Unit cell structure of the wurtzite phase of GaP nanowires: X-ray diffraction studies and density functional theory calculations

We present structural characterization of the wurtzite crystal structure of GaP nanowires, which were recently shown to have a direct electronic band gap. The structural parameters of the wurtzite phase do consist of two lattice parameters and one internal degree of freedom, determining the Ga-P bond length along the $c$ direction. Using density functional theory calculations, we study the influence of the internal degree of freedom on the band structure. By synchrotron x-ray diffraction studies near the Ga-$K$ edge we determine the lattice parameters $a=3.8419$ Å and $c=6.3353$ Å as well as the internal degree of freedom $u=0.37385$ with high accuracy. We find that different Ga-P bond lengths are not equal, in contrast to the case in the zinc blende bulk phase. As a result, a spontaneous polarization is predicted for wurtzite GaP.

Figure 1

Panel (a) shows a sketch of the WZ unit cell with stacking sequence ABAB$\cdots$ of III-V bilayers. Shown are the lattice parameters $a$ and $c$ as well as the internal parameter $u$, specifying the bond length along the $c$ direction within the III-V bilayer, and the bond angles $\alpha$ and $\beta$. Also indicated, however, exaggerated, is the deformation of the unit cell with respect to what would be expected from bulk ZB bond lengths. In panel (b) scanning electron micrographs recorded in side view (left) and top view (right) are shown. The homogeneous length and areal density of the nanowires can be seen. The scale bars correspond to 1 $\mu$m.

Figure 2

Panel (a) shows a sketch of the diffraction setup with the primary and diffracted wave vectors ${\mathbf{k}}_{i,f}$ as well as their difference the momentum transfer $\mathbf{q}$. The detector area $A$, which is perpendicular to the diffracted wave vector is indicated by a plane marked in green color. The diffraction positions of some $(h0\overline{h}.\overline{l})$ Bragg positions along [000.1] crystal truncation rods are illustrated. In panel (b) a three-dimensional contour plot of the measured intensity pattern of the $(10\overline{1}.\overline{5})$ WZ Bragg reflection is shown. The contour plot is cut open to show the intensity variation inside as a semitransparent color plot. Dotted lines indicate the facet streaks due to the hexagonal cross section of the nanowires.

Figure 3

Variation of the diffracted intensity of the $(10\overline{1}.\overline{7})$ WZ Bragg reflection in dependence of the x-ray energy and the internal parameter around the Ga-K absorption edge. Panel (a) shows the variation of the total intensity (for $\varepsilon =0$) vs the x-ray energy, and panel (b) shows the variation of the normalized intensity with respect to the internal parameter $\varepsilon$. In the vicinity of the absorption edge the diffracted intensity is most sensitive to the internal parameter. Arrows indicate the two energies used in the experiment.

Figure 4

Integrated detector signal during a sample x-ray rocking scan at the $(10\overline{1}.\overline{5})$ Bragg reflection of GaP. For comparison two different simulations of the intensity are shown, one depicting the modeled detector signal for a perfectly monodisperse ensemble of nanowires (dashed red line) and one showing the simulated signal including ensemble averaging (solid blue line), which accounts for a Gaussian size distribution of $80\pm 3$ nm with and tilt distribution with sigma $\sigma =0.{016}^{\circ }$ of the nanowires. The curves are shifted vertically for clarity.

Figure 5

Total energy per unit cell determined by DFT calculations in dependence of the lattice paramters $(a,c)$. From the minimum in the energy surface (marked by a black dot) the optimal lattice parameters are determined. The optimized lattice parameter clearly show a $c/a$ ratio larger than $\sqrt{8/3}$, which is indicated by a dashed line.

Figure 6

Influence of the internal cell parameter $\varepsilon$ on the total energy per unit cell (a) and fundamental energy gap (b) derived from DFT calculations. The equilibrium band gap $E_g^{\left(0\right)}$ at the minimum of the total energy is indicated.