Chaotic Transport in Antidot Lattices

Transcription

1 Journal Chaotic of ELECTRONIC Transport MATERIALS, in Antidot Vol. Lattices 29, No. 5, Special Issue Paper Chaotic Transport in Antidot Lattices TSUNEYA ANDO and SEIJI URYU University of Tokyo, Institute for Solid State Physics, Roppongi, Minato-ku, Tokyo , Japan A review is given of quantum effects and roles of chaos in transport in antidot lattices mainly from a theoretical point of view. The topics include diffusive orbits combined with a magnetic focusing effect as the origin of the commensurability peaks, semiclassical quantization of periodic orbits as the origin of the Aharonov-Bohm type oscillation superimposed, and the importance of inherent disorder in the antidot potential itself. Key words: Antidot lattice, chaos, magnetotransport, quantum transport INTRODUCTION The two-dimensional (2D) system modulated by a periodic strong repulsive potential is called an antidot lattice. The transport in this system is ballistic, i.e., electrons are scattered from the antidot potential itself rather than from impurities. Various interesting phenomena have been observed in magnetic fields. The purpose of this paper is to give a brief review on transport properties in antidot lattices in the presence of a magnetic field mainly from a theoretical point of view. In the next section, antidot lattices are introduced along with commensurability peaks in the magnetoresistivity. This is followed by a section on the origin of the commensurability peaks is discussed with emphasis on the roles of classical chaotic motion. Next is the section on the close connection between quantum oscillation superimposed on the commensurability peaks and semiclassical quantization of periodic orbits is discussed. The next section covers the roles of inherent disorder in the antidot potential arising during fabrication processes are emphasized. A brief summary is given in the final section. ANTIDOT LATTICE AND COMMENSURABILITY PEAK As shown in Fig. 1, antidot lattices are usually fabricated from high mobility GaAs/AlGaAs heterostructures by making an array of shallow holes with the use of electron beam lithography and etching (Received November 12, 1999; accepted December 21, 1999) techniques. Because the chemical potential of electrons is always pinned near the center of the band gap at the surface, the 2D electron gas is completely depleted in the antidot region of the fabricated holes. Figure 2 shows a schematic illustration of the potential for the 2D electron system. The antidot lattice is characterized by the period a and the diameter d of each antidot. We have typically d > ~ 1000 Å which is larger than the Fermi wavelength of the 2D electron system, λ F ~ 500 Å, for a typical electron concentration less than cm 2 in a GaAs/AlGaAs single heterostructure. Correspondingly, the period of the antidot is usually a 2000 Å. This means that the antidot lattice is in the boundary region between quantum and semiclassical regimes. The classical electron motion is known to be fully chaotic in this system. The model antidot potential for a square antidot, used frequently and used also in this paper, is given by U(r) = U 0 F(r R) within a Wigner-Seitz cell, where U 0 is a potential maximum located at point r = R and πx πy Fr ( ) = cos cos a a 2/ β The parameter β characterizes its steepness. The antidot diameter can be defined as d/2 = r with U(r) = E F, where E F is the Fermi energy and r is chosen on a line connecting two nearest-neighbor points. A self-consistent calculation in quantum wires fabricated at GaAs/AlGaAs heterostructures suggests that the potential is nearly parabolic for a wire with small width and consists of a flat central region and a para- (1) 557

