ELECTRICAL CONDUCTIVITY OF METALS

Electrical Conductivity in Metals

• Drift velocity (Vd): Defined as the average velocity that a particle (electrons, in the case of metals) attains due to an electric field.

• Thermal velocity (uav): The average velocity due to thermal energy, which is a more rapid movement of free electrons before any electric field is applied.

• Mean free path (l): Refers to the average distance a particle travels between collisions.

Quantitative Relationships

• Current density (J):

  • Represents the current per unit area flowing through a conductor.

  • Defined as: $J = rac{dQ}{dA imes dt}$

  • Average drift velocity: $Vd = rac{1}{N} imes (V<em>{x1} + V</em>{x2} + … + V_{xN})$

  • Time dependent current is indicated with the average drift represented in terms of time and electric field.

Drift Velocity Calculations

• Formula for drift velocity under a varying electric field:

  • $Vdx = rac{1}{N} imes [U<em>{x1} + U</em>{x2} + … + U_{xN}]$

  • Drift mobility expression:

  • $U<em>{dx} = rac{\bar{ au} E</em>x}{m_e}$

  • Drift current density: $J<em>x = en\bar{U} E</em>x$

  • Electric field (E): A factor affecting drift velocity.

Electron Drift Mobility in Metals

• Example Calculation:

  • Given parameters:

  • Conductivity of copper ($\bar{ ext{s}} = 5.9 imes 10^5 ext{ S cm}^{-1}$)

  • Density ($ρ_{Cu} = 8.96 ext{ g cm}^{-3}$)

  • Atomic mass ($M_{at} = 63.5 ext{ g mol}^{-1}$)

  • Performing Calculations:

  • Number density of conduction electrons:

    • $n = rac{ρ<em>{Cu} imes N</em>A}{M_{at}}$

    • Result: $n = 8.5 imes 10^{22} ext{ electrons cm}^{-3}$

• Drift Mobility Calculation:

  • $\bar{ ilde{μ}}_d = rac{e imes n}{\bar{s}}$

  • $\bar{ ilde{μ}}_d = rac{5.9 imes 10^5 ext{ S cm}^{-1}}{e imes n}$

  • Computed mobility: $\bar{ ilde{μ}}_d = 43.4 ext{ cm}^2 ext{ V}^{-1} ext{ s}^{-1}$

Relaxation Time and Mean Free Path

• Mean free path (λ):

  • Formulated as: $λ = \bar{u} imes \bar{t}$

  • Where $\bar{u}$ is average velocity and $\bar{t}$ is average time between collisions.

• Relaxation time (τ):

  • Represents the average time interval between collisions:

  • $τ = rac{M_d}{en}$

  • This is crucial in understanding conductivity and mobility.

Drift Velocity and Mean Speed

• Calculating applied electric field for a given drift velocity:

  • Required conditions: Drift velocity of electrons to equal 0.1% of their mean speed ($u ext{ approximately } 10^6 ext{ m s}^{-1}$)

  • So, $Vdx = 10^3 ext{ m s}^{-1}$

  • Applied electric field: $E_x = rac{2.3 imes 10^5 ext{ V m}^{-1}}{43.4 imes 10^{-4} ext{ m}^2 ext{ V}^{-1} ext{ s}^{-1}}$

• Current Density Calculation:

  • $J<em>x = σ E</em>x = (5.9 imes 10^7 ext{ S m}^{-1})(2.3 imes 10^5 ext{ V m}^{-1})$

  • Resulting current density: $1.4 imes 10^{13} ext{ A m}^{-2}$

  • Converted: $1.4 imes 10^{7} ext{ A mm}^{-2}$

Temperature Dependence of Resistivity

• Causes:

  • Scattering due to thermal vibrations of atoms.

  • Increased temperature increases average electron velocity and atomic vibrations (1)

• Mean free path under varying temperatures:

  • $S = rac{Ь}{Υ} imes A$

  • Where N represents concentration; for a scattering center: $S imes U imes T_{N} = 1$

Thermal Vibrations and Scattering

• Average kinetic energy of oscillating atoms:

  • Boltzmann approximation: $rac{1}{2} M a^2 ω^2 ext{ approximately equals } rac{3}{2} kT$

  • This links energy associated with thermal vibration to resistivity.

• Lattice scattering-limited conductivity: Occurs due to electron scattering via thermal vibrations of lattice atoms.

Resistance Change with Temperature

• Resistance formula:

  • $R ∝ ρ = A T$

• Example scenario: Resistance change calculation from summer $20° C$ to winter $-30° C$:

  • $rac{R<em>{summer} - R</em>{winter}}{R<em>{summer}} = rac{T</em>{summer} - T<em>{winter}}{T</em>{summer}}$

  • Result: 0.171 ext{ or } 17 ext{%}

Drift Mobility Estimates Under Conditions

• Estimating drift mobility using fundamental physical constants and density of copper:

  • Drift mobility equation in use:

  • $σ = e n \bar{μ}_d$

• Calculating mean speed and other parameters:

  • $S = rac{2πkT}{ ext{Atmospheric pressure}}$

Matthiessen's Rule for Resistivity and Temperature Coefficient

• Conception of Matthiessen's Rule:

  • This principle states that the total resistivity ($ρ$) is the sum of intrinsic resistivity due to perfect lattice + additional resistivity due to impurities.

• Temperature coefficient of resistivity (α):

  • Defined as the fractional change in resistivity per unit temperature increase: $rac{Δρ}{ρ_0ΔT}$

  • Where $ρ0$ is the resistivity at a reference temperature $T0$.

• Apply this concept in determining how resistivity varies with temperature, indicating both temperature independent constants A & B.