Bandstructure E?ects in Multiwall Carbon Nanotubes
Bernhard Stojetz and Christoph Strunk
Institute of Experimental and Applied Physics, University of Regensburg, 93040 Regensburg, Germany
arXiv:cond-mat/0410764v1 [cond-mat.mes-hall] 29 Oct 2004
Csilla Miko and Laszlo Forr? o
Institute of Physics of Complex Matter, FBS Swiss Federal Institute of Technology (EPFL), CH-1015 Lausanne, Switzerland (Dated: February 2, 2008) We report conductance measurements on multiwall carbon nanotubes in a perpendicular magnetic ?eld. A gate electrode with large capacitance is used to considerably vary the nanotube Fermi level. This enables us to search for signatures of the unique electronic band structure of the nanotubes in the regime of di?usive quantum transport. We ?nd an unusual quenching of the magnetoconductance and the zero bias anomaly in the di?erential conductance at certain gate voltages, which can be linked to the onset of quasi-one-dimensional subbands.
Quantum transport in multiwall carbon nanotubes has been intensely studied in recent years [1, 2]. Despite some indications of ballistic transport even at room temperature [3, 4], the majority of experiments revealed typical signatures of di?usive quantum transport in a magnetic ?eld B such as weak localization (WL), universal conductance ?uctuations (UCF) and the h/2e-periodic Altshuler-Aronov-Spivak (AAS) oscillations [2, 5, 6, 7]. These phenomena are caused by the Aharonov-Bohm phase, either by coherent backscattering of pairs of timereversed di?usion paths (WL and AAS) or by interference of di?erent paths (UCF). In addition, zero bias anomalies caused by electron-electron interactions in the di?erential conductance have been observed . In those experiments, the multiwall tubes seemed to behave as ordinary metallic quantum wires. On the other hand, bandstructure calculations for singlewall nanotubes predict strictly one-dimensional transport channels, which give rise to van Hove singularities in the density of states. Experimental evidence for this has been obtained mainly by electron tunneling spectroscopy on single wall nanotubes . In this picture of strictly one-dimensional transport a quasiclassical trajectory cannot enclose magnetic ?ux and no low-?eld magnetoconductance is expected. Hence, the question arises how the speci?c band structure is re?ected in the conductance as well as in its quantum corrections and how those on ?rst glance contradictory approaches can be merged into a consistent picture of electronic transport. In this experiment, we use a strongly coupled gate, which is e?cient enough to shift the Fermi level through several quasi-onedimensional subbands. At certain gate voltages, which can be associated with the bottoms of the subbands, we observe a strong suppression of both the magnetoconductance and the di?erential conductance. The samples were produced on top of thermally oxidized Silicon wafers. First, Aluminium strips of 10 ?m width and 40 nm thickness were evaporated. Exposure to air provides an insulating native oxide layer of a few nm thickness. These strips serve as a backgate for the individual nanotubes, which are deposited from a chlo-
0.6 0.4 0.5 0.0 0.5 0.0 -1.0
-1 0 1 Gate Voltage (V)
Gate Voltage (V)
FIG. 1: (A) SEM image of a typical sample: individual multiwall nanotubes are deposited on a prestructured Al gate electrode and contacted by four Au ?ngers, which are deposited on top of the tube. The electrode spacing is 300 nm. For the measurements, only the two inner electrodes are used. (B) Room temperature conductance of sample A as a function of gate voltage in units of the conductance quantum 2e2 /h. The estimated position of the charge neutrality point corresponds to the minimum of conductance and is indicated by a grey line. (C) Same as in Fig. 1B, but for 10 K, 1 K and 30 mK (top to bottom). For the 10K curve, both the positions of the charge neutrality point (grey line) and the regions of quenched magnetoconductance (black lines) as observed in Fig. 2 are indicated.
