Electrode-side impedance nonlinearity in polycarbazole bioelectrodes quantifiable by a quaternion formalism

Nonlinear response to sinusoidal electrification is a phenomenon rarely observed for conductive interfaces. We report nonlinear response as an electrode phenomenon in a conjugated polymer electrode polycarbazole. While other semiconductors manifest an impedance quantifiable in the complex field ($e.g.$ Warburg where complex Z = Z$_{W\infty }$ and resistive with Z = Z$_{Hl}$ [Hl-Halbleiter]) the polycarbazole manifests no definable impedance due to essential nonlinearity. There is no description available for this form of pseudoconductivity. We introduce a quaternion formalism Z$_{\subset T}$=a+bi+cj+dk [where a,b,c,d are real and i$^{2}$=j$^{2}$=k$^{2}$=-1 and jk=i] that successfully describes all conductivity (c=d=0) and pseudoconductivity presently known as a normed ring, and reduces to the complex field for conductivity. In this formalism, the normalized impedance of a capacitor is Z=i, the experimentally determined polycarbazole pseudoimpedance Z$_{\subset T}$=k, that of a resistance is Z=1 and that of the Z$_{W\infty }=\surd $i=i$^{\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} }$. The non-Abelian character of Z$_{\subset T}$ implies that the Onsager relation fails for some interfaces. Remarkably, certain Kramers-Kronig relations (Hilbert transformation in not only the complex but also the [j,k] plane) still hold for certain experimental setups. Computation of the energy integral $\smallint $\textbf{D}$^{.}$\textbf{E}dt reveals that charge transport is lossless, similar to conduction in an ordinary capacitance.