Unit 5 - Notes

PHY109

Unit 5: Solid State Physics

1. Free Electron Theory

1.1 Introduction

The free electron theory is used to explain the physical properties of metals, such as electrical and thermal conductivity. There are two main stages in the development of this theory:

Classical Free Electron Theory (Drude-Lorentz Model):
- Assumes that metals contain free electrons that move inside the metal like gas molecules in a container (an "electron gas").
- The mutual repulsion between electrons and the attraction between electrons and positive ions are ignored.
- It successfully explains Ohm's law and electrical conductivity but fails to explain the specific heat of metals and the temperature dependence of conductivity accurately.
Quantum Free Electron Theory (Sommerfeld Model):
- Treats electrons as quantum particles obeying Fermi-Dirac statistics.
- Electrons move in a constant potential within the metal boundaries.
- Energy levels are quantized (discrete) rather than continuous.
- It resolves many discrepancies of the classical model, particularly regarding specific heat.

1.2 Drift and Diffusion Current (Qualitative)

In solid-state devices, current flow arises from two distinct mechanisms:

Drift Current:
- Definition: The directed motion of charge carriers (electrons or holes) under the influence of an applied external electric field.
- Mechanism: When an electric field is applied, carriers accelerate but collide with lattice atoms, resulting in a constant average velocity called drift velocity ( $v_d$ ).
- Relationship: Current density $J_{drift} \propto$ Electric Field ( $E$ ).
Diffusion Current:
- Definition: The movement of charge carriers from a region of higher concentration to a region of lower concentration.
- Mechanism: This is a statistical process resulting from random thermal motion. No external electric field is required.
- Relationship: Current density $J_{diff} \propto$ Concentration Gradient ( $\frac{dn}{dx}$ ).

2. Statistical Mechanics in Solids

2.1 Fermi Energy ( $E_F$ )

Fermi energy is a critical concept in quantum mechanics referring to the energy of the electrons in a metal.

Definition at 0K: It is the maximum energy possessed by an electron at absolute zero temperature ( $0K$ ).
Physical Significance: It represents the boundary between occupied and unoccupied energy states at absolute zero. All states below $E_F$ are filled; all states above $E_F$ are empty.
Magnitude: In metals, $E_F$ is typically in the range of 3 eV to 10 eV.

2.2 Fermi-Dirac Distribution Function

The Fermi-Dirac distribution function, $f(E)$ , gives the probability that an energy state $E$ is occupied by an electron at a given temperature $T$ .

The Formula:

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

Where:

$E$ = Energy of the state
$E_F$ = Fermi Energy
$k_B$ = Boltzmann constant
$T$ = Absolute temperature

Behavior of $f(E)$ :

At T = 0K:
- If $E < E_F$ , the exponent is $-\infty$ , so $f(E) = 1$ (100% probability of occupation).
- If $E > E_F$ , the exponent is $+\infty$ , so $f(E) = 0$ (0% probability).
At T > 0K:
- If $E = E_F$ , $f(E) = \frac{1}{1+e^0} = \frac{1}{2}$ .
- Significance: At any temperature above absolute zero, the probability of finding an electron exactly at the Fermi level is 50%. Electrons near $E_F$ are thermally excited to higher energy levels.

2.3 Density of States (Qualitative)

The Density of States, $g(E)$ , represents the number of available electron energy states per unit volume per unit energy interval.

Before an electron can occupy an energy level, a state must exist at that energy.
Relationship: For a free electron gas in 3D, the density of states is proportional to the square root of the energy:
$g(E) \propto \sqrt{E}$
This implies that as energy increases, the available "seats" for electrons to occupy become more numerous.

3. Band Theory of Solids

3.1 Formation of Energy Bands

Isolated Atoms: Electrons occupy discrete, sharp energy levels.
Crystal Formation: When atoms are brought close together to form a solid, the outer electron orbits overlap. Due to the Pauli Exclusion Principle, discrete energy levels split into a range of closely spaced levels called Energy Bands.

3.2 Band Structure Terminology

Valence Band (VB): The band occupied by valence electrons (outermost shell). It is usually filled or partially filled.
Conduction Band (CB): The band above the valence band. Electrons here are free to move and contribute to conduction.
Forbidden Energy Gap ( $E_g$ ): The energy difference between the top of the valence band and the bottom of the conduction band. No electrons can exist in this region.

3.3 Classification of Solids

Based on the band gap ( $E_g$ ), solids are classified into:

Conductors (Metals):
- VB and CB overlap.
- $E_g \approx 0$ .
- Electrons can easily move to the conduction band.
- Example: Copper, Silver.
Semiconductors:
- Small forbidden gap.
- $E_g \approx 1$ eV (e.g., Silicon $\approx 1.1$ eV, Germanium $\approx 0.7$ eV).
- Behave as insulators at 0K but conduct at higher temperatures.
Insulators:
- Large forbidden gap.
- $E_g > 3$ eV (e.g., Diamond $\approx 5.4$ eV).
- Electrons cannot jump from VB to CB under normal conditions.