2 558 Ando and Uryu a a b Fig. 1. (a) Schematic picture of the antidot array structure (a thin GaAs cap layer at the top is not shown here). (b) The top view of the overall structure of the 2D system with an antidot array. bolic increase near the edge for a wider wire. 1 4 The width of the region where the potential increases from the bottom to the Fermi energy is of the same order as the Fermi wavelength for typical electron concentrations. This leads to roughly β ~ 1 for d/a ~ 0.5, β ~ 2 for d/a ~ 0.4, and β ~ 4 for d/a ~ 0.3 in the case d~1000 Å. A similar potential can be chosen for a triangular lattice. 5,6 We sometimes use the following potential also: b Fig. 2. A model antidot potential in a square lattice. (a) d/a = 0.5 and β = 1. (b) d/a = 0.3 and β = 4. The potential is cut off at the Fermi energy. Ur E d / 2 + () r = ( d / r) F θ 2 + (2) 2 where r is the distance from the center and θ(t) is the step function defined by θ(t) = 1 for t > 0 and 0 for t ~ 0. The parameter describes the steepness of the antidot potential (typically /a ~ 0.24 for d/a = 0.5). In experiments performed in a square antidot lattice, 7 prominent peaks were observed in the diagonal resistivity ρ xx in weak magnetic fields as shown in Fig. 3. They are called commensurability peaks. At certain magnetic fields electrons can move on the commensurate classical orbit encircling a specific number of antidots (pinned orbits). The magnetoresistance was expected to increase at this magnetic field. 7 A classical simulation showed that the change in the volume of the pinned orbits in the phase space is not enough to cause the commensurability oscillation. 8 The importance of the runaway orbit, which skips regularly from an antidot to its neighboring antidot in the same direction, was also proposed. 9 Experiments on rectangular and/or disordered Fig. 3. An example of ρ xx and ρ xy observed in patterned (solid lines) and unpatterned (dotted lines) 2D systems. 7 Two prominent peaks and step structures appear at ~0.6 and ~0.2 T in the diagonal and Hall resistivity, respectively, in antidot lattices.

3 Chaotic Transport in Antidot Lattices 559 Fig. 6. The region (shaded) of the center of the cyclotron orbits passing through two nearest neighbor antidots. Fig. 4. Schematic illustration of magnetic focusing leading to the fundamental commensurability peak. Fig. 7. The area of the region where cyclotron orbits pass through two neighboring antidots. The diagonal conductivity is roughly proportional to the area. Fig. 5. Some examples of diffusive cyclotron orbits contributing to the migration of the guiding center. The dashed circles correspond to pinned orbits (the number denotes antidots in the orbit). antidots and numerical simulations provided pieces of evidence showing the importance of such orbits. A better way of understanding the origin of the commensurability peaks is through diffusive orbits and magnetic focusing as discussed below. MAGNETIC FOCUSING AND DIFFUSIVE ORBITS Consider first the limit of the small aspect ratio d/a << 1. In this case, the electron loses its previous memory of the direction of the velocity when it collides with an antidots, and therefore successive scattering with antidots can be generally regarded as independent of each other. 17 This means that the antidots are nothing but independent scatterers. In magnetic fields, transport is possible through migration of the center of the cyclotron motion and therefore the conductivity σ xx vanishes in the absence of scattering. 18 In fact, the conductivity is written as σ xx = e 2 D(E F )D*, where D* is the diffusion coefficient and D(E F ) is the density of states at the Fermi level. In the absence of scattering, the center of the cyclotron orbit stays at the same position and the diffusion coefficient vanishes. In the antidot system in the classical regime, the density of states does not have any sizable structure and therefore commensurability peaks arise from those of the diffusion coefficient. When the cyclotron diameter is smaller than the period, i.e., 2R c < a where R c is the cyclotron radius, the scattering of an electron from an antidot cannot give rise to diffusion or conduction because the electron is trapped by the antidot. Scattering from antidots starts to contribute to the conductivity when 2R C > a. The migration of the center of the cyclotron orbit occurs most frequently due to successive scattering from nearest-neighbor antidots at the magnetic field corresponding to 2R C = a. In fact, as is shown in Fig. 4, an electron emitted from an antidot is focused onto a neighboring antidot. This leads to an increase in the phase-space volume of the orbits contributing to the increase of the diffusion coefficient at 2RC ~ a. In magnetic fields, the resistivity is given as ρ xx = σ xx / ( σ 2 xx + σ 2 xy) in terms of the diagonal conductivity σ xx and the Hall conductivity σ xy. In high fields, in particular, we usually have σ xx >>σ ξξ and therefore 2 ρ xx ~σ ξξ / σ xy which shows that a peak in the resistivity is given by a peak in σ ξξ.

4 560 Ando and Uryu Fig. 8. A schematic illustration of scattering from two adjacent antidots. This leads to an enhancement of σ xy for 2R c < a and a reduction for 2R c > a. Fig. 10. An example of observed quantum oscillations superimposed on the fundamental commensurability peak at around 0.9 T. 20 The oscillation becomes clear in the difference of the resistivity at T=4.2 K and that at T= 1.5 K (shifted vertically by 1 kω). Fig. 9. Some examples of resistivity and conductivities calculated using classical mechanics for square antidot lattices. 17 As shown in Fig. 5, the orbit corresponding to 2R c = a can be denoted as (n x,n y ) = (±1,0) or (0,±1), where the line segment connecting the point (n x,n y ) and the origin constitutes the diameter of the circle. With a further decrease in the magnetic field, successive scattering with next nearest-neighbor antidots becomes possible and the conductivity has a peak around 2R c = 2a corresponding to (±1,±1). This contribution becomes less prominent, however, because the orbit passes through the position of a nearest-neighbor antidot. The next peak arises from (±2, 0) or (0,±2) and (±2,±1) or (±1,±2), which lie close to each other. The latter contribution should be larger because its measure is twice as large as that of the former and therefore the peak occurs around 2R c ~ 5a. This peak should be weaker, however, as these orbits also pass through the position of other antidots. The next prominent peak is given by (±2,±2), (±3,0), (0,±3), (±3,±1), and (±1,±3). Actually, the peak is at 2R c 3a, because the orbits (±3,0) and (0,±3) are not disturbed by other antidots. In this way we can explain most of the commensurability peaks observed experimentally. The reason why the peak positions can also be explained by the pinned orbits can be seen in Fig. 5, which shows that the orbits specified by (n x, n y ) correspond approximately to certain pinned orbits by a slight shift in the position of the guiding center. When the diameter d is no longer negligible in comparison with the period a, the fundamental commensurability peak in σ xx deviates from the condition 2R c = a. A cyclotron orbit starting at an antidot and colliding with a neighboring antidot has a center in the region schematically illustrated in Fig. 6, to which the phase-space volume of such orbits is proportional. Figure 7 shows the area as a function of the magnetic field. The area has a maximum at 2R c = a + fd with f ~ 0.3 being a constant weakly dependent on d/a. This

5 Chaotic Transport in Antidot Lattices 561 Fig. 11. Critical values of the aspect ratio at which the periodic orbit encircling an antidot disappears. A/a=0.18 (solid line), 0.24 (dotted line, realistic), and 0.36 (dashed line). The right upper and lower insets show the periodic orbit (solid line) giving the AB type oscillation and the other (dotted line) that is unstable. These orbits merge and disappear when the aspect ratio approaches a critical value. means that the commensurability peak is shifted to the weak-field side roughly in proportion to d/a. For a nonnegligible d/a, the correlation among successive scattering from antidots becomes important. In fact, it manifests itself most directly in the Hall conductivity. As illustrated in Fig. 8, it tends to enhance the left circular motion of electrons in the weak-field region 2R c < ~ a and reduce it in the high-field region 2R c < ~ a. This leads to an enhancement of σ xy for 2R c > ~ a and a reduction for 2R c < ~ a, giving rise to a step-like structure around 2R c a. With the increase of d/a, the measure of the diffusive orbits increases as shown in Fig. 7, and the diagonal conductivity becomes larger and comparable to σ xy (numerical calculations show that σ xx ~ σ xy around d/a ~ 0.5). 17 Therefore, the commensurability peaks in the resistivity become different from those in σ xx. Let σ xx and σ xy be a small change around the field corresponding to the commensurability peak and σ xx and σ xy be the average ( σ xx << σ xx σ xy << σ xy ). Then, we have ρ xx ( ) 2 2 σ σ σ 2σ σ σ = ( σxx + σxy) xy xx xx xx xy xy (3) This means that σ xy determines the structure of ρ xx Fig. 12. Calculated magnetoresistance for /a = The origin of the magnetoresistance is shifted for each aspect ratio. when σ xx ~ σ xy, i.e., the reduction of σ xy in the high field side gives the peak in ρ xx for large d/a. Numerical simulations were performed. 17,19 Figure 9 shows some examples of the results. The results show that the peak in σ xx is broadened considerably with the increase of d/a and at the same time shifted to the lower magnetic field side. A step structure corresponding to the first derivative of σ xx with the magnetic field appears in σ xy. It is interesting, however, that the commensurability peak in the resistivity ρ xx remains around 2R c = a independent of d/a. For d/a 0.5, the structures in σ xx and σ xy disappear almost completely and ρ xx does not show a clear commensurability peak. PERIODIC ORBITS AND SEMICLASSICAL QUANTIZATION A fine Aharonov-Bohm (AB) type oscillation was observed superimposed on commensurability peaks of the magnetoresistance Figure 10 shows an example. 20 The period is roughly given by the magnetic flux quantum Φ 0 = ch/e per unit cell. The density of states in classically chaotic systems is given semiclassically in the so-called periodic orbit theory According to the trace formula the density of states is given by the sum of the classical contribution and the quantum correction. The latter is determined by semiclassically quantized energy levels associated with each periodic orbit and the former is proportional to the classical phase-space volume at a given energy.

6 562 Ando and Uryu Fig. 13. Calculated resistance as a function of the total flux Φ passing through a unit cell divided by the flux quantum Φ 0 for varying Fermi wavelength. Short vertical lines represent semiclassically quantized energy levels associated with the periodic orbit encircling an antidot (solid and dotted lines for stable and unstable orbits, respectively). The density of states has been shown to be correlated well with the semiclassical energy levels. 29,30 The same is true for the resistivity as shown in Fig. 11. A semiclassical expression can be obtained for the conductivity tensor, 29,31,32 but has proved to be unsuccessful for the parameter range corresponding to the experiments mentioned above. 29 As discussed in the previous section, the diffusive orbits have a dominant contribution to the transport in the magnetic-field region near the commensurability peak. These orbits are perturbed by the presence of quantized periodic orbits, which is likely to lead to a quantum oscillation. The semiclassical expression for the conductivity does not take into account such effects. Quite recently, a clear correlation between the disappearance of a periodic orbit encircling an antidot and that of the AB type oscillation was demonstrated. 33 Figure 12 shows the boundary of the existence of the periodic orbit encircling an antidot for a model potential given by Eq. 2. The insets in the right hand side of the figure show the periodic orbit (solid line) for d/a = 0.6 and With the increase of d/a it merges into another orbit (dotted line), which is unstable and has a longer trajectory, and disappears. It contains the boundary in the case of /a = 0.18 (steeper) and 0.36 (softer) as well. The critical aspect ratio becomes smaller with the increase of /a, i.e., as the antidot potential becomes broader. Fig. 14. Calculated magnetoresistance in the case of dominant impurity scattering characterized by the mean free path l e. The arrows indicate the field corresponding to 2R c /a = 1 for two different values of the Fermi wave length λ F. Short vertical lines represent semiclassically quantized energy levels associated with the periodic orbit encircling an antidot (solid and dotted lines for stable and unstable orbits, respectively). Figure 13 shows the calculated results of the magnetoresistance for various values of the aspect ratio. Short vertical lines indicate the magnetic fields where the quantized levels of a periodic orbit encircling an antidot cross the Fermi energy. Solid lines mean stable orbits and dotted ones unstable. The AB type oscillation changes as a function of the magnetic field in an irregular manner, but the dependence on the field is, on the whole, in good agreement with that of the position of quantized levels of the periodic orbit. The oscillation disappears or becomes unrecognizable near the field corresponding to 2R c = a for large aspect ratio d/a = 0.7. This field corresponds to the disappearance of the periodic orbit. In fact, the periodic orbit is absent in the hatched region. DISORDER IN ANTIDOT POTENTIAL A kind of Altshuler-Aronov-Spivak 34 (AAS) oscillation characterized by the period given by a half of the flux quantum has been observed in antidot lattices in weak magnetic fields. 35,36 The amplitude of the oscillation turned out to be much larger in triangular lattices. 36 A numerical study has demonstrated that the AAS oscillation presents a clear indication that real antidot lattices have a sizable amount of disorder in the antidot potential itself. 5,6 In fact, in the absence of disorder in the antidot

