What to write in the email for sending PhD application to a research group's head?
Your CV has your background. Hopefully you have written an SOP (Statement of purpose) giving your goals for grad study and beyond. In the email you can probably best help yourself writing about why you want to work with him, but make it honest and not too "over the top". But a bit more than just your last suggestion. How does this group fit your SOP? 

If he gets a lot of applications he will quickly sort them into two piles. You want to be in the pile for "further consideration".