Non Equillibrium Green’s Function Technique

Non Equillibrium Green’s Function Technique NON EQUILLIBRIUM GREEN’S FUNCTION TECHNIQUE USED FOR THE METAL-INSULATOR-METAL DIODES ANSHUMAN Electronics and Communication Dept. NIT Kurukshetra Abstract – In this paper theoretical analysis of NEGF method, including the transport equation and Poisson equation, is done followed by the derivation of an analytical model using NEGF tunnelling probability through any number of insulating layers. Numerical NEGF simulator are shown matching with the AF-TMM simulator results. INTRODUCTION THE STUDY OF tunnelling phenomena in Metal Insulator Metal (MIM) is an important topic for the aim of the development of rectennas for energy harvesting and infrared detectors applications. Although the interest in Metal-Insulator-Metal (MIM) diodes dates back to 1950s [1]–[4], but they attracted the attention again in the last few years due to its applications, energy harvesting [5]–[8] and infrared/terahertz detectors [9]–[11]. Earlier, various analytical expressions for the tunnelling transmission probability through MIM diodes were developed based on WKB approximation [2]–[4]. However, the WKB does not take into consideration the wave function reflections at the interface between different layers [14]. Therefore, there came the need for other models to simulate the tunnelling probability. Non Equilibrium Green Function (NEGF) [12] numerical method is one of the methods used to calculate the tunnelling transmission probability [15]–[18]. It is an accurate numerical method, but it needs long time of calculations on a PC in comparison to other analytical models. Any program used for the simulation of a device performs a solution of transport equation and â€Å"Poisson† equation [19].The transport equation gives the electron density, n(r) and the current, I for a known potential profile U(r), while Poisson equation gives the effective U(r), felt by an electron due to the presence of other electron in its vicinity. Here, in this paper the Quantum transport, Green’s functions and its various equations under non equilibrium condition are discussed and a detailed quantum mechanical modeling of the tunnelling current through MIM diodes is presented. An analytical expression for the tunnelling transmission probability is presented using the NEGF equations for any number of insulator layers between the two metals. Fig.1. Transport of electrons for single energy level device The paper is organized as follows: in Section II, the transport equations are discussed. In section III NEGF equations for MIM Diode is described in detail. The governing equations and numerical implementation of it is outlined. The material parameters used in the simulation are also summarized. GENERAL TRANSPORT EQUATION Let’s consider the model for a single device sandwiched between two metals 1and 2ION THE METAL-INSULATOR.ce of other electron in its vicinity.port and 111111111111111111111111111111111111111111, as shown in fig. 1.The device is assumed to be having a single energy level, Ô . Our first aim is to find the number of electrons, N in the device. Let Ef be the Fermi level set by the work function of the two metal contacts under the equilibrium condition. On applying the bias voltage, Vb between metal 1 and 2, the Fermi- energies of two metals gets modified to  µ1 and  µ2 respectively and given as [19]: (1) This difference in Fermi-energy gives rise to a non-equilibrium condition and hence two different Fermi-functions for the two contacts. If device is in equilibrium with metal 1, then number of electrons will be f ­1 but if it is in equilibrium with metal 2, number of electrons will be f ­2, where (2) Let and be the rate of escape of electron from device into metal 1 and metal 2 respectively. Therefore the currents I1 and I2 crossing metal1 and 2 interfaces are given as[20]: And (3) For I1 = I2 = I, we get steady-state number of electrons N and current I as: (4a) (4b) Due to the applied bias voltage one of the reservoir keeps pumping the electron trying to increase the number while the other keeps emptying it trying to lower the number. Ultimately, there is a continuous flow of current, I (eq. 4b) in the external circuit. Assuming à °Ã‚ Ã…“†¡1 > à °Ã‚ Ã…“â‚ ¬ > ÃŽ ¼2 and the temperature is low enough that f1 (ÃŽ µ) ≠¡ f0 (ÃŽ µ − ÃŽ ¼1) ≈ 1 and f2 (ÃŽ µ) ≠¡ f0 (ÃŽ µ − ÃŽ ¼2) ≈ 0, the Eq. 4b simplifies to [21]: If = (5) Eq.5 suggests that we can flow an unlimited current through this one level device if we increase, i.e. by