{ "cells": [ { "cell_type": "markdown", "id": "52686ff2", "metadata": {}, "source": [ "# Bandes d'énergie dans un réseau cristallin" ] }, { "cell_type": "markdown", "id": "93433dbc", "metadata": {}, "source": [ "**Modèle de Kronig-Penney**: Un électron se déplace dans un potentiel périodique 1-dimensionnel $V(x)$ de période $a$. Le potentiel est zéro, sauf pour des barrières de potentiel rectangulaires de largeur $b$ et de hauteur $V_0$ aux intervalles réguliers:\n", "\n", "" ] }, { "cell_type": "markdown", "id": "a502ada5", "metadata": {}, "source": [ "Ce modèle donne lieu à une **structure de bandes** pour les énergies électroniques. Objectif: trouver les **bandes d'énergies** permises et interdites.\n", "\n", "Selon le **théorème de Bloch**, la fonction d'onde pour un tel potentiel périodique prendra la forme\n", "$$\\psi(x)=e^{ikx}\\,u(x)$$\n", "avec $k$ réel et $u(x)$ une fonction périodique de période $a$. L'équation de Schrödinger est $H\\psi=E\\psi$, ou bien\n", "$$\n", "-\\frac{\\hbar^2}{2m}\\psi''(x)+V(x)\\psi(x)=E\\psi(x)\\,.\n", "$$\n", "Sa solution dans les domaines où $V(x)=0$ est donnée par des ondes planes,\n", "$$\\psi_1(x)=A e^{i\\alpha x}+Be^{-i\\alpha x}=e^{ikx}u_1(x)\\,,\\qquad u_1(x)=Ae^{i(\\alpha-k) x}+Be^{-i(\\alpha+k) x}\\,,\\qquad \\alpha=\\sqrt{\\frac{2mE}{\\hbar^2}}\\,.$$\n", "Dans les barrières où $V(x)=V_0$, elle est donnée par\n", "$$\\psi_{2}(x)=C e^{\\beta x}+De^{-\\beta x}=e^{ikx}u_{2}(x)\\,,\\qquad u_{2}(x)=Ce^{(\\beta-ik) x}+De^{-(\\beta+ik) x}\\,,\\qquad \\beta=\\sqrt{\\frac{2m(V_0-E)}{\\hbar^2}}\\,.$$\n", "Choisissons la coordonnée $x$ comme suit:\n", "\n", "\n", "\n", "La fonction d'onde doit être continue en $x=0$, $\\psi_1(0)=\\psi_{2}(0)$, alors \n", "$$A+B=C+D\\,.$$ \n", "Sa dérivée doit être continue en $x=0$, $\\psi_1'(0)=\\psi_{2}'(0)$, ce qui donne \n", "$$i\\alpha(A-B)=\\beta(C-D)\\,.$$ \n", "Les fonctions $u(x)$ et $u'(x)$ doivent être périodiques selon le théorème de Bloch: $u_{2}(-b)=u_1(a-b)$ et $u_{2}'(-b)=u_1'(a-b)$. On obtient ainsi\n", "$$e^{-(\\beta-ik)b}C+e^{(\\beta+ik)b}D=e^{i(\\alpha-k)(a-b)}A+e^{-i(\\alpha+k)(a-b)}B$$ \n", "et \n", "$$(\\beta-ik)e^{-(\\beta-ik)b}C-(\\beta+ik)e^{(\\beta+ik)b}D=i(\\alpha-k)e^{i(\\alpha-k)(a-b)}A-i(\\alpha+k)e^{-i(\\alpha+k)(a-b)}B\\,.$$\n", "Tout ensemble, on a quatre équations linéaires pour les quatre coefficients $A$, $B$, $C$ et $D$. En forme matricielle:\n", "$$\n", "\\left(\\begin{array}{cccc}1&1&-1&-1 \\\\ i\\alpha &-i\\alpha&-\\beta&\\beta \\\\ e^{i(\\alpha-k)(a-b)}&e^{-i(\\alpha+k)(a-b)}&-e^{-(\\beta-ik)b}&-e^{(\\beta+ik)b}\\\\i(\\alpha-k)e^{i(\\alpha-k)(a-b)}&-i(\\alpha+k)e^{-i(\\alpha+k)(a-b)}&-(\\beta-ik)e^{-(\\beta-ik)b}&(\\beta+ik)e^{(\\beta+ik)b}\\end{array}\\right)\\left(\\begin{array}{c} A\\\\B\\\\C\\\\D\\end{array}\\right)=\\left(\\begin{array}{c}0\\\\0\\\\0\\\\0\\end{array}\\right)\\,.