Armchair nanoribbons of silicon and germanium honeycomb structures

We present a first-principles study of bare and hydrogen passivated armchair nanoribbons of the puckered single layer honeycomb structures of silicon and germanium. Our study includes optimization of atomic structure, stability analysis based on the calculation of phonon dispersions, electronic structure, and the variation in band gap with the width of the ribbon. The band gaps of silicon and germanium nanoribbons exhibit family behavior similar to those of graphene nanoribbons. The edges of bare nanoribbons are sharply reconstructed, which can be eliminated by the hydrogen termination of dangling bonds at the edges. Periodic modulation of the nanoribbon width results in a superlattice structure which can act as a multiple quantum well. Specific electronic states are confined in these wells. Confinement trends are qualitatively explained by including the effects of the interface. In order to investigate wide and long superlattice structures we also performed empirical tight-binding calculations with parameters determined from ab initio calculations.

Figure 1

(Color online) (a) Atomic structure of fully relaxed bare armchair Si nanoribbon of width $n=9$ (ASiNR-9). (b) Atomic structure and bond-length distribution of hydrogen saturated ASiNR-9. Similar pattern is observed in hydrogen saturated AGeNR-9. The primitive unit cell of ASiNR-9 is shaded. The zigzag chain of Si atoms perpendicular to the nanoribbon axis is delineated by the dashed lines.

Figure 2

(Color online) (a) Phonon dispersions calculated for hydrogen saturated ASiNR-9 and for 2D honeycomb structure of Si. States that appear above $600{\text{cm}}^{-1}$ are related to Si-H bonds of hydrogen saturated ASiNR-9 and were not shown. (b) Phonon DOSs of the hydrogen saturated ASiNR-9 projected to Si atoms at the center and at the edges and also to H atoms at the edges. DOS of 2D honeycomb structure of Si is also presented for comparison.

Figure 3

(Color online) Calculated (a) energy gap and (b) effective mass versus ribbon width, $n$, for hydrogen saturated Si and Ge armchair nanoribbons. Filled circles indicate the ab initio results while empty circles stand for the results of the tight-binding fitting. The fitting is performed using only the energy gap data. Parameters found from this fitting were used to generate the tight-binding effective mass data. In each panel three branches are observed and named in increasing order of band gaps and effective masses as families I, II, and III.

Figure 4

(Color online) Band structure and charge density isosurface profiles for several superlattice structures. Solid (green) and dashed (red) lines stand for the ab initio and tight-binding calculation results, respectively. Structures are labeled, as defined in the text, and are given on top of each structure. Top and bottom panels present the results for superlattice structures with $\Delta N=2$ and 4, respectively. Constituent nanoribbons of superlattices that appear in the same column are members of the same family. Charge density isosurfaces calculated by ab initio technique around the $\Gamma$ point for two conduction and two valence band edge states are ordered in the similar manner as they appear on the energy band structure.

Figure 5

(Color online) Variation in confinements and energy gaps of superlattices with the lengths of constituent parts. Results are derived by the tight-binding model described in the text. Numbers labeling the curves correspond to $n_1$ and $n_2$. The lengths, $l$, of the constituent parts are equal and shown in the logarithmic scale.

Figure 6

(Color online) Model representation of three different superlattice structures. (a) Conduction band edge profile when interface effects are not included. Here the height of the potential in each side is determined by $E_n$, which is the difference between the conduction band edge minimum and Fermi level of a nanoribbon with a width $n$. Fermi levels of each structure are shown by the dashed lines. The family name corresponding to each side is written under these lines. (b) Magnitude squares $\left({\left|\Psi \right|}^2\right)$ of solutions. Dark (blue) and light (green) lines represent solutions for the cases where interface effects are included and are not included, respectively. The thin line joining the panels (a) and (b) corresponds to the geometric interface of the superlattices. (c) Conduction band edge profile when interface effects are included. Here the height of the potential at the interface is determined by $E_{n_1-n_2}$, which is the difference between the conduction band edge minimum and Fermi level of $\text{SL}\left(n_1,n_2;1\right)$ structure. The interfaces act as if they are composed of the families I, II, or III as written in that region.