6. Summary and Conclusions

EVER SINCE the first demonstration of carbon nanotube FETs (CNT-FETs), their performance is improving very rapidly and the understanding of such devices is evolving. The one-dimensional nature of CNTs severely reduces the phase space for scattering, allowing CNTs to operate close to the ballistic limit even at room temperature. The low scattering probability and high mobility are responsible for high on-current of CNT-FETs.

Furthermore, the chemical stability and perfection of the CNT structure suggests that the carrier mobility at high gate fields may not be affected by processing and roughness scattering as it is the case in the conventional semiconductor channel. Electrostatic control is improved as well. The fact that there are no dangling bond states at the surface of CNTs allows for a much wider choice of gate insulators beyond the conventional $ \mathrm{SiO_2}$. Also, the strong one-dimensional electron confinement of the single-wall CNTs (typically $ 1-2~\mathrm{nm}$ diameter) should lead to a suppression of short-channel effects in transistor devices [5].

As far as integration is concerned, semiconducting CNTs benefit from their band structure which gives essentially the same effective mass for electrons and holes. This should enable similar mobilities and performance of n-type and p-type transistors, which is necessary for a complementary metal-oxide semiconductor (CMOS)-like technology. The most important appeal of this approach is the ability to fabricate one of the critical device dimensions (the CNT diameter) reproducibly using synthetic chemistry.

The purposes of this work are to develop a simulation approach and tools for CNT-FETs and apply them to understand device physics and explore device issues, which are crucial for improving device performance. We employed the non-equilibrium GREEN's function (NEGF) technique for modeling transport phenomena in CNT-FETs. The NEGF technique allow one to study the time evolution of a many-particle quantum system. Knowledge of the single-particle GREEN's function provides properties of the system and the excitation energies of the systems containing one more or one less particle. The many-particle information about the system is cast into self-energies, parts of the equations of motion for the GREEN's functions. GREEN's functions can be expressed as a perturbation expansion, which is the key to approximate the self-energies. GREEN's functions provide a very powerful technique for evaluating properties of many-particle systems both in thermodynamic equilibrium and also in non-equilibrium situations.

We solve the coupled system of transport and POISSON equations self-consistently. A tight-binding HAMILTONian is used to describe transport phenomena in CNT-FETs. The mode-space transformation used in this work reduces the computational cost considerably. The mode-space approach takes only a relatively small number of transverse modes into consideration. To reduce the computational cost even further, we used the local scattering approximation [112]. In this approximation the scattering self-energy terms are diagonal in coordinate representation. We show that the local approximation is well justified for electron-phonon scattering caused by deformation potential interaction.

The carrier concentration is related to the diagonal elements of the GREEN's function. The calculation of the current requires only the nearest off-diagonal elements of the GREEN's function. Furthermore, by using a nearest tight-binding HAMILTONian and assuming the local scattering approximation the achieved matrix is tridiagonal. Considering these factors we employed the efficient recursive GREEN's function method to calculate only the required elements of the GREEN's functions.

We also investigated methods of generating energy grids for numerical integration and their effects on the convergence behavior of the self-consistent iteration. Our results indicate that for accurate and fast convergent simulations the energy grid must be carefully adapted. All methods were implemented into the multi-purpose quantum-mechanical solver VSP.

Employing the described model, we investigated both the static and dynamic response of CNT-FETs. Based on the result we propose methods to improve the functionality and performance of such devices. The ambipolar conduction of CNT-FETs, which limits the performance, is studied in detail. We propose a double-gate structure to suppress this behavior. The first gate controls carrier injection at the source contact and the second one controls carrier injection at the drain contact, which can be used to suppress parasitic carrier injection.

We also considered single-gate devices. Scaling of the gate-source and gate-drain spacer length of single-gate CNT-FETs is studied in this work. By increasing the gate-drain spacer length the ambipolar conduction decreases and the  $ \ensuremath{I_\mathrm{on}}/\ensuremath{I_\mathrm{off}}$ ratio increases. Furthermore, the parasitic capacitances are reduced which results in a decrease of the switching time. By increasing the gate-source spacer length both the on-current and parasitic capacitances decrease. We show that by appropriately selecting this spacer length the performance of the device can be significantly enhanced. The results indicate that these effects can be very different from that in conventional MOSFETs.

Finally, the effect of electron-phonon interaction on the device characteristics is discussed in detail. In agreement with experimental data, our results indicate that electron phonon interaction affects the DC current of CNT-FETs only weakly, whereas the switching response of such devices can be significantly affected.

The implementation of these techniques allows the simulation and analysis of nano-electronic devices where quantum effects are either a parasitic effect or deliberately used as a part of the device functionality. Future work will concentrate on using these techniques to study transport in novel devices such as multiple gate MOSFETs, silicon nano-wires, and molecular devices. Furthermore, scattering processes can be more rigorously approximated. One can consider electron-electron interaction beyond the HARTREE approximation. Regarding electron-phonon interaction, one can relax the assumption of equilibrium phonons and calculate the renormalization of the phonon GREEN's function due to interaction with electrons.

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