Electronic Materials – Study Notes (BEEE 201L)

Module 1: Physics of Materials – Electronic Materials Notes

• Core topics covered in the transcript:
  • Atomic structure and atomic number
  • Electron spin and Pauli’s exclusion principle
  • Bonding types in solids
  • Concepts of Fermi level and energy bands in solids
  • Classification of materials: metals, semiconductors, insulators
  • Crystal directions and planes, crystal properties, defects and vacancies
  • Potential barrier problems and related concepts (not deeply expanded here)

Atomic Structure and Atomic Number

• Mass is concentrated in the nucleus, which contains protons and neutrons.
• Protons are positively charged; neutrons are neutral; both have similar masses.
• The nucleus is held together by the strong force, which acts at very short range: typically
  ext{range}
  oughly < 10^{-15} ext{ m}.
• Electromagnetic repulsion between protons is overcome by the strong force at very short distances, keeping the nucleus intact.
• Atomic number Z = number of protons in the nucleus.
• The nucleus contains the total mass of the atom; electrons orbit the nucleus at much larger distances than the nucleus is small.

Shell Model and Electron Structure

• Electrons occupy well-defined spherical regions (shells and subshells) around the nucleus, not the entire space.
• Shell/qn numbers:
  • Principal quantum number: $n = 1, 2, 3, \,\ldots$
  • Orbital angular momentum quantum number: $\ell = 0, 1, 2, \,\ldots, n-1$
• Shell labels (for integer n): $n=1,2,3,4,\ldots$ correspond to capital letters K, L, M, N, …
• Subshell labels corresponding to $\ell$: $\ell=0\to s, \ 1\to p, \ 2\to d, \ 3\to f, \ldots$
• Electron capacity of a subshell: \text{#electrons in subshell} = 2(2\ell + 1).
  • s subshell ((\ell=0)): 2 electrons
  • p subshell ((\ell=1)): 6 electrons
  • d subshell ((\ell=2)): 10 electrons
  • f subshell ((\ell=3)): 14 electrons
• Example: Carbon (Z=6) configuration example: $1s^2 2s^2 2p^2$; inert element helium has a full K shell: $1s^2\quad\text{(He)}$ or shorthand: $[\text{He}] 2s^2 2p^2$ when building beyond He.
• The general rule uses the nearest inert element in brackets ( Aufbau-like filling).

Aufbau Principle and Subshell Filling (Diagonal Rule)

• Electrons fill subshells in a specific order, influenced by energy levels and electron–electron interactions.
• Common filling order (Aufbau order / Diagonal Rule):
  $1s \rightarrow 2s \rightarrow 2p \rightarrow 3s \rightarrow 3p \rightarrow 4s \rightarrow 3d \rightarrow 4p \rightarrow 5s \rightarrow 4d \rightarrow 5p \rightarrow 6s \rightarrow 4f \rightarrow 5d \rightarrow 6p \rightarrow 7s \ldots$
• This results in the observed electron configurations (examples include Li: $1s^2 2s^1$; Ne: $1s^2 2s^2 2p^6$).

Quantum Numbers in Atoms

• Principal quantum number: $n \in {1,2,3,4,\ldots}$
• Orbital angular momentum quantum number: $\ell \in {0,1,2,3,\ldots, n-1}$
• Magnetic quantum number: $m_{\ell} \in {-\ell, -\ell+1, \ldots, \ell-1, \ell}$
• Spin quantum number: $m_s \in {+\tfrac{1}{2}, -\tfrac{1}{2}}$
• Pauli’s exclusion principle: no two electrons in an atom may have the same set of quantum numbers $(n, \ell, m<em>{\ell}, m</em>s)$.
  • If two electrons occupy the same orbital (same $(n,\ell,m_{\ell})$), they must have opposite spins (spin-paired).
  • A third electron cannot enter the same orbital because $m_s$ can take only two values.

Orbital Angular Momentum and Space Quantization

• Electron angular momentum is quantized; the orbital angular momentum magnitude is described by $\mathbf{L}$ with quantum number $\ell$.
• The component of angular momentum along the external field (z-direction) is quantized by $m_{\ell}$ as above.
• The electron also has intrinsic spin angular momentum $\mathbf{S}$ with spin quantum number $m_s\in{+\tfrac{1}{2}, -\tfrac{1}{2}}$.
• The total angular momentum is a combination of orbital and spin contributions; in many contexts, spin and orbital motions are treated separately (spin-orbit coupling is a separate topic).

