G.1 Matrix Truncation

the equation (4.11) ( ) can be written as,

where

corresponds to the left semi-infinite contact,

corresponds to the right semi-infinite contact, and

(G.5) |

corresponds to the device region.

The coupling between the left and right contacts and device are respectively given by

(G.6) |

and

It should be noted that , , and and ( , and ) are sparse matrices. Their only non-zero entry represents the coupling of the left (right) contact and device. From (G.2), one obtains

Substituting (G.8) and (G.9) in (G.10), one obtains a matrix equation with a dimension corresponding to the total number of grid points in device layers,

The second and third terms of (G.11) are self-energies due to coupling of the device region to left and right contacts, respectively.

The GREEN's functions of the isolated semi-infinite contacts are defined as

The