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We have investigated the highly nonlinear terahertz (THz) light-matter interaction in single-walled carbon nanotubes (SWNTs). The high-peak THz electric-field (∼0.7 MV/cm) and the low effective mass of carriers result in their ponderomotive energy exceeding the bandgap energy of semiconducting SWNTs. Under such an intense THz pulse irradiation, the interband excitation that results in the generation of excitons occurs, although the THz photon energy (∼4 meV) is much smaller than the gap energy of SWNTs (∼1 eV). The ultrafast dynamics of this exciton generation process is investigated by THz pump and optical probe spectroscopy. The exciton generation mechanism is described by impact excitation process induced by the strong THz E-field. Such intense THz pulse excitation provides a powerful tool to study nonlinear terahertz optics in non-perturbative regime as well as nonlinear transport phenomena in solids with ultrafast temporal resolution.
© 2011 Optical Society of America
Corrections
Shinichi Watanabe, Nobutsugu Minami, and Ryo Shimano, "Intense terahertz pulse induced exciton generation in carbon nanotubes: erratum," Opt. Express 19, 15388-15388 (2011)https://opg.optica.org/oe/abstract.cfm?uri=oe-19-16-15388
1. Introduction
The interaction between light and matter changes from the low-field perturbative regime, i.e., multiphoton transition regime, to the high-field tunnel-ionization regime with increasing laser field strength. Keldysh, in his seminal paper in 1965 [1], introduced a critical parameter in the description of the tunnel ionization process, defined as , where Ip is the atomic ionization potential and Up = e2E2/4meω2 is the ponderomotive energy with electron charge e, laser frequency ω, free electron mass me, and light electric-field (E-field) amplitude E. This γ corresponds to the ratio of laser frequency to tunnelling frequency, so γ < 1 gives the criterion for the light field strength to cause tunnel ionization. In this regime, the instantaneous E-field of the light is recognized as being quasi-static. This treatment has been successfully used to describe field-ionization processes in atomic systems under strong laser-pulse excitation [2, 3].
Since Up is inversely proportional to the square of the laser frequency, Up ∝ 1/ω2, such a non-perturbative regime can be more easily accessed by using a longer-wavelength light source [2,4,5]. With the recent progress of high-peak-power monocycle terahertz (THz) pulses, the peak E-field has reached above 100 kV/cm [6–8], and various nonlinear optical effects utilizing such intense THz pulses are investigated [4,7–15]. In order to study the THz light-matter interaction in a highly non-perturbative regime, γ << 1, terahertz pulse with much higher peak E-field amplitude should be required.
In this article, we investigated highly nonlinear THz light-matter interaction on single-walled carbon nanotubes (SWNTs). We realized an intense THz pulse with the peak E-field amplitude as large as 0.87 MV/cm using the optical rectification of near-infrared (IR) laser pulse with a LiNbO3 crystal [16]. The small effective mass of carriers ( ) in SWNTs further resulted in a large ponderomotive energy and a small Keldysh parameter γ ∼ 0.05 << 1. Such highly nonlinear THz light-matter interaction caused the interband excitation, in particular the exciton generation in SWNTs by the THz pulse, although the THz photon energy (∼4 meV) is much smaller than the gap energy of SWNTs (∼1 eV). The ultrafast creation dynamics of excitons was probed by THz-pump and optical-probe experiments with ∼ 90 fs temporal resolution. The exciton generation mechanism in SWNTs was described by impact excitation process induced by the strong THz E-field. The developed intense terahertz pulse opens an avenue to study nonlinear terahertz optics in non-perturbative regime as well as nonlinear transport phenomena with ultrafast temporal resolution.
2. Experimental setup
A schematic picture of the THz pump and near-IR optical probe (TPOP) experiment is shown in Fig. 1(a). We used SWNT samples produced by the CoMoCAT method [17,18]. The SWNT diameters were around 0.8 nm with a narrow size distribution. Photoluminescence excitation and emission mapping experiments have shown that the sample contained several kinds of semiconducting SWNTs such as (6,5), (7,5), (8,4) and (7,6) SWNTs [19]. Each nanotube was wrapped with sodium dodecylbenzene sulfonate (SDBS) to form a micelle, thus it was isolated with each other. The micelle-SWNTs solution was then mixed with an aqueous gelatin solution and dried to form a thin film with a thickness of l=30 μm. The composite film was mechanically stretched so that SWNTs were aligned in the stretching direction. Further details of the processing method are given in Ref. [20]. The absorption coefficient of the pure gelatin film was (45 ± 10) cm−1 at 1 THz, which results in about 10 % absorption of the incident THz power.
