On the calculation of the tunnelling probability in a heavily-doped p-n junction

Abstract

The one-dimensional Schrodinger's equation for a triangular potential barrier (appropriate to tunnel diodes) is solved directly to obtain an expression for the probability of tunnelling of an electron through it. The result has been compared with that of Kane and the W.K.B. method. It is concluded that the results based on this calculation predict the same functional dependence of the tunnelling probability on m?, E_g and F as that predicted by earlier methods.     

Persistent current and transmission probability in the Aharonov-Bohm ring with an embedded quantum dot

Persistent current and transmission probability in the Aharonov-Bohm (AB) ring with an embedded quantum dot (QD) are studied using the technique of the scattering matrix. For the first time, we find that the persistent current can arise in the absence of magnetic flux in the ring with an embedded QD. The persistent current and the transmission probability are sensitive to the lead-ring coupling and the short-range potential barrier. It is shown that increasing the lead-ring coupling or the short-range potential barrier causes the suppression of the persistent current and the increasing resonance width of the transmission probability. The effect of the potential barrier on the number of the transmission peaks is also investigated. The dependence of the persistent current and the transmission probability on the magnetic flux exhibits a periodic property with period of the flux quantum.     

Electronic transport for armchair graphene nanoribbons with a potential barrier

This paper studies the electronic transport property through a square potential barrier in armchair-edge graphene nanoribbon (AGNR). Using the Dirac equation with the continuity condition for wave functions at the interfaces between regions with and without a barrier, we calculate the mode-dependent transmission probability for both semiconducting and metallic AGNRs, respectively. It is shown that, by some numerical examples, the transmission probability is generally an oscillating function of the height and range of the barrier for both types of AGNRs. The main difference between the two types of systems is that the magnitude of oscillation for the semiconducting AGNR is larger than that for the metallic one. This fact implies that the electronic transport property for AGNRs depends sensitively on their widths and edge details due to the Dirac nature of fermions in the system.     

A note on tunneling in the presence of a superintense laser field

The influence of a superintense laser field on the transmission probability of a charged particle through a potential barrier is discussed. It is shown that the effect of the intense laser field is to weaken the potential barrier in such a way that its height decreases exponentially with the laser field. The implications to the ferro — to para - electric phase transition in BaTiO₃ under an intense CO₂ - laser field is also discussed.     

Quantum superarrivals and information transfer through a time-varying boundary

The time-dependent Schrödinger equation is solved numerically for the case of a Gaussian wave packet incident on a time-varying potential barrier. The time evolving reflection and transmission probabilities of the wave packet are computed for several different time-dependent boundary conditions obtained by reducing or increasing the height of the potential barrier. We show that in the case when the barrier height is reduced to zero, a time interval is found during which the reflection probability is larger (superarrivals) compared to the unperturbed case. We further show that the transmission probability exhibits superarrivals when the barrier is raised from zero to a finite value of its height. Superarrivals could be understood by ascribing the features of a real physical field to the Schrödinger wave function which acts as a carrier through which a disturbance, resulting from the boundary condition being perturbed, prpagates from the barrier to the detectors measuring reflected and transmitted probabilities. The speed of propagation of this effect depends upon the rate of reducing or raising the barrier height, thus suggesting an application for secure information transfer using superarrivals.     

Interband resonant tunneling in superconductor heterostructures in a quantizing magnetic field

The resonant tunneling of electrons through quasistationary levels in the valence band of a quantum well in double-barrier structures based on III–V materials with type-II heterojunctions is considered in a quantizing magnetic field directed perpendicularly to the interfaces. The transmission coefficients of the tunnel structure for transitions from states corresponding to different Landau levels are calculated using the Kane model. It is shown that transitions with a unit change in the Landau level index n as a result of mixing of the wave functions of states with opposite spin orientations are possible on the interfaces due to spin-orbit coupling. The probability of such transitions can be comparable to the probability of transitions without a change in the Landau level index for InAs/AlGaSb/GaSb resonant-tunneling structures. Fiz. Tverd. Tela (St. Petersburg) 40, 2121–2126 (November 1998)     

Quantum transmittance of barriers for composite particles

The transmission of two bound particles through a repulsive barrier is studied. A simple mechanism for the appearance of barrier resonances, which results in anomalous barrier transmittance as compared with the transmission probability for structureless objects, is demonstrated. It is shown that the probabilities for two interacting particles to tunnel from a false vacuum can be much higher than previously thought.     

