Semiempirical pseudopotential approach for nitride-based nanostructures and ab initio based passivation of free surfaces

We present a semiempirical pseudopotential method based on screened atomic pseudopotentials and derived from ab initio calculations. This approach is motivated by the demand for pseudopotentials able to address nanostructures, where ab initio methods are both too costly and insufficiently accurate at the level of the local density approximation, while mesoscopic effective-mass approaches are inapplicable due to the small size of the structures along, at least, one dimension. In this work, we improve the traditional pseudopotential method by a two-step process: First, we invert a set of self-consistently determined screened ab initio potentials in wurtzite GaN for a range of unit-cell volumes, thus determining spherically symmetric and structurally averaged atomic potentials. Second, we adjust the potentials to reproduce observed excitation energies. We find that the adjustment represents a reasonably small perturbation over the potential, so that the ensuing potential still reproduces the original wave functions, while the excitation energies are significantly improved. We furthermore deal with the passivation of the dangling bonds of free surfaces which is relevant for the study of nanowires and colloidal nanoparticles. We present a methodology to derive passivant pseudopotentials from ab initio calculations. We apply our pseudopotential approach to the exploration of the confinement effects on the electronic structure of GaN nanowires.

Figure 1

The LDA-derived contributions $v_{+}$ and $v_{-}$ for GaN as defined in the main text. We have obtained five sets of data points by varying the lattice constant $a$ as indicated in the legends. The $c/a$ ratio is kept constant at the ideal value. The lines represent a fit to the data points using cubic splines.

Figure 2

Screened atomic semiempirical pseudopotentials for gallium (solid line) and nitrogen (dashed line).

Figure 3

Band structure of GaN calculated with DFT in the LDA (solid lines), with the SEP method without correcting the band gap (dashed line), and with the SEP method correcting the band gap (dots). $A/2$ and $M/2$ indicate the path in the Brillouin zone. The LDA band gap is 2.7 eV, 1.6 eV for unshifted SEP, and 3.5 eV for shifted SEP.

Figure 4

Top panels: Band structure of a bare and passivated $\left[1\overline{1}00\right]$ GaN slab calculated with DFT in the LDA. Middle panels: Square of the wave functions of selected conduction ($c_1$, $c_2$), valence ($v_1$), and surface ($s_1$) states, along with the screened local effective DFT-LDA potential averaged along $\left(1\overline{1}00\right)$ planes. Lower panels: Schematic representation of the atomic positions..

Figure 5

Potential given by Eq. (16) for a 15-Å-thick GaN film. The three different fits to the data points represent three different values of $R_C=0.75$, $0.80$ and $0.85$ a.u. (where the cubic splines are connected to the Yukawa potential). We obtain the best results with $R_C=0.80$ a.u.

Figure 6

Passivant screened pseudopotentials for Ga (H-Ga) and N (H-N) in reciprocal space.

Figure 7

Charge densities of one conduction band state $c_1$ and two valence band states $v_{1,2}$ of a passivated GaN slab calculated with the SEPM. Lower panel: Atomic semiempirical pseudopotential across the slab. The quantities are averaged in successive planes perpendicular to the surface.

Figure 8

Ball-and-stick model of a free-standing wurtzite GaN nanowire, grown along the $\left[0001\right]$ direction. The lateral surfaces of the NW are $(1-100)$, and the corresponding surface atoms are passivated by pseudo-hydrogen atoms.

Figure 9

On the left scale, it is indicated the confinement energy of the CB NW states as a function of the NW size. The blue dashed line is the fit obtained with Eq. (21) in the text, whereas the red dots represent the confinement energy of the lowest CB NW. The solid line refers to the right scale and it represents the projection of the $c_1$ wave function onto the CB edge bulk state. The charge density of the CB state $c_1\left(s\right)$ for $S=6.5$ nm is shown as an inset.

Figure 10

Size dependence of the confinement energy of the first VB NW states as a function of the NW size. The charge density of the first VB state for a NW size of 6.5 nm, $v_1\left(p\right)$, is shown as an inset. The solid line connects the size dispersion of the hybrid $A$-$B$ state with $p$-type envelope symmetry, which is discussed in the main text.

Figure 11

The upper panel gives the single-particle energies for the VB NW states $v_1$, $v_2$, $v_3$, and $v_4$, relative to the highest VB state $\varepsilon \left(v_1\right)$ for various NW sizes. The corresponding charge densities are displayed in the lower panels.