# sisl.physics.RealSpaceSI

class sisl.physics.RealSpaceSI(semi, surface, k_axes, unfold=(1, 1, 1), **options)

Bases: SelfEnergy

Surface real-space self-energy (or Green function) for a given physical object with limited periodicity

The surface real-space self-energy is calculated via the k-averaged Green function:

$\boldsymbol\Sigma^\mathcal{R}(E) = \mathbf S^\mathcal{R} (E+i\eta) - \mathbf H^\mathcal{R} - \Big[\sum_{\mathbf k} \mathbf G_{\mathbf k}(E)\Big]^{-1}$

The method actually used is relying on RecursiveSI and Bloch objects.

Parameters
• semi (SemiInfinite) – physical object which contains the semi-infinite direction, it is from this object we calculate the self-energy to be put into the surface. a physical object from which to calculate the real-space self-energy. semi and surface must have parallel lattice vectors.

• surface (SparseOrbitalBZ) – parent object containing the surface of system. semi is attached into this object via the overlapping regions, the atoms that overlap semi and surface are determined in the initialize routine. semi and surface must have parallel lattice vectors.

• k_axes (array_like of int) – axes where k-points are desired. 1 or 2 values are required. The axis cannot be a direction along the semi semi-infinite direction.

• unfold ((3,) of int) – number of times the surface structure is tiled along each direction Since this is a surface there will maximally be 2 unfolds being non-unity.

• eta (float, optional) – imaginary part in the self-energy calculations (default 1e-4 eV)

• dk (float, optional) – fineness of the default integration grid, specified in units of Ang, default to 1000 which translates to 1000 k-points along reciprocal cells of length 1. Ang^-1.

• bz (BrillouinZone, optional) – integration k-points, if not passed the number of k-points will be determined using dk and time-reversal symmetry will be determined by trs, the number of points refers to the unfolded system.

• trs (bool, optional) – whether time-reversal symmetry is used in the BrillouinZone integration, default to true.

Examples

>>> graphene = geom.graphene()
>>> H = Hamiltonian(graphene)
>>> H.construct([(0.1, 1.44), (0, -2.7)])
>>> se = RecursiveSI(H, "-A")
>>> Hsurf = H.tile(3, 0)
>>> Hsurf.set_nsc(a=1)
>>> rsi = RealSpaceSI(se, Hsurf, 1, (1, 4, 1))
>>> rsi.green(0.1)


The Brillouin zone integration is determined naturally.

>>> graphene = geom.graphene()
>>> H = Hamiltonian(graphene)
>>> H.construct([(0.1, 1.44), (0, -2.7)])
>>> se = RecursiveSI(H, "-A")
>>> Hsurf = H.tile(3, 0)
>>> Hsurf.set_nsc(a=1)
>>> rsi = RealSpaceSI(se, Hsurf, 1, (1, 4, 1))
>>> rsi.set_options(eta=1e-3, bz=MonkhorstPack(H, [1, 1000, 1]))
>>> rsi.initialize()
>>> rsi.green(0.1) # eta = 1e-3
>>> rsi.green(0.1 + 1j * 1e-4) # eta = 1e-4


Manually specify Brillouin zone integration and default $$\eta$$ value.

Methods

 Clears the internal arrays created in initialize green(E[, k, dtype]) Calculate the real-space Green function Initialize the internal data-arrays used for efficient calculation of the real-space quantities real_space_coupling([ret_indices]) Real-space coupling surface where the outside fold into the surface real-space unit cell Fully expanded real-space surface parent scattering_matrix(*args, **kwargs) Calculate the scattering matrix by first calculating the self-energy Calculate the scattering matrix from the self-energy self_energy(E[, k, bulk, coupling, dtype]) Calculate real-space surface self-energy set_options(**options) Update options in the real-space self-energy
__init__(semi, surface, k_axes, unfold=(1, 1, 1), **options)[source]

Initialize real-space self-energy calculator

clear()[source]

Clears the internal arrays created in initialize

green(E, k=(0, 0, 0), dtype=None, **kwargs)[source]

Calculate the real-space Green function

The real space Green function is calculated via:

$\mathbf G^\mathcal{R}(E) = \sum_{\mathbf k} \mathbf G_{\mathbf k}(E)$
Parameters
• E (float/complex) – energy to evaluate the real-space Green function at

• k (array_like, optional) – only viable for 3D bulk systems with real-space Green functions along 2 directions. I.e. this would correspond to a circular real-space Green function

• dtype (numpy.dtype, optional) – the resulting data type, default to np.complex128

• **kwargs (dict, optional) – arguments passed directly to the self.surface.Pk method (not self.surface.Sk), for instance spin

initialize()[source]

Initialize the internal data-arrays used for efficient calculation of the real-space quantities

This method should first be called after all options has been specified.

