Nonlinear circuit foundations for nanodevices, part I: the four-element torus

Not all molecular and nanodevices are useful from an information technology perspective. Such devices are said to be inept in a precise technical sense that can be easily tested from an explicit mathematical criteria to be presented in this two-part tutorial review. Often an inept device can be redesigned into a smart device capable of computing and artificial intelligence by massaging the device's parameters, such as doping, concentration, geometrical profile, chemical moiety, etc., in accordance with the principle of local activity to be articulated in Part II. In particular, designing a smart nanodevice amounts to fine tuning the device parameters into a much smaller niche within the device's locally active parameter region called the edge of chaos where complexity abounds. Molecular and nanodevices will remain novelty toys for nanodevice specialists unless they possess realistic nonlinear circuit models so that future nano circuit designers can simulate their exotic designs as easily and accurately as current CMOS circuit designers. A mathematically consistent theory for modeling nonlinear, high-frequency nanodevices, specially those which exploited exotic tunneling and entanglement quantum mechanical effects, such as Coulomb blockade, quasi-particle dynamics, Kondo resonance, Aharonov-Bohm nonlocal interactions, etc., will require the introduction of a complete family of fundamental circuit elements as model building blocks. They are presented via a doubly periodic table of circuit elements somewhat reminiscent of Mendeleev's periodic table of chemical elements. These fundamental circuit elements can be compactly represented by a loop of four generic species of circuit elements wrapped around the surface of a torus where any higher order element having an arbitrarily high order of frequency dependence can be generated from one of them, modulo the integer 4, ad infinitum. The significance of this four-element torus is that realistic circuit models of all current and future molecular and nanodevices must necessarily build upon an appropriate subset of nonlinear circuit elements begotten from this torus.