Abstract
Recently, magicangle twisted bilayer graphene (MATBLG) has emerged with various interactiondriven novel quantum phases at the commensurate fillings of the moiré superlattice, while the charge neutrality point (CNP) remains mostly a trivial insulator. Here, we show an emerging phase of compensated semimetallicity at the CNP of twisted double bilayer graphene (TDBLG), a close cousin of MATBLG, with signatures of electronic correlation. Using electrical and thermal transport, we find two orders of magnitude enhancement of the thermopower at magnetic fields much smaller than the extreme quantum limit, accompanied by large magnetoresistance ( ~ 2500%) at CNP, providing strong experimental evidence of compensated semimetallicity at CNP of TDBLG. Moreover, at low temperatures, we observe unusual sublinear temperature dependence of resistance. A recent theory^{1} predicts the formation of an excitonic metal near CNP, where small electron and hole pockets coexist. We understand this sublinear temperature dependence in terms of critical fluctuations in this theory.
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Introduction
The field of twist angle engineered moiré heterostructures has emerged as the latest platform to study strongly correlated quantum matter in condensed matter physics. Recent advances in graphenebased moiré systems have unveiled a vivid spectrum of correlationdriven unconventional phases. For example, in magicangle twisted bilayer graphene (MATBLG), exotic phases and phenomena like superconductivity^{2,3,4,5,6,7,8}, correlatedinsulator^{4,8,9,10,11,12,13,14}, Cherninsulator^{15,16,17,18}, ferromagnetism^{19}, Dirac revival^{14,20,21}, and giant thermopower at low temperatures^{22,23} have been observed. Twisted double bilayer graphene (TDBLG) is another prominent member of graphenebased moiré heterostructures, where two sheets of Bernalstacked bilayer graphene are stacked on top of each other with a small twist angle between them (Fig. 1a). The resultant reconstruction of electronic levels into bands in the moiré Brillouin zone (mBZ) leads to the formation of low energy bands, whose bandwidth is sensitive to the twist angle^{24,25,26}. The bandwidth is minimum around an angle ~1. 2°. Unlike MATBLG, the flat bands in TDBLG survive over a broader range of twist angles (1. 1° − 1.35°), providing a robust foundation to study strong correlation effects^{27}. The band structure of TDBLG can also be tuned by a perpendicular electric field^{28,29,30,31}, which can drive the system from a metallic to an insulating state with interesting topological properties.
Theoretical predictions have shown that in TDBLG the low energy valence and conduction bands, though separated in momentum space, overlap in energy, allowing the coexistence of electronhole pockets near the CNP^{24,25,26}. The coexistence of electronhole pockets has fascinating implications like colossal magnetoresistance^{32}, large nonsaturating thermopower with applied magnetic field^{33} (these are also seen in Dirac and Weyl semimetals^{33,34,35,36,37}). A similarly enhanced magnetoresistance ( ~ 200%) has been reported for semimetallic bismuth and graphite^{38,39}. However, experimental demonstration of compensated semimetallic phase or the coexistence of electron and hole pockets in TDBLG has not been reported earlier. Electronic interactions in compensated semimetals can lead to formation of excitonic insulator^{40,41} driven by Coulomb attraction between the electrons and holes, or to exciton condensation in metallic^{42,43,44} backgrounds near the CNP^{1}. Experimentally, the effects of strong electronic correlations in TDBLG have been seen when the samples are subjected to strong perpendicular electric fields^{28,29,30,31} or magnetic fields^{27} at the commensurate fillings of moiré superlattice. However, without these perturbations, TDBLG has shown^{29,45} trivial metallic behaviour without any report of strong electronic correlations at CNP.
