Microwave spectroscopy of the low-filling-factor bilayer electron solid in a wide quantum well

At the low Landau filling factor termination of the fractional quantum Hall effect series, two-dimensional electron systems exhibit an insulating phase that is understood as a form of pinned Wigner solid. Here we use microwave spectroscopy to probe the transition to the insulator for a wide quantum well sample that can support single-layer or bilayer states depending on its overall carrier density. We find that the insulator exhibits a resonance which is characteristic of a bilayer solid. The resonance also reveals a pair of transitions within the solid, which are not accessible to dc transport measurements. As density is biased deeper into the bilayer solid regime, the resonance grows in specific intensity, and the transitions within the insulator disappear. These behaviours are suggestive of a picture of the insulating phase as an emulsion of liquid and solid components.

W ide quantum wells (WQWs) support bilayer as well as single-layer states, depending on the well width, w, and on the overall areal carrier density, n (refs [1][2][3][4][5][6]. A measure of the tendency of the charge in a WQW to separate into two layers is g(e 2 /4pE 0 Ec B )/D SAS (refs 4-6) where c B ¼ (n/2pn) 1/2 is the magnetic length, D SAS is the gap between the lowest and the first-excited sub-band and e 2 /4pE 0 Ec B is the Coulomb energy. Figure 1a shows a phase diagram in the g and filling factor, n, plane. The interpretation of the insulator as a bilayer solid was inferred in ref. 5 from the phase diagram and from the response of the insulator to asymmetric gate bias that produced mismatched layer densities. At small g the two-dimensional (2D) electron system (2DES) behaves like a single layer, exhibiting a one-component (1C) Wigner solid for n below the 1/5 fractional quantum Hall effect (FQHE), and also for a narrow re-entrant range 7 above it. At large g the 2DES behaves as two layers with weak interaction, and the solid occurs at nB2/5, or per-layer filling n L B1/5. At intermediate g, the insulator includes a n range re-entrant above the 1/2 FQHE 5,6 , which is known to be a bilayer, interlayer-correlated state; therefore, the nearby insulator is likely a bilayer two-component (2C) solid.
Microwave spectroscopy is ideal for studies of electron solids since these exhibit pinning mode resonances [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] , in which pieces of the solid oscillate within the potential of the residual disorder. This disorder also pins the solid, rendering it insulating. The resonance peak frequency, f pk , is sensitive to solid properties, such as shear modulus and the proximity of the carriers to disorder in the host. A recent study of WQW-pinning modes 22 focused on the small-g, single-layer regime in the neighbourhood of n ¼ 1. Earlier studies of bilayer pinning modes 16,20,21 used double quantum wells, for which the layers are defined by a barrier, and 1C-2C transitions were considered in refs 20,21. In this paper we use microwave pinning-mode spectroscopy on a WQW to probe the low-n bilayer insulator, which is understood as a bilayer electron solid. We observe two phase transitions within the solid from the pinning mode as jumps in f pk concurrent with dips in the amplitude. We find that with increasing density the bilayer resonance is strengthened concurrent with a disappearance of the transitions. Our observations are interpreted in terms of intermediate phases, possibly involving bilayer solid and nonresonant liquid components.

Results
Sample details. Our measurements were performed on a GaAs/ AlGaAs WQW of width w ¼ 80 nm with an as-cooled density of n ¼ 1.1 in units of 10 11 cm À 2 , which we use throughout the paper for brevity. The microwave technique 18,[20][21][22] is diagrammed in Fig. 2a,b. We maintained a symmetric growth-direction charge distribution about the well centre unless otherwise noted, and calculated g(n) from simulations (see Methods section). The sample was measured in a 60-mK bath.
Microwave spectroscopy. The strong pinning mode shown in Fig. 3 is the evidence of a solid. We interpret the two sharp increases in f pk as due to transitions between different solid configurations. Figure 4 gives an overview of the spectra for the n range we surveyed, as image plots of Re (s xx ) in the (n, f) plane. Figure 5a,b shows s pk (Re (s xx ) at the resonance maximum) and f pk versus n, respectively, for many n. We interpret the minima in s pk versus n as due to the transitions, and denote the lower and higher n of the transitions by n 1 and n 2 , respectively. At n 2 , the f pk jump is maximal for n ¼ 1.26, just above the transition to the insulator from the FQHE. This jump weakens gradually as n increases and is absent by n ¼ 1.41.
The transitions within the insulator at n 1 and n 2 are most pronounced for n near the transition to insulator from the FQHE. Except near n 1 or n 2 , the resonance s pk increases with n, as seen in Fig. 5a. At the same time, f pk changes little with n except near n 1 or n 2 . Pinning modes approximately obey a sum rule 15,24,25 , pne 2 /2h ¼ S/f pk , where S is the integrated Re (s xx ) versus f for the resonance. Figure 5c shows S/f pk versus n for n ¼ 0.4. Except near n ¼ 1.38, at which S/f pk is suppressed, this n is away from n 1 and n 2 . The curve saturates as n goes above 1.4 to a value in reasonable agreement with the sum rule; nC1.4 is approximately the largest density at which we observe the jump in f pk at n 2 , and the minima in s pk versus n are considerably weakened by this n.
An important result is that the solid appears to be a bilayer. Figure 5d shows s pk versus n for a fixed overall density of n ¼ 1.30, but varying the charge transfer, Dn, between layers. As layer imbalance grows, s pk is reduced and the transition minima shift only slightly in n. The interpretation is that excess carriers of the majority layer do not participate in the resonance, and reduce its strength by damping. The dc transport study in ref. 5 found the resistivity also to be sharply reduced on layer imbalance for a 75-nm WQW in this low n insulator.
Phase diagram in the c-n plane. Figure 6 shows the g-n plane with the shaded area marking the region in which a resonance is observed. Compared with the dc-transport boundary of Fig. 1a (ref. 5), the nearly vertical left boundary of the solid in Fig. 6 occurs at larger g. Some of this difference is likely due to the WQW in this paper being wider, 80 nm versus 75 nm for ref. 5. It may be that relative to the dc boundary, the resonance appears further into the insulating phase. This difference between the resonance boundary and the dc boundary is what reduces the  apparent depth of the rightward boundary indentation centred at n ¼ 1/3. We also find no resonance in the n41/2 re-entrant insulator adjacent to the 1/2 FQHE.

Discussion
As yet there is no theory of 2C solids in WQWs. Hartree-Fock 26,27 and classical 28 theories for double quantum wells (DQWs) admit phase transitions between different types of solids. The Hartree-Fock DQW theories 26,27 predict multiple phases of bilayer solid, as sketched in Fig. 1b-d. A 1C triangular solid exists at low n, large interlayer separation d or small g, while a weakly coupled, staggered triangular 2C solid exists at the opposite extreme. 2C staggered square, rhombic or rectangular lattices exist in between, when D SAS is small enough and d/l B , which measures the ratio of intralayer and interlayer interaction energies, is in a particular (n-dependent) range. The theories 26,27 predict that n of all transitions decreases with increasing g, in contrast to the rising n 1 and n 2 versus g in Fig. 6. Nonetheless, the predicted n, g of a solid-solid transition comes close to that of the observed transitions. For example, for nB1/3 our observed n 1 occurs for gB27.5. The lowest-g transition predicted 26 between 2C lattices (staggered square to staggered rhombic) is at gB33, for d/l B ¼ 3 and n ¼ 1/3. (ref. 26 calculates up to d/l B ¼ 3 and we extrapolate a larger g for d/l B B8, which applies to the transition we see at n 1 ¼ 1/3.) One source of differences between the theory and the results may be the different vertical confinements in WQWs and the DQWs for which the theory was made. Theories 13,29 of composite fermion 30 (CF) Wigner crystals have considered single-layer systems, but predict transitions between solids of different CF vortex number 2p. Such transitions may explain the evolution 18 of the pinning mode for no1/5 in lower n single-layer quantum well states, and may be related to the transitions of ref. 22. 2p is predicted to increase as n decreases, where a transition from 2p ¼ 2-4 should occur between 1/5 and 1/6. This is in the per-layer-n range of the transitions we see, but for the large g case, which corresponds to two weakly coupled parallel layers, the transitions disappear. This is contrary to the expectation for the single-layer-like CF vortex-number transitions, which calculations 13 indicate are driven by n, and whose phase diagram in a single-layer quantum well was described 13 as not sensitive to well width.
The strengthening of the resonance as n moves deeper into the insulator from the FQHE liquids suggests that the solid may be existing along with a component that does not produce a resonance, and that the solid fraction increases with g. In summary, microwave measurements of a WQW establish its low n insulator to be a bilayer solid, since it has a pinning mode resonance that is degraded by layer-density imbalance. The resonance reveals a pair of structural transitions within the insulator. Increases in resonance strength  Figure 4 | Electron density dependence of the transition. Plots of the microwave spectra, Re (s xx ), in the (n, f) plane at various densities, n. s max is the largest Re (s xx ) on the colour scale. Vertical lines mark n ¼ 1/3 and the solid white area at low n is outside the magnetic field measurement range. (a) For n ¼ 1.18, the FQHE around n ¼ 1/3 appears as a region with a flat spectrum and vanishing Re (s xx ) ¼ 0. A resonance is seen only for nr0.31, just below the FQHE state. At n ¼ 0.285, the resonance peak (s pk ) in Re (s xx ) has a sharp minimum, interpreted as a transition within the solid. Possibly coincidentally, this n is within error of 2/7. (b) On increasing to n ¼ 1.22, the resonance is re-entrant around the 1/3 FQHE and a minimum in s pk is located at n ¼ 0.300. (c) For n ¼ 1.26, a resonance has replaced the FQHE at n ¼ 1/3, while for n ¼ 0.315 and 0.340 there are s pk minima, which are accompanied by jumps in f pk . (d,e) Data for n ¼ 1.32 and n ¼ 1.38, respectively. Each plot shows two minima in s pk with jumps in f pk . The jump in f pk is larger for the higher-n (n 2 ) minimum in s pk . For n above n 2 the resonance is broader with lower s pk . The jump in f pk for the lower-n (n 1 ) transition at n ¼ 1.38 is much weaker than it is for lower n. (f) Data for n ¼ 1.52, at which both transitions have disappeared, and f pk and the resonance amplitude vary only marginally throughout the measurement range. The peak frequency f pk versus n for the same n as (a) successively offset by 0.5 GHz. Spectra were taken at n intervals of 0.005. We denote the lower and higher n minima, respectively, by n 1 and n 2 . (c) S/f pk versus n, where S is the integrated Re (s xx ) versus f for the resonance, for n ¼ 0.4 (where the integration is performed for 0.1ofo1.5 GHz). Conversion to g is plotted on the top axis. The error in S/f pk is calculated as the difference between the measured value and that obtained by fitting the resonance to a Gaussian. The dotted line marks the theoretical value 15 S/f pk ¼ pe 2 n/2h for full-charge carrier participation. Except near n ¼ 1.38, at which S/f pk is suppressed, the curve increases before saturating at B85% of the theoretical value. The observed increase in S/f pk demonstrates increase in the strength of the bilayer solid with increasing n as the available particles contribute to the solid. (d) The peak resonance s pk versus n at fixed n ¼ 1.30. Here the well is imbalanced by asymmetric gating to transfer charge Dn between the two layers.
with g and the disappearance of the transitions as S/f pk reaches full strength are qualitatively consistent with a picture of an inhomogeneous insulator containing resonant and nonresonant components.

Methods
Charge distribution in the growth direction. The density of the well was changed by front and back gates. The back gate was in direct contact with the bottom of the sample and the front gate was deposited on a piece of glass that was etched to be spaced from the sample surface to not interfere with the microwave transmission line. A symmetric, balanced, growth direction charge distribution was maintained by biasing the gates such that individually they would change the density by the same amount with equal and opposite electric fields. The asymmetric, imbalanced, distribution was obtained by first biasing one gate to get half the desired charge imbalance and then biasing the other in the opposite manner to get the same charge imbalance while maintaining the same total density with applied electric fields in equal amount and the same direction. d and D SAS were calculated from simulations. The calculated D SAS was used to obtain g and had an uncertainty of about ± 15% for this WQW.
Microwave spectroscopy. Our microwave spectroscopy technique 18,22 uses a coplanar waveguide (CPW) on the surface of a sample. A NiCr front gate was deposited on glass that was etched to space it from the CPW by B10 mm. A schematic diagram of the microwave measurement technique is shown in Fig. 2a, and a cutaway view of the sample is shown in Fig. 2b. In the high-frequency, lowloss limit, diagonal conductivity is approximated by s xx (f) ¼ (s/lZ 0 )ln(t/t 0 ), where s ¼ 30 mm is the distance between the centre conductor and ground plane, l ¼ 28 mm is the length of the CPW and Z 0 ¼ 50 O is the characteristic impedance without the 2DES. t is the amplitude at the receiver and t 0 is the normalizing amplitude. The normalizing amplitude was taken at n ¼ 1/2; the difference on using n ¼ 1 rather than 1/2 as a reference is less than 1 mS. Hence, s xx (f) is the difference between the conductivity and that for n ¼ 0.5. The microwave measurements were carried out in the low-power limit, in which the measurement is not sensitive to the excitation power.
Field modelling. We calculate the reported Re (s xx ) data from t using the results of modelling the fields and currents of the system of the CPW coupled to the WQW, without requiring a high-f low-loss limit. As in ref. 21, the model calculation is based on ref. 32. Let x be the distance from the CPW centre, measured perpendicular to the propagation direction. f 0 (x) is the potential with 1 V on the centre conductor and 0 V on the ground plane. G z0 ðx; x 0 Þ is the electrostatic potential Green's function. G z0 ðx; x 0 Þ and f 0 (x) are found by conformal mapping taking s xx 0. Assuming s xx to be wave vector-independent, the quasistatic potential, f(x), is found from the numerical solution of fðxÞ ¼ f 0 ðxÞ þ s xx io Z G z0 ðx; x 0 Þ @ 2 fðx 0 Þ @x 02 dx 0 : ð1Þ f(x) is then differentiated to obtain the nonequilibrium charge, from which we calculate the induced charge on the CPW. We then calculate the per-unit-length admittance Y L , the s xx -independent per-unit-length CPW series impedance Z L , and calculate the complex transmission coefficient s 21 of the line using lossy transmission-line models 32 . Finally, the measured s xx is found by iteratively comparing measured t/t 0 to the calculated s 21 (s xx , f)/s 21 (0, f). The result is a correction of at most a factor of two from the approximate formula. The correction had no effect on measured peak frequencies.