Using Tight-Binding Model Essay Example for Free

Using Tight-Binding Model Essay Abstract— In this study, using tight binding model a simple analytical approach has been proposed to investigate the energy dispersion of graphene under the conditions of different planner strain distribution. Here the change in the angle between the primitive unit vectors due to application of external strain has been taken into consideration to propose the approach. From our proposed model it is found that graphene under relaxed or symmetrical strain distribution is a zero bandgap semiconductor. However a band gap is opened as the asymmetrical strain is applied to it. It is seen that upto a certain level of strain (i.e. 12.2 % parallel to carbon-carbon bond and 7.3% perpendicular to carbon-carbon bond) the band gap of graphene increases and then begin to fall . So, four different assumptions have been made for angular change of primitive unit vectors for four different regions of applied strain (i.e. before and after the strain of 12.2 % parallel to carbon-carbon bond before and after the strain of 7.3% perpendicular to carbon-carbon bond). The result obtained in the present study are compared and found an excellent agreement, with more or less 96% accuracy with that of determined from first principle technique. Keywords—Graphene, planner strain, tight binding model, energy dispersion, band-gap. I. INTRODUCTION Graphene, a strictly two-dimensional material having unusual and interesting properties [1] is a rapidly rising star on the horizon of material science and condensed matter physics. It is a material of interest in semiconductor industry because of its exceptionally high crystal and electronic quality, excellent transport properties (i.e. high electron mobility [2] and high thermal conductivity), and as it is planner, it is capable of extreme device scaling comparing with silicon technology. However these excellent properties are associated with a major drawback; graphene is a zero bandgap semiconductor or semimetal [3]-[4]. For large scale manufacturing, the absence of bandgap is the most difficult engineering issue to solve. The zero bandgap revels that it is impossible to switch graphene based device from the conductive to the nonconductive state. So it can not be used in the logic circuit. As the zero bandgap property of graphene limits its application in practical fields, scientists are working to find out the methods to open the bandgap in graphene. To solve this problem several methods have been proposed, such as graphene nanoribbin using quantum confinement effect in its transverse direction [5]-[8], bilayer graphene introducing symmetry breaking between two carbon layers via an external electric field [9],[10] , by the process of doping [11]-[13] and by the process of external strain [14],[15]. To investigate the bandgap opening by the above methods, several techniques have been applied for calculating the band structure of graphene such as first principal calculation, tight binding modeling, k.p method etc. All of them are performed earlier using the software simulation or numerical techniques, which require a huge computational complexity and time consuming and need high capacity super computer. In our study we have proposed a simple analytical approach to investigate the energy dispersion of graphene under different planner strain condition. Using the proposed method the bandgap opening is calculated under the application of asymmetrical strain parallel and perpendicular to the carbon-carbon bond in graphene. The results obtained from the proposed method is compared with the result published by the first principle method and found to be in good agreement with more or less 96 % accuracy. II. METHOD Graphene is a honeycomb lattice of regular hexagonal structure. But it loses its regular hexagonal structural symmetry under uniaxial/shear strain. When planar stress is applied to graphene, the position of carbon atoms shift relative to each other. As a result the vector position of lattice point changes. To explain this, the angle between a1 and a2 is considered here as ÃŽ ¸ instead of assuming 60o which is true for ideal or relaxed graphene structure. The eï ¬â‚¬ect in the tight-binding Hamiltonian is that the parameters of tight-binding scales changes accordingly. The strained lattice structure of graphene is shown in Fig.1. We have used the simple nearest Neighbor tight binding model. Here each Carbon atom is ÏÆ' bonded with three of its nearest neighbor Carbon atoms. Fig.1 : The direct lattice structure of graphene under strained condition The primitive unit vectors can be represented by where The separation of the carbon atoms (A and B) can be represented by three vectors R1, R2, R3 From Tight-binding energy dispersion model the