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January 07, 2015

C 2015 American Chemical Society

A Phonon Scattering Bottleneck forCarrier Cooling in Lead ChalcogenideNanocrystalsPieter Geiregat,†,‡,§ Christophe Delerue,^ Yolanda Justo,‡,§ Michiel Aerts, ) Frank Spoor, )

Dries Van Thourhout,†,§ Laurens D. A. Siebbeles, ) Guy Allan,^ Arjan J. Houtepen, ),‡ and Zeger Hens*,‡

†Photonics Research Group, Ghent University, 9000 Ghent, Belgium, ‡Physics and Chemistry of Nanostructures, Ghent University, 9000 Ghent, Belgium,§Center for Nano and Biophotonics, Ghent University, 9000 Ghent, Belgium, ^IEMN, Département ISEN, UMR CNRS, 59046 Lille Cedex, France, and

)Optoelectronic Materials Section, TU Delft, 2628 BL Delft, The Netherlands

Absorption of photons by a semicon-ductor with an energy exceeding theband gap transition results in the

formation of hot electron�hole pairs thatquickly dissipate their excess free energy,resulting in (quasi) thermalized conductionband electrons and valence band holes. Forphotovoltaic solar energy conversion, thiscooling of the hot electron�hole pair is amajor loss channel that restricts the max-imum conversion efficiency of a single junc-tion solar cell to the Schockley�Queisserlimit.1,2 Harvesting this excess free energyby competing processes such as hot chargecarrier transfer has proven challenging dueto the high cooling rate, which can reach1 ps�1 or more due to electron�phononand/or carrier�carrier interactions.3 On theother hand, good use can be made of thehigh cooling rate for all-optical light mod-ulation, where response times of a few pshave been demonstrated using, e.g., hotcarriers generated by light absorption ingraphene4 or injected in ITO through aplasmonic nanoparticle as a sensitizer.5

Similar approaches of hot electron transferfollowing the optical excitation of surface

plasmons have recently been applied incatalysis,6 photovoltaics,7 photodetection7

and nanoscopic imaging.8

In the field of semiconductor nanocryst-als or quantum dots (QDs), electron�holepair cooling has attracted wide interestespecially since a slowing down of electronand hole cooling due to a phonon bottle-neck was predicted.9 The discrete energylevels in QDs may be separated by severalLO phonon energies, hence requiring multi-phonon emission for cooling, a processwhich is very unlikely. Whereas for speciallydesigned QDs, a slowing of the cooling ofthe conduction band electron from its 1P toits 1S state could be confirmed,10 carriercooling appeared to remain fast in mostQDs with rates still close to 1 ps�1, a resultoften attributed to additional cooling path-ways involving Auger processes or energytransfer to surface ligands.10 Nevertheless,it proved possible to harvest the excessenergy of hot excitons either by hot carriertransfer11,12 or the generation of multipleexcitons.13�15 Both processes involve hotelectron�hole pairs with a substantial ex-cess energy, at least the QD band gap in the

* Address correspondence tozeger.hens@ugent.be.

Received for review November 3, 2014and accepted January 7, 2015.

Published online10.1021/nn5062723

ABSTRACT The cooling dynamics of hot charge carriers in colloidal lead chalcogenide

nanocrystals is studied by hyperspectral transient absorption spectroscopy. We demonstrate a

transient accumulation of charge carriers at a high energy critical point in the Brillouin zone.

Using a theoretical study of the cooling rate in lead chalcogenides, we attribute this slowing

down of charge carrier cooling to a phonon scattering bottleneck around this critical point.

The relevance of this observation for the possible harvesting of the excess energy of hot

carriers by schemes such as multiexciton generation is discussed.

KEYWORDS: PbS . PbSe . quantum dots . transient absorption spectroscopy . electron�phonon interaction .multiple exciton generation

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case of multiple exciton generation, and their effi-ciency therefore depends on competition with coolingof electrons and holes occupying energy levels far fromthe band edges. Most studies of carrier cooling how-ever focus on the changing occupation of states closeto the band edge. Since filling of band-edge states isthe end point of the cooling process, this only gives aview on the rate-determining step of the entire coolingcascade. This corresponds very often to the 1P�1Stransition, which is the final stage of the coolingcascade. As a result, the thus obtained cooling ratesmay not be the most relevant for analyzing the com-petition between cooling and multiple exciton gen-eration or hot carrier transfer.Here, we analyze electron�hole pair cooling in PbS

and PbSe colloidal QDs after excitation with highenergy photons where, by means of white light tran-sient absorption spectroscopy, energy levels through-out the entire Brillouin zone are probed. PbS and PbSeare probably the most widely studied QDs for photo-voltaics and photodetection, showing effective multi-ple exciton generation at thresholds close to thetheoretical limits13,14,16 and yielding the most efficientQD-based solar cells.17 In bulk, both have a direct bandgap at the 4-fold degenerate L point of the Brillouinzone. At higher energy, there are a number of criticalpoints, most notably along the Σ andΔ direction in theBrillouin zone, which add specific, size-dependent

