Interpolation of the effect by the bias voltage

Since for large-scale systems it is very time-consuming to perform the SCF calculation at each bias voltage, an interpolation scheme is available to reduce the computational cost in the calculations by the NEGF method. The interpolation scheme is performed in the following way: (i) the SCF calculations are performed for a few bias voltages which are selected in the regime of the bias voltage of interest. (ii) when the transmission and current are calculated, linear interpolation is made for the Hamiltonian block elements, $H_{\sigma,C}^{(\bf k)}$ and $H_{\sigma,R}^{(\bf k)}$, of the central scattering region and the right lead, and the chemical potential, $\mu_{R}$, of the right lead by

$\displaystyle H_{\sigma,C}^{(\bf k)}$ $\textstyle =$ $\displaystyle \lambda H_{\sigma,C}^{({\bf k},1)}
+ (1-\lambda) H_{\sigma,C}^{({\bf k},2)},$  
$\displaystyle H_{\sigma,R}^{(\bf k)}$ $\textstyle =$ $\displaystyle \lambda H_{\sigma,R}^{({\bf k},1)}
+ (1-\lambda) H_{\sigma,R}^{({\bf k},2)},$  
$\displaystyle \mu_{R}$ $\textstyle =$ $\displaystyle \lambda \mu_{R}^{(1)} + (1-\lambda) \mu_{R}^{(2)},$  

where the indices $1$ and $2$ in the superscript mean that the quantities are calculated or used at the corresponding bias voltages where the SCF calculations are performed beforehand. In general, $\lambda$ should range from 0 to 1 for the moderate interpolation.

In the calculation of the step 3, the interpolation is made by adding the following keywords in the input file:

  NEGF.tran.interpolate         on               # default=off, on|off
  NEGF.tran.interpolate.file1  c1-negf-0.5.tranb
  NEGF.tran.interpolate.file2  c1-negf-1.0.tranb
  NEGF.tran.interpolate.coes    0.7 0.3          # default=1.0 0.0
When you perform the interpolation, the keyword 'NEGF.tran.interpolate' should be 'on'. In this case, files 'c1-negf-0.5.tranb' and 'c1-negf-1.0.tranb' specified by the keywords 'NEGF.tran.interpolate.file1' and 'NEGF.tran.interpolate.file2' are the results under bias voltages of 0.5 and 1.0 V, respectively, and the transmission and current at $V=0.7*0.5+0.3*1.0=0.65 [V]$ are evaluated by the interpolation scheme, where the weights of 0.7 and 0.3 are specified by the keyword 'NEGF.tran.interpolate.coes'.

Figure 44: (a) Currents of the linear carbon chain calculated by the SCF calculations (solid line) and the interpolation scheme (dotted line). (b) Transmission of the linear carbon chain under a bias voltage of 0.3 V, calculated by the SCF calculations (solid line) and the interpolation scheme (dotted line). The imaginary part of 0.01 eV and the grid spacing of 0.01 eV are used for the integration of the nonequilibrium term in the density matrix.

A comparison between the fully self consistent and the interpolated results is shown with respect to the current and transmission in the linear carbon chain in Figs. 44(a) and (b). In this case, the SCF calculations at three bias voltages of 0, 0.5, and 1.0 V are performed, and the results at the other bias voltages are obtained by the interpolation scheme. For comparison we also calculate the currents via the SCF calculations at all the bias voltages. It is confirmed that the simple interpolation scheme gives notably accurate results for both the calculations of the current and transmission. Although the proper selection of bias voltages used for the SCF calculations may depend on systems, the result suggests that the simple scheme is very useful to interpolate the effect of the bias voltage while keeping the accuracy of the calculations.