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Negative magnetoresistance and anomalous diffusion of two-dimensional electrons in a disordered array of antidots
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ELSEVIER
Surface Science 305 (1994) 443-447
Negative magnetoresistance and anomalous diffusion of two-dimensional electrons in a disordered array of antidots G.M. Gusev *71,a,P. Basmaji a, Z.D. Kvon b, L.V. Litvin b, Yu.V. Nastaushev b, A.I. Toropov b aInstitute de Fisica e Q&mica de &io Carlos, Universidade de Srio Paula, SGo Paula, Brazil b Institute of Semiconductor Physics, Russian Academy of Sciences, Siberian Branch, Novosibirsk, Russian Federation (Received 8 April 1993; accepted for publication 29 June 1993)
Abstract Negative linear magnetoresistance of two-dimensional (2D) electrons in a disordered array of antidots has been found. The existence in a magnetic field of trajectories which roll along the array of antidots was suggested. These trajectories have a mean free path larger than the average value for electrons with ordinary diffusion.
Magnetic field decreases electron mean free path, therefore the magnetoresistance of semiconductors and metals is positive. However, at low temperature and weak magnetic field negative magnetoresistance has been observed [l]. This phenomenon has been explained from a quantum standpoint. The crossed electron trajectories increase the backscattering probability due to the interference at the crossing point. Magnetic field suppresses this interference because of the Aharonov-Bohm effect, the backscattering probability decreases, and as a result negative magnetoresistance appears [l]. From a classical standpoint the negative magnetoresistance is a result of increase of electron elastic length or appear-
* Corresponding author. address: Institute of Semiconductor Physics, Russian Academy of Sciences, Siberian Branch, Novosibirsk, Russian, Federation.
’ Permanent
0039-6028/94/$07.00
0 1994 Elsevier Science
SSDI 0039-6028(93)E0590-Q
ante of the carriers with mean free path larger than the average value. In Ref. [2] Wagenhuber et al. have found theoretically that this probability exists in a 2D electron gas in a lateral two-dimensional lattice. At some value of lattice potential in a magnetic field the anomalous diffusion of electrons arises. This is characterized by a linear increase of the mean square displacement of carriers, (x2)“* = fl+rr, with (Y= 1. It is responsible for violation of the exponential distribution of a portion of particles with mean free path, and thus, electrons with anomalously large elastic length appear. However, the influence of this effect on the electron transport, in particular magnetoresistance, was not considered. One type of two-dimensional lateral superlattice with strong repulsive potential is an array of antidots [4], which is produced in a 2D electron gas by holes with submicron diameter fabricated by etching or irradiation by ions. This system has attracted attention also because it allows various
B.V. All rights reserved
444
G.M. Gusr~~ et cd. /Surfuce
arrangements of antidots to be simulated, square [3,4], hexagonal [.5], two-dimensional quasicrystals 161, and disordered [7,81. In this work the magnetoresistance of 2D electrons in a periodic and disordered lattice of antidots has been studied, and a negative magnetoresistance which grows with degree of disorder has been found. The test samples were Hall bridges based on GaAs/AlGaAs heterostructures with a 2D electron gas. The parameters of the initial heterostructures were electron density, n, = 5 X IO” cme2, and mobility, p = (2 - 5) x 10’ cm*/V . s. A lattice of antidots, produced by electron-beam lithography and reactive ion etching, covered a part of the sample between the potential probes. The antidot diameter was 0.15-0.2 Frn. The effective diameter, a, which consists of a lithographic diameter and a depletion length, was 0.2-0.3 pm. Samples in which the array of antidots is disordered with various degrees of disorder were made for the experiments (Fig. 1). The disordering of the lattice was accomplished in the following way. A random number generator determined the shift in the position of the antidots in the direction of the neighboring antidots. The deviations of antidots from their periodic arrangement in the lattice with period d = 0.7 pm at the peak were A = 0.0, 0.1, 0.25, and 0.35 pm. We also measured mesoscopic samples with size d = 0.6 pm, and 4X4 pm2, average periodicity A = 0.3 pm. Thus, the short-range order of the system was violated, but its long-range order was preserved. The magnetoresistance was measured by the four-terminal method at frequencies of 70-700 Hz in a magnetic field, B, up to 0.4 T, and at temperatures of 1.7-4.2 K. Electron scattering by the antidot lattice was dominant in our samples, from this the mean free path, 1, at B = 0 was found to be 0.4, 017, 0.21, and 0.4 pm for samples with different degrees of disorder. We see that the behavior of I is nonmonotonic with disorder. With an increase of A, the length 1 initially decreases, and increases again. Similar nonmonotonic behavior has been observed recently in samples with a quasiperiodic (Penrose tiling) lattice of antidots [6], which is an intermediate type of
Science 30.5 (1994) 443-447
......
.I
....... ‘.
.
.....
.’
.......
!. ...... I. ......
k_L~2-?2L I . . . . .*
. . .* =1 0.
..a. l
.*..... . . . . l . . . .*..
.d
. . . . . .I l * -I*
-i 0
0.2
04 B,T
Fig. 1. Magnetoresistance dependence on B for a sample with different degrees of antidot disorder and for a periodic lattice: d = 0.7 Frn: (1) A = 0.0; (2) A = 0.1: (3) A = 0.25; (4) A = 0.35 wrn; T = 4.2 K.
array between periodic and disordered. In this case the length I initially decreases with increase of the basic triangle size in the Penrose cell, and then increases again. This behavior has not been explained, and further theoretical analysis of the scattering in this system is required. Fig. 1 shows the magnetoresistance for samples with various degrees of disorder as a function of magnetic field. We see commensurability oscillations for all samples when the cyclotron to the lattice period. diameter, R,, is comparable The oscillation amplitude decreases with increase of A, and the position of the last peak shifts to lower magnetic field [8]. The preservation of long-range order in our sample is responsible for oscillations in the disordered antidot lattice. Fig. 1 shows that for the periodic lattice, the second peak maximum is higher than the peak maximum which lies at lower magnetic field. However, for the array with disorder degree A = 0.1 pm their amplitudes are equal, i.e., negative magnetoresis-
G.hf
0
Guseu
et af.
/&itface
01
B,T Fig. 2. Magnetoresistan~~ dependence on B for low magnetic field. (2) d = 0.1; (3) A = 0.25; (4) A = 0.35 pm; T = 4.2 K.
tance (nMR) appears, but it is not visible due to the oscillations. With increase of disorder degree nMR at low B is observed. Fig. 2 shows the low field part of the nMR for samples with a disordered array of antidots. We see that the magnetoresistance is linear for low magnetic field and
Fig. 3. Magnetoresistance of the mesoscopic samples with disordered lattice of antidots measured after different intervals of time: (1) 0 min, (2) 5 min, (3) 40 min; d = 0.6 pm, d = 0.3 pm, T = 1.7 K.
Science
.?&i
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increases with increasing disorder. We did not find this magnetoresistan~e to be temperature dependent at temperatures 1.7-4.2 K, which proves that this nMR has a classical origin. Fig. 3 shows the magnetoresistance of a mesoscopic sample with period d = 0.6 pm measured after different intervals of time. We also see linear negative magnetoresistance which does not depend on the temperature. Against the background of this magnetoresistance, there are fluctuations. The pattern of these fluctuations as a function of B is changed in time. We believe, that the fluctuations of magnetoresistance are due to the interference of electron trajectories scattered by the antidot lattice. However, more experiments are necessary in this system. We also believe that linear negative magnetoresistance has the same nature as for macroscopic samples. We should note that we measured all samples before the patterning of antidots and observed no negative magnetoresistance in this region of magnetic field. For Hail bridge samples pXX= ~~,/(a:~ + a&), where pXXis the resistivity, and o;rX, gXYare the diagonal and Hall parts of the conductivity. pXXis constant in magnetic field for one group of charge carriers. Thus, the appearance of negative magnetoresistance at low magnetic field is a result of an increase of the total conductivity; however, the coexisting Drude contribution decreases the total conductivity,
where w, is the cyclotron frequency, T is the elastic scattering time, and a0 is the Drude conductivity at zero magnetic field. In our case for weak magnetic field the Drude contribution to conductivity is small, and the observed negative magnetoresistance corresponds to a positive magneto~onductance, Aq,, > 0, although at stronger magnetic field da,, < 0. The Drude conductivity and a,, exactly compensate, and we observe only negative magnetoresistance. In a one-dimensional lateral superlattice with strongly modulated potential, a positive magnetoresistance has been observed [9,10]. It was analyzed from the classical standpoint and explained
446
G.M.
Gusel, et al. /Surface
by the formation of open electron orbits, which drift in the crossed external magnetic field and in periodic superlattice electric field. For the periodic antidot lattice, as mentioned above, at low magnetic field commensurability oscillations have been found. With increase of B, when 2 R, < d, negative magnetoresistance has been observed [3,4]. Increase of electrons scattered by antidots and localization around antidots are responsible for this magnetoresistance. In this case the magnetic field is strong, and pXl = pXY, where prY is the Hall resistivity. Thus, because of pXX= pxY, a decrease in resistivity signifies a decrease in the conductivity, i.e. localization, in contrast to the weak magnetic field case, where pXX= l/a,,. There are two models which explain commensurability oscillations in an antidot lattice. The first model considers electron orbits which are not scattered by antidots [4]. A portion of these as a function of magnetic orbits, fP, oscillates field, but the amplitude of these oscillations are too small to explain conductivity oscillations. In this case Fleischmann et al. [ll] calculated the contribution of the chaotic trajectories which lie in the phase volume near the regular pinned orbits. The resultant conductivity is 1111
CJ= (1 -&I/
dt e-‘/T,
where ( ci( t )u,(O)) is the velocity correlation function. Fleischmann et al. have found, that the main contribution to the conductivity is determined by chaotic orbits, which are localized around antidot groups. The other model also considered the electron motion in the antidot lattice from the standpoint of dynamical chaos theory [12]. But in this model stable trajectories which roll along the lattice row have been found. These trajectories, in contrast to localized orbits considered in the first model, increase the total conductivity. However, as in the first model, the contribution of these trajectories to conductivity is not sufficient to explain the oscillation amplitudes. Therefore, nearby chaotic trajectories with the same dynamics should be included in the electron diffusion coefficient calculation. The runaway trajectories contribution to conductance
Science 305 (1994) 443-447
can be found point contacts 2e’ u=pk,p-7. h
from the theory [ 121,
d-u rr
for resistance
of
L 2S dn--
where S is the fraction of phase space which is filled by runaway trajectories, L is the length of the sample, or L = I, and k, is the electron wave vector. The diffusion coefficient for quasi-runaway trajectories was calculated through numerical simulation. Thus, one model analyzes “islands of pinned trajectories” and a nearby sea of chaotic orbits with the same dynamics Ill], and the other “islands of runaway trajectories” one considers surrounded by a sea of electron orbits with anomalous diffusion [12]. Agreement of theory with experiment without adjustable parameters has been obtained in Ref. [ill; therefore, the contribution of chaotic orbits to conductivity is dominant. The existence of commensurability oscillations in the disordered antidot lattice (Fig. 11, as indicated above, gives evidence of the stability of pinned electron trajectories when short-range order is violated. The reason can be connected to the fact that the chaotic orbits are localized within several superlattice periods. In mesoscopic samples we observe no commensurability oscillations (Fig. 3). Therefore, more than 4 periods are necessary for these oscillations. In a disordered antidot array some long-range order is preserved, therefore these orbits are not destroyed in contrast to the regular trajectories which are determined by short-range order. It should be noted, that the runaway trajectories considered in model [121 have a diffusion coefficient divergence, since the relation (x(t)> >_f holds for these electrons. Similar trajectories with anomalous diffusion have been predicted in a superlattice with weak periodic potential [2]. These orbits appear in weak magnetic field. For the antidot array the same situation occurs if d/a > 1; however, at finite magnetic field the portion of runaway trajectories should oscillate as a function of B. For the samples investigated in our work, the ratio d/a = 3.5, and trajectories with anomalous diffusion at low magnetic field are not prevented by shadowing antidots in the next rows. The situation is differ-