If the user hasn’t specified the bz value as an option this method will update the internal integration Brillouin zone based on the dk option. The $$\mathbf k$$ point sampling corresponds to the number of points in the non-folded system and thus the final sampling is equivalent to the sampling times the unfolding (per $$\mathbf k$$ direction).

real_space_coupling(ret_indices=False)[source]

Real-space coupling surface where the outside fold into the surface real-space unit cell

The resulting parent object only contains the inner-cell couplings for the elements that couple out of the real-space matrix.

Parameters

ret_indices (bool, optional) – if true, also return the atomic indices (corresponding to real_space_parent) that encompass the coupling matrix

Returns

• parent (object) – parent object only retaining the elements of the atoms that couple out of the primary unit cell

• atom_index (numpy.ndarray) – indices for the atoms that couple out of the geometry, only if ret_indices is true

real_space_parent()[source]

Fully expanded real-space surface parent

Notes

The returned object does not obey the semi_bulk option. I.e. the matrix elements correspond to the self.surface object, always!

scattering_matrix(*args, **kwargs)

Calculate the scattering matrix by first calculating the self-energy

Any arguments that is passed to this method is directly passed to self_energy.

See self_energy for details.

This corresponds to:

$\boldsymbol\Gamma = i(\boldsymbol\Sigma - \boldsymbol \Sigma ^\dagger)$

Examples

Calculating both the self-energy and the scattering matrix.

>>> SE = SelfEnergy(...)
>>> self_energy = SE.self_energy(0.1)
>>> gamma = SE.scattering_matrix(0.1)


For a huge performance boost, please do:

>>> SE = SelfEnergy(...)
>>> self_energy = SE.self_energy(0.1)
>>> gamma = SE.se2scat(self_energy)


Notes

When using both the self-energy and the scattering matrix please use se2scat after having calculated the self-energy, this will be much, MUCH faster!

se2scat

converting the self-energy to the scattering matrix

self_energy

the used routine to calculate the self-energy before calculating the scattering matrix

static se2scat(SE)

Calculate the scattering matrix from the self-energy

$\boldsymbol\Gamma = i(\boldsymbol\Sigma - \boldsymbol \Sigma ^\dagger)$
Parameters

SE (matrix) – self-energy matrix

self_energy(E, k=(0, 0, 0), bulk=False, coupling=False, dtype=None, **kwargs)[source]

Calculate real-space surface self-energy

The real space self-energy is calculated via:

$\boldsymbol\Sigma^{\mathcal{R}}(E) = \mathbf S^{\mathcal{R}} E - \mathbf H^{\mathcal{R}} - \Big[\sum_{\mathbf k} \mathbf G_{\mathbf k}(E)\Big]^{-1}$
Parameters
• E (float/complex) – energy to evaluate the real-space self-energy at

• k (array_like, optional) – only viable for 3D bulk systems with real-space self-energies along 2 directions. I.e. this would correspond to circular self-energies.

• bulk (bool, optional) – if true, $$\mathbf S^{\mathcal{R}} E - \mathbf H^{\mathcal{R}} - \boldsymbol\Sigma^\mathcal{R}$$ is returned, otherwise $$\boldsymbol\Sigma^\mathcal{R}$$ is returned

• coupling (bool, optional) – if True, only the self-energy terms located on the coupling geometry (coupling_geometry) are returned

• dtype (numpy.dtype, optional) – the resulting data type, default to np.complex128

• **kwargs (dict, optional) – arguments passed directly to the self.surface.Pk method (not self.surface.Sk), for instance spin

set_options(**options)[source]

Update options in the real-space self-energy

After updating options one should re-call initialize for consistency.

Parameters
• semi_bulk (bool, optional) – whether the semi-infinite matrix elements are used for in the surface. Default to true.

• eta (float, optional) – imaginary part in the self-energy calculations (default 1e-4 eV)

• dk (float, optional) – fineness of the default integration grid, specified in units of Ang, default to 1000 which translates to 1000 k-points along reciprocal cells of length 1. Ang^-1.

• bz (BrillouinZone, optional) – integration k-points, if not passed the number of k-points will be determined using dk and time-reversal symmetry will be determined by trs, the number of points refers to the unfolded system.

• trs (bool, optional) – whether time-reversal symmetry is used in the BrillouinZone integration, default to true.