features to the QD absorption spectrum.18�20 Mostnotably, we observe a transient accumulation ofcharge carriers at the critical point along the Σ direc-tion, labeled as Σc, when the pump-photon energyexceeds the energy difference at Σc. We link this cool-ing bottleneck, which slows down carrier cooling to anet rate of ≈1 ps�1 that is typically observed whenmonitoring band-edge states, to the energy levelstructure around Σc and we show that cooling via Σc

is the dominant pathway for high energy chargecarriers. Moreover, the buildup rate of the carrierpopulation at Σc gives a direct way to probe coolingrates of high energy charge carriers, cooling rates thatare most relevant for competing processes such asmultiple exciton generation or hot carrier transfer.

EXPERIMENTAL RESULTS

Optical Properties of Lead-Chalcogenide Quantum Dots. Westart with results obtained on PbS nanocrystal disper-sions synthesized according to a procedure describedby Moreels et al.,23 with average diameters dNC that wedetermined from the peak wavelength of the firstexciton transition as indicated in Figure 1a.21 As shownin Figure 1b, various transitions between valence andconduction band states can be traced back by meansof the second derivative of the absorbance, where nextto thewell-known 1S�1S and 1P�1P transitions a thirdtransition at visible wavelengths stands out. In the case

Figure 1. (a) Overviewof absorption spectra of PbSQDs of different sizes, including (blue, orange, dark red) the three samplesused in this study. For each spectrum, the legend indicates the averagediameter as determined from themaximumof thefirstexciton absorption.21 (b) Absorption spectrumand secondderivative spectrumof 5.65 nmPbSQDswith an assignment of themost relevant absorption features as the 1S�1S, 1P�1P and Σc transition. (c) Representation of the PbS bulk band structurewith an indication of the critical points Σc and Δc along Σ and Δ. (d) Size-dependence of the 1S�1S and the Σc transitionenergy, togetherwith an indication (dashed lines) of the correspondingbulk values22 and the photon energies correspondingto the pump wavelengths used in this study. (e) Time-wavelength map of the differential absorbance near the bandgaptransition for 4.6 nm PbS nanocrystals.

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of PbSe QDs, a similar feature was attributed to elec-tron transitions close to the critical point in the bandstructure along the Σ direction,20 i.e., the point weindicate as Σc (see Figure 1c). Figure 1d demonstratesthat the feature appearing at visiblewavelengths in thesecond derivative spectra of PbS QDs shows weakerconfinement than the bandgap transition and extra-polates with increasing dNC to the Σc energy differenceof 1.94 eV as reported for bulk PbS.22 As this behavior ishighly similar to that of the Σc transition in PbSe QDs,we also attribute this visible feature in PbS QDs to theelectronic transitions around Σc as already suggestedby Cadmeratiri et al.19

Figure 1e shows the change of the absorptionspectrum of 4.6 nm PbS QDs around their bandgaptransition following a 180 fs laser pulse at 700 nm thatcreates on average 0.3 excitons per QD. The changein absorption, defined as the difference in absorbanceΔA = A* � A0 of the dispersion after and beforephotoexcitation, respectively, is represented by a colorcode as a function of probe wavelength and time delayafter the excitation pulse. The figure shows the typicalreduction of the bandgap absorption due to theoccupation of the band-edge states by photogener-ated electrons and holes.24 The buildup of the bleach

signal around the bandgap transition within the first1�2 ps after the pump pulse reflects the final step ofthe cooling of the photogenerated charge carriers tothe band-edge states, a process accompanied by astrong reduction of the sub-bandgap photoinducedabsorption related to intraband transitions.25

Above Band Gap Transient Absorption Spectroscopy. Beingthe result of a cascade of cooling steps via phononemission, the rate of band-edge state filling may notgive an accurate view on the initial cooling dynamics ofthe high energy electron and hole created by thepump pulse, which is the most relevant in view ofcompeting processes such as hot carrier extraction orcarrier multiplication.16 Therefore, we probed the tran-sient absorption at photon energies well above theQD bandgap. As shown in Figure 2a�c, such abovebandgap transient absorption maps are dominatedby photoinduced absorption (ΔA > 0) for all PbSQDs studied when using a pump wavelength of700 nm (pω = 1.77 eV). As outlined in the SupportingInformation (section S1), this photoinduced absorptionresults from a spectral redshift that is largely indepen-dent of the probe wavelength. Following the argu-mentation developed in ref 26, we attribute thisspectral shift to the net Coulomb interaction between

Figure 2. Contourmaps ofΔA illustrating the carrier dynamics of the high energy transitions (i.e., in the visible spectrum) for3 different sizes of PbSquantumdots: (a) 3.55, (b) 5.65 and (c) 10.25nm. Sampleswerepumpedat 700nm (top row) and450nm(bottom row). Color scale goes from negative (blue) to positive (red). The black dotted line is the second derivative of thecorresponding linear absorption spectrum. Theminimumcorresponds to the onset of the absorption of theΣ-point transitions.

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the exciton(s) resulting from the pump-pulse and theexciton newly created by the probe pulse.

When changing the excitation wavelength to450 nm, i.e., to photon energy well above the Σc

transition, the most notable difference in the transientabsorption maps is the appearance of a short-livednegativeΔA at around 500�600 nm. A similar negativefeature was observed by Cho et al., who used degen-erate pump�probe spectroscopy.27 The occurrence ofthis bleach feature in the first 1�2 ps after photoexci-tation, i.e., the typical cooling time, points to thetemporal blocking of electronic transitions due to theaccumulation of cooling charge carriers in particularelectronic states. The dashed line added to eachtransient absorption map in Figure 2 represents thesecond derivative of the respective absorption spectra,where the dip corresponds to the position of the Σc

transition (see Figure 1b). It can be clearly seen that foreach QD sample analyzed the wavelength range of thetransient bleach corresponds to that of the Σc transi-tion. We thus conclude that the cooling of chargecarriers can be shortly delayed in the electronic statesaround the critical point along the Σ direction in theBrillouin zone. As outlined in the Supporting Informa-tion (section S2), a similar feature is observed in thecase of PbSe QDs pumped at photon energies abovethe Σc transition.

Transient Bleach Kinetics at Σc. Taking vertical cutsthrough the 2D maps of Figure 2 at the maximum ofthis transient bleach feature, we obtain the kinetictraces displayed in Figure 3 for the 3.55 nm PbSnanocrystals. We clearly observe that excitation abovethe Σc-transition (at 450 nm pump) leads to a tracedistinctly different from excitation below Σc (at700 nm). While pumping at 700 nm only leads to therapid buildup of a photoinduced absorption, a bleachfeature develops and vanishes for the sample pumpedat 450 nm on a subpicosecond time scale. If weinterpret the transient absorption at 450 nm as com-posed of a bleach feature due to charge carrier

accumulation and photoinduced absorption due toexciton�exciton interaction, the trace recorded witha 700 nm pump can be seen as a measure for thephotoinduced absorption background. The resultingnet bleach, obtained by subtracting the (properlyrescaled) 700 nm from the 450 nm pump trace hasbeen added to Figure 3a.

We tentativelymodel this net bleach by considering3 different states, representing hot carriers, carriersoccupying states around Σc and cold carriers, wherehot carriers occupy the states around Σc at a rate kinand these cool down further at a rate kout (seeFigure 3b). As a result, the transient exciton populationÆNæΣ(t) around Σc can be described as

ÆNæΣ(t) ¼kin

kout � kinf ÆNæ(e�kint � e�koutt) (1)

Here, ÆNæ represents the average number of excitonscreated per QD by the pump pulse and f is the fractionof excitons cooling via Σc. A fit of the net bleach to eq 1is shown in Figure 3c, where the bleach signal has beenrescaled to represent the occupation probability σ(t)of the states around Σc (σ(t) = ÆNæΣ/fÆNæ). The corre-sponding rates kin and kout have been summarized inTable 1 for the different PbS QDs studied. One seesthat, regardless of the QD size, the rate of coolingtoward and away from Σc amounts to ≈3.8 and2.8 ps�1, which implies that σ(t) reaches a maximum

Figure 3. Kinetic analysis ofΔAof 5.65 nmQDs. (a) Carrier dynamics atΣc for excitationbelow (700 nm, solid black) and above(450 nm, solid blue) the Σc -transition. Empty markers indicate the netto bleach. (b) Schematic depicting simple 3-levelapproach tomodel the observednetto bleach atΣc: inflow from carriers fromhigher energy (hot) and outflow to lower energy(cold) levels. (c) Probability σ(t) to have1 exciton in the ensembleofΣ-states as a function of time (see text) for the 5.65 nmsizePbS QDs. A fit to eq 1 of the experimental data (empty circles) is shown (solid red line).