Resonant tunneling,eigenvalue and energy band calculation for potential and periodic potential structures

Recursion formulae for the reflection and the transmission probability amplitudes and the eigenvalue equation for multistep potential structures are derived. Using the recursion relations, a dispersion equation for periodic potential structures is presented. Some numerical results for the transmission probability of a double barrier structure with scattering centers, the lifetime of the quasi-bound state in a single quantum well with an applied field, and the miniband of a periodic potential structure are presented.     

Dwell time in one-dimensional graphene asymmetrical barrier

The authors have investigated theoretically the dwell time of Dirac fermions tunneling through electrostatic square barrier in monolayer graphene, including asymmetrical and symmetrical potential barriers. It is found that the incident angle determines the critical incident energy. When the incident energy is larger than the critical incident energy, the dwell time saturate with the increase of the barrier thickness. But when the incident energy is smaller than the critical incident energy, the dwell time oscillates with the increase of the barrier thickness. The behaviors of oscillation and saturation of the dwell time are related with the transmission probability. These results may be helpful for the basic physics and potential application of graphene based electronic devices.     

Particle tunneling and scattering in a three-dimensional potential with a barrier

The particle tunneling through a 3-D rectangular potential barrier has been studied. The simplest model for multiple internal reflections has been assumed. The explicit expression for all the transmission and reflection probability amplitudes have been derived, as well as the tunneling and reflection phase times.      

Transport properties through double-magnetic-barrier structures in graphene

We study electrons tunneling through a double-magnetic-barrier structure on the surface of monolayer graphene.The transmission probability and the conductance are calculated by using the transfer matrix method.The results show that the normal incident transmission probability is blocked by the magnetic vector potential and the Klein tunneling region depends strongly on the direction of the incidence electron.The transmission probability and the conductance can be modulated by changing structural parameters of the barrier,such as width and height,offering a possibility to control electron beams on graphene.     

Quantum scattering in one-dimensional systems satisfying the minimal length uncertainty relation

In quantum gravity theories, when the scattering energy is comparable to the Planck energy the Heisenberg uncertainty principle breaks down and is replaced by the minimal length uncertainty relation. In this paper, the consequences of the minimal length uncertainty relation on one-dimensional quantum scattering are studied using an approach involving a recently proposed second-order differential equation. An exact analytical expression for the tunneling probability through a locally-periodic rectangular potential barrier system is obtained. Results show that the existence of a non-zero minimal length uncertainty tends to shift the resonant tunneling energies to the positive direction. Scattering through a locally-periodic potential composed of double-rectangular potential barriers shows that the first band of resonant tunneling energies widens for minimal length cases when the double-rectangular potential barrier is symmetric but narrows down when the double-rectangular potential barrier is asymmetric. A numerical solution which exploits the use of Wronskians is used to calculate the transmission probabilities through the Pöschl–Teller well, Gaussian barrier, and double-Gaussian barrier. Results show that the probability of passage through the Pöschl–Teller well and Gaussian barrier is smaller in the minimal length cases compared to the non-minimal length case. For the double-Gaussian barrier, the probability of passage for energies that are more positive than the resonant tunneling energy is larger in the minimal length cases compared to the non-minimal length case. The approach is exact and applicable to many types of scattering potential.     

On the transmission coefficient for the double delta-function potential embedded in a crystalline lattice

The solution of the Schrödinger–Wannier equation is examined for the potential that is defined as a sum of the two delta functions of unequal strengths. The analytical expression for the transmission coefficient is derived from the solution. The transmission coefficient is shown to exhibit maxima and minima. Moreover, it is proved that the transmission coefficient in its maxima is larger and in its minima is smaller than the transmission coefficient for the corresponding single delta-function potential. Some differences between the transmission coefficient of particles with the quadratic energy spectrum and the transmission coefficient of particles with the Kane energy spectrum are presented.     

The effect of dissipation on quantum transmission resonance

Quantum transmissions of a free particle passing through a rectangular potential barrier with dissipation are studied using a path decomposition technique. Dissipative processes strongly suppress the transmission probability at resonance just above the barrier resulting in an unexpected reduction of the mean traversal time through the potential barrier.     

