Quantum Tunneling

4 min readNov 10, 2023

In the quantum world, particles can behave in ways that seem to violate the classical laws of physics — a phenomenon exemplified by quantum tunneling. This process allows subatomic particles like electrons to pass through potential barriers that they seemingly do not have the energy to overcome. This is a counterintuitive phenomenon where particles like electrons traverse barriers that, according to classical physics, they shouldn’t be able to cross due to insufficient energy. This principle is not just a theoretical curiosity; it has practical applications, particularly in the field of semiconductors.

Quantum tunneling reveals the limitations of classical mechanics at atomic and subatomic scales. Particles such as electrons are also waves, and their position is not defined by a single point but by a wave function. This wave function, when squared, gives the probability of the particle’s location.

Quantum tunneling is one of the differences between the classical world and the quantum realm.

When an electron encounters a potential energy barrier that it does not have the energy to overcome in the classical sense, its wave function doesn’t abruptly go to zero at the barrier. Instead, it decreases exponentially within the barrier and may continue on the other side. This non-zero probability on the other side of the barrier allows for the possibility of the electron ‘tunneling’ through the barrier.

Heisenberg’s uncertainty principle supports this phenomenon, allowing for a temporary and spontaneous ‘borrowing’ of energy, enabling the electron to overcome the barrier without violating energy conservation laws, as this borrowed energy is within the quantum uncertainty limits for energy and time.

The Scenario: Quantum Tunneling in Semiconductors

To illustrate, consider the following example problem that might be encountered in a quantum physics course:

An electron with a kinetic energy of 2 electron volts (eV) approaches a potential energy barrier with a height of 4 eV and a width of 0.5 nano-meters (nm). Classically, the electron cannot pass over the barrier since it has less energy than the height of the barrier. However, quantum mechanics tells us there’s still a chance the electron can tunnel through.

Solving the Quantum Conundrum

To calculate the probability of the electron tunneling through the barrier, physicists turn to a formula derived from the Schrödinger equation. This formula states that the tunneling probability, T, depends on the barrier’s width and the difference in energy between the electron and the barrier:

where α is given by:

Here, m is the mass of the electron, V is the potential energy of the barrier, E is the kinetic energy of the electron, and ℏ is the reduced Planck constant.

To solve this, the energies are converted from electron volts to joules, and the width from nano-meters to meters, since the standard units for these constants are kilograms (kg), joules (J), and meters (m).

The Calculation Step-by-Step

The code to calculate this probability is written in Python, using the NumPy library for its exponential function. Here’s the code used to solve this problem:

import numpy as np

# Constants
m = 9.11e-31  # mass of an electron in kg
h_bar = 1.055e-34  # reduced Planck's constant in J*s
eV_to_J = 1.602e-19  # conversion factor from eV to J

# Given values in electron volts
E_eV = 2  # kinetic energy of the electron in eV
V_eV = 4  # height of the potential barrier in eV
width_nm = 0.5  # width of the barrier in nanometers

# Convert eV to Joules
E_J = E_eV * eV_to_J
V_J = V_eV * eV_to_J

# Convert nm to meters
width_m = width_nm * 1e-9

# Calculate alpha (the decay constant of the wave function inside the barrier)
alpha = (2 * m * (V_J - E_J))**0.5 / h_bar

# Calculate the tunneling probability T
T = np.exp(-2 * alpha * width_m)

The Result: Probability Unveiled

The output of the code gives us the tunneling probability of approximately 0.000716, or 0.0716%. This small probability demonstrates the non-zero chance that the electron could tunnel through the barrier.

Implications in Technology

This quantum mechanical effect is not just theoretical; it has practical applications in modern technology. For example, it’s the principle behind the operation of tunnel diodes and flash memory. Understanding quantum tunneling is essential for developing smaller, faster, and more efficient semiconductor devices.

While quantum tunneling is harnessed in various applications, it also presents challenges. As semiconductor devices scale down to nano-meter sizes, unwanted tunneling can occur where it is not designed to, such as between transistor gates and channels in microprocessors. This leads to leakage currents, which can cause power loss and heat generation, compromising the efficiency and longevity of the device.

The phenomenon of quantum tunneling continues to inspire new technologies. Quantum dots, for example, are semiconductor particles small enough to exhibit quantum mechanical properties, including tunneling. They have potential applications in quantum computing, displays, and solar cells.

Researchers are also investigating tunnel field-effect transistors (TFETs) as an alternative to conventional transistors. TFETs utilize tunneling to switch on and off, promising to greatly reduce power consumption in electronic circuits.