coupling the device more and strongly to the metal contacts. But the maximum conductance of a one-level device is equal to [20], so there must be some reduction factor. This reduction is due to the broadening of the discrete level that occurs because of increased coupling of the device with the two metals. This broadened discrete level can be described by the distribution: With line-width of ÃŽ ³ and shift of level from ÃŽ µ to ÃŽ µ+à ¢Ã‹â€ Ã¢â‚¬  , where. This broadening phenomena modifies the Eqs. (4a, b) to include an integral over all energies weighted by the distribution D(E) [13]: (6a) (6b) Using algebraic manipulation Eqs. (6a, b) becomes: (7a) (7b) Where (8) (9) Till now we have discussed device with single energy level ÃŽ µ. But in practical situation (i.e. for real devices) there exist multiple energy levels. Any device, in general, can be represented by a Hamiltonian matrix, whose eigenvalues tells about the allowed energy levels. For example if we describe the device using an effective mass Hamiltonian H = then it can be represented with a (NxN) matrix by choosing a discrete lattice with N point and applying methods of finite-differences [13]. This corresponds to using a discretized real space basis. Similarly, we define self-energy matrices [∑ ­1,2] which describe the broadening and shift of energy levels due to coupling with the two metals. The required NEGF equations now can be obtained from Eqs(7a, b) by replacing scalar quantities ÃŽ µ and ÏÆ' ­1,2 with the corresponding matrices [H] and [∑ ­1,2], and is given as: , (10) , (11) The number of electrons N, in the device is replaced with the density matrix, given by: {} (12) Current is still represented by Eq. (7b). The transmission can be given as the trace of the analogous matrix quantity: (13) TRANSMISSION EQUATION FOR MIM DIODE USING NEGF EQUATIONS The 1D time-independent single-particle Schrà ¶dinger equation is given by [13]: Where, is the reduced Plank constant, ψ(x) is the electron wave-function, m is the effective mass and U(x) is the potential energy. If it is assumed that the insulator layers are divided into M grid points having uniform spacing, a, then finite difference discretization on the 1D grid is applied to Schrà ¶dinger equation Eq. (1) at each node i as follows [14]: (2) Where, represents the interaction between the nearest neighbour grid points i and i+1, Ui ≠¡ U (xi), and mi is the electron effective mass between the nodes i and i + 1. The coupling of the potential barrier to the left and right metal electrodes is taken into consideration by rewriting Eq. (1) for i =1 and i = M with open boundary conditions expressed at Metal1/Insulator and Insulator/Metal2 interfaces. So, Schrà ¶dinger equation now takes the following form [13]: (3) Where, H is the M Ãâ€" M Hamiltonian matrix of the insulator potential, I is the M Ãâ€" M identity matrix, ψ is the wavefunction M Ãâ€" 1 vector and S is M Ãâ€" 1 vector. ∑L and ∑R are the M Ãâ€" M self-energies of the left and right contacts respectively. Fig. 1. Potential of a stack of N insulator materials under applied bias voltage, Vb. Each insulator layer is characterized by a barrier height (Uj), a thickness (d j), a dielectric constant ÃŽ µ j, and an effective mass (m j). Now, under a tri-diagonal form H can be rewritten as: ∑L and ∑R are given as: The solution of Eq. (1) can be given in the terms of retarded Green’s function as where is MÃâ€"M retarded Green’s function [13]: The rate of escape of electron to either left or right metal from a given state can be taken into consideration by defining two quantities, ΓL and ΓR [14]. Hence, the tunnelling probability can now be computed as [14]: COMPARISION OF NEGF MODEL WITH OTHER MODELS A model of MIIM diode was simulated using NEGF, AF-TMM and WKB Approximation for a comparative analysis of their transmission probability vs. electron transmission energy curve. The parameter spacing, a, for the NEGF calculation was assumed equal to the hundredth of the insulator layer thickness. This was found adequate for reasonable simulation time. The effective mass was assumed equal to the free mass of the electron throughout the MIM structure. Fig.3 below shows theof the simulated MIIM of Nb/Nb2O5 (2nm)-Ta2O5 (1nm)/Nb at 0.1V of bias voltage. A complete matching between AF-TMM and NEGF results is observed. Fig.3. Transmission probability T (Ex) versus the electron transmission energy calculated using AF-TMM, NEGF, and WKB at Vb = 0.1 V for Nb/Nb2O5/Ta2O5/ Nb MIIM diode. Fig.4. Energy band diagram of the MIIM