Exactly solvable model for metal–insulator–metal stepped boundary tunnel junctions

Highly Efficient Midinfrared On-Chip Electrical Generation of Graphene Plasmons by Inelastic Electron Tunneling Excitation

“…The complex permittivity for Al is taken from Palik's handbook [22], while that for Ag is obtained from Johnson and Christy's paper [23]. The permittivity of Al2O3 is approximately 3, and has an effective tunnelling mass of 0.2×me [24]. The results of both simulated and experimental spectra are compared in Fig.…”

Transmission and Dwell Time Oscillations Caused by Coexistence of Tunneling and Propagating in a Trapezoidal Barrier

“…In our previous work, the biased junction was treated using an exactly solvable trapezoidal barrier potential model, wherein an effective resulting barrier potential is equal to the trapezoidal barrier potential plus the applied electric potential, and thus the distortion of the barrier by the bias voltage is included. [11] It means that the applied electric field is located inside the barrier insulator region of the junctions and the junctions have the ideal stepped boundaries. Thus, the exact analytic expressions for the electron wave functions and the transmission coefficients in the biased MIM tunnel junctions were obtained by solving Schrödinger equation strictly.…”

“…Thus, the exact analytic expressions for the electron wave functions and the transmission coefficients in the biased MIM tunnel junctions were obtained by solving Schrödinger equation strictly. On the basis of the three models, i.e., our exactly solvable model and two Brinkman's models, three groups of the barrier parameters were determined by fitting three calculated I-V curves to the experimental ones for three types of junctions (with Au, Ag, and Cu top electrodes) at temperature 77 K. [11] In this paper we present a refined one of our exactly solvable model, wherein the longitudinal kinetic energy and the effective mass of the electrons in the electrode on the left of the barrier distinguish from that in the electrode on the right, thus the effects of the two dissimilar electrodes on the electron effective mass are included. It is found that as the longitudinal kinetic energy of the incident electron, E xL , is greater than the shorter side of the resultant trapezoidal barrier potential, the electrons tun-nel through one barrier subregion where E xL is lower than the barrier, then propagate over the other where E xL is higher than the barrier.…”

“…Equation (11) with the upper and lower sign is a modified Bessel equation and a Bessel equation of one third order, respectively. A general solution for the modified Bessel equation of one third order in Eq.…”

“…In this case, the electron tunnels through the whole barrier and the solved electron wave function ψ 2 (x) satisfies the modified Bessel equation Eq. (11) with the upper sign and the solution has the form of Eq. ( 12) and is the combination of the I 1/3 and K 1/3 with the different constant C 1 , C 2 , R, and S. Similarly, using (i) the boundary conditions at x = 0 and d: ψ 1 (0) = ψ 2 (0) and dψ 1 (0)/dx = dψ 2 (0)/dx; ψ 2 (d) = ψ 4 (d) and dψ 2 (d)/dx = dψ 4 (d)/dx, and (ii) the recurrence relations of Bessel, Neumann, and modified Bessel functions, [15] the constant C 1 , C 2 , R, and S are determined as follows:…”