The application of the hybrid topological-quantum-mechanical method to the search of new allotropes of 14th group elements is demonstrated for silicon and germanium. Starting from the databases of hypothetical and real zeolite nets and subsequently apllying the geometrical and energetic selection criteria, we extract the most energetically favourable structures for the allotropic modifications of silicon and germanium, and study their optical properties. In the framework of density functional theory we calculate the frequency-dependent complex dielectric tensors, refraction and absorption coefficients of the selected allotropes and their electronic band gaps.
2.308-2.382 2.308-2.415 2.308-2.406 2.301-2.386 2.322-2.422 2.322-2.380 95.98-126.34 95.31-125.84 93.14-124.12 96.95-127.46 97.91-120.04 98.16-119.32 3.818 3.820 3.818 3.826 3.828 3.837 Germanium
The recent development of methods for quantum mechanical calculations of energy and the electron density distribution of
many-electron systems (atoms, molecules, crystals) is associated with the success of density functional theory (DFT) [6,7,8] and
combined or hybrid methods based both on DFT and the Hartree-Fock (HF) approach [9], which take into account the exchange
interaction more accurately than the DFT. Quantum mechanical calculations of the physical properties of crystals require a
significant computational resources, in comparison with atomic or molecular calculations, and can be performed only on
supercomputers or multiprocessor cluster systems. In our work, we use the most common and widely used licensed software
packages CRYSTAL [
To check the stability of the predicted silicon and germanium allotropes, the matrices of elastic constants and various elastic coefficients were calculated, and we showed the structures to be mechanically (energetically) stable at zero external pressure (see Fig. Tables 2 and 3). We also demonstrated the absence of imaginary frequencies in the phonon spectra of studied silicon and germanium allotropes. As an example, Fig. 1 shows the calculated phonon spectrum for the allotrope Si#50. All calculations were performed in the generalized gradient approximation (GGA) of the density functional theory with the exchange-correlation functional PBE [12].
Methods for calculating the optical properties of crystals depend on the chosen electromagnetic wavelength range. The properties of the complex dielectric tensor in the infrared region are determined in the semi-classical Drude-Lorentz theory by singularities of the crystal lattice vibration spectrum, which can be calculated in the quasi-harmonic approximation [13] with the transverse and longitudinal optical vibration modes. The spectrum of eigenfrequencies of allotropes allows one to calculate their Raman scattering spectra and absorption spectra in the IR range. Investigation of Raman spectra can be used for experimental search for new allotropic modifications, since the position of the Raman peaks is uniquely related to the structure of the crystal lattice, the forces and lengths of the chemical bonds. Unlike the diamond modification of silicon, the Raman spectra of its allotropic modifications have a more complex structure, which makes it difficult to identify them experimentally. At the same time, the Raman spectra of germanium allotropes, like the Raman spectrum of the diamond modification of germanium, have only one strong peak, which is significantly shifted relative to the peak at a frequency of about 300 sec-1, which is observed for the ground state of germanium, Fig. 2. It is well known that diamond and diamond-like crystals of silicon and germanium practically do not absorb in the IR range. The silicon and germanium allotopes predicted by us have narrow absorption bands in the IR range, which can also be used for their experimental search and identification, see Fig. 3.
The electronic band structure of a crystal determines the properties of its complex dielectric function, hence, the dependence of the absorption and refraction coefficients on the frequency of electromagnetic radiation in the visible and UV ranges. DFT allows to obtain a microscopic dielectric function in the random phase approximation within the theory of linear response, in the visible and ultraviolet frequency ranges. Neglecting the local field effects, we can derive from the microscopic dielectric function a macroscopic dielectric function. The imaginary part of the latter is a tensor and can be written in the form of a weighted sum over transitions between levels, and the real part is obtained from the imaginary one using the Kramers -Kronig relations. We can extract both the optical constants and the complex dielectric function averaged over the directions directly from the components of the macroscopic dielectric tensor, which allow us to determine the optical absorption and refraction spectra for the studied structures.
The electronic band gap of a semiconductor determines the energy boundary of absorption of optical photons. It is known that standard DFT methods do not allow to reproduce this gap correctly, in particular for materials with a narrow optical gap, such as germanium. Therefore, in our calculations we used the hybrid functional HSE06 [14], which allows to obtain results comparable to the experimental data. However, the standard functional PBE is more adequate in describing the position of the peaks of the complex dielectric function. Tables 2 and 3 show the calculated values of the band gaps (indirect and in the Gamma point of the reciprocal space) and the permittivity coefficients of silicon and germanium allotropes, respectively. We note that the small value of the band gap of the allotrope Ge#27, 0.23 eV, may indicate metallization of this structure at high temperatures and loss of semiconductor properties. In Fig. 4 we present an electronic band structure for the allotrope Si#27 and Ge#27.
In Fig. 5 we show the calculated (dash-dotted line – HSE06, dashed – PBE) curves in comparison with the experimental data (black line) of the frequency-dependent refractive indices n for diamond configurations Si (left) and Ge (right). The absorption spectra (k), together with the relative spectrum of solar irradiation at the reference air mass of 1.5, are shown in Fig. 6, respectively. Amorphous forms of Si and Ge, the so-called a-Si and a-Ge, as well as their various hydrogenated forms and some Si-Ge compounds [1] demonstrate promising properties for use in electronics and photovoltaics, especially in solar cells. 3rd International conference “Information Technology and Nanotechnology 2017” 13 Comparing the results for our allotropes for the refraction and absorption coefficients with the corresponding spectra of amorphous forms, we observe quantitative and qualitative agreement both for the position of the refraction/absorption peak and for its absolute value (see Figures 7 and 8). This can be an evidence that the predicted by us allotropes can be a counterpart of the corresponding amorphous forms.
Fig. 7. The refractive index for allotropes #28 of silicon (left) and germanium (right). Dash-dotted line – the results obtained for the functional HSE06, the dashed line – for the functional PBE, solid line – experimental data for the amorphous forms [1].
We investigated the six new low-energy allotropes of silicon and germanium [15], isostructural to the previously proposed
carbon allotropes [5], demonstrating an application of the hybrid topology-quantum-mechanical approach [5] to prediction of
new structures. Using the ab initio methods implemented in the CRYSTAL [