Fig. 1 Schematic diagram of the cylindrical crucible with platforms: a 3:1; b 2:1 (The red and the blue dotted lines indicate the observation positions of the cross and the longitudinal sections, respectively)

Fig. 2 Solidified microstructure and the EBSD images of the transverse section at 5 mm above the 3:1 platform (The red dotted lines mark the boundary between the stray-crystal and the matrix single-crystal in the figure. The black dotted lines represent the enlarged view of the stray-crystal and the matrix single-crystal areas.): a1-a2 without magnetic field; b1-b2 CMF; c1-c2 LMF

Fig. 3 Stray-crystal ratio in the cross section at 5 mm above the 3:1 platform

Fig. 4 Microstructure of longitudinal section under the different configurations of magnetic fields above the 3:1 platform (The red dotted lines mark the boundary between the stray-crystal and the matrix single-crystal. The black dotted lines represent an enlarged view of some of the stray-crystal areas.): a1-a2 without magnetic field; b1-b2 CMF; c1-c2 LMF

Fig. 5 EBSD images of the transverse section at 5 mm above the 2:1 platform: a without magnetic field; b CMF; c LMF

Fig. 6 Stray-crystal ratio in the cross section at 5 mm above the 2:1 platform

Fig. 7 Flow field in the melt when the directional solidification proceeds to the 3:1 platform: a without magnetic field; b CMF; c LMF

Fig. 8 Schematic diagram of magnetic line distribution of a CMF and b LMF (The white platform shape indicates the solidified sample)

Fig. 9 Schematic diagram of the TEMF acting on the liquid-solid interface at the 3:1 platform: a CMF and b LMF (θ is the angle between the thermoelectric current and magnetic field)

Fig. 10 Evolution of the temperature field on the 3:1 platform: a without magnetic field; b CMF; c LMF

Fig. 11 Temperature difference between the middle and the corner as the directional solidification reaches to the 3:1 platform

Fig. 12 Schematic diagram of the evolution of TEMF at the liquid–solid interface with different platform sizes: a CMF; b LMF (The solid black lines represent 3:1 sample; the red dotted lines represent 2:1 sample; the blue lines represent magnetic lines of force; the arrows represent the magnetic field direction; ${\overrightarrow{B}}_{{\text{H}}}$ represents the strength of the horizontal component, indicating that it is getting stronger)