TABLE 1. Summary of the Rates of Cooling toward (kin)

and Away from (kout) Σc as Obtained by Fitting the

Experimental Traces to eq 1 and the Energy Loss Rate

wE,in as Obtained from the Product of kin and the Energy

Per Charge Carrier to Be Dissipated for Cooling toward Σc

QD dNC (nm) kin (ps�1) kout (ps

�1) wE,in (eV ps�1) wph,in (ps

�1)

PbS 3.55 3.8 2.8 0.76 29PbS 5.65 3.9 2.9 1.2 45PbS 10.2 4.1 3.1 1.5 58PbSe 4.80 1.85 2.7 0.37 44

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value of ≈0.4. We see that passage via the Σ directioncritical point delays the cooling of charge carriers. Bymeans of kin, the 3-level model analysis provides a wayto estimate the initial cooling rate of hot charge carriersand not the final cooling rates which are typicallymeasured when looking at state filling of the band-edge states. Clearly, this initial cooling rate is whatmatters for most process aiming at harvesting the hotexciton energy. Moreover, this rate will also set a limitto applications such as light modulation that couldmake use of the rapid change of optoelectronic prop-erties that come with cooling. It should be noted thatthe figure of ≈4 ps�1 reported here most likely under-estimates the cooling rate since it approaches thereciprocal of the temporal width of the pump pulse.Even so, we find that the cooling rate for high excessenergy electron�hole pairs in PbSe QDs is significantlylower than for PbS QDs.

Transient Bleach Intensity atΣc. A key quantity in eq 1 isthe fraction f of the excited electron�hole pairs thatcool viaΣc. When f≈ 1,most charge carriers experiencea temporal slowing down of their cooling, whichincreases the probability for competing processes toharvest the excess free energy of the electron�holepair. On the other hand, when f, 1, the observation ofa slowdown of the cooling rate merely serves as a wayto estimate the initial hot carrier cooling rate yet haslittle influence on the overall carrier cooling rate.Considering again the valence- and conduction-bandstates aroundΣc as single levels with respective degen-eracies gV and gC provides us with a way to estimatef from the maximum bleach of the absorbance. Withinthis picture, the integrated absorption coefficient(related to the experimentally accessible absorbanceA throughA=RL/ln 10, with L the cuvette length) of theΣc transition can be written as

Rint,Σc¼ gVgCR0,Σc

(2)

Here, R0,Σc represents the average integrated absorp-tion coefficient of an individual transition betweenvalence and conduction band contributing to the Σc

absorbance. If an exciton is present in these levels, theabsorption coefficient changes due to state filling andthe possibility for stimulated emission. The resultingnonlinear absorption coefficient reads

R�int,Σc¼ (gV � 1)(gC � 1)R0,Σc

� R0,Σc(3)

As a result, we can write the normalized change intransmission (bleach) for an average occupation ÆNæ as

ΔRRint,Σc

¼ R�int,Σc� Rint,Σc

Rint,Σc

¼ gV þ gCgVgC

ÆNæΣc(4)

Hence, the maximum normalized bleach after integra-tion becomes

ΔRRint,Σc

¼ gV þ gCgVgC

fσmaxÆNæ (5)

To determine the fraction f of hot carriers coolingviaΣc from the experimental transient absorption data,the degeneracies gV and gC must be known. Sincethere are 12 equivalent Σ directions, the degeneraciesgV and gC should be 24 at least (including spin). On theother hand, gV and gC will be capped by the product ofthe density of states around Σc for bulk PbS and thespectral width of the Σc bleach. In Figure 4, we plot thedensity of states needed according to eq 5 to have f = 1or f = 0.1 together with the above-mentioned lowerand upper limits to the degeneracy, where we used adensity of states of 0.5 eV/PbS unit cell for bulk PbSaround Σc.

28 One sees that for all QD sizes studied, thedegeneracy needed to have f = 1 closely agrees withthat calculated from the bulk density of states while forthe smallest QDs, f = 0.1 corresponds to degeneraciesthat are physically impossible. We thus conclude that asizable fraction of the excitons, if not all excitons, coolvia Σc and thus experience a transient slow-down ofthe cooling rate.

THEORETICAL ANALYSIS

Energy Band Structure of Bulk PbS. To gain a betterunderstanding of the observed reduction of the cool-ing rate, we return to the band structure of bulk PbS,which we calculated within the tight-binding approx-imation. As shown in Figure 1c, the lowest conductionband minima and the highest valence band maximaare located at the 8-fold degenerate L points of theBrillouin zone. Other critical points can be found, forexample, along the Σ ([110]) and Δ ([100]) direction,where in particular around Σc, our results point towarda slowing down of charge carrier cooling. Importantly,bothΣc andΔc are saddle points, meaning that the firstderivatives of the energy Ek with respect to the threecomponents of k vanishes, giving rise to a high densityof states and amarked feature in the optical absorptionspectrum, whereas the band curvature is either posi-tive or negative depending on the direction in theBrillouin zone. For example, near k∑c = (π/a)[1,1,0], theconduction band can be approximated as

Ec, k � Ec,Σcþ p2

2δk2110m110

þ δk2110

m110

� δk2001m001

!(6)

Here, δk = (δk110,δk110,δk001) is defined as the differ-ence between the wavevector k of the actual electronstate and k∑c, projected on the directions indicated bythe subscript (see Figure 5a) and m110 (0.20 m0), m110

(0.17 m0) and m001 (0.80 m0) are the correspondingeffective masses. As shown in Figure 5b, the energydispersion is positive along the [110] (Σcf K) and [110](Σcf X) direction yet it is negative along [001] (Σcf L)(these directions have been indicated in Figure 5a).Moreover, the effective mass is much larger, i.e.,the energy surface is far less curved, along [001] thanalong the two others directions, which feature similar

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curvature. As discussed in the Supporting Information(section S3), this behavior can be traced back to thePb�S ppσ interaction dominating the formation ofenergy bands in PbS. As a result, a similar situationfeaturing an even more pronounced difference ineffective masses exists for the valence band (see Sup-porting Information, section S3).

