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Electrical Spin Injection and Transport in Semiconductors
Published in Evgeny Y. Tsymbal, Igor Žutić, Spintronics Handbook: Spin Transport and Magnetism, Second Edition, 2019
Discrete insulating layers may also be used to form a tunnel barrier between the FM metal and the semiconductor. These provide the more familiar and well-studied rectangular potential barrier, but also introduce an additional interface into the structure and raise issues of pinholes and uniformity of layer thickness. FM metal/oxide/metal structures have been extensively studied, since they form the basis for a tunneling spectroscopy used to determine the metal spin polarization [143], and for tunnel magnetoresistance devices being developed for nonvolatile memory [144–147]. Reviews of this area may be found in Chapters 10 and 11, Volume 1. Various oxides have been used as tunnel barriers for spin injection into a semiconductor, including Al2O3, SiO2, MgO, and Ga2O3. Examples of each are presented below.
Modeling and Analysis of Molecular Electronic Devices
Published in Sergey Edward Lyshevski, Molecular Electronics, Circuits, and Processing Platforms, 2018
For a rectangular potential barrier we have Π(x)={0for x<−L,Π0for−L≤x≤L,0for x>L.
Fundamentals of Quantum Nanoelectronics
Published in Razali Ismail, Mohammad Taghi Ahmadi, Sohail Anwar, Advanced Nanoelectronics, 2018
Jeffrey Frank Webb, Mohammad Taghi Ahmadi
We give a straightforward example to show how tunneling occurs in quantum theory by considering the one-dimensional rectangular potential barrier of Figure 1.19. The equation describing it is () V(x)={V0if0≤w,0ifx<0,x>w.
Spin-polarised wave-packets transport through one-dimensional nonlinear multiple barriers: Rashba and Dresselhaus Spin–Orbit interactions
Published in Philosophical Magazine, 2023
Maryam Sabzevar, Mohammad Hossien Ehsani, Mehdi Solaimani
Furthermore, almost the same research on the propagation properties of an incoming wave-packet on a double-barrier has been done [18]. Besides that, the influence of the effective mass differences between the barrier and the well on the wave-packet tunnelling phenomenon through double-barrier heterostructures having rectangular and trapezoidal-shaped potential profiles have been studied [19]. The effect of the slope width (the difference between the barrier top and the well bottom of the trapezoidal potential) on the static properties, as well as the dynamic properties of the wave-packet tunnelling, has also been checked. It has been shown that the trapping and the transmission coefficients are influenced significantly by the slope, width, and effective mass differences in the barriers. On the other hand, Luis et al. [20,21] have solved the effective-mass nonlinear Schrodinger equation numerically for an electron wave- packet in a coupled quantum-well system. They have shown that the wave-packet was dynamically trapped in both wells. They have also studied how electron–electron and exchange interactions can affect tunnelling phenomena. So far, the problems of wave-packet tunnelling through the time-modulated barrier [22–24] through a triple-barrier system [25], and through a reflection-less potential barrier [26] have also been examined in detail. Besides, the effect of electron–electron interaction, the barrier width, the barrier height, and the wave-packet velocity on the transmission, reflection, and trapping coefficients of a travelling wave-packet through a rectangular potential barrier have been investigated numerically [27]. However, the time- dependent tunnelling of wave-packets through asymmetric heterostructures has attracted considerably less attention, and more work is needed in this field of research.
The chemical potential and the work function of a metal film on a dielectric substrate
Published in Philosophical Magazine Letters, 2019
P. P. Kostrobij, B. M. Markovych
One of the methods for a theoretical study of thin metal films is the use of model potentials, which are simple enough both to solve the stationary Schrödinger equation analytically and qualitatively correctly reflect the physical picture, namely, they do not allow electrons to leave a metal film. The simplest model potential is the infinite rectangular potential well, which, in particular, was used in Refs. [1–9]. In Refs. [1–4] the chemical potential of non-interacting electrons in a film has been calculated with inadequate taking into account the electroneutrality condition. Namely, it has been assumed that the film thickness equals the width of the rectangular well. However, it was noted in Ref. [10] that the film thickness is less than the width of the rectangular potential well by some value. It depends on the width and height of the barrier of the potential well and has been obtained by Bardeen [11] for the semi-infinite jellium with the finite rectangular potential barrier. Moreover, an issue of why this distance is necessary to be taken into account was discussed in Refs. [12,13]. As a result of the procedure implemented in Refs. [1–4], the values of the chemical potential were obtained too big compared to the results of Ref. [8], and the bulk limit was not recovered upon increasing film thickness. In Ref. [14], the work function and the electronic elastic forces for the film in the model of non-interacting electrons have been calculated by using the finite rectangular potential well with the electroneutrality condition taken into account inadequately. As a result, the calculated values of the work function are too small compared to the experimental data and with increasing the film thickness do not tend to its bulk value. The same inaccuracy occurred in Ref. [15], in which the work function and the electronic elastic forces for the film have been also calculated, and in the recent article [16], in which the work function for films which are in the vacuum and on the dielectric substrate, has been calculated. The authors of Ref. [15] came to the inadequate conclusion that the work function of a low-dimensional metal structure is always less than the work function of a semi-infinite metal.