3-month progress Report
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1 3-month progress Report Graphene Devices and Circuits Supervisor Dr. P.A Childs
2 Table of Content Abstract Introduction Graphene gold rush Properties of graphene Semiconductor graphene structure Graphene bilayer Carbon nanoribbons (CNRs) The objectives of the project Progress of the first three months Further plan Web Page... 9 Reference... 11
3 Abstract This is the 3 month progress report in partial fulfillment of the requirements for the degree of philosophy doctor of with project title Graphene Devices and Circuits. The report presents the background of the new material graphene and the motivations for doing research in the field of graphene devices and circuits. The report is organized by following the guideline of the university. It starts from introduction and objectives of the project. Then, it summarise the first three month work, mainly on the simple tight-bonding calculation of the band structure of graphene; the study of using finite difference method and NEGF method on graphene device simulation. Finally, an outline work plan is given. 1. Introduction 1.1 Graphene gold rush Graphene is the name given to a flat monolayer of carbon atoms tightly packed into a two-dimensional honeycomb lattice, and is a basic building block for graphitic materials of all the other dimensionalities. It can be wrapped up into 0D buckyballs, rolled into 1D nanotubes or stacked into 3D graphite (Shown in Figure 1). Although it has been theoretically studied for sixty years[1], graphene was presumed not to exist in free state until Novoselov et.al in 2005 reported their discovery and the anomalous features it exhibit[2]. This experimental breakthrough has generated much excitement in physics community. This can be clearly seen by studying the number of papers appearing in the search result on Web of Science containing the word graphene in the abstract as shown in Figure 2. The reason for this graphene gold rush is the unique electronic and optical properties of graphene, and the potential to build the next generation ballistic transport nano-electronic devices. Before graphene, carbon nanotubes (CNTs) have attracted broad attention because some prototype structures of CNT transistors show great performance comparing to conventional silicon transistor. However, in order to use CNT MOSFETs in realistic IC applications, the electronic properties such as the band gap (Eg) should be well controlled. While the band gap of CNT is strongly affected by its chirality and there is currently no straightforward way to control the CNT chirality during growth. A possible way to overcome the CNT chirality control problem is to use some semiconductor graphene structure, such as graphene nanoribbons or bilayer graphene. Another advantage of graphene is that graphene material can be easily patterned using electron beams or other standard nano-electronics lithography methods. 1
4 Figure 1 Graphene is a 2D building material for carbon materials of all other dimensionalities. [3] Figure 2 Number of Papers published during containing the word graphene in the abstract. The result is obtained using Web of Science search engine. 2
5 1.2 Properties of graphene Graphene has a honeycomb lattice structure of carbon atoms in sp 2 hybridization state. Every unit cell of graphene lattice contains two carbon atoms and each atom contributes a free electron. The Brillouin zone of graphene is hexagonal as shown in Figure 3. Figure 3 Graphene reciprocal lattice defined by the primitive reciprocal lattice vectors K1 and K2. The central hexagonal Brillouin zone is depicted in several shades of gray to indicate regions related by reciprocal lattice vectors to corresponding regions of the primitive unit cell defined by K1 and K2. Two of the high-symmetry in equivalent K points (under translational symmetry) at the corners of the hexagonal zone are labeled as K and K, and these, as well as equivalent points in the lattice, are shown as black and white circles, respectively.[4] The band structure of graphene can be simply calculated from a nearest-neighbor tight-binding approximation[1](shown in Figure 4), and the result is unique comparing to other materials. The valence and the conduction band meet at a point at the six corners of the Brillouin zone. Near these crossing points (named as Dirac points), the electron energy is linearly dependent on the wave vector. 3
6 Figure 4 Band structure of Graphene [4] As a result of the linear energy-momentum dispersion relation, at the Dirac points an electron has an effective mass of zero and behaves more like a photon than a conventional massive particle whose energy-momentum dispersion is parabolic. The tight-binding calculation of the band structure of graphene is based on Schrödinger equation, at low energy level however, a more natural way to describe the carriers transport is using Dirac equation. 1.3 Semiconductor graphene structure From the band structure diagram, we can see that graphene is neither semiconductor nor metallic. In order to make semiconductor devices, for instance FET transistors, we need to make a small band gap. This can be done by two different ways: Use a bilayer graphene or tailor the bulk graphene sheet to a smaller graphene nanoribbon Graphene bilayer Graphene bilayer is a pair of coupled graphene monolayers. When on-site energy difference U between two layers is non-zero a band gap will open. By controlling the doping or external electric fields one can easily change U and open a band gap. Theoretical and experimental study also shows that bilayer graphene band structure is highly affected by the lattice symmetry (shown in Figure 5). 4