formula of energy dispersion is given by [13] (1) Where Here is a fitting parameter which is often called the nearest neighbor overlap energy or hopping integral. The value of varies from 2.7eV to 3.3eV. (2) This is the generalized equation for the energy dispersion of graphene. Here is the angle between the primitive unit vectors. For the unstrained or relaxed condition, the value of the angle = 60o. In this case the Ï€ bands overlap at direct point or K point of the two dimensional brillouin zone. (a) (b) Fig.2(a) energy dispersion of relaxed graphene and (b) the corresponding brillouin zone. We have investigated the electronic structure of graphene under different planar strain distributions by the tight-binding (TB) approach. The graphene has been strained in three different ways [12]. These are : (i) symmetrical strain distribution (keeping the hexagonal symmetry unchanged) as shown in fig. 3.1(a) , (ii) asymmetrical strain distribution parallel to C-C bonds as shown in Fig. 3.1(b) , (iii) asymmetrical distribution perpendicular to C-C bonds as shown in Fig.3.1(c). Fig 3(a) Graphene system with symmetrical strain distribution, (b) asymmetrical strain distribution perpendicular to C-C bonds, and (c) asymmetrical strain distribution parallel to C-C bonds. Corresponding primitive cells in black, reciprocal lattices in green dashed and Brillouin zones in green grey are illustrated below the deformed lattices. Γ, K, M, R and S are the high symmetrical points. Lx and Ly are the half of the diagonal lengths of the primitive cells in parallel and perpendicular direction of the carbon-carbon bond. As the strain is applied to the graphene, it causes the deformation of the regular hexagonal structure of it . It also causes the deformation in the primitive unit cell. If the strain is symmetric then the band property of the system does not change but for asymmetrical strain , the band property of the system changes due to symmetry breaking. When an asymmetrical strain parallel to C-C bond is applied, it causes a deformation in the primitive unit cell. This deformation is taken as a change in angle between the primitive unit vectors. Here the strain is applied upto 12.2 % and it is seen that with the increase in strain the angle between the primitive unit vectors is reduced by following a 3 degree polynomial with respect to Lx and Ly(where Lx and Ly are in nanometer). The equation of is (3) This value of is then put in equation (2) to calculate the band gap under different strain distribution . It is seen that up to Ly =0.2396 nm band gap of graphene increases then the bandgap begin to fall . For this region the assumption of is different and it is, (4) In case asymmetrical applied strain perpendicular to C-C bond , up to 7.3 % strain the angle between the primitive unit vectors is increased by following a 2 degree polynomial with respect Lx and Ly. The equation of is, (5) Now up to Lx = 0.1323 nm band gap of graphene increases and then the bandgap begins to fall. For this region the assumption of is, (6) III.RESULT Asymmetrical strain distribution results in the opening of the bandgap between the maximum of the valance band and the minimum of the conduction band in graphene. When an asymmetrical strain parallel to carbon-carbon bond is applied, Ly increases. Then for the system in order to come back to its lowest energy, Lx decreases during the structural relaxation. Due to change of Lx and Ly, the angle between the primitive unit vectors decreases and causes the symmetry breaking. This angular change is taken as the parameter of deformed primitive cell to calculate the electronic structure of graphene. For example, for Ly = 0.2196, 0.2236, 0.2396, and 0.2436 nm the corresponding optimized values of Lx are Lx= 0.1228, 0.1224, 0.1217 and 0.1216 nm. Then from our proposed model the corresponding angle between the primitive unit vectors are =59.47o, 58.91o, 54.79o and 57.75o. The corresponding electronic structure or band diagrams are shown in fig.4 with the extended view at K point (a) (b) (c) (d) Fig.4 Extended view of bandgap opening for (a) Ly=0.2196 nm and Lx=0.1228 nm (b) Ly=0.2236 nm and Lx=0.1224 nm (c) Ly=0.2396 nm and Lx=0.1217 nm (d) Ly=0.2436 nm and Lx=0.1216 nm. Similar behavior is obtained in the graphene system, when asymmetrical strain perpendicular to carbon-carbon bond is applied. In this case for example for Lx =0.1268, 0.1292, 0.1353 nm the corresponding optimized Ly are Ly=0.2126, 0.2120 and 0.2105 nm and the corresponding deformed angle are = 60.52o, 61.05oand 60.38o. The opening of bandgap corresponding to these deformed angle are shown in fig.5 (a) (b) (c) FIG.4 EXTENDED VIEW OF BANDGAP OPENING FOR (A) LX=0.1268 NM AND LY= 0.2126 NM (B) LX= 0.1292 NM AND LY=0.2120 NM (C) LY=0.1353 NM AND LX= 0.2105 NM . These results