\n", "$$\n", "\n", "Une solution non triviale n'existe que si la matrice des coefficients $M$ possède une valeur propre $0$. Il faut donc que $\\det M=0$, ce qui est le cas ssi\n", "$$\n", "\\cos(ka)=\\cosh(\\beta b)\\cos(\\alpha(a-b))-\\frac{\\alpha^2-\\beta^2}{2\\alpha\\beta}\\sinh(\\beta b)\\sin(\\alpha(a-b))\\,.\n", "$$\n", "Dans la limite $b\\rightarrow 0$, $V_0\\rightarrow\\infty$ avec $V_0 b=$const., cette équation se simplifie. On a \n", "$$\\cosh(\\beta b)\\rightarrow 1,\\quad \\cos(\\alpha(a-b))\\rightarrow\\cos(\\alpha a)\\,,\\quad\\sinh(\\beta b)\\rightarrow\\beta b\\,,\\quad\\sin(\\alpha(a-b))\\rightarrow\\sin(\\alpha a)\\,,\\quad(\\alpha^2-\\beta^2)\\rightarrow -2mV_0/\\hbar^2\\,;$$ ainsi on obtient la condition\n", "$$\\boxed{\n", "\\cos(ka)=\\cos(\\alpha a)+P\\frac{\\sin(\\alpha a)}{\\alpha a}\\qquad(\\star)}\n", "$$\n", "où on a posé\n", "$$\n", "\\qquad P=\\frac{abmV_0}{\\hbar^2}\\,.\n", "$$\n", "La dépendance de l'énergie est implicite dans $\\alpha=\\sqrt{\\frac{2mE}{\\hbar^2}}$. Pour que l'éq. ($\\star$) admette une solution avec $k$ et $E$ réels, il faut que son membre de droite soit compris entre $-1$ et $+1$, vu que son membre de gauche l'est. Cette condition détermine les **bandes d'énergies** permises:\n", "$$\n", "E \\text{ est énergie permise}\\quad\\Leftrightarrow\\quad -1\\leq \\cos(s)+P\\frac{\\sin{s}}{s}\\leq 1\\quad\\text{avec } s=\\alpha a = a\\sqrt{\\frac{2mE}{\\hbar^2}}\\,.\n", "$$\n", "\n", "Traçons le graphe de $\\cos(s) + P\\sin(s)/s$ pour $P=5$ en fonction de $s$:" ] }, { "cell_type": "code", "execution_count": 1, "id": "2a449c02", "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "# On trace la courbe pour s entre -20 et 20\n", "s = np.linspace(-20, 20, 500)\n", "\n", "P = 5\n", "\n", "# La fonction sin(s)/s, définie par continuité d'être 1 en s=0\n", "def sinc(s):\n", " return np.piecewise(s, [s == 0, s != 0], [1, lambda t: np.sin(t)/t])\n", "\n", "plt.plot(s, np.cos(s) + P * sinc(s))\n", "plt.fill_between(s, -1, 1, where=(np.abs(np.cos(s) + P * sinc(s)) <= 1), color='g', alpha=0.2) # bandes permises\n", "plt.fill_between(s, -1, 1, where=(np.abs(np.cos(s) + P * sinc(s)) > 1), color='r', alpha=0.2) # bandes interdites\n", "plt.plot(s, np.ones_like(s), 'k') # tracer une ligne horizontale à +1\n", "plt.plot(s, -np.ones_like(s),'k') # et une autre à -1\n", "plt.xlabel(\"s\")\n", "plt.ylabel(\"cos(s) + P sin(s)/s\")\n", "plt.ylim([-3,3])\n", "plt.title(\"Structure de bandes dans le modèle de Kronig-Penney, P = 5\")\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "a01e5628", "metadata": {}, "source": [ "Les valeurs de $s$ séparant les bandes permises (tracées en vert) des bandes interdites (rouge) sont déterminées par les solutions des équations\n", "$$\\boxed{\\cos(s)+P\\frac{\\sin(s)}{s}=\\pm 1\\,.