Photoelectric effect gif

Here are provided two simple gif images in order to illustrate the photoelectric effect.

Below the threshold energy, nothing append, whatever the light intensity.

No photoelectric effect
No photoelectric effect

Above the threshold energy, each photon bear enough energy in order to extract electrons from the material.

Photoelectric effect
Photoelectric effect

All pictures can be downloaded here : graphene.zip

Bands diagram using VASP and pymatgen

This article presents a python source code in order to plot the bands diagram of graphene calculated using VASP. The plot is done using pymatgen library and RGB coloring adapted from this example. Here is the output obtained with the script :

Graphene bands diagram
Graphene bands diagram

On the band diagram, the contribution of s, (px,py) and pz atomic orbital is map on a RGB scale. Red is associated to a s contribution, green to (px, py) contribution and blue to pz contribution. The same function or procedure can be used to map atomic contributions on the band diagram.

At the end of the article you can find a tarball containing all VASP input/ouput files and the python source code. This calculation is not accurate and is used only for band diagram plotting illustration. For example, you can see a short gap on the DOS which does not exist.

Edit 2016, March 15, update : source code is now python3 and pymatgen version 3.3.5 compatible. You can find the below scripts in an up to date version on github : bandstructureplots

#!/usr/bin/env python
# -*- coding=utf-8 -*-
 
import sys
 
import numpy as np
from numpy import array as npa
 
import matplotlib.pyplot as plt
from matplotlib.collections import LineCollection
from matplotlib.gridspec import GridSpec
 
import pymatgen as mg
from pymatgen.io.vasp.outputs import Vasprun, Procar
from pymatgen.symmetry.bandstructure import HighSymmKpath
from pymatgen.electronic_structure.core import Spin, Orbital
 
 
def rgbline(ax, k, e, red, green, blue, alpha=1.):
    # creation of segments based on
    # http://nbviewer.ipython.org/urls/raw.github.com/dpsanders/matplotlib-examples/master/colorline.ipynb
    pts = np.array([k, e]).T.reshape(-1, 1, 2)
    seg = np.concatenate([pts[:-1], pts[1:]], axis=1)
 
    nseg = len(k) - 1
    r = [0.5 * (red[i] + red[i + 1]) for i in range(nseg)]
    g = [0.5 * (green[i] + green[i + 1]) for i in range(nseg)]
    b = [0.5 * (blue[i] + blue[i + 1]) for i in range(nseg)]
    a = np.ones(nseg, np.float) * alpha
    lc = LineCollection(seg, colors=list(zip(r, g, b, a)), linewidth=2)
    ax.add_collection(lc)
 
if __name__ == "__main__":
    # read data
    # ---------
 
    # kpoints labels
    path = HighSymmKpath(mg.Structure.from_file("./opt/CONTCAR")).kpath["path"]
    labels = [r"$%s$" % lab for lab in path[0][0:4]]
 
    # bands
    bands = Vasprun("./bands/vasprun.xml").get_band_structure("./bands/KPOINTS", line_mode=True)
 
    # projected bands
    data = Procar("./bands/PROCAR").data
 
    # density of state
    dosrun = Vasprun("./dos/vasprun.xml")
 
    # set up matplotlib plot
    # ----------------------
 
    # general options for plot
    font = {'family': 'serif', 'size': 24}
    plt.rc('font', **font)
 
    # set up 2 graph with aspec ration 2/1
    # plot 1: bands diagram
    # plot 2: Density of State
    gs = GridSpec(1, 2, width_ratios=[2, 1])
    fig = plt.figure(figsize=(11.69, 8.27))
    fig.suptitle("Bands diagram of graphene")
    ax1 = plt.subplot(gs[0])
    ax2 = plt.subplot(gs[1])  # , sharey=ax1)
 
    # set ylim for the plot
    # ---------------------
    emin = 1e100
    emax = -1e100
    for spin in bands.bands.keys():
        for b in range(bands.nb_bands):
            emin = min(emin, min(bands.bands[spin][b]))
            emax = max(emax, max(bands.bands[spin][b]))
 
    emin -= bands.efermi + 1
    emax -= bands.efermi - 1
    ax1.set_ylim(emin, emax)
    ax2.set_ylim(emin, emax)
 
    # Band Diagram
    # ------------
 
    # sum up contribution over carbon atoms
    data = data[Spin.up].sum(axis=2)
 
    # sum up px and py contributions and normalize contributions
    contrib = np.zeros((bands.nb_bands, len(bands.kpoints), 3))
    for b in range(bands.nb_bands):
        for k in range(len(bands.kpoints)):
            sc = data[k][b][Orbital.s.value]**2
            pxpyc = data[k][b][Orbital.px.value]**2 + \
                data[k][b][Orbital.py.value]**2
            pzc = data[k][b][Orbital.pz.value]**2
            tot = sc + pxpyc + pzc
            if tot != 0.0:
                contrib[b, k, 0] = sc / tot
                contrib[b, k, 1] = pxpyc / tot
                contrib[b, k, 2] = pzc / tot
 
