The longitudinal and Hall magnetoresistance of random and periodic arrays of artificial scatterers, imposed on a high-mobility two-dimensional electron gas, were investigated in the vicinity of Landau level filling factor ν=1/2. In periodic arrays, commensurability effects between the period of the antidot array and the cyclotron radius of composite fermions are observed. In addition, the Hall resistance shows a deviation from the anticipated linear dependence, reminiscent of quenching around zero magnetic field. Both effects are absent for random antidot lattices. The relative amplitude of the geometric resonances for opposite signs of the effective magnetic field and its dependence on illumination illustrate enhanced soft wall effects for composite fermions

Bergmann, R.

Coleridge, P. T.

Scherer, A.

Schweizer, H.

Smet, J. H.

von Klitzing, K.

Wasilewski, Z. W.

Weiss, D.

English

The longitudinal and Hall magnetoresistance  of random and periodic arrays of artificial  scatterers, imposed on a high-mobility two- dimensional electron gas, were investigated  in the vicinity of Landau level filling  factor í=1/2. In periodic arrays,  commensurability effects between the period  of the antidot array and the cyclotron  radius of composite fermions are observed.  In addition, the Hall resistance shows a  deviation from the anticipated linear  dependence, reminiscent of quenching around  zero magnetic field. Both effects are absent  for random antidot lattices. The relative  amplitude of the geometric resonances for  opposite signs of the effective magnetic  field and its dependence on illumination  illustrate enhanced soft wall effects for  composite fermions

Smet, J.

Weiss, Dieter

Klitzing, K.

Coleridge, P.

Wasilewski, Z.

University of Regensburg Publication Server

PHYSICAL REVIEW B 15 AUGUST 1997-IVOLUME 56, NUMBER 7Composite fermions in periodic and random antidot latticesJ. H. Smet, D. Weiss,* and K. von KlitzingMax-Planck Institut fu¨r Festko¨rperforschung, Heisenbergstrabe 1, D-70569 Stuttgart, GermanyP. T. Coleridge and Z. W. WasilewskiInstitute for Microstructural Sciences, National Research Council of Canada, Ottawa, Ontario, Canada K1A OR6R. Bergmann and H. Schweizer4. Physikalisches Institut, Universita¨t Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, GermanyA. SchererDepartment of Physics, California Institute of Technology, Pasadena, California 91125~Received 27 March 1997!The longitudinal and Hall magnetoresistance of random and periodic arrays of artificial scatterers, imposedon a high-mobility two-dimensional electron gas, were investigated in the vicinity of Landau level filling factorn51/2. In periodic arrays, commensurability effects between the period of the antidot array and the cyclotronradius of composite fermions are observed. In addition, the Hall resistance shows a deviation from the antici-pated linear dependence, reminiscent of quenching around zero magnetic field. Both effects are absent forrandom antidot lattices. The relative amplitude of the geometric resonances for opposite signs of the effectivemagnetic field and its dependence on illumination illustrate enhanced soft wall effects for composite fermions.@S0163-1829~97!06231-0#In 1989, Jain,1 guided by the phenomenological similaritybetween the fractional and integer quantum Hall effect~FQHE and IQHE!, introduced the concept of composite fer-mions ~CF’s! as an elegant alternative theory to describe theprominent features of the FQHE at filling fractionsn5p/(2p61), where p is an integer. The electron system issubjected to a statistical gauge field transformation whichmaps it into a new fermionic system of composite particles,each consisting of an electron and two flux quanta and expe-riencing both the external magnetic field B and nucleatedflux. Subsequently, it is hoped for that a mean-field approxi-mation to the transformed Hamiltonian, such that the CF’sare subjected to an effective magnetic field Beff equal toB2Bn51/2 , captures the essential physics of the originalproblem. The series of FQHE liquids around half filling maythen be regarded as a manifestation of the IQHE of CF’s.Furthermore, the prediction of Halperin et al.2 of the exis-tence of a CF Fermi sea with a well-defined Fermi surface atn51/2 showed the way towards quasiclassical experiments,such as commensurability oscillations in 1D or 2D periodi-cally modulated systems and transverse magneticfocusing,3–6 to demonstrate the existence of these quasipar-ticles and their quasiclassical cyclotron motion.The strength of this class of experiments in providing sup-port for the composite-fermion picture stems from the avail-ability of an easily variable scaling parameter, either the pe-riodicity of periodically modulated systems or the center-to-center constriction spacing of focusing geometries. Since ourunderstanding of the asymptotic low-energy behavior of thecomposite-fermion dispersion is on a weak footing, they inaddition benefit from the advantage that the matching condi-tion between cyclotron radius and this scaling parameter isindependent of the effective mass. Here, square periodic and560163-1829/97/56~7!/3598~4!/$10.00fully random antidot superlattices have been investigated. Inaddition to the longitudinal resistivity, previously studied byKang et al.7 for periodic arrays, the Hall resistivity is alsoinvestigated. We find a quenching of the Hall resistivityaround n51/2, ascribed to ‘‘channeling’’ trajectories ofcomposite fermions. These commensurability features disap-pear in random arrays of antidots, demonstrating that theanomalies around n51/2 are solely due to the presence of aperiodic potential. This and previous experiments7–9 makethe case that a mean-field approximation to the Hamiltonianwith a Chern-Simons gauge field transformation embodiesmuch of the physics of the original electron problem. Fur-thermore, the influence of changes in the potential landscapedue to additional illumination are reported.The samples were prepared from two high-mobilityGaAs-Al xGa12xAs heterojunctions with the two-dimensional electron gas ~2DEG! located 160 nm ~sample I!and 200 nm ~sample II!, and the d doped layer 100 nm un-derneath the sample surface. The samples were shaped intoHall bar geometries using standard lithographic techniquesand either alloyed AuGe/Ni/Au or In contact pads. Prior toelectron-beam lithography and after illumination with a redlight-emitting diode ~LED!, the areal density ne and electronmobility m at zero magnetic field and 1.3 K were2.231011 cm22 and 3.43106 cm2/Vs ~sample I!, and1.331011 cm22 and 4.33106 cm2/Vs ~sample II!. Antidotlattices with periodicities a of 400 nm to 700 nm and aperi-odic arrays ~see lower inset of Fig. 2! with equal averageantidot densities were imposed by electron-beam lithographyand a subsequent, approximately 130 