We report a measurement on quantum capacitance of individual semiconducting
and small band gap SWNTs. The observed quantum capacitance is remarkably
smaller than that originating from density of states and it implies a strong
electron correlation in SWNTs

Cui, Xiaodong

Dai, Junfeng

Li, Jun

Zeng, Hualing

Applied Physics Letters

English

arXiv

Junfeng Dai

Jun Li

Hualing Zeng

Xiaodong Cui

Crossref

Measurements on quantum capacitance of individual single walled carbon nanotubes

We report measurements of the quantum capacitance of individual semiconducting and small band gap single walled carbon nanotubes (SWNTs). The observed quantum capacitance, 82 aF/μm for a semiconducting SWNT with chiral index (16,8) and 10.3 aF/μm for a small band gap SWNT upon Fermi level lying at the first subband are remarkably smaller than those originating from the density of states. We attribute the discrepancy to a strong electron correlation in SWNTs and derive the Luttinger parameter g of 0.25-0.3 for the (16,8) SWNT and of 0.32 for a small band gap SWNT. © 2009 American Institute of Physics.published_or_final_versio

Dai, J

Li, J

Zeng, H

Cui, X

HKU Scholars Hub

Title Measurements on quantum capacitance of individual singlewalled carbon nanotubesAuthor(s) Dai, J; Li, J; Zeng, H; Cui, XCitation Applied Physics Letters, 2009, v. 94 n. 9Issued Date 2009URL http://hdl.handle.net/10722/59600Rights Creative Commons: Attribution 3.0 Hong Kong LicenseMeasurements on quantum capacitance of individual single walledcarbon nanotubesJunfeng Dai, Jun Li, Hualing Zeng, and Xiaodong CuiaDepartment of Physics, The University of Hong Kong, Hong Kong, People’s Republic of ChinaReceived 23 October 2008; accepted 8 February 2009; published online 6 March 2009We report measurements of the quantum capacitance of individual semiconducting and small bandgap single walled carbon nanotubes SWNTs. The observed quantum capacitance, 82 aF /m fora semiconducting SWNT with chiral index 16,8 and 10.3 aF /m for a small band gap SWNTupon Fermi level lying at the first subband are remarkably smaller than those originating from thedensity of states. We attribute the discrepancy to a strong electron correlation in SWNTs and derivethe Luttinger parameter g of 0.25–0.3 for the 16,8 SWNT and of 0.32 for a small band gapSWNT. © 2009 American Institute of Physics. DOI: 10.1063/1.3093443In the classical regime, capacitance describes the ob-ject’s storage capability of electric charges, and it is solelydetermined by electrostatic force, eventually by the geometryand the medium’s dielectric constant. As the object’s spatialdimensions shrink to nanometer scale where quantum effectshave to be taken account, the states accommodating chargesare numbered and the electric charges are regulated by thenumber of available states as well as electrostatic force. ThePauli exclusion principle prohibits electrons/holes from oc-cupying the same states. Besides the modification stemmingfrom finite density of states DOS, electron-electron interac-tion may give a further correction on the quantum capaci-tance under some circumstance. For these nanometer scalesamples and low dimensional objects such as two-dimensional electron gas,1,2 nanotubes, nanowires,3–5nanojunctions,2,6 and quantum dots,7,8 their quantum capaci-tance is comparable to the classical one and therefore thequantum capacitance measurement as a probe to fundamentalaspects of electronic properties is possible and has been dis-cussed theoretically and experimentally since late 1980s. Re-cently Ilani et al. first reported their observation of quantumcapacitance of a top-gated individual carbon nanotube.4 Inthis letter, we present our experimental measurements on ca-pacitance of individual microscopically long single walledcarbon nanotubes SWNTs. The measured capacitance is re-markably less than the simulated one from DOS based ontight-binding calculation, and the difference may result fromthe strong electron correlation.Individual macroscopically long and diluted SWNTswere grown on a degenerately doped Si /SiO2 oxidethickness=300 nm substrate using the chemical vapordeposition method. The catalyst was prepared byselectively dipping the diluted solution of FeCl3 on thesilicon substrate and then by being reduced underAr /H2 400 SCCM /50 SCCM SCCM denotes standardcubic centimeters per minute at STP at 900 °C for 20 min.Individual SWNTs were formed on the substrates in ethanolvapor with the same gas mixture at 900 °C for 1 h. Electriccontacts to SWNTs were made by standard optical lithogra-phy, Ti/Al metal deposition on top of as-prepared SWNTs onsubstrates, and finally lift off. The spacing between metalelectrodes