**1. Introduction**

Since childhood, we all have an intuition of what a solid is. However, most properties we intuitively assign to solids come in a vast range. Diamonds—and some metals—are hard, and ordinary glasses are brittle; but vulcanized rubber is neither, and it is a solid too. Perhaps the best characterization is this: *at our human timescales, a solid does not flow*. That is why this category includes glasses and ice (which do flow but at least at geological timescales).

Regarding their structure, a huge class of solids are *crystalline*. This is so to such extent that *solid state* came to be synonymous of crystalline structure, and the more comprehensive category of *condensed matter* (which admittedly includes condensed fluids or liquids) came into fashion. The name *crystal* was assigned in the late antiquity to precious and semiprecious stones that outstood for their transparency and diaphaneity. In fact, the modern meaning of the term as "an almost perfectly ordered structure" explains easily those properties.<sup>1</sup>

Many solids we interact with—metals, stones, etc.—are random assemblies of grains, held together by strong adhesion forces. Like those of sand, quartz, or salt, those grains are very likely to be themselves crystals (which as said do not imply they are perfect: they may contain lots of impurities and defects). But there are two particular aspects of crystals we are concerned with here. The first is that

<sup>1</sup> For isolators like these, the bandgap is too large for visible light to be absorbed by creating electronhole pairs. Moreover, the absence of charge carriers rules out light scattering. Impurities provide localized midgap states, which favor two-step electron-hole pair creation by visible light.

unlike complex systems, which may display emergent structures at each scale (think, e.g., of mitochondria, cells, tissues, organs, etc.), crystals are very simple: they are huge assemblies of elementary building blocks (be they atoms, molecules, nanoclusters, or whatever). The second is that since the building blocks obey quantum mechanics, crystals inherit the quantum character (despite being themselves macroscopic).

Here we are indeed concerned with Pauli's principle. But we deal with it in the style of quantum field theory, by allowing *at most one* electron of each spin projection per atom. For an electron to move ("hop") one lattice site, it must be *annihilated* at its former host atom and *created* in its nearest neighbor one. The purpose of this section is to illustrate an efficient Monte Carlo scheme that implements this strategy to find the ground state of many-electron systems. Recognizing that electrostatic (Coulomb) interaction between electrons is *not* a weak effect but is simply overwhelmed by Pauli's principle, a popular model of itinerant magnetism (the Hubbard model) adds to its Hamiltonian a repulsion term whenever an atom hosts *two* (opposite spin

Section 4 explores the boundaries of the concept of solid. Perhaps, it should be regarded as a metaphor of this concept. We illustrate a non-equilibrium spatiotemporal pattern formation process, akin to resonant crystal structures, in arrays of

**2. Band spectra in the tight-binding approach: effects of the overlaps**

For the benefit of those readers who are unfamiliar with the standard formalism

• **Dynamical states are vectors**: one can account for the wavelike behavior of quantum objects (e.g., diffraction of single electrons by two slits) by letting their dynamical state ∣*ψ*i belong to a vector space over the complex numbers. In few problems (e.g., addition of angular momenta), this vector space is finite-dimensional. But most problems entail infinite sequences (e.g., energy spectrum of the hydrogen atom) or even a continuum of values (e.g., in the measurement of positions and momenta), so the notion of dimension is replaced by that of *completitude* (*any* state can be spanned in suitable "bases"). By assigning a complex number h i *φ*j*ψ* (their "internal product") to every pair of dynamical states ∣*ψ*⟩, ∣*φ*⟩, the complete vector space is made into a *Hilbert*

• **Probabilistic interpretation**: if ∣*ψ*⟩ ¼ ∑*<sup>I</sup> αI*j*ψI*i (be aware that the index set *I* may be infinite or may even be a patch of <sup>R</sup>*<sup>d</sup>*), then *<sup>α</sup><sup>I</sup>* j j<sup>2</sup> yields the probability to find an outcome represented by ∣*ψI*⟩, when the system is in state ∣*ψ*⟩. This

• **Dynamical magnitudes are linear operators** *L*, which take a vector into another vector. For instance, the projector *P<sup>φ</sup>* ≔ ∣*φ*⟩⟨*ψ*∣ projects state ∣*ψ*⟩ onto

eigenvalues *L*∣*lI*⟩ ¼ *lI*∣*lI*⟩. Also of interest is the mean (or expectation) value h i *ψ*j*L*j*ψ* of *L* in a generic state ∣*ψ*⟩. Correspondence with classical physics imposes that those eigenvalues be real, and thus dynamical magnitudes must be self-adjoint (Hermitian) operators (*P<sup>φ</sup>* is thus *not* a dynamical magnitude).

