Abstract
Vertical heterostructures combining different layered materials offer novel opportunities for applications and fundamental studies. Here we report a new class of heterostructures comprising a singlelayer (or bilayer) graphene in close proximity to a quantum well created in GaAs and supporting a highmobility twodimensional electron gas. In our devices, graphene is naturally holedoped, thereby allowing for the investigation of electron–hole interactions. We focus on the Coulomb drag transport measurements, which are sensitive to manybody effects, and find that the Coulomb drag resistivity significantly increases for temperatures <5–10 K. The lowtemperature data follow a logarithmic law, therefore displaying a notable departure from the ordinary quadratic temperature dependence expected in a weakly correlated Fermiliquid. This anomalous behaviour is consistent with the onset of strong interlayer correlations. Our heterostructures represent a new platform for the creation of coherent circuits and topologically protected quantum bits.
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Introduction
Solidstate dipolar quantum liquids are manyparticle systems of repulsively interacting dipoles made of electron–hole (e–h) pairs, which have been studied for the past 50 years (for recent reviews, see for example, refs 1, 2). These systems display a rich and intriguing phenomenology. Particular attention has been devoted to the Bose–Einstein condensation (BEC) of e–h pairs, that is, exciton condensation, with notable results^{1,2}. Optical signatures of BEC of excitons have been reported in optically excited exciton^{3,4,5} and exciton–polariton^{6} cold gases, where, however, nonequilibrium effects conflict, to some extent, with equilibrium thermodynamics. Spectacular implications of spontaneous coherence on transport have also been discovered in systems of permanent (interlayer) excitons^{7}. Quantum Hall fluids in highmobility GaAs/AlGaAs semiconductor double quantum wells display a large variety of transport anomalies^{8,9} due to spontaneous coherence of interlayer excitons.
There is an ongoing effort to find new systems that display spontaneous coherence and superfluidity of permanent excitons. In addition to the fundamental interest in understanding when these phenomena occur in nature, there is a practical interest in discovering manyparticle systems where these types of orders occur under less extreme physical conditions, in particular in the absence of strong magnetic fields and, possibly, at noncryogenic temperatures. Interest in interlayer excitons has been recently revitalized^{10,11,12} by theoretical predictions of hightemperature spontaneous coherence and superfluidity in electrically decoupled graphene layers^{13,14,15} and topological insulator thin films^{16,17,18,19}. The former systems are just examples of van der Waals heterostructures^{20} in which different layered materials are combined to offer novel opportunities for applications^{21,22,23} and fundamental studies^{10,11,12}.
In the following, we report a new class of vertical heterostructures comprising a singlelayer, SLG (or bilayer, BLG) graphene carrying a fluid of massless (massive) chiral holes in close proximity to a GaAs quantum well hosting a highmobility twodimensional (2D) electron gas. We focus on the Coulomb drag transport measurements, which are sensitive to manybody effects. We find that the Coulomb drag resistivity significantly increases for temperatures T<5–10 K, with a notable departure from the T^{2} temperature dependence expected in a weakly correlated Fermiliquid scenario^{24}. The lowtemperature data follow a logarithmic law, without the onset of saturation in the case of bilayer graphene/GaAs samples. This anomalous behaviour is consistent with the onset of strong interlayer correlations. These heterostructures may offer new routes for the exploration of a variety of e–h phenomena, including coherent circuits with minimal dissipation^{25,26,27} and nanodevices including analoguetodigital converters^{28} and topologically protected quantum bits^{29}.
Results
Sample design and characterization
Our vertical heterostructures are prepared as follows: SLG and BLG flakes are produced by micromechanical exfoliation of graphite on Si/SiO_{2} (ref. 30). The number of layers is identified by a combination of optical microscopy^{31} and Raman spectroscopy^{32,33}. The latter is also used to monitor the sample quality by measuring the D to G ratio^{34} and the doping level^{35}. Selected flakes are then placed onto a GaAsbased substrate at the centre of a prepatterned Hall bar by using a polymerbased wet transfer process^{22} (for details, see Supplementary Note 1). The GaAsbased substrates consist of modulationdoped GaAs/AlGaAs heterostructures hosting a 2D electron gas (2DEG) in the GaAs quantum well placed 31.5 nm below the surface. The heterostructures are grown by molecular beam epitaxy^{36} and consist of a ndoped GaAs cap layer, a AlGaAs barrier, a GaAs well and a thick AlGaAs barrier with a delta doping layer (see Supplementary Fig. 1). Two different samples are fabricated: sample A, with a 15nmthick quantum well, and sample B, with a 22nm quantum well. Hall bars (300 μm wide and 1,500 μm long) are fabricated by ultraviolet lithography. Ni/AuGe/Ni/Au layers are then evaporated and annealed at 400 °C to form Ohmic contacts to the 2DEG, to be used for transport and the Coulomb drag measurements (see Fig. 1). The Hall bar mesas are defined by conventional wet etching in acid solution. To ensure that the current in the 2DEG flows only in the region below the graphene flakes, channels with a width comparable to the transferred graphene flakes (typically ~30 μm) are defined in the Hall bar by electron beam lithography and wet etching (Fig. 1e,f). A SLG flake is transferred onto sample A and a BLG flake onto sample B. The integrity of the SLG and BLG flakes is monitored throughout the process by Raman spectroscopy. Supplementary Fig. 2 compares the Raman spectra of the asprepared SLG on Si/SiO_{2} and after transfer on GaAs. The analysis of G peak position, Pos(G), its full width at half maximum, FWHM(G), Pos(2D) and the area and intensity ratios of 2D and G peaks allow us to monitor the amount and type of doping^{33,35,37,38}. This indicates a small pdoping for the asprepared sample, decreasing to below 100 meV for the transferred sample^{33,35,37}. The absence of a significant D peak both before and after transfer indicates that the samples have negligible amount of defects^{33,34} and that the transfer procedure does not add defects. Similarly, no increase in defects is seen for the BLG samples.
To ensure that the 2D chiral hole gas in SLG/BLG and the 2DEG in GaAs are electrically isolated, we monitor the interlayer I_{I}–V_{I} characteristics in the T=0.25–50 K temperature range (see Supplementary Note 2), with I_{I} and V_{I} the interlayer (‘leakage’) current and interlayer voltage, respectively, and the layers being the SLG (or BLG) and the GaAs quantum well. In SLGbased devices, a negligible interlayer current <0.2 nA is measured for V_{I} up to −0.3 V for all values of T, leading to interlayer resistances ~1 GΩ. In the case of BLG, for T~45 K, I_{I} increases to 100 nA at V_{I}=−0.3 V, with the interlayer resistance increasing to several MΩ. In all cases, therefore, the interlayer resistance is much larger than the largest intralayer resistance for SLG, BLG and GaAs quantum well, which is ~10 kΩ.
Magnetotransport and the Coulomb drag measurements
To search for signatures of correlations between the 2DEG in the GaAs quantum well and the 2D chiral hole fluid^{39} in SLG or BLG, we measure the T dependence of the Coulomb drag resistance R_{D}. Experimentally, the Coulomb drag is routinely used as a sensitive probe of strong correlations including transitions to the superconducting state^{40}, metalinsulator transitions^{41} and Luttinger liquid correlations^{42} in quantum wires, and exciton condensation in quantum Hall bilayers^{9}. In a Coulomb drag experiment^{24,43,44}, a current source is connected to one of the two layers (the active or drive layer). The other layer (the passive layer) is connected to an external voltmeter so that the layer can be assumed to be an open circuit (no current can flow in it). The drive current I_{drive} drags carriers in the passive layer, which accumulate at the ends of the layer, building up an electric field. The voltage drop V_{drag} related to this field is then measured. The quantity R_{D} is defined as the ratio V_{drag}/I_{drive} and is determined by the rate at which momentum is transferred between quasiparticles in the two layers^{24}. Before the Coulomb drag experiments, we performed magnetotransport measurements at 4 K, as for Fig. 2a,b. In our setup, the 2DEG is induced in the quantum well by shining light from an infrared diode. In the SLG/2DEG device, we find a 2DEG with density n=1.2 × 10^{11} cm^{−2} from lowfield (below 1 T) classical Hall effect and a mobility μ_{e}=13,000 cm^{2} V^{−1} s^{−1} at 4 K. At T=45 K, the density decreases to 4.0 × 10^{10} cm^{−2} and μ_{e}=8,700 cm^{2} V^{−1} s^{−1}. Figure 2a) shows the quantum Hall effect in the 2DEG. The quantum Hall plateaus at h/(2e^{2}) and h/(4e^{2}) (blue trace) correspond to the first two spin degenerate Landau levels^{45}. In correspondence of the plateaus, minima are found^{45} in the longitudinal resistance R_{xx} (red trace).
