Role of the k-point in Transmission Calculation

[zhangguangping]

Dear all:
 
I now have a question related to the k-point in calculation.
 
As we know, if we use periodic boundary we should use k sampling in the brillouin zone of the xy direction. I learnt from the Solid State Physics, all the qauntities in the crystal is the function of k-points. So we should average over the k points for a certain qauntity. The more k points, the more accurate and the more time calculation costs.
 
As is pointed out that the Hamiltonian (density matrix) converge fast along the k points, however the transmission function much slower. So we should increase the k points in the transmission calculation.
 
Now I have a qestion that, as attached, the transmission function has a sawtooth-like above the Fermi energy  for a poor 4*4 k point sampling as seen in FIG1. When we increase the k point to 20*20, we see it becomes smoother. However, if we zoom it, as seen in FIG2, the sawtooth still remains there.
 
Q1: I wonder what these sawtooth mean?
 
Q2: What happens when we increase the k points? That is how k points affect the transmission? In more detail, I know for a special energy E, the transmission coefficient is averaged over all the sampled k points. If we say the sample k points are (k_x, k_y), what are the kx and ky related to E. Do there has the ralation E=(k_x*k_x+k_y*k_y)/2m_e?
 
 
With best regards.

[zh]

The dependence of transmission spectra calculation on the k point sampling has been studied in the paper:
M. Stilling , K. Stokbro & K. Flensberg (2007): Crystalline Magnetotunnel Junctions: Fe-mgo-fe, Fe-
feomgo-fe And Fe-aumgoau-fe, Molecular Simulation, 33:7, 557-561
http://www.tandfonline.com/doi/abs/10.1080/08927020600930508#.UbZ0dJzRL0d

[zhangguangping]

The dependence of transmission spectra calculation on the k point sampling has been studied in the paper:
M. Stilling , K. Stokbro & K. Flensberg (2007): Crystalline Magnetotunnel Junctions: Fe-mgo-fe, Fe-
feomgo-fe And Fe-aumgoau-fe, Molecular Simulation, 33:7, 557-561
http://www.tandfonline.com/doi/abs/10.1080/08927020600930508#.UbZ0dJzRL0d

Dear zh,

Thanks for your suggestion. I have read that paper. But I still puzzled why the electron with different k has different transmission coefficient. Is it related to the velocity for z component? Is it satisfy E=(k_x*K_x+k_y*k_y+k_z*k_z)/2m?
if E=(k_x*K_x+k_y*k_y)/2m then the tranmission will be zero ?

Can you do me a favor to give some explanation?

[Anders Blom]

k_z is not a good quantum number since we don't have periodic boundary conditions in the Z direction. And besides, your expressions only hold in the free electron gas approximation. The correct picture is quite different.

The conserved quantum numbers for elastic, coherent tunneling are k_x, k_y and the energy e (I use lower case here to separate it from E later on). The transmission coefficients are computed for each energy, k_x, k_y separately. Given a particular value of these, the inverse problem can be solved, to determine the possible values k_z such that

E(k_x, k_y, k_z)=e

where E is the "band structure function" for the electrodes (actually, the complex band structure, which can be computed by ATK and plotted). Some of the values k_z will be real (propagating), some will be complex (decaying modes). For each mode (both real and imaginary) the probability that it will be transmitted to the other electrode is computed, taking also into account whether there are any matching solutions in that electrode - remember the electrodes could made from different materials.

In reality this calculation is not carried out this way, the whole thing is done with Green's functions and self energies, but the result is the same.

The sum of the probabilities is the total transmission coefficient T(e,k_x,k_y), which can be >1 since it's a sum of possibly many probabilities, each one of which is <=1.

Finally, the sum (integral) over the Brillouin zone gives the total transmission

T(e) = int T(e,k_x,k_y) dk_x dk_y

which thus also can be >1.

Both the transmission coefficients T(e,k_x,k_y) and the probabilities for each mode (the so-called transmission eigenvalues) can be extracted from a calculation in ATK.

[zhangguangping]

k_z is not a good quantum number since we don't have periodic boundary conditions in the Z direction. And besides, your expressions only hold in the free electron gas approximation. The correct picture is quite different.

The conserved quantum numbers for elastic, coherent tunneling are k_x, k_y and the energy e (I use lower case here to separate it from E later on). The transmission coefficients are computed for each energy, k_x, k_y separately. Given a particular value of these, the inverse problem can be solved, to determine the possible values k_z such that

E(k_x, k_y, k_z)=e

where E is the "band structure function" for the electrodes (actually, the complex band structure, which can be computed by ATK and plotted). Some of the values k_z will be real (propagating), some will be complex (decaying modes). For each mode (both real and imaginary) the probability that it will be transmitted to the other electrode is computed, taking also into account whether there are any matching solutions in that electrode - remember the electrodes could made from different materials.

In reality this calculation is not carried out this way, the whole thing is done with Green's functions and self energies, but the result is the same.

The sum of the probabilities is the total transmission coefficient T(e,k_x,k_y), which can be >1 since it's a sum of possibly many probabilities, each one of which is <=1.

Finally, the sum (integral) over the Brillouin zone gives the total transmission

T(e) = int T(e,k_x,k_y) dk_x dk_y

which thus also can be >1.

Both the transmission coefficients T(e,k_x,k_y) and the probabilities for each mode (the so-called transmission eigenvalues) can be extracted from a calculation in ATK.

Thanks for your answering.