Magnetic semiconductors from functionalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Cr</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> </mml:math> MXenes

We report an ab initio investigation of functionalized and $3d$-electron-doped ${\mathrm{Cr}}_{2}\mathrm{C}$ MXenes. Upon functionalization, the ${\mathrm{Cr}}_{2}\mathrm{C}$ becomes chemically, dynamically, and mechanically stable, and it exhibits magnetic semiconducting behavior. ${\mathrm{Cr}}_{2}{\mathrm{CF}}_{2}$ stands out as a wide band gap semiconductor, possessing superexchange interaction mediated by F atoms within the layer; however, the applied strain transforms it from an indirect to a direct band gap semiconductor. Strong spin-phonon coupling found in ${\mathrm{Cr}}_{2}{\mathrm{CH}}_{2}$ is supported by the distorted Cr spin density due to the hydrogen environment. Two magnon branches, associated with two sublattice spins, are found in the ferromagnetic ${\mathrm{Cr}}_{2}{\mathrm{CO}}_{2}$ and antiferromagnetic ${\mathrm{Cr}}_{2}{\mathrm{CF}}_{2}$. Depending on the types of $3d$-electron dopants and functionalization, ${\mathrm{Cr}}_{2}\mathrm{C}$ MXenes (except for ${\mathrm{Cr}}_{2}{\mathrm{CO}}_{2}$) change from an indirect band gap magnetic semiconductor to different states of electronic and magnetic matter, including an exotic direct band gap magnetic bipolar semiconductor. In addition, we reveal a band inversion between the two highest valence bands in the Fe-doped ${\mathrm{Cr}}_{2}{\mathrm{CCl}}_{2}$.