VOLUME 48, NUMBER 22 PHYSICAL REVIEW LETTERS 31MAy1982 Two-Dimensional Magnetotransport in the Extreme Quantum Limit D. C. Tsui, (a), (b) H. L. Stormer, () and A. C. Gossard Bell Laboratories, Murray Hill, New Jersey 07974 (Received 5 March 1982) A quantized Hall plateau of pxy hle, accompanied by a minimum in, was observed charge-density-wave state with triangular symmetry is suggested as a possible explana- tion. PACS numbers:72.20.my,71.45.-d,73.40.lq,73.60.w In the presence of an intense perpendicular shows a dip at v=s, which becomes stronger at magnetic field B, system of two-dimensional lower T. For< Pxx follows an approximately (2D)electrons is expected to form a Wigner sol-exponential increase with inverse T. The Hall id12 at low temperatures (T). In the infinite-B resistivity Pxy on the other hand, approaches a limit, an analogy can be drawn to the classical electron gas on the su step of 3h at as rdecreases, but re- which crystallizes int ratio of this Hall plateau. These features of the data re- the electron's average pot to thermal semble those of the quantized Hall resistance and energy r=e2m1//ekT=13 the zero-resistance state expected exclusively for areal density). At fini fects be- integral values of v. We suggest that these strik- harge-density-wave ing results are evidence for a new electronic consistent with the no- sible at conside or a CDW state with Wigner crystallizati etry, is favored at v=s carried out on the S when the unit cell area of the lattice is multi- Sio2 interface. Kawa he area of a magnetic flux quantum. Tsui6 made high-B ments and observ consisting of 1-um undoped GaAs, 600-A Si-doped , dependences whic Si-doped GaAs single dependent-electron ly grown on insulating Subsequently, Kenn using molecular-beam-epitaxy cyclotron resonanc on gas, resulting from line narrowing, ide AlGaAs, 2is average electro aAs side of the GaAs- ron diameter. oles were cut into the dependence es and Ohmic contacts to the level filling fact e made with In at 400C. Low- pinned-CDW mo s were used to deter- account of the cye amples have from 1.1 10 in the range of n a rom8000to100000 performed, local The high-B measurements were per- SiO2 interface is formed at the Francis Bitter National Magnet the absence b en as peen possible t Figure shows p xy and xx of one specimen as f rom t te p e ra t ures. In this Letter he scale at the top of the figure shows the Lan- results on the tra dau level filling factor v, which gives the number electrons, in GaA of occupied levels. At integral values of v, the the extreme quantum limit (<1), when the low data sho haracteristic feature €9Un-€- est-energy, spin-polarized Landau level is par tized Hall plateaus and the vanishing of pxx, 13-15 tially filled. We found that at temperatures T<5 when the Fermi energy is pinned in the gap K, the diagonal part xx of the resistivity tensor between two adjacent levels. Removal of spin de- 1982 The American Physical Society 1559
VOLUME 48. NUMBER 22 PHYSICAL REVIEW LETTERS 31MAY1982 FILLING FACTOR v 231/2 L 1mm aOouNd 4·1.23x10"cm-2 1.38 10 cm- 048K 王10(ka) OOK 2 .65K o(k/口 415K MAGNETIC FIELD B (kG) TEMPERATURE T《K FIG. 2. T dependence of(a) the slope of Pxy at v=3, Q FIG. 1. pry and pxx vS B, taken from a GaAs-Alo 37 normalized to the slope at 30 K, (b)P at v=a,and V sec, using /=I uA. The Landau level filling factor ig (e)ex, at v=( Gao 7As sample with n=1.23x 10/cm2, H=90000 cm2/ defined by v=nh/eB values of v at higher T. Moreover, for v50 kG in Fig. 1), the sys- slope at high T( 30 K), for three samples with tem is completely spin polarized. For T>4.2K, slightly different n. Figure 2(b)shows the ac Prv=B/ne, and px shows also nearly linear de companying P xx minimum(at v=3), and Fig. 2(c) pendence on B, as expected from the free-elec shows pxr at v =0. 24 to illustrate its T dependence tron theory of Ando and Uemura ,L6 At lower T, for v<s, away from the Hall plateau. Several Pxy deviates from pxy=B/ne at v 3. This devia- points should be noted. First, the slope of pat tion becomes more pronounced as t decreases v =3 approaches zero at T-0. 4 K, indicative of a and approaches a plateau of Pxy=3h/e2, within an true quantized Hall plateau. Second, replotting accuracy better than 1% at 0. 42 K. The appear- the data in Fig. 2(a)on logarithmic slope versus ance of this plateau is accompanied by a minimum inverse T scale shows a linear portion for data in pxx, as apparent in the lower panel of Fig. 1. taken at Te1.1 K. This fact allows us to extrapo- The development of these features is similar to late the normalized slope to 1 at To=5 K, which that of the quantized Hall resistance and the con- we identify as the temperature for the onset of comitant vanishing of Pxx, observed at integral his phenomenon. Third, Pxx at v=3is w6 ks/u 1560
VOLUME 48. NUMBER 22 PHYSICAL REVIEW LETTERS 31MAY1982 at our lowest T of M0. 42 K, and much lower T is pxv=B/ne= 3h/e. As discussed by Baraff and needed to determine if pxx at v= 3 indeed vanishes Tsui, 23 observation of the quantized Hall plateau with vanishing T. Finally, if the data for pxx at may be attributed to the presence of donor states 0. 24 Fig. 2(c)] are replotted on logarithmic inside Al Ga,-x As. The thermal activation of P xx Prr versus inverse T scale, an exponential depen- at v=3 may result from activation of defects in dence on 1/T is seen for T>0.6K, with a preex- the condensate which give rise to dissipation. ponential factor of 13k&/O. This result may be Finally, our data also show weaker but similar nterpreted as due to thermally activated trans structures in Pxx near v=3 and near v=2,ac port with an activation energy of 0.94 K. The pre- companied by slight changes in the slope of p exponential factor is considerably lower than the These structures, though discernible in Fig. 1 maximum metallic sheet resistance of 40 k&/ are well resolved in the data taken from the sam of Landau levels, but comparable to that signify- sec at 1.2 K. In our picture the structure at ing a metal-insulator transition for 2D systems in =2 may be identified as due to the formation of a the absence of B. Moreover, the data for v <3 Wigner solid or CDw of holes, expected from suggest a state of localization, in which the elec- electron-hole symmetry. At v=2, the(0, t)Lar tron mobility is thermally activated, as seen in dau level is completely filled, but the(o, +)level Pxx. The electron density, as seen in the slope of is half filled. Consequently 3 of all the electrons Pxy VS B, remains essentially independent of T. occupy the low-lying(0, t)level, and only the re- The existence of quantized Hall resistance ac- maining 3 with spin may participate in the for companied by the vanishing of Pxx at integral val- mation of a collective ground state. Our data ap ues of v is now well known. Their observation is pear to suggest that this condition is also favored attributed to the existence of an energy gap be for a wigner solid or Cdw ground state. tween the extended states in two adjacent Landau In summary, we observed striking struct levels and the presence of localized states, which in the magnetotransport coefficients of high-u pin E F in the gap, to keep all the extended states 2D electrons in GaAs-Al Ga,.As heterojunctions in finite numbers of Landau levels completely oc- at v=3, and similar, but much weaker, struc cupid for finite regions of B or n. Laughlins tures at v=3 and 2. Their development as a argument based on gauge invariance demon function of T is reminiscent of the quantized p strated that the quantized Hall resistance, given and the concomitant vanishing of cted by Pxy=h/ie2(i=1, 2, ..) results from complete only for integral values of v. We suggest as a occupation of all the extended states in the Lan- possible explanation the formation of a new elec dau level, regardless of the presence of localiza- tronic state, such as a wigner solid or CDw tion, Our observation of a quantized Hall resis- state with a triangular sy mmetry tance of 3h/ e2 at v=3 is a case where Laughlins We thank P. M. Tedrow for the he refrigerator argument breaks down. If we attribute it to the R. B. Laughlin, P. A. Lee, V. Narayanamurti presence of a gap at e when 3 of the lowest Lan- and P. M. Platzman for discussions; and K.Bald dau level is occupied, his argument will lead to win, G. Kaminsky, and W. wiegmann for techni- quasiparticles with fractional electronic charge cal assistance. This work was supported in part of 3, as has been suggested for 3-filled quasi by the National Science Foundation. one-dimensional systems. 2 At the present, there is no satisfactory explana tion for all of phenomenon always occurs at v =3 and that it is a visiting scientist at the Francis Bitter National most striking in samples with the highest electron Magnet Laboratory, Cambridge, Mass. 02139 mobility suggest the formation of a new spin-po- (present address: Department of Electrical En- larized electronic state such as wigner solid gineering and Computer Science, Princeton University, CDw, with a triangular sy mmetry which is Princeton, N.. 08544 favored at v=3. In this picture, the observed E. P. wigne ev.46,1002(1934) features of Pxx and p xy may be attributed to trans 2YE.Lozovik 。日 Yudson, Pis ma Zh. Eksp. 2,11(1975)] SC C. Grimes and G. Adams, Phys. Rev. Lett. ransport is free of dissipation and px is expected 795 (1979) to vanish. Since the number of electrons in this H, Fukuyama, P. M. Platzman, and P. w. Anderson ground state is n=eB/3h, the Hall resistivity Phys.Rev.B19,5211(1979)
VOLUME 48. NUMBER 22 PHYSICAL REVIEW LETTERS 31MAY1982 S. Kawaji and J. Wakabayashi, Solid State Cor 550(1981) 22,87(1977 D. C. Tsui, Solid State Commun. 21, 675(1977) 15D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys Rev.B25,1405(1982) T. Ando and Y Uemura, J. Phys. Soc. Jpn 36,959 6D. C. Tsui, H. L Stormer, A.C. Gossard, and (1974) w. Wiegmann, Phys. Rev. 21, 1589(1980); Th. engle ST. A Kennedy, R.J. Wagner, B.D. McCombe, and D. C. Tsui, and A. C. Gossard, Surface Sci. 113, 295 D. C. Tsui, Solid State Commun. 22, 459(1977) ' B. A. Wilson, S.J. Allen, and D. C. Tsui, Phys. Rev 1M. A. Paalanen, D. C. Tsui, and A C. Gossard 44,479(1980) Phys.Rev,B25,5566(19 Fukuyama and P A, Lee, Phys. Rev. B 18, 6245 (1978) 1H. Aoki and H. Kamimura, Solid State Commun. 21 A, Y Cho and J. R. Arthur, Prog. Solid State Cher D.J. Bishop, D. C. Tsui, and R. C Dynes, Phys Rev. Lett. 44, 1153 H, L. Stormer, A. Pinczuk, A C. Gossard, and R. B. Laughlin, Rev,B23,5632(1981) W. Wiegm Appl. Phys. Lett. 38, 691(1981), T.J. 2w. P Su and J. R Schrieffer, Phys. Rev. Lett. 46, Morkoc, and A. Y. Cho, J. Appl. Phy 738(1981) 。-130091 2D. Yoshioka and H. Fukuyama, J. Phys. Soc.Jpn K. von Klitzing, G. Dorda, and M. Pepper, Phys 47,394(1979),and50,1560(1981) Rev.Lett.45,494(980) 2G. A Baraff and D. C. Tsui, Phys. Rev. B 24, 2274 AD. C. Tsui and A. C. Gossard, Appl. Phys. Lett.37 (1981) Test of High-Pr Scaling from Cosmic-Ray Interactions up to 400 TeV W. E. Hazen and A, Z. Hendel Randall Laboratory of Physics, University of Michigan, Ann Arbor Michigan 48109 A. G. Ash, J. M. Foster. A. L. Hodson. and M.R. porter Department of Physics, University of Leeds, Leeds, United Kingdom M. Bull Department of Physics, University of Nottingham, Nottingham, United Kingdom (Received 19 April 1982) he rate of subcores in cosmic-ray air showers has been measured near sea level wit a close-packed 35-m" array of spark chambers at Leeds, Auxiliary experiments demon strate that our transition effects are negligible. The rate of subcores versus shower size is translated into the rate of high-pr events versus energy of the interacting hadron. Com- parison with calculations of Halzen favors Pr over Pr scaling. The highest energy bin is centered on 350 Tev PAcS numbers: 94. 40.Rc Cosmic rays provide a hadron beam with inten In view of the conflict in results among previous sity sufficient for exploratory study of interac experiments, we first review the case in favor tions up to energies of a few 100 Tev, that is, of our results. We then describe the analysis of ten times higher than the energy from the super our data and compare to Halzen's scaling from proton synchrotron collider at CERN. There accelerator data. The results strongly favor the fore, the results can still be of interest in spite parton model with Pr scaling. of the limit to accuracy inherent in the cosmic In order to interpret any subcore data, a con nection must be made between high-energy sec This is a report of an experiment to test mod- ondary hadrons produced in the atmosphere and els for deep-inelastic interactions by measuring the subcore as observed in a detector The in the energy dependence of the cross section fo formation used to link the two comes either from high-Pr production. The results btained computer simulations, or from analytical solu from the observed rate of subcores in cosmic tions of cascade shower equations, or from a ray air showers combination. These have been carried out for 1562