Discrepancy in effective mass calculation in bulk silicon

[Thomas Joseph]

Hi,

Why does the effective mass calculated using MGGA show a large variation with respect to GGA for the hole effective mass? I was under the impression that MGGA will not change the band curvatures in comparison to LDA and GGA functionals. Below is the extracted effective mass after DFT calculation using respective functionals.

Effective mass of bulk silicon(MGGA):
Electrons(longitudnal): 0.917    Heavy Hole: 0.327, 0.240  Light Hole: 0.270

Effective mass of bulk silicon(GGA):
Electrons(longitudnal): 0.911    Heavy Hole: 0.259, 0.257 Light Hole: 0.159

Please find attached the script that was used to generate the above said results. Please let me know if there are any mistakes in my script that raises this discrepancy.

Thanks in advance,
Thomas

[Jess Wellendorff]

I see no obvious mistakes in your script. However, there is in general no guarantee that MGGA will simply push the GGA bands up or down in energy without any changes to the band curvature. MGGA essentially modifies the exchange-correlation potential in such a way that semiconductor band gaps get larger, but effective masses may also be affected by this, though the effect on effective masses is usually fairly small.

[Thomas Joseph]

Hi Jess Wellendorff,

I did understand your point. But for the MGGA scenario why is one of the heavy hole mass (0.240) lighter than the light hole mass (0.270). This doesn't confirm with the bandstructure plot. At first glance from MGGA bandstructure, band curvatures for both heavy hole bands shows similar curvatures while the light hole band has a larger one. But the effective mass calculation doesn't conform with this.

Please find attached the bandstructure plot for bulk Silicon calculated using MGGA.

Thank you for your time and consideration,
Thomas

[zh]

Who not put the band structures of MGGA and GGA together into one figure? It would give the difference of two band structures clearly.

[Thomas Joseph]

Hi zh,

What I am confused about is not the difference in the bandstructure plots or the difference in curvature between MGGA and GGA. I do agree there will be some difference. What I don't understand is why the effective mass calculated from MGGA doesn't conform with the same bandstructure plot. From the above bulk silicon bandstructure plot there are three degenerate valence band. Two heavy hole bands which should have a mass higher than the light hole band. One can make this clever guess from the bandstructure plot but the mass calculated says otherwise (Heavy Hole: 0.327, 0.240  Light Hole: 0.270). This is what confuses me.

Thanks in advance,
Thomas

[Petr Khomyakov]

If you zoom into the MGGA calculated band structure of Si around the top of the valence band, you will actually see that the heavy and light hole bands are split, and the two-fold degeneracy of heavy hole bands is lifted. This is certainly an artifact of the MGGA calculation.

This splitting can be reduced by increasing the mesh cut-off value in the ATK-DFT Calculator, making the real-space mesh denser compared to its default density, e.g., using 200 Hartree instead of 75 Hartree, which is the default value. However, the effective masses along G->X may still be inaccurate, and we are investigating this issue.  Note that calculating the second derivative (related to effective mass) of the band energy with respect to the electron quasimomentum, k, requires very accurate numerical calculation of the band energy, and that seems to be an issue within the framework of the MGGA approach.