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Quantum hall effect and additional oscillations of conductivity in weak magnetic fields
Volume 106A, number 4
PHYSICS LETTERS
3 December 1984
QUANTUM HALL E F F E C T AND ADDITIONAL OSCILLATIONS OF CONDUCTIVITY IN WEAK MAGNETIC FIELDS
D.E. KHMELNITSKII L.D. Landau Institute for Theoretical Physics, Chernogolovka, Moscow District 142432, USSR Received 23 July 1984
In the framework of the scaling hypothesis for localization of 2D electrons in a magnetic field B it is shown that at T = 0 the conductivity on has maxima at B(I,2) = (rnc/e~) (Ev/(2n + 1) +- ([ET:/(2n + 1) 12 _ (8/r)2 }!/2 ), i.e. besides oscillations of the Shubnikov type with maxima at Bn(t) ,~ (mc'/eli)EF/(n + 1/2)there is the same number of additional oscillations at B(2) ~ (mc/e)(~l/E,~r2)(n + 1/2). The Hall conductivity o_ = (e2/2nTl)n at B.(1) < B < B (1), and at B(r~ 2), < B < B(~) gives the number of delocalized states at E < E F with energies E n described by the interpolating relation E n = ~ 2 (n + 1/2)[ 1 + (12r)-2 ]. At B < B0, when r H > 1 (r H is the magnetic length, 1 is the mean free path), all states with E < E F are localized.
1. The quantization o f the Hall conductivity discovered experimentally in 1980 [ 1 ] was almost immediately explained as a result o f the Anderson localization o f electronic states in a random impurity potential [ 2 - 4 ] . It was necessary to suggest [3] that, provided disorder is weak, i.e. ~2r >> 1 (~2 = eB/mc is the cyclotron frequency, r is the mean free time), at least one state should be localized at each Landau level. The evident contradiction between this and the generally accepted suggestion that all states in a 2D system are localized [5] has been recently settled by Levin, Libby and Pruisken [6]. They have shown that the effective statistical sum Z o f this problem contains the Hall conductivity in the form exp [i Oxy (41r2h/e2)N] , where N is integer due to topological reasons, and therefore, Oxy changing by a magnitude multiple o f e2/2rr/i, Z does not change. They have also shown that at Oxy = (e2/21ffO(n + 1/2), where n is integer, the dissipative conductivity Oxx 4= O, and, under these conditions, the state with E = E F is delocatized. Thus, the necessary agreement has been achieved, but it has turned out that under the Fermi level there are n ~ EF/~I2 delocalized states ,1, n increasing with decreasing magnetic field. But it is evident that at B = 0 all states with E < E F should be localized. This
182
paper aims at solving this paradox. 2. In ref. [7] a scaling theory of electron localization in magnetic field has been developed. The dependencies o f Oxx and Oxy on the scale L are given by the renormalization group equations
doxx/d~ = 3xx (Oxx, Oxy) , doxy/d~=[Jxy (Oxx , Oxy ),
~ =lnL,
(1)
where 3xx and 3xy are periodic functions o f Oxy, their period equal to eZ/2n~. The phase diagram o f the system (1) is shown in fig. 1. Initial conditions, o(x° x) and o(°.~, at In L = 0 should be obtained with the help of ~y the kinetic equation. In particular, it is convenient to use the following interpolating formula for O(x~:
o ~ ) = (ne2r/m) ~ r / [ 1 + ( ~ . ) 2 ] .
(2)
The result o f the renormalization o f Oxy and Oxx, i.e. 4:1 It is not clear now whether there is (i) only one delocalized state at the Landau level, or (ii) a band o f levels, In the case (i) Oxx ~ 0 at discrete values of B, and in the case (ii) Oxx ~= 0 in a range o f B. In what follows the difference is not important, and it would be suggested that the case (i) is
realized as more probable. If it appears that the case (ii) takes place, then the words "a delocalized energy level" should be replaced by "a band of delocalized states".
