In [1]:
%pylab inline
import matplotlib.pyplot as plt 

#from IPython.core.display import HTML

Populating the interactive namespace from numpy and matplotlib

Cyclotron resonance

Electrons can absorb energy from the electric field when the mean time $\tau$ between collisions is much larger than the period of the field, ie, $\omega \tau \gg 1$.

The complex conductivity: $\sigma_{11} = \sigma_0\, \frac{1-i \omega \tau}{1+(\omega_c^2 - \omega^2)\tau^2-2 i \omega \tau }$,

where $\sigma_0 =n e^2 \tau/m_1^*$ is the conductivity of electrons of effective mass $m_1^*$ in the Drude model.

For details see Eq. (21.2.48) in Jenő Sólyom: Fundamentals of the Physics of Solids, Volume 2: Electronic Properties (A modern szilárdtest-fizika alpjai II. Fémek, félvezetők, szupravezetők).

Absorption depending on the magnetic field for various values of $\omega \tau$

Compare the result below with Fig. 21.11 in Sólyom's book.

In [2]:
# Az abra kimentesehez az alabbiakat a plt.show()  ele kell tenni!!! 

#savefig('fig_rainbow_p2_1ray.pdf');  # Abra kimentese
#savefig('fig_rainbow_p2_1ray.eps');  # Abra kimentese

# Abra es fontmeretek
xfig_meret= 9   #    12 nagy abrahoz
yfig_meret= 6    #   12 nagy abrahoz
xyticks_meret= 15  #  20 nagy abrahoz
xylabel_meret= 30  #  30 nagy abrahoz
legend_meret= 17   #  30 nagy abrahoz
In [3]:
# The dependence of absorption on the strength of the magnetic field for various values of ωτ

def sigma_11(om,omc):
    sigma11= (1-1j*om)/(1+omc**2-om**2-2*1j*om)  # the complex conductivity
    # real and imaginary part of the conductivity
    return (sigma11.real,sigma11.imag)
In [4]:
Npoints=100;
omc_max=3;
xx = linspace(0,omc_max,Npoints)
omtau=[0.5,1,3,4,50]
szin=['k','brown','g','b','r']

figsize(xfig_meret,yfig_meret)


[plot(xx,sigma_11(omtau[j],omtau[j]*xx)[0],label=r'$\omega \tau=$'+str(omtau[j]),color=szin[j]) \
 for j in range(len(omtau))];

legend(loc='upper right',fontsize=legend_meret)
#legend(loc='best')
#legend()

title("Absorption vs magnetic field",fontsize=20)

xticks(fontsize=xyticks_meret)
yticks(fontsize=xyticks_meret)

xlabel(r'$\omega_c/\omega \sim B$',fontsize=xylabel_meret)
ylabel(r'$\mathrm{Re}\left(\frac{\sigma_{11}}{\sigma_0}\right)$',fontsize=xylabel_meret)

#xlim(0,en_max*1.)
ylim(0,1.)

grid();

#savefig('Fig_mu_B.eps');  # Abra kimentese
In [5]:
Npoints=100;
omc_max=3;
xx = linspace(0,omc_max,Npoints)
omtau=[0.5,1,3,4,50]
szin=['k','brown','g','b','r']

figsize(xfig_meret,yfig_meret)


[plot(xx,sigma_11(omtau[j],omtau[j]*xx)[1],label=r'$\omega \tau=$'+str(omtau[j]),color=szin[j]) \
 for j in range(len(omtau))];

legend(loc='upper right',fontsize=legend_meret)
#legend(loc='best')
#legend()

title("Imaginary part of the conductivity",fontsize=20)

xticks(fontsize=xyticks_meret)
yticks(fontsize=xyticks_meret)

xlabel(r'$\omega_c/\omega \sim B$',fontsize=xylabel_meret)
ylabel(r'$\mathrm{Im}\left(\frac{\sigma_{11}}{\sigma_0}\right)$',fontsize=xylabel_meret)

#xlim(0,en_max*1.)
#ylim(0,.6)

grid();

#savefig('Fig_mu_B.eps');  # Abra kimentese
In [ ]: