Fig. 4: Theoretical calculations of the energy difference between the spin-polarized and non-polarized electron energy entering the HM layer.
a Scheme of an iDMI at the interface between the HM (blue) and FM (gray). The iDMI vector (D12) is related to the triangle composed of two magnetic sites and an atom with a large SOC. The scheme represents the oscillating magnitude of D12 with corresponding spin tilting. b Energy difference (equal to the absolute value of the iDMI surface energy) calculated for the Pt/MgO/CoFeSiB system. c Shift of the oscillation-like behavior due to the assumption of a quantization condition Eq. (4) of \({k}_{3,1}{L}_{3,1}+{k}_{3,2}{L}_{3,2}+{k}_{3,3}{L}_{3,3}+\ldots =2\pi q+\varphi\) expressed in terms of the out-of-plane components of the circular wavenumbers k3,n and thickness L3,n of all three constituent layers (n = 1, 2, 3). The black curve represents φ = 0˚, while the red and blue curves represent φ = π/4 and φ = −π/4, respectively. The energy difference is displayed in terms of absolute value. d Theoretically calculated influence of the spacer nature on iDMI, where the black and red curve represents the energy difference for our sample (EF –EMgO = 9 × 10−21 J) and more conductive spacer sample (EF–Econductive spacer = 12 × 10−21 J), respectively.
