Lattice Structure description:
ScAl has the same type of structure that CsCl, which is simple cubic cell. The cell vector could be (1, 0, 0), (0, 1, 0), (0, 0, 1). Fractional coordinates for Cs could be (0, 0, 0) and for Al could be (0.5, 0.5, 0.5). Diagram for this structure is shown in Fig 1.
Fig1 Golden spheres are Sc, the brown shere is Al.
Since ScAl has simple cubic structure, We just need to adjust lattice parameter a in order to predict its structure. Idea is calculate total free energy for different parameters and find out the energetically favorable one.
DFT calculation is adopted for this search.
Cutoff energy test:
A test is done to find a proper cutoff energy. Fixing other setting: functional: GGA PBE, k points: 6*6*6, lattice parameter a=4.0Å, we change cutoff energy and compare their total free energy results. Results are shown in Table 1. Energy difference between cutoff energy ‘460 eV’ and ‘560 eV’ is less than 0.02eV. If time cost in considered and total free energy resolution is controlled at 0.02 eV, using ‘460 eV’ for cutoff energy for following calculations is an acceptable choice.
| cutoff energy(eV) | total free energy(eV) |
| 60 | -1140.936931 |
| 160 | -1347.963162 |
| 260 | -1379.612688 |
| 360 | -1383.327654 |
| 460 | -1383.662277 |
| 560 | -1383.681389 |
| 660 | -1383.681690 |
Table 1
K point test:
functional: GGA PBE, cutoff energy: 460eV are fixed and lattice parameter a is changed.
K points in default will change with lattice parameter. (CASTEP tool is used here, ‘default’ meaning default number for k points in CASTEP tool)
Results are shown in Table 2 .
| lattice parameter a(Å) | k points | total free energy (eV) |
| 2.0000 | 14*14*14 | -1336.132886 |
| 2.6000 | 10*10*10 | -1376.299647 |
| 2.7000 | 10*10*10 | -1379.066514 |
| 2.8000 | 10*10*10 | -1381.168776 |
| 2.9000 | 10*10*10 | -1382.723256 |
| 3.0000 | 8*8*8 | -1383.831689 |
| 3.1000 | 8*8*8 | -1384.578369 |
| 3.2000 | 8*8*8 | -1385.031976 |
| 3.3000 | 8*8*8 | -1385.254431 |
| 3.3600 | 8*8*8 | -1385.298682 |
| 3.3700 | 8*8*8 | -1385.300556 |
| 3.3750 | 8*8*8 | -1385.300611 |
| 3.3800 | 8*8*8 | -1385.300591 |
| 3.3850 | 8*8*8 | -1385.299964 |
| 3.3900 | 8*8*8 | -1385.29928 |
| 3.4000 | 8*8*8 | -1385.296417 |
| 3.4200 | 8*8*8 | -1385.287388 |
| 4.0000 | 6*6*6 | -1383.662277 |
| 5.0000 | 6*6*6 | -1379.859133 |
| 6.0000 | 4*4*4 | -1377.783054 |
Table 2
If density of k points is defined as number of k points in one direction over k space parameter in that direction, this according change of k point might have a purpose of keeping density of k point unchanged. Since the lengths of lattice vector in cell and lattice vector in k space have inverse proportion relation. So in this simple cubic system, expectation would be that number of k points in one direction times lattice parameter ‘a’ should lead to a constant. Obviously, this expectation is not obeyed in this test.
K points will effect the precision and time cost of a calculation, so finding a balance point of precision and efficiency means we need to find a suitable k points. This ‘finding a balance’ situation occurs as well when we deal with cutoff energy.
So which k point choice is suitable for this calculation? We can discuss this based on calculation results.
Fig 2 and Fig 3 show the search for lattice parameter. Relatively, one is rough, the other is fine.
Fig 2
Fig 3
We can see the parameter range which is located at energy valley is (3.36, 3.40). At this range, the k point is set as ‘8*8*8’ and in this range the finest search step is 0.005Å.
In ‘cutoff energy test’, ‘460 eV’ is used for cutoff energy so that resolution for total free energy is set to ‘0.02 eV’. Please notice that the ‘0.02 eV’ resolution actually also includes the setting of k points as ‘6*6*6’. And in the range we care about most adopts ‘8*8*8’ k point setting which should give precise enough results for this search. Energy numbers in table 2 for range (3.36, 3.40) do have difference less than 0.02 eV, which actually is less than 0.002 eV. So we can say that if ‘460 eV’ is adopted for cutoff energy, ‘8*8*8’ k point setting is ‘safe enough’ for this calculation. Of course, accordingly, it will be dangerous to make a prediction for lattice parameter beyond the precision of ‘0.005Å’.
Convergence test, however, is still done for k points, at a=4.0Å , cutoff energy=460 eV. Results are shown in Table 3.
| k points | total free energy(eV) |
| 4*4*4 | -1383.589307 |
| 5*5*5 | -1383.619788 |
| 6*6*6 | -1383.662277 |
| 7*7*7 | -1383.635492 |
| 8*8*8 | -1383.633134 |
| 9*9*9 | -1383.639153 |
| 10*10*10 | -1383.636616 |
| 11*11*11 | -1383.63629 |
| 12*12*12 | -1383.637744 |
Table 3
From data in this table, total free energy’s difference between ‘8*8*8’ and ‘9*9*9’ is less than 0.02 eV, which supports the point that ‘8*8*8’ setting for k points is precise enough under resolution of 0.02 eV for total free energy. Consistent with expectation, with increasing number of k points, we have smaller energy difference.
‘8*8*8′ for k points is adopted for lattice parameters outside (3.36, 3.40) in order to constrain variables when comparing different parameters’ energy. And for parameters in (2.00, 2.90), calculations have larger k points so it would be meaningless to re-calculate these points. Just using ‘8*8*8’ k points re-calculate points with a=4.00, 5.00, 6.00 Å. Results and comparison are shown in Table 4.
| lattice parameter(Å) | total free energy with 8*8*8 k points(eV) | total free energy with default k points(eV) |
| 4.000 | -1383.633134 | -1383.662277 |
| 5.000 | -1379.864649 | -1383.662277 |
| 6.000 | -1377.763773 | -1377.783054 |
Table 4
From data in the table, we can see that with ‘8*8*8’ k points, total free energy for these points goes higher, which does not affect our search for lowest energy point.
Conclusion:
If ‘460 eV’ cutoff energy and ‘0.02 eV’ precision for total free energy is adopted, ‘8*8*8’ k points setting could provide precise enough for the search of lattice parameter. At the same time the precision of this parameter search is limited at ‘0.005Å’.
Based on the calculation results and just considering minimizing total free energy, ScAl should have a lattice parameter around 3.375Å.
If more decimal place is wanted for this prediction, larger cutoff energy and k points should be adopted.
Reference:
First principles methods using CASTEP. Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005) S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson, M. C. Payne