The 2D chiral hole fluids in SLG and BLG have their highest mobility when the 2DEG is not induced. This is shown in Fig. 2b for the SLGbased device (see also Supplementary Note 3). Figure 2a,b indicates that the sign of the Hall resistance R_{xy} in SLG is opposite to the 2DEG, thereby demonstrating that SLG is pdoped. At 4 K, the hole density is p=9.9 × 10^{11} cm^{−2} and μ_{h}=4,100 cm^{2} V^{−1} s^{−1}. At 45 K, the corresponding values are p=6.7 × 10^{11} cm^{−2} and μ_{h}=2,400 cm^{2} V^{−1} s^{−1}. LowT magnetotransport in SLG (Fig. 2b) reveals quantum Hall plateaus at h/(2e^{2}) and h/(6e^{2}), corresponding to the massless Dirac fermions with spin and valley degeneracy^{39}. On the contrary, when the 2DEG is optically induced, the hole density in SLG at 4 K is p=6.7 × 10^{11} cm^{−2} and μ_{h}=2,100 cm^{2} V^{−1} s^{−1}, thereby weakening the manifestations of the quantum Hall effect (see Supplementary Fig. 3). The degradation of the SLG transport properties in the presence of the 2DEG could be linked to the creation of ionized Si donors within the ndoped GaAs cap layer, acting as positively charged scatterers^{46}.
We now focus on the Coulomb drag experiments. These are performed in the configuration sketched in Fig. 1a,b and in a ^{3}He cryostat with a 240 mK–50 K range. Ten V_{drag}−I_{drive} curves in a d.c. configuration are acquired for each T and then averaged. We first address the SLG/2DEG case. Figure 2c reports a representative set of averaged drag voltage data taken in the 2DEG at T=42.5 K. In this configuration, the SLG gating effect and consequent carrier depletion in the 2DEG are avoided by applying a positive current, from 0 to +2 μA in the SLG channel. Figure 2c) shows that, at this representative T, the drag voltage is linear with the drive current, thereby allowing the extraction of R_{D} from the slope of a linear fit. Figure 2d shows the plots of R_{D} for 30 K≤T≤50 K, with the 2DEG used as the drive (black points) or passive (red points) layer. It also reports calculations of the temperature dependence of R_{D} in a hybrid Dirac/Schrödinger SLG/2DEG double layer within a Boltzmanntransport theory, which is justified in the Fermiliquid regime^{47,48}. This is done by generalizing the theory of ref. 47 to include effects due to the finite width of the GaAs quantum well (see Supplementary Note 4). This shows that the experimental results in this temperature range are consistent with the canonical Fermiliquid prediction^{24,43,44,45,46,47,48,49,50}, that is, R_{D}∝T^{2} (see also Fig. 3a), as constrained by the available phasespace of the initial and final states involved in the scattering process. The magnitude of the measured effect, however, is smaller than predicted by theory. Discrepancies of similar magnitude have been previously reported for the Coulomb drag measurements between two SLGs encapsulated in hexagonal boron nitride^{11}. Figure 2d demonstrates that the Onsager reciprocity relations^{51}, which in our case require that the resistance measured by interchanging drive and passive layers should not change, are satisfied in the 30 K≤T≤40 K range. A violation of reciprocity occurs for T>40 K. We ascribe this to the effect of the interlayer current I_{I}, leading to an additional contribution sensitive to the exchange of active and passive layers (see Supplementary Note 2 and Supplementary Fig. 4). Consistent with this interpretation, we observe larger violations of Onsager reciprocity in the BLG/GaAs sample, where I_{I} is three orders of magnitude larger than SLG/GaAs at T=45 K (see Supplementary Fig. 5). We now discuss the behaviour of R_{D} in the lowtemperature regime. We follow ref. 11 and use the lowest quality layer, in our case SLG, as the drive layer and measure the drag voltage in the 2DEG. In the reversed configuration, the drag voltage measured in SLG shows fluctuations^{11,12} as a function of the drive current, which hamper the extraction of R_{D}, see Supplementary Note 5). R_{D} measured in the 2DEG reveals an anomalous behaviour below 10 K. Figure 3 indicates that R_{D} deviates from the ordinary T^{2} dependence, as shown by a large upturn for T lower than an ‘upturn’ temperature T_{u}~5 K. The enhancement of R_{D} at low T is a very strong effect: the drag signal increases by more than one order of magnitude by decreasing T below T_{u}, where R_{D} is vanishingly small, in agreement with the Fermiliquid predictions (see Supplementary Note 4), down to T=240 mK.
Figure 3b is a zoom of the drag enhancement data in the lowT range together with a fit (solid line) of the type:
where R_{0} and A are two fitting parameters and T_{c} indicates the meanfield critical temperature of a lowT phase transition at which the drag resistance would display a weak singularity. This fitting procedure cannot predict T_{c}: for example, fixing T_{c}=10 mK, the bestfit yields A=(0.416±0.015) Ω and R_{0}=(2.66±0.08) Ω. We can well fit the experimental data by choosing any value of T_{c}, as long as this is substantially lower than the lowest investigated temperature, that is, T=240 mK. As it is clear from the functional form of the fitting function in equation (1), in this regime a change can be reabsorbed into a change of the background resistance . Despite this caveat, we note that our fitting procedure is in excellent agreement with the data for T_{c} in the range 10–100 mK.
We note, however, that the data point at the lowest measured T=240 mK deviates from the logarithmic trend, possibly pointing to the onset of saturation. To further investigate this, we explore a second device comprising of a holedoped exfoliated BLG deposited on the surface of a GaAs quantum heterostructure. The hole density in BLG is p=1.4 × 10^{12} cm^{−2} from the lowfield (below 1 T) classical Hall effect and the mobility is 670 cm^{2} V^{−1} s^{−1} at 4 K. The 2DEG has an electron density n=2 × 10^{11} cm^{−2} and a mobility 86,000 cm^{2} V^{−1} s^{−1} at 4 K. Contrary to the SLG/2DEG case, in the BLG/2DEG device, both electron and hole fluids have parabolic energy–momentum dispersions (Fig. 4b,c). A lower kinetic energy in BLG (vanishing like k^{2} (ref. 39) rather than like k for small values of momentum ℏk) compared with SLG is expected to enhance the relative importance of Coulomb interactions^{52}. It is therefore interesting to evaluate whether this enhancement manifests in the lowtemperature drag resistance. To probe this, we measure the evolution of R_{D} as a function of T using BLG as the drive layer (See Supplementary Figs. 6 and 7). Figure 4 again shows a significant departure from the Fermiliquid T^{2} dependence. Consistent with the expected larger impact of interactions^{52,53}, we get T_{u}~10 K, that is, twice the SLG/2DEG case, while the bestfit of R_{D} data based on equation (1) yields T_{c}=190 mK (to be compared with T_{c}=10–100 mK in the SLG/2DEG). The observed drag resistance remains two–three orders of magnitude smaller than the layer resistance. However, if we follow the logarithmic increase of the drag signal as T approaches T_{c}, the drag resistance can reach arbitrarily large values, of the order of 10^{2}–10^{3} Ω (that is, comparable to the isolatedlayer resistance).
Discussion
The observed upturn of the drag resistance as a function of T is puzzling and so is the fact that the experimental data in the upturn regime are well fitted by a logarithmic function, rather than an exponential or a power law. We now elaborate on possible explanations of this phenomenology. Ref. 54^{54} predicted a finite value of the drag resistivity at T=0 as a consequence of higherorder effects in perturbation theory (third order in the interlayer interaction). The predicted value of interactioninduced drag, however, is far too small (≈10^{−5} Ω per square) to explain our upturn. Mesoscopic effects can lead both to positive and negative fluctuations of the lowtemperature drag resistivity as a function of the electron density^{55}. In our experiments, however, the upturn is fully reproducible for different devices (SLG/GaAs and BLG/GaAs) and cool downs (yielding slightly different densities). We therefore exclude that this effect is responsible for the logarithmic increase. Finally, the disorder effects in the diffusive regime^{49,50} do lead to a logarithmic enhancement of the drag signal of the form R_{D}(T)∝−T^{2} log(T) for T→0, which is therefore a correction to the Fermiliquid behaviour and not an upturn.
The logarithmic enhancement of R_{D} described by equation (1) was theoretically predicted in refs 17, 18 on the basis of a Boltzmanntransport theory for e–h double layers, where the scattering amplitude is evaluated in a ladder approximation^{56}. Similar results were obtained on the basis of a Kuboformula approach^{57}. Within these frameworks, the enhancement is attributed to e–h pairing fluctuations extending above T_{c} for a phase transition into an exciton condensed phase^{1,17,18,57,58}. If this scenario is correct, Fig. 3b indicates that the T range above T_{c}, in which fluctuations are responsible for deviations from the Fermiliquid T^{2} dependence, is very large. This can be ascribed to the quasi2D nature of our SLG/2DEG heterostructure and shares similarities with other 2D systems where fluctuations play an important role, such as cuprate superconductors (see, for example, ref. 59) and ultracold Fermi gases^{60}.
The mismatch in the Fermi wave numbers , which is ~25% in the SLG/2DEG and Δk_{F}~30% in the BLG/2DEG case, raises concern on the validity of the excitoncondensate scenario. (Here and are the Fermi wave numbers in the layers with n and pdoping, respectively.) Indeed, such a mismatch is expected^{61} to weaken the robustness of the excitoncondensate phase in which the condensed e–h pairs have zero total momentum ℏK. However, preliminary calculations including screening in the condensed phase^{62,63,64} indicate that the K=0 excitoncondensate state persists even in the presence of these values of Δk_{F}, with T_{c} scales comparable to those reported here. On the other hand, a mismatch in the Fermi wave numbers of the two fluids may favour Fulde–Ferrell–Larkin–Ovchinnikov^{65} or Sarma^{66} phases. These are however rather fragile in dimensionality d>1, although evidence of a Fulde–Ferrell–Larkin–Ovchinnikov phase was reported, for example, in the layered heavyfermion superconductor CeCoIn_{5} (ref. 67).
We stress that a perfect matching condition would be essential for the excitoncondensate scenario if the system was in the BCS weakcoupling regime, where a small density imbalance leads to a chemical potential difference greater than the pairing gap, and this would indeed destroy superfluidity. However, it was shown theoretically that e–h superfluidity can only appear at experimentally reachable temperatures when the system is already in the socalled BCS–BEC crossover^{64}. In this regime, superfluidity is robust against density imbalance, and there is strong evidence from both theory and experiment^{61,68,69} that it is not killed by the Fermi surface mismatches up to 50%.
Finally, we recall that upturns of the Coulomb drag resistivity were reported in e–h doped GaAs/AlGaAs coupled quantum wells^{70,71,72}. However, the combination of 2D electron and hole gases in the same GaAs material required a large nanofabrication effort, and the reported magnitude of the drag anomalies was smaller than in our hybrid heterostructures.
Our observations establish a new class of vertical heterostructure devices with a potentially large flexibility in the design of band dispersions, doping and e–h coupling, where excitonic phenomena are accessible.
Additional information
How to cite this article: Gamucci, A. et al. Anomalous lowtemperature Coulomb drag in grapheneGaAs heterostructures. Nat. Commun. 5:5824 doi: 10.1038/ncomms6824 (2014).
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Acknowledgements
We thank R. Duine, R. Fazio, A. Hamilton, M. Katsnelson, A. MacDonald, D. Neilson, K. Novoselov, A. Perali, A. Pinczuk and G. Vignale for very useful discussions. We acknowledge funding from the EU Graphene Flagship (contract no. CNECTICT604391), the EC ITN project ‘INDEX’ Grant No. FP72011289968, the Italian Ministry of Education, University and Research (MIUR) through the programs ‘FIRB—Futuro in Ricerca 2010’ Grant No. RBFR10M5BT (‘PLASMOGRAPH’) and ‘Progetti Premiali 2012’ (Project ‘ABNANOTECH’), ERC grants NANOPOTS, Hetero2D, a Royal Society Wolfson Research Merit Award, EU projects GENIUS, CARERAMM, RODIN, EPSRC grants EP/K01711X/1, EP/K017144/1 and EP/L016087/1.
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Gamucci, A., Spirito, D., Carrega, M. et al. Anomalous lowtemperature Coulomb drag in grapheneGaAs heterostructures. Nat Commun 5, 5824 (2014). https://doi.org/10.1038/ncomms6824
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DOI: https://doi.org/10.1038/ncomms6824
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