Improvement of the Simmons model for tunnel junctions

TL;DR

This work revisits the Simmons model for metal–insulator–metal tunnel junctions and derives new analytical formulas for elastic tunneling current and conductance at finite voltage and temperature, achieving much closer agreement with direct WKB results than the classic Simmons results. The authors obtain a generalized current density $J(V,T)$ for arbitrary barrier shapes and a corrected, tractable conductance formula $G(V,T)$ for trapezoidal barriers, including a dimensionless temperature-correction factor $C$ that also alters the $G(V)$ curvature. A closed-form $G(V)$ for rectangular barriers under the WKB framework provides a rigorous benchmark. When applied to experimental $G-V$ data (notably AlOx barriers), the improved parabolic model yields barrier thicknesses and heights with notable differences from Simmons fits and reduced parameter uncertainties, demonstrating practical utility for barrier-property extraction in tunnel devices.

Abstract

The Simmons model is a well-known and widely used model for the elastic tunneling current of a metallic tunnel junction, and fitting it to electrical measurements can be used to estimate thicknesses and heights of the tunnel barriers. We present here an improvement of the Simmons model, deriving new more accurate analytical formulas for the tunneling current density and conductance at finite voltage and temperature. We demonstrate that our conductance-voltage formulas are much closer to the Wentzel-Kramers-Brillouin approximation than the Simmons model and its commonly used simplified parabolic approximation. In addition, we demonstrate the practical use of our model, by fitting it to experimental tunnel junction conductance-voltage data and showing a sizeable difference from the Simmons model.

Figures (9)

• Figure 1: A general tunnel barrier formed by an insulating film between two metal electrodes. Electrode 1 with chemical potential $\mu_1=\mu$ is negatively biased with voltage $V$ with respect to electrode 2 that has a chemical potential $\mu_2=\mu-eV$ shifted by the bias voltage $V$. The net flow of electrons takes place from electrode 1 to electrode 2.
• Figure 2: An unbiased trapezoidal tunnel barrier (dashed line) and a voltage-biased barrier (solid line) with well-defined electrode-insulator interfaces and physical thickness $d_\textrm{phys} = d$. Electrons tunnel in $x$ direction.
• Figure 3: Comparison of different $G-V$ models for a rectangular barrier with realistic barrier parameters $d = 9 \textrm{ Å}$ and $\phi_0=1 \textrm{ eV}$. The cyan curve shows the WKB numerical result Eq. \ref{['WKB G(V,0)']} with $\mu=11.7$ eV (the Fermi energy of Al), the red curve corresponds to our Eq. \ref{['ImprovedG(V,T)']} at $T=0$, the black curve shows its quadratic approximation Eq. \ref{['ParabolicApproximationOfImprovedG(V,T)']} at $T=0$, whereas the green curve illustrates the parabolic Simmons model of Eq. \ref{['SimmonsSimpleGVatT=0']}, and the blue curve shows the full Simmons model, Eq. \ref{['SimmonsGV']}.
• Figure 4: The correction factor $C$ as a function of $d$ and $\phi_0$ (color scale) together with fit parameters obtained for Ti-Au (blue points), Cu (red points) and Al devices (black points) by fitting Eq. \ref{['ParabolicApproximationOfImprovedG(V,T)']} to experimental $G-V$ data taken at room temperature.
• Figure 5: Experimental $G-V$ characteristic (blue dots) of a representative Ti-Au device with $A_{\text{j}}=(0.52\pm0.01)\text{ }\mu\text{m}^2$ at room temperature. The solid red line at $|V| \leq 0.13 \textrm{ V}$ shows the fit given by Eq. \ref{['ParabolicApproximationOfImprovedG(V,T)']}. From the fit, one can calculate barrier parameters $d=(8.38\pm0.03)\text{ Å}$ and $\phi_0=(1.00\pm0.01)\text{ eV}$. The dashed red line at $|V| > 0.13 \textrm{ V}$ shows the extrapolation of the fit.