An Embedded-Atom-Method model that is successful at describing the vibrational properties of alkali metals in the BCC crystal structure is applied to the HCP structure. Phonon dispersion curves, density of states, and Debye temperatures are calculalated for Li, Na, K, Rb, and Cs. Comparison of BCC, FCC and HCP Debye temperatures show that BCC is the preferred structure at higher temperatures, which agrees with experiment

Christensen, Jake

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Utah State University DigitalCommons@USU Physics Capstone Project Physics Student Research 5-4-2018 Phonon Dynamics of Alkali Metals in the HCP Lattice Structure Jake Christensen Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/phys_capstoneproject  Part of the Physics Commons Recommended Citation Christensen, Jake, "Phonon Dynamics of Alkali Metals in the HCP Lattice Structure" (2018). Physics Capstone Project. Paper 65. https://digitalcommons.usu.edu/phys_capstoneproject/65 This Report is brought to you for free and open access by the Physics Student Research at DigitalCommons@USU. It has been accepted for inclusion in Physics Capstone Project by an authorized administrator of DigitalCommons@USU. For more information, please contact digitalcommons@usu.edu. Phonon Dynamics of Alkali Metals in the HCP LatticeStructureJake ChristensenMay 4, 2018AbstractAn Embedded-Atom-Method model that is successful at describing the vibrational propertiesof alkali metals in the BCC crystal structure is applied to the HCP structure. Phonon dispersioncurves, density of states, and Debye temperatures are calculalated for Li, Na, K, Rb, and Cs.Comparison of BCC, FCC and HCP Debye temperatures show that BCC is the preferred structureat higher temperatures, which agrees with experiment.1 IntroductionEmbedded-atom-method (EAM) models have been used to study many material properties of FCCand BCC metals, and have been successful at describing vibrational dynamics [1]. In 2012 Wilsonand Rie constructed an EAM model that predicts phonon dispersion and Debye temperatures inexcellent agreement with experimental data for the alkali metals [2]. Although Wilson did someseparate calculations on FCC metals, the model was primarily used for BCC alkalis [3].It is well known that lithium and sodium undergo a phase transition to a close-packed structure atlow temperature [4, 5]. Thus, studying the behaviour of the alkali metals in a close packed structurecan reveal useful information about the Wilson-Rie model. Also, because of the success of the Wilson-Rie model, there is reason to believe that it will model close-packed structures accurately. Here, weexplore the alkali metals in a hexagonal close-packed (HCP) structure. Dispersion curves, density ofstates, and Debye temperatures have been calculated and plotted.2 The EAM ModelThe EAM model is a model of the potential energy of a crystal. The model incorporates pair-wiseinteractions of atoms, as well as an embedding energy, which accounts for interactions of the atomswith the electron charge density of the lattice. With these two pieces, the potential energy for the ithatom in the lattice can be expressed asUi =12∑jφ(rij) + F (ρi), (1)where j labels atoms in the solid, rij is the distance between the atoms i and j, φ is the pair potentialfunction, F is the embedding energy function, and ρi is the electron density at the location of atom i.The density ρi can be written as a sum of individual atomic densties f(rij) from the atoms surroundingit,ρi =∑jf(rij). (2)12.1 The Dynamical MatrixTo describe the dynamics of the lattice, we follow the formulation of Ibach and Luth [6] by expandingthe potential energy of atom i [Eq. (1)] in a Taylor series expansion in small displacements snαi fromequilibrium. Here, α labels atoms the basis, n labels unit cells, and i labels the Cartesian componentof displacement. The Taylor expansion isU(rnαi + snαi) = U(rnαi) +12∑mβj,nαi∂2U∂rnαi∂rmβjsnαismβj + ..., (3)where the sum runs over all atoms in all unit cells in the x, y, and z directions. By taking a harmonicapproach and ignoring higher order terms, this equation represents an extension of the harmonicoscillator potential in the case of many particles. The derivatives of the potential,Φmβjnαi ≡∂2U∂rnαi∂rmβj, (4)have dimensions of spring constants and are called the coupling constants. These take the role ofspring constants in the equation of motion for the i component of atom nα,Mαs̈nαi +∑mβjΦmβjnαi = 0. (5)Here Mα is the mass of atom α. This gives rise to 3rN dierectial equations from the N unit cellseach with r atoms. However, with periodic structures, signicant simplications can be achieved byusing a suitable ansatz. We write the displacements snαi as a plane wave that is only dened at theatom locations rnα,snαi =1√Mαuαi(q)ei(q·r−ωt), (6)Here u is the amplitude of a given wave with wave vector q, and ω is the associated frequency.Substituting (6) into (5) and performing some simplications gives us the dynamical matrix, whoseeigenvalues tell us the frequency squared of the vibrations, and the associated eigenvectors give us thepolarizations. The elements of the dynamical matrix take the formDβjαi =∑m1√MαMβΦmβjnαi eiq(rm−rn). (7)Diagonalizing the dynamical matrix for a series of wave vectors along various high-symmetry directionsgives us dispersion curves, like those shown in Fig. 1 for BCC lithium.Figure 1: Lithium BCC Dispersion Curves22.2 Wilson-Rie EAM ModelIn the Wilson-Rie EAM model, previously introduced forms of the pair potential, embedding energy,and atomic density are used. The pair potential model is taken from