When we see half moon, why is it always the lower half?

Actually, you sometimes can see the Moon illuminated by the Sun from the top, and therein lies the answer to your question - during day. When looking at the Moon during night time, the side of the Earth where you're standing and looking at it is pointed away from the Sun, while this is reversed during the day. The illuminated side of the Moon is also facing the Sun, of course. So we have:

So, the direction from which the Moon is illuminated from is the same as the direction the Earth is, both illuminated by the same celestial body - our Sun. So, when you see the Sun high above the skies, and if the Moon is visible during daytime, it's going to be illuminated from the same direction from which you are, and during night time, when you'd expect the Sun "below where you're standing", it's illuminated from that direction.