between orientation of mag-netic field and the QD anisotropy was 16 ◦ . (c) Schematic ofthe corresponding energy diagram with the main recombina-tion channels. admixture of optically allowed states is not sufficient tosuccessfully compete with other radiative channels. VI. SUMMARY
Basing on the set of different photoluminescence ex-periments we have extracted a key parameters describ-ing properties of excitonic complexes in CdTe/ZnTe dots.The values of the parameters were analyzed statisticallyover the large number of different dots. The averagedvalues of the main parameters are summarized in TableII.Despite large inhomogeneous broadening of QD emis-sion in our samples, we demonstrate an universality ofa single-dot emission spectrum with characteristic se-quence of emission lines: X, X + , X − , X − (two lines),XX, and XX − (two lines). The fine structures of thesetransitions were successfully reproduced using an exten-sion of the model developed by Bayer et al . The modelinvolves exchange interaction between s -shell electronand hole as well as between s -shell hole and p -shell elec-0 Parameter Value E XX − E X (13 . ± .
3) meV δ (0 . ± .
13) meV δ (0 . ± .
11) meV δ ps0 (0 . ± .
08) meV δ ps1 ( − . ± .
13) meVTABLE II: A summary of parameters describing CdTe/ZnTeQD emission spectrum. Symbols were explained in the text.Uncertainties of the listed values are related to the spreadbetween individual dots. They in each case exceeded the ex-perimental errors. tron. We determined average values of parameters ofthese two interactions separately by analysis of the finestructure of X and X − transition respectively.Independently, from the analysis of XX − transitionwe have obtained effective exchange parameters e δ =(0 . ± .
16) meV and e δ = ( − . ± .
08) meV. Wecompare these values with respective combination of δ i and δ psi parameters and find acceptable agreement be-tween them. Such a comparison is an important test ofthe consistency of the model based solely on spin Hamil-tonians.The second important result of our work is a verifi-cation of the model calculations of excitonic transitionsin the external magnetic field either in Faraday or Voigtconfiguration. Our findings clearly demonstrate that themeasured transition energies as well as polarization se-lection rules perfectly follow the theoretical predictions.This agreement firmly supports applicability of the spinHamiltonian model to the fine structure of states featur-ing also p -shell electrons. Acknowledgments
This work was supported by the Polish Ministry ofScience and Higher Education as research grants in years2009-2011, by the EuromagNetII, by the sixth ResearchFramework Programme of EU (Contract No. MTKD-CT-2005-029671) and by the Foundation for Polish Sci-ence. One of us (P.K.) was financially supported by theEU under FP7, Contract No. 221515 “MOCNA”.
Appendix A: Hamiltonian operators
Here we describe the Hamiltonians used for the calcu-lation of energies of excitonic states and transitions dis-cussed in the manuscript. For the sake of transparencyof the calculations, we have included only terms related to the observed effects. Namely, we included iso- andanisotropic e-h exchange interaction (described by δ and δ for interaction with s-shell and δ ps0 and δ ps1 for interac-tion with p-shell electron) and electron and hole g-factors(normal — g z e and g z h ; and in-plane — g x , y e and g x , y h ). Wehave neglected exchange interaction between ↑⇑ and ↓⇓ configurations ( δ term), heavy-light hole mixing, orbitaleffects related to the p-shell (zero-field degeneracy, or-bital angular momentum), anisotropy of e-e exchange in-teraction, and configuration mixing of exciton complexesdue to direct Coulomb interaction . Furthermore, in thestates containing two electrons (initial state of X − andfinal state of XX − transition) we have assumed dominantrole of electron-electron interaction and limited the anal-ysis to the subspace corresponding to electron triplet con-figuration. The orientation of the in-plane QD anisotropyis arbitrarily chosen along the x axis.Eigenstates and their energies were obtained by ana-lytical diagonalization . The only exception was the finalstate of XX − transition with in-plane magnetic field forwhich which the 6 × P + and P − corresponding to σ + and σ − polariza-tion. We were not interested in the absolute values of theoscillator strength and took P +( − ) = a s, ↑ ( ↓ ) b s, ⇓ ( ⇑ ) where a , b are annihilation operators for electron and hole re-spectively. Intensity (relative) of PL line related to tran-sition | i i → | f i was calculated as |h f | αP + + βP − | i i| where parameters α and β are defined by the polariza-tion used in detection (e.g. α = 1, β = 0 for σ + or α = β = √ for horizontal linear polarization). In suchapproach we did not analyze excitation dynamics nor thepopulation effect on the PL intensity.Below we explicitly present all matrices related to X − and XX − transitions. The base states are given using anotation (cid:12)(cid:12)(cid:12) AB C E , where A is related to p -shell electrons, Bis related to s -shell electrons, and C is related to s -shellholes.
1. Matrices related to X − transition Basis of the initial state: (cid:12)(cid:12)(cid:12)(cid:12) ↑↑↓ ⇑ (cid:29) , (cid:12)(cid:12)(cid:12)(cid:12) ↓↑↓ ⇑ (cid:29) , (cid:12)(cid:12)(cid:12)(cid:12) ↑↑↓ ⇓ (cid:29) , (cid:12)(cid:12)(cid:12)(cid:12) ↓↑↓ ⇓ (cid:29) . Basis of the final state: (cid:12)(cid:12)(cid:12)(cid:12) ↑↑ (cid:29) , √ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ↓↑ (cid:29) + (cid:12)(cid:12)(cid:12)(cid:12) ↑↓ (cid:29)(cid:19) , (cid:12)(cid:12)(cid:12)(cid:12) ↓↓ (cid:29) . − transition: H i = 12 − δ ps + ( g z e + g z h ) µ B B z g x e µ B B x − ıg y e µ B B y g x h µ B B x − ıg y h µ B B y g x e µ B B x + ıg y e µ B B y δ ps + ( − g z e + g z h ) µ B B z δ ps g x h µ B B x − ıg y h µ B B y g x h µ B B x + ıg y h µ B B y δ ps δ ps + ( g z e − g z h ) µ B B z g x e µ B B x − ıg y e µ B B y g x h µ B B x + ıg y h µ B B y g x e µ B B x + ıg y e µ B B y − δ ps + ( − g z e − g z h ) µ B B z Hamiltonian of the final state of X − transition: H f = g z e µ B B z √ g x e µ B B x − ı √ g y e µ B B y √ g x e µ B B x + ı √ g y e µ B B y √ g x e µ B B x − ı √ g y e µ B B y √ g x e µ B B x + ı √ g y e µ B B y − g z e µ B B z Transition operators: P + = − − √ P − = √
00 0 0 1
2. Matrices related to XX − transition Basis of the initial state: (cid:12)(cid:12)(cid:12)(cid:12) ↑↑↓ ⇑⇓ (cid:29) , (cid:12)(cid:12)(cid:12)(cid:12) ↓↑↓ ⇑⇓ (cid:29) . Basis of the final state: (cid:12)(cid:12)(cid:12) ↑↑ ⇑ E , √ (cid:16)(cid:12)(cid:12)(cid:12) ↓↑ ⇑ E + (cid:12)(cid:12)(cid:12) ↑↓ ⇑ E(cid:17) , (cid:12)(cid:12)(cid:12) ↓↓ ⇑ E , (cid:12)(cid:12)(cid:12) ↑↑ ⇓ E , √ (cid:16)(cid:12)(cid:12)(cid:12) ↓↑ ⇓ E + (cid:12)(cid:12)(cid:12) ↑↓ ⇓ E(cid:17) , (cid:12)(cid:12)(cid:12) ↓↓ ⇓ E . Hamiltonian of the initial state of XX − transition: H i = 12 g z e µ B B z g x e µ B B x − ıg y e µ B B y g x e µ B B x + ıg y e µ B B y − g z e µ B B z ! (A1)Hamiltonian of the final state of XX − transition: H f = − e δ + ξ z e + ξ z h √ ξ x e − ı √ ξ y e ξ x h − ı ξ y h √ ξ x e + ı √ ξ y e ξ z h √ ξ x e − ı √ ξ y e √ e δ ξ x h − ı ξ y h √ ξ x e + ı √ ξ y e e δ − ξ z e + ξ z h √ e δ ξ x h − ı ξ y e ξ x h + ı ξ y h √ e δ e δ + ξ z e − ξ z h √ ξ x e − ı √ ξ y e ξ x h + ı ξ y h √ e δ √ ξ x e + ı √ ξ y e − ξ z h √ ξ x e − ı √ ξ y e ξ x h + ı ξ y e √ ξ x e + ı √ ξ y e − e δ − ξ z e − ξ z h (A2)where ξ j i denotes g j i µ B B j and e δ i denotes ( δ i + δ psi ).Transition operators: P − = − − √ , P + = − √ −
