Monthly Archives: January 2018

Determination of the Lattice Parameter of ScAl in the CsCl Structure

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This Project aims to predict the lattice constant of ScAl with CASTEP calculation.

A. Project Description 

In this study, we focus on predicting the lattice constant of  ScAl based on the CsCl structure, and figuring out a converged energy cutoff and k-points in the CASTEP energy calculation. Atomic electron configuration for Al is 3s2 3p1, for Sc is 3s2 3p6 3d1 4s2. The energy calculation in CASTEP can provide a reasonable crystal structure for ScAl. The energy calculation in this study is based on the exchange correlation Perdew-Burke-Ernzerhof (PBE) density functionale, which is from the class of Generalized gradient approximation (GGA) functional. The relationships between energy and lattice parameters, energy cutoff, and k-points are discussed below.

B. Crystal Model

ScAl has CsCl structure, where Scandium (Sc) locates at the corner and Aluminum (Al) in the center of the unit cell. This structure belongs to the cubic system, the lattice parameter a=b=c, α=β=γ=90 degrees.

The cell vectors are along x, y, direction of the unit cell, with orthogonal \(a_i\) (1,0,0), \(a_j\)(0,1,0), and \(a_k\)(0,0,1) respectively. The equivalent fractional coordinate of the Sc is (0,0,0) whereas the Al is (1/2, 1/2, 1/2). The real coordinates of Sc and Al in the unit cell depend on the lattice parameter a, with Sc at (0,0,0), (a,0,0), (0,a,0), (0,0,a), (a,a,0), (a,0,a), (0,a,a), and (a,a,a); Al at (a/2,a/2,a/2). Figure 1 shows an example of Sc and Al positions with the lattice parameter a=b=c=3.379 Å.

Figure 1. Simple cubic structure of ScAl, where Aluminum locates in the center and Scandium in the corner of the unit cell.

C. Determine the Lattice Parameters of ScAl

In order to predict a reasonable lattice parameter of ScAl, the energy of the unit cell is calculated with the variation on lattice parameters. Given the structure is from the cubic system, the lattice parameters a, b, and C are equal and will be referred to “a” as below. Before the calculation, the lattice parameter a is estimated based on the atomic radius of Al (1.43 angstrom) and Sc (2.30 angstrom). To get a well packed structure along the body diagonal in (111) face, the lattice parameter should be smaller than 2(r(Al)+r(Sc))/√3, which is 4.3 Å.

As a starting point the energy of the ScAl structure was calculated witha lattice parameter of  4.3Å using the CASTEP calculation ( Energy cutoff 500 eV, k-points 10*10*10). With the decreased lattice parameter from the starting point, the free energy of the unit cell reached a minimum to some point and then increased with the lattice parameter decreased further (Figure 2). This shape of curve is caused by the relative atom positions, either too far or too close, generateing higher energy (less stable structure) than the minimum energy (the most stable structure by calculation). The lattice parameter \(a_0\)= 3.379 Å corresponds the minimum cohesive energy (Figure 2).

Figure 2. Cohesive energy for simple cubic ScAl, using 10*10*10 kpoints and 500eV energy cutoff, as a function of lattice parameter


D. The Energy Cutoff

Multiple calculations for ScAl structure were completed with a variation of energy cutoff \(E_{cut}\) from 200eV to 800eV. Lattice parameter and kpoints remained the same at 3.379 Å and 8*8*8 k-point grid for Brillouin zone integration respectively. An increase in the energy cutoff increases the number of plane-waves and improves the accuracy of ion cores, but costs longer computation time. Repeated calculations with higher energy cutoff aim to converge to a decent final free energy (Figure 3&4). Convergence is reached with an energy cutoff of 500eV providing an accuracy of the absolute energy better than 0.001eV (Figure 4).

Figure 3. Both Al and Sc atomic energy converged at 500eV with a lattice parameter 3.379 Å and 8*8*8 k-point grid for Brillouin zone integration.
Figure 4. Energy per cell and cohesive energy of ScAl as a function of energy cutoff with a lattice parameter 3.379 Å and 8*8*8 k-point grid for Brillouin zone integration.

The cohesive energy of a solid material is the energy to separate the condensed material into isolated free atoms.

\begin{equation}E_{coh}=(E_{total}-E_{atom})/N\end{equation}
with \(E_{total}\)is the total energy of the unit cell, \(E_{atom}\) is the total energy of the atom, and N as the number of atoms in the unit cell. Table 1 and Figure 4 indicate that the cohesive energy decreased with higher energy cutoff, converged to an accuracy of  0.01eV at energy cutoff of 500eV.

