3. kwantspectrum – Reference documentation

kwantspectrum.spectrum(syst, args=(), *, params=None, kmin=-3.141592653589793, kmax=3.141592653589793, orderpoint=0, tol=1e-08, match=<function _match_functions>)

Interpolate the dispersion function and provide methods to simplify curve sketching and analyzation the periodic spectrum.

Parameters

• syst (kwant.system.InfiniteSystem) – The low level infinite system for which the energies are to be calculated.

• args (tuple, defaults to empty) – Positional arguments to pass to the hamiltonian method. Mutually exclusive with ‘params’.

• params (dict, optional) – Dictionary of parameter names and their values. Mutually exclusive with ‘args’.

• kmin (float, optional) – Left-hand site of the momentum interval over which the band structure should be analyzed. Defaut = \(-\pi\)

• kmax (float, optional) – Right-hand site of the momentum interval over which the band structure should be analyzed. Defaut = \(\pi\)

• orderpoint (int or float, optional) – Momentum value where the band ordering is defined. Bands are ordered in energy, such that the band with the lowest energy at momentum orderpoint has index 0. Defaut = 0

• tol (float, optional) – Numerical tolerance of the interpolated function.

• match (callable, optional) – Matching algorithm

Returns

spec

Return type

BandSketching instance or a list of these

Notes

The tolerance \(tol\) is the required precision of the interpolated band structure. The matching algorithm which orders the bands should work correctly quite independently of the tolerance \(tol\). However, as the matching algorithm uses piecewise cubic interpolations for the error estimate internally, the cubic piecewise interpolation should be used for the subsequent interpolation of the spectrum for consistency.

kwantspectrum.spectra(syst, *args, **kwargs)

Interpolates a sequence of dispersion functions and provide methods to simplify curve sketching and analyzation the periodic spectra.

Parameters

syst (kwant.system.InfiniteSystem or sequence thereof) – The low level infinite systems.

Returns

specs

Return type

list of BandSketching

Notes

The function is similar to spectrum but works as well for a sequence of leads. It always returns a list of spectra. See spectrum for optional function arguments.

kwantspectrum.intersect_intervals(interval_a, interval_b, tol=1e-16)

Return the intersecting intervals between two lists of intervals.

class kwantspectrum.BandSketching(x, y, dy, mode_function, tol=1e-08, interpolation=<function _cubic_interpolation>)

Interpolate the dispersion function and provide methods to simplify curve sketching and analyzation the periodic spectrum.

Parameters

• x (numpy list, shape (n, )) – momentum (k) points

• y (numpy list, shape (n, nbands)) – energy values

• dy (numpy list, shape (n, nbands)) – velocity values

• mode_function (callable) – calculates the open modes for a given energy

• tol (float) – Numerical accuracy of the interpolated function.

• interpolation (object, optional) – Method to perform numerical interpolations. Default = cubic Hermite interpolation (piecewise cubic interpolation, 0. and 1. order of the exact and interpolation function are similar on the knots).

Notes

The routine designed for a periodic function that is represented as discrete values on a finite gridpoints. Finite accuracy of the data and possible unphysical results are taking into account by rounding value \(tol\).

energy_to_scattering_mode(energy, band, kmin, kmax)

Finds the scattering mode index for a band giving its energy.

Parameters

• energy (float) –

• band (int) – band index

• kmax (kmin,) – momentum interval [kmin, kmax] including the searched momentum, where the velocity does not change sign

Returns

mode – kwant scattering mode index The mode index fulfills: 0 <= mode < nbands where nbands is the total number of bands of the spectrum. If no open mode could be found, mode = -1 is returned.

Return type

int

Notes

An exception is raised if the momentum interval [kmin, kmax] is badly chose, such that either no mode / multiple modes are found, and therefore no unique (energy, band) -> mode mapping is possible, a warning is printed.

intersect(f, band, derivative_order=0, kmin=None, kmax=None, tol=None, ytol=None)

Returns all momentum (k) points, that solves the equation: \(\partial_k^{n} E(k) = f(k),\, k_{min} \leq k \leq k_{max}\).

Parameters

• f (scalar numerical value or callable) – Equation to solve \(\partial_k^{n} E(k) = f\). If f is callable, solve equation: \(\partial_k^{n} E(k) = f(k)\).

• band (int) – band index, requirement: 0 <= band < nbands.

• derivative_order (int, optional) – Derivative order (n) of the band dispersion. Default is zero.

• kmin (scalar numeric value, optional) – Lowest k point value. Default is kmin from initialization.

• kmax (scalar numeric value, optional) – Largest k point value. Default is kmax from initialization.

• tol (float, optional) – Numerical tolerance, k points closer tol are merged to the same point. Default is the tol from initialization.

• ytol (float, optional) – Numerical tolerance to remove noise if the spectrum \(\partial_k^{n} E(k)\) is almost flat. Values for the spectrum are set to thier mean value (averaged over all momentum points where the band is sampled), if they flucutate more than ytol. Default is the tol from initialization.

Returns

k – List of momentum (k) points, that solves the above equation.

Return type

numpy list.

Notes

kmin and kmin can be larger than the first Brillouin zone. In that case we return also the periodic images.

intervals(band, derivative_order=0, lower=None, upper=None, kmin=None, kmax=None, tol=None)

returns a list of momentum (k) intervals that solves the equation \(\text{lower} \leq \partial^m_k E_n(k) \leq \text{upper}\)

Parameters

• band (int) – band index n, requirement: 0 <= n < nbands.

• derivative_order (int, optional) – Derivative order m of the band dispersion. Default is zero.

• lower (int or float, optional) – Select intervals such that \(\partial_k^{m} E_n(k) \geq \text{lower}\)

• upper (int or float, optional) – Select intervals such that \(\partial_k^{m} E_n(k) \leq \text{upper}\)

• kmin (scalar numeric value, optional) – Lowest k point value. Default is kmin from initialization.

• kmax (scalar numeric value, optional) – Largest k point value. Default is kmax from initialization.

• tol (float, optional) – Tolerance of the interval selection, such that intervals smaller tol are neglected and two intervals closer tol are merged.

Returns

intervals – Ordered list of momentum intervals \([k_i, k_{i+1}]\). Each interval is given in form of a tuple.

Return type

list.

momentum_to_scattering_mode(k, band)

Finds the scattering mode index for a band with momentum k.

Parameters

• k (float) – momentum value

• band (int) – band index

Returns

mode – kwant scattering mode index. The mode index fulfills: 0 <= mode < nbands where nbands is the total number of bands of the spectrum. If no open mode could be found, mode = -1 is returned.

Return type

int

set_period(period)

Set periodicity of the spectrum

Parameters

period (2-tuple or False) – Perodic interval provided in the form (left_bound, right_bound) over which the band structure is assumed as periodic. If period is set to false, the dispersion function is evaluated as non-periodic fuction and the user has to perform periodic mapping herself if needed.