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The low temperature metal-insulator transition in 2-dimensional electron gas in the absence of a magnetic field I am not sure if this can be considered 'most' controversial but the origin of the metal-insulator transition at very low temperature in very pure samples of semiconductors like Si is still highly disputed.The theory of localization [1] predicts that in 2 dimensions, all the electrons or holes states of the system are localized in the absence of a magnetic field in the limit of zero temperature, no matter how clean the 2D system is. So, for any 2D system, as you cool the system down towards 0 K, the conductivity of the system approaches zero i.e. it becomes insulating. This was experimentally confirmed in the early eighties [2, 3] in electrical measurement taken in very high-quality Si MOSFETs (metal-oxide-silicon field effect transistors). The classic signature of an insulator is that its resistivity $\rho$ increases as you lower the temperature. [Image of a Si MOSFET taken from http://www.learnabout-electronic...]However, in such structures, you can 'tune' the carrier (electron or hole) density by modulating the gate voltage or the electric field perpendicular to the 2D system. When you apply a positive gate voltage, the electrons get attracted to the area just beneath the gate, and the electron density increases. At a sufficiently high carrier density, the system undergoes a metal goes a transition from insulating to conducting. In a conducting (or metallic) system, the resistivity $\rho$ decreases as you lower the temperature. This of course violates the predictions from the theory of localization [1].[Image taken from http://dx.doi.org/10.1590/S0103-...] The figure above shows the resistivity of a Si MOSFET in zero magnetic field as a function of temperature for electron densities $n_s$ =  8.58, 8.80, 9.02, 9.33, 9.52, 9.90, and 11.0E10 cm-2. The critical density for the metal-insulator transition is 9.02E10 cm-2. The controversy is over the origins of this 'transition'. Candidate explanations for this phenomenon include the formation of a metallic state in a strongly interacting 2D electron gas [4], strong spin-orbit interaction [5] as well as more conventional explanation along the lines of temperature-dependent screening of charged impurities [6]. This phenomenon was quite an active topic of research in the late 90's and early 2021's. However, with the discovery of graphene and other 2-dimensional layered crystals, a lot of attention was shifted away from the field since a lot of concepts and techniques developed for studying the metal-insulator transition could be more profitably applied to these newly discovered materials [7]. References:1. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979)2. M J Uren et al., J. Phys. C: Solid State Phys. 14, 5737 (1981)3. D. J. Bishop, D. C. Tsui, and R. C. Dynes, Phys. Rev. Lett. 44, 1153 (1980)4. A. M. Finkelshtein, Z. Phys. B 56, 189 (1984)5. Y. Lyanda-Geller, Phys. Rev. Lett. 80, 4273. 6. S. He and X. C. Xie, Phys. Rev. Lett. 80, 3324 (1998); B. L. Altshuler and D. L. Maslov, Phys. Rev. Lett. 82, 145 (1999)7.  K. S. Novoselov et al., Science 306, 666 (2021)