For years now, there has been a fairly heated debate about the nature of an apparent metal-insulator transition (as a function of carrier density) seen in various 2d electronic and hole systems. The basic observation, originally made in some Si MOSFETs of impressively high interface quality made in Russia, is that as the 2d carrier density is reduced, the temperature dependence of the sheet resistance changes qualitatively, from a metallic dependence (lower T = lower resistance) at high carrier concentration to an insulating dependence (lower T = higher resistance) at low concentration, with a separatrix in between with nearly T-independent resistance at some critical carrier density. A famous 1979 paper by the "Gang of Four" (Anderson, Abrahams, Licciardello, and Ramakrishnan) on the scaling theory of localization had previously argued that 2d systems of noninteracting carriers all become insulating at T=0 for arbitrarily weak disorder.So, there has been a long-simmering controversy about why some 2d systems (electrons or holes) seem to show a really metallic temperature dependence of their conductance at low temperatures. This dependence, where the conductivity apparently increases by, say, a factor of 2 from \(T =\) 4.2K down to 0.1 K, takes place over a temperature range where the scattering of electrons by lattice vibrations (the mechanism responsible for the increase in conductivity of ordinary metals as they are cooled from room temperature down to cryogenic temperatures) is supposed to be all finished. I mentioned this as an ongoing controversy in '06 and again in '12. What is going on here?

There is a new preprint from Bruce Kane and colleagues at Maryland that clarifies things considerably, in my view. Kane, probably best known for proposing a quantum computing scheme involving individual phosphorus donors in Si, is a very clever experimentalist. He has developed a method of creating field-effect transistors,where the conducting channel is the hydrogen-terminated surface of a Si wafer, and the gate dielectric is vacuum (!). Using these devices, his group has been able to look at the apparent metallicity in both electrons and holes in the same system. They find that the improvement in conduction at low temperatures has to do with the screening of charged impurities by the conducting system (and for the experts: in Si the electrons are able to do this better than the holes because there are 6 conduction band valleys, while there is no valley degeneracy for the holes). This doesn't directly get to the "fundamental" question about whether the true, zero-temperature ground state is insulating in a real, interacting system, but it does go a long way toward demonstrating why the conductivity still has a metallic change with temperature even though phonons should be out of the picture.

## 3 comments:

Hey Doug,

As I understand it, the 2D MIT in Si MOSFETS remains quite the can of worms. At zeroth order, the question is whether disorder is really playing an important role, since Kravchenko's samples at least had relatively high mobilities. It is possible that the physical effects of disorder would only set in at very low temperatures. If disorder is important, then before we get to scaling theory and localization, it's important to ask about the statistics of the potential. Weak and Anderson localization are wave interference effects that are important when particles would be able to freely propagate through the potential landscape. Even if there is no classical trapping, a potential problem that can occur at low carrier densities is long-ranged correlations in the potential landscape itself. The potential due to randomly distributed ionized dopants is logarithmically correlated in 2D. Of course, one typically has Thomas-Fermi screening which induces a finite (effectively short) correlation length. But at low carrier densities, screening can become poor and non-linear, and then it is not obvious what the range of correlations is, or if power-law what is the exponent. Too small an exponent means that you can't assume self-averaging, and then you can get a puddling/percolation scenario, which seems to happen in graphene on SiO2 near the Dirac point.

If disorder is important and the effective potential is sufficiently short-range correlated, then one should really consider the physics of the interacting diffusive electron gas. A correct version must at least incorporate Altshuler-Aronov (AA) corrections (which have been known almost as long as the scaling theory itself, and occur due to scattering of carriers off of impurity-induced density Friedel oscillations). One framework for that is the Finkel'stein non-linear sigma model (FNLsM), which gives the same results as careful calculations in disorder-averaged many body perturbation theory.

The FNLsM incorporates at least three effects not present in the non-interacting theory:

1) AA corrections to the conductance (additional terms in the "beta function" of scaling theory)

2) Renormalization of interactions 1: enhancement or suppression of matrix elements in the basis of exact eigenstates, due to disorder

3) Renormalization of interactions 2: non-linear effects

The issue with the 2D MIT is that it is observed in a system with little or no relevant spin-orbit coupling, and without magnetic impurities. Thus spin diffusion is a good hydrodynamic mode, and contributes to 1) through the spin triplet coupling strength. For the usual sign of this coupling (negative due to exchange), this gives an enhancement of the conductance, i.e. it is anti-localizing. This is well-established and not controversial.

The problem, however, is that in the FNLsM effect 3) takes over before one can determine the fate of the system, i.e. whether it flows to a metal or insulator. Basically, the spin coupling blows up. This has been taken as possible evidence of a magnetic instability that precedes the MIT. Punnoose and Finkel'stein have attempted to fit Kravchenko's data with RG predictions following from a large-N version of the original model, which suppresses the spin instability described above.

There are interacting MITs in 3D wherein the spin channel is suppressed (by a weak magnetic field, magnetic impurities, spin-orbit coupling, etc) such that one doesn't have the above problem. An example is the GaMnAs system studied by Yazdani in 2010. Despite some initial grumbling that the data didn't match the predictions of the non-interacting scaling theory (which it shouldn't), there is now qualitative agreement between the experimental data and Finkel'stein sigma model predictions. See e.g. Burmistrov, Gornyi, and Mirlin.

matt

Thanks, Matt!

Information given in this article is helpful for me too!

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