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