By Charles Bigelow
Abstract
A copper crystal was analyzed using DFT to observe the relaxation effects of three of its Miller indices to predict the optimal surface structure under zero degree Kelvin conditions. The three miller indices observed were the 1 1 1, 1 1 0, and the 1 0 0 lattices. The overall result of the calculations indicated that 1 1 1 Miller plane is the most stable structure of the copper surface.
Introduction
Oftentimes, a metallic or crystal compound will have several different configurations depending on which plane of reference that is being analyzed; these are referred to as miller indices, which are planes which bisect a given crystal lattice structure at particular coordinates in reference to the base structure. These various indices are of crucial importance to chemical and physical properties of a substance and its surface.
One can use Miller indices to generate and calculate surface energies and orientations of a particular lattice using DFT. This gives the ability to calculate a particular material’s surface structure theoretically; this gives added flexibility as simulations are able to analyze systems normally outside of the capabilities of current experimental methods with relative ease, and allows prediction of structures that have not yet been synthesized. This is often accomplished by constructing a surface of the crystal at a specific plane, a certain number of layers selected, and then a vacuum slab generated to produce a surface with periodic boundaries. The vacuum must be large enough to prevent interactions between successive slabs in the periodic structure.
Methodology
In order to observe the surface energy of the various copper Miller planes, surface slabs were generated using the materials studio software. Various slabs needed to be generated, each having multiple iterations with different thicknesses. The significance in varying the slab thickness is to ensure that the inner portion of the slab is significant enough in size to properly simulate bulk properties; too small of a slab would produce unconverged energies, while too large of a slab would significantly increase computational costs.
Slab generation was done by taking a FCC single unit lattice of copper and cutting along the miller indices of 111, 110, and 100 using the cleave crystal function found in the build section of Materials Studio. All three lattices were using a single unit cell. 111 was generated in integer steps of thickness, each layer adding another atom into the lattice. for 110 and 100, the slabs were generated using half integer steps, as their layers consist of 2 atoms per layer, versus the 1 atom per layer of 111; this is most likely due to the structure repeating after two atom’s depth for both 110 and 100. The vacuum distance used for all slabs was 10 angstroms to prevent interactions between the periodic slabs in the C axis of the lattice structure.
Figure 1: The 111 surface at a relative thickness of five layers
Figure 2: The 110 surface at a relative thickness of 2.5 layers
Figure 3: The 100 surface at a relative thickness of 2.5 layers
The cutoff energy was converged for a five layered slab of 111 copper; 360eV was found to be an effective value to ensure efficient and accurate calculations, as it only differed by roughly 0.1eV from a cutoff frequency of 380eV . Higher values were observed at 400eV and 440eV, however these oscillated around the same energy within a value of 0.1eV. Further optimization of this parameter was limited due to constraints of computational power. 360eV was used for all three miller indices.
All three indices had a basis set of 1s2 2s2 2p6 3s2 3p6 3d10 4s1. The SCF tolerance was set at 2.0e-6eV/atom, and a maximum of 130 SCF cycles to allow for more precise convergence. a 4x4x1 k-point lattice was used for all three of the indices and was kept constant for each system; it would be more optimal to individually optimize the k-point mesh for each lattice type, however this was not pursued due to lack of computational capabilities. The pseudopotential which was used was on-the-fly-generation ultrasoft [1]. In these calculations, a GGA PBE functional was used; These parameters were used for all systems[2].
The surface energy was calculated using equation 4.2 in chapter four of the “Density Functional Theory, A Practical Introduction” book by David S. Sholl.
1/2A*[E(slab) – n*E(bulk)]
where 2A is the surface area in angstroms (both top and bottom surface), E(slab) is the energy of the slab, n the number of atoms present, and E(bulk) is the FCC copper lattice energy for one atom.
Results
Observation of the 111 energies displayed a convergence at a slab thickness of four and five; it should be noted that beyond five, convergence issues were present, but due to the limits in computational capability and time allotted, these were unable to be addressed, and would require further investigation to the cause of the notable deviation present; even with higher level calculations, the deviations remained in the calculations. These observations were also noted in the 110 and 100 slabs above 3.5 layers (7 atoms).
For the 110 and 100 slabs, convergence was achieved at 3-3.5 and 3.5-4 layers, respectively, with 3.5 being used as energy reference for both 100 and 110. Larger slabs ran into the issues described above. Before the energy per surface area was calculated, the gross difference between the bulk and the surface energy was roughly
Before the energy per surface area was calculated, the gross difference between the bulk and the surface energy ranged between two to three electron volts in difference, which is a factor of 10 greater than the 0.1eV deviations observed with the various cutoff values.
The energy for the chosen slabs of 111, 110, and 100 were
0.0946eV/Angstrom^2, 0.146eV/Angstrom^2, and 0.116eV/Angstrom^2 respectively.
Conclusion
By comparing the three surfaces, it is predicted that the 111 surface is the most probable surface present in a sample of copper metal; it is theorized that this is due to the larger density of atoms on the surface of the 111 miller plane that causes this lower energy; this prediction is consistent between the 100 and 110 surfaces as well. This analysis was consistent with what was seen in experimental data.[3,4]
URLs for References
- https://www.tcm.phy.cam.ac.uk/castep/documentation/WebHelp/content/modules/castep/tskcastepsetelecpotentials.htm
- https://www.tcm.phy.cam.ac.uk/castep/documentation/WebHelp/content/modules/castep/tskcastepsetelecxc.htm
- D. E. Fowler and J. V. Barth, Structure and Dynamics of the Cu(001) surface Investigated by Medium-Energy Ion Scattering, Phys. Rev. B 52 (1995), 2117.
- F. R. de Boer, R. Boom, W. C. M. Mattens, A. R. Miedema, and A. K. Niessen, Cohesion in Metals , North-Holland, Amsterdam, 1988.