G.M. Guseu et al. /Surface
em for a disordered antidot lattice. In this case runaway trajectories could appear not only for commensurability conditions, as in the periodic system, but at any magnetic field. Two reasons are responsible for the existence of anomalous diffusion trajectories, correlation of the electron cyclotron radius, and distance between antidots in any magnetic field because of disorder and accidental absence of shadowing of these orbits. Thus, the negative magnetoresistance observed in the disordered antidot lattice could be due to the existence of trajectories with anomalous diffusion in the magnetic field. Also the probability of appearance of these orbits increases with disorder increase, and growth of the magnetoresistance value is expected, as observed in experiments (Fig. 2). The runaway trajectory is analogous to the edge states in classically strong magnetic field [13]. The conductance of these trajectories can be written in the form a=e2N/h,
where N = k,R,/2; however, conductivity decreases thus a=B-‘; with B. Recently, linear magnetoresistance in a quantum point contact has been observed, which was negative due to a difference in the value of the edge current inside and Hall current outside of the contact [14]. Our situation is similar, however, the difference between Hall and diagonal components of anomalous currents is not clear. In summary, the samples with disordered antidot lattice, in contrast to the periodic array, have revealed linear negative magnetoresistance which is temperature independent. Against the magnetoresistance background commensurability oscillations, which are due to chaotic orbit localization, have been found. We believe that the trajectories which roll along rows of the lattice and pinned orbits remain stable when the short-range lattice order is violated. The small ratio d/a is responsible for this stability. Further antidot lat-
Science 305 (1994) 443-447
447
tice experimental work with larger d/a ratios and theoretical analysis of the transition from order to disorder are required. This work is supported by CNPq and FAPESP (Brazil).
1. References 111P.
Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57 (1985) 287. 121J. Wagenhuber, T. Geisel, P. Niebauer and G. Obermair, Phys. Rev. B 45 (1992) 4372. 131 K. Ensslin and P. Petroff, Phys. Rev. B 41 (1990) 12307; G.M. Gusev, V.T. Dolgopolov, Z.D. Kvon, A.A. Shashkin, V.M. Kudryashov, L.V. Litvin and Yu.V. Nastaushev, JETP Lett. 54 (1991) 364; A. Lorke, J.P. Kotthaus and K. Ploog, Phys. Rev. B 44 (1991) 3447. [41 D. Weiss, M.L. Roukes, A. Menschig, P. Grambow, K. von Klitzing and G. Weimann, Phys. Rev. Lett. 66 (1991) 2790. J. Takahara, Y. Takagaki, K. Gamo, S. 151T. Yamashiro, Namba, S. Takaoka and K. Murase, Solid State Commun. 79 (1991) 885. 161 G.M. Gusev, P. Basmaji, D.I. Lubyshev, L.V. Litvin, Yu.V. Nastaushev and V.V. Preobragenskii, Phys. Rev. B 47 (1993) 9928. [71 G.M. Gusev, Z.D. Kvon, L.V. Litvin, Yu.V. Nastaushev, A.K. Kalagin and A.I. Toropov, JETP Lett. 56 (1992) 173. 181 G.M. Gusev, Z.D. Kvon, L.V. Litvin, Yu.V. Nastaushev, A.K. Kalagin and AI. Toropov, Abstracts of the Sixth International Conference on Superlattice and Microstructures, Sian, People’s Republic of China (1992) 198. [91 P.H. Beton, E.S. Alves, P.C. Main, L. Eaves, M.W. Dellow, M. Henini, O.H. Hughes, S.P. Beamont and C.D.W. Wilkinson, Phys. Rev. B 42 (1990) 9229. [lOI C.W.J. Beenakker, Phys. Rev. Lett. 62 (1989) 2020. 1111 R. Fleischmann, T. Geisel and R. Ketzmerick, Phys. Rev. Lett. 68 (1992) 1367. WI E.M. Baskin, G.M. Gusev, Z.D. Kvon, A.G. Pogosov and M.V. Entin, JETP Lett. 55 (1992) 678. 1131M. Buttiker, Phys. Rev. B 38 (1988) 9375. [14] H. van Houten, C.W.J. Beenakker, P.H. Loosdrecht, T.J. Thornton, H. Ahmed, M. Pepper, C.T. Foxon and J.J. Harris, Phys. Rev. B 37 (1988) 8534.
