7.3 Tunneling and barrier penetration
4 min read•july 30, 2024
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 , the 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 and (RTDs)
- In (STM), between a sharp probe tip and a sample surface allows for high-resolution imaging of the surface at the atomic level
- In nuclear physics, 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 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:
- , 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:
- , where e is the electron charge, h is , 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 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
- are superconducting devices that consist of two superconductors separated by a thin insulating barrier
- (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
Key Terms to Review (25)
Alpha Decay: Alpha decay is a type of radioactive decay in which an unstable atomic nucleus emits an alpha particle, consisting of two protons and two neutrons, effectively transforming into a different element. This process occurs because the original nucleus seeks stability by reducing its mass and atomic number, leading to the emission of the alpha particle. The phenomenon is closely related to tunneling and barrier penetration, as the alpha particle must overcome an energy barrier to escape the nucleus.
Barrier Penetration: Barrier penetration refers to the quantum phenomenon where a particle can pass through a potential energy barrier, even if its energy is lower than the height of the barrier. This effect is a direct result of the wave-like nature of particles in quantum mechanics, allowing for a probability of transmission rather than strictly adhering to classical mechanics where such an event would be impossible. Barrier penetration is crucial in understanding various processes like tunneling in quantum systems and has significant implications in fields such as nuclear physics and semiconductor technology.
Classical limit: The classical limit refers to the conditions under which quantum mechanical systems behave like classical systems, typically when quantum effects become negligible. In this limit, the probabilistic nature of quantum mechanics transitions into deterministic laws that govern classical physics, allowing for an intuitive understanding of physical phenomena such as tunneling and barrier penetration.
Cooper pairs: Cooper pairs are pairs of electrons that are bound together at low temperatures in a superconductor, allowing them to move without resistance. This phenomenon is a fundamental aspect of superconductivity, where these paired electrons form a collective ground state that enables the flow of electric current without energy loss. The concept of Cooper pairs was introduced by Leon Cooper in 1956 and is crucial for understanding the behavior of superconducting materials.
Decay Rate: Decay rate refers to the probability per unit time that a quantum particle will transition from a state of higher energy to a lower energy state, resulting in the emission of a particle or radiation. In the context of tunneling and barrier penetration, the decay rate is crucial for understanding how particles can pass through energy barriers that they classically shouldn't be able to cross, illustrating the non-intuitive nature of quantum mechanics.
Finite potential barrier model: The finite potential barrier model is a quantum mechanical concept that describes the behavior of particles encountering a potential barrier that is not infinitely high. It illustrates how particles can penetrate barriers of finite height and width, leading to phenomena such as tunneling, where particles have a probability of passing through barriers that would be classically forbidden. This model plays a crucial role in understanding the quantum behavior of particles at small scales, especially in contexts like semiconductor physics and nuclear fusion.
Josephson Junctions: Josephson junctions are quantum mechanical devices made from two superconductors separated by a thin insulating barrier, allowing for the phenomenon of quantum tunneling. This unique setup enables the flow of supercurrent without any voltage across the junction, which is pivotal in various applications such as superconducting qubits and sensitive magnetic field measurements. The behavior of Josephson junctions can be explained through the principles of tunneling and barrier penetration, where the quantum properties of particles allow them to cross energy barriers they would classically not be able to surmount.
Non-classical behavior: Non-classical behavior refers to phenomena in quantum mechanics that cannot be explained by classical physics, showcasing the unique and often counterintuitive aspects of quantum systems. This behavior includes effects such as superposition, entanglement, and tunneling, which challenge our conventional understanding of particles and their interactions, particularly when they encounter potential barriers.
Nuclear fusion: Nuclear fusion is the process where two light atomic nuclei combine to form a heavier nucleus, releasing a significant amount of energy in the process. This reaction is the source of energy for stars, including our sun, as it converts hydrogen into helium and emits energy that powers stellar processes. Fusion has the potential for clean energy production on Earth, but requires extreme conditions to overcome the repulsive forces between positively charged nuclei.
Planck's constant: Planck's constant is a fundamental physical constant that relates the energy of a photon to the frequency of its electromagnetic wave, represented as $$h \approx 6.626 \times 10^{-34} \, \text{Js}$$. This constant is crucial for understanding quantum mechanics and explains phenomena like tunneling, energy transitions, and molecular vibrations. Its significance extends across various areas, linking energy quantization to wave-particle duality in quantum systems.
Quantum barrier model: The quantum barrier model is a theoretical framework used to describe the behavior of particles as they encounter potential energy barriers that they classically should not be able to cross. This model is crucial for understanding tunneling phenomena, where particles can 'leak' through barriers due to their wave-like properties. The model illustrates how quantum mechanics allows for the probability of a particle being found on the other side of a barrier, even when it doesn't have enough energy to overcome it classically.
