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matthias@quintern.xyz 2025-03-29 10:19:49 +01:00
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commit a4dd8a3d1b
4 changed files with 114 additions and 57 deletions
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105
scripts/cm_optics.py Normal file
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@ -0,0 +1,105 @@
#!/usr/bin env python3
from formulary import *
from scipy.constants import Boltzmann as kB, hbar, electron_volt, elementary_charge, epsilon_0, electron_mass
# OPTICS
def eps_r(omega, omega0, gamma, chi, N):
return 1 + chi + N * elementary_charge**2 /(epsilon_0 * electron_mass) * (1/(omega0**2 - omega**2 - 1j*gamma*omega))
def eps_r_lim0(omega0, gamma, chi, N):
return 1 + chi + N * elementary_charge**2 /(epsilon_0 * electron_mass * omega0**2)
def eps_r_lim_infty(omega0, gamma, chi, N):
return 1 + chi
def dielectric_absorption():
# values taken from adv. sc. physics ex 10/1
fig, axs = plt.subplots(1, 2, figsize=size_formula_normal_default, sharex=True)
omega0 = 100e12 # 100 THz
gamma = 5e12 # 5 THz
omegas = np.linspace(60e12, 140e12)
chi = 5
N = 1e25
eps_complex = eps_r(omegas, omega0, gamma, chi, N)
eps_real = eps_complex.copy().real
eps_imag = eps_complex.copy().imag
eps_real_0 = eps_r_lim0(omega0, gamma, chi, N)
eps_real_infty = eps_r_lim_infty(omega0, gamma, chi, N)
omegas_THz = omegas / 1e12
anno_color = COLORSCHEME["fg2"]
axs[0].set_ylabel(r"$\epsreal(\omega)$")
axs[0].hlines([eps_real_0], omegas_THz[0], omegas_THz[omegas_THz.shape[0]//3], color=anno_color, linestyle="dotted")
axs[0].hlines([eps_real_infty], omegas_THz[omegas_THz.shape[0]*2//3], omegas_THz[-1], color=anno_color, linestyle="dotted")
axs[0].text(omegas_THz[-1], eps_real_infty, r"$\epsilon_\txr(\infty)$", ha="right", va="bottom", color=anno_color)
axs[0].text(omegas_THz[0], eps_real_0, r"$\epsilon_\txr(0)$", ha="left", va="top", color=anno_color)
axs[1].set_ylabel(r"$\epscomplex(\omega)$")
vals = [eps_real, eps_imag]
for i in range(2):
ax = axs[i]
val = vals[i]
ax.set_xlabel(r"$\omega\,[\si{\tera\hertz}]$")
ax.vlines([omega0/1e12], 0, 100, color=anno_color, linestyle="dashed")
ax.plot(omegas_THz, val)
vmax = val.max()
vmin = val.min()
margin = (vmax - vmin) * 0.05
ax.set_ylim(vmin - margin, vmax + margin)
return fig
# Free e-
@np.vectorize
def fn(omega: float, omega_p:float, eps_infty:float, tau:float):
eps_real = eps_infty * (1-(omega_p**2 * tau**2)/(1+omega**2 * tau**2))
eps_imag = eps_infty * omega_p**2 * tau/(omega*(1+omega**2 * tau**2))
eps_complex = (eps_real + 1j*eps_imag)
n_complex: complex = np.sqrt(eps_complex)
# n_real = n_complex.real
# n_imag = n_complex.imag
# return n_real, n_imag
return n_complex
def free_electrons_absorption():
# values taken from adv. sc. physics ex 10/1
fig, axs = plt.subplots(2, 2, figsize=size_formula_fill_default, sharex=True)
omega_p = 30e12 # 30 THz
eps_infty = 11.7
tau = 1e-6
omegas = omega_p * np.logspace(-1, 3, 1000, base=10, dtype=float)
# omegas = omega_p * np.linspace(1e-1, 1e3, 1000)
# print(omegas)
n_complex = fn(omegas, omega_p, eps_infty, tau)
n_real = n_complex.copy().real
n_imag = n_complex.copy().imag
R = np.abs((n_complex-1)/(n_complex+1))
alpha = 2*omegas*n_imag/3e8
for ax in axs[1,:]:
ax.set_xlabel(r"$\omega/\omega_\text{p}$")
ax.set_xscale("log")
ax.set_xticks([1e-1,1e0,1e1,1e2,1e3])
ax.set_xticklabels([0.1, 1, 10, 100, 1000])
omegas_rel = omegas/omega_p
axs[0,0].plot(omegas_rel, n_real)
axs[0,0].set_yticks([0,1,2,np.sqrt(eps_infty)])
axs[0,0].set_yticklabels([0,1,2,r"$\epsilon_\infty$"])
axs[0,0].set_ylim(-0.2,0.2+np.sqrt(eps_infty))
axs[0,0].set_ylabel(r"$n^\prime(\omega)$")
axs[1,0].plot(omegas_rel, n_imag)
axs[1,0].set_ylabel(r"$n^{\prime\prime}(\omega)$")
axs[1,0].set_yscale("log")
axs[0,1].plot(omegas_rel, R)
axs[0,1].set_ylabel(r"$R(\omega)$")
axs[1,1].plot(omegas_rel, alpha)
axs[1,1].set_ylabel(r"$\alpha(\omega)$")
axs[1,1].set_yscale("log")
return fig
if __name__ == '__main__':
export(free_electrons_absorption(), "cm_optics_absorption_free_electrons")
export(dielectric_absorption(), "cm_optics_absorption_dielectric")

52
scripts/cm_semiconductors.py
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@ -149,57 +149,6 @@ def test():
# OPTICS
@np.vectorize
def fn(omega: float, omega_p:float, eps_infty:float, tau:float):
eps_real = eps_infty * (1-(omega_p**2 * tau**2)/(1+omega**2 * tau**2))
eps_imag = eps_infty * omega_p**2 * tau/(omega*(1+omega**2 * tau**2))
eps_complex = (eps_real + 1j*eps_imag)
n_complex: complex = np.sqrt(eps_complex)
# n_real = n_complex.real
# n_imag = n_complex.imag
# return n_real, n_imag
return n_complex
def free_electrons_absorption():
# values taken from adv. sc. physics ex 10/1
fig, axs = plt.subplots(2, 2, figsize=size_formula_fill_default, sharex=True)
omega_p = 30e12 # 30 THz
eps_infty = 11.7
tau = 1e-6
omegas = omega_p * np.logspace(-1, 3, 1000, base=10, dtype=float)
# omegas = omega_p * np.linspace(1e-1, 1e3, 1000)
# print(omegas)
n_complex = fn(omegas, omega_p, eps_infty, tau)
n_real = n_complex.copy().real
n_imag = n_complex.copy().imag
R = np.abs((n_complex-1)/(n_complex+1))
alpha = 2*omegas*n_imag/3e8
for ax in axs[1,:]:
ax.set_xlabel(r"$\omega/\omega_\text{p}$")
ax.set_xscale("log")
ax.set_xticks([1e-1,1e0,1e1,1e2,1e3])
ax.set_xticklabels([0.1, 1, 10, 100, 1000])
omegas_rel = omegas/omega_p
axs[0,0].plot(omegas_rel, n_real)
axs[0,0].set_yticks([0,1,2,np.sqrt(eps_infty)])
axs[0,0].set_yticklabels([0,1,2,r"$\epsilon_\infty$"])
axs[0,0].set_ylim(-0.2,0.2+np.sqrt(eps_infty))
axs[0,0].set_ylabel(r"$n^\prime(\omega)$")
axs[1,0].plot(omegas_rel, n_imag)
axs[1,0].set_ylabel(r"$n^{\prime\prime}(\omega)$")
axs[1,0].set_yscale("log")
axs[0,1].plot(omegas_rel, R)
axs[0,1].set_ylabel(r"$R(\omega)$")
axs[1,1].plot(omegas_rel, alpha)
axs[1,1].set_ylabel(r"$\alpha(\omega)$")
axs[1,1].set_yscale("log")
return fig
@ -207,5 +156,4 @@ if __name__ == '__main__':
# export(test(), "cm_sc_charge_carrier_density")
export(schottky_barrier(), "cm_sc_devices_metal-n-sc_schottky")
# export(charge_carrier_density(), "cm_sc_charge_carrier_density")
# export(free_electrons_absorption(), "cm_sc_optics_free_electrons")

6
scripts/formulary.py
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@ -9,7 +9,11 @@ import matplotlib as mpl
mpl.rcParams["font.family"] = "serif"
mpl.rcParams["mathtext.fontset"] = "stix"
mpl.rcParams["text.usetex"] = True
mpl.rcParams['text.latex.preamble'] = r'\usepackage{amsmath}\usepackage{siunitx}'
mpl.rcParams['text.latex.preamble'] = f'''
\\usepackage{{amsmath}}
\\usepackage{{siunitx}}
\\input{{{os.path.abspath("../src/util/math-macros.tex")}}}
'''
if __name__ == "__main__": # make relative imports work as described here: https://peps.python.org/pep-0366/#proposed-change

8
src/cm/optics.tex
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@ -20,7 +20,7 @@
\complex{\epsilon}_\txr(0) &\to 1+\chi + \frac{Ne^2}{\epsilon_0 m_\txe \omega_0^2} \\
\complex{\epsilon}_\txr(\infty) &= \epsilon_\infty = 1+\chi
}
\TODO{plot}
\fig{img/cm_optics_absorption_dielectric.pdf}
\end{formula}
\begin{formula}{clausius-mosotti}
@ -59,7 +59,7 @@
\begin{formula}{absorption}
\desc{Free charge carrier absorption}{Exemplary values}{$\omega_\txp$ \fRef{::plasma_frequency}, \QtyRef{refraction_index_real}, \QtyRef{refraction_index_complex}, \QtyRef{absorption_coefficient}, $R$ \fRef{ed:optics:reflectivity}}
\desc[german]{Freie Ladungsträger}{Beispielwerte}{}
\fig{img/cm_sc_optics_free_electrons.pdf}
\fig{img/cm_optics_absorption_free_electrons.pdf}
\TODO{Include equations? aus adv sc ex 10/1c}
\end{formula}
@ -87,8 +87,8 @@
\eq{W_{1\to2} = \frac{2\pi}{\hbar} \abs{M_{12}}^2 \delta(E_1-E_2+\hbar\omega)}
\end{formula}
\begin{formula}{emission}
\desc{Absorption}{}{}
\desc[german]{Absorption}{}{}
\desc{Emission}{}{}
\desc[german]{Emission}{}{}
\eq{W_{2\to1} = \frac{2\pi}{\hbar} \abs{M_{12}}^2 \delta(E_1-E_2-\hbar\omega)}
\end{formula}
\TODO{stimulated vs spontaneous}