:.‘:‘.‘.‘.“...‘.:.‘.:
.,.,,,,/,..(,, Jo:,
. . . .. .. . . ...\...
” -:.:.. .\ . . . .. . . . . . .. .. . . . .?~:‘:w.F: . .........y... :.:.:.>.:.:.:.:.:~:.:.~,,,..:,:,..,,,
““““““.‘..(“‘.‘.......: ,,\,,.,,,,,, .::::‘:::j::::;:>~i;~:::::..::::Y.. .... . .,......,..,....,,,.,. ,.. .,...,~,,,:_:.::: :::ii::::i
;;i:$$::
.\.
“surface science‘
.:i::: . ...::::>::>.::.+ . _+.,::::;,:;i .,,.,.. .......x..:.:.: >..’ ....:.:‘:::::::::.:::::xi’i::~~~~~~:~.~ i:i.::::>::~i::.:::.:.:::~:.::~,;.>:.: ..,.:, ~,.,.,,:;, ,,,,,,:,,,,
ELSEVIER
Surface Science 305 (1994) 443-447
Negative magnetoresistance and anomalous diffusion of two-dimensional electrons in a disordered array of antidots G.M. Gusev *71,a,P. Basmaji a, Z.D. Kvon b, L.V. Litvin b, Yu.V. Nastaushev b, A.I. Toropov b aInstitute de Fisica e Q&mica de &io Carlos, Universidade de Srio Paula, SGo Paula, Brazil b Institute of Semiconductor Physics, Russian Academy of Sciences, Siberian Branch, Novosibirsk, Russian Federation (Received 8 April 1993; accepted for publication 29 June 1993)
Abstract Negative linear magnetoresistance of two-dimensional (2D) electrons in a disordered array of antidots has been found. The existence in a magnetic field of trajectories which roll along the array of antidots was suggested. These trajectories have a mean free path larger than the average value for electrons with ordinary diffusion.
Magnetic field decreases electron mean free path, therefore the magnetoresistance of semiconductors and metals is positive. However, at low temperature and weak magnetic field negative magnetoresistance has been observed [l]. This phenomenon has been explained from a quantum standpoint. The crossed electron trajectories increase the backscattering probability due to the interference at the crossing point. Magnetic field suppresses this interference because of the Aharonov-Bohm effect, the backscattering probability decreases, and as a result negative magnetoresistance appears [l]. From a classical standpoint the negative magnetoresistance is a result of increase of electron elastic length or appear-
* Corresponding author. address: Institute of Semiconductor Physics, Russian Academy of Sciences, Siberian Branch, Novosibirsk, Russian, Federation.
