General

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:

• SCF calculation of system with two leads under zero and finite bias voltage
• SCF calculation under gate bias voltage
• Compatible with the DFT+$U$ method
• Spin-dependent transmission and current
• k-resolved transmission and current along perpendicular to the current axis
• Calculation of current-voltage curve
• Accurate and efficient contour integration scheme
• Interpolation of the effect by the bias voltage
• Quick calculation for periodic systems under zero bias
• The eigen-channel analysis [141]
• The real-space charge and spin current [139]

The details of the implementation can be found in Ref. [73]. First the usage of the functionalities for the collinear case is explained in the following subsections. After that, the non-collinear case will be discussed.

System we consider

In the current implementation of OpenMX Ver. 3.9, a system shown in Fig. 39(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 an one-dimensional problem depending on the Bloch wave vector ${\bf k}$ shown in Fig. 39(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 39: (a) Configuration of the system, treated by the NEGF method, with the 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

• Step 1

  The band structure calculations are performed for the left and right leads using a program code 'openmx'. The calculated results will be used to represent the Hamiltonian of the leads in the NEGF calculation of the step 2.

• Step 2

  The NEGF calculation is performed for the structure shown in Fig. 39 under zero or a finite bias voltage using a program code 'openmx', where the result in the step 1 is used for the construction of the leads.

• Step 3

  By making use of the result of the step 2, the transmission, charge/spin current density, and the eigenchannel are calculated by a program code 'openmx'.

An example: carbon chain

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

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

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

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

  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
  negf-chain.tranec0_0_0_2_i.cube

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. 40.

Figure 40: Transmission of a carbon chain as a function of energy. The origin of energy is set to the chemical potential of the left lead.