I'm not really sure about the I-V and C-V characteristics of MOM diodes. I am familiar with MOM tunnel diodes with Al-Al2O3-Pt materials in a scenario of collecting power through rectification in simple dipole antennas. MOM tunnel diodes can be used as rectifiers as I'm sure you know. Previously it was thought that when implementing them to collect signals in dipole antennas, their oxide layer thermal effect is quantitatively measure power collected. Recently however it has been proved that its current rectification properties can also be measured for the same purpose.
I use ANSYS High Frequency Structure Simulator (HFSS) for this-material properties obtained from ellipsometry. Since the antenna operates in the IR region the oxide layer dimensions are around your 5nm specification. HFSS does allow plotting I-V characteristics versus frequency so it is possible. If this isn't exactly what you want then other Electromagnetic packages of ANSYS like Maxwell or SIWave etc could be explore although I'm not too familiar with them.
PS: Sorry for late reply