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Ayan Ghosh, Souvik Chakraborty, Unmesh Ghorai, Arup Kumar Paul, [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), Rajdeep Sensarma, Anindya Das

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[Evidence of compensated semimetal with electronic correlations at charge neutrality of twisted double bilayer graphene](https://mdr.nims.go.jp/datasets/c2df4eac-856c-46e5-b9e5-9016870a01f2)

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Evidence of compensated semimetal with electronic correlations at charge neutrality of twisted double bilayer grapheneARTICLEEvidence of compensated semimetal withelectronic correlations at charge neutrality oftwisted double bilayer grapheneAyan Ghosh1,4, Souvik Chakraborty1,4, Unmesh Ghorai2,4, Arup Kumar Paul1, K. Watanabe 3, T. Taniguchi 3,Rajdeep Sensarma2✉ & Anindya Das 1✉Recently, magic-angle twisted bilayer graphene (MATBLG) has emerged with variousinteraction-driven novel quantum phases at the commensurate fillings of the moiré super-lattice, while the charge neutrality point (CNP) remains mostly a trivial insulator. Here, weshow an emerging phase of compensated semimetallicity at the CNP of twisted doublebilayer graphene (TDBLG), a close cousin of MATBLG, with signatures of electronic corre-lation. Using electrical and thermal transport, we find two orders of magnitude enhancementof the thermopower at magnetic fields much smaller than the extreme quantum limit,accompanied by large magnetoresistance ( ~ 2500%) at CNP, providing strong experimentalevidence of compensated semimetallicity at CNP of TDBLG. Moreover, at low temperatures,we observe unusual sublinear temperature dependence of resistance. A recent theory1 pre-dicts the formation of an excitonic metal near CNP, where small electron and hole pocketsco-exist. We understand this sublinear temperature dependence in terms of critical fluc-tuations in this theory.https://doi.org/10.1038/s42005-023-01480-x OPEN1 Department of Physics, Indian Institute of Science, Bangalore 560012, India. 2 Department of Theoretical Physics, Tata Institute of Fundamental Research,Mumbai 400005, India. 3 National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 4These authors contributed equally: Ayan Ghosh,Souvik Chakraborty, Unmesh Ghorai. ✉email: sensarma@theory.tifr.res.in; anindya@iisc.ac.inCOMMUNICATIONS PHYSICS |           (2023) 6:360 | https://doi.org/10.1038/s42005-023-01480-x | www.nature.com/commsphys 11234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s42005-023-01480-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s42005-023-01480-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s42005-023-01480-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s42005-023-01480-x&domain=pdfhttp://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-6310-1576http://orcid.org/0000-0002-6310-1576http://orcid.org/0000-0002-6310-1576http://orcid.org/0000-0002-6310-1576http://orcid.org/0000-0002-6310-1576mailto:sensarma@theory.tifr.res.inmailto:anindya@iisc.ac.inwww.nature.com/commsphyswww.nature.com/commsphysThe field of twist angle engineered moiré heterostructures hasemerged as the latest platform to study strongly correlatedquantum matter in condensed matter physics. Recentadvances in graphene-based moiré systems have unveiled a vividspectrum of correlation-driven unconventional phases. For exam-ple, in magic-angle twisted bilayer graphene (MATBLG), exoticphases and phenomena like superconductivity2–8, correlated-insulator4,8–14, Chern-insulator15–18, ferromagnetism19, Diracrevival14,20,21, and giant thermopower at low temperatures22,23have been observed. Twisted double bilayer graphene (TDBLG) isanother prominent member of graphene-based moiré hetero-structures, where two sheets of Bernal-stacked bilayer graphene arestacked on top of each other with a small twist angle between them(Fig. 1a). The resultant reconstruction of electronic levels intobands in the moiré Brillouin zone (mBZ) leads to the formation oflow energy bands, whose bandwidth is sensitive to the twistangle24–26. The bandwidth is minimum around an angle ~1. 2°.Unlike MATBLG, the flat bands in TDBLG survive over a broaderrange of twist angles (1. 1°− 1.35°), providing a robust foundationto study strong correlation effects27. The band structure of TDBLGcan also be tuned by a perpendicular electric field28–31, which candrive the system from a metallic to an insulating state with inter-esting topological properties.Theoretical predictions have shown that in TDBLG the low energyvalence and conduction bands, though separated in momentumspace, overlap in energy, allowing the co-existence of electron-holepockets near the CNP24–26. The co-existence of electron-hole pocketshas fascinating implications like colossal magnetoresistance32, largenon-saturating thermopower with applied magnetic field33 (these arealso seen in Dirac and Weyl semimetals33–37). A similarly enhancedmagnetoresistance ( ~ 200%) has been reported for semimetallic bis-muth and graphite38,39. However, experimental demonstration ofcompensated semimetallic phase or the