SUPPLEMENTARY INFORMATION Phonon populations and electrical power dissipation in carbon nanotube transistors Supplemental Information Mathias Steiner 1, Marcus Freitag 1, Vasili Perebeinos 1, James C. Tsang 1, Joshua P. Small 1, Megumi Kinoshita 1, Dongning Yuan 2, Jie Liu 2 and Phaedon Avouris 1* * avouris@us.ibm.com 1 IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598, USA 2 Department of Chemistry, Duke University, North Carolina 27708, USA (Date 01/13/2009) 1. Device characterization, electrical and optical measurements Spatially isolated, single-walled carbon nanotubes (CNTs) were grown by chemical vapor deposition on degenerately doped silicon substrates with 1μm SiO 2 using methane and Fe/Mo catalyst similar to the method described in [1]. Pd/Ti source and drain electrodes were fabricated by e-beam lithography. Two devices, 1 (channel length L 1 =5μm) and 2 (L 2 =3μm), fabricated on the same CNT, were used for the measurements reported here. Throughout the manuscript, we normalized the electrical power by the respective device length L 1,2 (W/m). A device schematic is shown in Supplemental Figure 1 a, followed by a scanning electron microscope (SEM) image of device 1 (see Supplemental Figure 1 b). Electrical transport characteristics measured on device 1 using nature nanotechnology www.nature.com/naturenanotechnology 1
supplementary information an Ithaca current preamplifier are shown in Supplemental Figure 1 c. From Raman measurements on the CNT (see Supplemental Figure 1 d), the tube diameter has been determined to be 1.5nm. We measured the Raman spectra using a scanning optical microscope equipped with a standard microscope objective (100x, NA=0.8, Nikon) providing a focal spot diameter of about 0.5 micron and a power density of up to 150 kw/cm 2. A feedbackcontrolled piezoelectric scanning stage (P-527.2CL, PI) accomplished raster scanning of the CNT-device with respect to the microscope objective with nanometer-precision. As excitation light sources, we used an Ar + laser (Innova 300, Coherent) operated at 476.5 nm and 514.5 nm, respectively, a frequency-doubled Nd:YVO4 laser (Millenia, Spectra- Physics) at 532 nm, a tunable dye laser (CR-599 operated with R6G solution, Coherent), as well as a HeNe laser (1144 P, JDS Uniphase) operated at 632.8 nm. The laser light was tightly focused from the top onto the CNT close to the center of the device between source and drain electrodes (see Supplemental Figure 1 b). The Raman-scattered light was separated from the laser excitation light using suitable holographic notch filters (Kaiser) and spectrally analyzed using either (1) a spectrograph (Triax 322, Jobin Yvon/Horiba) equipped with gratings having a groove density of 300 mm -1 and 1200 mm -1 and a LN 2 -cooled charge coupled device (Spectrum One, Jobin Yvon/Horiba) or (2) a spectrographic stage (of the triple-stage spectrograph XY, Dilor) equipped with a grating having a groove density of 1200 mm -1 and a LN 2 -cooled and IR-enhanced deepdepleted charge coupled device (Jobin Yvon/Horiba). With (1), we achieve a spectral resolution of 8cm -1 (FWHM, 1200 line/mm grating) while with (2) we obtain 1.8cm -1 (see, for example, RBM spectra in Supplemental Figure 1 d). We chose acquisition times 2 nature nanotechnology www.nature.com/naturenanotechnology
supplementary information ranging between 30 seconds and 300 seconds to achieve adequate signal-to-noise-ratios in the measured Raman spectra. A corresponding background spectrum measured with the same set of acquisition parameters was subtracted from each measured Raman spectrum to improve data quality. 