Semi-classical orbits in a caterpillar like Sinaı̈ billiard

Physica E 7 (2000) 731–734

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Semi-classical orbits in a caterpillar like Sina billiard A. Pouydebasquea; b;∗ , M.V. Budantsevc , A.G. Pogosovc , Z.D. Kvonc , D.K. Maudea , J.C. Portala; b; d a Grenoble

High Magnetic Field Laboratory, MPI-FKF and CNRS, B.P. 166, F-38042 Grenoble, France b Institut National des Sciences Appliquà ees, F-31077 Toulouse Cedex 4, France c Institute of Semiconductor Physics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 630090, Russia d Institut Universitaire de France, France

Abstract The transport properties of a caterpillar-like Sinai billiard are investigated in the ballistic regime. The experimental kinetic coecients exhibit pronounced features, characteristic of this kind of billiard. A Monte-Carlo style simulation of the semi-classical electron motion is performed in order to model the transmission probabilities Tij of an ideal caterpillar structure. By visualising the spatial charge density distributions, the main features in the transmission probabilities are related to special electron orbits which dominate the electron dynamics at a given B. Finally, Poincare sections are calculated close to the commensurability condition. The appearance of stability islands surrounded by the stochastic sea characterises the presence of regular electron skipping orbits involved in the ballistic transport phenomena. ? 2000 Elsevier Science B.V. All rights reserved. PACS: 73.23.Ad; 73.61.Ey Keywords: Magnetotransport; Sina billiards; Dynamic chaos

The transport properties of a two-dimensional electron gas with arti cial scatterers have been widely investigated [1–3]. If the characteristic dimensions of the scatterers are smaller than the electron mean free path, which is limited by randomly distributed impurities, these systems can be considered as electron ballistic billiards. It was established that the main magnetotransport features in billiards, measured at liq∗ Corresponding author. Tel.: +33-4-76-88-78-60; fax: +33-4-76-85-56-10. E-mail address: [email protected] (A. Pouydebasque)

uid helium temperatures, can be described within the framework of the classical chaotic dynamics of an electron. The array of antidots is the system investigated in the greatest depth, and has revealed some pronounced peaks in the magnetoresistance, when the cyclotron radius Rc is commensurate with the lattice period d [1]. Two analyses have been put forward to explain these magnetoresistance maxima: electron orbits pinned around one or several antidots [2], and electron trajectories skipping along a row of antidots [3]. Both explanations have been widely considered as correct, even if it has not yet been possible to determine which is dominant. The numerical simulations

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Fig. 1. The experimental magnetoresistances of the caterpillar-like Sina billiard: the bend resistance (solid line), the Hall resistance (dashed), and the longitudinal resistance (dotted). The inset shows the schematic diagram of the billiard investigated. The period of the antidots is d = 0:6 m, the antidot radius a60:3 m, and the distance between antidots chains W = 0:9 m.

of the electron motion usually used to describe the observed anomalies of magnetoresistance average over the contributions of all the electron trajectories and thus fail to point out the special electron orbits responsible for the observed features. From this viewpoint the study of dierent types of billiards is meaningful, because electron trajectories that are essential for one type of billiard may be less important or absent in another type of billiard. Here we present experimental results and computer calculations obtained on a caterpillar like Sina billiard. The computer simulations were based on the formalism proposed by Landauer and Buttiker [4,5]. This model allows to relate the resistances to the transmission coecients between the contacts of the structure. The samples were fabricated from a 2DEG formed at an AlGaAs=GaAs heterojunction with electron density ns = (2:5–4) × 1011 cm−2 and mobility  = (6–8) × 105 cm2 =V s. The caterpillar-like billiard was created by electron beam lithography and subsequent reactive plasma etching (see inset of Fig. 1). All the dimensions of the billiard were smaller than the electron mean free path lp = 5–7 m. Somewhat similar systems have been investigated [6,7]. The device involved here has two additional Hall contacts. This enables the measurement of three dierent four-terminal resistances: RL = R12; 12 ; RH = R12; 43 ; RB = R13; 42 (respectively, longitudinal, Hall and bend resistance). Four-terminal resistances Rij; kl of the structure were

