Solving rate equations for electron tunneling via discrete quantum states

We consider the form of the current-voltage curves generated when tunneling spectroscopy is used to measure the energies of individual electronic energy levels in nanometer-scale systems. We point out that the voltage positions of the tunneling resonances can undergo temperature-dependent shifts, leading to errors in spectroscopic measurements that are proportional to the temperature. We do this by solving the set of rate equations that can be used to describe electron tunneling via discrete quantum states, for a number of cases important for comparison to experiments, including (1) when just one spin-degenerate level is accessible for transport, (2) when two spin-degenerate levels are accessible, with no variation in electron-electron interactions between eigenstates, and (3) when two spin-degenerate levels are accessible, but with variations in electron-electron interactions. We also comment on the general case with an arbitrary number of accessible levels. In each case we analyze the voltage positions, amplitudes, and widths of the current steps due to the quantum states.