co-existence of electron andhole pockets in TDBLG has not been reported earlier. Electronicinteractions in compensated semimetals can lead to formation ofexcitonic insulator40,41 driven by Coulomb attraction between theelectrons and holes, or to exciton condensation in metallic42–44backgrounds near the CNP1. Experimentally, the effects of strongelectronic correlations in TDBLG have been seen when the samplesare subjected to strong perpendicular electric fields28–31 or magneticfields27 at the commensurate fillings of moiré superlattice. However,without these perturbations, TDBLG has shown29,45 trivial metallicbehaviour without any report of strong electronic correlations at CNP.This work presents a comprehensive study of temperature,carrier density and magnetic field-dependent resistance, andthermopower of TDBLG with a twist angle 1. 2°. At zero magneticfield, the thermopower is almost zero around the CNP due tocompensation from opposite charge carriers (electrons andholes). Upon application of a small magnetic field, the thermo-power at low temperatures ( < 3K) increases rapidly till 100− 400times before saturating to 10− 15 μV/K within ~ 0.25 T. Simi-larly, the magnetoresistance (MR) at the CNP increases quiterapidly with the application of a small magnetic field and satu-rates before 1T with an enhancement of 2500%. The-4 -2 0 2 402468-2 0 2g(ε)valence bandn (cm-2) 1012c)d) e).02.614581014203240-1 -0.5 0 0.5 1-5051K2K4K5KR(K�)S(μVK   )-1-0.5 0.5n/nsT (K)T TCH-10-50510E(meV)valence bandb)K � M K’a)-4 -2 0 2 4n (cm-2) 1012n (cm-2) 1012n/nsFig. 1 Thermal and electrical transport in Twisted double bilayer graphene (TDBLG) at zero magnetic field. a Schematic of TDBLG with temperaturegradient (TH and TC respectively are the hot and cold terminal temperatures). b Band dispersion of TDBLG with twist angle 1. 2° along the high symmetryaxis in the moiré Brillouin zone (mBZ). The band structure shows an energy overlap (shaded region) of ~ 4.5 meV between valence (blue) and conduction(red) bands. c Density of states of the valence (blue) and conduction (red) bands as a function of normalised carrier density (n/ns), where n is the chargecarrier density induced by the gate voltage and ns is the carrier density required to fulfill the flat band. The shaded region indicates the presence of bandoverlap. d Resistance (R) versus carrier density (n) as function of increasing temperature. Several resistance peaks can be observed at full-filling of bandand near the charge neutrality point (CNP). The shaded region indicates the presence of band overlap. e Evolution of thermopower as a function ofnormalized density n/ns at several temperatures.ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-023-01480-x2 COMMUNICATIONS PHYSICS |           (2023) 6:360 | https://doi.org/10.1038/s42005-023-01480-x | www.nature.com/commsphyswww.nature.com/commsphysenhancement of thermopower by two orders of magnitude andlarge MR at the CNP of TDBLG is quite striking and is not seenfor MATBLG in our study. A two-band particle-hole asymmetricmodel of TDBLG band structure with the co-existence of electronand hole pockets24–26 is invoked to qualitatively and semi-quantitatively explain the results. The compensated semimetalallows a quantum transport regime with small Hall angles36,38,where the electric current is dominated by Drude dissipativeprocesses; however, the thermal current is dominated by thespiraling of the charge carriers in crossed electric and magneticfields. These two different mechanisms lead to a strongenhancement of thermopower and magnetoresistance. We notethat our data provide the first clinching evidence for compensatedsemimetallic phase near the CNP in TDBLG.Further, the metallic behaviour (resistance decreasing withdecreasing temperature) around the compensated semimetallic phaseis quite striking. Near the CNP, we find an unusual sublinear tem-perature dependence of the resistance (R ~Tα, α ~ 0.67− 0.83) below10K, whereas the temperature dependence is superlinear (R ~Tα,α ~ 2− 2.5) away from the region of co-existing electron-holepockets on either side of CNP. Note that the temperature dependenceof the resistance becomes linear around the CNP above 10K. Here, weinvoke the recent theory of excitonic metal1 at the CNP of TDBLG toexplain our unusual temperature dependence of the resistance.Results and DiscussionDevice and setup. The TDBLG device is realized by following thetear and stack technique2. Two AB-stacked bilayer graphene sheetsare stacked with a relative twist angle (1. 