This work presents a comprehensive study of temperature, carrier density and magnetic fielddependent resistance, and thermopower of TDBLG with a twist angle 1. 2°. At zero magnetic field, the thermopower is almost zero around the CNP due to compensation from opposite charge carriers (electrons and holes). Upon application of a small magnetic field, the thermopower at low temperatures ( < 3K) increases rapidly till 100 − 400 times before saturating to 10 − 15 μV/K within ~ 0.25 T. Similarly, the magnetoresistance (MR) at the CNP increases quite rapidly with the application of a small magnetic field and saturates before 1T with an enhancement of 2500%. The enhancement of thermopower by two orders of magnitude and large MR at the CNP of TDBLG is quite striking and is not seen for MATBLG in our study. A twoband particlehole asymmetric model of TDBLG band structure with the coexistence of electron and hole pockets^{24,25,26} is invoked to qualitatively and semiquantitatively explain the results. The compensated semimetal allows a quantum transport regime with small Hall angles^{36,38}, where the electric current is dominated by Drude dissipative processes; however, the thermal current is dominated by the spiraling of the charge carriers in crossed electric and magnetic fields. These two different mechanisms lead to a strong enhancement of thermopower and magnetoresistance. We note that our data provide the first clinching evidence for compensated semimetallic phase near the CNP in TDBLG.
Further, the metallic behaviour (resistance decreasing with decreasing temperature) around the compensated semimetallic phase is quite striking. Near the CNP, we find an unusual sublinear temperature dependence of the resistance (R ~ T^{α}, α ~ 0.67 − 0.83) below 10K, whereas the temperature dependence is superlinear (R ~ T^{α}, α ~ 2 − 2.5) away from the region of coexisting electronhole pockets on either side of CNP. Note that the temperature dependence of the resistance becomes linear around the CNP above 10K. Here, we invoke the recent theory of excitonic metal^{1} at the CNP of TDBLG to explain our unusual temperature dependence of the resistance.
Results and Discussion
Device and setup
The TDBLG device is realized by following the tear and stack technique^{2}. Two ABstacked bilayer graphene sheets are stacked with a relative twist angle (1. 2°) and encapsulated by hexagonal boron nitride (hBN). The details of the device fabrication and measurement setup are mentioned in the supplementary information (Supplementary Note 1). For the twoprobe resistance measurement, the standard lockin technique (13 Hz) with current bias has been used. For the thermopower measurement, an isolated gold line is placed parallel to our device, serving as a heater. Passing a current through the heater creates the temperature gradient across the device (Fig. 1a shows the temperature profile as a color gradient). The generated thermometric voltage across the device is measured using standard V_{2ω} technique^{46,47,48,49,50,51,52}. For the temperature gradient, we have employed Johnson noise thermometry for the precise measurement of ΔT, which gives an accurate value of Seebeck coefficient, S = V_{2ω}/ΔT. The noise measurement is elaborated in our previous work^{22} and shown in the supplementary information (Supplementary Note 2–6). All the measurements are performed in the linear regime (see Supplementary Note 7). We note that for accurate measurement of the ΔT, a linear falloff of the temperature is necessary. Purposefully a simple twoprobe geometry is implemented instead of having multiple metal leads (that can act as constant temperature heat sinks) which would have heavily altered the linear temperature falloff to a more complicated form. We have solved Fourier heat diffusion equations for a multilayer stack using finite element calculations in Comsol (Supplementary Note 14) to validate our assumption of a linear temperature profile for a twoprobe geometry.
Electrical and Thermal Transport
Carrier densitydependent resistance measured at several temperatures (between 20 mK − 40 K), is shown in Fig. 1d. The carrier density, n, is obtained from the gate voltage applied to the Si/SiO_{2} back gate by assuming an effective capacitance of the device. At low temperature, the resistance shows two strong peaks at large positive and negative densities of Fig. 1d with associated insulating behaviour (resistance increasing with decreasing temperature). We identify these peaks with the occurrence of moiré gaps, i.e. with carrier densities n = ± n_{s}, where the moiré conduction (valence) band of TDBLG is completely filled (emptied out). This corresponds to having 4 electrons (holes) per moiré unit cell. Using the expression^{2,9}\({n}_{s}=\frac{4}{A}\approx \frac{8{\theta }^{2}}{\sqrt{3}{a}^{2}}\), where A is the area of the moiré unit cell and a is the graphene lattice constant; this translates to a twist angle of θ ≃ 1. 2°. This confirms that the twist angle is within the range of observable flat bands for TDBLG.^{29,30,53}.
The resistance exhibits metallic behaviour at all densities other than the vicinity of the moiré insulator, including at the CNP. This is consistent with earlier work on magic angle TDBLG^{27,30,31} and is in stark contrast with MATBLG, which behaves like an insulator^{4,5,9} near the CNP. Theoretically, this is explained by the fact that MATBLG shows protected Dirac nodes with zero density of electronic states at the Fermi level^{54}, while the valence and conduction bands of magic angle TDBLG overlap in energy^{24,25,26}, leading to formation of a compensated semimetal at the CNP with electron and hole pockets. The overlap of the valence (blue) and the conduction (red) bands of TDBLG can be seen from the theoretical band dispersions of TDBLG (Fig. 1b), calculated at a twist angle of 1. 2°. A detailed model of the moiré bands, which breaks particlehole symmetry and includes trigonal warping, is needed to obtain this energy overlap of ~ 4.5 meV between the bands (see Supplementary Note 10). Figure 1c shows the variation of the density of states (DOS) at the Fermi level with the carrier density n (separately for these two bands). The electron and hole pockets coexist between carrier densities n_{−} ~ − 0.7 × 10^{12} cm^{−2} (n_{−}/n_{s} ~ − 0.2) and n_{+} ~ + 0.4 × 10^{12} cm^{−2} (n_{+}/n_{s} ~ 0.12). This density span is marked by grey shaded region in Fig. 1c–e. The resistance at a fixed temperature shows additional peaks/dips as a function of density visible between the CNP and the moiré insulator. These features survive up to ~ 20K and become more prominent with increasing temperature. However, these features do not appear exactly at commensurate fillings and are not caused by strong electronic correlations; rather, they may be correlated with VanHove singularities crossing the Fermi level and are consistent with the reported results on TDBLG^{27}.
The thermopower or Seebeck coefficient is defined as a generation of electric voltage due to a temperature difference (\(S=\frac{\Delta V}{\Delta T}\)). Alternatively, using Onsager relation, S can be written in terms of the Peltier coefficient (the ratio of heat current, J_{Q} produced by an applied electrical current, J_{e}) as: \(S=\frac{1}{T}\frac{{J}_{Q}}{{J}_{e}}\). At the CNP or in compensated semimetals, the heat carried by the opposite charge carriers flows in opposite directions. Thus at a given applied current, S will be very small and proportional to: \(\frac{{J}_{Q}}{{J}_{e}} \sim \frac{({n}_{e}{n}_{h})}{({n}_{e}+{n}_{h})}\). Here, n_{e} and n_{h} are the carrier concentrations for electron and hole, respectively. The densitydependence of thermopower at different temperatures from 1K to 5K is shown in Fig. 1e. As expected, S reverses sign at the CNP as well as at n = ± n_{s}. We also observe several interesting features in the density dependence of thermopower away from CNP. These include additional change of sign at n/n_{s} ~ + 0.3, − 0.4, dips around n/n_{s} ~ ± 0.75 and peaks in between. A comparison between measured S and thermopower predicted by Mott formula has been discussed in the supplementary information (Supplementary Note 15). The sign change is related to change of topology of the Fermi 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, leaving the explanation of these features for future work.
Thermopower enhancement with lowmagnetic field
The metallic resistivity, together with the sign change of thermopower at CNP, strongly suggests that the system is a semimetal. However, electrons and holes have opposite charges and respond to magnetic fields in different ways. Hence, for clear evidence of ambipolar transport, we now consider thermal transport in our sample in the presence of a magnetic field B applied perpendicular to the plane of the sample. Figure 2a shows the variation of the measured S with B for three values of doping near CNP at 1K. S is almost zero around the CNP in the absence of magnetic fields; it increases rapidly with the application of few milliTesla (mT) magnetic field and saturates around ~ 10 − 14 μV/K beyond 0.3T. The 2Dcolour plot in Fig. 2c shows this large enhancement of thermopower restricted to the vicinity of the CNP.
