Surface doping of ZnO nanowires with Bi: Density-functional supercell calculations of defect energetics

Defect calculations using the density and hybrid functionals in combination with the supercell approach are employed to characterize the electrical properties of a number of ZnO nanowires of various thicknesses doped with Bi atoms occupying surface sites. The variation of the differences between the total energies of charged and neutral supercells with the supercell size is studied, which led the authors to devise an extrapolation procedure to obtain reliable defect energetics in the dilute defect limit. The calculated defect formation energies indicate that although the substitution of Bi into Zn or O sites can take place spontaneously under suitable thermodynamic conditions, the substitution into Zn sites is generally more likely. The defect (charge-state) transition energies are computed and parameterized as a function of the nanowire thickness. It is revealed that the substitution of Bi into O (Zn) sites on the surface of ZnO nanowires yields deep acceptor (shallow donor) levels (except for extremely thin nanowires). It is therefore concluded that the incorporation of Bi into the surface of ZnO nanowires results in $n$-type doping.

Figure 1

Top view of the supercell containing a $\text{[}{\text{(ZnO)}}_{96}{\text{]}}_n$ nanowire. The nanowire diameter $D$ is defined in terms of the inradius $r$ and circumradius $R$.

Figure 2

The lowest-energy configurations for Bi-doped and undoped ZnO nanowires. The sticks representing the Zn-O and Bi-O or Bi-Zn bonds are colored to indicate the respective bond lengths values, i.e., $d_{\text{Zn-O}}$ (the upper colorbar) and $d_{\text{Bi-O}}$ or $d_{\text{Zn-O}}$ (the lower colorbar).

Figure 3

(a) The variation of the energy differences $\mathrm{\Delta }{E}_0^{+}, \mathrm{\Delta }{E}_0^{-}, \mathrm{\Delta }E\left({\text{Bi}}_{\text{Zn}}^{+}\right)$, and $\mathrm{\Delta }E\left({\text{Bi}}_{\text{O}}^{-}\right)$ with the lateral dimension $L_{\perp }$ of tetragonal supercells with $\text{[}{\text{(ZnO)}}_{24}{\text{]}}_n$ nanowire for a set of values of $L_{\parallel }=nc$ (with $n=5,6,7,8,9$), where $c$ denotes the length of periodicity along the wire axis. (b) The extrapolated values of the foregoing energy differences corresponding to $L_{\perp }\to \infty$ as a function of $L_{\parallel }$. The solid curves in each panel represent the results of fitting according to Eq. (5).

Figure 4

The variation of the energy differences (a) $\mathrm{\Delta }{E}_0^{+}$, (b) $\mathrm{\Delta }{E}_0^{-}$, (c) $\mathrm{\Delta }E\left({\text{Bi}}_{\text{Zn}}^{+}\right)$, and (d) $\mathrm{\Delta }E\left({\text{Bi}}_{\text{O}}^{-}\right)$ with the edge length $L$ of the used cubic supercells. The solid curves in each panel represent the results of fitting according to Eq. (6).

Figure 5

(a) The variation of the extrapolated values of the valence and conduction band edge energies ($E_{\text{V}}$ and $E_{\text{C}}$, respectively) with the inverse diameter $1/D$. (b) An energy diagram showing $E_{\text{V}}, E_{\text{C}}$ and the VBM and CBM eigenvalues (${\epsilon }_{\text{v}}$ and ${\epsilon }_{\text{c}}$, respectively) for ZnO nanowire with $N=24$.

Figure 6

(a) The formation energies of ${\text{(ZnO)}}_N{\text{:Bi}}_{\text{Zn}}$ and ${\text{(ZnO)}}_N{\text{:Bi}}_{\text{O}}$ as a function of the Fermi level $E_{\text{F}}$ in the dilute defect limit under O-rich as well as O-poor conditions. The latter two conditions are maintained by setting ${\mu }_{\text{O}}=0$ and ${\mu }_{\text{O}}=\mathrm{\Delta }{H}_N$, respectively, where $\mathrm{\Delta }{H}_N$ denotes the heat of formation (per formula unit) of the nanowire made of $N$ Zn–O pairs [15]. The value of ${\mu }_{\text{Bi}}$ is set to correspond to an equilibrium with the atomic Bi gas adsorbed on the surface of the host nanowire [12]. In each graph, the vertical shading is employed to highlight the ranges of Fermi level: $E_{\text{F}}\le E_{\text{V}}$ (the left-hand side), $E_{\text{V}}\le E_{\text{F}}\le E_{\text{C}}$ (unshaded), and $E_{\text{C}}\le E_{\text{F}}$ (the right-hand side). (b) An energy diagram showing the charge-state transition energies $\varepsilon (+/0)$ and $\varepsilon (0/-)$ for ${\text{Bi}}_{\text{Zn}}$ and ${\text{Bi}}_{\text{O}}$, respectively. (c) The variation of the energy differences $E_{\text{C}}-\varepsilon (+/0)$ and $\varepsilon (0/-)-E_{\text{V}}$ with the inverse diameter $1/D$; the solid curves are given by Eqs. (12) and (13).

Figure 7

Comparison of ${\epsilon }_{\text{c}}-\varepsilon (+/0)[\varepsilon (0/-)-{\epsilon }_{\text{v}}]$ (the empty symbols) obtained via Eq. (3) to $E_{\text{C}}-\varepsilon (+/0)[\varepsilon (0/-)-E_{\text{V}}]$ (the filled symbols) obtained via Eq. (7). The dashed and solid curves represent fits to the empty and filled symbols, respectively.