In this thesis, I discuss my use of a millikelvin scanning tunneling microscope (STM) to investigate tunneling phenomena in superconductors. As part of an effort to construct an STM to measure the superconducting phase difference, I first describe how I modified a dual-tip scanning tunneling microscope by electrically connecting the two tips together with a short (3 mm) strip of flexible 25 µm thick Nb foil. I also discuss the technique I developed for keeping each tip in feedback when only the total tunnel current through both tips can be measured. I then describe simultaneous room-temperature imaging with both tips on samples of Au/mica and highly oriented pyrolytic graphite (HOPG).

Next, I report single-tip results from scanning tunneling microscopy of 25 nm and 50 nm thick films of superconducting TiN at 0.5 K. I found large variations in the tip-sample conductance-voltage characteristics in these samples. At some locations the characteristics showed a clear superconducting gap, as expected for superconductor-normal (S-I-N) tunneling through a high barrier height. At other locations there was a distinct zero-voltage conductance peak, as expected for S-N Andreev tunneling through a low barrier height. I compare the data to the BlonderTinkham-Klapwijk (BTK) theory and the Dynes model of tunneling into a superconductor with broadened density of states. I find that the BTK model provides better fits and reveals a remarkable correlation between the superconducting gap ∆, the temperature T and the barrier height Z. Possible causes for this correlation, including local heating and surface contamination, are discussed.

Finally, I describe measurements of I(V) characteristics of a Josephson junction formed by a scanning tunneling microscope with a Nb sample and a Nb tip at 50 mK and 1.5 K. To better understand the physics of this system, I generalized the multiple Andreev reflection (MAR) theory of Averin and Bardas to describe junctions having electrodes with different superconducting gaps. For tunneling resistance Rn between 10 MΩ and 100 kΩ, there was no observable supercurrent at 50 mK or 1.5 K. For Rn between 100 kΩ and about 10 kΩ, the junctions showed hysteretic behavior, with the forward-sweep switching current Is larger than the reverse-sweep retrapping current Ir. In this regime, the critical current I0 was suppressed and the current-voltage characteristics showed a relatively small non-zero resistance R0 at V = 0 that scaled with . For Rn less than the quantum resistance (∼ 12 kΩ), the I-V characteristics deviate from single channel MAR theory. In this limit, the tip makes contact with the sample, as revealed by the dependence of the junction conductance curves on the tip-sample separation. By fitting my two-gap MAR theory to the I(V) data, I obtain superconducting gaps of the tip and sample as a function of the tunnel resistance Rn. I find the sample has nearly the full gap of bulk Nb (∆∼ 1.5 meV), but the tip gap is only about 0.67 meV, and decreases for Rn ≤ 10 kΩ.