- Email: [email protected]
Antidot shape dependence of the commensurability oscillation of magnetoresistance in two-dimensional antidot arrays
Physica B 256±258 (1998) 397±400
Antidot shape dependence of the commensurability oscillation of magnetoresistance in two-dimensional antidot arrays Takashi Azuma a, Toshihito Osada a
b,a,*
Research Center for Advanced Science and Technology, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan b Institute for Solid State Physics, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106-8666, Japan
Abstract We have studied commensurability oscillations in magnetoresistance (MR) of antidot arrays by changing the antidot shape. For the bar-shaped antidot array, we found the clear dependence of the amplitude of commensurability oscillations on the current direction. This feature was successfully reproduced by numerical calculations. Based on these results, we discuss the origin of the commensurability oscillations. We have also studied antidot arrays with other antidot shapes experimentally and numerically. Ó 1998 Elsevier Science B.V. All rights reserved. Keywords: Antidot array; Antidot shape; Commensurability oscillation; Magnetotransport
1. Introduction Various interesting phenomena have been observed in the antidot array system under magnetic ®elds [1,2]. Among them, ``the commensurability oscillation'' of magnetoresistance (MR) is the most popular phenomenon in this system. Weiss et al. observed that the oscillation peaks appear when the cyclotron orbit can encircle a speci®c number of antidots 1; 2; 4; 9; . . . [1]. They ascribed the origin of this phenomenon to the existence of these ``pinned'' orbits. Baskin et al. proposed an alternative interpretation for the fundamental peak of oscillations [3]. They claimed that the peak is due to an enhancement of the diusion coecient produced by ``runaway'' orbits which skip antidots in succession. Recently, Tsukagoshi et al. experimentally showed that the commensurability oscil-
*
Corresponding author. Fax: 81 3 3478 5472; e-mail: [email protected]
lations are determined only by the period along the direction perpendicular to the current ¯ow [4]. Their results support strongly the runaway orbit model. On the other hand, based on numerical calculation, Fleischmann et al. presented a model that the chaotic orbits of the electron make a major contribution to the commensurability oscillation [5]. In this paper, in order to clarify the origin of the commensurability oscillation, we have changed the conventional circular antidot shape to various shapes and studied the eect on the commensurability oscillation. As to the non-circular antidots, Lorke et al. have already studied a triangular antidot to introduce the asymmetry in the system [6]. 2. Experiment The samples were prepared by processing the high-mobility AlGaAs/GaAs heterostructure using the electron beam lithography and the wet etching
0921-4526/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 6 7 8 - 4
398
T. Azuma, T. Osada / Physica B 256±258 (1998) 397±400
technique. From the Shubnikov-de Haas (SdH) oscillations at T 0.3 K, the carrier density, mobility, and electron mean free path of the unpatterned two dimensional electron gas (2DEG) region were estimated as ne 4 ´ 1011 cmÿ2 , m 1.0 ´ 106 cm2 /V s, and le 10 mm, respectively. We studied three types of the antidot shape; the bar-type ()), the cross-type (+), and the diamondtype (¨). These antidots were arrayed onto a square lattice where the lattice constant was ®xed at a 1 mm. Their SEM images are shown in Fig. 1. The MR was measured at T 0.3 K under magnetic ®elds perpendicular to the 2D plane using standard dc technique. Fig. 2 shows the MR traces for the three types of antidot arrays. As to the barshaped antidot array, we measured the MR for two current directions; parallel and perpendicular to the antidot bar. Above B 0.3 T, the SdH oscillations are seen in both cases, however, the amplitude of the commensurability oscillations strongly depends on the current direction. When the current direction is perpendicular to the antidot bar,
commensurability oscillations are clearly observed (trace (a)). In contrast, they almost vanish when the current is parallel to the bar (trace (b)). In trace (a), the fundamental peak at B 0.2 T satis®es the condition 2Rc a, where 2Rc is the cyclotron diameter and a is the lattice constant. The broad peak at B 0.4 T satis®es 2Rc 1/2a, and the peaks below 0.2 T correspond to the condition that cyclotron orbits encircling a speci®c number of antidots, 2; 4; 9; . . .. On the other hand, in the cross antidot array and the diamond antidot array, we observed unidenti®ed commensurability peaks above 0.3 T in addition to the ordinary commensurability oscillations. This fact means that the detail of the antidot shapes aects the commensurability oscillations in the ®eld region above the fundamental peak. 3. Discussion The observed dependence on current direction in the bar-shaped antidot array apparently seem
Fig. 1. SEM image of (a) bar-shaped antidot array, (b) cross-shaped antidot array, and (c) diamond-shaped antidot array. In (a) relevant electron orbits are shown, (A) is a pinned orbit, (B) is a runaway orbit when the current direction is perpendicular to the antidot bar, (C) is a runaway orbit when the current direction is parallel to the antidot bar.
T. Azuma, T. Osada / Physica B 256±258 (1998) 397±400 b
399 b
U U0 cos px x cos py y ;
Fig. 2. Magnetoresistance of (a), (b) the bar-shaped antidot array, (c) cross-shaped antidot array, and (d) diamond-shaped antidot array. In bar-shaped antidot array (a): The current direction is perpendicular to the antidot bar and in (b): The current direction is parallel to the antidot bar. The data in (b) are scaled by 3 for clarity.
2
where bx 64, by 4. The calculated MR of the bar-shaped antidot array for two current directions is shown in Fig. 3. The observed features mentioned in the experimental results are well reproduced by the numerical calculation. Then, to clarify the responsible orbit, we classi®ed all orbits into the localized orbits and extended ones, and evaluated the MR for each set of orbits. The results are summarized as follows: (1) Generally, both sets of orbits are responsible for the commensurability oscillations. (2) The contribution of localized (pinned) orbits is dominant in the low ®eld region below the fundamental peak, while that of the extended orbit is dominant in the high ®eld region. (3) The contribution of extended orbits is controlled by the current direction. The observed features are well explained by this picture.
to suggest that as origin of the commensurability oscillations the runaway orbit mechanism is more plausible than the pinned orbit model, since only runaway orbits depend on the current direction as shown in Fig. 1(a). In order to clarify the most responsible orbit, we carried out the numerical simulation on the commensurability oscillations. Since the Fermi length is smaller than the lattice constant, the problem can be dealt with in the classical approximation. The conductivity can be evaluated by the following Kubo type formula: Z rij
of E dE ÿ oE
Z1
dteÿt=s vi tvj 0 ;
1
0
where f(E) is the Fermi distribution function, vi (t) is the velocity of the electron moving along the orbit, and s is the average scattering time. The ensemble average is taken over the initial state of all possible orbits. The present method is dierent from Fleischmann's one, which considered only the contribution of chaotic orbits. For the barshaped antidot, we employed the potential
Fig. 3. Numerical calculated magnetoresistance of (a), (b) barshaped antidot arrays, (c) cross-shaped antidot array, and (d) diamond-shaped antidot array (dotted line). In (a) the current direction is perpendicular to the antidot bar. In (b) the current direction is parallel to the antidot bar. Magnetic ®eld is normalized by B0 where 2Rc a. In this experiments B0 0.2 T. At each magnetic ®eld where peaks are observed the relevant electron orbits are sketched.
400
T. Azuma, T. Osada / Physica B 256±258 (1998) 397±400
We also performed the numerical calculation for the cross antidot array and the diamond antidot array. The observed unidenti®ed peaks are successfully reproduced. According to the above conclusion, these unidenti®ed peaks in high ®eld region should originate mainly from the extended orbits re¯ected by the antidot edge, so that it is reasonable that they are sensitive to the detail of the antidot shapes. 4. Conclusion We experimentally found that the amplitude of the commensurability oscillations in the system with anisotropic antidot shape depends on the current direction distinctly, and that the characteristic commensurability peaks, which appear in the high ®eld region, depend on the antidot shape. The numerical calculation concludes that the commensu-
rability peaks in the high ®eld region originate from the skipping electron motion which is sensitive to the antidot shape.
References [1] D. Weiss, M.L. Roukes, A. Menschig, P. Grambow, K. von Klitzing, G. Weimann, Phys. Rev. Lett. 58 (1991) 2960. [2] D. Weiss, K. Richter, A. Menschig, R. Bergmann, H. Schweizer, K. von Klitzing, G. Weimann, Phys. Rev. Lett. 70 (1993) 4118. [3] E.M. Baskin, G.M. Gusev, Z.D. Kvon, A.G. Pogosov, M.V. Entin, Pisma Zh. Eskp. Teor. Fiz. 55 (1992) 649; JETP Lett. 55 (1992) 679. [4] K. Tsukagoshi, S. Wakayama, K. Oto, S. Takaoka, K. Murase, K. Gamo, J. Phys. Soc. Jpn. 65 (1996) 811. [5] R. Fleischmann, T. Geisel, R. Ketzmerick, Phys. Rev. Lett. 68 (1992) 1367. [6] A. Lorke, S. Wimmer, B. Jager, J.P. Kotthaus, Surf. Sci. (to be published).
