# Si_{0}_{.}_{5}Ge_{0}_{.}_{5}¶

This document is part of the QuantumWise Semiconductor Whitepapers. Here we investigate
computational methods for ATK-DFT calculations for a simple Si_{0}_{.}_{5}Ge_{0}_{.}_{5} alloy in the
diamond crystal structure, i.e., with an atom of each element in the basis
(in this document also denoted SiGe for brevity).
We apply the SG15 pseudopotentials and a number of exchange-correlation
functionals; PBE, PBEsol, MGGA, and PBE with the Pseudopotential Projector-Shift method.
A limited number of HSE calculatios have been done for comparison. See the section
Introduction for more information.

The following **quantities** have been calculated for each computational method,
both at the experimental Si_{0}_{.}_{5}Ge_{0}_{.}_{5} lattice constant and at the theoretically predicted
ones:

- Elastic constants: Bulk modulus, Poisson ratio, and Young’s modulus.
- Band structure along the X–Γ–L Brillouin zone path.
- Direct and indirect band gaps.
- Effective masses in the conduction band minimum.
- Static dielectric constant.

**Convergence studies** indicate that a 8x8x8 *k*-point grid and a 100 Hartree
density mesh cutoff yields well-converged values for the Si_{0}_{.}_{5}Ge_{0}_{.}_{5} lattice constant,
total energy and indirect band gap. These settings were used for computing the
ground state electronic structure used in all the ATK-DFT analyses listed above.

The projector shifts used for the **pps-PBE** method are 11.23 eV for Si *s*-orbitals
and -1.09 eV for Si *p*-orbitals, while shifts of 15.0 eV are used for Ge *s*-orbitals,
0.2 eV for Ge *p*-orbitals, and \(-\)2.0 eV for Ge *d*-orbitals. Note that
these PPS parameters should only be used with SG15 pseudopotentials.

This document is organized as follows. The section
Convergence briefly presents the convergence studies
mentioned above, and the section Timing compares CPU
timings and scalability of different SG15 basis sets for Si_{0}_{.}_{5}Ge_{0}_{.}_{5}. Next, the section
Results contains figures illustrating the performance of
the different computational methods in predicting the materials properties listed
above. Finally, the Appendix gives a template ATK Python
script for setting up ATK-DFT calculations for Si_{0}_{.}_{5}Ge_{0}_{.}_{5} similar to the ones presented
here.

## Summary¶

The following list summarizes the main conclusions from this study.

- The PBEsol, and to a lesser degree pps-PBE, methods both yield very accurate
Si
_{0}_{.}_{5}Ge_{0}_{.}_{5}**lattice constants**, and also give the best results for elastic constants. - All tested methods yield a qualitatively correct Si
_{0}_{.}_{5}Ge_{0}_{.}_{5}**band structure**, but with significant differences between the different methods. The method which is closest to the HSE reference is pps-PBE. - The pps-PBE method accurately reproduces the fundamental (indirect)
**band gap**. There is a large spread in the results for the direct gaps, so care is needed to describe these accurately. **Electron effective masses**in the \(\Delta\)-point CBM are quite well captured by all tested methods.- Even though PBE and PBEsol do not describe the band gaps as accurately as pps-PBE,
they are actually significantly better for the
**static dielectric constant**. It is underestimated by the other methods.

## Convergence¶

The type of Monkhorst–Pack k-point grid and the density mesh cutoff energy are essential ATK-DFT parameters that affect the computational efficiency and precision: A dense k-point grid and high cutoff energy usually give high precision, but may also be computationally intense. It is therefore important to investigate the trade-off between computational precision and cost. Fig. 159 illustrates how the bulk total energy, fundamental band gap, and lattice constant depend on the mesh cutoff and k-point grid.

Note that a standard
Monkhorst–Pack k-point grid is Γ-centered for odd numbers of points (e.g. 3x3x3),
but does not include the Γ point for even numbers of points (e.g. 4x4x4). The blue
lines in Fig. 159 are for k-point grids where even grids are shifted to
Γ (such that both 3x3x3 and 4x4x4 grids **do** include Γ), while the red lines are for
calculations where odd grids are shifted away from Γ (such that both 3x3x3 and 4x4x4
grids **do not** include Γ).

For the SG15 pseudopotential with both Medium and High basis sets, a 100 Hartree
mesh cutoff gives band gaps and lattice constants that are converged to within
10^{-4} eV and roughly 10^{-4} Å, respectively.

It is also clear that an off-Γ k-point grid (non-centered, red) gives faster convergence of the total energy and lattice constant than the k-point grid that includes the Γ point (centered, blue). We see that the non-centered 8x8x8 Monkhorst–Pack grid yields highly converged lattice constants.

Note

All remaining ATK-DFT calculations in this study therefore use a 100 Hartree mesh cutoff energy and a standard (non-centered) 8x8x8 Monkhorst–Pack k-point grid for both elastic properties and electronic properties.

## Timing¶

The computational cost of a calculation may depend critically on the choice
of basis set, particularly for calculations on large systems with many atoms.
Fig. 160 shown below illustrates this for the
present case of Si_{0}_{.}_{5}Ge_{0}_{.}_{5} with the Medium, High, and Ultra SG15 basis sets.

The SiGe primitive cell (containing 2 atoms) was systematically repeated in steps of one along all three lattice vectors, resulting in rapidly increasing systems sizes. PBE calculations were then performed for each system, using Γ-point sampling only, to avoid any influence of k-point sampling on the recorded CPU times.

It is quite clear from the figure that the computational loads of the SG15 Medium and High basis sets are not much different for relatively small system sizes, but the High basis set becomes significantly more demanding for larger system sizes. For the 432-atom unit cell, the CPU time difference is roughly a factor of 2. Moreover, the Ultra basis set appears in comparison extremely demanding, and will not be used in the remaining calculations in this study.

## Results¶

This section presents results obtained with the PBE, PBEsol, pps-PBE, and MGGA
methods using Medium and High basis sets with the SG15 pseudopotentials for Si_{0}_{.}_{5}Ge_{0}_{.}_{5}.

The cited experimental reference values are in general for a 50/50 mix of Si and Ge, but with no local order in the distribution of Si and Ge on the lattice. There may therefore be a slight difference between the experimentally investigated system and the very small model system used here.

The list below gives direct links to all subsections:

### Lattice constant¶

The PBEsol lattice constant for Si_{0}_{.}_{5}Ge_{0}_{.}_{5} is very close to the experimental value, 5.538 Å, but the pps-PBE method is also quite accurate. The bare PBE functional (without any pseudopotential projector shifts) is known to overestimate lattice constants in general, and in this case it overestimates by more than 1%. This is in contrast to pure silicon where PBE only overestimates the lattice constant by 0.6%. HSE performs better than PBE, but slightly worse than pps-PBE. However, HSE is expected to perform much better for the details of the electronic structure.

In all cases, except for PBEsol, we see a significantly smaller error for the better basis set (High or Tight).

### Elastic constants¶

Both PBESol and pps-PBE hit close to the experimental bulk modulus, while regular PBE underestimates it. The same trend is observed for Young’s modulus, but without a reference value for comparison. The Poisson ratio is approximately the same for all methods.

### Band structure¶

All tested methods correctly predict a valence band maximum (VBM) at the Brillouin zone Γ point, and most also predict a conduction band minimum (CBM) close to the X point, along the Γ–X path. However, the position of the CBM depends slightly on the method applied, and the band energy in the CBM significantly so (see also section Band gaps). In fact, for PBE the CBM may shift to the L point depending on basis set.

There is also a qualitative difference between HSE and pps-PBE at the X point. HSE has a gap of about 0.3 eV between the two lowest conduction bands, while they (almost) touch for pps-PBE.

Changing the basis set from Medium to High gives a shift of about 0.1 eV at the X point, and quickly decreasing towards Γ. In all cases, the shift brings the result closer to the HSE values. For HSE, there is almost no difference between Light and Tight basis sets.

### Band gaps¶

The SiGe fundamental band gap (\(E_\Delta\)) is especially well reproduced by the pps-PBE method, but also HSE, regular PBE and PBESol give reasonable agreement with experiment. Note, however, that the MGGA c-parameter was calculated self-consistently from the electronic structure (no fitting), while the pps-PBE projector shifts were essentially fitted to the silicon and germanium band gaps and lattice constants. On the other hand, the MGGA functional cannot be used for geometry optimization, which the pps-PBE method is well suited for.

There is much less agreement between the methods when it comes to the direct gaps, with MGGA consistently giving larger values, PBE and PBESol giving smaller values, and pps-PBE and HSE in between.

We also evaluate the effect of using HSE for band gap calculations. The figure below shows GGA and HSE band gaps of SiGe computed at the GGA lattice constants. The HSE indirect (fundamental) gaps are all roughly the same, while the direct gap at the \(\Gamma\) point does vary with lattice constant. The latter is probably due to strain effects. The PBE and PBEsol gaps are all significantly smaller than the corresponding HSE ones, as expected. On the other hand, the pps-PBE gaps are quite similar to the corresponding HSE gaps.

### Effective masses¶

The SiGe electron effective masses in the \(\Delta\)-point CBM are fairly close to experimental values for all applied methods.

### Dielectric constant¶

The static (\(\omega=0\)) dielectric constant of the SiG lattice is most reliably reproduced by the PBE and PBEsol functionals, with the pps-PBE method a bit off, and MGGA more than 30% off. The latter is probably related to the fact that MGGA severely overestimates the direct band gap, while PBE and PBEsol are closer to the experimental direct gap. However, the fact pps-PBE gives a very good direct gap, but is somewhat off for the static dielectric constant, may indicate that the overall bandstructure is not as accurately described as the direct gap itself. We see almost no dependence of the dielectric constant on the basis set quality.

## Appendix¶

The ATK Python script shown below may be used as a template for ATK-DFT
calculations for Si_{0}_{.}_{5}Ge_{0}_{.}_{5} with the SG15 pseudopotential. The script defines
the SiGe bulk configuration and then sets up the ATK-DFT calculator
with PBE exchange-correlation. The script blocked named `Basis Set`

shows various options for the SG15 basis set; ordinary PBE and PBE with
pseudopotential projector shifts, both with Medium and High basis sets.

The script is available for direct download: `si50ge50.py`

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# -------------------------------------------------------------
# Bulk Configuration
# -------------------------------------------------------------
# Set up lattice
lattice = FaceCenteredCubic(5.533*Angstrom)
# Define elements
elements = [Silicon, Germanium]
# Define coordinates
fractional_coordinates = [[ 0. , 0. , 0. ],
[ 0.25, 0.25, 0.25]]
# Set up configuration
bulk_configuration = BulkConfiguration(
bravais_lattice=lattice,
elements=elements,
fractional_coordinates=fractional_coordinates
)
# -------------------------------------------------------------
# Calculator
# -------------------------------------------------------------
#----------------------------------------
# Basis Set
#----------------------------------------
# Ordinary GGA: Medium or High basis set for SG15.
basis_set = [BasisGGASG15.Silicon_Medium,
BasisGGASG15.Germanium_Medium]
#basis_set = [BasisGGASG15.Silicon_High,
# BasisGGASG15.Germanium_High]
# GGA with pseudopotential projector-shift method: Medium or High basis set for SG15.
#shift_si = PseudoPotentialProjectorShift(s_orbital_shift=11.23*eV,
# p_orbital_shift=-1.09*eV)
#shift_ge = PseudoPotentialProjectorShift(s_orbital_shift=15.0*eV,
# p_orbital_shift=0.2*eV,
# d_orbital_shift=-2.0*eV)
#basis_set = [BasisGGASG15.Silicon_Medium(projector_shift=shift_si),
# BasisGGASG15.Germanium_Medium(projector_shift=shift_ge)]
#basis_set = [BasisGGASG15.Silicon_High(projector_shift=shift_si),
# BasisGGASG15.Germanium_High(projector_shift=shift_ge)]
#----------------------------------------
# Exchange-Correlation
#----------------------------------------
exchange_correlation = GGA.PBE
k_point_sampling = MonkhorstPackGrid(
na=8,
nb=8,
nc=8,
)
numerical_accuracy_parameters = NumericalAccuracyParameters(
k_point_sampling=k_point_sampling,
density_mesh_cutoff=100.0*Hartree,
)
iteration_control_parameters = IterationControlParameters(
damping_factor=0.5,
)
calculator = LCAOCalculator(
basis_set=basis_set,
exchange_correlation=exchange_correlation,
numerical_accuracy_parameters=numerical_accuracy_parameters,
iteration_control_parameters=iteration_control_parameters,
)
bulk_configuration.setCalculator(calculator)
nlprint(bulk_configuration)
bulk_configuration.update()
nlsave('SiGe_5050.nc', bulk_configuration)
``` |

## References¶