Brillouin zone classes¶
The Brillouin zone objects are all special classes enabling easy manipulation of an underlying physical quantity.
Quite often a physical quantity will be required to be averaged, or calculated individually over a number of k-points. In this regard the Brillouin zone objects can help.
The BrillouinZone object allows direct looping of contained k-points while invoking particular methods from the contained object. This is best shown with an example:
>>> H = Hamiltonian(...)
>>> bz = BrillouinZone(H)
>>> bz.apply.array.eigh()
This will calculate eigenvalues for all k-points associated with the BrillouinZone and
return everything as an array. The dispatch property of
the BrillouinZone object has several use cases (here array is shown).
This may be extremely convenient when calculating band-structures:
>>> H = Hamiltonian(...)
>>> bs = BandStructure(H, [[0, 0, 0], [0.5, 0, 0]], 100)
>>> bs_eig = bs.apply.array.eigh().T
>>> plt.plot(bs.lineark(), bs_eig)
and then you have all eigenvalues for all the k-points along the path.
Sometimes one may want to post-process the data for each k-point. As an example lets post-process the DOS on a per k-point basis while calculating the average:
>>> H = Hamiltonian(...)
>>> mp = MonkhorstPack(H, [10, 10, 10])
>>> E = np.linspace(-2, 2, 100)
>>> def wrap_DOS(eigenstate):
... # Calculate the DOS for the eigenstates
... DOS = eigenstate.DOS(E)
... # Calculate the velocity for the eigenstates
... v = eigenstate.velocity()
... V = (v ** 2).sum(1)
... return DOS.reshape(-1, 1) * v ** 2 / V.reshape(-1, 1)
>>> DOS = mp.apply.average.eigenstate(wrap=wrap_DOS, eta=True)
This will, calculate the Monkhorst pack k-averaged DOS split into 3 Cartesian
directions based on the eigenstates velocity direction. This method of manipulating
the result can be extremely powerful to calculate many quantities while running an
efficient BrillouinZone average. The eta flag will print, to stdout, a progress-bar.
The usage of the wrap method are also passed optional arguments, parent which is
H in the above example. k and weight are the current k-point and weight of the
corresponding k-point. An example could be to manipulate the DOS depending on the k-point and
weight:
>>> H = Hamiltonian(...)
>>> mp = MonkhorstPack(H, [10, 10, 10])
>>> E = np.linspace(-2, 2, 100)
>>> def wrap_DOS(eigenstate, k, weight):
... # Calculate the DOS for the eigenstates and weight by k_x and weight
... return eigenstate.DOS(E) * k[0] * weight
>>> DOS = mp.apply.sum.eigenstate(wrap=wrap_DOS, eta=True)
When using wrap to calculate more than one quantity per eigenstate it may be advantageous
to use oplist to handle cases of BrillouinZone.apply.average and BrillouinZone.apply.sum.
>>> H = Hamiltonian(...)
>>> mp = MonkhorstPack(H, [10, 10, 10])
>>> E = np.linspace(-2, 2, 100)
>>> def wrap_multiple(eigenstate):
... # Calculate DOS/PDOS for eigenstates
... DOS = eigenstate.DOS(E)
... PDOS = eigenstate.PDOS(E)
... # Calculate velocity for the eigenstates
... v = eigenstate.velocity()
... return oplist([DOS, PDOS, v])
>>> DOS, PDOS, v = mp.apply.average.eigenstate(wrap=wrap_multiple, eta=True)
Which does mathematical operations (averaging/summing) using oplist.
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A class to construct Brillouin zone related quantities |
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Create a Monkhorst-Pack grid for the Brillouin zone |
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Create a path in the Brillouin zone for plotting band-structures etc. |