MM 765: Semiconductor Devices

Quantum Theory of Solids

"There is a geometry in the humming of the strings, there is a music in the spacing of the spheres."

In previous lectures, we successfully utilized Schrödinger's Equation to determine the allowed energy states, quantum numbers, and probability density functions of a single, non-interacting electron in a hydrogen atom.

The Ultimate Goal

To determine the properties of electrons within a crystal lattice.
To determine the statistical characteristics of a very large number of interacting electrons in a solid material.

Formation of Electronic Energy Bands

Extrapolating results from a single one-electron atom to a macroscopic crystal involves understanding wave function overlap. When two atoms are brought close together, their wave functions begin to interact. Due to Pauli's Exclusion Principle, the discrete energy state splits to accommodate the electrons without sharing the exact same quantum state. When several one-electron atoms are brought together into a periodic arrangement, this splitting cascades into a band of allowed energies.

The Quasi-Continuous Approximation

Consider a system containing \( 10^{19} \) one-electron atoms. If the width of the resulting energy band at equilibrium is 1 eV, the separation between individual levels is \( \sim 10^{-19} \text{ eV} \).

Because this energy difference \( 10^{-19} \text{ eV} \ll k_B T \) (ambient thermal energy, \( \sim 25 \text{ meV} \)), the distribution within a band is treated as quasi-continuous.

Isolated Atoms (Large r) Crystal (Small r)

Interactive: Drag slider to decrease interatomic distance \( r \). Discrete states split into continuous energy bands.

The Kronig-Penney Model

To solve Schrödinger's equation for a macroscopic crystal, we need a mathematical model of the potential energy \( V(x) \).

Potential Approximations

Potential of a single non-interacting atom: \( V(r) \propto -\frac{1}{r} \).

Mathematical Formulation

The Kronig-Penney model approximates the periodic crystal potential as a series of rectangular potential wells. Region I (\(0 < x < a\)) represents the space between ion cores where \(V(x)=0\). Region II (\(-b < x < 0\)) represents the repulsive potential barrier of the ion core where \(V(x)=V_0\).

Bloch's Theorem

Wave Vector (k):

All one-electron wave functions in a periodic potential must be a product of a plane wave and a function with the lattice periodicity. The periodic envelope \( u_k(x) \) explicitly depends on \( k \), causing the probability density \( |\psi|^2 \) to morph as the wave vector changes:

\psi(x) = u_k(x)e^{jkx} \implies |\psi(x)|^2 = |u_k(x)|^2

Interactive Waveform & Probability Density

Valence band state: Electron minimizes potential energy by concentrating charge density ON the atoms.

+ Ion Cores \( u_k(x) \) Envelope \( Re(\psi) \) Wave \( |\psi|^2 \) Probability

Applying boundary conditions (continuity of wave amplitude and derivative) yields a transcendental equation that reveals only a band of allowed energies.

                    \[ \frac{\gamma^2 - \alpha^2}{2\alpha\gamma} \sin(\alpha a)\sinh(\gamma b) + \cos(\alpha a)\cosh(\gamma b) = \cos(k(a+b)) \]
                

Graphical Solution (Dirac Delta Approximation)

Kronig and Penney approximated the barriers as Dirac delta functions. Letting barrier width \( b \to 0 \) and height \( V_0 \to \infty \) such that the product \( bV_0 \) remains finite simplifies the complex determinant into:

P' \frac{\sin(\alpha a)}{\alpha a} + \cos(\alpha a) = \cos(ka)

Where \( P' = \frac{mV_0 b a}{\hbar^2} \) is a measure of the "binding strength" or "scattering power" of the potential barrier.

Adjust Binding Strength (P')

The RHS is \(\cos(ka)\), which is strictly bounded between +1 and -1. Therefore, energy values (represented by \(\alpha a\)) are only allowed when the LHS function \(f(\alpha a)\) falls within the shaded blue regions.

