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We construct a four-leg spin-1/2  t – J  type model to simulate a doped two-leg spin-1 antiferromagnetic Heisenberg ladder. Employing renormalized mean-field theory with simple Gutzwiller factors, we obtain three degenerate superconducting states with different pairing symmetry. Through improving the Gutzwiller factors, we find that the state C with inter-layer modified    d y 2 − z 2 -wave pairing has the lowest energy in a large doping range. Besides, we use the density matrix renormalization group method to solve the model. The negative binding energy reveals the pairing tendency, and the pair–pair correlation functions exhibit a slowly decaying behavior on certain types of bonds. From the pair correlations, we confirm the inter-layer modified    d y 2 − z 2 -wave superconducting state as the ground state of the model.

Inter-layer modified d y 2 − z 2 -wave superconductivity in an effectively doped spin-1 ladder