Monthly Archives: February 2018

Comparing ScAl in CsCl and NaCl Structures and Determining the Optimal Lattice Parameter of the Preferred Structure

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The goal of this post is to identify if ScAl, which has AB stoichiometry, exists in the CsCl structure (figure 1) or NaCl Structure (figure 2 and figure 3). To determine this primitive cells of ScAl were produced for both types of structures. The plan is to plot the cohesive energy of the structure as a function of the volume per ScAl dimer. From these plots the optimal lattice parameters for each structure can be determined. Then by comparing the cohesive energies of the two structures with optimal lattice parameters we can determine which structure is preferred by ScAl. All energy calculations were carried out using plane-wave based DFT. The GGA based PBE functional was used to treat the exchange-correlation effects. The ion and core were treated using ultrasoft pseudopotentials generated on the fly (OTFG ultrasoft) with Koelling-Harmon relativistic treatments. Pseudo atomic calculations for Sc treated the 3s2 3p6 3d1 4s2 electrons as valence electrons and the electrons in lower levels were treated as part of the frozen core. Pseudo atomic calculations for Al treated the 3s2 3p1 electrons as the valence electrons.

Figure 1 : Primitive unit cell of ScAl in the CsCl structure. This a simple cubic structure with a two atom basis.

Figure 2 : (a) Conventional unit cell of ScAl in the NaCl structure. This a face centered cubic structure with a two atom basis. (b) Primitive unit cell of ScAl in the NaCl structure.

However before we begin calculating the data points to populate the plots we described in the previous paragraph, we need to select appropriate cutoff energy and \(\vec{k}\) point mesh for our calculations. The constraints on our choice of cutoff energy and \(\vec{k}\) point mesh are (1) computational cost, (2) convergence of results to a desired tolerance and (3) ensuring that we have approximately the same \(\vec{k}\) point density for each structure.

To determine the cutoff energy we use for our calculations we first determine the approximate lattice parameter of CsCl structure primitive unit cell. For this we use a cutoff energy of 460 eV and \(\vec{k}\) point mesh of \(8\times8\times8\) . The cutoff energy and \(\vec{k}\) point mesh chosen here are the default settings for the “ultra-fine” quality energy calculation using the CASTEP tool implemented in Material Studio. The cohesive energy vs volume per  ScAl dimer plot (figure 4) indicates that cohesive energy is minimized when the lattice parameter is \(\sim 3.5\;\mathring{A}\).

Figure 3 : Plot of Energy per ScAl dimer vs lattice parameter for ScAl in the CsCl structure, used to determine the approximate value for the optimal lattice parameter.

Next we investigated the convergence of the total energy of a primitive unit cell of ScAl in the CsCl structure with respect to the cutoff energy used for the calculation. The results were plotted as shown in figure 5.  From this plot note that we get a convergence of \(\sim\;1\;m\,eV\) for a cutoff energy of \(500\;m\,eV\).

Figure 4 : Calculated total energy of a unit cell of ScAl in the CsCl structure (lattice constant = 3.5 \(\mathring{A}\)) vs the cutoff energy used for the calculation.

Next to obtain the most suitable \(\vec{k}\) point mesh, we plot the energy per ScAl dimer for ae a fixed cutoff energy of \(500\;eV\) while varying the number of \(\vec{k}\) points used to sample the first brillouin zone. Since all the reciprocal lattice vectors ( and real space lattice vectors) have the same length, we can specify the \(\vec{k}\) mesh by specifying the number of \(k\)-points used along each reciprocal lattice vector. Figure 6 shows this plot; and we can see that a \(8\times8\times8\) \(\vec{k}\) point is sufficient for the energy per ScAl dimer to have a convergence of \(\sim\;10\;m\,eV\). The resulting spacing between sampled \(k\) points is \(0.0357\;\mathring{A}^{-1}\). To ensure our subsequent calculations have the same degree of convergence, we will impose that the separation between two adjacent \(k\) points that are sampled along a reciprocal lattice vector is at most \(0.0357\;\mathring{A}^{-1}\).

Figure 5 : Plot of energy per ScAl dimer vs volume per ScAl dimer.

Now we are ready to calculate the energy per ScAl dimer and the corresponding volume per dimer, for both structures and various lattice parameters. Figure 6 shows the plot of energy per ScAl dimer vs volume per dimer, for ScAl in the CsCl and NaCl structure. From the plots in figure 6 it is clear that ScAl prefers CsCl structure over the NaCl structure. From the best fit line we obtain the optimal lattice parameter in the CsCl structure to be \(3.38\;\mathring{A}\). If ScAl were to be found in the NaCl structure the optimal lattice parameter would be \(4.00\;\mathring{A}\).

Experimentally ScAl has been verified to exist in CsCl structure with a lattice parameter of \(3.450\;\mathring{A}\) [1]. Our results verify this and estimate the lattice parameter within \(\sim\;2%\) of the experimentally determined lattice constant.

 

[1] O. Schob and E. Parthe. Ab Compounds with Sc Y and Rare Earth Metals. I. Scandium and
Yttrium Compounds with Crb and Cscl Structure. Acta Crystallographica, 19:214-&, 1965.

 

Determining the Lattice Constants of Hafnium

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Experimental results have reported that hafnium is a hcp metal with a lattice parameter ratio of c/a=1.58. [1] We verify this result with ab initio calculations; our calculations suggest that the lattice parameters of hafnium are a=3.12Å, c=4.94Å, assuming that hcp is the stable crystal structure of hafnium.

Figure 1 – The unit cell of hafnium

Methods and Results

We used the CASTEP code with the CA-PZ functional (Ceperley and Alder 1980, Perdew and Zunger 1981) for density functional theory calculations in the local density approximation (LDA). Ultrasoft pseudopotentials with core radii of 2.096Å were generated on-the-fly, and calculations in reciprocal space used a cutoff energy of 500eV and a 13×13×10 k-point mesh.

Atomic calculations were performed for the following core orbitals: 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 4f14 5s2 5p6 5d2 6s2

Pseudopotentials were used for the following valence orbitals: 4f14 5s2 5p6 5d2 6s2

Using twice the empirical atomic radius (1.59Å) as a starting point, we began searching for a minimum in the free energy with an energy cutoff of 435.4eV and 9×9×6 k-point mesh until we found an approximate minimum at a=3.12Å, to three significant figures. c was initially fixed with respect to a by the ratio c/a=1.58 and was updated as a varied to maintain this ratio. Once we had found this minimum, we increased energy cutoff and k-point sampling fineness until the final free energy had converged with a spead of less than ~0.001eV. We checked convergence by increasing cutoff energy first, then by increasing k-point sampling, and then checked that the cutoff energy had not been consequentially altered by the change in k-point sampling. We then returned to the minimization process to check that the same value of a minimized free energy. Having confirmed this, we minimized free energy by varying c with fixed a. After finding a minimum at c=4.94Å, as shown in Figure 2, we checked whether a=3.12Å still minimized free energy, and we confirmed that it did up to the level of precision established earlier. This is shown in Figure 3.

Figure 2 – Final minimization of the free energy over c

 

Figure 3 – Final minimization of the free energy over a

Discussion

Previously published results [3] indicate that a=3.20Å and c/a=1.582; our results corroborate the latter ratio (our c/a=1.583) but disagree significantly with the actual magnitudes of the lattice parameters. Given that our colleagues matched published values much more closely by using different functionals and parameters, such as the generalized-gradient approximation, we suspect that the LDA functional is inadequate for precision calculations, but a closer examination of LDA parameters reveal a way for this functional to be comparatively useful.

[1] D. Sholl and J. Steckel, Density Functional Theory: A Practical Introduction. (Wiley 2009)
[2] S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson, M. C. Payne, “First principles methods using CASTEP”, Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005)
[3] K. Hermann, Crystallography and Surface Structure: An Introduction for Surface Scientists and Nanoscientists. (Wiley-VCH, 2011)

Status of Posts 1 due on Feb. 15th

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(Reviewer > Author)

