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Electric Field-Dependent Charge-Carrier Velocity in Semiconducting Carbon Nanotubes

Charge transport in semiconducting single-walled nanotubes (SWNTs) with Schottky-barrier contacts has been studied at high bias. We observe nearly symmetric ambipolar transport with electron and hole currents significantly exceeding 25 micron-ampere, the reported current limit in metallic SWNTs due to optical phonon emission. Four simple models for the field-dependent velocity (ballistic, current saturation, velocity saturation, and constant mobility) are studied in the unipolar regime; the high-bias behavior is best explained by a velocity saturation model with a saturation velocity of 2 x 10^7 cm/s.

One of the most striking aspects of carbon nanotubes is that they can carry extremely large current densities [1][2][3], exceeding 10 9 A/cm 2 , orders of magnitude larger than those at which metal wires fail by electromigration [4]. Zone-boundary phonon emission at energy hf = 160 meV explains the current limit in metallic single-wall carbon nanotubes (SWNTs) of (4e/h)(hf) ≈ 25 µA [1,5,6]. By using electrical breakdown of multi-wall carbon nanotubes (MWNTs), it was found that metallic and semiconducting shells carry similar saturation currents [2,7]. However, there is no reason to assume that the limit in semiconducting nanotubes is identical, since the electronic band structure is different: The equivalent zero-momentum phonon emission process in semiconducting nanotubes involves relaxation of electrons across the band gap. Bourlon, et al. proposed a model of competition between electron-phonon scattering and Zener tunneling to explain the geometrical dependence of saturation currents in metallic and semiconducting single shells in MWNTs [8], but experiments on single semiconducting SWNTs are lacking.
In this letter we investigate high-bias transport in single Schottky-barrier (SB)contacted semiconducting SWNTs up to bias voltages of 10 V, corresponding to average electric fields up to 5 kV/cm. We examine four simple models for the electric-field dependence of the carrier velocity: ballistic, current saturation, velocity saturation, and constant mobility to understand the current at unipolar bias voltage regimes. The results are directly compared with the experimental data, and we find the best agreement with the velocity saturation model, with a saturation velocity v s of 2 × 10 7 cm/s. Heavily n-doped Si chips with a SiO 2 layer of thickness t = 500 nm are used as our device substrates. SWNTs were grown using a chemical vapor deposition (CVD) process with catalyst prepared as in Ref. 9 and carbon feedstock gases as introduced as in Ref. 10. A scanning electron microscope (SEM) was used to locate the nanotubes [11], and electron-beam lithography followed by thermal evaporation of 1.5 nm Cr and 100 nm Au formed the electrodes. Devices were annealed at 400 o C under Ar and H 2 flow to lower the contact resistance. The diameters of the nanotubes were checked by atomic force microscope (AFM).
We have investigated several semiconducting SWNT devices with similar transport behavior [12]. The measured nanotube diameters d range from 2 to 2.4 nm, and the nanotube lengths between the contacts (channel length L) range from 10 to 20 µm.
The measured nanotube diameters indicate they are SWNTs or small MWNTs. (We expect that in the case of a small MWNT with all semiconducting shells that the transport occurs primarily in the outermost shell with the lowest bandgap.) We focus on the transport data from one particular device, with d = 2.4 nm and L = 20 µm.
Electrical measurements were performed by grounding the source electrode (V s = 0) and applying V d to the drain, and V g to the gate, while measuring the drain current I d . Figure 1 shows I d as a function of V g at V d = -1 V. This device behaves as an ambipolar semiconductor although p-type (hole) conduction is slightly better than n-type (electron) conduction. Because of the ambipolar behavior, we assume the presence of SBs at the electrodes for electrons and holes, but a smaller SB for holes. The maximum transconductances g m for holes and electrons are ~ 2.5 and 2.0 µA/V, respectively, which correspond to field-effect mobilities [13]   The saturation is followed by an increase in the current as V d becomes greater than 2V g .
We interpret this increase in I d as the change from majority electron to majority hole current or vice versa. V d = 2V g corresponds to the symmetric bias condition [14,15], equivalent to holding V g = 0 and applying equal and opposite voltages to source and drain; at this point electron and hole currents should be equal. The inset of Figure 2 shows the current is minimum at V d = 2V g . The curves for electrons and holes are similar, with slightly better conduction for holes, consistent with the low-bias data of Figure 1.
We will focus here on the conduction at unipolar bias regions (V g is larger than both V d and V s for electrons, or V g is smaller than both V d and V s for holes). Under these bias conditions, there are very few minority carriers in the channel. (The ambipolar case, where the current contribution from minority carriers can not be ignored, will be discussed elsewhere.) First we examine the expected behavior for a ballistic nanotube FET. We calculate the current using some assumptions and simplifications: (1) The nanotube is undoped. (2) The electrode Fermi level is aligned with the middle of the nanotube gap.
(3) The subbands are approximated by a hyperbolic band , where the upper (lower) sign is for the valence (conduction) bands, V NT (x) is the potential of the nanotube at position x, k is the wave vector, v F,m = 9.35 × 10 5 m/s is the Fermi velocity of metallic nanotubes, for second subbands, and ∆ is a half of nanotube bandgap.
(4) The device capacitance is dominated by capacitance to the gate since the channel length is fairly long compared with gate oxide thickness, and the charge is the carrier density (the 4 from spin and subband degeneracies), k > (k < ) means k is in the range of k > 0 (k < 0).
h is the velocity of state k. Here, only the lowest conduction bands or highest valence bands are considered to contribute the current conduction.
The result of the ballistic model for holes as majority carriers is plotted in Figure   3 The ballistic model reproduces many of qualitative features of the experimental data in Figure 3(a), but the current and conductance are significantly higher than those in the experiment. This is reasonable: since the channel length is tens of microns, we do not expect ballistic transport. We next examine two models for the field-dependent velocity in the nanotube. In the first, the current saturation model, we assume that the maximum difference in the left-and right-moving quasi-Fermi levels is set by optical phonon scattering at hf ≈ 160 meV. The current is then limited to 4ef ≈ 25 µA as observed in metallic nanotubes [1], and as has been suggested for semiconducting nanotubes [17].
where µ 0 is the zero field mobility and n is the carrier density.
Again using the assumptions above, we calculate the current ) (The zero-field mobility µ 0 , determined using the data in the range -0.1 V < V d < 0.1 V, is slightly larger than the field effect mobility µ FE discussed above due to finite V d = -1 V used there.) Figure 3(a) compares the calculation with experiment; disagreement is seen in several aspects. First, rather than saturating at 25 µA at high positive drain bias, the measured current increases with roughly constant slope. Second, the current at negative bias is larger than that in experiment. I 0 = 25 µA was chosen assuming only the first valence subbands participate in the conduction and the physics is similar to metallic SWNT; however, no choice of I 0 gives a good fit. Specifically, I 0 = 50 µA, which might correspond to two contributing subbands, is significantly worse.
The qualitative agreement of the ballistic model with the measured current suggests that the charge-controlled model of the nanotube transistor, namely I d ∝ q, is correct. However, the charges must not move with Fermi velocity, but somewhat slower.
In conventional semiconductors, typically the electric-field-dependent carrier velocity is observed to saturate to a constant value at high electric field. Empirically, the carrier mobility often follows where v s is the saturation velocity. Eq. (2) means at low E-field regime, the carrier velocity v increases linearly with E-field with slope µ 0 ; at high E-field regime, v saturates