Förster energy transfer of dark excitons enhanced by a magnetic field in an ensemble of CdTe colloidal nanocrystals
Feng Liu, A. V. Rodina, D. R. Yakovlev, A. A. Golovatenko, A. Greilich, E. D. Vakhtin, A. Susha, A. L. Rogach, Yu. G. Kusrayev, M. Bayer
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec F¨orster energy transfer of dark excitons enhanced by a magnetic fieldin an ensemble of CdTe colloidal nanocrystals
Feng Liu, , A. V. Rodina, D. R. Yakovlev, , A. A. Golovatenko, A. Greilich, E. D. Vakhtin, A. Susha, A. L. Rogach, Yu. G. Kusrayev and M. Bayer , Experimentelle Physik 2, Technische Universit¨at Dortmund, 44221 Dortmund, Germany Department of Physics and Astronomy, University of Sheffield, Sheffield, S3 7RH, United Kingdom Ioffe Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia St. Petersburg State Polytechnical University, 195251 St. Petersburg, Russia and City University of Hong Kong, Hong Kong (Dated: August 26, 2018)We present a systematic experimental study along with theoretical modeling of the energy transferin an ensemble of closely-packed CdTe colloidal nanocrystals identified as the F¨orster resonant energytransfer (FRET). We prove that at low temperature of 4.2 K, mainly the ground dark exciton statesin the initially excited small-size (donor) nanocrystals participate in the dipole-dipole FRET leadingto additional excitation of the large-size (acceptor) nanocrystals. The FRET becomes possible dueto the weak admixture of the bright exciton states to the dark states. The admixture takes placeeven in zero magnetic field and allows the radiative recombination of the dark excitons. An externalmagnetic field considerably enhances this admixture, thus increasing the energy transfer rate by afactor of 2 − PACS numbers: 73.21.La, 78.47.jd, 78.55.Et, 78.67.Hc
I. INTRODUCTION
Colloidal semiconductor nanocrystals (NCs), and espe-cially their optical properties are attracting a lot of atten-tion in very different fields.
A strong motivation hereis related to promising applications ranging from lightabsorbers in photovoltaics and light emitters in optoelec-tronics to medicine and biology, where they can serveas efficient luminescent markers.
Due to the strongcarrier confinement, NCs are appealing objects for ba-sic research. They offer great variety of the structuralparameters and engineering of the band gap profiles. Itis possible to grow them as core-shell NCs of type-I ortype-II band alignment, or synthesize NCs with differentshapes, such as rods or platelets, which in turn can becombined in dot-in-rod or dot-in-plate structures. Dif-ferent potential applications of NCs are related to effectsbased on energy transfer in NC ensembles and in theirvarious hybrid structures . At the same time, nonra-diative energy transfer in an ensemble of closely spacednanocrystals often leads to reduction of the photolumi-nescence (PL) quantum yield as compared to samples insolution. Colloidal NCs are often based on II-VI semiconduc-tors, e.g. CdSe, CdS, CdTe, ZnS, and their optical prop-erties are dominated by the band edge excitons. Theexciton fine structure is controlled by the anisotropy ofthe crystal lattice and the nanocrystal shape as well asby the strong electron–hole exchange interaction that isenhanced due to the carrier confinement. The lowest exciton state is optically–forbidden in electric-dipole ap-proximation for a one-photon process and is thereforereferred to as a ”dark exciton”. The optically-allowed”bright” exciton is shifted to higher energy by the ex-change energy, which can be as large as a few meV. Asa result, the recombination dynamics of NCs especiallyat low temperatures is nontrivial being dependent on thepopulation of the dark and bright exciton states, the spinrelaxation between them and their mixing, e.g., in exter-nal magnetic fields.In ensembles of closely packed NCs the phenomenol-ogy of recombination dynamics becomes even richer dueto the energy transfer between the neighboring NCs. Dueto the inhomogeneous broadening of the optical tran-sitions caused by variations in NC size and shape, theNCs of smaller size emit in the high-energy flank of thespectrum and can serve as donor NCs as their energycan be transferred to the NCs of larger size (acceptorNCs), whose emission is shifted to the lower energy sideof the spectrum. Experimentally the energy transfer isusually evidenced by the observation of spectral diffu-sion (i.e., a red shift of the PL spectrum with time) orby the observation of a spectral dependence of the pho-toluminescence dynamics, where the emission decay isshortened for the donor NCs. These findings have beenreported for CdSe , CdS , CdTe , PbS andSi based nanostructures. It is well established and com-monly accepted that the most relevant mechanism ofenergy transfer (ET) in an ensemble of NCs with aver-age diameter of 4 − The F¨orster energy transfer rate Γ ET canbe described by the following equation :Γ ET = 2 π ~ ( µ d µ a κ ) R n Θ . (1)Here µ d and µ a are the transition dipole moments of ex-citons in the donor and acceptor NCs, respectively, κ isan orientational factor (it accounts for the distributionof angles between the donor and the acceptor dipole mo-ments; for random dipole orientation κ = 2 / R da is the distance between donors and acceptors, n is therefractive index of the medium, and Θ is the overlap in-tegral between the donor emission and the acceptor ab-sorption spectra which are normalized to µ and µ , re-spectively. It can be seen from Eq. (1), that Γ ET ∝ /R is extremely sensitive to the distance between donor andacceptor NCs. The most efficient energy transfer cor-responds to the situation for which two nanocrystals ofdifferent sizes are located in an immediate vicinity of eachother and the ground state emitting level of the donor NCinteracts resonantly with a higher lying absorbing levelof the acceptor NC. The nonradiative dipole-dipole en-ergy transfer represents an additional effective channelfor the decay and shortening of the PL lifetime in thedonor NCs, which is caused by the interaction of the ra-diative dipoles in donor and acceptor NCs without emis-sion and reabsorption of real photons. The radiative partof this interaction may also lead to energy transfer viaemission and reabsorption of real photons as well as toa radiative correction to the donor radiative recombina-tion rate. However, as it was shown recently, the ra-diative corrections to the radiative and energy transferrates can be neglected for the typical spatial separationbetween the donor and acceptor dipoles at which the en-ergy transfer is effective.Thus, the important (dominating) role of the FRETfor colloidal NCs is well documented and has been stud-ied by continuous-wave and time-resolved photolumines-cence. Most experiments have been performed at roomtemperature , with some low-temperature data be-ing also available . However, the details of thetransfer mechanism at low temperatures, when the PLis governed by the emission from dark exciton state, aswell as the possibilities to affect the efficiency of the en-ergy transfer by external electric or magnetic fields, arenot yet clarified and still open for detailed investigations.So far, the FRET from semiconductor CdSe/CdSnanorods to dye molecules controlled by an electric fieldwas demonstrated . The responsible mechanism isbased on the quantum confined Stark effect that enablea shift of the nanorod emission spectrum into resonancewith the dye absorption band.The effect of an external magnetic field on the FRETefficiency in colloidal NCs has not been studied systemat-ically. There are two experimental papers related to thisissue. Furis et al. studied CdSe NCs in high magneticfields up to 45 T. Pronounced exciton transfer via FRET was found in the spectrally-resolved recombination dy-namics. The magnetic-field-induced circular polarizationdegree of the PL was insensitive to the FRET process andthe authors suggested that this may provide evidence fora spin-conserving FRET. The magnetic field effect on theFRET efficiency was not discussed, no conclusion in thisrespect can be drawn from the presented experimentaldata. Blumling et al. studied the effect of temperatureand magnetic field on the energy transfer in CdSe NC ag-gregates by measuring the steady-state PL. The authorsclaimed that both temperature and magnetic field canenhance the energy transfer due to the population of thebright exciton state.Theoretically, the dipole-dipole interaction mechanismfor energy transfer and the influence of such factors asthe donor-acceptor separation, the spectral overlap andthe effect of the surrounding are well established.In contrast to multipole and exchange mechanisms, the direct dipole-dipole coupling conserves the spin of theparticipating charge carries. As the energy transfer ratedepends on the dipole moments of the excitons in thedonor and acceptor NCs, the dipole-dipole energy trans-fer has been considered to take place only between thebright exciton states. At the same time it is well estab-lished that the dark exciton states in colloidal nanocrys-tals are activated due to an admixture of bright excitonstate wave functions. This also means that the darkexciton states possess a nonzero dipole moment that isproportional to the admixed bright state dipole moment µ and the radiative life time of the dark exciton state is τ F ∝ /µ . Up to now, the possibility of FRET betweendark excitons has been analyzed neither theoretically norexperimentally.A quantitative analysis of the energy transfer ratesfrom experimental data is complicated by several fac-tors. Usually, the rates are extracted from the compari-son of luminescence decay curves for small NCs (donors)that participate and do not participate in the transferprocess. However, the donor decay curves are usu-ally strongly nonexponential due to the inhomogeneousspatial distribution of acceptors around the donors. The-oretical modeling of the time evolution of donor and ac-ceptor populations in a mixed ensemble of CdSe NCs hasbeen done in Refs. 18,19 and reproduced the observedspectra well. However, this analysis has not taken intoaccount the exciton fine structure and the possibility ofthe dark exciton states to contribute to the energy trans-fer process.In this paper we report an experimental and theoreti-cal study of the energy transfer in an ensemble of closelypacked CdTe NCs. The energy transfer is evidenced ex-perimentally by the time-dependent shift of the PL max-imum after pulsed excitation. The observed shift is welldescribed theoretically by the energy transfer betweensmall and large NCs. An important observation concernsthe time scale of the process - at low temperatures the en-ergy shift is observed during times much longer than thelife time of the bright exciton that is shortened by the fastthermalization due to the relaxation to the dark excitonstate. Spectrally and temporally resolved photolumines-cence measurements reveal a strong spectral dispersion ofthe exciton life time and give clear evidence of a secondrise of the PL intensity (after the initial decay) at thelow energy side of the spectrum that becomes enhancedin an applied magnetic field. Our theoretical modeling ofthe donor and acceptor population takes into account theexciton fine structure and the possibility of dipole-dipoleenergy transfer from the dark exciton state. The resultsof the simulations reproduce the observed emission de-cays of the acceptor NCs and allow us to determine allparameters of the energy transfer process and their de-pendence on the magnetic field. The energy transfer ratefrom the dark exciton increases in the external magneticfield due to the increase of the dipole moments in boththe donor and acceptor NCs.The paper is organized as follows: Sec. II gives adescription of the samples and experimental conditions.The experimental data with an emphasis on evidencingthe energy transfer process are presented in Sec. III. Thetheoretical considerations and the resulting modeling ofthe experimental data are presented in Sec. IV and Sec.V, respectively. The main conclusions are summarized inSec. VI.
II. EXPERIMENTALS
Thiol-capped CdTe colloidal NCs were synthesized inwater as described in Ref. 45. The exciton recombinationand spin relaxation dynamics in such NCs were reportedin Ref. 46. In this work, two samples with average corediameters of 3.4 and 3.7 nm were studied. For the opticalexperiments at cryogenic temperatures aqueous solutionsof CdTe NCs were drop-casted on a glass slice and dried.The resulting films consist of areas of NCs with varying(high and low) in-plane densities of NCs, which are char-acterized by high and low total PL intensities, respec-tively. These inhomogeneous films allow us to study andcompare the effect of the energy transfer on the ensemblePL spectrum and PL dynamics for areas with differentaverage spatial separation between NCs and thus differ-ent values of R da determining the transfer rate.The samples were inserted into a cryostat equippedwith a 15 T superconducting magnet. The magnetic field, B , was applied in the Faraday geometry, it was orientedperpendicular to the glass slice and parallel to the lightwave vector. The sample was in contact with helium gasand the bath temperature was varied from T = 4 . to avoid anymultiexcitonic contribution to the emission spectra. III. EXPERIMENTAL RESULTSA. Steady state PL spectra and spectrallyintegrated recombination dynamics
Steady-state PL spectra of the 3.4 and 3.7 nm CdTeNCs measured under CW excitation at T = 4 . ∼
120 meV, whichevidences the considerable NC size dispersion of about7%.The exciton recombination dynamics in the NCs can becharacterized by the spectrally integrated time-resolvedPL. Figure 1(b) shows the integral PL decay obtainedby summing up the PL decays measured at 20 energiesdistributed across the whole PL band of the 3.4 nm NCs.At room temperature the PL decay can be describedby a biexponential function with decay times of 6 and22 ns. The longer component originates from the ther-mally mixed bright and dark exciton states. With de-creasing temperature down to 4.2 K the decay shows amultiexponential behavior due to the exciton thermal-ization into the optically-forbidden (dark) state, which isthe lowest exciton state in NCs. It is well established thatthe very fast initial decay with a time of approximately2 ns is related to the optically-allowed (bright) exciton.Its decay is dominated by the fast scattering from thebright to the dark state and has some contribution fromthe radiative recombination of bright excitons . Theslow component with a decay time of about 260 ns cor-responds to the lifetime of the dark (optically forbidden)exciton τ F , whose recombination becomes partially al-lowed due to a weak mixing of the dark and the brightexciton states caused, e.g., by the magnetic moments ofdangling bonds at NC imperfections and surface states.An external magnetic field induces a mixing of thebright and the dark exciton states, which results in thevanishing of the fast decay component and the shorten-ing of the slow component. Such a behavior, well es-tablished for CdSe and CdTe NCs, is observed alsofor the studied sample. In a magnetic field of 15 T theamplitude of the fast component ultimately vanishes and (b)(a) T = 4.2 KB = 0 T 15 T P L i n t en s i t y ( a r b . un i t s ) Energy (eV)
T = 4.2 KB = 0 T4.2 K15 T 300 K0 T P L i n t en s i t y ( a r b . un i t s ) Time (ns)
FIG. 1: (a) Normalized steady-state PL spectra of the 3.4 nmand 3.7 nm CdTe NCs measured at T = 4 . B = 0 T (black lines) and at B = 15 T (red line).The arrows indicate the energies at which the PL dynamicsshown in Figs. 7 and 8 are measured. (b) Recombination dy-namics of the spectrally-integrated PL intensity in the 3.4 nmCdTe NCs measured at T = 4 . B = 0T (black line) and B = 15 T (red line), and at T = 300 K(blue line). the slow component is shortened down to about 100 ns,compare the red and black curves in Fig. 1(b). B. Evidence for the energy transfer process:spectral diffusion
We now turn to the experimental results that evidenceon the energy transfer in the studied CdTe NC solids.Figure 2(a) compares the steady-state (solid lines) andthe time-resolved (lines with dots) PL spectra of the3.4 nm NCs measured at zero time delay just after theexcitation pulse. The red and black lines show the resultsmeasured at two different sample areas with different NCdensities, high and low, respectively. The PL intensitiesat these points differ by a factor of 2.5. The time-resolvedspectra from these high- and low-density areas are simi-lar to each other, indicating the same NC size dispersionin these areas. Note, that these spectra, measured right after the pulse, are not contributed by the energy trans-fer and, therefore, give us information on the density ofstates in the NC ensemble.The steady-state PL spectra in Fig. 2(a) are shiftedto lower energies compared to the time-resolved spectra.This shift is larger in the area with higher NC densitiesreaching 50 meV compared to 29 meV in the high- andlow-density areas, respectively. In general, such shiftsmay be induced by the energy transfer, but also the spec-tral dependence of the PL dynamics across the emissionband may be a possible origin. As the second reason isnot relevant for CdTe NCs , we attribute the shift solelyto the energy transfer. The higher density of NCs corre-sponds to a smaller separation between them. Therefore,for the higher NC density the FRET is more efficientand, consequently, a lager shift between time-resolvedand steady-state PL spectra is expected.A systematic correlation between the NC density andthe energy shift of the steady-state PL spectra measuredat different sample areas is shown in Fig. 2(b). Here thePL intensity from the sample areas with different NCdensities is plotted against the peak energy. One cansee that stronger PL intensities correspond to the areaswith lower peak energy, i.e. larger shift. This correlationagrees with the expectations from the F¨orster mechanismfor the energy transfer and is in line with the experimen-tal data reported in Ref. 30.The time evolution of the PL spectra measured in highdensity areas at different time delays is presented in Fig. 3for temperatures of 4.2 and 300 K. The strong spectralshift with increasing delay is prominent at both temper-atures. For comparison, the steady-state PL spectra arealso shown by the solid red lines. At T = 4 . t = 0 ns down to 2.01 eV at 70 ns, seeFig. 3(a). A similar behavior is observed at room temper-ature, where the time-resolved spectra shift from 1.96 eVat 0 ns down to 1.92 eV at 70 ns. An important fea-ture related to the energy transfer is seen at room tem-perature in Fig. 3(b). Namely, the PL intensity variesnon-monotonically at the low energy tail of the emissionband. It increases during several nanoseconds after theexcitation pulse and only then starts to decay. This be-havior will be shown in more detail below where the PLdynamics measured at different spectral energies are pre-sented.Figures 4(a)-4(c) show shifts of the PL maxima withtime, measured at two sample areas having similar PLintensities. Due to the sample inhomogeneity several ex-perimental data sets were measured at different sampleareas. The characteristic behavior is, however, well re-producible for all these areas. For area 1 the PL spectrumshifts by 47 meV during the first 70 ns at T = 4 .
