.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "examples_generated/tutorials/defects.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_examples_generated_tutorials_defects.py: .. _defects: ============================== Defect calculations: ASE Tools ============================== This section gives an (incomplete) overview of features in ASE that help in the preparation and analysis of supercell calculations as most commonly employed in the computation of defect properties. .. contents:: .. GENERATED FROM PYTHON SOURCE LINES 14-41 Supercell creation ================== Background ---------- Defect properties are most commonly investigated in the so-called dilute limit, i.e. under conditions, in which defect-defect interactions are negligible. While alternative approaches in particular embedding techniques exist, the most common approach is to use supercells. To this end, one creates a supercell by a *suitable* (see below) repetition of the primitive unit cell, after which a defect, e.g., a vacancy or an impurity atom, is inserted. The calculation thus corresponds to a periodic arrangement of defects. Accordingly, care must be taken to keep the interactions between defects as small as possible, which generally calls for large supercells. Thus the typical goal for generating the simulation supercell for defect calculations is to maximize the defect-defect separation in *all* directions, for a reasonable number of atoms (and thus computational cost). In principle, we can do a good job of this by using a supercell with a suitable shape. To illustrate this for different lattices, we build and plot three 2D lattices with identical unit cell area but different lattice symmetry. We build and visualize (3,3,1) supercells by repeating the unit cell. .. GENERATED FROM PYTHON SOURCE LINES 41-162 .. code-block:: Python import matplotlib.pyplot as plt from ase.build import bulk from ase.geometry import get_distances from ase.visualize.plot import plot_atoms def add_decoration(ax, conf, centralidx, neighborcutoff, arrow_offset=(0, 0)): # function for adding arrows to neighboring atoms vectors, distances = get_distances( conf.positions[centralidx], p2=conf.positions ) neighboridx = [ i for i, j in enumerate(distances[0]) if (neighborcutoff >= j) ] # we normalize the size of the arrows for illustration purposes norm = neighborcutoff / 5 for idx in neighboridx: ax.arrow( conf.positions[centralidx][0] + arrow_offset[0], conf.positions[centralidx][1] + arrow_offset[1], vectors[0][idx][0] / norm, vectors[0][idx][1] / norm, width=0.1, color='k', ) lattice_constant = 8 centralidx = 4 neighborcutoff = 8 # square lattice # build structure conf = bulk('Po', a=lattice_constant) # moving the atom in the middle of the cell positions = conf.get_positions() positions += conf.cell[0][0] / 2 conf.set_positions(positions) conf = conf.repeat((3, 3, 1)) # plot cs structure fig, ax = plt.subplots() add_decoration(ax, conf, centralidx, neighborcutoff, arrow_offset=(0.5, 0.5)) plot_atoms(conf, ax, offset=(0, 0)) # , rotation=('-80x,0y,0z')) ax.set_axis_off() ax.text( 0.1, -0.1, 'square lattice: r$_1$=a, Z$_1$=4', transform=ax.transAxes, fontsize=16, ) plt.show() # rectangular lattice # build structure conf = bulk( 'C', crystalstructure='orthorhombic', a=lattice_constant, b=lattice_constant / 2, c=lattice_constant / 2, ) # moving the atom in the middle of the cell positions = conf.get_positions() positions += [conf.cell[0][0] / 2, conf.cell[0][0] / 4, conf.cell[0][0] / 4] conf.set_positions(positions) conf = conf.repeat((3, 3, 1)) # plot orc structure fig, ax = plt.subplots() add_decoration(ax, conf, centralidx, neighborcutoff, arrow_offset=(0.5, 0.5)) plot_atoms(conf, ax, offset=(0, 0)) ax.set_axis_off() ax.text( 0, -0.2, 'rectangular lattice with a 2:1 aspect ratio:\n r$_1$=a/2, Z$_1$=2', transform=ax.transAxes, fontsize=16, ) plt.show() # hexagonal lattice # build structure conf = bulk('Be', a=lattice_constant) # here, we slice the cell to have one a 2D layer of atoms confmask = [i.index for i in conf if i.position[2] < 1] conf = conf[confmask] conf = conf.repeat((3, 3, 1)) positions = conf.get_positions() positions -= conf.positions[0] conf.set_positions(positions) # plot hpc structure fig, ax = plt.subplots() add_decoration( ax, conf, centralidx, neighborcutoff, arrow_offset=(-conf.cell[1][0] + 1, 1.5), ) plot_atoms(conf, ax, offset=(0, 0), rotation=('0x,0y,0z')) ax.set_axis_off() ax.text( 0.1, -0.1, 'hexagonal lattice: r$_1$=1.075a, Z$_1$=6', transform=ax.transAxes, fontsize=16, ) plt.show() .. rst-class:: sphx-glr-horizontal * .. image-sg:: /examples_generated/tutorials/images/sphx_glr_defects_001.png :alt: defects :srcset: /examples_generated/tutorials/images/sphx_glr_defects_001.png :class: sphx-glr-multi-img * .. image-sg:: /examples_generated/tutorials/images/sphx_glr_defects_002.png :alt: defects :srcset: /examples_generated/tutorials/images/sphx_glr_defects_002.png :class: sphx-glr-multi-img * .. image-sg:: /examples_generated/tutorials/images/sphx_glr_defects_003.png :alt: defects :srcset: /examples_generated/tutorials/images/sphx_glr_defects_003.png :class: sphx-glr-multi-img .. GENERATED FROM PYTHON SOURCE LINES 163-270 In the case of the square lattice, each defect has :math:`Z_1=4` nearest neighbors at a distance of :math:`r_1=a_0`, where :math:`a_0=\sqrt{A}` with :math:`A` being the unit cell area. By comparison in a rectangular lattice with an aspect ratio of 2:1, the defects are much closer to each other with :math:`r_1 = a_0/\sqrt{2}` and :math:`Z_1=2`, where again :math:`a_0` = :math:`\sqrt{A}` (the 'effective cubic length'). The largest defect-defect distance (at constant unit cell area) is obtained for the hexagonal lattice, which also correponds to the most closely packed 2D arrangement. Here, one obtains :math:`r_1=\sqrt{2}/\sqrt[4]{3}=1.075 a_0` and :math:`Z_1=6`. For defect calculations, supercells corresponding to hexagonal or square lattices have thus clear advantages. This argument can be extended to 3D: Square lattices in 2D correspond to cubic lattices (supercells) in 3D with :math:`r_1=a_0` and :math:`Z_1=6`. The 3D analogue of the hexagonal 2D lattice are hexagonal and cubic close packed structures (i.e. FCC, HCP), both of which yield :math:`r_1 = \sqrt[6]{2} a_0 \approx 1.1225 a_0` and :math:`Z_1=12`. It is straightforward to construct cubic or face-centered cubic (fcc, cubic closed packed) supercells for cubic materials (including e.g, diamond and zincblende) by using simple repetitions of the conventional or primitive unit cells. For countless materials of lower symmetry the choice of a supercell is, however not necessarily so simple. The algorithm below represents a general solution to this issue. In the case of semiconductors and insulators with small dielectric constants, defect-defect interactions are particularly pronounced due to the weak screening of long-ranged electrostatic interactions. While various correction schemes have been proposed, the most reliable approach is still finite-size extrapolation using supercells of different size. In this case care must be taken to use a sequence of self-similar supercells in order for the extrapolation to be meaningful. To motivate this statement consider that the leading (monopole-monopole) term :math:`E_{mp}`, which scales with :math:`1/r` and is proportional to the (ionic) dielectric constant :math:`\epsilon_0`. The :math:`E_{mp}` term is geometry dependent and in the case of simple lattices the dependence is easily expressed by the Madelung constant. The geometry dependence implies that different (super)cell shapes fall on different lines when plotting e.g., the formation energy as a function of :math:`N^{-1/3}` (equivalent to an effective inverse cell size, :math:`L^{-1} \propto N^{-1/3}`. For extrapolation one should therefore only use geometrically equivalent cells or at least cells that are as self-similar to each other as possibly, see Fig. 10 in [Erhart]_ for a very clear example. In this context there is therefore also a particular need for supercells of a