Density functional theory (DFT) modeling has become an indispensable tool for understanding piezoelectric effects in nanostructures, offering insights into polarization mechanisms, strain-response behavior, and quantum confinement effects. Piezoelectricity arises from the coupling between mechanical strain and electric polarization, and DFT provides a first-principles approach to quantify these phenomena at the nanoscale. This article focuses on DFT methodologies for piezoelectric property calculations, with particular emphasis on ZnO nanowires and MoS2 monolayers, while comparing their responses to bulk counterparts.

Polarization calculations in piezoelectric nanostructures rely on the Berry phase formalism, which captures the electronic contribution to polarization under strain. The Berry phase approach evaluates the geometric phase of occupied electronic states, directly linking wavefunction evolution to macroscopic polarization. For ZnO nanowires, DFT simulations reveal that polarization along the [0001] direction is dominated by ionic displacements of Zn and O sublattices under uniaxial strain. The piezoelectric coefficient d33 in ZnO nanowires with diameters below 10 nm can reach 12-15 pm/V, significantly higher than the bulk value of 9.93 pm/V due to surface effects and reduced screening. In MoS2 monolayers, breaking inversion symmetry enables out-of-plane piezoelectricity absent in bulk MoS2, with computed d31 values around 3.65 pm/V under biaxial strain.

Strain-response coefficients are derived from DFT by applying incremental deformations and monitoring changes in polarization. The piezoelectric stress tensor eij and strain tensor dij are obtained through linear response theory or finite-difference methods. For wurtzite ZnO nanowires, the e33 coefficient exhibits strong diameter dependence, increasing from 1.12 C/m² in bulk to 1.45 C/m² in 3 nm nanowires due to surface charge redistribution. In MoS2, the in-plane e11 coefficient shows anisotropic behavior, with values of 2.9×10⁻¹⁰ C/m under armchair strain versus 3.2×10⁻¹⁰ C/m along zigzag directions. These coefficients are sensitive to edge termination in finite nanostructures, with sulfur-terminated edges showing 18% higher response than molybdenum-terminated edges.

Size-dependent piezoelectricity emerges from quantum confinement and surface effects. DFT studies demonstrate that ZnO nanowires below 5 nm diameter exhibit nonlinear piezoelectric enhancement, deviating from classical continuum models. The enhancement follows a power-law relationship with diameter (d^-α, where α≈0.7), attributed to increased bond ionicity at surfaces. For MoS2 monolayers, layer thickness modulation shows that the piezoelectric coefficient decays exponentially with layer number, dropping by 65% from monolayer to bilayer due to restored inversion symmetry. Thickness-dependent screening effects also play a role, with dielectric constant variations modifying the effective piezoelectric response.

Symmetry breaking is fundamental to piezoelectricity in nanostructures. Bulk ZnO belongs to the P63mc space group with inherent polarity, while nanostructures exhibit additional symmetry reduction at surfaces. DFT reveals that surface reconstructions in ZnO nanowires lower local symmetry to C3v, enhancing shear piezoelectric components. In MoS2, the transition from centrosymmetric bulk (P63/mmc) to non-centrosymmetric monolayer (P6m2) creates entirely new piezoelectric tensor components. Defects such as sulfur vacancies further break symmetry, increasing piezoelectric coefficients by up to 30% at vacancy concentrations of 2%.

Quantum confinement effects modify piezoelectric responses through several mechanisms. In ZnO nanowires, DFT shows bandgap widening reduces electronic screening, amplifying strain-induced polarization fields. The confinement energy scales with diameter as Econf ∝ d^-1.8, correlating with enhanced d33 coefficients. For MoS2 monolayers, quantum confinement creates discrete electronic states that modify deformation potentials. The piezoelectric response becomes sensitive to Fermi level position, with n-doped systems showing 22% lower response than intrinsic monolayers due to charge screening.

Comparisons with bulk materials highlight nanoscale advantages. Bulk ZnO exhibits isotropic piezoelectricity in the basal plane, while nanowires develop anisotropic responses with e31/e33 ratio increasing from 0.45 in bulk to 0.68 in 4 nm wires. Bulk MoS2 shows no piezoelectricity, whereas monolayers achieve responses comparable to traditional piezoelectrics. Temperature-dependent DFT simulations indicate nanostructures maintain piezoelectric coefficients up to higher temperatures than bulk, with ZnO nanowires retaining 85% of room-temperature response at 500 K versus 60% for bulk.

Berry phase approaches require careful implementation in nanostructures. The modern theory of polarization necessitates careful k-point sampling, with nanowires typically requiring 8×8×16 meshes for convergence. For 2D materials like MoS2, vacuum spacing must exceed 15 Å to prevent spurious interactions. The Berry phase is computed by integrating the imaginary part of the logarithm of overlap matrices between adjacent k-points, with convergence thresholds below 10^-4 for polarization quanta.

DFT modeling has revealed several unexpected phenomena in nanostructure piezoelectricity. ZnO nanowires exhibit negative piezoelectric coefficients under certain surface functionalizations, with hydroxylation reversing the sign of d33. MoS2 monolayers show strain-gradient-induced flexoelectricity contributing up to 15% of total response at 2% strain gradients. These effects are only accessible through first-principles simulations.

Challenges remain in DFT modeling of piezoelectric nanostructures. Van der Waals corrections are essential for layered materials, with the D3 method providing accurate interlayer interactions. For large nanowires, hybrid functionals like HSE06 are necessary to properly describe surface states, though computational cost scales steeply. Future directions include time-dependent DFT for dynamic piezoelectric responses and machine learning potentials for larger-scale simulations.

The predictive power of DFT has guided experimental work on nanostructure piezoelectricity. Calculated size-scaling laws have been verified for ZnO nanowires down to 3 nm diameter. MoS2 monolayer predictions spurred measurements using piezoresponse force microscopy, confirming theoretical values within 10%. These successes demonstrate DFT's role as a fundamental tool for nanoscale piezoelectric phenomena understanding.