7 Chaotic Transport in Antidot Lattices 563 Fig. 15. Calculated resistance in the case of dominant scattering from antidot disorder. The parameters are same as in Fig. 14. potential, the conductance exhibits only irregular interference oscillations depending sensitively on parameters such as the Fermi wavelength, the aspect ratio, the steepness of the potential, etc. A regular AAS oscillation appears only when the mean free path is much smaller than the antidot period. However, this corresponds to the usual AAS oscillation in metallic diffusive systems. Because the present antidot system is in the ballistic regime rather than in the diffusive regime, we have to seek other mechanisms which average out irregular oscillations. The numerical result shows that an introduction of small but sizable fluctuations in the antidot diameter lying in the range < ~ d f /a < ~ 0.07 for d/a ~ 0.5 destroys irregular oscillations and leads to a regular AAS oscillation, where d f is the root mean square fluctuation of d. 5,6 Actual systems are likely to have this amount of disorder, although the exact estimation is difficult. This disorder in the antidot potential has strong effects also on the commensurability peak and the superimposed AB type oscillation. Figures 14 and 15 show some examples of the resistance of a square antidot lattice with a large but finite size at two different values of the Fermi energy calculated in an S matrix formalism In the case of the dominant impurity scattering, the commensurability peak changes sensitively when the Fermi energy is varied. This is due to anomalous behavior of the Hall conductivity due to the complicated band mixing. 41 In the case of the dominant antidot disorder, on the other hand, the commensurability peak changes smoothly if the Fermi energy is varied. This suggests that such band structure is destroyed almost completely by the presence of the antidot disorder. Many periodic orbits can contribute to the AB type oscillation in ideal antidot lattices. In the presence of disorder in the antidot potential, the shortest orbit encircling a single antidot is expected to be less influenced and its quantized energy levels can survive, while those of other orbits are strongly disturbed and smeared out. In other words, the antidot disorder tends to enhance the contribution of the energy levels of this orbit relative to those of other orbits. This explains the fact that the AB type oscillation is sensitive to the Fermi energy for impurity scattering but not for scattering from antidot disorder. SUMMARY AND CONCLUSION A review of magnetotransport in antidot lattices has been given mainly from a theoretical point of view. The commensurability peaks appearing in the magnetoresistivity can be understood quite well in terms of an enhancement in the measure of diffusive orbits contributing to the diffusion coefficient due to a magnetic focusing effect. The Aharonov-Bohm type oscillation superimposed on the commensurability peak is closely related with semiclassically quantized energy levels associated with the periodic orbit encircling an antidot. It has also been shown that realistic antidot lattices have a sizable amount of disorder in the potential. ACKNOWLEDGEMENTS This work is supported in part by the Japan Society for the Promotion of Science ( Research for the Future Program JSPS-RFTF96P00103). One of the authors (S.U.) has been supported by Research Fellowships from the Japan Society for the Promotion of Science for Young Scientists. Some numerical calculations were performed on FACOMVPP500 in the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. REFERENCES 1. A. Kumar, S.E. Laux, and F. Stern, Phys. Rev. B 42, 5166 (1990). 2. T. Suzuki and T. Ando, J. Phys. Soc. Jpn. 62, 2986 (1993). 3. T. Suzuki and T. Ando, Physica B 201, 345 (1994). 4. T. Suzuki and T. Ando, Physica B 227, 46 (1996). 5. T. Nakanishi and T. Ando, Phys. Rev. B 54, 8021 (1996). 6. T. Nakanishi and T. Ando, Physica B 227, 127 (1996). 7. D. Weiss, M.L. Roukes, A. Menschig, P. Grambow, K. von Klitzing, and G. Weimann, Phys. Rev. Lett. 66, 2790 (1991). 8. R. Fleischmann, T. Geisel, and R. Ketzmerick, Phys. Rev. Lett. 68, 1367 (1992). 9. E.M. Baskin, G.M. Gusev, Z.D. Kvon, A.G. Pogosov, and E.V. Entin, JETP Lett. 55, 678 (1992). 10. K. Ensslin, S. Sasa, T. Deruelle, and P.M. Petroff, Surf. Sci. 263, 319 (1992). 11. J. Takahara, A. Nomura, K. Gamo, S. Takaoka, K. Murase, and H. Ahmed, Jpn. J. Appl. Phys. 34, 4325 (1995). 12. K. Tsukagoshi, S. Wakayama, K. Oto, S. Takaoka, K. Murase,

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