Electronic transport properties of molecules, nano-wires, and bulks such as superlattice structures can be calculated based on a non-equilibrium Green function (NEGF) method within the collinear and non-collinear DFT methods. The features and capabilities are listed below:

The details of the implementation can be found in Ref. [58]. First the usage of the functionalities for the collinear case is explained in the following subsections. After then, the non-collinear case will be discussed.

System we consider

In the current implementation of OpenMX Ver. 3.8, a system shown in Fig. 31(a) is treated by the NEGF method. The system consists of a central region connected with infinite left and right leads, and the two dimensional periodicity spreads over the bc-plane. Considering the two dimensional periodicity, the system can be cast into a one-dimensional problem depending on the Bloch wave vector ${\bf k}$ shown in Fig. 31(b). Also, the Green function of the region $ C (\equiv L_0\vert C_0\vert R_0)$ is self-consistently determined in order to take account of relaxation of electronic structure around the interface between the central region $C_0$ and the region $L_0(R_0)$. It should be noted that the electronic transport is assumed to be along the a-axis in the current implementation. Thus, users have to keep in mind the specification when the geometrical structure is constructed. See also the subsection 'Step 1: The calculations for leads'.

Figure 31: (a) Configuration of the system, treated by the NEGF method, with infinite left and right leads along the a-axis under a two dimensional periodic boundary condition on the bc-plane. (b) One dimensional system compacted from the configuration of (a) by considering the periodicity on the bc-plane, where the region $C$ is an extended central region consisting of $C_{0}$, $L_{0}$, and $R_{0}$.

Computational flow

The NEGF calculation is performed by the following three steps:

Step 1 $\to $ Step 2 $\to $ Step 3

Each step consists of

An example: carbon chain

As a first trial, let us illustrate the three steps by employing a carbon chain. Before going to the illustration,

Step 1

      %./openmx Lead-Chain.dat | tee lead-chain.std 

A file 'negf-chain.hks' is generated by the step 1.

Step 2

      %./openmx NEGF-Chain.dat | tee negf-chain.std 

A file 'negf-chain.tranb' is generated by the step 2.

Step 3

openmx starts step 3 immediately after it finishes step 2. If we perform separately the step2 and the step 3, we run openmx as follows:

      %./openmx Lead-Chain.dat | tee lead-chain.std 

In the step 3, openmx reads a file 'negf-chain.tranb' and calculates the transmission, current, and eigen channels.

  negf-chain.conductance            negf-chain.tranec0_0_0_2_r.cube
  negf-chain.current                negf-chain.tranec0_0_0_3_i.cube
  negf-chain.tran0_0                negf-chain.tranec0_0_0_3_r.cube
  negf-chain.tranec0_0_0_0_i.cube   negf-chain.tranec0_0_0_4_i.cube
  negf-chain.tranec0_0_0_0_r.cube   negf-chain.tranec0_0_0_4_r.cube
  negf-chain.tranec0_0_0_1_i.cube   negf-chain.traneval0_0_0
  negf-chain.tranec0_0_0_1_r.cube   negf-chain.tranevec0_0_0
are generated by the step 3.

The calculations can be traced by using the input files stored in a directory of 'work/negf_example'. By plotting the sixth column in 'negf-chain.tran0_0' as a function of the fourth column, you can see a transmission curve as shown in Fig. 32.

Figure 32: Transmission of a carbon chain as a function of energy. The origin of energy is set to the chemical potential of the left lead.