Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$?

$$\Gamma=\left( {\begin{array}{cc} I & B \\ B^{*} & I \\ \end{array} } \right)$$ Where $I$ is the identity matrix and the the moduli of entries of $B$ and those of its conjugate $B^{*}$ are all equal to $\frac{1}{\sqrt{d}}$. Also the blocks are all $\frac{d}{2}$by$\frac{d}{2}$. $\Gamma$ is a Gram matrix and further assume that the rows and columns are linearly independent. Hence we know that the lower bound is larger than zero but can we say anything more?

For simplicity we can assume the field of the matrix is real. Hence the entries of the off-diagonal blocks ($B$ and $B^{T}$) are $\pm\frac{1}{\sqrt{d}}$.

I appreciate any input very much!