NE 225: Structure and Properties of Nanomaterials
Estimated study time: 9 minutes
Table of contents
Sources and References
- Cao and Wang, Nanostructures and Nanomaterials: Synthesis, Properties, and Applications, 2nd ed., World Scientific.
- Kittel, Introduction to Solid State Physics, 8th ed., Wiley.
- Atkins and de Paula, Physical Chemistry, 11th ed., Oxford University Press.
- Klimov (ed.), Nanocrystal Quantum Dots, 2nd ed., CRC Press.
- Dresselhaus, Dresselhaus, and Eklund, Science of Fullerenes and Carbon Nanotubes, Academic Press.
Chapter 1: Electronic Structure of Atoms and Molecules
The properties of nanomaterials arise from the behaviour of electrons in matter at scales where quantum confinement, surface effects, and finite cluster size distinguish them from bulk solids. The chain of reasoning starts with single atoms and progresses through molecules to extended structures.
1.1 Atomic Orbitals
Solutions of the time-independent Schrödinger equation for the hydrogen atom give orbitals labelled by quantum numbers \( n, \ell, m_\ell, m_s \). For multi-electron atoms, the orbital picture is approximate but retains the ordering s, p, d, f, and the Aufbau, Pauli, and Hund rules that dictate ground-state configurations.
1.2 Molecular Orbitals
When atoms bond, atomic orbitals combine to form molecular orbitals — bonding and antibonding. The linear combination of atomic orbitals (LCAO) approximation is the starting point. In diatomic molecules the energy levels split into \( \sigma \), \( \pi \), \( \sigma^* \), \( \pi^* \). In larger molecules, molecular orbitals delocalize over multiple atoms; benzene’s six-membered ring exhibits three bonding \( \pi \) orbitals filled by six electrons and three antibonding \( \pi^* \) orbitals empty at ground state.
1.3 From Molecules to Solids
As the number of atoms increases, discrete molecular-orbital energies broaden into bands. A chain of \( N \) hydrogen atoms produces \( N \) states spread over an energy width set by the nearest-neighbour hopping integral. For a three-dimensional crystal of \( 10^{23} \) atoms, the bands are continuous for all practical purposes. Crucially, this picture — discrete in small clusters, continuous in the bulk — is exactly the transition that nanomaterials occupy.
Chapter 2: Band Structure and the Solid State
2.1 Bloch’s Theorem
In a periodic potential, one-electron wavefunctions take the form \( \psi_{n\mathbf{k}}(\mathbf{r}) = u_{n\mathbf{k}}(\mathbf{r}) e^{i\mathbf{k}\cdot\mathbf{r}} \), where \( u \) has the lattice periodicity. The allowed energies form bands \( E_n(\mathbf{k}) \) over the Brillouin zone.
2.2 Metals, Semiconductors, Insulators
Metals have partially filled bands crossing the Fermi level; they conduct at any temperature. Semiconductors have a filled valence band separated from an empty conduction band by a gap \( E_g \) small enough (about 0.1–3 eV) that thermal or optical excitation populates the conduction band appreciably. Insulators have gaps large enough (> ~3 eV) to suppress excitation at ordinary conditions.
2.3 Effective Mass and Density of States
Near band extrema, \( E(\mathbf{k}) \) is quadratic, and the carrier dynamics are captured by an effective mass \( m^* \). The density of states in three dimensions is
\[ g_{3D}(E) = \frac{1}{2\pi^2}\left(\frac{2 m^*}{\hbar^2}\right)^{3/2} (E - E_c)^{1/2}. \]Reduced-dimensionality analogues give \( g_{2D} \) constant and \( g_{1D} \) diverging as \( (E-E_c)^{-1/2} \); \( g_{0D} \) is a sum of delta functions. These functional forms drive the dramatic differences in optical and electronic behaviour between bulk, quantum-well, quantum-wire, and quantum-dot materials.
Chapter 3: Covalent and Ionic Solid Nanoparticles
3.1 Ionic Nanoparticles
Ionic compounds (metal oxides, halides) form nanoparticles with structures inherited from the bulk but with distinct surface relaxation. CeO₂, TiO₂, ZnO, and Fe₂O₃ nanoparticles are workhorses for catalysis and sensing. Surface defects — cation and anion vacancies — govern reactivity and often dominate performance in catalytic applications.
3.2 Covalent Nanoparticles
Silicon nanoparticles and nanocrystalline diamond exemplify covalent nanomaterials. Their properties combine the covalent rigidity and wide bandgap of the bulk with surface-dominated behaviour at the nanoscale. Silicon nanocrystals show size-tunable photoluminescence arising from quantum confinement and from surface Si–O and Si–H states; nanodiamond shows exceptional hardness, biocompatibility, and unique surface chemistry.