nm deep, reactive ionetch ~RIE! with SiCl4. The cell areas of the Wigner-Seitz cellconstruction for each of the antidots of a random lattice forma Poisson distribution with a mean s¯ equal to a2 and3598 © 1997 The American Physical Society56 3599BRIEF REPORTSs/ s¯50.53, where s is the standard deviation. Clustering ofantidots is not prevented and therefore the local antidot den-sity may deviate considerably from the average value. Usingstandard ac lock-in techniques with the external magneticfield applied perpendicular to the 2DEG, four-point magne-toresistance measurements were carried out in a 30 T Bittermagnet with 3He-insert or an 18 T superconducting magnetwith dilution refrigerator.Figure 1 presents the magnetoresistance Rxx at a tempera-ture of 450 mK near zero magnetic field and around halffilling for antidot superlattices with periods a of 400 and 500nm and for a reference section devoid of antidots on one ofFIG. 1. The magnetoresistance Rxx on sample I of antidot su-perlattices with lattice periods a equal to 500 ~a! and 400 nm ~b!near n51/2 ~left axes, thick solid curves! and near zero magneticfield ~right axes, thin solid curves! at 450 mK. Also shown in ~c! isthe magnetoresistance near n51/2 and B50 for an unpatternedreference section of the Hall bar. For ease of comparison, the mag-netic field scale of the CF curves has been divided by A2 andshifted horizontally to make Bn51/2 coincide with B50(Beff5B2Bn51/2). The carrier concentrations, obtained after illu-mination with a red LED, are, respectively, 2.1431011 ~a!,2.3031011 ~b!, and 2.1931011 cm-2. The inset shows a scanningelectron micrograph of a 400 nm antidot array.the Hall bars. Low field maxima in the electron traces ~rightaxes! are anticipated whenever the electron cyclotron radiusis commensurate with the lattice, such that a fraction of thecarriers follow pinned trajectories that encircle an integernumber of antidots and do not contribute to the transportprocess.3 Both lattices exhibit two prominent electron com-mensurability peaks, labeled 1 and 4 in Fig. 1~a!.Accepting the validity of the composite-fermion–Chern-Simons theory, the ground state and low-energy excitationsat filling factor n51/2 can be described by a modified Fermi-liquid theory.2 For a magnetic field B deviating a smallamount Beff from the field Bn51/2 corresponding to half fill-ing, composite fermions follow quasiclassical circular orbitswith a cyclotron radiusRc ,CF5\kF ,CFeBeff, ~1!where the Fermi wave vector kF ,CF is related to the electrondensity by kF ,CF5A4pne and differs by a factor A2 from theelectron Fermi wave vector due to complete spin polarizationof the composite fermion sea at these high magnetic fields.The magnetic field value Bn51/2 at half filling was deter-mined from the well-developed fractional quantum Hallminimum at n52/3. This value was in all cases within lessthan 100 mT of the local minimum in Rxx near n51/2. Thegeometric resonance of CF’s around one antidot is indeedobserved in Fig. 1. Its magnetic field position properly scaleswith the lattice period and lines up, after taking into accountcomplete spin polarization, with the electron resonances. Thedensity inhomogeneity and the much smaller CF mean freepath have been pointed to as the main culprits for the broad-ening of the geometric resonance features.7,8The periodic arrangement of antidots also causes the Hallresistance Rxy , shown in Fig. 2 for a periodic and randomlattice, to deviate from its expected linear dependence aroundhalf filling. Its behavior reminds us of the quenching of theHall resistance around zero magnetic field in antidot lattices3and in small mesoscopic junctions.10 In the former, the Hallfield is impaired due to trajectories channeling over manyperiods along the main axes of the antidot array. Besides aslope change at n51/2, a shift to higher resistance values isnoticeable and has been predicted in Ref. 11. It is noteworthythat the emperical relationship Rxx}BdRxy /dB ~Ref. 12!also reproduces the exact position of the geometric reso-nances.Both this and the previously reported antidot experiment7consistently yield an asymmetry around half filling with areduced strength of the commensurability peak for positiveeffective magnetic field. It has been attributed to soft walleffects and originates from the CF property that the nonuni-form electrostatic potential in the vicinity of soft walls trans-lates into a spatially modulated effective magnetic field.9,11The 2D modulation potential may be influenced by addi-tional illumination of the sample with a red LED. Figure 3demonstrates the evolution of the commensurabilty peaks forelectrons and CF’s with successive illumination steps. Theasymmetry gradually becomes more pronounced. For a trueantidot potential, illumination merely steepens the soft wallsand according to the quasiclassical theory for the motion ofCF’s outlined in Ref. 11 one should recover the symmetry of3600 56BRIEF REPORTSthe geometric resonances around zero effective magneticfield. This apparent contradiction between theory and experi-ment may however be resolved by considering the low fielddata in Fig. 3~a!. Limitations of the RIE-etching process inproducing antidots with large depth to diameter aspect ratiosprevented us from reaching the 200 nm deep 2DEG. As aresult, we hypothesize that upon sufficient illumination oneno longer retains a true antidot potential but, although pre-sumably still relatively strong, 2D modulation not necessar-ily with any fully depleted zones @see inset of Fig. 3~b!