Legal Custody and Physical Custody

Alright. Long story short. My son is 13 months old. When I found out I was pregnant, the father and I were actually split up. When I told him I was pregnant, he made no effort to have any part. He never came to ANY of the appointments except ONE when they determined the sex. After Draven was born, he was around for about 2 months, then just up and took off for 3 months, and repeated this for most of Dravens life. I've kept a journal of times he's done things that parents just shouldnt do. (i.e. leaving us stuck at the house with no car and no formula because he was mad, or failing to pick Draven up from his grandmothers when he had already said he was going to... etc) He's always been extremely irresponsible when it came to being a parent to Draven, and he only seems to be getting worse. His mother pays more attention to Draven and important things like his doctors appointments etc. I don't want to keep him from seeing or having a relationship from his son, but I dont think he's mature enough or should have any say in certain things in Dravens life. I was wondering if it were possible to have Sole Legal Custody with Shared physical....