Author:
Mihir Parekh
Abstract:
In this project, ‘Material Studio’ has been used to study the crystal structure of lithium by using Density Functional Theory (DFT). Optimum lattice parameters have been obtained for BCC, SCC, and FCC lattices by using plane-wave basis set density functional theory methods [1]. A comparison of 0 K lattice energies at the optimum lattice parameter suggests that the energy difference between BCC and FCC is lower than 0.005 eV, with BCC having lower energy. However, SCC has a significantly higher energy compared to BCC and FCC.
Introduction:
Lithium is a very important component of batteries today. Today, lithium ion batteries are being used in variety of applications such as electric vehicles, laptops, mobile phones, grid storage etc. and lithium metal batteries are amongst the most promising future energy storage technologies. Hence, lithium is one of the most important metals and so the crystal structure of lithium was studied in this project. Density Functional Theory (DFT) was used for studying the crystal structure as it is much cheaper than using experimental techniques such as X-ray diffraction. Moreover, repeating calculations in a software is also much cheaper than repeating an experiment.
Methods:
In order to study the crystal structure of lithium, energy was minimized for different crystal structures by using CASTEP in Material studio. For all calculations Generalized Gradient Approximation (GGA) and Perdew-Burke-Ernzerhof (PBE) exchange correlation functionals were used with On The Fly Generated (OTFG) ultrasoft pseudopotential. A core radius of 1 Bohr and an electronic configuration of 1s2 2s1 was used for calculations. Minimum energy of different crystal structures was compared to obtain the optimum crystal structure and the corresponding lattice parameter.
Fig. 1: Optimization of no. of k points
However, before varying the lattice parameter for a given crystal structure, the optimum number of k points and the cutoff energy to be used in calculations were obtained. The number of k points was optimized for BCC lattice with a lattice constant of 346 pm. BCC lattice was chosen because at room temperature lithium crystallizes as BCC with a lattice constant of 351 pm [2]. Since 351 pm is the lattice constant of BCC at room temperature, varying the lattice constant from 346 pm to 356 pm would be sufficient to get an optimum lattice constant at which the energy is minimized. Moreover, lowest lattice constant requires the highest number of k points and hence 346 pm was chosen for optimizing the number of k points. Fig. 1 shows that a 12x12x12 grid in k-space is sufficient for an accuracy of 0.005 eV. The values plotted on y axis of Fig. 1 are the differences between energies of the current no. of k points and previous no. of k points. While optimizing number of k points, the cut-off energy was held constant at the default value of 408 eV.
Note: For Figs. 2,3,4,5 the energy values on y axis are relative to the lowest energy obtained during cutoff energy optimization.
Fig. 2: Cutoff energy optimization
Cut-off energy was then optimized for 12x12x12 grid on a BCC lattice with 346 pm as the lattice constant. Fig. 2 shows that a cutoff energy of 775 eV is good enough to obtain a roughly constant energy. The difference between energies for a cutoff energy of 775 eV and 900 eV is roughly 0.00057 eV. The energy difference for cut-off energy of 900 eV and 1000 eV is roughly 0.00056 eV. This shows is that the set tolerance criteria of 0.005 eV is satisfied if 775 eV is chosen as the cut-off energy.
Results:
For a fair comparison of the crystal energies, using a constant cut-off energy of 775 eV and a 12x12x12 grid in the reciprocal space, lattice parameters were varied for BCC, FCC and SCC lattices. However, for an SCC lattice a step change was seen in the energy vs lattice parameter plot. Hence, a higher cut-off energy of 1200 eV was used to obtain the optimum lattice parameters. The variation of energy versus lattice constant for BCC, SCC, and FCC lattices has been shown in Figs. 3, 4, 5 respectively. Thus, BCC with a lattice constant of 343 pm seems to have the least energy at 0 K. However, the difference between lowest energies of FCC and BCC is lower than the tolerance limit (0.005 eV) and hence it is difficult to distinctly predict the crystal structure. In Fig. 4 of their paper, Orlov et al. [3] mark a phase transition boundary for temperatures greater than 50 K. For any given pressure, Orlov et al. [3] predict FCC lattice at lower temperatures. So, the obtained result does not agree with the result reported by Orlov et al. [3].
Fig. 3: Lattice parameter optimization for BCC
Fig. 4: Lattice parameter optimization for SCC
Fig. 5: Lattice parameter optimization for FCC
Limitations:
- Initially the number of k points were optimized for a lattice constant of 346 pm. However, clearly 346 pm is not the lowest amongst the various lattice constants used for BCC lattices. Thus. the number of k points used have not been optimized for the lowest lattice constant.
- The cut-off energy and number of k-points were not optimized for FCC and SCC lattices.
References:
[1] Clark SJ, Segall MD, Pickard CJ, Hasnip PJ, Probert MI, Refson K, Payne MC. First principles methods using CASTEP. Zeitschrift für Kristallographie-Crystalline Materials. 2005 May 1;220(5/6):567-70.
[2] Mark Winter, U. (2020). WebElements Periodic Table » Lithium » the essentials. [online] Webelements.com. Available at: https://www.webelements.com/lithium/ [Accessed 31 Jan. 2020].
[3] Orlov AI, Brazhkin VV. Electron transport properties of lithium and phase transitions at high pressures. JETP letters. 2013 May 1;97(5):270-3.
