MonkhorstPack¶

class sisl.physics.brillouinzone.MonkhorstPack(parent, nkpt, displacement=None, size=None, centered=True, trs=True)[source]¶

Create a Monkhorst-Pack grid for the Brillouin zone

Parameters:

parent : object or array_like: An object with associated parent.cell and parent.rcell or an array of floats which may be turned into a SuperCell
nktp : array_like of ints: a list of number of k-points along each cell direction
displacement : float or array_like of float, optional: the displacement of the evenly spaced grid, a single floating number is the displacement for the 3 directions, else they are the individual displacements
size : float or array_like of float, optional: the size of the Brillouin zone sampled. This reduces the boundaries of the Brillouin zone around the displacement to the fraction specified. I.e. size must be of values \(]0 ; 1]\). Defaults to the entire BZ. Note that this will also reduce the weights such that the weights are normalized to the entire BZ.
centered : bool, optional: whether the k-points are \(\Gamma\)-centered (for zero displacement)
trs : bool, optional: whether time-reversal symmetry exists in the Brillouin zone.

Examples

>>> sc = SuperCell(3.)
>>> MonkhorstPack(sc, 10) # 10 x 10 x 10 (with TRS)
>>> MonkhorstPack(sc, [10, 5, 5]) # 10 x 5 x 5 (with TRS)
>>> MonkhorstPack(sc, [10, 5, 5], trs=False) # 10 x 5 x 5 (without TRS)

Attributes

`cell`
`k`	A list of all k-points (if available)
`rcell`
`weight`	Weight of the k-points in the `BrillouinZone` object

Methods

`__init__`(parent, nkpt[, displacement, size, …])	Initialize self.
`asarray`()	Return self with `numpy.ndarray` returned quantities
`asaverage`()	Return self with k-averaged quantities
`asgrid`()	Return self with Grid quantities
`aslist`()	Return self with list returned quantities
`asnone`()	Return self with None, this may be done for instance when wrapping the function calls.
`assum`()	Return self with summed quantities
`asyield`()	Return self with yielded quantities
`copy`()	Create a copy of this object
`grid`(n[, displ, size, centered, trs])	Create a grid of n points with an offset of displ and sampling size around displ
`in_primitive`(k)	Move the k-point into the primitive point(s) ]-0.5 ; 0.5]
`iter`([ret_weight])	An iterator for the k-points and (possibly) the weights
`param_circle`(sc, N_or_dk, kR, normal, origo)	Create a parameterized k-point list where the k-points are generated on a circle around an origo
`parametrize`(sc, func, N, args, *kwargs)	Generate a new `BrillouinZone` object with k-points parameterized via the function func in N separations
`replace`(k, mp)	Replace a k-point with a new set of k-points from a Monkhorst-Pack grid
`set_parent`(parent)	Update the parent associated to this object
`tocartesian`(k)	Transfer a k-point in reduced coordinates to the Cartesian coordinates
`toreduced`(k)	Transfer a k-point in Cartesian coordinates to the reduced coordinates
`write`(sile, args, *kwargs)	Writes k-points to a `tableSile`.

asarray()¶

Return self with numpy.ndarray returned quantities

This forces the __call__ routine to return a single array.

See also

asyield: all output returned through an iterator
asaverage: take the average (with k-weights) of the Brillouin zone
assum: return the sum of values in the Brillouin zone
aslist: all output returned as a Python list

Notes

All invocations of sub-methods are added these keyword-only arguments:

eta : bool, optional: if true a progress-bar is created, default false.
wrap : callable, optional: a function that accepts the output of the given routine and post-process it. Defaults to lambda x: x.

Examples

>>> obj = BrillouinZone(...) 
>>> obj.asarray().eigh(eta=True) 

asaverage()¶

Return self with k-averaged quantities

This forces the __call__ routine to return a single k-averaged value.

See also

asarray: all output as a single array
asyield: all output returned through an iterator
assum: return the sum of values in the Brillouin zone
aslist: all output returned as a Python list

Notes

All invocations of sub-methods are added these keyword-only arguments:

eta : bool, optional: if true a progress-bar is created, default false.
wrap : callable, optional: a function that accepts the output of the given routine and post-process it. Defaults to lambda x: x.

Examples

>>> obj = BrillouinZone(Hamiltonian) 
>>> obj.asaverage().DOS(np.linspace(-2, 2, 100)) 

>>> obj = BrillouinZone(Hamiltonian) 
>>> obj.asaverage() 
>>> obj.DOS(np.linspace(-2, 2, 100)) 
>>> obj.PDOS(np.linspace(-2, 2, 100), eta=True) 

asgrid()[source]¶

Return self with Grid quantities

This forces the __call__ routine to return all k-point values in a regular grid.

The calculation of values on a grid requires some careful thought before running the calculation as the returned grid may be somewhat difficult to comprehend.

