Electrical Conductors and Electronic Band Theory

Table of Contents

Classification of Materials by Electrical Conductivity

Fundamental Categories

Materials can be classified into four primary categories based on their electrical conductive properties:

1. Solids

• Metals: High conductivity ($10^6 - 10^8$ S/m)
• Semiconductors: Variable conductivity ($10^{-6} - 10^3$ S/m)
• Insulators: Very low conductivity ($< 10^{-10}$ S/m)

2. Liquids

• Electrolytes: Ionic conductivity
• Liquid metals: Electronic conductivity
• Pure liquids: Generally insulating

3. Gases

• Ionized gases: Can be highly conductive
• Most powerful electrical switches due to rapid ionization/deionization
• Plasma: Fourth state of matter with free electrons and ions

4. Plasma

• Fully ionized gas with equal numbers of positive and negative charges
• Extremely high conductivity approaching that of metals

Historical Significance: The Germanium Case Study

The development of germanium semiconductors revolutionized electronics and radar technology:

• Purity Requirements: Small amounts of impurities (parts per billion) dramatically alter conductivity
• Doping Effect: Controlled addition of impurities can increase conductivity by orders of magnitude
• Radar Technology: Germanium diodes became essential components in radar receivers during WWII
• Foundation of Electronics: Led to the development of transistors and modern semiconductor devices

Conductivity Range: Pure germanium has resistivity ~0.5 Ω·m, but doped germanium can achieve resistivities as low as $10^{-4}$ Ω·m.

Quantum Mechanical Description of Electrons in Crystals

Fundamental Assumptions

The quantum mechanical treatment of electrons in crystalline solids is based on several key assumptions:

1. Non-Free Electron Model

Electrons in crystals are not free particles but are constrained by the crystal structure and interactions.

2. Electrostatic Interactions

Electrons experience complex electrostatic potentials from:

• Lattice ions: Coulomb attraction to positive nuclei
• Other electrons: Coulomb repulsion and exchange interactions
• Core electrons: Screening effects from inner shell electrons

3. Periodic Potential

The total electrostatic potential has the same periodicity as the crystal lattice: $U(\mathbf{r}) = U(\mathbf{r} + \mathbf{R})$ where $\mathbf{R}$ is any lattice vector.

4. Band Structure Solutions

Solutions to the Schrödinger equation in this periodic potential yield electronic band structures, which determine all electronic properties of the solid.

The Tight-Binding Approximation

Physical Foundation

The tight-binding model provides intuitive understanding of how atomic orbitals combine to form electronic bands:

Isolated Atoms: When atoms are far apart (>> 1 nm), electrons are localized around individual nuclei with discrete energy levels determined by atomic quantum mechanics.

Crystal Formation: As atoms approach each other to form a solid (typical spacing ~0.3 nm for semiconductors), several phenomena occur:

1. Wavefunction Overlap: Electron wavefunctions from neighboring atoms begin to overlap significantly
2. Energy Level Splitting: Each atomic energy level splits into multiple closely-spaced levels
3. Band Formation: With $N$ atoms, each atomic level splits into $N$ closely-spaced levels forming a continuous band

Mathematical Description

For a system of $N$ identical atoms, an atomic orbital $\phi(\mathbf{r} - \mathbf{R}_i)$ centered at site $\mathbf{R}_i$ combines to form Bloch states:

$\psi_k(\mathbf{r}) = \frac{1}{\sqrt{N}} \sum_{i=1}^{N} e^{i\mathbf{k} \cdot \mathbf{R}_i} \phi(\mathbf{r} - \mathbf{R}_i)$

Energy Dispersion: The energy becomes k-dependent: $E(k) = E_{\text{atomic}} - t\sum_{\delta} e^{i\mathbf{k} \cdot \boldsymbol{\delta}}$ where $t$ is the hopping integral and $\boldsymbol{\delta}$ represents nearest-neighbor vectors.

Critical Interatomic Distance

Equilibrium Spacing: Most semiconductors form stable crystals at approximately 3 Å (0.3 nm) interatomic spacing, representing an optimal balance between:

• Attractive forces: Coulomb attraction, covalent bonding
• Repulsive forces: Pauli exclusion, core electron repulsion

Mathematical Framework: From Atoms to Bands

Potential Energy Landscapes

Single Atom Potential

For an isolated atom, the potential energy is: $U_{\text{atom}}(r) = -\frac{Ze^2}{4\pi\varepsilon_0 r}$ where $Z$ is the nuclear charge.

One-Dimensional Crystal Potential

For a 1D chain of atoms with lattice constant $a$: $U_{\text{crystal}}(x) = \sum_{n=-\infty}^{\infty} U_{\text{atom}}(x - na)$

Simplified Model Potential

At the limit of zero interatomic distance, the potential approaches a constant value rather than diverging to $-\infty$, leading to the nearly free electron model.

Wavefunction Solutions

The transition from atomic to crystalline wavefunctions involves:

1. Atomic Wavefunctions: $\psi_{\text{atomic}}(r) = R_{nl}(r)Y_l^m(\theta,\phi)$
2. Linear Combinations: Form Bloch states as linear combinations
3. Exponential Modulation: Include plane wave factors for momentum

The Bloch Theorem: Foundation of Band Theory

Statement of the Theorem

For a periodic potential $U(\mathbf{r}) = U(\mathbf{r} + \mathbf{R})$, where $\mathbf{R}$ is any lattice vector, the solutions to the Schrödinger equation have the form:

$\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} u_{n\mathbf{k}}(\mathbf{r})$

where $u_{n\mathbf{k}}(\mathbf{r})$ has the same periodicity as the lattice: $u_{n\mathbf{k}}(\mathbf{r} + \mathbf{R}) = u_{n\mathbf{k}}(\mathbf{r})$

Derivation and Physical Meaning

Schrödinger Equation with Periodic Potential

$-\frac{\hbar^2}{2m}\nabla^2\psi + U(\mathbf{r})\psi = E\psi$

Periodicity Condition

The key insight is that $U(\mathbf{r}) = U(\mathbf{r} + \mathbf{R})$ leads to: $\psi(\mathbf{r} + \mathbf{R}) = e^{i\mathbf{k} \cdot \mathbf{R}}\psi(\mathbf{r})$

Decomposition of the Wavefunction

The Bloch wavefunction contains two independent components:

1. Plane Wave Component: $e^{i\mathbf{k} \cdot \mathbf{r}}$

   • Represents the momentum of the electron
   • Provides translational symmetry
   • Slowly varying on the scale of the lattice

2. Periodic Component: $u_{n\mathbf{k}}(\mathbf{r})$

   • Has the periodicity of the lattice
   • Rapidly varying on atomic scales
   • Determines the band index $n$

Free Electron Limit

Special Case: Zero Potential

When $U(\mathbf{r}) = 0$, the Schrödinger equation becomes: $-\frac{\hbar^2}{2m}\nabla^2\psi = E\psi$

Solution

The solution is a pure plane wave: $\psi(\mathbf{r}) = A_0 e^{i\mathbf{k} \cdot \mathbf{r}}$

Substituting back: $-\frac{\hbar^2}{2m}(-k^2)A_0 e^{i\mathbf{k} \cdot \mathbf{r}} = EA_0 e^{i\mathbf{k} \cdot \mathbf{r}}$

Dispersion Relation

This yields the free electron dispersion relation: $E = \frac{\hbar^2 k^2}{2m} = \frac{p^2}{2m}$

de Broglie Relationship

The momentum-wavelength relationship is fundamental: $p = \frac{h}{\lambda} = \frac{h \cdot k}{2\pi} = \hbar k$

Kinetic Energy

For free electrons, all energy is kinetic: $E_{\text{kinetic}} = \frac{p^2}{2m} = E - U = E \quad \text{(since } U = 0\text{)}$

This dispersion relation connects momentum (or k-vector) to energy and is fundamental to understanding electronic properties.

Kronig-Penney Model: Exact Solution for Periodic Potentials

Model Setup

The Kronig-Penney model provides an exactly solvable example of electrons in a periodic potential, consisting of:

Periodic Square Well Potential:

\begin{tikzpicture}[scale=0.8]
\draw[thick,->] (-4,0) -- (4,0) node[right] {$x$};
\draw[thick,->] (-4,0) -- (-4,3) node[above] {$U(x)$};

% Draw the periodic potential
\draw[thick,blue] (-4,0) -- (-3,0) -- (-3,2) -- (-2,2) -- (-2,0) -- (-1,0) -- (-1,2) -- (0,2) -- (0,0) -- (1,0) -- (1,2) -- (2,2) -- (2,0) -- (3,0) -- (3,2) -- (4,2);

% Labels for potential heights
\node[left] at (-4,2) {$U_0$};
\node[left] at (-4,0) {$0$};

% Width labels
\draw[<->] (-2,-0.3) -- (-1,-0.3);
\node[below] at (-1.5,-0.3) {$a$};
\draw[<->] (-3,-0.5) -- (-2,-0.5);
\node[below] at (-2.5,-0.5) {$b$};
\draw[<->] (-3,-0.7) -- (-1,-0.7);
\node[below] at (-2,-0.7) {$P = a + b$};

% Potential labels
\node[blue] at (-2.5,1) {$U_0$};
\node[blue] at (-0.5,1) {$U_0$};
\node[blue] at (1.5,1) {$U_0$};
\end{tikzpicture}

Model Parameters:

• Potential Wells: Width $a$, potential $U = 0$
• Potential Barriers: Width $b$, potential $U = U_0 > 0$
• Period: $P = a + b$

Region-Specific Schrödinger Equations

Region I: Inside Wells ($0 < x < a$)

$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi$

Defining $\alpha^2 = \frac{2mE}{\hbar^2}$: $\frac{d^2\psi}{dx^2} + \alpha^2\psi = 0$

General Solution: $\psi_I(x) = A\sin(\alpha x) + B\cos(\alpha x)$

Region II: Inside Barriers ($-b < x < 0$)