2. Data analysis and fit procedure We improved the fit quality for the experimental RBM resonance Raman excitation spectra by correcting for systematic errors due to optical realignment. The corrections include a constant scaling of the Raman intensities for both RBM Stokes and anti-stokes signals for a given laser excitation energy, such that the resulting set of fit parameters (E 33, γ 33 ) in Eq. (1) and (4) of our manuscript reproduce both RBM and G-band resonance Raman excitation spectra with reduced noise. The corrections are up to 10% and do not affect the reported values of the RBM temperature as well as the reported (E 33, γ 33 )- values, but improve significantly the fit quality (see Fig. 1 b of the manuscript). The uncertainty in the relative anti-stokes/stokes intensity ratio X ( I A IS) ( I I ) Δ = translates A S into an uncertainty of the measured phonon temperature T ph : ( kt B ph ) Δ = X ( kt ) 2 B ph hω ph (S1) For example, as the temperature of the RBM phonon increases from 300 K to 700 K, the absolute uncertainty of the RBM temperature increases by a factor of 5.4. Correspondingly, the absolute uncertainties of the measured G-phonon temperatures are expected to be one order of magnitude smaller due to the much larger phonon energy. nature nanotechnology www.nature.com/naturenanotechnology 3
supplementary information 3. E 33 excited state decay pathways and line widths We calculate the line width of the E 33 excited state using [2]: 2 γ ph( E ) Ψ33 Vexc ph Ψ q (1 + nq) ρ( E hωq) + nqρ( E + h ωq) (S2) Here, ψ 33 and ψ q are the wave functions of the optically active exciton originating from the third CNT subband (E 33 ) and the finite momentum q-excitonic state, respectively, which are coupled by the exciton-phonon interaction V exc-ph. The joint exciton-phonon density of states is labeled ρ. The temperature dependence is captured by the phonon occupation numbers n q. The squared exciton-phonon coupling matrix element of the zone boundary K-optical phonon (Raman D-mode) is twice as large as that of the zone center G-phonon mode [3, 4] and, therefore, the phonon decay of the excited states is dominated by the K-optical phonon [5]. We performed numerical calculations in order to evaluate the phonon broadening γ ph at E=E 33. The joint density of states ρ is evaluated using the same γ ph which is determined self-consistently in Equation (S2). As in reference [2], we use a tight-binding approach for the electronic states assuming t 0 =3.0 ev and the Su-Schriefer-Heeger model for the electron-phonon coupling assuming g=5.3 ev/å. Note, that the solution of Equation (S2) in the self-consistent Born-approximation implies an energy dependence of the γ ph while the data in Figure 1 b of the main manuscript has been fitted assuming an energy-independent width. Hence, we expect that inclusion of the energy dependence of the width and the polaronic shift [4] in the fitting function would significantly improve the fit quality in Fig. 1b of the manuscript. However, the ratios of the RBM-intensities that have been used to extract RBM phonon temperatures are not affected. 4 nature nanotechnology www.nature.com/naturenanotechnology
supplementary information The E 33 decay pathways are shown in Supplemental Figure 2. The zone-boundary phonons at the K-point dominate the decay. We find that acoustic phonons contribute about 4% to the total width of the E 33 state at room temperature while acoustic zone boundary phonons contribute about 8%, zone boundary optical phonons (K) about 50%, and G-phonon about 35%, respectively, almost independently of the nanotube s chirality as shown in Supplemental Figure 3. The average value of the calculated lifetime broadening due to the phonons is γ ph 30.5 mev as shown in Supplemental Figure 3. In addition to the phonon broadening, the E 33 -state can also decay into the continuum of free electron-hole states in the first and second electronic bands. We estimate the electronic contribution to line-width broadening γ el by calculating absorption spectra based on solutions of the Bethe-Salpeter equation (see Supplemental Figure 4). The electronic width γ el depends on the alignment of the E 33 resonance with respect to the bottom of the Δ 11 and Δ 22 electron-hole continuum bands. The latter depends on the chirality of the tube, giving rise to γ el -variations around the average value of γ el 5 mev, as shown in Supplemental Figure 5. As a result, the total width of γ 33 =γ ph +γ el 35 mev is in very good agreement with the measured value of 37 mev. We find that the broadening of the E 33 resonance due to the electric field F effect which facilitates exciton ionization [6] is rather small: For F=2 V/μm, we obtain an additional electronic broadening of only about 1 mev (see Supplemental Figure 4). nature nanotechnology www.nature.com/naturenanotechnology 5
supplementary information Supplemental figures Supplemental Figure 1. The carbon nanotube (CNT) field effect transistor. a, Device schematic. b, Scanning electron microscopy (SEM) image of device 1 c, Electrical transport characteristics and gate dependence of device 1. d, High-resolution anti-stokes and Stokes Raman spectra, respectively, reveal the transition bands associated with the radial breathing mode (RBM) of the CNT imaged in b. From the measured center frequency Ω RBM =166cm -1, the CNT-diameter d t is determined to be d t =248/Ω RBM =1.5nm [7]. 6 nature nanotechnology www.nature.com/naturenanotechnology
supplementary information x 1/4 x 1/4 K TO E 33 decay path K' acoustic at Γ RBM Γ K acoustic zone boundary G 0.00 0.05 0.10 E ph (ev) 0.15 0.20 Supplemental Figure 2. Decay channels of the E 33 state. Phonon decay pathways for the E 33 resonance in a (19, 0)-CNT. The decay into the zone center acoustic phonon and the RBM, the zone boundary acoustic phonon, the zone boundary optical K-phonon, and G-phonon contributes to the total decay rate by about 4%, 8%, 53%, and 35%, respectively. The inset shows pockets of the Brillouin zone of the phonons participating in the E 33 decay (the decay probability is proportional to the symbol size that has been reduced by a factor of four for the strongest contributions). nature nanotechnology www.nature.com/naturenanotechnology 7
supplementary information γ ph (mev) phonon decay path (%) 40 35 30 25 20 36 32 55 50 45 40 4 2 G phonon K TO phonon acoustic phonon 0 1.40 1.45 1.50 d (nm) 1.55 1.60 Supplemental Figure 3. Chirality-dependence of the E 33 phonon decay rate γ ph. Top: Chirality-dependence of the E 33 phonon decay rate (blue circles). The red square indicates the experimental value. The solid line shows an average decay rate of 30.5 mev. Three bottom panels: Chirality-dependence of the E 33 decay into G-phonon, K TO phonon and acoustic phonon plus RBM. The solid lines indicate the mean values around 35.6 %, 50.8%, and 4.1% respectively. 8 nature nanotechnology www.nature.com/naturenanotechnology
supplementary information 1.6 1.4 ε 2 (arb. units) 1.2 1.0 0.8 0.6 0.4 F = 0 V/μm F = 2 V/μm 0.2 0.0 1.98 2.00 2.02 2.04 2.06 2.08 E (ev) Supplemental Figure 4. Absorption spectra of the E 33 state accounting only for electronic decay. Absorption spectra of a (19, 0)-CNT accounting for the electronic broadening mechanism introduced by the E 33 exciton coupling to the continua of Δ 11 and Δ 22 free electron-hole states. The circles correspond to solutions of the Bethe-Salpeter equation for F=0.0 V/μm (red) and F=2.0 V/μm (green), respectively, while the solid curves are Fano-fits with γ el =5.3 mev and γ el =6.2 mev, respectively. nature nanotechnology www.nature.com/naturenanotechnology 9
supplementary information 8 6 γ el (mev) 4 2 0 1.40 1.45 1.50 1.55 1.60 d (nm) Supplemental Figure 5. Chirality-dependence of the E 33 electronic decay rate γ el. Chirality-dependence of the E 33 electronic lifetime broadening (red circles). The solid line shows the average lifetime broadening of 5.1 mev. References [1] Huang, S., Cai, X. & Liu, J. Growth of Millimeter-Long and Horizontally Aligned Single-Walled Carbon Nanotubes on Flat Substrates. J. Am. Chem. Soc. 125, 5636-5637 (2003). [2] Perebeinos, V., Tersoff, J. & Avouris, Ph. Effect of Exciton-Phonon Coupling in the Calculated Optical Absorption of Carbon Nanotubes. Phys. Rev. Lett. 94, 027402 (2005). [3] Piscanec, S., et al. Kohn Anomalies and Electron-Phonon Interactions in Graphite. Phys. Rev. Lett. 93, 185503 (2004). [4] Perebeinos, V., Tersoff, J. & P. Avouris Ph. Electron-phonon interaction and transport in semiconducting carbon nanotubes. Phys. Rev. Lett. 94, 086802 (2005). [5] Hertel, T., et al. Intersubband decay of 1-D exciton resonances in carbon nanotubes. Nano Lett. 8, 87-91 (2008). [6] Perebeinos, V. & Avouris, Ph. Exciton ionization, Franz-Keldysh, and Stark effects in carbon nanotubes. Nano Lett. 7, 609-613 (2007). [7] Jorio, A., et al. Structural (n, m) Determination of Isolated Single-Wall Carbon Nanotubes by Resonant Raman Scattering. Phys. Rev. Lett. 86, 1118-1121 (2001). 10 nature nanotechnology www.nature.com/naturenanotechnology