Why one can expect large rectification in molecular junctions based on alkane monothiols and why rectification is so modest

The Stark effect plays a key role in understanding why, against expectation, alkane thiols are not high-performance molecular rectifiers.

: Schematic representation of the potentiometer rule.

Potentiometer rule
The potentiometer rule assumes that screening effects within the junction are altogether ineffective, and that the contacts (i.e., the electrode-molecule interfaces) do in no way affect the electric potential. Then, the potential profile V(z) across the junction is simply that of a region characterized by a constant electric field, varying linearly (cf. Figure S1) along the junction Here the coordinate z is the measured from the center, and VVt-Vs=V/2-(-V/2) is the difference between the potentials Vt,s of the tip (t) and substrate (s). Within this picture, the energy shift of the nearly point-like MO strongly localized around z=zMO caused by the applied bias V can be expressed as Eqs. 2 and S3 indicate a linear dependence on zMO of the asymmetry parameter  within the potentiometer rule framework That is, the closer to an electrode, the larger is the magnitude || of the asymmetry parameter, whose maximum value, corresponding to zMO=±d/2 (cf. eq. S2), is reached in the case of an MO located very close to the molecule-electrode interface. According to eq S4 the maximum value is equal to |max|=1/2. 3 The idea underlying Figure S1 is based on the above eq. S3, which visualizes the physical content of the potentiometer rule.  Charge Transport in Low Bias Range. Our results for the low bias resistance are presented in Figure S3. In discussing these results, we will separately consider the impact of the molecular length (n) and the contact (metal) nature. For homologous molecular series, the two aforementioned effects can be conveniently disentangled by expressing the low bias resistance = analyzed as follows 1 Here is the effective contact resistance, is the tunneling decay parameter, L0≈1.2 Å is the repeat unit length, and n is the number of repeating units. The exponential length dependence shown in the equation represents a general feature of off-resonant tunneling.
From the slope of the semilogarithmic plot of = versus n, one can determine the tunneling attenuation factor β, while its intercept at n = 0 gives the effective contact resistance . Low bias resistances of CnT as well as the values of and for the various types of junctions are shown in Table 1 of the main text.
Tunneling Attenuation Factor. Figure

Determination of the Model Parameters.
HOMO-Fermi energy offset h. In line with the philosophy underlying TVS, 12 we use the Vt± values (cf. Table 1, Figure S5) extracted from I-V measurements to estimate the energy offset h=-0 via eq. 4 deduced earlier 13 for the asymmetric single level model.
The calculated energy offsets h of CnT junctions with different metal contacts are listed in Table 1. As shown in Figure 3A, h are independent of the length of the molecule (n). This agrees with our quantum chemical calculations (not shown here) indicating that the HOMO energies of the isolated alkane thiols are practically independent of the molecular size n. On the other hand, the HOMO energy offset slightly decreases with increasing work function of the contact metals, Figure 3B. Specifically, Table 1 Figure S7; average values and statistical deviations are presented in Table 1 in the main text. model underlying eq. 1 and 2 of the main text, in addition to the curves depicted in Figure   5 of the main text, further examples are presented in Figure S8.

OVGF results of LUMO energies of the alkanethiol vs. applied electric field Є / bias
Vm between the molecular ends. In Figure S9A, we present results for the LUMO energy for alkanethiol molecules in external field. They clearly invalidate the requirement imposed by eq. S6; the bias-driven LUMO energy shifts ( Figure S9B) are almost independent of the bias polarity.