Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Breakdown of semiclassical description of thermoelectricity in near-magic angle twisted bilayer graphene

Abstract

The planar assembly of twisted bilayer graphene (tBLG) hosts multitude of interaction-driven phases when the relative rotation is close to the magic angle (θm = 1.1). This includes correlation-induced ground states that reveal spontaneous symmetry breaking at low temperature, as well as possibility of non-Fermi liquid (NFL) excitations. However, experimentally, manifestation of NFL effects in transport properties of twisted bilayer graphene remains ambiguous. Here we report simultaneous measurements of electrical resistivity (ρ) and thermoelectric power (S) in tBLG for several twist angles between θ ~ 1.0 − 1.7. We observe an emergent violation of the semiclassical Mott relation in the form of excess S close to half-filling for θ ~ 1.6 that vanishes for θ 2. The excess S (≈2 μV/K at low temperatures T ~ 10 K at θ ≈ 1.6) persists upto ≈40 K, and is accompanied by metallic T-linear ρ with transport scattering rate (τ−1) of near-Planckian magnitude τ−1 ~ kBT/. Closer to θm, the excess S was also observed for fractional band filling (ν ≈ 0.5). The combination of non-trivial electrical transport and violation of Mott relation provides compelling evidence of NFL physics intrinsic to tBLG.

Introduction

In moiré systems with twisted bilayer graphene (tBLG), the amplification of Coulomb correlation effects at low twist angles (θ) is a result of nearly flat low-energy electronic bands1,2 and divergent density of states (DOS) at van Hove singularities (vHS)3. In addition to superconductivity4, ferromagnetism5, the strong correlation effects in tBLG manifest in a cascade of broken symmetry phases at integer band filling factor (ν) close to θ = θm6,7. Near half-filling (ν = ± 2) of the four-fold spin-valley degenerate conduction and valence bands, a linear T-dependence of the resistivity (ρ) seems to indicate an interaction-related absence of a well-defined quasiparticle spectrum, which is concomitant with non-Fermi liquid (NFL) excitations8,9. The persistence of the linearity in ρ for θ well away from θm, e.g. for θ ~ 1.5 − 2, however, has been interpreted in terms of a contrary scenario that is addressable within the non-interacting framework10. The uncertainty persists even in scanning tunneling microscopy experiments11,12,13, where the possibility of an interaction-driven magnetic order has been claimed close to the vHS for θ as high as 1.6, although the spontaneous breaking of C6 lattice symmetry to nematic orbital order has not been observed for θ > θm. Thus a comprehensive understanding of the impact of correlation in tBLG requires a complementary experimental probe that is capable of identifying the departure from non interacting physics in an unambiguous manner.

Here we have carried out simultaneous electrical and thermoelectric measurements in tBLG for twist angles varying from θ ~ 1.0 − 1. 7. The dependence on T and on the carrier density (n) of the thermoelectric power (S), or the Seebeck coefficient, is used as an independent and sensitive probe of the correlation effects. Thermoelectric power is often interpreted as a thermodynamic entity that represents the entropy carried by each charge carrier. Within the degenerate quasiparticle description in the Boltzmann transport regime (TTF, where TF is the Fermi temperature), S is related to the resistance (R) through the semiclassical Mott relation (SMR),

$${S}_{{{{{{{{\rm{Mott}}}}}}}}}=\frac{{\pi }^{2}{k}_{{{{{{{{\rm{B}}}}}}}}}^{2}T}{3| e| }{\left.\frac{{{{{{{{\rm{dln}}}}}}}}R(E)}{{{{{{{{\rm{d}}}}}}}}E}\right|}_{{E}_{{{{{{{{\rm{F}}}}}}}}}},$$
(1)

where R(E), e and EF are the energy-dependent resistance, electronic charge and Fermi energy, respectively. Eq. (1) is valid for a quasiparticle description of transport using semiclassical Boltzmann equation under the assumption that scattering is elastic close to Fermi surface. Remarkably, this simple assumption of elastic quasiparticle scattering remain valid in a wide variety of systems, such as disordered metals/semiconductors14,15, organic materials16, monolayer graphene17 and topological insulators18. The SMR effectively arises from the quasiparticles carrying heat and charge under identical constraints, imposed by the momentum conservation. Thus, the validity of SMR in Eq. (1) provides a definitive probe into the nature of the scattering mechanisms and energy distribution of the charge carriers near the Fermi surface, and it breaks down when strong correlation effects become important15,19.