  1. Bryan Brasile -> Ran Chen (Review done – Post Published)
  2. Mark Bronson -> Si Chen (Review done – Post Published)
  3. Ran Chen -> Zihao Chen (Review done – Post Published)
  4. Si Chen -> Garrett DuCharme (Review done – Post Published)
  5. Zihao Chen -> Brett Green (Review done – Post Published)
  6. Garrett DuCharme -> Mark Hagemann (Review done – Post Published)
  7. Brett Green -> Haoran He (Review done – Post Published 2018-02-19)
  8. Mark Hagemann -> Jeffrey Rable (Review done – Post Published)
  9. Haoran He -> Jonathan Schirmer (Project not finished)
  10. Jeffery Rable -> Mohamed Umar (Review done – Post Published)
  11. Jonathan Schirmer -> Run Xiao (Review done – Post Published)
  12. Mohamed Umar -> Tongzhou Zhao (Review done – Post Published)
  13. Run Xiao -> Bryan Basile (Review done – Post Published)
  14. Tongzhou Zhao -> Mark Bronson (Review done – Post Published)

Determination of the crystal structure with optimal lattice constant for Pt

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  1. Description of the problem

For this first project, we aim to predict the most-favored crystal structure of Pt and calculate the optimal lattice parameters for these structures.

Usually, the metal crystals can have simple cubic (sc), face centered cubic (fcc), and hexagonal close packed (hcp) structures. In this work, we will firstly examine the optimal lattice parameter for sc Pt based on the energy of bulk Pt, followed by the tests of on the fcc structure. Both optimal lattice parameter a and the ratio a/c will be determined for hcp Pt. Convergence tests will be done with respect to the number of k-points and the cutoff energy for all studies.

The Vienna Ab initio Simulation Package (VASP) is used to perform the periodic DFT calculations,1-3 employing the projected augmented-wave (PAW) pseudopotentials,4,5 as well as generalized gradient approximation with the exchange-correlation functional by Perdew, Burke, and Ernzerhof (PBE).6

  1. Simple cubic

The sc structure of Pt is built using the software Material Studio as shown in Figure 1. In each unit cell, there is one Pt atom.

               Figure 1. Unit cell of sc Pt                       Table 1. Results of k-points convergence for sc

Before we can calculate the energy of this whole system, the convergence tests are required with respect to the number of k-points and the cutoff energy. The criteria for choosing the number of k-points and energy cutoff in the following calculation is set to have the energy difference within 0.01 eV.

Firstly, in order to test the convergence of the number of k-points, we initially set the cutoff energy to 400 eV. The results of using number of k-points ranging from 1 to 120 are summarized above in Table 1. Figure 2 shows the trend of bulk energy as well as the computational cost versus the number of k-points.

Figure 2. Bulk energy and computational cost versus the number of k-points for sc

We can see the bulk energy becomes more stable with increased number of k-points while the computational cost keeps increasing. Considering the balance between higher accuracy and cost, the k-points sampling of 14x14x14 is chosen for further calculations.

The convergence of the cutoff energy is also tested as summarized in Table 2 with fixed k-points sampling shown above. An interesting thing is that we can see the (pseudo) atomic energy keeps becoming lower while the cutoff energy is increased (later we will see the same behavior for all other structures). Figure 3 shows the relationship of bulk energy and computational costs depending on the cutoff energy.

Table 2. Results of the cutoff energy convergence for sc

Figure 3. Bulk energy and computational cost versus the cutoff energy for sc

Similar as the cases for k-points sampling, the energy becomes stable with increased cutoff energy while the cost keeps increasing. The energy cutoff of 400 eV is chosen for further calculations.

In order to determine the optimal lattice parameter, we vary the lattice parameter to find the one resulting in lowest bulk energy. We firstly do a rough search using step size of 0.1 Å from 1.50 to 3.00 Å. With such rough search, we are able to determine the interval where the optimal value lies in and based on that, we can do a more precious search with step size of 0.01 Å. All the results are shown in Figure 4.

(a)                                                                            (b)

Figure 4. Bulk energy versus the lattice parameter for sc. (a) for rough search and (b) for detailed search.

From the first plot in Figure 4, we can see that the optimal value is in the interval from 2.50 to 2.70 Å. The detailed search is done in this interval with step size of 0.01Å. Finally, the optimal lattice parameter is determined to be 2.62 Å, which gives the lowest bulk energy of -5.655 eV for the simple cubic structure.

  1. Face centered cubic

The search for the optimal lattice parameter of fcc Pt is very similar to the study of sc Pt. The unit cell of fcc Pt is built with Material Studio as shown in Figure 5, containing on Pt atom.