revels that the zero bandgap or semi-metallic behavior of graphene sheet gets modified or a bandgap is opened when asymmetrical strain is applied to it. Now the question is what is the reason behind this? We know that planner graphene consists of strong bonds and delocalized pz electrons. Here orbitals are formed by overlapping the pz orbitals of the carbon atoms in the hexagonal lattice. These and bands touches each other at the K point causing zero bandgap. When the strain is applied the carbon-carbon bonds of graphene get elongated. Due to this elongation of carbon-carbon bonds, the electron clouds get localized on the corresponding carbon atoms. Therefore a restriction is imposed on movement of the electrons which causes the opening the bandgap at the K point. In this way, under strained condition, graphene loses its semimetal characteristics and turns into a direct bandgap semiconductor. IV.COMPARISON WITH ESTABLISHED METHOD: In this study, we have calculated the band gap of graphene under the application of asymmetrical strain by an analytical approach. The calculated value is found to be in great agreement with the measured value obtained by first principle calculation [14]. It is found that in case of asymmetrical strain distribution parallel and perpendicular to C-C bond, the percentage of error is more or less 4 % for most of the data which is shown In this study an analytical model has been proposed for investigating the energy dispersion of strained graphene under the distribution of the planner strain in parallel and perpendicular to carbon-carbon bond. Using the proposed model the energy dispersion for different planner strain has been calculated. From our study it is seen that for relaxed or symmetrically strained graphene, the band gap of the system is zero. When asymmetrical strain parallel or perpendicular to carbon-carbon bond is applied then a finite bandgap is opened. And it is also seen that tuning the percentage strain, the band gap can also be tuned. Here it is also seen that comparing the perpendicular strain, parallel strain parallel to the carbon -carbon bond more induces more bandgap. Finally we have compared our results with the results obtained from the established method and found good agreement with around 96 % accuracy. References [1]A.K. Geim and K.S. Novoselov, â€Å"The rise of graphene ,† Nat. Mater, vol.6, pp.183-191, 2007. [2]Ryutaro Sako, Hideaki Tsuchiyaand Matsuto Ogawa, â€Å"Influence of bandgap opening on ballistic electron transport in bilayer graphene and graphene nanoribbon FETs,† IEEE Trans. Electronic Devices., vol. 58, no. 10, pp. 3300–3306, Oct. 2011. [3]K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva and A.A. Firsov, Science 306, 666 (2004). [4]K.S. Novoselov, D. Jiang, F.Schedin, T.J. Bhoot, V.V. Khot-kevich, S.V. Morozov and A.K Geim, Proc.Natl.Acad. Sci. U.S.A. 102,10451 (2005). [5]M.Y. Han, B. Ozylmaz, Y. Zhang, and P. Kim, â€Å"Energy band gap engineering of grapheme nanoribbons,† Phys. Rev. Lett. ,vol. 98, no. 20, P. 206805, May 2007. [6]G. Liang, N. Neophytou, D.E. Nikonov, and M.S. Lundstrom, â€Å" Performance projections for ballistic graphene nanoribbon ï ¬ eld-effect transistors,† IEEE Trans. Electr on Devices, vol. 54, no. 4, pp. 677–682, Apr. 2007. [7]Y. W. Son, M. L. Cohen, and S. G. Louie, â€Å"Energy gaps in graphene nanoribbons,† Phys. Rev. Lett., vol. 97, no. 21, p. 216803, Nov. 2006. [8]X. Li, X. Wang, L. Zhang, S. Lee, and H. Dai, â€Å"Chemically derived, ultrasmooth graphene nanoribbon semiconductors,† Science, vol. 319, no. 5867, pp. 1229–1232, Feb. 2008. [9]T. Ohta, A. Bostwick, T. Seyller, K. Horn, and E. Rotenberg, â€Å"Controlling the electronic structure of bilayer graphene,† Science, vol. 313, no. 5789, pp. 951–954, Aug. 2006. [10]Y. Zhang, T.-T. Tang, C. Girit, Z. Hao, M. C. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, â€Å"Direct observation of a widely tunable bandgap in bilayer graphene,† Nature, vol. 459, no. 7248, pp. 820–823, Jun. 2009. [11]Jun Ito, Jun Nakamura, and Akiko Natori, â€Å"Semiconducting nature of the oxygen-adsorbed graphene sheet ,† Journal of applied phys. 103,113712 (2008). [12]Paolo Marconcini, Gianluca Fiori, Alessandro Ferretti, Giuseppe Iannaccone, and Massimo Macucci, â€Å"Numerical analysis of transport properties ofboron-doped graphene FETs†. [13]Zhipping Xu and Kun Xue, â€Å"Engeneering grap hene by oxidation: a first principle study,† Nanotechnology 21 (2010) 045704 [14]Gui Gui, Jin Li, and Jianxin Zhong ‘Band structure engineering of graphene by strain: First-principles calculations’, PHYSICAL REVIEW B 78, 075435 2008. [15]V.J. Surya, K. Iyakutti, H. Mizuseki, and Y. Kawazoe, â€Å"Tuning electronic structure of Graphene: A first-principle calculation,† IEEE Trans. Nanotechnology, vol.11, No.3,pp.534-541, May 2012.

