6. In a cube $C$ with side length 1, construct a large inscribed sphere $O_{1}$, and then in one of the corners of $C$, construct a small sphere $O_{2}$, such that it is externally tangent to the large sphere and simultaneously tangent to three faces of the cube. Then the surface area of sphere $O_{2}$ is ( ).
(A) $(7-4 \sqrt{3}) \pi$
(B) $(7+4 \sqrt{3}) \pi$
(C) $\frac{2-\sqrt{3}}{2} \pi$
(D) $\frac{2+\sqrt{3}}{2} \pi$
6.A.

As shown in Figure 4, let the radius of sphere $\mathrm{O}_{2}$ be $r$, and assume sphere $O_{2}$ is located within $\angle D^{\prime}$. Then $O_{1}$ and $O_{2}$ lie on the diagonal $B D^{\prime}$. Let $\angle A D^{\prime} B=\theta$, then $\sin \theta=\frac{1}{\sqrt{3}}$. In $\triangle D^{\prime} E O_{2}$, $D^{\prime} O_{2}=\frac{r}{\sin \theta}=\sqrt{3} r, O_{1} O_{2}=r+\frac{1}{2}$. Therefore,
$$
\begin{array}{l}
2\left[\sqrt{3} r+\left(r+\frac{1}{2}\right)\right]=B D^{\prime}=\sqrt{3} \\
r=\frac{2-\sqrt{3}}{2}, 4 \pi r^{2}=(7-4 \sqrt{3}) \pi
\end{array}
$$