The Fractional Quantum Hall Effect H
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Proc. Nati. Acad. Sci. USA Vol. 84, pp. 4696-4697, July 1987 Symposium Paper This paper was presented at a symposium "Interfaces and Thin Films," organized by John Armstrong, Dean E. Eastman, and George M. Whitesides, held March 23 and 24, 1987, at the National Academy of Sciences, Washington, D.C. The fractional quantum Hall effect H. L. STORMERt AND D. C. Tsuit tAT&T Bell Laboratories, Murray Hill, NJ 07974-2070; and tPrinceton University, Princeton, NJ 08544 Introduction quired quasi-perfect planar transport unobstructed by impu- rities and interfacial defects, and MBE-grown, modulation- The fractional quantum Hall effect (FQHE) is an example of doped GaAs-(AlGa)As heterostructures provided an ideal the new physics that has emerged in recent years as a result medium. of research in quantum-confined carriers in semiconductor heterostructures. It was first observed (1) in a high-mobility, Present Status two-dimensional, modulation-doped (2) GaAs-(AlGa)As heterostructure prepared by molecular-beam epitaxy (MBE). At the present time the FQHE is well documented (6). Many The experimental facts are simple. The Hall resistance px, is research groups have confirmed the initial data and are found to be quantized to pxy = h/ve2, where v is a rational progressing in documenting the characteristics of this new fraction with exclusively odd denominators (3). electronic state. High-field, low-temperature transport ex- Concomitant with the quantization ofpxy, the resistivity Pxx periments remain the preferred experimental method of of the specimen vanishes as the temperature T approaches T research, although more recently optical tools (7) have been = 0. The FQHE is presently understood as the manifestation applied to the FQHE. The transport experiments have been ofthe existence of a series of new electronic ground states (4) pursued in magnetic fields as high as 30 T at temperatures as resulting from the strong correlation of the electronic motion low as -0.06 K, uncovering an increasing number ofrational in a high magnetic field. The experimentally observed trans- fractions (8, 9). At the time of this writing, minima in the port phenomena are taken as evidence for the formation of resistivity p,, have been observed in the vicinity of: fractionally charged quasi-particles separated in energy from the condensed ground state by finite gaps. Forefront theo- v= 1/3, 2/3, 4/3, 5/3, 7/3, 8/3 retical many-particle physics and state-of-the-art experimen- tal efforts are presently trying to unravel the remarkable v 1/5, 2/5, 3/5, 4/5, 7/5, 8/5 properties of the novel electronic state. v = 2/7, 3/7, 4/7, 9/7,9iLZ Application of a magnetic field perpendicular to a two- dimensional electron system quantizes the carriers' in-plane v =4/9, 5/9,1 motion and transforms their energy spectrum into a set of discrete, highly degenerate levels. In the lowest of these Except for the underlined fractions, these minima are asso- Landau levels, the carriers' kinetic energy becomes com- ciated with plateaus in the Hall resistance quantized to py = pletely suppressed and their mutual Coulomb interaction h/ve2. It appears that only fractions of odd denominators are dominates. Under such extreme circumstances, intriguing allowed. There is presently no strong evidence for fractions possibilities for the formation of an ordered ground state are of even denominator (8). to be expected. Experimental search for it was initially More recently, experimental efforts have focused on a conducted in silicon MOSFETs (metal oxide semiconductor determination of the strength of the FQHE characterized by field-effect transistors) but did not become successful until the value of the excitation gap above the ground state. the invention of MBE and the discovery of the integral Activation energy measurements on the most pronounced quantum Hall effect (IQHE) (5) in silicon MOSFETs. minima reveal a strong magnetic field dependence (9, 10). MBE allowed the fabrication of nearly perfect semicon- From the experiments alone, the following statements may be ductor interfaces that, when combined with modulation- made about the electronic states underlying the FQHE. doping, brought about two-dimensional carrier systems with mobilities in (i) The states are formed when a Landau level is partially unprecedentedly high ranging presently the filled to a fraction v = p/q, where q is always odd. several 106 cm2/V sec (J. English, A. C. Gossard, H.L.S., (ii) The states are sensitive to disorder and nonexistent in and K. Baldwin, unpublished data). The discovery of the sufficiently disordered systems. IQHE led to a better understanding of the physics of (iii) The Hall resistance is quantized to py = h/ye2. two-dimensional systems in a high magnetic field and the (iv) The