Results

The tBLG devices we study were created using standard van der Waals stacking20, which consists of two graphene layers aligned at either 60 + θ or at θ, where θ is the effective twist angle, and encapsulated within two sheets of hexagonal boron nitride (hBN) (See Supplementary Note 1). A local top-gate tunes n in the overlap region where the moiré super-lattice is formed. Figure 1a shows the four terminal resistance R measured across the tBLG devices as a function of band filling factor ν and T for four different θ. The recurring features in R across the tBLG devices can be identified as the maxima in R at the charge neutrality point (CNP) and at the full-filling of the moiré band (ν = ±4). In addition, near θm, the device with θ ≈ 1.01 exhibit additional maxima in R at integer values of ν, whereas for θ ≈ 1.24 resistance peaks are shifted slightly away from integer fillings. For θ ≈ 1.24, we observe a substantial shift Δν ~ 0.25 of resistance peaks near ν = +1 and +3 from 7 K to 35 K, suggesting the possibility of isospin-polarization in the system21,22 (See Supplementary Note 4). We speculate that the noticeable asymmetry in the doping dependence of R on the electron and hole sides is most likely related to the particle-hole asymmetry of the band structure since in both tBLG devices near θm, the correlated states are more pronounced at electron doping.

Fig. 1: Electrical transport in twisted bilayer graphene.
figure 1

a Temperature-dependent resistance R as a function of band filling ν for four different devices with twist angle θ ≈ 1.01, 1.24, 1.6 and 1.7. The inset in the bottom panel shows the optical image of a typical device with current and voltage leads marked for resistivity measurements. The scale bar represents a length of 5 μm. ρ as a function of T for a few representative values of ν for (b) θ ≈ 1.01, (c) 1.24 and (d) 1. 6. The various curves for fractional ν in (c) represent the T-dependence of the CI/CS states as marked in the second panel of (a). e Comparison of T-dependence of ρ at ν = −2 for different twist angles. The solid lines represent T-linearity.

For 0 < ν < 4, the T-dependence for all devices was found to be generally metallic at low temperatures  40 K (Fig. 1b–d). However, the resistance peaks near integer ν exhibit either weak insulating T-dependence, i.e, a correlated insulating phase (CI), or T-linear resistivity i.e, a correlated semimetallic (CS) phase. In the metallic regime, ρ can expressed as ρ = ρ0 + AT, where ρ0 is the residual resistivity. The values of A (~10–100 Ω/K) are at least two orders of magnitude greater compared to that of the monolayer graphene, and are consistent with the earlier transport measurements in tBLG devices10. From the comparison of ρ(T) at half-filling (ν = −2) in Fig. 1e, we find that the resistivity is linear in T for all four low-angle devices. However, at θ ≈ 1.24 and θ ≈ 1.01, the linearity persists only upto ~ 40 K, which could be due to the smaller bandwidth and smaller bandgap10. We also note that the value of A (~40–100 Ω/K) at ν = −2 is much larger near θ ~ θm than that of the devices away from θm (See Supplementary Note 4). The ubiquitous T-linearity across all tBLG devices at low temperature is a clear departure from ρ ~ T2 dependence associated with electron-electron scattering, or the ρT4 behavior, expected due to electron-acoustic phonon scattering below the Bloch-Grüneisen temperature (TBG)23. While this suggests the continuum of correlation-driven metallic states across the tBLG devices, an alterate scenario has been proposed10,24 to view the tBLG in this regime as a two dimensional, weakly (or non-) interacting metal with largely reduced TBG.

To complement the electrical transport, we have performed thermoelectric measurements on the same devices. Briefly, a sinusoidal current (Iω) is allowed to flow between two contacts of the monolayer branch outside the top gated region, setting up a temperature gradient (ΔT) across the tBLG region (Fig. 2a, b). The resulting second-harmonic thermovoltage (V2ω) is recorded on the tBLG region as a function of doping and heating current17,20. The linear response was ensured from \({V}_{2\omega }\propto {I}_{\omega }^{2}\) for the range of heating current used (Fig. 2c–e). We begin with the results in tBLG devices closer to θm. Figure 2c exhibits the ν-dependence of normalized V2ω for tBLG device θ ≈ 1.01 at low temperature (5 K), which exhibits multiple sign-reversals when EF is varied across the lowest energy band. While the sign reversals near the CNP and the super-lattice gaps at ν = ±4 are due to changes in the quasiparticle excitations, those near integer values of 0 < ν < 4 can be attributed to the correlated states. The sign-reversal of V2ω near each correlated states is fascinating since it indicates a change in the topology of the Fermi surface, which is naturally associated with the Lifshitz transition25,26. Although the correlated states are metallic in nature (Fig. 1a), the concomitant Lifshitz transitions depicts the interaction-driven occurrence of diverging DOS at each integer values of ν. This is in stark contrast to the charge-inversions at ν = 0, ±4, and hints at the topological facet of the lowest energy band when filled with integer number of charge carriers7,27.

Fig. 2: Thermoelectric transport in twisted bilayer graphene.
figure 2

a The cross-sectional view of the device, showing the constituent layers, electrical contacts, and the gate assembly. b In-plane heating and measurement schematic for thermovoltage V2ω in the tBLG region. Density dependence of \({V}_{2\omega }/{I}_{\omega }^{2}\) (in the units of VA−2 × 106) measured at 5 K for (c) θ ≈ 1.01, d 1.24, and at 3 K for (e) 1. 6 device. The different curves in each panel represent the \({V}_{2\omega }/{I}_{\omega }^{2}\) measured at different Iω. The bottom graphs in each panel show the numerically calculated α = (1/R)dR/dn (in the units of m2) for comparison.

To establish the connection between the two different types of transports, we rewrite Eq. (1) as,

$${S}_{{{{{{{{\rm{Mott}}}}}}}}}=\frac{{\pi }^{2}{k}_{{{{{{{{\rm{B}}}}}}}}}^{2}T}{3| e| }\frac{1}{R}\frac{{{{{{{{\rm{d}}}}}}}}R}{{{{{{{{\rm{d}}}}}}}}{V}_{{{{{{{{\rm{tg}}}}}}}}}}\frac{{{{{{{{\rm{d}}}}}}}}{V}_{{{{{{{{\rm{tg}}}}}}}}}}{{{{{{{{\rm{d}}}}}}}}n}{\left.\frac{{{{{{{{\rm{d}}}}}}}}n}{{{{{{{{\rm{d}}}}}}}}E}\right|}_{{E}_{{{{{{{{\rm{F}}}}}}}}}},$$
(2)

where (1/R)dR/dVtg is measured experimentally, and dn/dE is the DOS (dVtg/dn = e/ChBN, where ChBN is the known topgate capacitance per unit area). The difficulty in accurate estimation of DOS in the presence of strong correlation effects near θm prohibits us from accurately estimating SMott, in particular close to the integer fillings for θ ≈ 1.01 and ≈ 1.24. Although a qualitative correspondence in the oscillations and sign-reversals of the measured V2ω and α = (1/R)dR/dn can be seen in these devices including at the CNP and the superlattice gap, absence of accurate knowledge of the DOS prohibits a quantitative estimation of the deviation of the measured thermopower from that expected from the semiclassical model. However, for the device with θ ≈ 1.24 (Fig. 2d), we detect an excess V2ω near ν ~ 0.5 which has no analogue in α. While this indicates a clear violation of the Mott relation and highlights the possible manifestation of electron-correlation effects at fractional band filling27, the exact origin of the excess V2ω at ν ≈ 0.5 is not clear at present.

Although the interaction-effects are expected to be weaker when θ is away from θm, the devices θ ≈ 1. 6 and 1. 7 provide a better quantitative comparison with SMR as the non-interacting DOS can be calculated with greater accuracy. The qualitative comparison of V2ω with α at θ ≈ 1. 6 exhibits a discrepancy at low temperature (3 K), where two additional extrema, consisting of a maximum at ν = +2 and minimum at ν = −2, are distinctly absent in α (Fig. 2e). Figure 3a shows the tight binding calculation for the electronic band structure and the corresponding DOS for θ ≈ 1. 6 (See Methods and Supplementary Note 8 for more details on the band structure calculations). Using ΔT as the single fitting parameter, we obtain excellent agreement between the measured V2ω and Eq. (2) at the CNP (ν ~ 0), ν ~ ±4, and also in the higher energy dispersive band (ν > ±4) simultaneously (See Supplementary Fig. 15). For fitting Eq. (2), we also note that TTF is maintained throughout almost the entire temperature and gate voltage range shown in Fig. 3b, except very close to the CNP (ν = 0) and ν = ±4 (See Supplementary Note 6). While the SMR explains the observed V2ω over almost the entire doping regime ( −4 ν +4) at high temperatures ( 40 K) (bottom panel of Fig. 3b), the excess thermopower centered around ν = ±2, becomes evident at lower T. We also find that the excess thermovoltage is intrinsically particle-hole asymmetric, however, on the electron doped side, the excess thermopower is closer to the commensurate filling (ν = +2) as seen for two devices (Figs. 3b and c). We also detect evidence of small excess V2ω between ν = −3 and −4. This could also be due to electron-correlation effects, however, the exact origin is not clear as we do not observe any evidence of such anomalous thermopower near same filling factor in the other devices (see e.g. Fig. 3c for the device with θ ≈ 1. 7). Using the ΔT extracted from the fitting of V2ω, we show the T-dependence of S = V2ωT in Fig. 3d for different ν (See Supplementary Note 6). Evidently, S exhibits a linear dependence on T at all doping including the higher-energy dispersive band, except in the vicinity of ν = ±2, thus validating the estimation of ΔT from Mott fitting25. The ST behavior is expected in a degenerate weakly or non-interacting metal within the semiclassical framework, and has been verified for monolayer graphene17 as well as tBLG at slightly larger θ (2θ 5)28. Close to ν = ±2, however, we find an unexpected increase in S when temperature is decreased below ~ 40 K, in contrast to the expectation of S ≈ 0 (inset of Fig. 3d) from SMR and approaches S ≈ ±2μV/K for ν = ±2 respectively, at low T (Figs. 3d and 4b). This is remarkable because, (1) at low T, the observed sign of V2ω can not be assigned to the electron(hole)-like bands any more, and (2) the excess S persists to a temperature scale ( ~40 K) that is much higher than the superconducting transition (Tc ~ 1.7 K) in tBLG at θ = θm or the temperature scale for correlated insulator ( 4 K)4,6,11, suggesting a very distinct nature of the ground state. The absolute magnitude of the excess thermopower at ν = ±2 decreases with increasing θ, as illustrated for a device with θ = 1. 7 in Fig. 3c, and becomes undetectable for θ 2.

Fig. 3: Comparison with semiclassical Mott relation at ~ 1.6.
figure 3

a Electronic band structure and density of states (DOS) of tBLG (θ = 1. 6) calculated using tight binding model. The bands shown in red are the low energy active bands. b Comparison between the measured V2ω (black lines) and that calculated (orange line) from the semiclassical Mott relation (Eq. (2)) for θ ≈ 1. 6 at three representative temperatures. ΔT is obtained as a fitting parameter to match SMR with the experimental V2ω at the CNP. c Doping dependence of S for θ ≈ 1. 7 compared to that of the SMR at 5 K. d Temperature dependence of S at various band filling factors. The dashed lines show the ST dependence. The inset shows the T dependence of \(S/{S}_{\max }\) at ν = ±2.

Fig. 4: Breakdown of semiclassical Mott relation and scattering rate.
figure 4

a Surface plot of \((S-{S}_{{{{{{{{\rm{Mott}}}}}}}}})/{S}_{\max }\) as a function of T and ν for θ ≈ 1. 6. b (S − SMott) at 10 K for θ ≈ 1. 6 and θ ~ 4. c dρ/dT extracted in the T-linear regime at different ν for the three twist angles. d The estimated dimensionless pre-factor C of the scattering rate Γ = CkBT/ as a function of ν. e Seebeck coefficient S computed in DMFT with U = 38 meV as a function of filling ν for the four lowest bands at three temperatures T = 2 K, 14 K and 26 K, respectively.