3.3 Catalytic Properties
Many nanoparticles excel as heterogeneous catalysts because a large fraction of atoms sit at accessible surface sites. Gold nanoparticles below 3 nm catalyze CO oxidation at low temperature — a reaction inert on bulk gold. Platinum-group nanoparticles in automotive catalytic converters oxidize hydrocarbons and reduce NOₓ. Size, shape, support interaction, and surface capping all modulate activity and selectivity.
3.4 Electrochemical, Electrical, Optical, and Magnetic Properties
Nanoparticles exhibit distinctive behaviours in each domain:
- Electrochemical: high surface-to-volume ratio boosts capacity in battery and supercapacitor electrodes.
- Electrical: Coulomb blockade in single-nanoparticle junctions quantizes charge transport.
- Optical: surface-plasmon resonances in metallic nanoparticles produce strong size- and shape-dependent colour; quantum confinement in semiconductor nanoparticles produces size-tunable absorption and emission.
- Magnetic: superparamagnetism replaces ferromagnetism below a critical size (about 15–25 nm for Fe₃O₄), enabling magnetic nanoparticles to be manipulated by external fields without remanence.
Chapter 4: Semiconductors and Carbon/Silicon-Based Nanomaterials
4.1 Bulk Semiconductor Review
Silicon (indirect gap 1.12 eV), GaAs (direct 1.42 eV), CdSe (direct 1.74 eV), and CdS (direct 2.42 eV) form the workhorse semiconductors at different wavelengths and applications. Direct-gap materials emit and absorb efficiently; indirect-gap materials do not.
4.2 Quantum Confinement
Confining carriers in one, two, or three dimensions quantizes their energy. For a spherical semiconductor nanocrystal of radius \( R \), the effective-mass approximation gives
\[ E(R) = E_g + \frac{\hbar^2 \pi^2}{2 R^2}\left(\frac{1}{m_e^*} + \frac{1}{m_h^*}\right) - \frac{1.8 e^2}{4\pi\varepsilon R}, \]with Brus’s Coulombic correction. Nanocrystal bandgap thus depends on size: 5 nm CdSe emits green, 3 nm emits blue, 7 nm emits red. This tunability underlies quantum-dot displays, imaging, and photovoltaic applications.
4.3 Carbon Nanomaterials
Fullerenes (C₆₀, C₇₀) are closed-cage molecules with delocalized π systems. Carbon nanotubes are seamless cylinders of graphene, characterized by chiral indices (n, m); they are metallic when (n − m) is divisible by three and semiconducting otherwise. Their bandgap scales inversely with diameter for semiconducting tubes,
\[ E_g \approx \frac{2 a_{C-C} \gamma_0}{d}, \]with \( a_{C-C} \) the bond length and \( \gamma_0 \) the hopping integral. Graphene is a zero-gap semimetal with linear dispersion at the K points; electrons behave as massless Dirac fermions and exhibit extraordinarily high mobility.
4.4 Silicon Nanostructures
Silicon nanowires, fabricated by vapour-liquid-solid growth or by top-down etching, exhibit size-dependent electronic properties and support applications from field-effect sensors to photovoltaics. Porous silicon emits visible photoluminescence through confinement and surface states and has been explored for biodegradable biomaterials.
Chapter 5: Dendrimers, Micelles, and Quantum-Dot Case Study
5.1 Dendrimers
Dendrimers are hyperbranched molecules with well-defined generations. Poly(amidoamine) (PAMAM) dendrimers have controlled size, surface functionality, and encapsulation capacity. Applications include drug delivery (internal cavities load payload), imaging (surface labelling with fluorophores or MRI contrast), and catalysis (metal nanoparticles templated within dendritic interior).
5.2 Micelles and Supramolecular Assemblies
Amphiphilic block copolymers self-assemble in selective solvents into micelles, vesicles, and cylindrical aggregates. Critical micelle concentration, aggregation number, and core-shell architecture are controlled by block composition. These assemblies carry hydrophobic drugs in aqueous circulation, act as nanoreactors for synthesis, and template nanostructured materials.
5.3 Quantum-Dot Case Study
Colloidal CdSe/ZnS core–shell quantum dots exemplify the interplay of structure and properties. The CdSe core provides size-tunable bandgap through confinement; the ZnS shell passivates surface states and raises quantum yield. Exchange of surface ligands transforms hydrophobic dots from synthesis into water-soluble imaging agents. Blinking — intermittent fluorescence — and photo-darkening are characteristic single-dot phenomena. Applications include displays, cell and animal imaging, solar concentrators, and single-photon sources.