Table 1 Results from computing the total energy, atomic energy, and cohesive energy of simple cubic ScAl with 8*8*8 kpoints in different energy cutoffs

 

E. k-points

As k-point varied but energy cutoff (500 eV) and lattice parameter (3.379 Å) unchanged in the following calculation, the total energy of the cell converged with more kpoints.  The cell volume and atomic energy remained the same, because the cell volume only depends on lattice parameter and atomic energy depends on the energy cut-off.

Figure 5. Energy per cell and cohesive energy of ScAl as a function of M*M*M k-points, with a lattice parameter 3.379 Å and energy cutoff 500eV.

The used k points are reduced by symmetry operations using Monkhorst-Pack approach with M*M*M k points in ScAl cubic structure. Table 2 and Figure 5 shows the number of k poins in irreducible Brillouin zone (IBZ) and correspondent total energy and binding energy. Both the odd (2n+1) and even (2n+2) values of M have the same number of k points in IBZ, but even values (2n+2) of M converges better than odd values in regard to the same computational time. In our cases, both the 7*7*7 and 8*8*8 k points required 8.08 seconds to finish the calculation, but 8*8*8 converging better because all k points are inside of the IBZ (Sholl and Steckel, 2011). In ScAl structure, 12*12*12 k points are enough to get accurate energy.

Table 2 Energy calculation with k points varies, energy cutoff 500eV, lattice parameter 3.379 Å.

 

Conclusion:

The total energy of simple cubic ScAl minimized at lattice parameter a=3.379 Å. Energy per cell converges when energy cutoff is 500eV, k-points are 12*12*12.

 

Reference

[1] Sholl, D. and Steckel, J.A., 2011. Density functional theory: a practical introduction. John Wiley & Sons.

[2] BIOVA, 2014. CASTEP guide , Material Studio. http://www.tcm.phy.cam.ac.uk/castep/documentation/WebHelp/content/pdfs/castep.htm

 

Searching for the Lattice Parameter of ScAl

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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

 

 

 

 

 

Determining the Structure and Lattice Constant of Platinum

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In this project,  the lattice constant and structure of crystalline platinum were found by performing CASTEP energy calculations and finding where the energy was minimized [1]. These calculations were performed in Materials Studio using the GGA Perdew Burke Ernzerhof (PBA) functional. The calculations used an energy cutoff of 321.1eV for the rougher calculations and 420eV for the finer calculations. The pseudopotential was solved using the Koelling-Harmon atomic solver with the interior shells 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10, giving an energy of -13041.2296 eV. The outer shells used in the calculations consisted of the 4f14 5s2 5p6 5d9 6s1 electrons.

1. Simple Cubic Lattice (P23 Point Group)

Fig. 1. A primitive cell of the simple cubic lattice

For the simple cubic lattice rough calculations, 6x6x6 k points were sampled with 0.1eV Gaussian smearing and no shift. After applying symmetry, this sampling reduced to 11 total k points.

Table 1. The roughly calculated energy of the simple cubic lattice for varying values of the lattice constant.

Fig. 2. The roughly calculated energy of the simple cubic lattice configuration for different lattice constants. The minimum occurs at 2.6 Angstroms, which will help determine a starting value for both the finer, more time consuming simple cubic calculations and the rough calculations for other lattice structures. The lines connecting the data points act as a guide to the eye and are not data.

Fig. 3. K space sampling vs. energy (eV) for the P23 simple cubic lattice. The energy converges to 0.01eV beyond 176 points.

Upon finding the approximate lattice constant with an 11 point k space sampling and a 321.1eV energy cutoff, we can perform the same calculations around that point with a higher k space sampling and cutoff energy to find a more accurate result. For these calculations, 176 k points were sampled (16x16x16) and a cutoff of 420eV was used. These results should converge to 0.01eV. Ultimately, these calculations resulted in a lattice constant of approximately 2.62 Angstroms.

Table 2. The finer lattice constant vs. energy calculations.

Fig 4. Finer lattice constant vs. energy graph. The lines connecting the data points act as a guide to the eye and are not data.

 

2. Hexagonal Close-Packed Lattice (D3H-3 Point Group)

Fig. 5. A primitive cell of the hexagonal close-packed lattice.

For the hexagonal close-packed lattice, 24x24x18 k points were sampled with 0.1eV Gaussian smearing and a shift of (0.021inverse Angstroms ,0.021 inverse Angstroms,0). The calculation was ultimately performed using 882 k points. For this lattice structure, only calculations with a 420eV cutoff were performed resulting in approximately a 2.60 Angstrom lattice constant. For these calculations, the ratio c/a was set at 1.633 because it gives ideal hard sphere close packing.

Table 3. Table of the HCP lattice constant (angstroms) and corresponding energy. The minimum lies at approximately 2.60 and 2.61 Angstroms.

Fig. 6. Graph of the Pt HCP lattice constant (Angstroms) vs. energy (eV). The lines connecting the data points act as a guide to the eye and are not data.

Fig. 7. K space sampling vs. energy (eV) for the HCP lattice. The energy converges to 0.01eV beyond 882 points.

3. Face Centered Cubic Lattice

Fig. 8. A conventional cell of the FCC cubic lattice.

3a. F23 Point Group

For the first F23 FCC lattice calculations, 8x8x8 k points were sampled with 0.1eV Gaussian smearing and no shift. This resulted in the rough calculation being performed on 88 total k points and the finer calculations being performed on 176 k points. The finer calculation were performed on 16x16x16 k points, resulting in a total sample of 688 points, with an energy cutoff of 420eV. Both the rough and fine calculations resulted in a lattice constant of 2.80 Angstroms.

Table 4. Shows the roughly calculated energy of the F23 FCC lattice compared to the lattice constant.

Fig. 9. Shows a rough calculation of the energy of the F23 FCC lattice vs. the lattice constant. The minimum occurs around 2.8 Angstroms, which will be used as a starting point for the finer calculations. The lines connecting the data points act as a guide to the eye and are not data.

Fig. 10. K space sampling vs. energy (eV) for the F23 FCC lattice. The energy converges to 0.01eV beyond 76 points.

Table 5. Shows the lattice constant (Angstroms) vs. energy for the F23 FCC lattice with a finer k space sampling and energy cutoff. The minimum lies at approximately 2.8 Angstroms.

Fig. 11. Graph of the lattice constant (Angstroms) vs. the finely calculated energy (eV) of the configuration. The lines connecting the data points act as a guide to the eye and are not data.

3b. FM-3M Point Group

For the second FCC lattice calculation, the FM-3M point group was used, as platinum has been shown to form an FCC lattice under this point group in nature [2]. For the rough calculation, 8x8x8 k points were sampled with a 0.1eV Gaussian smearing and no shift, for a total of 20 k points. For the finer calculations, 16x16x16 k points were sampled, for a total of 120 points.

Table 6. Shows the energy of the FM-3M FCC lattice compared to the lattice constant.

Fig. 12. Shows the lattice constant vs. the energy of the FM-3M FCC Lattice for a rough calculation. The minimum occurs at 3.95 Angstroms, which will be used as a starting point for finer calculations.The lines connecting the data points act as a guide to the eye and are not data.

Fig. 13. K space sampling vs. energy (eV) for the F23 FCC lattice. The energy converges to 0.01eV beyond 35 points.

Table 7. Results for the fine calculations of the energy of the FM-3M FCC lattice at various lattice constants.

Fig. 14. Graph of the fine calculations of the lattice constant (Angstroms) vs. energy (eV) for the FM-3M FCC Lattice. The lines connecting the data points act as a guide to the eye and are not data.

For the FM-3M FCC lattice, the fine calculations were performed with a 420eV energy cutoff and a k space sampling of 16x16x16, resulting in a total of 120 k points being used. These results gave a minimum energy of -52203.84eV at 3.96 Angstroms.

Final Results

(a)

(b)

Fig. 15. (a) Comparing the rough results from the F23 FCC lattice, the P23 simple cubic lattice, and the D3H-3 HCP lattice with c/a=1.633, the F23 lattice has the lowest energy minimum. (b) Comparing all four lattice structures, the FM-3M FCC lattice energy is approximately 4 times lower and reaches a minimum at about 1.16 Angstroms higher lattice constant. The lines connecting the data points act as a guide to the eye and are not data.

For the first three lattice configurations, the F23 FCC lattice had the lowest energy minimum at lattice constant 2.8 Angstroms. However, the FM-3M FCC lattice has a lower energy by approximately a factor of 4, with a minimum at 3.96 Angstroms. This lattice configuration closely matches with experimental data, which shows that platinum forms a lattice in the FM-3M point group with lattice constant 3.92 Angstroms [2].

Appendix: Energy Cutoff Convergence

For the finer energy vs. lattice constant calculations, a 420eV cutoff energy was chosen based on the energy’s convergence to 0.01eV beyond this point.

Fig. 16. Graph of the energy cutoff (eV) vs. energy (eV) for platinum in an F23 FCC lattice. The energy converges beyond a cutoff of approximately 390eV and 420eV was ultimately used for the finer energy calculations.

