Electron resonances and Savitzky–Golay methods: How can the tunneling current be properly calculated?

Abstract

We deal with the calculation of the current vs. applied bias, I=I(V) , through a double-barrier resonant tunneling heterostructure in the approximation of one-electron states. Because this calculation implies to numerically integrate the quantum-mechanical electron-transmission probability function with its maxima associated to resonances, we face the general problem of computing the integral of a continuous and derivable physical function which presents sharp local maxima at a few locations within the integration-variable range. Adequate energy integration of the maxima of the transmission probability function, several orders of magnitude larger than the local average of the function itself, is absolutely necessary because they represent the principal contribution to the current. Our approach to the problem consists, firstly, in determining – to the desired precision – the energy interval where each maximum occurs. Savitzky–Golay derivation techniques with decreasing energy interval size are well suited for this purpose. In a second stage before energy integration, the transmission is evaluated at energy-intervals of increasing size around the centered maximum until the original energy-interval size is regained. To obtain the tunneling current through the heterostructure for each bias, energy integration is undertaken using the vector of variable size intervals previously determined when analyzing the shape of the transmission function at that bias. Following this procedure, the contribution of sharp energy resonances to the I(V) curve is accurately included. The here described method of integration of functions with very few but sharp extrema is universal and could be applied to other physical problems.