6.2 From Bulk to Zero-Dimensional Structures
191
(3D) value. Accordingly, the binding energy should be raised by a factor of four
compared to bulk. However, practical quantum wells have a finite thickness and
finite potential height, which prevent them to reach that ideal value, as discussed in
numerous studies from the 1980s on quantum-well systems [18–20]. Note that for
practical quantum well systems, the thickness L plays an important role: a trade-off
has to be found between an increased oscillator strength through a decrease of L and
an undesired increased penetration of the wave-function into the barrier materials for
finite barrier heights, as well as a rise of energy levels closer to the unbound energies
of the surrounding materials with decreasing L (facilitating leakage of particles out
of the potential well region). Indeed, monolayer 2D semiconductors of the TMDC
family with their in-plane excitons strongly confined to the layer of transition metals
offer a model system as natural nearly-ideal quantum wells.
6.2.2 Density of States
For quantum structures, dimensionality plays an important role, as it influences the
distribution of energies of those systems. This distribution is described for any lowdimensional structure by its respective density of states (DOS), sketched in Fig. 6.1.
In a periodic system, the DOS can be mathematically derived using periodic boundary conditions and the reciprocal relationship between real space and phase space
(i.e. momentum space or, short, k space). To obtain this distribution for a given
dimensionality d (considering integer positive numbers), one seeks to determine
how many particle states around an energy E exist within the energy interval δ E per
k-space volume (2π)
d
/V d per real-space volume V d = L
d in a (box-like) system of
length L for each available direction. The typically-derived expression for the DOS,
D d (E)δ E, is a characteristic for the dimensionality of the system and has a peculiar
trend as a function of energy, which in a text-book manner can be described for the
4 distinct cases at the example of electronic states as follows.
The cubic 3D electronic system has an energy distribution (DOS) following a
square-root-of-E proportionality (beginning at the ground state E 0 , see first column
of Fig. 6.1). The film-like 2D system exhibits a stairways-like DOS with a constant
value added for each confinement energy level of the system
6 (second column of
Fig. 6.1). The wire-like 1D system features spikes at every quantisation energy with
inverse-square-root proportionality, i.e. E
−1/2 , leading to a decrease of energy states
6 Amendment: At the example of the quantum well, the determination of the DOS D(E) can be
summarised as follows. Taking into account spin (an energy state can host two fermions of opposite
spin), a factor of two will be carried in D(E). With the increment of dk in 2D k space, N 2D (k) dk =
2π k dk states situated on the circumference of a circle with radius k add up at E(k) per energy
increment dE. Then, the DOS can be obtained through D 2D (E) dE = 2(N 2D (k)δk/δ E)/(2π/L) 2 ·
(1/L 2 ) dE = 2(2π k)/(2π) 2 · (δk/δ E) dE. Due to dE/dk = 2 k/m, one obtains D 2D (E) dE =
2m/(2π 2 ) dE. Thus, D 2D (E) shows no dependence on E. In the 3D and 1D case, N 3D (k) dk =
4π k 2 dk (surface of a sphere with radius k) and N 1D (k) dk = 2 dk (2 for both propagation directions
on a line), respectively.
191
(3D) value. Accordingly, the binding energy should be raised by a factor of four
compared to bulk. However, practical quantum wells have a finite thickness and
finite potential height, which prevent them to reach that ideal value, as discussed in
numerous studies from the 1980s on quantum-well systems [18–20]. Note that for
practical quantum well systems, the thickness L plays an important role: a trade-off
has to be found between an increased oscillator strength through a decrease of L and
an undesired increased penetration of the wave-function into the barrier materials for
finite barrier heights, as well as a rise of energy levels closer to the unbound energies
of the surrounding materials with decreasing L (facilitating leakage of particles out
of the potential well region). Indeed, monolayer 2D semiconductors of the TMDC
family with their in-plane excitons strongly confined to the layer of transition metals
offer a model system as natural nearly-ideal quantum wells.
6.2.2 Density of States
For quantum structures, dimensionality plays an important role, as it influences the
distribution of energies of those systems. This distribution is described for any lowdimensional structure by its respective density of states (DOS), sketched in Fig. 6.1.
In a periodic system, the DOS can be mathematically derived using periodic boundary conditions and the reciprocal relationship between real space and phase space
(i.e. momentum space or, short, k space). To obtain this distribution for a given
dimensionality d (considering integer positive numbers), one seeks to determine
how many particle states around an energy E exist within the energy interval δ E per
k-space volume (2π)
d
/V d per real-space volume V d = L
d in a (box-like) system of
length L for each available direction. The typically-derived expression for the DOS,
D d (E)δ E, is a characteristic for the dimensionality of the system and has a peculiar
trend as a function of energy, which in a text-book manner can be described for the
4 distinct cases at the example of electronic states as follows.
The cubic 3D electronic system has an energy distribution (DOS) following a
square-root-of-E proportionality (beginning at the ground state E 0 , see first column
of Fig. 6.1). The film-like 2D system exhibits a stairways-like DOS with a constant
value added for each confinement energy level of the system
6 (second column of
Fig. 6.1). The wire-like 1D system features spikes at every quantisation energy with
inverse-square-root proportionality, i.e. E
−1/2 , leading to a decrease of energy states
6 Amendment: At the example of the quantum well, the determination of the DOS D(E) can be
summarised as follows. Taking into account spin (an energy state can host two fermions of opposite
spin), a factor of two will be carried in D(E). With the increment of dk in 2D k space, N 2D (k) dk =
2π k dk states situated on the circumference of a circle with radius k add up at E(k) per energy
increment dE. Then, the DOS can be obtained through D 2D (E) dE = 2(N 2D (k)δk/δ E)/(2π/L) 2 ·
(1/L 2 ) dE = 2(2π k)/(2π) 2 · (δk/δ E) dE. Due to dE/dk = 2 k/m, one obtains D 2D (E) dE =
2m/(2π 2 ) dE. Thus, D 2D (E) shows no dependence on E. In the 3D and 1D case, N 3D (k) dk =
4π k 2 dk (surface of a sphere with radius k) and N 1D (k) dk = 2 dk (2 for both propagation directions
on a line), respectively.