We investigate some possible symmetries of the superconducting state that emerges in three-dimensional altermagnets in the presence of spin-orbit coupling. We demonstrate within a weak-coupling approach that these altermagnets, which naturally possess an order modulated by a vector form factor ${\mathit{g}}_{\mathbf{k}}$, favor spin-triplet superconductivity described by gap functions given by $\mathit{d}\left(\mathbf{k}\right)=\mathit{u}\left(\mathbf{k}\right)\times {\mathit{g}}_{\mathbf{k}}$, where $\mathit{u}\left(\mathbf{k}\right)=-\mathit{u}(-\mathbf{k})$. Consequently, this singles out $f$-wave spin-triplet superconductivity as the most favorable pairing state to appear in the vicinity of $d$-wave altermagnetism. Furthermore, we obtain that the combination of spin-singlet superconducting states with altermagnetism gives rise to Bogoliubov-Fermi surfaces, which are protected by a ${\mathbb{Z}}_2$ topological invariant. Using a Ginzburg-Landau analysis, we show that, for a class of spin-orbit coupled altermagnetic models, a superconducting phase is expected to appear at low temperatures as an intertwined $d+if$ state, thus breaking time-reversal symmetry spontaneously.

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• Geometric & topological phases
• Phase diagrams
• Quantum phase transitions
• Spin-orbit coupling
• Spin-singlet pairing
• Spin-triplet pairing
• Superconducting phase transition
• Symmetry protected topological states
• Topological superconductors
• Altermagnets
• Unconventional superconductors
• Bogoliubov-de Gennes equations
• Landau-Ginzburg theory