CurrentDensity

	CurrentDensity
OpticalSpectrum	ChemicalPotential

Name

CurrentDensity — Class for representing the current density for a given configuration and calculator.

Synopsis

Namespace: NanoLanguage

CurrentDensity(
configuration,
kpoints,
energies,
energy_weights,
spin,
electrode_temperatures,
self_energy_calculator,
energy_zero_parameter,
infinitesimal
)

Description

Constructor for the CurrentDensity object.

CurrentDensity Arguments

configuration

The configuration to calculate the current density of.

Type: DeviceConfiguration with attached calculator.

Default:


          None

kpoints

The kpoints to integrate over.

Type: MonkhorstPackGrid

Default:


          MonkhorstPackGrid(nx,ny), where nx, ny is the sampling used for the self consistent calculation.

energies

The energies to integrate over.

Type: List with PhysicalQuanties of type energy.

Default:


          An energy range that covers the bias window as specified by the DoubleContourIntegralParameters on the calculator attached to the configuration.

energy_weights

The weight of each energy point. The keyword Current gives an energy dependent weight, f_R(E) -f_L(E), corresponding to the current integral. Left/Right gives an energy independent weight of 1, for current carrying states from Left/Right.

Type: Current | Left | Right | List of floats

Default:


          Current

spin

The spin for which the current density should be calculated.

Type: Spin.Sum | Spin.Up | Spin.Down

Default:


          Spin.Sum

electrode_temperatures

The electrode temperatures to be used the current calculation.

Type: Must be given as 2 temperatures - for example, [100, 200]*Kelvin.

Default:


          The temperature used for the self-consistent calculation.

self_energy_calculator

The SelfEnergyCalculator to be used for the current density.

Type: DirectSelfEnergy() | RecursionSelfEnergy() | KrylovSelfEnergy()

Default:


          DirectSelfEnergy()

energy_zero_parameter

Specifies the choice for the energy zero.

Type: AverageFermiLevel | AbsoluteEnergy.

Default:


          AverageFermiLevel

infinitesimal

Small energy, used to move the self energy calculation away from the real axis. This is only relevant for recursion-style self-energy calculators.

Type: PhysicalQuantity with type energy.

Default:


          1.0e-6*eV

CurrentDensity Methods

A CurrentDensity object provides the following methods:

electrodeFermiLevels(): Return the electrodes Fermi levels in absolute energies.
electrodeFermiTemperatures(): Return the electrodes Fermi temperatures.
energies(): Return the energies used for this current density .
energyWeights(): Return the energy weights used for this current density .
energyZero(): Return the energy zero used for the energy scale in this current density.
evaluate(x, y, z): Evaluate in the point x,y,z

x

The cartesian x coordinate.

Type: physical quantity with type length.

y

The cartesian y coordinate.

Type: physical quantity with type length.

z

The cartesian z coordinate.

Type: physical quantity with type length.
findFingerPrint(): Return the finger print of the object.
gridCoordinate(i, j, k): Return the grid coordinate for a given grid index.

i

The grid index in the A direction.

Type: integer

j

The grid index in the B direction.

Type: integer

k

The grid index in the C direction.

Type: integer
infinitesimal(): Return the infinitesimal used for calculating the current density.
nlprint(stream): Print a string containing an ASCII representation of the data.

stream

The stream the data should be written to.
Type: A stream that supports strings being written to using 'write'.
Default: sys.stdout
scale(scale): Scale the field with a double.

scale

The parameter to scale with.

Type: float
shape(): Query function for getting the shape of the data.
spin(): Return the spins.
toArray(): Return the values of the grid as a numpy array slicing off any units.
unit(): Get the unit of the data in the grids.
unitCell(): Return the unit cell for the grids.
volumeElement(): Get the volume element of the grid.

Usage Examples

Load a current density object and plot the current density along z, integrated over the x,y plane

# Read current density from a file
cd = nlread('filename.nc',CurrentDensity)[0]

# Calculate the volume element and x,y integration area
dX, dY, dZ = cd.volumeElement()
dAXY = numpy.linalg.norm( numpy.cross(dX,dY) )

# Calculate the current density along z integrated over x,y
shape = cd.shape()
cd_z = [ cd[2,:,:,i].sum() * dAXY for i in range(shape[3]) ]

# Plot the data
import pylab
pylab.figure()
dz = dZ.norm()
z = numpy.arange(shape[3])*dz.inUnitsOf(Bohr)
pylab.plot(z,cd_z)
pylab.show()

current_density1.py

Calculate the current density from right to left going states at the Fermi level. Plot the current density in the x-z plane as a contour plot.

# Calculate the current density at the fermi level from right to left going states
cd = CurrentDensity(
    configuration,
    energies = [0.0]*eV,
    energy_weights = Right,
    kpoints=MonkhorstPackGrid(1,11),
    )

# Calculate the volume element
dX, dY, dZ = cd.volumeElement()
dx = dX.norm()
dy = dY.norm()
dz = dZ.norm()

# Calculate the current density in the xz plane integrated over y
cd_xz = numpy.apply_over_axes(numpy.sum, cd[2,:,:,:] ,[2]).flatten()

# Make 2-d arrays for contour plot of the data
shape = cd.shape()
x = dx.inUnitsOf(Ang)*numpy.arange(shape[1])
z = dz.inUnitsOf(Ang)*numpy.arange(shape[3])
Z, X = numpy.meshgrid(z,x)
F = numpy.array(cd_xz).reshape(numpy.shape(Z))

#plot the current density in the (z,x) plane
import pylab
pylab.xlabel('z (Angstrom)',fontsize=12,family='sans-serif')
pylab.ylabel('x (Angstrom)',fontsize=12,family='sans-serif')
pylab.contourf(Z, X, F)
pylab.colorbar()
pylab.savefig('cd.png',dpi=100)
pylab.show()

current_density2.py

Notes

The current density is given by

$\displaystyle {\bf J}(r,E) = -\frac{e \hbar}{4 \pi m } \int dE w(E) \sum_{\mu, \nu} G^<_{\mu, \nu}(E) \phi_\nu {\bf \nabla} \phi_\mu,$

where $G^<$ is the lesser greens function from left/right dependent on the current direction[44], and $w_E = f_R(E) -f_E(E)$ is an energy weight given by the left, $f_L$ , and right, $f_R$ , Fermi occupations.

The positive current direction is from left to right (corresponding to electron propagation from right to left).
Through the energy_weights keyword, it is possible to obtain a spectral resolution of left and right going states. This is particular useful when using the current density to analyze zero bias calculations. Setting the energy weights to Right is identical to setting the energy_weights to 1 for all states, since a positive number corresponds to calculating right to left going modes (positive current). Setting the energy weights to Left is identical to setting the energy_weights to -1 for all states, since a negative number corresponds to calculating left to right going modes (negative current).
The current density does not include the non-local potential correction[44] and spill in contributions from the electrodes. Thus, the area integrated current density is not constant along z, but has a small fluctuating component from the non-local potential and larger errors near the electrodes from spill in components.


OpticalSpectrum		ChemicalPotential