3d-lattice-tunneling

name	3D Lattice Tunneling
description	Calculate electron transmission probability through planar barriers in 3D crystals by applying momentum conservation principles. Use when analyzing tunneling in three-dimensional lattices, when perpendicular momentum conservation must be accounted for, or when converting 1D tunneling results to 3D scenarios involving crystal structures.

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).