Structural and electronic properties of $\alpha$-tin nanocrystals from first principles

The $\alpha$ phase of tin is a zero-gap semiconductor with an inverted band structure with respect to other group-IV elements like Ge. The ${\Gamma }_{6c}$ states lie energetically below the ${\Gamma }_{8v}$ levels. How these unique electronic properties transform in nanostructures with spatial confinement has not been studied. We apply density-functional theory within the local density approximation to investigate the energetic, structural, and electronic properties of bulk $\alpha$-Sn and its nanocrystals (NCs) up to a size of 363 Sn atoms. For NCs with larger diameters up to 14 nm the tight-binding method is applied for the electronic states. Spin-orbit coupling is taken into account. The clusters are modeled in such a way that the $T_d$ symmetry of the bulk system is conserved. Their surfaces are passivated with hydrogen. We show that the spatial confinement causes not only a decrease of the fundamental gap for increased NC size but also a topological transition where the ordering of $s$- and $p$-like highest-occupied molecular orbital and lowest-unoccupied molecular orbital states is interchanged. The influence of quasiparticle and excitonic effects on the lowest pair excitation energies is investigated within approximations based on the hybrid exchange-correlation functional by J. Heyd, G. E. Scuseria, and M. Ernzerhof [J. Chem. Phys. 118, 8207 (2003)] (HSE) and the $\Delta$SCF method.

Figure 1

Primitive unit cell and atomic positions of bulk $\alpha$-Sn.

Figure 2

Band structure of $\alpha$-Sn calculated using SOC and the hybrid functional HSE06.[46] Additionally, the angular-momentum resolved DOS is given to describe the symmetry of the electronic states in a certain energy interval. The irreducible representations and their degeneracy are indicated. The ${\Gamma }_{8v}$ level is used as energy zero.

Figure 3

Atomic geometries of the studied nanocrystals. Tin atoms are marked by large red spheres, while the passivating hydrogen atoms are represented by small white dots. The resulting $\left\{001\right\}$ and $\left\{111\right\}$ facets are clearly visible.

Figure 4

(a) Radially averaged bond length versus the bond distance to the NC center and (b) globally averaged bond length depending on the NC diameter. The horizontal solid lines indicate the bulk bond length.

Figure 5

Enthalpy of formation (a) depending on the chemical potential of the tin reservoir for a H${}_2$ gas as the hydrogen reservoir or (b) on the NC diameter for Sn-rich preparation conditions for the two choices of the hydrogen reservoir.

Figure 6

Energy-level schemes of Kohn-Sham eigenvalues of $\alpha$-Sn nanocrystals passivated with hydrogen. The fundamental energy gaps are indicated by the corresponding energy eigenvalues. The energy scales of the different NCs have been aligned by the vacuum levels derived from the spatially averaged electrostatic potentials.

Figure 7

Energy levels and angular-momentum resolved density of states of the Sn${}_{17}$H${}_{36}$ NC around the fundamental gap. The bulk states that form the HOMO, LUMO, and the first occupied spin-orbit split-off NC state are labeled. The level degeneracy is indicated by numbers 2 and 4.

Figure 8

Isosurfaces of probabilities to find holes or electrons (left panels) and radially averaged wave function squares (right panels) of the (a) HOMO and for the (b) LUMO for the Sn${}_5$H${}_{12}$ cluster. The arrow indicates the nominal radius of the NC.

Figure 9

Lowest pair excitation energies of the hydrogenated Sn nanocrystals from three different ab initio approximations. The insert displays the HOMO-LUMO gaps taken from the TB calculations. The agreement with the DFT-LDA values is shown for small NC diameters.

Figure 10

Diameter dependence of the HOMO and LUMO level positions are plotted. The respective bulk orbital symmetries ${\Gamma }_{6c}$ and ${\Gamma }_{8v}$ are indicated. In addition, the energy position of the first occupied state below the HOMO with ${\Gamma }_{7v}$ symmetry is also given. (a) The KS eigenvalues from the DFT-LDA calculation with SOC. The vacuum level of the electrostatic potentials is used as energy zero. (b) Eigenvalues derived from the diagonalization of the TB Hamiltonian using the parameters listed in Table .