Semiconductor Physics: Fermi Energy & Carrier Concentration

Semiconductor Bandgap and Temperature

• At a given temperature, electrons in a semiconductor with a larger bandgap are less likely to jump to the conduction band due to the higher energy barrier
• Fewer electrons present in the conduction band leads to lower electrical conductivity.

Germanium Energy Gap

• The estimated energy gap of germanium at room temperature is approximately 0.67 eV

Hall Effect

• The Hall Effect refers to the voltage difference that develops across a conducting material when a magnetic field is applied perpendicular to the direction of current flow.

Hall Effect: Distortion of Charge Carrier Flow

• The distortion of charge carrier flow in the Hall Effect is primarily due to the Lorentz force acting on moving charges in a magnetic field. This force deflects the charge carriers, creating an accumulation of charge on one side of the conductor.

Hall Voltage Equation

• Hall voltage (VH) is mathematically expressed as: VH = (IB)/ (nqd)
  • Where:
    • I is the current
    • B is the magnetic field strength
    • n is the charge carrier density
    • q is the charge of a single carrier
    • d is the thickness of the conductor

Hall Effect on a Plate

• When a magnetic field is applied to a charge-carrying plate, a Hall voltage develops across the plate perpendicular to both the current flow and the magnetic field.

Edwin Hall

• The Hall Effect was discovered by Edwin Hall in 1879.

Electrons in Conduction Band: Germanium

• The number of electrons excited to the conduction band in germanium is directly related to its energy gap, and increases exponentially with increasing temperature.

Fermi Energy and Electrical Conductivity

• Fermi energy represents the highest energy level occupied by electrons at absolute zero.
• Materials with higher Fermi energies generally have higher electrical conductivity, as more electrons are available for conduction.

Temperature and Electron Distribution

• Increasing temperature causes the electrons in conductors to occupy higher energy levels, leading to increased thermal energy and possibly increased electrical conductivity.

Carrier Concentration in Semiconductors

• Carrier concentration in semiconductors is directly influenced by temperature and doping levels.
• The number of charge carriers in an intrinsic semiconductor increases with temperature due to the excitation of electrons to the conduction band.

Electron Fraction in the Conduction Band

• The fraction of electrons in the conduction band of an intrinsic semiconductor can be described by the Fermi-Dirac distribution function, which depends on the energy gap and the temperature.

Highest Electron Fraction at 300 K

• At 300 K, the material with the highest fraction of electrons in the conduction band would be the semiconductor with the smallest energy gap.

Significance of Fermi Energy in Semiconductor Physics

• Fermi energy is a crucial parameter in semiconductor physics, as it determines the energy level at which the probability of an electron being present is 50%.
• It helps define the intrinsic carrier concentration, which is a key factor in determining the electrical conductivity of semiconductors.

Low Fermi Energy Implication

• Materials with a low Fermi energy typically have poorer electrical conductivity, as fewer electrons are available for transport.

Temperature's Role in Electrical Properties

• Temperature significantly affects the electrical properties of materials, influencing:
  • The number of charge carriers
  • Their mobility
  • The overall conductivity.

Hall Coefficient Purpose

• The Hall coefficient is a measure of the charge carrier density and its sign in semiconductor materials.
• It provides valuable insights into the type of doping (n-type or p-type) and the concentration of charge carriers.

Negative Hall Coefficient

• A negative Hall coefficient indicates that the majority charge carriers in a semiconductor are electrons (n-type semiconductor).

Hall Voltage Calculation

• The Hall voltage (VH) can be calculated using the formula: VH = (IB)/(nqd)
  • Where:
    • I is the current
    • B is the magnetic field strength
    • n is the charge carrier density
    • q is the charge of a single charge carrier
    • d is the thickness of the conductor

Magnetic Flux Density

• Magnetic flux density measures the strength of a magnetic field and is defined as the number of magnetic field lines passing per unit area, perpendicular to the direction of the field.

Hall Coefficient Unit

• The unit of the Hall coefficient (RH) is typically m^3/C.

Hall Voltage Implication

• If a conductor experiences a Hall voltage of 37µV under a current of 20 mA in a magnetic field of 0.5 Wbm−2, this implies that the semiconductor has a specific charge carrier density based on the Hall Effect equation.

Magnetic Dipole Moment

• Magnetic dipole moment is measured in Ampere-meter squared (Am^2).

Hall Effect Applications

• Applications of the Hall effect include:
  • Magnetic field sensing (e.g., in compasses, speedometers)
  • Current measurement (e.g., Hall effect sensors)
  • Material characterization (e.g., determining carrier density and mobility)

Ferromagnetic Material

• A ferromagnetic material exhibits spontaneous magnetization due to strong interactions between electron spins, leading to a permanent magnetic moment even without an external magnetic field.

Magnetic Susceptibility

• Magnetic susceptibility is a parameter that determines how easily a material can be magnetized in response to an external magnetic field. It describes the strength of the material's magnetic response.

Diamagnetic Magnetization

• Diamagnetic materials exhibit induced magnetization in the opposite direction of the applied magnetic field, which disappears when the external field is removed, leaving the material unmagnetized.

Thickness in Hall Voltage Calculation

• In a semiconductor, the thickness (d) in the Hall voltage calculation represents the distance between the two faces across which the Hall voltage is measured.

N in Magnetic Moment Equation

• In the equation m = NIA, N represents the total number of turns in a coil.

Paramagnetic Materials

• Paramagnetic materials exhibit weak magnetization in the same direction as the applied magnetic field. This magnetization disappears when the external field is removed.

Magnetic Field Strength

• Magnetic field strength is also known as magnetic intensity or magnetizing force.

Antiferromagnetic Materials

• Antiferromagnetic materials have magnetic moments that align in opposite directions, resulting in zero net magnetization.

Type I Superconductor in Magnetic Field

• When the magnetic field exceeds the critical field (HC) in a Type I superconductor, the superconductivity is destroyed, and the material reverts to its normal conducting state.

Type II Superconductor Example

• Niobium-titanium (NbTi) is an example of a type II superconductor.

Niobium Transition Temperature

• The behavior of niobium transitioning from superconducting to normal conductors is scrutinized in the temperature range around 9.3 K.

Magnetization in Type I Superconductors

• In Type I superconductors below the critical field (HC), perfect diamagnetism occurs. This means that the magnetic field is completely expelled from the superconductor, resulting in zero magnetization.

Type II Superconductor Critical Fields

• Type II superconductors are characterized by two critical fields: HC1 and HC2.
• HC1 is the field at which partial penetration of the magnetic field begins.
• HC2 is the field at which superconductivity is completely destroyed.

HC2 Range for Type II Superconductors

• The critical field HC2 for type II superconductors typically reaches values in the range of several Tesla (T).

Magnetization in Type II Superconductors

• Between HC1 and HC2, type II superconductors exhibit partial magnetization, where some magnetic field lines penetrate the material, creating vortices.

Type I Superconductor Transition

• As a Type I superconductor transitions out of superconductivity, it becomes a normal conductor, with finite resistance and allowing magnetic field penetration.