Analyzing the Intercepts: The boundaries of these allowed bands occur precisely where \(f(\alpha a) = \pm 1\). As shown in the annotations below, the right edge of the \(n\)-th allowed band always occurs exactly at \(\alpha a = n\pi\) (corresponding to \(ka = n\pi\)). The left edge occurs where the oscillating function crosses the bound, mapped to \(ka = (n-1)\pi\).

P' = 0 (Free Electron) Tight Binding

Current P' 5.0

The E-k Diagram & Brillouin Zones

Schematic Generation of E-k Diagram

To understand the origin of the band structure, we dynamically map the allowed regions from the Kronig-Penney model directly to the E-k space. By plotting the Kronig-Penney function with energy \( E \propto (\alpha a)^2 \) on the shared vertical axis, we can horizontally project the allowed states (\( -1 \le f(\alpha a) \le 1 \)) directly to construct the Brillouin zones.

Note on Energy Gaps: In real crystal lattices, the higher Fourier components of the periodic potential decay. Consequently, as energy increases, the allowed bands rapidly widen, and the forbidden energy gaps strictly shrink (\( E_{g1} > E_{g2} > E_{g3} \)). Let the animation finish below to view the exact measurements proving this phenomenon.

Press Play to construct bands

By plotting the allowed energy values \( E \) against the wave vector \( k \), we visualize the completed band structure of the material.

The Free Particle (\( V_0 = 0 \))

If there is no potential barrier, \( P' = 0 \). The Kronig-Penney equation simplifies to \( \cos(\alpha a) = \cos(ka) \implies \alpha = k \). Since \( \alpha^2 = \frac{2mE}{\hbar^2} \):

E = \frac{\hbar^2 k^2}{2m} = \frac{p^2}{2m}

This maps to a continuous, gapless parabola in the E-k space.

Periodic Lattice (\( V_0 > 0 \))

Discontinuities (band gaps) occur exactly at boundaries where \( ka = \pm n\pi \). At these points, the curve flattens such that \( dE/dk = 0 \).
The variable \( p = \hbar k \) represents the crystal momentum—a constant of motion incorporating crystal interactions, not just mechanical momentum.
Because \( \cos(ka) = \cos(ka \pm 2n\pi) \), periodicity allows "folding" all states back into the First Brillouin Zone \( [-\frac{\pi}{a}, \frac{\pi}{a}] \).

k-Space Representations

Free Particle: Continuous parabolic relationship \( E = \hbar^2 k^2 / 2m \). No band gaps.

Electric Conduction & Drift Current

At T = 0 K

Valence band is completely full; conduction band is empty. Electrons are entirely immobile (unless a massive breakdown field is applied).

At T > 0 K

Thermal energy prompts electrons to jump into the CB, leaving positively charged "holes" in the VB. Both act as mobile charge carriers.

Drift Under Electric Field

With external field \( \mathcal{E} \), carriers acquire a net drift velocity (\( v_d \)). The net current density is:

J = q \sum_{i=1}^{N} v_{i} = -e \sum_{i=1}^{N} v_{i} = q N v_d

Thermal Energy (T) 0 K

Electric Field (E) 0 V/cm

Conduction Band (CB)

Valence Band (VB)

References & Web Resources

Textbooks

Semiconductor Physics and Devices (4th Ed) Donald A. Neamen. (Refer to Chapter 3 for detailed diagrams).
Solid State Electronic Devices (7th Ed) Ben G. Streetman & S. K. Banerjee. Pearson, 2016.

Web Links

Wikipedia: Kronig-Penney Model Detailed mathematical formulation and variations.
HyperPhysics: Energy Bands Excellent conceptual overview of energy bands and Bloch's Theorem.

Practice Assessment

Lecture 3 Knowledge Check

Put your understanding of the Quantum Theory of Solids, Kronig-Penney Model, E-k diagrams, Brillouin zones and Drift Current to the test.

Start Assessment