6. Let the right directrix of the hyperbola intersect the two asymptotes at points $A$ and $B$. The circle with diameter $AB$ passes through the right focus $F$. Then the eccentricity of the hyperbola is ( ).
(A) $\sqrt{2}$
(B) $2 \sqrt{2}$
(C) $\sqrt{3}$
(D) $2 \sqrt{3}$
6.A.

Let the equation of the hyperbola be $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. Then the equation of the right directrix is $x=\frac{a^{2}}{c}$, and the equations of the asymptotes are $y= \pm \frac{b}{a} x$.
From this, the coordinates of points $A$ and $B$ are $\left(\frac{a^{2}}{c}, \frac{a b}{c}\right)$ and $\left(\frac{a^{2}}{c},-\frac{a b}{c}\right)$, respectively.
Thus, $|A B|=\frac{2 a b}{c}$.
Let the intersection of the right directrix with the $x$-axis be point $K$. Then $|K F|=c-\frac{a^{2}}{c}=\frac{b^{2}}{c}$.
Also, $|K F|=\frac{1}{2}|A B|$, so $\frac{b^{2}}{c}=\frac{a b}{c}$.
Therefore, $a=b$, which means $e=\sqrt{2}$.