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

from scipy.integrate import * # az integráló rutinok betöltése

import warnings
warnings.filterwarnings('ignore')

Populating the interactive namespace from numpy and matplotlib

The resistivity due to electron-phonon interaction as a function of $T /\Theta_\mathrm{D}$:

$\varrho(T)= K {\left(\frac{T}{\Theta_\mathrm{D}}\right)}^5 \int_0^{\Theta_\mathrm{D}/T}\, \frac{x^5 e^x}{{\left(e^x-1\right)}^2}\, dx$,

where $K$ is a constant.

See Eq. (24.4.16) and Fig. 24.7 in Jenő Sólyom: Fundamentals of the Physics of Solids, Volume 2: Electronic Properties (A modern szilárdtestfizika alpajai II. Fémek, félvezetők, szupravezetők) see here on page 394.

For low/high temperature:

[\varrho(T) \sim \begin{cases} T^5 \text{ if $T\ll \Theta_\mathrm{D}$,} & \ T \text{ if $T\gg \Theta_\mathrm{D}$.} \end{cases} ]

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= 20  #  30 nagy abrahoz
legend_meret= 20   #  30 nagy abrahoz
In [3]:
def J5(t):
    expt= exp(t)
    res = t**5*expt/(expt-1)**2
    return res
In [4]:
Np = 100

Tmax=0.7
pici = 0.001

T = linspace(pici,Tmax,Np)
Tlow = linspace(pici,0.17,Np/2)
Tm = 0.4
Thigh = linspace(Tm,Tmax,Np/2) 

resist=[]
for tt in T:
    resist.append(tt**5*quad(J5,pici,1/tt)[0])


figsize(xfig_meret,yfig_meret)
plot (T, resist,color='r')
plot (Tlow,124.43*Tlow**5, color='g',
      label=r'$\varrho(T) \sim T \quad \mathrm{ if} \quad T\gg \Theta_\mathrm{D}$')
plot (Thigh,1/4*Thigh-1/72/Thigh, color='b',
      label=r'$\varrho(T) \sim T^5 \quad \mathrm{ if} \quad T\ll \Theta_\mathrm{D}$')

xlabel(r'$T/\Theta_{\mathrm{D}}$',fontsize=xylabel_meret)
ylabel(r'$\varrho(T)$',fontsize=xylabel_meret)

legend(loc='upper left',fontsize=legend_meret)

title(r'The resistivity as a function of T',fontsize=20)


xlim(0,Tmax)
#ylim(0,0.1)

grid();

#savefig('Fig_resistivity.png');  # Abra kimentese
In [ ]: