Quantum theory of the piezoelectric effect in rare-earth ferroborates

Quantum theory of piezoelectric effect in rare-earth ferroborates has been developed. It was found that single-ion mechanisms are predominantly responsible for the effect: electric dipole moments are induced directly in the electronic 4$f$ shells of rare-earth ions in the presence of mechanical stresses in a crystal, which leads to the polarization of the crystal. The effect is quantitatively considered using the example of neodymium and samarium ferroborates, in particular, in the temperature range from 0 K to 300 K, the temperature dependences of piezoelectric modules are obtained. It is additionally shown that it is also possible for the crystal to become polarized along the crystallographic $c$-axis at the temperatures below the Néel point upon deformation of the crystal in a magnetic field. The temperature, field and orientation dependences of magnetically induced piezomodules are calculated.