18 On Converse Piezoelectricity
339
are the overlap, Fock (or Kohn-Sham), and dipole matrices, respectively. It is important to note that k remains a good quantum number, as indicated in Eq. (18.24),
and thus translational symmetry is preserved. On the other hand, due to the
∂
∂k term
on the left hand side, Eq. (18.24) is not a simple matrix-eigenvalue relation. Since
C j (k) may contain essentially random, j - and k-dependent phase factors, it is a
non-trivial problem to solve Eq. (18.24). During the last few years, however, we
have presented a numerically stable and accurate procedure for doing so [8, 18–21].
18.2.4 The Total Energy
For the large finite system exposed to an electrostatic field, the total energy can be
written as
E tot = E kin
{ψ j }
+ E J
ρ(r)
+ E xc
{ψ j }
− E · μ
≡ E tot,0 − E · μ.
(18.26)
In principle, the above expression is exact, provided one uses a set of occupied
orbitals {ψ j } that yield the exact density. In practice, we will assume that one is
using either the Hartree-Fock or the Kohn-Sham independent particle model. The
first term in the top line of the right hand side is the kinetic energy contribution;
the second term is the classical Coulomb interaction between all charged particles
(i.e., electrons and nuclei); and the third term is either the exchange energy for a
Hartree-Fock treatment or the exchange-correlation energy for a Kohn-Sham treatment. The last term describes the interaction between the system and the electrostatic field. Since the electronic orbitals, and thereby the electron density, obtained
from Eq. (18.24) depend upon the electrostatic field each term in the total energy
expression will depend upon E including E tot,0 as well as μ.
By considering large finite systems of different (sufficiently large) size it is possible to calculate the total energy per unit, ¯
E tot . Alternatively, one may consider the
infinite periodic system directly. In that case the quantity μ per unit cell is calculated
using the expression of Eq. (18.6) with the nuclear part given by Eq. (18.10) and the
electronic part by Eq. (18.14).
18.3 Results for Semiconductors
18.3.1 A Model System
As discussed in the previous section, for the infinite periodic system, ¯
μ contains an
unknown additive term, i.e.
¯
μ = ¯
μ 0 − ˜
n · a,
(18.27)
339
are the overlap, Fock (or Kohn-Sham), and dipole matrices, respectively. It is important to note that k remains a good quantum number, as indicated in Eq. (18.24),
and thus translational symmetry is preserved. On the other hand, due to the
∂
∂k term
on the left hand side, Eq. (18.24) is not a simple matrix-eigenvalue relation. Since
C j (k) may contain essentially random, j - and k-dependent phase factors, it is a
non-trivial problem to solve Eq. (18.24). During the last few years, however, we
have presented a numerically stable and accurate procedure for doing so [8, 18–21].
18.2.4 The Total Energy
For the large finite system exposed to an electrostatic field, the total energy can be
written as
E tot = E kin
{ψ j }
+ E J
ρ(r)
+ E xc
{ψ j }
− E · μ
≡ E tot,0 − E · μ.
(18.26)
In principle, the above expression is exact, provided one uses a set of occupied
orbitals {ψ j } that yield the exact density. In practice, we will assume that one is
using either the Hartree-Fock or the Kohn-Sham independent particle model. The
first term in the top line of the right hand side is the kinetic energy contribution;
the second term is the classical Coulomb interaction between all charged particles
(i.e., electrons and nuclei); and the third term is either the exchange energy for a
Hartree-Fock treatment or the exchange-correlation energy for a Kohn-Sham treatment. The last term describes the interaction between the system and the electrostatic field. Since the electronic orbitals, and thereby the electron density, obtained
from Eq. (18.24) depend upon the electrostatic field each term in the total energy
expression will depend upon E including E tot,0 as well as μ.
By considering large finite systems of different (sufficiently large) size it is possible to calculate the total energy per unit, ¯
E tot . Alternatively, one may consider the
infinite periodic system directly. In that case the quantity μ per unit cell is calculated
using the expression of Eq. (18.6) with the nuclear part given by Eq. (18.10) and the
electronic part by Eq. (18.14).
18.3 Results for Semiconductors
18.3.1 A Model System
As discussed in the previous section, for the infinite periodic system, ¯
μ contains an
unknown additive term, i.e.
¯
μ = ¯
μ 0 − ˜
n · a,
(18.27)