Lectures 17-21

Lecture 17: Resistivities of Pure Metals and Alloys

• Resistance (R)
    - Defined as $R = S$, where:
      - R (Ω): resistance of the wire
      - S (Ω·m): resistivity is a material property

• Matthiessen's Rule
    - Expression: $S_T = S_{th} + S_{imp} + S_{def}$
      - $S_{th}$: is resistivity due to phonon scattering
      - At 288K for Cu: $S_{th} ext{ ≈ } 10^{-6} ext{ Ω·m}$
      - At 7812 for some other element: $S_{th} ext{ ≈ } 10^{-7} ext{ Ω·m}$
      - For other materials: $S_{th} ext{ ≈ } 10^{-12} ext{ Ω·m}$

• Temperature Dependence of Resistivity
    - General formula: S_{2} = S_{1} [1 + eta (T_{2} - T_{1})]
      - Where:
          - eta: temperature coefficient of resistivity

• Alloy Resistivity Increase
    - Resistivity of alloys increases with solute content.
      - In dilute single-phase alloys: increases with the square of valence difference between solute and solvent elements.
      - This relationship is known as Linde's Rule.

• Valence Information
    - Valence is determined for:
      - Group 1 (Valence 1)
      - Group 2 (Valence 2)
      - Group 13 (Valence 3)
      - Group 14 (Valence 4)
      - Group 15 (Valence 5)
      - Group 16 (Valence 6)
      - Group 17 (Valence 7)
      - Group 18 (Valence 8)

• Nordheim's Rule
    - $S = X_A S_A + X_B S_B + C imes X_A X_B$
      - Where:
          - $X_A, X_B$: volume fractions of elements A and B
          - $S_A, S_B$: resistivities of A and B elements
          - C: material constant associated with interaction effects

Temperature Dependence of Resistivity for Selected Copper Alloys

• Preparation of resistivity curves for various copper alloys demonstrating specific atomic percentages can be visualized.
      - Data Example:
          - $S_{Cu} أطلس = 10^6$
          - Temperature range exemplified by the diagram from Linde, Ann. d. Physik. 15 (1932), 219.

Resistivities Due to Lattice Defects

• According to Matthiessen's Rule:
  $S_T = S_{th} + S_V + S_{int} + S_{disl} + S_{SF} + S_{solute}$
      - $S_V$: resistivity due to vacancies
          - Sv (Ωar{m}) = C_v (13) imes S_v (cm^3)
      - $C_v$: concentration of vacancies
      - $S_{int}$: resistivity due to interstitials
          - $S_{int} = C_{int} imes S_{int}$
      - $S_{disl}$: resistivity due to dislocations
          - $S_{disl} = C_d N_B IS_{disl}$
          - $C_d$: dislocation density in m²
      - $S_{SF}$: resistivity due to surface defects such as grain boundaries or stacking faults
          - $S_{SF} = C_{SF} S_{SF}$
          - $S_{SF} =10^{-1} (5^2cm^2)$
      - $S_{solute}$: resistivity due to solute atoms

Change in Specific Resistance Due to Solute Atoms

• The change in specific resistance per atomic percent is summarized as follows:
      - Ratio of atomic volumes of the first-named metal to the second defines the resistivity change
      - Examples:
          - $S_0 = 0.54$ (Na in K)
          - $S_0 = 0.70$ (K in Na)
          - The case of Mg and Cd is noteworthy due to conductivity changes arising from differing electron availability in transition metals.

Superconductors and Their Resistivity

• Resistivity of superconductors drops below a critical temperature (T_c): $Tc ext{ causes resistivity to approach zero}$.

• The superconducting state can be disrupted by magnetic fields or critical currents.
   - The relationship is expressed by:
  $H_c = H_0(1 - T^2)$

Semiconductor Properties

• Work Function: Energy needed for an electron to escape the metal.
      - This energy barrier represents how easily electrons can contribute to current flow for semiconductors.

Junctions Between Two Metals

• When two metals are in contact, their Fermi levels will equilibrate. This leads to potential differences measured across the junction. The potential drop can be expressed as:
  $V_{eg} = ext{Work Function of Metal 1} - ext{Work Function of Metal 2}$.

Thermoelectric Effects

• Seebeck Effect: Explains how a temperature gradient across a material induces an electric potential difference.

• Peltier Effect: Describe how applying an electric current through a junction of dissimilar materials can create cooling or heating.

Lecture 18: Thermal Properties of Solids

• Based on the first law of thermodynamics:
  $ext{DE} = ext{W} + ext{Q}$

• Where:
      - W = work done on the system
      - Q = heat supplied to the system
      - DE = change in energy of a closed thermodynamic system

• Heat Capacity: Amount of energy needed to raise the substance by 1°C or 1K.

• Distinctions in heat capacity under constant volume (C_v) and constant pressure (C_p).
      - C_v = rac{dQ}{dT} igg|{V ext{ const}}     - C_p = rac{dQ}{dT} igg|{P ext{ const}}

• Dulong-Petit Law: Molar heat capacity approaches a value near: $C_v ext{ ≈ } 25 ext{ J mol}^{-1} K^{-1}$ at room temperature for metals and solids.

Classical and Quantum Mechanical Theories of Heat Capacity

Classical Theory

• Molar heat capacity estimated through harmonic motion for vibrating atoms:
      - $E = rac{3}{2}K_B T$
      - $C_v = rac{3N_0k_B}{M}$ where M is molar mass.

Quantum Mechanical Theory

• Einstein Model: Energy quantized for oscillating atoms; phonons involved in thermal properties.
     - Derived equations for heat capacity that agree with classical models at high temperatures.

• Debye Model: Accounts for more complex vibrational modes, allowing for increased accuracy in predicting heat capacities at lower temperatures.

Thermal Conductivity of Solids

• Described by heat flux proportional to temperature gradient:
  $J = -K rac{dT}{dx}$

• Total thermal conductivity combines contributions from both electrons and phonons. High purity metals conduct heat best, typically involving mechanisms of both phonon and electron dynamics.

Semiconductors and P-N Junctions

• Band theory distinguishes semiconductor types:
      - Intrinsic: Pure semiconductors without dopants.
      - Extrinsic: Modified properties through the introduction of dopants.

• Encountering P-N junctions:
      - Movement of charge carriers creates depletion zones and built-in potentials affecting conductivity behavior.

Summary of Key Equations and Principles

• Matthiessen's Rule

• Nordheim's Rule

• Seebeck and Peltier Effects

• Thermal and Electrical Conductivity Relations

• Importance of temperature effects in mines and realized properties of materials across various conditions.

• These concepts lay foundational understanding of materials within electrical and semiconductor engineering applications.