Nobel Lecture: Interplay of disorder and interaction in two-dimensional electron gas in intense magnetic fields* Daniel C. Tsui Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544 [S0034-6861(99)00804-1] In this lecture, I would like to briefly go through the was able to demonstrate experimentally the electric-field physics that I have learned in the years since I ventured quantization of the surface space-charge layer, first pro- into what we nowadays call research of semiconductor posed by Schrieffer in the fifties, by doing a tunneling electronics in low dimensions, or in my case, more sim- experiment on InAs to observe directly the quantized ply the electronic properties of two-dimensional systems energy levels and the Landau levels of the resulting two- (Ando et al., 1983). To summarize, electrons confined to dimensional (2D) electrons (Tsui, 1970). However, the the interface of two different semiconductors normally most exciting part of this effort was my discovery, in behave like an ordinary gas of particles in two dimen- writing the paper on this work, of the beautiful work on sions. But, when taken to extreme conditions of low the Si metal-oxide-semiconductor field-effect transistor temperature and high magnetic field, they show new (Si MOSFET) done by the IBM group in Yorktown physics phenomena manifesting the interplay of Heights (Fang et al., 1983). They laid a solid foundation electron-electron interactions and the interaction of the for the development and growth of 2D electron physics electrons with imperfections in the semiconductors. Let in the subsequent decades. me first recall the events in my earlier research that led Based on the IBM work, Jim Allen and I made a tem- me to the journey that Art Gossard, Horst Stormer, and perature dependence study of the inversion layer con- I took in our adventure towards the discovery of the ductance in Si MOSFETs to look for the 2D Anderson fractional quantum Hall effect (FQHE) (Tsui et al., localization-delocalization transition. By varying the 1982). gate voltage on the device, we were able to move the Fermi level into the band tail and observe directly a PROLOGUE transition from the two-dimensional electron gas (2DEG) behavior to the behavior of an insulator. But, I joined Bell Laboratories in the spring of 1968. I was to our disappointment, we were unable to obtain quan- sufficiently naive that I foolhardily convinced myself to titative agreement with theory. We thought interaction leave behind the more familiar band structures and might be the cause and came to the conclusion that to Fermi surfaces of metal physics, which I had become enhance the interaction we should apply a magnetic comfortable with through my years of research as a field, which would change the individual electron’s ki- graduate student, and to try something different, e.g., netic energy into the cyclotron energy. This should be surfaces or interfaces, and preferably some many-body especially effective in the extreme quantum limit when interaction physics. I read about Anderson localization, the cyclotron diameter was less than the average the Mott transition, and the notion that disorder and electron-electron separation. electron-electron interaction were the richest and most In a 1976 paper, Kawaji and Wakabayashi (1976) re- challenging problems in solid-state physics. But I did not ported the observation of localized states in the energy have the foggiest idea on how to do what to get started. gap between two Landau levels. This discovery was a Fortunately, I was advised to talk to John Rowell, most important milestone on the path to the quantized who had at the time just completed the Rowell- Hall effect. In response to their work, I studied at the McMillan electron-phonon interaction work using tun- Frances Bitter National Magnet Laboratory on the MIT neling in superconductivity. John told me to look into campus the conductance in the extreme quantum limit, when all electrons occupy the lowest Landau level, and point-contact tunneling into the high-Tc superconduct- ors of those days and suggested that, instead of using saw some structures. Phil Anderson, after hearing this tungsten whiskers, I should experiment with semicon- from John Rowell, asked to see the data. But by the ductor tips which have built-in surface depletion layers time I showed them to him in the Bell Labs tearoom, I as tunnel barriers. My effort to tunnel into the supercon- had repeated the experiment and found them to be ductors of niobium and vanadium compounds was un- sample specific. I told this to Phil and he made a cryptic successful, but the project forced me to learn some phys- remark under his breath that there should be