As a light source, we used a regenerative amplified laser system with pulse energy of 1 mJ, center wavelength of 800 nm, pulse duration of 90 fs, and repetition of 1 kHz. We generate intense THz pulses by optical rectification of near-IR laser pulses in a LiNbO3 crystal with the pulse front tilted by a diffraction grating [16]. We placed a pair of lens ( f = −200 mm, and f = 300 mm) and a pair of cylindrical lens ( f = 80 mm, and f = −40 mm) in the optical pass to modify the laser beam profile in the elongated shape (∼4 mm in height and ∼2 mm in width) before entering the crystal surface. Since the angle between the propagation directions of the laser beam and the THz pulse is γ = 62.7° in order to fulfill the phase matching condition in LiNbO3 crystal [21], the horizontal width of the THz beam became 1/cosγ ∼ 2 times larger compared to that of the laser beam. Therefore, the output THz beam became in a round shape with a diameter of ∼4 mm. We firstly focused the THz beam by a parabolic mirror with 10 mm in diameter and 10 mm effective focal length. The THz beam was then further lead to another focal point using two other parabolic mirrors with 50.8 mm in diameter and 50.8 mm effective focal length. The careful control of the THz beam profile allowed us to tightly focus the THz pump beam at the sample position with a diffraction limited spot size of 300 μm in diameter for 1 THz. The waveform of the THz E-field pulse was measured by the free space electro-optic sampling method using a (110) GaP crystal with a thickness of 0.376 mm as shown in Fig. 1(b). Since the THz E-field is so strong, we inserted 6 high-resistivity silicon wafers to attenuate the amplitude before the GaP detection crystal. The absolute value of the E-field of the attenuated THz pulse was evaluated by using the EO coefficient of the GaP crystal, r41=0.88 pm/V [22]. The peak E-field amplitude of the generated THz pulse without the silicon attenuators was estimated to be 0.87 MV/cm and the total pulse energy estimated by integrating the square of the waveform both temporally and spatially was ∼0.4 μJ. We also measured the THz pulse energy by a pyroelectric detector (SPH-THz, Spectrum Detector Inc.) and obtained a value 0.4 μJ. In order to avoid the effect of the stray light of the IR pulse on the sample, we put a white paper, and cut the IR pulse before the SWNT sample. This reduces the maximum THz peak amplitude at the sample position to 0.69 MV/cm. To change the THz pump fluence continuously, we inserted three wire-grid polarizers into the THz optical path.
As a probe light source, we used a small portion of the laser beam separated by a beam splitter and focused it on a 1 mm thick sapphire plate to generate a white light probe beam. The time-integrated spectra of the white light probe pulse that transmitted the sample with and without the THz pump pulse were measured by a spectrometer equipped with a liquid-nitrogen cooled InGaAs photodiode array as a function of the delay time Δt of the probe pulse to the THz pump pulse. Since the white light probe pulse was chirped, we compensated for the effect of the chirp after measuring the transmission spectra. The polarizations of the THz pulse and the near-IR pulse were both set parallel to the stretching direction of the SWNT/SDBS/gelatin film. The THz setup was purged with dry air to avoid the effects of water vapor on the THz beam and also on the sample. All the measurements were performed at room temperature. Further details of the THz pump and optical probe experiments are given in Ref. [23].
