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
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
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.
Quantum tunneling, a quintessential quantum phenomenon, challenges classical intuitions by allowing particles to pass through seemingly insurmountable barriers. This study delves into the intricate dynamics of electron tunneling, focusing on the critical role of sub-barrier interactions and complex time processes. Through a combination of experimental measurements and simulations based on Feynman’s path integral of complex time trajectories, we reveal that the sub-barrier potential interaction introduces an asymmetric electron wave packet probability distribution along the real time axis and a marked delay at the tunneling exit. The energy-dependent behavior of this delay in electronic tunneling agrees with that observed in the previous atomic tunneling studies, suggesting a universal mechanism underlying quantum tunneling across different scales. Our findings underscore the necessity of incorporating both the wave and particle aspects of quantum entities to fully comprehend tunneling dynamics, which holds promise for an intuitive interpretation of complex-number quantum processes.
In a recent review paper [Phys. Rpeorts 214 (1992) 339] we proposed, within conventional quantum mechanics, new definitions fot the subarrier tunnelling and reflection times. Aims of the present paper are: (i) presenting and analysing the results of various numerical calculations (based on our equations) on the penetration and return times $\\left \\langle tau_P_e_n \\right \\rangle $ , $\\left \\langle tau_R_e_t\\right \\rangle $ during tunnelling inside a rectangular potential barrier, for various penetration depths xf ; (ii) putting forth and discussing suitable definitions. Besides of the mean values, also of the variances (or dispersions)$D\\tau_T\\ $ and $ D\\tau_R\\ $ for the tima durations of trasmission and reflection processes; (iii) mentioning that our definition $\\left \\langle tau_T \\right \\rangle $ for the average transmission time results to constituite an improvemente of the ordinary dwell-time $ \\bar tau^D^W\\ $ formula: (iv) commenting upon some recent criticism by C.R. Leavens, on the basis of our new numerical results. We stress that our numerical evaluations confirm that our approach implied, and implies, the existence of the Hartman effect: an effect that in these days (due to the theoretical connections between tunnelling and evanescent-wave propagation) is receiving - at Cologne, Berkeley, Florence and Wien - indirect, but quite interesting, experimental verifications. At last we briefly analyze some other definitions of tunnelling times.