Multiple dimensions of spin-gapless semiconducting states in tetragonal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Sr</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>CuF</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:mrow></mml:math>

Spin-gapless semiconductors (SGSs), with intrinsic magnetism, $100%$ spin polarization, and zero-gap band crossing points, have attracted great scientific interest owing to their potential applications in spintronics. In this Letter, using first-principles calculations and symmetry analysis, we demonstrate that the realistic material ${\mathrm{Sr}}_{2}{\mathrm{CuF}}_{6}$ is a spintronic material with multiple dimensions of spin-gapless semiconducting states. Tetragonal ${\mathrm{Sr}}_{2}{\mathrm{CuF}}_{6}$ has a zero-dimensional zero-gap nodal point, a one-dimensional zero-gap nodal line, and a two-dimensional nearly zero-gap nodal surface in one spin direction. Moreover, it hosts a wide band gap in the other spin direction. Our results extend the SGS members from nodal point SGSs and nodal line SGSs to nodal surface SGSs. Furthermore, we report a SGS candidate in an experimentally realized material exhibiting different dimensions of zero-gap points in momentum space. It is hoped that spintronic materials with multiple dimensions of spin-gapless semiconducting states may have significant applications in new-generation spintronics.

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