Biexciton, single carrier, and trion generation dynamics in single-walled carbon nanotubes
Bertrand Yuma, Stéphane Berciaud, Jean Besbas, Jonah Shaver, Silvia M. Santos, Saunab Gosh, R. Bruce Weisman, Laurent Cognet, Mathieu Gallart, Marc Ziegler, Bernd Hönerlage, Brahim Lounis, Pierre Gilliot
BBiexciton, single carrier, and trion generation dynamics in single-walled carbonnanotubes
B. Yuma, S. Berciaud, J. Besbas, J. Shaver, S. Santos, S. Ghosh, R.B. Weisman, L. Cognet, M. Gallart, M. Ziegler, B. H¨onerlage, B. Lounis, and P. Gilliot IPCMS, CNRS and Universit´e de Strasbourg, 23, rue du Lœss, F-67034 Strasbourg, France LP2N, Universit´e de Bordeaux, Institut d’Optique Graduate School,and CNRS, 351 cours de la Lib´eration, F-33405 Talence, France Department of Chemistry and R.E. Smalley Institute for Nanoscale Science and Technology,Rice University, 6100 Main Street, Houston, Texas 77005, USA (Dated: November 5, 2018)We present a study of free carrier photo-generation and multi-carrier bound states, such as biex-citons and trions (ionized excitons), in semiconducting single-walled carbon nanotubes. Pump-and-probe measurements performed with fs pulses reveal the effects of strong Coulomb interactionsbetween carriers on their dynamics. Biexciton formation by optical transition from exciton popula-tion results in an induced absorption line (binding energy 130 meV). Exciton-exciton annihilationprocess is shown to evolve at high densities towards an Auger process that can expel carriers fromnanotubes. The remaining carriers give rise to an induced absorption due to trion formation (bindingenergy 190 meV). These features show the dynamics of exciton and free carriers populations.
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INTRODUCTION
The excitonic nature of optical resonances in single-walled carbon nanotubes (SWCNTs) [1, 2] has stimulated nu-merous research studies on the influence of Coulomb interaction on the photophysics of these quasi one-dimensionalsystems. Enhanced Coulomb interaction and reduced dielectric screening in SWCNTs give rise to considerable many-body effects/bandgap renormalization and binding energies of excitons ( X ), as high as 700 meV . nm for freestandingsemiconducting SWCNTs[3, 4], a value that is a large fraction of the single particle bandgap. Linear optical spec-troscopy has revealed the richness of the exciton manifold [5, 6]. For sufficiently high excitation rates, many bodybound states such as biexcitons[7–10] ( XX ) or charged excitons [11, 12] (trions, X ∗ ) should form. Because of thestrong Coulomb interaction between photogenerated charge carriers, such SWNT states have been predicted to havesignificant binding energies in the 60 - 250 meV range for biexcitons [7].Biexcitons are bound states of two excitons similar to a hydrogen molecule. They usually arise from the collisionbetween two excitons, but they can be also generated from an exciton population by one-photon absorption, resultingin an induced absorption line [13–15]. In carbon nanotubes, however, Coulomb interaction also results in highlyefficient Auger processes such as exciton-exciton annihilation (EEA) [16–18], that are expected to bypass biexcitonformation. This is in agreement with the complete saturation behavior reported in air-suspended nanotubes [19].At high exciton densities, Coulomb interactions may also lead to exciton ionization and the subsequent generationof a carrier gas, whose signature is the appearance of a new transition toward trion states. Trions have been firstpredicted and observed in inorganic semiconductors [20]. They can be seen as the bound state of three carriers: anoptically excited electron-hole (e-h) pair and an additional carrier (electron or hole) that is already present. Thissimple picture works mainly for strongly confined zero-dimensional systems ( e.g. quantum dots), that can onlyaccommodate a few carriers. In extended systems ( e.g. et al. on intentionally hole-doped nanotube ensembles [26]. In thatcase, the presence of the additional carriers allowing trion formation was expected, whereas the observation of charged a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b many-body bound states in pristine samples through carrier photogeneration was singular and less anticipated [25].Interestingly, no evidence of bi-exciton formation could be found in the emission spectra.In this letter, we report a detailed transient absorption study that allows to characterize the dynamics of theprocesses that result from Coulomb interactions: many-body bound state formation and carrier generation originatingfrom exciton collisions. Our data show evidence for biexciton and free carrier formation in chirality sorted (6,5)SWCNTs. Following the generation of several excitons by a pump pulse, a probe pulse allows the observation of twoinduced absorption (IA) features. These are red-shifted with respect to the main induced transmission (IT) featureassigned to the S exciton and are attributed to the formation of biexcitons and trions. We show that the X IT and X ∗ IA dynamics reflect the evolutions of the exciton and free carrier populations, respectively. From analysis of thesedynamics, we deduce the processes that allow photogeneration of carriers through exciton-exciton collisions and theirspatial separation.
