Geometric crystal and tropical R for $D_n^{(1)}$

We construct a geometric crystal for the affine Lie algebra $D_n^{(1)}$ in the sense of Berenstein and Kazhdan. Based on a matrix realization including a spectral parameter, we prove uniqueness and explicit form of the tropical R, the birational map that intertwines products of the geometric crystals. The tropical R commutes with geometric Kashiwara operators and satisfies the Yang-Baxter equation. It is subtraction-free and yields a piecewise linear formula of the combinatorial R for crystals upon ultradiscretization.