• **Unitary evolution**: in order to conserve the probabilistic interpretation, the dynamic evolution of the state is accomplished by a unitary operator. Again, correspondence with classical physics (already implicit in Schrödinger's

∣*φ*⟩. Measuring a dynamical magnitude thus means finding one of its

projection) electrons.

*Issues in Solid-State Physics*

*DOI: http://dx.doi.org/10.5772/intechopen.84367*

FitzHugh-Nagumo cells.

space.

**5**

**between neighboring orbitals**

**2.1 Quantum mechanics in a nutshell**

of quantum mechanics, we review its main facts:

obviously requires normalization: h i *ψ*j*ψ* ¼ 1.

equation) forces this operator to be exp ð Þ �*iH* .

As recent experiments have shown, whereas most interactions (but gravity) are effectively short-ranged, there is no limit for quantum correlations; and this fact makes them the most important fact to account for in modeling. Quantum correlations manifest themselves in many ways, but the by far dominant one comes from the indistinguishability of identical particles. Unless the crystal is a monolayer, the state vector of a system of many indistinguishable particles must be either *totally symmetric* or *totally antisymmetric* (a determinant) under exchange. In the first case, the particles obey Bose-Einstein statistics and are called *bosons*. In the second, the particles obey Fermi-Dirac statistics and are called *fermions*. The requirement that the state vector of a system with many fermions be totally antisymmetric is the celebrated *exclusion principle*, postulated by Pauli.

At present, there is no question that atoms are distinguishable. They can even be individually manipulated.<sup>2</sup> Since in modeling crystals, it suffices to take atoms as building blocks (we resolve up to the nanoscale), it does not matter that they are themselves composed of other indistinguishable particles (i.e., protons and neutrons, confined to <sup>10</sup><sup>6</sup> nm) besides electrons. Instead, considering the typical effective masses of electrons in metals and semiconductors, their thermal lengths at room temperature can reach the μm, so they are highly delocalized. The following two sections illustrate two different ways of dealing with Pauli's exclusion principle when modeling crystalline solids, corresponding to two radically different ways of doing quantum mechanics.

Section 2 keeps within the framework of *first quantization*. It is assumed that neither electrons (we mean *crystal* electrons, with effective masses) nor holes can be either created or destroyed. There is only one electron in the whole crystal, submitted to a potential which is mainly the juxtaposition of shielded Coulomb terms, due to atomic orbitals located at the crystal's lattice sites. The way Pauli's principle is dealt with is by comparing the one-electron band spectrum with the Fermi level of an ideal free-electron gas (see Nomenclature). The Fermi level is the chemical potential of such a gas. The exclusion principle can make it so high that for white dwarfs and neutron stars, the pressure it generates prevents the system from becoming a black hole. But the quantum correlation we are concerned with in this section is not Pauli's principle but the overlap between atomic orbitals, usually neglected in simple tight-binding calculations of band structure. The main assumption of the *tight-binding approach* to band spectra is that atoms in a crystal interact only very weakly. As a consequence, the electron's state vector should not differ very much from that of the plain juxtaposition of atomic orbitals located at the crystal's lattice sites. However, neglecting almost all interaction terms and overlap integrals (atomic states at different lattice sites need not be orthogonal to each other) may be too drastic an approximation. Thus Section 2 is devoted to a thorough discussion of the issue.

Instead, the framework of Section 3 is that of *second quantization*. Again, our view of the crystal is that of tight-binding (atoms do not lose their identities).

<sup>2</sup> Sadly, the generalized disbelief in the mere existence of atoms just one century ago may have contributed to Ludwig Boltzmann's suicide.

### *Issues in Solid-State Physics DOI: http://dx.doi.org/10.5772/intechopen.84367*

unlike complex systems, which may display emergent structures at each scale (think, e.g., of mitochondria, cells, tissues, organs, etc.), crystals are very simple: they are huge assemblies of elementary building blocks (be they atoms, molecules, nanoclusters, or whatever). The second is that since the building blocks obey quantum mechanics, crystals inherit the quantum character (despite being them-

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

As recent experiments have shown, whereas most interactions (but gravity) are effectively short-ranged, there is no limit for quantum correlations; and this fact makes them the most important fact to account for in modeling. Quantum correlations manifest themselves in many ways, but the by far dominant one comes from the indistinguishability of identical particles. Unless the crystal is a monolayer, the state vector of a system of many indistinguishable particles must be either *totally symmetric* or *totally antisymmetric* (a determinant) under exchange. In the first case, the particles obey Bose-Einstein statistics and are called *bosons*. In the second, the particles obey Fermi-Dirac statistics and are called *fermions*. The requirement that the state vector of a system with many fermions be totally antisymmetric is the

At present, there is no question that atoms are distinguishable. They can even be individually manipulated.<sup>2</sup> Since in modeling crystals, it suffices to take atoms as building blocks (we resolve up to the nanoscale), it does not matter that they are themselves composed of other indistinguishable particles (i.e., protons and neutrons, confined to <sup>10</sup><sup>6</sup> nm) besides electrons. Instead, considering the typical effective masses of electrons in metals and semiconductors, their thermal lengths at room temperature can reach the μm, so they are highly delocalized. The following two sections illustrate two different ways of dealing with Pauli's exclusion principle when modeling crystalline solids, corresponding to two radically different ways of

Section 2 keeps within the framework of *first quantization*. It is assumed that neither electrons (we mean *crystal* electrons, with effective masses) nor holes can be either created or destroyed. There is only one electron in the whole crystal, submitted to a potential which is mainly the juxtaposition of shielded Coulomb terms, due to atomic orbitals located at the crystal's lattice sites. The way Pauli's principle is dealt with is by comparing the one-electron band spectrum with the Fermi level of an ideal free-electron gas (see Nomenclature). The Fermi level is the chemical potential of such a gas. The exclusion principle can make it so high that for white dwarfs and neutron stars, the pressure it generates prevents the system from becoming a black hole. But the quantum correlation we are concerned with in this section is not Pauli's principle but the overlap between atomic orbitals, usually neglected in simple tight-binding calculations of band structure. The main assumption of the *tight-binding approach* to band spectra is that atoms in a crystal interact only very weakly. As a consequence, the electron's state vector should not differ very much from that of the plain juxtaposition of atomic orbitals located at the crystal's lattice sites. However, neglecting almost all interaction terms and overlap integrals (atomic states at different lattice sites need not be orthogonal to each other) may be too drastic an approximation. Thus Section 2 is devoted to a thorough

Instead, the framework of Section 3 is that of *second quantization*. Again, our view of the crystal is that of tight-binding (atoms do not lose their identities).

<sup>2</sup> Sadly, the generalized disbelief in the mere existence of atoms just one century ago may have

selves macroscopic).

doing quantum mechanics.

discussion of the issue.

**4**

contributed to Ludwig Boltzmann's suicide.

celebrated *exclusion principle*, postulated by Pauli.

Here we are indeed concerned with Pauli's principle. But we deal with it in the style of quantum field theory, by allowing *at most one* electron of each spin projection per atom. For an electron to move ("hop") one lattice site, it must be *annihilated* at its former host atom and *created* in its nearest neighbor one. The purpose of this section is to illustrate an efficient Monte Carlo scheme that implements this strategy to find the ground state of many-electron systems. Recognizing that electrostatic (Coulomb) interaction between electrons is *not* a weak effect but is simply overwhelmed by Pauli's principle, a popular model of itinerant magnetism (the Hubbard model) adds to its Hamiltonian a repulsion term whenever an atom hosts *two* (opposite spin projection) electrons.

Section 4 explores the boundaries of the concept of solid. Perhaps, it should be regarded as a metaphor of this concept. We illustrate a non-equilibrium spatiotemporal pattern formation process, akin to resonant crystal structures, in arrays of FitzHugh-Nagumo cells.
