A decrease of the critical current density J$_{c}$ with increasing film thickness d was observed in thin superconducting films of YBa$_{2}$Cu$_{3}$O$_{7}$$_{−}$$_{δ}$ on both single 1,2 and polycrystalline 3–5 sub- strates. Possible explanations are a degradation of the film or a change in the defect structure with thickness, but also the self-field of the sample causes a significant reduction if J$_{c}$ de- pends on the low magnetic induction B generated by the transport current $^{6}$. Identifying the true cause of the decrease is vital for the development of coated conductors, which must be grown to higher thicknesses to enhance their performance. 

We employ numerical calculations of the current density distribution in a thin film to derive a practical approximation allowing the evaluation of the self-field depression of J$_{c}$ for a given J$_{c}$ ( B ) dependence. Therefore, it is essential to distinguish between the intrinsic J$_{c}$ ( B ) of the material, i.e., the (local) dependence of the critical current density on the magnetic induction B in the film, and the average critical current density as a function of the external applied field J$_{c}$ ( H ), which is determined in a transport measurement of the critical current. Note, that J$_{c}$ ( B ) provides information on pinning in the material and bridges theory and experiment. 

The procedure we employ is similar to the iterative algorithm used by Rostila et al. 6 At the beginning the external applied field determines the starting current density distribution J$_{c}$ ( B ( x,y ) = µ$_{0}$H ), which is constant across the cross-section of the film. After calculat- ing the self-field distribution B$_{sf}$ ( x,y ) in the sample the current densities are updated to J$_{c}$ ( B ( x,y ) = µ$_{0}$H + B$_{sf}$ ( x,y )), which results in a new self-field distribution. This step is iterated until current density and magnetic induction satisfy the (arbitrary) material law J$_{c}$ ( B ) at every position in the film. For comparison with experiment J$_{c}$ ( H ) is computed by calculating the average current density flowing through the conductor at a range of external applied fields. 

Earlier 7 we used this procedure to extrapolate the power-law J$_{c}$ ( B ) ∝ B − α dependence, which is commonly observed in thin films of YBCO at high fields, to the lowest fields and showed that despite the divergence of J$_{c}$ ( B ) the film carries a finite critical current at zero applied field because of the self-field of the sample. This result (see Fig. 1) is particularly suited for our purpose, because the clear deviation between J$_{c}$ ( B ) and J$_{c}$ ( H ) at low fields emphasizes important features of the J$_{c}$ ( H ) curve: Starting from zero applied field, J$_{c}$ ( H ) remains approximately constant at the self-field critical current density J$_{sf}$ ≡ J$_{c}$ ( H = 0) as long as the applied field is negligible compared to the self-field of the sample. At high 

