Electronic excitations in single-walled GaN nanotubes from first principles: Dark excitons and unconventional diameter dependences

We present the first ab initio predictions of electronic and excitonic states in gallium nitride nanotubes. Electron-hole interactions dramatically affect optical properties. Low-energy excitons are dark (dipole forbidden) in all nanotubes with key ramifications for applications. We describe an unusual decrease of energy gaps with decreasing nanotube diameter, opposing expectations from quantum confinement. This stems from a combination of nanoscale curvature and gallium chemistry, should apply to other systems, and furnishes an interesting way to reduce excitation energies with decreasing dimension.

Figure 1

(Color online) Quasiparticle bands for the hexagonal GaN sheet based on the $GW$ approximation. The valence band maximum at K is the zero of energy. The bands outlined with red circles are the $\pi$ and ${\pi }^{*}$ bands arising from out-of-plane $p_z$ orbitals. The other bands are in-plane $\sigma$ and ${\sigma }^{*}$ states.

Figure 2

(Color online) Imaginary part of the frequency-dependent dielectric function ${\epsilon }_2\left(\omega \right)$ for the hexagonal GaN sheet based on the RPA approximation with $GW$ band energies (red circles) and based on the BSE which includes excitonic effects (solid blue). The first BSE peak corresponds to the bright exciton at $4.3\mathrm{eV}$.

Figure 3

(Color online) Radial breathing mode frequencies in ${\mathrm{cm}}^{-1}$ versus nanotube diameter $d$ in Å. The upper set of data is for the antisymmetric mode ${\nu }_a$, and the lower set of data is for the symmetric mode ${\nu }_s$. Open (blue) triangles are for $(n,0)$ nanotubes, while (green) circles are for $(n,n)$ nanotubes. The upper continuous dashed curve is ${\nu }_a=288+2070∕d^2$, and the lower solid continuous curve is ${\nu }_s=1040∕d$.

Figure 4

(Color online) Imaginary part of the dielectric function ${\epsilon }_2\left(\omega \right)$ for four GaN nanotubes and the GaN sheet (from Fig. 2). The top panel has plots for (4,0), (5,0), and the sheet; the bottom panel has plots for (3,3), (4,4), and the sheet. Thick red circles are results based on the RPA with $GW$ band energies, and the thinner solid blue lines are BSE results including excitonic effects. The vertical scale in each plot is arbitrary. For each nanotube, two arrows mark the energies of lowest dark and bright excitons from Table . The leftmost arrow is for the dark exciton, and the other is for the bright exciton and necessarily coincides with the peak in the BSE spectrum.

Figure 5

(Color online) Isosurfaces of probability density (in green/medium gray) of finding an electron when the hole is placed at the position of the red open circle (close to a N atom) for the lowest dark exciton of the (3,3) nanotube. The isosurface is at 25% of its maximum value. The top view is down the nanotube axis and shows the distribution around the nanotube circumference. The bottom view shows the distribution along the nanotube axis. Larger purple/dark gray balls are Ga, and smaller cyan/light gray balls are N atoms.

Figure 6

(Color online) Valence band maximum (left) and conduction band minimum (right) energies, both at $\Gamma$, for single-walled GaNNTs versus diameter and with respect to the vacuum. Triangles are for $(n,0)$, and circles for $(n,n)$ tubes.

Figure 7

(Color online) Curved segment of a $(n,0)$ GaNNT. Large purple spheres are Ga, smaller cyan are N, and black lines are bonds. Signs indicate phases of $s$ orbitals for the antibonding CBM state. The arrow highlights the second-neighbor Ga $s\text{−}s$ interaction discussed in the text.