A two-dimensional (2D) electron gas formed in a modulation-doped GaAs/AlGaAs single quantum well undergoes a first-order transition when the first excited subband is occupied with electrons, as the Fermi level is tuned into resonance with the excited subband by applying a dc voltage. Direct evidence for this effect is obtained from low-temperature photoluminescence spectra which display the sudden renormalization of the intersubband energy $E_{01}$ upon the abrupt occupation of the first excited subband. Calculations within density-functional theory, which treat the 2D exchange potential {it exactly}, show that this thermodynamical instability of the electron system is mainly driven by {it intersubband} terms of the exchange Coulomb interaction. From temperature-dependent measurements the existence of a critical point at $T_c = 35pm 5$ K is inferred.
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