29 meV50 meV P L i n t en s i t y ( a r b . un i t s ) Energy (eV) x0.4 (b)(a)
T = 300 K P L i n t en s i t y ( a r b . un i t s ) Energy (eV)
FIG. 2: (a) Normalized time-resolved (lines with dots) andsteady-state (solid lines) PL spectra measured at high-density(red) and low-density (black) areas of the sample with 3.4 nmCdTe NCs. (b) PL intensity versus peak position of thesteady-state PL spectra measured at areas with different den-sities of CdTe NCs. The line is a linear interpolation. ally slowed down with a tendency to saturated. A similarshift of the PL maximum was presented in Refs. 20,21,25and attributed to the energy transfer process at roomtemperature. It should be noted, that the spectral shift(the spectral diffusion) can be also observed in systemswithout any energy transfer, for example in the donor-acceptor pair recombination in bulk semiconductors. In this case, the spectral diffusion is caused by the spec-tral dependence (dispersion) of the recombination rates.However, for CdTe NCs no dispersion of the recombina-tion dynamics was found in an ensemble of noninteractingNCs, i.e. in the absence of energy transfer. In addition,we observed the same correlation between the temporalshift of the PL maxima and the PL intensity as for the
T = 300 K P L i n t en s i t y ( a r b . un i t s ) Energy (eV) (b)(a)
T = 4.2 K P L i n t en s i t y ( a r b . un i t s ) Energy (eV)
60 meV
Time delay: 0 ns 0.6 ns 1 ns 2 ns 3 ns 5 ns 12 ns 30 ns 70 nsTime delay: 0 ns 0.5 ns 2 ns 5 ns 9 ns 22 ns 48 ns 70 ns
50 meV
FIG. 3: Steady-state PL spectra (solid red lines) and time-resolved PL spectra (lines with dots) of the 3.4 nm CdTe NCsmeasured at (a) T = 4 . shift of the cw spectral maxima: the areas with largerPL intensity and thus higher NC density demonstrate alarger temporal shift of the PL maximum. Therefore, weattribute the observed spectral shift solely to the effectof the energy transfer.It is important to note, that at T = 4 . B = 15 T (51 meV) is larger than that measured at0 T (40 meV). This is in contrast to the shift of theCW spectra which is independent of the magnetic field,as can be seen from comparison of the CW spectra at T = 300 K area 2area 1 (b)(a)
T = 4.2 K P ea k po s i t i on ( e V ) Time (ns)B = 0 T
B = 15 T B = 0 T
T = 4.2 K P ea k po s i t i on ( e V ) Time (ns)
FIG. 4: Temporal shift of the PL maximum in 3.4 nm CdTeNCs measured for two sample areas at different temperaturesand magnetic fields. The experimental data are shown bysymbols. Lines are fits according to Eq. (2). E ( t ) = E + ∆ E ( e − t Γ ∆ E −
1) + ∆ E ( e − t Γ ∆ E − . (2)The parameters of the fits, the shift energies ∆ E , andthe shift rates Γ ∆ E , are given in Table I. We note thatthe fitting with only one characteristic shift rate does notwork well: in all cases the first initial fast energy shift isfollowed by a slow shift. We comment on the physicalmeaning of the different shift rates in the theory sectionV C. area E ∆ E Γ ∆ E ∆ E Γ ∆ E condition (eV) (meV) (ns − ) (meV) (ns − )area 14.2 K, 0 T 2.055 22 0.38 32 0.025area 1300 K, 0 T 1.967 34 0.41 48 0.031area 24.2 K, 0 T 2.035 16 0.45 27 0.027area 24.2 K, 15 T 2.036 14 0.23 45 0.033TABLE I: Fit parameters for modeling the time evolution ofthe PL peak maximum according to Eq. (2). C. Evidence for the energy transfer process:spectrally–resolved PL dynamics influenced by anexternal magnetic field
Figure 5shows the spectrally-resolved PL dynamics ofthe 3.4 nm CdTe NCs measured at T = 4 . B = 15 T at the same energy of 1.88 eV,where the low energy tail of the 3.4 nm ensemble overlapswith the high energy tail of the 3.7 nm ensemble. Dueto the NC size dispersion in the ensembles the specificenergy corresponds to the specific NC size. While the PLdynamics differ drastically being strongly contributed by (b)(a) T = 4.2 KB = 15 T1.88 eV2.08 eV T = 4.2 KB = 0 T
Time (ns) P L i n t en s i t y ( a r b . un i t s ) FIG. 5: Spectrally-resolved PL dynamics of the 3.4 nm CdTeNCs measured at (a) B = 0 T and (b) 15 T. The detectionenergy is varied in the range from 1.88 to 2.08 eV. The signalsare normalized on their intensity at zero delay, i.e., right afterthe excitation pulse. the energy transfer.Figure 7(a)- 7(c) show the PL dynamics measured atdifferent magnetic fields for three energies on the 3.4 nmNCs, compare with Fig. 1. For clarity, Figs. 7(d-f) areclose-ups of the initial 50 ns of the PL dynamics andshow their amplitude on a linear scale. The PL decay atthe high energy position shows the behavior typical forcolloidal NCs, see Figs. 7(a,d). With increasing magneticfield, the fast component vanishes and the slow compo-nent shortens due to the mixing of bright and dark ex-citon states. However, a different behavior is observedat the maximum of PL spectra in Figs. 7(b,e). The PLdynamics at B = 0 T shows a decay in the time frame t ∈ (5 ns ,
15 ns), which slows down with increasing mag-netic field and turns into a rise at B = 15 T. Such unusualbehavior is even more prominent at the low energy posi-tion, see Figs. 7(c,f). At B = 15 T the PL intensity at t = 12 ns is larger than the intensity at t = 0 ns.A qualitatively very similar behavior was found for the3.7 nm NCs, whose PL dynamics during the initial 50 nsmeasured at the PL maximum of 1.82 eV is presentedin Figure 8. In panels (a-c) the PL dynamics measuredat fixed temperature are compared for different magneticfields. At T = 4 . B = 0 T. At higher fields the PL intensity in the time P L i n t en s i t y ( a r b . un i t s ) Time (ns)
FIG. 6: Spectrally-resolved PL dynamics of the 3.4-nm and3.7 nm CdTe NC ensembles measured at the energy of 1.88 eV,see Fig.1(a). (c) P L i n t en s i t y ( a r b . un i t s ) (d) (b) Low energy 1.93 eVMaximum 1.99 eVLow energy 1.93 eVMaximum 1.99 eV
Time (ns)
High energy 2.04 eVHigh energy 2.04 eV
Time (ns)
0T 5T 10T 15T (a)
T = 4.2 K P L i n t en s i t y ( a r b . un i t s ) (e)(f) FIG. 7: (Color online)(a-c) PL dynamics of the 3.4 nm CdTeNCs measured at different spectral energies (see Fig. 1) anddifferent magnetic fields. Panels (d-f) detail the PL dynamicsduring the initial 50 ns. The green lines show fits to thesecurves with the function described in Appendix. interval t ∈ (5 ns ,
15 ns) starts to grow, and at 15 T thedecay turns into a rise. This behavior is more pronouncedat higher temperatures, see Fig. 8(b). At 10 K, the decayturns to a rise already at B = 5 T. At even higher tem-perature of 15 K the fast decay component within thefirst 5 ns disappears, which can be seen in Figs. 8(a,b),and only a slow rise is visible during the initial 10 ns, seeFig. 8(c). The rise time increases for stronger magneticfields.Besides the magnetic field, the shape of the PL decayis also strongly influenced by temperature. As shown (c) P L i n t en s i t y ( a r b . un i t s )
10 T5 T0 T (d) (b)
Maximum 1.82 eV
B = 15 T4.2 K10 KT = 15 K
B = 10 T5 T0 T
B = 5 TT = 15 KT = 10 K
Time (ns)
B = 0 T
Time (ns)
T = 4.2 K (a)
10 T5 T0 T P L i n t en s i t y ( a r b . un i t s ) (e) (f) FIG. 8: PL dynamics of the 3.7 nm CdTe NCs measured atthe maximum of the PL band at 1.82 eV for various magneticfields and temperatures. in Fig. 8(d), the PL dynamics at 4.2 K and 10 K startwith a fast decay component, while this fast componentcannot be seen at 15 K. Instead, the PL dynamics startswith a rise of the PL intensity. At B = 5 T the rise ofthe PL intensity becomes more prominent, see Fig. 8(e).At B = 15 T the PL dynamics show this rise during theinitial 15 ns at all studied temperatures. The fast decaycomponent presents at 4.2 K and 10 K, but disappearsat 15 K, see Fig. 8(f).Figure 9 shows the spectral dependence of the PL de-cay times of the 3.4 nm CdTe NCs for B = 0 T and 15 Tmeasured at T = 4 . τ Aai observedat the very beginning of the PL decay. We relate thisdecay to the relaxation of excitons from the bright stateto the dark state. The black closed diamonds correspondto the longest component τ Fai , related to the decay of thedark excitons that are not involved in the energy transferprocess as donors. The blue open triangles correspond tothe component τ Fd related to the dark excitons involved inthe energy transfer process as donors having, therefore,a shortened lifetime. The amplitude of this componentis small at the low energy side of the PL spectra, wherethe FRET does not contribute to the PL dynamics, andincreases towards the high energy side of the PL spec-tra. The fast component marked by red circles, whichappears only at the high energy side of the PL spectra, is FIG. 9: Spectral dependence of the PL decay and rise timesof the 3.4 nm CdTe NCs for (a) B = 0 T and (b) 15 T at T = 4 . most likely related to the energy transfer from the brightexciton states. One can see, that at B = 0 T the charac-teristic times of all these components show no apparentspectral dependence, while their relative amplitudes arespectrally dependent.For the PL dynamics at a magnetic field of 15 T thecharacteristic times are collected in Fig. 9(b). Here theopen squares correspond to the rising component inducedby the energy transfer. This component is visible only atthe low energy side of the PL spectra. Compared with thezero-field case, two new components (the red stars andthe orange circles) appear at the high energy side of thePL spectra, indicating that new energy transfer paths areactivated by the magnetic field or additional NCs becomeinvolved in the FRET. On the basis of these character-istic decay times, we developed a theoretical model de-scribing the observed experimental results. Description FIG. 10: Magnetic field dependence of the PL decay and risetimes of the 3.4 nm CdTe NCs measured at 1.93 eV. The sizeof symbols (except for the closed diamonds) is proportionalto the amplitude of the corresponding component. of the model and presentation of the simulation resultswill be done in the following sections.Figure 10 shows the magnetic field dependence of thecharacteristic times for the 3.4 nm CdTe NCs measuredat the low energy side of the PL spectra at 1.93 eV for T = 4 . τ Aa,i . Thelongest component τ Fa,i (black closed diamonds) corre-sponding to the dark exciton decay shortens by a factorof ∼ IV. THEORETICAL CONSIDERATIONS
In this section we present the theoretical model, thatdescribes the PL intensity of an ensemble of NCs takinginto account the energy transfer process between them.We start from the general assumptions about the proper-ties of the NC ensemble and consider the donor and theacceptor NCs participating in the energy transfer processas well as the independent NCs (not participating in theFRET process). We discuss the possible recombination,relaxation and energy transfer pathways for excitons andwrite down a system of rate equations for the excitonpopulations in donor, acceptor and independent NCs.
