Crystal Structures and Classical Theory of Conduction

Classification of Solids by Atomic Arrangement

• Solids are categorized into three distinct groups based on the spatial arrangement of atoms:

  • Amorphous Solids: Atoms are positioned at random with no discernible pattern.

  • Crystalline Solids: Atoms exhibit a long-range order, forming a predictable structure across the material.

  • Polycrystalline Solids: Characterized by localized short-range order. They consist of many small crystalline regions, known as grains, which have random orientations and are separated by interfaces called grain boundaries.

Fundamental Properties of Crystal Structures

• Periodicity: The most critical property of a crystal. It provides long-range order where local bonding geometry is repeated at regular intervals, resulting in a periodic array of atoms.

• Predictability: Due to periodicity, the location of any atom within the crystal is predictable.

• Prevalence: Nearly all metals, semiconductors, and ceramics exist in a crystalline state.

• Constituents of a Crystal:

  • Lattice: A mathematical abstraction; a regular array of points in space showing distinguishable periodicity.

  • Basis: An identical group of atoms or molecules (identical in composition, arrangement, and orientation) placed at each lattice point.

  • Formula: $Crystal = Lattice + Basis$.

The Unit Cell and Lattice Parameters

• Unit Cell: The smallest convenient volume of a crystal that retains all the characteristics of the crystal structure. Repetition of the unit cell in three dimensions generates the entire crystal.

• Representation: Usually represented by a parallelepiped.

• Lattice Parameters:

  • The edges of the unit cell are defined along the $x, y, z$ axes as lengths $a, b, c$.

  • The angles between these edges are defined as $\beta$ (between $a$ and $c$), $\beta$ (between $a$ and $c$), and $\beta$ (between $a$ and $c$). Note: The transcript explicitly labels parameters as sides $a, b, c$ and angles $\beta, \beta, \beta$. (Correction: usually $\beta, \beta, \beta$ are used, transcript shows $\beta, \beta, \beta$ diagrammatically for angles).

• Bravais Lattices: There are $14$ distinct lattices possible in 3D space, grouped into $7$ crystal systems.

The Seven Crystal Systems

• Cubic System: $a = b = c$; $\beta = \beta = \beta = 90^°$. Examples include Al, Cu, Fe, Pb, NaCl, CsCl, LiF, Si, and GaAs.

• Tetragonal System: $a = b \neq c$; $\beta = \beta = \beta = 90^°$. Examples: In, Sn, Barium Titanate, $TiO_2$.

• Orthorhombic System: $a \neq b \neq c$; $\beta = \beta = \beta = 90^°$. Examples: S, U, Pl, Ga (below $30^°C$), Iodine, Cementite ($Fe_3C$), Sodium Sulfate.

• Rhombohedral System: $a = b = c$; $\beta = \beta = \beta \neq 90^°$. Examples: Arsenic, Boron, Bismuth, Antimony, Mercury (below $-39^°C$).

• Hexagonal System: $a = b \neq c$; $\beta = \beta = 90^°$; $\beta = 120^°$. Examples: Cadmium, Magnesium, Zinc, Graphite.

• Monoclinic System: $a \neq b \neq c$; $\beta = \beta = 90^°$; $\beta \neq 90^°$. Examples: $\alpha$-Selenium, Phosphorus, Lithium Sulfate, Tin Fluoride.

• Triclinic System: $a \neq b \neq c$; $\beta \neq \beta \neq \beta \neq 90^°$. Example: Potassium dichromate.

Metallic Crystal Structures: FCC and BCC

• Face-Centered Cubic (FCC):

  • Atoms are located at each corner of the cube and at the center of each face.

  • Effective Number of Atoms: $(8 \times 1/8) + (6 \times 1/2) = 4$ atoms.

  • Geometry: The face diagonal is $a\beta(2) = 4R$, leading to $a = 2\beta(2)R$.

  • Atomic Packing Factor (APF): Calculated as $[4 \times (4/3)\beta R^3] / a^3 ≈ 0.74$ ($74\%$ occupied).

  • Examples: Ag, Al, Au, Cu.

• Body-Centered Cubic (BCC):

  • Atoms are located at each corner and one atom at the center of the unit cell.

  • Effective Number of Atoms: $(8 \times 1/8) + 1 = 2$ atoms.

  • Geometry: The body diagonal is $4R$, and $(4R)^2 = 3a^2$, leading to $a = (4/\beta(3))R$.

  • Atomic Packing Factor (APF): Calculated as $[2 \times (4/3)\beta R^3] / a^3 ≈ 0.68$ ($68\%$ occupied).

  • Examples: Alkali metals (Li, Na, K, Rb), Cr, Mo, W, Mn, $\alpha$-Fe (below $912^°C$), $\beta$-Ti (above $882^°C$).

• Atomic Concentration ($n_{at}$): defined as $\frac{\text{Number of atoms in unit cell}}{\text{Volume of unit cell}}$.

  • For Ag (FCC): $n_{at} = 4/a^3 = 5.87 \times 10^{22} cm^{-3}$.

  • For Fe (BCC): $n_{at} = 2/a^3 = 8.5 \times 10^{22} cm^{-3}$.

Diamond and Zinc Blende Structures

• Diamond Structure:

  • Space lattice is FCC.

  • Basis consists of two identical atoms: one at $(0,0,0)$ and one at $(1/4, 1/4, 1/4)$.

  • Bonding is directional tetrahedral covalent bonding.

  • Coordination: Each atom has $4$ nearest neighbors and $12$ next-nearest neighbors.

  • Packing Efficiency: Very low; only $34\%$ filled.

  • Examples: C, Si, Ge.

• Zinc Blende Structure:

  • Similar to diamond lattice but uses a basis of two non-identical atoms.

  • Examples: ZnS, GaAs.

Miller Indices for Planes and Directions

• Importance: Properties like elastic modulus, electrical resistivity, and magnetic susceptibility are directional (anisotropic). Processes like oxidation and etching depend on specific planes.

• Calculating Miller Indices for Planes ($hkl$):

  1. Find intercepts of the plane with the axes (as multiples of $a, b, c$).

  2. Take the reciprocals of these intercepts.

  3. Reduce to the smallest set of integers $h, k, l$.

  • Note: If a plane is parallel to an axis, the intercept is infinity and the index is $0$. Negative intercepts are denoted with a bar above the number (e.g., $\bar{h}$).

• Family of Planes: Denoted by ${hkl}$, representing equivalent planes (e.g., ${100}$ includes $(100)$, $(010)$, and $(001)$).

• Crystal Directions: Denoted by $[hkl]$, based on vector components. Families of directions are denoted by $\langle hkl \rangle$.

Planar Concentration of Atoms

• Definition: The number of atoms per unit area on a specific $(hkl)$ plane.

• FCC Planar Densities (e.g., for Silver, $a = 0.4084 nm$):

  • (100) Plane: $n(100) = 2 / a^2 = 12 \times 10^{18} atoms/m^2$.

  • (110) Plane: $n(110) = 2 / (\beta(2)a^2) = 8.5 \times 10^{18} atoms/m^2$.

  • (111) Plane: Most densely packed. $n(111) = 2 / [a^2(\beta(3)/2)] = 13.8 \times 10^{18} atoms/m^2$.

• Diamond Planar Densities ($a = 0.5659 nm$):

  • (110) Plane: $n(110) = 8.8 atoms/nm^2$.

  • (111) Plane: $n(111) = 7.2 atoms/nm^2$.

  • (100) Planes: Least planar concentration in diamond.

Crystalline Defects

• Point Defects (0D):

  • Vacancies: Missing atoms at lattice points. They exist in thermal equilibrium (thermodynamic defects). Concentration formula: $n_v = N e^{-E_v / kT}$, where $E_v$ is activation energy, $k$ is Boltzmann’s constant ($1.38 \times 10^{-23} J/K$), and $T$ is temperature in Kelvin.

  • Impurities: Substitutional (occupy lattice sites, e.g., As in Si) or Interstitial (occupy spaces between sites, e.g., C in Fe). Doping is the intentional addition of impurities to alter electrical properties.

• Line Defects (1D) - Dislocations:

  • Edge Dislocation: An extra half-plane of atoms terminates within the crystal. Creates strain fields (compression above, tension below).

  • Screw Dislocation: Result of shearing; the crystal lattice follows a helical path.

  • Nature: Non-equilibrium defects arising from stress or growth.

• Planar Defects (2D):

  • Grain Boundaries: Interfaces between grains in polycrystalline solids. They are high-energy regions with broken (dangling) bonds, voids, and strained bonds. Atoms diffuse more easily along these boundaries.

Real Crystal Surfaces

• Real surfaces lack infinite periodicity and exhibit unique features:

  • Dangling Bonds: Incomplete bonds at the surface (e.g., Si surface atoms have one unpaired electron).

  • Reconstruction: Surface atoms rearrange to share dangling bonds, causing slight displacement.

  • Chemisorption: Foreign atoms (e.g., O, H) form primary chemical bonds with dangling bonds.

  • Physisorption: Molecules (e.g., polar water) form secondary physical bonds with the surface.

Classical Theory of Electrical Conduction: The Drude Model

• Drude Model (1900): Assumes metals contain free electrons that move randomly like particles in a gas, obeying Newton's laws.

