Frozen electron solid in the presence of small concentrations of defects J. S. Thakur School of Physics, The University of New South Wales, Sydney 2052, Australia D. Neilson School of Physics, The University of New South Wales, Sydney 2052, Australia and Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy Received 11 March 1996; revised manuscript received 31 May 1996  We investigate the freezing of a low-density electron liquid for a two-dimensional electron layer in the presence of low levels of defects typical of a high-quality semiconductor interface. We use a memory function approach with mode-coupling approximation and include the effect of strong electron-electron correlations, which we find are crucial for the transition. For a range of low impurity concentrations we find a stable frozen solid with a liquidlike short-range order. At higher impurity concentrations the electrons localize separately and there is no short-range order. Our electron-density vs peak-mobility phase diagram at zero temperature is in agreement with recent metal-insulator transition experiments in silicon heterostructures. S0163-18299608935-7 Numerical simulation studies of a two-dimensional elec- tron liquid predict that it would only condense into a Wigner crystal at exceedingly low densities r s 375. 1 However, the energy differences between the liquid and crystalline states are already very small for densities r s 10. This sug- gests that low levels of defects might be sufficient to induce a transition to a solid phase at a much higher density than r s =37. Pudalov et al. 2 working with electron inversion lay- ers at silicon metal-oxide-semiconductor field-effect transis- tor MOSFET interfaces, observed a collective metal- insulator transition at densities as high as r s 8. The nature of the coherent insulating state is still not fully clear. We have found that low levels of disorder typical of those in state-of-the-art semiconductor substrates acting in concert with strong correlations between electrons can cause a tran- sition to a coherent solid of localized electrons at densities as high as r s 7 . The levels of disorder needed for this typically correspond to far fewer impurities than electrons. The solid is not a Wigner crystal but a frozen macroscopi- cally coherent state with liquidlike short-range order. It is quite different from another frozen state obtained from local- ized electrons interacting with a disordered medium which has been termed an electron glass. 3–5 The vanishing of a soft Coulomb gap in the single-particle density of states or a nonzero value of the Edwards-Anderson–like order param- eter provides the signature in that case. Our mechanism for electron localization is also different from the mechanism discussed earlier for free-electron scat- tering from randomly distributed impurities. 6 There localiza- tion was obtained by increasing the strength of the impurity- potential fluctuations see also the discussion in Ref. 7. Our localization, in contrast, is mainly driven by the strong effect of electron correlations at large r s . The mecha- nism for the transition is associated with the increasing rela- tive size of the correlation hole surrounding each electron as r s increases. From Ref. 1 we know that for r s 10 the ex- change correlation hole excludes all other electrons from a central region surrounding it as if the electron had a hard core. For r s 10 the electrons with their exchange correla- tion holes resemble hard disks. With decreasing electron density the fraction of the total area occupied by excluded regions approaches the close packing limit 8 and at this stage it becomes increasingly difficult for electrons to pass by each other. A small amount of impurity disorder introduces pin- ning centers and breaks the translational invariance of the system. The localization is quasiclassical because it is driven by the increasing size of the exchange-correlation hole. We find that with a small amount of disorder included the elec- trons can freeze for r s 7. We investigate the transition to a nonergodic phase using a model of the glass transition originally constructed to ac- count for the freezing of dense classical systems. 9,10 We have adapted this to a quantum system. We study the limiting behavior of the Kubo relaxation function lim t →   ( q, t ) „N ( q, t ) N ( q,0) … , defined on the normalized density fluctuation basis, N ( q, t ) =  ( q, t )/   ( q), where  ( q) is the electron liquid static sus- ceptibility. At lower electron densities the density fluctuation operators  ( q, t ) are an appropriate choice for the dynamical variables of the system rather than single-electron wave functions since the shape of the exchange correlation hole is determined mainly by the strong repulsive interactions. Ex- change effects play only a secondary role here and interfer- ence effects between single-electron waves are not expected to produce solidification. In the liquid phase  ( q, t ) tends to zero for times greater than the macroscopic relaxation time of the system. When a freezing point is approached the time decay in  ( q, t ) be- comes very slow and eventually stops, implying a complete arrest of density fluctuations. We use a memory function approach with mode-coupling approximation to investigate the existence of the transition. 9 Being a quantum system we are not able to look at dynamic properties of the liquid as the transition is approached since, unlike the classical case, there exists no unique relation between the dynamic structure fac- tor and response function. However, even in the quantum case the relation does exist if the frequency is first set equal to zero, which enables one to write quantum-correlation PHYSICAL REVIEW B 15 SEPTEMBER 1996-I VOLUME 54, NUMBER 11 54 0163-1829/96/5411/76744/$10.00 7674 © 1996 The American Physical Society