7 Figure 5 Electronic structure of a single (A), symmetric double layer (B), and asymmetric double layer (C) of graphene. [5] Carbon nanoribbons (CNRs) When tailored to less than 100nm wide nanoribbons, graphene may open a band gap due to the electron confinement. There are multiple types of Graphene nanoribbons (also called carbon nanoribbons). Like CNTs CNR can be classified by the shape of their edges. Figure 6 (a) (b)[6] shows the armchair and zigzag with the width of N atoms respectively and Figure 6 (c) shows some general edges CNRs Figure 6 different types of graphene nanoribbons. Tight-binding calculations show that the conductivity of CNRs highly depended on their width and edge types. For armchair CNRs when N=3M-1 where M is an integer the CNRs will become metallic otherwise they are semiconductor, and the band gap decrease as N growth. Unlike armchair CNRs, zigzag CNRs are all metallic, this is mainly because the additional energy state appear in their zigzag edge[6]. Figure 7(a)-(c) and Figure 8(a)-(c) shows the Calculated band structure of armchair and zigzag nanoribbons of various widths N=4, 5, 6 respectively. Figure 7 (d) and Figure 8 (d) are the calculated band structures of an armchair nanoribbon of N=30 and a zigzag nanoribbon of N=25. The projected band structure of 2D graphite onto an armchair axis is also shown in Figure 7(e) and Figure 8(e). Dashed lines indicate the boundary of the first Brillouin zone. 5
8 Figure 7 Band structure of armchair CNRs[6] Figure 8 Band structure of zigzag CNRs[6] 6
9 2. The objectives of the project The aim of this project is to study the electronic properties of graphene and develop graphene based next generation semiconductor device. To archive that we must first know the band structure of bulk graphene and other different graphene structure, currently most of the calculation are done using nearest-neighbor tight-bonding approximation which is perfect in bulk graphene and graphite. However, when the geometry size goes to nanometers such as graphene nanoribbons the boundary condition is no longer periodic. Therefore, first-principles calculation becomes necessary. Once we have the electronic properties of graphene the next step is to design new device structure and study the carrier transport properties of graphene in different kind of devices. 3. Progress of the first three months During the first three month of the project, I have already studied the nearest-neighbor tight-binding method followed S.Datta s book [7], and a simple calculation of the bulk graphene has been performed. Using a single slater-koster parameter t=3ev, the hopping matrix element between Pz orbitals associated with nearest neighboring C-C atoms in the graphene sheet, the Hamiltonian of the system can be reduced to a summation of a series of 2 2 matrixes 0 t 0 t exp( ik a1) 0 t exp( ik a [ h( k )] t t t exp( ik a2) 0 t exp( ik a1) 0 2 ) (1) Where k is the wave vector and a 1 and a 2 are the two real space lattice parameters of a unit cell in graphene sheet (shown in figure 9). The E(k) relation can be obtained by solving the Eigen-value of h(k ) : 2 E t 1 4cos k bcos k a 4cos k b (2) y x y 7
10 Figure 9 Real space lattice parameters of graphene This calculation has its own limitation: it can be only used for periodic 2-D crystal lattice, which means that when calculating the band structure of graphene nanoribbons, the non-periodic boundary conditions should be considered. In the field of device simulation, I spent a lot of time studying solving 3-D Schrödinger equation and Poisson equation self-consistently. This is a common approach of today s semiconductor device simulation. The approach starts with writing down the Schrödinger equation and Poisson equation of the system using finite difference matrix, then guess the initial charge density of every grids inside the device (usually zero). Next, by solving the Poisson equation we can know the electron potential of every grid inside the device. Using the non-equilibrium Green s function method (NEGF) we can then get a new charge density distribution inside the device from the Schrödinger equation. Use this new charge density to solve the Poisson equation again and repeat the steps many times until new value of the charge density matches the old one, we can at last get the final solution of the charge density. Hence, the current and other characteristic of the device can be found. A flow chart of this method is shown in Figure 10. Figure 10 Flow Chart of the approach used in device simulation 4. Further plan In the field of graphene band structure calculation, my next plan is to perform first-principle calculations on graphene nanoribbon to overcome the limitation of nearest-neighbor tight-binding approximation. Density functional theory (DFT) has been proved to be an efficient and accurate first-principle method, and is used in lots of quantum chemistry calculation 8
