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PHYSICAL REVIEW B 71, 155328 2005

Spin splitting in the quantum Hall effect of disordered GaAs layers with strong overlap of the spin subbands
S. S. Murzin,1,2 M. Weiss,2 D. A. Knyazev,1 A. G. M. Jansen,2,3 and K. Eberl4
of Solid State Physics RAS, 142432, Chernogolovka, Moscow District, Russia Grenoble High Magnetic Field Laboratory, Max-Planck-Institut fÝr FestkÆrperforschung and Centre National de la Recherche Scientifique, BoÍte Postale 166, F-38042, Grenoble Cedex 9, France Service de Physique Statistique, MagnÈtisme, et SupraconductivitÈ, DÈpartement de Recherche Fondamentale sur la MatiÕre CondensÈe, CEA-Grenoble, 38054 Grenoble Cedex 9, France 4 Max-Planck-Institut fÝr FestkÆrperforschung, Postfach 800 665 D-70569, Stuttgart, Germany Received 27 December 2004; published 29 April 2005
2 1Institute

3

The spin-resolved quantum Hall effect is observed in the magnetotransport data of strongly disordered GaAs 2000 cm2 / V s, in spite of the fact that the spin-splitting energy is much layers with low electron mobility smaller than the level broadening. Experimental results are explained in the frame of scaling theory of the quantum Hall effect, applied independently to each of the two spin subbands. DOI: 10.1103/PhysRevB.71.155328 PACS number s : 71.70.Ej, 71.30. h, 73.43. f

In an electron system with a small g factor, strong disorder broadens and suppresses the spin-splitted structure in the electron spectrum in an applied magnetic field. Therefore, spin splitting with energy separation Es = g BB B is the Bohr magneton and B the magnetic field does not show up in the kinetic and thermodynamic properties of strongly disordered three-dimensional 3D bulk electron systems. However, for a 2D system, the scaling theory for diffusive interference i.e., localization effects leads to a quite unexpected conclusion: spin splitting can arise in the magnetoquantum transport data even when spin splitting is absent in the density of states for spin-splitting energy Es very small compared to the Landau-level broadening . For this situation with Es , the spin-splitted quantum Hall effect QHE with odd integer Hall-conductance plateaus at Gxy = 2i +1 e2 / h and corresponding minima in the diagonal conductance per square Gxx could appear at sufficiently low temperature due to the localization of the electronic states in between the extended states of the strongly overlapping spin-up and spin-down Landau levels.1,2 This localization should occur at very low temperatures when the coherence length of the electrons becomes larger than their localization length. However, spin splitting due to localization effects was not observed in disordered 2D GaAs systems with mo, because even bilities 1000 cm2 /V s Ref. 3 when Es lower temperatures than those used in the experiments would have been necessary. Higher mobility samples generally do show spin splitting4,5 because of an exchange enhanced g factor.6,7 Strong disorder should suppress this enhancement of spin-splitting8 as observed experimentally.10 In the present work, we observed the manifestation of spin splitting due to localization effects in the magnetoconductance of a strongly disordered system, a heavily Si-doped 2000 cm2 /V s. GaAs layer with a low electron mobility For these layers, essential exchange enhancement of the g factor is not expected. The special interest of our samples resides in the fact that the Zeeman energy, with Es / kB = g BB / kB 4 K using g = -0.5 for GaAs at a magnetic field of B = 12 ­ 13 T, is much smaller than the level broad1098-0121/2005/71 15 /155328 5 /$23.00

ening / kB 100 K resulting in a strong overlap of the two spin subbands. We analyzed the scaling properties of the transport data of this electron system assuming that the conductances of the different spin subbands are renormalized independently for variations due to diffusive interference effects. Such an approach is justified in the absence of spin-flip scattering, at least for noninteracting electrons. Our experimental data are in accordance with such an analysis. The investigated heavily Si-doped n-type GaAs layers sandwiched between undoped GaAs were prepared by molecular-beam epitaxy. The number given for a sample corresponds to the thickness d of the conducting doped layers with d = 34, 40, and 50 nm. The Si-donor concentration is 1.5 1017 cm-3. Hall bar geometries of width 0.2 mm and length 2.8 mm were etched out of the wafers. A phasesensitive ac technique was used for the magnetotransport measurements down to 40 mK with the applied magnetic field up to 20 T perpendicular to the layers. The electron densities per square as derived from the slope of the Hall resistance Rxy in weak magnetic fields 0.5 ­ 3 T at T = 4.2 K are Ns = 3.9, 4.6, and 5.0 1011 cm-2 for samples 34, 40, and 50, respectively. The bare hightemperature mobilities 0 are about 2000, 2200, and 2400 cm2 / V s. Because of the rather large quantum corrections to the conductance, even in zero magnetic field at 4.2 K, we used for determining the mobility the approximate relation 0 = Rxy / BRxx at the intersection point of the Rxx B curves for different temperatures. The characteristic energy scales of our samples with not more than two size-quantized energy levels are as follows. The Fermi energy at zero magnetic field EF / kB 200 K, the / d 2 /2mkB splitting of the size quantization Esq / kB =3 100 ­ 200 K for our thinnest sample with Esq / kB 200 K, the second subband is occupied due to disorder , / kB 100 K is the transport relaxation time at zero magnetic / kB field , the Landau-level energy broadening / kB 100 K, and cyclotron energy / kB 250 K at the magc netic field B = 12 ­ 13 T. For our layers thicker than the magnetic length lB = / eB = 7 nm at 13 T, the Coulomb energy
©2005 The American Physical Society

