1 Thermodynamic Properties at Landau Levels of Two – dimensional electron gas S. Bouzgarrou 1, 2 , A. Kalboussi 1 1 Laboratoire de Microélectronique et Instrumentation (UR 03/13-04), Faculté des Sciences de Monastir, Avenue de l'Environnement 5000 Monastir, Tunisia. 2 Qassim University, Faculty of Sciences, Saoudi Arabia b_sonia3@yahoo.fr Abstract: A critical anomaly of a new specific heat capacity   of InAlAs heterostructure of two - dimensional electron gas (2DEG) , has been theoretically investigations as a function of temperature T. This study has shown the presence of a large and continuous anomaly. This anomaly is explained through a theory based on the general assumption. The present theoretical research is based essentially on the characteristic of specific heat capacity extending over a large temperature, but we underline a good agreement with results of the relation with chemical potential μ and Broadening parameter Γ as a function of temperature. It is found that the specific heat capacity observed by a peak at low temperature, at a critical temperature   , is directly linked to Schottky anomaly and unveiling the existence of phase transition in InAlAs. Our results are completed by the study of the dependence of the heat capacity on the spin as a function of temperature. This study confirms the same behavior with result without spin. Key words: Two – dimensional electron gas (2DEG), Landeau levels, Specific heat capacity, anomaly, critical temperature   , chemical potential μ , Broadening parameter Γ. Introduction: