Piezoelectric properties of zinc oxide nanowires: an ab initio study

Zinc oxide (ZnO) has attracted extensive attention for half a century because of its excellent performance in optoelectronics, and spintronics [1, 2]. Most recently, we have been witnessing a significant increase of interest in ZnO nanostructures due to their unique physical, chemical, and electronic properties with potential applications in a wide range of technological fields, which include fundamental building blocks for fabrication of devices for energy harvesting [3], ultrasensitive gas sensors [4], nanoresonators [5], and nanocantilevers [6]. It has been recently proposed that ZnO nanowires (NWs) can be successfully employed for converting mechanical energy into electric power via atomic force microscopy [7], although this result has been questioned [8]: the underlying mechanisms indeed still await a microscopic description to enhance and take full advantage of the structural and electric properties of this material at the nanoscale size. Beyond this, a better knowledge of the elastic response of nanostructures to external deformations would be beneficial to prevent failure and fractures of novel nanodevices. The working principle of nanogenerators relies on a beneficial coupling between piezoelectricity and semiconducting properties to produce a net current through Schottky contacts between the wide-gap semiconductor NWs and a metal coated AFM tip. Nonetheless, controversial results appear in the literature, depending on the material, on the nanostructure form, and on the growth direction [8–10]. From the experimental point of view, indeed, the characterization of mechanical properties of NWs can be severely affected by artifacts linked to the specific experimental setup, such as AFM on a single sample or a network, measurements performed on clamped or free standing samples, on conductive versus insulating substrates, etc (see e.g. Espinosa et al [9]).

On the theoretical front, as well, while it has been assessed that spontaneous polarization of wires strongly depends on the nanostructure size [11], the question whether size affects piezoelectric properties still remains open. In particular, Cicero et al [11] established the existence of a minimum diameter below which the NW polarization field is inverted with respect to bulk because of large surface effects in small nanostructures. As for the piezoelectric response of ZnO NWs, Xiang et al [12] proposed that NWs with diameter larger than 2.8 nm tend to have almost constant effective piezoelectric constants (close to the bulk value), while for small NWs, they found a non-trivial dependence of electromechanical coupling on the radius as a result of two competitive effects, i.e., increase of the lattice constant along with decrease of the NW radius. In contrast a giant piezoelectric response was predicted by Agrawal et al [13] of approximately two orders of magnitude larger than bulk for NWs with diameter lower than 1 nm; such an increase should be attenuated for NWs with diameters exceeding 1.5 nm. It is noteworthy that effective piezoelectric coefficients of ZnO NWs reported in the above mentioned studies [12–14] were determined using similar computational approaches, yet leading to dramatically different results.

The origin of the above mentioned debate is due to the sources of uncertainty, that, although often neglected, severely alter the volume definition of a nanostructure: these are, namely, surface relaxation (dx) and ionic radius (see figure 1). While the first source of error can be easily accounted for via accurate calculations or dedicated experimental techniques, the second issue is more tricky. The volume definition used to calculate polarization of nanowires is usually not mentioned in previously published papers. However, the nanowire size can be defined through its diameter either by considering the atoms as point charges or as finite spheres (e.g. via the ionic radius): this is enough to obtain an uncertainty of few Angstroms on its value (≃2 Å in the case of ZnO). This uncertainty on the diameter has increasing relevance at the nanometer scale, and indeed leads to a large uncertainty in the volume occupied by the nanostructure (up to 70% in the case of the smallest NW considered in this work): the smaller the structures the larger is the uncertainty. As such, all the physical quantities normalized to volume, like spontaneous polarization and piezoelectric constants, present large variability depending on the particular volume definition employed in the calculations⁴.

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In this paper, we propose a rigorous method to overcome the problem: in our study we adopt a scheme that passes through a normalization in terms of number of formula units per supercell (a ZnO pair in the case of zinc oxide nanosystems) without loss of generality. This approach was successfully employed in [11] to describe the size effect on the spontaneous polarization of nanowires, while correctly reproducing bulk values at increasing diameter. In particular, we take advantage of the Wannier functions (WFs) definition that allows one to calculate local contributions (one for each ZnO couple) to the average dipole of a system. In this way, we show that the piezoelectric properties of NWs are only mildly influenced by size reduction to the nanoscale. The procedure allows us to define in a unique rigorous way the different contributions (bulk versus surface) relevant to evaluate the nanostructure response to external deformation, and thus finally opens the possibility of an optimized design of novel piezo-devices at the nanoscale.

Density functional theory (DFT) in conjunction with maximally localized Wannier function (MLWF) was used to perform a detailed study of piezoelectric properties of ZnO NWs. All calculations were performed with reference to the theoretical DFT equilibrium lattice parameters of the ZnO wires [11], using the ultrasoft pseudopotential plane wave implementation of DFT in the Quantum Espresso code [15], with energy cutoff of 28 Ryd (280 Ryd) for the wavefunctions (charge density); Wannier functions for ZnO bulk and NWs were evaluated using the wanT [16] code. The exchange–correlation energies were calculated by using the Perdew–Burke–Ernzerhof [17] approximation of the generalized gradient approach. The NWs were modeled in periodic supercells with the z direction corresponding to the wire axis (equilibrium c lattice constants are reported in table 1) and lateral lattice parameters (along the x and y axes) large enough to avoid interaction between periodic replicas. The convergence of k-space sampling was studied and a Monkhorst pack grid of 1 × 1 × 6 was used for the DFT calculations. All atoms were allowed to move, until forces were less than 0.02 eV Å⁻¹. This procedure leads to fully relaxed nanostructures, including surface contribution⁵. In this way, we were able to study NWs with diameter in the range 0.38–2.3 nm containing from 1 to 4 ZnO shells (NW1–NW4) as shown in figure 1.

According to the modern theory [18, 19], the macroscopic polarization of a periodic structure, P, is defined only with respect to a reference system; for example, in the case of wurtzite (WZ) one can choose the zincblende (ZB) phase, since the two structures are equivalent up to third neighbors⁶. In terms of WFs [20] P can be written as

$$\begin{eqnarray} &&P={P}_{\mathrm{WZ}}-{P}_{\mathrm{ZB}}=\frac{e}{\Omega }\mathop{\sum }\limits _{I}{Z}_{I}\Delta {R}_{I}-\frac{2 e}{\Omega }\mathop{\sum }\limits _{i}\Delta r_{i}^{\mathrm{W}} \end{eqnarray} \tag{ 1 }$$

where we indicate with ${r}_{i}^{\mathrm{W}}$ the positions of the Wannier centers, with R_I (Z_I) the ionic positions (charges) and with Ω the volume of the system. For a periodic structure, P is defined modulo 2eR_l/Ω,R_l being a direct lattice vector. We stress again that in the presence of a surface, or for the case of nanostructures, the volume of the system is an ill-defined quantity, thus a different definition is required. In the previous formula, if one divides by the number of ZnO pairs that compose the system instead of dividing by Ω, one gets a dipole averaged over the whole structure (〈LD〉) [11]: in the case of a nanowire, this is a quantity that contains both bulk and surface effects. More specifically, local contributions (local dipoles—LDs) can be obtained by partitioning the WF set and ionic charges into neutral units (ZnO couples) in such a way to obtain zero polarization for the reference bulk ZB phase structure [11]. Thus, the polarization of the system can be featured simply by variations of local dipoles. Indeed, from a radial analysis of LDs, obtained by averaging LD contributions of ZnO pairs contained in concentric cylindrical shells at increasing distance from the NW center, one can see that for the largest wires considered in this study (NW3 and NW4) bulk behavior (〈LD〉 =− 0.24 D) is quickly recovered after the outer-most shell (see bottom panel of figure 1). In particular, for these two wires, one can clearly identify a 'core' part in which ZnO pairs have bulk-like behavior (points of the flat region of figure 1) and a surface shell that strongly deviates from bulk-like behavior. In this way, one eliminates inconsistencies and arbitrariness linked to the common relations that use diameter or volume of the system: the scheme provides a method to define in an unambiguous and rigorous way a normalization useful in particular for nanostructures, although recovering bulk limits at increasing size [11].