2°) and encapsulated byhexagonal boron nitride (hBN). The details of the device fabricationand measurement setup are mentioned in the supplementaryinformation (Supplementary Note 1). For the two-probe resistancemeasurement, the standard lock-in technique (13 Hz) with currentbias has been used. For the thermopower measurement, an isolatedgold line is placed parallel to our device, serving as a heater. Passing acurrent through the heater creates the temperature gradient acrossthe device (Fig. 1a shows the temperature profile as a color gradient).The generated thermometric voltage across the device is measuredusing standard V2ω technique46–52. For the temperature gradient, wehave employed Johnson noise thermometry for the precise mea-surement of ΔT, which gives an accurate value of Seebeck coefficient,S=V2ω/ΔT. The noise measurement is elaborated in our previouswork22 and shown in the supplementary information (Supplemen-tary Note 2–6). All the measurements are performed in the linearregime (see Supplementary Note 7). We note that for accuratemeasurement of the ΔT, a linear fall-off of the temperature isnecessary. Purposefully a simple two-probe geometry is imple-mented instead of having multiple metal leads (that can act asconstant temperature heat sinks) which would have heavily alteredthe linear temperature fall-off to a more complicated form. We havesolved Fourier heat diffusion equations for a multi-layer stack usingfinite element calculations in Comsol (Supplementary Note 14) tovalidate our assumption of a linear temperature profile for a two-probe geometry.Electrical and Thermal Transport. Carrier density-dependentresistance measured at several temperatures (between 20 mK− 40K), is shown in Fig. 1d. The carrier density, n, is obtained fromthe gate voltage applied to the Si/SiO2 back gate by assuming aneffective capacitance of the device. At low temperature, theresistance shows two strong peaks at large positive and negativedensities of Fig. 1d with associated insulating behaviour (resis-tance increasing with decreasing temperature). We identify thesepeaks with the occurrence of moiré gaps, i.e. with carrier densitiesn= ± ns, where the moiré conduction (valence) band ofTDBLG is completely filled (emptied out). This corresponds tohaving 4 electrons (holes) per moiré unit cell. Using theexpression2,9ns ¼ 4A � 8θ2ffiffi3pa2, where A is the area of the moiré unitcell and a is the graphene lattice constant; this translates to a twistangle of θ≃ 1. 2°. This confirms that the twist angle is within therange of observable flat bands for TDBLG.29,30,53.The resistance exhibits metallic behaviour at all densities otherthan the vicinity of the moiré insulator, including at the CNP.This is consistent with earlier work on magic angle TDBLG27,30,31and is in stark contrast with MATBLG, which behaves like aninsulator4,5,9 near the CNP. Theoretically, this is explained by thefact that MATBLG shows protected Dirac nodes with zero densityof electronic states at the Fermi level54, while the valence andconduction bands of magic angle TDBLG overlap in energy24–26,leading to formation of a compensated semimetal at the CNPwith electron and hole pockets. The overlap of the valence (blue)and the conduction (red) bands of TDBLG can be seen from thetheoretical band dispersions of TDBLG (Fig. 1b), calculated at atwist angle of 1. 2°. A detailed model of the moiré bands, whichbreaks particle-hole symmetry and includes trigonal warping, isneeded to obtain this energy overlap of ~ 4.5 meV between thebands (see Supplementary Note 10). Figure 1c shows the variationof the density of states (DOS) at the Fermi level with the carrierdensity n (separately for these two bands). The electron and holepockets co-exist between carrier densities n− ~− 0.7 × 1012 cm−2(n−/ns ~− 0.2) and n+ ~+ 0.4 × 1012 cm−2 (n+/ns ~ 0.12). Thisdensity span is marked by grey shaded region in Fig. 1c–e. Theresistance at a fixed temperature shows additional peaks/dips as afunction of density visible between the CNP and the moiréinsulator. These features survive up to ~ 20K and become moreprominent with increasing temperature. However, these featuresdo not appear exactly at commensurate fillings and are not causedby strong electronic correlations; rather, they may be correlatedwith Van-Hove singularities crossing the Fermi level and areconsistent with the reported results on TDBLG27.The thermopower or Seebeck coefficient is defined as ageneration of electric voltage due to a temperature difference(S ¼ � ΔVΔT). Alternatively, using Onsager relation, S can be writtenin terms of the Peltier coefficient (the ratio of heat current, JQproduced by an applied electrical current, Je) as: S ¼ 1TJQJe. At theCNP or in compensated semimetals, the heat carried by theopposite charge carriers flows in opposite directions. Thus at agiven applied current, S will be very small and proportional to:JQJe� ðne�nhÞðneþnhÞ. Here, ne and nh are the carrier concentrations forelectron and hole, respectively. The density-dependence ofthermopower at different temperatures from 1K to 5K is shownin Fig. 1e. As expected, S reverses sign at the CNP as well as atn= ± ns. We also observe several interesting features in the densitydependence of thermopower away from CNP. These includeadditional change of sign at n/ns ~+ 0.3,− 0.4, dips aroundn/ns ~ ± 0.75 and peaks in between. A comparison betweenmeasured S and thermopower predicted by Mott formula has beendiscussed in the supplementary information (SupplementaryNote 15). The sign change is related to change of topology of theFermi surface (from electron (hole) like to a hole (electron) like)while peaks and dips may be related to possible Lifshitz transitions.However, in this article, we