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ISSN 0021 3640, JETP Letters, 2010, Vol. 91, No. 3, pp. 155­157. © Pleiades Publishing, Inc., 2010. Original Russian Text © S.S. Murzin, 2010, published in Pis'ma v Zhurnal èksperimental'no i Teoretichesko Fiziki, 2010, Vol. 91, No. 3, pp. 166­168.

On the Phase Boundaries of the Integer Quantum Hall Effect. Part II
S. S. Murzin
Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow region, 142432 Russia e mail: murzin@issp.ac.ru
Received January 11, 2010

It has been shown that the observation of the transitions between the dielectric phase and the integer quan q tum Hall effect phases with the quantized Hall conductivity xy 3e2/h announced in a number of works is unjustified. In these works, the crossing points of the magnetic field dependence of the diagonal resistivity xx at different temperatures T and c = 1 have been misidentified as the critical points of the phase transi tions. In fact, these crossing points are due to the sign change of the derivative dxx/dT owing to the quantum corrections to the conductivity. Here, c = eB/m is the cyclotron frequency, is the transport relaxation time, and m is the effective electron mass. DOI: 10.1134/S0021364010030112

The phase diagram of two dimensional systems in the magnetic field has attracted the attention of both theorists and experimentalists for many years. Treating the integer quantum Hall effect (IQHE) in the context of the two parameter scaling theory [1], which is graphically represented as a flow diagram [2, 3], yields the solution of the problem disregarding the electron­ electron interaction. The further development of the scaling theory showed that the electron­electron interaction does not affect the position of the IQHE phase boundaries of a spin polarized electron system [4, 5]. According to the scaling theory, the boundary between two IQHE phases is possible only if the quan tized values of the Hall conductivity q of these two xy phases at zero temperature differ by e2/h or (in the case of the spin degeneracy of the Landau levels) 2e2/h. However, a number of works reported on the observa q tion of the transitions between the dielectric ( xy = 0) phase and the IQHE phases with the quantized Hall q conductivity xy 3e2/h. Song et al. [6] reported on xy = the observation of the transition xy = 0 2/h in two dimensional hole systems in a strained 3e Ge quantum well. The observation of the transitions q q xy = 0 xy 3e2/h in the two dimensional hole systems in a strained Ge quantum well was also announced in [7]. Lee et al. [8] claimed the observation q q q xy = 6e2/h and xy = of the transitions xy = 0 0
q xy q q

quantum wells. Huang et al. [9] also reported on the q observation of the transitions from the state with xy = 0 to the states with xy = 6­16e2/h in GaAs/AlGaAs heterojunctions. In all of these works [6­9], the cross ing points of the magnetic field dependence of the diagonal resistivity xx at c 1 and different temper atures T (c = eB/m is the cyclotron frequency, is the transport relaxation time, and m is the effective elec tron mass) were considered as the critical points Bc of the phase transitions (see Fig. 1). In this case, xx weakly depends on the magnetic field and temperature near Bc. In this work, it is shown that the above statements of the observation of the transitions between the dielectric phase and the IQHE phases with the quan q tized Hall conductivity xy 3e2/h are unjustified. In fact, the crossing point of the magnetic field depen dence of the diagonal resistivity xx(T) at different temperatures is caused by the sign change of the deriv ative dxx/dT owing to the quantum corrections [10] to the conductivity. The classical diagonal and Hall conductivity in the magnetic field take the form and
0 xy 0 xx q

Ne 1 =s m 1 + ( c )
2

2

2

(1)

= 8e2/h in doped AlGaAs/GaAs/AlGaAs
155

=

Ns e c , m 1 + ( c )2

(2)


156

MURZIN

0.3 K 3.2 K

3.2 K
B
c

(e2/h)

c

0.3 K

Fig. 1. Diagonal resistivity xx and the Hall resistivity xy of the doped AlGaAs/GaAs/AlGaAs quantum well. The electron density is Ns = 1.04 â 1016 m­2. The temperatures for different xx values are 0.3, 0.5, 0.8, 1.2, 1.7, 2, and 3.2 K. The spin splitting is small and therefore invisible in xx and xy. The figure is taken from [8].

(e2/h)
Fig. 2. Schematic scaling flow diagram for the quantum well with the parameters given in Fig. 1. The spin splitting is negligible. The solid lines are the separatrices of the dia gram. The dashed lines are the bare conductivities ) at c < 1 for the sample with the zero field bare conductivity 0 = 7.52e2/h. The dotted lines are the scal ing flow lines. The dash­dotted straight line corresponds to c = 1. xy (
0 0 xx

where Ns is the electron density. The quantum correc xx to the diagonal tions xx(T) = xx(T) ­ xx conductivity decrease with temperature. At T 0, xx(T) 0 except for the critical points, where = (i + 1/2) (see Fig. 2). Excluding the weak localization region (B 1 T), the Hall conductivity xy is independent of the temperature down to the temperatures at which xx ~ e2/h. At lower tempera tures, the Hall conductivity depends on the tempera ture and approaches the nearest quantized integer value xy(Bi)= ie2/h with Bi < Bc at c < 1. Taking into account the quantum corrections, the diagonal and Hall resistivity of the two dimensional electron system are given by the expressions e2/h xx ( T ) = xx + [ ( xy ) ­ ( xx ) ] xx ( T ) and xy ( T ) = xy ­ 2 xx xy xx ( T ) . Here, we took into account that (
0 02 xy ) ] 0 xx 0 0 0 0 0 2 0 2 0 xy 0 0

of the conductivity and resistivity, which correspond to the diffusion motion of electrons without the inter ference (localization) effects at distances longer than the diffusion step length. The temperature depen dence of xx changes its sign in the magnetic field B such that xx (B) = xy (B). In the classical treatment, xx (B) = xy (B) at c = 1. At xy e2/h and c < 1, the diagonal resistivity xx first increases with a decrease in the temperature, reaching the value
xx, max 0 0 0 0

(3)

=

1 , 0 2 xy

(5)

(4)
0 /[( xx )2 + 0 0 xx , xy ,

=
0

0 xx

and
0 xy

0 xy

= /[(

0 xy

02 xx )

+ ( xy )2];

xx , and

are the bare (non renormalized) values

0 excluding and then decreases and vanishes at T 0 the critical magnetic fields in which xy = (i + 1/2)e2/h. Thus, the negative value of the derivative dxx/dT within the experimental range does not imply that the electron system is in a dielectric phase.
JETP LETTERS Vol. 91 No. 3 2010


ON THE PHASE BOUNDARIES OF THE INTEGER QUANTUM HALL EFFECT

157

Note that the magnetic field position of the IQHE phases at c 1 is not determined by the filling factor . Rather, it is given by the magnitude of xy h/e2. This quantity is different from at c 1 [11]. Thus, we have shown that the statements [6­9] of the observation of the transitions between the dielec tric phase and the IQHE phases with the quantized q Hall conductivity xy 3 are unjustified. In fact, the crossing point of the magnetic field dependence of the diagonal resistivity xx(T) at different temperatures is caused by the sign change of the derivative dxx/dT at c 1 owing to the quantum corrections to the con ductivity. This work was supported by the Russian Founda tion for Basic Research. REFERENCES
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0

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, M. A. Baranov, and I. S. Burmis trov, Pis'ma Zh. Eksp. Teor. Fiz. 82, 166 (2005) [JETP Lett. 82, 150 (2005)]. 5. A. M. M. Pruisken and I. S. Burmistrov, Pis'ma Zh. Eksp. Teor. Fiz. 87, 252 (2008) [JETP Lett. 87, 220 (2008)]. 6. S. H. Song, D. Shahar, D. C. Tsui, et al., Phys. Rev. Lett. 78, 2200 (1997). 7. M. Hilke, D. Shahar, S. H. Song, et al., Phys. Rev. B 62, 6940 (2000). 8. C. H. Lee, Y. H. Chang, Y. W. Suen, and H. H. Lin, Phys. Rev. B 58, 10629 (1998). 9. C. F. Huang, Y. H. Chang, C. H. Lee, et al., Phys. Rev. B 65, 045303 (2001). 10. B. L. Al'tshuler and A. G. Aronov, in Electron­Electron Interaction in Disordered Systems, Ed. by A. L. Efros and M. Pollak (North Holland, Amsterdam, 1987). 11. S. S. Murzin, Pis'ma Zh. Eksp. Teor. Fiz. 89, 347 (2009) [JETP Lett. 89, 298 (2009)].

Translated by A. Safonov

JETP LETTERS

Vol. 91

No. 3

2010