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Conversion of mechanical forces to electric signal is possible innon-centrosymmetric materials due to linear piezoelectricity. The extraordinarymechanical properties of two-dimensional materials and their high crystallinitymake them exceptional platforms to study and exploit the piezoelectric effect.Here, the piezoelectric response of non-centrosymmetric hexagonaltwo-dimensional crystals is studied using the modern theory of polarization and${\bm k} \cdot {\bm p}$ model Hamiltonians. An analytical expression for thepiezoelectric constant is obtained in terms of topological quantities such asthe {\it valley Chern number}. The theory is applied to semiconductingtransition metal dichalcogenides and hexagonal Boron Nitride. We find goodagreement with available experimental measurements for MoS$_2$. We furthergeneralise the theory to study the polarization of samples subjected toinhomogeneous strain (e.g.~nanobubbles). We obtain a simple expression in termsof the strain tensor, and show that charge densities $\gtrsim 10^{11} {\rmcm}^{-2}$ can be induced by realistic inhomogeneous strains, $\epsilon \approx0.01 - 0.03$.

Piezoelectricity and valley Chern number in inhomogeneous hexagonal 2D crystals