Unit5 - Subjective Questions

Classical Free Electron Theory (Drude-Lorentz Theory):

This theory assumes that a metal consists of a lattice of positive ion cores surrounded by a 'gas' of free electrons that move randomly.

Key Assumptions:

• Valence electrons are free to move throughout the volume of the metal like gas molecules.
• Electrons collide with positive ions and other electrons (elastic collisions).
• Between collisions, electrons move in straight lines.
• The motion follows classical Maxwell-Boltzmann statistics.

Merits:

• It successfully explains Ohm's Law (Electrical Conductivity).
• It explains high thermal conductivity of metals.
• It derives the Wiedemann-Franz law (relation between thermal and electrical conductivity).

Drawbacks/Failures:

• Specific Heat: It predicts the specific heat of free electrons to be , but experimental values are much smaller ().
• Temperature Dependence: It fails to explain the correct dependence of electrical conductivity on temperature (Theory says , Experiment shows ).
• Paramagnetism: It fails to explain the magnetic susceptibility of free electrons.

Fermi Energy ():
It is defined as the energy of the highest occupied quantum state at absolute zero temperature (). It represents the maximum kinetic energy an electron can possess at .

Fermi-Dirac Distribution Function ():
It gives the probability that a quantum state with energy is occupied by an electron at temperature .

Where:

• = Boltzmann constant
• = Absolute temperature

Variation with Temperature:

1. At :
   • If , the exponential term is $0$, so (All levels below are filled).
   • If , the exponential term is , so (All levels above are empty).
   • Graph is a step function.
2. At :
   • Electrons near gain thermal energy and move to higher levels.
   • The probability curve smooths out.
   • At , . The probability of finding an electron at the Fermi level is exactly 50%.

Formation of Energy Bands:

1. Isolated Atoms: In an isolated atom, electrons occupy discrete, sharp energy levels defined by quantum mechanics.
2. Interaction in Solids: When atoms are brought close together to form a solid crystal, the interatomic distance decreases.
3. Pauli's Exclusion Principle: As wave functions of outer electrons overlap, Pauli's principle forbids electrons from occupying the exact same quantum state.
4. Splitting of Levels: To accommodate the N electrons in the crystal, each discrete atomic energy level splits into N closely spaced energy levels.
5. Band Formation: Since N is very large (), these closely spaced levels form a continuous range of energies called an Energy Band.
6. Allowed vs. Forbidden:
   • Allowed Bands: Ranges of energy that electrons can occupy (e.g., Valence Band, Conduction Band).
   • Forbidden Gap (Band Gap ): The energy range between the top of the valence band and the bottom of the conduction band where no electron states exist.

1. Conductors (Metals):

• Band Structure: The Valence Band (VB) and Conduction Band (CB) overlap, or the CB is partially filled.
• Band Gap (): Zero ( eV).
• Conductivity: Electrons can easily move from VB to CB. Very high conductivity.

2. Semiconductors:

• Band Structure: There is a small forbidden energy gap between a completely filled VB and an empty CB (at 0K).
• Band Gap (): Small ( eV). E.g., Silicon ($1.1$ eV), Germanium ($0.7$ eV).
• Conductivity: At , they act as insulators. At room temp, thermal energy excites some electrons to CB, allowing conduction.

3. Insulators:

• Band Structure: Large forbidden energy gap separates a filled VB and an empty CB.
• Band Gap (): Large ( to $6$ eV). E.g., Diamond ($5.4$ eV).
• Conductivity: Thermal energy is insufficient to bridge the gap; hence, conductivity is negligible.

Concept: The mass of an electron appears to change when it moves inside a periodic crystal lattice due to interaction with the lattice field. This modified mass is called Effective Mass ().

Derivation:
Group velocity of an electron wave packet is .
Since , we have .

Acceleration

From external force doing work: , so .
Substituting this into acceleration:

Comparing with Newton's law or :

Significance:

• Effective mass depends on the curvature () of the E-k band.
• High curvature (steep curve) Small mass High mobility.
• Flat curve Large mass Low mobility.

Concept of Holes:

• In a semiconductor valence band, when an electron is excited to the conduction band, it leaves behind a vacancy.
• Near the top of the valence band, the E-k curve is concave downwards, meaning is negative.
• Consequently, the effective mass of an electron at the top of the valence band is negative.
• Physically, a vacancy moving with a negative mass behaves mathematically and physically exactly like a particle with a positive mass and a positive charge.
• This imaginary positive charge carrier is called a Hole.
• Holes respond to electric fields by moving in the direction of the field (unlike electrons), contributing to current.

Hall Effect:
When a magnetic field () is applied perpendicular to a current-carrying conductor (), a voltage is developed across the conductor perpendicular to both the current and the magnetic field. This voltage is called the Hall Voltage ().

Derivation:
Consider a slab of width and thickness .

1. Lorentz Force: Charge carriers (electrons) experience a downward magnetic force: .

2. Charge Accumulation: Electrons accumulate at the bottom face, creating an electric field pointing downwards (Hall Field).

3. Equilibrium: The electric force balances the magnetic force.
   

4. Current Density (): .
   Substituting into :
   

5. Hall Coefficient (): Defined as the ratio of Hall field to the product of Current density and Magnetic field.
   

Therefore, (for electrons, often written with a negative sign: ).

