Non Equillibrium Green’s Function Technique
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NON EQUILLIBRIUM GREEN’S FUNCTION TECHNIQUE USED FOR THE METALINSULATORMETAL 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 AFTMM 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 MetalInsulatorMetal (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 “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 METALINSULATOR.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 E_{f} be the Fermi level set by the work function of the two metal contacts under the equilibrium condition. On applying the bias voltage, V_{b} between metal 1 and 2, the Fermi energies of two metals gets modified to µ_{1} and µ_{2} respectively and given as [19]:
(1)
This difference in Fermienergy gives rise to a nonequilibrium condition and hence two different Fermifunctions 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 I_{1} and I_{2} crossing metal1 and 2 interfaces are given as[20]:
And (3)
For I_{1} = I_{2} = I, we get steadystate 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 ðœ‡_{1} > ðœ€ > μ_{2} and the temperature is low enough that f_{1} (ε) ≡ f_{0} (ε − μ_{1}) ≈ 1 and f_{2} (ε) ≡ f_{0} (ε − μ_{2}) ≈ 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 onelevel 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 linewidth of γ and shift of level from ε to ε+âˆ†, 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 ε. 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 finitedifferences [13]. This corresponds to using a discretized real space basis.
Similarly, we define selfenergy 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 ε and σ_{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 timeindependent singleparticle Schrödinger equation is given by [13]:
Where, is the reduced Plank constant, ψ(x) is the electron wavefunction, 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 Schrödinger 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, U_{i} ≡ U (x_{i}), and m_{i} 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, Schrödinger 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, ψ is the wavefunction M × 1 vector and S is M × 1 vector. ∑_{L} and ∑_{R} are the M × M selfenergies of the left and right contacts respectively.
Fig. 1. Potential of a stack of N insulator materials under applied bias voltage, V_{b}. Each insulator layer is characterized by a barrier height (U_{j}), a thickness (d _{j}), a dielectric constant ε _{j}, and an effective mass (m _{j}).
Now, under a tridiagonal 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 M×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, Γ_{L} and Γ_{R} [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, AFTMM 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/Nb_{2}O_{5} (2nm)Ta_{2}O_{5} (1nm)/Nb at 0.1V of bias voltage. A complete matching between AFTMM and NEGF results is observed.
Fig.3. Transmission probability T (Ex) versus the electron transmission energy calculated using AFTMM, NEGF, and WKB at V_{b} = 0.1 V for Nb/Nb_{2}O_{5}/Ta_{2}O_{5}/ Nb MIIM diode.
Fig.4. Energy band diagram of the MIIM diode used for simulation
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SYMBOLS
Symbol 
Units 

q 
electronic charge 
1.602 × 10^{−}^{19} C 
H 
Planck constant 
6.626 × 10^{−}^{34} J s 
â„ 
h/2π 
1.055 × 10^{−}^{34} J s 
M 
free electron mass 
9.11 × 10^{−}^{31} kg 
ðœº_{0} 
permittivity of free space 
8.854 × 10^{−}^{12} F/m 
q^{2}/h 
Conductance quantum 
38.7 × 10^{−}^{6} S 
t_{0} 
â„ ^{2}/2m*a^{2} 
electronvolts (eV) 
N 
number of electrons or number of photons 
dimensionless 
ð†

density matrix 
dimensionless 
ε

energy level 
eV 
H

Hamiltonian 
eV 
U

selfconsistent potential 
eV 
E

energy 
eV 
μ

electrochemical potential 
eV 
f(E) 
Fermi function 
dimensionless 
D(E)

density of states (DOS) 
/eV 
A(E) 
spectral function 
/eV 
G(E)

Green’s function (retarded) 
/eV 

transmission function 
dimensionless 
T (E) 
transmission probability (<1) 
dimensionless 
ðœ¸ ,ðšª(E)

broadening 
eV 
ðšº(E) 
selfenergy (retarded) 
eV 
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