5 Enabling Materials By Dimensionality: From 0D to 3D Carbon-Based. . .
139
The former represents a tool for coupling and transferring energy. This mechanism
depends on the dimensional scale. Indeed, contrary to ordinary surface plasmons
of bulk materials, in low-dimensional structures, plasmon dispersion goes to zero
in the long wavelength limit, covering an energy range from terahertz to near
infrared. Moreover, in layered materials their in-plane dispersion, such as in the
case of transition metal dichalcogenides, can show a negative in-plane plasmon
dispersion [19]. Quantum Hall effect in low-dimensional materials, e.g. in graphene,
is different from the spin Hall effect found in 3D systems, as it leads to a phase which
is topologically distinct from a band insulator [20].
So far, according to the abovementioned Anderson’s conjecture, we have shown
examples in which the dependence on dimensionality of the particle-particle
correlation plays a crucial role. Nevertheless, in some occasions a dimensional
change can be enough to modify dramatically the behaviour of physical systems,
despite correlation among constituents is switched off.
For example, let us analyse the effect that a dimensional change has on the
electronic properties of a system of free (noninteracting) electrons. This model,
which is called Fermi gas, is appropriate to study the conduction in simple systems,
such as alkali and noble metals, even though rigorously the electron motion is also
influenced by the periodic potential created by the ions in the lattice. By solving the
Schrödinger equation for one electron in a box of edge L where periodic boundary
conditions are introduced, a set of discrete energy levels E n = ¯
h 2
2m (
2πn x,y,z
L
) 2 =
¯
h 2
2m (k x,y,z ) 2 emerge, where n x,y,z is an integer number, m is the electron mass, ¯
h
is the reduced Plank constant, and k x,y,z is the corresponding wavevector along the
three Cartesian directions x, y, z. By neglecting the electron-electron interactions,
we can build up the N -electron ground state of the system by accommodating the
charges into the allowed one-electron levels starting from the bottom, provided that
the same state cannot be occupied by more than two electrons, one with spin up and
one with spin down orientation. Indeed, electrons are fermions, following the FermiDirac statistics, and obey the Pauli’s exclusion principle. The occupied orbitals are
represented by a point in the k-space inside a sphere of radius k F =
2mm F / ¯
h 2 ,
which is the highest momentum an electron can have within the box. F is the
Fermi energy, which defines the so-called Fermi sphere, within which all occupied
one-electron levels lay. The number of electronic states per unit energy range
(DOS, D(()) in the solid is a quantum observable: for Al, for example, one has
F = 11.6 eV and D(( F ) = 0.39 (eV atom) −1 .
Furthermore, the kinetic energy of the electron gas increases with temperature,
and some energy levels, which were vacant at 0 K, start to be populated. The
distribution of electrons among the levels is described by the Fermi distribution
function, f (E) = 1/(exp ((−μ)/k B T +1), which gives the probability that the energy
level E is occupied by fermions. Multiplying the DOS by the latter function, one
can obtain the DOS analytic expressions for electrons at temperature T confined
in their motion by infinite barriers into (i) a cube box of side L (3D), (ii) a square
surface of side L (2D) and (iii) a wire of length L (1D), as follows:
139
The former represents a tool for coupling and transferring energy. This mechanism
depends on the dimensional scale. Indeed, contrary to ordinary surface plasmons
of bulk materials, in low-dimensional structures, plasmon dispersion goes to zero
in the long wavelength limit, covering an energy range from terahertz to near
infrared. Moreover, in layered materials their in-plane dispersion, such as in the
case of transition metal dichalcogenides, can show a negative in-plane plasmon
dispersion [19]. Quantum Hall effect in low-dimensional materials, e.g. in graphene,
is different from the spin Hall effect found in 3D systems, as it leads to a phase which
is topologically distinct from a band insulator [20].
So far, according to the abovementioned Anderson’s conjecture, we have shown
examples in which the dependence on dimensionality of the particle-particle
correlation plays a crucial role. Nevertheless, in some occasions a dimensional
change can be enough to modify dramatically the behaviour of physical systems,
despite correlation among constituents is switched off.
For example, let us analyse the effect that a dimensional change has on the
electronic properties of a system of free (noninteracting) electrons. This model,
which is called Fermi gas, is appropriate to study the conduction in simple systems,
such as alkali and noble metals, even though rigorously the electron motion is also
influenced by the periodic potential created by the ions in the lattice. By solving the
Schrödinger equation for one electron in a box of edge L where periodic boundary
conditions are introduced, a set of discrete energy levels E n = ¯
h 2
2m (
2πn x,y,z
L
) 2 =
¯
h 2
2m (k x,y,z ) 2 emerge, where n x,y,z is an integer number, m is the electron mass, ¯
h
is the reduced Plank constant, and k x,y,z is the corresponding wavevector along the
three Cartesian directions x, y, z. By neglecting the electron-electron interactions,
we can build up the N -electron ground state of the system by accommodating the
charges into the allowed one-electron levels starting from the bottom, provided that
the same state cannot be occupied by more than two electrons, one with spin up and
one with spin down orientation. Indeed, electrons are fermions, following the FermiDirac statistics, and obey the Pauli’s exclusion principle. The occupied orbitals are
represented by a point in the k-space inside a sphere of radius k F =
2mm F / ¯
h 2 ,
which is the highest momentum an electron can have within the box. F is the
Fermi energy, which defines the so-called Fermi sphere, within which all occupied
one-electron levels lay. The number of electronic states per unit energy range
(DOS, D(()) in the solid is a quantum observable: for Al, for example, one has
F = 11.6 eV and D(( F ) = 0.39 (eV atom) −1 .
Furthermore, the kinetic energy of the electron gas increases with temperature,
and some energy levels, which were vacant at 0 K, start to be populated. The
distribution of electrons among the levels is described by the Fermi distribution
function, f (E) = 1/(exp ((−μ)/k B T +1), which gives the probability that the energy
level E is occupied by fermions. Multiplying the DOS by the latter function, one
can obtain the DOS analytic expressions for electrons at temperature T confined
in their motion by infinite barriers into (i) a cube box of side L (3D), (ii) a square
surface of side L (2D) and (iii) a wire of length L (1D), as follows: