Podcast Beta
Questions and Answers
What is the Lorentz Force acting on an electron in an electric field?
eE
Thermal conductivity is defined as the amount of heat flowing per unit time through the material having unit area of cross section and maintaining at unit temperature gradient (dT/dx). The formula is Q = __________.
KdT
Which statement is true about the classical free electron theory?
Quantum theory explains the phenomenon of photo-electric effect better than the classical theory.
Signup and view all the answers
What is the term used to describe the number of electrons per unit volume?
Signup and view all the answers
What is the formula to calculate the carrier concentration of electrons in energy bands?
Signup and view all the answers
What is Fermi energy level at 0 Kelvin also known as?
Signup and view all the answers
In the free electron approximation, the potential energy of the electron is assumed to be greater than its total energy.
Signup and view all the answers
Which approximation assumes the atoms are free while the electrons are tightly bounded?
Signup and view all the answers
What does equation (10) represent?
Signup and view all the answers
In which special case is the effective mass positive?
Signup and view all the answers
True or False: In the upper energy band, the effective mass is negative.
Signup and view all the answers
The electron with negative effective mass is called _______.
Signup and view all the answers
What is the concept of forbidden energy levels?
Signup and view all the answers
How are materials classified based on the width of the forbidden band gap?
Signup and view all the answers
Calculate the temperature at which there is 1% probability of a state with energy 0.5 eV above Fermi energy.
Signup and view all the answers
Calculate the carrier concentration of electrons in an energy interval of 0.01 eV above the Fermi level of sodium metal.
Signup and view all the answers
Write the Fermi - Dirac distribution function.
Signup and view all the answers
Explain how Fermi function varies with temperature.
Signup and view all the answers
What is the density of states for a metal?
Signup and view all the answers
How is the density of states used to calculate the Fermi energy of metals?
Signup and view all the answers
What happens to the density of states for non-zero temperature?
Signup and view all the answers
Discuss the average energy of electrons at zero Kelvin.
Signup and view all the answers
Calculate the number of electrons per unit volume in copper.
Signup and view all the answers
Find the relaxation time in the given scenario.
Signup and view all the answers
Calculate the mobility of electrons in copper.
Signup and view all the answers
Determine the electrical conductivity of copper at 300K.
Signup and view all the answers
Calculate the electrical conductivity of copper based on mean free path, electron density, and thermal velocity.
Signup and view all the answers
Compute the electrical conductivity, thermal conductivity, and Lorentz number for a metal with a given relaxation time.
Signup and view all the answers
Calculate the drift velocity and thermal velocity of conduction electrons in copper at 300K.
Signup and view all the answers
Determine the drift velocity of free electrons in a copper wire with a specific cross-sectional area.
Signup and view all the answers
Calculate the average drift velocity and the mean collision time in a metallic wire with given parameters.
Signup and view all the answers
Find the probability of an energy level being occupied by an electron above the Fermi level.
Signup and view all the answers
Evaluate the Fermi function for an energy kT above the Fermi energy.
Signup and view all the answers
Calculate the Fermi energy of copper at 0 K.
Signup and view all the answers
Study Notes
Electrical Properties of Materials
- Metals are composed of atoms with electrons in permissible orbits, which are free to move in all directions like molecules in a perfect gas container
- The force between conduction electrons and ion cores is neglected, and the total energy of the electron is assumed to be kinetic energy (potential energy is zero)
- When an electric field is applied, the free electrons move in a direction opposite to the direction of the applied field with a drift velocity (Vd)
- The Lorentz force acting on the electron is F = eE, which accelerates the electrons, causing them to collide with positive ion cores and other free electrons elastically
- The relaxation time (Ï„) is the time between collisions, and the drift velocity (Vd) is proportional to the electric field (E) and inversely proportional to the relaxation time (Ï„)
Electrical Conductivity
- Electrical conductivity (σ) is the amount of electrical charge (Q) conducted per unit time (t) per unit area (A) of a solid along a unit applied electrical field (E)
- Expression for electrical conductivity: σ = ne²τ / m, where ne is the density of conduction electrons, e is the charge of an electron, τ is the relaxation time, and m is the mass of an electron
Thermal Conductivity
- Thermal conductivity (K) is the amount of heat (Q) flowing per unit time through a material having unit area of cross-section and maintaining a unit temperature gradient (dT/dx)
- Expression for thermal conductivity: K = nvkλ / 2, where nv is the density of conduction electrons, k is the Boltzmann constant, λ is the mean free path of electrons, and v is the velocity of electrons
- Wiedemann-Franz law: K / σ = LT, where L is the Lorentz number, a constant that depends on the material
Fermi-Dirac Distribution Function
- The probability (F(E)) of an electron occupying a given energy level at absolute temperature T is given by the Fermi-Dirac distribution function: F(E) = 1 / (1 + e^(E-Ef)/kT), where E is the energy of the level, Ef is the Fermi energy, k is the Boltzmann constant, and T is the absolute temperature
- Effect of temperature on Fermi function:
- At T = 0 K, F(E) = 1 for E < Ef, and F(E) = 0 for E > Ef
- At T > 0 K, F(E) varies with E, and the Fermi function reduces to the classical Boltzmann distribution at high temperatures
Density of States
- The density of states (Z(E)) is the number of available electron states per unit volume in an energy interval E and E+dE
- Expression for density of states: Z(E) = (1/8) * (4πn³/3) / (V * Ea³), where V is the volume of the metal piece, Ea is the energy of the electron, and n is the quantum number
- Importance of density of states:
- It gives the probability of electron occupation at a given energy state at a given temperature
- It is used to calculate the number of free electrons per unit volume at a given temperature
- It is used to calculate the Fermi energy of the metal### Density of States
- The density of states (N(E)) is the number of energy states between the shell of radius 'n' and 'n+dn'
- The formula for N(E) is: N(E) = (1/8) × (4πn^2 dn) / (3) = (π/2) × (n^2 dn) / (3)
- Neglecting higher powers of dn, N(E) = (π/2) × (4n^2 dn) / (3)
- The density of states is used to calculate the carrier concentration in metals and semiconductors
Carrier Concentration
- The carrier concentration (n_c) is the number of electrons per unit volume
- The formula for carrier concentration is: n_c = ∫[Z(E)F(E)dE]
- The Fermi distribution function (F(E)) is used to calculate the probability of filling an electron state
- The carrier concentration is calculated by integrating the product of the density of states (Z(E)) and the Fermi distribution function (F(E)) over the entire energy range
Fermi Energy
- The Fermi energy (E_f) is the maximum energy level that can be occupied by an electron at 0K
- The Fermi energy level is dependent on the density of states and the Fermi distribution function
Bloch's Theorem
- Bloch's theorem states that the wave function of an electron in a crystal can be written as: ψ(x) = e^(ikx)u_k(x)
- The theorem is used to solve the Schrödinger equation for an electron in a periodic potential
- The solution leads to the formation of allowed energy bands separated by forbidden energy gaps
Energy Bands
- The energy bands are allowed energy ranges for an electron in a crystal
- The energy bands are separated by forbidden energy gaps
- The width of the allowed energy bands increases with the increase in the interaction between the electrons and the lattice points
Free Electron Approximation
- The free electron approximation assumes that the potential energy of an electron is less than its total energy
- The approximation is used to describe the behavior of electrons in metals and semiconductors
- The approximation is valid when the interaction between the electrons and the lattice points is weak
Tight Binding Approximation
- The tight binding approximation assumes that the potential energy of an electron is almost equal to its total energy
- The approximation is used to describe the behavior of electrons in insulators
- The approximation is valid when the interaction between the electrons and the lattice points is strong
Brillouin Zones
- Brillouin zones are the boundaries marked by the values of the propagation vector k
- The Brillouin zones are used to describe the energy bands of an electron in a crystal
- The first Brillouin zone is the range of allowed energy values between -π/a and π/a
Effective Mass of an Electron
- The effective mass of an electron (m*) is the mass of an electron in a periodic potential
- The effective mass is dependent on the energy of the electron and the curvature of the energy band
- The effective mass can be positive or negative depending on the energy band
Concept of a Hole
- A hole is a positively charged particle with a negative effective mass
- The concept of a hole is used to describe the behavior of electrons in the upper band of the energy spectrum
- The presence of a hole is attributed to an empty state in the energy band### Valence Electrons and Energy Levels
- Valence electrons possess more energy than inner orbit electrons, and the larger the orbit, the greater is its energy.
- Energy levels (E1, E2, E3, ...) represent the energy of different orbits (K, L, M, ... shells).
- Electrons can only revolve in certain permitted orbits (r1, r2, r3, ...) and not in arbitrary orbits, resulting in forbidden energy levels.
Energy Bands
- When atoms are brought together, interatomic force of attraction modifies the energy levels of a solid, forming energy bands.
- Energy levels of single free atoms split into multiple levels, forming energy bands in solids.
- Energy bands can be defined as the range of energies possessed by an electron in a solid.
Forbidden Gap
- Energy bands are separated by small regions, called forbidden gaps or forbidden energy gaps (Eg), which do not allow any energy levels.
Classification of Materials
- Based on band theory and the presence of forbidden band gaps, materials are classified into:
- Metals (or) Conductors: no forbidden band gap, valence and conduction bands overlap.
- Semiconductors: small forbidden band gap (0.5 to 1.5 eV), free electrons in valence band are relatively less.
- Insulators: wide forbidden band gap (3 to 5.47 eV), requiring a large energy for conduction.
Calculations
- Mobility (μ) and average time of collision (τ) can be calculated using the Wiedemann-Franz (WKF) law.
- Electrical conductivity (σ) can be calculated using the Lorentz number (L) and thermal conductivity (K).
- Drift velocity (vd) and thermal velocity (v) can be calculated using the electron density (n), charge (e), and mass (m).
Problems and Solutions
- Several problems and their solutions are provided, covering topics such as electrical conductivity, thermal conductivity, Lorentz number, drift velocity, and Fermi energy.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Solve problems related to energy and force using the given formulas. Practice calculating energy and force with the help of these formulas and examples.