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FIG. 1. Magnetic field dependence of the diagonal Rxx, per square and Hall Rxy resistance, of the diagonal Gxx and Hall Gxy conductance, and of the derivative dGxy / dB for sample 40 in a magnetic field perpendicular to the heavily doped GaAs layer thickness 40 nm at different temperatures, showing spin splitting at 13 T where Gxy 1. For the lowest temperature Rxy is shown for two directions of the magnetic field.

FIG. 2. Flow diagram of the (Gxx T , Gxy T ) data points for sample 34 with decreasing temperature arrows from 12 down to 0.1 K. Different symbols connected by solid lines are for different magnetic fields from 9 to 14 T. The dashed line follows the Gxx , Gxy data points from 9 to 14 T at the lowest temperature. The dotted lines show the flow of Gxx , Gxy with increasing coherence length for a totally polarized electron system according to scaling theory Ref. 16 and the dash-dotted line a two times larger semicircle than Eq. 2 .

The spin-splitting energy Es in our experiments can be estimated from the magnetic field difference B between the spin-splitting maxima of Gxx B and Gxy / B . For the case of constant electron density Ns, one finds Ns Ns B =0 , Es + EF B

scale EC / kB = e2 / dkB 30 ­ 40 K is d / lB times smaller than EC, the exchange enhancefor thin 2D systems.6 With ment of the g factor should be suppressed completely by disorder8 leading to Es . Magnetotransport data for the three samples are rather similar. In Fig. 1, the diagonal Rxx, per square and Hall Rxy resistance both given in units of h / e2 and the diagonal Gxx and Hall Gxy conductance as calculated from resistance have been plotted for sample 40 at the two opposite field orientations. The absolute values of the Hall resistance Rxy differ strongly for the two orientations at the highest fields, probably because of an admixture of Rxx to Rxy when Rxx Rxy. The Hall conductance Gxy in the field range of B = 0.5 ­ 4 T does not depend on temperature, unlike the Hall resistance Rxy. This is in accordance with the theory of quantum corrections in the diffusive transport due to electronelectron interaction.9 At low temperatures, the curves Rxy B and Gxy B of Fig. 1 show a wide QHE plateau from 6 up to 11.5 T with Gxy = 2 accompanied by exponentially small values of Rxx and Gxx at low temperatures T 0.3 K.11 At the lowest temperature, the derivative Rxx / B, the diagonal conductance Gxx, and the derivative Gxy / B show minima at B 13 T, which we ascribe to development of the spin-resolved QHE. The minima occur at a field where, at all temperatures, the Hall conductance Gxy 1. However, at B 13 T, the filling Nsh / eB 1.5, which is much larger than the exfactor pected value 1 for the QHE of high-mobility 2D electron systems. Above 13 T, Rxy reveals an additional plateau at a value close to 1 at the lowest temperature where Rxx Rxy, possibly indicating a quantized Hall-insulator state.12,13

1

where EF is the Fermi energy, Ns / EF Ns / , and Ns / B -Ns / B. Therefore, Es / kB B / BkB 10 K is much smaller than / kB 100 K, leading to a vanishing spin-splitted structure in the density of states. Moreover, an additional argument for the absence of a spin-splitting structure in the density of states which would explain the spinsplitting QHE structure is given by the fact that, if one would with a minimum in the density of states, the have Es spin-splitting structure should become apparent already in the conductance at temperatures of the order of / kB 100 K, whereas experimentally it develops only at T 0.1 K. The very low temperatures for the observation of the spin splitting point to a different mechanism such as localization for the explanation of the phenomenon. The scaling treatment of the QHE is graphically presented by a flow diagram for the coupled evolution of the diagonal Gxx and Hall Gxy conductance components with increasing coherence length.14,15 Recent developments16 of the scaling theory based on symmetry arguments resulted in a calculation of the exact shape of the flow lines Gxx Gxy for a totally spin-polarized electron system as plotted in Fig. 2 with dotted lines for 0 Gxy 2. The different quantum Hall phases i =0,1, ... in the flow diagram are separated by the vertical lines Gxy = i +1/2. At sufficiently low temperatures, the Gxx , Gxy data flow on a separatrix in the form of a semicircle

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2 Gxx + Gxy - i + 1/2 2

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= 1/4 .

