Exact Analytic Formula for Conductance Predicting a Tunable Sommerfeld–Arrhenius Thermal Transition within a Single‐Step Tunneling Mechanism in Molecular Junctions Subject to Mechanical Stretching

Abstract

It is shown that the conductance G of molecular tunnel junctions wherein the charge transport is dominated by a single energy level can be expressed in closed analytic form which is exact and valid at arbitrary temperature T and model parameter values. On this basis, it is shown that the single-step tunneling mechanism is compatible with a continuous thermal transition from a weakly T-dependent G at low T (Sommerfeld regime) to a nearly exponential 1∕T-dependent G at high T (Arrhenius-like regime). This Sommerfeld-Arrhenius transition can be observed in real molecular junctions and can be continuously tuned, for example, via mechanical stretching.

Estimating the Number of Molecules in Molecular Junctions Merely Based on the Low Bias Tunneling Conductance at Variable Temperature

“…The real part of the tetragamma function ψ(2; z) ≡ d 2 dz 2 ψ(z) with z = 1/2 + iy (real y) entering above in the RHS is not available in closed analytic form, but we found that it can be very accurately approximated (Figure 1) via elementary functions as follows [43] Re ψ 2; For parameter ranges virtually covering all experimental situations of interest wherein a T-dependent G can be expected, the parameter…”

“…Therefore, departures of equation ( 7) from equation ( 4) become substantial when |ε 0 |, Γ a , and k B T have comparable values. For this reason, for temperatures around T c (see equation (11a) below), equation ( 4) better quantifies the gradual transition between an Arrhenius-type (high T) and a Sommerfeld (low T) regime [43] than equation (7).…”

“…As shown recently [43], the RHS of equation ( 1) can be integrated out analytically. The result for the conductance per molecule expressed via the real part of Euler's trigamma function of complex argument function ψ (z) [55] reads…”

“…According to the general Keldysh formalism [44][45][46][47], the low bias conductance G ≡ ∂I(V)∂V| V→0 of a single molecule tunneling junction at finite temperature β = (k B T) −1 can be expressed as [43,48]…”

“…It is merely based on low bias conductance data collected on large area junctions at variable temperature. The exact formula for the temperature dependent conductance deduced recently by us [43] constitutes the theoretical framework of this methodology, which will be considered in the next section.…”