5.5.2 Phonon Energy

Figure 5.22-a shows the dependence of the ballisticity with respect to the phonon energy. With increasing phonon energy the effect of phonon scattering on the current is reduced, because scattered electrons lose more kinetic energy and the probability for traveling back to the source contact decreases. The considerable decrease of ballisticity for low energy phonons is due to the phonon absorption process.

The right part of Fig. 5.20-b shows an electron absorbing energy from a phonon and scattering into a higher energy state. In this case, the probability for arriving at the source contact increases. This process can severely reduce the total current.

Fig. 5.22-b separately shows the effects of the phonon emission and absorption processes on the ballisticity. As the phonon energy decreases, the phonon occupation number (C.18) increases exponentially, and the self-energy contributions of these two components increase. However, due to the higher probability for back-scattering of electrons in the case of phonon absorption, this component reduces the total current more effectively than the phonon emission process does.

Fig. 5.23-a shows the ratio of the gate-delay time (Section 5.3.1) in the presence of electron-phonon interaction to that in the ballistic case, $\tau_\mathrm{Sc}/\tau_\mathrm{Bl}$, as a function of the electron-phonon coupling strength. As the phonon energy increases the gate-delay time increases. This behavior can be attributed to the average electron velocity in the channel, which is high for ballistic electrons and low for electrons scattered to lower energy states.

Figure 5.22: a) Ballisticity versus phonon energy for a CNT of 50 nm length. Results for inelastic scattering with different electron-phonon couplings are shown. $V_\textrm {G}$= $V_\textrm {D}$=1 V. b) Ballisticity versus phonon energy with D=10$^{-1}$ eV$^2$ at the bias point $V_\textrm {G}$= $V_\textrm {D}$=1 V. The contributions due to phonon absorption and emission are shown.

Figure 5.23: a) The ratio of the gate-delay time in the presence of electron-phonon interaction to the gate-delay time in the ballistic case, $\tau _\textrm {Sc}/\tau _\textrm {Bl}$, as a function of the electron-phonon coupling strength. For comparison, the ratio $I_\textrm {Sc}/I_\textrm {Bl}$ is also shown. As the phonon energy increases the gate-delay time increases. This behavior is due to the reduction of the electron velocity in the channel and the resulting charge pile up. b) The spectra of the source and drain currents. The effect of inelastic scattering with different phonon energies is shown. The electron-phonon coupling strength is D = 2$\times$10$^{-1}$ eV$^2$. The figure shows a considerable increase of the electron population close to the conduction band-edge as the phonon energy increases.

Figure 5.24: a) The profile of the electron velocity near the source contact. b) The profile of the electron concentration along the device. The results for the ballistic case and for electron-phonon interaction are shown. As the phonon energy increases the electrons scatter to lower energy states. Therefore, the electron velocity decreases and the carrier concentration increases. The electron-phonon coupling strength is D = 10$^{-1}$ eV$^2$ and the bias point is $V_\textrm {G}$ = $V_\textrm {D}$ = 1 V.

Fig. 5.23-b shows the spectra of the source and drain currents for different inelastic phonon energies. Electrons can emit a single phonon or a couple of phonons to reach lower energy states. The probability of multiple phonon emissions decreases as the number of interactions increases. Therefore, as the phonon energy increases, the occupation of electrons at lower energy states increases.

As shown in Fig. 5.23-b, the electron population close to the conduction band-edge considerably increases as the phonon energy increases. Therefore, as the phonon energy increases the mean velocity of electrons decreases and the carrier concentration in the channel increases (Fig. 5.24). The increased charge in the channel results in an increased gate-delay time.

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