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The temperature T s of spontaneous creation of vortex lines in Josephson coupled layered superconductors is obtained by taking into account the entropy contribution to the free energy due to thermal distortions. For Bi-and Tl-based high-T c superconductors and superlattices, the superconducting critical temperature T s lies noticeably below the mean-field transition temperature T c o (T c o -T s « 4 K or more). The contribution of thermal distortions to the free energy causes significant changes in the temperature and field dependences of magnetization below T s seen in Bi compounds.PACS numbers: 74.60.GeThe importance of topological excitations in the thermodynamics of two-dimensional (2D) magnets and superconductors was first discussed by Berezinskii and Kosterlitz and Thouless [1]; for a review see [2]. Above the Kosterlitz-Thouless (KT) transition temperature TKT> which lies below the mean-field transition temperature T c o, thermally induced free vortices are present in 2D superconductors giving rise to dissipation and persistentcurrent decay. The spontaneous creation of free vortices above TKT occurs due to the entropy contribution to the free energy F = e-Ta, where e = (<^d/167r 2 A 2 ) ln(iJ/£) and a = ln(i2 2 /£ 2 ) are the vortex energy and its entropy (cj)o is the flux quantum, A is the London penetration depth, £ is the coherence length, TTR 2 and d are the system area and thickness). Above TKT? which is given implicitly by ld 32TT 2 A 2 (T)'(1) the free energy is negative, and spontaneous creation of vortices is favorable.Generalizations of the KT approach to layered (but still three-dimensional) high-T c superconductors [3][4][5] were stimulated by experiments that show the KT scaling behavior of resistivity; Bi-2:2:1:2 and superlattices YBaCuO/PrBaCuO are perhaps the best example (see review [6] and references therein). The idea behind the generalizations was brought about by the observation that in the absence of interlayer superconducting coupling, 2D vortices of different layers interact only via the magnetic field, which provides the same logarithmic dependence of the vortex energy e on the sample size as in 2D superconductors [5,7]. This made feasible the KT transition [3-5], i.e., spontaneous creation of 2D vortices and antivortices within the same layer above the temperature obtained by solving Eq. (1) with d replaced by the interlayer distance s. However, even in the case of weak Josephson coupling [8], e increases linearly with R [9], while the entropy a of a 2D vortex still changes only logarithmically. In these circumstances, thermally induced spontaneous creation of free 2D vortices (in the low concentration limit) is virtually impossible.In the following, taking into account the entropy contribution to the free energy due to thermal distortions of vortices, we show that thermally induced spontaneous creation of vortex lines should occur in Josephson coupled layered materials above some temperature T s < T c $. We estimate the difference T c o-T s to be as high as 5-10 K for Bi-and T...