Carrier Cooling in Bulk PbS. The cooling of hot carriersin polar materials like PbS and PbSe is mainly due toscattering by longitudinal optical (LO) phonons,29

where the matrix element describing the couplingbetween electrons and LO phonons has been derivedby Fröhlich.30,31 Assuming dispersionless LO phononsof energy pωLO (26meV in PbS,32 17meV in PbSe33), weaccordingly define the rate of energy losswE as( pωLO

multiplied by the scattering rate w (in 1/s) for emission(þ) or absorption (�) of a phonon. Using the Fermigolden rule, the energy loss rate for a carrier in a state|kæ can then be determined by summing over allpossible phonon wavevectors q:30

wE(k) ¼ (2πepωLO)2

pV

1ε(¥)

� 1ε(0)

� �∑q

jÆk � qje�iq 3 rjkæj2q2

� :::

f(Nþ 1)δ(Ek�q � Ek þ pωLO) � Nδ(Ek�q � Ek � pωLO])g(7)

Here, N = 1/[exp(pωLO/kT) � 1] is the average numberof phonons at temperature T in a givenmode, ε0 and ε¥are the static and high frequency dielectric constantand V is the crystal volume. The first term in the secondline of eq 7 describes the emission of phonons, thesecond the absorption.

We used eq 7 to calculate the energy loss rate for hotcharge carriers using the bulk PbS electron statescalculated in tight-binding (see Methods section), con-sidering 109 different q vectors in the Brillouin zone.Both intra- and interband processes are considered, yetthe dependence on 1/q2 in eq 7 implies that polarscattering has a much greater role in intraband than ininterband processes.29 Figure 6 represents the resultingrate at 300 K for electrons and holes near Σc. Forelectrons with a relatively high excess energy relativeto E∑c (E� E∑c. pωL) the rate ismore or less constant, inthe eV ps�1 range. This is a typical result for hot carrierrelaxation in polar materials29 and its order of magni-tude agrees with the rate wE,in derived here for carriercooling toward Σc (see Table 1). However, when thecarrier energy is progressively reduced, the rate firstslightly increases but then suddenly drops, within anenergy rangemorenarrow thanpωL, to reach very small,even negative values. Decreasing the energy below

Figure 4. Estimation of fraction f of hot carriers cooling via Σc. (a) Netto bleach spectrum used to calculate the integratedabsorption spectrum δR. (b) Bulk prediction and lower limit for number of states involved in the Σc transition. Markersindicate the experimentally obtained values for f = 0.1 (emptymarkers) and f = 1 (filled markers). The coincidence of the f = 1markers with the bulk number of states indicates that f is of the order of unity, indicating that also the fraction of the hotcarriers cooling via Σc is of the order of unity.

Figure 5. (a) Representation of (left) the Brillouin zone of the fcc lattice with an indication of the most prominent symmetrypoints and directions and (right) the local coordinate system introduced to analyze the dispersion relation around Σc. (b) 2Dplot of the energy dispersion as a function of δk110 and δk001 showing pronounced positive curvature (light effective mass)along (110) and a relatively small negative curvature (heavy effective mass) along (001).

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E∑c results again in a saturation of the cooling rate, yetat a smaller value than at high energy. The behavior forholes near Σc is similar, but even more pronounced.Negative rates are predicted in an energy window ofabout 70 meV at 300 K, which means that the hole hasa higher probability to move up, i.e., absorb a photon,than down. This is a direct consequence of the almostflat valence-band energy surface in the [001] direction,which implies a very high hole effective mass in thatdirection (see Supporting Information, section S3).

These findings can be understood more qualita-tively by returning to eq 7, which shows that twoelements favor high loss rates. First, nonzero contribu-tions to the loss rate are only obtained for thosephonons where the phonon energy and momentummatches the energy difference between the initial anda final electron state. Hence for a given phonondispersion relation, the larger the electron density ofstates, the higher the cooling rate. Second, having q2 =|q|2 in the numerator implies that high loss rates resultfrom electron scattering by phonons that meet energyand momentum conservation with small magnitudephonon or scattering wavevectors q, i.e., when thegradient rkE is large (see Supporting Information,section S4). Returning to the dispersion relation aroundΣc, it follows that both elements favor rapid coolingtoward Σc yet hamper cooling away from Σc. Whilethere are two directions of positive curvature with loweffective masses, there is only one of negative curva-ture with a high effective mass. This results in a higherdensity of states and steeper energy gradients abovethan below Σc, which makes that the change of direc-tion of the scattering wavevector necessary to pass Σc

constitutes a significant bottleneck for cooling ofelectrons and holes toward the band edges (seeSupporting Information, section S4).

