When are the join-irreducibles in a complete lattice join-dense?

A subset $$A$$ of a complete lattice $$L$$ is said to be join-dense if $$L=\{\bigvee R|R\subseteq A\}$$. An element $$a\in L$$ is said to be join-irreducible if $$a\neq 0$$ and if $$a=x\vee y$$ then $$a=x$$ or $$a=y$$. Is there a nice necessary and sufficient condition for when the join-irreducible elements of a complete lattice are join-dense, or is there a representation theorem for such lattices?

Of course, if $$L$$ is a complete lattice and $$A$$ is the collection of all join-irreducibles, then we may ''shrink'' the lattice $$L$$ to the lattice $$\{\bigvee R|R\subseteq A\}$$ so that the join-irreducibles are join-dense. I know that the distributive complete lattices where the join-irreducibles are join-dense are precisely the spatial coframes (the spatial coframes are the lattices isomorphic to the closed sets in some topological space). Furthermore, if $$L$$ is a complete lattice satisfying DCC, then the join-irreducible elements in $$L$$ are join-dense in $$L$$. However, none of these ideas characterizes the complete lattices where the join-irreducibles are join-dense.