focus on transport near CNP, leavingthe explanation of these features for future work.Thermopower enhancement with low-magnetic field. Themetallic resistivity, together with the sign change of thermopowerat CNP, strongly suggests that the system is a semimetal. How-ever, electrons and holes have opposite charges and respond tomagnetic fields in different ways. Hence, for clear evidence ofCOMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-023-01480-x ARTICLECOMMUNICATIONS PHYSICS |           (2023) 6:360 | https://doi.org/10.1038/s42005-023-01480-x | www.nature.com/commsphys 3www.nature.com/commsphyswww.nature.com/commsphysambipolar transport, we now consider thermal transport in oursample in the presence of a magnetic field B applied perpendi-cular to the plane of the sample. Figure 2a shows the variation ofthe measured S with B for three values of doping near CNP at 1K.S is almost zero around the CNP in the absence of magneticfields; it increases rapidly with the application of few milli-Tesla(mT) magnetic field and saturates around ~ 10− 14 μV/K beyond0.3T. The 2D-colour plot in Fig. 2c shows this large enhancementof thermopower restricted to the vicinity of the CNP.The enhancement of S at relatively low-magnetic fields forcompensated semimetals can be understood in the followingway, which has been quantitatively explained by Feng et. al36. Inpresence of crossed electric (E) and magnetic (B) fields, the driftvelocity of charged particles have two components: a Drude00.51-0.4 -0.2 0 0.2 0.40.80.60.40.20-0.2B (T) n/(n e+nh)b)c)0 0.2 0.4 0.6 0.8 102468101214 0.030-0.030-0.002a)n/nse)g)d)BEVspHehEXEtotE x Btotθ��������������H�5101520250 0.5 1B (T)00.030n/ns0 0.5 1 1.5B (T)02.557.51012.515 -0.0800.0390.0570.0980.1780 5 10 15T (K)024681001R (K�)B (T)R (K�)-0.4 -0.2 0 0.2 0.41.8.6.4.202 4 6 8R (K�) 20mKB (T)f)-5 0 5 10S( μVK   )-1-4 -3 -2 -1 0 1 2 3 4Density (cm-2) X10120.250.20.150.10.050B/  X1015-1 -0.5 0 0.5 10(m-2)2-16 -14 -12 -10 -8 -6 -4 -2 2 4 6 8 124 6 8 10 12S(μVK   )-1S(μVK   )-1B (T) n/nsn/nsn/nsn/nsS( μVK   )-1-5 0 5 10Fig. 2 Finite magnetic field thermopower and resistance measurements. a Magnetic field dependence of measured thermopower atn/ns=+ 0.03,− 0.002,− 0.03 at 1K, where ns is the carrier density (n) at full filling of the flat band. The inset shows the theoretically predictedthermopower for compensated semimetallic band. b Theoretically extracted normalized effective charge density (ðne�nhÞðneþnhÞ) (ne(nh) being the electron(hole)density) as a function of n/ns. c 2D color plot of thermopower as a function of perpendicular magnetic field and n/ns at 1K. (d) Cartoon illustration of the(Etot × B) drift on the carriers in the limit when magnetic field (B) approaches BH (i.e at magnetic field where Hall angle θH→ 90o). The Hall voltage alongthe y direction leads to an electric field EH, where ∣EH∣ ≫ ∣Ex∣ (Ex is the applied electric field) in the limit θH≈ 90o. Electrons (labeled e) and holes (labeled h)both drift alongside in presence of crossed electric (Etot= Ex+ EH≈ EH) and magnetic field contributing additively to the heat current36. e 2D color plot ofthermopower plotted over a wide range of density / n/ns (bottom axis / top axis) and magnetic field (divided by flux quanta Φ0) with red (yellow) dashedlines marking the landau levels emanating from n/ns= 0 (− 1). f Resistance as a function of perpendicular magnetic field and n/ns at 20 mK.g Perpendicular magnetic field dependence of measured resistance at several n/ns at 1 K. The presence of semimetallic band is further reflected throughmetal-insulator transition of resistance versus temperature curve with increasing perpendicular magnetic field, as demonstrated in the inset.ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-023-01480-x4 COMMUNICATIONS PHYSICS |           (2023) 6:360 | https://doi.org/10.1038/s42005-023-01480-x | www.nature.com/commsphyswww.nature.com/commsphysresponse, vd ¼ ± μE1þðμBÞ2 (μ being the mobility of electrons andholes) and a E ×B spiralling component (shown in Fig. 2d). For theelectric current, the Drude response is proportional to nT= ne+ nh,while the response from the spiral component is proportional toΔn= ne− nh, where ne and nh are electron and hole densities. Notethat Δn= ne− nh is the effective charge carrier density of the deviceas defined as n earlier. Since Δn≪ nT, one can have a situation wherethe transport is in the quantum regime36 (μB≫ 1) for B > B1 (B1 ≈ 1/μ), but the Hall angle tan θH ¼ σxyσxx¼ μBΔn=nT is still small due tothe compensation from the ratio of densities; i.e. the electrictransport is dominated by the Drude response. However, for thethermal current, the Drude response is proportional to Δn, while theresponse from the spiral component is proportional to nT, and hencethermal transport is dominated by the drift coming from the spiralcomponent. In this regime, the thermopower Sxx= JQ/TJe is given bySxx � k2BTeϵFΔnnTμ2B2. This rapid quadratic rise of the thermopower isseen in our data in Fig. 2a. At higher magnetic fields B > BH(BH � B1nTΔn), where μB≫ nT/Δn, one enters the regime of extremequantum transport with large Hall angles, where the electric currentis also dominated by the drift coming from the spiral component. Inthis case, the thermopower saturates and is given by Sxx � k2BTeϵFnTΔn.The inset of Fig. 2a shows the theoretically calculated (mentioned inSupplementary Note 11) thermopower for compensated semime-tallic band36, which resembles very well with our experimental data.In Fig. 2b, we plot j ΔnnT j, obtained from theoretical band dispersions,as a function of the carrier density. We find that this theoretical j ΔnnT jfalls to zero (ne ≈ nh, hence Δn= ne− nh→ 0) in the region wherethe large saturation value of thermopower is seen, corresponding tothe region of co-existence of both the electrons and holes. Outsidethis region, the theoretical j ΔnnT j remain close to 1, and thus noenhancement of thermopower is expected as seen in the experi-mental data (Fig. 2c). It should be noted the enhanced thermopowershould change its sign around the CNP. However, as can be seen inFig. 2c, the sign change happens around n/ns ~+ 0.05. Thisdiscrepancy may arise due to differences in mobility for electronsand holes, which was assumed to be the same in the simple model36as described before. Also, note that for n/ns >+ 0.05, the enhance-ment is negative but with a smaller saturation value. In Fig. 2c weobserve a distinct region near CNP (particularly spanning more inthe hole side) where quantum oscillations in S are suppressed. Whileaway from this region (both in electron and hole-doped regions)clear Landau fans can be observed. This suggests the apparentsuppression may be connected to the co-existence of electron-holepockets. One possible reason may be, that the slightly mismatchedeffective mass of either carrier can form landau levels (L.L) withmarginally different L.L gaps. This small misalignment in energygives rise to effective L.L energy gaps much smaller than L.Ls ofeither carrier, hence the oscillations are smeared out.Although the enhancement of thermopower for semimetals likebismuth38,55 and tantalum phosphide56 in the bulk form have beenreported earlier, the predicted saturation of thermopower with themagnetic field previously has not been observed experimentally.Our work on TDBLG provides the first experimental evidence ofthe saturation of thermopower for compensated semimetalsaccurately; we also demonstrate the tunability of the thermopowerwith carrier concentration because of the two-dimensional natureof our system. Additionally, for comparison we have studied thethermopower response for MATBLGwith the magnetic field, and itbarely changes around the CNP (Supplementary Note 8), asexpected for non-compensated semimetals. It should be noted thatthe enhancement decreases as we increase the temperature andvanishes beyond 10K (Supplementary Note 9) where quantumeffects are destroyed due to increased scattering rate and the ideaμB≫ 1 is no longer valid.We also have measured the thermopower over a wider range offilling and magnetic field (at 1K) as shown in Fig. 2e. Alongside thepreviously mentioned thermopower enhancement around CNP wealso observe clear signatures of Landau fans emanating fromn/ns= 0,− 1. Using the Diophantine equation57 for Landau levels(L.L) we find a two-fold degenerate L.L sequence at both fillings athigh magnetic fields. We observe a further decrement in thermo-power to occur beyond 2T. At these higher values of the appliedperpendicular magnetic field, symmetry breaking causes a gapopening at CNP. The lack of DOS (due to gap opening) causesdecay in thermopower with any further increase in a magnetic field.At even higher fields, Landau levels start emerging resulting inoscillations in thermopower along the crossings of the Landau fans.Large magnetoresistance at CNP. To further investigate thedistinct footprints of electron-hole pockets in TDBLG, weexamine the magnetoresistance (MR) of the system and its tem-perature dependence near the compensated region. In Fig. 2f alarge enhancement of MR (measured at 1K) confined within thevicinity of CNP can be observed in the 2D-colour plot of R as afunction of B and n/ns. Figure 2g shows resistance (measured at20 mK) as a function of the magnetic field at different carrierdensities. Close to CNP, the MR monotonically increases with Band saturates around a magnetic field of 1T with a maximumincrement of 2500%. The rapid rise of the MR with a magneticfield can be understood from the quantum limit of electricaltransport33,58, where it is still dominated by the Drude response,while the saturation behaviour is dominated by the spiral com-ponent of current. Similar to thermopower, the maxima of MRappears slightly away from the CNP at n/ns ~+ 0.05 (see Fig. 2f).This could also be due to the mobility mismatch between elec-trons and holes. Note that the position of the CNP was identifiedby looking at the resistance peak in Fig. 1d, the thermopowercrossing point at zero magnetic field in Fig. 1e, and the LandauFan emerging point in Fig. 2e.The high MR and the saturation is a distinctive behaviour ofmany compensated semimetals38,59. Normal metals, on the otherhand, have higher scattering rates which limit their