  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 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 2 spin-degenerate levels are
accessible, with no variation in electron-electron interactions between
eigenstates, and (3) when 2 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.
