Competing charge density wave and antiferromagnetism of metallic atom wires in GaN($10\overline{1}0$) and ZnO($10\overline{1}0$)

Low-dimensional electron systems often show a delicate interplay between electron-phonon and electron-electron interactions, giving rise to interesting quantum phases such as the charge density wave (CDW) and magnetism. Using the density-functional theory (DFT) calculations with the semilocal and hybrid exchange-correlation functionals as well as the exact-exchange plus correlation in the random-phase approximation (EX + cRPA), we systematically investigate the ground state of the metallic atom wires containing dangling-bond (DB) electrons, fabricated by partially hydrogenating the GaN($10\overline{1}0$) and ZnO($10\overline{1}0$) surfaces. We find that the CDW or antiferromagnetic (AFM) order has an electronic energy gain due to a band-gap opening, thereby being more stabilized compared to the metallic state. Our semilocal DFT calculation predicts that both DB wires in GaN($10\overline{1}0$) and ZnO($10\overline{1}0$) have the same CDW ground state, whereas the hybrid DFT and EX + cRPA calculations predict the AFM ground state for the former DB wire and the CDW ground state for the latter one. It is revealed that more localized Ga DB electrons in GaN($10\overline{1}0$) prefer the AFM order, while less localized Zn DB electrons in ZnO($10\overline{1}0$) the CDW formation. Our findings demonstrate that the drastically different ground states are competing in the DB wires created on the two representative compound semiconductor surfaces.

Figure 1

Top and side views of the optimized (a) NM-$p(1\times 1)$, (b) CDW-$p(1\times 2)$, (c) CDW-$p(2\times 2)$, (d) AFM-$p(1\times 2)$, and (e) AFM-$p(2\times 2)$ structures of GaN($10\overline{1}0)$-1H, obtained using the PBE functional. The $x$, $y$, and $z$ axes point along the [0001], $\left[11\overline{2}0\right]$, and $\left[10\overline{1}0\right]$ directions, respectively. The red, blue, and black circles represent Ga, N, and H atoms, respectively. For distinction, the surface Ga or N atoms are drawn with relatively larger circles compared to the subsurface atoms. The unit cell of each structure is indicated by the solid line. In (a), the charge density of DB electrons, obtained by integrating the occupied half-filled band, is drawn with an isosurface of 0.005 electrons/${\AA }^3$. The surface Brillouin zones of the $1\times 1$ and $2\times 2$ unit cells are also drawn in (a). In (d) and (e), the spin density is drawn with an isosurface of $\pm 0.02$ electrons/${\AA }^3$.

Figure 2

Calculated band structures for the NM-$p(1\times 1)$ structures of (a) GaN($10\overline{1}0)$-1H and (b) ZnO($10\overline{1}0)$-1H. The energy zero represents the Fermi level.

Figure 3

Calculated band structures for the NM-$p(1\times 1)$ structures of (a) GaN($10\overline{1}0)$-1H and (b) ZnO($10\overline{1}0)$-1H, obtained using the VASP code. The bands projected onto the $s$ and $p_{\mathrm{z}}$ orbitals of the Ga (Zn) surface atoms are also displayed with circles whose radii are proportional to their weights, together with the PDOS projected onto the corresponding Ga and neighboring N (or Zn and O) orbitals.

Figure 4

Calculated total energy difference $\mathrm{\Delta }{E}_{\text{CDW-AFM}}$ between CDW-$p(2\times 2)$ and AFM-$p(2\times 2)$ for (a) GaN($10\overline{1}0)$-1H and (b) ZnO($10\overline{1}0)$-1H. In (a) and (b), the EX + cRPA results are indicated by yellow lines.

Figure 5

Calculated HSE band structures for CDW-$p(2\times 2)$ and AFM-$p(2\times 2)$ of (a) GaN($10\overline{1}0)$-1H and (b) ZnO($10\overline{1}0)$-1H. The energy zero represents the Fermi level.

Figure 6

(a) Calculated band structure of FM-$p(1\times 1)$. The spin-polarized local DOS projected onto two neighboring Ga surface atoms (${\mathrm{Ga}}_1$ and ${\mathrm{Ga}}_2)$ in the AFM-$p(2\times 2)$ structure of GaN($10\overline{1}0)$-1H is displayed in (b). The charge characters of the spin-up and spin-down states for the highest occupied and the lowest unoccupied bands are taken at the $\mathrm{\Gamma }$ point with an isosurface of 0.01 electrons/${\AA }^3$. The energy zero represents the Fermi level.