EXPERIMENTAL SETUP AND SAMPLES
Since the characteristic timescales associated with EEA processes are in the picosecond range [16–18], femtosecondpump-probe spectroscopy is the method of choice to study optical excitations in nanotubes. Such pump-probeexperiments have often been performed on heterogeneous ensembles of SWCNTs composed of many different species.In these conditions, experimental fingerprints from many-body bound states may be masked by unwanted residualsignals from non-resonantly excited species. To overcome this issue, we performed transient absorption spectroscopymeasurements on an ensemble of chirality-sorted (6,5) SWCNTs. Enrichment in the (6,5) chirality was obtained bynonlinear density gradient ultracentrifugation (DGU) [27]. The nanotubes are dispersed in an aqueous solution ofsodium cholate in a fused silica cell. As shown in Fig. 1, the linear absorption spectrum is largely dominated bycontributions from (6,5) SWCNTs. ( 8 , 6 )S = 1 . 0 6 e V ( 6 , 4 ) - S = = ( 6 , 5 ) - S = Absorbance (arb. units)
P h o t o n E n e r g y ( e V ) ( 6 , 5 ) - S = FIG. 1: Linear absorption spectrum of enriched (6,5) SWCNTs. The S exciton and S exciton of (6,5) SWCNTs, as well ascontributions from residual species, are indicated by arrows. For the time-resolved measurements, we used an amplified Ti:Sapphire laser that produces ultrashort pulses (100 fs)at a repetition rate of 200 kHz. The energy of pump photons was tuned into resonance with the S or S transitions(X lines) at 1 . eV or 2 . eV , respectively, by means of an optical parametric amplifier (OPA). The pump fluencerange corresponds to 5 × − . × photons per cm . Ultrashort continuum probe pulses were generated in asapphire crystal. Their spectral range (0.95-1.35 eV) covered the region where the exciton, trion, and biexciton linesare observed. Transmission spectra of the probe beam were acquired as a function of the time-delay between the pumpand probe pulses using a InGaAs linear diode array at the output of a spectrometer, and differential transmissionspectra were calculated from the data. In order to study the dynamics of the various spectral features, we extractedtheir contributions as a function of pump-probe delay. S e x c i t a t i o nS e x c i t a t i o n D T/T (x10-4) D T/T (x10-4) ( a ) S e x c i t a t i o n ( 6 , 5 ) - X X ( c )( b ) ( 6 , 5 ) - S ( 6 , 5 ) - X * P h o t o n e n e r g y ( e V )
T i m e d e l a y : < 0 7 0 0 f s 3 p s 1 2 p s 1 0 0 p s
FIG. 2: (a)-(b) Differential transmission spectra obtained after excitation at S excitonic transition. The IT at 1.26 eV is thesignature of the S excitonic transition from (6,5) nanotubes. The pump fluence is 6 . × photons / pulse / cm .(b) Zoom onthe spectral region below the energy of the exciton of the (6,5) nanotube. At 1.06 eV, an IT is present at very short timesand attributed to the S excitonic transition from a residual fraction of (8,6) nanotubes. The IA at 1.08 eV is attributed totrion ( X ∗ ) formation from a charge level. At 1.13 eV, another IA, attributed to biexciton ( XX ) formation is visible from shortdelays up to 50 ps approximately. (c) Differential transmission spectra in the same near infrared region after excitation at theS excitonic transition. The pump fluence is 1 . × photons / pulse / cm . TRANSIENT ABSORPTION SPECTRA
Figure 2 shows differential transmission spectra that were obtained after excitation by pump pulses tuned to 2.19 eVnear the S excitonic absorption line of (6,5) nanotubes. From comparison with the linear absorption spectrum, thestrong IT peak that appears in Fig. 2(a) at 1.26 eV is attributed to the S excitonic transition from (6,5) nanotubes.Three other features are also observed in the near-infrared region that lies below, as shown in Figure 2(b): an IT at ∼ exciton line at1.26 eV, with a laser fluence of 1 . × photons / pulse / cm . Fig. 2(c) presents the corresponding differential trans-mission spectra. We observed no major spectral differences from the spectra obtained by pumping at S . Thesesimilar behaviors are due to the extremely fast ( ∼ to S intersubband decay[28] which is not resolved inour measurements. EXCITON BLEACHING AND DYNAMICS
