5 Enabling Materials By Dimensionality: From 0D to 3D Carbon-Based. . .
137
mechanics to be solved, the two-dimensional one is highly nontrivial. To date
furthermore, the three- or higher-dimensional Ising problems remain unsolved
although there exist different approaches related to quantum field theory to tackle
this issue.
In more realistic systems, one needs to look no further than studies of the 2D
electron gas in semiconductor heterostructures [3] or to the rich physics of graphene
[4] and of layered hybrid materials [5] to find examples of remarkable as well often
unexpected behaviour of low-dimensional systems. Indeed monolayers, which can
be obtained by mechanical exfoliation of bulk crystals, have generally distinctive
properties from their bulk counterpart. For example, while bulk MoS 2 in the 2H
phase is an indirect band gap semiconductor, MoS 2 monolayer shows a direct band
gap.
Moreover, the synthesis of stable cylindrical shapes in several material families,
such as nanotubes and nanowires [6], has driven the discovery of completely
novel extraordinary thermal, mechanical and electrical properties. Owing to their
monodimensional shape, nanotubes and nanowires can be easily integrated in
nanoscale devices and used efficiently in electron charge transport and optical excitations with potential applications in nanoelectronics, in composites and functional
nanomaterials to enhance their mechanical properties, as well as drug delivers or in
photodynamic therapy for cancer cure [7]. Carbon nanotubes in particular [8], owing
to the material’s exceptional strength and stiffness, have been synthesized with a
length-to-diameter ratio of up to 132 ×10 6 : 1 [9]. Their electronic and optical
properties are determined by the tube’s chirality, which is a feature emerging from
the 1D geometry inducing an exceptionally high excitonic binding energy [10–12].
By further miniaturizing a device, so to obtain quantum dots (0D) [13], one
can observe several phenomena such as the Coulomb blockade due to the strong
Coulomb repulsion in charge confinement and the electron tunneling which led
to the concept of single-electron transistors [14]. Quantum dots have also been
suggested as a possible mean of implementations of qubits for quantum information
processing [15].
Another feature of low-dimensional systems is to show energetically discrete
molecular-like bands due to confinement and, thus, a sharper density of states
(DOS) with respect to higher-dimensional structures. In particular, they exhibit DOS
singularities, thus having the potential for superior transport and optical properties
with respect to their higher-dimensional counterparts [16]. For example, we sketch
the typical DOS of 0D (dots), 1D (wires), 2D (wells) and 3D crystals in left panel of
Fig. 5.1, where spikes emerge in the spectrum descending the dimensional ladder.
The fingerprint of dimensionality can be also found in the optical properties
of solids. In particular, limiting the discussion to the interband transitions in a
semiconductor at zero temperature, a radiation field impinging on a crystal can be
absorbed at energies equal to the difference between valence and conduction bands,
whereby an electron is excited to a higher energetic level with respect to its ground
state. This information is encoded in the energy-dependent absorption coefficient,
which in turn is proportional to the imaginary part of the energy-dependent dielectric
function [17]:
137
mechanics to be solved, the two-dimensional one is highly nontrivial. To date
furthermore, the three- or higher-dimensional Ising problems remain unsolved
although there exist different approaches related to quantum field theory to tackle
this issue.
In more realistic systems, one needs to look no further than studies of the 2D
electron gas in semiconductor heterostructures [3] or to the rich physics of graphene
[4] and of layered hybrid materials [5] to find examples of remarkable as well often
unexpected behaviour of low-dimensional systems. Indeed monolayers, which can
be obtained by mechanical exfoliation of bulk crystals, have generally distinctive
properties from their bulk counterpart. For example, while bulk MoS 2 in the 2H
phase is an indirect band gap semiconductor, MoS 2 monolayer shows a direct band
gap.
Moreover, the synthesis of stable cylindrical shapes in several material families,
such as nanotubes and nanowires [6], has driven the discovery of completely
novel extraordinary thermal, mechanical and electrical properties. Owing to their
monodimensional shape, nanotubes and nanowires can be easily integrated in
nanoscale devices and used efficiently in electron charge transport and optical excitations with potential applications in nanoelectronics, in composites and functional
nanomaterials to enhance their mechanical properties, as well as drug delivers or in
photodynamic therapy for cancer cure [7]. Carbon nanotubes in particular [8], owing
to the material’s exceptional strength and stiffness, have been synthesized with a
length-to-diameter ratio of up to 132 ×10 6 : 1 [9]. Their electronic and optical
properties are determined by the tube’s chirality, which is a feature emerging from
the 1D geometry inducing an exceptionally high excitonic binding energy [10–12].
By further miniaturizing a device, so to obtain quantum dots (0D) [13], one
can observe several phenomena such as the Coulomb blockade due to the strong
Coulomb repulsion in charge confinement and the electron tunneling which led
to the concept of single-electron transistors [14]. Quantum dots have also been
suggested as a possible mean of implementations of qubits for quantum information
processing [15].
Another feature of low-dimensional systems is to show energetically discrete
molecular-like bands due to confinement and, thus, a sharper density of states
(DOS) with respect to higher-dimensional structures. In particular, they exhibit DOS
singularities, thus having the potential for superior transport and optical properties
with respect to their higher-dimensional counterparts [16]. For example, we sketch
the typical DOS of 0D (dots), 1D (wires), 2D (wells) and 3D crystals in left panel of
Fig. 5.1, where spikes emerge in the spectrum descending the dimensional ladder.
The fingerprint of dimensionality can be also found in the optical properties
of solids. In particular, limiting the discussion to the interband transitions in a
semiconductor at zero temperature, a radiation field impinging on a crystal can be
absorbed at energies equal to the difference between valence and conduction bands,
whereby an electron is excited to a higher energetic level with respect to its ground
state. This information is encoded in the energy-dependent absorption coefficient,
which in turn is proportional to the imaginary part of the energy-dependent dielectric
function [17]: