Suppression of Magneto-Intersubband Resistance Oscillations by Large-Scale Fluctuations of the Intersubband Energy Splitting

CONDENSED MATTER

where is the characteristic (average) fluctuation of the Fermi energy corresponding to the δn value. It can be seen that the Dingle plot acquires an additive proportional to 1/ in addition to the term linear in 1/ . This additive takes into account the inhomogeneous broadening of Landau levels caused by density fluctuations δn.

Owing to intersubband electron scattering, which becomes resonant when the Landau levels belonging to different subbands coincide with each other, magneto-intersubband (MIS) oscillations arise in magnetic field dependences of the resistance in multiband electron systems along with SdH oscillations [12][13][14][15]. In a two-subband electron system, the position of the maxima of MIS oscillations in a magnetic field is determined by the equality (3) where and E 2 are the positions of the bottoms of the first and second subbands, respectively, and k is a positive integer. Magneto-intersubband oscillations, as well as SdH oscillations, are periodic in 1/ , and their period is determined by the ratio , where

The amplitude of MIS resistance oscillations for charge carriers in the quantum well with two occupied energy subbands is expressed by the equality [16][17][18] (4) where , is the transport scattering time, is the effective intersubband scattering time, , and τ q1 and τ q2 are the electron quantum lifetimes in the first and second subbands, respectively. It follows from Eq. (4) that the dependence on 1/ is linear, and its slope is determined by the value; i.e., Eq. ( 4) predicts the linear behavior of Dingle plots for MIS oscillations.

Dingle plots for SdH oscillations in the GaAs/AlGaAs heterojunction with two occupied sub-

MISO q τ bands and were studied in [14]. It was shown that they are described by linear dependences, and the quantities τ q1 and τ q2 , which are calculated from their slope, are significantly different. Dingle plots for MIS oscillations were studied only in GaAs quantum wells with AlAs/GaAs lateral superlattice barriers [19][20][21][22]. It was also established that the dependences on 1/ in such heterostructures at K are nonlinear. The origins for the discovered nonlinearity are still debatable. Here, we report the experimental results for dependences in GaAs quantum wells with the value varying from 22.7 to 1.44 meV. These data are analyzed within a model that takes into account the role of large-scale fluctuations in the suppression of MIS oscillations.

In this work, we studied symmetrically doped GaAs quantum wells with widths of 22, 26, 30, 36, and 46 nm. Short-period AlAs/GaAs superlattices were used as side barriers to quantum wells [23,24]. The heterostructures were grown by molecular beam epitaxy on (100) GaAs substrates. Samples for magnetotransport measurements were Hall bars with the length μm and width μm equipped with Schottky field-effect gates. The studies were carried out at a temperature of K in magnetic fields of T. The resistances and were measured in a linear mode with an alternating electric current with a frequency of 888 Hz and an amplitude below 1 μA. The total electron density in quantum wells n T was calculated from the resistance in a magnetic field of T. The mobility μ was calculated from n T and . The parameters of studied samples are given in Table 1.

Figure 1 shows a typical dependence for a "narrow" quantum well ( nm), which is a "single-layer" two-subband electron system. The value in this case is mainly determined by the width of the quantum well. The classical positive magnetoresistance is manifested in the narrow quantum well in the fields of T [20]; the MIS oscillations then follow, which coexist with SdH oscillations in the fields

<. 01 B T. An insignificant modulation of the amplitude of MIS oscillations in the range from 0.1 to 0.5 T is associated with the interference of magnetophonon and MIS oscillations [25,26]. The Fourier analysis of such dependences gives three frequencies corresponding to the periods of SdH oscillations in the first and second subbands, as well as the period of MIS oscillations. The electron densities in subbands in narrow quantum wells differ strongly; therefore, the quantum times τ q1 and τ q2 are not equal in the general case. In accordance with Eq. ( 4), the slope of the Dingle plot in this case is determined by the quantity , and the value at 1/ is .

Figure 2 shows a typical dependence at K for a "wide" quantum well ( nm), which is a "two-layer" two-subband electron system. The intersubband splitting in this case is mainly determined by the tunneling coupling between the electron layers separated by a smooth barrier arising due to the electrostatic repulsion of electrons to the heterointerfaces of the quantum well [27]. In wide quantum wells, in contrast to narrow ones, the classical positive magnetoresistance is not exhibited because the electron density and mobility are approximately the same in both subbands. In such a system, the quantum times in the subbands can be considered to be close, . The slope of the Dingle plot for MIS oscillations in this case is determined by the value , and the value at 1/ is [18].

Figure 3a demonstrates the behavior of oscillating components of experimental dependences for wide and narrow quantum wells in the region of 1/B > 4 T -1 . Only MIS oscillations are present in this region. It can be seen that MIS oscillations decay faster with the increase in 1/B in the narrow quantum well. According to Eq. ( 4), this means that in the wide quantum well should be longer than that in the narrow one. Figure 3b shows the results of Fourier analysis of dependences in the region of 1/B > 4 T -1 . In this region, there is only a peak for MIS oscillations whose frequencies are determined by the value. It can be seen that the width of the peak for the narrow quantum well is larger than that for the wide one. This is in agreement with the assumption that in the narrow quantum well is smaller than that in the wide one. Figure 4 shows dependences for the (a) wide and (b) narrow quantum wells. The experimental dependences are nonlinear, which is inconsistent with Eq. ( 4). This means that the different behaviors of dependences for the narrow and wide quantum wells cannot be explained only by the difference of values in them.

