Nonlinear Photoluminescence Excitation Spectroscopy of Carbon Nanotubes: Exploring the Upper Density Limit of One-Dimensional Excitons
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Nonlinear Photoluminescence Excitation Spectroscopy of Carbon Nanotubes:Exploring the Upper Density Limit of One-Dimensional Excitons
Yoichi Murakami
1, 2 and Junichiro Kono ∗ Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, USA Department of Chemical System Engineering, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
We have studied emission properties of high-density excitons in single-walled carbon nanotubesthrough nonlinear photoluminescence excitation spectroscopy. As the excitation intensity was in-creased, all emission peaks arising from different chiralities showed clear saturation in intensity. Eachpeak exhibited a saturation value that was independent of the excitation wavelength, indicating thatthere is an upper limit on the exciton density for each nanotube species. We developed a theoreticalmodel based on exciton diffusion and exciton-exciton annihilation that successfully reproduced thesaturation behavior, allowing us to estimate exciton densities. These estimated densities were foundto be still substantially smaller than the expected Mott density even in the saturation regime, incontrast to conventional semiconductor quantum wires.
PACS numbers: 78.67.Ch,71.35.-y,78.55.-m
Linear and nonlinear optical processes in one-dimensional (1-D) semiconductors are affected by strongCoulomb interactions and have been the subject of manytheoretical [1, 2, 3, 4] and experimental [5, 6, 7, 8, 9,10, 11] studies. Early studies of lasing in semiconductorquantum wires (QWRs) [5, 6] stimulated much interestin the fundamental properties of high-density e - h ) pairs, especiallyin their stability against biexciton formation, band-gaprenormalization (BGR), and dissociation. Although ithas been established that excitons in QWRs are stable upto very high densities (10 -10 cm − ), unanswered ques-tions still remain. As the density increases, an insulatingexciton gas is expected to become unstable and eventu-ally transform into a metallic e - h plasma at the Mottdensity, where the inter-exciton distance approaches theexciton size. At what density gain should appear andwhether a clear Mott transition exists in 1-D systems areopen questions both theoretically [2, 3, 4] and experi-mentally [7, 8, 9, 10, 11].Single-walled carbon nanotubes (SWNTs) [12] haverecently emerged as novel 1-D solids with very strongquantum confinement. The exciton binding energiesreported for semiconducting SWNTs are very large( ∼
400 meV [13, 14]) compared to typical GaAs QWRs( ∼
20 meV [2, 6]). Despite much recent progress in un-derstanding their basic optical properties, so far only alimited number of studies have been performed under thecondition of high carrier/exciton densities [15, 16, 17]. Inparticular, there have been no reports quantifying exci-ton densities in relation to the Mott density.Here we describe detailed characteristics of photolumi-nescence (PL) and photoluminescence excitation (PLE)spectra of SWNTs at high exciton densities. AlthoughPL intensity rapidly saturated with increasing pump flu-ence, PL spectra were stable, indicating that there isno BGR, screening, or dissociation into an e - h plasma.However, we observed significant broadening and even- tual complete flattening of PLE spectra at high laserintensities. We show that this originates from strongexciton-exciton annihilation (EEA) that provides an up-per limit for the exciton population in the lowest-energy( E ) state. Our model, taking into account the diffusionlength of excitons in the EEA process, allowed us to esti-mate exciton densities from experimental PL saturationcurves. The estimated densities for the highly saturatedregime were on the order of 1 × cm − , which remainsmore than one order of magnitude smaller than the ex-pected Mott density and explains the observed stabilityof PL spectra even in the saturation regime.The sample we studied was a centrifuged supernatantof CoMoCAT SWNTs dispersed by 1 wt% sodium cholatein D O. The solution was held in a 1-mm-thick quartz cu-vette. The optical density in the E region was below0.2, which helped avoid inhomogeneous excitation andre-absorption of PL. Our excitation source was a 1 kHz, ∼
