We provide an explicit construction of finite 4-regular graphs (Γk)k∈N with girthΓk→∞ as k→ ∞ and diamΓkgirthΓk⩽D for some D> 0 and all k∈ N. For each fixed dimension n⩾ 2 , we find a pair of matrices in SL_n(Z) such that (i) they generate a free subgroup, (ii) their reductions modp generate SL_n(F_p) for all sufficiently large primes p, (iii) the corresponding Cayley graphs of SL_n(F_p) have girth at least c_nlog p for some c_n> 0. Relying on growth results (with no use of expansion properties of the involved graphs), we observe that the diameter of those Cayley graphs is at most O(log p). This gives infinite sequences of finite 4-regular Cayley graphs of SL_n(F_p) as p→ ∞ with large girth and bounded diameter-by-girth ratio. These are the first explicit examples in all dimensions n⩾ 2 (all prior examples were in n= 2). Moreover, they happen to be expanders. Together with Margulis’ and Lubotzky–Phillips–Sarnak’s classical constructions, these new graphs are the only known explicit logarithmic girth Cayley graph expanders.