5.5.4 Discussion

In general the electron-phonon interaction parameters depend on the diameter and the chirality of the CNT (see Section 2.6.1). CNTs with a diameter $ d_\mathrm{CNT}>\mathrm{2~nm}$ have a band gap $ E_\mathrm{G}<\mathrm{0.4~eV}$ (2.10), which render them unsuitable as channel for transistors. Since the fabrication of devices with a diameter $ d_\mathrm{CNT}<\mathrm{1~nm}$ is very difficult, we limit our study to zigzag CNTs with diameters in the range of $ d_\mathrm{CNT}=\mathrm{1-2~nm}$. Scattering with acoustic phonons is treated as an elastic process. The electron-phonon coupling is also weak for acoustic phonons ( $ D_\mathrm{AP}<\mathrm{10^{-3}~eV^{2}}$), which implies that elastic back-scattering of carriers is weak. Inelastic scattering is induced by $ \mathrm{OP}$, $ \mathrm{RBM}$, and $ \mathrm{K}$-point phonons (Section 2.5.2). Considering the class of CNTs discussed above, the energies of the these phonons are $ \mathrm{\hbar\omega_\mathrm{OP}\approx 200~meV}$, $ \mathrm{\hbar\omega_\mathrm{RBM}\approx 25~meV}$, and $ \mathrm{\hbar\omega_\mathrm{K_1}\approx 160~meV}$ and $ \mathrm{\hbar\omega_\mathrm{K_2}\approx 180~meV}$ [278,276]. The corresponding coupling coefficients are $ D_\mathrm{OP}\approx\mathrm{40\times 10^{-3}~eV^{2}}$, $ D_\mathrm{RBM}\approx\mathrm{10^{-3}~eV^{2}}$, and $ D_\mathrm{K_1}\approx\mathrm{10^{-4}~eV^{2}}$, and $ D_\mathrm{K_2}\approx\mathrm{10^{-3}~eV^{2}}$ [276].

As discussed in Section 5.5.2, high energy phonons such as $ \mathrm{OP}$ and $ \mathrm{K}$-point phonons reduce the on-current only weakly, but can increase the gate-delay time considerably due to charge pileup in the channel. Low energy phonons such as the RBM phonon can reduce the on-current more effectively, but have a weaker effect on the gate-delay time. However, due to strong coupling, scattering processes are mostly due to electron-phonon interaction with high energy phonons. Therefore, the on-current of short CNT-FETs can be close to the ballistic limit [279] (see Fig. 5.26), whereas the gate-delay time can be significantly below that limit [280,89]. The intrinsic (without parasitic capacitances) gate-delay time for the ballistic case can be approximated as $ \tau
\approx1.7~\mathrm{ps/\mu m}$, or equivalently $ f_{T}\approx100~\mathrm{GHz/\mu
m}$ [281]. The highest reported cutoff frequency for a device with a length of less than $ \mathrm{1\mu m}$ is $ f_{T}\approx\mathrm{10~GHz}$ [90], which is far below the ballistic limit. Apart from parasitic capacitances, inelastic electron-phonon interaction with high energy phonon has to be considered to explain the results.

Figure 5.26: Comparison of the simulation results and experimental data for the a) output and b) transfer characteristics. Lines show the simulation results and symbols show experimental data. The result for $ V_\textrm {G}$=-1.3 V is compared with the ballistic limit. Experimental data have been adopted from [279].
\includegraphics[width=0.43\textwidth]{figures/IVD-Comp.eps} \includegraphics[width=0.45\textwidth]{figures/IVG-Comp.eps}

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors