Change in the Phonon Frequency Spectra of Xenes due to an Isotopic Impurity

Xenes are new mono-elemental two-dimensional (2D) materials (e.g., silicene and germanene) that are “2D materials beyond graphene”. They represent a revolutionary new development in the science and technology of materials. Compared to normal 3D solids, the electronic and phononic characteristics of these materials are highly unusual.<br/>Some Xenes have a strong spin-orbit coupling that gives rise to quantum spin Hall effect. They also have an electrically tunable band gap. These properties make Xenes potentially very useful in the design of exciting new solid-state devices such as heat management, thermoelectric energy conversion, quantum computing, field effect transistors and spintronics. Hence, there is strong contemporary interest in exploring silicene and other Xenes for diverse industrial and defense applications. Silicene is also of special interest because silicon-based device technology is already highly developed and may be transferred to silicene.<br/>In general, the performance of a device, is sensitive to the phonon spectrum of its material. For example, thermal conductivity of materials is an important design parameter for the devices. It is sensitive to the DOS (density of states) of low-frequency phonons. For energy conversion devices, the thermal conductivity of the material should be low but the electrical conductivity high. Hence, for designing thermal devices, it is desirable to be able to vary the electrical and thermal conductivities independently. However, in natural materials there is usually a strong positive correlation between the two conductivities.<br/>One way of satisfying this contradictory requirement is to introduce isotopic impurities in otherwise pure crystals. Isotopes of an atom have different masses but same electronic structure. A heavier isotopic impurity results into a lower phonon frequency with no effect on its electronic properties. Thus, in principle, isotopic impurities can tune the thermal conductivity of the host without affecting its electrical conductivity. For design purposes, therefore, it is important to study the effect of isotopic impurities on phonons in silicene.<br/>Modeling the effect of isotopic impurities is also needed for characterization of silicene samples. The common isotope of silicon is ²⁸Si. Its abundance is about 92.2 %. The abundance of its two heavier isotopes is 4.7% for ²⁹Si and 3.1% for ³⁰Si. Hence a normal sample of silicene is likely to contain almost 8% isotopic impurities. This explains the strong topical interest in mathematical modeling of the effect of isotopic impurities on phonon spectra of Xenes, which is the objective of our research.<br/>The modeling challenge in such calculations arises because the 2D materials have a strong size effect. Hence, a large crystallite needs to be simulated, which makes the conventional techniques based upon molecular dynamics or density functional theory computationally very expensive. Moreover, low frequency phonons are needed to model the large time behavior of the material system, for which the conventional techniques are poorly convergent.<br/>In our calculations, we write phonon Green’s functions (GF) in complex frequency space. This representation yields an analytical expression for the perturbed phonon modes of the lattice. A major advantage of this technique is that the imaginary part of the GF directly gives the full phonon DOS. Further, our calculations show a very interesting fact that even isotopic impurities in a lattice can interact. Intuitively, isotopic impurities should be non-interacting, because they do not induce a strain field in the solid. We show that they have a thermodynamic interaction at non-zero temperatures.<br/>In this talk, we will briefly describe our phonon GF method and its application to calculation of phonon modes in Xenes containing an isotopic impurity. Numerical results will be presented for silicene.