Within this approach, the piezoelectric response of the ZnO wires grown along the [0001] axis, which is also the direction of spontaneous polarization, can be calculated in terms of variation of 〈LD〉 with respect to strain. The results for longitudinal strain ranging from −3% to 3% are reported in figure 2(a). It is apparent that apart from the smallest wire, which structurally does not have any bulk-like ZnO pair, the behavior is well approximated by a linear fit whose slope is related to the piezoelectric response of the wire (see table 1). Also the analysis in terms of 〈LD〉/strain reveals that NWs and bulk ZnO have similar piezoelectric behavior. Only NW1 is characterized by a fairly large increase of the slope, but this system is most probably unphysical. This close similarity between bulk and nanostructure piezoelectric responses reveals that the high enhancement of piezoelectric constants reported in [13] should not occur, and it is possibly related to different effects, other than the NW size.

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Taking advantage of the local analysis previously discussed, for NW3 and NW4 it is possible to separate the surface and core contribution to 〈LD〉 as a function of strain. The surface contribution is obtained by averaging LD on the ZnO pairs that deviate from bulk-like behavior and are unambiguously identified by the radial analysis reported in the bottom panel of figure 1, while the core contribution is obtained by averaging over bulk-like inner ZnO shells (results are reported in panel (c) and panel (b) of figure 2 respectively). Interestingly, also the surface contribution to the wire response is almost linear and the values of 〈LD〉/strain have values of 1.79 D and 1.25 D for NW3 and NW4 surfaces respectively (to be compared with the values reported in table 1). This result indicates that the surface response to deformation is about one order of magnitude smaller than the bulk-like term and only slightly dependent on size; furthermore surface contributions become quickly negligible and the surface/volume ratio decreases. (See the supplementary data section (available at stacks.iop.org/Nano/24/475401/mmedia) for a comparison with the more standard approach, in terms of bond deformations). These results are comparable to those reported in [12, 14], but significantly smaller than those reported in [13], probably due to the use of incorrect volume values in their calculated quantities: this supports the idea that different operating conditions motivate the spread in experimental results (see Espinosa et al [9] and references therein), and that surface defects may hinder a more trivial interpretation.

Since in the core (bulk-like) region of the wires the volume that each ZnO couple occupies is a well defined quantity, one can recall the usual definition of polarization in terms of volume, and write the polarization change induced by strain, in terms of the bulk piezoelectric constants. For a WZ structure elongated along the spontaneous polarization axis [0001], and free to contract or expand along the perpendicular axes, as in the case of NWs, one has

$$\begin{eqnarray} &&P-{P}_{\circ }=2{e}_{3 1}{\epsilon }_{1}+{e}_{3 3}{\epsilon }_{3}=({e}_{3 3}-2{e}_{3 1}\nu ){\epsilon }_{3}, \end{eqnarray} \tag{ 2 }$$

where P indicates the polarization for a strained structure, P_∘ is the equilibrium structure polarization, e₃₁ and e₃₃ are the piezoelectric constants, _i are the strains along the three lattice directions and ν is the Poisson ratio. In this case, the appropriate quantity to describe the response to strain is the effective piezoelectric constant, ${e}_{3 3}^{\mathrm{eff}}=({e}_{3 3}-2{e}_{3 1}\nu )$. Multiplying equation (2) by the volume, V, occupied by a bulk ZnO pair, and expressing V as a function of the strained lattice parameters, it is possible to write an analytical formula for the dependence of 〈LD〉 on the applied strain, ₃, for the NW cores,

$$\begin{eqnarray} \langle \mathrm{LD}\rangle &=&P V=({P}_{\circ }+{\epsilon }_{3}({e}_{3 3}-2{e}_{3 1}\nu ))\nonumber\\ &&\times ~{V}_{0}(1-{\epsilon }_{3}\nu )^{2}(1+{\epsilon }_{3}). \end{eqnarray} \tag{ 3 }$$

V₀ is the equilibrium volume/ZnO pair, i.e. at zero strain. We then use this expression to fit the data reported in figure 2(b) letting V₀,P_∘,e₃₁,e₃₃ and ν act as free parameters. From the latter values it is possible to estimate ${e}_{3 3}^{\mathrm{eff}}$ for the wire cores: for NW3 and NW4 we obtain Poisson ratio, ν, values of 0.27 and 0.28 respectively that well compare to the value of 0.29 calculated for ZnO bulk. Correspondingly, the effective piezoelectric constants of NW3 and NW4 thus estimated are 1.19 and 1.21 C m⁻² respectively, very close to the DFT bulk value of 1.28 C m⁻² (experimental 1.22 C m⁻² [21]).

In conclusion, our analysis shows that piezoelectric constants expressed in terms of 〈LD〉 are negligibly influenced by the NW size: this is further demonstrated by a quick recovery of the bulk value. Surface effects and quantum confinement which are normally anticipated to dramatically modify properties of nanostructures seem to have minor influence especially on piezoelectric properties. Similar findings have been reported by Dai et al [22], who, using DFT, showed that for thin films the piezoelectric constant converges rapidly with increase in film thickness, in contradiction of classical molecular dynamics results where slow convergence was predicted [13].

In order to explain the energy harvesting mechanism observed in ZnO NWs [7], and to describe why it is expected that nanostructures perform better than bulk systems, we analyzed how nanowires respond to external mechanical deformation and evaluated their effective spring constants. In the elastic regime the strain energy, U, required to elongate the lattice parameter of a wire along its main axis of an amount Δc, can be written as

$$\begin{eqnarray} U(\Delta c)&=&\frac{1}{2}k \Delta {c}^{2}=\frac{1}{2}k \Delta {c}^{2}\left (\frac{{c}_{0}}{{c}_{0}}\right )^{2} \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} &=&\frac{1}{2}k{c}_{0}^{2}{\epsilon }_{3}^{2}=\frac{1}{2}K{\epsilon }_{3}^{2} \end{eqnarray} \tag{ 5 }$$

where k is the elastic constant of the nanostructure, $K=k{c}_{0}^{2}$ is the effective spring constant, c₀ is the NW equilibrium lattice parameter and is the strain applied along the wire axis. From the above equations we obtain the strain energy per ZnO pair, U_s,

$$\begin{eqnarray} &&{U}_{\mathrm{s}}=\frac{U(\Delta c)}{n}=\frac{1}{2 n}K{\epsilon }_{3}^{2}=\frac{1}{2}{U}^{\prime\prime}{\epsilon }_{3}^{2} \end{eqnarray} \tag{ 6 }$$

where n is the number of ZnO pairs in the simulation supercell and ${U}^{\prime\prime}=\frac{K}{n}$ is called the effective strain energy of the system. The latter quantity gives a quantitative estimate of the amount of energy required to deform a finite structure and contains both bulk and surface effects. As reported in table 1, the effective strain energy decreases with decreasing the size of the NWs, and for the smallest wire it is reduced by about 50%. The above results contribute to solving an existing debate on the mechanical properties of ZnO NWs still present in literature (see e.g. [9] and references therein), and support the conclusion that the Young's modulus of ZnO wires decreases with decreasing size. This piece of information is relevant to open the way for the design and fabrication of efficient energy scavenging devices based on nanostructures. Furthermore, the results show that the energy requirements of these smart devices belong to the range of environmental noise, such as air, vibration, fluid flow and others [23].

In summary, in this work we solve the existing debate about the 'size effects' in the piezoelectric response of ZnO NWs with respect to bulk revealing that they are negligible. In contrast, effective strain energies of NWs are more sensitive to size reduction and are much lower compared to bulk. In particular, our theoretical predictions indicate that the advantage of using NWs for energy harvesting is due to their sensitivity to small mechanical agitation, thus making them ideal candidates for building efficient energy scavenging devices irrespective of the fact that NW nanogenerators (with typical size of 40 nm) are expected to have piezoelectric properties similar to bulk ones.

Acknowledgments