4. Semiconductors and Fermi Levels

4.1 Intrinsic Semiconductors

Definition: Pure semiconductors without impurities (e.g., pure Si).
Carriers: Number of electrons ( $n$ ) = Number of holes ( $p$ ).
Fermi Level: Located exactly in the middle of the forbidden energy gap.
$E_F = \frac{E_C + E_V}{2}$

4.2 Extrinsic Semiconductors

Doping (adding impurities) creates extrinsic semiconductors and shifts the Fermi level.

A. N-Type Semiconductor:

Dopant: Pentavalent impurity (Phosphorus, Arsenic). Donates extra electrons.
Donor Level: A new energy level ( $E_D$ ) forms just below the conduction band.
Fermi Level: Shifts upward, closer to the conduction band.
- At 0K, $E_F$ is exactly between $E_C$ and $E_D$ .

B. P-Type Semiconductor:

Dopant: Trivalent impurity (Boron, Gallium). Creates holes.
Acceptor Level: A new energy level ( $E_A$ ) forms just above the valence band.
Fermi Level: Shifts downward, closer to the valence band.
- At 0K, $E_F$ is exactly between $E_V$ and $E_A$ .

5. Advanced Band Concepts

5.1 Direct vs. Indirect Band Gap Semiconductors

The classification depends on the alignment of the minimum energy of the conduction band ( $E_{c,min}$ ) and the maximum energy of the valence band ( $E_{v,max}$ ) in "k-space" (momentum space).

Feature	Direct Band Gap	Indirect Band Gap
Momentum ( $k$ ) alignment	$k_{min}$ of CB aligns with $k_{max}$ of VB.	$k_{min}$ of CB does not align with $k_{max}$ of VB.
Electron Recombination	Electron falls directly to VB, releasing energy as a photon.	Electron needs a change in momentum (phonon interaction) to fall to VB.
Efficiency	High probability of light emission.	Energy is mostly released as heat (lattice vibration).
Application	LEDs, Laser Diodes.	Rectifiers, Transistors (not optical).
Examples	Gallium Arsenide (GaAs), InP.	Silicon (Si), Germanium (Ge).

5.2 Concept of Effective Mass ( $m^*$ )

An electron in a crystal moves under the influence of the periodic potential of the lattice atoms, not just the external electric field. It responds to external forces as if it had a mass different from the free electron mass ( $m_e$ ).

Definition: The mass that a particle seems to carry when responding to an applied force in a periodic lattice potential.
Mathematical Relation: It depends on the curvature of the Energy () vs. Wave vector () curve.
- High curvature (steep parabola) $\rightarrow$ Small effective mass (highly mobile).
- Low curvature (flat parabola) $\rightarrow$ Large effective mass (heavy).

Electrons vs. Holes:

Electrons: Near the bottom of the conduction band, curvature is positive, so $m^*$ is positive.
Holes: Near the top of the valence band, curvature is negative. A hole acts like a positive charge with a positive effective mass (representing the collective behavior of the remaining electrons).

6. Hall Effect

6.1 Introduction and Principle

Definition: When a magnetic field is applied perpendicular to a current-carrying conductor, a voltage is developed across the specimen in a direction perpendicular to both the current and the magnetic field. This is called the Hall Effect.
Generated Voltage: Hall Voltage ( $V_H$ ).
Significance: It determines whether a semiconductor is N-type or P-type and measures carrier concentration and mobility.

6.2 Derivation

Consider a rectangular slab of N-type semiconductor.

Width: $w$ (along y-axis)
Thickness: $t$ (along z-axis)
Current ( $I$ ): Flowing along X-axis.
Magnetic Field ( $B$ ): Applied along Z-axis.

Forces at play:

Lorentz Force ( $F_L$ ): The magnetic field exerts a downward force on drifting electrons.

$F_L = -e (v_d \times B)$
Magnitude: $F_L = B e v_d$ (acting downwards).
This causes electrons to accumulate at the bottom face, creating a negative charge, while the top face becomes positive.
Hall Electric Force ( $F_E$ ): The separation of charge creates an internal electric field ( $E_H$ ) directed from top to bottom.

$F_E = e E_H$ (acting upwards).

Equilibrium Condition:
Accumulation stops when the electric force balances the magnetic force.

F_E = F_L

e E_H = B e v_d

E_H = B v_d

--- (Equation 1)

Relationship to Hall Voltage:
The Hall field is related to Hall voltage $V_H$ across the width $w$ :

E_H = \frac{V_H}{w}

Substituting into Eq 1:

\frac{V_H}{w} = B v_d \implies V_H = B v_d w

Relationship to Current Density:
Current density $J$ is given by:

J = \frac{I}{Area} = \frac{I}{w \cdot t}

Also,

J = -n e v_d

(where

n

is electron density).

v_d = -\frac{J}{ne} = -\frac{I}{w t n e}

Final Hall Voltage Formula:
Substitute $v_d$ back into the $V_H$ equation:

V_H = B \left( \frac{I}{w t n e} \right) w

V_H = \frac{B I}{n e t}

6.3 Hall Coefficient ( $R_H$ )

The Hall coefficient is defined as the Hall field per unit current density per unit magnetic induction.

R_H = \frac{1}{ne}

Thus, the Hall Voltage can be written as:

V_H = R_H \frac{BI}{t}

For N-type, $R_H$ is negative.
For P-type, $R_H$ is positive.