Bibliography

2. Povarennych, A. & Povarennyck, A. Crystal chemical classification of minerals. 192 (Plenum Press, 1972).

Determining the Lattice Parameters of Hf

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DFT and Lattice Parameters

For many metals (simple cubic, body-centered cubic, and face-centered cubic) using DFT to calculate the lattice parameter for a metal or crystal is rather straightforward as there is only one parameter to vary. One may begin by assuming the total energy of the system is a Taylor expansion of the lattice parameter, a, as below:

\begin{equation}E_{tot}(a) = E_{tot}(a_{0}) + \alpha (a-a_{0}) + \beta (a-a_{0})^2\end{equation}

Following Sholl (1), we may reduce this equation to:

\begin{equation}E_{tot}(a) = E_{tot}(a_{0}) + \beta (a-a_{0})^2\end{equation}

Thus, the lattice parameter for many metals and crystals can be determined by sampling various values of a. The energy of the system can then be calculated at each value of a and should look like a quadratic with a minimum at the true value of the lattice parameter.

Lattice Parameters for hcp Crystals

For hexagonal-close packed (hcp) structures, determining the lattice parameters is not so straightforward. For sc, fcc, and bcc metals there is only one parameter to optimize the energy. When considering an hcp metal, there are two lattice parameters on which the total energy depends: a, and c.

It is still possible, however, to manage this multi-variable minimization using our technique from above. First, we fix the ratio c/a to some value, r and sample various values of a at which to calculate the energy. With c/a fixed we will also know the value of c and we may construct our energy curves at varying values of r. The curve with the lowest total energy at its minima will be considered the “theoretical values” of the lattice parameters.

Lattice Parameters of Hf

Hafnium, element 72 on the periodic table, is a d-block transition metal with an hcp crystal structure. We wish to use DFT, as outlined above, to determine the equilibrium (ground state) lattice parameters of Hf.

Below are results obtained from CASTEP single point calculations for r = 1.40, 1.48, 1.58, 1.72, and 1.85 with an energy cut-off of 290 eV and a k-point grid of 8x8x6.

From this plot we can determine that the optimum value of r is about 1.58. Using this, a refined set of calculations may be performed at r = c/a=1.58 for various a to calculate an accurate estimate for both a and c. The results of these calculations are found below.

Fig 2. Refined determination of a.

In the above plot the blue points are CASTEP results and ther orange line is the harmonic approximation of the total system energy around the true lattice parameter. From this data we can observe a couple of things. First, we can say that the lattice parameters of Hf are approximately:

$$a_{0} = 3.22$$ $$c_{0} = 5.10$$

We may also see that the behavior of the energy is not truly quadratic with respect to the lattice parameter(s). The harmonic potential is a very good approximation for a near a0, but for a > a0 the harmonic potential is an overestimate of the energy and an underestimate for a < a0. This is because at larger separations (a > a0) the energy decreases as it should approach the dissociation energy of Hf (a → ∞) while at shorter separations (a < a0) the energy increases due to strong repulsive forces.

Fig. 3 Optimized Hf Cell

 

 

WebElements, an online reference for chemical elements, reports the lattice constants for Hafnium as (2):

$$a_{0}^{ref} = 3.20$$

$$c_{0}^{ref} =  5.05$$

Thus our calculations agree quite well with available data and we are satisfied.

Convergence

The above results were found using a relatively small energy cut-off (290 eV) and k-point grid (9x9x6) to allow for quick calculations that give an idea of the behavior or the energy with respect to the lattice constant. We now wish to see if we were converged with respect to the energy cut-off and k-point grid.

For the energy cut-off:

Fig 4. Energy Cut Off Convergence

While it appears that a rather high energy cut-off (~600 eV) is needed for convergence, it is important to realize that the energy differences, even between Ecut=250-270 are within chemical accuracy (~0.04 eV) and therefore using an Ecut of 290 should produce rather reliable results (as we have confirmed with literature values). Chemical accuracy is a standard used by computational chemists as a benchmark for making reliably accurate chemical predictions. Essentially, if energies are within ~0.04 eV then we can make confident predictions; this is exactly the case for our Ecut convergence.

Similarly for the k-points (where we have fixed the ratio kx/kz = ky/kz = 4/3) we find the following. Here our convention for “# of KPoints” is just to add the # of KPoints in each direction (e.g. 8x8x6 = 22 KPoints in the plot)

Fig. 5 K-Point Convergence

Again, we see that not very many k-points are needed (~20) before our energies are within chemical accuracy. This means that our use of an 8x8x6 k-point grid is sufficient for our needs.