10 00 00 0 . Appendix B: Polarization of XX − lines in Faradayconfiguration Below we give the analytical expressions describing thedegree of circular polarization for each spectral line ofXX − transition in magnetic field in Faraday configura-tion. The Hamiltonians used to obtain these expressions2were presented in Appendix A (Eq. A1 and A2). P E , P F , P G , and P H denote circular polarization P = I σ + − I σ − I σ + + I σ − forthe lines E, F, G, and H of XX − transition as defined inFig. 10. P E = (cid:18)q β + ( η − + ( η − (cid:19) − β (cid:18)q β + ( η − + ( η − (cid:19) + β (B1) P F = (cid:18)q β + ( η + 1) + ( η + 1) (cid:19) − β (cid:18)q β + ( η + 1) + ( η + 1) (cid:19) + 4 β (B2) P G = (cid:18)q β + ( η + 1) − ( η + 1) (cid:19) − β (cid:18)q β + ( η + 1) − ( η + 1) (cid:19) + 4 β (B3) (B4) P H = (cid:18)q β + ( η − − ( η − (cid:19) − β (cid:18)q β + ( η − − ( η − (cid:19) + β (B5)(B6)where β = e δ / e δ (B7) η = ( g z e − g z h ) µ B B/ e δ (B8) ∗ Electronic address: [email protected] † deceased D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer,and D. Park, Phys. Rev. Lett. , 3005 (1996). O. Benson, C. Santori, M. Pelton, and Y. Yamamoto,Phys. Rev. Lett. , 2513 (2000). R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper,D. A. Ritchie, and A. J. Shields, Nature , 179 (2006). N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky,J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff,Phys. Rev. Lett. , 130501 (2006). L. Besombes, Y. L´eger, L. Maingault, D. Ferrand, H. Ma-riette, and J. Cibert, Phys. Rev. Lett. , 207403 (2004). M. Goryca, T. Kazimierczuk, M. Nawrocki, A. Golnik,J. A. Gaj, P. Kossacki, P. Wojnar, and G. Karczewski,Phys. Rev. Lett. , 087401 (2009). M. Goryca, P. Plochocka, T. Kazimierczuk, P. Wojnar,G. Karczewski, J. A. Gaj, M. Potemski, and P. Kossacki,Phys. Rev. B , 165323 (2010). B. Urbaszek, R. J. Warburton, K. Karrai, B. D. Gerar-dot, P. M. Petroff, and J. M. Garcia, Phys. Rev. Lett. ,247403 (2003). I. A. Akimov, K. V. Kavokin, A. Hundt, and F. Hen-neberger, Phys. Rev. B , 075326 (2005). N. I. Cade, H. Gotoh, H. Kamada, H. Nakano, andH. Okamoto, Phys. Rev. B , 115322 (2006). M. Ediger, G. Bester, B. D. Gerardot, A. Badolato, P. M.Petroff, K. Karrai, A. Zunger, and R. J. Warburton, Phys.Rev. Lett. , 036808 (2007). P. Hawrylak, Phys. Rev. B , 5597 (1999). F. Tinjod, B. Gilles, S. Moehl, K. Kheng, and H. Mariette,Applied Physics Letters , 4340 (2003). P. Wojnar, J. Suffczy´nski, K. Kowalik, A. Golnik,M. Aleszkiewicz, G. Karczewski, and J. Kossut, Nanotech-nology , 235403 (2008). J. Suffczy´nski, T. Kazimierczuk, M. Goryca, B. Piechal,A. Trajnerowicz, K. Kowalik, P. Kossacki, A. Golnik, K. P.Korona, M. Nawrocki, et al., Phys. Rev. B , 085319(2006). Y. L´eger, L. Besombes, L. Maingault, and H. Mariette,Phys. Rev. B , 045331 (2007). H. S. Lee, A. Rastelli, M. Benyoucef, F. Ding, T. W.Kim, H. L. Park, and O. G. Schmidt, Nanotechnology ,075705 (2009). Y. L´eger, L. Besombes, J. Fern´andez-Rossier, L. Main-gault, and H. Mariette, Phys. Rev. Lett. , 107401 (2006). T. Kazimierczuk, J. Suffczynski, A. Golnik, J. A. Gaj,P. Kossacki, and P. Wojnar, Phys. Rev. B , 153301(2009). S. Cortez, O. Krebs, S. Laurent, M. Senes, X. Marie,P. Voisin, R. Ferreira, G. Bastard, J.-M. G´erard, andT. Amand, Phys. Rev. Lett. , 207401 (2002). R. I. Dzhioev, V. L. Korenev, B. P. Zakharchenya, D. Gam-mon, A. S. Bracker, J. G. Tischler, and D. S. Katzer, Phys.Rev. B , 153409 (2002). I. A. Akimov, D. H. Feng, and F. Henneberger, Phys. Rev.Lett. , 056602 (2006). T. Kazimierczuk, M. Goryca, M. Koperski, A. Golnik,J. A. Gaj, M. Nawrocki, P. Wojnar, and P. Kossacki, Phys.Rev. B , 155313 (2010). K. Haas, T. Kazimierczuk, P. Wojnar, A. Golnik, J. Gaj,and P. Kossacki, Acta Phys. Pol. A , 896 (2009). A. Kudelski, A. Golnik, J. A. Gaj, F. V. Kyrychenko,G. Karczewski, T. Wojtowicz, Y. G. Semenov, O. Krebs,and P. Voisin, Phys. Rev. B , 045312 (2001). M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gor-bunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L.Reinecke, et al., Phys. Rev. B , 195315 (2002). E. Poem, Y. Kodriano, C. Tradonsky, N. H. Lindner, B. D.Gerardot, P. M. Petroff, and D. Gershoni, Nature Physics , 993 (2010). A. Kudelski et al., in
Proceedings of the 25th Interna-tional Conference on Physics of Semiconductors (Springer,2000), p. 1249. K. Kowalik, O. Krebs, A. Golnik, J. Suffczy´nski, P. Woj-nar, J. Kossut, J. A. Gaj, and P. Voisin, Phys. Rev. B ,195340 (2007). T. Kazimierczuk, A. Golnik, M. Goryca, P. Wojnar, J. Gaj, and P. Kossacki, Acta Phys. Pol. A , 882 (2009). E. Poem, J. Shemesh, I. Marderfeld, D. Galushko,N. Akopian, D. Gershoni, B. D. Gerardot, A. Badolato,and P. M. Petroff, Phys. Rev. B , 235304 (2007). J. J. Finley, P. W. Fry, A. D. Ashmore, A. Lemaˆıtre, A. I. Tartakovskii, R. Oulton, D. J. Mowbray, M. S. Skolnick,M. Hopkinson, P. D. Buckle, et al., Phys. Rev. B63