18 On Converse Piezoelectricity
347
We also sketched our theoretical approach for including an external electrostatic
field in calculations on infinite periodic systems, as is necessary to determine the
converse piezoelectric coefficient. This approach involves development of an appropriate dipole moment operator containing both ‘charge’ and ‘current’ terms. It
also leads to the introduction of an (unknown) integer multiple of the lattice constant
into the phases of the crystal orbitals. Many details are non-trivial and could not be
discussed here, but for a complete and more thorough description the interested
reader is referred to [16].
Results are shown from earlier model calculations on semiconductor chains that
verify and illustrate our theoretical treatment. More recent calculations for real materials are included as well. They involve some approximations but, nonetheless,
confirm that the surface effect is significant, though not very large for the particular
materials studied.
The infinite periodic treatment we have presented is applicable only for semiconductors and insulators. It is shown to be inapplicable to metals because the electronic
charge cannot be localized to a central region and, by the same token, the electronic
dipole moment may take any arbitrary value. For a large finite system the electronic
dipole moment grows supralinearly with chain length.
Based on our analysis of a metallic system we studied an idealized device where
a semiconductor (or insulator) is sandwiched between two metallic capacitor plates
and short-circuited by connecting the plates through the metal. If the free-standing
semiconductor is spontaneously polarized (i.e. has a permanent dipole moment) it
will have an associated internal electric field. When attached to the metal the latter
will respond by creating an (external) field that exactly compensates the internal
one. The external field, in turn, will induce a change in the polarization of the semiconductor, and so forth. This leads ultimately to a total energy for the semiconductor
that depends upon the square of its self-consistent polarization. Using the results for
the real material, we were able to estimate the relative change in the lattice constant,
i.e. the strain, due to short-circuiting the semiconductor.
Acknowledgements One of the authors (MS) is very grateful to the International Center for Materials Research, University of California, Santa Barbara, for generous hospitality. Another author
(JV) is very grateful to CONACYT, Mexico, and DAAD, Germany, for financial support. Moreover, this work was supported financially by the German Research Council (DFG) through project
Sp439/37. We thank Adlane Sayede, Université Lille Nord de France, France, for the ab initio results behind our discussion in Sect. 18.3.2 and Stanislaw Krukowski, Polish Academy of Sciences,
Warsaw, Poland, for discussions about the system of Sect. 18.5.
References
1. Woo JWF (1971) Phys Rev B 4:1218
2. Martin RM (1972) Phys Rev B 5:1607
3. Martin RM (1972) Phys Rev B 6:4874
4. Woo JWF, Landauer R (1972) Phys Rev B 6:4876
5. Nelson DF, Lax M (1976) Phys Rev B 13:1785
347
We also sketched our theoretical approach for including an external electrostatic
field in calculations on infinite periodic systems, as is necessary to determine the
converse piezoelectric coefficient. This approach involves development of an appropriate dipole moment operator containing both ‘charge’ and ‘current’ terms. It
also leads to the introduction of an (unknown) integer multiple of the lattice constant
into the phases of the crystal orbitals. Many details are non-trivial and could not be
discussed here, but for a complete and more thorough description the interested
reader is referred to [16].
Results are shown from earlier model calculations on semiconductor chains that
verify and illustrate our theoretical treatment. More recent calculations for real materials are included as well. They involve some approximations but, nonetheless,
confirm that the surface effect is significant, though not very large for the particular
materials studied.
The infinite periodic treatment we have presented is applicable only for semiconductors and insulators. It is shown to be inapplicable to metals because the electronic
charge cannot be localized to a central region and, by the same token, the electronic
dipole moment may take any arbitrary value. For a large finite system the electronic
dipole moment grows supralinearly with chain length.
Based on our analysis of a metallic system we studied an idealized device where
a semiconductor (or insulator) is sandwiched between two metallic capacitor plates
and short-circuited by connecting the plates through the metal. If the free-standing
semiconductor is spontaneously polarized (i.e. has a permanent dipole moment) it
will have an associated internal electric field. When attached to the metal the latter
will respond by creating an (external) field that exactly compensates the internal
one. The external field, in turn, will induce a change in the polarization of the semiconductor, and so forth. This leads ultimately to a total energy for the semiconductor
that depends upon the square of its self-consistent polarization. Using the results for
the real material, we were able to estimate the relative change in the lattice constant,
i.e. the strain, due to short-circuiting the semiconductor.
Acknowledgements One of the authors (MS) is very grateful to the International Center for Materials Research, University of California, Santa Barbara, for generous hospitality. Another author
(JV) is very grateful to CONACYT, Mexico, and DAAD, Germany, for financial support. Moreover, this work was supported financially by the German Research Council (DFG) through project
Sp439/37. We thank Adlane Sayede, Université Lille Nord de France, France, for the ab initio results behind our discussion in Sect. 18.3.2 and Stanislaw Krukowski, Polish Academy of Sciences,
Warsaw, Poland, for discussions about the system of Sect. 18.5.
References
1. Woo JWF (1971) Phys Rev B 4:1218
2. Martin RM (1972) Phys Rev B 5:1607
3. Martin RM (1972) Phys Rev B 6:4874
4. Woo JWF, Landauer R (1972) Phys Rev B 6:4876
5. Nelson DF, Lax M (1976) Phys Rev B 13:1785