On the basis of the picture of preferential cooling ofelectrons along �rkE, i.e., the direction of steepest

descent, it may seem surprising that a sizable fractionof the charge carriers experience a slowing of theircooling rate. Indeed, since the conduction band energydispersion around Σc corresponds to a hyperbolicparaboloid, with two directions of high positive curva-ture and one of low, negative curvature, only chargecarriers created at the principle axes with positivecurvature will cool toward Σc. However, the strongdifference in curvature makes the conduction bandenergy dispersion resemble a gutter (see Figure 5b),where the direction of steepest descent will be almostperpendicular to the direction of negative curvaturealso for points off the two principle axes with positivecurvature. As a result, also carriers created at thesepoints will be forced to change direction abruptlywhen reaching the axis of negative curvature, whichbrings about the above-discussed slowing down oftheir cooling rate. Hence, even if the fraction of elec-trons and holes that exactly passΣc during coolingmaybe small, a far larger number will by slowed downduring cooling at energies close to that of Σc.

Importantly, Figure 6 shows that the energy rangeof slow cooling around Σc is one to two phononenergies wide. This results in a picture of high energycarriers that (1) cool toward Σc by the rapid, successiveemission of LO phonons, (2) slow down around Σc dueto the reorientation of the scattering wavevector,which involves one or two slow phonon emission stepsand (3) continue cooling toward the band edge statesby rapid, successive phonon emission. As discussed inthe Supporting Information (section S5), this leads to amore precise interpretation of the rates kin and kout inthe phenomenological 3-level model we introduced todescribe the transient bleach around Σc. The incomingrate kin is the effective rate of the first part of thecooling process. As such, it is equal to the ratiobetween the energy loss rate wE and the energy theelectron and the hole dissipate before reaching Σc or,

Figure 6. (a) Energy loss rate at 300K calculated for an electron (solid blue line) (hole, solid red line) placed in thelowest (highest) conduction (valence) band near the Σc (k0 = π/a[110]) point where the zero of energy corresponds to theenergy of Σc. The wavevector of the initial electron state is oriented along [110] with respect to k0 for E > 0 and along [001] forE < 0 (b) A depiction of a hot carrier cooling through a process of fast phonon emission (with quantized energy pωLO) until abottleneck transition is reached, denoted as ∑c

u f ∑cd.

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phrased differently, to the product of the phononemission rate wph,in and the number of phonons tobe emitted before reaching Σc. Both wE,in and wph,in

thus calculated have been added to Table 1. Impor-tantly, since the measured cooling rate kin is close tothe width of the pump pulse, the highest rates calcu-lated should be seen as lower limits to the true energyloss and phonon emission rate. The outgoing rate kouton the other hand corresponds to the slow emission ofone or two phonons around Σc. Especially the compar-ison of the phonon emission rate for cooling toward(wph,in > 50 ps�1) and around Σc (wph,out≈ 2.5�5 ps�1)makes clear how considerable a bottleneck Σc is forcarrier cooling.

From Bulk to Quantum Dots. Electron states in nano-crystals can be seen as linear combinations of bulkstates, which essentially results in a quantization of theelectron wavevectors k. Taking the case of cubicnanocrystals with an edge length L for example, theenergies E of the quantized states around Σc read

E ¼ Ec,Σcþ p2

2m110

n110π

L

� �2

þ p2

2m110

n110π

L

� �2

� p2

2m001

n001π

L

� �2

(8)

Here, n110, n110 and n001 are nonzero, positive integersthat quantize the incremental wavevector δk in thedirections indicated by the subscript. The experimentspresented here show that at energies well above Σc,the cooling of hot carriers remains as efficient as inbulk, with very similar characteristic energy loss rates inthe eV ps�1 range. We thus conclude that size quanti-zation does not really limit the cooling of carriersin these energy ranges. Since the energy splittingbetween states is in practice never exactly equal to aLO phonon energy, multiphonon processes are re-quired. The stronger coupling to acoustic phonons innanocrystals compared to bulk certainly contributes tocompensate the energy mismatch.34,35 However, thedriving force for the relaxation remains the coupling toLO phononswhich follows the same rules as in the bulkbut taking into account quantized components.