Quantum Mechanics: Quantum mechanics is the fundamental theory in physics that describes the behavior of matter and energy at the smallest scales, such as atoms and subatomic particles. It introduces concepts like wave-particle duality and quantization, fundamentally altering our understanding of physical phenomena and allowing for advanced applications in various fields, including atomic structures, tunneling effects, and the intrinsic properties of particles like spin.
Quantum Tunneling: Quantum tunneling is a phenomenon where a particle passes through a potential energy barrier that it classically should not be able to cross due to insufficient energy. This occurs because, at a quantum level, particles exhibit wave-like behavior, allowing them to exist in a superposition of states and have a probability of being found on the other side of the barrier, even when their energy is lower than the barrier's height.
Reduced Planck's Constant: Reduced Planck's constant, denoted as \(\hbar\), is a fundamental physical constant that is equal to Planck's constant divided by \(2\pi\). It plays a critical role in quantum mechanics, particularly in the formulation of wave functions and the uncertainty principle, allowing for the description of particle behavior at the quantum level, including phenomena such as tunneling and barrier penetration.
Resonant tunneling diodes: Resonant tunneling diodes (RTDs) are semiconductor devices that exploit the quantum mechanical phenomenon of tunneling to allow current to flow through a potential energy barrier at specific energy levels. These diodes feature a double-barrier structure that creates quantized energy states, enabling electrons to tunnel through the barriers when their energy matches these states, leading to negative differential resistance and unique current-voltage characteristics.
Scanning Tunneling Microscopy: Scanning tunneling microscopy (STM) is a powerful imaging technique that allows scientists to visualize surfaces at the atomic level by measuring the tunneling current that occurs when a sharp metallic tip is brought very close to a conductive or semiconductive surface. This method relies on the quantum mechanical phenomenon of tunneling, where electrons can 'tunnel' through a barrier instead of going over it, enabling detailed examination of the surface topography and electronic properties.
Schrödinger Equation: The Schrödinger Equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is central to understanding wave functions, which represent the probabilities of finding a particle in various states, and it plays a crucial role in explaining phenomena such as tunneling and the behavior of particles in potential barriers, as well as in approximations used to simplify complex molecular systems.
Superposition: Superposition is a fundamental principle in quantum mechanics stating that a quantum system can exist in multiple states simultaneously until it is measured or observed. This idea is essential in understanding how particles can behave like both waves and particles, and it plays a crucial role in various phenomena and applications in quantum physics.
Transmission Coefficient: The transmission coefficient is a measure of the likelihood that a particle will tunnel through a potential barrier rather than reflecting off it. This concept is crucial in understanding quantum tunneling, where particles can penetrate barriers they classically shouldn't be able to cross. It is mathematically expressed as the ratio of the probability current of transmitted particles to that of incident particles.
Tunnel Diodes: Tunnel diodes are a type of semiconductor device that exhibit negative resistance due to quantum tunneling, allowing them to conduct electricity in the reverse direction. This unique property enables tunnel diodes to operate at very high speeds and makes them useful in high-frequency applications, such as oscillators and amplifiers. The ability to tunnel through potential energy barriers is crucial in understanding how these devices function in electronic circuits.
Tunneling current: Tunneling current refers to the phenomenon where charged particles, such as electrons, pass through a potential energy barrier that they classically shouldn't be able to surmount. This occurs due to the wave-like properties of particles in quantum mechanics, allowing them to 'tunnel' through barriers, leading to measurable currents in devices like tunnel diodes and scanning tunneling microscopes.
Uncertainty Principle: The uncertainty principle is a fundamental concept in quantum mechanics that states it is impossible to precisely measure certain pairs of properties, such as position and momentum, simultaneously. This principle reflects the intrinsic limitations of measurement at the quantum level, emphasizing the wave-particle duality of matter and how measurement affects the state of a system.
Wave Function: A wave function is a mathematical function that describes the quantum state of a particle or system of particles. It encodes all the information about a system's position, momentum, and other properties, allowing us to predict the probabilities of various outcomes when measurements are made. The wave function is central to understanding how quantum systems evolve over time and how they behave under different conditions.
Wave-particle duality: Wave-particle duality is the concept in quantum mechanics that describes how every particle or quantum entity exhibits both wave-like and particle-like properties. This duality is fundamental to understanding phenomena like interference and tunneling, where particles can behave as waves, passing through barriers they would typically not be able to cross, and also exhibit distinct particle characteristics when observed or measured.
WKB approximation: The WKB approximation is a semi-classical method used to find approximate solutions to quantum mechanical problems, particularly in cases where the potential changes slowly. This technique connects classical mechanics and quantum mechanics, allowing for the analysis of phenomena like tunneling, where particles can pass through barriers they classically shouldn't be able to, and calculating scattering processes in slowly varying potentials.