Spin-polarized tunneling from a Kondo insulator: equilibrium surface magnetism versus magneto-electric response in SmB$_6$

We theoretically analyze the spin-polarized tunneling between SmB$_6$ tip and the bicollinear antiferromagnet Fe$_{1+x}$Te. The emergence of magnetic contrast in tunneling requires time-reversal symmetry breaking on the surface of the tip. The magnetic contrast part of the tunneling conductance then has two contributions: the first one originates from the magnetic moment due to the equilibrium long-range order on the surface, while the other is governed by the magnetic moment induced by the electric field of the junction through the surface magneto-electric effect. We show that the ratio of the equilibrium moment to the response extracted near zero bias equals, up to the sign, twice the bias at which the contrast changes sign, independently of the relative strength of the two mechanisms, so that this ratio itself does not measure the strength of the magneto-electric effect. We find that within the experimental energy window $|eV|\lesssim10$~meV the {bias-dependent} field-induced contribution is at most a few percent of the equilibrium one, and the two become comparable only at biases of order $100$~meV or higher. The ratio of the field-induced to the equilibrium contribution is inversely proportional to the distance between the tip and the substrate.