In [1]:
%pylab inline
import matplotlib.pyplot as plt
from IPython.core.display import HTML
Electrons in strong magnetic field, Landau levels and related quantities¶
For details see Chapter 22. in Jenő Sólyom: Fundamentals of the Physics of Solids, Volume 2: Electronic Properties (A modern szilárdtes-tfizika alpajai II. Fémek, félvezetők, szupravezetők).
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= 21 # 30 nagy abrahoz
In [3]:
def nmax_func(x):
nn=floor(x)
if(x-nn <= 1/2):
nmax=nn-1
else:
nmax=nn
#print ("nmax+1/2 < x < nmax+3/2", nmax+1/2, x, nmax+3/2)
return (nmax)
In [4]:
def mu_func(x):
nmax = nmax_func(x)
nlist=arange(nmax+1)
tmp=sum(sqrt(x-(i+1/2)) for i in nlist)
#print("sum=",res)
#print(nmax,nn,xx)
res=(3/2*tmp)**(2/3)
return res
In [5]:
def dos(x):
nmax = nmax_func(x)
nlist=arange(nmax+1)
res=0.5*sum(1/sqrt(x-(i+1/2)) for i in nlist)
return res
In [6]:
def dos_Fermi(x):
# x ---> hbar omega_c/E_F
nmax = floor(1/x-1/2)
nlist=arange(nmax+1)
#print("nmax= ",nmax)
res=0.5*sqrt(x)*sum(1/sqrt(1/x-(i+1/2)) for i in nlist)
return res
In [7]:
# energy in 2D
def en_2D(B):
n=floor(1/B)
res=2*B*(n+1/2-1/2*n*(n+1)*B)
return res
In [8]:
# M in 2D
def M_2D(B):
n=floor(1/B)
res=2*B*(n+1/2-1/2*n*(n+1)*B)
return res
In [9]:
# energy in 3D
def En_Landau_dia(x):
# energy in units of N_e * E_F
# x ---> hbar omega_c/E_F
# x <= 2
nmax = floor(1/x-1/2)
nlist=arange(nmax+1)
#print("nmax= ",nmax)
res=1-x/2*sum((1-(i+1/2)*x)**(3/2) for i in nlist)
return res
In [10]:
# M in 3D
def M_Landau_dia(x):
# x ---> hbar omega_c/E_F
nmax = floor(1/x-1/2)
nlist=arange(nmax+1)
#print("nmax= ",nmax)
res=1/2*sum((1-(i+1/2)*x)**(3/2) for i in nlist)
return res
In [11]:
# chi in 3D, susceptibility
def chi_Landau_dia(x):
# x ---> hbar omega_c/E_F
nmax = floor(1/x-1/2)
nlist=arange(nmax+1)
#print("nmax= ",nmax)
res=-3/2*sum((i+1/2)*(1-(i+1/2)*x)**(1/2) for i in nlist)
return res
In [12]:
def M_osc(B,T,params):
# Solyom konyv II. (22.4.86)
# B---> hbar omega_c/E_F
# T---> k_B T/hbar omega_c
# mcs ---> m^*/m)e
# ge ---> g_e
ge,mcs,lmax = params
llist=arange(1,lmax+1)
sum = 0;
for l in llist:
x=1/sqrt(l)*(-1)**(l+1)*cos(pi*l*ge/2*mcs)*sin(pi/4-2*pi*l/B)/(sinh(2*pi*pi*l*T))**2
sum = sum + x
res=(ge/2)**2-1/3*(1/mcs)**2 + 2*T*sqrt(2/B)*sum
return res
Density of states (DOS) $\varrho(E)$ in 3D¶
$\frac{\varrho(E)}{\varrho_0(E)} = \frac{1}{2}\, \sqrt{\frac{\hbar \omega_c}{E}}¶
\sum{n=0}^{n{\mathrm{max}}}\, \frac{1}{\sqrt{\frac{E}{\hbar \omega_c}-(n+\frac{1}{2})}}$,
where $\omega_c = \frac{eB}{m}$, $(n_{\mathrm{max}}+1/2)\,\hbar \omega_c \le E$ and $\varrho_0(E)$ is the DOS with zero magnetic field.
