Transmission ratio through single barrier (Python3)

Transmission ratio (t) of a particle through single barrier is calculated and shown by Python3. This is preparation for another case of double barriers and a quantum well shown in the next page.

Following equations used here are a set of solution obtained by one-dimensional Schrödinger equation with typical initial condition. Height (V₀) and thickness (a = D_b) of the barrier are 0.5 eV and 2.8 nm, respectively. Effective mass (m) of a particle is 0.35 m₀ (m₀: electron mass in vacuum).

#   Tunneling_single_barrier_01.py #   on Python 3.4.1, Nov. 29, 2014 #   by Masao Sakuraba import math import numpy import scipy import pylab def main():     meff = 0.35   # effective mass ratio of a particle     V0 = 0.5   # barrier height (eV)     Db = 2.5e-9   # barrier thickness (m)     q = 1.602e-19    # elementary charge (C)     hbar = 1.0546e-34   # reduced Planck constant (J s)     hbar2 = hbar ** 2     m = meff * 9.1095e-31   # absolute mass of a particle (kg)     # parameters for lower energy (0 < E < V0)     M = 50   # number of energy step     Alpha_d = []     Alpha_d2 = []     Beta_d = []     Beta_d2 = []     Sinh = []     Sinh2 = []     t_d = []     # parameters for higher energy (E > V0)     N = 100   # number of energy step     Alpha_u = []     Alpha_u2 = []     Beta_u = []     Beta_u2 = []     Sin = []     Sin2 = []     t_u = []     E_d = numpy.linspace(V0 / M, V0 - V0 / M, M - 1)     for i in range(M - 1):         Alpha_d2.append(2 * m / hbar2 * E_d[i] * q)         Alpha_d.append(math.sqrt(Alpha_d2[i]))         Beta_d2.append(2 * m / hbar2 * (V0 - E_d[i]) * q)         Beta_d.append(math.sqrt(Beta_d2[i]))         Sinh.append(numpy.sinh((Beta_d[i]) * Db))         Sinh2.append(Sinh[i] ** 2)         t_d.append(             4 * Alpha_d2[i] * Beta_d2[i] /                          ((Alpha_d2[i] + Beta_d2[i]) ** 2 * Sinh2[i]                           + 4 * Alpha_d2[i] * Beta_d2[i])             )     E_u = numpy.linspace(V0 + V0 / N, V0 * 3.5 - V0 / N, N - 1)     for j in range(N - 1):         Alpha_u2.append(2 * m / hbar2 * E_u[j] * q)         Alpha_u.append(math.sqrt(Alpha_u2[j]))         Beta_u2.append(2 * m / hbar2 * (E_u[j] - V0) * q)         Beta_u.append(math.sqrt(Beta_u2[j]))         Sin.append(math.sin(Alpha_u[j] * Db))         Sin2.append(Sin[j] ** 2)         t_u.append(             4 * Alpha_u2[j] * Beta_u2[j] /                          ((Alpha_u2[j] - Beta_u2[j]) ** 2 * Sin2[j]                           + 4 * Alpha_u2[j] * Beta_u2[j])             )     # 'option' : color (r,g,b,c,m,y,k,w) & line style (-,--,:,-.,.,o,^)     pylab.plot(E_d, t_d, 'g:.', label='0 < E < V0')     pylab.plot(E_u, t_u, 'r:.', label='E > V0')     pylab.legend(loc='upper right')     pylab.title('Transmission through single barrier')     pylab.xlabel('Energy (eV)')     pylab.ylabel('Transmission Ratio')     pylab.xlim(-0.1, V0 * 4)     pylab.yscale('log')    # Delete this line, if linear y. #    pylab.ylim(-0.1, 1.4)     pylab.show() if __name__ == '__main__':     main()