Convert image color mode to monochrome (bitmap mode) with transparent Background

I searched and tried, but without success.
I want to convert linedrawing images to black and white/monochrome (in photoshop called "bitmap mode") with a threshold of 50% to create images like woodcuts that have a transparent background.

A conversion to a png image would support that, but this is no solution, as long as the color mode of the image is still greyscale instead of "bitmap mode/monochrome" or with another kind of transparent background outside the line drawings.

So how can I convert images into that real black/white color mode and save them as png files or get that result in another way?