$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + U_0\psi = E\psi$

Defining $\beta^2 = \frac{2m(U_0 - E)}{\hbar^2}$ (assuming $E < U_0$): $\frac{d^2\psi}{dx^2} - \beta^2\psi = 0$

General Solution: $\psi_{II}(x) = C e^{\beta x} + D e^{-\beta x}$

Boundary Conditions and Continuity

Continuity Requirements

1. Wavefunction Continuity: $\psi$ must be continuous at all interfaces
2. Derivative Continuity: $\frac{d\psi}{dx}$ must be continuous (ensuring current conservation)
3. Bloch Periodicity: $\psi(x + P) = e^{ikP}\psi(x)$
4. Derivative Periodicity: $\frac{d\psi}{dx}\Big|_{x+P} = e^{ikP}\frac{d\psi}{dx}\Big|_x$

Boundary Condition at $x = 0$

From continuity: $B = C + D$ $\alpha A = \beta(C - D)$

Periodic Boundary Conditions

These lead to a system of four linear equations in four unknowns $(A, B, C, D)$.

The Kronig-Penney Dispersion Relation

Final Result

After applying all boundary conditions and requiring non-trivial solutions, we obtain:

$-\frac{\alpha^2 + \beta^2}{2\alpha\beta}\sin(\alpha a)\sinh(\beta b) + \cos(\alpha a)\cosh(\beta b) = \cos(k(a+b))$

Physical Interpretation

• Left Side: Depends only on energy $E$ through $\alpha$ and $\beta$
• Right Side: Depends only on crystal momentum $k$
• Constraint: The right side must be between -1 and +1 for real solutions

Dimensionless Parameters

Introducing dimensionless variables:

• Reduced Energy: $\varepsilon = \frac{E}{U_0}$
• Characteristic Parameter: $\alpha_0 = \sqrt{\frac{2mU_0}{\hbar^2}}$

The dispersion relation becomes: $P(\varepsilon) = \cos(k(a+b))$

where $P(\varepsilon)$ is the left-hand side expressed in terms of $\varepsilon$.

Energy Band Structure

Allowed and Forbidden Bands

Allowed Bands: Energy ranges where $|P(\varepsilon)| \leq 1$

• Electrons can propagate with real $k$ values
• Form continuous energy bands

Forbidden Bands (Band Gaps): Energy ranges where $|P(\varepsilon)| > 1$

• No real solutions for $k$
• Electrons cannot exist at these energies

Band Edge Conditions

• Band Edge: Where $P(\varepsilon) = \pm 1$
• Band Center: Where $P(\varepsilon) = 0$
• Effective Mass: Determined by curvature $\frac{d^2E}{dk^2}$ near band edges

Connection to Real Materials

Limiting Cases

1. Free Electron Limit ($U_0 \to 0$): $E = \frac{\hbar^2 k^2}{2m}$

2. Strong Potential Limit ($U_0 \gg E$):

   • Tight-binding behavior
   • Narrow bands with large gaps

3. Nearly Free Electrons ($U_0$ small):

   • Band gaps open at Brillouin zone boundaries
   • Explains metal-insulator transitions

Physical Significance

The Kronig-Penney model demonstrates how:

• Periodic potentials naturally lead to band structures
• Energy gaps arise from wave interference effects
• Electronic properties depend on the relationship between Fermi energy and band structure

Brillouin Zones and Band Structure

Energy-Momentum Relations in Crystals

The energy-momentum relation $E(\mathbf{k})$ is only valid within specific regions of momentum space known as Brillouin Zones.

Brillouin Zone Definition

A Brillouin Zone is the primitive unit cell of the reciprocal lattice, constructed by:

1. Placing the origin at a reciprocal lattice point
2. Drawing vectors to nearest neighbor lattice points
3. Constructing perpendicular bisecting planes
4. The first Brillouin zone is the smallest volume enclosed by these planes

Crystallographic Symmetry Points and Directions

Notation Convention

Roman Letters: Denote high-symmetry points in k-space Greek Letters: Denote high-symmetry directions (lines) in k-space

FCC Brillouin Zone Structure

For face-centered cubic lattices (common in semiconductors), important symmetry points include:

• Γ (Gamma): Zone center at $\mathbf{k} = (0,0,0)$
• X: Zone boundary points along principal axes
• L: Zone boundary points along face diagonals
• K: Zone boundary points along edges

Direct vs. Indirect Band Gap Semiconductors

Direct Band Gap

Condition: The valence band maximum and conduction band minimum occur at the same k-value

• Usually both extrema are located at the Γ point
• Examples: GaAs, InP, GaN
• Optical Properties: Strong light absorption and emission

Indirect Band Gap

Condition: The valence band maximum and conduction band minimum occur at different k-values

• Silicon: Valence band maximum at Γ, conduction band minimum near X point
• Germanium: Valence band maximum at Γ, conduction band minimum at L point
• Optical Properties: Weaker optical transitions (require phonon assistance)

Significance for Electronic Devices

Energy Bands: Exist for all semiconductors, determining electrical and optical properties

Band Gaps:

• Present in semiconductors and insulators
• Absent in metals - this is why metals don’t have energy gaps like semiconductors