obtained by passing the current through probes i and j and measuring the voltage drop from probes k and l. Magnetoresistance measurements were performed using a standard lock-in technique in a magnetic eld up to 1 T, and in a range of temperatures from 40 mK to 4.2 K. Typical experimental magnetoresistance curves are presented in Fig. 1. The electron density determined from the Shubnikov–de Haas oscillations was ns = 2:7 × 1011 cm−2 . The longitudinal resistance RL presents a positive value in a weak range of magnetic eld, for B ¡ 0:1 T, similar to the positive magnetoresistance observed in wires with rough boundaries. This can be explained by the destruction of the channeling trajectories connecting opposite contacts. For higher magnetic elds, a decrease of RL is observed, featuring a characteristic plateau around B ≈ 0:24 T. The presence of the plateau close to the commensurability condition d = 2Rc corresponding to Bc ≈ 0:29 T, seems to be a characteristic feature of a regular spaced billiard [6]. Looking at the Hall resistance RH , a quenching for very low magnetic elds (B ¡ 0:1 T) can be observed at rst. This can be considered as a classical feature of a ballistic cross-junction, and con rms the existence of collimation in this regime [8,9]. RH presents a noticeable dimple at B ≈ 0:27 T, and nally a plateau, which can be compared to the classical “last Hall plateau” [8]. Finally, the bend resistance exhibits a huge negative value for zero magnetic eld (typical for ballistic cross-junction [9]), which disappears for higher B. An additional negative peak is then apparent at B ≈ 0:27 T. As previously stated, the usual approach to characterise transport phenomena in ballistic systems is to analyse the behaviour of the transmission probabilities in the system. A Monte-Carlo style simulation of the electron transport was performed in order to calculate the transmission probabilities Tij of a hard wall caterpillar structure presenting the same geometrical features as the experimental sample (Fig. 2). The only tting parameter used in the calculations was the antidot radius a. The value of a was estimated from the resistance of the Hall contacts, which was approximately equal to h=2e2 thus giving a = 0:29 m for ns = 2:7 × 1011 cm−2 . The calculations were based on a semi-classical model [8,9]. The symmetries of the device and reciprocity relations Tij (B) = Tji (−B) enable to reduce the transmission probabilities to four

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Fig. 2. Transmission probabilities between leads 4 and 1 (dashed line), 4 and 2 (solid), 4 and 3 (dotted) and 1 and 2 (dash–dotted).

independent coecients: T14 ; T24 ; T34 and T21 . Since the calculations were performed using a semi-classical method, the features observed in the transmission coecients are closely related to electron trajectories which connect, or not, the dierent leads, depending on the magnetic eld. Subsequently, our attention was focused on the most pronounced properties of the transmission probabilities, supposing that they are characteristic of the special regular electron orbits connecting dierent leads. A brief observation of the coecients allows us to point out some characteristic features: T14 presents a noticeable peak at B=Bc ≈ 0:76 and T24 a deep dimple at B=Bc ≈ 0:73: T34 exhibits a strong peak at B=Bc ≈ 0:73 and T21 , a peak at B=Bc ≈ 0:87. A basic analysis of the results shows that the main characteristics are observed in the ratio 0:756 B=Bc 61:20. The main features in the experimental curves correspond to the condition 0:76B=Bc 61, which is compatible with the calculations. Subsequently, it would appear reasonable to assume that the features observed in our calculations are closely related to those present in the experimental curves. However, the analysis of the transmission probabilities in the structure does not allow us to relate directly special electron orbits to magnetotransport features. For this purpose we have performed semi-classical simulations of spatial charge density distributions in the billiard, which are usually considered as a useful tool for the visualisation of the spatial repartition of the electron transport [8]. Fig. 3(a) shows a charge distribution diagram calculated for the

Fig. 3. Spatial charge density distributions. The distributions are calculated for electrons injected via the lead 4 (a, b), and 1 (c). The darkest regions in the patterns correspond to the higher potentials.

condition B=Bc = 0:73. The pattern clearly exhibits striking features resembling electron trajectories connecting leads 3 and 4 (dashed line in Fig 3(a)). The corresponding transmission probability T34 presents a noticeable peak at the same B=Bc value which is obviously formed by the regular orbits, emphasised in Fig. 3(a). In the same way, Fig. 3(b) reveals the existence of a more sophisticated kind of electron orbit connecting leads 4 and 1. The distribution was calculated for the ratio B=Bc = 0:76, which corresponds to the highest peak in T14 . As for T34 , this type of orbit can be considered as closely related to the presence of the T14 peak. Finally, the charge density distribution was calculated for the ratio B=Bc = 0:87. Due to the vicinity of the condition B = Bc , one can expect to observe the in uence of runaway trajectories. The charge density distribution picture exhibits a higher density in a region close to the bottom of each antidot in the upper row. On the contrary, the density is lower close to the top of the antidots in the lower row of the caterpillar structure. This can be seen as the manifestation of regular electron orbits skipping on

