/:..:y .:.,:,.,. ,,
.’ ,. .:. ,>; ;: ,: ..
.: :. .
.: : ;: ‘.
;. -..‘:!-. ‘.,. -._.: .. :.. .,: _ .: ::.:.:._::.; ‘,: : ;::.) ,I.,.; __; ..::,:‘:.->_( )::I,‘. ,\ Surface
Science .705 f 1994) X- I7
New collective quantum Hall states in double quantum wells G.S. Boebinger
*, S.Q. Murphy,
K.W. West, Song He
.4 T& T &ll ~ub~)r~~(~~~~.s, Mztnzly Hiil. h!J 07Vld. C.SA (Received
for public~lti~~n 30 August
Abstract Transport studies on double-layer two-dimensional electron systems in &As double quantum wells have revcalcd quantum Hall states at total filling fraction v = I /2 and v = 1. We conclude that these states arise from intcrlayct Coulomb correlations, based on studies of samples with different densities and layer separations and cxperimcnts utilizing tilted magnetic fields.
1. lntroductjon The standard mode1 of the fractional quantum Hall effect (FQHE) describes a myriad of observed odd-denominator FQHE states in terms of a fully spin-polarized two-dimensional electron system (2DES) [l]. Recent work has revealed novel unpolarized spin configurations for the FQHE states at various Landau-level filling fractions [2,3]. If the spin-flip energy is not too large, the 2DES can exploit the spin degree of freedom and form FQHE states not contained within the standard model. In particular, the only known even-denominator FQHE in a conventiona single-layer 2DES (at Landau-level filling fraction v = j/2), is thought to be an example of such an unpolarized state . By anaIogy, a double-layer 2DES possesses an extra degree of freedom, the layer index, that is expected to give rise to new FQHE states not present in single-layer systems [S-71.
Yoshioka et al. 171 systematically outlined the p~~ssibilities for new FQHE states in double-layer systems using the two-component Jastrow wavefunction proposed by Halperin :
vf,)(,,)I,,,, = lJ( i; -Z,y(W, -
w,,)““( -I-q)“. zI,
In this wavefunction, the positions of electrons in one well are given by 2;. positions in the othcl well by wj. The exponents wz and M’ govern the intra-layer correlations, while the cross-term exponent, n, governs the inter-layer correlations. Note that only 171and M’ are constrained to be odd by the requirement for an odd-parity wavcfunction. In the regime where inter-layc1 Coulomb interactions are comparable to the ordinary intra-layer interactions. the P’j,i,, and P,~,,, states have been proposed as candidate ground states for a double-layer FQHE at v = l/2 and v = 1, respectively. Since the magnetic length I, = ((I/&)‘/~ sets the length scale for intra-layrl Coulomb interactions, these new FQHE states are most stable in doubte-layer ‘DES with ci/i,; = 1. where d is the inter-layer spacing.
G.S. Boebinger et al. /Surface
Insofar as intra-layer correlations are conis quite similar to cerned, the PX,3,, wavefunction the original Laughlin l/3-state and the Vi,,,, wavefunction is quite similar to the integral QHE state at v = 1. However, each state contains additional inter-layer correlations which, in effect, keep electrons in adjacent layers from occupying the same position in the 2D plane. Since interlayer tunneling will suppress this anticorrelation, these new FQHE states weaken in a double-layer 2DES if the symmetric-antisymmetric tunneling is not substantially smaller than chargap, ASAS, acteristic Coulomb energies, e*/&l, . In this paper we review the discovery of a new FQHE state at total filling factor v = l/2 in a double-layer 2DES [lo]. We present evidence that both this and the quantized state at v = 1 arise from an interplay of intraand inter-layer Coulomb interactions.
For these experiments a series of modulationdoped GaAs/AlGaAs double quantum well (DQW> structures were grown by molecular beam epitaxy. These samples were designed to have minimal tunneling and yet still to be coupled via Coulomb interactions. The details of the sample growth are given in Ref. [lo]. The electron density was balanced between the two wells and standard low-frequency magnetotransport techniques were employed. We define the Landaulevel filling fraction as v = hN,,,/eB, with N,,, the total carrier density in the structure and B the magnetic field. (Thus, a widely-spaced doublelayer system will exhibit only even-integer QHE states and fractional QHE states with even numerators.) Unless otherwise noted, all data presenteod in this paper come from a sample with two 1800A wide GaAs quantum wells separated by a 31 A wide u!doped pure AlAs barrier layer (giving d = 211 A and ASAS= 0.9 K). Fig. 1 shows the diagonal resistivity pXX and Hall resistivity pXY at T = 150 and 430 mK, respectively. Four quantum Hall states, both integral and fractional, are noted by their total filling fraction v as defined above. As the figure shows,
Science 305 (1994) 8-12
150m K! 200m ’
‘; / \ ‘j K "~'~400m 1
Fig. 1. Diagonal resistivity at T = 150 mK and Hall resistivity at T = 430 mK. Note the v = l/2 fractional quantum Hall state. Temperature dependence of pxx near v = l/2 is also shown. The pxx trace for B < 7 T has been amplified ten-fold. Inset: Schematic conduction-band diagram of the double quantum well.