3. Experimental results
3.1. Absorption spectrum and its spectral decomposition
Figure 2 shows the near-IR absorption spectrum of the SWNT/SDBS/gelatin film sample without the THz pump, where three peaks (A–C) are discerned. The spectrum above 1 eV (Peak A and B) can be reproduced by sum of four Lorentzian line shapes as
where the fitting parameters Ai, Ei, γi, and corresponding (n, m) structures are summarized in Table 3.1. In this spectral decomposition, the broad background component, presumably the tail of π-plasmon resonance which centered at about 5 eV [24,25] was subtracted. The exciton absorption peak energies of (8,4) and (7,6) SWNTs were almost the same, and thus they exhibited a single absorption peak. The large uncertainty in the (8,3) peak energy was due to its small contribution in the absorption spectrum. The relative absorbance between the nanotube compositions was consistent with that reported in Ref. [19], where the narrow distribution of tube diameters was clearly observed by the photoluminescence excitation and emission mapping experiments. The absorption peak energy of every SWNT was red-shifted by 10–30 meV from that of the CoMoCAT SWNT in aqueous suspension [26], possibly because of the different environmental conditions [19]. The broad absorption peak below 1 eV [Peak C] was assigned to be 0.96 eV from the photobleaching spectrum using a terahertz-pump and optical-probe experiment as discussed later. From this exciton energy, the possible tube chiralities that contribute to the peak C were estimated as (8,6), (8,7), and (9,5).
3.2. THz pump and optical probe experiments
Figures 3(a)–3(d) show the transmission change (ΔT/T) and absorption spectrum change (Δαl ∼ −ΔT/T) induced by the THz pump as a function of Δt under peak THz E-fields of 0.22 MV/cm (Figs. 3(b) and 3(d)), and 0.69 MV/cm (Figs. 3(a) and 3(c)) around the spectral peak A in Fig. 2. Under a relatively low pump field in Fig. 3(b), the temporal behavior of the transmission change nearly coincided with the square of the THz E-field amplitude, showing an instantaneous response to the THz oscillation. Such an ultrafast absorption change can be assigned to the THz-field-induced Stark effect of excitons that shows a quadratic dependence on the THz E-field [23, 27]. The behavior became dramatically different when the peak THz E-field amplitude was increased to 0.69 MV/cm: a photobleaching (PB) signal sustaining even after the THz pulse passed through the sample distinctly appeared (Figs. 3(a) and 3(c)). It should be noted here that the long-lived PB signal was observed for the temporal region where the THz pulse arrives much earlier than the probe optical pulse (positive delay region). This result firmly indicates that the long-lived PB signal was not caused by the THz-field induced ionization of excitons created by the earlier arriving optical probe pulse. Although the nonnegligible long-lived PB signal can be also seen in Fig. 3(b), the signal showed a highly nonlinear dependence on the THz E-field amplitude which we discuss later in detail.
Fig. 2 Near-infrared absorption spectrum of a SWNT/SDBS/gelatin film (solid line). Dashed line shows the fitting curve obtained by Eq. (1). The spectral components used in the fitting is also shown with the corresponding tube chirality (n, m). π-plasmon absorption is also included as a broad background.
Fig. 3 Transmission spectrum change (ΔT/T) ((a) and (b)) and absorption spectrum change (Δαl) ((c) and (d)) as a function of delay time (Δt) under low ((b) and (d)) and high ((a) and (c)) peak THz pump intensity. The peak THz E-field amplitude is 0.22 MV/cm ((b) and (d)), and 0.69 MV/cm ((a) and (c)) at Δt = 0 ps. The left panels in Figs. 3(a) and (b) show the temporal profiles of the squared THz E-field (solid line). The THz temporal profiles in (a) and (b) are different because of the slightly different day-to-day alignment of the THz generation. The transmission change signals at probe photon energy of 1.24 eV as indicated by the vertical lines in the right panels are also shown by open circles. The right panels in Figs. 3(a) and (b) show image plots of ΔT/T spectra versus Δt. Figures 3(c) and (d) show the spectral profile of Δαl at Δt = 0 ps and Δt = 1.6 ps, respectively.
To consider the origin of this long-lived PB signal induced by the THz pump, we also performed optical-pump and optical-probe (OPOP) experiments, as shown in Figs. 4(a)–4(c). Here, we used a near-IR pulse (1.55 eV, 90 fs) as the pump light source to generate excitons directly by one-photon absorption. PB signals with peak energies corresponding to (7,5) and (6,5) SWNTs can be seen in Fig. 4(a). These signals can be attributed to the exciton generation, which causes absorption saturation via the phase space filling effect [28, 29]. Figure 4(c) shows the delay time dependence of the absorption change signal, which represents the dynamics of the exciton population in SWNTs after the photoexcitation. The temporal profile shows a short-decay component (τ1 ∼ (0.6 ± 0.1) ps) and a long one (τ2 ∼ (85 ± 30) ps). The short one can be fitted by a bimolecular decay function (1 + Δt/τ1)−1, which is consistent with previous reports [30,31], where the mechanism has been attributed to exciton annihilation through exciton-exciton collisions in SWNTs. The long-decay component (τ2) can be fitted by a single exponential function exp(−Δt/τ2), and perhaps ascribed to the relaxation of excitons in trapped states [30].
Fig. 4 Results for the near-IR-pump (h̄ωpump = 1.55 eV) ((a)–(c)) and THz-pump experiments (h̄ωpump ∼ 4 meV) ((d)–(f)). (a) and (d): Absorption spectrum change around exciton resonance of (7,5) and (6,5) SWNTs for various pump fluence. Time delay Δt was set to 1 ps after photoexcitation. (b) and (e): Pump fluence dependence of absorption changes at 1.24 eV and 1 ps after photoexcitation. Circles represent experimental results and lines represent fitting curves: a line in Fig. 4(b) and −Δαl=0.3· exp(−Eth/ETHz(Δt=0 ps)) (Eth=2.3 MV/cm) in Fig. 4(e). (c) and (f): Delay time dependence of the absorption change at 1.24 eV. Insets show semi-log plots of the absorption change in the longer time range of Δt =10–40 ps. Circles are experimental results; lines are fitting curves. Details of the fitting are given in the text.
Now we consider the long-lived PB signal observed in the TPOP experiments. THz-pump-induced absorption spectrum changes for different excitation powers are shown in Fig. 4(d). Two peaks corresponding to excitons in (7,5) and (6,5) SWNTs can be discerned, similar to the OPOP experiments (Fig. 4(a)), so we attribute this PB signal to THz-pump-induced exciton generation. A very small photoinduced absorption (Δαl > 0) observed below 1.17 eV in Fig. 4(d) as well as in Fig. 4(a) may be attributed to the biexciton absorption [32]. A striking difference from the near-IR pump case was observed in the pump fluence dependence, as plotted in Figs. 4(b) and 4(e). Whereas the PB signal showed linear pump fluence dependence in the OPOP experiments, it showed a threshold-like behavior in the TPOP case and was well fitted by the relation |Δαl|=0.3 ·exp(−Eth/E THz(Δt = 0)), where Eth = 2.3 (± 0.2) MV/cm is the threshold E-field amplitude.
The delay time dependence of the absorption change signal is shown in Fig. 4(f). After the instantaneous response induced by the THz E-field, the PB signal showed short-decay (τ1) and long-decay (τ2) components.
To view the temporal profile of the THz-pump-induced absorption change in more detail, we plotted the results for various THz pump intensities in Fig. 5(b). All the experimental curves are well reproduced by,
with parameters: A=0.07(MV/cm)−2, B=0.2, C=0.2, Eth=(2.3 ± 0.2) MV/cm, τ1=(0.7 ± 0.3) ps, and τ2= (85 ± 30) ps. A waveform of the THz pump pulse ETHz(Δt) is shown in Fig. 5(a). The first term on the right hand side of Eq. (2) represents the instantaneous component, and the rest represent the PB signal due to excitons created by the THz pulse. The short-decay component (τ1) can be expressed by an exponential function in this case presumably because of the scattering of excitons by optical phonons mediated by the THz E-field acceleration of carriers, as discussed later in more detail. The long-decay component (τ2) has a similar value to that in the OPOP experiments, so we ascribed it to exciton relaxation. Note that the fitting curves in Fig. 5(b) as well as that in Fig. 4(c) and 4(f) are convoluted by the finite pulse width of the probe pulse (∼ 90 fs). Here, we assumed that the exciton formation time is much shorter than the temporal resolution of our measurements, and the exciton production rate can be expressed by the instantaneous THz E-field amplitudes, i.e. Bexp(−Eth/ETHz(t)). This assumption is consistent with the recent near-IR pump and mid-IR probe experiments on SWNTs [31], where a short exciton formation was observed in the above-gap one-photon excitation condition.3.3. Discussions