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Fig. 4. Poincare section of an ideal in nite caterpillar billiard. For each collision electron–antidot ’ is the angular position of the collision on the antidot, and is the emergent angle of the electron orbit. The electrons start out from an arbitrary point situated in the upper chain of antidots. The Poincare section was calculated for B=Bc = 0:87. The black points correspond to electrons bouncing on the same row of antidots, while a collision from a side to another is represented in grey. Stability islands are only present in the right side of the pattern because of the initial conditions.

the upper antidots along the structure, for electrons injected in the left lead in an anticlockwise cyclotron motion. The low charge density observed on the top of the lower chain of antidots is due to the absence of electrons injected in the right lead. To verify the assumption about the existence of runaway trajectories, the system was considered within the framework of the dynamic chaos theory. The Sina billiards are well known as the experimental source for the study of classical dynamical chaos phenomena. Baskin et al. [10] considered Poincare sections calculations to demonstrate that delocalised regular trajectories play an important role in the ballistic transport phenomena in an antidot lattice. In the same way, Poincare sections of an in nite caterpillar billiard for various magnetic elds close to the commensurability condition were calculated. The electron motion was simulated and, for each collision electron– antidot, the parameters ’ and were obtained, ’ being the angular position of the collision on the antidot, and the emergent angle of the electron orbit. For 0:7 ¡ B=Bc ¡ 1 regular orbits can be observed. A regular trajectory form a close contour (stability island) while the chaotic motion lls the (’; ) space randomly. The stability islands are situated in the vicinity

of the condition (’ = 3=2; = 0), which obviously corresponds to an electron orbit skipping on the upper row of antidots. Even if the dynamical variables (’; ) do not preserve the area of the phase space, the topology of the phase portrait does not change. Thus, it is worth mentioning that the Poincare sections present the most pronounced stability islands at the condition B=Bc ≈ 0:87 (Fig. 4), which is also the one where the main feature in both the T21 curve and the experimental resistances are observed. Therefore, it seems to be reasonable to argue that the peak in T21 at B=Bc = 0:87 is formed by the regular skipping trajectories underlined in the Poincare section patterns. To summarise, performing semi-classical simulations of the electron motion, we obtained transmission probabilities, spatial charge density distributions and Poincare sections that enabled us to attribute experimentally observed features to speci c electron orbits in the system. Acknowledgements This work was supported by CNRS-PICS 628, NATO HTECH-LG 971304 and RFBR (99-02-16756). References [1] K. Ensslin, P.M. Petro, Phys. Rev. B 41 (1990) 12 307. [2] D. Weiss, M.L. Roukes, A. Menschig, P. Grambow, K. von Klitzing, G. Weimann, Phys. Rev. Lett. 66 (1991) 123. [3] E.M. Baskin, G.M. Gusev, Z.D. Kvon, A.G. Pogosov, M.V. Entin, JETP Lett. 55 (1992) 678. [4] R. Landauer, IBM J. Res. Dev. 1 (1957) 233. [5] M. Buttiker, Phys. Rev. Lett. 57 (1986) 1761. [6] M.V. Budantsev, Z.D. Kvon, A.G. Pogosov, N.T. Moschegov, A.E. Plotnikov, A.I. Toropov, Surf. Sci. 361=362 (1996) 739. [7] Y. Ochiai, A.W. Widjaja, N. Sasaki, K. Yamamoto, R. Akis, D.K. Ferry, J.P. Bird, K. Ishibashi, Y. Aoyagi, T. Sugano, Phys. Rev. B 56 (1997) 1073. [8] C.W.J. Beenaker, H. van Houten, Phys. Rev. Lett. 63 (1989) 1857. [9] H.U Baranger, D.P. DiVicenzo, R.A. Jalabert, A.D. Stone, Phys. Rev. B 44 (1991) 10 637. [10] E.M. Baskin, A.G. Pogosov, M.V. Entin, JETP 83 (1996) 1135.