1. First-Principles Computational Study of the Properties of Some Silicon-based Type II Clathrate Compounds Dong Xue and Charles W. Myles
Introduction
The Type II silicon-based clathrate structure is characterized by the cubic space group of symmetry pm3n. The lattice framework of these materials contains 136 atoms in the cubic unit cell. This empty clathrate lattice is formed by 20- and 28-atom polyhedron cages connected periodically in a 2:1 ratio. Because of this “cage-structured” configuration, the unfilled framework is able to encapsulate guests such as alkali metal or alkaline earth atoms, helping to provide a novel guest-framework system for further investigation.
Clathrate compounds are of practical interest because they can play a crucial role in seeking efficient thermoelectric (TE) materials. High-performance TE devices made of clathrate compounds possess very good electrical transport properties and glass-like thermal transport properties. They thus fall into the category of “phonon-glass electron-crystal” (PGEC) materials as suggested originally by Slack [1]. Previous work on Type-I and Type-II clathrate materials have shown that the low-lying “rattling” modes of the loosely bound guests in the polyhedral cages (Si24, Si28) results in reduced lattice thermal conductivity, contributing to a large, dimensionless figure-of-merit (ZT value) [2].
Methods
Our first principles calculations are based on the local density approximation (LDA) to density functional theory and have utilized the Vienna ab-initio Simulation Package (VASP). Starting with structural optimization, we determine the lattice constant at the equilibrium geometry through a conjugate gradient method.
Our approach to obtain our predicted phonon-dispersion curves, as well as the effective potential energy diagrams for the guest “rattlers” in the Si hexakaidecahedra cage is summarized as follows. The first step is to obtain the dynamical matrix D(q) after moving each guest atom initially located at Si28 cage center by a small finite displacement (U0 = 0.02Å), in the presence of a 222 k-point grid as well as for wave vectors in the vicinity of gamma (Γ) point [q = (0,0,0)]. Secondly, diagonalization of the dynamical matrix D(q) allows us to determine the vibrational eigenvalues (squared frequencies) and eigenvectors. In addition, we have used the quasiharmonic approximation (QHA) method to explore the mode Grüeneisen parameters for transverse acoustic (TA) and longitudinal acoustic (LA) phonons along different high-symmetry directions in the first Brillouin Zone.
Abstract
Type II Si-based clathrate materials with alkali metal atom guests in the lattice cages have shown great promise for use as high-performance thermoelectric materials. Studying these materials can also reveal some very interesting basic physics. Here, we report the results of a systematic, first principles, theoretical and computational study of the structural, electronic, thermal and vibrational properties of some Type II Si-based clathrate materials. Our calculations are based on density functional theory and utilize the VASP code. Our predictions include the guest atom composition (x) dependence of the lattice parameters in the binary compounds AxSi136 (A=alkali metal atom, 0<x≤24). For these materials, we find that as x increases, the lattice contracts for 0<x<8 and expands for 8<x<24, in agreement with recent observations. We also present results for the x dependence of the elastic constants, the electronic densities of states, and the vibrational modes in these binary materials. The results of our investigation of guest-framework coupling in the compounds Rb8Al8Si128, Cs8Al8Si128 are also presented. Finally, we discuss results for the composition (x) dependence of the volume thermal expansion coefficient in the alloy clathrate Si136-xGex (x = 8, 32, 96) (with empty cages). The mode Grüneisen parameters for the transverse acoustic (TA) and longitudinal acoustic(LA) phonons along different high-symmetry lattice directions are also explored for these alloys. For these same materials, by using a quasiharmonic approximation method to treat the lattice vibrations, we find a negative thermal expansion coefficient for some temperatures.
Conclusions
Our calculations predict a lattice contraction as x increases for 0 < x < 8 and an expansion as x increases for x > 8 in NaxSi136 (0 < x ≤ 24) [3]. This is consistent with X-ray diffraction (XRD) data [4], which has shown that the larger Si28 cages are filled first for x < 8. However, if Na is replaced by K, Rb, or Cs, we find no such unusual lattice response to filling the cages. Hence, the incorporation of heavier guests into Si28 cages causes a much smaller change in the lattice constant than we have found in NaxSi136 (x = 4,8,12).
We have compared our calculated Cs guest “rattling” mode frequencies to their Rb counterparts and have examined how the mass of the guest atom effects the guest-framework coupling strength in clathrates of the form Rb8Al8Si128 and Cs8Al8Si Since Cs is a heavier atom than Rb, it might be expected that the Cs ratting mode frequency would be below that for Rb. However, we find that the Rb mode lies in the range cm-1 which is below the rattling mode frequency we find for Cs guests (40-42 cm-1).
Given the calculated effective potential of A8Si136 (A = K, Rb, Cs), we have found that an increased mass ratio of guest to host is intrinsically associated with weak anharmonic effects.
The results of our calculation of the temperature dependence of the lattice thermal expansion coefficient for the empty Type II Si clathrate lattice predict a negative thermal expansion for (NTE) below T = 200 K. The calculated band-index and wave-vector dependent mode Grüneisen parameters (γi) below T = 200 K are also negative, which is consistent with an NTE phenomenon. We also have found that γi values for the TA and LA modes along the L-Γ-X line are negative for Si136-x Gex (x = 8, 32, 96) alloys.
Results
Figure 4: Predicted Volume Expansion Coefficient for Si136
and for Diamond Si
Figure 3a: Predicted Phonon Dispersion Curves for Cs8Al8Si128
Direction
Mode
Si128Ge8
(ωi , cm-1)
Si104Ge32
(ωi , cm-1)
Si40Ge96
(γi)
[001] TA 99 91 76
-0.96
-0.83
-0.38 LA
129
104 80
-0.74
-0.63
-0.56
[111] 92
-0.97
-0.57
Figure 1: Predicted x Dependence of the Lattice Constant for the
Type II Clathrate-Based Compounds AxSi136 (A = Na, K, Rb, Cs)
Table I: Predicted Mode Gruneisen Parameters for Si136-xGex
Figure 3b: Predicted Phonon Dispersion Curves for Rb8Al8Si128 b
References:
1) Slack, G.A. Materials Research Society: Pittsburgh, PA, USA, 1997; Volume 478, p. 47.
2) Charles W. Myles, Jianjun Dong, and Otto F. Sankey. phys. stat. sol. (b) 239, No. 1, 26-34, (2003).
3)Dong Xue, Charles W. Myles, and Craig Higgins. materials, (2016).
4)Beekman, M.; Hermann, R.P.; Möchel, A.; Juranyi, F.; Nolas, G.S. J. Phys. Cond. Matter 2010,22, a c d
Figure 5: Predicted Mode
Gruneisen Parameters for
a) Si136, b) Si128Ge8,
c) Si104Ge32, d)Si40Ge96
Figure 2: Predicted Effective Potential Curves for Alkaline Atom
Guests in the Si28 Cages in Si136.