The extreme quantum limit in lightly-doped SrTiO$_{3}$*

When a three dimensional electron gas is placed in a sufficiently strong magnetic field, it is said to be in the quantum limit when the cyclotron energy $\hbar \omega_{c}$ \textgreater $\varepsilon_{F}$ \textgreater \textgreater kT, and all of the electrons occupy the lowest Landau level. Achieving this limit in a material requires a small Fermi energy relative to the applied magnetic field, and a weak disorder potential such that magnetic freeze-out is avoided. We present an experimental study of lightly-doped single crystals of SrTiO$_{3}$, which remain good bulk conductors in temperatures down to 25 mK and magnetic fields up to 45 T. Our measurements probe deep into the quantum limit, where $\hbar \omega _{c}$ \textgreater \textgreater $\varepsilon_{F}$ and theory has long predicted that electron-electron interactions can drive the system into a charge density wave or Wigner crystal like state. A number of interesting features arise in electrical transport in this regime, including a striking re-entrant nonlinearity in the current-voltage characteristics. We discuss these features in the context of possible correlated electron states, and present a picture based on magnetic field induced puddling of electrons in a disorder potential landscape.