varies from 10 m to hundreds of micrometers.For these semiconducting SWNTs contacting to two or moreelectrodes, field effect transistors FETs are formed with thedegenerated silicon substrate as a back gate electrode. Elec-tric characterization of these long channel up to 1000 mFETs shows standard p-type or ambipolar behaviors at roomtemperature. It implies that structural discontinuity or highelectric barriers due to defects are rare along SWNTs up to1000 m. The capacitance measurement was carried out byadding a small sinusoidal bias on the back gate and monitor-ing the displace current through the SWNT connected elec-trodes. To get rid of contribution from the electrodes’ capaci-tance and the parasite capacitance of the background, wefeed both displace currents from a SWNT connected elec-trode and from a suspended electrode of the identical geom-etry to a balanced amplifier, as sketch in Fig. 1. So the cur-rent arising from the capacitance of electrodes was canceledout and the metal-oxide semiconductor capacitor like behav-ior of silicon substrates was eliminated as well. Finally thebalanced signal was fed to a lock-in amplifier, which extractsthe signal amplitude and phase information, and the capaci-tance and resistance were analyzed. All electric measure-ments were carried out in vacuum condition 10−2 Pa atthe temperature range of 77–300 K. The geometric structureof the specific SWNTs was in situ identified by resonantmicro-Raman scattering where 532 and 633 nm lasers wereselected as excitation sources.9aAuthor to whom correspondence should be addressed. Electronic mail:xdcui@hku.hk.FIG. 1. Color online a Schematic of the capacitance measurement. Basedon the equivalent circuit model b, the capacitance can be derived. c ASEM image of devices. Inset is a zoom-in of an electrode connected with anindividual SWNT white.APPLIED PHYSICS LETTERS 94, 093114 20090003-6951/2009/949/093114/3/$25.00 © 2009 American Institute of Physics94, 093114-1Downloaded 25 Nov 2011 to 147.8.21.150. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsFigure 2 shows the measured capacitance of a singleSWNT 16,8 as a function of Vg at various temperatures.The curves exhibit a zero baseline at low gate bias, corre-sponding to the “off” state of standard SWNT-FETs. In thiscase, the Schottky barrier between electrodes and the SWNTprevents electrons/holes from charging and meanwhile theFermi level lies amid the SWNT band gap and therefore nostates available to accommodate charges. While Vg increasestoward either negative or positive, the Schottky barrier atelectrode contacts becomes thinner and lower and the Fermilevel approaches to the first subband and therefore electrons/holes start to charge the SWNT. Consequently the capaci-tance exhibits remarkable increase within the gate voltagelimits of 10 V until a plateau of 8 pF appears at Vg−6 V, where the first subband is fully involved. Noticeablythe capacitance at “on” state keeps unchanged at the tem-perature range of 77–160 K.Comparing with the signal from the capacitance the Xcomponent of the lock-in out-out, the quadrature componentexhibits negligible change in the whole gate voltage range at77 K. It implies the resistance of the whole system is negli-gibly small compared to 1 /C=1 kHz, which is consis-tent with independent electric characterization, and thereforeSWNTs are under equilibrium at the testing frequency eventhough electrons propagate diffusively along the SWNT.Since the capacitance is a measure of charging effect,whether electron transport is ballistic or diffusive does notaffect the capacitance measurement as long as time scale islong enough for charging. For the presented SWNT of330 m long measured by scanning electron microscopySEM, it is however questionable that there exist defectsand local disorders, which act as local barriers to low energyelectrons/holes. This question was examined by studying thecapacitance dependence on the ac bias amplitude. Figure 2cshows that the on state capacitance measured at various acamplitudes keeps constant in the amplitude range of 0.8–2 V.While the ac amplitude is lower than 0.7 V, the capacitancedrops significantly due to existence of barriers either againstpropagating electrons/holes or against charge injection andconsequently the effective length of SWNT shrinks. Once theac amplitude is higher than 0.8 V, barriers vanish and theeffective SWNT length expands to the whole physical lengthof 330 m. The fact that the capacitance at on state keepsunchanged at various temperature also supports this scenario.Another possible factor affecting our measurement onintrinsic SWNT capacitance lies in the interface capacitancebetween