136
S. Taioli
experimental procedure that ended up in the production of graphene flakes by C 60
impact on copper surfaces.
Keywords Dimensionality · Carbon-based materials ·
Optical, electronic, andmechanicalproperties ·
Multiscalesimulations, synthesis, characterization
5.1 Introduction: The Course of Dimensionality
Dimensionality affects dramatically the physical properties of materials owing to the
different way that Coulomb repulsion acts upon the electrons in three-dimensional
(3D), two-dimensional (2D), one-dimensional (1D), and molecular (0D) structures.
Indeed, the presence of constraints on the particle’s motion in one or more degrees
of freedom leads to remarkable consequences, such as quantum confinement,
anisotropic characteristics and new phases. These effects can completely modify
the properties that low-dimensional physical systems exhibit with respect to their
bulk counterparts.
Additionally, quantum objects do interfere with one another, so that the quantum
state of a many-body system is the result of the interaction between its constituent
particles. This many-body potential depends on dimensionality and confinement
and thus is much more than the simple sum of the interaction between its building
blocks. This concept was masterly described by Philip W. Anderson in his article
“More is different” [1], where he argues that “the behaviour of large and complex
aggregations of elementary particles, it turns out, is not to be understood in terms
of a simple extrapolation of the properties of a few particles. Instead at each level
of complexity entirely new properties appear. . . ”. In this regard, for example, while
at angstrom scale it is hard to differentiate between 0D point-like atomic species,
such as tantalum or niobium, at 3D macroscale the former is a lustrous transition
metal, and the latter undergoes a phase transition to a BCS-type II superconductor
at 9.26 K. This means that at the time we reach the microscale, electrons of Nb pair
up in Cooper pairs and condensate, transforming the material in a superconductor
characterized by zero-resistance conductivity.
While intuitively one may think that increasing the number of degrees of freedom
generally results in higher levels of complexity, at odds in physics and chemistry,
the curse of dimensionality can act in one or another direction. In this book chapter,
we will analyse quite a few examples of this “unconventional” scaling.
For instance, a relative simple approach to describe magnetic materials is
provided by the Ising model [2]. This simplified mathematical model of solids
represents real systems as made of atomic spins interacting with their neighbours on
a lattice and can be used to identify phase transitions. According to the traditional
solutions, there is no magnetization in the one-dimensional Ising model in the
absence of external magnetic fields, while the two-dimensional square lattice is the
simplest statistical problem to show a phase transition. In this respect, while the
one-dimensional Ising lattice represents a relatively simple toy model in statistical
S. Taioli
experimental procedure that ended up in the production of graphene flakes by C 60
impact on copper surfaces.
Keywords Dimensionality · Carbon-based materials ·
Optical, electronic, andmechanicalproperties ·
Multiscalesimulations, synthesis, characterization
5.1 Introduction: The Course of Dimensionality
Dimensionality affects dramatically the physical properties of materials owing to the
different way that Coulomb repulsion acts upon the electrons in three-dimensional
(3D), two-dimensional (2D), one-dimensional (1D), and molecular (0D) structures.
Indeed, the presence of constraints on the particle’s motion in one or more degrees
of freedom leads to remarkable consequences, such as quantum confinement,
anisotropic characteristics and new phases. These effects can completely modify
the properties that low-dimensional physical systems exhibit with respect to their
bulk counterparts.
Additionally, quantum objects do interfere with one another, so that the quantum
state of a many-body system is the result of the interaction between its constituent
particles. This many-body potential depends on dimensionality and confinement
and thus is much more than the simple sum of the interaction between its building
blocks. This concept was masterly described by Philip W. Anderson in his article
“More is different” [1], where he argues that “the behaviour of large and complex
aggregations of elementary particles, it turns out, is not to be understood in terms
of a simple extrapolation of the properties of a few particles. Instead at each level
of complexity entirely new properties appear. . . ”. In this regard, for example, while
at angstrom scale it is hard to differentiate between 0D point-like atomic species,
such as tantalum or niobium, at 3D macroscale the former is a lustrous transition
metal, and the latter undergoes a phase transition to a BCS-type II superconductor
at 9.26 K. This means that at the time we reach the microscale, electrons of Nb pair
up in Cooper pairs and condensate, transforming the material in a superconductor
characterized by zero-resistance conductivity.
While intuitively one may think that increasing the number of degrees of freedom
generally results in higher levels of complexity, at odds in physics and chemistry,
the curse of dimensionality can act in one or another direction. In this book chapter,
we will analyse quite a few examples of this “unconventional” scaling.
For instance, a relative simple approach to describe magnetic materials is
provided by the Ising model [2]. This simplified mathematical model of solids
represents real systems as made of atomic spins interacting with their neighbours on
a lattice and can be used to identify phase transitions. According to the traditional
solutions, there is no magnetization in the one-dimensional Ising model in the
absence of external magnetic fields, while the two-dimensional square lattice is the
simplest statistical problem to show a phase transition. In this respect, while the
one-dimensional Ising lattice represents a relatively simple toy model in statistical