Is it permissible to correspond with a relative via email and speak with him over the phone with the intention of brotherhood if he is younger?

What you are doing is a sin, for righteousness is good character, and sin is what wavers in the breast and what you would dislike for people to know about. The fundamental principle is for a woman to remain in her home and avoid mixing with men, even through writing and correspondence, as Allah the Almighty says: "And abide in your houses and do not display yourselves as [was] the display of [the former] ignorance." Therefore, it is not permissible for her to mix with non-mahram men in any form, even through speech, except out of necessity and with security from temptation; due to His saying, "Then do not be soft in speech [to men], lest he in whose heart is disease should covet." It does not matter whether he is a relative, younger, or older; what matters is that he is a non-mahram man.