magnetore-sistance. The temperature dependence of resistance at variousmagnetic fields at a fixed density near the charge neutrality isshown in the inset of Fig. 2g. The behaviour is very similar to thatof previously reported compensated semimetals38,39,60. We canclearly see that the system exhibits a field-induced metal-to-insulator transition around a magnetic field of 0.2T.Sublinear temperature scaling of resistance around CNP.As seen in Fig. 1d, we observe a monotonic increment of resis-tance with temperature in the whole density range, suggestingmetallic transport throughout the flat band. In Fig. 3a, we studythe temperature dependence of the resistance of the sampleat several densities on either side of CNP. At large positive ornegative densities, when there is only one type of carrier, wesee that the resistance has a superlinear (with exponent between2 and 2.5) behaviour with a temperature below 10K (markedby blue solid line). However, the situation changes dramaticallynear CNP, where both carriers are present. The resistance atthese densities (n/ns=− 0.051, − 0.021, 0.00, 0.02) show asublinear behaviour with temperature in the range 200 mK− 10K marked by blue solid line, and a linear dependence above 10 K.The sublinear dependence can be fitted using R= aTα+ R0(where a, α and R0, respectively are the proportionality constant,exponent and zero temperature resistance), and obtain α betweenCOMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-023-01480-x ARTICLECOMMUNICATIONS PHYSICS |           (2023) 6:360 | https://doi.org/10.1038/s42005-023-01480-x | www.nature.com/commsphys 5www.nature.com/commsphyswww.nature.com/commsphys0.67− 0.83. The insets show the low temperature regime(blue fitting) in a log-log scale to better represent the low-temperature behaviour. The evolution of α with n/ns is shown inFig. 3b (in blue circles with error-bar). It can be noticed fromFig. 3b that the sublinear temperature dependence is prominentin the region where the electron and hole pockets co-exist, whichis marked by the vertical dashed lines in Fig. 3b. As describedbefore, in this measurement scheme (optimized for thermo-power) we are limited to two-terminal resistance measurements,which has finite contributions from the contact resistance. Itshould be noted that beyond the full-filling in Fig. 1d, the resis-tance barely changes with increasing temperature, where aswithin the full-filling the resistance increase monotonically(metal-like) with increasing temperature (Figs. 3a and 1d) andchanges by ~ 200Ω/Kelvin. This suggests that the contact resis-tance at the measured low temperature range (200 mK− 10 K)barely changes, and our data is predominantly governed by thechannel resistance.The observed sublinear temperature dependence of the resis-tance around the CNP in Fig. 3a and b is quite unusual. Note that ingraphene61 and MATBLG4,62, one finds insulating behaviour nearthe CNP due to vanishing density of states. In contrast, smallelectron and hole pockets are formed in TDBLG near CNP, (seeFig. 3c) which leads to metallic behaviour. However one wouldexpect a T2 behaviour of resistance at the lowest temperatures and alinear temperature dependence above Bloch Gruneissen tempera-ture, which is within a few Kelvin in TDBLG near CNP. Indeed, wefind linear dependence of resistivity for T > 10K. Though, beyond0 20 4002460 20 4002460 20 4002460 20 4002460 20 4002460 20 4002460 20 400246810n/n =0.020sn/n =-0.100sn/n =-0.150sn/n =0.025sn/n =-0.051 n/n =-0.021sn/n =0.000sn/n =0.095sT(K) T(K) T(K)T(K)s�1.15�0.77�0.97�1.15�0.82�1.16�0.71�2.27�1.11�0.62c) d)a)�1.13�1.39�1.13 �1.17�0.65�1.161 101 10 1 10 1 101 10 1 101 101.2 10e hIndirect ExcitonValence bandQ0Q1Q2Q3h-pockete-pocketkykx0 20 40012345100 10010-110-210-210-310-1.2 .2.2.2.2 .2.210-110010-110010-110010-110010-1100 10010-210-4R (K�)R (K�)b)11.522.5-0.4 0.4-0.2 0 0.2n/ns<10K>10KFig. 3 Electronic correlations in Twisted double bilayer graphene (TDBLG) near charge neutrality point (CNP). a Power-law fitting of resistance as afunction of temperature at different densities. Away from the CNP, at normalized carrier densities n/ns ~− 0.150,− 0.100, 0.095, the best fit forR= R0+ aTα (R0, a and α respectively are the zero temperature resistance, proportionality constant and exponent) is shown as the blue (red) line at lowtemperature (high temperature) range. The low temperature exponent shows a clear superlinear behaviour. In contrast near CNP, atn/ns ~− 0.05,− 0.02, 0.0, 0.02, the resistance shows a sublinear behaviour (blue line is the fit) at low temperature (0.02− 10 K). The insets show thedata in log-log scale. The behaviour of R above 10 K at all density ranges is almost linear with T. b Fitted exponent (α) both for low temperature (<10K) andhigher temperature fitting ( > 10K and < 40K) depicted by black circles (with blue error bars) and red circles (with grey error bars) respectively, the errorbar represents 95% confidence bound of the fitting for cofficient α, the blue vertical dashed lines frame the region of electron and hole co-existence. Thelow temperature exponent shows nontrivial sublinearity inside the aforementioned co-existence. c Calculated Fermi surfaces in TDBLG at CNP showing theelectron (blue) and the hole (red) pockets. Note that the centres of the pockets are shifted in momentum space (Q1,Q2 and Q3 being the respective shifts).(d) Schematic representation of indirect excitons as pairing between electrons and holes with momentum offset Q0.ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-023-01480-x6 COMMUNICATIONS PHYSICS |           (2023) 6:360 | https://doi.org/10.1038/s42005-023-01480-x | www.nature.com/commsphyswww.nature.com/commsphysn/ns= ± 0.2 the fitting tends to move marginally away from theexpected linearity as seen in Fig. 3b. It should be noted that, beyondn/ns= ± 0.2 the Fermi surface increases resulting in higher BlochGruneissen temperature, as a result to see the expected linearity onemay need to go higher temperature than the measured temperaturerange. To understand the sublinear dependence around the CNP,we rely on the recent theory1, which predicts the formationof excitonic condensate due to Coulomb attraction between theelectron and hole pockets (see Supplementary Note 12, 13). Sincethe electron and hole pockets are shifted inmomentum (see Fig. 3c),indirect excitons with momenta connecting the center of thepockets are formed, as shown in Fig. 3d. This leads to an excitonicmetal at low temperatures. The fermions in this metal are scatteredby Landau-damped critical fluctations of the excitonic order. Thisleads to a non-Fermi liquid behaviour1,63–67, where the scatteringrate and hence the resistance, R ~ T2/3. Note that the hole pocket,when shifted by the momentum of the exciton, lies on top of theelectron pocket and hence this is similar to the scenario for an orderparameter with zero momentum63, which is known to lead to T2/3scaling of scattering rates1. Based on the theory, we believe that ourexperimental data with sublinear temperature dependence of theresistance with exponent 0.67− 0.83 around the CNP shows thesignature of excitonic metal in TDBLG.ConclusionsWe have reported strong enhancement of thermopower and mag-netoresistance in TDBLG at low temperatures near the CNP forrelatively modest magnetic fields. This behaviour is understood interms of electric and heat transport in a compensated semimetal andprovides clear evidence of simultaneous electron and hole pocketsin this system. The resistance at low temperatures shows a sublineardependence, attributed to the formation of an excitonic metaldescribed in recent theoretical work1. Note that the presence ofdisorders around the CNP can not explain the simultaneous obser-vation of large metallicity (~200 Ω/Kelvin), orders of magnitude ofenhancement of thermopower, magneto resistance, as well asquantum oscillations observed with a few hundred mT of the mag-netic field. It will be interesting to see how these features evolve witha perpendicular displacement field, which is left for future studies.MethodsDevice fabrication and measurement setup. For assembling thehBN encapsulated TDBLG, we have used the standard ‘tear andstack’ technique2,9. The encapsulated device is placed on aSi/SiO2 substrate acting as an electrostatic gate. The fabricationprocess is explained in much greater details in supplementaryinformation (Supplementary Note 1). The length and width ofthe representative device are approximately 6 μm and 3 μm,respectively. An optical image of the measured device is pro-vided in SI-Fig. 3. An isolated thin gold line, placed ~ 3 μmaway from the source probe acts as a heater. During thermo-power measurement, upon injecting a current (Iω) in the heaterline a temperature gradient arise across the length of the device.The source contact neighbouring the heater gets hotter (Th)while the drain is maintained at constant bath temperature ofthe mixing chamber (m.c) plate due to cold ground. Thevoltage (V2ω) generated across the channel is measured usingstandard Lock-in amplifier (Supplementary Note 2). For resis-tance measurement low-frequency ( ~ 13 Hz) Lock-in technique(Supplementary Note 2) is employed. To measure the tem-perature difference (ΔT), we employ Johnson noise thermo-metry. The noise thermometry circuit consists of LC resonant(fr ~ 720 kHz) tank circuit, followed by a cryogenic amplifierand a room temperature amplifier (see SI-Fig. 2d). A detailedgain calculation of the amplifier chain is mentioned inthe supplementary information (Supplementary Note 8). Asdepicted in SI-Fig. 2a a relay situated on the mixing chamberplate is used to switch between high-frequency (ΔT) and low-frequency (Resistance and V2ω) measurement scheme.Theory. Twisted Double Bilayer Graphene consists of two Bernalstacked (AB) bilayer graphene (BLG) sheets with a relative twistangle θ between them. Here, we work with the ABAB stacking, sothat the B sublattice of the top interface layer sits on top of the Asublattice of the bottom interface layer. Here we consider the bandstructure of TDBLG following Ref.s24,26. The details of the Hamil-tonian construction can be found in the supplementary information(Supplementary Note 10). For this work, we have taken the followingcoupling parameters24, ℏv0/a= 2.1354 eV (the nearest neighbourtunneling amplitude along the monolayer graphene sheet),γ1= 400meV (the c-axis inter-layer hopping between the dimersites), γ3= 320meV (the inter-layer hopping between the nondimersites), γ4= 44meV (the coupling between dimer and non-dimersites), and Δ0 ¼ 50 meV (the potential difference between dimerizedand non-dimerized sites). For the AA/BB and AB tunnelingamplitudes across the twisted layers, we have used24, u= 79.7meVand u0 ¼ 97:5 meV respectively in our calculations. In this work, wehave taken a 184 dimensional matrix which gives an error of < 1% inthe band dispersions at the magic angle of 1. 2°.The Coulomb attraction between the electron and hole pocketslead to formation of indirect exciton condensates in TDBLG nearCNP. In this calculation we will replace the Coulomb potentialbetween electrons and holes by a screened short range potential.In fact we will use an effective momentum independent potentialwith the energy scale V0 ~ 10.8 meV. Note that there are threeelectron pockets separated from the three hole pockets bywavevectors Q1(2)(3). The mean field Hamiltonian describing theexcitonic condensate is given byHðQiÞ ¼13 ϵck � μ ΔΔ 13 ϵvkþQi� μ" #ð1Þwhere, the ϵcðvÞk represents the non-interacting conduction (valence)band dispersion and the chemical potential is denoted as μ. Notethe order parameter Δ is same for all the pockets and is determinedself-consistently. We can then write the modified quasi-particledispersion relation in presence of the excitonic condensate,E ±QiðkÞ ¼ϵck þ ϵvkþQi6� μ±ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ϵck � ϵvkþQi�236þ Δ2sð2ÞThe above energy spectrum generates a finite Fermi surface nearCNP, which leads to metallic transport in presence of the condensate.Data availabilityAll data needed to evaluate the conclusions in the paper are present in the paper and inan online repository (https://doi.org/10.6084/m9.figshare.24573631). Additional datarelated to this paper will be available upon reasonable request to the correspondingauthor.Received: 17 August 2023; Accepted: 23 November 2023;References1. Ghorai, U., Ghosh, A., Chakraborty, S., Das, A. & Sensarma, R. Excitonicmetal and non-fermi liquid behavior in twisted double bilayer graphene nearcharge neutrality. Phys. Rev. B 108, 045117 (2023).2. Cao, Y. et al. 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Recent developments in non-fermi liquid theory. Ann. Rev. Condens.Matter Phys. 9, 227–244 (2018).AcknowledgementsA.D. thanks the Department of Science and Technology (DST) and Science and Engi-neering Research Board (SERB), India for financial support (DSTO-2051) andacknowledges the Swarnajayanti Fellowship of the DST/SJF/PSA-03/2018-19. K.W. andT.T. acknowledge support from the Elemental Strategy Initiative conducted by theMEXT, Japan and the CREST (JPMJCR15F3), JST. UG and RS acknowledge computa-tional facilities at the Department of Theoretical Physics, TIFR Mumbai. R.S. acknowl-edges support of the Department of Atomic Energy, Government of India, under ProjectIdentification No. RTI 4002.ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-023-01480-x8 COMMUNICATIONS PHYSICS |           (2023) 6:360 | https://doi.org/10.1038/s42005-023-01480-x | www.nature.com/commsphyshttps://www.sciencedirect.com/science/article/pii/S0081194708607407https://www.sciencedirect.com/science/article/pii/S0081194708607407www.nature.com/commsphysAuthor contributionsA.G., S.C. and A.K.P. contributed to device fabrication and data acquisition. A.G. contributedto analysis. A.D. contributed in designing the experiment, data interpretation and analysis.K.W. and T.T. synthesized the hBN single crystals. U.G., and R.S. contributed in developmentof theory, data interpretation, and all the authors contributed in writing the. manuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version contains supplementary materialavailable at https://doi.org/10.1038/s42005-023-01480-x.Correspondence and requests for materials should be addressed to Rajdeep Sensarma orAnindya Das.Peer review information Communications Physics thanks Ludwig Holleis and the other,anonymous, reviewer(s) for their contribution to the peer review of this work. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2023COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-023-01480-x ARTICLECOMMUNICATIONS PHYSICS |           (2023) 6:360 | https://doi.org/10.1038/s42005-023-01480-x | www.nature.com/commsphys 9https://doi.org/10.1038/s42005-023-01480-xhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/www.nature.com/commsphyswww.nature.com/commsphys Evidence of compensated semimetal with electronic correlations at charge neutrality of twisted double bilayer graphene Results and Discussion Device and�setup Electrical and Thermal Transport Thermopower enhancement with low-magnetic�field Large magnetoresistance�at CNP Sublinear temperature scaling of resistance around�CNP Conclusions Methods Device fabrication and measurement�setup Theory Data availability References References Acknowledgements Author contributions Competing interests Additional information