Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions
(E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673), 2446 citations
This is the fourth in a series of articles by James Riordon. The first article appeared in the November 2002 issue.
The seventh paper in our list of the ten most cited Physical Review Letters revolutionized understanding of the electrical properties of materials in certain, important regimes. APS News interviewed three of the paper's authors, and some of their thoughts are reprinted below.
Elihu Abrahams is the Bernard Serin Professor of Physics (emeritus) at Rutgers, Nobel Laureate Philip Anderson is the Joseph Henry Professor of Physics (emeritus) at Princeton, and T. V. "Rama" Ramakrishnan is a newly elected Fellow of the Royal Society and a physics professor at the Indian Institute of Science in Bangalore. Donald Licciardello left Princeton shortly after the publication of the Scaling Theory of Localization PRL to found Princeton Telecom, and was not available for an interview.
APS News: What are the important findings in your 1979 PRL?
Abrahams: There were two significant results from our paper. One is the new conclusion that in a two- dimensional, disordered electronic system (i.e. defects are present) where the electrons do not interact with each other (or the interaction, for some reason, can be neglected) the ground state, which is the state at zero temperature, is always insulating. Thus, there are no true metallic states in two-dimensions. The other important conclusion is that in dimensions greater than two, where you could have a metal-insulator transition by changing some parameter (such as the degree of disorder), the transition is continuous. This means that, as you pass from metal to insulator at zero temperature, the conductivity goes continuously to zero, rather than jumping from some minimum value to zero.
APS News: Why has the paper acquired so many citations?
Ramakrishnan: The initial attention was I think due to the fact that we made experimentally testable, unusual predictions based on a new quantum, many-body, interference process; one which explored the process of localization, i.e. the nature of transition from a metal to insulator. At that time (1978-79), the question of the nature of this transition and its experimental manifestations was beginning to attract a lot of attention, since it represents a total change in the nature of electronic states (from extended to localized) not as a result of binding to a strong enough attractive potential as in one-electron quantum mechanics, but as a consequence of increasing disorder.
Abrahams: In recent years, the experimental discovery of what appear to be metallic states at zero temperature in 2-D systems has resulted in many experimental and theoretical papers which also reference our earlier work.
APS News: What led you to study this particular problem?
Anderson: It was partly a lecture I gave in an advanced course which set me thinking, plus a lunch table conversation with Rama and Abrahams, where Rama recalled some puzzling old work which we generalized. We were all aware of Sir Neville Mott's idea [regarding resistivity and disorder] and experimental data confirming it, and Licciardello had written a paper with Thouless which kind of foreshadowed scaling.
Ramakrishnan: . . . Anderson had shown, in a famous paper published in 1957, that if the medium in which the electron moves is sufficiently disordered, the electronic state is spatially localized and is not extended throughout the system (free electron ).The system is then an insulator. For small disorder, one intuitively feels that the system is a metal with a small residual resistivity. What is the nature of the transition from metal to insulator as disorder is increased? There was a very significant conjecture with considerable experimental support, due to Sir Neville Mott which went as follows: as disorder increases, the residual resistivity continues to increase to a maximum value (which he estimated) and then, for a slightly larger disorder, one has an insulator with an infinite resistivity (at T=0). This focused attention on the fact that none of the theories of localization at that time calculated a physically measured quantity such as resistivity; they concentrated on whether the state was localized or not, e.g. by trying to calculate the localization length, or asking whether an electron in the random medium diffused away or not.
APS News: Were you surprised by the theoretical results of the work?
Ramakrishnan: Yes, very much so. We had uncovered a new quantum mechanical process of interference between paths for the electron starting and ending at the same point, diffusing through the random medium, but in opposite directions. This is the localizing process which increases the amplitude for an electron to come back where it started. It leads to a nonclassical scale-dependent resistivity and is behind the unexpected result that there are no metals in two dimensions, at absolute zero.
Anderson: No, it seemed to fit right in with what we knew or guessed.
APS News: How has your career been affected by being the author of such a highly cited paper? Has it limited your options as a researcher? Has it expanded them?
Abrahams: Briefly-it was good for the career, hasn't limited research options and expanded them somewhat.
Ramakrishnan: It was a wonderful experience, especially after the discovery of the localizing process, to take part in the creative interaction between theory and experiment, and to realize the variety of new phenomena which 'emerge' from a new idea.
Anderson: This is my nth highly cited paper so it couldn't make that much difference. It was a little important to me because it bolstered my self-confidence at that particular time. I had read things about how the Nobel prize would destroy my productivity, which I now know is hogwash-it depends on what you want to do, and I wanted to go on producing physics. But I also wrote a highly cited paper, more important than this one, in 1977 (TAP, or Thouless-Anderson-Palmer), and continued to do important physics until now.
APS News: Is there anything else you'd like to say?
Anderson: I was delighted with the prominence of this paper because El Abrahams had been important in following up my original localization work and hadn't been recognized for it, and just for alphabetical reasons we put him first-he, Rama and I contributed pretty equally. Another nice thing was that it was followed by a spate of work worldwide which was very cooperative and collegial, with everyone cooperating with everyone else - Russians, Japanese, Germans, Americans, Indians and others. It was a marvelous contrast to what has happened with other discoveries.
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