Direct tunneling current model for MOS devices with ultra-thin gate oxide including quantization effect and polysilicon depletion effect

An efficient direct tunneling current model is presented for the ultra thin gate dielectric MOS structure. The tunneling current is modeled by including the inversion layer quantization effect with a finite potential barrier height as the boundary condition and the modified WKB method for calculating the transmission probability. The model is in good agreement with the full quantum calculation and the experiments. The results indicate that the finite boundary condition has to be considered for the ultra thin gate dielectric and the gate dielectric materials with lower barrier height. This model is accuracy and computational efficient and suitable to be used in characterized the sub-100 nm MOSFET with gate oxide below 2.0 nm.     

Escape by diffusion from a square well across a square barrier

The problem of escape of a particle by diffusion from a square potential well across a square barrier is studied on the basis of the one-dimensional Smoluchowski equation for the space- and time-dependent probability distribution. For the model potential the Smoluchowski equation is solved exactly by a Laplace transform with respect to time. In the limit of a high barrier the rate of escape is given by an asymptotic result similar to that derived by Kramers for a curved well and a curved barrier. An approximate analytic formula is derived for the outward time-dependent probability current in terms of the width and depth of the well and the width and height of the barrier. A similar expression holds for the complete probability distribution.     

The effect of the crystal structure of the field emitter on field ion energy distributions

The field ionization probability of an atom as a function of distance from the field emitter is discussed in terms of the atomic arrangement and the electron scattering properties of the ion cores of the emitter in the immediate neighborhood of the atom to be ionized, and the electron transmission properties of the potential barrier between the emitter and that atom. This approach to field ionization calculations is somewhat similar to field ionization calculations based on low energy electron diffraction (LEED) procedure in that it takes into account electron scattering from the first few atomic layers of the emitter. It differs from LEED type calculations, because it considers the highly localized nature of the ionization near a surface atom. This localization makes the ionization probability relatively insensitive to the two-dimensional periodicity of the emitter surface. A one-dimensional calculation, in which only the potential barrier and three ion core scatterers in line with the field are considered, shows secondary structure in the predicted field ion energy distributions near the critical energy deficit, as well as the well known, primary field induced resonance peaks. The surface orientation dependence of these distributions arises naturally from this model because the secondary structure depends strongly upon the crystal parameter along a line parallel to the field. This one-dimensional calculation can be no more than an approximation to a complete calculation. It is interesting, however, that such a simple physical model, in which scattering from the image potential and only two or three ion cores is considered, rather than scattering from a complete crystal, can give prodicted field ion onergy distributions which are similar to those experimentally observed.     

Tunneling through a disordered potential barrier

Tunnelling through a weakly disordered potential barrier is studied analytically. A perturbative approach is developed to calculate all statistical moments of the tunnelling transmission coefficient, and its probability distribution function. It is shown that on average disorder enhances the tunnelling conductance, resistance, and the coherent component of the transmitted field.On leave from the Institute of Low-Temperature Physics and Engineering, Kharkov 310164, Ukraine.     

Anomalous photon-assisted tunneling in graphene

We investigated the transmission of Dirac electrons through a potential barrier in the presence of circularly polarized light. An anomalous photon-assisted enhanced transmission is predicted and explained. It is demonstrated that the perfect transmission for nearly head-on collision in infinite graphene is suppressed in gapped dressed states of electrons, which is further accompanied by a shift of peaks as a function of the incident angle away from head-on collision. In addition, the perfect transmission is partially suppressed by a photon-induced gap in illuminated graphene. After the effect of rough edges of the potential barrier or impurity scattering is included, the perfect transmission with no potential barrier becomes completely suppressed and the energy range for the photon-assisted transmission is reduced at the same time.     

Diffusion in a bistable potential

The problem of diffusion of a particle in a bistable potential is studied on the basis of the one-dimensional Smoluchowski equation for the space- and time-dependent probability distribution. The potential is modeled as two parabolic wells separated by a parabolic barrier. For the model potential the Smoluchowski equation is solved exactly by a Laplace transform with respect to time for the initial condition that at time zero the probability distribution is given by a thermal equilibrium distribution in one of the wells. In the limit of a high barrier the rate of transition to the other well is given by an asymptotic result due to Kramers. For a potential barrier of moderate height there are significant corrections to the asymptotic result.