<span xmlns:dct="http://purl.org/dc/terms/" property="dct:title"></span> The following notes written by <span xmlns:cc="http://creativecommons.org/ns#" property="cc:attributionName">Sergio Gutiérrez Rodrigo (sergut@unizar.es) </span>. Distributed under License Creative Commons Atribución-NoComercial-CompartirIgual 4.0 Internacional

Departamento de Física Aplicada
Universidad de Zaragoza
Instituto de Nanociencia y Materiales de Aragón (INMA)
C/ Pedro Cerbuna, 12, 50009, Zaragoza, España

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Photonic devices and systems - 2D materials dispersion relation (hexagonal Boron Nitride)

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Definitions: van der Waals polariton dispersion relation

'''
Thin Film Dispersion Relation hBN
Eq.1 in Tunable Phonon Polaritons in Atomically Thin van der Waals Crystals of Boron Nitride
S. Dai et al. Science 343, 1125 (2014);
DOI: 10.1126/science.124683
'''
def psi(lambda_):
    return np.sqrt(eps_z(omega(lambda_))) / (1j * np.sqrt(eps_t(omega(lambda_))))

# Modes for a given qy
qy = 0
def qx(lambda_):
    return -(psi(lambda_) / d) * (np.arctan(eps_a / (eps_t(omega(lambda_)) * psi(lambda_))) + np.arctan(eps_s(omega(lambda_)) / (eps_t(omega(lambda_)) * psi(lambda_))) + np.pi * l)

def qz(lambda_):
    return np.sqrt(eps_t(omega(lambda_)) / eps_z(omega(lambda_)) * ((qx(lambda_))**2 + (qy)**2) - eps_t(omega(lambda_)) * ((2.0 * np.pi) / lambda_)**2)

Optical properties of hBN, substrate and cover

import numpy as np
import matplotlib.pyplot as plt
import scipy.optimize as opt

# Constants
h = 6.626e-34  # J*s
c = 3e8  # m/s
e = 1.6021773e-19  # J/eV
flw = 1239.8283  # (h*c)/e * 10^9

def omega(x):
    return flw / x

'''
Dielectric functions for hBN
'''
# eps_per
eps_r1 = 4.90
lambda1 = flw / 0.16939
Lambda_eps1 = 1.93872
Omega1 = flw / lambda1
Gamma1 = 8.67879831e-4
def eps_t(omega):
    return eps_r1 - (Lambda_eps1 * Omega1**2) / (omega**2 - Omega1**2 + 1j * omega * Gamma1)

# eps_par
eps_r2 = 2.95
lambda2 = flw / 0.09423
Lambda_eps2 = 0.52618
Omega2 = flw / lambda2
Gamma2 = 2.47966e-4
def eps_z(omega):
    return eps_r2 - (Lambda_eps2 * Omega2**2) / (omega**2 - Omega2**2 + 1j * omega * Gamma2)

# Dielectric constant for superstrate
eps_a=1.0

# Dielectric constant for substrate
def eps_s(omega):
    return 1.88422  # CaF2 (fixed value)  # CaF2

Wavelength range

# Wavelength range (nm)
lambda_o = 6200.0
lambda_f = 7400.0
N_lambda = 6000
lambda_range=np.linspace(lambda_o,lambda_f,N_lambda)

Example of van der Waals polariton dispersion relation

d = 256.0  # Thicknesses (nm)
l = 0      # Mode index

plt.plot(1e3*np.real(qx(lambda_range)),1e7/lambda_range,label=r'$Re(q_x)$')
plt.plot(1e3*np.imag(qx(lambda_range)),1e7/lambda_range,label=r'$Im(q_x)$')
plt.xlabel(r'$q (10^5 cm^{-1})$')
plt.ylabel(r'$\omega (cm^{-1})$')
plt.xlim(0,50)
plt.ylim(1350.0,1600.0)
plt.legend()
plt.show()

plt.plot(lambda_range,np.real(qz(lambda_range)),label=r'$Re(q_z)$')
plt.plot(lambda_range,np.imag(qz(lambda_range)),label=r'$Im(q_z)$')
plt.xlabel(r'wavelength (nm)')
plt.legend()
plt.show()