Plotting the solution of the central equation using python

[patric44]

hi guys

i was trying to came up with a basic code that could show me the band gaps in a solid after adding the periodic potential term to my solution :
$$ E = \frac{ħ^2q^2}{2m} \pm Vg $$
where Vg is my periodic potential , q is the k values in the first billion zone
from my understanding if i set the potential = 0 , i will get the normal parabolic dispersion for the free electron model , but once i introduce the potential it will create a separation at certain values of k which will indicate the forbidden gaps .
i tried to create that in python :

from scipy import signal
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.style
import matplotlib.colors
m = 1
ħ  = 1
G = 50
K = 200
E1 = []
E2 = []
E3 = []
x1 = []
x2 = []
x3 = []
def energy1(q):
    erg1 = ((ħ**2 * (q-G)**2 )/(8*m))+ np.cos(q)
    return erg1
def energy2(q):
    erg2 = ((ħ**2 * q**2 )/(8*m))+ np.cos(q)
    return erg2
def energy3(q):
    erg3 = ((ħ**2 * (q+G)**2 )/(8*m))+ np.cos(q)
    return erg3

for z in range(-K,K):
    x1.append(z)
    E1.append(energy1(z))
for z in range(-K,K):
    x2.append(z)
    E2.append(energy2(z))
for z in range(-K,K):
    x3.append(z)
    E3.append(energy3(z)) 
plt.plot(x1,E1,'r')
plt.plot(x1,E2,'g')
plt.plot(x1,E3,'b')
plt.axvline(x=G)
plt.axvline(x=-G)
plt.grid()
plt.show()

and it gives that :

i was just trying to see the effect of basic periodic function like the sin , but there is no gaps! ,
i must be missing something or doing something wrong ?
any help will be appreciated , thanks

 

[Fred Wright]

Your model doesn't properly provide for how the periodicity of the lattice effects the electron wave function. Try studying the Kronig-Penny model, for example here, Electron in periodic lattice.

 

[patric44]

Your model doesn't properly provide for how the periodicity of the lattice effects the electron wave function. Try studying the Kronig-Penny model, for example here, Electron in periodic lattice.

i don't understand what you mean ? do you mean that the potential function that i picked will not give arise
to a band gap at the zone bounders , ( is it must be a square wave like the Kronig-Penny model ?!)
my solution was taken from this book :

how would i modify this code to show the band gap at the zone boundaries ?

 

[Fred Wright]

how would i modify this code to show the band gap at the zone boundaries ?

In the link I provided the following dispersion relation is derived,$$
\cos (ka)=\cosh (\alpha a)+P\frac{\sinh (\alpha a)}{\alpha a}\\
\alpha^2 =\frac{2m| E |}{\hbar^2}\\
P=\frac{mV_0 ba}{\hbar^2}$$where $a$ is the lattice constant, $b$ is the width of the well, and $V_0$ is the depth of the well. Your algorithm must pick an E value for the r.h.s. of the equation, then take the arccosine and divide by $a$ to determine the corresponding k. It's a two step process, choose $E$ then compute $k$.

 

[patric44]

In the link I provided the following dispersion relation is derived,$$
\cos (ka)=\cosh (\alpha a)+P\frac{\sinh (\alpha a)}{\alpha a}\\
\alpha^2 =\frac{2m| E |}{\hbar^2}\\
P=\frac{mV_0 ba}{\hbar^2}$$where $a$ is the lattice constant, $b$ is the width of the well, and $V_0$ is the depth of the well. Your algorithm must pick an E value for the r.h.s. of the equation, then take the arccosine and divide by $a$ to determine the corresponding k. It's a two step process, choose $E$ then compute $k$.

thanks for helping , but i have a question :
why the paper from the book that i provided doesn't mention this , i mean the book just says
$$ E=\frac{ħ^2q^2}{ 2m} \pm Vg $$
why the book didn't mention the kronig model , does the book has another approach or it was just being
very hand wavy about it ?!

 

[Fred Wright]

thanks for helping , but i have a question :
why the paper from the book that i provided doesn't mention this , i mean the book just says
$$ E=\frac{ħ^2q^2}{ 2m} \pm Vg $$
why the book didn't mention the kronig model , does the book has another approach or it was just being
very hand wavy about it ?!

I don't have your book so I can't answer your question. I suspect that the chapter you are studying introduces Bloch's theorem but stops short of deriving a dispersion relation for a particular model...maybe in a subsequent chapter it will.