In particular, it appears that all the conclusionsabout the dimensionality of the parameter space forthe final state and the directionality ofqwith respect tok remain valid. In bulk, the threshold for the change inenergy loss rate, linked to the necessary change in thedirection of the scattering wavevector is near theenergy gap ΔE∑c at Σc. Given the difference in effectivemasses for the directions of positive and negativecurvature, the threshold will be close to ΔE∑c þ p2π2/L2((1/m110)þ (1/m110)) in nanocrystals. Further coolingrequires an increase of the quantum number n001, i.e.,the quantized equivalent of a change in direction ofthe scattering wavevector. Hence, we indeed expectthe slow down of carrier cooling to shift to higher

energy for smaller nanocrystals and follow, as ob-served, the quantization of energy levels around Σc.eq 8 is only an approximation due to its dependenceon a parabolic band structure and a particular nano-crystal geometry. More elaborate calculations showthat the eigenstates are somewhat spread in k space.In addition, since there are 12 Σc points, intervalleycouplings mix all the states. Including the spin degreeof freedom, this leads to a high density of states in thevicinity of Σc. These effects should contribute to broad-en the energy range where the carrier accumulationoccurs. However, the experimental results show thatthe underlying effects originating from the saddlepoints of the bulk remain efficient.

Cooling Around Other Saddle Points. The band structureof bulk PbS (PbSe) is characterized by other saddlepoints where a similar analysis could be applied toassess their influence on the cooling rate. Most nota-bly, a saddle point Δc is present along the Δ direction(Γf X), whose energy gap of≈3 eV falls outside of therange of the TA setup used here. Calling this directionthe parallel one, the band dispersion around kΔc

≈0.82(π/a)(1,0,0) can be approximated as

Ec, k � Ec,Δcþ p2

2

δk2)

m )

� δk2^m^

!(9)

Here, we introduce the effective masses m ) (≈0.16m0)along Δ and m^ (≈ 0.51m0) for the two directionsperpendicular to Δ. In contrast to Σc, we calculate astrong enhancement of the cooling rate near Δc (seeSupporting Information S6). This can be readily under-stood from the band dispersion where now two direc-tions instead of one have a negative dispersion insteadof one. Therefore, when the carrier is close to thesaddle point, the parameter space for the final stateis determined by one additional degree of freedomthat considerably increases the density of allowedprocesses and thus the cooling rate. We thus predictthat, opposite from Σc,Δc does not constitute a bottle-neck for electron cooling.

DISCUSSION

The relaxation rate of hot electron�hole pairs is ofparticular importance for QD-based photovoltaic cellsthat exploit multiple-exciton generation (MEG), wherea single hot electron�hole pair creates multiple elec-tron�hole pairs with lower energy.15,40 Several authorshave proposed that the efficiency of MEG is deter-mined by the competition between the generation ofmultiple excitons via impact ionization and hot carrierrelaxation.16,40�42 Therefore, the characterization ofhot carrier relaxation is of great importance for under-standing this MEG efficiency. The relevant rate in thiscontext is the cooling rate of carriers at high excessenergy, above the energy threshold for MEG Eth. In thisrespect, the results shown here raise the question as to

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whether the cooling bottleneck around Σc is respon-sible for the efficient MEG that is observed in PbS andPbSe QDs.If indeed the occurrence of MEG is linked to slow

cooling around Σc, one would expect the MEG thresh-old, i.e., the energy where the yield of excitons starts toexceed 100%, to be related to E∑c. Figure 7 thereforerepresents both E∑c and the few available data on theMEG threshold for PbS36,37 and PbSe36 QDs36,38,39 as afunction of the reciprocal QD diameter 1/dNC, wherethe bulk values are also included at 1/dNC = 0.39 In thecase of bulk PbS and PbSe, E∑c/Eg amounts to 4.36 and5.75, respectively. Interestingly, both numbers corre-spond closely to the experimental bulkMEG thresholdsas reported by Pijpers et al.39 As a result of the highereffective masses at Σc compared to the band edgeeffective masses, E∑c/Eg decreases with decreasingdiameter. For both PbS and PbSe QDs, the MEG thresh-old follows the same trend, yet the available sets ofdata show considerable discrepancies. Whereas theMEG threshold as determined by Nootz et al. closelyfollows Σc,

37 Midgett et al. report MEG thresholds thatexceed E∑c/Eg by more than the QD bandgap for 4 nmPbS and PbSe QDs. Figure 7 also indicates that for PbSand PbSe QDs smaller than ≈3.5 nm, E∑c/Eg dropsbelow 2. This is below the energy conservation thresh-old, which implies that the slow cooling around Σc isirrelevant for MEG observed in these QDs.Since MEG by impact ionization is a process that

competes with charge carrier cooling, the MEG effi-ciency in PbS and PbSe QDs is closely related to thecooling rate, where higher MEG efficiencies implyslower cooling.41�43 Focusing on PbS QDs, we haveestimated the MEG efficiencies assuming that hotcarrier relaxation results from this competition (seeSupporting Information S7). Following the methodol-ogy described in ref 44, we have calculated theenergy-dependent impact ionization rates in bulkPbS and PbS QDs. The phonon cooling is consideredas a parameter independent of the carrier energy.Figure S10 shows that we theoretically retrieve MEG