The (6,5)-exciton bleaching at 1.26 eV can be attributed to absorption saturation by the pump beam that givesrise to an IT feature in the probe spectrum. It is due to both phase-space filling (PSF) and Coulomb interaction (CI)between excitons [29–32]. On the high energy side of the S exciton line, at 1.3 eV, an IA rises together with the IT.We suggest that this blue-shifted IA arises from the strong exciton-exciton interaction that occurs in one-dimensionalsystems. In particular, screening of the electron-hole interaction within the high density exciton gas leads to a decreaseof their binding energy and therefore to a blue shift of the X line. As seen in figure 2, the exciton IT signal risesrapidly, within a few hundred femtoseconds. This time scale is close to the temporal resolution of our setup. Thesignal mostly decreases within ∼
10 ps, although there remains a slowly decaying contribution after 100 ps [33, 34].This fast initial decay has been shown to result from exciton-exciton annihilation (EEA) [16–18] that destroys theexciton population within a few picoseconds. The dynamics of the IT observed at ∼ exciton transition. It will not be discussed further but has been taken into account whenfitting the differential transmission spectra. BIEXCITON FORMATION
We now consider the IA feature observed at 1.13 eV, which is red-shifted by about 130 meV from the S brightexciton IT feature. This spectral position coincides with the emission of the (6,5)-S phonon sideband that couplesa K-momentum dark exciton to near zone edge optical phonons [35, 36]. However, the signal observed in our pump-probe experiments is an induced absorption, whereas can emission would appear as an induced transmission. Thus,it cannot be the fingerprint of a phonon sideband. Another possibility would be the emission of a brightened tripletexciton state appearing in photo-damaged SWCNTs, as observed under strong laser irradiation [35]. Such defect-induced processes should however appear in the linear absorption spectra, which is not the case in our experiments.In addition, their signature in differential transmission measurements should also be an IT. D T/T (normalized on (8,6) exciton )
P h o t o n E n e r g y ( e V )
P u m p p h o t o n e n e r g y : ( 8 , 6 ) - S ( 6 , 5 ) - B i e x c i t o n
FIG. 3: Transient spectra showing IT at 1.06 eV ((8,6)-SWCNT exciton) and IA at 1 . line. The delay between pump and probe pulses is fixed at 700fs.The spectra have been normalized to the exciton signal for ease of comparison. A more compelling interpretation for this new IA feature is the creation by the probe pulse of a biexciton ( XX ),from the exciton population previously generated by the pump pulse, i.e. to the X → XX optical transition. Theseparation of 130 meV observed in our experiments between the line at 1.13 eV and the exciton line at 1 . eV is large compared to thermal energy. Its magnitude order matches theoretical calculations [7]. To verify that thisfeature arises from (6,5) SWCNTs, we performed differential transmission measurements using different pump photonenergies at and below the S transition. The spectra shown in Fig. 3 are normalized with respect to the amplitude ofthe IT feature due to the residual population of (8,6) SWCNTs at a given time delay. While the exciton IT signal ofthe (8,6) nanotubes is not expected to change, as they are excited above the transition, the amplitude of the XX IAfeature decreases strongly when the energy of the pump photons is tuned below the S resonance of (6,5) SWCNTsand the photoinduced population of excitons in those nanotubes becomes smaller.The transient dynamics of the XX feature, represented in Figure 4 along with that of the X IT, bolsters ourinterpretation. The dynamics of the X and XX features are quite similar, with a signal decay that occurs mainlyin the first 5 ps. Then, even if the exciton population undergoes EEA, the corresponding IA absorption should beobserved, as long as an exciton population remains. The signal dynamics should thus follow that of the excitonpopulation, and not the biexciton state lifetime, which may be very short. Measurements performed at higher pumpfluence (not shown here) show a faster decay of the XX IA feature that follows the enhanced exciton decay due toEEA. We emphasize that our observation is not contradictory with previous PL experiments [25, 37] that have ruledout the formation of biexcitons because of the competition with highly efficient EEA. Indeed, the process that involvesa collision between two excitons to form a biexciton, is highly improbable if those collisions also give exciton-exciton
