  The electrical conductivity of a material can feature subtle, nontrivial, and
spatially-varying signatures with critical insight into the material's
underlying physics. Here we demonstrate a conductivity imaging technique based
on the atom-sized nitrogen-vacancy (NV) defect in diamond that offers local,
quantitative, and noninvasive conductivity imaging with nanoscale spatial
resolution. We monitor the spin relaxation rate of a single NV center in a
scanning probe geometry to quantitatively image the magnetic fluctuations
produced by thermal electron motion in nanopatterned metallic conductors. We
achieve 40-nm scale spatial resolution of the conductivity and realize a
25-fold increase in imaging speed by implementing spin-to-charge conversion
readout of a shallow NV center. NV-based conductivity imaging can probe
condensed-matter systems in a new regime, and as a model example, we project
readily achievable imaging of nanoscale phase separation in complex oxides.
