7.3 Tunneling and barrier penetration

Quantum tunneling is a mind-bending phenomenon where particles sneak through barriers they shouldn't be able to cross. It's like a magic trick that defies classical physics, allowing electrons to teleport through walls and alpha particles to escape atomic nuclei.

This weird quantum behavior has huge real-world impacts. From making electronics work to enabling radioactive decay, tunneling is everywhere. Understanding it helps us grasp the bizarre quantum world and harness its power in technology.

Quantum Tunneling Phenomenon

Quantum Mechanical Phenomenon

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• Quantum tunneling is a quantum mechanical phenomenon where a particle can penetrate through a potential barrier that it classically could not surmount
• In classical mechanics, a particle with energy less than the potential barrier height is unable to pass through the barrier
• However, in quantum mechanics, the wave function of the particle extends through the barrier, allowing for a non-zero probability of the particle being found on the other side

Implications in Physical Systems

• Quantum tunneling has significant implications in various physical systems
  • In semiconductors, tunneling enables the operation of devices like tunnel diodes and resonant tunneling diodes (RTDs)
  • In scanning tunneling microscopy (STM), tunneling current between a sharp probe tip and a sample surface allows for high-resolution imaging of the surface at the atomic level
  • In nuclear physics, alpha decay occurs due to the tunneling of alpha particles (two protons and two neutrons) through the potential barrier of the nucleus
  • In chemical reactions, tunneling can enable reactions that would be classically forbidden, leading to deviations from the Arrhenius equation and faster reaction rates

Tunneling Probability Calculation

WKB Approximation

• The Wentzel-Kramers-Brillouin (WKB) approximation is a semi-classical method for calculating the transmission probability and tunneling current through potential barriers
• The WKB approximation assumes that the potential barrier varies slowly compared to the wavelength of the particle, allowing for a semi-classical treatment
• The transmission probability, T, through a potential barrier V(x) in the WKB approximation is given by:
  • $T ≈ exp(-2/ħ ∫√(2m(V(x)-E))dx)$, where m is the particle mass, E is the particle energy, and the integral is taken over the classically forbidden region
• The WKB approximation provides accurate results for slowly varying potential barriers but breaks down for rapidly varying or high-energy barriers

Tunneling Current Calculation

• The tunneling current, I, through a potential barrier can be calculated using the Landauer formula:
  • $I = (2e/h) ∫T(E)dE$, where e is the electron charge, h is Planck's constant, and the integral is taken over the energy range of interest
• The Landauer formula relates the tunneling current to the transmission probability, T(E), which can be calculated using the WKB approximation or by solving the Schrödinger equation numerically for specific barrier shapes

Tunneling Probability Dependence

Barrier Height and Width

• The tunneling probability through a potential barrier is strongly dependent on the barrier height and width
• Barrier height: As the barrier height increases, the tunneling probability decreases exponentially since a higher barrier requires more energy for the particle to penetrate through, reducing the likelihood of tunneling
• Barrier width: The tunneling probability decreases exponentially with increasing barrier width because a wider barrier means the particle has to traverse a longer distance in the classically forbidden region, reducing the probability of tunneling

Particle Energy

• The tunneling probability increases with increasing particle energy
• A higher-energy particle has a shorter wavelength, allowing it to penetrate more easily through the barrier
• The dependence of tunneling probability on particle energy can be quantified using the WKB approximation or by solving the Schrödinger equation numerically for specific barrier shapes
• Understanding the dependence of tunneling probability on barrier characteristics and particle energy is crucial for designing and optimizing devices that rely on quantum tunneling, such as tunnel diodes and scanning tunneling microscopes (STMs)

Applications of Quantum Tunneling

Scanning Tunneling Microscopy (STM)

• STM uses the principle of quantum tunneling to image surfaces at the atomic level
• A sharp conducting tip is brought close to the sample surface, and a bias voltage is applied
• The resulting tunneling current between the tip and the surface is measured, providing information about the surface topography and electronic structure with sub-angstrom resolution
• STM has revolutionized the field of surface science by enabling the visualization and manipulation of individual atoms and molecules on surfaces

Radioactive Decay and Alpha Decay

• Alpha decay is a type of radioactive decay in which an atomic nucleus emits an alpha particle (two protons and two neutrons)
• The alpha particle is initially confined within the nucleus by the strong nuclear force but can tunnel through the potential barrier and escape the nucleus
• The probability of alpha decay depends on the barrier height and width, which are determined by the nuclear properties
• Quantum tunneling plays a crucial role in understanding the stability and decay rates of radioactive isotopes

Semiconductor Devices

• Tunnel diodes are semiconductor devices that exploit quantum tunneling to achieve negative differential resistance
• They consist of heavily doped p-n junctions, where electrons can tunnel through the narrow depletion region
• Tunnel diodes find applications in high-frequency oscillators, amplifiers, and switching circuits
• Resonant tunneling diodes (RTDs) are another type of semiconductor device that utilizes quantum tunneling through double potential barriers, resulting in resonant tunneling and negative differential resistance characteristics

Superconducting Devices

• Josephson junctions are superconducting devices that consist of two superconductors separated by a thin insulating barrier
• Cooper pairs (paired electrons in a superconductor) can tunnel through the barrier, leading to the Josephson effect
• Josephson junctions are used in superconducting quantum interference devices (SQUIDs) for sensitive magnetic field measurements and in quantum computing as qubits
• Quantum tunneling is a fundamental mechanism in superconducting devices and plays a crucial role in their operation and applications