metal electrode and the SWNT, which can betreated as a serial capacitor connecting to the SWNT. Toestimate this capacitance, we performed a direct measure-ment on several SWNT-FET structures, as sketched in Fig.3a. A small low frequency ac bias was fed into the drainelectrode. The current through the source electrode wasmonitored by a lock-in amplifier. The system capacitanceand the resistance could be extracted from the current ampli-tude and phase information according to LRC circuit model.Figure 3b shows the conductance and phase through aSWNT-FET with a source-drain channel of 10 m as afunction of Vg. Note that the phase signal at low gate voltagecorresponding to off state is overwhelmed by electronicnoise. Since the inductance of SWNTs is estimated at tens ofnH /m Kinetic inductance,10 RL, its contribution tothe impedance is negligible at low frequency. Based on thesimplified RC circuit analysis, we can obtain the interfacecapacitance around 10 pF at maximum at on state, where theSchottky barrier becomes thinnest. Measurements over sev-eral SWNT-FET structures with various channel length showsimilar results on the interface capacitance. As the interfacecapacitance ~pF is three orders of magnitude greater thanthat the whole capacitance ~fF, as shown in Fig. 2, itscontribution to the total capacitance is negligible.To extract the quantum capacitance of single SWNT16,8, we assume that 1 Ctotal−1=Cg−1+CQ−1, where Cg andCQ are the geometric and quantum capacitance, respectively,and 2 Cg=2L /cosh−12h /d2L / ln4h /d, where dand L are the SWNT diameter of 1.67 nm and length of330 m, h=300 nm further confirmed with SEM imagingis the thickness of oxide layer and the dielectric permittivity=3.90 for SiO2, and 3 the whole capacitance is sum ofFIG. 2. Color online a The measured capacitance of a SWNT of 330 mas a function of gate voltage at various temperatures. b The G-mode Ra-man scattering of the SWNT with the excitation laser of 532 nm. Radialbreathing mode RBM cannot be obtained owing to off resonance. TwoLorentzian fittings in green are used to find the G− and G+ position.The diameter is obtained by the relation of d=G+ −G− /C, where C=47.7 cm−1 for semiconducting SWNTs Ref. 9. With the aid of Katauraplot, we identified its chiral index as 16,8 Ref. 9. c The measuredcapacitance as a function of ac bias amplitude at 77 K. Error bars give thesignal range averaged from many runs. d A simulated DOS of a SWNT16,8 and the corresponding capacitance arising from DOS as a function ofFermi energy, based on a zone folded tight-binding calculation.FIG. 3. Color online a Schematic of the measurement of the capacitancebetween electrodes and the SWNT. b The measured current amplitude andphase. c The extracted interface capacitance and contact resistance.093114-2 Dai et al. Appl. Phys. Lett. 94, 093114 2009Downloaded 25 Nov 2011 to 147.8.21.150. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionssegments in parallel. Accordingly we obtained the quantumcapacitance CQ for the SWNT 16,8 and the quantum ca-pacitance rises from zero at low gate bias to 82 aF /m atFermi level sitting at its first subband. It is however signifi-cantly smaller than the simulated value arising from DOScalculated from the zone folded tight-binding simulation,11as shown in Fig. 2d, where we defineCDOSu =dQudu= e0  fV − uu	 DOSVdV .Here fV−u denotes the Fermi–Dirac distribution andnearby carbon-carbon interaction 0=2.9 eV and overlap in-tegral s=0.129 were taken in the DOS calculation.11Figure 4a shows the measured capacitance as a func-tion of Vg of another individual SWNT. Its Raman signalcould not be obtained as its optical transition does not reso-nate with the excitation laser, and the exact geometric indexwas unknown. The electric characteristic curve conductanceas a function of gate voltage, not shown shows metallic atroom temperature but semiconducting at below 160 K.Therefore we can identify the SWNT with small band gapSWNTs which has geometric index n ,m satisfying n−m=3	 integer.12,13 This kind of SWNTs are expected to bemetallic by tight binding calculation but are in fact semicon-ducting due to curvature effect that couples P and  orbitsto open a small gap. The capacitance measurements shown inFig. 