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ISSN 0021 3640, JETP Letters, 2009, Vol. 89, No. 6, pp. 298­300. © Pleiades Publishing, Ltd., 2009. Original Russian Text © S.S. Murzin, 2009, published in Pis'ma v Zhurnal èksperimental'no i Teoretichesko Fiziki, 2009, Vol. 89, No. 6, pp. 347­349.

On the Phase Boundaries of the Integer Quantum Hall Effect
S. S. Murzin
Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow region, 142432 Russia e mail: murzin@issp.ac.ru
Received February 6, 2009; in final form, February 13, 2009

It has been pointed out that, according to the two parameter scaling theory, the magnetic field position of the phases of the integer quantum Hall effect (IQHE) at c 1 is not determined by the filling factor = nh/eB. The position of the IQHE phases is given by the bare Hall conductivity 0 . In this regard, it has been xy shown that the diagonal resistivity in the magnetic field measured by Sakr et al. [Phys. Rev. B 64, 161308 (2001)] does not exhibit transitions between the xy = 3, 4 and 6 IQHE states on the one hand and the dielec tric state on the other hand in contrast to the assertion by Sakr et al. PACS numbers: 71.30.+h, 73.43. f DOI: 10.1134/S0021364009060083

The phase diagram of the integer quantum Hall effect (IQHE) for many years has attracted the interest of both theorists and experimentalists. Interpretation of the IQHE on the basis of the two parameter scaling theory [1], which is graphically represented as a flow diagram [2, 3], suggests the solution of the problem disregarding the electron­electron interaction. Fur ther development of the scaling theory showed that the electron­electron interaction [4­7] does not affect the position of the IQHE phase boundaries. According to the scaling approach, the boundary between two IQHE phases is possible only when the q quantized values of the Hall conductivity xy in these two phases at zero temperature differ by 1 or (in the case of spin degeneracy of the Landau levels) 2 in units of e2/h. Sakr et al. [8] reported the observation of dielectric phases in p SiGe heterostructures between the IQHE phases with the filling factors = 2 and 3, 3 q and 4, and 4 and 6. They supposed that = xy and, consequently, the changes in xy at the 3 0, 4 0, and 6 0 phase transitions (0 denotes the dielec tric state) are greater than 2, implying that the two parameter scaling theory interpretation of the IQHE is incorrect. In this work, it is pointed out that, according to the two parameter scaling theory, the magnetic field posi tion of the IQHE phases at c 1 is not determined by the filling factor . Here, c = eB/m is the cyclotron frequency, is the transport relaxation time, and m is the electron effective mass. The positions of the phases are determined by the bare (see below) Hall conduc 0 tivity xy . In this regard, it is shown that the depen dences of the diagonal resistivity xx on the magnetic
q

field B presented in [8] do not exhibit the presence of q the IQHE phases with xy = 3, 4, and 6 and, the more so, the 3 0, 4 0, and 6 0 transitions. The bare (non renormalized) Hall conductivity 0 xy corresponds to the diffusive motion of an electron without the interference (localization) effects at dis tances longer than the diffusion step length. It exhibits Shubnikov­de Haas oscillations with a change in the magnetic field [3]. According to the scaling theory, different IQHE phases of spinless electrons [1­3] are separated in the magnetic fields Bi , for which xy ( B i ) = i + 1/2,
0

(1)

where i = 0, 1, 2, 3, .... The Shubnikov­de Haas oscil 0 lations of xy are small at c 1 and c 1, where xy . For this reason, the crude consideration involves the classical expression for the bare Hall con ductivity (in units of e2/h)
0 xy 0

=

( c ) n h c = . 2 2 m 1 + ( ) 1 + ( c ) c
0

2

(2)

It follows from Eq. (2) that at c 1, xy and, consequently, the magnetic field positions of the IQHE phase boundaries are different from the posi tions of the half integer values at which the maxima of the Shubnikov­de Haas oscillations of the bare 0 diagonal resistivity xx occur. The positions of the IQHE phase boundaries are found from the solution of Eqs. (1) and (2). As an example, consider the two dimensional sys tem with the resistivity in zero magnetic field 0 =

298


ON THE PHASE BOUNDARIES OF THE INTEGER QUANTUM HALL EFFECT

299

0.133 (in units of h/e2), the Fermi energy EF = 7.5 K, and the effective mass m = 0.3m0 (m0 is the free elec tron mass). In this case, the transport relaxation time is = /(0EF) = 7.64 â 10­12 s and n = mEF/2 2 = 4.05 â 1014 m­2. The figure shows the schematic draw 0 ings of (dotted line) the bare diagonal resistivity xx ; (dashed line) the diagonal resistivity xx, T with the inclusion of localization effects at finite temperature; and (solid line) the diagonal resistivity xx,0 at zero temperature as functions of c. The upper scale in the figure is inversely proportional to . The lower scale c is constructed from the upper one with known n, , and m: c = nh 1 = 7.5 . m
0

(3)

(h/e2)

The bare diagonal resistivity xx is plotted so that the Shubnikov­de Haas oscillation maxima and minima appear at half integer and integer values, respec tively. The oscillation amplitude is drawn arbitrarily. The diagonal resistivity at zero temperature xx,0 is zero everywhere except the IQHE phase boundaries. The positions of the peaks that appear at the phase boundaries were calculated according to Eqs. (1) and q (2) and the amplitudes at the xy xy + 1 transi tions [9] are 1/2 xy ( xy + 1). The positions of the peaks are different from the positions of the Shubni kov­de Haas oscillation maxima of the bare resistivity 0 0 xx . In particular, the xx maxima at zero temperature and = 3.5 and 4.5 appear in the IQHE region with q 0 xy = 3 and xx,0 = 0. The xx minimum at zero tem perature and = 6 lies in the minimum of the IQHE q q region with xy = 4 and the IQHE region with xy = 6 is absent. To show how the curve (c) is transformed into xx,0(c) with an increase in the localization effects, the diagonal resistivity at finite temperature xx, T is plotted. In strong magnetic fields, the curve xx, T(c) consists of the broadened peaks at the phase boundaries. In weak magnetic fields at c < 1, xx, T > xx due to quantum corrections. In the intermediate region (1.5 < c < 2.5), the of minima and maxima Shubnikov­de Haas xx, T are shifted downward with respect to the xx extrema. Reasoning from the aforesaid, consider the results of Sakr et al. [8], who presented the magnetic field dependence of the diagonal resistivity xx for p SiGe heterostructures. They interpret the weak minima of xx as indications of the IQHE phases with the quan tized values
q xy 0 0 0 xx q q

(Dotted line) The bare diagonal resistivity xx ; (dashed line) the diagonal resistivity xx, T with the inclusion of localization effects at finite temperature; and (solid line) the diagonal resistivity xx,0 at zero temperature schemat ically plotted as functions of c. The upper scale in the figure is inversely proportional to . The numbers between the peaks of xx,0 are the values of the Hall conductivity at zero temperature
q xy

0

in different IQHE phases.

dielectric phases between the IQHE phases with = 2 and 3, 3 and 4, and 4 and 6. At the same time, they assume that the IQHE phases are characterized by the q filling factor and that xy in these phases are equal to the nearest integer values. According to our consid eration of the problem, this is not the case. At c 1, xy < (see Eq. (2)). Although Sakr et al. [8] quoted only the xx(B) curves, it may be shown that xy < 1.7 always when > 1.2, in particular, at = 3, 4, and 6. Indeed, max [ xy ] = max
2 xy 2
xx

q

( xx ) + ( xy )

= 1/2 xx .
= const

(4)

For > 1.2, xx > 0.29 and thus xy < 1.7. The value 1.7 is significantly smaller than 3, 4, and 6; therefore, there are no grounds to speak about the IQHE with q xy = 3, 4, and 6 and the transitions 3 0, 0, 4 and 6 0. According to the scaling theory, only the q IQHE with xy = 1 and 2 is possible in the samples under consideration. To conclude, it has been showed that the magnetic field position of the integer quantum Hall effect 0 phases is determined by the bare Hall conductivity xy rather than the filling factor. For c 1, xy . The experimental results reported in [8] do not indicate the direct transitions from the dielectric phase to the inte q ger quantum Hall effect phases with xy 3e2/h.
0

= and claim the observation of
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MURZIN 5. A. M. M. Pruisken, M. A. Baranov, and I. S. Burmis trov, Pis'ma Zh. Eksp. Teor. Fiz. 82, 166 (2005) [JETP Lett. 82, 150 (2005)]. 6. A. M. M. Pruisken and I. S. Burmistrov, Ann. Phys. (N.Y.) 322, 1265 (2007). 7. A. M. M. Pruisken and I. S. Burmistrov, Pis'ma Zh. Eksp. Teor. Fiz. 87, 252 (2008) [JETP Lett. 87, 220 (2008)]. 8. M. R. Sakr, Maryam Rahimi, S. V. Kravchenko, et al., Phys. Rev. B 64, 161308 (2001). 9. S. S. Murzin, Pis'ma Zh. Eksp. Teor. Fiz. 88, 374 (2008) [JETP Lett. 88, 326 (2008)].

I am grateful to V.F. Gantmakher for fruitful dis cussions. This work was supported by the Russian Foundation for Basic Research and INTAS. REFERENCES
1. H. Levine, S. B. Libby, and A. M.M. Pruisken, Phys. Rev. Lett. 51, 1915 (1983). 2. D. E. Khmel'nitskii, Pis'ma Zh. Eksp. Teor. Fiz. 38, 454 (1983) [JETP Lett. 38, 552 (1984)]; Phys. Lett. A 106, 182 (1984); Helvetica Phys. Acta 65, 164 (1992). 3. A. M. M. Pruisken, in The Quantum Hall Effect, Ed. by R. E. Prange and S. M. Girvin (Springer, Berlin, 1990). 4. A. M. M. Pruisken and I. S. Burmistrov, Ann. Phys. (N.Y.) 316, 285 (2005).

Translated by A. Safonov

JETP LETTERS

Vol. 89

No. 6

2009