This letter presents a detailed study of the electron relaxation lifetimes from the excited p-like state into the ground s-like state via Auger cooling in positively and negatively charged CdSe nanocrystals, where the dependence of Auger cooling rates on size, temperature and carrier-carrier interaction effects is investigated. A nearly two-orders-of-magnitude reduction of Auger rates is

found in small nanocrystals populated by two electrons and one hole at room temperature. This effect increases with decreasing temperature leading to a total lifetime increase of up to 4 orders of magnitude for T ∼ 10 K. Such a giant suppression of Auger cooling rates appears to be a general

property of semiconductor nanocrystals. A similar reduction of Auger rates at room T is found in negatively charged PbSe nanocrystals as well

Califano, M.

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promoting access to White Rose research papers    White Rose Research Online   Universities of Leeds, Sheffield and York http://eprints.whiterose.ac.uk/    This is an author produced version of a paper due to be published in Applied Physics Letters.   White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/3543/    Published paper Califano, M. (2007) Giant suppression of Auger electron cooling in charged nanocrystals, Applied Physics Letters 91 (17), 172114.    eprints@whiterose.ac.uk  Giant suppression of Auger electron cooling in chargedsemiconductor nanocrystalsMarco CalifanoInstitute of Microwaves and Photonics,School of Electronic and Electrical Engineering,University of Leeds, Leeds LS2 9JT, United KingdomAbstractThis letter presents a detailed study of the electron relaxation lifetimes from the excited p-likestate into the ground s-like state via Auger cooling in positively and negatively charged CdSenanocrystals, where the dependence of Auger cooling rates on size, temperature and carrier-carrierinteraction effects is investigated. A nearly two-orders-of-magnitude reduction of Auger rates isfound in small nanocrystals populated by two electrons and one hole at room temperature. Thiseffect increases with decreasing temperature leading to a total lifetime increase of up to 4 orders ofmagnitude for T ∼ 10 K. Such a giant suppression of Auger cooling rates appears to be a generalproperty of semiconductor nanocrystals. A similar reduction of Auger rates at room T is found innegatively charged PbSe nanocrystals as well.PACS numbers: 71.15.-m, 71.55.-i1Understanding and manipulating carrier dynamics in nanomaterials is of paramount im-portance for their device application. An excellent example is the quantum cascade laser [1]where accurate design of the width and composition of the component quantum wells, leadsto an impressive level of control over sub-band populations, carrier dynamics and emissionenergies. All this however comes at the expense of a fabrication process involving expensivehigh precision growth of the semiconductor structure and a lengthy processing of the wafer.Semiconductor nanocrystals (NCs), in contrast, are inexpensive, easy to produce throughchemical synthesis and can be manufactured in large quantities. Furthermore, due to theirchemical flexibility, they can easily be ordered in 3D superlattices, prepared as close-packedfilms or incorporated with high densities into sol gel, glasses or polymers. NCs with sizessmaller than the excitonic Bohr radius (strong confinement regime) offer wide tunability oftheir electronic and optical properties coupled with strong Coulomb interaction between thecharge carriers forced to coexist within a small volume due to spatial confinement. In CdSecolloidal NCs, the intraband transition energies between the p-like and the s-like conductionstates are of the order of hundreds of meV, and can be tuned by varying the NC size tocover most of the infrared (IR) energy window. Since the CdSe bulk optical phonon energyis ~ωLO = 26 meV, according to the adiabatic approximation, the p-like state should havea very long lifetime, as the multi-phonon emission process associated with its decay has asmall probability which decreases as the number of phonon emitted increases (this limitationis known as the phonon bottleneck). This is in contrast with the experimental observationof sub-picosecond lifetimes [2]. The absence of a phonon bottleneck has been explained [3, 4]in terms of an Auger-like process whereby the electron excess energy is transferred to thephotogenerated hole which is excited to deep valence levels. The hole then undergoes a fastrelaxation (with typical times ≤ 1 ps) to the band edge through the denser valence bandenergy ladder. This hypothesis, although still controversial, has gained increasing evidenceculminating with the very recent direct observation of electron-to-hole energy transfer byHendry and co-workers [5], where time-resolved luminescence and terahertz spectroscopyrevealed that the rate of cooling of the