2

c Critical points can be found at Gc , Gxx = i +1/2,1/2 . The xy same critical positions were found in microscopic descriptions of the QHE for the case of noninteracting electrons.17,18 In Fig. 2, we have plotted the experimental flow lines showing the temperature evolution of the points (Gxy T , Gxx T ) of the conductances for sample 34 at different magnetic fields with temperature ranging from 10 down to 0.1 K. For sample 40, the flow diagram is rather similar to the one for sample 34. For these samples, at the magnetic fields where the spin splitting is observed, the flow lines move upwards and then downwards for decreasing temperatures. The lines cross each other for data at different magnetic fields, in contrast to the theoretical prediction for the case of a totally spin-polarized electron system. For sample 50, the flow lines do not show the upward trend and are not crossing each other. For low temperatures below 3 K , the flow diagrams are very similar for all three samples: the flow lines approach the semicircles according to Eq. 2 . Linear extrapolation of Gxx T and Gxy T from 0.5 to 0 K at the two fields where Gxx B has maxima for sample 40 at 12.5 and 13.7 T, see Fig. 1 results in values Gxx = 0.5 ± 0.02 and Gxy = 0.5 ± 0.05 and 1.5 ± 0.05. These critical values are the same as predicted for a totally spinpolarized electron system. At the lowest temperatures, the magnetic-field driven dependence Gxx Gxy is not far from the semicircles Eq. 2 , as shown in Fig. 2. In the absence of spin-flip scattering, the conductances of the different spin subbands are renormalized independently, at least for the case of noninteracting electrons. Since the temperature dependence of the magnetoconductance is not known for a single spin-polarized band, it is impossible to estimate accurately the flow lines for the total conductance from the flow lines for the single polarized bands because the summation Gij T = G- T + G+ T involves different posiij ij tions on the spin-polarized flow lines at the same temperature. The indices and correspond to the majority and minority spin subsystems with larger and smaller Hall conductances, respectively. Nevertheless, we can draw some conclusions on the scaling properties of the total conductance Gij. For weak spin splitting g BB / EF, the bare nonrenormalized conductances G0± for the two spin subbands ij as measured at high temperatures,

G0± = ij

G 0 g BB G 0 ij ij , ± 2 4 E

3

differ weakly from each other because g BB G0 / E ij G0 . Here G0 = G0- + G0+. The QHE with toG 0 g BB / ij ij ij ij ij tal Hall conductance Gxy = 1 should arise when one sub- system is in the insulator state G- , Gxx 0,0 for T 0 xy + + and the other is in the QHE state Gxy , Gxx 1,0 . This occurs in a narrow magnetic field range where G+0 1/2 but xy G-0 1 / 2. At the critical value of G-0 =1/2 and G+0 1/2, xy xy xy - + G- , Gxx 1/2,1/2 and G+ , Gxx 1,0 , therefore the xy xy total conductance Gxy , Gxx 3/2,1/2 . Similarly, at the critical value of G+0 =1/2, the total conductance Gxy , Gxx xy

1/2,1/2 . Thus, the critical points are the same as for the case of a totally spin-polarized electron system, in accordance with experimental results. This differs from the results of Ref. 19 predicting essentially different positions of the critical points whose exact position depends on the amount of spin splitting. Note that these results19 have been obtained on the basis of a postulated symmetry group in order to include spin splitting, without giving any microscopic picture for the scaling behavior. At low enough temperatures, when the spin-splitted QHE is rather well developed so that in the QHE minimum Gxx 0, one can argue that the flow lines should follow the lines derived for the case of a totally spin-polarized electron sys- tem. In the minimum of Gxx holds G- , Gxx = 0,0 and xy + + Gxy , Gxx = 1,0 , i.e., the minority subsystem does not contribute to conductance and the majority subsystem contributes only the quantum value Gxy = 1 to the Hall conductance. At lower magnetic fields, the subsystem contributes only to the Hall conductance the value 1 as before, and the total - conductance Gxy , Gxx = G- +1, Gxx . Similarly, at higher xy + + magnetic fields Gxy , Gxx = Gxy , Gxx . At the lowest temperatures, the total conductance Gxy , Gxx is expected to flow along the same lines as derived for a single spin-polarized electron system. Therefore, Gxx as a function of Gxy flows for a changing magnetic field close to the semicircles given by Eq. 2 , in accordance with experimental data below 0.1 K. With G0± far away from 1 / 2, the conductances of the xy different spin subbands flow approximately in the same way, and, therefore, the flow lines for the total conductance should be twice as elongated compared to those calculated for a spin-polarized electron system.16 In agreement with this, the outermost experimental flow lines approach the large semicircle in Fig. 2 and all data are inside this semicircle. With G0± close to 1 / 2 or G0 close to 1 at the spin-splitted xy xy QHE structure, the temperature dependence of the total conductance can be essentially different depending on the value 0 0 0± of Gxx. For Gxx 1 or Gxx 1 / 2, the two spin-polarized flow lines go down and the total conductance Gxx decreases for 0 0± decreasing temperature. For Gxx 1 or Gxx 1/2, Gxx stays constant followed by a decrease at lower temperatures, as 0 0± shown in Fig. 3 for sample 50. For Gxx 1 or Gxx 1/2, the 0± conductances Gxx first increase and then decrease for decreasing temperature resulting in the observed nonmonotonic temperature dependence of Gxx for samples 34 and 40 in Fig. 3. As mentioned above, for the totally spin-polarized electron system, the flow lines should not cross each other15 in contrast to our experimental data. For the case of two different spin projections, we will show now that the flow lines starting in the region in between the large semicircle and the two smaller semicircles shown in Fig. 2 can cross each other. Consider the line ending at the critical point 1/2,1/2 as a reference line. Starting slightly at the left from the starting point of this line, and knowing that this line should end at 0,0 , the crossing is unavoidable. Starting at the right from the starting point of this reference line, and knowing that this line should end at 1,0 , leads also to a crossing point. The spin-resolved QHE should develop at very low temperatures, where the phase-coherence length increases above

