There's no reason I can come up with that it shouldn't be possible to do through an attribute. It might be in Microsoft's backlog. Who knows.
The best solution I have found is to use the defaultValueSql parameter in the code first migration.
CreateTable(
"dbo.SomeTable",
c => new
{
TheDateField = c.DateTime(defaultValueSql: "GETDATE()")
});
I don't like the often reference solution of setting it in the entity class constructor because if anything other than Entity Framework sticks a record in that table, the date field won't get a default value. And the idea of using a trigger to handle that case just seems wrong to me.