Non-Equilibrium Non-Crossing Approximation with VIBrations
This code uses the equations by Ned S. Wingreen and Yigal Meir Phys. Rev. B 49, 11040 – Published 15 April 1994 to account for the non-equilibrium Kondo effect between two electrodes. The impurity is in the present version treated as a single orbital and its total spin si fixed to 1/2. The code is very efficient and converges with great accuracy by using convolutions coded in FFT (needs fftw3). The code uses the self-consistent Born Approximation to deal with a local vibration of the impurity. As is done, we can add as many vibrations as needed. This follwos the paper by P. Roura-Bas, L. Tosi, and A. A. Aligia Phys. Rev. B 93, 115139 – Published 24 March 2016. We have run the code using the Kondo efect with various e-vib couplind and it converges excedeengly well even in the strong coupling regime. Moreover the code is totally able to recover the vibronic regime in the presence of a bias drop, which is very interesting.
The input is in the file WMnca.input. And example is:
-5.0 !Omega_ini (eV) (the bandwidth extends -2*Omega_ini)
0.00001 !step_omega (eV)
0.0001 !Broadening eta (eV)
- !B_field (Teslas)
2.0 !gyromagnetic factor
5.0 !Temperature (K)
0.01 !Convergence tolerance
20 !Maximum number of loops
-0.5 !Bias ini (Volts)
0.5 !Bias fin (Volts)
1 !Number of bias
1.0 !Fraction of bias drop at the tip (1-Tip drops at the sample)
Hamiltonian.dat !name of the file containing the level energy (rpt Fermi)
Gamma.dat !file with Go, Omega_0, Delta_0 for tip and substrate
Current.dat !file with V and Current OUTPUT
9 !N_vib number of modes
8.3 ! Freq meV
8.8 ! Freq meV
20.26 ! Freq meV
20.48 ! Freq meV
26.72 ! Freq meV
31.47 ! Freq meV
35.82 ! Freq meV
42.1 ! Freq meV
43.66 ! Freq meV
1.0 ! e-ph coupling meV
1.0 ! e-ph coupling meV
2.0 ! e-ph coupling meV
2.0 ! e-ph coupling meV
1.0 ! e-ph coupling meV
1.0 ! e-ph coupling meV
5.0 ! e-ph coupling meV
10.0 ! e-ph coupling meV
10.0 ! e-ph coupling meV
.true. ! convergence loops on phonon?
the code uses two more files:
The Gamma.dat file that reads as:
G ! two options G (Gaussian= or L (Lorentzian) for the Hybridisation function 0.0 0 1.0 ! G0, E0, D0 for left electrode (eV) 0.1 0 2.0 ! same for right electrode 8eV)
where if it is a gaussian the hybridization function looks like G0*exp (-(omega-E0)^2/D0^2), a Lorentzian like G0/(omega-E0)^2+D0^2) so that D0 is like an effective bandwidth and G0 is the value of the hybridisation (times 2 pi would be the width of the level) at the energy E0.
The other file is the Hamiltonian that here it is just a level referred to the Fermi energy. Hamiltonian.dat ::
-0.65
in eV.
ENJOY