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How to Use NEGF Matrix Formulas: Step-by-Step Tutorial in Pictures Branislav K. Nikolić Department of Physics & Astronomy, University of Delaware, Newark, DE 19716, U.S.A. https://wiki.physics.udel.edu/phys824 PHYS824: Nanophysics and Nanotechnology NEGF in pictures Step 1: Construct Matrix Representation of the Tight-Binding Hamiltonian of Central Region eV =μμ− bLR 3x3 lattice of atoms in the central region rix I II III t 3x3 submatrix describing n ma 0 second row of atoms nia o t mil a 3x3 submatrix describing H 0 hoppings from third to 9x9 second row of atoms PHYS824: Nanophysics and Nanotechnology NEGF in pictures Step 2: Construct Self-Energy Matrices for the Left and the Right Semi-Infinite Lead 9x9 self-energy matrix 9x9 self-energy matrix for the left electrode for the right electrode 0 0 3x3 submatrix of 0 0 0 non-zero elements 0 0 0 which is returned 0 0 0 by the Function 0 0 0 self_energy.m 0 0 Since self-energies are summed with the Hamiltonian before inversion to get the Green function of an open system, they MUST be defined on the same Hilbert space of 9 orbitals H(1:3,1:3)=H1; MATLAB This self-energy (i.e., its non-zero 3x3 H(7:9,7:9)=H3; for energy=-4:0.02:4 block returned by the function) is added onto sigma=self_energy(3,1,energy,t_lead,t_couple); the corners (representing edge layers of H(1:3,1:3)= H(1:3,1:3)+sigma; the 3x3 sample) of the ORIGINAL H(7:9,7:9)= H(7:9,7:9)+sigma; Hamiltonan recorded outside of the loop Gr=inv(energy*eye(Hsize,Hsize)-H); PHYS824: Nanophysics and Nanotechnology NEGF in pictures Step 3: Construct Retarded Green Function for an Open Quantum System This 3x3 submatrix is I II III composed of quantum- mechanical probability 3 6 9 rix amplitudes for electron to This 3x3 submatrix is t propagate between I and ||| composed of quantum- ma row of atoms. This, it must mechanical probability n be related to the scattering 2 5 8 amplitudes for electron to io matrix concept in the propagate between ||| and I t Landauer formalism (which row of atoms. For example, unc contains probability matrix element G gives amplitude to propagate from 29 n f a conducting channel in left 1 4 7 probability amplitude to e led to a conducting channel propagate from atom 9 to Gre in the right lead) atom 2. d rde This 3x3 submatrix allows a t us to compute the local density of states on each atom from row | using −ImG/π 9x9 reThe total density of states is then obtained by summing all elements of on the −ImG/π diagonal of PHYS824: Nanophysics and Nanotechnology NEGF in pictures
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