Thermo-tunneling of hot electrons across a few nanometer gap has application to vacuum electronics, flat panel displays, and holds great potential in thermo-electric cooling and energy generation. However development of such applications requires formation of dynamically balanced gap separating the two surfaces. One such approach is the use of Lorentz (repulsive) and Coulomb (attractive) forces to obtain an equilibrium gap between two elastic electrodes. The present paper describes the application of the Differential Quadrature Method (DQM) to the solution of a clamped-clamped Euler-Bernouli beam subject to the combined action of Lorentz and Coulomb forces. The results show that due to non-local action of the Lorentz force, the shape of the tunneling electrode is inherently non-uniform with Coulomb forces acting primarily at one end of the beam while the Lorentz force distributed along the remaining part. DQM method also allows analysis of the stability of the tunneling current as a function of the applied external potential and magnetic field. In addition to the classical electrostatic pull-in instability with no-tunneling, a second regime with non-zero tunneling current is also identified. To the best of our knowledge this is the first attempt to analyze this phenomenon under the effect of both Electrostatic and Lorenz forces in this particular case. Linear stability analysis of the tunneling regime indicates the appearance of a saddle-saddle bifurcation indicating unstable tunneling regime.

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