Barb your question is a little confusing. The only mulches that wouldn't break down would be inorganic ones. All mulches act as a barrier but none are perfect. If you are looking for a mulch that breaks down quickly to add organic matter to your surrounding soil I would suggest a leaf mulch. A wood chip mulch takes longer to break down but requires less maintenance.

I think I know what you mean, Barb.

IMO, mulch that acts as a barrier is mulch that is too thick.

For instance, I had a spot in my garden where I put very thick loads of leaves. Lovely leaves, but I overdid it. And instead of planting things, I went beserk. Must. Have. More. Leaves. A year later, the beautiful, amazing soil I was expecting was neither beautiful nor amazing. I had two distinct layers: the soil, which was not particularly great, and then the broken down leaves which were water repellent due to drought. Very disappointing.

It would be easier to point out which mulch material does not decompose, things such as stone. Otherwise all carbon based, organic, material would be digested (decomposed) provided the soil they were laid on had an active Soil Food Web.

Shredded leaves, grass clippings, wood chips, bark, etc. are all materials used for mulching and the all will be digested, eventually, if the soil is a good healthy soil with an active Soil Food Web. To be an effective "weed" barrier a mulch needs to be relatively thick, 4 to 6 inches not just 1 or 2 inches, and to do the other things mulches do the same depth applies. Of course, at the appropriate depth a mulch might then become a barrier to a light application of water, and if applied too thickly that mulch could also be a barrier to the soil needed air and then become an anaerobic soil which is not good to have.

Common sense always applies, enough mulch to aid in unwanted plants suppression, moisture retention, soil temperature control, and feeding the Soil Food Web without being detrimental to the health of the soil.

Well, if we are talking about plant-based material - the woodier and bulkier the material the longer it will take to break down. For example:

Wood chips last longer than shredded wood (more bulk)

Shredded wood lasts longer than leaves (woodier)

Whole leaves last longer than shredded leaves (more bulk)

If you want to encourage a woodier material to break down faster mix a nitrogen-rich material into it, like coffee grounds, grass clippings, blood meal or urine.

Thanks so much for the help. I got something from each post. Appreciate them. Barb