roform suspension in the next step. Electric contacts are de?ned by electron beam lithography. After application of an oxygen plasma, 80 nm of Gold are deposited. In this way we achieve typical resistances between 10 k? and 30 k? at 4.2 K. The samples were operated by a low frequency ac bias voltage and application of a dc gate voltage UGate to the Aluminium layer. Up to gate voltages of 3 V no leakage current between the gate and the tube was observed (ILeak < 100 fA). Typical breakdown voltages of the gate oxide were 3-4 V. Two-terminal resistance measurements were carried out for two samples, A and B. The lengths of the samples are 5 ?m and 2 ?m and their diameters are 19 nm and 14 nm, respectively. A scanning electron micrograph of a typical sample is presented in Fig. 1A. In order to characterize the dependence of the conductance of sample A on UGate , a small ac bias voltage of
2 2 ?V ? kB T was applied and the current was measured at several temperatures T (Fig. 1B,C). Fig. 1B shows the conductance G as a function of gate voltage at 300 K. The corresponding curves for 10 K, 1 K and 30 mK are presented in Fig. 1C. The conductance at room temperature exhibits a shallow minimum located at UGate ≈ ?0.2 V. When the Fermi level is tuned away from the charge neutrality point, more and more subbands can contribute to the transport and an increase of the conductance is expected. Thus we attribute the position of the conductance minimum to the charge neutrality point, where bands with positive energy are unoccupied while those with negative energies are completely ?lled . This reveals the high e?ciency of the gate as well as an intrinsic n-doping of the tube. The location of the minimum varied from sample to sample. We observed p- as well as n-doping at UGate = 0 V in several samples. The G(UGate ) curves in Fig. 1C show an increasing amplitude of the conductance ?uctuations as the temperature is lowered, while the average conductance decreases. This can be interpreted as a gradual transition from a coexistence of band structure e?ects, UCFs and charging e?ects at 10 K and 1 K to the dominance of Coulomb blockade at 30 mK. In contrast to experiments on clean single wall nanotubes, no periodic Coulomb oscillations are found. Instead, irregular peaks in conductance occur. It is likely that disorder induces a nonuniform series of strongly coupled quantum dots and that transport is governed by higher order tunneling processes . Next, conductance traces G(UGate ) were recorded at several temperatures and in magnetic ?elds perpendicular to the tube axis. The result at a temperature of 10 K is displayed as a color plot in Fig. 2A. We have checked for several gate voltages that G(B) is symmetric with respect to magnetic ?eld reversal as required in a two point con?guration (not shown). In addition, most of the curves show a conductance minimum at zero magnetic ?eld. A closer look at the data reveals that both the amplitude and the width of the conductance dip vary strongly with gate voltage. In order to make this variation more visible, we subtracted the curve at zero magnetic ?eld (see Fig. 1C) from all gate traces at ?nite ?elds. The deviation from the zero-?eld conductance is presented as a color plot in Fig. 2B. The most striking observation is that the magnetoconductance (MC) disappears at certain gate voltages U ? , as indicated by arrows. These voltages U ? are grouped symmetrically around the conductance minimum at UGate ≈ ?0.2 V in Fig. 1B, which we have assigned to the charge neutrality point. The position of the latter, as well as the gate voltages of MC quenches have been indicated also in the linear response conductance curve (Fig 1C) by red and black vertical lines, respectively. The latter always coincide with conductance maxima. These observations lead us to the conjecture that the quenched MC may occur at the onset of subbands of the outermost nanotube shell, which is believed to carry the major part of the current at low temperatures . To con?rm this idea, we applied a simple bandstructure model. The black line in Fig. 3A shows the density of states of a single wall (140,140) armchair nanotube, which matches to the diameter of sample A (19 nm). Typical van Hove singularities arise at the energies, where the subband bottoms are located . By integration over energy one obtains the number ?N of excess electrons on the tube, plotted as a red line in Fig. 3A. In this way, we can determine the number ?N ? of electrons at the onset of the nanotube subbands. If we assume as usual a capacitative coupling between the gate and the tube, ?N can be converted into a gate voltage via CUGate = e?N . In Fig. 3B the measured gate voltages U ? of quenched MC are plotted versus the calculated ?N ? for both samples. Both data sets ?t very well into straight lines, which demonstrates that most of the positions U ? of the quenched MC agree very well with the expected subband onsets. In addition, the gate capacitances C are provided by the slope of U ? vs. ?N ? . The capacitances per length are nearly identical, i.e. 120 aF/?m and 129 aF/?m for samples A and B, respectively. These values agree within a factor of 2 with simple geometrical estimates of C, indicating the consistency of the interpretation. From the capacitance C and the calculated dependence of the number of electrons N on energy one can convert the gate voltage into an equivalent Fermi energy. This energy scale is shown in Fig. 2F. The typical dip in the MC at B = 0 in Fig. 2A has been observed earlier and can be explained in terms of weak localization in absence of spin-orbit scattering [2, 6, 13]. The weak localization correction ?GWL to conductance provides information on the phase coherence length L? of the electrons. With W being the measured diameter and L = 300 nm the electrode spacing of the nanotube, ?GWL is given in the quasi-one-dimensional case (L? > W ) by ?GWL = ?(e2 /π L)·(L?2 +W 2 /3?4 )?1/2 , ? m where ?m = ( /eB)1/2 is the magnetic length. In Fig. 2B each row displays a dip around zero magnetic ?eld, where both the amplitude and the width of the dip vary strongly with gate voltage. We have used the weak localization expression above to ?t the low ?eld MC with L? and G(B = 0) as free parameters. The conductance ?GWL as calculated using the ?t parameters is plotted in Fig. 2C. We ?nd that conductance traces are reproduced very well by the ?t for ?elds up to 2 T. For higher ?elds, deviations occur, most probably due to residual universal conductance ?uctuations. In this way we obtain an energy dependent phase coherence length L? (EF ), which is plotted in Fig. 2D. L? varies from 20 to 60 nm and displays pronounced minima which correspond to the regions of nearly ?at MC in Fig. 2B. From the preceding discussion, we can say that weak localization seems to be suppressed at the onset of nanotube subbands. In order to con?rm the validity of our interpretation in terms of weak localization, we have studied the temperature dependence of the phase coherence length. As the dominating dephasing mechanism, quasielastic electronelectron scattering has been identi?ed [2, 6, 14]. De-
-0.8 Gate Voltage (V) -0.4 0.0 0.4 0.8
-200 -150 -100 0 100 150 200 250 Energy (meV)
-2 0 2 4 6 -2 0 2 4 6 B (T) B (T)
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0 30 60 -5 0 5 0 Lf (nm) VBias (mV)
FIG. 2: (A) Grey scaled conductance G of sample A as a function of gate voltage U and perpendicular magnetic ?eld at a temperature of 10 K. (B) Deviation of G from the zero-?eld conductance G(U, B) ? G(U, 0). White arrows indicate the regions of quenched magnetoconductance. (C) Reproduction of the magnetoconductance by 1D weak localization ?ts. The parameters L? and G(B = 0) are used as obtained by ?tting the data on Fig. 2A. (D) Phase coherence length L? vs. gate voltage as obtained from the ?t. The positions of the charge neutrality point (grey line) and the regions of quenched magnetoconductance (black lines) are indicated. (E) Di?erential conductance of sample A as a function of gate voltage and dc bias voltage VBias at T =10 K. (F) Exponent α vs. gate voltage as obtained from ?tting a power law V α to the di?erential conductance in the range eV ? kB T . Right: scale conversion of the gate voltage into a (nonlinear) energy scale using the gate capacitance as obtained from Fig. 3B.
phasing by electron-phonon scattering is negligible since the corresponding mean free path exceeds 1 ?m even at 300 K [15, 16]. The theory by Altshuler, Aronov and Khmelnitzky  predicts L? = (GDL 2 /2e2 kB T )1/3 , where G is the conductance, D is the di?usion constant, L is the length of the tube. The dominance of electron-electron-scattering can be con?rmed by studying the temperature dependence of L? . Therefore, the MC measurements have been repeated for temperatures ranging from 1 K to 60 K. In order to eliminate the contribution of the universal conductance ?uctuations, the MC curves have been averaged over all gate voltages. The result is plotted in Fig. 4A. For the comparison of the curves with theory, one has to bear in mind that the average runs also on curves with suppressed MC. Hence, for the ?t an averaged weak localization contribution of the form ?G? = A · ?GWL with a scaling factor 0 < A < 1 WL has been taken into account. The ?tted curves are included in Fig. 4A. They match the data very well, up to magnetic ?elds of 7 T. In Fig. 4B the resulting L? (T ) are presented. The contribution of the universal conductance ?uctuations is completely suppressed by ensemble averaging. The temperature dependence matches a power law with exponent -0.31, which is close to the theoretical prediction of -1/3. Another quantum correction to the conductance is in-
