Fig. 3: Comparison between experiment and theory.

a Interfacial nuclear-spin current and thermal equilibration of nuclear spins in MnCO₃. An interfacial spin current, J_ne, is mediated by the Korringa process through the hyperfine interaction between nuclear spins of ⁵⁵Mn and electron spins in the metal at the Pt/MnCO₃ interface. J_ne arises in proportion to the effective temperature difference between the electrons in Pt (T_e) and nuclei in MnCO₃ (T_n): \({J}_{{\rm{ne}}}={\Gamma }_{{\rm{ne}}}{k}_{\mathrm{B}}({T}_{\mathrm{e}}-{T}_{\mathrm{n}})\). Here, the difference T_e − T_n may be triggered by the interfacial temperature drop T_e − T_p between the Pt and MnCO₃ (T_p: phonon temperature in MnCO₃ close to the interface) and the thermalization between nuclei and phonons in MnCO₃, whose rate is given by \({J}_{{\rm{np}}}={\Gamma }_{{\rm{np}}}{k}_{\mathrm{B}}({T}_{\mathrm{n}}-{T}_{\mathrm{p}})\). b B dependence of the calculated temperature difference \({T}_{\mathrm{e}}-{T}_{\mathrm{n}}\) normalized by the interfacial temperature drop \({T}_{\mathrm{e}}-{T}_{\mathrm{p}}\) at T = 100 mK and 1 K. In the steady state, \({J}_{{\rm{ne}}}={J}_{{\rm{np}}}\)³¹, which gives \(({T}_{\mathrm{e}}-{T}_{\mathrm{n}})/({T}_{\mathrm{e}}-{T}_{\mathrm{p}})={\Gamma }_{{\rm{np}}}/({\Gamma }_{{\rm{np}}}+{\Gamma }_{{\rm{ne}}})\). B_c indicates the crossover field, where \({\Gamma }_{{\rm{ne}}}={\Gamma }_{{\rm{np}}}\). c Comparison between the B dependence of the experimental V/I²_rms (blue plots) for the Pt/MnCO₃ Device 2 and the calculated V/I²_rms for the nuclear SSE \({{\mathscr{S}}}_{{\rm{n}}}\) (red solid curve) and for the electronic SSE \({{\mathscr{S}}}_{{\rm{m}}}\) (gray solid curve) at T = 101 mK (see Supplementary Note 9 for details). The inset shows a blowup of the calculated V/I²_rms for the electronic SSE \({{\mathscr{S}}}_{{\rm{m}}}\) (multiplied by 10³). d Comparison between the B dependence of the experimental V/I²_rms (blue plots) and the calculated V/I²_rms for the nuclear SSE \({{\mathscr{S}}}_{{\rm{n}}}\) (red solid line) at 100 mK < T < 1 K. e Comparison between the T dependence of the experimental V/I²_rms (green rhombus) and the calculated V/I²_rms for the nuclear SSE \({{\mathscr{S}}}_{{\rm{n}}}\) (red solid curve). The error bar represents the SD.