The Death Penalty is a Just and Proper Punishment :: Pro Capital Punishment Death Penalty Crime Essays

The Death Penalty is a Just and Proper Punishment The idea of putting another human to death is hard to completely fathom. The physical mechanics involved in the act of execution are easy to grasp, but the emotions involved in carrying out a death sentence on another person, regardless of how much they deserve it, is beyond my own understanding. I know it must be painful, dehumanizing, and sickening. However, this act is sometimes necessary and it is our responsibility as a society to see that it is done. Opponents of capital punishment have basically four arguments. The first is that there is a possibility of error. However, the chance that there might be an error is separate from the issue of whether the death penalty can be justified or not. If an error does occur, and an innocent person is executed, then the problem lies in the court system, not in the death penalty. Furthermore, most activities in our world, in which humans are involved, possess a possibility of injury or death. Construction, sports, driving, and air travel all offer the possibility of accidental death even though the highest levels of precautions are taken. These activities continue to take place, and continue to occasionally take human lives, because we have all decided, as a society, that the advantages outweigh the unintended loss. We have also decided that the advantages of having dangerous murderers removed from our society outweigh the losses of the offender. The second argument against capital punishment is that it is unfair in its administration. Statistics show that the poor and minorities are more likely to receive the death penalty. Once again, this is a separate issue. It can't be disputed sadly, the rich are more likely to get off with a lesser sentence, and this bias is wrong. However, this is yet another problem of our current court system. The racial and economic bias is not a valid argument against the death penalty. It is an argument against the courts and their unfair system of sentencing. The third argument is actually a rebuttal to a claim made by some supporters of the death penalty. The claim is that the threat of capital punishment reduces violent crimes. Opponents of the death penalty do not agree and have a valid argument when they say, "The claims that capital punishment reduces violent crime is inconclusive and certainly not proven." I am not refuting this accusation. In fact, statistics show that the death penalty neither lowers or raises the incidence of violent crimes.

Fast food †Harry Snyder Essay

Have you ever had a food that you could never forget? You still remember the perfect combination of taste and mouthwatering appearance as you took your first bite. For many, In-N-Out Burger’s famous burgers are unforgettable; â€Å"Known for its made-to-order hamburgers, fresh ingredients, and efficient service. † The simple beginning, tidy appearance, basic menu, and customer satisfaction each contribute to the popularity of this restaurant. One reason the restaurant is so well-known is because of its beginning. Founded in Baldwin Park, California in 1948 by Harry and Esther Snyder, In-N-Out Burger was. California’s first drive through restaurant. The couple had the idea of customers being able to drive up and order their food through a two-way speaker unit, which they ultimately accomplished. Many new proposals to open their business elsewhere then arose, and offers were being accepted at a rather rapid pace. The successful opening contributed to the quickly gained popularity. Another reason In-N-Out Burger is rather popular is a result of the notably clean appearance. As part of the promise of Harry and Esther Snyder, every venue has cleanliness worth remembering as there is a policy to maintain tidy quarters. Charlie Palmer states, â€Å"The places are incredibly clean. I’m impressed by that†. Even as a restaurant entrepreneur himself, Palmer took note of the pristine premises. Well-kept In-N-Out locations often contribute to the widespread popularity. Also, the simple menu of In-N-Out has remained the same from the very beginning, providing yet another reason for customers to come back. The basic burgers, fries, pop, and shakes displayed on the In-N-Out menu make it very easy to order from. There are three combos to choose from that each include a different burger with fries and a drink, or you can simply choose just one or two of the food choices. Without the complications of numerous added food items, people are able to quickly decide what they’d like to eat and drink. The straightforward menu contributes to the amount of incoming customers in the popular In-N-Out Burger chain. Their mission statement was simple, â€Å"Give customers the freshest, highest quality foods you can buy and provide them with friendly service in a sparkling clean environment. † They still use this philosophy in every running In-N-Out open today. Quality, freshness, and good service are the core ingredient to In-N-Out’s success. Harry and Esther’s two sons, Guy and Rich, learned the business from working in their parents store when they were young. When Harry Snyder passed away in 1976, there were 18 drive-through locations running. Rich Snyder, at age 24, took over as President. Guy Snyder helped him to establish what like a cafeteria at the Baldwin Park Headquarters. This new facility allowed In-N-Out to have total quality control over all In-N-Out ingredients including the secret sauce. They also started an In-N-Out â€Å"University†, where new managers are trained and the In-N-Out philosophies for Alyissa Balderama M/W 