Discussion

Although the Mott formula has been verified in a range of graphene-based devices17,25, it can be violated in the hydrodynamic regime29 and due to phonon drag in cross-plane thermoelectric transport in tBLG at θ > 620. While the hydrodynamic regime is expected to appear at higher temperatures (>100 K), we eliminate the possibility of phonon drag from the observation of ST (away from ν = ±2, Fig. 3d). Furthermore, as shown in Fig. 4a, the occurrence of excess S, normalized as \((S-{S}_{{{{{{{{\rm{Mott}}}}}}}}})/{S}_{\max }\), where \({S}_{\max }\) is the maximum value of S at a given T, is concentrated in the low T dome-like regions around ν = ±2 in the T − (ν, n) phase diagram. Since neither adiabatic (static) nor dynamical phonon effects can violate Mott formula30,31, the enhancement of thermopower beyond the SMR limit suggests the possibility of a many body ground state similar to NFL phases in correlated oxides32 and heavy Fermions33. A near-ubiquitous feature of the NFL regime in itinerant Fermionic systems, ranging from cuprates34, ruthanates35, pnictides36 to heavy Fermions33, is the ‘strange metal’ phase, characterized by the absence of well defined quasiparticles and linear T dependence of ρ. Theoretical work also suggests possibilities of excess entropy, analogous to Bekenstein-Hawking entropy in charged black holes, in this regime, that remains finite down to vanishingly small T37.

To check the mutuality between the excess thermopower and the strange metallic behaviour, we compare the ν-dependence of excess S at T = 10 K (Fig. 4b), and the scattering rate obtained from the slope dρ/dT in the T-dependence of ρ (Fig. 4c). For reference, we also present the results from another device at θ ≈ 4, where we find no violation of SMR over the experimental range of n. In the NFL state, the incoherent scattering rate is τ−1 = CkBT/, where the dimensionless coefficient C is of the order of unity for Planckian dissipation. In Fig. 4c we plot the ν-dependence of dρ/dT and C (Fig. 4d), where C is computed from dρ/dT assuming Drude-like resistivity in accordance to Refs. 8,9 (See Supplementary Note 7). Away from the CNP, both 1. 6 and 1. 7 devices show dρ/dT ≈ 10 Ω/K near ν ≈ ±2, which is nearly two orders of magnitude larger than dρ/dT ≈ 0.2–0.3 Ω/K for the tBLG device at θ ~ 4, implying that the individual layers are essentially decoupled in the latter8,10. Intriguingly, for tBLG at θ = 1. 6 and 1. 7, we find C to approach the order of unity in the vicinity of ν → ±2, raising the possibility of a common physical origin for the violation of SMR. Notably, the excess thermopower was found largely unaffected in the in-plane magnetic field (See Supplementary Fig. 20), and thus unlikely to arise from an underlying spin/magnetic texture5. Theoretically a dynamical mean field theory (DMFT)3,38,39 calculation shows qualitative agreement in the density dependence of excess thermopower at ν = ±2 but fails to capture its finite magnitude at low temperature (Fig. 4e). This is because the particular single site DMFT framework used in our calculation would invariably lead to the FL phase as T → 0, even though some excess thermopower can be observed in the intermediate temperature range (See Supplementary Note 8). Particle-hole asymmetry due to Ω/T (Ω is the energy of an excitation counted from Fermi level) scaling in certain non-Fermi liquid Planckian metals, on the other hand, may not only cause a logarithmically divergent S at low T (Fig. 3d, inset), but also a sign reversal in S for the electron- and hole-type bands40,41.

In summary, we have measured the electrical resistivity and thermopower in twisted bilayer graphene over a broad range of low-twist angles. At larger θ (~1. 6 − 1. 7), our experimental results show concurrent T-linear resistivity at Planckian dissipation scales and emergent excess thermopower below T 40 K near ν = ±2 signifying the breakdown of the semiclassical Mott relation. The thermopower near ν = ±2 approaches a finite magnitude (≈2 μV/K at 1. 6) at low T providing a new facet to the strongly correlated ‘strange metal’ phase in tBLG. Our experimental results point to a truly non-Fermi liquid (NFL) metallic state in tBLG at low twist angle that carry strong similarities to those observed in cuprates or heavy-Fermion materials with low coherence temperatures.

Methods

Device fabrication

All devices in this work were fabricated using a layer-by-layer mechanical transfer method20. Monolayer graphene and hexagonal boron nitride (hBN) were exfoliated on SiO2/Si wafers and graphene flakes were identified using optical microscopy and Raman spectroscopy. For θ ≈ 1. 6, the edges of the graphene flakes were aligned under an optical microscope and encapsulated within two hBN layers. Other tBLG devices were fabricated using tear and stack method42. Electron beam lithography was used to define Cr/Au top gate for tuning the number density in the tBLG region. Finally, the electrical contacts were patterned by electron-beam lithography and reactive ion etching followed by metal deposition (5 nm Cr/50 nm Au) using thermal evaporation technique.