250 fs optical parametric amplifier (OPA), tunable inthe visible and near-infrared, pumped by a chirped pulseamplifier (Clark-MXR, CPA2010). The OPA beam wasfocused onto the sample with a spot size of 300-400 µ m.The PL from the sample was recorded with a liquid-nitrogen-cooled InGaAs array detector.Figure 1(a) compares two PL spectra. The black curverepresents excitation by the OPA with a wavelength of654 nm (or 1.90 eV) and a pulse energy of 29 nJ, whilethe red curve represents excitation by a weak (100 µ W)CW laser with a wavelength of 658 nm (or 1.88 eV). Itis seen that the relative intensities of different PL peaksare drastically different between the two curves. The in-set confirms that the two spectra actually coincide veryaccurately when the OPA pulse energy is kept very low(300 pJ). Figure 1(b) shows PL spectra for pulse energiesof 1 nJ (curves P L I n t en s i t y ( a r b . un i t s ) OPA, 654 nm CW laser diode, 658 nm (8,3)(6,5) (7,5) (a) P L I n t en s i t y ( a r b . un i t s ) Wavelength (nm) low powerhigh power
OPA, 654 nm
Wavelength (nm)
FIG. 1: (color online). Pump-intensity-dependent PL spec-tra. (a) Black — spectrum obtained with OPA (654 nm,29 nJ). Red curve — spectrum obtained with a CW laserdiode (658 nm, 100 µ W). Inset shows that the two spectracoincide when the OPA pulse energy is very low (300 pJ).(b) Change of PL spectra versus OPA pulse energy between1 nJ and 30 nJ (in the order of ∼ × photons/cm ). curves were taken in the order of ∼
15% from curve ∼ × photons/cm with pumpwavelengths of 570, 615, and 654 nm, showing clearsaturation behaviors at high pump fluences. The E -resonant wavelengths of these tube types are 570, 647,and 673 nm, respectively. The integrated PL intensi-ties were obtained through spectral decomposition anal-ysis by assuming 50% Lorentzian + 50% Gaussian, whilekeeping all the peak positions fixed. Figure 2 indicatesthat saturation starts at a lower (higher) fluence when thesample is resonantly (non-resonantly) excited. The unex-pectedly fast saturation of the (7,5) peak with 570 nm ex-
570 nm excitation 615 nm excitation 654 nm excitation (6,5) (a) I n t en s i t y ( a r b . un i t s )
570 nm excitation 615 nm excitation 654 nm excitation (7,5) (b)
Fluence (× 10 photons/cm )
570 nm excitation 615 nm excitation 654 nm excitation (8,3) (c)
FIG. 2: (color online). Integrated PL intensity versus pumpfluence for (6,5), (7,5), and (8,3) SWNTs. Pump wavelengthswere 570 nm (circles), 615 nm (squares), and 654 nm (trian-gles). The error bars account for ± citation (which is non-resonant) is likely due to its prox-imity to the phonon sideband at 585 nm [18]. The solidand dashed curves are theoretical fits to the experimentalcurves based on our model to be discussed later.Figures 3(a)-3(d) show PLE maps taken with vari-ous pump fluences. The step size for the pump pho-ton energy was 20 meV. The data taken with the low-est fluence (1.2 × photons/cm ) [Fig. 3(a)] is es-sentially the same as that taken with low-intensity CWlight. However, as the fluence is increased [Figs. 3(b)-3(d)], the E excitation peaks gradually broaden andeventually become completely flat at the highest fluence(1.2 × photons/cm ) — i.e., PL intensities becomeindependent of the excitation wavelength . The corre-sponding PLE spectra are shown in Figs. 3(e)-3(h) forthree PL wavelengths at 983, 1034, and 1125 nm.To provide further insight into the nature of the ob-served broadening and flattening of PLE spectra, wealso performed absorption measurements in the E region using OPA pulses. Figure 4 compares twotransmission spectra measured with the OPA at flu-ences of 1.0 × photons/cm (linear regime) and1.0 × photons/cm (saturation regime), respec-tively, and a transmission spectrum taken with weak CWwhitelight. It is clear from the figure that the E absorp-tion peaks do not exhibit any broadening and shifts even cm -2 (6,5) (7,5)(8,3) (7,6)(8,4) (a) cm -2 (b) cm -2 (c) Emission Wavelength (nm) cm -2 (d) I n t en s i t y ( a r b . un i t s )
983 nm1034 nm1125 nm (6,5) (7,5)(7,6)(8,4)1.2 (cid:215) 10 cm -2 (e) I n t en s i t y ( a r b . un i t s ) cm -2 (f) I n t en s i t y ( a r b . un i t s ) cm -2 (g) I n t en s i t y ( a r b . un i t s ) Excitation Energy (eV) cm -2 (h) E xc i t a t i on E ne r g y ( e V ) FIG. 3: (color online). Evolution of PLE data with increasingpump pulse fluence: (a) 1.2 × , (b) 1.2 × , (c) 4.1 × , and (d) 1.2 × photons/cm . (e-h): PLE spectracorresponding to (a)-(d) at emission wavelengths of 983 nm(circles), 1034 nm (squares), and 1125 nm (triangles). in the saturation regime, strongly suggesting that theseintensities are not high enough to cause light-inducednonlinear effects such as state filling, carrier-density-dependent dephasing, and non-perturbative light-mattercoupling (or “dressing”) on states in the E range.We interpret these observations as results of very ef-ficient EEA [17, 19]. We assume that the formationof E excitons occurs in a very short time scale (e.g., ∼
40 fs [20]), shorter than our pulse width ( ∼
250 fs).Thus, excitons quickly accumulate in the E state dur-ing and right after photo-creation of e - h pairs. However,the number of excitons that can be accommodated in the E state is limited by EEA. As the exciton density, n x ,approaches its maximum value, EEA begins to preventa further increase by efficiently removing excitons non-radiatively , which explains the PL saturation behavior.Since EEA serves as a bottleneck for the exciton density,the PL intensity becomes insensitive to whether the ex-citons were created resonantly or non-resonantly aroundthe E level and independent of the pump wavelength. T r an s m i tt an c e ( % ) Wavelength (nm)
CW white light 1.0 × 10 cm -2 cm -2 (6,5) (7,5)(8,3) FIG. 4: (color online). Transmission spectra mea-sured with high-fluence OPA pulses (filled circles: 1.0 × photons/cm , open circles: 1.0 × photons/cm )compared with a transmission spectrum taken with a weakCW white-light beam (dashed line). Namely, at very high pump fluences, the PL intensityis determined not by how efficiently excitons are cre-ated but by how many E excitons can be accommo-dated within a particular type of SWNT as well as by therelative abundance of that type of SWNT in the sample.To determine exciton densities n x from the PL satura-tion curves, we have developed a theoretical model [21].Existing EEA models [17, 19] assume the annihilaitonrate to be ∝ n , a probability of finding two excitons atthe same position in space. However, its validity becomesquestionable when one deals with high density excitonsin 1D where the dynamics and/or size of the excitons areexpected to influence the annihilation rate. We introducea dimensionless exciton population (0 ≤ ζ <
1) definedas ζ ≡ N l x /L NT , where N is the number of excitons ina SWNT, L NT is the length of the SWNT, and l x is thecharacteristic length scale each exciton “occupies” in theSWNT. In a static limit, l x should simply be the excitonsize. However, recent experiments have indicated the im-portance of exciton diffusion [22, 23, 24]. In particular,Cognet et al . [24], via micro-PL studies on single micelle-suspended SWNTs of various chiralities, found that exci-tons diffusively traverse a substantial distance ( ∼
90 nm)within their lifetime. Therefore, we assume that l x is de-termined by the diffusion length and any two excitonsformed within l x undergo EEA. We also assume that anexciton promoted to a higher energy level as a result ofEEA returns to the E level with 100% probability dueto ultrafast and efficient E -to- E relaxation; i.e., twoexcitons become one exciton through EEA.Two limiting cases are considered, which we refer to asthe “instantaneous” and “steady-state” limits. In the in-stantaneous limit, the initial creation of excitons is com-pleted before EEA follows, while the steady-state limitcorresponds to CW excitation. The actual experimentalsituation is considered to be closer to the former. Assum-ing L NT ≫ l x , the PL saturation equation is given [21]for the instantaneous limit ψ = 1 e (cid:26) Ei (cid:18) ζ − ζ (cid:19) − Ei(1) (cid:27) , (1)and for the steady-state limit ψ = ζ − ζ exp (cid:18) ζ − ζ (cid:19) (cid:20) ζ ≡ I PL c , ψ ≡ I pump c (cid:21) . (2)These are implicit equations relating the PL intensity( I PL ) and pump intensity ( I pump ) in terms of their re-spective dimensionless parameters, ζ and ψ . They con-tain no fitting parameters other than the two linear scal-ing factors c and c and simply become a linear rela-tionship ( ψ = ζ ) in the low density limit ( ζ → ζ were obtained through this analysis forrespective data points shown in Fig. 2. Finally, the exci-ton density can be obtained through n x = ζ / l x .Using l x = 45 nm (one half of the excursion rangein [24]), we estimated n x for (6,5) to be 1.7 × , 1.3 × , and 1.1 × cm − for excitation with 570,615, and 654 nm, respectively, at a fluence of 1.02 × photons/cm in the instantaneous limit. Theseare much smaller than the expected Mott density n ∗ x ( ∼ × cm − , assuming exciton size ∼ as-created e - h pairs is estimated to be 1-2 × cm − in the cases ofresonant excitation, which is similar to n ∗ x . These obser-vations appear qualitatively different from GaAs QWRswhere PL saturation is not obvious until the formationof biexcitons and an e - h plasma [11]. Highly efficient and rapid EEA in SWNTs, which is consistent with the re-ported absence of biexciton signatures [26], is probablythe direct reason for such characteristic differences.In summary, we have studied PL and PLE spectra ofSWNTs at high exciton densities. Complete flatteningof PLE spectra and clear PL saturation were observed,indicating the existence of an upper density limit. Wedeveloped a model that reproduced the observed satura-tion behavior and allowed us to estimate exciton densi-ties, which remained more than one-order-of-magnitudelower than the expected Mott density and explained thestability of PL spectra in the saturated regime.The authors thank the Robert A. Welch Foundation(Grant No. C-1509) and NSF (Grant No. DMR-0325474)for support and H. Akiyama and K. Matsuda for valu-able discussions. One of us (Y.M.) thanks T. Okubo andS. Maruyama for their support for the fulfillment of hisJSPS program ∗ [email protected]; corresponding author. [1] T. Ogawa and T. Takagahara, Phys. Rev. B , 14325(1991); ibid. , 8138 (1991).[2] F. Rossi and E. Molinari, Phys. Rev. Lett. , 3642(1996).[3] F. Tassone and C. Piermarocchi, Phys. Rev. Lett. , 843(1999); C. Piermarocchi and F. Tassone, Phys. Rev. B , 245308 (2001).[4] S. Das Sarma and D. W. Wang, Phys. Rev. Lett. , 2010(2000); D. W. Wang and S. Das Sarma, Phys. Rev. B ,195313 (2001).[5] E. Kapon, D. M. Hwang, and R. Bhat, Phys. Rev. Lett. , 430 (1989).[6] W. Wegscheider, L. N. Pfeiffer, M. M. Dignam,A. Pinczuk, K. W. West, S. L. McCall, and R. Hull,Phys. Rev. Lett. , 4071 (1993).[7] R. Ambigapathy, I. Bar-Joseph, D. Y. Oberli, S. Haacke,M. J. Brasil, F. Reinhardt, E. Kapon, and B. Deveaud,Phys. Rev. Lett. , 3579 (1997).[8] J. Rubio, L. Pfeiffer, M. H. Szymanska, A. Pinczuk,S. He, H. U. Baranger, P. B. Littlewood, K. W. West,and B. S. Dennis, Solid State Commun. , 423 (2001).[9] H. Akiyama, L. N. Pfeiffer, M. Yoshita, A. Pinczuk, P. B.Littlewood, K. W. West, M. J. Matthews, and J. Wynn,Phys. Rev. B , 041302 (2003).[10] T. Guillet, R. Grousson, V. Voliotis, M. Menant, X. L.Wang, and M. Ogura, Phys. Rev. B , 235324 (2003).[11] Y. Hayamizu, M. Yoshita, Y. Takahashi, H. Akiyama,C. Z. Ning, L. N. Pfeiffer, and K. W. West, Phys. Rev.Lett. , 167403 (2007).[12] S. Iijima and T. Ichihashi, Nature , 603 (1993).[13] F. Wang, G. Dukovic, L. E. Brus, and T. F. Heinz, Sci-ence , 838 (2005).[14] J. Maultzsch, R. Pomraenke, S. Reich, E. Chang,D. Prezzi, A. Ruini, E. Molinari, M. S. Strano, C. Thom-sen, and C. Lienau, Phys. Rev. B , 241402 (2005).[15] F. Wang, G. Dukovic, E. Knoesel, L. E. Brus, and T. F.Heinz, Phys. Rev. B , 241403 (2004).[16] G. N. Ostojic, S. Zaric, J. Kono, V. C. Moore, R. H.Hauge, and R. E. Smalley, Phys. Rev. Lett. , 097401(2005).[17] Y.-Z. Ma, L. Valkunas, S. L. Dexheimer, S. M. Bachilo,and G. R. Fleming, Phys. Rev. Lett. , 157402 (2005).[18] Y. Miyauchi and S. Mauyama, Phys. Rev. B , 035415(2006).[19] T. W. Roberti, N. J. Cherepy, and J. Z. Zhang, J. Chem.Phys. , 2143 (1998).[20] C. Manzoni, A. Gambetta, E. Menna, M. Meneghetti,G. Lanzani, and G. Cerullo, Phys. Rev. Lett. , 207401(2005).[21] Y. Murakami and J. Kono, in preparation.[22] C.-X. Sheng, Z. V. Vardeny, A. B. Dalton, and R. H.Baughman, Phys. Rev. B , 125427 (2005).[23] R. M. Russo, E. J. Mele, C. L. Kane, I. V. Rubtsov,M. J. Therien, and D. E. Luzzi, Phys. Rev. B , 041405(2006).[24] L. Cognet, D. A. Tsyboulski, J. R. Rocha, C. D. Donyle,J. M. Tour, and R. B. Weisman, Science , 1465(2007).[25] V. Perebeinos, J. Tersoff, and P. Avouris, Phys. Rev.Lett. , 257402 (2004).[26] K. Matsuda, T. Inoue, Y. Murakami, S. Maruyama, andY. Kanemitsu, Phys. Rev. B77