Fig. S6. Numerical analysis for the normalized sixth-DOF torque, \(\left|d_{3}\right|\), of MMRs with different \(\varphi\). (A) Plotting the MMRs’ \(\left|d_{3,\text{highest}}\right|\) with respect to different \(\varphi\), whereby \(\left|d_{3,\text{highest}}\right|\) is seen to be maximized when \(\varphi\) = ±90° (indicated by the red dots). From the graph, the  \(\left|d_{3,\text{highest}}\right|\) of our MMR is 1.27×10-7 A m3. For existing MMRs with a harmonic magnetization profile that have  \(\varphi\) of 45°  \cite{Hu2018,Ren2021,sitti2021,Culha2020} and 0° \cite{sitti2014a,Zhang2018,diller2015,diller2016}, their computed  \(\left|d_{3,\text{highest}}\right|\) are 9.06 ×10-8 A m3 and 1.67× ×10-9 A m3 , respectively. (B) Evaluating the MMRs’ \(\left|d_{3}\right|\) when they deform under different \(B_{z,\{L\}}\). Three different MMRs have been analyzed: (i) our proposed MMR, and existing MMRs with a harmonic magnetization profile that have  \(\varphi\) of (ii) 45°  \cite{Hu2018,Ren2021,sitti2021,Culha2020} and (iii) 0° \cite{sitti2014a,Zhang2018,diller2015,diller2016}. The average  \(\left|d_{3}\right|\) of each MMR can be computed by finding their corresponding area under the curve and subsequently dividing those areas across the entire domain of \(B_{z,\{L\}}\).