Letter pubs.acs.org/NanoLett
Ultrafast Relaxation Dynamics via Acoustic Phonons in Carbon Nanotubes Olga A. Dyatlova,*,† Christopher Köhler,‡ Ermin Malic,‡ Jordi Gomis-Bresco,⊥ Janina Maultzsch,§ Andrey Tsagan-Mandzhiev,† Tobias Watermann,∥ Andreas Knorr,‡ and Ulrike Woggon† †
Institut für Optik und Atomare Physik, Technische Universität Berlin, 10623 Berlin, Germany Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany § Institut für Festkörperphysik, Technische Universität Berlin, 10623 Berlin, Germany ∥ Dahlem Center for Complex Quantum Systems, Physics Department, Free University Berlin, Arnimallee 14, 14195 Berlin, Germany ⊥ Catalan Institute of Nanotechnology, Campus UAB, 08193 Bellaterra, Barcelona, Spain ‡
ABSTRACT: Carbon nanotubes as one-dimensional nanostructures are ideal model systems to study relaxation channels of excited charged carriers. The understanding of the ultrafast scattering processes is the key for exploiting the huge application potential that nanotubes offer, e.g., for light-emitting and detecting nanoscale electronic devices. In a joint study of twocolor pump−probe experiments and microscopic calculations based on the density matrix formalism, we extract, both experimentally and theoretically, a picosecond carrier relaxation dynamics, and ascribe it to the intraband scattering of excited carriers with acoustic phonons. The calculated picosecond relaxation times show a decrease for smaller tube diameters. The best agreement between experiment and theory is obtained for the (8,7) nanotubes with the largest investigated diameter and chiral angle for which the applied zone-folded tight-binding wave functions are a good approximation. KEYWORDS: Carbon nanotubes, acoustic phonons, pump−probe spectroscopy, intraband scattering
S
lation and low-energy phonons in the carrier relaxation has been discussed, e.g., in refs 13 and 14. In spite of the remarkable advances in ultrafast spectroscopic methods, the microscopic origin of the ultrafast carrier relaxation dynamics in carbon nanotubes is often controversially discussed, and a microscopic model going beyond a phenomenological picture is still missing. The experimental data need to be complemented by theoretical studies on a microscopic footing, thus allowing one to track the way of excited carriers toward equilibrium via different relaxation channels. From previous theoretical studies, it is known that the intraband scattering via Coulomb interaction takes place within the first 100 fs,15,16 while the scattering of excited carriers with optical phonons occurs on a subpicosecond time scale.17 In this paper, we present a joint study of experiment and theory, addressing the relaxation dynamics of optically excited carriers in various types of SWCNTs, focusing in particular on the impact of intraband scattering with acoustic phonons. In the two-color pump−probe experiment, the pump and the probe pulses are tuned resonantly to the E22 and E11 transition energies, respectively, of four different chiral index (n,m) carbon nanotubes. Eii denotes the excitonic transitions from the
ingle-walled carbon nanotubes (SWCNTs) are cylinders constructed by rolling up a layer of graphene. They are one-dimensional structures with diameters in the range of a few nanometers and a length of up to several micrometers.1,2 Their exceptional optical and electronic properties have sparked interest in both fundamental research and industrial applications.2−5 The key for designing and engineering novel carbon nanotube-based optoelectronic devices is a thorough microscopic understanding of the ultrafast relaxation dynamics of nonequilibrium charge carriers. Experiments on carrier dynamics in carbon nanotubes6−12 have been performed showing different relaxation behaviors with various explanations about the observed decay times. Photoluminescence lifetimes vary in the range from 100 fs to a few nanoseconds. Manzoni et al.6 investigated the intersubband relaxation dynamics, finding a decay time of 40 fs. Ostojic et al.7 measured two decay times: one in the 5 to 20 ps range, ascribed to the interband carrier recombination, and a faster component around 0.3 to 1.2 ps, which they ascribe to the intraband carrier relaxation. Recently, two-dimensional nonlinear-optical experiments on carbon nanotubes were reported by Graham et al.,8 showing biexponential decay with two components of 120 fs and 1.25 ps. Yang et al.9 observed a decay time in the fewpicosecond time range, which they tentatively ascribed to bundling effects. The involvement of exciton−exciton annihi© XXXX American Chemical Society
Received: December 13, 2011 Revised: February 24, 2012
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Figure 1. (a) Illustration of the E11 and E22 transition energies within PLE spectroscopy. The high intensity peak is assigned to the (8,7) tube. The profiles are obtained by binning the intensity between the respective dashed lines. The excitation (emission) energy corresponding to the maximum of intensity of the blue (red) profile is ascribed to the E22 (E11) transition energy. (b) Schematic of the density of electronic states, from which the excitonic transitions are derived, for the (8,7) nanotube. The arrows represent the pumping and probing energies in the two-color pump−probe experiment.
i-valence band to the i-conduction band. As an example, in Figure 1a, the definition of Eii transition energies for the (8,7) tube within photoluminescence excitation (PLE) spectroscopy is shown (for details of the method, see below). The density of electronic states of the (8,7) tube is schematically shown in Figure 1b. Experiments are performed on samples of SWCNTs suspended in sodium dodecyl sulfate (SDS) and deuterium oxide. The tubes are grown with a high-pressure carbon monoxide (HiPco) method.18 The theoretical investigation is based on the microscopic density-matrix formalism, an established technique for the quantum-mechanical treatment of the dynamics for many-particle systems.19,20 It allows one to study different relaxation channels resolved in time and momentum. We identify a decay component on a picosecond time scale and assign it to scattering processes with acoustic phonons. To perform energy-selective pump−probe experiments, we first determine the E11 and E22 transition energies of the SWCNTs in solution by PLE spectroscopy based on a tunable Ti:Sa laser excitation. The excitation energy is tuned stepwise through the range from 1.40 to 1.75 eV, and the luminescence is recorded in the spectral window from 0.85 to 1.25 eV, as shown in Figure 2. From the PLE map we can assign eight tube species and define their E11 and E22 transition energies with the help of the empirical functions obtained by Weisman et al.21 We assume intertube interactions to be a secondary effect in our studies, since we do not observed any pronounced intertube energy transfer peaks on the PLE map, which are expected in the case of strong tube−tube interaction.22 The wavelength tuning of the pump and probe pulse is achieved by using two synchronized laser systems: a titanium sapphire laser serves as the master laser and is used for the pump arm of the experiment (tunability range from 700 nm to 1.1 μm with a full width at half-maximum (fwhm) of about 6 nm, pulse duration of 150 fs, and repetition rate of 75.4 MHz). A fiber laser serves as the slave laser and is used for the probe arm (tunability range from 900 nm to 1.3 μm, fwhm 10 nm, pulse duration of 200 fs). The control of the temporal pulse shape of the fiber laser is performed by an external pulse shaper. The used synchronization unit provides a short time jitter below 0.5 ps. In order to obtain a high sensitivity of the setup, the lock-in technique combined with balance detection is applied. The modulation of
Figure 2. Three-dimensional contour plot of photoluminescence intensity versus excitation and emission energies for the SWCNTs suspended in SDS and deuterium oxide. Eight high-intensity spots are assigned to specific (n,m) tubes (black points). The color-coded photoluminescence intensity scale is linear.
the pump beam is performed by an acousto-optical modulator (AOM) at a frequency of about 127 kHz to suppress noise. The ultrafast carrier relaxation is investigated for a selected set of SWCNTs, i.e., the (8,7), (10,2), (11,3), and (12,1) carbon nanotube species (see Figure 2). This set represents both different semiconducting families (n − m)mod3 = ± 1 as well as three Kataura branches (determined by 2n + m = const.)1,2 and covers a diameter range between 0.88 and 1.03 nm. To excite nonequilibrium charge carriers, high-power pump pulses of an energy resonant to the E22 transition are chosen in two-color pump−probe experiments. Using weak probe pulses resonant to the E11 transition energy of the same tube, the decay time of the differential transmission (DT) signal DT = ΔT/T is measured as a function of pump−probe delay time. The results are shown in Figure 3. The observed fast increase of the DT signal reflects the fast rise in carrier population in the lower subband, resulting in a bleaching of the absorption at the probe energy. The optically excited carriers injected into the upper subband reach the energetically lower subband via Coulomb- and optical-phonon-induced inter- and intrasubband scattering processes, which all occur on a femtosecond B
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slower component τ2 around 50 to 100 ps (see Table 1). An increase in pump power by about 1 order of magnitude did not result in a significant change in the picosecond relaxation dynamics. Therefore, we expect electron−phonon processes to determine the relaxation dynamics. To obtain a better understanding of the observed relaxation dynamics, we perform microscopic calculations based on the density matrix formalism.19 We resolve the nonequilibrium carrier relaxation in momentum and time by deriving Boltzmann equations driven by scattering of electrons with acoustic phonons. From earlier work, we know that the scattering via Coulomb interaction and optical phonons takes place on a femtosecond time scale15,17 and therefore cannot explain the decay times observed in the presented two-color pump−probe experiment. Within the second-order Born− Markov approximation,5,19,24 we obtain a Boltzmann equation for the subband occupation ρλk = ⟨a+λkaλk⟩, where a+λk creates and aλk annihilates an electron in the subband λ with the wave vector k:
Figure 3. Two-color pump−probe curves of four different nanotubes labeled by the chiral indices (n,m): their family and Kataura branch. Curves obtained with pump and probe energies are resonant to E22 and E11 transition energies of the specific (n,m) tube, respectively. The inset shows a fitting curve (red curve) of a two-color pump−probe signal (black curve) for the exemplary (8,7) nanotube and for zero background constant.
