Non Equillibrium Green’s Function Technique

Non Equillibrium verdancys Function techniqueNON EQUILLIBRIUM GREENS FUNCTION TECHNIQUE USED FOR THE METAL-INSULATOR-METAL DIODESANSHUMANElectronics and Communication Dept. NIT Kurukshetra twitch In this publisher theoretical analysis of NEGF method, including the shift par and Poisson equation, is do followed by the derivation of an analytical mock up utilise NEGF tunnelling prospect by any issuing of insulating layers. Numerical NEGF simulator are shown matching with the AF-TMM simulator results.INTRODUCTIONTHE strike OF tunnelling phe noena in Metal nonconductor Metal (MIM) is an important topic for the beat posterior of the development of rectennas for aptitude harvesting and infrared detectors applications. Although the interest in Metal-Insulator-Metal (MIM) diodes dates back to 1950s 14, but they attracted the attention again in the last hardly a(prenominal) years due to its applications, energy harvesting 58 and infrared/terahertz detectors 911.Earlier, d ifferent analytical nerves for the tunnelling transmission probability through MIM diodes were developed based on WKB approximation 24. However, the WKB does not take into consideration the wave go reflections at the interface amid different layers 14. indeed, in that respect came the need for other models to mold the tunnelling probability. Non Equilibrium putting greenish Function (NEGF) 12 numeral method is one of the methods apply to calculate the tunnelling transmission probability 1518. It is an accurate numeral method, but it needfully long time of calculations on a PC in compare to other analytical models.Any program used for the simulation of a machination performs a solution of apotheosis equation and Poisson equation 19.The transport equation gives the electron density, n(r) and the on-line(prenominal), I for a known potential compose U(r), while Poisson equation gives the telling U(r), felt by an electron due to the presence of other electron in its vi cinity.Here, in this paper the Quantum transport, Greens affairs and its various equations under(a) 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 in the midst of the two metallic elements.Fig.1. Transport of electrons for single energy level machinationThe paper is organized as follows in Section II, the transport equations are discussed. In section III NEGF equations for MIM Diode is hear in detail. The governing equations and numerical implementation of it is outlined. The material disputations used in the simulation are excessively summarized.GENERAL TRANSPORT EQUATIONLets consider the model for a single subterfuge sandwiched between two metals 1and 2ION THE METAL-INSULATOR.ce of other electron in its vicinity.port and 111111111111111111 111111111111111111111111, as shown in fig. 1.The gismo is assumed to be having a single energy level, . 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 potency, Vb between metal 1 and 2, the Fermi- energies of two metals gets modified to 1 and 2 respectively and given as 19 (1)This difference in Fermi-energy gives demonstrate 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 f1 but if it is in equilibrium with metal 2, number of electrons will be f2, where (2)Let and be the rate of hightail it of electron from device into metal 1 and metal 2 respectively. Therefore the currents I1 and I2 crossing metal1 and 2 interfaces are given as20 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 invariable flow of current, I (eq. 4b) in the external circuit.Assuming 1 2 and the temperature is low enough that f1 () f0 ( 1) 1 and f2 () f0 ( 2) 0, the Eq. 4b simplifies to 21 If = (5)Eq.5 suggests that we stop flow an unlimited current through this one level device if we increase, i.e. by twin the device more and strongly to the metal contacts. But the maximum conductance of a one-level device is rival to 20, so there must be some reduction factor.This reduction is due to the broadening of the separate level that occurs because of increased twin of the device with the two metals. This broadened discrete level can be described by the distributionWith line-width of and shift of level from to +, where. This broadening phenomena modifies the Eqs. (4a, b) to i nclude an constitutive(a) over all energies weighted by the distribution D(E) 13 (6a) (6b)Using algebraical manipulation Eqs. (6a, b) becomes (7a) (7b)Where (8) (9)Till now we have discussed device with single energy level . 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 numberive 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 aloofness 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 and 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 quasi(prenominal) matrix quantity (13)TRANSMISSION EQUATION FOR MIM DIODE USING NEGF EQUATIONSThe 1D time-independent single-particle Schrdinger equation is given by 13Where, is the reduced Plank constant, (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 gridiron points having uniform spacing, a, then finite difference discretization on the 1D grid is applied to Schrdinger 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 responsibility met al electrodes is interpreted 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, Schrdinger equation now takes the following form 13 (3)Where, H is the M M Hamiltonian matrix of the insulator potential, I is the M M identity matrix, is the wavefunction M 1 vector and S is M 1 vector. L and R are the M 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. distributively insulator layer is characterized by a barrier height (Uj), a thickness (d j), a dielectric constant j, and an effective mass (m j).Now, under a tri-diagonal form H can be rewritten asL and R are given asThe solution of Eq. (1) can be given in the terms of retarded Greens function as where is MM retarded Greens function 13The rate of escape of electron to either left or right metal from a given state can be taken i nto consideration by defining two quantities, L and R 14.Hence, the tunnelling probability can now be computed as 14COMPARISION OF NEGF MODEL WITH OTHER MODELSA model of MIIM diode was simulated using NEGF, AF-TMM and WKB Approximation for a relative 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. 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