Bonding and Types of Solids

• When atoms come together, their valence electrons interact with neighboring nuclei to form bonds, leading to solid structures.
• Key bond types (primary bonds):
  • Covalent bonds: sharing of valence electrons to complete subshells (directional).
  • Ionic bonds: electrostatic attraction between oppositely charged ions (often Na+ and Cl− in salts).
  • Metallic bonds: delocalized electrons form an electron gas that bonds a lattice of positive ion cores (non-directional).
• Secondary bonds (van der Waals): weaker attractions, including induced-dipole–induced-dipole interactions; important in molecular solids and low-temperature behavior.
• Mixed bonding: many materials (ceramics) exhibit a mixture of bonding types (covalent + ionic, etc.).

Covalently Bonded Solids: Diamond

• Diamond as a classic covalently bonded solid:
  • Each carbon atom forms covalent bonds with four nearest neighbors (coordination number CN = 4).
  • Three-dimensional network of strong covalent bonds yields extremely high melting temperature and hardness; diamond is one of the hardest known materials.
  • Electrical conductivity is poor because valence electrons are localized in bonds and not free to move.
• Key properties:
  • High hardness, high melting point
  • Insolubility in most solvents
  • Brittle (nonductile, brittle fracture)
• Coordination number for carbon in diamond: CN = 4.

Metallic Bonding: Copper

• In metals, valence electrons are delocalized and form an electron gas that moves freely through the lattice of positively charged ion cores.
• Non-directional bonding leads to high ductility and the ability to deform without fracture.
• Electrical conductivity: the electron gas can respond readily to applied electric fields, enabling high conductivity.
• Thermal conductivity: free electrons transfer energy efficiently, giving metals high thermal conductivity.
• The non-directional nature of metallic bonds allows metals to flow under stress, aided by crystal defects such as dislocations.

Ionically Bonded Solids: Salt (NaCl)

• Example: NaCl consists of Na+ cations and Cl− anions held together by Coulombic attraction.
• Ionic bonding is common in solids with metal + nonmetal constituents.
• Na+ resembles inert Ne configuration; Cl− resembles inert Ar configuration after gaining an electron.
• Structure and stability:
  • Each Na+ neighbor is coordinated by Cl− ions (and vice versa).
  • The total coordination in NaCl is 6 (FCC-like ionic lattice).
• Key properties:
  • Strong, brittle, high melting points
  • Soluble in polar solvents
  • Generally electrical insulators due to lack of free electrons; ions are held tightly in lattice
  • Lower thermal conductivity compared to metals because phonons or lattice vibrations transfer energy less efficiently than free-electron conduction
• Atomic cohesive energy is the energy required to separate the solid into isolated atoms.

Secondary Bonding: Van der Waals Forces

• Covalent, ionic, and metallic bonds are primary bonds; van der Waals (vdW) forces are secondary bonds.
• vdW forces arise from transient dipole moments and synchronized fluctuations of electron distributions between atoms/molecules.
• Induced-dipole–induced-dipole interactions exist between nonpolar molecules (e.g., Ne, Ar; H2, CH4).
• Molecular solids (ice, dry ice CO2, O2, CH4) are held together largely by vdW forces and are typically softer and have lower melting points than ionic/covalent solids.
• vdW forces are weaker than primary bonds but crucial for understanding many materials, especially molecular crystals and layered materials.

Mixed Bonding in Ceramics

• Many ceramics exhibit a mix of covalent and ionic bonding (polar bonds).
• Examples and bonding characteristics:
  • Si3N4: predominantly covalent
  • MgO: predominantly ionic
  • Al2O3: mixture of ionic and covalent bonding
• Consequences: brittleness, high melting temperature, electrical insulation.

Energy Bands in Solids

• In conductors (metals), semiconductors, and many insulators, electronic conduction arises from electronic states arranging into bands.
• Band formation: as atoms come close, their atomic states split and broaden into bands; the outermost shells are most affected first.
• At equilibrium spacing, some subshells may not form bands; gaps can exist between adjacent bands (band gaps).
• Number of states in a band equals the total states contributed by all atoms. For example:
  • An s band contains N states; a p band contains 3N states.
• Each energy state can hold two electrons with opposite spins (Pauli principle).
• Occupancy and conductivities depend on the band structure:
  • The arrangement of bands and their filling determines electrical properties.
  • Fermi energy, Ef, is the energy of the highest occupied state at 0 K.
• Types of band structures at 0 K:
  • Partially filled outermost band (typical of some metals like Cu with a single s valence electron: each atom has one 4s electron; solid 4s band can hold 2N electrons; only half filled)
  • Overlapping bands: e.g., Mg where 3s and 3p bands overlap; Fermi energy lies below the energy where N states are filled (two electrons per state)
  • Valence band fully filled and separated by a band gap from the conduction band: insulators and semiconductors
• Band gap, Eg:
  • Wide band gap (> ~2 eV): insulators
  • Narrow band gap (< ~2 eV): semiconductors
  • Ef lies within the gap for insulators/semiconductors
• Conductivity and carriers:
  • Free electrons above Ef contribute to conduction (electrons/holes in semiconductors)
  • Holes contribute to conduction in semiconductors; conductivity increases with more free carriers

Metals, Insulators, and Semiconductors

• Metals:
  • Low resistance to electron excitation; vacant states near Ef; electrons can be promoted into empty states with relatively little energy, enabling high conductivity.
  • Electron gas model: valence electrons behave as a free electron gas; high electrical and thermal conductivity.
• Insulators:
  • Large band gaps; empty states adjacent to valence band are not accessible; conduction requires energy to cross the gap Eg.
  • Typically strong ionic or covalent bonding with highly localized valence electrons; poor conductivity.
• Semiconductors:
  • Covalent bonding with moderate band gaps; conductivity increases with temperature due to more thermal excitations across Eg.
  • More tunable properties via doping, impurities, and defects.

Virial Theorem (Brief Concept)

• Virial theorem relates average kinetic energy (KE), average potential energy (PE), and total energy (E) for systems bound by certain forces.
• For Coulombic systems (e.g., atomic electrons):
  • $\text{KE} = -E$
  • $\text{PE} = 2E$
  • $E = \text{KE} + \text{PE}$
• Alternative expression: $2\,\text{KE} = -\text{PE}$

Numerical Examples (Hydrogen Atom and Related Problems)

• Example 1: Hydrogen in the ground state (1s) with ionization energy (IE) = $13.6\text{ eV}$.

  • Energy reference: zero at infinity; total energy of the electron in H is $E = -13.6\text{ eV}.$
  • Using Virial relations:
  • $\text{PE} = 2E = -27.2\text{ eV}$
  • $\text{KE} = -E = 13.6\text{ eV}$

• Bohr radius (radius of 1s orbit) calculation:

  • Coulombic potential energy between charges:
    $\text{PE} = \dfrac{Q<em>1 Q</em>2}{4\pi\varepsilon<em>0 r</em>0} = -\dfrac{e^2}{4\pi\varepsilon<em>0 r</em>0}$
  • With given numbers, the Bohr radius is:
    $r_0 = 5.29\times 10^{-11}\text{ m} = 0.0529\text{ nm}.$

• Electron velocity in ground state:

  • Using KE and mass me: $\text{KE} = \tfrac{1}{2} m</em>e v^2$
  • Solve for v: $v = \sqrt{\dfrac{2\,\text{KE}}{m_e}} = 2.19\times 10^6\ \text{m s}^{-1}.$

• Orbital frequency: $f = \dfrac{v}{2\pi r_0} = 6.59\times 10^{15}\ \text{s}^{-1}.$

• Orbital period: $T = \dfrac{1}{f} = 1.52\times 10^{-16}\ \text{s}.$

• Summary of hydrogen results:

  • Ground state energy: $E = -13.6\text{ eV}$
  • PE: $-27.2\text{ eV}$
  • KE: $13.6\text{ eV}$
  • Bohr radius: $r_0 = 0.0529\text{ nm}$
  • Electron velocity: $v = 2.19\times 10^{6}\ \text{m s}^{-1}$
  • Frequency: $f = 6.59\times 10^{15}\ \text{Hz}$