               Figure 5. Unit cell of fcc Pt                    Table 3. Results of k-points convergence for fcc

As we did before, the convergence tests are firstly made and the results on k-points are summarized in Table 2. Figure 6 shows the relationship of bulk energy as well computational time to the number of k-points. Similarly, the k-points sampling of 12x12x12 is finally used.

Figure 6. Bulk energy and computational cost versus the number of k-points for fcc

The same tests on cutoff energy is done as summarized in Table 4 and Figure 7. The cutoff energy we use for further calculations of fcc Pt remains 400 eV.

Table 4. Results of the cutoff energy convergence for fcc

Figure 7. Bulk energy and computational cost versus the cutoff energy for fcc

The search for the optimal lattice parameter is carried out in a similar way, but roughly ranging from 3.10 to 4.30 Å. The results are shown in Figure 8. We can see from the left side of Figure 8 that the optimal lattice parameter lies in the interval from 3.90 to 4.10 Å. Thus, the optimal lattice parameter is determined with the detailed search using step size of 0.01Å. The optimal lattice parameter is 3.97 Å, which gives the lowest bulk energy of -6.097 eV for the face centered cubic structure.

(a)                                                                            (b)

Figure 8. Bulk energy versus the lattice parameter for fcc. (a) for rough search and (b) for detailed search.

  1. Hexagonal close packed

The difference in determining the optimal lattice parameter for hcp Pt is that there will be different optimums for different c/a. So in this section, we will compare the opmital lattice parameter for several potential c/a ratios (in this section, cases of c/a=1.57, 1.60, 1.63, 1.67, 1.70, 1.73 will be studied) and find the most-favored one which gives us the lowest bulk energy among them. Similarly, the hcp unit cell is consturcted using Material Studio as shown in Figure 9 containing 2 Pt atoms.

Figure 9. Unit cell of hcp Pt

The case of c/a=1.60 is chosen to test the convergence. The results are summarized below in Table 5. Figure 10 and 11 show the trends of bulk energy as well as computational cost with respect to the number of k-points and cutoff energy, respectively.

Table 5. Results of the convergence tests for hcp

Figure 10. Bulk energy and computational cost versus the number of k-points for hcp

Figure 11. Bulk energy and computational cost versus the cutoff energy for hcp

Accordingly, the k-points sampling is chosen to be 10x10x6 and the cutoff energy is 400 eV.

The search for optimal lattice parameter is achieved using the same method, but with different c/a values roughly ranging from 2.00 to 3.30 Å. Figure 12 shows the results of rough search for different c/a values.

Figure 12. Bulk energy versus the lattice parameter for hcp (rough search)

We can see that, for the case of c/a=1.56, the optimum lies in the interval from 2.80 to 3.00 Å. For all other cases, the optimal parameter is in the interval from 2.70 to 2.90 Å. The results for corresponding detailed search is shown in Figure 13 below. The final results reveal that the optimal lattice parameter for hcp Pt is 2.76 Å with c/a=1.73, giving the lowest bulk energy of -6.046 eV.

Figure 13. Bulk energy versus the lattice parameter for hcp (detailed search)

  1. Conclusion

According to the results above, we know that the optimal lattice parameters for these three different structures are 2.62, 3.97, and 2.72 Å, respectively. Among them, the fcc structure with lattice parameter of 3.97 Å gives the lowest bulk energy of -6.097 eV (-5.655 and -6.046 eV for sc and hcp). The experimental lattice constant is 3.92 Å. Our result is about 1.01% larger than the experimental observation value, which is commonly seen while using PBE functional as PBE tends to overestimate the lattice constant.

Reference

  1. Kresse, G. and J. Hafner, Ab initio molecular dynamics for liquid metals. Physical Review B, 1993. 47(1): p. 558.
  2. Kresse, G. and J. Hafner, Ab initio molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium. Physical Review B, 1994. 49(20): p. 14251.
  3. Kresse, G. and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Physical review B, 1996. 54(16): p. 11169.
  4. Kresse, G. and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method. Physical Review B, 1999. 59(3): p. 1758.
  5. Blöchl, P.E., Projector augmented-wave method. Physical review B, 1994. 50(24): p. 17953.
  6. Perdew, J.P., K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple. Physical review letters, 1996. 77(18): p. 3865.