We have derived an expression of the Dzyaloshinskii–Moriya interaction (DMI), where all the three components of the DMI vector can be calculated independently, for a general, non-collinear magnetic configuration. The formalism is implemented in a real space—linear muffin-tin orbital—atomic sphere approximation (RS-LMTO-ASA) method. We have chosen the Cr triangular trimer on Au(111) and Mn triangular trimers on Ag(111) and Au(111) surfaces as numerical examples. The results show that the DMI (module and direction) is drastically different between collinear and non-collinear states. Based on the relation between the spin and charge currents flowing in the system and their coupling to the non-collinear magnetic configuration of the triangular trimer, we demonstrate that the DMI interaction can be significant, even in the absence of spin-orbit coupling. This is shown to emanate from the non-collinear magnetic structure, that can induce significant spin and charge currents even with spin-orbit coupling is ignored.

我们导出了Dzyaloshinskii-Moriya相互作用（DMI）的表达式，其中对于一般的非共线磁性配置，可以独立计算DMI向量的所有三个分量。形式主义是在真实空间中（线性松饼-锡轨道-原子球近似）（RS-LMTO-ASA）方法实现的。作为数值示例，我们选择了Au（111）上的Cr三角形三聚体和Ag（111）和Au（111）上的Mn三角形三聚体。结果表明，共线状态和非共线状态的DMI（模块和方向）截然不同。根据系统中的自旋电流和电荷电流之间的关系及其与三角形微调器的非共线磁性配置的耦合关系，我们证明DMI相互作用非常重要，即使没有自旋轨道耦合。事实证明，这是由于非共线磁性结构产生的，即使忽略了自旋-轨道耦合，该结构也可能引起明显的自旋和充电电流。