matrix – Search function of the nth min value UNIQUE in a list?

I have this matrix …

mat = Table[{k, 0}, {k, 0, limit}];

After a series of dynamic iterative operations, some of the 0 initialized inputs (that is, those of the 2nd column of matte, ie matte[[All, 2]]) are replaced by positive integers, for example,

mat = {{0,78}, {1,0}, {2,0}, {3,11}, {4,11}, {5,11}, ..., {limit, 0}}

I realize that there are several standard methods for recovering the nth minimum value of matte[[All, 2]]but, since mat has so many 0, they do not help me.

I am looking for a standard / elegant / efficient way to find the nth UN value min of mat, along its associated line index, for example, if limit = 6 & i wanted to find the second value / index pair, by running .. .

Impression[UniqueMin[carpet[UniqueMin[mast[UniqueMin[tapis[UniqueMin[mat[[All, 2]], 2]];

would be out …

{4,11}

Bc 11 was the second smallest single value of the carpet and row 4 was the one where it was first.