Electrical Conductivity and Resistivity

Electrical Conductivity in Solids

• Classical Free Electron Theory (CFET):
  • Assumptions:
    • Metal is a 3D network of positive ions with free electrons moving about
    • Free electrons obey kinetic theory of gases
    • Electric potential due to ionic core is constant throughout the metal
    • Attractions and repulsions between electrons and ions are insignificant
  • Failures:
    • Can't explain specific heat, temperature dependence of conductivity, and dependence of electrical conductivity on electron concentration
    • Can't explain Hall coefficient of some metals

Drift Velocity and Electrical Conductivity

• Drift Velocity:
  • Definition: Net displacement of electrons per unit time in a direction opposite to the direction of the applied electric field
  • Formula: vd = eEτ / m
  • Importance: Accounts for the current in the direction of the field
• Electrical Conductivity:
  • Definition: Ability of a material to conduct electricity
  • Formula: σ = nevd / A
  • Importance: Characterizes the conducting ability of a material

Resistivity and Conductivity

• Resistivity:
  • Definition: Measure of opposition to electric current in a material
  • Formula: ρ = RA / L
  • Importance: Property of a material
• Conductivity:
  • Definition: Reciprocal of resistivity
  • Formula: σ = 1 / ρ
  • Importance: Also a property of a material

Phonons

• Definition: Units of vibrational energy in a solid, arising from oscillating atoms within a crystal
• Importance: Explain the electrical resistivity of metals, especially at low temperatures

Matthiessen's Rule

• Definition: States that the total resistivity of a metal is the sum of resistivity due to phonon scattering and resistivity due to scattering by impurities
• Formula: ρ = ρph + ρi
• Importance: Helps understand the dependence of resistivity on temperature

Quantum Free Electron Theory (QFET)

• Assumptions:
  • Energy values of free electrons are quantized
  • Distribution of electrons in energy levels follows Pauli's Exclusion Principle
  • Distribution of electrons in energy levels obeys Fermi-Dirac statistics
  • Free electrons travel in a constant potential inside the metal but stay confined within its boundaries
  • Attractions and repulsions between electrons and ions are ignored
• Fermi Level and Fermi Energy:
  • Definition: Highest occupied energy level at 0K, and the corresponding energy value, respectively
  • Importance: Characterize the energy distribution of electrons in a metal

Fermi-Dirac Statistics

• Definition: Statistical rule that describes the distribution of electrons among available energy states
• Fermi Factor:
  • Formula: f(E) = 1 / (1 + exp((E - EF) / kT))
  • Importance: Represents the probability that a quantum state with energy E is occupied by an electron

Dependence of Fermi Factor on Temperature

• At 0K: All levels below EF are occupied, and all levels above EF are vacant
• At T > 0K: Probability of occupation of Fermi level is 0.5, and Fermi energy is the average energy possessed by electrons participating in conduction### Quantum Physics for Engineers

Electrical Conductivity of Solids

• Definition of density of states: The number of available states per unit volume per unit energy interval.
• Formula for density of states: $g(E)dE = \frac{4\pi}{3}\frac{(2m)^{3/2}}{h^3}E^{1/2}dE$

Carrier Concentration in Metals and Fermi Energy at 0K

• Number of free electrons/unit volume: $n = \int_{0}^{E_F} g(E)f(E)dE$
• Fermi energy at 0K: $E_F = \frac{h^2}{8m}\left(\frac{3n}{\pi}\right)^{2/3}$
• Expression for concentration of electrons in a metal at 0K: $n = \frac{3}{2}\frac{(2m)^{3/2}}{3h^3}(E_F)^{3/2}$

Success of Quantum Free Electron Theory

• Explains: specific heat capacity of metals, temperature dependence of electrical conductivity, dependence of electrical conductivity on electron concentration, and photoelectric effect, Compton effect, Black body radiation, and Zeeman effect.

Band Theory of Solids

• Energy band structure: determines whether a solid is a conductor, insulator, or semiconductor.
• Energy bands: formed by overlapping energy levels of individual atoms in a solid.
• Valence band: energy band formed by energy levels of valence electrons.
• Conduction band: energy band immediately above the valence band.
• Forbidden energy gap: separation between the upper level of valence band and the bottom level of conduction band.

Semiconductors

• Definition: materials with electrical conductivity greater than insulators but significantly lower than conductors at room temperature.
• Conductivity range: 10^-4 to 10^4 S/m.
• Bipolar: current is transported by two types of charge carriers of opposite sign (electrons and holes).
• Doping: enhances conductivity by introducing impurities.

Intrinsic Semiconductors

• Definition: extremely pure semiconductor.
• Intrinsic carriers: electrons and holes generated by breaking of bonds.
• Intrinsic concentration: equal to the number of electrons and holes generated.

Carrier Concentration in Intrinsic Semiconductors

• Formula for electron concentration: $n = \int_{E_c}^{\infty} f(E)g_c(E)dE$
• Formula for hole concentration: $p = N_ve^{-(E_F-E_v)/kT}$

Fermi Level in Intrinsic Semiconductors

• Fermi level: lies at the center of the energy gap.
• Formula for Fermi level: $E_F = \frac{E_c+E_v}{2}$### Intrinsic and Extrinsic Semiconductors
• Intrinsic semiconductors have no impurities, while extrinsic semiconductors have controlled amounts of impurities introduced into the material.
• Doping is the process of introducing impurities into an intrinsic semiconductor to modify its electrical conductivity.
• There are two types of extrinsic semiconductors: p-type and n-type.

N-type Semiconductors

• N-type semiconductors are produced by doping an intrinsic semiconductor with pentavalent impurity atoms, such as Phosphorous or Arsenic.
• The electron concentration in an n-type semiconductor depends on temperature:
  • At 0K, donor levels are fully occupied, meaning all donor electrons are bound to donor atoms.
  • At low temperatures (region I), there is not enough energy to ionize all donors, and the electron concentration is low.
  • At higher temperatures (region II), donor atoms are ionized, and the electron concentration increases.
  • At even higher temperatures (region III), the electron concentration due to donor impurities is exceeded by the intrinsic electron concentration.

P-type Semiconductors

• P-type semiconductors are produced by doping an intrinsic semiconductor with trivalent impurity atoms, such as Aluminum or Boron.
• The hole concentration in a p-type semiconductor also depends on temperature:
  • At 0K, acceptor levels are empty, meaning all valence band electrons are bound to acceptor atoms.
  • At low temperatures (ionization region), electrons from the valence band jump into acceptor levels, increasing the hole concentration.
  • At higher temperatures (saturation region), the hole concentration remains constant, as thermal energy is not sufficient to cause electron transitions from the valence band to the conduction band.
  • At even higher temperatures (intrinsic region), the hole concentration due to acceptor impurities is exceeded by the intrinsic hole concentration.

Fermi Level in Extrinsic Semiconductors

• The Fermi level in an n-type semiconductor varies with temperature:
  • At low temperatures, the Fermi level lies between the donor level and the bottom edge of the conduction band.
  • As temperature increases, the Fermi level shifts downward, approaching the intrinsic Fermi level.
• The Fermi level in a p-type semiconductor also varies with temperature:
  • At low temperatures, the Fermi level rises with increasing temperature from below the acceptor level to the intrinsic level.

Effect of Impurity Concentration

• Increasing the impurity concentration in an n-type semiconductor causes the donor level to broaden into a band, eventually overlapping with the conduction band.
• Similarly, increasing the impurity concentration in a p-type semiconductor causes the acceptor level to broaden into a band, eventually overlapping with the valence band.

Hall Effect

• The Hall effect is a phenomenon where a transverse voltage is developed in a current-carrying conductor when a magnetic field is applied perpendicular to the current flow.
• The Hall effect helps to:
  • Determine the sign of charge carriers in a material
  • Determine the charge carrier concentration
  • Determine the mobility of charge carriers if the conductivity of the material is known
• The Hall voltage (VH) is given by VH = BI/ne, where B is the magnetic field, I is the current, n is the charge carrier concentration, and e is the elementary charge.

Hall Effect in Metals and Semiconductors

• The Hall effect in metals is similar to that in semiconductors, with the majority charge carriers being electrons in metals and n-type semiconductors, and holes in p-type semiconductors.
• The Hall voltage is negative for metals and n-type semiconductors, and positive for p-type semiconductors.
• The Hall coefficient (RH) is given by RH = 1/ne, and the carrier concentration can be calculated from the Hall voltage and Hall coefficient.