ShaneLogic/3D Lattice Tunneling

3D Lattice Tunneling

When to Use This Skill

Apply this skill when:

• Analyzing electron tunneling through planar barriers in 3D crystal structures
• Converting 1D tunneling transmission probabilities to 3D lattice scenarios
• Accounting for momentum conservation effects on tunneling rates
• Calculating reduction factors due to perpendicular momentum distribution

Prerequisites

• Understanding of momentum conservation in crystals
• Foundation in 1D tunneling theory
• Planar barrier geometry

Constraints

• Assumes planar barrier geometry only
• Valid for electrons in 3D crystalline materials

Procedure

Step 1: Apply Momentum Conservation Principle

Recognize that in a 3D lattice, momentum components perpendicular to the tunneling direction are conserved. Only the momentum component in the tunneling direction decreases exponentially through the barrier.

Step 2: Calculate Perpendicular Momentum Energy

Compute the energy associated with momentum perpendicular to tunneling:

E_perp = (ℏ² × k_perp²) / (2m)

where:

• ℏ = reduced Planck constant
• k_perp = wavevector component perpendicular to tunneling
• m = electron mass

Step 3: Calculate Mean Tunneling Energy

Determine the mean energy of the tunneling electron:

E_bar = e × F × a

where:

• e = electron charge
• F = electric field strength
• a = lattice constant

Step 4: Compute Reduction Factor

Calculate the reduction factor η representing the fraction of electrons with favorable momentum distribution:

η = 1 / (1 + (e × F × a) / (4 × E))

This integrated result accounts for the distribution of perpendicular momenta across the electron population.

Step 5: Calculate 3D Transmission Probability

Combine the reduction factor with the 1D transmission probability:

Te_3D = η × Te_1D

For a flat plate parabolic barrier, the complete expression becomes:

Te_3D = [1 / (1 + (e × F × a) / (4 × E))] × exp(-π × (ΔE)² / (2 × ℏ × e × F))

where ΔE is the energy difference across the barrier.

Key Variables

| Variable | Type | Description | |----------|------|-------------| | η | Factor | Reduction factor due to momentum distribution | | E_perp | Energy | Energy of perpendicular momentum component | | E_bar | Energy | Mean energy of tunneling electron | | k_perp | Wavenumber | Wavevector perpendicular to tunneling direction | | Te_3D | Probability | 3D transmission probability | | Te_1D | Probability | 1D transmission probability |

Output

Returns the reduced transmission probability Te_3D that accounts for 3D momentum distribution effects in crystal lattice tunneling.

Reference

Based on calculations by Moll (1964).