efficiencies that closely follow E∑c/Eg, as reported byNootz et al., for phonon emission rates of ≈2 ps�1.On the other hand, we obtain the lower efficienciesfound by Midgett et al. if the phonon emission rate israised to ≈25 ps�1. Based on the figures provided inTable 1, it follows that both possibilities can beconsistent with our experimental data. If the MEGthreshold effectively follows E∑c/Eg, it requires a slow-ing down of the cooling to a rate that corresponds towhat we find around Σc. On the other hand, if MEG isonly effective for excitations well above E∑c, it com-petes with amuch faster coolingwhere again, the ratewe measure can reasonably account for the experi-mental thresholds. It thus appears that the under-standing of the mechanism behind MEG, and thepossible role of the slow cooling around Σc requiresamore precise determination of theMEG threshold asa function of QD size. Moreover, any theoreticalmodelattempting to describe the MEG efficiency shouldstart from the experimental results on hot chargecarrier cooling presented in this manuscript, i.e., cool-ing by phonon emission at rates of≈50 ps�1 (or more)for electron (hole) states above (below)Σc and at ratesof the order of kout around Σc.

CONCLUSION

In summary, we have shown that PbS and PbSequantum dots exhibit a transient absorption bleach atphoton energies corresponding to the Σc transition.This is interpreted in terms of a temporal accumula-tion of charge carriers due to a slowing down ofcharge carrier cooling. This temporal accumulationof cooling carriers enables us to quantify the energyloss rate of hot electron�hole pairs and to estimatethat the fraction of hot carriers cooling via thisΣc bottleneck is of the order of 1. Importantly, wedemonstrate that this cooling bottleneck is intrinsi-cally linked to the properties of the band structure ofbulk PbS and PbSe around the critical point Σc. Thisindicates that cooling bottlenecks at critical points inthe band structure present a general opportunity to

Figure 7. Comparison of Σc and threshold energy for MEG (normalized to the band gap energy Eg) for (a) PbS fromMidgett36

and Nootz37 and (b) PbSe38 as a function of the inverse particle size 1/d. Note that the bulk limit39 corresponds to 1/d = 0.

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use hot charge carriers more efficiently. In this re-spect, an intriguing, very recent finding is that critical

points can be created on purpose in twisted bilayergraphene.45

METHODSMaterials. Oleylamine (OLA) capped PbS quantum dots

(QDs) were synthesized using the procedure described byCademartiri et al.19 and modified by Moreels et al.23 Aftersynthesis, the (oleylamine) OLA ligand shell is substituted byoleic acid (OA). An exchange to OA is typically performed byadding OA to a toluene suspension of PbS Qdots in a ratio of1.5:10 OA/toluene. After precipitation with ethanol and centri-fugation, the QDs are resuspended in toluene and the exchangeis repeated.

White Light Pump�Probe Spectroscopy. Wepresent a hyperspec-tral analysis of the hot carrier decay in lead chalcogenidenanocrystals (NCs) using broadband and nondegeneratepump�probe spectroscpy. As such, 180 fs pulses are generatedin a YKGBW oscillator (Light Conversion, Pharos SP) at 1028 nmand amplified. Themajority of this fundamental beam is used togenerate pump light at 450 and 700 nm using an OPA andsecond harmonics module (Light Conversion, Orpheus). A smallfraction of the fundamental beam is split off to generatebroadband 'white' light in a sapphire crystal. The probe lightis delayed using an automated delay stage to achieve probedelays up to 3 ns. The colloidal NCs are diluted in tetrachlor-oethylene to achieve optical densities of 1 at the shortestexcitation wavelength of 450 nm.

The fluence is chosen sufficiently low (ÆNæ = Jph � σa = 0.1)such that, at most, single excitons are present. Assuming aPoissonian distribution of the carrier population after photo-excitation, the probability PN to have N excitations in one NC isgiven by

PN ¼ e�ÆNæÆNæN

N!(10)

Here, ÆNæ is determined from ÆNæ = J � σpump, where J is thepump fluence in photons/second and σpump is the photonabsorption cross section at the pump wavelength λpump. Wemake sure P2 < 0.01.

Conflict of Interest: The authors declare no competingfinancial interest.

Acknowledgment. The authors acknowledge GhentUniversity (BOF scholarship, GOA Detavernier-Hens), the FWO-Vlaanderen (G.0760.12), BelSPo (IAP 7.35, photonics@be), EU-FP7(Strep Navolchi), FOM and ADEM.

Supporting Information Available: Information on the spec-tral shifts, the four-level approximation, the valence banddispersion around Σc, scatter rates and the analysis of the Δc

critical point. Data for carrier cooling in PbSe nanocrystals is alsoprovided. This material is available free of charge via theInternet at http://pubs.acs.org.

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