    # plot bands using rgb mapping
    for b in range(bands.nb_bands):
        rgbline(ax1,
                range(len(bands.kpoints)),
                [e - bands.efermi for e in bands.bands[Spin.up][b]],
                contrib[b, :, 0],
                contrib[b, :, 1],
                contrib[b, :, 2])
 
    # style
    ax1.set_xlabel("k-points")
    ax1.set_ylabel(r"$E - E_f$   /   eV")
    ax1.grid()
 
    # fermi level at 0
    ax1.hlines(y=0, xmin=0, xmax=len(bands.kpoints), color="k", lw=2)
 
    # labels
    nlabs = len(labels)
    step = len(bands.kpoints) / (nlabs - 1)
    for i, lab in enumerate(labels):
        ax1.vlines(i * step, emin, emax, "k")
    ax1.set_xticks([i * step for i in range(nlabs)])
    ax1.set_xticklabels(labels)
 
    ax1.set_xlim(0, len(bands.kpoints))
 
    # Density of state
    # ----------------
 
    ax2.set_yticklabels([])
    ax2.grid()
    ax2.set_xticks(np.arange(0, 1.5, 0.4))
    ax2.set_xticklabels(np.arange(0, 1.5, 0.4))
    ax2.set_xlim(1e-6, 1.5)
    ax2.hlines(y=0, xmin=0, xmax=1.5, color="k", lw=2)
    ax2.set_xlabel("Density of State")
 
    # s contribution
    ax2.plot(npa(dosrun.pdos[0][Orbital.s][Spin.up]) +
             npa(dosrun.pdos[1][Orbital.s][Spin.up]),
             dosrun.tdos.energies - dosrun.efermi,
             "r-", label="s", linewidth=2)
 
    # px py contribution
    ax2.plot(npa(dosrun.pdos[0][Orbital.px][Spin.up]) +
             npa(dosrun.pdos[1][Orbital.px][Spin.up]) +
             npa(dosrun.pdos[0][Orbital.py][Spin.up]) +
             npa(dosrun.pdos[1][Orbital.py][Spin.up]),
             dosrun.tdos.energies - dosrun.efermi,
             "g-",
             label="(px, py)",
             linewidth=2)
 
    # pz contribution
    ax2.plot(npa(dosrun.pdos[0][Orbital.pz][Spin.up]) +
             npa(dosrun.pdos[1][Orbital.pz][Spin.up]),
             dosrun.tdos.energies - dosrun.efermi,
             "b-", label="pz", linewidth=2)
 
    # total dos
    ax2.fill_between(dosrun.tdos.densities[Spin.up],
                     0,
                     dosrun.tdos.energies - dosrun.efermi,
                     color=(0.7, 0.7, 0.7),
                     facecolor=(0.7, 0.7, 0.7))
 
    ax2.plot(dosrun.tdos.densities[Spin.up],
             dosrun.tdos.energies - dosrun.efermi,
             color=(0.6, 0.6, 0.6),
             label="total DOS")
 
    # plot format style
    # -----------------
 
    ax2.legend(fancybox=True, shadow=True, prop={'size': 18})
    plt.subplots_adjust(wspace=0)
 
    # plt.show()
    plt.savefig(sys.argv[0].strip(".py") + ".pdf", format="pdf")

graphene.zip

Hereafter there are two more examples for Cu and Si. The script may be simpler as all data can be extracted using pymatgen methods.

Si_bands.zip

Cu_bands.zip

Bands diagram of copper

Bands diagram of silicon

Réseaux de bravais

Voici un document regroupant les 14 réseaux de bravais avec les schémas dessinés avec tikz. Le document tableau_bravais.pdf contient uniquement un tableau qui résume les 14 réseaux de bravais.

Bravais : Réseau monoclinique

Fichier source : bravais.tex

Fichier pdf : bravais.pdf

Fichier pdf contenant un tableau résumant les 14 réseaux de bravais : tableau_bravais.pdf

Relations entre les coordonnées cartésiennes et les coordonnées réduites

Voici un document pdf (et le fichier tex source) qui donne les relations entre les coordonnées cartésiennes d’un repère orthonormé (Oijk sur la figure) et les coordonnées réduites dans le repère cristallin (Oabc). Ces relations sont assez utiles, en pratique, pour passer des coordonnées réduites aux coordonnées cartésiennes.

Coordonnées réduites et cartésiennes

coord_abc_xyz.pdf

Fichier sources si vous souhaitez l’adapter : coord_abc_xyz.tex