Electrical transport measurements were performed in a four-terminal geometry with typical ac current excitations of 10–100 nA using a standard low-frequency lock-in amplifier at 226 Hz, in a dilution refrigerator and a 1.5-K cryostat. For thermoelectric measurements, local Joule heating was employed to create a ΔT across the tBLG channel. A range of sinusoidal currents (2–5 μA) at excitation frequency ω = 17 Hz were used for Joule heating and the resulting 2nd harmonic thermal voltage (V2ω) was recorded using a lock-in amplifier. Thermoelectric measurements were conducted in a 1.5-K cryostat/20 mK dilution refrigerator with magnetic field.

Tight binding calculation of DOS

The rigid bilayer structures were generated using the Twister code43. The structures were subsequently relaxed in LAMMPS44,45 using REBO46 as the intralayer potential and DRIP47 as the interlayer potential. These relaxed structures were used for performing all the calculations.

The electronic band structures were calculated by approximating the tight binding transfer integrals under the Slater Koster formalism48. A more detailed discussion of the calculations is available in the Supplementary Note 8.

Data availability

Source data are available for this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.

References

  1. Trambly de Laissardière, G., Mayou, D. & Magaud, L. Localization of Dirac electrons in rotated graphene bilayers. Nano Lett. 10, 804–808 (2010).

    Article  ADS  Google Scholar 

  2. Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. 108, 12233–12237 (2011).

    CAS  Article  ADS  Google Scholar 

  3. Yuan, N. F. Q., Isobe, H. & Fu, L. Magic of high-order van Hove singularity. Nat. Comm. 10, 5769 (2019).

  4. Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43 (2018).

    CAS  Article  ADS  Google Scholar 

  5. Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019).

    CAS  Article  ADS  Google Scholar 

  6. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80 (2018).

    CAS  Article  ADS  Google Scholar 

  7. Zondiner, U. et al. Cascade of phase transitions and dirac revivals in magic-angle graphene. Nature 582, 203–208 (2020).

    CAS  Article  ADS  Google Scholar 

  8. Cao, Y. et al. Strange metal in magic-angle graphene with near Planckian dissipation. Phys. Rev. Lett. 124, 076801 (2020).

    CAS  Article  ADS  Google Scholar 

  9. Bruin, J., Sakai, H., Perry, R. & Mackenzie, A. Similarity of scattering rates in metals showing T-linear resistivity. Science 339, 804–807 (2013).

    CAS  Article  ADS  Google Scholar 

  10. Polshyn, H. et al. Large linear-in-temperature resistivity in twisted bilayer graphene. Nat. Phys. 15, 1011–1016 (2019).

    CAS  Article  Google Scholar 

  11. Kerelsky, A. et al. Maximized electron interactions at the magic angle in twisted bilayer graphene. Nature 572, 95–100 (2019).

    CAS  Article  ADS  Google Scholar 

  12. Jiang, Y. et al. Charge order and broken rotational symmetry in magic-angle twisted bilayer graphene. Nature 573, 91–95 (2019).

    CAS  Article  ADS  Google Scholar 

  13. Liu, Y.-W. et al. Magnetism near half-filling of a van hove singularity in twisted graphene bilayer. Phys. Rev. B 99, 201408 (2019).

    CAS  Article  ADS  Google Scholar 

  14. Rowe, D. M. Materials, preparation, and characterization in thermoelectrics (CRC press, 2017).

  15. Behnia, K. Fundamentals of thermoelectricity (OUP Oxford, 2015).

  16. Watanabe, S. et al. Validity of the Mott formula and the origin of thermopower in π-conjugated semicrystalline polymers. Phys. Rev. B 100, 241201 (2019).

    CAS  Article  ADS  Google Scholar 

  17. Zuev, Y. M., Chang, W. & Kim, P. Thermoelectric and magnetothermoelectric transport measurements of graphene. Phys. Rev. Lett. 102, 096807 (2009).

    Article  ADS  Google Scholar 

  18. Kim, D., Syers, P., Butch, N. P., Paglione, J. & Fuhrer, M. S. Ambipolar surface state thermoelectric power of topological insulator Bi2Se3. Nano Lett. 14, 1701–1706 (2014).

    CAS  Article  ADS  Google Scholar 

  19. Arsenijević, S. et al. Signatures of quantum criticality in the thermopower of Ba(Fe1−xCox)2As2. Phys. Rev. B 87, 224508 (2013).

