344
M. Springborg et al.
to fix ¯
μ ze (up to an integer multiple of the lattice constant). As a result the electronic
dipole moment per unit cell given by Eq. (18.8) may take any value.
A similar result is obtained when attempting to use Wannier functions [see
Eq. (18.12)] to calculate ¯
μ ze . That approach is not applicable because Wannier functions cannot be constructed for fractionally occupied bands.
One could try to calculate ¯
μ ze from Eq. (18.7). For a semiconductor/insulator the
S ± matrices contain K blocks of dimension B × B (B is the number of occupied
bands) with non-vanishing matrix elements between band orbitals of neighboring
k values [see Eq. (18.9) and Fig. 18.3] whereas all other matrix elements are zero.
Thus, det(S ± ) will equal the product of the K determinants of each block. For
a metallic system, since there are partially occupied bands, some of these square
blocks will become rectangular and det(S ± ) will vanish. Thus, as above, ¯
μ ze can
take any value.
Next we turn to the case of a very large finite system, for which the energy
gap between occupied and unoccupied orbitals approaches zero as the system size
grows. As pointed out by Resta [24], one can construct localized orbitals for such
small band gap semiconductors even though it is not possible to do so for a metal.
Indeed, in our model zero field calculations these long finite chains behaved similarly to ordinary semiconductors. However, as soon as a field was turned on, it
became no longer possible to identify a central region even for the longest chains,
which were more than one order of magnitude longer than those discussed earlier
in Sect. 18.3.1). Moreover, the dipole moment per unit increased supralinearly with
chain length. Upon fitting the latter property to a power series in the applied field,
¯
μ = ¯
μ
(0)
+ ¯
αE +
1
2
¯
βE
2
+
1
6
¯
γ E
3
+ · · · ,
(18.36)
we found that the polarizability per unit, ¯
α, as well as the hyperpolarizabilities per
unit, ¯
β and ¯
γ , also grow rapidly with system size in accord with the behavior for a
perfect metal. Thus, the field causes a charge redistribution throughout the complete
system and, as a result, the charge associated with the chosen surface region can take
any value.
18.5 Short-Circuited Semiconductor
The results of the last two sections will now be used to study a system (‘device’)
like that of Fig. 18.5. This system contains a semiconductor placed between two
metal capacitor plates connected through a metal so that the semiconductor is shortcircuited. We assume that there is a non-vanishing polarization P inside the semiconductor in the direction perpendicular to the capacitor plates. P is related to the
dipole moment per unit cell through
P =
1
Ω
¯
μ
(18.37)
M. Springborg et al.
to fix ¯
μ ze (up to an integer multiple of the lattice constant). As a result the electronic
dipole moment per unit cell given by Eq. (18.8) may take any value.
A similar result is obtained when attempting to use Wannier functions [see
Eq. (18.12)] to calculate ¯
μ ze . That approach is not applicable because Wannier functions cannot be constructed for fractionally occupied bands.
One could try to calculate ¯
μ ze from Eq. (18.7). For a semiconductor/insulator the
S ± matrices contain K blocks of dimension B × B (B is the number of occupied
bands) with non-vanishing matrix elements between band orbitals of neighboring
k values [see Eq. (18.9) and Fig. 18.3] whereas all other matrix elements are zero.
Thus, det(S ± ) will equal the product of the K determinants of each block. For
a metallic system, since there are partially occupied bands, some of these square
blocks will become rectangular and det(S ± ) will vanish. Thus, as above, ¯
μ ze can
take any value.
Next we turn to the case of a very large finite system, for which the energy
gap between occupied and unoccupied orbitals approaches zero as the system size
grows. As pointed out by Resta [24], one can construct localized orbitals for such
small band gap semiconductors even though it is not possible to do so for a metal.
Indeed, in our model zero field calculations these long finite chains behaved similarly to ordinary semiconductors. However, as soon as a field was turned on, it
became no longer possible to identify a central region even for the longest chains,
which were more than one order of magnitude longer than those discussed earlier
in Sect. 18.3.1). Moreover, the dipole moment per unit increased supralinearly with
chain length. Upon fitting the latter property to a power series in the applied field,
¯
μ = ¯
μ
(0)
+ ¯
αE +
1
2
¯
βE
2
+
1
6
¯
γ E
3
+ · · · ,
(18.36)
we found that the polarizability per unit, ¯
α, as well as the hyperpolarizabilities per
unit, ¯
β and ¯
γ , also grow rapidly with system size in accord with the behavior for a
perfect metal. Thus, the field causes a charge redistribution throughout the complete
system and, as a result, the charge associated with the chosen surface region can take
any value.
18.5 Short-Circuited Semiconductor
The results of the last two sections will now be used to study a system (‘device’)
like that of Fig. 18.5. This system contains a semiconductor placed between two
metal capacitor plates connected through a metal so that the semiconductor is shortcircuited. We assume that there is a non-vanishing polarization P inside the semiconductor in the direction perpendicular to the capacitor plates. P is related to the
dipole moment per unit cell through
P =
1
Ω
¯
μ
(18.37)