The enhancement of S at relatively lowmagnetic fields for compensated semimetals can be understood in the following way, which has been quantitatively explained by Feng et. al^{36}. In presence of crossed electric (E) and magnetic (B) fields, the drift velocity of charged particles have two components: a Drude response, \({v}_{d}=\pm \frac{\mu E}{1+{(\mu B)}^{2}}\) (μ being the mobility of electrons and holes) and a E × B spiralling component (shown in Fig. 2d). For the electric current, the Drude response is proportional to n_{T} = n_{e} + n_{h}, while the response from the spiral component is proportional to Δn = n_{e} − n_{h}, where n_{e} and n_{h} are electron and hole densities. Note that Δn = n_{e} − n_{h} is the effective charge carrier density of the device as defined as n earlier. Since Δn ≪ n_{T}, one can have a situation where the transport is in the quantum regime^{36} (μB ≫ 1) for B > B_{1} (B_{1} ≈ 1/μ), but the Hall angle \(\tan {\theta }_{H}=\frac{{\sigma }_{xy}}{{\sigma }_{xx}}=\mu B\Delta n/{n}_{T}\) is still small due to the compensation from the ratio of densities; i.e. the electric transport is dominated by the Drude response. However, for the thermal current, the Drude response is proportional to Δn, while the response from the spiral component is proportional to n_{T}, and hence thermal transport is dominated by the drift coming from the spiral component. In this regime, the thermopower S_{xx} = J_{Q}/TJ_{e} is given by \({S}_{xx} \sim \frac{{k}_{B}^{2}T}{e{\epsilon }_{F}}\frac{\Delta n}{{n}_{T}}{\mu }^{2}{B}^{2}\). This rapid quadratic rise of the thermopower is seen in our data in Fig. 2a. At higher magnetic fields B > B_{H} (\({B}_{H}\approx {B}_{1}\frac{{n}_{T}}{\Delta n}\)), where μB ≫ n_{T}/Δn, one enters the regime of extreme quantum transport with large Hall angles, where the electric current is also dominated by the drift coming from the spiral component. In this case, the thermopower saturates and is given by \({S}_{xx} \sim \frac{{k}_{B}^{2}T}{e{\epsilon }_{F}}\frac{{n}_{T}}{\Delta n}\). The inset of Fig. 2a shows the theoretically calculated (mentioned in Supplementary Note 11) thermopower for compensated semimetallic band^{36}, which resembles very well with our experimental data. In Fig. 2b, we plot \(\vert \frac{\Delta n}{{n}_{T}}\vert\), obtained from theoretical band dispersions, as a function of the carrier density. We find that this theoretical \(\vert \frac{\Delta n}{{n}_{T}}\vert\) falls to zero (n_{e} ≈ n_{h}, hence Δn = n_{e} − n_{h} → 0) in the region where the large saturation value of thermopower is seen, corresponding to the region of coexistence of both the electrons and holes. Outside this region, the theoretical \(\vert \frac{\Delta n}{{n}_{T}}\vert\) remain close to 1, and thus no enhancement of thermopower is expected as seen in the experimental data (Fig. 2c). It should be noted the enhanced thermopower should change its sign around the CNP. However, as can be seen in Fig. 2c, the sign change happens around n/n_{s} ~ + 0.05. This discrepancy may arise due to differences in mobility for electrons and holes, which was assumed to be the same in the simple model^{36} as described before. Also, note that for n/n_{s} > + 0.05, the enhancement is negative but with a smaller saturation value. In Fig. 2c we observe a distinct region near CNP (particularly spanning more in the hole side) where quantum oscillations in S are suppressed. While away from this region (both in electron and holedoped regions) clear Landau fans can be observed. This suggests the apparent suppression may be connected to the coexistence of electronhole pockets. One possible reason may be, that the slightly mismatched effective mass of either carrier can form landau levels (L.L) with marginally different L.L gaps. This small misalignment in energy gives rise to effective L.L energy gaps much smaller than L.Ls of either carrier, hence the oscillations are smeared out.
