Thermodynamic stability and electronic structure of pristine wurtzite $\mathrm{ZnO}\left\{0001\right\}$ inversion domain boundaries

Recently, it was demonstrated that the conductivity of wurzite (wz) ZnO bicrystal samples containing $\left\{0001\right\}$ inversion domain boundaries (IDB) can be massively and reversibly tuned by mechanical loading. As a step towards a detailed microscopic understanding of this effect, we systematically investigate the atomic structure and chemical composition of such IDBs using density functional theory calculations. In total, 92 model geometries that differ in structure and/or chemical composition are constructed, optimized, and compared thermodynamically. The lack of higher symmetries in wz ZnO prohibits a straightforward calculation of individual grain boundary (GB) excess energies. However, we show that, in nonperiodic slab models of wz $\left\{0001\right\}$ IDBs, the additional surface contribution to the total energy may be approximated by that of corresponding zincblende (zb) surfaces; the latter can be obtained by a series of prism calculations. Subtracting these surface energies allows us to construct absolute GB energy diagrams for wz IDBs and compare their thermodynamic stability with other GBs known from the literature. We find that thermodynamically favored IDBs are characterized by fully (4-fold) coordinated atoms and possess relatively low excess energies that range from 45 to $95\mathrm{meV}/{\AA }^2$, depending on the termination (Zn/Zn or O/O) of the IDB and the exchange-correlation functional used in the calculation (LDA, GGA, or $\text{GGA}+U$). The electronic properties of the GB deviate only weakly from those of the bulk and are rather insensitive towards compressive and tensile strains. Our results thus indicate that experimentally observed piezotronic properties of wz bicrystals are not an intrinsic property of the pristine GB itself, but originate, for example, from externally supplied trapped charges, defects, impurities, or dopants. Low-energy structure models identified here may also be transferable to other wz- or zb-type IDBs (e.g., GaN, AlN, SiC, etc.).

Figure 1

Structure and composition of inversion domain boundaries (IDB). (Left) All grain-boundary models are constructed by contacting two $\mathrm{wz}\left\{0001\right\}$ grains. The first grain is aligned along the (0001) or along the $\left(000\overline{1}\right)$ direction and terminated by a three-fold coordinated layer. By definition the stacking is $...BABA$. Different sites are distinguished by different colors; we do not distinguish between Zn/O. The second grain is obtained by mirroring grain 1 at the $\left\{0001\right\}$ plane; in addition the second grain may be rotated by ${180}^{\circ }$ around the $c$ axis through an $A$ site and translated by $1/3$ or $2/3$ along the diagonal of the $1\times 1$ cell. (Right) An interlayer (indicated by dark green spheres) is added between the two grains in various sites. This interlayer may be fully or partially occupied, leading to chemically different GBs. Partial occupancy is achieved using different in-plane repeat units.

Figure 2

Surface energies (average over both terminations/sides) of wz(0001) and zb(111) calculated using LDA (left), PBE (middle), and $\text{RPBE}+U$ (right). Independent of the XC functional, both surfaces show almost identical excess energies.

Figure 3

Zincblende prisms containing two Zn terminated (111) and one O terminated (001) surface. Overall the prisms contain no Zn excess or deficiency. The total excess energy of such a prism consists of the excess of the surfaces and of the three edges. As the size of the prism increases, the surface excess becomes more and more dominant and can be extracted for the limit of an infinitely large prism.

Figure 4

Excess energies of zb prisms as obtained from a series of calculations using the $\text{RPBE}+U$ XC functional. Prisms containing Zn terminated (111) facets (Zn-O-Zn prisms, open black circles) and prisms containing O terminated (111) facets (O-Zn-O prisms, open gray circles) are considered individually. The solid black/gray lines correspond to the $1/l_{\left(111\right)}$ fit (excluding smaller prisms). The dashed black/gray lines indicate $\mathrm{\Delta }{\epsilon }^{\infty }$ for the Zn-O-Zn and O-Zn-O prisms; the dashed orange line shows their average. The solid orange line finally shows the expected value obtained from the known average surface energies, see Eq. (8).

Figure 5

Surface energy of Zn (solid lines) and O terminated (dashed lines) zb(001) calculated using LDA, PBE, and $\text{RPBE}+U$ (from top to bottom). Convergence relative to the slab thickness has been performed. The insets detail two differently terminated slabs.

Figure 6

Variation of the surface energy of Zn and O terminated $\mathrm{zb}\left(111\right)$ and $\mathrm{wz}\left(0001\right)$ ZnO surfaces calculated using LDA, PBE, and $\text{RPBE}+U$ (from top to bottom).

Figure 7

Grain boundary excess energies for Zn/Zn (left) and O/O (right) IDBs with various coverages of O and Zn interlayers, calculated using LDA, PBE, and $\text{RPBE}+U$ (from top to bottom). Independent of the XC functional, the $\theta =1/2$ IDBs are thermodynamically favored over almost the entire range of the O chemical potential.

Figure 8

Atomic and electronic structure of thermodynamically favored Zn/Zn (columns 1–3) and O/O IDBs (column 4). At $\theta =1/2$ GBs, all atoms are ideally 4-fold coordinated. At Zn/Zn IDBs, systems with overcoordinated Zn ($\theta =4/7,1$) may be stabilized at high O chemical potential, compare to Fig. 7. Electronic properties are presented for the stress-free state and at 2500 MPa compressive/tensile stress. In the PDOS, effects of mechanical strain are indicated in black (compressive) and gray (tensile) but are hard to resolve by the eye. In particular, no gap states arise at the IDBs. Similarly, strain has hardly any effect on Bader excess electrons or on effective potentials. Color coding: O and Zn atoms are represented as large light and small dark gray spheres; overcoordinated atoms are colored in red, undercoordinated atoms are not present. Bader charges/effective potential of strain-free (compressively/tensile strained) systems are shown in green (blue/orange).