11:00. Professor Read success are consistently reinforced. While Rich was President, In-N-Out grew from those 18 locations in 1976 to 93 locations. Rich Snyder remained true to his parents’ goal of serving only the freshest foods available. None of the ingredients were frozen, and no microwaves were used. All orders were made to order, contributing to what some in the food industry considered a long wait for a fast-food hamburger. The milk shakes were made with real ice cream, and the burgers were 100 percent beef. The beef was ground and formed into patties by In-N-Out workers at the Baldwin Park facility. The lettuce was broken into leaves by hand, and the buns were baked fresh using old-fashioned sponge dough that took six to eight hours to rise. The potatoes for the French fries were shipped in burlap sacks to the outlets, where associates cut them by hand. In-N-Out has used southern California-grown Kennebec potatoes, which are said to be ideal for frying. The French fries have always been fried in cholesterol-free vegetable oil. Guy Snyder then took over and became Chairman of the Board and CEO in 1993. As Chairman, H. Guy Snyder, led In-N-Out into the future with continued expansion throughout California, Nevada and Arizona. Guy carried on the same tradition that was set in 1948 by his parents, stressing the same basic values that helped make In-N-Out so successful. While Guy was Chairman of the Board, In-N-Out grew from 93 locations to 140 at the time of his death in 1999. The atmosphere of enthusiasm for serving customers the freshest quality hamburgers and French fries can be seen all the way from the many store locations to the office Associates. Though times have changed, little has changed at In-N-Out. The menu-burgers, fries and drinks-is still the same basic menu customers have enjoyed since 1948. Everything is still made fresh to order. There are no microwaves or freezers. Customers may observe French fries being made from hand-diced, fresh, whole potatoes. And the shakes are made from real ice cream. There have been a few modifications in recent years. The original In-N-Out offered only drive-thru and walk up service. Most of the newer In-N-Out Burger locations provide indoor and outdoor seating. Aside from building improvements, though, In-N-Out has retained the basic traditions that have made it a favorite for 60 years. Lastly, customer satisfaction makes a huge difference for the better in the stature of In-N-Out. Consumers are often very dedicated and spread the word of their fast food favorite. Even chef reviews are completely positive according to Esquire’s chef survey. Alton Brown is reminded of his childhood when he drinks the shakes and Thomas Keller claims it’s an ideal illustration of classic American fast food. And although some who go for a burger find it as just average, their number is much outweighed by the majority of favorable feedback. Customer’s enjoyment in In-N-Out Burger surely helps with the positive standing amongst other fast food places. Alyissa Balderama M/W 11:00 Professor Read. Overall, In-N-Out Burger’s marketable beginning, clean restaurant appearance, simple menu, and customer satisfaction all help maintain a good reputation. Every location follows the Snyder’s promise to keep the restaurant as clean as possible and the never-changing menu has stayed as easy to order from as it was in the beginning. So the next time you’re in the area of an In-n-Out Burger, make sure to stop by and try something there. Alyissa Balderama M/W 11:00 Professor Read Works Cited â€Å"In-N-Out Burger. † In-N-Out Burger. N. p. , n. d. Web. 19 Nov. 2013. Wikipedia. Wikimedia Foundation, n. d. Web. 18 Nov. 2013.

Sample Resume For A Public School Program - 816 Words

Having such a unique nature regarding the budget, the recommendations for involving staff in conducting operations and setting priorities is challenging. Our budget analyst is extremely helpful in guiding us through the maze of proper procedure when making request to access funds. Having taught in public school for many years, it was a challenging transition because there are funds available for expenditures that are impossible in public schools, however accessing them is requires planning and careful attention to detail. The most effective recommendation that can be made for staff in conducting operations is to be organized. Knowing what will be needed for educational and extracurricular activities is essential in receiving approval for expenditures. A solution to this would be to include quarterly updates for staff informing them of any changes in the process, addition of new TERO vendors, restricted vendor access, or obtaining permission to utilize a non-TERO vendor. This re commendation would assist the staff in prioritizing expenditures and receiving requests in a timely fashion. Another recommendation would be to provide new teachers a training session that outlines the nature of the school budget and the process used to access funds. In our school we have coaches, class sponsors and club sponsors that need to access their funds many times throughout the year. Providing these new individuals a flow chart, sample requests, and a list of TERO vendors would beShow MoreRelatedThe For Intellectual Property Law Essay1617 Words   |  7 Pagesin Intellectual Property Law, will require that I first graduate with a four-year degree. The degree with which one takes is not specific, in fact, law schools admit individuals with degrees from almost every field of study (LearnHowToBecome.org, 2016). The next step is going through law school which typically takes three years. 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