  Tang-Zhang, Choe-Hoppe, showed independently that one can produce minimal
submanifolds in spheres via Clifford type minimal product of minimal
submanifolds. In this note, we show that the minimal product is immersed by its
first eigenfunctions (of its Laplacian) if and only if the two beginning
minimal submanifolds are immersed by their first eigenfunctions. Moreover, we
give estimates of Morse index and nullity of the minimal product. In
particular, we show that the Clifford minimal submanifold
$\left(\sqrt{\frac{n_1}{n}}S^{n_1},\cdots,\sqrt{\frac{n_k}{n}}S^{n_k}\right)\subset
S^{n+k-1}$ has index $(k-1)(n+k+1)$ and nullity $(k-1)\sum_{1\leq i<j\leq
k}(n_i+1)(n_j+1)$ (with $n=\sum n_j$).