Antidot shape dependence of the commensurability oscillation of magnetoresistance in two-dimensional antidot arrays Takashi Azuma a, Toshihito Osada a
b,a,*
Research Center for Advanced Science and Technology, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan b Institute for Solid State Physics, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106-8666, Japan
Abstract We have studied commensurability oscillations in magnetoresistance (MR) of antidot arrays by changing the antidot shape. For the bar-shaped antidot array, we found the clear dependence of the amplitude of commensurability oscillations on the current direction. This feature was successfully reproduced by numerical calculations. Based on these results, we discuss the origin of the commensurability oscillations. We have also studied antidot arrays with other antidot shapes experimentally and numerically. Ó 1998 Elsevier Science B.V. All rights reserved. Keywords: Antidot array; Antidot shape; Commensurability oscillation; Magnetotransport
1. Introduction Various interesting phenomena have been observed in the antidot array system under magnetic ®elds [1,2]. Among them, ``the commensurability oscillation'' of magnetoresistance (MR) is the most popular phenomenon in this system. Weiss et al. observed that the oscillation peaks appear when the cyclotron orbit can encircle a speci®c number of antidots 1; 2; 4; 9; . . . [1]. They ascribed the origin of this phenomenon to the existence of these ``pinned'' orbits. Baskin et al. proposed an alternative interpretation for the fundamental peak of oscillations [3]. They claimed that the peak is due to an enhancement of the diusion coecient produced by ``runaway'' orbits which skip antidots in succession. Recently, Tsukagoshi et al. experimentally showed that the commensurability oscil-
*
Corresponding author. Fax: 81 3 3478 5472; e-mail: [email protected]
lations are determined only by the period along the direction perpendicular to the current ¯ow [4]. Their results support strongly the runaway orbit model. On the other hand, based on numerical calculation, Fleischmann et al. presented a model that the chaotic orbits of the electron make a major contribution to the commensurability oscillation [5]. In this paper, in order to clarify the origin of the commensurability oscillation, we have changed the conventional circular antidot shape to various shapes and studied the eect on the commensurability oscillation. As to the non-circular antidots, Lorke et al. have already studied a triangular antidot to introduce the asymmetry in the system [6]. 2. Experiment The samples were prepared by processing the high-mobility AlGaAs/GaAs heterostructure using the electron beam lithography and the wet etching
0921-4526/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 6 7 8 - 4
398
T. Azuma, T. Osada / Physica B 256±258 (1998) 397±400
technique. From the Shubnikov-de Haas (SdH) oscillations at T 0.3 K, the carrier density, mobility, and electron mean free path of the unpatterned two dimensional electron gas (2DEG) region were estimated as ne 4 ´ 1011 cmÿ2 , m 1.0 ´ 106 cm2 /V s, and le 10 mm, respectively. We studied three types of the antidot shape; the bar-type ()), the cross-type (+), and the diamondtype (¨). These antidots were arrayed onto a square lattice where the lattice constant was ®xed at a 1 mm. Their SEM images are shown in Fig. 1. The MR was measured at T 0.3 K under magnetic ®elds perpendicular to the 2D plane using standard dc technique. Fig. 2 shows the MR traces for the three types of antidot arrays. As to the barshaped antidot array, we measured the MR for two current directions; parallel and perpendicular to the antidot bar. Above B 0.3 T, the SdH oscillations are seen in both cases, however, the amplitude of the commensurability oscillations strongly depends on the current direction. When the current direction is perpendicular to the antidot bar,
commensurability oscillations are clearly observed (trace (a)). In contrast, they almost vanish when the current is parallel to the bar (trace (b)). In trace (a), the fundamental peak at B 0.2 T satis®es the condition 2Rc a, where 2Rc is the cyclotron diameter and a is the lattice constant. The broad peak at B 0.4 T satis®es 2Rc 1/2a, and the peaks below 0.2 T correspond to the condition that cyclotron orbits encircling a speci®c number of antidots, 2; 4; 9; . . .. On the other hand, in the cross antidot array and the diamond antidot array, we observed unidenti®ed commensurability peaks above 0.3 T in addition to the ordinary commensurability oscillations. This fact means that the detail of the antidot shapes aects the commensurability oscillations in the ®eld region above the fundamental peak. 3. Discussion The observed dependence on current direction in the bar-shaped antidot array apparently seem
Fig. 1. SEM image of (a) bar-shaped antidot array, (b) cross-shaped antidot array, and (c) diamond-shaped antidot array. In (a) relevant electron orbits are shown, (A) is a pinned orbit, (B) is a runaway orbit when the current direction is perpendicular to the antidot bar, (C) is a runaway orbit when the current direction is parallel to the antidot bar.