Single Point Calculations

For the single point calculations performed for Hf with varying lattice constants the following (ultra-fine) calculation settings were used:

Exchange Correlation Functional Type: Generalized Gradient Approximation (GGA)

Exchange Correlation Functional: PBE

Plane-Wave Energy Cut-Off: 290.0 eV

K-Point Grid: 9x9x6

Pseudopotentials: Ultrasoft

Electronic Energy Convergence Criteria: 5.0 10-7 eV/atom

Geometry Optimization

For the geometry optimization of Hf, the following (ultra-fine) calculation settings were used:

Exchange Correlation Functional Type: Generalized Gradient Approximation (GGA)

Exchange Correlation Functional: PBE

Plane-Wave Energy Cut-Off: 290.0 eV

K-Point Grid: 9x9x6

Pseudopotentials: Ultrasoft

Electronic Energy Convergence Criteria: 5.0 10-7 eV/atom

Ionic Energy Convergence Criteria: 5.0 10-6 eV/atom

Ionic Force Convergence Criteria: 0.01 eV/Å

Future Work:

In the future it would be of interest to investigate the effect of the type of pseudo-potential used (e.g. ultrasoft, soft, etc.) as well as the psuedo-potential cut-off for treating the core and valence electrons. Also of interest would be how our results change using different XC Functionals.

References:

(1) Sholl, D. & Steckel, J. A. Density Functional Theory: A Practical Introduction. (John Wiley & Sons, 2011).

(2) https://www.webelements.com/hafnium/crystal_structure.html

(3) 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

Determining the Lattice Constants of Hf in an hcp Crystal Structure

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Purpose of Calculation

“Hf is experimentally observed to be an hcp metal with c/a = 1.58. Perform calculations to predict the lattice parameters for Hf and compare them with experimental observations.” [1]

Fig. 1 The unit cell for hafnium in the hexagonal close-packed crystal structure

Calculation Methodology

The lattice parameter ratio (c/a) and the lattice constant a are predicted for hafnium in the hcp unit cell, by calculating the minimum energy of the system.

All calculations are performed with the Perdew-Burke-Ernzerhof (PBE) [2] exchange-correlation functional, a Generalized Gradient Approximation (GGA) functional. Pseudopotentials were  calculated on the fly, with the cutoff the 4f electrons and above used as the interacting electrons (4f14 5s2 5p6 5d2 6s2), while lower energy electrons were designated as core electrons (1s2 2s2 2sp 3s2 3p6 3d10 4s2 4p6 4d10). The Koelling-Harmon relativistic treatment was used for atomic solutions. [3]0.1 eV Gaussian smearing was used. Calculations were performed with Castep [4].

Convergence Calculations

Calculations to check the convergence of the minimum energy output with respect to the energy cutoff and the k-point mesh were performed, as shown in figures 2 and 3.

Fig. 2 Convergence calculations of the energy with respect to the k-point number

Fig. 3 Convergence calculations of the energy with respect the energy cutoff

Calculations performed with respect to the energy cutoff were performed with a = 3.1946 angstroms, c = 5.0511 angstroms, and a k-point mesh of 9x9x6. Between 480 eV and 500 eV, the free energy varies less than 0.005 eV, so we select 480 eV as our cutoff energy. Similarly, while varying the k-point mesh, we hold the energy cutoff at 480 eV. At a k-point mesh of 9x9x6, the free energy is similarly converged to 0.005 eV. The remaining calculations were performed at 480 eV and a k-point mesh of 9x9x6.

Calculation Results

Figure 4: Calculation of the internal energy of hcp hafnium as a function of lattice constant a for different ratios of the lattice constants, c/a. Included straight lines are meant to guide the eye and aid in identifying data curves. They do not represent the data themselves.

The minimization of the energy with respect to variation in the lattice constant ratio occurs at c/a = 1.581, which matches experimental observations, and helps support that these calculations are properly converged. The value of a that minimizes the free energy for this ratio is 3.20 angstroms, giving a value of 5.059 angstroms for the predicted value of c. These values for the lattice constants predict the expected unit cell for hafnium in the hcp crystal structure, and agree well with reference values from experiment [5].

References

[1]  D. Sholl and J. Steckel, Density Functional Theory: A Practical Introduction. (Wiley 2009)

[2]  John P. Perdew, Kieron Burke, and Matthias Ernzerhof, “Generalized Gradient Approximation Made Simple”, Phys. Rev. Lett. 77, 3865 – Published 28 October 1996; Erratum Phys. Rev. Lett. 78, 1396 (1997)

[3]  D D Koelling and B N Harmon 1977 J. Phys. C: Solid State Phys. 10 3107

[4]  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)

[5]  https://www.webelements.com/hafnium/crystal_structure.html