# Python API introduction¶

IFermi is a fully featured Python 3.6+ library for the generation, analysis, and visualisation of Fermi surfaces and Fermi slices. The goal of the library is to provide full featured `FermiSurface`

and `FermiSlice`

objects that allow for easy manipulation and analysis. The main features include:

Interpolation of electronic band structures onto dense k-point meshes.

Extraction of Fermi surfaces and Fermi slices from electronic band structures.

Projection of arbitrary properties on to Fermi surfaces and Fermi slices.

Tools to calculate Fermi surface dimensionality, orientation, and averaged projections.

Interactive visualisation of Fermi surfaces and slices, with support for mayavi, plotly and matplotlib.

Generation and visualisation of spin-texture.

Many of the options provided in the Python API are also accessible from the command-line. This notebook gives a demonstration of how to use the Python library.

**Warning**

IFermi currently only works with VASP calculations but support for additional DFT packages will be added in the future.

The overall workflow for using IFermi can be summarised as:

Load DFT calculation outputs to create a

`BandStructure`

object.Interpolate the band structure onto a dense k-point mesh using the

`FourierInterpolater`

class that is based on BoltzTraP2.Extract the Fermi surface at a given energy level to create a

`FermiSurface`

object.(Optionally) slice the Fermi surface along a plane to create a

`FermiSlice`

object.The library provides a rich set of functions for the analysis and visualisation of

`FermiSurface`

and`FermiSlice`

objects.

Below, we provide details for each of these steps. This page can also be run as an interactive jupyter notebook.

## Loading VASP outputs¶

The first step is to load the VASP calculation outputs needed for the plot. This is achieved using the pymatgen package, which contains classes for parsing outputs and representing band structure calculations.

The only input required is a vasprun.xml file. Here, we use the MgB\(_2\) calculation data in the `examples/MgB2/`

folder. We first load the vasprun.xml using the `Vasprun`

class.

```
[2]:
```

```
from pymatgen.io.vasp.outputs import Vasprun
vr = Vasprun('MgB2/vasprun.xml')
```

Next, we extract the band structure information into a `BandStructure`

object. This class contains information on the eigenvalues, k-points, and reciprocal lattice vectors.

```
[3]:
```

```
bs = vr.get_band_structure()
```

## Interpolating onto a dense k-point mesh¶

There are currently two issues with our extracted band structure:

It only contains the irreducible portion of the Brillouin zone (since symmetry was used in the calculation) and therefore does not contain enough information to plot the Fermi surface across the full reciprocal lattice.

It was calculated on a relatively coarse k-point mesh and therefore will produce a rather jagged Fermi surface.

Both issues can be solved be interpolating the band structure onto a denser k-point mesh. This can be achieved using the `FourierInterpolator`

class. Internerally, this class uses the BoltzTraP2 package to Fourier interpolate the eigenvalues on to a denser mesh that covers the full Brillouin zone.

The interpolater can be initialized using a `BandStructure`

object as input.

```
[4]:
```

```
from ifermi.interpolate import FourierInterpolator
interpolator = FourierInterpolator(bs)
```

The band structure can be interpolated using the `interpolate_bands()`

function. The degree of interpolation is controlled by the `interpolation_factor`

argument. A value of `5`

, roughly indicates that the interpolated band structure will contain 5x as many k-points. Increasing the interpolation factor will result in smoother Fermi surfaces.

**Warning**

As the interpolation increases, the generation of the Fermi surface and plot will take a longer time and can result in large file sizes.

```
[5]:
```

```
dense_bs = interpolator.interpolate_bands(interpolation_factor=5)
```

The `interpolate_bands()`

function returns both the band structure on a dense k-point mesh.

To include the group velocity as a Fermi surface property, the velocities can be calculated during interpolation by setting the `return_velocities`

option.

```
[6]:
```

```
dense_bs, velocities = interpolator.interpolate_bands(interpolation_factor=5, return_velocities=True)
```

## Generating Fermi surfaces¶

The next step is to extract the Fermi surface from the band structure. The `FermiSurface.from_band_structure()`

function generates a `FermiSurface`

object from the dense band structure we just created.

This function has a number of options that are detailed in the FermiSurface API Reference page. The two key options are:

`mu`

(default:`0.0`

) which controls the energy offset at which the Fermi surface is calculated.`wigner_seitz`

(default:`False`

) which controls whether the Fermi surface is calculated in the Wigner–Seitz cell or the parallelepiped reciprocal lattice cell.

MgB2 is metallic, so we are mainly interested in the Fermi surface at the Fermi level.

