Phonon populations and electrical power ... - Semantic Scholar

LETTERS PUBLISHED ONLINE: 1 MARCH 2009 | DOI: 10.1038/NNANO.2009.22

Phonon populations and electrical power dissipation in carbon nanotube transistors Mathias Steiner1, Marcus Freitag1, Vasili Perebeinos1, James C. Tsang1, Joshua P. Small1, Megumi Kinoshita1, Dongning Yuan2, Jie Liu2 and Phaedon Avouris1 * Carbon nanotubes and graphene are candidate materials for nanoscale electronic devices1,2. Both materials show weak acoustic phonon scattering and long mean free paths for lowenergy charge carriers. However, high-energy carriers couple strongly to optical phonons1,3, which leads to current saturation4–6 and the generation of hot phonons7. A non-equilibrium phonon distribution has been invoked to explain the negative differential conductance observed in suspended metallic nanotubes8, while Raman studies have shown the electrical generation of hot G-phonons in metallic nanotubes9,10. Here, we present a complete picture of the phonon distribution in a functioning nanotube transistor including the G and the radial breathing modes, the Raman-inactive zone boundary K mode and the intermediate-frequency mode populated by anharmonic decay. The effective temperatures of the high- and intermediate-frequency phonons are considerably higher than those of acoustic phonons, indicating a phonon-decay bottleneck. Most importantly, inclusion of scattering by substrate polar phonons is needed to fully account for the observed electronic transport behaviour. Using an individual carbon nanotube as the active channel of a field-effect transistor (FET) and spectrally analysing the Raman scattered light from it, we are able to monitor the changes in both the nanotube phonon spectrum and its electronic resonances while passing an electrical current through it. The Raman spectra of nanotubes are dominated by transitions associated with its radial breathing mode (RBM) and the tangential C–C stretching G mode11–13. From the Stokes/anti-Stokes intensities of the G and RBM we extract the respective phonon populations at different current levels. We also use the resonant enhancement of the Raman scattering of the RBM to map the E33 electronic resonance14–16. Electronic states such as E33 under current decay primarily due to their efficient coupling to zone-boundary (K) phonons17,18. The E33 width can, therefore, be used as a measure of the K-phonon population. In addition, the population of intermediate-frequency phonons (IFPs) is extracted from the current-induced broadening of the G phonons19. Finally, transport measurements provide new information regarding energy transfer from the nanotube to the substrate. Details regarding the nanotube device and the measurements are given in the Supplementary Information. The Raman anti-Stokes intensity, in which an existing phonon imparts its energy to an incoming photon, is proportional to the initial phonon population n, and the Stokes process, where a phonon is created, is proportional to n þ 1: IA ðEL Þ / nðEL þ hvph Þ3 jxA ðEL Þj2 IS ðEL Þ / ð1 þ nÞðEL hvph Þ3 jxS ðEL Þj2

ð1Þ

where IA and IS are the anti-Stokes and Stokes intensities, EL and hvph are the laser and phonon energies, and x S and x A are the Stokes and anti-Stokes Raman susceptibilities. The ratio of the two intensities can be used to measure the phonon temperature20. Using time reversal symmetry, x S(E) ¼ x A(E – hvph), we obtain the phonon population ni or an effective phonon temperature Ti of an individual mode i using 3 IS ðEL Þ 1 þ ni ðEL hvph Þ ¼ IA ðEL hvph Þ ni EL3 ðEL hvph Þ3 hvph ¼ exp EL3 kB T i

ð2Þ

For the G band (Fig. 1a), we find that the phonon temperature is proportional to the electrical power P (Fig. 1d): Ti ¼ Tsub þ P=gi

ð3Þ

The electrical power is P ¼ IVDS/L, where VDS is the source-drain voltage, IDS is the current and L 5 mm is the nanotube-FET channel length. Tsub is the ambient (substrate) temperature and 1/gi depends on the nanotube/substrate interface thermal coupling and the detailed phonon dynamics. The best fit in Fig. 1d gives gG ¼ 0.063 + 0.001 W m21 K21. Figure 1b shows integrated Stokes and anti-Stokes intensities for the RBM as a function of the laser excitation energy EL and electrical power levels P. From the frequency VRBM ¼ 166 cm21 (Supplementary Fig. S1), the nanotube diameter dt is determined to be dt ¼ 248/VRBM ¼ 1.5 nm (ref. 14). Using the Katauraplot13, we identify the electronic resonance observed in the Raman excitation spectra as the E33 state (2.1 eV) of a semiconducting nanotube. We fit each resonance Raman profile with a single electronic transition, characterized by its energy E and lifetime broadening g (see Supplementary Information for an extension of this form):

