# Experimental Aspects of the Fractional Quantum Hall Effect

## Abstract

The fractional quantum Hall effect, FQHE, is one of the most exciting phenomena to be discovered recently in solid state physics. The FQHE is observed in high-mobility (µ > 10^{5}cm^{2}/Vs) two-dimensional systems at low temperatures (T < 2 K) and high magnetic fields (B > 5 T). The FQHE is phenomenologically similar to the integral quantum Hall effect, IQHE: Plateaus are observed in the Hall resistivity, p_{xy}, concomitant with minima in the diagonal resistivity, p_{xx} (see Figure 1.1). While the IQHE exists at magnetic fields corresponding to integral Landau-level filling, v, the FQHE is observed at fractional Landau-level filling v = p/q where q is always odd (v = nh/eB, n = area density, and eB/h = Landau-level degeneracy). In the T → 0 limit, for both the IQHE and the FQHE, the plateaus in p_{xy} are accurately quantized to h/ve^{2} and the minima in p_{xx} reveal the presence of zero resistance states in the two-dimensional system. This low temperature behavior indicates the existence of energy gaps above the ground state of the system. While the IQHE can be phenomenologically understood through consideration of the single particle density of states, the FQHE requires a many-body approach to account for the phenomena.

## Keywords

Filling Factor High Magnetic Field Landau Level Magnetic Field Dependence Dilution Refrigerator## Preview

Unable to display preview. Download preview PDF.

## References

- Arovas, Daniel, J.R. Schrieffer, and Frank Wilczek, 1984, Fractional statistics and the quantum Hall effect, Phys. Rev. Lett. 53:722.ADSCrossRefGoogle Scholar
- Boebinger, G.S., A.M. Chang, H.L. Stormer, and D.C. Tsui, 1985a, Competition between neighboring minima in the fractional quantum Hall effect, Phys. Rev. B32:4268.ADSGoogle Scholar
- Boebinger, G.S., A.M. Chang, H.L. Stormer, and D.C. Tsui, 1985b, Magnetic field dependence of activation energies in the fractional quantum Hall effect, Phys. Rev. Lett. 55:1606.ADSCrossRefGoogle Scholar
- Boebinger, G.S., A.M. Chang, H.L. Stormer, D.C. Tsui, J.C.M. Hwang, G. Weimann, A. Cho, and C. Tu, 1986, Activation energies of fundamental and higher order states in the fractional quantum Hall effect, Surf. Sci. 170:129.ADSCrossRefGoogle Scholar
- Boebinger, G.S., D.C. Tsui, H.L. Stormer, G. Weimann, J.C.M. Hwang, A. Cho, and C. Tu, unpublished results on the v =1/5 minimum in pxx and studies on low-mobility samples.Google Scholar
- Chakraborty, T., 1985, Elementary excitations in the fractional quantum Hall effect, Phys. Rev. B31:4026.ADSGoogle Scholar
- Chakraborty, T., P. Pietiläinen, and F.C Zhang, 1986a, Elementary excitations in the fractional quantum Hall effect and the spin-reversed quasiparticles, Phys. Rev. Lett. 57:130.ADSCrossRefGoogle Scholar
- Chakraborty, T., 1986b, Spin-reversed quasiparticles in the fractional quantum Hall effect -many body approach, Phys. Rev. B34:2926.ADSGoogle Scholar
- Chang, A.M., M.A. Paalanen, D.C. Tsui, H.L. Stormer, and J.C.M. Hwang, 1983, Fractional quantum Hall effect at low temperatures. Phys. Rev. B28:6133.ADSGoogle Scholar
- Chang, A.M., P. Berglund, D.C. Tsui, H.L. Stormer, and J.C.M. Hwang, 1984a, Higher-order states in the multiple-series, fractional, quantum Hall effect, Phys. Rev. Lett. 53:997.ADSCrossRefGoogle Scholar
- Chang, A.M., M.A. Paalanen, H.L. Stormer, J.C.M. Hwang, and D.C. Tsui, 1984b, Fractional quantum Hall effect at low temperatures, Surf. Sci. 142:173.ADSCrossRefGoogle Scholar
- Clark, R.G., R.J. Nicholas, A. Usher, C.T. Foxon, and J.J. Harris, 1986, Odd and even fractionally quantized states in GaAs-GaAlAs hetero-junctions, Surf. Sci. 170:141.ADSCrossRefGoogle Scholar
- Ebert, G., K. von Klitzing, C. Probst, E. Schuberth, K. Ploog, and G. Weimann, 1983, Hopping conduction in the Landau level tails in GaAs-Alx Ga1-x As heterostructures at low temperatures, Solid State Commun. 45:625.ADSCrossRefGoogle Scholar
- Ebert, G., K. von Klitzing, J.C. Maan, G. Remenyi, C. Probst, G. Weimann, and W. Schlapp, 1984, Fractional quantum Hall effect at filling factors up to v =3, J. Phys. C17:L775.ADSGoogle Scholar
- Fano, G., F. Ortolani, and E. Colombo, 1986, Configuration-interaction calculations on the fractional quantum Hall effect, Phys. Rev. B34: 2670.ADSGoogle Scholar
- Girvin, S.M., A.H. MacDonald, and P.M. Platzman, 1985, Collective-excitation gap in the fractional quantum Hall effect, Phys. Rev. Lett. 54:581.ADSCrossRefGoogle Scholar
- Girvin, S.M., A.H. MacDonald, and P.M. Platzman, 1986, Magneto-roton theory of collective excitations in the fractional quantum Hall effect, Phys. Rev. B33:2481.ADSGoogle Scholar
- Gold, A., 1986, Disorder effects on the transport properties in the fractional quantized Hall regime, Europhysics Lett. 1:241.ADSCrossRefGoogle Scholar
- Haldane, F.D.M., 1983, Fractional quantization of the Hall effect: a hierarchy of incompressible quantum fluid states. Phys. Rev. Lett. 51:605.MathSciNetADSCrossRefGoogle Scholar
- Haldane, F.D.M. and E.H. Rezayi, 1985, Finite-size studies of the incompressible state of the fractionally quantized Hall effect and its excitations, Phys. Rev. Lett. 54:237.ADSCrossRefGoogle Scholar
- Halperin, B.I., 1984, Statistics of quasiparticles and the hierarchy of fractional quantized Hall states, Phys. Rev. Lett. 52:1583.ADSCrossRefGoogle Scholar
- Halperin, B.I., Z. Tesanovic, and F. Axel, 1986, Compatibility of crystalline order and the quantized Hall effect, Phys. Rev. Lett. 57:922(c).ADSCrossRefGoogle Scholar
- Hurkx, G.A.M. and W. van Haeringern, 1985, Self-consistent calculations on GaAs-Alx Ga1-x As heterojunctions, J. Phys. C18:5617.ADSGoogle Scholar
- Hwang, J.C.M., A.Kastalsky, H.L. Stormer, and V.G. Keramidas, 1984, Transport properties of selectively doped GaAs-(AlGa)As hetero-structures grown by molecular beam epitaxy, Appl. Phys. Lett. 44:802.ADSCrossRefGoogle Scholar
- Ihm, J. and J.C. Phillips, 1985, Activation energies and localization in the fractional quantized Hall effect, J. Phys. Soc. Jpn. 54:1506.ADSCrossRefGoogle Scholar
- Kittel, C., 1976, Introduction to Solid State Physics, 5th ed., John Wiley & Sons, Inc., New York.Google Scholar
- Kivelson, S., C. Kallin, D.P. Arovas, and J.R. Schrieffer, 1986, Cooperative-ring-exchange theory of the fractional quantized Hall effect, Phys. Rev. Lett. 56:873.ADSCrossRefGoogle Scholar
- Kukushkin, I.V. and V.B. Timofeev, 1986, Activation gaps sin the energy spectrum and influence of disorder on the fractional quantum Hall effect in silicon MOSFETS, Surf. Sci. 170:148.ADSCrossRefGoogle Scholar
- Lam, Pui K., and S.M. Girvin, 1984, Liquid-solid transition and the fractional quantum-Hall effect, Phys. Rev. B30:473.ADSGoogle Scholar
- Laughlin, R.B., 1981, Quantized Hall conductivity in two dimensions, Phys. Rev. B23:5632.ADSGoogle Scholar
- Laughlin, R.B., 1983, Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50:1395.ADSCrossRefGoogle Scholar
- Laughlin, R.B. 1984a, The gauge argument for accurate quantization of the Hall conductance, in “Two-Dimensional Systems, Heterostructures, and Superlattices,” G. Bauer, F. Kuchar, H. Heinrich, eds., Springer-Verlag, Berlin, p.272.Google Scholar
- Laughlin, R.B., 1984b, Primitive and composite ground states in the fractional quantum Hall effect, Surf. Sci 142:163.ADSCrossRefGoogle Scholar
- Laughlin, R.B., 1984c, Excitons in the fractional quantum Hall effect, Physica (Amsterdam) 126B:254.ADSGoogle Scholar
- Laughlin, R.B., M.L. Cohen, J.M. Kosterlitz, H. Levine, S.B. Libby, and A.M.M. Pruisken, 1985, Scaling of conductivities in the fractional quantum Hall effect, Phys. Rev. B32:1311.ADSGoogle Scholar
- Laughlin, R.B., 1986, Destruction of the fractional quantum Hall effect by disorder, Surf. Sci. 170:167.ADSCrossRefGoogle Scholar
- Levesque, D., J.J. Weis, and A.H. MacDonald, 1984, Crystallization of the incompressible quantum fluid state of a two-dimensional electron gas in a strong magnetic field, Phys. Rev. B30:1056.ADSGoogle Scholar
- MacDonald, A.H., K.L. Liu, S.M. Girvin, and P.M. Platzman, 1986, Disorder and the fractional quantum Hall effect: activation energies and the collapse of the gap, Phys. Rev. B33:4014.ADSGoogle Scholar
- Mendez, E.E., L.L. Chang, M. Heiblum, L. Esaki, M. Naughton, K. Martin, and J. Brooks, 1984, Fractionally quantized Hall effect in two-dimensional systems of extreme electron concentration, Phys. Rev. B30:7310.ADSGoogle Scholar
- Morf, R., and B.I. Halperin, 1986, Monte-Carlo evaluation of trial wave functions for the fractional quantized Hall effect: disk geometry, Phys. Rev. B33:2221.ADSGoogle Scholar
- Mott, N.F., and E.A. Davis, 1979, “Electronic Properties in Non-Crystalline Materials,” 2nd ed., Clarendon, Oxford.Google Scholar
- Ono, Y., 1982, Localization of electrons under strong magnetic fields in a two-dimensional system, J. Phys. Soc. Jpn. 51:237.ADSCrossRefGoogle Scholar
- Paalanen, M.A., D.C. Tsui, and A.C. Gossard, 1982, Quantized Hall effect at low temperatures, Phys. Rev. B25:5566.ADSGoogle Scholar
- Paalanen, M.A., D.C. Tsui, A.C. Gossard, and J.C.M. Hwang, 1984, Disorder and the fractional quantum Hall effect, Solid State Commun. 50:841.ADSCrossRefGoogle Scholar
- Platzman, P.M., S.M. Girvin, and A.H. MacDonald, 1985, Conductivity in the fractionally quantized Hall effect, Phys. Rev. B32.-8458.ADSGoogle Scholar
- Rezayi, E.H., and F.D.M. Haldane, 1985, Incompressible states of the fractionally quantized Hall effect in the presence of impurities: a finite-size study, Phys. Rev. B32:6924.ADSGoogle Scholar
- Stern, F., and S. Das Sarma, 1984, Electron energy levels in GaAs-Ga1-x AlxAs heterojunctions, Phys. Rev. B30:840.ADSGoogle Scholar
- Stormer, H.L., A.C. Gossard, and W. Wiegmann, 1981, Backside-gated modulation-doped GaAs-(AlGa)As heterojunction interface, Appl. Phys.Lett. 39:493.ADSCrossRefGoogle Scholar
- Stormer, H.L., A. Chang, D.C. Tsui, J.C.M. Hwang, A.C. Gossard, and W. Wiegmann, 1983a, Fractional quantization of the Hall effect, Phys.Rev. Lett. 50:1953.ADSCrossRefGoogle Scholar
- Stormer, H.L., 1983b, Electron mobilities in modulation-doped GaAs-(AlGa) As heterostructures, Surf. Sci. 132:519.ADSCrossRefGoogle Scholar
- Tsui, D.C., A.C Gossard, B.F. Field, M.E. Cage, and R.F. Dziuba, 1982a, Determination of the fine-structure constant using GaAs-AlxGa1-xAs heterostructures, Phys. Rev. Lett. 48:3.ADSCrossRefGoogle Scholar
- Tsui, D.C., H.L. Stormer, and A.C. Gossard, 1982b, Two-dimensional magne-totransport in the extreme quantum limit, Phys. Rev. Lett. 48:1559.ADSCrossRefGoogle Scholar
- Tsui, D.C., H.L. Stormer, and A.C. Gossard, 1982c, Zero-resistance state of two-dimensional electrons in a quantizing magnetic field, Phys.Rev. B25.-1405.ADSGoogle Scholar
- Tsui, D.C., H.L. Stormer, J.C.M. Hwang, J.S. Brooks, and M.J. Naughton, 1983, Observation of a fractional quantum number, Phys. Rev. B28:2274.ADSGoogle Scholar
- von Klitzing, K, G. Dorda, and M. Pepper, 1980, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45:494.ADSCrossRefGoogle Scholar
- Wakabayashi, J., S. Kawaji, J. Yoshino, and H. Sakaki, 1986, Activation energies of the fractional quantum Hall effect in GaAs-AlGaAs heterostructures, J. Phys. Soc. Jpn. 55:1319.ADSCrossRefGoogle Scholar
- Yoshioka, D., B.I. Haperin, and P.A. Lee, 1983, Ground state of two-dimensional electrons in strong magnetic fields and 1/3 quantized Hall effect, Phys. Rev. Lett. 50:1219.ADSCrossRefGoogle Scholar
- Yoshioka, D., 1984, Effect of the Landau level mixing on the ground state of two-dimensional electrons, J. Phys. Soc. Jpn. 53:3740.ADSCrossRefGoogle Scholar
- Yoshioka, D., 1986, Excitation energies of the fractional quantum Hall effect, J. Phys. Soc. Jpn. 55;885.ADSCrossRefGoogle Scholar
- Zhang, F.C., V.Z. Vulovic, Y. Guo, and S. Das Sarma, 1985, Effect of charged impurity on the fractional quantum Hall effect: exact numerical treatment of finite system, Phys. Rev. B32.-6920.ADSGoogle Scholar
- Zhang, F.C. and S. Das Sarma, 1986, Excitation gap in the fractional quantum Hall effect: finite layer thickness corrections, Phys. Rev. B33:2903.ADSGoogle Scholar