Thermally activated domain wall (DW) motion in magnetic insulators has been considered theoretically, with a particular focus on the role of Dzyaloshinskii-Moriya Interaction (DMI) and thermomagnonic torques. The thermally assisted DW motion is a consequence of the magnonic spin current due to the applied thermal bias. In addition to the exchange magnonic spin current and the exchange adiabatic and the entropic spin transfer torques, we also consider the DMI-induced magnonic spin current, thermomagnonic DMI field-like torque and the DMI entropic torque. Analytical estimations are supported by numerical calculations. We found that the DMI has a substantial influence on the size and the geometry of DWs, and that the DWs become oriented parallel to the long axis of the nanostrip. Increasing the temperature smoothes the DWs. Moreover, the thermallyinduced magnonic current generates a torque on the DWs, which is responsible for their motion. From our analysis it follows that for a large enough DMI the influence of DMI-induced field-like torque is much stronger than that of the DMI and the exchange entropic torques. By manipulating the strength of the DMI constant, one can control the speed of the DW motion, and the direction of the DW motion can be switched, as well. We also found that DMI not only contributes to the total magnonic current, but also it modifies the exchange magnonic spin current, and this modification depends on the orientation of the steady state magnetization. The observed phenomenon can be utilized in spin caloritronics devices, for example in the DMI based thermal diodes. By switching the magnetization direction, one can rectify the total magnonic spin current.
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