photoexcited hole depended critically on the electronexcess energy. Whilst such a fast relaxation would benefit devices based on interband las-ing (emitting or absorbing in the visible range), it is detrimental to applications exploitingintraband transitions, such as IR sources and detectors.Auger cooling (AC) rates have been shown to depend on: (i) the NC size [2, 4] (the2observed lifetimes increased with size remaining however in the sub-picosecond range forNC radii up to 40 Å); (ii) the surface termination (i.e., the nature of the capping groups)[2, 6, 7] (a lifetime increase of up to 2 orders of magnitude in charge-separated, pyridine-capped dots was observed by Sionnest and co-workers [6], whereas the increase observed byKlimov et al. [2, 7] in pyridine-capped dots was of only about one order of magnitude); (iii)the NC temperature [6] (only a weak dependence was observed).A detailed theoretical investigation into the effects of (i) and (iii) on Auger electronrelaxation rates in CdSe NCs in the presence of a single additional unbalanced spectatorcarrier delocalized within the dot (i.e., not trapped in a surface state), is presented usingthe formalism developed in Ref. [4]. The electron relaxation in the different configurationsis schematically presented in Fig. 1. Four CdSe NCs with sizes R1 = 10.3 Å, R2 = 14.6 Å,and R3 = 19.2 Å, and R4 = 28 Å, are considered, spanning the regimes from very strongto intermediate confinement, and a PbSe NC with R = 15.3 Å for comparison. The Augerrates are calculated as [4]:1/τi =Γ~∑n| < i|∆H|fn > |2(Efn − Ei)2 + (Γ/2)2, (1)where |i > and |fn > are the initial and final Auger electronic states, Efn and Ei are theireigen-energies, and ∆H is the Coulomb interaction. For T 6= 0 a Boltzmann average iscalculated over the initial states. The single-particle energy levels εi are calculated using theplane-wave semiempirical pseudopotential method described in Ref. [8], including spin-orbiteffects.If carrier-carrier interaction effects are neglected (single-particle, SP, approximation), thepresence of an additional spectator electron in the conduction band minimum (CBM, thees state in Fig. 1) leading to a reduction of the number of available final states by a factorof 1/2 compared with the neutral NC, results in a decrease of the Auger relaxation rate1/τ(2e, 1h) by the same amount. Similarly, the presence of a hole in one of the two hsstates in Fig. 1, by doubling the number of final states, yields an electron relaxation rate1/τ(1e, 2h) twice as large as in a neutral NC. Both effects are shown in Figs. 2 (b) and (c)(circles), which also illustrate the very weak temperature dependence exhibited by the SPlifetimes.When the interactions between carriers are taken into account within the configurationinteraction (CI) formalism [9], however, large departures from the above intuitive predictions3are found (see Fig. 2 (b) and (c), squares). The AC lifetimes of a negatively charged dotshow a size-dependent increase, compared with the neutral configuration, of about two ordersof magnitude for the smallest NC, and of a factor of ∼ 10 for the largest dot considered.Furthermore they also exhibit a huge temperature dependence, further increasing by 2-3orders of magnitude, depending on the NC size, from room temperature to T = 10 K, withlifetimes of the order of hundreds of ps already at T = 100 K (see Fig. 2 (a), squares). Thepresence of an additional hole leads instead to less dramatic effects on the AC lifetime, bothin terms of magnitude and in terms of temperature dependence (see Fig. 2 (c) and (d)). Therest of this work will therefore concentrate on the more intriguing τ(2e, 1h).The very different behaviour between SP and CI results may seem artificial but a simpleargument can provide a qualitative explanation for it. The three particles in the initial Augerstate (es, ep, hs) can be in either of 8 spin configurations, in four of which the electrons spinsare aligned and in the remaining 4 have opposite direction. The two electrons in the finalAuger state (es, es, hn) are both in the CBM, therefore they have opposite spins. Transitionsbetween initial states with aligned electron spins and final states with opposite spins wouldrequire a spin flip and the corresponding matrix elements in (1) will be small, as Coulombinteractions do not change spin, yielding long lifetimes. On the other hand, transitionsnot requiring the electron spin to flip can have large matrix elements with correspondinglyshort lifetimes. The difference between SP and CI results is due to the different nature ofmulti-particle states in the two approaches. With no interparticle interactions taken intoaccount, the SP initial state consists of a multiplet of 8 degenerate levels each of whichis a pure spin state with the electrons having either the same or opposite spin; the CIinitial states are instead only two-fold degenerate and, most importantly, resulting from asuperposition of different excitonic configurations, are not pure spin states, but receive alllarge contributions from spin-aligned configurations (such contribution is calculated to be43% in the case of the lowest energy multiplet). The two electrons in the final Auger statehave opposite spins in both SP and CI approaches, the only difference being that, again,in the SP description the states are pure spin states (i.e., they are contributed to 100%by a single spin configuration), whereas CI states receive contributions from many differentstates, all of which have, however, two electrons with opposite spins (the hole can be indifferent spin and ”orbital” configurations). It is now clear why the two approaches yieldsuch different lifetimes: Auger SP rates contain matrix elements of transitions between pure4spin states with same electron spins, for each initial state energy, whereas CI matrix elementsonly couple mixed-spin (opposite and aligned) initial states with pure opposite spins finalstates for all transition energies. The large temperature dependence of the CI rates shownin Fig. 2(a) and (b) originates from the large contribution of spin-aligned configurations tothe lowest energy initial state mentioned above, leading to a small rate at low temperature,whilst higher energy levels receive larger contributions from opposite-spin configurations,yielding larger transition rates, which are dominant at room temperature. This is illustratedin Fig. 3. The same arguments can of course not be applied to the case of a NC populatedby a single electron-hole pair, as in that configuration there is no restriction on the spin ofthe particles in the final states. As a consequence CI and SP yield similar AC lifetimes [4].The first question that needs to be addressed now is whether configuration (a) in Fig. 1 isrealistic, i.e., can be achieved in laboratory. Experimentally there are two different ways toexcite an electron to the p state: (i) by resonantly photoexciting it directly into the excitedp state [5] (or to a state above it), or (ii) by photoexciting it first to the ground (s) stateand then re-exciting it using an IR pump pulse [6, 7]. If an unpaired ground state electronis present during procedure (i), it will have a low probability of absorbing the pump photonand will remain in the s state. The configuration considered here will be achieved directly (orafter relaxation of the high-energy electron-hole pair, in the case of a larger photoexcitationenergy). If procedure (ii) is applied, then, for high enough IR pump fluences, the spectatorelectron may be excited to the p level together with the photogenerated electron. Wecalculate a very short lifetime for this configuration, with one of the electrons decaying viaAC to the s state in times of the order of tens of fs, bringing the system in the desiredlong-lived configuration.The next question that needs to be considered regards the alternative decay channels ex-pected to be most effective. Due to the presence of the additional electron, another relaxationmechanism, namely Auger recombination (AR), where an electron-hole pair recombines nonradiatively via energy transfer to a third particle (the electron in the present case), whichis excited to high energy states, becomes possible. AR was both experimentally observed[11] and theoretically predicted [4] to occur on time scales of the order of tens of ps at roomtemperature when two electrons and two holes are present, and is therefore expected to pro-vide an upper limit to the lifetime of the p electron. The fact that only one hole is presenthere (which together with the two electrons forms a negative trion instead of the neutral5bi-exciton investigated in Refs. [11] and [4]), is of no consequence since, as mentioned above,the recombination process is a three-particle process and there is a simple relationship [4]between the lifetime of the state with 2 electrons and 2 holes and that of the states with 2electrons and 1 hole and 1 electron and 2 holes. The important difference, however, is thatboth measurements and theoretical calculations for AR were done in a configuration wherethe two electrons and the two holes occupy the band edge (i.e., are in the respective s states).This turns out to be a fundamental difference with the configuration assumed in the presentwork, where one of the electrons occupies the excited p level. In fact our calculated ARlifetimes in this case are of the order of hundreds of ps at room temperature, therefore largerthan the corresponding AC decay times. This means that AC is more efficient than AR, andour calculations show that p state electron lifetimes of the order of ∼ 100 ps are achievable innegatively charged NCs at temperatures lower than 150 K (see Fig. 2(a)). A slow relaxationcomponent with lifetimes about two order of magnitude longer than the fast component,representing less than 10% of the total bleaching signal for TOPO- or thiol-capped samples,was indeed observed in CdSe NC at room temperature [6], but was attributed to decay incharge-separated quantum dots. However, a similar slow background was also subsequentlyobserved [7] in ZnS-capped NCs, where electron-hole charge separation is inhibited, and wasexplained in terms of accumulation of electrons in some long-lived state. In the light of theresults presented here it may alternatively be interpreted as having been due to a fraction ofthe NCs being negatively charged. In fact it is not uncommon for NC ensembles to containsome percentage of charged dots. Furthermore the lifetimes of such slow decay measured inRef. [6] at 80 K are of the order of a few hundreds of ps, in very good agreement with ourresults in that temperature range (Fig. 2(a)).Finally the generality of the predicted suppression of AC rates found in CdSe NCs isinvestigated by calculating p electron Auger lifetimes in PbSe NCs with R = 15.3. PbSeand CdSe NC have fundamentally different crystal and electronic structure, therefore thereare no a priori simple arguments suggesting any similarity in this effect. A similar increasein the AC lifetime is found in the presence of a spectator electron in the s state at roomtemperature [12] (see Fig. 2b empty squares), suggesting this effect to be a general propertyof semiconductor NCs.In summary, negatively charged CdSe semiconductor NCs with an unpaired electron inthe s state exhibit a 3-4 orders of magnitude reduction of AC rates at low T with a long6lived excited electron state lasting ∼ 100 ps at temperatures as high as 150 K in small dots.I’d like to thank A. Franceschetti and A. Zunger for supplying many codes used in thiswork and for their comments on the manuscript. I also gratefully acknowledge the RoyalSociety for financial support.7[1] J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, M. S. Hybertsen, A. Y. Cho,Phys. Rev. Lett. 76, 411 (1996).[2] V. I. Klimov, J. Phys. Chem. B 104, 6112 (2000).[3] Al. L. Efros, V. A. Kharchenko, and M. Rosen, Solid State Comm. 93, 281 (1995).[4] L.-W. Wang, M. Califano, A. Zunger, and A. Franceschetti, Phys. Rev. Lett. 91, 056404(2003).[5] E. Hendry, M. Koeberg, F. Wang, H. Zhang, C. de Mello Donega, D. Vanmaekelbergh, andM. Bonn, Phys. Rev. Lett. 96, 057408 (2006).[6] P. Guyot-Sionnest, M. Shim, C. Matranga, and M. Hines, Phys. Rev. B 60, R2181 (1999).[7] V. I. Klimov, A. A. Mikhailovsky, D. W. McBranch, C. A. Leatherdale, and M. G. Bawendi,Phys. Rev. B 61, R13 349 (2000).[8] L.-W. Wang and A. Zunger, Phys. Rev. B 51, 17 398 (1995).[9] A. Franceschetti, H. Fu, L.-W. Wang and A. Zunger, Phys. Rev. B 60, 1819 (1999).[10] As in Ref. [4], AC lifetimes are calculated as a function of the electron sp energy splitting∆esp = εep − εes. The values displayed here and referred to in this work represent averages overa range of ∆esp corresponding to a 10% dispersion in the NC size.[11] V. I. Klimov, A. A. Mikhailovsky, D. W. McBranch, C. A. Leatherdale, and M. G. Bawendi,Science 287, 1011 (2000).[12] The temperature dependence in this case is however much weaker (the lifetimes increasing bya factor of 2 between T=300 K and T=10 K) compared to that found for CdSe NCs.8hseshiepNeutral Positively chargedNegatively charged(c)(a) (b)FIG. 1: Schematics of Auger decay in the multi-particle configurations considered in this work: (a)negative trion X−, (b) neutral exciton X, (c) positive trion X+.9110010000τ (2e,1h) (ps)CISP0 50 100 150 200 250T (K)00.10.20.30.4τ (1e,2h) (ps)110010000τ (2e,1h)/τ (1e,1h)I0 50 100 150 200 250 300T (K)00.511.52τ (1e,2h)/τ (1e,1h)R=10.3 Å(a) (b)(d) (c)R=14.6 ÅR=19.2 ÅPbSe R=15.3 ÅR=28.0 ÅFIG. 2: (Color online): The lifetimes of negatively (a) and positively (d) charged CdSe NCs [10]and their relative increase [(b) and (c), respectively], compared to that of neutral NCs, in theconfiguration interaction (CI, squares) and single-particle (SP, circles) approaches, as a function oftemperature (T) for all sizes considered. All SP curves overlap in (b) and (c). In (b) our calculatedCI lifetime as a function of T for a PbSe NC with R=15.3 Å is shown for comparison (emptysquares).101×10-51×10-41×10-31×10-21×10-11×1001×1011×1021×1031/τ i (ps-1 )SPCI0 10 20 30 40 50Initial state index i0.20.40.60.81Boltzman weights300KFIG. 3: (Color online) Comparison between the rates 1/τi(2e, 1h) (upper panel) in Eq. (1), andthe Boltzmann factors at T=300 K (lower panel) calculated, for each initial state i, within the CIand SP approaches (the two sets of results have been offset along the x axis for clarity), for a CdSeNC with R = 14.6 Å. The only appreciable Boltzmann weights at T=10 K are those with value=1 at room temperature.11

Giant suppression of Auger electron cooling in charged nanocrystals

Marco Califano

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Giant suppression of Auger electron cooling in charged nanocrystals

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