SemiCircleContour

class SemiCircleContour(integral_lower_bound=None, circle_eccentricity=None, logarithmic_bunching=None, circle_points=None, fermi_line_points=None, fermi_function_poles=None)

An equilibrium contour using the semi-circle contour points and weights defined in [gBMO+02].

Parameters:
  • integral_lower_bound (PhysicalQuantity of type energy) – The distance between the lowest Fermi-level and the lowest energy circle contour point.
    Default: An energy determined on the chosen pseudopotentials or 1.5 Hartree for the semi-empirical calculators.
  • circle_eccentricity (float) – The eccentricity of the circle contour. This should be a float between 0 and 1. 0 is a circle, 1 is a line.
    Default: 0.3
  • logarithmic_bunching (float) – Logarithmic bunching of the circle contour around the Fermi level. This should be a float between 0 and 1. 0 means no bunching, equidistant points, 1 means all bunched, centred on the Fermi Level.
    Default: 0.3
  • circle_points (int > 2) – The number of points on the circle contour.
    Default: 30
  • fermi_line_points (int) – The number of points on the straight line from the Fermi energy level up to infinity. This should be an integer in the range 1 to 11 (inclusive).
    Default: 10
  • fermi_function_poles (int > 0) – The number of poles of the Fermi function to include. Determines the imaginary shift of the straight line from the Fermi level to infinity.
    Default: 8
circleEccentricity()
Returns:The eccentricity of the circle contour.
Return type:float
circlePoints()
Returns:The number of circle points.
Return type:int
fermiFunctionPoles()
Returns:The number of poles for the Fermi function.
Return type:int
fermiLinePoints()
Returns:The number of points on the line from the Fermi energy level up to infinity.
Return type:int
integralLowerBound()
Returns:The distance between the lowest Fermi-level to the lowest energy circle contour point.
Return type:PhysicalQuantity of type energy
logarithmicBunching()
Returns:The logarithmic bunching.
Return type:float

Usage Example

One can use the SemiCircleContour by defining it as an equilibrium contour

equilibrium_contour = SemiCircleContour()

which constructs a SemiCircleContour with all defaults. Alternatively, more parameters that alter the accuracy of the approximation can be specified, e.g.

equilibrium_contour = SemiCircleContour(
        circle_eccentricity=0.1,
        logarithmic_bunching=0.2,
        circle_points=100,
        fermi_line_points=11,
        fermi_function_poles=10)

To use it in a calculation of the equilibrium density matrix,

the equilibrium contour explicitly on the ContourParameters object, the EquilibriumContour object is passed to the ContourParameters object,

contour_parameters = ContourParameters(equilibrium_contour=equilibrium_contour)

and saved on the calculator

device_calculator = DeviceLCAOCalculator(contour_parameters=contour_parameters)

Notes

The SemiCircleContour is a method to calculate the equilibrium density matrix \(D\) by performing an integration of the Greens Function \(G\).

\[D = \Im\left[-\frac{2}{\pi}\lim_{R\rightarrow\infty} \int_{-R}^{+R} dE f(E - \mu) G(E + i0^+)\right]~,\]

in which \(f(E)\) is the Fermi-Dirac distribution and \(\mu\) the Fermi level.

This integral can be solved by using the residue theorem:

\[\oint dz G(z) f(z) = - 2\pi i kT \sum_{z_\nu} G(z_\nu)\]

in which the sum on the right hand side runs over the poles of the integrant included in the contour.

The SemiCircleContour defines the contour in the upper-half of the imaginary plane and is comprised of a semicircle, a semi-infinite line segment and a finite number of Fermi poles.

The semicircle \(C\) starts from the lower bound \(E_B\) controlled by integral_lower_bound. The parameter circle_eccentricity defines the eccentricity of the semi circle, while adjusting logarithmic_bunching alters the distribution of the semicircle’s contour points around the fermi level. The end point of the semicircle is defined at a distance \(\gamma\) below the Fermi energy \(\mu\) and \(\Delta\) above the real axis. The line segment \(L\) runs from the semicircle’s end point to \(+\infty\). The distance \(\Delta\) between the line segment and the real axis is determined by the number of fermi_poles_poles included in the contour.

The precision of the contour integration improves as circle_points, the number of contour points on the semicircle, fermi_line_points, the number of contour points on the line segment, or fermi_poles_poles increases.

../../../_images/semicirclecontour.png

More information about this approach can be found in [gBMO+02]

[gBMO+02](1, 2) M. Brandbyge, J.-L. Mozos, P. Ordejón, J. Taylor, and K. Stokbro. Density-functional method for nonequilibrium electron transport. Phys. Rev. B, 65:165401, Mar 2002. doi:10.1103/PhysRevB.65.165401.