#. Theelectron mean free path, extracted from the zero field resis-tance, increases from 0.75 mm to more than 12 mm uponillumination. For an antidot potential, the electron mean freepath ought to remain comparable to the lattice period. Ini-tially the electron commensurabilty features hardly shift onthe magnetic field axis @top four curves in Fig. 3~a!#, butbecome weaker and get more and more submerged inShubnikov–de Haas oscillations. The two lower traces nolonger display the characteristic antidot peaks, but showstructure reminiscent of the commensurability oscillationsobserved in weakly modulated 2DEG’s.5,6 The minima of theFIG. 2. The Hall and longitudinal resistance on sample II of a400 nm periodic antidot array ~thick lines! and a random lattice~thin lines! of equal average antidot density at 30 mK. A shift tohigher values and quenching of the Hall resistivity in the periodiclattice is apparent. The cross marks filling factor n51/2. Insetsshow scanning electron micrographs of the lattices. The resistancemaximum adjacent to the commensurability peaks on the high mag-netic field side appears to be the precursor of the resistance peakbetween filling factors n52/5 and n53/7. This interpretation issupported by the illumination dependent behavior shown in Fig.3~b!.oscillations are close to the expected values given by2Rc5a(l21/4) with l an integer. We project that whentuning the potential landscape from a true antidot potentialtowards weak 2D modulation the asymmetry may increaserather than decrease.In summary, commensurability oscillations of compositefermions in antidot lattices have been observed, confirmingprevious results.7 In addition, the Hall resistance exhibitsFIG. 3. The dependence of the longitudinal magnetoresistancefor the 400 nm periodic antidot array of Fig. 2 on the amount ofillumination with a red LED at low ~a! and high ~b! magnetic fields.Commensurability features encircling 1 and 4 antidots are markedwith solid circles and rectangles, respectively. The inset showsschematically the expected evolution of the potential profile fromthe low magnetic field data in ~a!.56 3601BRIEF REPORTSquenching at n51/2. While they are absent for random ar-rays, both effects can be ascribed to the periodic arrangementof the antidots. The experiments, in particular the asymmet-ric antidot peaks around n51/2 as well as the deviations inthe Hall resistance, emphasize the role of soft wall potentials,as has been previously pointed out in recent theoreticalwork.11We are indebted to R. Fleischmann for enlightening dis-cussions, P. Zawadski for initial sample characterization, andH. Gra¨beldinger, M. Riek, B. Scho¨nherr, and U. Waizmannfor help with sample preparation. This work has been sup-ported by the Bundesministerium fu¨r Bildung und Fors-chung, the European Community project ‘‘Human Capitaland Mobility, Physics in High Magnetic Fields,’’ the Euro-pean Canadian Mesoscopics Initiative, and a NATO Col-laborative Research Grant. The transport measurements athigh magnetic fields were carried out in part at the NationalHigh Magnetic Field Laboratory sponsored by the NationalScience Foundation and the state of Florida.*Also with Institut fu¨r Experimentelle und Angewandte Physik,Universita¨t Regensburg, D-93040 Regensburg, Germany.1 J. K. Jain, Phys. Rev. Lett. 63, 199 ~1989!.2 B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, 7312~1993!.3 D. Weiss, M. L. Roukes, A. Menschig, P. Grambow, K. vonKlitzing, and G. Weimann, Phys. Rev. Lett. 66, 2790 ~1991!.4 H. van Houten, C. W. J. Beenakker, J. G. Williamson, M. E. I.Broekaart, P. H. M. van Loosdrecht, B. J. van Wees, J. E. Mooij,C. T. Foxon, and J. J. Harris, Phys. Rev. B 39, 8556~1989!.5 D. Weiss, K. von Klitzing, K. Ploog, and G. Weimann, Europhys.Lett. 8, 179 ~1989!; R. R. Gerhardts, D. Weiss, and K. vonKlitzing, Phys. Rev. Lett. 62, 1173 ~1989!.6 R. W. Winkler, J. P. Kotthaus, and K. Ploog, Phys. Rev. Lett. 62,1177 ~1989!.7 W. Kang, H. L. Sto¨rmer, L. N. Pfeiffer, K. W. Baldwin, and K.W. West, Phys. Rev. Lett. 71, 3850 ~1993!.8 R. L. Willett, R. R. Ruel, K. W. West, and L. N. Pfeiffer, Phys.Rev. Lett. 71, 3846 ~1993!.9 J. H. Smet, D. Weiss, R. H. Blick, G. Lu¨tjering, K. von Klitzing,R. Fleischmann, R. Ketzmerick, T. Geisel, and G. Weimann,Phys. Rev. Lett. 77, 2272 ~1996!.10 M. L. Roukes, A. Scherer, S. J. Allen, Jr., H. G. Craighead, R. M.Ruthen, E. D. Beebe, and J. P. Harbison, Phys. Rev. Lett. 59,3011 ~1987!.11 R. Fleischmann, T. Geisel, C. Holzknecht, and R. Ketzmerick,Europhys. Lett. 36, 167 ~1996!.12 S. H. Simon and B. I. Halperin, Phys. Rev. Lett. 73, 3278 ~1994!,and references therein.

Composite Fermions in Periodic and Random Anti-Dot Lattices

Caltech Authors - Main

PHYSICAL REVIEW B 15 AUGUST 1997-IVOLUME 56, NUMBER 7Composite fermions in periodic and random antidot lattices
J. H. Smet, D. Weiss,* and K. von Klitzing
Max-Planck Institut fu¨r Festkörperforschung, Heisenbergstrabe 1, D-70569 Stuttgart, Germany
P. T. Coleridge and Z. W. Wasilewski
Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, Ontario, Canada K1A OR6
R. Bergmann and H. Schweizer
4. Physikalisches Institut, Universita¨t Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany
A. Scherer
Department of Physics, California Institute of Technology, Pasadena, California 91125
~Received 27 March 1997!
The longitudinal and Hall magnetoresistance of random and periodic arrays of artificial scatterers, imposed
on a high-mobility two-dimensional electron gas, were investigated in the vicinity of Landau level filling factor
n51/2. In periodic arrays, commensurability effects between the period of the antidot array and the cyclotron
radius of composite fermions are observed. In addition, the Hall resistance shows a deviation from the antici-
pated linear dependence, reminiscent of quenching around zero magnetic field. Both effects are absent for
random antidot lattices. The relative amplitude of the geometric resonances for opposite signs of the effective
magnetic field and its dependence on illumination illustrate enhanced soft wall effects for composite fermions.
@S0163-1829~97!06231-0#ty
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~FQHE and IQHE!, introduced the concept of composite fe
mions~CF’s! as an elegant alternative theory to describe
prominent features of the FQHE at filling fraction
n5p/(2p61), wherep is an integer. The electron system
subjected to a statistical gauge field transformation wh
maps it into a new fermionic system of composite particl
each consisting of an electron and two flux quanta and ex
riencing both the external magnetic fieldB and nucleated
flux. Subsequently, it is hoped for that a mean-field appro
mation to the transformed Hamiltonian, such that the C
are subjected to an effective magnetic fieldBeff equal to
B2Bn51/2, captures the essential physics of the origin
problem. The series of FQHE liquids around half filling m
then be regarded as a manifestation of the IQHE of CF
Furthermore, the prediction of Halperinet al.2 of the exis-
tence of a CF Fermi sea with a well-defined Fermi surfac
n51/2 showed the way towards quasiclassical experime
such as commensurability oscillations in 1D or 2D perio
cally modulated systems and transverse magn
focusing,3–6 to demonstrate the existence of these quasip
ticles and their quasiclassical cyclotron motion.
The strength of this class of experiments in providing s
port for the composite-fermion picture stems from the av
ability of an easily variable scaling parameter, either the
riodicity of periodically modulated systems or the center-
center constriction spacing of focusing geometries. Since
understanding of the asymptotic low-energy behavior of
composite-fermion dispersion is on a weak footing, they
addition benefit from the advantage that the matching co
tion between cyclotron radius and this scaling paramete
independent of the effective mass. Here, square periodic560163-1829/97/56~7!/3598~4!/$10.00t
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fully random antidot superlattices have been investigated
addition to the longitudinal resistivity, previously studied b
Kang et al.7 for periodic arrays, the Hall resistivity is als
investigated. We find a quenching of the Hall resistiv
around n51/2, ascribed to ‘‘channeling’’ trajectories o
composite fermions. These commensurability features dis
pear in random arrays of antidots, demonstrating that
anomalies aroundn51/2 are solely due to the presence of
periodic potential. This and previous experiments7–9 make
the case that a mean-field approximation to the Hamilton
with a Chern-Simons gauge field transformation embod
much of the physics of the original electron problem. Fu
thermore, the influence of changes in the potential landsc
due to additional illumination are reported.
The samples were prepared from two high-mobil
GaAs-AlxGa12xAs heterojunctions with the two
dimensional electron gas~2DEG! located 160 nm~sample I!
and 200 nm~sample II!, and thed doped layer 100 nm un
derneath the sample surface. The samples were shaped
Hall bar geometries using standard lithographic techniq
and either alloyed AuGe/Ni/Au or In contact pads. Prior
electron-beam lithography and after illumination with a r
light-emitting diode~LED!, the areal densityne and electron
mobility m at zero magnetic field and 1.3 K wer
2.231011 cm22 and 3.43106 cm2/Vs ~sample I!, and