We next discuss a possible origin of the exciton generation in SWNTs by the THz pulse irradiation. Since the THz photon energy (∼ 4 meV) is far below the gap energy (Ip ∼ 1 eV) of semiconducting SWNTs, the multiphoton transition cannot be considered as the origin of the exciton generation; perhaps the THz E-field induced ionization could be the origin. Indeed, the field amplitude ETHz ∼ 0.7 MV/cm at 1 THz gives the Keldysh parameter , the field-induced ionization regime, for Ip ∼ 1 eV and the reduced mass of electrons and holes μ* ∼ 0.049me [33]. One possible mechanism for the THz-field-induced exciton generation in SWNTs is the Landau-Zener tunnelling of electrons from the valence band to conduction band. In the dc E-field case, the probability has been evaluated [34] as , where E is the applied E-field and vF is the Fermi velocity ∼ 108 cm/s. If we recognize the THz E-field as being static, P is estimated as ∼ 10−8 for Ip ∼ 1 eV and ETHz ∼ 0.7 MV/cm, which is too small to account for the experimental results. Another possible mechanism is impact excitation (IE) of excitons, which has been investigated [35–38] under the application of dc E-fields. In the dc IE process, an electron (or hole) seeded from electrodes into the first subband level (E1) of SWNTs is accelerated and then scattered to a lower kinetic energy state, transferring the energy difference to the exciton generation process. Taking into account the momentum conservation rule, a microscopic theory [38] has shown that the carriers should be accelerated to reach the third subband level with energy E3 to cause IE scattering, resulting in a threshold-like exciton production rate expressed approximately by P ∝ exp{−(E3 – E1)/eEλ}, where λ is the mean free path of carriers. By considering the optical phonons mean free path λop ∼ 15 × d [36] (d: tube diameter) and subband energy difference (E3 – E1(eV) ∼ 1.22/d(nm)), we obtain P ∝ exp(−Eth/E) with Eth ∼ 0.81/d 2(nm2) (MV/cm). For d ∼ 0.8 nm, Eth is estimated to be ∼1.4 MV/cm, which is consistent with the experimentally observed threshold value.
Fig. 5 (a) Temporal profile of the pump’s THz E-field pulse. (b) Delay time dependence of the absorption changes (open circles) at 1.24 eV at various THz pump fluence. The peak THz E-field amplitude at Δt = 0 is indicated in each panel. Red lines are fitting curves using equation (2).
To view such a tube diameter dependence, we plot in Fig. 6(a) the peak THz E-field dependence of the PB signal normalized by the linear absorption (−Δα/α), for several kinds of SWNTs. The signal evolves much faster for the (7,5) tubes than the (6,5) tubes with increasing the THz pump fluence. This signature can be seen also in Fig. 4(d) where the PB signal of the (7,5) SWNTs evolves faster than that of the (6,5) SWNTs. This result can be explained by the lower IE threshold for the (7,5) SWNTs because of the larger tube diameter. It should be noted here that, in the OPOP experiments (Fig. 4 (a)), we do not find any chirality dependence of the evolution of the PB signal, because the excitons are generated through the one-photon absorption process. The tube diameter (d) dependence of the threshold E-field (Eth) is plotted in Fig. 6(b). Clearly, Eth is smaller for larger d and scaled as Eth(MV/cm) = 1.4/d2(nm2), which reasonably agrees with the theoretical prediction.
Fig. 6 (a) Peak THz E-field amplitude dependence of the bleaching signal normalised by the linear absorption, −Δα/α, at Δt = 1 ps for absorption peak A (1.20 and 1.24 eV), B (1.07 eV), and C (0.97 eV) in Fig. 2. The corresponding tube chiralities (n, m) and their diameters are indicated. Lines are fitting curves to the function A · exp(−Eth/ETHz), where A and Eth are fitting parameters. The arrows indicate one fifth of the threshold E-field amplitude (Eth/5). (b) Log-log plot of the tube diameter dependence of the threshold THz E-field amplitude. Vertical error bars represent the fitting uncertainty in Fig. 6(a). Horizontal error bars represent the uncertainty of the tube diameter due to the mixing of two or three (n, m) SWNT structures. The solid line represents the result of fitting with the regression function Eth = A/d2.