**Note**

For gapped materials, mu must be adjusted so that it falls within the conduction band or valence band otherwise no Fermi surface will be displayed.

```
[7]:
```

```
from ifermi.surface import FermiSurface
fs = FermiSurface.from_band_structure(dense_bs, mu=0.0, wigner_seitz=True)
```

To include a property in the Fermi surface, we can supply the `property_data`

and `property_kpoints`

options during Fermi surface generation. The only requirement is that the propery k-points cover the entire Brillouin zone. For example:

```
[8]:
```

```
from ifermi.kpoints import kpoints_from_bandstructure
dense_kpoints = kpoints_from_bandstructure(dense_bs)
fs = FermiSurface.from_band_structure(
dense_bs,
property_data=velocities,
property_kpoints=dense_kpoints,
)
```

To calculate Fermi surface properties such as the dimensionality and orientation, the `calculate_dimensionality`

option can be used.

```
[9]:
```

```
fs = FermiSurface.from_band_structure(
dense_bs,
wigner_seitz=True,
property_data=velocities,
property_kpoints=dense_kpoints,
calculate_dimensionality=True
)
```

## Analyzing Fermi surfaces¶

The `FermiSurface`

object includes a rich API that can be used to calcualate many properties of the Fermi surface. The principle attribute is `isosurfaces`

. This contains an an `Isosurface`

object for each fully-connected surface. The isosurfaces are given for each spin channel in the band structure. The FermiSurface API Reference page gives a list of all available attributes.

The number of isosurfaces can be obtained using:

```
[10]:
```

```
fs.n_surfaces
```

```
[10]:
```

```
10
```

The number of isosurfaces in each spin channel is available using:

```
[11]:
```

```
fs.n_surfaces_per_spin
```

```
[11]:
```

```
{<Spin.up: 1>: 5, <Spin.down: -1>: 5}
```

The total area of the Fermi surface:

```
[12]:
```

```
fs.area
```

```
[12]:
```

```
32.63033435193189
```

And the area of each isosurface in each spin channel.

```
[13]:
```

```
fs.area_surfaces
```

```
[13]:
```

```
{<Spin.up: 1>: array([1.9257198 , 4.35767606, 2.94401577, 3.54389684, 3.54389612]),
<Spin.down: -1>: array([1.92566855, 4.35765367, 2.94401459, 3.54389684, 3.54389612])}
```

The isosurfaces for each spin channel can be accessed using the pymatgen `Spin`

object.

```
[14]:
```

```
from pymatgen.electronic_structure.core import Spin
fs.isosurfaces[Spin.up]
```

```
[14]:
```

```
[Isosurface(nvertices=524, nfaces=808, band_idx=5, dim=2D, orientation=(0, 0, 1)),
Isosurface(nvertices=1560, nfaces=1914, band_idx=6, dim=quasi-2D, orientation=(0, 0, 1)),
Isosurface(nvertices=736, nfaces=1136, band_idx=6, dim=2D, orientation=(0, 0, 1)),
Isosurface(nvertices=1392, nfaces=1680, band_idx=7, dim=quasi-2D, orientation=(0, 0, 1)),
Isosurface(nvertices=1396, nfaces=1682, band_idx=7, dim=quasi-2D, orientation=(0, 0, 1))]
```

The `Isosurface`

object has several utility methods for accessing and manipulating the isosurface properties. The Isosurface API Reference page gives the full documentation.

First, we select the first isosurface in the Spin up channel.

```
[15]:
```

```
isosurface = fs.isosurfaces[Spin.up][0]
```

The area of the isosurface is obtained through:

```
[16]:
```

```
isosurface.area
```

```
[16]:
```

```
1.9257197973874718
```

If the Fermi surface was generated using `calculate_dimensionality=True`

, the dimensionality of the isosurface (i.e., whether the surface cross periodic boundaries) can be obtained using.

```
[17]:
```

```
isosurface.dimensionality
```

```
[17]:
```

```
'2D'
```

For 2D and 1D isosurfaces, their orientation is available through:

```
[18]:
```

```
isosurface.orientation
```

```
[18]:
```

```
(0, 0, 1)
```

There are a number of functions for manipulating the face properties of isosurfaces.

```
[19]:
```

```
# does the surface have face properties
isosurface.has_properties
# calculate the norms of the properties
isosurface.properties_norms
# calculate scalar projection of properties on to [0 0 1] vector
isosurface.scalar_projection((0, 0, 1))
# get the average property across the surface
isosurface.average_properties()
# get the average of the property norms across the surface
isosurface.average_properties(norm=True)
```

```
[19]:
```

```
708736.4609522179
```

