From 5f8b98f45320b0d4d1f7b416c39d3525f913342e Mon Sep 17 00:00:00 2001 From: Lars kleyn Winkel <l.kleynwinkel@student.tudelft.nl> Date: Mon, 10 Feb 2020 21:41:06 +0000 Subject: [PATCH] Update src/7_tight_binding_model_sol.md --- src/7_tight_binding_model_sol.md | 60 -------------------------------- 1 file changed, 60 deletions(-) diff --git a/src/7_tight_binding_model_sol.md b/src/7_tight_binding_model_sol.md index ec85e1e7..b349c13d 100644 --- a/src/7_tight_binding_model_sol.md +++ b/src/7_tight_binding_model_sol.md @@ -81,63 +81,3 @@ The Schrödinger equation is given as: $|\Psi\rangle = \sum_n \phi_n |n\rangle$ Solving the Schrödinger equation yields dispersion: $$E(k) = E_0 -t\cos(ka) -t'\cos(2ka)$$ -### Subquestiion 3 - -$$m_{eff} = \frac{\hbar^2}{2a^2}\frac{1}{t\cos(ka)+4t'\cos(2ka)}$$ - -Plot for t=t': - -```python -k1 = np.linspace(-pi, -pi/2-0.01, 300); -k2 = np.linspace(-pi/2+0.01, pi/2-0.01, 300); -k3 = np.linspace(pi/2+0.01, pi, 300); - -pyplot.plot(k1, 1/(5*np.cos(k1)),'b'); -pyplot.plot(k2, 1/(5*np.cos(k2)),'b'); -pyplot.plot(k3, 1/(5*np.cos(k3)),'b'); -pyplot.xlabel('$k$'); pyplot.ylabel('$m_{eff}(k)$'); -pyplot.xticks([-pi,0,pi],[r'$-\pi/a$',0,r'$\pi/a$']); -pyplot.yticks([],[]); -pyplot.tight_layout(); -``` - -### Subquestion 4 - -Plots for 2t'=t, 4t'=t and 10t'=t: - -```python -def m(k,t): - return 1/(np.cos(k)+4*t*np.cos(2*k)) - -k1 = np.linspace(-1.6, -0.83, 300); -k2 = np.linspace(-0.826, 0.826, 300); -k3 = np.linspace(0.83, 1.6, 300); - -pyplot.plot(k1, m(k1,2),'b'); -pyplot.plot(k2, m(k2,2),'b'); -pyplot.plot(k3, m(k3,2),'b',label='t=2t\''); -pyplot.xlabel('$k$'); pyplot.ylabel('$m_{eff}(k)$'); -pyplot.xticks([-pi,0,pi],[r'$-\pi/a$',0,r'$\pi/a$']); -pyplot.yticks([-10,10],[]); -pyplot.tight_layout(); - -k1 = np.linspace(-1.58, -0.81, 300); -k2 = np.linspace(-0.804, 0.804, 300); -k3 = np.linspace(0.81, 1.58, 300); - -pyplot.plot(k1, m(k1,4),'r'); -pyplot.plot(k2, m(k2,4),'r'); -pyplot.plot(k3, m(k3,4),'r',label='t=4t\''); - -k1 = np.linspace(-1.575, -0.798, 300); -k2 = np.linspace(-0.790, 0.790, 300); -k3 = np.linspace(0.798, 1.575, 300); - -pyplot.plot(k1, m(k1,10),'k'); -pyplot.plot(k2, m(k2,10),'k'); -pyplot.plot(k3, m(k3,10),'k',label='t=10t\''); - -pyplot.legend() -``` - - -- GitLab