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FIG. 3. Temperature dependence of the diagonal Gxx conductance, for sample 34 diamonds , 40 squares , and 50 circles at magnetic fields B = 11.8, 13.1, and 13.1 T, correspondingly, where the minimum due to spin splitting is observed.

the localization lengths ± of the two spin systems, supposing that the samples are sufficiently homogeneous with the variation of G0 along the samples essentially smaller than the xy difference g BB G0 / E between spin-up and spin-down xy contributions. For the Gxy = 1 QHE, the localization lengths ± are getting very large because the electronic states at the Fermi level are not far from the extended states of both spin systems 1/2- G0± g BB G0 / E 1 . Therefore, for the xy ij observation of the QHE with Gxy = 1, a much lower temperature or a larger coherence length is necessary than for the QHE with Gxy = 2. In previous experiments mostly at smaller magnetic fields,3,20 the spin splitting was not observed, probably because smaller g BB G0 / E leads to a larger ±. In ij our samples, the spin splitting is observed only at low temperatures T 0.1 K, which is much smaller than / kB 100 K, and even than Es / kB 4 K. Usually, for small Landau-level broadening c, the integer QHE effect with Gxy = occurs at magnetic fields corresponding to an integer filling factor = Nsh / eB for the free-electron Hall conductance G0 = eNs / B / e2 / h normalxy

ized with respect to e2 / h. For the spin-splitted structure at B 13 T, the filling factor 1.5 is essentially larger compared to the expected value of 1. When becomes compa0 rable with c, Gxy will be smaller than the normalized Hall conductance eNs / B / e2 / h Ref. 21 leading to G0 and xy 1 at the explaining the observation of a filling factor spin-splitting minimum where G0 should be close to 1 acxy cording to the scaling theory. The assumption about independent renormalization of the conductances of the two spin subbands is undoubtedly valid for noninteracting electrons in the absence of spin-flip scattering. Although electron-electron interaction is important in real systems, the experimental study of the flow diagram on samples 34, 40, and other thinner layers20 shows good quantitative agreement with the predicted flow lines16 for half the measured conductance values in the field range below 6 T, where there is not any manifestation of spin splitting and, therefore, Gij /2= G+ = G- . This gives support for our model ij ij of independent spin-band contributions. In summary, we observed the spin-resolved quantum Hall effect in heavily doped n-type GaAs layers with disorder broadening much larger than the spin splitting energy, without any signature of spin splitting in the density of states. Our results are in accordance with the scaling treatment of the quantum Hall effect, applied independently to the two spin subbands. Namely, the magnetic field position for the QHE is imposed by the occurrence of the Hall quantum value Gxy 1, where at all temperatures the Hall conductance Gxy 1 while the filling factor 1. Several features in the Gxy , Gxx flow diagrams, like the observed critical c values Gxx = 0.5 ± 0.02 and Gc = 0.5 ± 0.05 and 1.5 ± 0.05, and xy the anomalous shapes of the flow lines, can be deduced from an independent summation of the contributions of the two spin bands. The spin splitting is well observed at temperatures T 0.1 K much smaller than all other energy scales determining the electron spectrum. Therefore, probably localization is at the origin of the observed spin-resolved QHE. This work is supported by the Russian Foundation for Basic Research. We would like to thank B. Lemke for her help in the preparation of the samples.

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