Volume 106A, number 4
PHYSICS LETTERS
cides with the Fermi level, Oxx has a peak. At [2r < 1 broadening of Landau levels is larger than the distance between the levels. Under these conditions, the energies of delocalized states increase with decreasing field and once more coincide with EF, resulting in additional oscillations. The number of delocalized states with E < E F is governed by the value of the Hall conductivity measured in the units of e2/2nR. Using eq. (2) and expressing electronic concentration through EF, we obtain the following interpolating formula for the energy of nth delocalized state * 2:
2.S" 2
I.SI
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6.S" 1
3 December 1984
2
Fig. 1. the field dependence of observed values of Oxy and Oxx at T = 0, is shown in figs. 2a and 2b. As seen from fig. 2b, besides the Shubnikov-type oscillations with maxima at fields B (1) ~ (mc/etOEF/(n + 1/2),
(3)
there is the same number o f additional oscillations with maxima o f Oxx at B (2) -~ (mc~i/e)(l/EFr2)(n + I / 2 ) . This result may be interpreted as follows: at I2r >~ 1 a delocalized level is associated with each Landau subband. When the energy of the delocalized level coin-
E n = tiI2(n + 1/2)[1 + ( ~ r ) - 2 ] .
(5)
The lowest delocalized level crosses E F at such fields, when r n = (ch/eB) 1/2 ~ l= OFt. At l < r n all states with E < E F are localized. 3. The additional oscillations are very difficult to observe since it is necessary that due to weak localization the conductivity Oxx ~ (EFrflOe2/ti should decrease to the value of order e2/h. However, if axy and axx are measured in a 2D system, for which EFt~ h ~ 1 and Oxy has a maximum of order e2fli at T 1 0 - 2 0 K, the function axy(B ) is expected to exhibit a plateau. The beginning and end of this plateau should coincide with the maxima (of the Shubnikov type at large B and additional at small B).
2 ~'F-I~1~/e ~ ,2 This relation has also been suggested by R.B. Laughlin.
2.5 2 ' I.S
References
I
8.S
b
H
[ 1] K.V. Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45 (1980) 494. [2] R.B. Laughlin, Phys. Rev. B23 (1981) 5632. [31 B.I. Halperin, Phys. Rev. B25 (1982) 2185. [4] H. Aohi and T. Ando, Solid State Commun. 38 (1981) 1079. [5] E. Abrahams, P.W. Anderson, D.C. Licardello and T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673. [6] H. Levin, S.B. Libby and A.M.M. Pruisken, Phys. Rev. Lett. 53 (1983) 1915. [71 D.E. Khmelnitskii, Pisma ZhETP 38 (1983) 454.
Fig. 2.
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PHYSICS LETTERS
3 December 1984
QUANTUM HALL E F F E C T AND ADDITIONAL OSCILLATIONS OF CONDUCTIVITY IN WEAK MAGNETIC FIELDS
D.E. KHMELNITSKII L.D. Landau Institute for Theoretical Physics, Chernogolovka, Moscow District 142432, USSR Received 23 July 1984
In the framework of the scaling hypothesis for localization of 2D electrons in a magnetic field B it is shown that at T = 0 the conductivity on has maxima at B(I,2) = (rnc/e~) (Ev/(2n + 1) +- ([ET:/(2n + 1) 12 _ (8/r)2 }!/2 ), i.e. besides oscillations of the Shubnikov type with maxima at Bn(t) ,~ (mc'/eli)EF/(n + 1/2)there is the same number of additional oscillations at B(2) ~ (mc/e)(~l/E,~r2)(n + 1/2). The Hall conductivity o_ = (e2/2nTl)n at B.(1) < B < B (1), and at B(r~ 2), < B < B(~) gives the number of delocalized states at E < E F with energies E n described by the interpolating relation E n = ~ 2 (n + 1/2)[ 1 + (12r)-2 ]. At B < B0, when r H > 1 (r H is the magnetic length, 1 is the mean free path), all states with E < E F are localized.
1. The quantization o f the Hall conductivity discovered experimentally in 1980 [ 1 ] was almost immediately explained as a result o f the Anderson localization o f electronic states in a random impurity potential [ 2 - 4 ] . It was necessary to suggest [3] that, provided disorder is weak, i.e. ~2r >> 1 (~2 = eB/mc is the cyclotron frequency, r is the mean free time), at least one state should be localized at each Landau level. The evident contradiction between this and the generally accepted suggestion that all states in a 2D system are localized [5] has been recently settled by Levin, Libby and Pruisken [6]. They have shown that the effective statistical sum Z o f this problem contains the Hall conductivity in the form exp [i Oxy (41r2h/e2)N] , where N is integer due to topological reasons, and therefore, Oxy changing by a magnitude multiple o f e2/2rr/i, Z does not change. They have also shown that at Oxy = (e2/21ffO(n + 1/2), where n is integer, the dissipative conductivity Oxx 4= O, and, under these conditions, the state with E = E F is delocatized. Thus, the necessary agreement has been achieved, but it has turned out that under the Fermi level there are n ~ EF/~I2 delocalized states ,1, n increasing with decreasing magnetic field. But it is evident that at B = 0 all states with E < E F should be localized. This
182
paper aims at solving this paradox. 2. In ref. [7] a scaling theory of electron localization in magnetic field has been developed. The dependencies o f Oxx and Oxy on the scale L are given by the renormalization group equations
doxx/d~ = 3xx (Oxx, Oxy) , doxy/d~=[Jxy (Oxx , Oxy ),
~ =lnL,
(1)
where 3xx and 3xy are periodic functions o f Oxy, their period equal to eZ/2n~. The phase diagram o f the system (1) is shown in fig. 1. Initial conditions, o(x° x) and o(°.~, at In L = 0 should be obtained with the help of ~y the kinetic equation. In particular, it is convenient to use the following interpolating formula for O(x~:
o ~ ) = (ne2r/m) ~ r / [ 1 + ( ~ . ) 2 ] .
(2)
The result o f the renormalization o f Oxy and Oxx, i.e. 4:1 It is not clear now whether there is (i) only one delocalized state at the Landau level, or (ii) a band o f levels, In the case (i) Oxx ~ 0 at discrete values of B, and in the case (ii) Oxx ~= 0 in a range o f B. In what follows the difference is not important, and it would be suggested that the case (i) is
realized as more probable. If it appears that the case (ii) takes place, then the words "a delocalized energy level" should be replaced by "a band of delocalized states".
Volume 106A, number 4
PHYSICS LETTERS
cides with the Fermi level, Oxx has a peak. At [2r < 1 broadening of Landau levels is larger than the distance between the levels. Under these conditions, the energies of delocalized states increase with decreasing field and once more coincide with EF, resulting in additional oscillations. The number of delocalized states with E < E F is governed by the value of the Hall conductivity measured in the units of e2/2nR. Using eq. (2) and expressing electronic concentration through EF, we obtain the following interpolating formula for the energy of nth delocalized state * 2:
2.S" 2
I.SI
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6.S" 1
3 December 1984
2
Fig. 1. the field dependence of observed values of Oxy and Oxx at T = 0, is shown in figs. 2a and 2b. As seen from fig. 2b, besides the Shubnikov-type oscillations with maxima at fields B (1) ~ (mc/etOEF/(n + 1/2),
(3)
there is the same number o f additional oscillations with maxima o f Oxx at B (2) -~ (mc~i/e)(l/EFr2)(n + I / 2 ) . This result may be interpreted as follows: at I2r >~ 1 a delocalized level is associated with each Landau subband. When the energy of the delocalized level coin-
E n = tiI2(n + 1/2)[1 + ( ~ r ) - 2 ] .
(5)
The lowest delocalized level crosses E F at such fields, when r n = (ch/eB) 1/2 ~ l= OFt. At l < r n all states with E < E F are localized. 3. The additional oscillations are very difficult to observe since it is necessary that due to weak localization the conductivity Oxx ~ (EFrflOe2/ti should decrease to the value of order e2/h. However, if axy and axx are measured in a 2D system, for which EFt~ h ~ 1 and Oxy has a maximum of order e2fli at T 1 0 - 2 0 K, the function axy(B ) is expected to exhibit a plateau. The beginning and end of this plateau should coincide with the maxima (of the Shubnikov type at large B and additional at small B).
2 ~'F-I~1~/e ~ ,2 This relation has also been suggested by R.B. Laughlin.
2.5 2 ' I.S
References
I
8.S
b
H
[ 1] K.V. Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45 (1980) 494. [2] R.B. Laughlin, Phys. Rev. B23 (1981) 5632. [31 B.I. Halperin, Phys. Rev. B25 (1982) 2185. [4] H. Aohi and T. Ando, Solid State Commun. 38 (1981) 1079. [5] E. Abrahams, P.W. Anderson, D.C. Licardello and T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673. [6] H. Levin, S.B. Libby and A.M.M. Pruisken, Phys. Rev. Lett. 53 (1983) 1915. [71 D.E. Khmelnitskii, Pisma ZhETP 38 (1983) 454.
Fig. 2.
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