\\qquad(\\star\\star)}$$ \n", "Il est évident que tout $s=n\\pi$ avec $n$ entier et $\\neq 0$ en est une solution, car $\\sin(n\\pi)=0$ et $|\\cos(n\\pi)|=1$. Selon la figure, ces valeurs de $s$ marquent les fins des bandes permises et les débuts des bandes interdites. Mais visiblement il y a encore d'autres solutions qui marquent les débuts des bande permises. Vu que l'on ne peut pas les trouver analytiquement, on va employer une méthode numérique.\n", "\n", "Définissons\n", "$$f_+(s)=\\cos(s)+P\\frac{\\sin(s)}{s}+1\\,,\\qquad f_-(s)=\\cos(s)+P\\frac{\\sin(s)}{s}-1\\,.$$\n", "Les zéros de $f_{\\mp}$ sont les solutions de $(\\star\\star)$. On va appliquer la **méthode de Newton** pour trouver les zéros manquants entre $n\\pi$ et $(n-1)\\pi$. Notons que les dérivées de $f_+$ et de $f_-$ sont identiques et données par\n", "$$f'(s)=-\\sin(s)+P\\frac{s\\cos(s)-\\sin(s)}{s^2}\\,.$$\n", "Si on choisit comme point de départ de la méthode de Newton $s_0=n\\pi-1$, on trouvera un zéro nontrivial de $f_+$ si $n$ est pair et de $f_-$ si $n$ est impair. Grace à la symétrie du problème, on peut se limiter aux $n$ positifs. " ] }, { "cell_type": "code", "execution_count": 2, "id": "73fca554", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Bande permise entre s = 2.28445 et s = 3.14159\n", "Bande permise entre s = 4.76129 et s = 6.28319\n", "Bande permise entre s = 7.46368 et s = 9.42478\n", "Bande permise entre s = 10.3266 et s = 12.5664\n", "Bande permise entre s = 13.2862 et s = 15.708\n", "Bande permise entre s = 16.3031 et s = 18.8496\n" ] } ], "source": [ "def f_plus(s):\n", " return np.cos(s) + P * sinc(s) + 1\n", "\n", "def f_minus(s):\n", " return np.cos(s) + P * sinc(s) - 1\n", "\n", "def f_prime(s):\n", " return - np.sin(s) + P * (s * np.cos(s) - np.sin(s)) / s**2 # pas besoin de spécifier f'(0) pour ce problème\n", "\n", "def newton(s0, f, max_it=10, tolerance = 1.E-5):\n", " # trouve un zéro de la fonction f (qui sera f_plus ou f_minus) avec dérivée f_prime\n", " # s0 = point de départ\n", " # max_it = nombre maximal d'itérations avant abandon\n", " # tolerance = delta s pour tester convergence ~ précision numérique minimale\n", " i = 0\n", " s = s0\n", " while i < max_it:\n", " s, s_old = s - f(s) / f_prime(s), s \n", " if np.abs(s - s_old) < tolerance:\n", " return s\n", " i += 1\n", " print(\"Erreur: la méthode de Newton n'a pas convergée pour s0 = {}\".format(s0))\n", " \n", "for n in range(1, 7):\n", " if n % 2 == 0: # n pair: on cherche un zéro de f_plus\n", " zero = newton(n * np.pi - 1, f_plus)\n", " else: # n impair: on cherche un zéro de f_minus\n", " zero = newton(n * np.pi - 1, f_minus)\n", " print(\"Bande permise entre s = {:.6} et s = {:.6}\".format(zero, n*np.pi))" ] }, { "cell_type": "markdown", "id": "3b1dcd79", "metadata": {}, "source": [ "Si on le souhaite, les énergies correspondantes peuvent ensuite être trouvées avec la formule\n", "$$\n", "E=\\frac{\\hbar^2 s^2}{2ma^2}\\,.\n", "$$" ] }, { "cell_type": "code", "execution_count": null, "id": "92620dba", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.12.3" } }, "nbformat": 4, "nbformat_minor": 5 }