4a tell the band gap disappears above or at 160 K.Following the similar treatment as described earlier, one canextract the quantum capacitance of 10.3 aF /m for thisSWNT upon the Fermi level lying in the first subband. Thequantum capacitance of metallic SWNTs arising from theDOS can be estimated as follows. The energy for accommo-dating one electron is supposed to be equal to that of electronat Fermi level, say e2 /CDOS=VFK=VF2 /L, then onegets the quantum capacitance per unit length considering thespin degeneracy CDOS /L=2e2 /hVF96 aF /m, whereVF is the Fermi velocity and is set to be 8	105 m /s formetallic SWNTs.The quantum capacitance obtained in the experiments isremarkably smaller than that from the DOS simulated withzone folded tight-binding calculation. It implies that a singleparticle model cannot well describe the situation here andstrong electron correlation must be considered. Theoreticallya one-dimensional interacting electron gas system is de-scribed by Tomonaga–Luttinger liquid instead of Fermi gas,where electron-electron interaction qualitatively instead ofperturbatively modifies electric properties due to the reduceddimensionality. Its validation on carbon nanotubes wasproved by independent experiments.14,15 In Luttinger liquidmodel, charges propagate in form of charge density wavewith a velocity that is different from Fermi velocity and isdetermined by the strength of the interaction, saying VC=VF /g, where VC denotes the charge velocity and g is adimensionless parameter describing the electron-electron in-teraction strength and therefore the compressibility of theliquid.16 As the electron correlation affect the compressibilityof the liquid, the velocity and consequently the capacitance,the factor g can be extracted by g=CQ /CDOS.17 Taking thismodel in mind, we estimate at 77 K g in range of 0.25–0.3for SWNT 16,8 and g around 0.32 for the small band gapSWNT in Fig. 4 upon the Fermi level lying at the first sub-band. The values agree with previously reportedexperiments.4,15,18,19In conclusion, we developed an experimental method tostudy the capacitance of individual carbon nanotubes. Weobtained the quantum capacitance of 82 aF /m for a semi-conducting SWNT with chiral index of 16,8 and of10.3 aF /m for a small band gap SWNTs upon Fermi levelsitting around the first subband. The observed quantum ca-pacitance is remarkably smaller than the capacitance origi-nating from DOS and it implies a strong electron correlationin SWNTs. The Luttinger parameter g of 0.25–0.3 for theSWNT of 16,8 and g=0.32 for the small band gap SWNTcan be derived.We thank Guanhua Chen, Jian Wang, and Liwei Chenfor helpful discussion. The work was supported by HongKong GRF under Grant No. HKU 701909.1S. Luryi, Appl. Phys. Lett. 52, 501 1988.2M. Buttiker, J. Phys.: Condens. Matter 5, 9361 1993.3J. Wang, H. Guo, J.-L. Mozos, C. C. Wan, G. Taraschi, and Q. Zheng,Phys. Rev. Lett. 80, 4277 1998.4S. Ilani, L. A. K. Donev, M. Kindermann, and P. L. McEuen, Nat. Phys. 2,687 2006.5R. Tu, L. Zhang, Y. Nishi, and H. Dai, Nano Lett. 7, 1561 2007.6J. G. Hou, B. Wang, J. Yang, X. R. Wang, H. Q. Wang, Q. Zhu, and X.Xiao, Phys. Rev. Lett. 86, 5321 2001.7P. L. McEuen, E. B. Foxman, U. Meirav, M. A. Kastner, Y. Meir, N. S.Wingreen, and S. J. Wind, Phys. Rev. Lett. 66, 1926 1991.8R. C. Ashoori, H. L. Stormer, J. S. Weiner, L. N. Pfeiffer, S. J. Pearton, K.W. Baldwin, and K. W. West, Phys. Rev. Lett. 68, 3088 1992.9A Jorio, R. Saito, J. H. Hafner, C. M. Lieber, M. Hunter, T. D. McClure,G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. Lett. 86, 1118 2001.10P. J. Burke, IEEE Trans. Nanotechnol. 1, 129 2002.11R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 61, 29812000.12N. Hamada, S. Sawada, and A. Oshiyama, Phys. Rev. Lett. 68, 15791992.13C. Zhou, J. Kong, and H. Dai, Phys. Rev. Lett. 84, 5604 2000.14S. J. Tans, M. H. Devoret, R. J. A. Groeneveld, and C. Dekker, NatureLondon 394, 761 1998.15H. Ishii, H. Kataura, H. Shiozawa, H. Yoshioka, H. Otsubo, Y. Takayama,T. Miyahara, S. Suzuki, Y. Achiba, M. Nakatake, T. Narimura, M. Higash-iguchi, K. Shimada, H. Namatame, and M. Taniguchi, Nature London426, 540 2003.16C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 68, 1220 1992.17Y. M. Blanter, F. W. J. Hekking, and M. Buttiker, Phys. Rev. Lett. 81,1925 1998.18M. Bockrath, D. H. Cobden, J. Lu, A. G. Rinzler, R. E. Smalley, L.Balents, and P. L. McEuen, Nature London 397, 598 1999.19Z. Yao, H. W. C. Postma, L. Balents, and C. Dekker, Nature London402, 273 1999.-10 -5 0 5 100.00.51.01.52.02.53.0Capacitance(fF)Vg (V)77 K120 K160 KFIG. 4. Color online The measured capacitance of the small band gapSWNT at various temperatures.093114-3 Dai et al. Appl. Phys. Lett. 94, 093114 2009Downloaded 25 Nov 2011 to 147.8.21.150. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions

http://hub.hku.hk/bitstream/10722/59600/1/content.pdf

Observation of Quantum Capacitance of individual single walled carbon
  nanotubes

Observation of Quantum Capacitance of individual single walled carbon nanotubes

Abstract

Similar works

Full text

Available Versions

Crossref

HKU Scholars Hub