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CRC Press eBooks

Tunneling

David K. Ferry

Published Aug 19, 2026• 0 citations
Publisher

Abstract

When we dealt with the finite potential well in the last chapter, it was observed that the wave function penetrated into the barriers. This process does not occur in classical mechanics, since a particle will actually bounce off the barrier. However, when we treat the particle as a wave, then the wave nature of barrier penetration can occur. This is familiar in electromagnetic waves, where the decaying wave (as opposed to a propagating wave) is termed an evanescent wave. For energies below the top of the barrier, the wave is attenuated, and it decays exponentially. Yet, it takes a significant distance for this decay to eliminate the wave completely. If the barrier is thinner than this critical distance, the evanescent wave can excite a propagating wave in the region beyond the barrier. In optics, this effect is termed frustrated total internal reflection . We are familiar with total internal reflection just from a swimming pool. Water has a higher dielectric constant than free space, so when we are under water, there is critical angle beyond which we can no longer see out (from under the water) of the pool. In optics, we can use a prism, and there will be reflection from the back surface for angles greater than a critical angle. The wave in back of the prism is attenuated and decaying. But, if we place another high dielectric constant material close to the prism, the decaying wave can excite a wave in this new material. 1 Thus, the wave can penetrate the barrier and continue to propagate, with an attenuated amplitude, in the trans-barrier region. For particles, this process is termed tunneling , with analogy to the miners who burrow through a mountain in order to get to the other side! In fact, it is known that the nuclear fusion process by which our sun produces its power would not occur were there no tunneling. The intra-nuclear repulsion is just too strong and tunneling is required for the nuclei to find one another.

