Band Structure and Edge modes of a Graphene Nano Ribbon
Quick Summary
We are interested in the edge modes of the Graphene Nano Ribbon. A nano ribbon is a 2D structure in which the molecules extend infinitely in one of the directions and finite in the other. One more visualizations of the above is forming a chain of atoms(Periodic boundary conditions) in one of the directions and finite chain(Open boundary conditions) in the other direction.
Importing the required libraries
import numpy as np
import matplotlib as mp
from matplotlib import pyplot as plt
import ipywidgets as ipy
plt.style.use('seaborn')
mp.rcParams['figure.figsize'] = (9, 7)
%matplotlib notebook
Modules for building the model and computing the eigenenergies and values
def gzig_zag(N = 25, t = 1.0, kval = 0.5):
Hmat = np.zeros([2*N,2*N], dtype = complex)
for i in range(len(Hmat)-1):
Hmat[i][i+1] = (1-i%2)*t*(1+np.exp(1j*kval)) + (i%2)*t #0.5*np.random.random()
Hmat[i+1][i] = np.conj(Hmat[i][i+1])
#Hmat[i][i] = np.random.random()
return Hmat
def compute(Hmat):
eigenenergies, eigenstate = np.linalg.eigh(Hmat)
return eigenenergies, eigenstate
Energy spectrum as a function of ‘k’
N = 15; t = 1.0
k = np.linspace(-4,4,1001)
eig = np.zeros([len(k),2*N])
for _ in range(len(k)):
eig[_] = compute(gzig_zag(N,t,k[_]))[0]
plt.figure()
plt.plot(k,eig)
plt.xlabel("$k$"); plt.ylabel("$E(k)$")
plt.title("Energy Spectrum for the Zig-Zag Graphene Structure")
plt.show()
Visualization of the Edge modes
vecs = np.zeros([2*N,2*N], dtype = complex)
vecs = compute(gzig_zag(N,t,-3))[1]
plt.figure()
plt.plot(range(2*N),vecs[:,N], label="Zero energy edge mode")
plt.plot(range(2*N),vecs[:,0], label="Ground state")
plt.title("Eigenvectors of the Zig-Zag Structure of Graphene")
plt.legend()
plt.show()
Checking the Robustness of the edge modes
def gzig_zag_r(N = 25, t = 1.0, kval = 0.5):
Hmat = np.zeros([2*N,2*N], dtype = complex)
for i in range(len(Hmat)-1):
Hmat[i][i+1] = (1-i%2)*(t-np.random.random())*(1+np.exp(1j*kval)) + (i%2)*(t+np.random.random()) #0.5*np.random.random()
Hmat[i+1][i] = np.conj(Hmat[i][i+1])
#Hmat[i][i] = np.random.random()
return Hmat
plt.figure()
N = 25; t = 1.0
k = np.linspace(-4,4,1001)
eig = np.zeros([len(k),2*N])
for _ in range(len(k)):
eig[_] = compute(gzig_zag_r(N,t,k[_]))[0]
plt.plot(k,eig)
plt.xlabel("$k$"); plt.ylabel("$E(k)$")
plt.title("Energy Spectrum for the Zig-Zag Graphene Structure")
plt.show()
plt.figure()
vecs = np.zeros([2*N,2*N], dtype = complex)
vecs = compute(gzig_zag_r(N,t,-3))[1]
plt.plot(range(2*N),vecs[:,N-1], label="Zero energy edge mode")
plt.plot(range(2*N),vecs[:,0], label="Ground state")
plt.title("Eigenvectors of the Zig-Zag Structure of Graphene")
plt.legend()
plt.show()
Chakradhar Rangi
Physics PhD Student
My research interests include Computational Condensed Matter Physics and developing scientific codes.