See also

asarray: all output as a single array
asyield: all output returned through an iterator
asaverage: take the average (with k-weights) of the Brillouin zone
aslist: all output returned as a Python list

Notes

All invocations of sub-methods are added these keyword-only arguments:

eta : bool, optional: if true a progress-bar is created, default false.
wrap : callable, optional: a function that accepts the output of the given routine and post-process it. Defaults to lambda x: x.
data_axis : int, optional: the Grid axis to put in the data values in. Has to be specified if the subsequent routine calls return more than 1 data-point per k-point.
grid_unit : {‘b’, ‘Ang’, ‘Bohr’}, optional: for ‘b’ the returned grid will be a cube, otherwise the grid will be the reciprocal lattice vectors (for any other value) and in the given reciprocal unit (‘Ang’ => 1/Ang)

Examples

>>> obj = MonkhorstPack(Hamiltonian, [10, 1, 10]) 
>>> grid = obj.asgrid().eigh(data_axis=1) 

aslist()¶

Return self with list returned quantities

This forces the __call__ routine to return a list with returned values.

See also

asarray: all output as a single array
asyield: all output returned through an iterator
assum: return the sum of values in the Brillouin zone
asaverage: take the average (with k-weights) of the Brillouin zone

Notes

All invocations of sub-methods are added these keyword-only arguments:

eta : bool, optional: if true a progress-bar is created, default false.
wrap : callable, optional: a function that accepts the output of the given routine and post-process it. Defaults to lambda x: x.

Examples

>>> obj = BrillouinZone(...) 
>>> def first_ten(es):
...    return es.sub(range(10))
>>> obj.aslist().eigenstate(eta=True, wrap=first_ten) 

asnone()¶

Return self with None, this may be done for instance when wrapping the function calls.

This forces the __call__ routine to return None. This usage is mainly intended when creating custom wrap function calls.

See also

asyield: all output returned through an iterator
asaverage: take the average (with k-weights) of the Brillouin zone
assum: return the sum of values in the Brillouin zone
aslist: all output returned as a Python list

Notes

All invocations of sub-methods are added these keyword-only arguments:

eta : bool, optional: if true a progress-bar is created, default false.
wrap : callable, optional: a function that accepts the output of the given routine and post-process it. Defaults to lambda x: x.

Examples

>>> obj = BrillouinZone(...) 
>>> obj.asnone().eigh(eta=True) 

assum()¶

Return self with summed quantities

This forces the __call__ routine to return all k-point values summed.

See also

asarray: all output as a single array
asyield: all output returned through an iterator
asaverage: take the average (with k-weights) of the Brillouin zone
aslist: all output returned as a Python list

Notes

All invocations of sub-methods are added these keyword-only arguments:

eta : bool, optional: if true a progress-bar is created, default false.
wrap : callable, optional: a function that accepts the output of the given routine and post-process it. Defaults to lambda x: x.

Examples

>>> obj = BrillouinZone(Hamiltonian) 
>>> obj.assum().DOS(np.linspace(-2, 2, 100)) 

>>> obj = BrillouinZone(Hamiltonian) 
>>> obj.assum() 
>>> obj.DOS(np.linspace(-2, 2, 100)) 
>>> obj.PDOS(np.linspace(-2, 2, 100), eta=True) 

asyield()¶

Return self with yielded quantities

This forces the __call__ routine to return a an iterator which may yield the quantities calculated.

See also

asarray: all output as a single array
asaverage: take the average (with k-weights) of the Brillouin zone
assum: return the sum of values in the Brillouin zone
aslist: all output returned as a Python list

Notes

All invocations of sub-methods are added these keyword-only arguments:

eta : bool, optional: if true a progress-bar is created, default false.
wrap : callable, optional: a function that accepts the output of the given routine and post-process it. Defaults to lambda x: x.

Examples

>>> obj = BrillouinZone(Hamiltonian) 
>>> obj.asyield().eigh(eta=True) 

cell¶

copy()[source]¶: Create a copy of this object

classmethod grid(n, displ=0.0, size=1.0, centered=True, trs=False)[source]¶

Create a grid of n points with an offset of displ and sampling size around displ

The \(k\)-points are \(\Gamma\) centered.

Parameters:	n : int number of points in the grid. If trs is `True` this may be smaller than n displ : float, optional the displacement of the grid size : float, optional the total size of the Brillouin zone to sample centered : bool, optional if the points are centered trs : bool, optional whether time-reversal-symmetry is applied
Returns:	k : np.ndarray the list of k-points in the Brillouin zone to be sampled w : np.ndarray weights for the k-points

static in_primitive(k)¶

Move the k-point into the primitive point(s) ]-0.5 ; 0.5]

Parameters:	k : array_like k-point(s) to move into the primitive cell
Returns:	k : all k-points moved into the primitive cell

iter(ret_weight=False)¶

An iterator for the k-points and (possibly) the weights

Parameters:	ret_weight : bool, optional if true, also yield the weight for the respective k-point
Yields:	kpt : k-point weight : weight of k-point, only if ret_weight is true.

k¶: A list of all k-points (if available)

classmethod param_circle(sc, N_or_dk, kR, normal, origo, loop=False)¶

Create a parameterized k-point list where the k-points are generated on a circle around an origo

The generated circle is a perfect circle in the reciprocal space (Cartesian coordinates). To generate a perfect circle in units of the reciprocal lattice vectors one can generate the circle for a diagonal supercell with side-length \(2\pi\), see example below.

Parameters:

sc : SuperCell, or SuperCellChild: the supercell used to construct the k-points
N_or_dk : int: number of k-points generated using the parameterization (if an integer), otherwise it specifies the discretization length on the circle (in 1/Ang), If the latter case will use less than 4 points a warning will be raised and the number of points increased to 4.
kR : float: radius of the k-point. In 1/Ang
normal : array_like of float: normal vector to determine the circle plane
origo : array_like of float: origo of the circle used to generate the circular parameterization
loop : bool, optional: whether the first and last point are equal

Returns:

BrillouinZone : with the parameterized k-points.

Examples

>>> sc = SuperCell([1, 1, 10, 90, 90, 60])
>>> bz = BrillouinZone.param_circle(sc, 10, 0.05, [0, 0, 1], [1./3, 2./3, 0])

To generate a circular set of k-points in reduced coordinates (reciprocal

>>> sc = SuperCell([1, 1, 10, 90, 90, 60])
>>> bz = BrillouinZone.param_circle(sc, 10, 0.05, [0, 0, 1], [1./3, 2./3, 0])
>>> bz_rec = BrillouinZone.param_circle(2*np.pi, 10, 0.05, [0, 0, 1], [1./3, 2./3, 0])
>>> bz.k[:, :] = bz_rec.k[:, :]

classmethod parametrize(sc, func, N, *args, **kwargs)¶

Generate a new BrillouinZone object with k-points parameterized via the function func in N separations

Generator of a parameterized Brillouin zone object that contains a parameterized k-point list.

Basically this generates a new BrillouinZone object as:

>>> def func(sc, frac):
...    return [frac, 0, 0]
>>> bz = BrillouinZone.parametrize(1, func, 10)
>>> len(bz) == 10
True
>>> np.allclose(bz.k[-1, :], [9./10, 0, 0])
True

Parameters:

sc : SuperCell, or SuperCellChild: the supercell used to construct the k-points
func : callable: method that parameterizes the k-points, must at least accept two arguments, sc (super-cell object containing the unit-cell and reciprocal cell) and frac (current parametrization fraction, between 0 and (N-1)/N. It must return a k-point in 3 dimensions.
N : int: number of k-points generated using the parameterization
args: list of arguments: arguments passed directly to func
kwargs: dictionary of arguments: keyword arguments passed directly to func

rcell¶

replace(k, mp)[source]¶

Replace a k-point with a new set of k-points from a Monkhorst-Pack grid

This method tries to replace an area corresponding to mp.size around the k-point k such that the k-points are replaced. This enables one to zoom in on specific points in the Brillouin zone for detailed analysis.

Parameters:	k : array_like k-point in this object to replace mp : MonkhorstPack object containing the replacement k-points.
Raises:	SislError : if the size of the replacement `MonkhorstPack` grid is not compatible with the k-point spacing in this object.

Examples

This example creates a zoomed-in view of the \(\Gamma\)-point by replacing it with a 3x3x3 Monkhorst-Pack grid.

>>> sc = SuperCell(1.)
>>> mp = MonkhorstPack(sc, [3, 3, 3])
>>> mp.replace([0, 0, 0], MonkhorstPack(sc, [3, 3, 3], size=1./3))

This example creates a zoomed-in view of the \(\Gamma\)-point by replacing it with a 4x4x4 Monkhorst-Pack grid.

>>> sc = SuperCell(1.)
>>> mp = MonkhorstPack(sc, [3, 3, 3])
>>> mp.replace([0, 0, 0], MonkhorstPack(sc, [4, 4, 4], size=1./3))

This example creates a zoomed-in view of the \(\Gamma\)-point by replacing it with a 4x4x1 Monkhorst-Pack grid.

>>> sc = SuperCell(1.)
>>> mp = MonkhorstPack(sc, [3, 3, 3])
>>> mp.replace([0, 0, 0], MonkhorstPack(sc, [4, 4, 1], size=1./3))

set_parent(parent)¶

Update the parent associated to this object

Parameters:	parent : object or array_like an object containing cell vectors

tocartesian(k)¶

Transfer a k-point in reduced coordinates to the Cartesian coordinates

Parameters:	k : list of float k-point in reduced coordinates
Returns:	k : in units of 1/Ang

toreduced(k)¶

Transfer a k-point in Cartesian coordinates to the reduced coordinates

Parameters:	k : list of float k-point in Cartesian coordinates
Returns:	k : in units of reciprocal lattice vectors ]-0.5 ; 0.5] (if k is in the primitive cell)

weight¶: Weight of the k-points in the BrillouinZone object

write(sile, *args, **kwargs)¶

Writes k-points to a tableSile.

This allows one to pass a tableSile or a file-name.