DOI
10.1201/9781003717195-3

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When we dealt with the finite potential well in the last chapter, it was observed that the wave function penetrated into the barriers. This process does not occur in classical mechanics, since a particle will actually bounce off the barrier. However, when we treat the particle as a wave, then the wave nature of barrier penetration can occur. This is familiar in electromagnetic waves, where the decaying wave (as opposed to a propagating wave) is termed an evanescent wave. For energies below the top of the barrier, the wave is attenuated, and it decays exponentially. Yet, it takes a significant distance for this decay to eliminate the wave completely. If the barrier is thinner than this critical distance, the evanescent wave can excite a propagating wave in the region beyond the barrier. In optics, this effect is termed frustrated total internal reflection . We are familiar with total internal reflection just from a swimming pool. Water has a higher dielectric constant than free space, so when we are under water, there is critical angle beyond which we can no longer see out (from under the water) of the pool. In optics, we can use a prism, and there will be reflection from the back surface for angles greater than a critical angle. The wave in back of the prism is attenuated and decaying. But, if we place another high dielectric constant material close to the prism, the decaying wave can excite a wave in this new material. 1 Thus, the wave can penetrate the barrier and continue to propagate, with an attenuated amplitude, in the trans-barrier region. For particles, this process is termed tunneling , with analogy to the miners who burrow through a mountain in order to get to the other side! In fact, it is known that the nuclear fusion process by which our sun produces its power would not occur were there no tunneling. The intra-nuclear repulsion is just too strong and tunneling is required for the nuclei to find one another.
This chapter discusses energy levels for more realistic systems. It looks at a different class of problems involving the scattering of free particles from a potential. The chapter uses the free-particle wave functions to describe an important physical process – the scattering of a steady beam of particles from a barrier. To get the first one note that cannot step down for ever, for eventually the ground state will be reached. To construct the excited wave functions one must substitute a full polynomial for F in equation and equate the coefficients of all the powers to zero. The rapidly declining tail on a wave function which penetrates into a classically forbidden region of a barrier could, if the barrier were of finite thickness, emerge from the remote side and become wave-like again, representing the tunnelling of particles through the barrier.
Tunneling is an outstanding physical process. The observation that particles surmount a high barrier in spite of the fact that they don't have the necessary energy can not be explained by classical physics. However, this so called tunneling became allowed by quantum mechanics. Experimental tunneling studies with different photonic barriers from microwave frequencies up to ultraviolet frequencies point towards a universal tunneling time [1, 2]. Tunneling can be described by virtual, that is, by unobservable particles [3]. In the case of tunneling there is a virtual particle between the real incident and the real transmitted particles. Tunneling modes are solutions of the Schrödinger and of the Helmholtz equations. The most prominent example of the occurrence of tunneling modes in optics is frustrated total internal reflection (FTIR) at double prisms. In 1949 Sommerfeld [4] pointed out that this optical phenomenon represents the analogy of quantum mechanical particle tunneling. Recent experimental and theoretical data confirmed the conjecture that the tunneling process is characterized by a universal tunneling time presumably independent of the kind of field. The observed tunneling processes proceeded at a time of the order of magnitude of the reciprocal frequency of the wave packets.
N 1923, during the infancy of the quantum theory, de Broglie [ 11 introducted a new fundamental hypothesis that matter may be endowed with a dualistic natureparticles may also have the characteristics of waves. This hypothesis, in the hands of Schrodinger [2] found expression in the definite form now known as the Schrodinger wave equation, whereby an electron or a particle is assumed to be represented by a solution to this equation. The continuous nonzero nature of such solutions, even in classically forbidden regions of negative kinetic energy, implies an ability to penetrate such forbidden regions and a probability of tunneling from one classically allowed region to another. The concept of tunneling, indeed, arises from this quantum-mechanical result. The subsequent experimental manifestations of this concept can be regarded as one of the early triumphs of the quantum theory. In 1928, theoretical physicists believed that tunneling could occur by the distortion, lowering or thinning, of a potential barrier under an externally applied high electric field. Oppenheimer [3] attributed the autoionization of excited states of atomic hydrogen to the tunnel effect: The coulombic potential well which binds an atomic electron could be distorted by a strong electric field so that the electron would see a finite potential barrier through which it could tunnel. Fowler and Nordheim [4] explained, on the basis of electron tunneling, the main features of the phenomenon of electron emission from cold metals by high external electric fields, which had been unexplained since its observation by Lilienfeld [SI in 1922. They proposed a one-dimensional model. Metal electrons are confined by a potential wall whose height is determined by the work function 9 plus the Fermi energy E/, and the wall thickness is substantially decreased with an externally applied high electric field, allowing electrons to tunnel through the potential wall, as shown in Fig. 1. They successfully derived the well-known Fowler-Nordheim formula for the current as a function of electric field F: J = AF* exp [-4(2nr)1'*t$a'*/3kF]. An application of these ideas which followed almost immediately came in the model for a decay as a tunneling process put forth by Gamow [6] and Gurney and Condon [7]. Subsequently, Rice [8] extended this theory to the description of molecular dissociation. The next important development was an attempt to invoke tunneling in order to understand transport properties of electrical contacts between two solid conductors. The problems of metal-to-metal and semiconductor-to-metal contacts are important technically, because they are directly related to electrical switches and rectifiers or detectors.
N 1923, during the infancy of the quantum theory, de Broglie [ 11 introducted a new fundamental hypothesis that matter may be endowed with a dualistic natureparticles may also have the characteristics of waves. This hypothesis, in the hands of Schrodinger [2] found expression in the definite form now known as the Schrodinger wave equation, whereby an electron or a particle is assumed to be represented by a solution to this equation. The continuous nonzero nature of such solutions, even in classically forbidden regions of negative kinetic energy, implies an ability to penetrate such forbidden regions and a probability of tunneling from one classically allowed region to another. The concept of tunneling, indeed, arises from this quantum-mechanical result. The subsequent experimental manifestations of this concept can be regarded as one of the early triumphs of the quantum theory. In 1928, theoretical physicists believed that tunneling could occur by the distortion, lowering or thinning, of a potential barrier under an externally applied high electric field. Oppenheimer [3] attributed the autoionization of excited states of atomic hydrogen to the tunnel effect: The coulombic potential well which binds an atomic electron could be distorted by a strong electric field so that the electron would see a finite potential barrier through which it could tunnel. Fowler and Nordheim [4] explained, on the basis of electron tunneling, the main features of the phenomenon of electron emission from cold metals by high external electric fields, which had been unexplained since its observation by Lilienfeld [SI in 1922. They proposed a one-dimensional model. Metal electrons are confined by a potential wall whose height is determined by the work function 9 plus the Fermi energy E/, and the wall thickness is substantially decreased with an externally applied high electric field, allowing electrons to tunnel through the potential wall, as shown in Fig. 1. They successfully derived the well-known Fowler-Nordheim formula for the current as a function of electric field F: J = AF* exp [-4(2nr)1'*t$a'*/3kF]. An application of these ideas which followed almost immediately came in the model for a decay as a tunneling process put forth by Gamow [6] and Gurney and Condon [7]. Subsequently, Rice [8] extended this theory to the description of molecular dissociation. The next important development was an attempt to invoke tunneling in order to understand transport properties of electrical contacts between two solid conductors. The problems of metal-to-metal and semiconductor-to-metal contacts are important technically, because they are directly related to electrical switches and rectifiers or detectors.
Quantum tunneling is a phenomenon in which an atom or a subatomic particle appears on the other side of a potential barrier that should be impenetrable to the particle, given the difference between the kinetic energy of the particle and the potential energy of the barrier, the former being significantly lesser. However, if the barrier shrinks enough the particle will be able to tunnel right through it. This is among the most well-known quantum physics phenomena. Historically, physicists struggled to grasp the wavelike and particle-like duality of light and subatomic particles. The aforementioned phenomenon occurs due to the wavelike behavior of photons. Quantum phenomena, such as quantum tunneling, might appear quite convoluted, therefore a simple experimental demonstration was performed to highlight the basic mechanisms of the phenomenon in a manner comprehensible to the wide masses. The demonstration, however, is just what we are in fact observing instead of quantum tunneling – “optical tunneling”, perhaps more commonly known as Frustrated Total Internal Reflection (FTIR). This mathematically analogous phenomenon serves as a mere tool for explaining quantum tunneling while aiding our understanding of it and facilitating its visualization. The experimental demonstration shall prove that quantum tunneling is a daily occurring phenomenon and that it is possible once the barrier is thin enough. If these arguments are indeed met with a successful visual demonstration a wavelike character of the photons happens to be confirmed as well.