E x c i t o n , S - e x c i t e d D T/T (x10-2)
T i m e D e l a y ( p s )
B i e x c i t o n , S - e x c i t e d D T/T (x10-5)
FIG. 4: Normalized transient spectra of S exciton IT at 1 . eV (blue curve), and biexciton IA at 1.13 eV (green line).The IA have been plotted using a reversed Y-axis for ease of comparison with the IT of the exciton. The laser fluence was2 . × photons / pulse / cm . The excitation was performed on the S transition. annihilation with a much higher probability. SINGLE CHARGE CARRIER GENERATION AND TRION FORMATION
As explained in our previous report [25], comparison with PL spectra of an ensemble of intentionally doped SWC-NTs [26] or pristine single SWCNT under intense laser excitation demonstrates that the IA at 1.08 eV can beattributed to trion (X ∗ ) formation from a charge carrier level, with a binding energy of 190 meV. Remarkably, asshown in Fig. 5(a), the trion IA rises within ∼ S excitonic IT feature. This shows that there is a common mechanism responsible for the decay of the exciton andthe rise of the trion feature. This supports the following scheme [25]: exciton-exciton collisions, which are responsibleof the decay of their population at short delays, lead to the formation of a population of charge carriers that isevidenced by the induced absorption toward the trion state. We note that the sign of the trion cannot be determinedhere: since electron and holes in nanotubes have very similar effective masses, the binding energies of positive andnegative trions are expected to be nearly equal, giving the same spectral signature for negative and positive trions [38].We now turn to a more quantitative analysis of our data and first examine the impact of the interplay between theexciton and charge carrier populations on the differential transmission signals. As mentioned above, in the absence ofmany-body bound states, the existence of the S exciton IT feature can be understood in terms of phase space filling(PSF) and Coulomb interactions (CI) between excitons [31]. Here, we assume that some of the photoexcited e-h pairsdissociate to produce single carriers that assist trion formation. Consequently, part of the exciton oscillator strengthis transferred to the trion transition. This effect should result in an IT (absorption decrease) at the X photon energytogether with an IA at the X ∗ photon energy. On the other hand, because trions and excitons share the same electronand hole band states from which their wave function is built, they should undergo the same saturation effects (PSFand CI), and the X ∗ IA amplitude should diminish, while the X IT amplitude should increase. Strictly speaking,the saturation effects and the oscillator strength transfer are thus involved at once in both X and X ∗ differentialtransmission. But their respective impact can be evaluated through the maximum amplitude of the signals. As shownin Fig. 2 and 6, the differential transmission of the exciton does not exceed 5%, while the trion IA is two orders ofmagnitude smaller, limited to 0.07%. Thus, the X ∗ signal is mainly due to oscillator strength transfer and the X signal originates essentially from e-h population effects. Consequently, one can reasonably consider that the X signalamplitude measures the exciton density n X , while the X ∗ signal is proportional to the density n c of single chargecarriers. ∆ TT (cid:12)(cid:12)(cid:12)(cid:12) X ∝ n X ( t ) ∆ TT (cid:12)(cid:12)(cid:12)(cid:12) X ∗ ∝ n c ( t ) (1)Nevertheless, the X IT signal shows a component that persists after 200 ps, with an amplitude roughly three times D T/T (x10-4)
E x c i t o n , S - e x c i t e d D T/T (x10-2)
E x c i t o n , S - e x c i t e d T r i o n , S - e x c i t e d D T/T (x10-4)
T i m e d e l a y ( p s )
T r i o n , S - e x c i t e d D T/T (x10-4) ( c )( a )
T r i o n , S - e x c i t e d ( b )
T r i o n , S - e x c i t e d D T/T (x10-4) D T/T (x10-2)