Although the enhancement of thermopower for semimetals like bismuth^{38,55} and tantalum phosphide^{56} in the bulk form have been reported earlier, the predicted saturation of thermopower with the magnetic field previously has not been observed experimentally. Our work on TDBLG provides the first experimental evidence of the saturation of thermopower for compensated semimetals accurately; we also demonstrate the tunability of the thermopower with carrier concentration because of the twodimensional nature of our system. Additionally, for comparison we have studied the thermopower response for MATBLG with the magnetic field, and it barely changes around the CNP (Supplementary Note 8), as expected for noncompensated semimetals. It should be noted that the enhancement decreases as we increase the temperature and vanishes beyond 10K (Supplementary Note 9) where quantum effects are destroyed due to increased scattering rate and the idea μB ≫ 1 is no longer valid.
We also have measured the thermopower over a wider range of filling and magnetic field (at 1K) as shown in Fig. 2e. Alongside the previously mentioned thermopower enhancement around CNP we also observe clear signatures of Landau fans emanating from n/n_{s} = 0, − 1. Using the Diophantine equation^{57} for Landau levels (L.L) we find a twofold degenerate L.L sequence at both fillings at high magnetic fields. We observe a further decrement in thermopower to occur beyond 2T. At these higher values of the applied perpendicular magnetic field, symmetry breaking causes a gap opening at CNP. The lack of DOS (due to gap opening) causes decay in thermopower with any further increase in a magnetic field. At even higher fields, Landau levels start emerging resulting in oscillations in thermopower along the crossings of the Landau fans.
Large magnetoresistance at CNP
To further investigate the distinct footprints of electronhole pockets in TDBLG, we examine the magnetoresistance (MR) of the system and its temperature dependence near the compensated region. In Fig. 2f a large enhancement of MR (measured at 1K) confined within the vicinity of CNP can be observed in the 2Dcolour plot of R as a function of B and n/n_{s}. Figure 2g shows resistance (measured at 20 mK) as a function of the magnetic field at different carrier densities. Close to CNP, the MR monotonically increases with B and saturates around a magnetic field of 1T with a maximum increment of 2500%. The rapid rise of the MR with a magnetic field can be understood from the quantum limit of electrical transport^{33,58}, where it is still dominated by the Drude response, while the saturation behaviour is dominated by the spiral component of current. Similar to thermopower, the maxima of MR appears slightly away from the CNP at n/n_{s} ~ + 0.05 (see Fig. 2f). This could also be due to the mobility mismatch between electrons and holes. Note that the position of the CNP was identified by looking at the resistance peak in Fig. 1d, the thermopower crossing point at zero magnetic field in Fig. 1e, and the Landau Fan emerging point in Fig. 2e.
The high MR and the saturation is a distinctive behaviour of many compensated semimetals^{38,59}. Normal metals, on the other hand, have higher scattering rates which limit their magnetoresistance. The temperature dependence of resistance at various magnetic fields at a fixed density near the charge neutrality is shown in the inset of Fig. 2g. The behaviour is very similar to that of previously reported compensated semimetals^{38,39,60}. We can clearly see that the system exhibits a fieldinduced metaltoinsulator 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 resistance with temperature in the whole density range, suggesting metallic transport throughout the flat band. In Fig. 3a, we study the temperature dependence of the resistance of the sample at several densities on either side of CNP. At large positive or negative densities, when there is only one type of carrier, we see that the resistance has a superlinear (with exponent between 2 and 2.5) behaviour with a temperature below 10K (marked by blue solid line). However, the situation changes dramatically near CNP, where both carriers are present. The resistance at these densities (n/n_{s} = − 0.051, − 0.021, 0.00, 0.02) show a sublinear behaviour with temperature in the range 200 mK − 10 K marked by blue solid line, and a linear dependence above 10 K. The sublinear dependence can be fitted using R = aT^{α} + R_{0} (where a, α and R_{0}, respectively are the proportionality constant, exponent and zero temperature resistance), and obtain α between 0.67 − 0.83. The insets show the low temperature regime (blue fitting) in a loglog scale to better represent the lowtemperature behaviour. The evolution of α with n/n_{s} is shown in Fig. 3b (in blue circles with errorbar). It can be noticed from Fig. 3b that the sublinear temperature dependence is prominent in the region where the electron and hole pockets coexist, which is marked by the vertical dashed lines in Fig. 3b. As described before, in this measurement scheme (optimized for thermopower) we are limited to twoterminal resistance measurements, which has finite contributions from the contact resistance. It should be noted that beyond the fullfilling in Fig. 1d, the resistance barely changes with increasing temperature, where as within the fullfilling the resistance increase monotonically (metallike) with increasing temperature (Figs. 3a and 1d) and changes by ~ 200Ω/Kelvin. This suggests that the contact resistance at the measured low temperature range (200 mK − 10 K) barely changes, and our data is predominantly governed by the channel resistance.
The observed sublinear temperature dependence of the resistance around the CNP in Fig. 3a and b is quite unusual. Note that in graphene^{61} and MATBLG^{4,62}, one finds insulating behaviour near the CNP due to vanishing density of states. In contrast, small electron and hole pockets are formed in TDBLG near CNP, (see Fig. 3c) which leads to metallic behaviour. However one would expect a T^{2} behaviour of resistance at the lowest temperatures and a linear temperature dependence above Bloch Gruneissen temperature, which is within a few Kelvin in TDBLG near CNP. Indeed, we find linear dependence of resistivity for T > 10K. Though, beyond n/n_{s} = ± 0.2 the fitting tends to move marginally away from the expected linearity as seen in Fig. 3b. It should be noted that, beyond n/n_{s} = ± 0.2 the Fermi surface increases resulting in higher Bloch Gruneissen temperature, as a result to see the expected linearity one may need to go higher temperature than the measured temperature range. To understand the sublinear dependence around the CNP, we rely on the recent theory^{1}, which predicts the formation of excitonic condensate due to Coulomb attraction between the electron and hole pockets (see Supplementary Note 12, 13). Since the electron and hole pockets are shifted in momentum (see Fig. 3c), indirect excitons with momenta connecting the center of the pockets are formed, as shown in Fig. 3d. This leads to an excitonic metal at low temperatures. The fermions in this metal are scattered by Landaudamped critical fluctations of the excitonic order. This leads to a nonFermi liquid behaviour^{1,63,64,65,66,67}, where the scattering rate and hence the resistance, R ~ T^{2/3}. Note that the hole pocket, when shifted by the momentum of the exciton, lies on top of the electron pocket and hence this is similar to the scenario for an order parameter with zero momentum^{63}, which is known to lead to T^{2/3} scaling of scattering rates^{1}. Based on the theory, we believe that our experimental data with sublinear temperature dependence of the resistance with exponent 0.67 − 0.83 around the CNP shows the signature of excitonic metal in TDBLG.
Conclusions
We have reported strong enhancement of thermopower and magnetoresistance in TDBLG at low temperatures near the CNP for relatively modest magnetic fields. This behaviour is understood in terms of electric and heat transport in a compensated semimetal and provides clear evidence of simultaneous electron and hole pockets in this system. The resistance at low temperatures shows a sublinear dependence, attributed to the formation of an excitonic metal described in recent theoretical work^{1}. Note that the presence of disorders around the CNP can not explain the simultaneous observation of large metallicity (~200 Ω/Kelvin), orders of magnitude of enhancement of thermopower, magneto resistance, as well as quantum oscillations observed with a few hundred mT of the magnetic field. It will be interesting to see how these features evolve with a perpendicular displacement field, which is left for future studies.
Methods
Device fabrication and measurement setup
For assembling the hBN encapsulated TDBLG, we have used the standard ‘tear and stack’ technique^{2,9}. The encapsulated device is placed on a Si/SiO_{2} substrate acting as an electrostatic gate. The fabrication process is explained in much greater details in supplementary information (Supplementary Note 1). The length and width of the representative device are approximately 6 μm and 3 μm, respectively. An optical image of the measured device is provided in SIFig. 3. An isolated thin gold line, placed ~ 3 μm away from the source probe acts as a heater. During thermopower measurement, upon injecting a current (I_{ω}) in the heater line a temperature gradient arise across the length of the device. The source contact neighbouring the heater gets hotter (T_{h}) while the drain is maintained at constant bath temperature of the mixing chamber (m.c) plate due to cold ground. The voltage (V_{2ω}) generated across the channel is measured using standard Lockin amplifier (Supplementary Note 2). For resistance measurement lowfrequency ( ~ 13 Hz) Lockin technique (Supplementary Note 2) is employed. To measure the temperature difference (ΔT), we employ Johnson noise thermometry. The noise thermometry circuit consists of LC resonant (f_{r} ~ 720 kHz) tank circuit, followed by a cryogenic amplifier and a room temperature amplifier (see SIFig. 2d). A detailed gain calculation of the amplifier chain is mentioned in the supplementary information (Supplementary Note 8). As depicted in SIFig. 2a a relay situated on the mixing chamber plate is used to switch between highfrequency (ΔT) and lowfrequency (Resistance and V_{2ω}) measurement scheme.
Theory
Twisted Double Bilayer Graphene consists of two Bernal stacked (AB) bilayer graphene (BLG) sheets with a relative twist angle θ between them. Here, we work with the ABAB stacking, so that the B sublattice of the top interface layer sits on top of the A sublattice of the bottom interface layer. Here we consider the band structure of TDBLG following Ref.s^{24,26}. The details of the Hamiltonian construction can be found in the supplementary information (Supplementary Note 10). For this work, we have taken the following coupling parameters^{24}, ℏv_{0}/a = 2.1354 eV (the nearest neighbour tunneling amplitude along the monolayer graphene sheet), γ_{1} = 400 meV (the caxis interlayer hopping between the dimer sites), γ_{3} = 320 meV (the interlayer hopping between the nondimer sites), γ_{4} = 44 meV (the coupling between dimer and nondimer sites), and \({\Delta }^{{\prime} }=50\,\,{{\mbox{meV}}}\) (the potential difference between dimerized and nondimerized sites). For the AA/BB and AB tunneling amplitudes across the twisted layers, we have used^{24}, u = 79.7 meV and \({u}^{{\prime} }=97.5\,\,{{\mbox{meV}}}\) respectively in our calculations. In this work, we have taken a 184 dimensional matrix which gives an error of < 1% in the band dispersions at the magic angle of 1. 2°.
The Coulomb attraction between the electron and hole pockets lead to formation of indirect exciton condensates in TDBLG near CNP. In this calculation we will replace the Coulomb potential between electrons and holes by a screened short range potential. In fact we will use an effective momentum independent potential with the energy scale V_{0} ~ 10.8 meV. Note that there are three electron pockets separated from the three hole pockets by wavevectors Q_{1(2)(3)}. The mean field Hamiltonian describing the excitonic condensate is given by
where, the \({\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{c(v)}\) represents the noninteracting conduction (valence) band dispersion and the chemical potential is denoted as μ. Note the order parameter Δ is same for all the pockets and is determined selfconsistently. We can then write the modified quasiparticle dispersion relation in presence of the excitonic condensate,
The above energy spectrum generates a finite Fermi surface near CNP, which leads to metallic transport in presence of the condensate.
Data availability
All data needed to evaluate the conclusions in the paper are present in the paper and in an online repository (https://doi.org/10.6084/m9.figshare.24573631). Additional data related to this paper will be available upon reasonable request to the corresponding author.
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Acknowledgements
A.D. thanks the Department of Science and Technology (DST) and Science and Engineering Research Board (SERB), India for financial support (DSTO2051) and acknowledges the Swarnajayanti Fellowship of the DST/SJF/PSA03/201819. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan and the CREST (JPMJCR15F3), JST. UG and RS acknowledge computational facilities at the Department of Theoretical Physics, TIFR Mumbai. R.S. acknowledges support of the Department of Atomic Energy, Government of India, under Project Identification No. RTI 4002.
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A.G., S.C. and A.K.P. contributed to device fabrication and data acquisition. A.G. contributed to 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 development of theory, data interpretation, and all the authors contributed in writing the. manuscript.
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Ghosh, A., Chakraborty, S., Ghorai, U. et al. Evidence of compensated semimetal with electronic correlations at charge neutrality of twisted double bilayer graphene. Commun Phys 6, 360 (2023). https://doi.org/10.1038/s4200502301480x
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DOI: https://doi.org/10.1038/s4200502301480x
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