Master's Thesis of Electrical-Communication Engineering
Abstract
Production and amplification of radio frequencies [1] is the heart of satellite communication and optical electronics applications. The telecommunications industry is looking for smaller and more efficient RF amplifiers at higher frequencies. Nanostructures meet these needs due to their unique characteristics. In this thesis, we describe the characteristics of the structure of graphene and how carbon nanotubes are formed from it, examine the similarities and differences between the structure of carbon nanotubes [2] and the waveguide tube amplifier [3] and compare the physical cause of the amplification in these two structures. We examine the Boltzmann equation, which is used for carbon nanotubes with simultaneous AC and DC bias, and deal with the physical analysis of the negative differential conductivity [4] created in the obtained graphs. Considering the lack of impedance matching that occurs in the use of carbon nanotubes in the real world, we should design a suitable platform to reduce the impedance mismatch. In this design, the coplanar waveguide is used because of its advantages such as high frequency capacity, the ability to manufacture in sub-micro dimensions. we use In the coplanar waveguide signal path, we create an empty space for embedding carbon nanotubes, we try to make this empty space as small as possible to reduce the number of carbon nanotubes used. The proposed structure reduced the impedance mismatch.
Keywords: carbon nanotubes, amplification in biased carbon nanotubes, Boltzmann equation, negative differential conductance.
1-1-Preface
Carbon nanotubes[1] for the first time by Ijima[2] They were discovered in 1991 and since then many attempts have been made to predict their electronic structure. Due to their unique features such as: high conductivity, flexibility, strength and hardness, they were much considered [1]. In this chapter, we examine the structure of carbon nanotubes and how they are made from graphene. After explaining the types of carbon nanotubes and how they are formed, we express the very important physical issues in nanostructures. We also examine the reinforcing structure of waveguide tubes [3].
1-2-Graphene and how to make carbon nanotubes from graphene
Graphene is a single layer of graphite. As shown in Figure (1-1), carbon-carbon bonding in graphene is formed by bonding orbitals, 2sp, s bonds, and the remaining orbitals, zp, form ? bonds. ? and s connections are defined as follows:
s forms in-plane connections, while ? connections are out-of-plane connections that do not interact with the nucleus. S junctions in graphene and carbon nanotubes create strong mechanical properties. In other words, electron conduction is largely through ? bonds. According to the figure (1-1), this feature can be understood. As can be seen, there are no zeros [4] in the ? bonding orbitals, the electrons move freely around the network, which are called delocalized [5] and form a connected network that explains the conductivity of graphene and carbon nanotubes [1]. style="direction: rtl;">The two-dimensional real space network of graphene is shown in figure (1-2). Graphene unit cell is composed of two separate atoms with an interatomic distance. Its unit vectors are as follows: (1-1) where is the grid constant. The unit cell consists of two lattice vectors, which are gray in figure (1-2) [1].
Figure (1-2) Graphene real space lattice. The unit cell is gray in color [1].
The two-dimensional network of k-space is shown in figure (1?3).The corresponding unit vectors 1b and 2b can be obtained by the following equation: (1?2) where dij is the Kronecker delta. As a result:
(1-3) is the constant of the reciprocal network. The first Brillouin zone [6] of graphene is shown in gray in figure (1-3).
Graphene k-space lattice. The Brillouin zone is shown in gray [1].
The tight junction model[7] is commonly used to obtain the analytical form of electron energy scattering or the E-band structure of graphene. Because it is practically impossible to solve the Schr?dinger equation in large systems, many approximate models are available with increasing complexity. Tight coupling approximation is known as one of the simplest methods. In this part, a brief explanation about how to achieve the electron sputtering relationship of graphene is discussed. We consider the following basic assumptions:
We ignore the electron-electron interaction. This is a one-electron model.
Only ? connections have an effect on conductivity.
The structure of graphene is infinitely large, completely alternating, and has no defects.
To reach the scattering function of graphene, the Schr?dinger equation must be solved for an electron subjected to a lattice potential, as below:
(1-4)
H is the lattice Hamiltonian, U is the lattice potential, m is the electron mass, jE is the eigenfunction and Yj is the eigenenergy for the jth band with wave vector k. Since this is an alternating problem, the eigenfunction (or Bloch function [8]) must satisfy the Bloch theory, which is given as follows: (1-5) is the Braweis lattice vector [9], r1 and r2 are integers [1]. Therefore, the wave function in the reciprocal space is alternating with the vector of the reciprocal lattice, where q1 and q2 are integers:
(1-6)
Finally, the graphene band structure is approximated as follows
(formulas are available in the main file)
Abstract
Radio frequency (RF) generation and amplification is at the heart of telecommunication, satellite and optoelectronics applications. The electronics industry is in constant search for RF amplifiers with smaller size, more efficient, and operating at higher frequencies. Nano materials with unique properties promise to fulfill these characteristics.
In this thesis, we represent the properties of graphene structure and the way of forming carbon nanotube (CNT) by graphene. We investigate the similarities and differences between traveling wave tube (TWT) and CNT and compare the physical mechanism of amplification in them. We use Boltzmann equation for CNTs under DC and AC fields and analyze the negative differential conductivity (NDC).
Due to the inherent impedance mismatch between carbon nanotube and the macro world through which they are probed, we designed a proper structure to solve this mismatch. In our design we use tapered coplanar waveguide (TCPW) because of some properties of CPW such as: guidance of much higher frequencies compared to other planar structures and easy design and fabrication with feature sizes less than one micrometer.
A gap in the central strip allows the CNTs to be aligned across for characterization and transmission measurements.