8 In triangle $A B C$, $D$ is the midpoint of side $B C$. If $\overrightarrow{A D} \cdot \overrightarrow{A C}=0$, then the minimum value of $\tan C-\cot A$ is $\qquad$ .
8 Take the midpoint $E$ of $AB$, then $DE \parallel AC, \angle ADE = 90^{\circ}, \angle DAE = A - 90^{\circ}$, let $AC = b, AD = m$, then $\tan C = \frac{m}{b}, \tan \left(A - 90^{\circ}\right) = \frac{b}{2m}$, $-\cot A = \frac{b}{2m}$.
So
$$
\tan C - \cot A = \frac{m}{b} + \frac{b}{2m} \geqslant \sqrt{2},
$$

with equality if and only if $m = \frac{\sqrt{2}}{2} b$. Therefore, the answer is $\sqrt{2}$.