some com- ics of semiconductor surfaces and interfaces. In fact, I mensuration energy anyway. I reasoned: given that n is the 2D electron density and that the magnetic field B (applied perpendicular to the 2D plane) is expressed in ϭ ␸ ␸ ϭ *The 1998 Nobel Prize in Physics was shared by Robert B. terms of average flux density n␾ B/ 0 (and 0 h/e is Laughlin, Horst L. Stormer, and Daniel C. Tsui. This lecture is the Dirac flux quantum), the ratio n/n␸ is the Landau- the text of Professor Tsui’s address on the occasion of the level filling factor ␯. For nϾn␸ , an integer i number of award. Landau levels are filled at commensuration and the cy- Reviews of Modern Physics, Vol. 71, No. 4, July 1999 0034-6861/99/71(4)/891(5)/$16.00 © The Nobel Foundation 1998 891 892 Daniel C. Tsui: 2D electron gas in intense magnetic fields ␳ ␳ FIG. 1. Magneto-transport coefficients ␳ and ␳ ofa2DEG FIG. 2. xx and xy of a relatively low-mobility 2DEG in xx xy ␳ in GaAs/Al Ga Ϫ As at 0.35 K in moderately low B. The in- GaAs/AlxGa1ϪxAs. The plateaus in xy are quantized in the x 1 x 2 sert shows the measurement geometry. The magnetic field B is natural conductance unit e /h with integer quantum numbers ϭ perpendicular to the plane of the 2DEG and to the current I. i 1,2, . Data taken by H. P. Wei. The voltages V and VH are measured along and perpendicular ␳ ϭ to I, respectively, xx (V/L)/(I/W) is the resistance across a ferent physical regimes. The first is the disorder- ␳ ϭ square, independent of the square size, and xy VH /I is the dominant regime, when the sample is dirty with low 2D Hall resistance independent of the sample width. Data taken electron mobility (e.g., ␮Ͻ105 cm2/V sec in the case of by A. Majumdar. GaAs). The striking features in the data constitute the integral quantum Hall effect (IQHE) (von Klitzing et al., clotron energy, separating the filled from the empty lev- 1980), which is understood in terms of the physics of els, is the commensuration energy. I assumed he meant: independent electrons and their localization in the pres- In the nϽn␸ extreme quantum limit, at commensuration ence of random impurities in the semiconductors. The when ␯ϭn/n␸ϭ1/i, some interaction energy might be- fractional quantum Hall effect (FQHE) is observed in come dominant to drive the 2D system into some new high-mobility samples in the regime where the electron- ground state. I was not brave enough to ask him: ‘‘What electron interaction dominates. It manifests the many- do you mean?’’ But I felt affirmed that I should continue body interaction physics of the 2DEG in the intense B to concentrate on the extreme quantum limit. field. Furthermore, even in the cleanest samples, the Indeed, with the advent of molecular beam epitaxy FQHE series terminates into an insulator in the high-B (Cho, 1995) and the invention of modulation doping to limit. This insulator is believed to be an electron crystal produce highly perfect 2D electron systems (Stormer pinned by defects to the semiconductor. The third re- et al., 1979), it soon became quite clear to Art Gossard, gime is this high-␮ and high-B-field limit, where disorder Horst Stormer, and me that where we wanted to go to and interaction play equally important roles and need to look for new many-body interaction physics should be a be treated on an equal footing. highest-mobility 2DEG sample placed in a most intense magnetic field. QUANTUM PHASE TRANSITIONS IN IQHE TWO-DIMENSIONAL MAGNETO-TRANSPORT Quantization of the Hall resistance in the natural con- ductance unit e2/h is currently understood in terms of In the presence of a perpendicular magnetic field, the the existence of an energy gap, separating the excited energy levels of a two-dimensional electron collapse, as states from the ground state, and localized states inside a result of Landau quantization of its cyclotron orbits, the gap. In the IQHE case, where the quantum numbers into discrete Landau levels separated by the cyclotron are integers identified with the number of completely energy quantum. Scattering broadens the Landau levels filled Landau levels, the energy gap is the Landau gap of and gives rise to 2D magneto-transport described by the a cyclotron energy quantum. The accurate quantization Ando-Uemura theory (Ando and Uemura, 1979). Fig- was shown by Laughlin, using a gedanken experiment, ure 1 is an example showing the quantum oscillations in to be a consequence of charge quantization. He showed ␳ the diagonal resistivity xx , reflecting the broadened that the experiment in effect measures the charge car- Landau-level structure of the 2DEG, and the Hall resis- ried by the excited electron. The localized states arise ␳ tance xy , well known from the Drude model.