In [13]:
Npoints=100;
en_max=7;
enlist = linspace(0,en_max,Npoints)
gyoken = sqrt(enlist)
dd = [];
for i in enlist:
dd.append(dos(i))
plot(enlist,dd,label=r'$\varrho(E)$',color='r')
plot(enlist,gyoken,color='b')
#legend(loc='upper left',fontsize=legend_meret)
title("DOS",fontsize=20)
xticks(fontsize=xyticks_meret)
yticks(fontsize=xyticks_meret)
xlabel(r'$\frac{E}{\hbar \omega_c}$',fontsize=xylabel_meret)
ylabel(r'$\varrho(E)$',fontsize=xylabel_meret)
xlim(0,en_max*1.)
ylim(0,en_max*1.)
grid();
#savefig('Fig_mu_B.eps'); # Abra kimentese
B dependence of the DOS at the Fermi energy¶
$\frac{\varrho(E_\mathrm{F})}{\varrho0(E\mathrm{F})} = \frac{1}{2}\, \sqrt{\frac{\hbar \omegac}{E\mathrm{F}}}¶
\sum{n=0}^{n{\mathrm{max}}}\, \frac{1}{\sqrt{\frac{E_\mathrm{F}}{\hbar \omega_c}-(n+\frac{1}{2})}}$, where $\omega_c = \frac{eB}{m}$ and $\varrho0(E\mathrm{F})$ is the DOS at the Fermi energy with zero magnetic field.
In [14]:
Npoints=200;
B_max=1;
Blist = linspace(0.1,B_max,Npoints)
dd = [];
for i in Blist:
dd.append(dos_Fermi(i))
plot(Blist,dd,label=r'$\varrho(E_\mathrm{F})$',color='r')
#plot(enlist,gyoken,color='b')
#legend(loc='upper left',fontsize=legend_meret)
title("DOS at the Fermi energy",fontsize=20)
xticks(fontsize=xyticks_meret)
yticks(fontsize=xyticks_meret)
xlabel(r'$\frac{\hbar \omega_c}{E_\mathrm{F}} \sim B$',fontsize=xylabel_meret)
ylabel(r'$\varrho(E_\mathrm{F})$',fontsize=xylabel_meret)
xlim(0,B_max*1.)
#ylim(0,1.5)
grid();
#savefig('Fig_mu_B.eps'); # Abra kimentese
1/B dependence of the DOS at the Fermi energy¶
equidistant peaks
In [15]:
Npoints=300;
B_max=5;
Blist = linspace(0.01,B_max,Npoints)
dd = [];
for i in Blist:
dd.append(dos_Fermi(1/i))
plot(Blist,dd,label=r'$\varrho(E_\mathrm{F})$',color='r')
#plot(enlist,gyoken,color='b')
#legend(loc='upper left',fontsize=legend_meret)
title("DOS at the Fermi energy",fontsize=20)
xticks(fontsize=xyticks_meret)
yticks(fontsize=xyticks_meret)
xlabel(r'$\frac{E_\mathrm{F}}{\hbar \omega_c} \sim 1/B$',fontsize=xylabel_meret)
ylabel(r'$\varrho(E_\mathrm{F})$',fontsize=xylabel_meret)
xlim(0,B_max*1.)
ylim(0,5)
grid();
#savefig('Fig_mu_B.eps'); # Abra kimentese
B dependence of the chemical potential $\mu(B):$¶
$\eta = \frac{\mu(B)}{\hbar \omega_c}$ and $\eta_0 = \frac{E_\mathrm{F}}{\hbar \omega_c}$¶
$\sum_{n=0}^{n_{\mathrm{max}}}\, \sqrt{\eta-(n+\frac{1}{2})} = \frac{2}{3}\, \eta_0^{3/2}$¶
where $n_{\mathrm{max}} +1/2 < \eta < n_{\mathrm{max}} +3/2$
For each values of $\eta$ we calculate $\eta_0$ from the above equation then $\eta$ is plotted as a function of $\eta_0$.