A. General assumptions
For the sake of clarity, let us consider an ensemble ofclosely–packed prolate CdTe NCs with the ground exci-ton state split into the bright (optically-allowed A exci-ton with spin projection ± ±
2) exciton states. Since the probability of theF¨orster energy transfer decreases proportional to R -6da ,where R da is the distance between the centers of thedonor and acceptor NCs, it is the largest for the near-est neighbor NC pairs and already much smaller for thenext nearest neighbors. Thus, in the model we con-sider the energy transfer process only between the near-est neighbors. In this case the energy transfer rate isdetermined by the magnitude of exciton dipole momentsin the donor and acceptor NCs and by the overlap inte-gral between the donor emission and acceptor absorptionspectra, while the average distance can be estimated asthe mean size of the NCs: R da ≈ d . We consider the lowexcitation regime, which is defined by assuming that theportion of initially excited NCs at each energy ( N ( E ))within the NC ensemble is negligible. This allows us toconsider the energy transfer only to initially unexcitedNCs. In turn, initially excited NCs with the ground stateexciton energy E may act as donors (with probability f d ( E )) and transfer their excitation to the nearest accep-tor NCs (see Fig. 11(a)), or as independent NCs whichdo not participate in the energy transfer process (seeFig. 11(b)). Thus, the total number of initially exciteddonor and independent NCs is N ( E ) = N ( E ) f d ( E ) and N ( E ) = N ( E )(1 − f d ( E )), respectively. The proper-ties of the energy-dependent probability function f d ( E )will be discussed later. Also, we neglect all cascade pro-cesses, so that one NC cannot act as donor and acceptorsimultaneously. FIG. 11: Scheme of possible processes initiated by opticalexcitation in a NC ensemble: (a) energy transfer between thenearest donor and acceptor NCs, and (b) independent NCsfor which no energy transfer occurs.
After these general remarks on the assumptions relatedto the ensemble of NCs, let us consider the energy trans-fer mechanism between the donor and acceptor NCs inmore detail. Let us consider a NC with the ground stateexciton energy E a at the low energy side of the PL bandarising from the acceptors. A smaller NC with higherground state exciton energy E d = E a + E da may playthe role of a donor for the chosen acceptor if there is0 FIG. 12: Schematic of the energy transfer process between adonor NC with energy E d and an acceptor NC with energy E a . The energy transfer is shown by the green arrows, therelaxation processes by the blue arrows, the recombinationpathes by the red arrows, and the laser pumping by the blackarrow. a nonzero overlap between the donor emission spectrumand the acceptor absorption spectrum. We assume forsimplicity that for each E d there is only one excited level E a + E da in the nearest acceptor NCs and consider theenergy difference E da as a parameter in our model.In case of initial nonresonant excitation with the rate G using photon energies well above the ground exciton stateenergy the relaxation of the hot excitons to the groundstate is a fast process assisted by optical phonons and theinteraction with the surface. After relaxation of the ex-citon to the ground state it can be resonantly transferredto another (acceptor) NC. The energy transfer processbetween the donor NC with ground state exciton energy E d and the acceptor NC with ground state exciton en- ergy E a is shown schematically in Fig. 12. We assumethat the bright (optically-allowed A) and dark (optically-forbidden F) exciton states have approximately the sameenergy splitting ∆ E AF in the donor and acceptor NCs.The energy transfer process consists of two steps. Ini-tially the resonant transfer from the ground energy stateof the donor NC to the first excited energy state of theacceptor NC takes place. We assume that the bright anddark ground exciton states in the donor NC are in reso-nance with the first excited bright and dark states in theacceptor NC, respectively. After the energy transfer, fastenergy relaxation from the excited to the ground state ofthe acceptor NC occurs. This relaxation process may oc-cur in two different ways, either spin conserving and thusallowing relaxation only from the bright to the brightstates or from the dark to the dark state or spin nonconserving. For the sake of clarity, we assume here theenergy difference E da to be large enough for the fast nonconserving relaxation processes, similar to those takingplace after the initial excitation of all NCs. In this casethe relaxation from the bright and dark excited statesoccurs with equal probability to the ground bright anddark states without any ”memory” effects. However, themodel can be easily modified for the case of spin conserv-ing relaxation in the acceptor NCs. B. Rate equations for the bright and dark excitonpopulations
To be able to describe the experimental data with ourmodel we express the PL intensity in terms of excitonpopulations in the NCs. The total population of excitons N ( E, t ) with a given ground state exciton energy E canbe written as the sum of exciton populations in the donorNCs, N d , the independent NCs, N i , and the acceptorNCs, N a , NCs. In turn, the population of excitons ineach type of NCs consists of the populations of excitonsin the bright, N Ad , i , a , and dark, N Fd , i , a , states: N ( E, t ) = N i ( E, t ) + N d ( E, t ) + N a ( E, t ) , (3) N d,i,a ( E, t ) = N Ad,i,a ( E, t ) + N Fd,i,a ( E, t ) . (4)Using the populations of the bright and dark excitonsin each type of NCs, one obtains for the PL intensity ofa NC ensemble at a given exciton energy EI ( E, t ) = I i ( E, t ) + I d ( E, t ) + I a ( E, t ) , (5) I d,i,a ( E, t ) = Γ radA ( E ) N Ad,i,a ( E, t ) + Γ radF ( E ) N Fd,i,a ( E, t ) . (6)Here Γ radA,F are the radiative recombination rates.Thus, knowing the time evolution of the populations N A,Fd,i,a ( E, t ), one can fully describe the time evolution ofthe PL intensity from a NC ensemble.The pair of the bright-dark exciton rate equations forthe donor and the independent NCs, can according to1Fig. 12, be written as: dN Ad,i ( E, t ) dt = − N Ad,i (Γ A + γ + γ th + Γ AET )+ N Fd,i γ th + 12 G d ( E, t ) , (7) dN Fd,i ( t ) dt = − N Fd,i ( t )(Γ F + γ th + Γ FET )+ N Ad,i ( t )( γ + γ th ) + 12 G i ( E, t ) . (8)Here the recombination rates Γ A,F = Γ dA,F (which mayinclude both radiative and on nonradiative paths otherthan the energy transfer), as well as the relaxation rates γ = γ d0 and γ th = γ dth (thermally induced relaxationgiven for the case of a one-phonon process by γ th = γ / [exp(∆ E AF /k B T ) − E d . The energy transfer rates are denoted asΓ AET and Γ
FET for the bright and dark exciton states inthe donor NCs, respectively, and have to be set to zero inthe independent NCs. The rates of pumping of the brightand dark exciton states are denoted as G d,i ( E, t ) andare equal because of the spin non conserving relaxationof the hot excitons, as mentioned previously.The rate equations for the acceptor NCs, according toFig. 12, can be written as: dN Aa ( E, t ) dt = − N Aa (Γ A + γ + γ th ) + N Fa γ th + 12 ( N Ad Γ AET + N Fd Γ FET ) , (9) dN Fa ( E, t ) dt = − N Fa ((Γ F + γ th ) + N Aa ( γ + γ th )+ 12 ( N Ad Γ AET + N Fd Γ FET ) . (10)Here the population N a should be considered at theexciton energy E a (as well as the recombination ratesΓ A,F = Γ aA,F and the relaxation rates γ = γ a0 as wellas γ th = γ ath rates), while the population N d has to betaken at the energy E a + E da . Since we consider the lowexcitation regime, a pumping term is not included in therate equations for the acceptor NCs. V. MODELING OF THE EXPERIMENTALDATAA. Determination of decay times for donor,acceptor and independent NCs
The system of rate equations described in subsectionIV B can be solved numerically or analytically. The ana-lytical solution can be simplified in the low temperaturelimit, assuming γ a(d)th ≈
0. Additionally, we consider thefollowing hierarchy of rates γ ≫ Γ A ≫ Γ F and also as-sume that all rates do not depend on energy. This allowsus to obtain simple approximate expressions for the time evolution of the populations N A,Fd,i,a ( E, t ), which dependon the energy E only via the energy dependent initialconditions N A,Fd,i,a ( E, t = 0). These solutions can be writ-ten for donor NCs and independent NCs as: N Ad,i ( E, t ) = N Ad,i ( E, t = 0) exp( − t/τ Ad,i ) , (11) N Fd,i ( E, t ) ≈ − N Ad,i ( E, t ) + N ( E ) exp( − t/τ Fd,i )(12)and for the acceptor NCs as N Aa ( E, t ) ≈ N Ad ( E + E da , t = 0) × (cid:2) exp( − t/τ Aa ) − exp( − t/τ Ad ) (cid:3) , (13) N Fa ( E, t ) = − N Aa ( E, t ) + N ( E + E da ) (cid:2) exp( − t/τ Fa ) − exp( − t/τ Fd ) (cid:3) . (14)Here we use N ( E ) = N Ad,i ( E, t = 0) + N Fd,i ( E, t = 0)and N ( E ) = N Aa ( E, t = 0) + N Fa ( E, t = 0) = 0. Thecharacteristic times are1 τ Ad = Γ A + γ + Γ AET , (15)1 τ Fd = Γ F + Γ FET (16)1 τ Aa,i = Γ A + γ , (17)1 τ Fa,i = Γ F . (18)The equations allow us to reveal the decay times for eachkind of NCs. By comparing these times with the timesextracted from the multi-exponential fit of the experi-mental decay curves we evaluate the rates Γ F,A , Γ
F,AET and γ . We assume that the recombination rate of thedark exciton Γ F and its magnetic field dependence canbe directly associated with the longest decay componentΓ F = 1 /τ Fa,i obtained by fitting the experimental decaycurves and shown by the black diamonds in Figs. 9 and10. One sees that this rate indeed very weakly depends onthe energy but increases nearly by a factor of two with in-creasing magnetic field up to 15 T. This increase is causedby the fact, that despite of the cubic symmetry of theCdTe crystal lattice, the NCs possess an anisotropic axisrelated to their nonspherical shape. As a result, a mag-netic field having nonzero projection on the anisotropyaxis mixes the bright and dark exciton states similar tothe well known situation in hexagonal CdSe NCs. Furthermore, the bright exciton recombination rateΓ A = 0 . − and the value of the bright-dark split-ting ∆ E AF = 2 . We determine the relaxation rate γ and its magneticfield dependence from the bright exciton lifetime τ Aa,i at E a = 1 .
93 eV (the green triangles in Fig. 10) as γ = 1 /τ Aa,i − Γ A . The energy transfer rate Γ FET can befound from the difference of the dark exciton lifetimesΓ
FET = 1 /τ Fd − /τ Fa,i . However, attempts to estimate2 multi-exponential fit rate equations
B γ Γ F Γ FET Γ F Γ FET w (T) (ns − ) (ns − ) (ns − ) (ns − ) (ns − )0 0.45 0.004 0.0035 0.055 0.235 0.72 0.005 0.023 0.0040 0.055 0.1710 1.0 0.007 0.069 0.0065 0.070 0.1315 1.3 0.009 0.090 0.0085 0.090 0.11TABLE II: Magnetic–field dependent parameters determinedfrom the analysis of the multi-exponential fit and from thesimulation of the PL dynamics with the rate equations. Γ FET is the energy transfer rate from the dark exciton state, Γ F isthe recombination rate of the dark exciton state, and γ isthe relaxation rate from the bright to the dark exciton statesat T = 0 K (taken to be the same for all donor and acceptorNCs). The parameter w is the fraction of bright excitons inindependent NCs at time t = 0. this rate from the decay components at the high energy(donor) side of the spectrum are complicated by the factthat there are more components than just one in highmagnetic field and their rates depend on the spectral po-sition, see Fig. 9(b). For this reason we use the rise rates(the open squares in Figs. 10 and 9(b)) observed at thelow energy side of the spectrum at B ≥ τ Fd averaged over the spectral position and toobtain the magnetic field dependence of the Γ FET . Thedetermination of the energy transfer rate Γ
AET from thebright exciton is more difficult because of its short life-time. We estimate that the energy transfer from thebright exciton state is faster than the relaxation to thedark exciton state in the donor NCs. The parametersdetermined from the analysis of the multi-exponential fitfor magnetic fields of B = 0, 5, 10 and 15 T are summa-rized in Table II. B. Shift of the CW PL spectrum due to thenonradiative energy transfer
The PL spectrum measured at time t = 0 can be ap-proximated by a Gaussian form with a peak energy at E = E : I ( E, t = 0) = Iσ √ π exp (cid:20) − ( E − E ) σ (cid:21) , (19)where I is the total spectrally-integrated intensity at t =0 and σ corresponds to the linewidth at half maximumaccording to 2 σ √ A and Γ F as well as their relative populationsat t = 0 we assume that the shape of the time-resolvedPL spectrum at t = 0 reflects the initial energy dispersionof the excitons populations: I ( E, t = 0) ∝ N ( E ).To simulate the temporal shift of the CW PL peakposition compared to the peak position at time t = 0we solve the system of rate equations in the steady state regime. Considering the low temperature limit and con-stant generation rates G d,i = GN , i ( E ) we obtain thefollowing solutions for the populations in the differentkinds of NCs: N Ad,i ( E ) = 12 Gτ Ad,i N ( E ) , (20) N Fd,i ( E ) = τ Fd,i (cid:2) GN ( E ) + γ N Ad,i ( E ) (cid:3) , (21) N Aa ( E ) = τ Aa (cid:2) Γ AET N Ad ( E d ) + Γ FET N Fd ( E d ) (cid:3) , (22) N Fa ( E ) ≈ N Aa ( E ) τ Fa τ Aa . (23)where E d = E + E da . The final equation describing theCW spectrum is rather cumbersome but can be simpli-fied by taking into account the ratio of bright and darkexciton lifetimes. From Eqs. (20-23) one can see that thepopulations of the bright and dark excitons are propor-tional to their characteristic lifetimes. As the lifetime ofthe dark excitons is two orders of magnitude longer thanthe bright exciton lifetime we can neglect the contribu-tion of the bright excitons. In this case a simple equationfor the shift of the CW spectrum can be obtained: I CWPL ( E ) = I ( E, t = 0) [1 − K ET T d ( E )] , (24) T d ( E ) = f d ( E ) − f d ( E + E da ) N ( E + E da ) N ( E ) , (25)where K ET = Γ FET Γ FET + Γ F (26)describes the efficiency of the energy transfer processfrom the dark exciton state. The physical meaning ofthe transfer function T d is clearly seen: the excitationis transferred from the spectral region T d ( E ) > T d ( E ) < f d ( E ) a Gaussian form withthe peak at E + E da . To keep the energy distance E da atthe same (average) value over the spectrum, we allow thedispersion σ d of the f d ( E ) to be larger than σ . Compari-son of the CW spectrum with the time-resolved spectrumat t = 0 allows us to determine the dispersion σ d = 90meV and the average value of E da = 106 meV. The com-parison of the exact solution with the solution given byEq. (24) shows no difference. Physical meaning is thatthe fast relaxation of excitons from the bright state withthe rate γ results in the accumulation of excitons in thedark state. In case of the CW excitation this is equivalentto the direct generation of dark excitons only.In the absence of the bright excitons in the donor NCsthe shift of the PL peak in the CW regime is determinedby the energy transfer efficiency K ET from the dark ex-citons. For modeling the spectral shift in Fig. 13(a) weused K ET ≈ . B = 15 T with the ratesgiven in Table II. Figure 13(b) shows the dependencies ofthe PL maximum shift E − E m on the ET efficiency K ET FIG. 13: (a) Modeling of the PL spectra: (1) Experimentallymeasured time-resolved PL spectrum of the 3.4 nm NCs at t =0; (2) Gaussian-shaped time-resolved PL spectrum I ( E, t =0) with the peak at E ; (3)Experimentally measured CW PLspectrum with the peak at E m ; (4) Donor probability function f d ; (5) Modeled CW PL spectrum given by the solution ofthe rate equations in the steady-state regime. (b) Modeleddependence of the PL shift maximum E − E m on the energytransfer efficiency K ET . for different values of the ratio σ d /σ . One sees, that inthe range of large K ET close to unity, the PL maximumshift E − E m changes insignificantly. We remind also,that the experimentally measured unpolarized CW spec-tra in different magnetic fields differ insignificantly as well(compare, for example, the CW spectra for B = 0 (theblack line) and B = 15 T (the red line) in Fig. 1(a)).The nonlinear dependence of E − E m on K ET can beobtained from Eq. (24) as E − E m = ∆( E m ) K ET , (27)∆( E ) = σ N ( E ) ∂ [ N ( E ) T d ( E )] ∂E . (28)Note, that the recombination rate Γ F in Eq. (26) mayinclude also the nonradiative decay path other than theenergy transfer. The presence of fast nonradiative re-combination could prevent the observation of the spectralshift of the PL line caused by the energy transfer due todecrease of the K ET value. It is not the case for the stud-ied samples, where K ET very close to unity is evaluated. To simplify the following modeling of the recombinationdynamics we neglect the nonradiative mechanisms otherthan ET and assume hereafter the recombination ratesto be purely radiative: Γ A,F = Γ radA,F . C. Spectral dependence of the recombinationdynamics
To simulate the recombination dynamics of the NCensemble, we combine the solution of the system of rateequations for a donor-acceptor NC pair in the transientregime with the determined probability function f d ( E )for the energy transfer. However, the initial conditionsat t = 0 for the bright excitons, N Ad,i ( E, t = 0), andthe dark excitons, N Fd,i ( E, t = 0), should be determinedfirst. The relative populations of the bright and the darkexcitons, N Ad,i ( E, t = 0) and N Fd,i ( E, t = 0), and their cor-responding contributions to the PL may depend on theconditions of excitation and detection. For photoexcita-tion with short laser pulses, the relaxation, recombina-tion and energy transfer processes start simultaneously.For our simulations at all energies, we choose as the ini-tial time t = 0 the time when the initial growth of thePL intensity after the pumping pulse turns into a decay.Even if the exciton relaxation to the ground state occurswith an equal probability to the dark and bright states,their populations at t = 0 might be not equal. The reasonis that the pulse duration (see Appendix) is comparablewith the relaxation time between the bright and dark ex-citon states and with the energy transfer rate from thebright excitons. Hence, we introduce the initial condi-tion for the bright exciton population in the independentNCs as N Ai ( E, t = 0) = wN ( E ) and consider w as anenergy independent parameter. The additional growth ofthe PL intensity after t = 0 is observed at the low energyside of the spectrum at 4 . N Ad ( E, t = 0) = 0.Using these initial conditions, a consistent modelingof the PL decay I ( E, t ) for NCs with emission energiesat E = E d = 2 .
04 eV and E = E a = 1 .
93 eV can beachieved. This modeling allows us to refine the values ofthe rates Γ F and Γ FET by comparing the simulated decaycurves with the experimental data measured at T = 4 . B = 0 −
15 T at the emission ener-gies E a = 1 .
93 eV and E d = 2 .
04 eV, see Figs. 7(d)and 7(f). The refined parameters used in the modelingare listed in Table II. The results of the calculations arepresented in Fig. 14. One sees that a good agreementwith the experimental data is achieved for the PL de-cay of the NCs at E a (see the red curves). For the PLdecay at E d (blue curves) the difference between the sim-ulated and the experimental decay curves increases withthe increasing magnetic field. Apparently, this difference4 FIG. 14: Experimental data (the black curves) and calcula-tions of the PL dynamics of acceptor NCs at E a = 1 .
93 eV(the red curves) and donor NCs at E a = 2 .
04 eV (the bluecurves) at T = 4 . B = 0, 5, 10 and 15 T. The calcula-tion results were achieved from solution of the rate equationsEqs. (8) and (10) using the parameters listed in the Table II.The green curve for B = 15 T corresponds to the solution ofthe rate equations with an additional non radiative process(see the description in text). is caused by neglected additional non radiative processes(for example, additional energy transfer to NCs otherthan those emitting at E a = 1 .
93 eV), which are indi-cated by the additional decay times shown in Fig. 9(b)for the high energy emission of the PL in a magnetic fieldof B = 15 T. Accounting for the additional process witha decay rate of 0 .
035 ns − corresponding to the orangecircles in Fig. 9(b) for the NCs at E d = 2 .
04 eV allowsus to simulate the decay curves (see the green curve inFig. 14(d) with better accuracy.Using the refined parameters for the recombinationand energy transfer rates we can also simulate the PLdynamics of the acceptor NCs at different temperatures.The calculations results at energy E = E a = 1 .
93 eVfor magnetic fields of B = 0, 10 and 15 T are shownin Fig. 15. The effect of the temperature is due to theincrease of the bright exciton state population with in-creasing temperature.Constructing the time-dependent intensities I ( E, t )obtained from Eq. (5) using the solutions of the rate equa-tions in the time-resolved regime for all energies E , it ispossible to model the time evolution of the spectrum as awhole and to describe the time evolution of the PL max-imum E ( t ). Using the approximate analytical solutionsgiven by Eqs. (12,14) and neglecting the initial popula-tion of the bright excitons in the donor NCs we obtainthe following expression: E − E ( t ) = ∆[ E ( t )][1 − exp( − t Γ FET )] , (29)where ∆[ E ( t )] given by Eq. (28) depends on energy andthus on time. In Eq. (2) used for fitting the experimen-tally observed temporal shift E ( t ) in Fig. 4, two energy N o r m a li z ed P L I n t en s i t y B = 0 T15 K4.2 K10 K N o r m a li z ed P L I n t en s i t y B = 10 T15 K4.2 K10 K15 K4.2 K N o r m a li z ed P L I n t en s i t y B = 15 T
Time (ns)
10 K15 K4.2 K10 K15 K4.2 K10 K
FIG. 15: Modeling of the PL dynamics of the 3.4 nm NCsat energy E = E a = 1 .
93 eV for the temperatures T = 4 . B = 0 (a), 10 T (b) and 15 T(c). The parameters used for the modeling are given in textand in Table II. shifts ∆ E , and two characteristic shift rates Γ ∆ E , were used. It is clear from Eq. (29), that the time evo-lution of the PL maximum can not be described by asingle exponential function even for the case when onlyone type of the energy transfer takes place. We can asso-ciate the slow component Γ ∆ E used in the fit in Fig. 4according to Eq. (2) with the energy transfer rate fromthe dark exciton Γ FET . The fast component Γ ∆ E is mostprobably caused by the energy dependence of the relax-ation rate from the bright to the dark exciton states γ or by the energy transfer from the bright exciton statethat is neglected in our simulations.5 VI. DISCUSSION AND CONCLUSIONS
From the time-resolved PL data it is clearly seen thatthe typical time scales during which the energy shift ofthe PL maximum and the growth of the PL intensityat the low energy part of the spectrum take place aresignificantly longer than the lifetime of the bright exci-tons. This fact points on the important role of the darkexciton, but does not provide any insight on the under-lying mechanisms. According to the experimental dataand modeling results the reason of the prominent role ofthe dark exciton in the ET process is that the initial op-tical pumping is accompanied by the relaxation processfrom the bright to the dark state and by the fast energytransfer process from the bright state. As a result, whenwe start observation at the certain conditional moment t = 0, the populations of the dark and bright excitonsare already redistributed and the bright exciton is nearlydepopulated in the donor NCs at low temperatures.As mentioned above, the FRET occurs via dipole-dipole interaction. Therefore, the efficiency of FRETdepends on the oscillator strength of the dipoles. In col-loidal NCs the dark exciton state is not completely dark- it has some finite dipole moment and participates inthe radiative recombination even at zero magnetic fielddue to the admixture of the bright exciton state. Thesame admixture allows the dipole-dipole energy transferfrom or to the dark exciton state in a zero magnetic field.An external magnetic field mixes additionally the brightand the dark exciton states, thus increasing the dipolemoment µ of the dark exciton and its radiative rate. This additional admixture leads to the enhancement ofthe energy transfer rate from the dark exciton state.The enhancement of the energy transfer by the mag-netic field can be seen already from Fig. 4(b). The shiftof the peak position during the first 70 ns at B = 15 T(51 meV) is larger than that at 0 T (40 meV). Since theshift of the peak position is related to the energy transfer,we can conclude that the energy transfer is enhanced bythe magnetic field. The importance of the energy trans-fer from the dark exciton state and its acceleration in themagnetic field is directly demonstrated by the simulationof the PL decay curves for a pair of donor-acceptor NCs.By adjusting the rates of excitons for a better matchingof the simulated decay curves with the experimental de-pendencies of these rates on magnetic field, refined rateswere achieved.In Fig. 16 the dependencies of the recombination rateΓ F and the ET rate Γ FET on magnetic field are presented.The black circles show the rates obtained directly fromthe multi-exponential fitting and the red circles give therates obtained from fitting the experimental PL decayswith the solution of the rate equations. One sees that therecombination rates obtained by both methods almostcoincide. From the results of the modeling a significantcorrection to magnetic field dependence of the Γ
FET rate(red triangles versus black triangles in Fig. 16(b)) in lowmagnetic fields is obtained. According to the modeling of F ( n s - ) (a) (b) F E T ( n s - ) (c) N o r m a li z ed r a t e s F and F E T Magnetic Field (T)
FIG. 16: Magnetic field dependence of (a) the radiative re-combination rate Γ F and (b) the energy transfer rate of thedark exciton state Γ FET , determined from fitting the experi-mental decay curves at the energy E = E a = 1 .
93 eV using amulti-exponentional form (the black symbols) and using thesolution of the rate equations (the red symbols). (c) Magneticfield dependence of the normalized rates Γ
FET ( B ) / Γ FET (0) andΓ F ( B ) / Γ F (0), determined by using the solution of the rateequations. The dashed and solid lines give fits with the mag-netic field strength dependencies Γ F ( B ) / Γ F (0) = 1 + ( B/ and Γ FET ( B ) / Γ FET (0) = 1 + ( B/ + ( B/ , respectively. the PL decays of donor and acceptor NCs, the ET rate isconstant in magnetic fields up to 6 T (the red triangles inFig. 16(b)). The magnetic field dependencies of the nor-6malized rates are fitted by Γ F ( B ) / Γ F (0) = 1 + ( B/ and Γ FET ( B ) / Γ FET (0) = 1 + ( B/ + ( B/ . The B dependence for the dark exciton recombination rate cor-responds to the linear in B increase of the dark exci-ton dipole moment µ ∝ B , caused by the magnetic-field-induced admixture of the bright exciton, so that∆Γ F ∝ µ ∝ B . When the FRET occurs from the darkto the bright exciton state, its rate will be enhanced bythe magnetic field according to Eq. (1) as ∝ µ ∝ B .In the case of FRET between two dark exciton states,its rate will be enhanced by the magnetic field accordingto Eq. (1) as ∝ µ µ ∝ B . Therefore, the B and B dependencies for the energy transfer rate show that theenergy transfer from the dark exciton state in the donorNC may take place to both bright and dark exciton statesin the acceptor NC.The rate of the energy transfer from the dark excitonstate Γ FET ( B ) is enhanced in the magnetic field. Thisenhancement is evidenced in the time-resolved studiesof the PL dynamics and the time evolution of the PLmaximum in the ensemble (see Fig. 4(b)). However, theshift of the PL maximum in the CW spectrum, E − E m , does not change in the external magnetic field. Thisfact is related to the nearly constant value of the energytransfer efficiency K ET in the magnetic field and to theweak nonlinear dependence of E − E m on K ET in therange of large K ET values (see Fig. 13(b)). The effect ofthe magnetic field on the CW spectrum might becomemore significant in the range with smaller K ET values.It is worth to remind that the energy transfer rateΓ ET ∝ ( R /R da ) (see Eq. (1)) decreases as the sixthpower of the distance between the donor and acceptorNCs. Here R is the characteristic F¨orster radius, corre-sponding to the FRET efficiency K ET = ( R /R da ) / [1 +( R /R da ) ] = 0 .
5. That is why the energy transfer effi-ciency is large in the areas with high density of NCs anddecreases strongly with decreasing NC density. Thereforethe areas with higher integral PL intensity correspond toareas with larger K ET values and vice versa. The cal-culated dependence of the PL maximum shift E m − E on K ET in Fig. 13(b) thus explains the correlation be-tween the value of the PL maximum shift and the inte-gral PL intensity. For example, from the PL maximumshift observed in two areas (see Fig. 2) we can estimatethe change of K ET from 0.9 in the high density area to0.5 in the low density area. This corresponds to the ratioof the R da values in the two areas of about 1.5. In thecase when the NCs form only one layer on the substrate, this would correspond to the PL intensity ratio betweenthe high and low density areas of about 2.25. This agreeswell with the 2.5 times ratio shown in Fig. 2(a). Explor-ing further the modeling assumption R da ≈ d in the highdensity area, we can estimate the F¨orster radius in theensemble of the CdTe NCs as R ≈ − T d ( E ) constructed from the initialdistribution function N ( E ) and the donor probabilityfunction f d ( E ) according to Eq. (25). For the parame-ters used in Fig. 13, T d ( E ) < E < .
02 eV. There-fore, the effect of the additional PL rise caused by theenergy transfer might be observed already for energiesbelow 2.02 eV. Indeed, application of the external mag-netic field allows us to observe this effect not only at lowerenergy part of the spectrum, but also at the centrum ofthe CW spectrum at 1.99 eV as can be seen in Fig. 7(e).In conclusion, CdTe colloidal NCs have been studiedby time-resolved photoluminescence in external magneticfields. We prove that the spectral diffusion observed inemission spectra is induced by the F¨orster energy trans-fer. The energy transfer rate Γ ET of an ensemble of ran-domly oriented CdTe NCs can be enhanced by a magneticfield. The fast relaxation of excitons from the bright todark state as well as the admixing of the bright to thedark exciton states caused by the magnetic field resultsin a dominant role of the dark excitons in the FRET atlow temperatures. Acknowledgments
The authors are thankful to R. A. Suris, Al. L. Efrosand A. N. Poddubny for helpful discussions and to D. N.Vakhtin for the help with developing the C ++ code. Thework was partly supported by the Deutsche Forschungs-gemeinschaft and the Russian Foundation of Basic Re-search in the frame of the ICRC TRR 160, by the Merca-tor Research Center Ruhr, by the Russian Foundation forBasic Research (Grant No. 13-02-00888), by the Govern-ment of Russia (project number 14.Z50.31.0021, leadingscientist M. Bayer), and by the Research Grant Councilof Hong Kong S.A.R. (project T23-713/11). V. I. Klimov (Ed.),
Semiconductor and Metal Nanocrystals (Marcel Dekker, New York, 2004). A. L. Rogach (Ed.),
Semiconductor Nanocrystal QuantumDots (Springer, Wien, 2008). D. V. Talapin, J. S. Lee, M. V. Kovalenko, and E. V.Shevchenko, Prospects of colloidal nanocrystals for elec-tronic and optoelectronic applications, Chem. Rev. , 389 (2010). A. L. Rogach, N. Gaponik, J. M. Lupton, C. Bertoni,D. E. Gallardo, S. Dunn, N. L. Pira, M. Paderi, P.Repetto, S. G. Romanov, C. O’Dwyer, C. M. S. Torres,and A. Eychm¨uller, Light-emitting diodes with semicon-ductor nanocrystals, Angew. Chem. Intern. Ed. , 6538(2008). I. L. Medintz, H. T. Uyeda, E. R. Goldman, and H. Mat-toussi, Quantum dot bioconjugates for imaging, labellingand sensing, Nature Materials , 435 (2005). P. V. Kamat, Quantum dot solar cells. Semiconductornanocrystals as light harvesters, J. Phys. Chem. C ,18737 (2008). S. R¨uhle, M. Shalom, and A. Zaban, Quantum-dot-sensitized solar cells, ChemPhysChem 11, 2290 (2010). doi:10.1002/cphc.201000069. C. d. M. Donega, Synthesis and properties of colloidalheteronanocrystals, Chemical Society Reviews , 1512(2011). N. Cicek, S. Nizamoglu, T. Ozel, E. Mutlugun, D. U.Karatay, V. Lesnyak, T. Otto, N. Gaponik, A. Eych-muller, and H. V. Demir, Structural tuning of color chro-maticity through nonradiative energy transfer by inter-spacing CdTe nanocrystal monolayers, Appl. Phys. Lett. , 061105 (2009). T. Franzl, D. S. Koktysh, T. A. Klar, A. L. Rogach, J.Feldmann, and N. Gaponik, Fast energy transfer in layer-by-layer assembled CdTe nanocrystal bilayers, Appl. Phys.Lett. , 2904 (2004). C.-H. Wang, C.-W. Chen, C.-M. Wei, Y.-F. Chen, C.-W.Lai, M.-L. Ho, and P.