SnGe with mGGA

[conor.odonnell]

Hi,

So I'm trying to use meta-GGA to describe the band structure of tin germanium.

My first step was to look at the individual elements.  I achieved a realistic band structure using the HGH solution suggested by Umberto in this thread: http://quantumwise.com/forum/index.php?topic=3202.0
Band gaps:
Γ-L: 0.630 eV
Γ-Γ: 0.846 eV

For tin, using these same parameters (c value and basis sets) I achieved a semimetallic tin structure.

I then created a 16 atom cell of 15 germanium atoms and 1 tin atom. (again using the same c value and basis sets), but the obtained band structure showed semimetallic behavior, instead of showing the band gap which is expected at this low (6.25%) concentration of tin.

Is this a limitation of meta GGA, or am I missing something?

All help would be much appreciated.

Regards,
Conor

[zh]

What you calculated Ge15Sn1 is the tin-doped Ge, the semimetallic feature may be reasonable because the impurity state of Sn dopant  appear in the band gap of Ge.

[conor.odonnell]

In the literature, SnGe isn't supposed to become semimetallic until a much higher (20-25%) concentration of tin. For values of 6.5 %, the band gap is not even expected to transition from indirect to direct.

[Anders Blom]

Did you optimize the structure for positions and lattice constants?

[conor.odonnell]

Yes, I did.

[Jess Wellendorff]

It would be easier to advise you if you attach the script(s) you have used (perhaps even log files).

Not knowing the details of your calculations, I tried the simplest possible:  16-atom Ge unit cell, substituted one Ge for a Sn atom, no relaxation, HGH and MGGA with self-consistently determined c-parameter, calculated the band structure. See attached script and PNG. I do not find semi-metallic behavior. Don't know what the effect of geometry optimization would be.

[conor.odonnell]

I've attached the input script for the calculation, as well as the standard output file.

Relaxation aside, I'm not sure it's wise to allow the c parameter to be determined self consistently,  as when this is done with germanium an inaccurate representation of the band structure is obtained.
There's more info on that in this forum post: http://quantumwise.com/forum/index.php?topic=3202.0

Thanks for the attention so far!

[Jess Wellendorff]

The germanium band structure is a tough one, even for pure Ge After performing a few different test calculations, I suggest you try out the newly implemented PseudopotentialProjectorShift (PPS) method along with SG15 pseudopotentials, see http://docs.quantumwise.com/manuals/Introduction.html#atk-2016 and http://docs.quantumwise.com/manuals/Types/PseudoPotentialProjectorShift/PseudoPotentialProjectorShift.html#NL.Calculators.DensityFunctionalTheory.LCAOCalculator.BasisSet.PseudoPotentialProjectorShift, and http://docs.quantumwise.com/manuals/ATKDFT.html#sec-sg15. For germanium, the following projector-shift parameters should work well:

Code

projector_shift = PseudoPotentialProjectorShift(s_orbital_shift=15.0*eV,
                                                p_orbital_shift=0.2*eV,
                                                d_orbital_shift=-2.0*eV)

See attached script on how to use the PPS method. The script relaxes the Sn1Ge15 bulk and performs band structure + DOS analysis. A fairly small gap opens up at the Gamma points, see attahced PNG. BTW: I can think of 2 reasons why this system may be so hard: 1) germanium is hard to describe with simple methods, such as GGA and MGGA. Perhaps specialized methods must be used, e.g. PPS or HSE. 2) perhaps relaxation effects are very important, i.e., perhaps the lattice constant needs to be just right in order to get the band gap. Just a thought...

[Anders Blom]

Jess, any thoughts on why the Fermi level is not in the middle of the gap? Maybe 5x5x5 k-points are needed, to include the Gamma point?

[ramkrishna]

From my experience, high k-point sampling is necessary to get Fermi level in the middle of gap for an intrinsic semiconductor.

[Jess Wellendorff]

Yes, an odd k-point grid may be needed to get the correct alignment of the Fermi level.

[conor.odonnell]

Thank you all for the assistance! I'll look in to the methods you've suggested. 

[faxer92]

Jess's predictions seems not match with experiments' data:
1. in J. Appl. Phys. 112, 103715 (2012) the lowest conduction band for doping x=0.05
didn't not make much different from pure Ge's curve.
2. Eg_L is smaller than Eg_G at x=0.06
any suggestive basis or potential can we perform to fit experiment by ATK? 

[Petr Khomyakov]

To the best of my knowledge, experimental x-values reported for the indirect to direct band gap transition are quite scattered in the literature. So, I would not see 11% in the cited JAP paper as the reference number because the calculations are based on a highly empirical method with many parameters fitted to a particular experiment.

There is a comprehensive computational study of SnGe alloy given in Phys. Rev. B 89, 165201 (2014), where DFT-MGGA has been adopted. In the paper, one may find many computational details such as supercell size, k-point sampling and many other. Note that in the paper there exist references to recent experiments giving x for indirect to direct gap transition in the range of 0.06 < x < 0.08. 

In ATK, you can also use DFT-MGGA for SnGe. I would suggest doing geometry optimization with LDA and the band structure calculation with MGGA, using the OMX-HIGH basis set. Note that MGGA might need to be tuned by changing the c-parameter to have band energies of bulk Ge closer to experiment. For example, the self-consistent MGGA (without spin-orbit interaction included) gives the c-parameter of 1.10028 for bulk Ge (for LDA optimized lattice constant of 5.641 Angs), and E_gap=E_L=0.88 eV, E_G = 0.98 eV in consistency with the PRB results, 0.82 and 0.97 eV, respectively.

Changing c-parameter to 1.0 will give E_gap=E_L=0.66 eV, E_G = 0.76 eV, and E_X (at the X-valley minimum) = 0.88 eV in agreement with room temperature experimental values E_gap=E_L=0.66 eV, E_G = 0.80 eV, and E_X (at the X-valley minimum) = 0.85 eV.

I would use the same c-parameter (of bulk Ge) for all Sn concentrations in SnGe alloy.

[faxer92]

Hi, Petr, could you kindly share one of your alloy BS results here? 
much appreciated.
Regards,