In [16]:
ss=[1.5,2.5,3.5]
kk=[];
ll=[];
pp=[];
dd=[];
for i in ss:
ll.append(nmax_func(i))
kk.append(mu_func(i))
pp.append(i/mu_func(i))
dd.append(dos(i))
print(ll)
print (kk)
print(pp)
print(dd)
In [17]:
Npoints=100;
eta_max=5;
eta = linspace(0,eta_max,Npoints)
eta_0 = [];
for i in eta:
eta_0.append(mu_func(i))
plot(eta_0,eta,label=r'$\eta = \frac{\mu(B)}{\hbar \omega_c}$',color='r')
plot(eta,eta,color='b')
legend(loc='upper left',fontsize=legend_meret)
title(r'$\mu(B)$',fontsize=xylabel_meret*0.8)
xticks(fontsize=xyticks_meret)
yticks(fontsize=xyticks_meret)
xlabel(r'$\eta_0 = \frac{E_\mathrm{F}}{\hbar \omega_c}$',fontsize=xylabel_meret)
ylabel(r'$\eta$',fontsize=xylabel_meret)
xlim(0,eta_max*1.1)
ylim(0,eta_max*1.1)
grid();
#savefig('Fig_mu_B.eps'); # Abra kimentese
B dependence of the chemical potential $\frac{\mu(B)}{E_\mathrm{F}}$¶
$\frac{\mu(B)}{E_\mathrm{F}}= \frac{\eta}{\eta_0}$
The result shows that $\mu(B)$ is weakly depend on B and practically equals to $E_\mathrm{F}$ for not too large magnetic field.
In [18]:
Npoints=200;
eta_max=5;
eta = linspace(0.01,eta_max,Npoints)
unit=linspace(1,1,Npoints)
etap = [];
eta_0 = [];
for i in eta:
eta_0.append(mu_func(i))
etap.append(i/mu_func(i))
plot(eta_0,etap,label=r'$\frac{\mu(B)}{E_\mathrm{F}}$',color='r')
plot(eta_0,unit,color='b')
legend(loc='upper right',fontsize=legend_meret)
title(r'$\mu(B)$',fontsize=xylabel_meret)
xticks(fontsize=xyticks_meret)
yticks(fontsize=xyticks_meret)
xlabel(r'$\eta_0 = \frac{E_\mathrm{F}}{\hbar \omega_c}$',fontsize=xylabel_meret)
ylabel(r'$\frac{\mu(B)}{E_\mathrm{F}}$',fontsize=xylabel_meret)
xlim(0.,eta_max)
ylim(0,2)
grid();
#savefig('Fig_mu_B.eps'); # Abra kimentese
Energy in 2D as a function of B¶
In [19]:
Npoints=300;
B_max=1.1;
Blist = linspace(0,B_max,Npoints)
dd = [];
for i in Blist:
dd.append(en_2D(i))
plot(Blist,dd,label=r'$\varrho(\epsilon)$',color='r')
#plot(enlist,gyoken,color='b')
#legend(loc='upper left',fontsize=legend_meret)
title(r'$E_0$',fontsize=20)
xticks(fontsize=xyticks_meret)
yticks(fontsize=xyticks_meret)
xlabel(r'$\frac{B}{B_0}$',fontsize=xylabel_meret)
ylabel(r'$E_0$',fontsize=xylabel_meret)
xlim(0,1.1)
#ylim(0,eta_max*1.)
grid();
#savefig('Fig_mu_B.eps'); # Abra kimentese
Energy in 2D as a function 1/B¶
In [20]:
Npoints=400;
B_max=6.27;
Blist = linspace(0.5,B_max,Npoints)
dd = [];
for i in Blist:
dd.append(en_2D(1/i))
plot(Blist,dd,label=r'$\varrho(\epsilon)$',color='r')
#plot(enlist,gyoken,color='b')
#legend(loc='upper left',fontsize=legend_meret)
title(r'$E_0$',fontsize=20)
xticks(fontsize=xyticks_meret)
yticks(fontsize=xyticks_meret)
xlabel(r'$\frac{B_0}{B}\sim 1/B$',fontsize=xylabel_meret)
ylabel(r'$E_0$',fontsize=xylabel_meret)
xlim(0,B_max)
ylim(1,1.15)
grid();
#savefig('Fig_mu_B.eps'); # Abra kimentese
Total energy in 3D as a function of the magnetic field at $T=0$¶
In [21]:
En_Landau_dia(0.27)
Out[21]:
In [22]:
Npoints=200;
B_max=2.; # hbar omega_c/E_F --> B_max <= 2
Blist = linspace(0.01,B_max,Npoints)
dd = [];
for i in Blist:
dd.append(En_Landau_dia(i))
plot(Blist,dd,label=r'$E_0(B)$',color='r')
#plot(enlist,gyoken,color='b')
#legend(loc='upper left',fontsize=legend_meret)
title("Total energy $E_0(B)$ at $T=0$",fontsize=20)
xticks(fontsize=xyticks_meret)
yticks(fontsize=xyticks_meret)
xlabel(r'$\frac{\hbar \omega_c}{E_\mathrm{F}} \sim B$',fontsize=xylabel_meret)
ylabel(r'$E_0(B)$',fontsize=xylabel_meret)
xlim(0,B_max*1.)
#ylim(0,0.5)
grid();
#savefig('Fig_mu_B.eps'); # Abra kimentese
Landau diamagnetization in 3D as a function of $B$ at $T=0$¶
In [23]:
M_Landau_dia(0.27)
Out[23]:
In [24]:
Npoints=200;
B_max=1; # hbar omega_c/E_F --> B_max <= 2
Blist = linspace(0.1,B_max,Npoints)
ddm = [];
for i in Blist:
ddm.append(M_Landau_dia(i))
plot(Blist,ddm,label=r'$M_(B)$',color='b')
#plot(enlist,gyoken,color='b')
#legend(loc='upper left',fontsize=legend_meret)
title("Total magnetization $M(B)$ at $T=0$",fontsize=20)
xticks(fontsize=xyticks_meret)
yticks(fontsize=xyticks_meret)
xlabel(r'$\frac{\hbar \omega_c}{E_\mathrm{F}} \sim B$',fontsize=xylabel_meret)
ylabel(r'$M(B)$',fontsize=xylabel_meret)
xlim(0,B_max*1.)
#\ylim(0,7)
grid();
#savefig('Fig_mu_B.eps'); # Abra kimentese
Landau diamagnetic contribution to the susceptibility $\chi(B)$ at $T=0$ and in 3D¶
In [25]:
Npoints=200;
B_max=1; # hbar omega_c/E_F --> B_max <= 2
Blist = linspace(0.1,B_max,Npoints)
ddm = [];
for i in Blist:
ddm.append(chi_Landau_dia(i))
plot(Blist,ddm,label=r'$M_(B)$',color='b')
#plot(enlist,gyoken,color='b')
#legend(loc='upper left',fontsize=legend_meret)
title("Landau dia, susceptibility $\chi(B)$ at $T=0$",fontsize=20)
xticks(fontsize=xyticks_meret)
yticks(fontsize=xyticks_meret)
xlabel(r'$\frac{\hbar \omega_c}{E_\mathrm{F}} \sim B$',fontsize=xylabel_meret)
ylabel(r'$\chi(B)$',fontsize=xylabel_meret)
xlim(0,B_max*1.)
#\ylim(0,7)
grid();
#savefig('Fig_mu_B.eps'); # Abra kimentese
Magnetization oscillation with B at finite temperature T and in 3D¶
In [26]:
B=0.17;
T=0.027;
ge=2;
mcs=2;
lmax =10;
print("B, T, ge, mcs, lmax = ",B, T, ge, mcs, lmax)
params=[ge, mcs, lmax]
print(params)
M_osc(B,T,params)
Out[26]:
In [27]:
T=0.01;
ge=2;
mcs=1;
lmax =10;
print("T, ge, mcs, lmax = ", T, ge, mcs, lmax)
params=[ge, mcs, lmax]
Npoints=200;
B_max=0.5; # hbar omega_c/E_F --> B_max <= 2
Blist = linspace(0.05,B_max,Npoints)
ddm = [];
for i in Blist:
ddm.append(M_osc(i,T,params))
plot(Blist,ddm,label=r'$M_(B)$',color='b')
#plot(enlist,gyoken,color='b')
#legend(loc='upper left',fontsize=legend_meret)
title("Oscillating magnetization $M(B)$ at $T$",fontsize=20)
xticks(fontsize=xyticks_meret)
yticks(fontsize=xyticks_meret)
xlabel(r'$\frac{\hbar \omega_c}{E_\mathrm{F}} \sim B$',fontsize=xylabel_meret)
ylabel(r'$M(B)$',fontsize=xylabel_meret)
xlim(0,B_max*1.)
#\ylim(0,7)
grid();
#savefig('Fig_mu_B.eps'); # Abra kimentese
In [ ]: