Python implementation of flexoelectric and piezoelectric coupling in a bended 2D material

It computes how sinusoidal bending (corrugation) of a MoS₂ monolayer induces both flexoelectric and piezoelectric polarization via strain and strain‐gradient couplings.

1. Strain tensor (u_{ij}(x)) arises from atomic displacements (U_i(x)).
2. Flexoelectricity: polarization (P_i) ≃ (f_{ijkl},\partial_j u_{kl}) — coupling of strain gradients (\partial_j u_{kl}).
3. Piezoelectricity: polarization (P_i) ≃ (e^s_{ijk},u_{jk}) — coupling of strains (u_{jk}) (surface‐induced for centrosymmetric monolayers).
4. Total polarization (neglecting electrostatic potential gradient term): [ P_i(x) ;=; f_{ijkl},\frac{\partial u_{kl}}{\partial x_j}
   • e^s_{ijk},u_{jk} ]

We impose a sinusoidal corrugation along (x): $$ u_z(x) = U_0 ,\cos(k,x), \qquad u_x(x) = V_0,x + W_0,\cos(2,k,x). $$

From (u_i(x)), the nonzero strain components are: $$ u_{zx}(x) = \tfrac12\bigl(\partial_x u_z + \partial_z u_x\bigr) = -\frac{U_0,k}{2},\sin(k,x), $$ $$ u_{xx}(x) = \partial_x,u_x = V_0 + 2,W_0,k,\cos(2,k,x). $$ Their derivatives (strain gradients) are $$ \frac{\partial u_{zx}}{\partial x} = -\frac{U_0,k^2}{2},\cos(k,x), \qquad \frac{\partial u_{xx}}{\partial x} = -4,W_0,k^2,\sin(2,k,x). $$

Each unit cell acquires an effective dipole (\mathbf{d}(x)) with projections $$ d_z(x) = e_{zzx},u_{zx}(x)

• e_{zxx},u_{xx}(x)
• f_{zxzx},\frac{\partial u_{zx}}{\partial x}
• f_{zxxx},\frac{\partial u_{xx}}{\partial x}, $$ $$ d_x(x) = e_{xzx},u_{zx}(x)
• e_{xxx},u_{xx}(x)
• f_{xxxz},\frac{\partial u_{xz}}{\partial x}
• f_{xxxx},\frac{\partial u_{xx}}{\partial x}. $$

Fitting the computed (d_i(x)) against the known sinusoidal forms yields: $$ \begin{aligned} f_{3131} &= -\frac{2,d^{(z)}\mathrm{cos}}{U_0,k^2}, &\quad e{331} &= -\frac{2,d^{(z)}\mathrm{sin}}{U_0,k}, \[4pt] f{1111} &= -\frac{d^{(x)}\mathrm{sin}}{4,W_0,k^2}, &\quad e{111} &= \frac{d^{(x)}_\mathrm{cos}}{2,W_0,k}. \end{aligned} $$