• Current Density ($J$): $J = I/A = q/(At)$.

• Drift Velocity ($v_d$): The average velocity gained by electrons in the direction of an applied electric field $E$.

  • Relation: $v_{dx} = (e τ / m_e) E_x$, where $τ$ is the mean scattering time (relaxation time).

• Resistivity ($ρ$): $ρ = 1 / σ$.

• Conductivity ($σ$): $\u03C3 = e n μ_d$.

• Drift Mobility ($μ_d$): $\u03BC_d = e τ / m_e$.

• Problem (Copper at Room Temp):

  • Given: $\u03C3 = 5.9 \times 10^5 Ω^{-1}cm^{-1}$, Density $= 8.96 g/cm^3$, Atomic mass $= 63.5 g/mol$.

  • Calculated concentration $n = 8.5 \times 10^{22} cm^{-3}$.

  • Mobility $\u03BC_d = 43.4 cm^2/Vs$.

Temperature Dependence of Resistivity

• Ideal Pure Metals: Scattering is caused by thermal lattice vibrations.

  • Mean free path $l = u τ$.

  • Scattering frequency depends on cross-sectional area $S = π a^2$, where $a$ is vibration amplitude.

  • Since average kinetic energy $(1/4) M a^2 ω^2 ≈ (1/2) kT$, the area $a^2 ∝ T$.

  • Therefore, $\u03C1 ∝ T$ (for pure metals at room temp).

• Metallic Alloys and Matthiessen’s Rule: Total resistivity is the sum of independent scattering processes:

  • $\u03C1 = ρ_T + ρ_I + ρ_D$.

  • $\u03C1_T$: Temperature-dependent (lattice vibrations).

  • $\u03C1_I$: Temperature-independent (impurities).

  • Matthiessen's Rule: $\u03C1 = AT + B$.

• Temperature Coefficient of Resistivity (TCR):

  • $\u03B1_0 = Δρ / (ρ_0 ΔT)$.

  • $\u03C1 = ρ_0 [1 + α_0(T - T_0)]$.

  • Below the Debye temperature, $\u03C1$ decreases rapidly because the number of phonons energetic enough to scatter electrons drops off.

The Hall Effect

• Phenomenon: When a magnetic field $B_z$ is perpendicular to current density $J_x$, a transverse electric field $E_H$ (Hall field) is generated in the $y$-direction.

• Hall Field: $E_H = R_H J_x B_z$.

• Hall Coefficient ($R_H$): For electrons, $R_H = -1/en$.

• Hall Voltage ($V_H$): $V_H = (R_H I_x B_z) / D$, where $D$ is the thickness.

• Applications: Measuring charge carrier concentration ($n$) and identifying carrier type. Semiconductors have larger $R_H$ than metals, making them more suitable for Hall devices.

• Hall-Effect Wattmeter: Uses a semiconductor sample to measure electrical power ($P = V_L I_L$). Hall voltage $V_H ∝ I_L B_z$ and $B_z ∝ V_L$, thus $V_H ∝ P$.

Thermal Conduction

• In Metals: Heat is primarily transported by the free electron gas. Energetic electrons at the hot end transfer energy to the cold end via collisions with atoms.

• Fourier’s Law: $Q' = κ A (δT / δx)$, where $κ$ is thermal conductivity.

• Wiedemann-Franz-Lorenz Law: Relates electrical ($σ$) and thermal ($κ$) conductivity:

  • $\u03BA / σ T = C_{WFL}$.

  • Lorenz number $C_{WFL} = 2.44 \times 10^{-8} WΩK^{-2}$.

• In Non-Metals: Heat is conducted via lattice vibrations (phonons). Atomic vibrations are coupled through bonds. Stronger bonds (e.g., Diamond) result in higher thermal conductivity.

Thermal Resistance ($θ$)

• Analogy: Thermal conduction is analogous to Ohm’s Law.

  • Electrical: $I = V / R_{elec}$, where $R_{elec} = L / (σ A)$.

  • Thermal: $Q' = ΔT / θ$, where $\u03B8 = L / (κ A)$.

• Problem (Brass Disk):

  • Given: $\u03C1 = 50 nΩm$, $Q' = 10 W$, diameter $= 20 mm$, thickness $= 30 mm$.

  • Step 1: Find thermal conductivity $\u03BA$ using Wiedemann-Franz Law: $\u03BA ≈ 146 W m^{-1}K^{-1}$.

  • Step 2: Calculate thermal resistance $\u03B8 = L / (κ A) = 0.65 K/W$.

  • Step 3: Temperature drop $\u0394T = Q' θ = 10 W \times 0.65 K/W = 6.5 °C$.