xS ðEL Þ ¼

A0 ðEL E33 þ ig33 ÞðEL E33 hvph þ ig33 Þ

ð4Þ

where A0 / p2ikMe2ph, and pik is the dipole matrix element for the optical transition and Me–ph is the electron–phonon coupling. Using equation (1) and equation (4), we fit the experimental resonance Raman excitation spectra (see Fig. 1b) and obtain the desired energies E33 and widths g33 (Fig. 1c) along with the relative intensities at the different electrical power levels (see Supplementary Information for details). From the intensities we extract the RBM phonon ‘temperature’, which is also found to be proportional to

1

IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598, USA, 2 Department of Chemistry, Duke University, North Carolina 27708, USA; * e-mail: [email protected]

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Figure 1 | Non-equilibrium temperature of G, RBM and K phonons in a carbon nanotube as a function of electrical power. a, G-band Stokes and anti-Stokes intensities for laser excitation energies of 2.156 and 1.959 eV, respectively, as a function of electrical power. b, RBM Stokes and anti-Stokes intensities as a function of the laser excitation energy for various electrical power levels. The Raman excitation spectra reveal the E33 resonance of the semiconducting nanotube. The dissipation of electrical power causes an excess population of optical phonons in the nanotube that effectively redshifts and broadens the E33 state. c, Experimental energies (E33) and widths (g33) of the electronic resonance as well as the Raman intensity A02 extracted from the data shown in b as a function of the dissipated electrical power per unit length. The simulation of the experimental decay rate (green solid curve in the middle panel of c) is based on the electronic width gel ¼ 5 meV and the calculated gph values that account for the coupling to all phonons at temperatures given by equation (3), with gRBM ¼ 0.11 W m21 K21 for acoustic phonons, gG ¼ 0.063 W m21 K21 for G phonons and gK’ ¼ 0.028 W m21 K21 for optical K phonons (fit parameter). As the phonon energies in the simulations do not include electron–phonon renormalization (Kohn-anomaly), we need to correct the fitted parameter gK ¼ gK’(180/160) ¼ 0.032 W m21 K21. d, As the total dissipated power along the nanotube increases, the linear temperature characteristics of all phonon modes reveal different slopes. The nanotube heats up in a mode-specific manner and is not in thermal equilibrium with the SiO2 substrate. The effective temperature of the K phonons (dashed line) has been extracted from a simulation of the experimental g33 , while the effective temperatures of G-band and RBM phonons have been derived by measuring the respective Stokes/anti-Stokes Raman scattering intensities.

the electric power. The best fit to equation (3) shown in Fig. 1d corresponds to gRBM ¼ 0.11 + 0.01 W m21 K21. The decay of high-energy excited states such as E33 involves both vibrational (temperature-dependent) and electronic (temperatureindependent) channels. Changes in E33 energy and width due to current flow result from an excess population of phonons excited by the current. The E33 peak position depends on both interband21 and intraband17 electron–phonon couplings. In addition, thermal lattice expansion modifies the E33 energy. In Fig. 1c (upper panel), we plot

the experimental E33 energies as a function of the power per unit length P and obtain a linear energy downshift. The downshift of the E33 energy with RBM temperature dE33/dTRBM ¼ –0.092 meV K21 obtained (see Fig. 1c, upper panel) is of similar size as the shift of the E22 under equilibrium thermal heating22. The width of the E33 peak of semiconducting nanotubes depends on the intraband exciton–optical phonon coupling. In particular, the optical K phonon (0.16 eV) is found to be primarily responsible for the decay of the higher-energy excited states17,18. We calculate


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Figure 2 | Raman shifts and spectral widths of carbon nanotube G-band phonons reveal the phonon temperatures of their decay products. a, Highresolution G-band spectra showing shifts and broadening of Gþ and G2 components as a function of electrical power injected into the nanotube. The G-band components are fitted individually by single Lorentzians. b, Raman shift (upper panel) and line widths (middle panel), respectively, of the Gþ and G2 transitions extracted from a and plotted as a function of dissipated electrical power per unit length for a L ¼ 3 mm channel length device fabricated from the same nanotube. The solid black curve is a fit to the anharmonic phonon decay model with the best fit corresponding to the IFP temperature TIFP ¼ Tsub þ P1.2/g* with g* ¼ 0.073, P in W m21 and Tsub in K. Lower panel: Effective temperature of the IFP populated by the decay of G-band phonons. The solid green curve represents the IFP temperature that gives the best fit of the G-phonon spectral width and the dashed black line is a linear fit to TIFP with gIFP ¼ 0.032 W m21 K21.