Ionic Bonding in NaCl: Equilibrium and Cohesive Energy

• Interionic potential in NaCl is modeled as:
  $E(r) = -\dfrac{A}{r^{6}} + \dfrac{B}{r^{12}}$
  with constants given in the problem (for a specific FCC lattice).
• Equilibrium separation (bond length) ro is found from the condition (\dfrac{dE}{dr} = 0):
  • Result: $r_0 = 0.28\times 10^{-9}\text{ m} = 0.28\text{ nm}.$
• Ionic bonding energy (cohesive energy) per NaCl pair (negative of E(ro)) is found to be:
  • The energy to break the crystal into Na+ and Cl− ions is $E_{ ext{bond}} = 7.84\text{ eV}$ per pair (as per the calculation combining ionization energy of Na and electron affinity of Cl).
• Additional energy accounting:
  • Ionization energy of Na: $I_{Na} = 5.14\text{ eV}$
  • Electron affinity of Cl: $EA_{Cl} = 3.61\text{ eV}$
  • Net cohesive energy per NaCl pair: $7.84\text{ eV}$ (taken as the energy required to separate into isolated ions, considering electron transfer energies).

van der Waals Bonding: Argon (Lennard–Jones Model)

• Lennard–Jones potential:
  $E(r) = -\dfrac{A}{r^{6}} + \dfrac{B}{r^{12}}.$
• Given constants: $A = 8.0 \times 10^{-77}\ \text{J m}^6,\quad B = 1.12 \times 10^{-133}\ \text{J m}^{12}.$
• Bond length (minimum energy) occurs where (\dfrac{dE}{dr} = 0):
  • Result: $r_0 = 3.75 \times 10^{-10}\ \text{m} = 0.375\ \text{nm}.$
• Bond energy (minimum of E(r)):
  • $E<em>{ ext{bond}} = |E(r</em>0)| = 1.43 \times 10^{-20}\ \text{J} = 0.089\ \text{eV}.$
• Implications: vdW bonds are much weaker than ionic/covalent/metallic bonds; argon solid forms at very low temperatures.

Mixed Bonding in Ceramics: Examples and Implications

• Ceramics like Si3N4, MgO, and Al2O3 exhibit varying mixtures of covalent and ionic bonding.
• Consequences: high melting temperatures, brittleness, electrical insulation; conduction is minimal due to localized electrons or lack of free carriers.
• Understanding the balance of covalent and ionic character helps explain material properties in ceramics.

Conceptual Connections and Real-World Relevance

• How bonding types influence macroscopic properties:
  • Covalent solids (e.g., diamond): hard, high melting point, poor electrical conduction due to localized electrons.
  • Metallic solids (e.g., copper): high electrical and thermal conductivity due to free electron gas; ductility due to non-directional bonding and defects.
  • Ionic solids (e.g., NaCl): high melting points, brittle, insulating, often soluble in polar solvents; strong Coulombic bonding and coordination tendencies.
  • vdW solids (e.g., Ar, Ne, molecular solids): low melting points; soft; minimal electrical conductivity; rely on weak intermolecular forces.
• Energy band perspective links microscopic bonding to macroscopic conductivity:
  • Conductors require partially filled bands or overlapped bands allowing free-carrier states.
  • Insulators possess wide band gaps; semiconductors have narrower gaps enabling tunable conduction with temperature/doping.
• Practical implications for technology:
  • Semiconductors form the backbone of electronics; doping controls carrier concentration.
  • Metals are essential for interconnects and heat management due to superior conductivity.
  • Ceramics are key for insulators, high-temperature components, and structural materials due to high melting points and hardness.

Practice Questions (MCQ Style) and Quick Solutions

• Q1: Which of the following sets of quantum numbers is NOT permissible for an electron in an atom?
  • a) $n=2, \ell=1, m<em>{\ell}=0, m</em>s=+\tfrac{1}{2}$
  • b) $n=3, \ell=2, m<em>{\ell}=-2, m</em>s=-\tfrac{1}{2}$
  • c) $n=4, \ell=0, m<em>{\ell}=-1, m</em>s=+\tfrac{1}{2}$
  • d) $n=1, \ell=0, m<em>{\ell}=0, m</em>s=-\tfrac{1}{2}$
  • Correct: c) is not permissible because for (\ell=0) (an s subshell) there is only one value of (m_{\ell}=0).
• Q2: Which quantum number governs the spatial orientation of an atomic orbital?
  • (a) Spin quantum number, (b) Magnetic quantum number, (c) Azimuthal quantum number, (d) Principal quantum number
  • Correct: (b) Magnetic quantum number, $m_{\ell}$.
• Q3: What is the maximum number of electrons in a shell with principal quantum number n?
  • a) $n$, b) $2n^2$, c) $n^2$, d) $2n$
  • Correct: b) $2n^2$.
• Q4: For (n + \ell = 4) cases, what is the maximum number of electrons having the same spin that can be accommodated in the corresponding subshells?
  • Explanation: possible values are (n=3, \ell=1) => 3p (6 electrons max) and (n=4, \ell=0) => 4s (2 electrons). Total = 8 electrons, same spin count = 8/2 = 4.

Key Formulas and Symbols to Remember

• Subshell electron capacity:
  $\text{electrons in subshell} = 2(2\ell + 1)\,.$
• Shell and subshell labels:
  • $n = 1,2,3,4,\ldots$
  • $\ell = 0,1,2,3,\ldots, n-1$
  • $K,L,M,N,\ldots$ correspond to $n=1,2,3,4,\ldots$
  • Subshell labels: $s,p,d,f$ correspond to $\ell = 0,1,2,3$
• Quantum numbers:
  • Principal: $n$
  • Azimuthal: $\ell$
  • Magnetic: $m_{\ell} \in [-\ell, -\ell+1, …, \ell]$
  • Spin: $m_s \in {+\tfrac{1}{2}, -\tfrac{1}{2}}$
• Pauli exclusion principle: no two electrons share the same quadruple of quantum numbers $(n, \ell, m<em>{\ell}, m</em>s)$.
• Bonding and properties summaries:
  • Diamond: CN = 4; covalent network; hardness; insulating.
  • Copper (metallic): electron gas; high electrical/thermal conductivity; ductility.
  • NaCl (ionic): CN = 6; insulating; polar solvents dissolve many salts.
  • Van der Waals solids: low binding energy; low temperatures; molecular solids.
• Energy bands and conduction concepts:
  • Band formation yields states: s-band (N states), p-band (3N states), etc.
  • Two electrons per state (spin-paired).
  • Fermi energy: $E_f$, top of filled states at 0 K.
  • Band gap: $E_g$; wide gaps in insulators, narrow gaps in semiconductors.
  • Conductivity depends on free electrons and holes; temperature can increase carrier density.
• Hydrogen and Bohr model results:
  • Ionization energy: $13.6\ \text{eV}$; total energy $E = -13.6\ \text{eV}$ for 1s.
  • PE: $-27.2\ \text{eV}$; KE: $13.6\ \text{eV}$.
  • Bohr radius: $r_0 = 5.29\times 10^{-11}\ \text{m}$; velocity $v = 2.19\times 10^6\ \text{m s}^{-1}$; frequency $f = 6.59\times 10^{15}\ \text{s}^{-1}$.
• NaCl and ionic cohesion specifics:
  • ro ≈ $0.28\ \text{nm}$; cohesive energy per pair ≈ $7.84\ \text{eV}$.
• Lennard–Jones case for Argon:
  • Potential: $E(r) = -\dfrac{A}{r^{6}} + \dfrac{B}{r^{12}}$
  • Minimum at $r<em>0 = 0.375\ \text{nm}$; energy at minimum $E</em>{ ext{bond}} \approx 0.089\ \text{eV}$.

Quick Real-World Takeaways

• Bond type strongly influences material properties (mechanical, thermal, electrical).

• Band theory explains why metals conduct well and many ceramics/insulators do not; doping and temperature tweak semiconductor behavior.

• Simple models (Coulombic hydrogen atom, Lennard–Jones potential) provide intuition for binding energies, bond lengths, and how materials respond to external stimuli.

• Theoretical tools like the Pauli principle, Aufbau filling, and the Virial theorem connect microscopic quantum properties to macroscopic material behavior.

• End of Module 1 notes.