Vacua by Derivative Corrections in $\mathcal{N}=1$ Supergravity with Matter Multiplets

We study the vacuum structures of four-dimensional $\mathcal{N}=1$ old minimal supergravity with higher derivative corrections. We find that $\mathcal{N}=1$ supergravity with Riemann curvature square corrections and higher derivative matter chiral multiplets induces a non-trivial de Sitter vacuum, even in the absence of superpotentials. This vacuum generically breaks supersymmetry. We show that the auxiliary fields in the gravity and the chiral multiples play important roles to generate a potential in supersymmetric higher derivative theories.