According to Eq. ( 3), the period of MIS oscillations is determined by the value. If large-scale fluctuations of are present in the system, they should lead to additional suppression of the amplitude of MIS oscillations because of averaging over the sample area. Following the approach proposed in [11], we assume that the distribution of large-scale fluctuations of is Gaussian. In this case, the inhomogeneous damping factor appears as an additional factor on the left-hand side of Eq. ( 4), and the amplitude of MIS oscillations is expressed by the formula (5) where is the average amplitude of fluctuations of . Formula (5) shows that Dingle plots for MIS oscillations are nonlinear when inhomogeneous broadening is taken into account.

It is seen in Fig. 4a that the theoretical dependence calculated by Eq. ( 5) and shown by line 1 is in good agreement with experimental data for the wide quantum well ( nm). Theoretical dependences 2 and 3 in this figure demonstrate the role of the quantity in the suppression of the amplitude of MIS oscillations. When the value, which describes the homogeneous (collisional) broadening of Landau levels, considerably exceeds the

, the suppression of with increasing 1/B is mainly determined by large-scale fluctuations . When , the role of large-scale fluctuations in the suppression of the amplitude of MIS oscillations can be ignored. Figure 5a demonstrates the effect of the gate voltage V g on the behavior of dependences . The application of the negative gate voltage V g leads to an increase in the classical positive magnetoresistance, as well as to a decrease in the amplitude and an increase in the frequency of MIS oscillations. Such effect of V g on the behavior of is quite expectable. First of all, the negative gate voltage V g squeezes out X electrons localized in AlAs layers adjacent to the Si-δdoped layer located between the Schottky gate and the quantum well [8]. The decrease in the concentration of compact dipoles, which are formed by positively charged donors in the Si-δ-doped layer and X electrons in AlAs layers, increases the scattering of electrons on the random potential of the dopant, which suppresses MIS oscillations. The increase in the frequency of MIS oscillations under the effect of V g indicates the increase in due to the "tilt" of the quantum well. Moreover, the negative gate voltage V g increases the difference between the mobilities in the subbands, which is a reason for an increase in the classical positive magnetoresistance.

Figure 5b demonstrates the effect of V g on the dependence of on 1/B. Experimental Dingle plots for the quantum well with the width of 26 nm for different V g values are in complete agreement with theoretical dependences of on 1/B calculated by Eq. ( 5). The observed agreement indicates that V g in the studied quantum well changes only the value, but does not affect the average amplitude of large-scale fluctuations of the intersubband energy splitting (

). The values obtained from the comparison of experimental and calculated dependences are given in Table 1. These data indicate that the quantity increases with the decrease in . Such behavior makes it possible to suppose that one of the main origins of large-scale fluctuations of is the presence of large-scale fluctuations of .

It is reasonable to assume that the average value of width fluctuations ( ) of the studied GaAs quantum wells, which have the same design of side barriers and were grown in the same technological modes, is the same for different values. In this case, increases with decreasing . Thus, large-scale fluctuations of with the same average value lead to the higher values in the wells with the smaller width. This dependence of on is really observed. There is no effect of V g on the quantity , despite the increase in with |V g |, which is in agreement with the fact that large-scale fluctuations in narrow quantum wells are primarily responsible for the inhomogeneous broadening of Landau lev- quantum wells, and in accordance with Eq. ( 5), this is a reason for the nonlinear behavior of Dingle plots for MIS oscillations.

The values calculated from the values are given in Table 1. It follows from Table 1 that nm for the wide quantum well does not agree with the values for narrower wells. This means that the physical origins of large-scale fluctuations of are different in narrow and wide quantum wells. We believe that this behavior is natural because the splitting in these wells is due to different physical mechanisms.

One of the origins of large-scale fluctuations of in the studied wide quantum well can be the presence of large-scale "lakes" of X electrons, which can arise in AlAs layers adjacent to the Si-δ-doped layers. Since the spatial distribution of these lakes in the left-and right-hand barriers to GaAs quantum well is random, the value in this case is nonzero, despite the symmetric location of δ-doping layers. The random distribution of large lakes in the left-and right-hand barriers leads to large-scale fluctuations of the tilt of the wide quantum well and, accordingly, to large-scale fluctuations of . To summarize, we have studied the effect of the inhomogeneous broadening of Landau levels on the amplitude of magneto-intersubband oscillations in quantum wells with two occupied energy subbands. It has been shown that large-scale fluctuations of the intersubband splitting are responsible for the nonlinearity of Dingle plots for MIS oscillations. An analytical expression has been obtained to take into account large-scale fluctuations of intersubband splitting in dependences of the amplitude of MIS oscillations on the inverse magnetic field. The comparison of theory and experiment has made it possible to determine the average amplitude of fluctuations of the intersubband splitting in the studied GaAs quantum wells with AlAs/GaAs lateral superlattice barriers.