a new FQHE state, with the requisite deep minimum in pXX and flat Hall plateau, is observed at filling fraction v = l/2. This state has also been observed in single quantum wells which are sufficiently wide that the electrons concentrate near the two interfaces [ill. Three additional samples, listed in Ref. [lo] were studied in order to clarify the role of interlayer correlations in the observed v = l/2 FQHE. The v = l/2 state was found to monotonically weaken as the density increases. This is an unusual observation since the single-layer FQHE generally becomes stronger at higher magnetic fields owing to the increasing Coulomb energy (e*/sl, a B’/*). For a double-layer system, however, the relative magnitude of intra- and interlayer Coulomb energies, d/l,, plays a critical role. The data span the range of d/l, increasing from 2.4 to 2.9 as the v = l/2 state weakens. Further increasing d/l, to 3.6 (by increasing the barrier thickness to 99 A) obliterates all evidence of a FQHE at v = l/2. This provides compelling evidence that the v = l/2 FQHE derives from inter-layer electron-electron interactions. Fig. 1 contains several QHE features in addition to the new state at v = l/2. The states observed at v = 2 and 2/3 both have well-known
counterparts in single-layer systems. For I’ = 2. this is the ordinary integral QHE obtained when the Fermi level lies in the spin gap of the lowest Landau level. Similarly, the FQHE seen at I’ = 2/.3 corresponds to the canonical Laughlin l/3state in each single layer. Finally, the data in Fig. I also reveal a strong QHE at 11= 1, for which there is no similar single-layer analog. As previously discussed, this II = 1 state could arise from the inter-layer correlations built into the Yf,,,,, state [S-7]. A v = 1 state could also arise. however, if the two layers arc close enough to hc coupled by substantial tunneling. If the tunneling gap between the l~)west-lying symmetric and antisymmetric states is experimentally resolved, then odd-integer QHE states result when the Fermi energy lies in this single-electron energy gap [ 121. Thus, the observed QHE state at L’= I could have two possible origins: single-particle tunncling or inter-layer many-body effects. The lowtemperature resistivity data shown in Fig. 2 supports the many-body alternative. Consider the lower trace taken with the magnetic field perpendicular to the ZDES (6 = 0”). Strong QHE minima are observed at ZJ= Y. 7. 5. and I. (We will discuss the absence of the P = 3 QHE later.) Application of an in-plane magnetic field, Ri,, has
-, 0 0
Fig. 2. L.ow-field component
statcb arc destroyed /I =
I i\ not. This
discussed in the text.
field for magnetic
to the ZDES. by an in-plane
at 7‘ = 70 mK wrs~s
of the magnetic
and 00” from normal
I’ r- 9. 7. and 5
field. while the
that the u =
field tilted 0’
I state arises
missing 18= 3 state is
hecn shown to destroy 19= odd QHE states which arise from tunneling [l?]. This effect is due to :t reduction of the tunneling matrix cfcment atemming from the r~~lliren~cnt of’ conserved canonical m~~rn~nturn. It assists us in ~~iscrimiIlatin~ bctwcen the single-particle and collective variants of the 17= I QHE. The upper trace in Fig. 3 shows the destruction of the QHE states at I’ = 0, 7. and 5 upon tilting the magnetic field by 00”. We therefore attribute these states to the finite tunneling gap in this sample. The 11= I. by contrast, remains strong, even in the presence of ;I 7.6 ‘I’ in-plane magnetic field. Activation energies measured at v = 1 as a function of H, are shown in Fig. 3. The dashed line (n~)rmalizcd to the data at Bii = 0) is the calculated ~~,-dep~nd~ncc of a tunneling gap at v = 1 . Although the sharp cnhanccmont of the activation energy very neat H, = 0 is no1 understood, the relative indepcndence of the activation energy over the range 1
G.S. Boebinger et al. /Surface
PARTICLE 0.05 ’
given sample (fixed d, AsAs, and N,,,). Filled/ open circles denote observed/missing quantum Hall states from Ref. [121, labelled by filling factor Y. Filled/open diamonds denote observed/ missing quantum Hall states from the 0 = 0” data of Fig. 2. The solid lines are schematic phase boundaries representing the Coulomb-driven destruction of the single-particle tunneling QHE and re-emergence of a many-body QHE in the regime of d/l, = 1 and A,,, + 0. In the regime of large ASAS, the observed v = odd QHE arises from the single-particle tunneling gap. The new phase in Fig. 4, labelled “MANY BODY QHE”, includes the I, = 1 state observed in Figs. 1 and 2.
Science 305 (1994) 8- 12
ASAS / (e%dd
Fig. 4. Schematic phase diagram of the quantized states at v = odd in a double quantum well, adapted from Refs. [9,15], including data from Ref.  (circles) and new data from Fig. 2 (diamonds). Since the axes are each scaled by the magnetic increasing magnetic length, I,, the dashed arrows represent field for a given sample (fixed d, A,,,, and N,,,). Filled/open symbols denote observed/missing quantum Hall states labelled by filling factor v. The solid lines are schematic phase boundaries representing the Coulomb-driven destruction of the single-particle tunneling QHE and the re-emergence of a many-body QHE in the regime of d/l, - 1 and AsAs --t 0.
systems [12,14]. These studies find a phase transition in which inter-layer Coulomb interactions destroy the single-particle tunneling gap. A welldefined boundary [15,16] exists between the larger odd-integral filling factors for which the singleparticle tunneling gap, and, hence, the QHE, survives, and the smaller odd integers for which it is collapsed and the QHE is quenched. Within this picture then, inter-layer interactions have destroyed the tunneling gap in our sample for all I/ 5 3. Thus, the reappearance of a QHE at v = 1 suggests these same inter-layer interactions are now strong enough to create a new, many-body gap to replace the tunneling gap. We therefore propose in Fig. 4 a refinement  of a previously proposed phase diagram for the QHE at v = odd [1.5]. The two axes (interwell distance, d, and single-particle tunneling gap, A,,,) are each scaled by the magnetic length, I,. The dashed arrows represent increasing magnetic field for a
3. Conclusions Two new quantized states have been observed in a double-layer 2D electron system at filling fractions v = l/2 and v = 1. Studies of samples with different densities and layer separations and experiments with in-plane magnetic fields establish a strong argument that these two states arise from an interplay between intra- and inter-layer Coulomb interactions.
4. References [l] For reviews, see: R.E. Prange and S.M. Girvin, Eds., The Quantum Hall Effect (Springer, New York, 1987); T. Chakraborty and P. Pietilainen, The Fractional Quantum Hall Effect, Vol. 85 of Springer Series in Solid State Physics (Springer, Berlin, 1988).  J.P. Eisenstein, H.L. Stormer, L.N. Pfeiffer and K.W. West, Phys. Rev. Lett. 62 (1989) 1540.  R.G. Clark, S.R. Haynes, A.M. Suckling, J.R. Mallett, P.A. Wright, J.J. Harris and C.T. Foxon, Phys. Rev. Lett. 62 (1989) 1536.  R.L. Willett, J.P. Eisenstein, H.L. Stormer and D.C. Tsui, Phys. Rev. Lett. 59 (1987) 1776.  E.H. Rezayi and F.D.M. Haldane, Bull. Am. Phys. Sot. 32 (1987) 892.  T. Chakraborty and P. Pietilainen, Phys. Rev. Lett. 59 (1987) 2784.  D. Yoshioka, A.H. MacDonald and S.M. Girvin, Phys. Rev. B 39 (1989) 1932.  B.I. Halperin, Helv. Phys. Acta 56 (1983) 75.  Song He, S. Das Sarma and X.C. Xie, Phys. Rev. B 47 (1993) 4394.
 J.P. Eisenstein, G.S. Boebinger, L.N. Pfeiffer, K.W. West and Song He, Phys. Rev. Lett. 68 (1992) 1383. [II] Y.W. Suen, L.W. Engel. M.B. Santos, M. Shayegan and D.C. Tsui, Phys. Rev. Lett. 68 (1992) 1379. [I21 G.S. Boebinger, H.W. Jiang, L.N. Pfeiffer and K.W. West, Phys. Rev. Lett. 64 (1990) 1793; G.S. Boebinger. in: High Magnetic Fields in Semiconductor Physics III, Ed. G. Landwehr, Vol. 101 of Springer Series in Solid State Physics (Springer, Berlin, 1992) p, 155.
[I31 J. Ilu and 12554.
[I41 G.S. Boebinger, L.N. Pfeitfer B 45 (1992) Il391.
B If) (lYY)2)
and K.W. West. Phy\. Kc\.
[151 A.II. MacDonald, P.M. Platzman and G.S. Phys. Rev. Lett. 65 (1900) 775.  L. Brey. Phys. Rev. Lctt. 65 (lY900 003.