ρ̇ kλ = −
2π ℏ
∑ ∑ |gqγ |2 δ(εkλ′′ − εkλ ± ℏωqγ )Fkkλλ′q′γ
(1)
q , γ , ∓ λ′
λλ ′ γ Here, F kk′q = [n qγ + (1/2) ± (1/2)]ρ kλ (1 − ρ k′λ′ ) −
time scale15,17 and are beyond the resolution of our experiment. Looking at the following recovery process of the optical transmission, i.e., the decay of the absorption bleaching signal, we observe a picosecond decay of the DT signal in Figure 3 pointing to the existence of a slower relaxation channel acting on carriers once they have reached the probed energetic region. Low-energy acoustic phonons might be involved in the further relaxation process down to the equilibrium. The impact of acoustic phonon scattering on picosecond dephasing processes was already emphasized in the literature for other types of semiconducting nanostructures, e.g., for InAs quantum dots (see, e.g., ref 23 and the references therein). To determine the decay times from the experimental data, we use a three exponential fitting routine, which is exemplarily shown for the (8,7) nanotube in the inset of Figure 3. We compare two different fitting routines that either (i) take into account a zero background constant of the DT signal or (ii) set the DT signal to its value at negative time delay. That different background level has only a small effect on the results obtained for the first two decay times, whereas it considerably changes the slowest nanosecond component. This third, nanosecond component is usually assigned to the recombination time back to the SWCNT ground state, which is much slower than all scattering channels and thus not further considered in our study. The values obtained for the τ1 and τ2 decay times are presented in Table 1 and were calculated as an average of the results from both fitting routines. As the main result, we find in all four investigated tubes two different time constants: a fast component τ1 in the range between 6 and 15 ps followed by a
[nγq + (1/2) ∓ (1/2)]ρλk′′(1 − ρλk) contains the processes of absorption and emission of phonons (±) with the momentum transfer k′ = k ∓ q. Furthermore, it contains the contributions stemming from Pauli blocking [∝ρλk′′(1 − ρλk)]. nγq denotes the phonon occupation number, which is given by the Bose− Einstein distribution at room temperature within the bath approximation. For intraband scattering, the sum over the subband indices contains only the term λ′ = λ. We assume longitudinal acoustic phonons, i.e., Γ−LA, to give the strongest relaxation channel25 after the completion of electron-opticalphonon scattering. The corresponding carrier-phonon coupling 2 2 element |gLA q | = (ℏ/L)(D /πdmνph)|q| is taken from ref 25 and adapted to carbon nanotubes.26 It depends on the absolute value of the phonon wave vector |q| and the deformation potential D. Here m is the mass density of graphene, d the diameter of the tube, and vph is the phonon velocity. The coupling does not depend on the nanotube length L, since it cancels through the sum over k. The dispersion relation for ΓLA phonons ℏωLA q = ℏvph|q| is approximated to be linear in the vicinity of the Γ-point, where ℏvph = 0.013 eV nm.25 The electronic bandstructure ελk of carbon nanotubes is calculated within the zone-folded nearest-neighbor tight-binding approach1 and parametrized by parabolas around the optically relevant region close to the band minimum. To model the described pump−probe experiment, we create a nonequilibrium carrier distribution via optical excitation of charge carriers into the second conduction subband corresponding to the E22 transition. We find an ultrafast intra- and
Table 1. List of the Decay Times Derived from the Experimental Dataa
a
CNT; family
diam. (nm)
E11 (eV)
E22 (eV)
τexp 1 (ps)
τexp 2 (ps)
τth 1 (ps)
(8,7); +1 (11,3); −1 (12,1); −1 (10,2); −1
1.032 1.014 0.995 0.884
0.978 1.035 1.050 1.172
1.687 1.558 1.556 1.685
6.1 ± 0.4 13.2 ± 0.5 15 ± 0.5 8.5 ± 0.5
57 ± 6 71 ± 8 88 ± 10 64 ± 10
4.4 3.5 3.1 2.5
The last column contains the theoretically calculated relaxation times describing the scattering of excited carriers with acoustic phonons. C
