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Free electron band structure (#321)
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* Some modifications to the text of the free electron notebook and its associated theory notebook.

* Added warning that figure may take some time to load when changing lattice type.

* Further modifications to text. Brought layout into standard format.

* Fixing the question number 3 and the lattice parameter of the SC lattice

---------

Co-authored-by: Taylor <taylor.baird@epfl.ch>
Co-authored-by: Kristjan Eimre <kristjaneimre@gmail.com>
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@giovannipizzi @Taylor-96 @eimrek
3 people authored Apr 30, 2024
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34 changes: 23 additions & 11 deletions notebook/band-theory/free_electron.ipynb
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"\n",
"**Source code:** https://github.com/osscar-org/quantum-mechanics/blob/master/notebook/band-theory/free_electron.ipynb\n",
"\n",
" The main objective of this notebook is to demonstrate the bandstructure for the\n",
" free-electron model in a periodic lattice.\n",
" The main objective of this notebook is to demonstrate the electronic bandstructure within the free-electron model for a periodic crystalline lattice of a metal.\n",
" \n",
" \n",
"Throughout the notebook, we employ the empty lattice (free-electron) approximation for the electrons in a periodic \n",
"solid system. Using it, we compute and plot the electronic band structure for three \n",
"types of Bravais lattice: simple cubic (SC), face-centered cubic (FCC) and body-centered cubic (BCC). We get the path in reciprocal space for the band structure \n",
"from the <a href=\"https://seekpath.readthedocs.io/en/latest/index.html\">seekpath</a>\n",
"package.\n",
"\n",
"<hr style=\"height:1px;border:none;color:#cccccc;background-color:#cccccc;\" />"
]
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"source": [
"## **Goals**\n",
"\n",
"* Familiarize oneself with the free-electron model.\n",
"* Examine the electronic band structure of the free electron model for different crystal structures.\n"
"* Familiarize yourself with the free-electron model of a metallic solid.\n",
"* Examine the electronic band structure of the free-electron model for different crystalline structures.\n"
]
},
{
Expand All @@ -52,19 +58,19 @@
" <li> Can you describe the shape of the band structure in the 1st Brillouin zone?\n",
" <details>\n",
" <summary style=\"color: red\">Solution</summary>\n",
" In the free electron model, the dispersion relation between electronic energy and wavevector is given by $E=\\frac{\\hbar^2k^2}{2m}$. Consequently, the shape of the bands is parabolic. \n",
" In the free electron model, the dispersion relation between electronic energy and wavevector is given by $E=\\frac{\\hbar^2k^2}{2m}$. Accordingly, the shape of the bands is parabolic. \n",
" </details> \n",
" </li>\n",
" <li> What properties of a material shall be best captured by the free electron model?\n",
" <li> What properties of a material shall be best captured by the free-electron model?\n",
" <details>\n",
" <summary style=\"color: red\">Solution</summary>\n",
" As the free-electron model neglects the effect of the ionic potential on the electrons, material properties which are primarily dependent on the kinetic energy of the conduction electrons are those which shall be best described by the model.\n",
" </details> \n",
" </li>\n",
" <li> Look at the bandstructure plots for different crystal structures by toggling the \"Cell type\" radio buttons. Why is the bandstructure associated with the BCC crystal structure much denser than that of the simple cubic cell (i.e., why is there so many more bands in the energy range considered for BCC compared to SC).\n",
" <li> Consider the simple cubic lattice, and consider the lowest energy band from Γ to the X point, from Γ to the R point, and from Γ to the M point. The curvature of those bands seem to be the same. Is this to be expected? What about the Γ-L, Γ-X, and Γ-K in the FCC case? Or the Γ-H, Γ-N, and Γ-P in the BCC case?\n",
" <details>\n",
" <summary style=\"color: red\">Solution</summary>\n",
" Recalling that the energy eigenvalues are given by $\\large E = \\frac{\\hbar^2(\\vec{k}+\\vec{G})^2}{2m}$, we can see that the origin of the increased density of bands for BCC is due to its Brillouin zone giving rise to a larger number of G-vectors with small magnitudes. This in turn increases the number of low-lying energy bands.\n",
" For a free electron case, the bands are isotropic (i.e., they are the same, independent of the direction in k space), and the effective mass is just the free-electron mass: $m^*=m_0$. Therefore, we expect all curvatures (of the lowest energy band starting from Γ) to be the same.\n",
" </details> \n",
" </li>\n",
"</ol>\n",
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"outputs": [],
"source": [
"def get_bands(real_lattice_bohr, reference_distance = 0.05, g_vectors_range = 3):\n",
" \"\"\"Function to return the band structure of a free-electron model.\n",
