# Quantum Simulation of Carbon Nanotube and Graphene Nanoribbon Field-Effect Transistors

In recent years, device simulation has been an indispensable tool to predict the performance and extract relevant parameters of nanoelectronics transistors. However, the dimensions of field-effect transistors (FETs) have become so small that quantum effects inevitably play an increasingly important role. While most commercially available simulation tools are based on (semi-)classical transport models that are enhanced with quantum mechanical add-ons, the so-called non-equilibrium Greens function method (NEGF) has become one of the premier methods for true quantum mechanical simulations of nanoscale FETs. The aim of the course is the development of a simple tool for the simulation of quantum transport in nanoscale field-effect transistor devices based on a self-consistent solution of the Poisson and Schrödinger equations (using NEGF). In addition, students will elaborate on a code for the computation of the band structure of carbon nanotubes/graphene nanoribbons using tight binding calculations.

Contents of the lecture:

The lecture starts with solving the Poisson equation in a FET based on a one-dimensional model of the electrostatics of a nanotube/-wire transistor. Using the Landauer approach and the so-called top-of-the-barrier model, the drain current in such a transistor is computed, next. Subsequently, the non-equilibrium Green’s function formalism is introduced and the carrier density is computed. A typical example of a local density of states within a FET is shown in the figure above. To realize convergence in the self-consistent calculation, the Newton-Raphson method is employed. Eventually, the current through the device is computed with the Fisher Lee relation. In a next step, the band structure of carbon nanotubes (CNTs) is computed with the tight-binding approach using only the indices (n,m) as input parameters. From the band structure, parameters such as the diameter, effective mass and band gap are extracted and used in the NEGF device simulations. Hence, the simulation tool enables students to study the dependence of the electrical behavior of nanoscale field-effect transistors on their geometry, applied voltages and materials in use. Finally, an inverter consisting of n- and p-type CNT FETs is computed with a floating output terminal and voltage transfer curves can be calculated as a function of the device parameters.

Organization of the Lecture:

In contrast to a usual lecture, this course combines theory and practice: after a brief lecture part where novel material and homework assignments are discussed, students will use the remainder of the course time to continue coding their own MATLAB program with one-to-one support from and supervision by the instructor. The final examination is a written test where students have to carry out simulation experiments employing their own simulation tool.

The lecture will be offered regularly during summer terms. Prerequisites to take part are a profound knowledge of nanoelectronics devices (such as the course “Nanoelectronics Devices- Physics, Modeling, Simulation”, see above). Knowledge in programming in MATLAB is beneficial but not compulsory.

Recommended literature includes: S. Datta: “Quantum Transport - Atom to Transistor”, S.M. Sze: “Physics of Semiconductor Devices”, and J. Knoch: “Nanoelectronics – Device Physics, Fabrication, Simulation”