The Hall Effect is used to determine:

1. Type of Semiconductor: The sign of the Hall Voltage (or Hall Coefficient ) determines if the material is N-type (negative ) or P-type (positive ).
2. Carrier Concentration (): From the formula , the density of charge carriers can be calculated if is measured.
3. Mobility (): Since conductivity and , mobility can be calculated as .
4. Magnetic Field Meter: Hall sensors are used to measure the strength of unknown magnetic fields (Gaussmeters).
5. Power Meters: It can be used to measure power in an electromagnetic wave.

In an intrinsic semiconductor, the number of electrons in the conduction band () equals the number of holes in the valence band ().

Concentration Formulas:

At Equilibrium ():
Assuming effective density of states (implies ):

Taking natural log () on both sides:

Conclusion: The Fermi level lies exactly at the center of the band gap for an intrinsic semiconductor at .

Position:
In an N-type semiconductor, donor impurities are added. At absolute zero, the Fermi level () lies very close to the bottom of the Conduction Band (), just below the donor energy level ().

Variation with Temperature:

1. Very Low Temp: is between and .
2. Moderate Temp (Extrinsic Range): As temperature increases, donor electrons move to the conduction band. The Fermi level starts moving downward away from .
3. High Temp (Intrinsic Range): At very high temperatures, thermally generated electron-hole pairs (intrinsic carriers) dominate over the fixed number of donor electrons. The material behaves like an intrinsic semiconductor, and the Fermi level shifts towards the middle of the band gap ().

Position:
In a P-type semiconductor, acceptor impurities create holes. At , the Fermi level () lies very close to the top of the Valence Band (), just above the acceptor energy level ().

Variation with Temperature:

• Low Temperature: The Fermi level is located between the acceptor level () and the valence band edge ().
• Increasing Temperature: As temperature rises, acceptor states get filled, and holes are generated in the valence band. begins to move upward, away from the valence band.
• High Temperature: Intrinsic carrier generation becomes dominant. The concentration of electrons roughly equals the concentration of holes. Consequently, the Fermi level approaches the center of the forbidden energy gap (intrinsic Fermi level).

Principle: Solar cells operate on the Photovoltaic Effect, which is the generation of a voltage difference (and electric current) in a material upon exposure to light.

Construction: It is essentially a P-N junction diode with a large surface area, often coated with an anti-reflection coating.

Working:

1. Absorption: Light photons with energy
   strike the solar cell and are absorbed in the depletion region.
2. Generation: The absorbed energy excites electrons from the valence band to the conduction band, generating Electron-Hole pairs (EHPs).
3. Separation: The built-in electric field at the P-N junction depletion layer sweeps electrons towards the N-side and holes towards the P-side.
4. Collection: This accumulation of charges creates a voltage (Open Circuit Voltage ). If an external load is connected, electrons flow through the wire from N to P, creating a photocurrent.

I-V Characteristics:

• The Solar cell acts as a battery but does not follow Ohm's law linearly.
• The curve is drawn in the fourth quadrant (as it delivers power).
• (Open Circuit Voltage): Voltage when current .
• (Short Circuit Current): Current when voltage .
• : The point on the curve where the product is maximum ().

Fill Factor (FF):
It is a measure of the quality of the solar cell. It is the ratio of the maximum obtainable power to the theoretical product of and .

A higher Fill Factor indicates a better solar cell (closer to an ideal rectangle in the I-V curve).

Definition: Mobility () is defined as the drift velocity () acquired by a charge carrier per unit applied electric field ().

Relation to Conductivity:

1. Current Density is given by , where is carrier concentration and is charge.
2. From Ohm's Law in point form: , where is conductivity.
3. Equating the two: .
4. Substitute :
   
   
   

For a semiconductor with both electrons and holes:

Metals:

• Conductivity , where is the relaxation time (time between collisions).
• As Temperature () increases, the lattice ions vibrate more vigorously.
• This increases the frequency of collisions, decreasing the mean free path and the relaxation time .
• Since (electron density) is constant, the decrease in causes conductivity to decrease.

Semiconductors:

• Conductivity .
• As increases, the primary effect is the breaking of covalent bonds, which exponentially increases the charge carrier density ( and ).
• Although mobility () decreases slightly due to lattice scattering, the exponential increase in carrier concentration dominates.
• Therefore, conductivity increases significantly.

The Fermi-Dirac distribution is given by:

If we consider an energy state exactly at the Fermi Level, i.e., :

Significance:
This implies that at any temperature , the Fermi Level () represents the energy state that has a 50% probability of being occupied by an electron. It serves as a reference level to determine the probability of occupation for other energy states.

Common Materials:

1. Crystalline Silicon (Si): Most common (monocrystalline or polycrystalline). Mature technology.
2. Gallium Arsenide (GaAs): High efficiency, used in space applications.
3. Cadmium Telluride (CdTe): Used in thin-film solar cells.

Criteria for Selection:

1. Band Gap (): The material should have a band gap between $1.1$ eV and $1.7$ eV. This matches the solar spectrum to absorb maximum optical energy. (Si is $1.1$ eV, GaAs is $1.4$ eV).
2. Optical Absorption: High optical absorption coefficient () so that thin layers can absorb light effectively.
3. Carrier Mobility: High mobility to ensure charge carriers reach the electrodes before recombining.
4. Cost and Availability: Materials like Silicon are abundant and relatively cheaper to process.

Density of States, :
It is defined as the number of available quantum states per unit energy interval per unit volume of the crystal.

Significance:

• The Fermi-Dirac function only tells us the probability that a state is filled.
• To find the actual number of electrons () in a band, we must multiply the number of available seats (states) by the probability that the seat is occupied.
• Formula:
• Without knowing the density of states (how many energy levels exist at a specific energy), we cannot calculate the carrier concentration or the position of the Fermi level.