duced by the electron-electron-interaction and reduces the density of states near the Fermi energy . This leads to zero bias anomalies in the di?erential conductance dI/dV , from which information on the strength of the electron-electron-interaction can be extracted. In the case of tunneling into an interacting electron system with an ohmic environment, the di?erential conductance dI/dV is given by a power law, i.e. dI/dV ∝ V α for eV ? kB T , where the exponent α depends both on the interaction strength and the sample geometry . In order to obtain complementary information, we have examined the dependence of the ZBA on the gate voltage UGate . The di?erential conductance has been measured as a function of UGate and VBias . The result is presented in Fig. 2E. For each gate voltage, the conductance shows a dip at zero bias. The zero bias anomaly has a strongly varying width with gate voltage and nearly vanishes at the same gate voltages UGate = U ? as the magnetoconductance. For each value of the gate voltage, a power law ?t for the bias voltage dependence of the di?erential conductance has been performed. The resulting exponent α(UGate ) is plotted in Fig. 2F. α varies between 0.03 and 0.3 and shows pronounced minima at the gate voltages U ?. We thus observe experimentally a strong correlation between the single particle interference e?ects (expressed by
4 explanation of the observed interplay between bandstructure e?ects and quantum corrections to the conductance
2000 5 0
G (2e /h)
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1 0 -1 -2 -2000 0
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FIG. 3: (A) Calculated π-orbital density of states (DOS) for a (140,140) armchair nanotube of diameter of 19 nm (grey) as a function of energy. Number of excess electrons N (E) (black) as obtained from the integration of the DOS from 0 to E. The subband spacing for this diameter is 66 meV. (B) Measured gate voltage values U ? of nanotube subband onsets vs. calculated numbers of electrons ?N ? at subband onsets for sample A (circles, diameter 19 nm) and B (triangles, diameter 14 nm). The lines correspond to linear ?ts of the data. The slopes of the lines correspond to gate capacitances per length of 120 aF/?m and 129 aF/?m for sample A and B, respectively.
FIG. 4: (A) Averaged magnetoconductance of sample A (circles) at temperatures of 60 K, 10 K, 3 K and 1K (top to bottom) and ?ts of 1D weak localization behavior (lines). (B) Double-logarithmic plot of the temperature dependence of the phase coherence length L? as obtained from the weak localization ?t (black dots). The line corresponds to a power law ?t with an exponent -0.31.
L? ) and the interaction e?ects (expressed by α). Both are strongly reduced at certain positions of the Fermi level, which match well the positions of the van Hove singularities estimated from simple bandstructure models. What is the e?ect of the bandstructure? Numerical calculations by Triozon et al.  indicate that the di?usion coe?cient D is not a constant as a function of EF , but displays pronounced minima at the onset of new subbands. At these points strong scattering occurs, resulting from the opening of a highly e?cient scattering channel. This has a direct e?ect on L? = D(EF )τ? . Of course, τ? may also be a?ected. Can the energy dependence of D(EF ) also explain the suppression of the interaction e?ects? This question has already been raised by Kanda et al. , who also observed a pronounced gate modulation of α. For weak electron-electron-interaction the theory of Ref.  predicts α ∝ 1/?el, where ?el is the elastic mean free path. This is de?nitely incompatible with the observed suppression of α at Fermi levels where di?usion is slow. The observed strong modulations of L? and α are accompanied by a rather weak modulation of the zero bias conductance at 10 K (see Fig. 1B). One may thus ask, whether the assumption of weak interactions is valid. Taking the simple Drude formula σ = e2 N (EF )D(EF ) as an orientation, this can be explained by a partial compensation of the variation of N and D with EF . However, a quantitative
requires a realistic model calculation for a thick, e.g., (140,140) nanotube including disorder and interaction effects. The simple model of strictly one-dimensional conductance channels is obviously incompatible with the observed weak-localization-like magnetoconductance close to the charge neutrality point. The disorder must be strong enough to mix the channels without completely smearing the density of states. In conclusion, our electronic transport measurements on multiwall carbon nanotubes reveal an interplay of bandstructure e?ects originating from the geometry of the tube and quantum interference induced by disorder. The results demonstrate the necessity of a systematic theoretical approach which can account both for disorder and geometrical e?ects on the same level.
We have bene?tted from inspiring discussions with A. Bachtold, V. Bouchiat, G. Cuniberti, H. Grabert, M. Grifoni, K. Richter, S. Roche, R. Sch¨fer, a C. Sch¨nenberger and F. Triozon. o Funding by the Deutsche Forschungsgemeinschaft within the Graduiertenkolleg 638 is acknowledged. The work in Lausanne was supported by the Swiss National Science Foundation.
            
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