Geometry and diameter dependence of the electronic and physical properties of GaN nanowires from first principles

We present a comprehensive first-principles investigation of the atomic and electronic structures of gallium nitride nanowires, and examine the dependence on nanowire diameter and shape. We consider nanowires in the [0001] growth direction, with diameters ranging from $8\text{to}35\mathrm{\AA }$, and investigate the influence of saturating the dangling bonds at the edges of nanowires. We find that unsaturated nanowires are semiconducting and contain dangling bond states in the region of the band gap, the positions of which remain rather constant with varying diameter. Saturating the nanowires with hydrogen removes these states, and the band gap decreases with increasing nanowire diameter. For the unsaturated wires there is a considerable contraction of the Ga-N bond lengths at the edge of the wires of 6.0%–7.4%, while for saturated wires it is 1.5%. We also calculate the heat of formation of the nanowires and find that as the diameter of the nanowire increases, the average relative stability of the nanowire increases, as intuitively expected.

Figure 1

Band structure of wurtzite GaN calculated using (A) SIESTA and (B) ${\mathrm{DMOL}}^3$.

Figure 2

(Color online) Nanowires in the [0001] direction that have (A) hexagonal and (B) triangular shaped cross sections, viewed along the wire direction, with perspective views of the 108 and 66 atom nanowires included. The number of atoms used in the calculations for each nanowire, and the diameter (in parentheses) are also labeled. Nitrogen and gallium atoms are indicated by the dark (blue) and light (aqua) spheres, respectively.

Figure 3

(Color online) Section of a hexagonal GaN nanowire (108 atoms), illustrating the change in outer edge Ga-N bonds and corresponding angles, where nitrogen and gallium atoms are represented by the dark (blue) and light (aqua) spheres, respectively.

Figure 4

The band structures for unsaturated hexagonal nanowires with diameters of (A) 9.5 (48 atoms) and (B) 28.6 (300 atoms) Å, and triangular nanowires with diameters of (C) 12.7 (66 atoms) and (D) 25.5 (194 atoms) Å. Calculations are performed using the ${\mathrm{DMOL}}^3$ code, and the energy zero is set at the highest occupied level. “Hex” and “Tri” represent “hexagonal” and “triangular,” respectively.

Figure 5

The band structures for saturated hexagonal nanowires with diameters of (A) 9.5 (72 atoms) and (B) 28.6 (360 atoms) Å, and triangular nanowires with diameters of (C) 12.7 (96 atoms) and (D) 25.5 (248 atoms) Å. Calculations are performed using the ${\mathrm{DMOL}}^3$ code, and the energy zero is set at the highest occupied level. “Hex” and “Tri” represent “hexagonal” and “triangular,” respectively.

Figure 6

(Color online) Spatial distribution of the HOMO and LUMO states at the gamma point for unsaturated nanowires, from ${\mathrm{DMOL}}^3$ calculations. The HOMO states (upper panel) are shown for hexagonal wires with diameters of (A) 9.5 (48 atoms) and (C) 28.6 (300 atoms) Å, and for triangular nanowires with diameters of (E) 12.7 (66 atoms) and (G) 25.5 (194 atoms). The LUMO states (lower panel) are shown for hexagonal wires with diameters of (B) 9.5 and (D) $28.6\mathrm{\AA }$, and for triangular nanowires with diameters of (F) 12.7 and (H) $25.5\mathrm{\AA }$. Nitrogen and gallium atoms are indicated by dark (blue) and light (aqua) spheres, respectively, and the orbitals are pale gray (yellow).

Figure 7

(Color online) Spatial distribution of the HOMO and LUMO states at the gamma point for saturated nanowires, from ${\mathrm{DMOL}}^3$ calculations. The HOMO states (upper panel) are shown for hexagonal wires with diameters of (A) 9.5 (72 atoms) and (C) 28.6 (360 atoms) Å, and triangular nanowires with diameters of (E) 12.7 (96 atoms) and (G) 25.5 (248 atoms). The LUMO states (lower panel) are shown for hexagonal wires with diameters of (B) 9.5 and (D) $28.6\mathrm{\AA }$, and triangular nanowires with diameters of (F) 12.7 and (H) $25.5\mathrm{\AA }$. Nitrogen and gallium atoms are dark (blue) and light (aqua) spheres, respectively, hydrogen atoms are represented by very small light gray spheres, and the orbitals are pale gray (yellow).

Figure 8

(Color online) Partial density of states (PDOS) plots for unsaturated hexagonal nanowires with diameters of (A) 9.5 (48 atoms) and (B) 28.6 (300 atoms) Å, and triangular nanowires with diameters of (C) 12.7 (66 atoms) and (D) 25.5 (194 atoms) Å, calculated using the ${\mathrm{DMOL}}^3$ code. The black, light gray (orange), and dark gray (blue) lines correspond to $s$-, $p$- and $d$-state contributions, respectively. The energy zero corresponds to the highest occupied state.

Figure 9

(Color online) Partial density of states (PDOS) plots for saturated hexagonal nanowires with diameters of (A) 9.5 (72 atoms) and (B) 28.6 (360 atoms) Å, and triangular nanowires with diameters of (C) 12.7 (96 atoms) and (D) 25.5 (248 atoms) Å, calculated using the ${\mathrm{DMOL}}^3$ code. The black, light gray (orange), and dark gray (blue) lines correspond to $s$-, $p$- and $d$-state contributions, respectively. The energy zero corresponds to the highest occupied state.

Figure 10

Relative band gap as a function of nanowire diameter as obtained using the (A) SIESTA and (B) ${\mathrm{DMOL}}^3$ codes. Band gaps are relative to the calculated bulk GaN band gap. “Hex” and “Tri” represent “hexagonal” and “triangular,” respectively, and “unsat,” “sat,” and “sat-frac. H” represent “unsaturated,” “saturated,” and “saturated with fractional charge hydrogen,” respectively.

Figure 11

Relative band gap as a function of $1∕d^x$ for saturated (A) hexagonal and (B) triangular nanowires with SIESTA, and saturated (C) hexagonal and (D) triangular nanowires with ${\mathrm{DMOL}}^3$, where $d$ is the nanowire diameter. The dashed lines indicate the linear regions for each of the data sets. “Hex” and “Tri” represent “hexagonal” and “triangular,” respectively. Band gaps are relative to the calculated bulk GaN band gap.

Figure 12

Relative band gap as a function of $1∕d^2$ for saturated (A) hexagonal and (B) triangular nanowires, where $d$ is the nanowire diameter. The dashed lines indicate the linear regions for each of the data sets. “Hex” and “Tri” represent “hexagonal” and “triangular,” respectively. Band gaps are relative to the calculated bulk GaN band gap.

Figure 13

Relative heat of formation (per atom) as a function of nanowire diameter for unsaturated and saturated nanowires in the [0001] direction, from (A) SIESTA and (B) ${\mathrm{DMOL}}^3$. Energies are relative to the heat of formation (per atom) of bulk GaN. “Hex” and “Tri” represent “hexagonal” and “triangular,” respectively, and “unsat” and “sat” represent “unsaturated” and “saturated,” respectively.