Dear Adel, Thank You very much for your answer. It seems the problem is in my deriving Landauer-Buttiker formula. I should find out why the approach stated in the jpg file doesn't work. I need to derive this formula by myself because my next step is to chek the chain with second- and third-neighbour hoppings. I want to be certain that I set the topology of hoppings in Kwant correctly.
Thank You once again, suppose I'll be back with new questions in some time) Sincerely, Jambulat 2017-06-01 1:14 GMT+03:00 Abbout Adel <[email protected]>: > Dear Jambulat, > > Your programs are well written but with no comments it is heavy to follow > them. > > For the system you propose, the analytical result is straightforward using > the Landauer -Buttiker formula: > > After simplification, we get: > T= 4*Gamma1*Gamma2/(Gamma1+Gamma2)**2 > > with Gamma the broadening. > You can check this as follows > > import kwant > from matplotlib import pyplot > from numpy import * > > t1,t2=1,1.5 > > lat = kwant.lattice.square(a) > sys = kwant.Builder() > sys[lat(0,0) ] = 0 ###system one site x==0 > > #define and attach the leads > lead1 = kwant.Builder(kwant.TranslationalSymmetry((-a, 0))) > lead1[lat(0, 0) ] = 0 > lead1[lat.neighbors()] = -t1 > > lead2 = kwant.Builder(kwant.TranslationalSymmetry((+a, 0))) > lead2[lat(1, 0) ] = 0 > lead2[lat.neighbors()] = -t2 > > sys.attach_lead(lead1) > sys.attach_lead(lead2) > > #finalize system > syst=sys.finalized() > energies=linspace(-1.999,1.999,50) > > #transmission calculated by Kwant > transmission=[] > for E in energies: > smatrix=kwant.smatrix(syst,E) > transmission.append(smatrix.transmission(0,1)) > > #the linewidth Gamma (Landauer Buttiker formula) > def Gamma(E,t): > return sqrt(4*t**2-E**2) > > #analytical result of the conductance using Landauer Buttiker formula > def T(E,t1,t2): > return 4*Gamma(E,t1)*Gamma(E,t2)/abs(Gamma(E,t1)+Gamma(E,t2))**2 > > > T=[T(E,1.,1.5) for E in energies] > > pyplot.plot(energies,T) #kwant > pyplot.plot(energies,transmission,'ro') #analytical result > pyplot.ylabel('Transmission') > pyplot.xlabel('Energy') > pyplot.show() > > > Hope this helps. > Adel > > > > > On Wed, May 31, 2017 at 7:34 PM, Jambulat Basaev <[email protected]> > wrote: > >> Dear Kwant Community, >> >> In my research work I've come to the simple test problem which can be >> solved both using Kwant and analytically. It's about scattering of the >> electron wave in the chain of atoms in which hopping integral is t1 for x < >> 0 and t2 for x > 0. Almost all the solution is in the picture in the >> attachments. Realization of this analytical solution is presented in >> testChain.py which plots the dependence on scattering energy of the >> transmission coefficient >> |tau|**2 * v2/v1, vj = -2*tj*sin(phi_j) - velocity of the electron. >> The testChain-kwant.py also plots transmission coefficient versus energy >> in the same system, but the results are not coincide. >> >> So, my question, is it possible to reach the accordance between the >> solutions in both cases? >> >> P.S. Analytical solution seems to be correct, at least >> |rho|**2 + |tau|**2 * v2/v1 is 1 identically. >> >> P.P.S. Is it possible to draw the different hopping kinds with different >> colors and line styles? It can be useful for controlling errors when >> building the geometries with hoppings between second-neighbors... >> >> Sincerely, >> Jambulat >> > > > > -- > Abbout Adel >