’ Permanent
0039-6028/94/$07.00
0 1994 Elsevier Science
SSDI 0039-6028(93)E0590-Q
ante of the carriers with mean free path larger than the average value. In Ref. [2] Wagenhuber et al. have found theoretically that this probability exists in a 2D electron gas in a lateral two-dimensional lattice. At some value of lattice potential in a magnetic field the anomalous diffusion of electrons arises. This is characterized by a linear increase of the mean square displacement of carriers, (x2)“* = fl+rr, with (Y= 1. It is responsible for violation of the exponential distribution of a portion of particles with mean free path, and thus, electrons with anomalously large elastic length appear. However, the influence of this effect on the electron transport, in particular magnetoresistance, was not considered. One type of two-dimensional lateral superlattice with strong repulsive potential is an array of antidots [4], which is produced in a 2D electron gas by holes with submicron diameter fabricated by etching or irradiation by ions. This system has attracted attention also because it allows various
B.V. All rights reserved
444
G.M. Gusr~~ et cd. /Surfuce
arrangements of antidots to be simulated, square [3,4], hexagonal [.5], two-dimensional quasicrystals 161, and disordered [7,81. In this work the magnetoresistance of 2D electrons in a periodic and disordered lattice of antidots has been studied, and a negative magnetoresistance which grows with degree of disorder has been found. The test samples were Hall bridges based on GaAs/AlGaAs heterostructures with a 2D electron gas. The parameters of the initial heterostructures were electron density, n, = 5 X IO” cme2, and mobility, p = (2 - 5) x 10’ cm*/V . s. A lattice of antidots, produced by electron-beam lithography and reactive ion etching, covered a part of the sample between the potential probes. The antidot diameter was 0.15-0.2 Frn. The effective diameter, a, which consists of a lithographic diameter and a depletion length, was 0.2-0.3 pm. Samples in which the array of antidots is disordered with various degrees of disorder were made for the experiments (Fig. 1). The disordering of the lattice was accomplished in the following way. A random number generator determined the shift in the position of the antidots in the direction of the neighboring antidots. The deviations of antidots from their periodic arrangement in the lattice with period d = 0.7 pm at the peak were A = 0.0, 0.1, 0.25, and 0.35 pm. We also measured mesoscopic samples with size d = 0.6 pm, and 4X4 pm2, average periodicity A = 0.3 pm. Thus, the short-range order of the system was violated, but its long-range order was preserved. The magnetoresistance was measured by the four-terminal method at frequencies of 70-700 Hz in a magnetic field, B, up to 0.4 T, and at temperatures of 1.7-4.2 K. Electron scattering by the antidot lattice was dominant in our samples, from this the mean free path, 1, at B = 0 was found to be 0.4, 017, 0.21, and 0.4 pm for samples with different degrees of disorder. We see that the behavior of I is nonmonotonic with disorder. With an increase of A, the length 1 initially decreases, and increases again. Similar nonmonotonic behavior has been observed recently in samples with a quasiperiodic (Penrose tiling) lattice of antidots [6], which is an intermediate type of
Science 30.5 (1994) 443-447
......
.I
....... ‘.
.
.....
.’
.......
!. ...... I. ......
k_L~2-?2L I . . . . .*
. . .* =1 0.
..a. l
.*..... . . . . l . . . .*..
.d
. . . . . .I l * -I*
-i 0
0.2
04 B,T
Fig. 1. Magnetoresistance dependence on B for a sample with different degrees of antidot disorder and for a periodic lattice: d = 0.7 Frn: (1) A = 0.0; (2) A = 0.1: (3) A = 0.25; (4) A = 0.35 wrn; T = 4.2 K.
array between periodic and disordered. In this case the length I initially decreases with increase of the basic triangle size in the Penrose cell, and then increases again. This behavior has not been explained, and further theoretical analysis of the scattering in this system is required. Fig. 1 shows the magnetoresistance for samples with various degrees of disorder as a function of magnetic field. We see commensurability oscillations for all samples when the cyclotron to the lattice period. diameter, R,, is comparable The oscillation amplitude decreases with increase of A, and the position of the last peak shifts to lower magnetic field [8]. The preservation of long-range order in our sample is responsible for oscillations in the disordered antidot lattice. Fig. 1 shows that for the periodic lattice, the second peak maximum is higher than the peak maximum which lies at lower magnetic field. However, for the array with disorder degree A = 0.1 pm their amplitudes are equal, i.e., negative magnetoresis-
G.hf
0
Guseu
et af.