" \n",
" :param real_lattice_bohr: 3x3 lattice vectors of the crystal, in bohr\n",
" :param reference_distance: distance between consecutive k-points in the band structure.\n",
" :param g_vectors_range: integer indicating how many G vectors should be included\n",
" to compute the band structure.\n",
" \"\"\"\n",
" \n",
" # Simple way to get automatically the band path:\n",
" # I go back to real space, just put a single atom at the origin,\n",
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"\n",
"alat_bohr = 7.72\n",
"\n",
"lattices = np.zeros((3, 3, 3));\n",
"lattices = np.zeros((3, 3, 3))\n",
"\n",
"lattices[0] = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 1]]) * alat_bohr / 2.0\n",
"lattices[1] = np.array([[0, 1, 1], [1, 0, 1], [1, 1, 0]]) * alat_bohr / 2.0\n",
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"cell_type": "markdown",
"metadata": {},
"source": [
"<details open>\n",
" <summary style=\"font-size: 22px;\"><b>Legend</b></summary>\n",
"## Legend\n",
"\n",
"<p style=\"text-align: justify;font-size:15px\">\n",
" The 1st Brillouin zone of the selected cell is shown on the left. \n",
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30 changes: 12 additions & 18 deletions notebook/band-theory/theory/theory_free_electron.ipynb
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"source": [
"## **Introduction**\n",
" \n",
"Here, we employ the empty lattice approximation for the electrons in a periodic \n",
"solid system. Using it, we compute and plot the electronic band structure for three \n",
"type of cells (simple cubic, FCC and BCC). We get the path for the band structure \n",
"from the <a href=\"https://seekpath.readthedocs.io/en/latest/index.html\">seekpath</a>\n",
"package. "
"This notebook gives a brief overview of the empty lattice approximation (free-electron model) of electrons in a metallic solid."
]
},
{
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"source": [
"## Empty lattice approximation\n",
"\n",
"In the empty lattice approximation, the electrons move \"freely\" in the \n",
"periodic potential. There is no electron-electron interaction. \n",
"In the empty lattice approximation, the electrons move \"freely\" in a weak, periodic potential. Electron-electron interactions are neglected.\n",
"The eigenfunctions of the Schrödinger equation for free electrons are:\n",
"\n",
"$$\\large \\psi(\\vec{r}) = e^{i\\vec{k} \\vec{r}}$$ \n",
"$$\\large \\psi(\\vec{r}) = e^{i\\vec{k} \\cdot \\vec{r}}$$ \n",
"\n",
"When $\\vec{k'}$ lies outside the 1st Brillouin zone, the plane wave \n",
"When $\\vec{k}$ lies outside the 1st Brillouin zone, the plane wave \n",
"can be written as:\n",
" \n",
"$$\\large \\psi(\\vec{r}) = e^{i\\vec{k} \\vec{r}}e^{i\\vec{G} \\vec{r}} = e^{i(\\vec{k}+\\vec{G})\\vec{r}}$$ \n",
"$$\\large \\psi(\\vec{r}) = e^{i\\vec{k}\\cdot \\vec{r}}e^{i\\vec{G} \\cdot\\vec{r}} = e^{i(\\vec{k}+\\vec{G})\\cdot \\vec{r}}$$ \n",
" \n",
"where, $\\vec{k}$ vector is inside the first Brillouin zone and $\\vec{G}$ \n",
"where $\\vec{k}$ vector is inside the first Brillouin zone and $\\vec{G}$ \n",
"is a reciprocal lattice vector. The dispersion is:\n",
" \n",
"$$\\large E = \\frac{\\hbar^2(\\vec{k}+\\vec{G})^2}{2m}$$\n",
"\n",
"Please read more at the [Wikipedia](https://en.wikipedia.org/wiki/Empty_lattice_approximation)."
"You can read more about this model on [Wikipedia](https://en.wikipedia.org/wiki/Empty_lattice_approximation)."
]
},
{
Expand All @@ -74,12 +69,11 @@
" band structure. Adpoted from Wikipedia</figcaption>\n",
"</div>\n",
" \n",
"Molecular diagrams can present the discrete energy levels for the \n",
"molecular systems. In contrast, solid system always have a very large \n",
"number of the orbitals. It leads to the energy levels to close together. Hence, the energy levels in solid are considered as continuous energy bands.\n",
"Molecular diagrams can represent the discrete energy levels in\n",
"molecular systems. In contrast, solid-state systems always have a macroscopic number of orbitals present. This leads to the energy levels merging together and forming a continuum. Hence, the energy levels in solid are considered to be continuous energy bands.\n",
"\n",
"Since the wavevector is in three dimensions ($k_x$, $k_y$ and $k_z$), \n",
"it is difficult to plot the bands as a function of the wavevector (4 dimensional plotting). Usually, the bands are plotted along the straight lines, which connects high symmetry points (see, e.g. Figure 2).\n",
"Since the wavevector is three dimensional, having components $k_x$, $k_y$ and $k_z$, \n",
"it is difficult to plot the bands as a function of the wavevector. Therefore, the energy bands are typically plotted along straight lines, connecting high symmetry points of the Brillouin zone (see, e.g. Figure 2).\n",
"\n",
"<div style=\"text-align:center\">\n",
"<image src=\"../images/1st_brillouin_zone.svg\" width=\"250\"></image>\n",
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"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.9.12"
"version": "3.10.9"
}
},
"nbformat": 4,
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