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But this does not explain why the Hall resistivity continues to be quantized even when the field is changed somewhat, so that there is a plateau of the quantized value, as seen in the data. Laughlin:!: offers the important insight that the integer effect is due to the phase coherence of the electronic wave function over the entire sample, and that the effect of impurities are important in producing the observed plateau. Consider a sample in the form of a ribbon forming a closed loop, as shown in Fig. 11.13. A magnetic field H pierces the ribbon everywhere normal to its surface, and a voltage V is applied across the edges of the ribbon. Our object is to deduce the relation between the Hall current I and V. The Hall current produces a magnetic moment /l = lA/c, where A is the area enclosed by the ribbon loop. Imagine that a small amount of magnetic flux 15cI> is introduced through the loop, corresponding to an increase in the magnetic
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*K. V. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980). tD. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48,1559 (1982). *R. B. Laughlin, Phys. Rev. B 23, 5632 (1981). See also B. I. Halperin, Phys. Rev. B 25,2185 (1982).
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-<'-- Filling fraction v
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Fig. 11.12 Quantized Hall effect: Schematic representation of experimental data. The filling fraction p is the fraction of degenerate states in the lowest Landau levels occupied by electrons. The Hall resistivity exhibits plateau of value lip, at p = 1, i, (in units of hle 2 .) The conventional resistivity becomes very small at these values. The quantization is accurate to at least one part in 10 4 .
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Fig. 11.13 Hall effect in idealized geometry.
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field 8H = 8cI>/ A normal to the plane of the loop. The energy of the system increases by 8E = /l 8H = (fA / e)( 8cI>/ A). Hence we can find the current from the formula
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We recall from our discussion of flux quantization in Section 11.3 that the "localized" electrons will not respond to the flux, but the "extended" ones may. Electrons in Landau levels do have extended wave functions, and thus will respond to the flux and contribute to the Hall current. Across the ribbon, in the y direction, the wave function of an electron is peaked about some value of y, say Yo' The allowed values of Yo extends from one edge of the ribbon to the other. (We are using here the "strip" representation of the wave functions, as shown in Fig. l1.9a.) Consider now a completely filled Landau level. The electron density across the ribbon may be represented schematically as in Fig. 11.14. The electrons lying closer to the right edge have a higher electrostic energy because of the applied voltage. Now imagine that the flux through the loop is increased slowly from zero. The electrons will respond to the change until the flux reaches the quantum value he/ e, at which point they cannot feel the flux. During the slow increase, the energy of the electrons must rise by the transfer of electrons from one edge of the ribbon to the other. When the flux reaches one quantum, the electron distribution must look exactly the same as before. Overall, therefore, the electrons play musical chairs, moving up one position per quantum of flux penetration, as indicated in Fig. 11.14. Since the gain in energy is 8E = e V, and the change in flux is 8cI> = he/e, we have from (11.97) 1= (e 2 /h)V, whence
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------Edgesof ribbon-------
Fig. 11.14 Schematic representation of electron density across the ribbon in Fig. 11.13, when the lowest Landau level is completely filled. The electrons move to the right by one "musical chair," when one unit of test flux pushes through the loop in Fig. 11.13.