  3D topological insulators are characterized by an insulating bulk and
extended surface states exhibiting a helical spin texture. In this work, we
investigate the hyperfine interaction between the spin-charge coupled transport
of electrons and the nuclear spins in these surface states. Previous work has
predicted that in the quantum spin Hall insulator phase, work can be extracted
from a bath of polarized nuclear spins as a resource. We employ nonequilibrium
Green's function analysis to show that a similar effect exists on the surface
of a 3D topological insulator, albeit rescaled by the ratio between electronic
mean free path and device length. The induced current due to thermal relaxation
of polarized nuclear spins has an inductive nature. We emphasize the inductive
response by rewriting the current-voltage relation in harmonic response as a
lumped element model containing two parallel resistors and an inductor. In a
low-frequency analysis, a universal inductance value emerges that is only
dependent on the device's aspect ratio. This scaling offers a means of
miniaturizing inductive circuit elements. An efficiency estimate follows from
comparing the spin-flip induced current to the Ohmic contribution. The
inductive effect is most prominent in topological insulators which have a large
number of spinful nuclei per coherent segment, of which the volume is given by
the mean free path length, Fermi wavelength and penetration depth of the
surface state.