-T. Chou, Resonant Energy Transferbetween CdSe/ZnS Type I and CdSe/ZnTe Type II Quan-tum Dots, J. Phys. Chem. C , 15548 (2009). A. L. Rogach, Fluorescence energy transfer in hybrid struc-tures of semiconductor nanocrystals, Nano Today , 355(2011). S. Chanyawadee, R. T. Harley, D. Taylor, M. Henini, A. S.Susha, A. L. Rogach, and P. G. Lagoudakis, Efficient lightharvesting in hybrid CdTe nanocrystal/bulk GaAs p-i-nphotovoltaic devices, Appl. Phys. Lett. , 233502 (2009). S. Chanyawadee, R. T. Harley, M. Henini, D. V. Talapin,and P. G. Lagoudakis, Photocurrent enhancement in hy-brid nanocrystal quantum-dot p-i-n photovoltaic devices,Phys. Rev. Lett. , 077402 (2009). I. L. Medintz, A. R. Clapp, H. Mattoussi, E. R. Gold-man, B. Fisher, and J. M. Mauro, Self-assembled nanoscalebiosensors based on quantum dot FRET donors, NatureMaterials ,630 (2003). W. K. Bae, S. Brovelli, and V. I. Klimov, Spectroscopic in-sights into the performance of quantum dot light-emittingdiodes, MRS Bulletin , 721 (2013). A. L. Efros, M. Rosen, M. Kuno, M. Nirmal, D. J. Norris,and M. Bawendi, Band-edge exciton in quantum dots ofsemiconductors with a degenerate valence band: Dark andbright exciton states, Phys. Rev. B , 4843 (1996). C. R. Kagan, C. B. Murray, and M. G. Bawendi, Long-range resonance transfer of electronic excitations in close-packed CdSe quantum-dot solids, Phys. Rev. B , 8633(1996). C. R. Kagan, C. B. Murray, M. Nirmal, and M. G.Bawendi, Electronic energy transfer in CdSe quantum dotsolids, Phys. Rev. Lett. , 1517 (1996). S. A. Crooker, J. A. Hollingsworth, S. Tretiak, and V. I.Klimov, Spectrally resolved dynamics of energy transfer inquantum-dot assemblies: Towards engineered energy flowsin artificial materials, Phys. Rev. Lett. , 186802 (2002). M. Achermann, M. Petruska, S. A. Crooker, and V. I.Klimov, Picosecond energy transfer in quantum dotLangmuir-Blodgett nanoassemblies, J. Phys. Chem. B , 13782 (2003). J. Miyazaki, S. Kinoshita, Site-selective spectroscopic study on the dynamics of exciton hopping in an array ofinhomogeneously broadened quantum dots, Phys. Rev. B , 035303 (2012). F. Xu, X. Ma, C. Haughn, J. Benavides, Efficient excitonfunneling in cascaded PbS quantum dot superstructures,J. ACS Nano , 9950 (2011). L. V. Poulikakos, F. Prins, W. A. Tisdale, Transition fromthermodynamic to kinetic-limited excitonic energy migra-tion in colloidal quantum dot solids, J. Phys. Chem. C ,7894 (2014). G. M. Akselrod, F. Prince, L. V. Poulikakos, E. M. Y. Lee,M. C. Weidman, A. J. Mork, A. P. Willard, V. Bulovic, andW. A. Tisdale, Subdiffusive exciton transport in quantumdot solids, Nano Letters , 3556 (2014). A. J. Mork, M. C. Weidman,F. Prins, and W. A. Tisdale,Magnitude of the F¨orster radius in colloidal quantum dotsolids, J. Phys. Chem. C 118, 13920 (2014). D. G. Kim, S. Okahara, M. Nakayama, and Y. G. Shim,Experimental verification of F¨orster energy transfer be-tween semiconductor quantum dots, Phys. Rev. B ,153301 (2008). S. F. Wuister, R. Koole, C. de Mello Donega, and A.Meijerink, Temperature-dependent energy transfer in cad-mium telluride quantum dot solids, J. Phys. Chem. B ,5504 (2005). R. Osovsky, A. Shavel, N. Gaponik, L. Amirav, A. Ey-chm´uller, H. Weller, and E. Lifshitz, Electrostatic and co-valent interactions in CdTe nanocrystalline assemblies, J.Phys. Chem. B , 20244 (2005). M. Lunz, A. L. Bradley, W.-Y. Chen, V. A. Gerard, S. J.Byrne, Y. K. Gun’ko, V. Lesnyak, and N. Gaponik, In-fluence of quantum dot concentration on F¨orster resonantenergy transfer in monodispersed nanocrystal quantum dotmonolayers, Phys. Rev. B , 205316 (2010). V. Rinnerbauer, H.J. Egelhaaf, K. Hingerl, P. Zimmer,S. Werner, T. Warming, A. Hoffmann, M. Kovalenko, W.Heiss, G. Hesser, and F. Schaffler, Energy transfer in close-packed PbS nanocrystal films, Phys. Rev. B , 085322(2008). O. B. Gusev, A. A. Prokofiev, O. A. Maslova, E. I.Terukov, and I. N. Yassievich, Energy transfer betweensilicon nanocrystals, JETP Letters , 147 (2011). A. L. Rogach, T. A. Klar, J. M. Lupton, A. Meijerink,and J. Feldmann, Energy transfer with semiconductornanocrystals, J. Mater. Chem , 1208 (2009). D. L. Dexter, A theory of sensitized luminescence in solids,J. Chem. Phys. , 836 (1953). Th. F¨orster, Zwischenmolekulare Energiewanderung undFluoreszenz, Annalen der Physik , 55 (1948). J. R. Lakowicz,
Principles of Fluorescence Spectroscopy (Kluwer Academic, New York, 1999). G. Allan and C. Delerue, Energy transfer between semi-conductor nanocrystals: Validity of F¨orsters theory, Phys.Rev. B , 195311 (2007). A. N. Poddubny and A. V. Rodina, Nonradiative andradiative F¨orster energy transfer between quantum dots,arXiv:1511.03557v2, to be published in JETP 122 (3),(2016). M. Furis, J. A. Hollingsworth, V. I. Klimov, and S. A.Crooker, Time- and polarization-resolved optical spec-troscopy of colloidal CdSe nanocrystal quantum dots inhigh magnetic fields, J. Phys. Chem. B , 15332 (2005). D. E. Blumling, T. Tokumoto, K. L. Knappenberger,and S. McGill, Temperature- and field-dependent en- ergy transfer in CdSe nanocrystal aggregates studied bymagneto-photoluminescence spectroscopy, Phys. Chem.Chem. Phys. , 11053 (2012). K. Becker, J. M. Lupton, J. M¨uller, A. L. Rogach, D. V.Talapin, H. Weller, and J. Feldmann, Electrical control ofF¨orster energy transfer, Nature Materials , 777 (2006). R. Vincent, and R. Carminati, Magneto-optical control ofF¨orster energy transfer, Phys. Rev. B , 165426 (2011). S. Y. Kruchinin, A. V. Fedorov, A. V. Baranov, T. S. Per-ova, and K. Berwick, Resonant energy transfer in quantumdots: Frequency-domain luminescent spectroscopy, Phys.Rev. B , 125311 (2008). A. Rodina and Al. L. Efros, Magnetic properties of non-magnetic nanostructures: Dangling bond magnetic po-laron in CdSe nanocrystals, Nano Lett. , 4214 (2015). A. L. Rogach, T. Franzl, T. A. Klar, J. Feldmann, N.Gaponik, V. Lesnyak, A. Shavel, A. Eychmu, Y. P.Rakovich, and J. F. Donegan, Aqueous synthesis of thiol-capped CdTe nanocrystals: State-of-the-art, J. Phys.Chem. C , 14628 (2007). F. Liu, A. V. Rodina, D. R. Yakovlev, A. Greilich, A. A.Golovatenko, A. S. Susha, A. L. Rogach, Yu. G. Kusrayev,and M. Bayer, Exciton spin dynamics of colloidal CdTenanocrystals in magnetic fields, Phys. Rev. B , 115306(2014). J. H. Blokland, V. I. Claessen, F. J. P. Wijnen, E. Groen-eveld, C. de Mello Donega, D. Vanmaekelbergh, A. Mei-jerink, J. C. Maan, and P. C. M. Christianen, Exciton life-times of CdTe nanocrystal quantum dots in high magneticfields, Phys. Rev. B , 035304 (2011). O. Labeau, P. Tamarat, and B. Lounis, Temperature de-pendence of the luminescence lifetime of single CdSe/ZnSquantum dots, Phys. Rev. Lett. , 257404 (2003). L. Biadala, Y. Louyer, Ph. Tamarat, and B. Lounis, Directobservation of the two lowest exciton zero-phonon linesin single CdSe/ZnS nanocrystals, Phys. Rev. Lett. ,037404 (2009). F. Liu, L. Biadala, A. V. Rodina, D. R. Yakovlev,D. Dunker, C. Javaux, J.-P. Hermier, Al. L. Efros,B. Dubertret, and M. Bayer, Spin dynamics of nega-tively charged excitons in CdSe/CdS colloidal nanocrys-tals, Phys. Rev. B , 035302 (2013). D. Thomas, J. Hopfield, and W. Augustyniak, Kineticsof radiative recombination at randomly distributed donorsand acceptors, Phys. Rev. B , A202 (1965). V. Klimov, A. Mikhailovsky, D. McBranch, C. Leatherdale,and M. Bawendi, Quantization of multiparticle Auger ratesin semiconductor quantum dots, Science , 1011 (2000).
Appendix: Fitting function for the PL decay
If the optical excitation with a δ ( t )-shaped short pulsecreates at t = 0 the populations N i ( i = 1 , ..., n ) in the n exciton states, the resulting PL decay with time is givenby the multi-exponential function I δ ( t ) = P i I i ( t ) = P i C i τ i exp( − tτ i ), where τ i is the characteristic decay time,and the amplitudes C i are proportional to the quantumefficiency of the i -th exciton state.The shape of the laser pulse exciting the NCs in ourexperiment was fitted by a Gaussian function I pulse ( t ) = 1 σ √ π exp (cid:20) − ( t − t ) σ (cid:21) , (30)with σ = 0 .
34 ns.We assume at least n = 6 exciton states contributing tothe observed PL (among them one is the upper excitedstate pumped by the laser pulse and one is the long-living trap state giving the background tail). The fittingfunction for the PL decay was obtained as the result ofthe following convolution: I PL ( t ) = Z t −∞ I pulse ( T ) I δ ( t ) dT = X i =0 C i τ i Z t −∞ I pulse ( T ) exp( − ( t − T ) /τ i ) dT = X i =0 A i τ i exp( − tτ i ) (cid:20) (cid:18) t − t √ σ − σ √ τ i (cid:19)(cid:21) , (31)where Erf( x ) is the error function and A i = C i exp(1 + σ / τ i ).In the fitting, the negative amplitude A correspondingto the initial fast rise of the PL intensity, and 1 /τ = 13ns − were fixed to the same values for all decay curves.The small 1 /τ < .
005 ns − was fixed for each decaycurve individually to fit the tale of the curve in the range I PL ( t ) /I PL (0) < . t = 0 was set foreach curve to correspond to the PL maximum, t < | t | < I PL ( t ) = 0. The four rates 1 /τ i > A i ( i = 2 , , ,
5) varied within thefitting procedure which was performed with a speciallydeveloped C ++ code. Positive amplitudes A i > A i <<