Wang and Boerker [7]φ(r) =6∑n=0Kn(rr1− 1)nexp(− nα( rr1− 1)2), (8)and the embedding energy and atomic density are taken from Johnson and Oh [8],F (ρ) = −(Ecoh − EUF1v )[1− λ ln( ρρe)](ρρe)λ(9)andf(r) = f1 exp(− β( rr1− 1)). (10)Using this model to derive our dynamical matrix, and evaluating the matrix at certain wave vectors,allows us to express the values of Kn in terms of experimental data. λ, α, and β are left as adjustableparameters. The combination of experimental inputs and free parameters allows theWilson-Rie modelto produce highly accurate results for phonon dispersion curves of the alkalis in the BCC structure.3 Code DevelopmentIn order to apply the Wilson-Rie model to alkali metals in HCP structures several changes had tobe made to the code. When I began the project, the code was only able to perform calculations forcrystal structures with one atom per unit cell. The one atom per unit cell scenario allows severalsimplications to be made in calculating the dynamical matrix, and it was these simplications thatwere coded in. However, for structures with more atoms per unit cell, no general simplications canbe made. Thus, I came up with a new scheme to eciently calculate vibrational properties for anarbitrary lattice structure with an arbitrary number of atoms per unit cell.The scheme is very straightforward. The biggest computational hurdle is calculating the dynamicalmatrix many times, once for each wave vector you wish to analyze. In order to streamline this I simplyinserted a step in which the coupling constants Φmβjnαi are calculated once and stored in a le, and thenread into memory during computation time. With this, the work required to calculate an elementof the dynamical matrix is O(N), where N is the number of atoms being considered. This is a vastimprovement from the O(N2) work required if one were to calculate the coupling constants for eachwave vector to be analyzed.3.1 Dispersion CurvesWith the appropriate algorithms developed, dispersion curves for HCP alkali metals were able to becalculated. Since the HCP structure has 2 atoms per unit cell, the dispersion curves have 6 branchesin any direction. The dispersion curves for Li, Na, K, Rb, and Cs are shown in Fig. 2 below.3Figure 2: (In order of top-left, top-right, middle left, etc.) Dispersion Curves for HCP Li, Na, K, Rb,and Cs.3.2 Density of StatesThe Density of States (DOS) is an incredibly useful function to calculate when working out thermaland vibrational properties of crystal structures. From the DOS one can work out Debye temperatures,which in turn can be used to calculate specic heat, thermal expansion coecients, entropy, and more.The DOS measures how many available modes are in some small width of frequency. To calculatethis, one must create a mesh of k-space points in the rst brillouin zone of the lattice and solve thedynamical matrix at each point. The rst brillouin zone for the HCP lattice is a fairly simple hexaganol4zone, shown in Fig. 3 below.Figure 3: First Brillouin zone of the HCP latticeFrom these solved frequencies, one can create a histogram of frequencies which is used as the DOS.Thus, creating as ne a mesh as possible is essential. For our calculations, we used on the order of 106k-space points.Using the same scheme to calculate the dynamical matrix that was used for the dispersion curves,we generated DOSs for Li, Na, K, Rb, and Cs, and these results are shown in Fig. 4 and Fig. 5 below.Figure 4: Density of states for HCP Li, Na, K, and Rb.5Figure 5: Density of states for Cs.3.3 Debye TemperaturesWith an accurate density of states, denoted be g(ω) below, the Debye temperatures ΘD(n) can becalculated,ΘD(n) =~kB(n+ 3n´ωng(ω)dω´g(ω)dω) 1n. (11)This is a straightforward calculation, and the results for each of the alkalis are shown below in Fig.6, Fig. 7, and Fig. 8, each compared with their BCC and FCC values.Figure 6: Debye temperatures for Li in HCP, BCC and FCC structures.6Figure 7: Debye temperatures for Na, K, and Rb in HCP, BCC and FCC structures.7Figure 8: Debye temperatures for Cs in HCP, BCC and FCC structures.Notice that the BCC metals tend to have lower Debye temperatures. This explains why BCC isthe prefered structure: lower Debye temperatures indicate higher entropy and thus lower free energyat higher temperatures.4 ConclusionThe Wilson-Rife model shed some interesting light onto the vibrational properties of close-packedstructures. The Debye temperatures of lithium and sodium suggest that BCC is the prefered structureat higher temperatures, which is consistent with experiment. Further investigation into close-packedstructures will reveal even more. Moving forward, we plan to apply the model to the 9R structure, aswell as performing surface calculations for FCC, HCP and 9R.References[1] M. S. Daw, S. M. Folles, and M. I. Baskes, Material Science Reports 9, 251 (1993)[2] R. B. Wilson and D. M. Rie, Condensed Matter 24, 335401 (2012)[3] R. B. Wilson, An embedded atom method investigation into the lattice dynamics of metallicsurfaces (2011)[4] A. H. Daane, R. E. Rundle, H. G. Smith, and F. H. Spedding, Acta Cryst. 7, 532 (1954)[5] C. S. Barret, Acta Cryst. 9, 671 (1956)[6] H. Ibach and H. Luth, Solid State Physics (New York, 1993)[7] Y. R. Wang and D. B. Boerker, Journal of Applied Physics 78, 122 (1995)[8] R. A. Johnson and D. J. Oh, Journal of Material Research 4, 1195 (1989)8

Phonon Dynamics of Alkali Metals in the HCP Lattice Structure

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Phonon Dynamics of Alkali Metals in the HCP Lattice Structure

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