particular shape. Algorithm for finding optimal supercell shapes ---------------------------------------------- The above considerations illustrate the need for a more systematic approach to supercell construction. A simple scheme to construct "optimal" supercells is described in [Erhart]_. Optimality here implies that one identifies the supercell that for a given size (number of atoms) most closely approximates the desired shape, most commonly a simple cubic or fcc metric (see above). This approach ensures that the defect separation is large and that the electrostatic interactions exhibit a systematic scaling. The ideal cubic cell metric for a given volume :math:`\Omega` is simply given by :math:`\Omega^{1/3} \mathbf{I}`, which in general does not satisfy the crystallographic boundary conditions. The :math:`l_2`-norm provides a convenient measure of the deviation of any other cell metric from a cubic shape. The optimality measure can thus be defined as .. math:: \Delta_\text{sc}(\mathbf{h}) = ||\mathbf{h} - \Omega^{1/3} \mathbf{1}||_2, Any cell metric that is compatible with the crystal symmetry can be written in the form .. math:: \mathbf{h} = \mathbf{P} \mathbf{h}_p where :math:`\mathbf{P} \in \mathbb{Z}^{3\times3}` and :math:`\mathbf{h}_p` is the primitive cell metric. This approach can be readily generalized to arbitrary target cell metrics. In order to obtain a measure that is size-independent it is furthermore convenient to introduce a normalization, which leads to the expression implemented here, namely .. math:: \bar{\Delta}(\mathbf{Ph}_p) = ||Q\mathbf{Ph}_p - \mathbf{h}_\text{target}||_2, where `Q = \left(\det\mathbf{h}_\text{target} \big/ \det\mathbf{h}_p\right)^{1/3}` is a normalization factor. The matrix :math:`\mathbf{P}_\text{opt}` that yields the optimal cell shape for a given cell size can then be obtained by .. math:: \mathbf{P}_\text{opt} = \underset{\mathbf{P}}{\operatorname{argmin}} \left\{ \bar\Delta\left(\mathbf{Ph}_p\right) | \det\mathbf{P} = N_{uc}\right\}, where :math:`N_{uc}` defines the size of the supercell in terms of the number of primitive unit cells. Implementation of algorithm --------------------------- For illustration consider the following example. First we set up a primitive face-centered cubic (fcc) unit cell and visualize it. .. GENERATED FROM PYTHON SOURCE LINES 271-284 .. code-block:: Python import matplotlib.pyplot as plt from ase.build import bulk from ase.visualize.plot import plot_atoms conf = bulk('Au') fig, ax = plt.subplots() plot_atoms(conf, ax) ax.set_axis_off() .. image-sg:: /examples_generated/tutorials/images/sphx_glr_defects_004.png :alt: defects :srcset: /examples_generated/tutorials/images/sphx_glr_defects_004.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 285-289 Then, we call :func:`~ase.build.find_optimal_cell_shape` to obtain a :math:`\mathbf{P}` matrix that will enable us to generate a supercell with 32 atoms that is as close as possible to a simple cubic shape: .. GENERATED FROM PYTHON SOURCE LINES 289-297 .. code-block:: Python import numpy as np from ase.build import find_optimal_cell_shape from ase.build.supercells import eval_length_deviation P1 = find_optimal_cell_shape(conf.cell, 32, 'sc') print(P1) .. rst-class:: sphx-glr-script-out .. code-block:: none [[-2 2 2] [ 2 -2 2] [ 2 2 -2]] .. GENERATED FROM PYTHON SOURCE LINES 298-310 More nicely rendered, this yields .. math:: \mathbf{P}_1 = \left(\begin{array}{rrr} -2 & 2 & 2 \\ 2 & -2 & 2 \\ 2 & 2 & -2 \end{array}\right) \quad \mathbf{h}_1 = \left(\begin{array}{ccc} 2 a_0 & 0 & 0 \\ 0 & 2 a_0 & 0 \\ 0 & 0 & 2 a_0 \end{array}\right), where :math:`a_0` =4.05 Å is the lattice constant. This is indeed the expected outcome as it corresponds to a :math:`2\times2\times2` repetition of the *conventional* (4-atom) unit cell. On the other hand repeating this exercise with: .. GENERATED FROM PYTHON SOURCE LINES 311-315 .. code-block:: Python P2 = find_optimal_cell_shape(conf.cell, 495, 'sc') print(P2) .. rst-class:: sphx-glr-script-out .. code-block:: none [[-5 5 5] [ 5 -4 5] [ 5 5 -4]] .. GENERATED FROM PYTHON SOURCE LINES 316-328 yields a less obvious result, namely .. math:: \mathbf{P}_2 = \left(\begin{array}{rrr} -6 & 5 & 5 \\ 5 & -6 & 5 \\ 5 & 5 & -5 \end{array}\right) \quad \mathbf{h}_2 = a_0 \left(\begin{array}{ccc} 5 & 0 & 0 \\ 0.5 & 5 & 0.5 \\ 0.5 & 0.5 & 5 \end{array}\right), which indeed corresponds to a reasonably cubic cell shape. One can also obtain the optimality measure :math:`\bar{\Delta}` by executing: .. GENERATED FROM PYTHON SOURCE LINES 328-333 .. code-block:: Python dev1 = eval_length_deviation(np.dot(P1, conf.cell)) dev2 = eval_length_deviation(np.dot(P2, conf.cell)) print(f'The length deviation for P_1 is {dev1}') print(f'The length deviation for P_2 is {dev2}') .. rst-class:: sphx-glr-script-out .. code-block:: none The length deviation for P_1 is 3.8459253727671276e-16 The length deviation for P_2 is 0.019161199641064268 .. GENERATED FROM PYTHON SOURCE LINES 334-382 which yields :math:`\bar{\Delta}(\mathbf{P}_1)=0` and :math:`\bar{\Delta}(\mathbf{P}_2)=0.0192`. Since this procedure requires only knowledge of the cell metric (and not the atomic positions) for standard metrics, e.g., fcc, bcc, and simple cubic one can generate series of shapes that are usable for *all* structures with the respective metric. For example the :math:`\mathbf{P}_\text{opt}` matrices that optimize the shape of a supercell build using a primitive FCC cell are directly applicable to diamond and zincblende lattices. For illustration, the :math:`\bar{\Delta}` values for supercells of SC, FCC and BCC lattices with SC/FCC target shapes are shown as a function of the number of unit cells :math:`N_{uc}\leq2000` in the panel below (taken from :mr:`3404`). The algorithm is, however, most useful for non-cubic cell shapes, for which finding several reasonably sized cell shapes is more challenging, as illustrated for a hexagonal material (LaBr\ :sub:`3`) in [Erhart]_. .. image:: https://gitlab.com/-/project/470007/uploads/5c52f1b09cfd8f82c3b8453f45762d4f/image.png .. note:: For unit cells with more complex space groups, this approach can be cumbersome due to the implementation which loops over many possible transformation matrices. The `find_optimal_cell_shape `_ function in `doped `_ implements the same algorithm with some efficiency improvements (~100x compute time speedup), and offers an efficient `algorithm `_ for *directly* optimising the periodic defect-defect distance (~10-50% improvements); see [Kavanagh]_ or the ``doped`` `tutorials `_. Generation of supercell ----------------------- Once the transformation matrix :math:`\mathbf{P}` it is straightforward to generate the actual supercell using e.g., the :func:`~ase.build.cut` function. A convenient interface is provided by the :func:`~ase.build.make_supercell` function, which is invoked as follows: .. GENERATED FROM PYTHON SOURCE LINES 382-392 .. code-block:: Python from ase.build import make_supercell conf = bulk('Au') P = find_optimal_cell_shape(conf.cell, 495, 'sc') supercell = make_supercell(conf, P) fig, ax = plt.subplots() plot_atoms(supercell, ax) ax.set_axis_off() .. image-sg:: /examples_generated/tutorials/images/sphx_glr_defects_005.png :alt: defects :srcset: /examples_generated/tutorials/images/sphx_glr_defects_005.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 393-403 .. [Erhart] P. Erhart, B. Sadigh, A. Schleife, and D. Åberg. First-principles study of codoping in lanthanum bromide, Phys. Rev. B, Vol **91**, 165206 (2012), :doi:`10.1103/PhysRevB.91.165206`; Appendix C .. [Kavanagh] S. R. Kavanagh et al. doped: Python toolkit for robust and repeatable charged defect supercell calculations J. Open Source Softw, 9(**96**), 6433 (2024), :doi:`10.21105/joss.06433` .. _sphx_glr_download_examples_generated_tutorials_defects.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: defects.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: defects.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: defects.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_