Scaling of the microwave magneto-impedance in Tl${}_2$Ba${}_2$CaCu${}_2$O${}_{8+\delta }$ thin films

We present measurements of the magnetic field-induced microwave complex resistivity changes at 47 GHz in Tl${}_2$Ba${}_2$CaCu${}_2$O${}_{8+\delta }$ (TBCCO) thin films, in the ranges 58 K$<T<T_c$ and 0$<{\mu }_{0}H<$0.8 T. The large imaginary part $\mathrm{\Delta }{\rho }_2(H)$ points to strong elastic response, but the data are not easily reconciled with a rigid vortex model. We find that, over a wide range of temperatures, all the pairs of curves $\mathrm{\Delta }{\rho }_1(H)$ and $\mathrm{\Delta }{\rho }_2(H)$ can be collapsed on a pair of scaling curves $\mathrm{\Delta }{\rho }_1[H\mathrm{/}{H}^{\ast }(T)]$, $\mathrm{\Delta }{\rho }_2[H\mathrm{/}{H}^{\ast }(T)]$, with the same scaling field $H^{\ast }(T)$. We argue that $H^{\ast }(T)$ is related to the loss of vortex rigidity due to a vortex transformation.