A note on the non-abelian tensor square of p-gorups

‎Let $G$ be a non-abelian $d$-generator finite $p$-group of order $p^{n}$ with $|G^{ }|=p^{k}$‎. ‎In 1991‎, ‎Rocco prove that $|G\otimes G|\leq p^{n(n-k)}$ that depends on $n‎, ‎k$‎. ‎In 2016‎, ‎Jafari proved that $|G\otimes G|\leq p ^{(n-1)d+2}$ that depends on $n,d$‎. ‎In this paper‎, ‎we obtain a new upper bound in terms of $n‎, ‎k,$ and $d$‎. ‎In fact‎, ‎we prove $|G\otimes G|\leq p^{(n-k)^2+d(n+k-2)+4}$‎, ‎for $p\neq 2$ and find the structure of all $p$-groups that attains the mentioned bound‎.