Unit 5 - Notes

PHY110

Unit 5: Solid State Physics

1. Free Electron Theory (Introduction)

The free electron theory attempts to explain the physical properties of metals (such as electrical and thermal conductivity) based on the behavior of valence electrons.

Classical Free Electron Theory (Drude-Lorentz Model)

Concept: Metals are visualized as a lattice of positive ions surrounded by a "gas" of free electrons.
Assumptions:
- Valence electrons move freely throughout the volume of the metal like gas molecules.
- Electrostatic attraction between electrons and positive ions is neglected (constant potential).
- Electrons collide only with positive ions (elastic collisions).
Successes: Explains Ohm’s Law and high electrical/thermal conductivity.
Failures: Cannot explain the specific heat of metals, the temperature dependence of conductivity, or magnetic susceptibility.

Quantum Free Electron Theory (Sommerfeld Model)

Modification: Retains the concept of free electrons but applies Quantum Mechanics (Pauli Exclusion Principle and Fermi-Dirac statistics) instead of classical Maxwell-Boltzmann statistics.
Key Insight: Electrons are fermions; only two electrons (spin up/down) can occupy a single energy state.

2. Transport Mechanisms: Diffusion and Drift Current

Drift Current

Drift current is the flow of charge carriers driven by an applied electric field.

Mechanism: When an electric field ( $E$ ) is applied, charge carriers experience a force ( $F = qE$ ) and acquire a net average velocity called drift velocity ( $v_d$ ).
Relationship:
$J_{drift} = \sigma E$
Where $J$ is current density and $\sigma$ is conductivity.
$v_d = \mu E$
Where $\mu$ is mobility.

Diffusion Current

Diffusion current is the flow of charge carriers driven by a concentration gradient (variation in carrier density).

Mechanism: Charge carriers move from a region of higher concentration to a region of lower concentration due to random thermal motion.
Fick's Law: The flux is proportional to the concentration gradient.
$J_{diffusion} = -q D \frac{dn}{dx}$
Where $D$ is the diffusion coefficient and $dn/dx$ is the concentration gradient.

Total Current Density:

J_{total} = J_{drift} + J_{diffusion}

3. Fermi Energy and Fermi-Dirac Distribution

Fermi Energy ( $E_F$ )

Definition: Fermi energy is the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature ( $0K$ ).
Physical Significance: It represents the maximum kinetic energy an electron can possess at $0K$ . It acts as a reference level that determines the probability of particle occupancy at temperatures $T > 0K$ .

Fermi-Dirac Distribution Function

This function describes the probability $f(E)$ that a specific energy state $E$ is occupied by an electron at a given absolute temperature $T$ .

Formula:

f(E) = \frac{1}{1 + e^{(E - E_F) / k_B T}}

$E$ : Energy of the state.
$E_F$ : Fermi Energy.
$k_B$ : Boltzmann constant.
$T$ : Absolute Temperature.

Behavior:

At T = 0K:
- If $E < E_F$ , the exponential term is $e^{-\infty} \to 0$ , so $f(E) = 1$ (State is filled).
- If $E > E_F$ , the exponential term is $e^{+\infty} \to \infty$ , so $f(E) = 0$ (State is empty).
At T > 0K:
- At $E = E_F$ , $f(E) = \frac{1}{1 + e^0} = \frac{1}{2}$ .
- There is a 50% probability of finding an electron exactly at the Fermi energy level at any temperature above absolute zero.

4. Theory of Solids: Band Theory

Formation of Energy Bands

Isolated Atoms: Electrons occupy discrete, sharp energy levels.
Solid Formation: When atoms are brought close together to form a crystal lattice, the wave functions of electrons in neighboring atoms overlap.
Splitting: Due to the Pauli Exclusion Principle, discrete energy levels split into a cluster of closely spaced levels, forming a Band.
Allowed Bands: Ranges of energy that electrons can possess (e.g., Valence Band, Conduction Band).
Forbidden Gap ( $E_g$ ): The energy gap between the valence band and conduction band where no electron states exist.

Classification of Solids

Insulators:
- Structure: Full Valence Band, Empty Conduction Band.
- Band Gap: Very large ( $E_g > 3 \text{ eV}$ , typically $5-6 \text{ eV}$ like Diamond).
- Conductivity: Electrons cannot jump to the conduction band at room temperature.
Semiconductors:
- Structure: Almost full Valence Band, almost empty Conduction Band.
- Band Gap: Small ( $E_g \approx 1 \text{ eV}$ , e.g., Silicon $1.1 \text{ eV}$ ).
- Conductivity: Thermal energy at room temperature is sufficient to excite some electrons across the gap.
Conductors (Metals):
- Structure: Valence Band and Conduction Band overlap OR the Conduction Band is partially filled.
- Band Gap: Zero ( $E_g = 0$ ).
- Conductivity: Electrons are free to move with minimal energy supply.

5. Concept of Effective Mass

An electron moving in a crystal lattice is subjected to internal periodic potentials from atoms, not just external fields. The effective mass ( $m^*$ ) is a parameter that accounts for these internal forces so the electron can be treated semi-classically.

Mathematical Definition: It is related to the curvature of the $E-k$ (Energy vs. Wave vector) curve.
$m^* = \hbar^2 \left( \frac{d^2E}{dk^2} \right)^{-1}$
Implication:
- High Curvature (Steep curve): Low effective mass (electron moves easily).
- Low Curvature (Flat curve): High effective mass (electron moves sluggishly).

Electrons and Holes

Electrons: Move in the conduction band. They have a positive effective mass near the bottom of the band.
Holes: Vacancies in the valence band. Near the top of the valence band, the effective mass of an electron is negative. To simplify, we treat this "missing electron" as a particle with positive mass and positive charge called a Hole.

6. Fermi Levels in Semiconductors

Intrinsic (Pure) Semiconductors

The number of electrons ( $n$ ) equals the number of holes ( $p$ ).
Fermi Level ( $E_{Fi}$ ): Lies exactly in the middle of the forbidden energy gap.
$E_{Fi} \approx \frac{E_c + E_v}{2}$

Extrinsic (Doped) Semiconductors

N-Type (Pentavalent impurity):
- Donates excess electrons.
- Fermi Level: Shifts upward, towards the Conduction Band ( $E_c$ ).
- Reason: Probability of finding an electron near the conduction band increases.
P-Type (Trivalent impurity):
- Creates excess holes.
- Fermi Level: Shifts downward, towards the Valence Band ( $E_v$ ).
- Reason: Probability of finding a hole (empty state) near the valence band increases.

7. Direct and Indirect Band Gap Semiconductors

The distinction lies in the conservation of crystal momentum ( $k$ ) during electron transition.

Feature	Direct Band Gap	Indirect Band Gap
Band Alignment	The minimum of the Conduction Band and maximum of the Valence Band occur at the same value of wave vector $k$ .	The minimum of the CB and maximum of the VB occur at different values of $k$ .
Transition	Electron falls directly from CB to VB.	Electron needs a change in momentum to fall (requires interaction with a phonon/lattice vibration).
Energy Release	Energy released mostly as Light (Photons).	Energy released mostly as Heat (Phonons).
Applications	LEDs, Laser Diodes.	Rectifiers, Transistors (not suitable for light emission).
Examples	Gallium Arsenide (GaAs), InP.	Silicon (Si), Germanium (Ge).

8. The Hall Effect

Definition: When a magnetic field is applied perpendicular to a current-carrying conductor, a potential difference (voltage) is developed across the conductor transverse to both the current and the magnetic field. This is called the Hall Voltage ( $V_H$ ).

Derivation

Consider a slab of semiconductor:

Current ( $I$ ) flows in $+x$ direction.
Magnetic Field ( $B$ ) is applied in $+z$ direction.
Electrons drift with velocity $v_d$ in $-x$ direction.

Lorentz Force: The magnetic force exerts a downward force on electrons:

$F_m = -e (v_d \times B)$
This causes electrons to accumulate on the bottom face, creating an electric field $E_H$ (Hall Field) in the $-y$ direction.
Equilibrium: The accumulation continues until the electric force balances the magnetic force.

$F_e = F_m$
$e E_H = e v_d B$
$E_H = v_d B$
Current Density ( $J$ ):

$J = n e v_d \implies v_d = \frac{J}{n e}$
Hall Coefficient ( $R_H$ ):
Substitute $v_d$ into the $E_H$ equation:

$E_H = \left( \frac{J}{n e} \right) B$
$E_H = R_H J B$
Therefore:
$R_H = \frac{1}{n e}$ (For electrons, $R_H$ is negative; for holes, positive).
Hall Voltage ( $V_H$ ):
If $w$ is the width of the specimen:

$V_H = E_H \cdot w$
$V_H = \frac{B I}{n e t}$
(Where $t$ is thickness).

Applications

Identify the type of semiconductor (N-type or P-type) based on the sign of $V_H$ .
Calculate carrier concentration ( $n$ ).
Calculate carrier mobility ( $\mu$ ).

9. Solar Cell Basics

A solar cell is a P-N junction diode that converts optical energy (light) directly into electrical energy via the Photovoltaic Effect.

Working Principle

Illumination: Light (photons with energy $h\nu > E_g$ ) strikes the P-N junction.
Generation: The energy is absorbed by electrons in the valence band, exciting them to the conduction band. This creates Electron-Hole Pairs (EHPs) in the depletion region.
Separation: The built-in electric field of the depletion region separates the charges:
- Electrons are swept toward the N-side.
- Holes are swept toward the P-side.
Collection: This accumulation of charges creates an EMF (Photo-voltage). If an external load is connected, current flows.

Key Characteristics (I-V Curve)

The solar cell operates in the fourth quadrant of the I-V characteristics (power generation).

Short Circuit Current ( $I_{sc}$ ): The current flowing when terminals are shorted ( $V=0$ ). Depends linearly on light intensity.
Open Circuit Voltage ( $V_{oc}$ ): The maximum voltage when terminals are open ( $I=0$ ).
Fill Factor (FF): A measure of the cell's quality (squareness of the I-V curve).