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energies of the bigger diameter SWCNTs more accurately than those of the smaller ones. Our theoretical investigations of intraband scattering of excited carriers with acoustic phonons do not describe a second, slower decay time τ2 around 50−100 ps. Its physical origin still needs to be explained and requires further research. In conclusion, we have performed two-color pump−probe experiments for different types of semiconducting carbon nanotubes, which were optically characterized by measuring a PLE map. The E11 and E22 transition energies and chirality were assigned and used for pump (in resonance to E22) and probe (in resonance to E11) experiments. For all investigated SWCNTs, we find that the carrier relaxation is characterized by a fast component τ1 in the range between 6 and 15 ps, which is in agreement with theoretically predicted intraband scattering times with acoustic phonons. The predicted diameter dependence of this picosecond relaxation time is a very interesting result that should stimulate further experimental and theoretical investigations. The new insights gained in our joint experiment−theory studies contribute to a better understanding of the ultrafast relaxation dynamics in carbon nanotubes.
intersubband carrier relaxation due to the scattering with optical phonons.17 From previous calculations,15 we know that the Coulomb-induced intraband scattering also occurs within the first 100 fs. Therefore, to model the current experiment showing a slower picosecond dynamics, we focus on intraband relaxation processes driven by acoustic phonons. The Coulomb coupling for nanotubes in media with large dielectric screening is considerably reduced,27,28 which allows us to focus on the investigation of the phonon-induced relaxation dynamics of excited electrons. It is important to mention here that, for SWCNTs in air, the Coulomb interaction including the formation of excitons plays an important role and cannot be neglected.29−31 We start with a nonequilibrium distribution resulting from ultrafast interband scattering processes driven by optical phonons17 (see the black line in Figure 4). Then, we calculate
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We acknowledge financial support by the Deutsche Forschungsgemeinschaft through SFB 658 and GRK 1558. E.M. acknowledges funding by the Einstein Foundation Berlin. J.M. acknowledges support from the European Research Council, ERC Grant No. 259286.
Figure 4. Occupation probability ρk of the lowest subband for the exemplary (8,7) nanotube as a function of energy at different relaxation times. To model the experimentally measured decay times, we exponentially fit the decay of the occupation close to the band minimum; see Table 1.
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the relaxation of optically excited electrons toward the energetically lower states via emission of acoustic phonons. In Figure 4, we show the occupation probability in the lower conduction subband as a function of the energy at different relaxation times. After approximately 20 ps, the carriers reach the band minimum, resulting in a Fermi distribution. The appearing oscillations result from the strict energy conservation within the Markov approximation. We calculate the relaxation times close to the band minimum by applying an exponential fit. We obtain times of a few picoseconds, as shown in Table 1. They are in agreement with the measured fast component in the experimental DT decay curves. In particular, the (8,7) nanotube with the larger diameter and chiral angle shows a good agreement, since here the applied zone-folded tightbinding wave functions are a very good approximation. The theoretical calculations result in a decrease of the relaxation time with decreasing tube diameter. The experimental results do not precisely reproduce that behavior. The mismatch between theoretical and experimental results can be explained as follows: (i) The sample, investigated in our experiment, is not chirality enriched and thus has an inhomogeneous broadening, in particular for the tubes (11,3) and (12,1) with similar Eii transition energies. (ii) Decay channels, such as the exciton−exciton annihilation,13 which are efficient for large pump fluences and slow down the carrier relaxation dynamics, are not considered in our theoretical model. (iii) The used tight-binding wave functions describe the
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