that acoustic phonons contribute only about 4% of the total E33 width at room temperature, whereas optical K phonons contribute 50% and G phonons about 35%. We can, therefore, extract an effective temperature of the Raman-inactive K phonon from the width of E33 in Fig. 1c by simulating the theoretical curve assuming that the K-phonon temperature follows equation (3), while acoustic phonon and G-phonon temperatures are taken from our Stokes and anti-Stokes measurements. We calculate the electronic contribution to the width to be about gel 5 meV (see Supplementary Information) almost independently of nanotube chirality and the electric field of up to 2 V mm21. The phonon contribution also shows little chirality dependence with an average gph 30 meV. The calculated room temperature width of gel þ gph 35 meV agrees remarkably well with the measured width g33 37 meV. The simulated width in Fig. 1c uses the measured values for the acoustic and the G-phonon temperatures, whereas gK ¼ 0.032 W m21 K21 is a fit parameter. The resulting K-phonon temperature as a function of electric power is shown in Fig. 1d. The reduction of the Raman susceptibility with electrical power (proportional to the A0 values, see Fig. 1c, lower panel) is likely to result from a loss of exciton oscillator strength with increasing nanotube screening under high field conditions23. 322

The observation that effective temperatures of electrically pumped optical (K and G) phonons are significantly higher than those of the acoustic phonon bath (RBM) can be explained by a bottleneck in their anharmonic decay. Specifically, their decay into long-lived (and Raman-inactive) IFPs inhibits full thermalization with the acoustic phonons of the nanotube. Figure 2a shows Raman spectra of the Gband where both Gþ and G2 transitions are clearly resolved. As the electrical power increases, both Gþ and G2 bands shift and broaden. The shifts (Fig. 2b) are linear with electric power. The corresponding slopes dhvGþ/dTRBM ¼ –0.021 cm21 K21 and dhvG2/dTRBM ¼ –0.017 cm21 K21 using the RBM temperatures from equation (3) are consistent with thermal heating results24. However, the widths of the Gþ and G2 bands as a function of electrical power (Fig. 2b, middle panel) show activated behaviour, as expected for decay into optical phonons19. We therefore fitted the data by accounting for the anharmonic phonon decay. Ab initio calculations of the G-phonon decay in graphene showed that it decays primarily into intermediatefrequency optical phonon bands centred at about 600 (IFP1) and 1,000 cm21 (IFP2) by 56%, and 800 cm21 (IFP3) by 44% (ref. 19). We assume that the same decay paths of G phonons operate in nanotubes. The decay rate depends on the IFP population nIFP as



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Figure 3 | Decay pathways of electrically excited phonon populations and electronic transport characteristics of a carbon nanotube on a substrate. a, Phonon decay paths for the E33 resonance in a (19, 0)-nanotube (dt ¼ 1.5 nm). The decay into the zone-centre 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 E33 decay (the decay probability is proportional to the symbol size which has been reduced by a factor of four for the strongest contributions). The excess population of the primarily excited optical K phonon (orange) decays as indicated by arrows into other phonon modes including the IFP (red), optical G-band phonons (green) and acoustic phonons such as the RBM (blue). Owing to the long-lived nature of the IFP with decay time in the range of 11 ps (ref. 19), the effective temperatures of acoustic phonon populations as represented by the RBM remain significantly below those of the optical phonons. b, Experimental drain–source current IDS as a function of the electric field EDS ¼ VDS/L (black circles) and simulated current assuming either scattering with only the nanotube phonons at elevated temperatures (red squares) or with hot nanotube phonons plus substrate surface polar phonons (SPP) kept at ambient temperatures (blue circles). We calculated the drift velocity in a (19, 0)-nanotube assuming a constant doping level of 0.1 e nm21.