FIG. 5: (a) and (b) Normalized transient S exciton IT at 1 . eV (blue line) and trion IA at 1 . eV (red line). TheIA have been plotted using a reversed Y-axis. The excitation is performed on the S transition. The laser fluence is 1 . × photons / pulse / cm . (c) Normalized transient IA signal at the trion transition after excitation at 1.26 eV (S ) (gray line)and after excitation at 2.19 eV (S ) (red line). The IA have been plotted using a reversed Y-axis. The laser fluence for the S excitation is 1 . × photons / pulse / cm . larger than that of the X ∗ IA ( (see Fig. 5(b)). As evidenced in static absorption of doped nanotubes [34], a populationof carriers induces an exciton bleaching that is not completely balanced by the transfer of oscillator strength towardsthe trion transition. This shows that part of the exciton IT signal, observed at times significantly longer than theexciton lifetime (that is in the range [39–41] of 10 ps to 100 ps), is due to band filling effects by carrier populationsthat decay slowly, as shown below.Figure 6 represents the maximum induced transmission of the X feature, which is proportional to the excitonnumber n X , initially created by the pump pulse, along with the maximum induced absorption of the X ∗ feature,that is attained after a few ps (cf. Fig. 5a) and is proportional to the maximum density of charge carriers n c . Theyscale identically as a function of the pump fluence ( i.e. as a function of the number of excitons). This could seemsurprising, considering that the derivative of the population n c of single carriers results from Auger processes andthus scales quadratically with the exciton population n X . However, since the creation of carriers results from theannihilation of excitons, the same term appears, with an opposite sign, in the differential rate equations that describeexciton decay and the photo-induced generation of charge carriers, respectively. At times shorter than the excitonlifetime, where recombination processes other than EEA can be neglected, this results in: dn c dt ∝ − dn X dt (2)which gives after integration n c ∝ n X . This implies that regardless of the number of prepared excitons n X (orequivalently the laser fluence), one should find a density n c that is proportional to n X (as observed in Fig. 6). Theseconclusions hold not only for linear or quadratic processes, but also for any other nonlinear transfer law that connectsthe two populations n c and n X and in particular if there is a threshold for the carrier generation from the excitonicpopulation.We can determine the actual initial exciton number per unit length n X ∝ ∆ T /T | max as a function of the pumpfluence using the absorption cross-section value of 1 × − cm per carbon atom [40] at low excitation. As shownin Fig. 6, it shows a saturation behavior, with a maximum exciton density of 55 µ m − , i.e. a separation of 18 nmbetween two excitons. This distance can be compared to the exciton Bohr radius [1, 2] in (6,5) SWCNT a B = 1 . µ m − at which EEA begins to saturate matches the threshold at which the trion signal is observed (Fig. 6). It is evenmore remarkable that the threshold for observing trion emission in PL experiments performed with a cw excitationon single nanotubes is of the same order [25]. In these experiments, the trion line emerges from the background at N X (cid:39) µ m − . ns − that gives, for an exciton lifetime of 100 ps, a mean density n X (cid:39) µ m − . As mentionedabove, trion formation implies the creation of single carriers following exciton ionization. We thus propose that fordensities larger than a few tenths of an exciton per micron, where trions start to be observed in transient absorptionexperiments, the mechanism involved during an exciton-exciton collision changes and gives other educts. While EEAresults in the transfer of the remaining exciton as a whole, for higher X densities, when Coulomb interaction withinthe exciton is strongly screened, exciton-exciton collision would ionize the remaining exciton, transferring the electronor the hole to high energy levels. This process should be closer to the usual Auger mechanism and differ from EEA inthe sense that it implies exciton dissociation into free electron and hole. The promoted carrier could then gain enoughenergy to be ejected from the nanotube, in its vicinity or at trapping sites on the surface, following a mechanismthat has been largely identified for semiconductor nanocrystals [42, 43]. The other carrier remaining in the nanotubeis thus not subject to recombination. It may localize at local fluctuations of the electrostatic potential along thenanotube and be involved in the binding with an exciton to form a trion.
