Context. One of the main scenarios of planet formation is the core accretion model where a massive core forms first and then accretes a gaseous envelope. This core forms by accreting solids, either planetesimals or pebbles. A key constraint in this model is that the accretion of gas must proceed before the dissipation of the gas disc. Classical planetesimal accretion scenarios predict that the time needed to form a giant planet\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'s core is much longer than the time needed to dissipate the disc. This difficulty led to the development of another accretion scenario, in which cores grow by accretion of pebbles, which are much smaller and thus more easily accreted, leading to more rapid formation. Aims: The aim of this paper is to compare our updated pebble-based planet formation model with observations, in particular the well-studied metallicity effect. Methods: We adopt the Bitsch et al. (2015a, A&A, 575, A28) disc model and the Bitsch et al. (2015b, A&A, 582, A112) pebble model and use a population synthesis approach to compare the formed planets with observations. Results: We find that keeping the same parameters as in Bitsch et al. (2015b, A&A, 582, A112) leads to no planet growth due to a computation mistake in the pebble flux (2018b). Indeed a large fraction of the heavy elements should be put into pebbles (Zpeb/Ztot = 0.9) in order to form massive planets using this approach. The resulting mass functions show a huge amount of giants and a lack of Neptune-mass planets, which are abundant according to observations. To overcome this issue we include the computation of the internal structure for the planetary atmosphere in our model. This leads to the formation of Neptune-mass planets but no observable giants. Furthermore, reducing the opacity of the planetary envelope more closely matches observations. Conclusions: We conclude that modelling the internal structure for the planetary atmosphere is necessary to reproduce observations.

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