1.331011 cm22 and 4.3 106 cm2/Vs ~sample II!. Antidot
lattices with periodicitiesa of 400 nm to 700 nm and aperi
odic arrays~see lower inset of Fig. 2! with equal average
antidot densities were imposed by electron-beam lithogra
and a subsequent, approximately 130 nm deep, reactive
etch~RIE! with SiCl4. The cell areas of the Wigner-Seitz ce
construction for each of the antidots of a random lattice fo
a Poisson distribution with a means̄ equal to a2 and3598 © 1997 The American Physical Society
of
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56 3599BRIEF REPORTSs/ s̄50.53, wheres is the standard deviation. Clustering
antidots is not prevented and therefore the local antidot d
sity may deviate considerably from the average value. Us
standard ac lock-in techniques with the external magn
field applied perpendicular to the 2DEG, four-point magn
toresistance measurements were carried out in a 30 T B
magnet with3He-insert or an 18 T superconducting magn
with dilution refrigerator.
Figure 1 presents the magnetoresistanceRxx at a tempera-
ture of 450 mK near zero magnetic field and around h
filling for antidot superlattices with periodsa of 400 and 500
nm and for a reference section devoid of antidots on one
FIG. 1. The magnetoresistanceRxx on sample I of antidot su-
perlattices with lattice periodsa equal to 500~a! and 400 nm~b!
nearn51/2 ~left axes, thick solid curves! and near zero magneti
field ~right axes, thin solid curves! at 450 mK. Also shown in~c! is
the magnetoresistance nearn51/2 andB50 for an unpatterned
reference section of the Hall bar. For ease of comparison, the m
netic field scale of the CF curves has been divided byA2 and
shifted horizontally to make Bn51/2 coincide with B50
(Beff5B2Bn51/2). The carrier concentrations, obtained after ill
mination with a red LED, are, respectively, 2.1431011 ~a!,
2.3031011 ~b!, and 2.1931011 cm-2. The inset shows a scannin
electron micrograph of a 400 nm antidot array.n-
g
ic
-
er
t
lf
of
the Hall bars. Low field maxima in the electron traces~right
axes! are anticipated whenever the electron cyclotron rad
is commensurate with the lattice, such that a fraction of
carriers follow pinned trajectories that encircle an integ
number of antidots and do not contribute to the transp
process.3 Both lattices exhibit two prominent electron com
mensurability peaks, labeled 1 and 4 in Fig. 1~a!.
Accepting the validity of the composite-fermion–Cher
Simons theory, the ground state and low-energy excitati
at filling factorn51/2 can be described by a modified Ferm
liquid theory.2 For a magnetic fieldB deviating a small
amountBeff from the fieldBn51/2 corresponding to half fill-
ing, composite fermions follow quasiclassical circular orb
with a cyclotron radius
Rc,CF5
\kF,CF
eBeff
, ~1!
where the Fermi wave vectorkF,CF is related to the electron
density bykF,CF5A4pne and differs by a factorA2 from the
electron Fermi wave vector due to complete spin polarizat
of the composite fermion sea at these high magnetic fie
The magnetic field valueBn51/2 at half filling was deter-
mined from the well-developed fractional quantum H
minimum atn52/3. This value was in all cases within les
than 100 mT of the local minimum inRxx nearn51/2. The
geometric resonance of CF’s around one antidot is ind
observed in Fig. 1. Its magnetic field position properly sca
with the lattice period and lines up, after taking into accou
complete spin polarization, with the electron resonances.
density inhomogeneity and the much smaller CF mean
path have been pointed to as the main culprits for the bro
ening of the geometric resonance features.7,8
The periodic arrangement of antidots also causes the
resistanceRxy , shown in Fig. 2 for a periodic and random
lattice, to deviate from its expected linear dependence aro
half filling. Its behavior reminds us of the quenching of th
Hall resistance around zero magnetic field in antidot lattic3
and in small mesoscopic junctions.10 In the former, the Hall
field is impaired due to trajectories channeling over ma
periods along the main axes of the antidot array. Beside
slope change atn51/2, a shift to higher resistance values
noticeable and has been predicted in Ref. 11. It is notewo
that the emperical relationshipRxx}BdRxy /dB ~Ref. 12!
also reproduces the exact position of the geometric re
nances.
Both this and the previously reported antidot experime7
consistently yield an asymmetry around half filling with
reduced strength of the commensurability peak for posit
effective magnetic field. It has been attributed to soft w
effects and originates from the CF property that the nonu
form electrostatic potential in the vicinity of soft walls tran
lates into a spatially modulated effective magnetic field.9,11
The 2D modulation potential may be influenced by ad
tional illumination of the sample with a red LED. Figure
demonstrates the evolution of the commensurabilty peaks
electrons and CF’s with successive illumination steps. T
asymmetry gradually becomes more pronounced. For a
antidot potential, illumination merely steepens the soft wa
and according to the quasiclassical theory for the motion
CF’s outlined in Ref. 11 one should recover the symmetry
g-
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f
d
s
3600 56BRIEF REPORTSthe geometric resonances around zero effective magn
field. This apparent contradiction between theory and exp
ment may however be resolved by considering the low fi
data in Fig. 3~a!. Limitations of the RIE-etching process i
producing antidots with large depth to diameter aspect ra
prevented us from reaching the 200 nm deep 2DEG. A
result, we hypothesize that upon sufficient illumination o
no longer retains a true antidot potential but, although p
sumably still relatively strong, 2D modulation not necess
ily with any fully depleted zones@see inset of Fig. 3~b!#. The
electron mean free path, extracted from the zero field re
tance, increases from 0.75mm to more than 12mm upon
illumination. For an antidot potential, the electron mean f
path ought to remain comparable to the lattice period.
tially the electron commensurabilty features hardly shift
the magnetic field axis@top four curves in Fig. 3~a!#, but
become weaker and get more and more submerged
Shubnikov–de Haas oscillations. The two lower traces
longer display the characteristic antidot peaks, but sh
structure reminiscent of the commensurability oscillatio
observed in weakly modulated 2DEG’s.5,6 The minima of the
FIG. 2. The Hall and longitudinal resistance on sample II o
400 nm periodic antidot array~thick lines! and a random lattice
~thin lines! of equal average antidot density at 30 mK. A shift
higher values and quenching of the Hall resistivity in the perio
lattice is apparent. The cross marks filling factorn51/2. Insets
show scanning electron micrographs of the lattices. The resist
maximum adjacent to the commensurability peaks on the high m
netic field side appears to be the precursor of the resistance
between filling factorsn52/5 and n53/7. This interpretation is
supported by the illumination dependent behavior shown in F
3~b!.tic
ri-
d
s
a
e
-
-
s-
e
i-
in
o
w
s
oscillations are close to the expected values given b
2Rc5a(l21/4) with l an integer. We project that when
tuning the potential landscape from a true antidot potenti
towards weak 2D modulation the asymmetry may increas
rather than decrease.
In summary, commensurability oscillations of composite
fermions in antidot lattices have been observed, confirmin
previous results.7 In addition, the Hall resistance exhibits
c
ce
g-
ak
.
FIG. 3. The dependence of the longitudinal magnetoresistan
for the 400 nm periodic antidot array of Fig. 2 on the amount o
illumination with a red LED at low~a! and high~b! magnetic fields.
Commensurability features encircling 1 and 4 antidots are marke
with solid circles and rectangles, respectively. The inset show
schematically the expected evolution of the potential profile from
the low magnetic field data in~a!.
r-
e
e
n
al
ic
is
an
p-
al
-
l-
s at
al
al
56 3601BRIEF REPORTSquenching atn51/2. While they are absent for random a
rays, both effects can be ascribed to the periodic arrangem
of the antidots. The experiments, in particular the asymm
ric antidot peaks aroundn51/2 as well as the deviations i
the Hall resistance, emphasize the role of soft wall potenti
as has been previously pointed out in recent theoret
work.11
We are indebted to R. Fleischmann for enlightening d
cussions, P. Zawadski for initial sample characterization,nt
t-
s,
al
-
d
H. Gräbeldinger, M. Riek, B. Scho¨nherr, and U. Waizmann
for help with sample preparation. This work has been su
ported by the Bundesministerium fu¨r Bildung und Fors-
chung, the European Community project ‘‘Human Capit
and Mobility, Physics in High Magnetic Fields,’’ the Euro
pean Canadian Mesoscopics Initiative, and a NATO Co
laborative Research Grant. The transport measurement
high magnetic fields were carried out in part at the Nation
High Magnetic Field Laboratory sponsored by the Nation
Science Foundation and the state of Florida.n,
.
k,*Also with Institut für Experimentelle und Angewandte Physik
Universität Regensburg, D-93040 Regensburg, Germany.
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Composite fermions in periodic and random antidot lattices

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Composite fermions in periodic and random antidot lattices

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