Furthermore, it is possible to sample the isosurface faces to a consistent density using:

```
[20]:
```

```
# returns the indices of the faces that give uniform density
isosurface.sample_uniform(0.1)
```

```
[20]:
```

```
array([795, 735, 253, 239, 242, 259, 261, 263, 793, 731, 212, 214, 217,
202, 204, 206, 773, 718, 345, 347, 370, 369, 367, 365, 362, 354,
353, 349, 383, 402, 399, 397, 395, 780, 238, 241, 243, 522, 525,
527, 528, 533, 535, 538, 539, 541, 256, 258, 260, 693, 225, 116,
215, 544, 546, 551, 553, 555, 557, 559, 563, 512, 479, 481, 200,
113, 110, 751, 723, 496, 499, 501, 504, 506, 509, 561, 495, 494,
484, 487, 488, 490, 783, 721, 470, 471, 432, 474, 475, 477, 802,
445, 434, 436, 442, 444, 426, 424, 407, 409, 412, 414, 416, 419,
420, 635, 636, 727, 654, 656, 658, 677, 687, 682, 650, 679, 637,
639, 642, 644, 646, 649, 675, 661, 652, 787, 730, 91, 93, 124,
127, 772, 712, 223, 265, 95, 320, 100, 104, 176, 174, 172, 170,
168, 107, 109, 119, 335, 336, 338, 797, 8, 10, 12, 14, 16,
300, 301, 303, 305, 285, 283, 728, 308, 310, 33, 314, 316, 341,
26, 42, 20, 50, 88, 85, 83, 81, 78, 76, 75, 769, 732,
393, 391, 318, 343, 3, 190, 188, 340, 334, 337, 386, 777, 724,
592, 590, 588, 585, 583, 580, 18, 146, 142, 15, 577, 575, 574,
571, 569, 567, 804, 706, 622, 620, 617, 155, 153, 151, 147, 139,
136, 134, 132, 615, 625, 629, 628, 786, 716, 287, 289, 306, 293,
277, 279, 281, 776])
```

## Fermi slices¶

IFermi can also generate two-dimensional slices of the Fermi surface along a specified plane. Planes are defined by their miller indices (j k l) and a distance from the plane, d. `FermiSlice`

objects can be generated directly from `FermiSurface`

objects.

```
[21]:
```

```
import numpy as np
fermi_slice = fs.get_fermi_slice(plane_normal=(0, 0, 1), distance=0)
```

The `FermiSlice`

object includes a rich API that can be used to calcualate many properties of the Fermi slice. The principle attribute is `isolines`

. This contains an an `Isoline`

object for each fully-connected line in the slice. The isolines are given for each spin channel in the band structure. The FermiSlice API Reference page gives a list of all available attributes. Below we showcase a few of the
functions available.

```
[22]:
```

```
# number of isolines in the slice
fermi_slice.n_lines
# do the lines have segment properties
fermi_slice.has_properties
# get the first isoline in the Spin up channel
isoline = fermi_slice.isolines[Spin.up][0]
# calculate the norms of the properties
isoline.properties_norms
# calculate scalar projection of properties on to [0 0 1] vector
isoline.scalar_projection((0, 0, 1))
# sample the line at a consistent density
# returns the indices of the segments that give the desired spacing
isoline.sample_uniform(0.1)
```

```
[22]:
```

```
array([ 50, 159, 266, 373, 479, 588, 693])
```

## Visualising Fermi surfaces and slices¶

IFermi includes a rich set of visualisation tools for plotting `FermiSurface`

and `FermiSlice`

objects.

Fermi surfaces can be visualised using the `FermiSurfacePlotter`

class. This object supports plotting using matplotlib, plotly and mayavi (provided the `mayavi`

package is installed). The detailed documentation is available on the FermiSurfacePlotter API Reference
page.

First, the plotter is initialised from a `FermiSurface`

object.

```
[23]:
```

```
from ifermi.plot import FermiSurfacePlotter
fs = FermiSurface.from_band_structure(dense_bs, mu=0.0, wigner_seitz=True)
plotter = FermiSurfacePlotter(fs)
```

Next, we can use the `get_plot()`

function, to get a plot object. The plotting backend is controlled using the `plot_type`

option. The available options are:

`plotly`

(default): We use this for all plots in this notebook.`matplotlib`

`mayavi`

: Requires`vtk`

and`mayavi`

to be installed.

The plot can be displayed directly in a jupyter notebook by calling the `show()`

function.

```
[24]:
```

```
plot = plotter.get_plot(plot_type="plotly")
plot.show()
```

### Selecting spin channels¶

In the plot above, the spins are degenerate (the Hamiltonian does not differentiate between the up and down spins). This is why the surface looks dappled - the plotter is plotting two redundant sufaces. To stop it from doing this, we can specify that only one component of the spin should be plotted.

*Note: from now on we omit the ``plot_type`` argument.*

```
[25]:
```

```
from pymatgen.electronic_structure.core import Spin
plot = plotter.get_plot(spin=Spin.up)
plot.show()
```