Anisotropic ferromagnetism in carbon-doped zinc oxide from first-principles studies

A density functional theory study of substitutional carbon impurities in ZnO has been performed, using both the generalized gradient approximation (GGA) and a hybrid functional (HSE06) as exchange-correlation functional. It is found that the nonspinpolarized C${}_{\mathrm{Zn}}$ impurity is under almost all conditions thermodynamically more stable than the C${}_{\mathrm{O}}$ impurity which has a magnetic moment of $2{\mu }_{\mathrm{B}}$, with the exception of very O-poor and C-rich conditions. This explains the experimental difficulties in sample preparation in order to realize $d^0$ ferromagnetism in C-doped ZnO. From GGA calculations with large 96-atom supercells, we conclude that two C${}_{\mathrm{O}}$-C${}_{\mathrm{O}}$ impurities in ZnO interact ferromagnetically, but the interaction is found to be short-ranged and anisotropic, much stronger within the hexagonal $ab$ plane of wurtzite ZnO than along the $c$ axis. This layered ferromagnetism is attributed to the anisotropy of the dispersion of carbon impurity bands near the Fermi level for C${}_{\mathrm{O}}$ impurities in ZnO. From the calculated results, we derive that a C${}_{\mathrm{O}}$ concentration between 2$\%$ and 6$\%$ should be optimal to achieve $d^0$-ferromagnetism in C-doped ZnO.

Figure 1

Supercells and impurity configurations used in the present study. The small (indigo) spheres represent Zn, and among the large spheres, the darker ones represent carbon on the O sites while light spheres represent oxygen atoms in the ZnO lattice. The supercells S122 (a), S221 (b), and S222 (c) contain a single impurity and are used to model impurity concentrations of 12.5$\%$ for S122 and S221 and 6.25$\%$ for S222. The C${}_{\mathrm{O}}$-C${}_{\mathrm{O}}$ interaction is studied with two C${}_{\mathrm{O}}$ in S226 (d) and S622 (e): One of them fixed at the position C0 and A0 in S226 and S622, respectively, and another placed on one of the C1–C13/A1–A13 sites, which corresponds to a concentration of 4.17$\%$.

Figure 2

The formation energy of C${}_{\mathrm{O}}$ and C${}_{\mathrm{Zn}}$ defects as a function of ${\mu }_{\mathrm{O}}$. The thick vertical line (blue colored line) shows the narrow range where the C${}_{\mathrm{O}}$ impurity can compete with C${}_{\mathrm{Zn}}$.

Figure 3

Density of states (DOS) of pure ZnO [(a)–(c)] and C${}_{\mathrm{O}}$ impurity in ZnO [(d)–(f)]. The DOS in different rows are obtained from GGA, GGA+$U_d$ on Zn, and from HSE06 functional. The spin-up DOS and the spin-down DOS are plotted as positive and negative $y$ axis. The DOS of C${}_{\mathrm{O}}$ impurity in ZnO in S222 geometry consists of the atomic contribution of nearest neighbor Zn, O to C in units of states/atom, while the total DOS is scaled to the number of formula units of ZnO.

Figure 4

Effective spin density ($\Delta \rho ={\rho }_{\uparrow }-{\rho }_{\downarrow }$) plotted along the plane perpendicular to $c$ axis and passing through C${}_{\mathrm{O}}$. The light (magenta), dark (turquoise), and black colored balls are used to represent Zn, O, and C atoms (Ref. 42). The isolines (and also the color gradient) are plotted from 0.003 $e{\AA }^{-3}$ to 0.028 $e{\AA }^{-3}$ at intervals of 0.005 $e{\AA }^{-3}$. One can clearly observe that the spin-polarization arising from C${}_{\mathrm{O}}$ extends to a relatively large spatial region.

Figure 5

DOS of C${}_{\mathrm{O}}$ in supercells S122 (a), S221 (b), and S222 (c), respectively. The DOS consists of the atomic contribution of nearest neighbor Zn and O to C (in units of states/atom), while the total DOS is scaled to the number of formula units of ZnO.

Figure 6

Magnetic polarization energy (E${}_{\mathrm{sp}}-E_{\mathrm{nsp}}$) (sp = spin polarized and nsp = nonspin polarized) as a function of the magnetic moment for different supercells from fixed-spin-moment (FSM) calculations. The minima in the magnetic polarization energy indicate the most stable magnetic moment in the supercell. The magnetic polarization energies obtained from vasp and that of fplo are in close agreement with each other.

Figure 7

Exchange constant extracted from the total energy using Eq. (3) plotted against the relaxed C${}_{\mathrm{O}}$-C${}_{\mathrm{O}}$ distance. C1–C13 and A1–A13 represent the different positions of C${}_{\mathrm{O}}$ in supercell S226 and S622 (represented as $i$) as in Figs. 1d and 1e, with the other C${}_{\mathrm{O}}$ positioned at C0 and A0 (represented as 0), respectively. The range of interaction values is bounded by two exponential curves. The $J_{0i}$ for the C${}_{\mathrm{O}}$-C${}_{\mathrm{O}}$ oriented along the $ab$-hexagonal plane, which are shown as filled symbols, are larger than those along the $c$ axis. From the minority-spin band structure of a C${}_{\mathrm{O}}$ impurity in ZnO (in GGA) shown in the inset one finds that the C${}_{\mathrm{O}}$ band crosses the Fermi energy in the hexagonal plane (A–L, $\Gamma$–M, A–H and $\Gamma$–K directions) imparting metallic character, and thus the magnetic interactions along the hexagonal plane are stronger.