/&itface
01
B,T Fig. 2. Magnetoresistan~~ dependence on B for low magnetic field. (2) d = 0.1; (3) A = 0.25; (4) A = 0.35 pm; T = 4.2 K.
tance (nMR) appears, but it is not visible due to the oscillations. With increase of disorder degree nMR at low B is observed. Fig. 2 shows the low field part of the nMR for samples with a disordered array of antidots. We see that the magnetoresistance is linear for low magnetic field and
Fig. 3. Magnetoresistance of the mesoscopic samples with disordered lattice of antidots measured after different intervals of time: (1) 0 min, (2) 5 min, (3) 40 min; d = 0.6 pm, d = 0.3 pm, T = 1.7 K.
Science
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(19%)
443-447
445
increases with increasing disorder. We did not find this magnetoresistan~e to be temperature dependent at temperatures 1.7-4.2 K, which proves that this nMR has a classical origin. Fig. 3 shows the magnetoresistance of a mesoscopic sample with period d = 0.6 pm measured after different intervals of time. We also see linear negative magnetoresistance which does not depend on the temperature. Against the background of this magnetoresistance, there are fluctuations. The pattern of these fluctuations as a function of B is changed in time. We believe, that the fluctuations of magnetoresistance are due to the interference of electron trajectories scattered by the antidot lattice. However, more experiments are necessary in this system. We also believe that linear negative magnetoresistance has the same nature as for macroscopic samples. We should note that we measured all samples before the patterning of antidots and observed no negative magnetoresistance in this region of magnetic field. For Hail bridge samples pXX= ~~,/(a:~ + a&), where pXXis the resistivity, and o;rX, gXYare the diagonal and Hall parts of the conductivity. pXXis constant in magnetic field for one group of charge carriers. Thus, the appearance of negative magnetoresistance at low magnetic field is a result of an increase of the total conductivity; however, the coexisting Drude contribution decreases the total conductivity,
where w, is the cyclotron frequency, T is the elastic scattering time, and a0 is the Drude conductivity at zero magnetic field. In our case for weak magnetic field the Drude contribution to conductivity is small, and the observed negative magnetoresistance corresponds to a positive magneto~onductance, Aq,, > 0, although at stronger magnetic field da,, < 0. The Drude conductivity and a,, exactly compensate, and we observe only negative magnetoresistance. In a one-dimensional lateral superlattice with strongly modulated potential, a positive magnetoresistance has been observed [9,10]. It was analyzed from the classical standpoint and explained
446
G.M.
Gusel, et al. /Surface
by the formation of open electron orbits, which drift in the crossed external magnetic field and in periodic superlattice electric field. For the periodic antidot lattice, as mentioned above, at low magnetic field commensurability oscillations have been found. With increase of B, when 2 R, < d, negative magnetoresistance has been observed [3,4]. Increase of electrons scattered by antidots and localization around antidots are responsible for this magnetoresistance. In this case the magnetic field is strong, and pXl = pXY, where prY is the Hall resistivity. Thus, because of pXX= pxY, a decrease in resistivity signifies a decrease in the conductivity, i.e. localization, in contrast to the weak magnetic field case, where pXX= l/a,,. There are two models which explain commensurability oscillations in an antidot lattice. The first model considers electron orbits which are not scattered by antidots [4]. A portion of these as a function of magnetic orbits, fP, oscillates field, but the amplitude of these oscillations are too small to explain conductivity oscillations. In this case Fleischmann et al. [ll] calculated the contribution of the chaotic trajectories which lie in the phase volume near the regular pinned orbits. The resultant conductivity is 1111
CJ= (1 -&I/
dt e-‘/T
where ( ci( t )u,(O)) is the velocity correlation function. Fleischmann et al. have found, that the main contribution to the conductivity is determined by chaotic orbits, which are localized around antidot groups. The other model also considered the electron motion in the antidot lattice from the standpoint of dynamical chaos theory [12]. But in this model stable trajectories which roll along the lattice row have been found. These trajectories, in contrast to localized orbits considered in the first model, increase the total conductivity. However, as in the first model, the contribution of these trajectories to conductivity is not sufficient to explain the oscillation amplitudes. Therefore, nearby chaotic trajectories with the same dynamics should be included in the electron diffusion coefficient calculation. The runaway trajectories contribution to conductance
Science 305 (1994) 443-447
can be found point contacts 2e’ u=pk,p-7. h
from the theory [ 121,
d-u rr
for resistance
of
L 2S dn--
where S is the fraction of phase space which is filled by runaway trajectories, L is the length of the sample, or L = I, and k, is the electron wave vector. The diffusion coefficient for quasi-runaway trajectories was calculated through numerical simulation. Thus, one model analyzes “islands of pinned trajectories” and a nearby sea of chaotic orbits with the same dynamics Ill], and the other “islands of runaway trajectories” one considers surrounded by a sea of electron orbits with anomalous diffusion [12]. Agreement of theory with experiment without adjustable parameters has been obtained in Ref. [ill; therefore, the contribution of chaotic orbits to conductivity is dominant. The existence of commensurability oscillations in the disordered antidot lattice (Fig. 11, as indicated above, gives evidence of the stability of pinned electron trajectories when short-range order is violated. The reason can be connected to the fact that the chaotic orbits are localized within several superlattice periods. In mesoscopic samples we observe no commensurability oscillations (Fig. 3). Therefore, more than 4 periods are necessary for these oscillations. In a disordered antidot array some long-range order is preserved, therefore these orbits are not destroyed in contrast to the regular trajectories which are determined by short-range order. It should be noted, that the runaway trajectories considered in model [121 have a diffusion coefficient divergence, since the relation (x(t)> >_f holds for these electrons. Similar trajectories with anomalous diffusion have been predicted in a superlattice with weak periodic potential [2]. These orbits appear in weak magnetic field. For the antidot array the same situation occurs if d/a > 1; however, at finite magnetic field the portion of runaway trajectories should oscillate as a function of B. For the samples investigated in our work, the ratio d/a = 3.5, and trajectories with anomalous diffusion at low magnetic field are not prevented by shadowing antidots in the next rows. The situation is differ-
G.M. Guseu et al. /Surface
em for a disordered antidot lattice. In this case runaway trajectories could appear not only for commensurability conditions, as in the periodic system, but at any magnetic field. Two reasons are responsible for the existence of anomalous diffusion trajectories, correlation of the electron cyclotron radius, and distance between antidots in any magnetic field because of disorder and accidental absence of shadowing of these orbits. Thus, the negative magnetoresistance observed in the disordered antidot lattice could be due to the existence of trajectories with anomalous diffusion in the magnetic field. Also the probability of appearance of these orbits increases with disorder increase, and growth of the magnetoresistance value is expected, as observed in experiments (Fig. 2). The runaway trajectory is analogous to the edge states in classically strong magnetic field [13]. The conductance of these trajectories can be written in the form a=e2N/h,
where N = k,R,/2; however, conductivity decreases thus a=B-‘; with B. Recently, linear magnetoresistance in a quantum point contact has been observed, which was negative due to a difference in the value of the edge current inside and Hall current outside of the contact [14]. Our situation is similar, however, the difference between Hall and diagonal components of anomalous currents is not clear. In summary, the samples with disordered antidot lattice, in contrast to the periodic array, have revealed linear negative magnetoresistance which is temperature independent. Against the magnetoresistance background commensurability oscillations, which are due to chaotic orbit localization, have been found. We believe that the trajectories which roll along rows of the lattice and pinned orbits remain stable when the short-range lattice order is violated. The small ratio d/a is responsible for this stability. Further antidot lat-
Science 305 (1994) 443-447
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tice experimental work with larger d/a ratios and theoretical analysis of the transition from order to disorder are required. This work is supported by CNPq and FAPESP (Brazil).
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