It has been observed experimentally that the tunneling current across pn tunnel diodes decreases when a magnetic field is applied to it at liquid helium temperatures. Such a decrease is noticed both for longitudinal and transverse applications of magnetic fields relative to the direction of the current flow. Classically, tunneling of particles is regarded to propagate with the speed of light. Therefore, on the basis of the postulates of special theory of relativity tunneling of a potential barrier by particles should be independent of the state of motion of an observer relative to the tunneling system. In other words, the tunneling rate should remain unaffected of the motion of the observer. Therefore, to explain tunneling of particles in a crossed configuration of electric and magnetic fields, a suitable frame of reference for an observer can always be chosen so that the magnetic flux density observed from the new frame reduces to zero. The tunneling along the reduced electric field intensity may then be studied on established laws. The resultant electric field intensity in such a case becomes a function of the original magnetic flux density and it shows a steady decline with the increase in the magnitude of the latter. Therefore, a reduction in the tunneling probability and the tunneling current density is expected when the magnetic flux density normal to the current flow is increased. Finally, at a particular magnetic field intensity the tunneling current density drops to zero. It then turns out that the cyclotron frequency corresponding to this cut off magnetic field is proportional to the reciprocal of the tunneling time derived on the ideas of quantum measurement and observations. Therefore, by recording the normal magnetic flux density required to cut off the tunneling current across pn tunnel diodes immersed in a liquid helium bath, it may be possible to make an estimate of the tunneling time across a potential barrier. The working expression of the calculation of the tunneling time also follows independently of the relativistic treatment outlined above. The identity of results derived on the two approaches seems to justify the fact that the speed Of tunneling is really high enough so that the relativistic treatments may be applied to it as an approximation although tunneling may not propagate with the exact speed of light.
It has been observed experimentally that the tunneling current across pn tunnel diodes decreases when a magnetic field is applied to it at liquid helium temperatures. Such a decrease is noticed both for longitudinal and transverse applications of magnetic fields relative to the direction of the current flow. Classically, tunneling of particles is regarded to propagate with the speed of light. Therefore, on the basis of the postulates of special theory of relativity tunneling of a potential barrier by particles should be independent of the state of motion of an observer relative to the tunneling system. In other words, the tunneling rate should remain unaffected of the motion of the observer. Therefore, to explain tunneling of particles in a crossed configuration of electric and magnetic fields, a suitable frame of reference for an observer can always be chosen so that the magnetic flux density observed from the new frame reduces to zero. The tunneling along the reduced electric field intensity may then be studied on established laws. The resultant electric field intensity in such a case becomes a function of the original magnetic flux density and it shows a steady decline with the increase in the magnitude of the latter. Therefore, a reduction in the tunneling probability and the tunneling current density is expected when the magnetic flux density normal to the current flow is increased. Finally, at a particular magnetic field intensity the tunneling current density drops to zero. It then turns out that the cyclotron frequency corresponding to this cut off magnetic field is proportional to the reciprocal of the tunneling time derived on the ideas of quantum measurement and observations. Therefore, by recording the normal magnetic flux density required to cut off the tunneling current across pn tunnel diodes immersed in a liquid helium bath, it may be possible to make an estimate of the tunneling time across a potential barrier. The working expression of the calculation of the tunneling time also follows independently of the relativistic treatment outlined above. The identity of results derived on the two approaches seems to justify the fact that the speed Of tunneling is really high enough so that the relativistic treatments may be applied to it as an approximation although tunneling may not propagate with the exact speed of light.
We apply the bosonization technique to the problem of tunneling in a Fermi liquid, and present a semiclassical theory of tunneling rates. To test the method, we derive and evaluate an expression for the tunneling current in the problem of two parallel Luttinger liquids. Next we study tunneling between parallel two dimensional Fermi liquids. In the absence of a magnetic field, the conductance I/V has a resonance peak at zero bias. Our estimate for the peak broadening agrees with the many-body theory results. In a magnetic field perpendicular to the plane, we find a tunneling gap linear in the field, in agreement with the experiment. In this regime the tunneling is spatially coherent over a distance of the order of the cyclotron radius. The coherence can be probed by a magnetic field parallel to the plane, which gives an Aharonov-Bohm phase to the tunneling amplitude. As a result, the current is an oscillatory function of the field.
This book presents a new approach to analyze quantum mechanical tunnelling of particles across potential barriers. The conventional concepts of this phenomenon, which are based on a time-in-dependent or a time-dependent perturbation approach are inadequate in furnishing explanations to a number of effects, e.g. (i) the limit of resolution of a field emission macroscope (FEM), (ii) Esaki integral, representing the I-V characteristics of solid-state junctions, (iii) Josephson effect, (iv) tunnelling time, (v) tunnelling current density etc. The new analysis presented here not only provides adequate explanations to all the above mentioned effects but also furnishes an appropriate expression for the tunnelling current density which yields results closer to experimentally observed values
The Lorentz covariance of relativistic Dirac equations serves as a fundamental principle underlying the laws of electromagnetism across different inertial frames. Exploiting the covariance, we obtain the general solutions for Dirac fermions under both the in-plane electric E and perpendicular magnetic B fields, which reduce to either a magnetic or electric field in the inertial frame with drift velocity along the E×B direction. This dichotomy defines the magnetic and electric regimes, separated by the critical field ratio E/B=vF with vF denoting the Fermi velocity of Dirac fermions. Using these solutions, we revisit Klein tunneling through a heterojunction with generalized electromagnetic potentials. In the magnetic regime, the transmission exhibits oscillations governed by the Fabry-Pérot interference. In the electric regime, perfect transmission occurs at normal incidence in the drifted frame. The interference phase is further analyzed in terms of the solid angles on the Bloch sphere, providing a geometric interpretation of Klein tunneling. Finally, we briefly discuss the relation between the tilting of Dirac cones and the in-plane electric field, establishing the correspondence of the undertilted and overtilted cases to the magnetic and electric regimes, respectively. Our findings reveal the manipulation of Klein tunneling by electromagnetic fields, offering a theoretical basis for designing novel electronic devices.
This book presents a new approach to quantum mechanical tunnelling and its applications to various fields of physics. The conventional concepts of this phenomenon, which are based on a time-dependent or time-independent approach, are inadequate in providing explanations for 1) the limit of resolution of field-emission microscope, 2) the Esaki-Integral representation the tunnelling I-V characteristics of tunnel diodes, 3) the Josephson effect, 4) the tunnelling time, etc. The new theory presented here not only provides adequate explanation for all the above-mentioned effects but also gives an accurate expression for tunnelling current density which predicts results closer to the experimentally observed values. The new presentation also provides a more comprehensive description of the results obtained using older treatment.
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.