Icosahedral silicon boride: A potential hybrid photovoltaic-thermoelectric for energy harvesting

Boron-rich compounds have attracted significant attention due to their promising and diverse physical properties, which include ultrahardness, resistance to oxidation and corrosion, and even superconductivity. Here, using a crystal structure search method based on first-principles calculations, we find a boron-rich silicon compound $\mathrm{Si}{\mathrm{B}}_{12}$ that is stable under moderate pressure of around 20 GPa, and which we predict is recoverable to ambient pressure. This silicon boride, with space group Pnnm, is structurally related to the $\gamma \text{−}{\mathrm{B}}_{28}$ boron phase. Specifically, the $\mathrm{Si}{\mathrm{B}}_{12}$ structure is formed by replacing the ${\mathrm{B}}_2$ pairs in $\gamma \text{−}{\mathrm{B}}_{28}$ with silicon atoms. Our calculations show that this Pnnm $\mathrm{Si}{\mathrm{B}}_{12}$ phase exhibits good thermal stability at moderate pressures above 20 GPa and temperatures to 900 K. We suggest this structure has dynamic stability at ambient pressure and remains stable to temperatures as high as 2000 K. Impressively, this $\mathrm{Si}{\mathrm{B}}_{12}$ phase possesses good light absorption and thermoelectrical properties, which are enhanced by its small and indirect band gap, doubly degenerate bands, and low lattice thermal conductivity. Our predictions should stimulate further investigations of this class of boron-rich semiconductors, especially in view of their superior photovoltaic and thermoelectric properties which may be beneficial in energy applications.

Figure 1

(a) The enthalpy ($H$) difference between the Pnnm $\mathrm{Si}{\mathrm{B}}_{12}$ structure and a mixture of $\beta$-Sn II-type silicon and $\gamma$ boron, or a mixture of $\mathrm{Si}{\mathrm{B}}_3$ and $\gamma$ boron, under moderate pressures $P$ of $10–40$ GPa. The dashed base line in (a) is the enthalpy base line for the equivalent mixture of elemental silicon and boron. (b)–(f) Thermal properties of $\mathrm{Si}{\mathrm{B}}_{12}$ across temperature-pressure space using the QHA. (b) The unit cell volume expansion with temperature $T$ at pressures of 20, 24, and 28 GPa. (c) The estimate of the difference in Gibbs free energy ($G$) of $\mathrm{Si}{\mathrm{B}}_{12}$ relative to the equivalent Si-II and $\gamma \text{−}{\mathrm{B}}_{28}$ mixture at different temperatures with constant pressures. (d) Volumetric thermal expansion coefficient $\beta$, and (e) heat capacity $C$, and (f) thermodynamics Grüneisen parameter $\gamma$ versus $T$. The black, red, and green solid lines in (b) and (c) represents thermal properties at pressures of 20, 24, and 28 GPa respectively. The ratio of $\beta /C$ is the inset in (f).

Figure 2

(a) Crystal structure for $\mathrm{Si}{\mathrm{B}}_{12}$ at 0 GPa (green for boron, and blue for silicon atoms). (b) The ELF map depicted by the plane cutting through boron atoms with yellow-crossing marks in (a). (c) The calculated two-dimensional (2D) graphic of noncovalent interactions (NCIs) by the reduced density gradient (RDG, $s$). (d) Three-dimensional (3D) NCIs counter plots with the isovalue $s=0.3$ arb. units as red dashed line shown in (c). To clearly visualize the NCIs regions, we did not draw the boron icosahedra in (d).

Figure 3

Dynamical and thermal stability of $\mathrm{Si}{\mathrm{B}}_{12}$ at 0 GPa and finite temperatures. (a) The harmonic phonon dispersion spectra (left panel) and partial phonon density of state (PDOS, right panel) at 0 GPa and 0 K. (b) The evolution for theoretical frequencies of Raman-active modes $\left({\mathrm{A}}_{\mathrm{g}}+{\mathrm{B}}_{1\mathrm{g}}+{\mathrm{B}}_{2\mathrm{g}}+{\mathrm{B}}_{3\mathrm{g}}\right)$ as a function of pressures. (c) The anharmonic phonon curves calculated by the SCPH method at 0 GPa and 300 K. (d)–(f) The broadened pair distribution functions and (g)–(i) thermal phonon spectra for $\mathrm{Si}{\mathrm{B}}_{12}$ structure, by performing the AIMD simulation at temperatures from 300 to 2000 K. Red curves in (a)–(c) represent the phonon dispersion of the highest frequency. We clearly observe the ${\mathrm{B}}_{\mathrm{p}}$ opposite atomic vibration behavior by hiding the silicon atoms, as the inset of (a) shows.

