Aims: The aim of this paper is to demonstrate that millimeter wave data can be used to distinguish between various atmospheric models of sunspots, whose temperature structure in the upper photosphere and chromosphere has been the source of some controversy. Methods: We use observations of the temperature contrast (relative to the quiet Sun) above a sunspot umbra at 3.5 mm obtained with the Berkeley-Illinois-Maryland Array (BIMA), complemented by submm observations from Lindsey & Kopp (1995) and 2 cm observations with the Very Large Array. These are compared with the umbral contrast calculated from various atmospheric models of sunspots. Results: Current mm and submm observational data suggest that the brightness observed at these wavelengths is low compared to the most widely used sunspot models. These data impose strong constraints on the temperature and density stratifications of the sunspot umbral atmosphere, in particular on the location and depth of the temperature minimum and the location of the transition region. Conclusions: A successful model that is in agreement with millimeter umbral brightness should have an extended and deep temperature minimum (below 3000 K). Better spatial resolution as well as better wavelength coverage are needed for a more complete determination of the chromospheric temperature stratification above sunspot umbrae.
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