Ganh ¼ G0(0.56(1 þ nIFP1 þ nIFP2) þ 0.44(1 þ 2nIFP3)), where G0 is the decay rate at T ¼ 0. From the width at zero bias, we find the intrinsic half-width of the G-phonons to be G0 ¼ 0.8 + 0.15 cm21. The main source of uncertainty is due to the instrumental width Winstr ¼ 1.8 + 0.3 cm21. The IFP population can now be extracted from the simulated curve that has been fitted to the measured line width, and the corresponding effective IFP temperature is shown in Fig. 2b. Although the IFP temperature does not scale linearly with electrical power it can still be well approximated by equation (3) with gIFP ¼ 0.032 + 0.004 Wm21 K21. We can now provide an overall picture of electrical power dissipation in a nanotube FET. Figure 3a shows how the electrically induced non-equilibrium populations of the directly pumped optical K and G phonons decay into long-lived IFPs. Subsequently, the IFP population decays into acoustic phonons, for example, the RBM. Also shown in Fig. 3a are a portion of the Brillouin zone and the computed spectra of the phonon bands contributing to the E33 decay. The effective temperatures of the optical phonons are elevated with respect to those in the acoustic phonon bath, which we assume is in thermal equilibrium with the RBM phonon. A key result is that the effective temperatures of all four phonon modes G, RBM, K and IFP show a linear electrical power dependence with different slopes gRBM . gG . gK gIFP (1/gRBM : 1/gG : 1/gK : 1/gIFP ¼ 1 : 1.7 : 3.4 : 3.4), which demonstrates clearly that they are driven out of thermal equilibrium. For a high power, P ¼ 40 W m21, the effective temperature of the K phonons and IFPs reach values as high as TK TIFP 1,500 K. At the same power, the G-band temperature approaches TG 950 K while the effective RBM temperature reaches values around TRBM 700 K. The latter result is in agreement with thermal dissipation studies on metallic nanotubes at high currents25. The elevation of the RBM temperature by dTRBM 400 K for P ¼ 40 W m21 indicates that the energy transport from the nanotube to the SiO2 substrate is far from being optimized. Variations in the interface thermal conductance can arise, for example, from differences in substrate structure and roughness, and can lead to significant variability of phonon populations along a nanotube. These can affect both the device operation and its stability. The hot phonons are expected to substantially affect nanotube electronic transport. In suspended nanotubes, they lead to negative differential conductance (NDC)8. The absence of NDC in our data,

together with our modelling, suggests that an additional scattering mechanism dominates transport. As we show below this mechanism involves energy transfer to the surface polar phonons (SPP) of the substrate26. We calculate the carrier drift velocity in the nanotube, udrift , as a function of the electric field27 using the power-dependent temperatures for acoustic, zone-boundary and zone-centre optical phonons determined from our experiments. We obtain the current using IDS ¼ Cg(Vg – Vth – 0.5VDS)udrift where Vth ¼ 6 V is the experimentally determined threshold voltage (see Supplementary Information) and Vg ¼ 0 V is the gate bias. The current–voltage characteristics shown in Fig. 3b are simulated with one adjustable parameter Cg in order to match the highest bias data point. The calculation that accounts for intra-nanotube electron–phonon scattering and hot phonons only (Fig. 3b, red squares) indeed shows NDC, which we do not observe experimentally (Fig. 3b, black circles). Therefore, in a second step, we perform simulations assuming both hot nanotube phonons and electron scattering by substrate surface optical phonons. Owing to the polar nature of the SiO2 gate dielectric, the carriers in the nanotube couple electrostatically to the long-range polarization field at the nanotube/SiO2 interface. The interaction occurs primarily with optical surface phonon modes28. We assume that the interaction term Vint (refs 26,29) is proportional to the dielectric ! polarization field P : ! ! R r !! Vint ¼ e P ð R Þ; ! !3 R r sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ! P / hvso 11 þ 1 10 þ 1

ð5Þ

! where R and ! r are substrate and nanotube coordinates, respectively, vso is the surface optical phonon frequency, and 10 and 11 are the low- and high-frequency dielectric constants of SiO2 , respectively. As a result, we find that NDC disappears (Fig. 3b, blue rings) and the drift velocity udrift saturates at 2 107 cm s21. Our experimental transport data are well reproduced assuming a gate coupling of Cg ¼ 0.1 pf cm21, in agreement with the geometrical gate capacitance.


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In conclusion, we have mapped the phonon populations and power dissipation pathways in a functioning carbon nanotube FET. We found a non-equilibrium distribution with hot optical, zone-boundary and intermediate-frequency phonons and relatively cool acoustic phonons. Furthermore, our transport data together with theoretical simulations implicate another, substrate-induced scattering mechanism, needed to successfully account for the electrical transport properties of supported nanotubes.

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Additional information Supplementary Information accompanies this paper at www.nature.com/ naturenanotechnology. Reprints and permission information is available online at http://npg. nature.com/reprintsandpermissions/. Correspondence and requests for materials should be addressed to P.A.