Figure 4

The stress-strain relationships for orthorhombic $\mathrm{Si}{\mathrm{B}}_{12}$. (a) The uniaxial ideal tensile strains, and (b) the biaxial shear strains under the Vickers-indentation deformations. (c) The structural snapshot at specific indentation strain of 0.06 for the $\left(1\overline{1}0\right)$ plane shearing along the $\left[110\right]$ direction as the blue circle-solid curve shown in (b). (d, left panel) The volume changes with indentation shear along the $\left[001\right]$, [$\overline{1}\overline{1}0],$ and $\left[110\right]$ direction for the $\left(1\overline{1}0\right)$ plane, which are subjected to the same-colored strain-stress curves in (b). The inset in (b) shows the total energy calculated under strains after fully optimizing the cells upon shearing in the $\left(1\overline{1}0\right)$ plane along the $\left[001\right], \left[\overline{1}\overline{1}0\right]$, and [110] directions. The solid circle and square lines in the right part of (d) exhibit the directional-bond lengths by projecting the ${\mathrm{B}}_{\mathrm{P}}–{\mathrm{B}}_{\mathrm{P}}$ pairs along the $\left(1\overline{1}0\right)\left[110\right]$ and its opposite shearing directions, respectively. Blue and green curves represent the bond length for B23–B24 pair as labeled in (c), red and magenta ones are for B21–B22 bond.

Figure 5

The band structures for (a)–(c) d-Si, (d)–(f) c-GaAs, and (g)–(i) $\mathrm{Si}{\mathrm{B}}_{12}$. The blue, red, and black band dispersion curves represent the VASP PBE, HSE06, and WIEN2k PBE-mBJ functional method, respectively.

Figure 6

The electronic structures of Pnnm $\mathrm{Si}{\mathrm{B}}_{12}$ and its absorption features in the visible-light range. (a) The electronic band structure (left panel) and partial density of state (pDOS, right panel), and (b) the optical properties compared to other typical optoelectronic materials, specifically, cubic GaAs (c-GaAs) and diamondlike silicon (d-Si) cells. We used the WIEN2k code to calculate the absorption efficiency with the mBJ potential. Removing the silicon atoms, dashed color lines in (a) and (b) show the corresponding band structure, pDOS and optical absorption for pure icosahedral ${\mathrm{B}}_{12}$ lattice ($B^{\prime }$). The black dashed line in (a) represents the maximum energy level of the valence bands. For comparison, we select the optical-absorption data for c-GaAs (red square) and d-Si (black square) cells at 300 K from the primary literature of Blakemore [12] and Green et al. [84], respectively. The yellow solid line in (b) describes the AM1.5 standard solar spectra in arbitrary units.

Figure 7

Electrical transport and lattice thermal properties for Pnnm $\mathrm{Si}{\mathrm{B}}_{12}$. (a) The Seebeck coefficient ($S$) and (b) power factor ($PF=S^2\sigma$) at constant relaxation time ($\tau$) are shown as a function of carrier concentration ($n$). (c) The lattice thermal conductivity (${\kappa }_L$) versus temperature ($T$) for $xx, yy$, and $zz$ components in Pnnm $\mathrm{Si}{\mathrm{B}}_{12}$. (d) The normalized cumulative thermal conductivity ${\kappa }_C/{\kappa }_L$ with phonon frequency ω at 300 K. (e) ${\kappa }_L$ for nanowires along [010] growth direction at temperatures from 300 to 900 K. (f) The calculated temperature-dependent figure of merit ($\mathit{ZT}$) versus carrier concentration $n$. The constant $\tau ={10}^{-14}\mathrm{s}$ is appropriate for applications in many systems [91, 92, 93, 94, 95, 96, 97].