resistivity Pxx is thermally activated and vanishes importance and relative strength ofelectron localization. The as the temperature approaches T = 0. IQHE is understood in terms of the motion of individual (v) The activation energy is strongly magnetic field de- carriers, reflecting electron-electron interaction. According- pendent and vanishes below a critical field that ly, the associated quantum numbers are integers reflecting increases with increasing disorder. the integrity of the single-particle picture. Research into (vi) Attempts to interpret the FQHE phenomenologically electron-electron correlation effects in two dimensions re- in analogy to Laughlin's argument (11) for the IQHE require the postulate of quasi-particles with fractional The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" Abbreviations: FQHE, fractional quantum Hall effect; IQHE, inte- in accordance with 18 U.S.C. §1734 solely to indicate this fact. gral quantum Hall effect; MBE, molecular-beam epitaxy. 4696 Downloaded by guest on September 30, 2021 Symposium Paper: Stormer and Tsui Proc. Natl. Acad. Sci. USA 84 (1987) 4697 charges and the existence of energy gaps at fractional exceptional phenomena, fractionally charged quasi-particles, Landau level filling. quasi-particle/quasi-hole excitations with roton character (vii) These energy gaps must be of many-body origin (17), the existence of an electron solid at very low Landau- because no gaps are expected at fractional filling level filling, and three-dimensional electron crystallization in within a single-particle picture. layered two-dimensional structures. None of these has yet been observed. To date only electrical measurements have Whereas the many-body origin of the FQHE was instantly been fully exploited. There exist tremendous opportunities recognized, a quantum-mechanical description of the highly for ingenious, though probably difficult, experiments to correlated ground state was not available. A novel many- probe directly into the structure of the ground states, the particle wave function able to account for much of the dispersion of the excitation spectra, and the dynamics of the experimental findings was finally constructed by Laughlin quasi-particles of the FQHE. (4). This many-electron state has the following properties. 1. Tsui, D. C., Stormer, H. L. & Gossard, A. C. (1982) Phys. (i) It is stable at Landau-level filling factors v = 1/m and Rev. Lett. 48, 1559-1562. v = 1 - 1/m with m = odd integers. 2. Stormer, H. L., Dingle, R., Gossard, A. C. Wiegmann, W. & (ii) Its pair correlation function indicates that it is a novel Sturge, M. D. (1979) Solid State Commun. 29, 705-709. quantum liquid. 3. Stormer, H. L., Chang, A. M., Tsui, D. C., Hwang, J. C. M., (iii) Its elementary excitations are separated from the ground Gossard, A. C. & Wiegmann, W. (1983) Phys. Rev. Lett. 50, state by a gap that decreases with increasing m. 1953-1957. (iv) These quasi-particles carry a fractional charge e* = 4. Laughlin, R. B. (1983) Phys. Rev. Lett. 50, 1395-1399. e/ml. 5. von Klitzing, K., Dorda, G. & Pepper, M. (1980) Phys. Rev. (v) The quantum liquid is incompressible and flows with- Lett. 45, 494-498. out dissipation at T = 0. 6. Prange, R. E. & Girvin, S. M., eds. (1987) The Quantum Hall (vi) For m 10, the quantum fluid is expected to crys- Effect (Springer, New York). tallize into a solid. 7. Kukushkin, I. V. & Timofeev, V. B. (1986) JETP Lett. 44, (vii) A hierarchical model independently developed by 228-230. 8. Clark, R. G., Nicholas, R. J., Usher, A., Foxon, C. T. & Haldane (12), Halperin (13), and Laughlin (14) is able Harris, J. J. (1986) Solid State Commun. 60, 183-187. to explain the higher-order FQI4E at filling factors v = 9. Boebinger, G. S., Chang, A. M., Stormer, H. L. & Tsui, p/q, where p and q are integers and q is odd. D. C. (1985) Phys. Rev. Lett. 55, 1606-1610. 10. Gavrilov, M. G., Kvon, Z. D., Kukushkin, I. V. & Timofeev, In summary, current theory can account for all striking V. B. (1904) JETP Lett. 39, 507-510. characteristics of the FQHE. Quantitative comparison with 11. Laughlin, R. B. (1981) Phys. Rev. B 23, 5623-5627. experiments requires a theoretical scheme that takes into 12. Haldane, F. D. M. (1983) Phys. Rev. Lett. 51, 605-609. account the effect of disorder, a problem that has only 13. Halperin, B. I. (1983) Helv. Phys. Acta 56, 75-82. recently been addressed (15, 16). 14. Laughlin, R. B. (1984) Surf. Sci. 142, 163-167. 15. McDonald, A. H., Liu, K. L., Girvin, S. M. & Platzman, Future Experimental Challenges P. M. (1985) Phys. Rev. B 33, 4014-4018. 16. Gold, A. (1986) Europhys. Lett. 1, 241-245. Recent theory has derived a detailed description of the 17. Girvin, S. M., McDonald, A. H. & Platzman, P. M. (1986) electronic processes underlying the FQHE that is rich in Phys. Rev. B 33, 2481-2485. Downloaded by guest on September 30, 2021.