diode used for simulation REFERENCES J. G. Simmons, â€Å"Electric tunnel effect between dissimilar electrodes separated by a thin insulating film,† J. Appl. Phys., vol. 34, no. 9, pp. 2581–2590, Mar. 1963. J. G. Simmons, â€Å"Generalized formula for the electric tunnel effect between similar electrodes separated by a thin insulating film,† J. Appl. Phys., vol. 34, no. 6, pp. 1793–1803, 1963. R. Stratton, â€Å"Volt-current characteristics for tunneling through insulating films,† J. Phys. Chem. Solids, vol. 23, no. 9, pp. 1177–1190, Sep. 1962. J. G. Simmons, â€Å"Potential barriers and emission-limited current flow between closely spaced parallel metal electrodes,† J. Appl. Phys., vol. 35, no. 8, pp. 2472–2481, Aug. 1964. S. Grover and G. Moddel, â€Å"Applicability of metal/insulator/metal (MIM) diodes to solar rectennas,† IEEE J. Photovoltaic, vol. 1, no. 1, pp. 78–83, Jul. 2011. K. Choi, M. Dagenais, F. Yesilkoy, A. N. Chryssis, and M. C. Peckerar, â€Å"Solar spectrum rectification using Nano-antenna and tunneling diodes,† Proc. SPIE, vol. 7605, pp. 1–12, Feb. 2010. M. R. Osgood, B. R. Kimball, and J. Carlson, â€Å"Nanoantenna-coupled MIM nanodiodes for efficient vis/nir energy conversion,† Proc. SPIE, vol. 6652, no. 1, pp. 1–11, Sep. 2007. R. Osgood, J. B. Carlson, B. R. Kimball, D. P. Ziegler, J. R. Welch, L. E. Belton, G. E. Fernandes, Z. Liu, and J. Xu â€Å"Plasmonic nanorectennas for energy conversion,† Proc. SPIE, vol. 7394, pp. 1–11, Aug. 2009. J. A. Bean, A. Weeks, and G. D. Boreman, â€Å"Performance optimization of antenna-coupled tunnel diode infrared detectors,† IEEE J. Quantum Electron., vol. 47, no. 1, pp. 126–135, Jan. 2011. B. Tiwari, J. A. Bean, G. Szakmany, G. H. Bernstein, P. Fay, and W. Porod, â€Å"Controlled etching and regrowth of tunnel oxide for antenna coupled metal-oxide-metal diodes,† J. Vacuum Sci. Technol. B, Microelectron. Nanometer Struct. , vol. 27, no. 5, pp. 2153–2160, 2009. J. A. Bean, B. Tiwari, G. H. Bernstein, P. Fay, and W. Porod, â€Å"Thermal infrared detection using dipole antenna-coupled metal-oxide-metal diodes,† J. Vacuum Sci. Technol. B, Microelectron. Nanometer Struct. , vol. 27, no. 1, pp. 11–14, 2009. Arun Goud Akkala, †NEGF Simulation of electron transport in resonant inteband tunneling diodes†, Purdue University, pp. 8-18, 2011. Supriyo Dutta, â€Å"Nanoscale device modeling: the Green’s function method,† Superlattices and Microstructures, vol. 28, no. 4, pp.253-272, 2000. Islam E. Hashem, Nadia H. Rafat and Ezzaldin A. Soliman, â€Å"Theoretical study of metal-insulator-metal tunneling diode figure of merit,† IEEE Journal of Quantum Electronics, vol. 9, no. 1, pp. 72-80, January 2013 M. Moreau, D. Munteanu, and J.-L. Autran, â€Å"Simulation of gate tunneling current in metal–insulator–metal capacitor with multilayer high-ÃŽ º dielectric stack using the non-equilibrium green’s function formalism,† Jpn. J. Appl. Phys., vol. 48, no. 11, pp. 111409-1–111409-8, Nov. 2009. D. Munteanu, J. L. Autran, M. Moreau, and M. Houssa, â€Å"Electron transport through high-ÃŽ º dielectric barriers: A non-equilibrium Green’s function (NEGF) study,† J. Non-Crystall. Sol., vol. 355, no. 21, pp. 1180–1184, Jul. 2009. S. Grover and G. Moddel, â€Å"Engineering the current-voltage characteristics of metal-insulator-metal diodes using double-insulator tunnel barriers,† Solid-State Electron., vol. 67, no. 1, pp. 94–99, Jan. 2012. A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, â€Å"A novel numerical technique for solving the one-dimensional Schrà ¶dinger equation using matrix approach-application to quantum well structures,† IEEE J. Quantum Electron., vol. 24, no. 8, pp. 1524–1531, Aug. 1988. Supriyo Dutta, â€Å"The Non-equilibrium green’s function (NEGF) formalism: An elementary introduction,† IEEE IEDM, pp. 703-706, 2002. Supriyo Dutta, Quantum transport: atom to transistor, 1st ed., Cambridge University Press, 2005, pp.1-30 SYMBOLS

Recent advances in medical technology

The two articles What is telecasters? And ‘Do we know too much? Illustrate what the evolution of medical technology is and how it improves people's life. Both articles provide medical studies to support the curative effects. Furthermore, although there are still some limitations or negative aspects toward the progressions, It Is believed that those will be solved with time. In the article What Is telecasters? ‘ the author elucidates how its operation differs from traditional surgery.It Is revealed that doctor an give a remote operation by using robotic arms connected to fiber-optic cables without distance and facilities restrictions. However, some people doubts If robots can be relied on; the author also states that It might not function well due to the network access and compatibility of computer. The second article alms to address the question of what happens If we can extract Information from our genes. By genetic tests, we can aware of the genetic disorder that we mig ht suffer from and prevent contracting certain diseases.Currently, large range of disorders can be detected by complex genetic tests such as newborn screening, diagnostic, carrier and predictive testing. By the progression of medical technology, we are able to lead a longer and healthier life, furthermore, tests can be used to blood relationship testing and applied to crack down on crime. However, some people worry about that it might result In some negative impact toward their living. As the remarkable development on therapy introduced in the essays, human beings benefit greatly from telecasters and genetic tests; though they are still some limitations and difficulties need to be coped with.

Princess Stories and Marriage: The Effect on Young Minds Essay

It is no secret that in modern day America more than half of marriages end in divorce, yet many still fantasize their fairytale weddings and happy endings knowing that the chances of staying together for the rest of their lives are slim. These early concepts of happy endings develop in the minds of young girls during the period of time when gender roles start to be enforced. They are not only introduced to parenting but in the media as well. In past princess movies and stories, men have always been the people in power. They are the kings and prince charming that are supposed to sweep every princess off their feet and provide them a comfy lifestyle in the castle with a happily ever after. The princesses, on the other hand, are always shy, quiet, demure, poised, and need some kind of saving. The resolution of these stories always ends in the prince marrying the princess or beautiful maiden, thus completing his conquest and receiving his â€Å"award.† Young girls, as they are more marketable towards these stories, have adopted these elements into their own romantic life giving them false expectations of real-life relationships. 1 in 4 women will be victims of severe violence by an intimate partner in their lifetimes. They are conditioned from a young age to want to have a perfect romance, but false expectations often lead them to be in relationships with the wrong people. Although these stories may seem harmless, they push an outdated agenda on to a society where most are trying to progress into an era where women are treated as equal individuals and as valued members in society, not just your average princess. In My Problem With Her Anger, by Eric Bartels, the husband and wife power dynamics in marriage is showcased in a different way than what is depicted in princess stories. Bartels starts off the piece. The wife, according to her husband, is condescending, never appreciates what he does, and fights with him constantly, which breaks the stereotypical traits of a princess and princely relationship. Charles Perrault’s Cinderella is one of the main princess fairy tales that will be showcased in my writing, as it includes many of the topics discussed, such as gender roles, power dynamics, which can be tied back to modern day domestic abuse. A question that may appear when researching a question like this may be, how exactly does the princess story influence children on certain topics. Today, we have tv, movies, books, and the internet. Children are getting more and more educated in the media as they are learning in school how to do so. They can easily access these stories through different outlets. Although I believe princess stories are harmless in intent. They carry old-fashioned values that children, mostly emulate during their play time. I too played princess as a child and have grown to know that’s not how life really is. In princess stories, The princess gets everything done for her, she has servants and she’s just there to wait around for the magical prince to come and make her life better. Many girls grow up thinking the same way until they’ve faced a hard reality, but if privileged enough are still able to think this way until the day they get married, and in marriage then realize itâ€℠¢s not what the princess stories had advertised. Women are told to â€Å"deal† with what their husband gives them in life. They’re supposed to take whatever comes at them, but still, manage to keep the image of a â€Å"good woman.† Someone who takes care of their husband, the children, someone who never seems sad, cooks, cleans, maintains a good body. These are unrealistic qualities that a woman should have to uphold. Princess stories further argue that. For example, we take Beauty and the Beast, a tale that is taken very lightly and is seen as a beautiful romance in the eyes of many, but in fact is in a way a promoter of abuse. The beast is manipulative and a horrible thing. It takes her father with no remorse. Belle deals with his temper and abusive and threatening behavior yet she still â€Å"falls in love† In real life, this would not work out. In this story, Belle believes she can change the monster from his abusive and threatening nature to someone prince-like and chivalrous, but in fact, you can’t really change someone that easily, but the movie says its possible. In real life, this would probably not cut it. Many girls go through their princess phase and wanting to have their prince, and it’s a little alarming to me how they already know what they want through just a story. To me, princess stories don’t promote diversity and keep up with the modern day which limits a child’s way of thought. Princess stories to this day have only been heterosexual relationships. We are changing as a society and love are possible in many different ways. If children aren’t taught to be more accepting of others, seeing only heterosexual relationships in media will make them close-minded and homophobic. I see this myself with my little sister. She’s never watched any gay princesses or princes, therefore the idea of that seems disgusting to her, even though I’ve never taught her to be that way.

Effects Of Racial Corruption In The Criminal Justice System

Racial Corruption in the Criminal Justice System It appears that the criminal justice system is racially biased. Instead of the criminal justice system giving all of the citizens of the United States of America the same civil treatment, it treats all individuals differently based upon the color of an individual’s skin. What does it mean to be racially biased? Being racially biased is known as to judge one by his or her physical appearance. Treating people of color unfairly is still common to this day. The criminal justice system is racially biased in its law enforcement, prisons, and court systems. To begin, when it comes to officers pulling over victims, they pull over more black and Hispanic Americans than they do Caucasians.†¦show more content†¦When it comes to holding drugs or weapons in a vehicle, law enforcement officers find more illegal items in white people’s vehicles. According to an article written by Jeff Guo, â€Å"Between 2009 and 2013, Chicago police stopped over 200,000 white drivers, but only searched 906, of which 237 had contraband. Of those searched, that’s a hit rate of 26 percent. In that same time period, they stopped over 300,000 black drivers, searched 6,593 of them, and found contraband in 1,232 cases. That’s a hit rate of 19 percent† (Guo). The statistics given above show that officers are more likely to check for illegal items in a minorities’ vehicle rather than a white person’s vehicle. Furthermore, the law enforcement here in the United States is extremely brutal to black and Hispanic Americans. There are law enforcement officers that go over the line when it comes to dealing with people of color. Officers tackle and beat black Americans down to the ground, but when it comes to a white American the police officers do not go so over board unless if needed. There are cases were both white Americans and black Americans talk back to officers, but for some reason officers seem to always show more cruelty to black Americans. According to â€Å"Interpersonal Relations and Group Processes†, â€Å"Black people may engage in more belligerent behavior, including â€Å"talking back† to police officers, and—in a vicious cycle—this belligerence may prompt more severe use of force by police† (Correll1006). AnShow MoreRelatedRacial Discrimination And The Criminal Justice System1725 Words   |  7 PagesRacial discrimination is defines as racism that implicates the credence in racial differences, which acts as a justification for non-equal treatment of members of that race. Also, this paper will be focusing on the race industry within the criminal justice system in all level law enforcement, courts and court system. 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