Condensed Matter Theories pp 149-155 | Cite as

# Self Consistent Model for Tunneling Across a One Dimensional Barrier in a Many Electron System

Chapter

## Abstract

We consider a model with two metals (modeled for the present as jellium) separated by a barrier described by a potential
where
This equation is solved self-consistently with boundary conditions appropriate to waves incident from the left and right (which form an orthonormal complete set when appropriately normalized). The result shows that, within this model, a finite current can pass through the barrier with no electrostatic potential drop. We will argue that this result does not depend on the local density approximation (which is the only computational approximation in the model) but is to be expected because there is no dissipative mechanism in the model.

*(***V***). We show that the current density***x***can be fixed by adding a term***j***to the energy which is of the form:***H’**$$H' = - m\int_{ - \infty }^\infty {v(x)j(x)d^3 x}$$

*(***v***) is a Lagrange multiplier function which depends on the position***x***. We show that***x***(***v***) may be interpreted as the hydrodynamic velocity of the electrons in the junction so that***x***(***v***) depends on the electron density***x***(***n***) as***x***= <***j***(***n***) >***x***(***v***). The expectation value of***x***(***j***) is independent of***x***by current conservation. The computational part of the tunneling problem is thus reduced to the self-consistent solution of the Schroedinger equation with one additional term:***x**$$\left( {H_o - mv(x)^2 /2} \right)\varphi _p (x) = E_p \varphi _p (x)$$

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## Copyright information

© Plenum Press, New York 1987