11 software. Among these software, CYSTAL06, has already been used to calculate the electronic structure and magnetic properties of graphene nanoribbons[8], I also plan to study and use this powerful software. This will be the major work in the next 9 months. Currently lots of groups are doing researches on graphene field effect transistors where graphene nanoribbon[9] and bilayer[10] are used as the channel material. However, to fabricate graphene field effect transistor, an important step is to build a metal-oxide-graphene structure, and the growth of an oxide layer and a metal gate on a 1-atom-thick graphene film is still a bottleneck in today s lab. In further step of this project we will not limited our simulation on graphene FET, other structure such as bipolar transistor, resonant tunneling device, single electronic transistor and quantum dots devices will also be studied. In order to do that the transport properties of different graphene junctions need to be done. A simulation work on one or two of this structure will be performed before the next report. Furthermore, the final objective of this project is to study the possibility of using current electron beam lithographic method to build next generation circuits based on graphene. Some papers already show this great potential [11] [12] of graphene. Based on the result of devices simulation, I will try to build some device model and use some circuits simulation tool (Pspice, matlab) to get a performance of graphene circuits after the 9 month report. 5. Web Page PhD student Group : Emerging Device Technology Supervisor : P.A. Childs wuyudong@gmail.com Personal : Biography Oct Present PhD student in Department of Electronic, Electrical and Computer Engineering, University of Birmingham, UK; Continue Bachelar degree in University of Birmingham, Oct July 2007 UK, Majored Electronic Engineering as an exchange student; Sep.2003-July 2006 BEng in Electronic Engineering, 9
12 Department of Electronic Science and Technology, Huazhong University of Science and Technology, China. Research interests Semiconductor devices, charge carrier transport, nanoscale devices Current project Graphene Devices and Circuits Graphene, a 2-D carbon based nanostructure discovered in 2005, is now the most interesting structures because of their rich variety of excellent physical properties. For instance, anomalous quantum hall effects (QHE) and massless Dirac electronic behavior have been discovered in the graphene systems. Graphene has also attracted the interest of technologists because of its potential to construct ballistic transistors and other semiconductor devices. The aim of this project is to study the electronic properties of graphene and develop graphene based next generation semiconductor device. To archive that we must first know the band structure of bulk graphene and other different graphene structure, currently most of the calculation are done using nearest-neighbor tight-bonding approximation which is perfect in bulk graphene and graphite. However, when the geometry size goes to nanometers such as graphene nanoribbons the boundary condition is no longer periodic. Therefore, first-principles calculation becomes necessary. Once we have the electronic properties of graphene the next step is to design new device structure and study the carrier transport properties of graphene in different kind of devices. 3 month report [ Personal Research Group School University ] Last updated by on Jan 7th 2008 Maintained by 10
13 Reference 1. Wallace, P.R., The Band Theory of Graphite. Physical Review, (9): p Novoselov, K.S., et al., Two-dimensional gas of massless Dirac fermions in graphene. Nature, (7065): p Novoselov, A.K.G.K.S., The Rise of Graphene. Nature Materials, 2007(6): p A White, C.A.M., JW, Fundamental properties of single-wall carbon nanotubes. J. Phys. Chem. B, (1): p Ohta, T., et al., Controlling the Electronic Structure of Bilayer Graphene. 2006, American Association for the Advancement of Science. p Nakada, K., et al., Edge state in graphene ribbons: Nanometer size effect and edge shape dependence. Physical Review B, (24): p Datta, S., Quantum Transport: Atom to Transistor. 2005: Cambridge University Press. 8. Pisani, L., et al., Electronic structure and magnetic properties of graphitic ribbons. Physical Review B, (6): p Liang, G.C., et al., Performance projections for ballistic graphene nanoribbon field-effect transistors. Ieee Transactions on Electron Devices, (4): p Oostinga, J.B., et al., Gate-tunable band-gap in bilayer graphene devices. eprint arxiv: , Xu, Z., Molecular circuits based on graphene nano-ribbon junctions. eprint arxiv: , Geim, A.K. and K.S. Novoselov, The rise of graphene. Nature Mater, : p
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