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ISSN 0021-3640, JETP Letters, 2008, Vol. 88, No. 11, pp. 745­746. © Pleiades Publishing, Ltd., 2008. Original Russian Text © S.S. Murzin, 2008, published in Pis'ma v Zhurnal èksperimental'nooe i Teoreticheskooe Fiziki, 2008, Vol. 88, No. 11, pp. 860­861.

Resistivity of Two-Dimensional Systems in a Magnetic Field at the Filling Factor n = 1/2
S. S. Murzin
Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow region, 142432 Russia e-mail: murzin@issp.ac.ru
Received October 10, 2008; in final form, October 27, 2008

Experimental data on the diagonal resistivity xx of GaAs/AlGaAs heterostructures in a magnetic field at the filling factor = 1/2 have been compared with the existing theoretical predictions [B. I. Halperin et al., Phys. Rev. B 47, 7312 (1993) and F. Evers et al., Phys. Rev. B 60, 8951 (1999)]. The experimental results have been found to follow the relation xx(1/2) n­2d­1.64, which disagrees with the predictions. PACS numbers: 71.30.+h, 73.43.-f DOI: 10.1134/S0021364008230094

The expression for the diagonal resistivity xx of two-dimensional systems in a magnetic field at the filling factor = nh/eB = 1/2 was obtained in [1] based on the composite fermion theory. The composite fermions are scattered by a random magnetic field induced by ionized impurities. The impurity concentration ni in the ideal selectively-doped two-dimensional sample is equal to the electron density n and, at = 1/2, 1h xx ( 1/2 ) ------- ---- . k F d e2 (1)

The experimental data are compared with Eq. (2) in Fig. 1 where we plotted the experimental and theoretical dependences of xx(1/2)kF on d (let us remind that kF = 4 n ). The experimental points exhibit a large spread not very far from the theoretical line. We tried to plot the experimental data in the form xx(1/2)np(d) with different integer and half-integer p values and to fit them by linear functions in the log­log scale. The best fit was obtained for p = 2 and the exponent of d equal to ­1.64 (see Fig. 2). This corresponds to the expression xx ( 1/2 ) = n d
­ 2 ­ 1.64

Here, kF = 4 n is the wavenumber of the composite fermions at the Fermi level and d is the spacer thickness. A more general and detailed analysis of xx(1/2) carried out in [2] yields the same but more accurate result for the ni = n case: (2)
(he­1 m­1)

,

(3)

1h xx ( 1/2 ) = 1.0 ------- ---- . k F d e2

In this work, Eq. (2) is compared with the published experimental data [3­16] on xx(1/2) of single GaAs/AlGaAs heterojunctions with one doped layer. We used the xx(1/2) data for the gateless samples with the mobility µ > 40 m2/V s, the electron density 6 â 1014 n < 5 â 1015 m­2, and the spacer thickness 20 d 240 nm. Some samples [3­8] were irradiated by light, the others [9­16] were not. The data for only two samples were not used. The fractional quantum Hall effect in these samples was substantially weaker than that in the other samples with the close parameters. For the sample used in [16] whose resistance depended on its prehistory, we used the data for the minimum-disorder case.
745

Fig. 1. xx(1/2)kF versus the spacer thickness d. The circles are the experimental data, and the straight line corresponds to Eq. (2).

F


746

MURZIN

tron density n. The length corresponding to is l = -1/2.36 = 5.2 â 10­8 m. Figure 2 indicates that the large spread in the data points in Fig. 1 does not result from the presence of random impurities or defects in the samples. The regular arrangement of the points in Fig. 2 calls for a new explanation of the electron transport in the magnetic field with = 1/2. Note that on average for non-irradiated samples, n decreases with an increase in d (see Fig. 3), but the relative spread of the data points in Fig. 3 is much larger than that in Fig. 2. This work was supported by the Russian Foundation for Basic Research and INTAS. REFERENCES

m

Fig. 2. xx(1/2)n2 versus the spacer thickness d. The gles are the experimental data, and the solid line is the fit of the data in the log­log scale. The (dashed line) and (dotted line) d ­2 dependences are also shown for parison.

trianlinear d ­1.5 com-

1 n (1015 m­2)

2

20

50 d (nm)

100

200

Fig. 3. Electron density n in non-irradiated samples versus the spaces thickness d.

where = 1.6 â 1017 m­2.36. The straight lines corresponding to d­1.5 and d­2 are also shown in Fig. 2. They display somewhat poorer agreement with the experimental data. The coefficient is independent of the magnetic field. At a given filling factor = nh/eB = 1/2, the magnetic field is unambiguously related to the elec-

1. B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, 7312 (1993). 2. F. Evers, A. D. Mirlin, D. G. Polyakov, and P. Wolfle, Phys. Rev. B 60, 8951 (1999). 3. J. R. Mallett, R. G. Clark, R. J. Nicholas, et al., Phys. Rev. B 38, 2200 (1988), Fig. 1. 4. D. R. Leadley, R. J. Nicholas, C. T. Foxon, and J. J. Harris, Phys. Rev. Lett. 72, 1906 (1994), Fig. 1. 5. S. Holmes, D. K. Maude, M. L. Williams, et al., Semicond. Sci. Technol. 9, 1549 (1994), Fig.1. 6. L. P. Rokhinson, B. Su, and V. J. Goldman, Phys. Rev. B 52, 11588 (1995), Fig. 3, sample A, at the maximum of the curve. 7. P. T. Coleridge, Z. W. Wasilewski, P. P. Zawadzki, et al., Phys. Rev. B 51, 11603 (1995), Fig. 1. 8. D. R. Leadley, M. van der Burgt, R. J. Nicholas, et al., Phys. Rev. B 53, 2057 (1996), Figs. 1a, 1b, 2. 9. D. C. Tsui, H. L. Stormer, J. C. M. Hwang, et al., Phys. Rev. B 28, 2274 (1983), Fig. 1. 10. E. E. Mendez, M. Heiblum, L. L. Chang, and L. Esaki, Phys. Rev. B 28, 4886 (1983), Fig. 1a. 11. R. J. Haug, K. von Klitzing, and R. J. Nicholas, Phys. Rev. B 36, 4528 (1987), Fig. 4. 12. R. G. Clark, J. R. Mallett, A. Usher, et al., Surf. Sci. 196, 219 (1988), Fig. 1c. 13. R. G. Clark, J. R. Mallett, S. R. Haynes, et al., Phys. Rev. Lett. 60, 1747 (1988). Fig. 1a. 14. A. J. Turberfield, S. R. Haynes, P. A. Wright, et al., Phys. Rev. Lett. 65, 637 (1990), Fig. 3b. 15. S. Koch, R. J. Haug, K. von Klitzing, and K. Ploog, Physica B 184, 72 (1993), Fig. 1. 16. I. V. Kukushkin, R. J. Haug, K. von Klitzing, and K. Eberl, Phys. Rev. B 52, 18045 (1995), Fig. 2.

Translated by A. Safonov

JETP LETTERS

Vol. 88

No. 11

2008