9.3. SIZE AND DIMENSIONALITY EFFECTS
241
The energy of a two-dimensional infinite rectangular square well
E, = (s)(n: + n:) = Eon2
(9.9)
depends on two quantum numbers, n, = 0, 1 , 2 , 3 , . . . and ny = 0, 1,2,3, . . . , where
n2 = n: + nj. This means that the lowest energy state El = Eo has two possibilities,
namely, n, = 0, ny = 1, and n, = 1, ny = 0, so the total degeneracy (including spin
direction) is 4. The energy state E, = 25E0 has more possibilities since it can have,
for example, n, = 0, ny = 5, or n, = 3 and n,, = 4, and so on, so its degeneracy is 8.
9.3.5. Partial Confinement
In the previous section we examined the confinement of electrons in various
dimensions, and we found that it always leads to a qualitatively similar spectrum
of discrete energies. This is true for a broad class of potential wells, irrespective of
their dimensionality and shape. We also examined, in Section 9.3.3, the Fermi gas
model for delocalized electrons in these same dimensions and found that the model
leads to energies and densities of states that differ quite significantly from each other.
This means that many electronic and other properties of metals and semiconductors
change dramatically when the dimensionality changes. Some nanostructures of
technological interest exhibit both potential well confinement and Fermi gas
delocalization, confinement in one or two dimensions, and delocalization in two
or one dimensions, so it will be instructive to show how these two strikingly different
behaviors coexist.
In a three-dimensional Fermi sphere the energy varies from E = 0 at the origin to
E = E, at the Fermi surface, and similarly for the one- and two-dimensional analogs.
When there is confinement in one or two directions, the conduction electrons will
distribute themselves among the corresponding potential well levels that lie below
the Fermi level along confinement coordinate directions, in accordance with their
respective degeneracies d,, and for each case the electrons will delocalize in the
remaining dimensions by populating Fermi gas levels in the delocalization direction
of the reciprocal lattice. Table 9.5 lists the formulas for the energy dependence of the
number of electrons N(E) for quantum dots that exhibit total confinement, quantum
wires and quantum wells, which involve partial confinement, and bulk material,
where there is no confinement. The density of states formulas D(E) for these four
cases are also listed in the table. The summations in these expressions are over the
various confinement well levels i.
Figure 9.15 shows plots of the energy dependence N ( E ) and the density of states
D(E) for the four types of nanostructures listed in Table 9.5. We see that the number
of electrons N(E) increases with the energy E, so the four nanostructure types vary
only qualitatively from each other. However, it is the density of states D(E) that
determines the various electronic and other properties, and these differ dramatically
for each of the three nanostructure types. This means that the nature of the
241
The energy of a two-dimensional infinite rectangular square well
E, = (s)(n: + n:) = Eon2
(9.9)
depends on two quantum numbers, n, = 0, 1 , 2 , 3 , . . . and ny = 0, 1,2,3, . . . , where
n2 = n: + nj. This means that the lowest energy state El = Eo has two possibilities,
namely, n, = 0, ny = 1, and n, = 1, ny = 0, so the total degeneracy (including spin
direction) is 4. The energy state E, = 25E0 has more possibilities since it can have,
for example, n, = 0, ny = 5, or n, = 3 and n,, = 4, and so on, so its degeneracy is 8.
9.3.5. Partial Confinement
In the previous section we examined the confinement of electrons in various
dimensions, and we found that it always leads to a qualitatively similar spectrum
of discrete energies. This is true for a broad class of potential wells, irrespective of
their dimensionality and shape. We also examined, in Section 9.3.3, the Fermi gas
model for delocalized electrons in these same dimensions and found that the model
leads to energies and densities of states that differ quite significantly from each other.
This means that many electronic and other properties of metals and semiconductors
change dramatically when the dimensionality changes. Some nanostructures of
technological interest exhibit both potential well confinement and Fermi gas
delocalization, confinement in one or two dimensions, and delocalization in two
or one dimensions, so it will be instructive to show how these two strikingly different
behaviors coexist.
In a three-dimensional Fermi sphere the energy varies from E = 0 at the origin to
E = E, at the Fermi surface, and similarly for the one- and two-dimensional analogs.
When there is confinement in one or two directions, the conduction electrons will
distribute themselves among the corresponding potential well levels that lie below
the Fermi level along confinement coordinate directions, in accordance with their
respective degeneracies d,, and for each case the electrons will delocalize in the
remaining dimensions by populating Fermi gas levels in the delocalization direction
of the reciprocal lattice. Table 9.5 lists the formulas for the energy dependence of the
number of electrons N(E) for quantum dots that exhibit total confinement, quantum
wires and quantum wells, which involve partial confinement, and bulk material,
where there is no confinement. The density of states formulas D(E) for these four
cases are also listed in the table. The summations in these expressions are over the
various confinement well levels i.
Figure 9.15 shows plots of the energy dependence N ( E ) and the density of states
D(E) for the four types of nanostructures listed in Table 9.5. We see that the number
of electrons N(E) increases with the energy E, so the four nanostructure types vary
only qualitatively from each other. However, it is the density of states D(E) that
determines the various electronic and other properties, and these differ dramatically
for each of the three nanostructure types. This means that the nature of the