Next, we roughly estimate the density of background carriers in the SWNTs which may act as the source of the IE process. Since the magnitude of Δα/α originates from the reduction of oscillator strength upon exciton creation, we can estimate the THz-induced exciton density at the most intense E-field irradiation condition in the (6,5) SWNTs (|Δα/α| ∼ 0.02) as (200 nm)−1 by using the size (∼2 nm) and the wavefunction of excitons in the (6,5) SWNTs [29]. By assuming that all the excitons are generated within 90 fs (time resolution of our measurement) at Δt = 0 ps, the density of the background carriers is estimated as (10 nm)−1 by taking into account the IE probability of a single seeded carrier [38]. This value should be viewed as an upper limit of the seeded carrier density since we assume that the carrier which generates an exciton at Δt = 0 ps does not contribute to another exciton generation at Δt = ≠ 0 ps. The estimated carrier density is small enough compared to the lowest doping density of the intentionally doped SWNT film samples reported in Ref. [39]. To confirm our specification on the mechanism of the THz pump induced exciton generation, however, further experiments such as doping dependence of the PB amplitude will be promising.
Next, we address on the effect of exciton dissociation and the Franz-Keldysh effect (FKE) under the intense THz E-field. Please note that the Keldysh parameter in our experiments is much smaller than unity, and thus the condition is in the FKE regime, but not in the dynamical FKE regime (γ ∼ 1) [40]. The FKE in carbon nanotubes under the static E-filed has been theoretically calculated by Perebeinos and Avouris [41], where the critical E-field for the exciton dissociation in the (14,0) nanotubes (d ∼1.11 nm in diameter) was estimated to be about 2 MV/cm. Therefore, the critical E-field of the (6,5) nanotubes (d ∼0.76 nm) should be much larger because of the smaller diameter and the larger exciton binding energy. Although the theory was formulated for the case of static E-field, we consider that the THz E-field in the present case can be recognized as the static field (see next paragraph), and thus the effect of the exciton dissociation due to the FKE is small within our amplitude range (ETHz < 0.7 MV/cm).
Finally, we consider why the THz E-field can be regarded as static in the IE process in SWNTs. First, we estimate the acceleration time t1 for carriers to gain the IE threshold energy E3 − E1 by using the relation , which leads to t1 ∼ 20 fs for ETHz ∼ 0.7 MV/cm in a (6,5) SWNT. The carrier scattering rate with optical phonons, t2, is roughly estimated to be fs for the same condition. Moreover, the IE scattering rate has been estimated to be much larger than [38]. Therefore, all the time scales involving the IE process are sufficiently shorter than the THz oscillation period (∼ 1 ps), so the THz E-field can be recognized as being quasi-static in the IE process in SWNTs.
4. Conclusions
We have investigated the interaction between intense THz pulses and SWNTs in the highly nonlinear regime of γ << 1, where we observed THz-pump-pulse-induced exciton generation. The ultrafast creation dynamics of excitons was probed in THz-pump and optical-probe experiments and was explained by the THz-E-field-induced impact excitation process. The experimental results demonstrated that the intense monocycle THz pulses with a well-defined carrier-envelope-phase enable a time-resolved study of quasi-static E-field-induced nonlinear THz light-matter interactions in solids. Even more intense THz pulse would induce Landau-Zener breakdown in SWNTs, where electron-hole pairs can be produced by the nonadiabatic tunneling of carries. While we investigate here the large-gap semiconducting SWNTs, further experiments in the narrow-gap or the metallic SWNTs with vanishing rest mass of electrons is highly interesting in terms of extremely nonlinear light-matter interactions in a Dirac electron system. The investigation of the intense THz nonlinear optics also provides a powerful tool to study high E-field effects in solids with ultrafast temporal resolution, making it possible to understand nonperturbative light-matter interactions in solids in a time-resolved manner.
Acknowledgments
The authors acknowledge S. Tsubota for his technical assistance in the development of intense THz pulses. This work was partially supported by Grants-in-Aid for Scientific Research (No. 18340082, 22684013, 22244036, 20110005) from MEXT, Japan.
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