T. Azuma, T. Osada / Physica B 256±258 (1998) 397±400 b
399 b
U U0 cos px x cos py y ;
Fig. 2. Magnetoresistance of (a), (b) the bar-shaped antidot array, (c) cross-shaped antidot array, and (d) diamond-shaped antidot array. In bar-shaped antidot array (a): The current direction is perpendicular to the antidot bar and in (b): The current direction is parallel to the antidot bar. The data in (b) are scaled by 3 for clarity.
2
where bx 64, by 4. The calculated MR of the bar-shaped antidot array for two current directions is shown in Fig. 3. The observed features mentioned in the experimental results are well reproduced by the numerical calculation. Then, to clarify the responsible orbit, we classi®ed all orbits into the localized orbits and extended ones, and evaluated the MR for each set of orbits. The results are summarized as follows: (1) Generally, both sets of orbits are responsible for the commensurability oscillations. (2) The contribution of localized (pinned) orbits is dominant in the low ®eld region below the fundamental peak, while that of the extended orbit is dominant in the high ®eld region. (3) The contribution of extended orbits is controlled by the current direction. The observed features are well explained by this picture.
to suggest that as origin of the commensurability oscillations the runaway orbit mechanism is more plausible than the pinned orbit model, since only runaway orbits depend on the current direction as shown in Fig. 1(a). In order to clarify the most responsible orbit, we carried out the numerical simulation on the commensurability oscillations. Since the Fermi length is smaller than the lattice constant, the problem can be dealt with in the classical approximation. The conductivity can be evaluated by the following Kubo type formula: Z rij
of E dE ÿ oE
Z1
dteÿt=s vi tvj 0 ;
1
0
where f(E) is the Fermi distribution function, vi (t) is the velocity of the electron moving along the orbit, and s is the average scattering time. The ensemble average is taken over the initial state of all possible orbits. The present method is dierent from Fleischmann's one, which considered only the contribution of chaotic orbits. For the barshaped antidot, we employed the potential
Fig. 3. Numerical calculated magnetoresistance of (a), (b) barshaped antidot arrays, (c) cross-shaped antidot array, and (d) diamond-shaped antidot array (dotted line). In (a) the current direction is perpendicular to the antidot bar. In (b) the current direction is parallel to the antidot bar. Magnetic ®eld is normalized by B0 where 2Rc a. In this experiments B0 0.2 T. At each magnetic ®eld where peaks are observed the relevant electron orbits are sketched.
400
T. Azuma, T. Osada / Physica B 256±258 (1998) 397±400
We also performed the numerical calculation for the cross antidot array and the diamond antidot array. The observed unidenti®ed peaks are successfully reproduced. According to the above conclusion, these unidenti®ed peaks in high ®eld region should originate mainly from the extended orbits re¯ected by the antidot edge, so that it is reasonable that they are sensitive to the detail of the antidot shapes. 4. Conclusion We experimentally found that the amplitude of the commensurability oscillations in the system with anisotropic antidot shape depends on the current direction distinctly, and that the characteristic commensurability peaks, which appear in the high ®eld region, depend on the antidot shape. The numerical calculation concludes that the commensu-
rability peaks in the high ®eld region originate from the skipping electron motion which is sensitive to the antidot shape.
References [1] D. Weiss, M.L. Roukes, A. Menschig, P. Grambow, K. von Klitzing, G. Weimann, Phys. Rev. Lett. 58 (1991) 2960. [2] D. Weiss, K. Richter, A. Menschig, R. Bergmann, H. Schweizer, K. von Klitzing, G. Weimann, Phys. Rev. Lett. 70 (1993) 4118. [3] E.M. Baskin, G.M. Gusev, Z.D. Kvon, A.G. Pogosov, M.V. Entin, Pisma Zh. Eskp. Teor. Fiz. 55 (1992) 649; JETP Lett. 55 (1992) 679. [4] K. Tsukagoshi, S. Wakayama, K. Oto, S. Takaoka, K. Murase, K. Gamo, J. Phys. Soc. Jpn. 65 (1996) 811. [5] R. Fleischmann, T. Geisel, R. Ketzmerick, Phys. Rev. Lett. 68 (1992) 1367. [6] A. Lorke, S. Wimmer, B. Jager, J.P. Kotthaus, Surf. Sci. (to be published).