## Coulomb Blockade and Transport Spectroscopy

For metallic nanotubes exhibiting high contact resistance with the electrical leads, the low temperature transport is dominated by the Coulomb blockade effect [4,5,21,22,23,24]. Figure 3a shows an example of the linear-response conductance versus gate voltage for a metallic nanotube rope measured at 100 mK in a dilution refrigerator. It exhibits a quasi-periodic sequence of sharp peaks separated by zero-conductance regions, which signifies Coulomb blockade single-electron charging behavior. Such phenomena have been well-studied in semiconductor quantum dots and small metallic grains [25,26]. Coulomb blockade occurs at low temperature when the total capacitance C of a conducting island (which is weakly coupled to source and drain leads through tunnel barriers with a resistance larger than the quantum resistance h/ee2 « 26 kQ), is so small that adding even a single electron requires an electrostatic energy Ec = e2/2C that is larger than the thermal energy kBT. For an estimate, the capacitance of a nanotube at a distance z away from a con-

ducting substrate is C = 2nere0L/ ln(2z/r), with er being the average dielectric constant of the environment, r and L the nanotube's radius and length. Using er « 2 (for comparison, er = 3.9 for SiO2), z = 300nm and r = 0.7nm, the charging energy is Ec « 5 meV/L (^m). Thus for a typical |im long tube, Coulomb blockade would set in below ~50K (kBT = 5meV). For small electrode spacing, the total capacitance is often dominated by the capacitance to the leads. Another relevant energy scale in the Coulomb blockade regime is the level splitting due to the finite size of the nanotube. A simple 'particle-in-a-box' estimate gives A E = hvF/4L « 1 meV/L (|m), where h is the Planck's constant, vF = 8.1 x 105 m/s the Fermi velocity in the nanotube and a factor of 2 has been introduced to account for the two subbands near the Fermi energy. Note that both Ec and A E scale inversely with length (up to a logarithmic factor), and the ratio Ec/A E is thus roughly independent of length, i.e., the level spacing is always a small but appreciable fraction of the charging energy.

In Fig. 3a, each conductance peak represents the addition of an extra electron. The peak spacing is given by A Vg = (2Ec + AE)/ea, where a = Cg/C, with Cg and C being the capacitance, to the gate and the total capacitance respectively, converts the gate voltage into the corresponding electrostatic potential change in the nanotube. At low temperatures, the height of the conductance peak varies inversely with temperature whereas the peak width is proportional to the temperature [4,5]. This shows that transport takes place via resonant tunneling through discrete energy levels and the electronic wave-functions are extended between the contacts. It is remarkable that metallic nanotubes are coherent quantum wires at mK temperature.

For a fixed gate voltage, the current shows a stepwise increase with increasing bias voltage, yielding the excited-state spectrum. Each step in the current (or peak in the differential conductance) is associated with a new higher-lying energy level that enters the bias window. A typical plot of the differential conductance dl/dV as a function of both bias voltage and gate voltage is shown in Fig. 3b. Within each of the diamonds (the full diamonds are not displayed due to the finite bias window), the number of electrons on the nanotube is fixed and the current is blockaded. The boundary of each diamond represents the transition between N and N +1 electrons and the parallel lines outside the diamonds correspond to excited states. Such a plot is well-understood within the constant-interaction model, in which the capacitance is assumed to be independent of the electronic states. However, Tans et al. reported significant deviations from this simple picture [22]. In their transport spectroscopy, changes in slopes or kinks are observed in transition lines bordering the Coulomb blockade region. These kinks can be explained by a gate-voltage-induced transition between different electronic states with varying capacitances, which could result from state-dependent screening properties or different charge density profiles for different single-particle states.

The ground-state spin configuration in a nanotube was revealed by studying the transport spectrum in a magnetic field B. Experiments by Cob-den et al. [21] showed that the level spectrum is split by the Zeeman energy g^BB, where is the Bohr magneton, and the g factor is found to be 2 indicating the absence of orbital effects as expected for nanotubes; the total spin of the ground state alternates between 0 and 1/2 as successive electrons are added, demonstrating simple shell-filling, or even-odd, effect, i.e., successive electrons occupy the levels in spin-up and spin-down pairs. In contrast, Tans et al. [22] found that the spin degeneracy has already been lifted at zero magnetic field and all the electrons enter the nanotubes with the same spin direction. This behavior can be explained by a model in which spin-polarized states can result from spin flips induced by the gate voltage. Indeed a microscopic model was proposed by Oreg et al. [27] considering nonuniform gate coupling to the nanotube due to the screening of the source and drain electrodes. Further experimental and theoretical work is needed to address this intriguing issue.

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