01 x 1 0 - 4 - 4 - 4 - 4 - 4 - 4 ( D T/T)max X x 10-2
P u m p f l u e n c e ( p h o t o n s . p u l s e - 1 . c m - 2 x 1 0 ) E x c i t o n T r i o n -( D T/T)max X* x 10-4
FIG. 6: Maximum of the IA signal at the photon energy of the trion (red triangles, right Y-axis) and maximum of the IT signalat the photon energy of the exciton (blue circles, left Y-axis) versus pump fluence.
DYNAMICS OF THE X ∗ FEATURE AND CARRIER POPULATION DECAY
The decay of the X ∗ IA feature at 1.08 eV, shown in Fig. 5b, reflects the dynamics of the population of chargecarriers that make the ground state of the single carrier-trion transition. After 900 ps, about 30% of the maximumtransient signal still remains. This slow decay indicates a long-lived feature [44], which cannot be fully resolved heresince our set-up is limited to pump-probe delays below 1 ns. This timescale is much longer than the lifetime of S -excitons [39–41]. This strengthens the hypothesis of carrier localization at trapping sites where they are protectedfrom collisions and their recombination is slowed down [25]. Nonetheless, no signal is observed at negative time delays,as seen in Fig. 2c. This proves that SWCNTs have relaxed down to their neutral ground state between two consecutive (a) X excitation(b) EEA (d) Auger photocarrier generationTrion Transition(c) XX excitation FIG. 7: Schemes summarizing the different processes involving multi-carrier states in SWCNTS. (a) exciton generation by lightabsorption, (b) EEA process due to exciton-exciton collisions,(c) biexciton formation from an exciton population by one-photonabsorption, (d) Auger carrier photogeneration and trion excitation. pump pulses. Thus the inverse of the repetition rate of our laser system sets an upper bound of 5 µ s for the lifetimeof the trapped charges. CONCLUSION
Our study uncovers the many-body processes that arise from Coulomb interactions in (6,5) semiconducting SWC-NTs, as well as their binding energies and the timescales associated with their dynamics. Figure 7 illustrates themechanisms that can lead to the formation of many-body bound states in the case of excitation by a pair of time-delayed, ultrashort pump and probe pulses. A population of bright singlet S excitons can be created upon resonantexcitation or following ultrafast intersubband decay of higher order excitons (e.g. S excitons) created with higherphoton energies (Fig. 7 (a)). Strong Coulomb interactions give rise to exciton-exciton annihilation, which results ina fast decay of the exciton population (Fig. 7 (b)). While a significant number of excitons exists, the probe photonsmay form biexcitons (more generally multi-exciton complexes, cf. Fig. 7 (c)). This process results in an inducedabsorption feature that is red shifted with respect to the bright exciton line by a biexciton binding energy of 130meV for (6,5) SWCNTs, in good agreement with theoretical predictions. This process obviously competes with EEA.Biexciton formation remains possible as long as there remains a population of excitons. Thus, the temporal dynamicsof the biexcitonic feature reflects exciton dynamics.For a large exciton density, exciton-exciton interaction predominantly results in Auger recombination, rather thanexciton-exciton annihilation, providing enough energy to dissociate an exciton and to expel a single carrier in thevicinity of the nanotube. The carrier remaining in the SWCNT can bind to another e-h pair to form a trion thatremain localized on the same sites (Fig. 7 (d)). This is evidenced in our study by another induced absorption featurethat is red shifted with respect to the bright exciton line by a binding energy of 190 meV for (6,5) SWCNTs. Thisfeature should be observed as long as a population of charge carriers persists in the nanotube. The characteristictimescales for expelled carriers to return into the nanotube are related to the distribution of trapping energies alongthe nanotube and its dielectric environment. Our measurements indicate that the de-trapping times are equal orlarger than the measured lifetime of ∼ ns and smaller than 5 µ s. With the prospect of designing opto-electronic andphotovoltaic nanotube-based devices, our work provides important insights into carrier photogeneration, and possiblycarrier multiplication [45, 46]. Acknowledgments
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