    Article  ADS  Google Scholar 

  20. Mahapatra, P. S., Sarkar, K., Krishnamurthy, H. R., Mukerjee, S. & Ghosh, A. Seebeck coefficient of a single van der Waals junction in twisted bilayer graphene. Nano Lett. 17, 6822–6827 (2017).

    CAS  Article  ADS  Google Scholar 

  21. Saito, Y. et al. Isospin pomeranchuk effect and the entropy of collective excitations in twisted bilayer graphene. arXiv preprint arXiv:2008.10830 (2020).

  22. Rozen, A. et al. Entropic evidence for a pomeranchuk effect in magic angle graphene. arXiv preprint arXiv:2009.01836 (2020).

  23. Efetov, D. K. & Kim, P. Controlling electron-phonon interactions in graphene at ultrahigh carrier densities. Phys. Rev. Lett. 105, 256805 (2010).

    Article  ADS  Google Scholar 

  24. Wu, F., Hwang, E. & Sarma, S. D. Phonon-induced giant linear-in-T resistivity in magic angle twisted bilayer graphene: Ordinary strangeness and exotic superconductivity. Phys. Rev. B 99, 165112 (2019).

    CAS  Article  ADS  Google Scholar 

  25. Jayaraman, A. et al. Evidence of lifshitz transition in the thermoelectric power of ultrahigh-mobility bilayer graphene. Nano Lett. 21, 1221–1227 (2021).

    MathSciNet  CAS  Article  ADS  Google Scholar 

  26. Buhmann, J. M. & Sigrist, M. Thermoelectric effect of correlated metals: Band-structure effects and the breakdown of Mott’s formula. Phys. Rev. B 88, 115128 (2013).

    Article  ADS  Google Scholar 

  27. Wu, S., Zhang, Z., Watanabe, K., Taniguchi, T. & Andrei, E. Y. Chern insulators, van hove singularities and topological flat bands in magic-angle twisted bilayer graphene. Nat. Mat. 20, 488–494 (2021).

    CAS  Article  Google Scholar 

  28. Mahapatra, P. S. et al. Misorientation-controlled cross-plane thermoelectricity in twisted bilayer graphene. Phys. Rev. Lett. 125, 226802 (2020).

    CAS  Article  ADS  Google Scholar 

  29. Ghahari, F. et al. Enhanced thermoelectric power in graphene: violation of the Mott relation by inelastic scattering. Phys. Rev. Lett. 116, 136802 (2016).

    Article  ADS  Google Scholar 

  30. Jonson, M. & Mahan, G. Mott’s formula for the thermopower and the Wiedemann-Franz law. Phys. Rev. B 21, 4223 (1980).

    MathSciNet  CAS  Article  ADS  Google Scholar 

  31. Jonson, M. & Mahan, G. Electron-phonon contribution to the thermopower of metals. Phys. Rev. B 42, 9350 (1990).

    CAS  Article  ADS  Google Scholar 

  32. Wang, Y., Rogado, N. S., Cava, R. J. & Ong, N. P. Spin entropy as the likely source of enhanced thermopower in NaxCo2O4. Nature 423, 425–428 (2003).

    CAS  Article  ADS  Google Scholar 

  33. Izawa, K. et al. Thermoelectric response near a quantum critical point: the case of CeCoIn5. Phys. Rev. Lett. 99, 147005 (2007).

    CAS  Article  ADS  Google Scholar 

  34. da Silva Neto, E. H. et al. Ubiquitous interplay between charge ordering and high-temperature superconductivity in cuprates. Science 343, 393–396 (2014).

    Article  ADS  Google Scholar 

  35. Rost, A., Perry, R., Mercure, J.-F., Mackenzie, A. & Grigera, S. Entropy landscape of phase formation associated with quantum criticality in Sr3Ru2O7. Science 325, 1360–1363 (2009).

    CAS  Article  ADS  Google Scholar 

  36. Lee, W.-C. & Phillips, P. W. Non-Fermi liquid due to orbital fluctuations in iron pnictide superconductors. Phys. Rev. B 86, 245113 (2012).

    Article  ADS  Google Scholar 

  37. Sachdev, S. Bekenstein-Hawking entropy and strange metals. Phys. Rev. X 5, 041025 (2015).

    Google Scholar 

  38. Georges, A., Kotliar, G., Krauth, W. & Rozenberg, M. J. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13–125 (1996).

    MathSciNet  CAS  Article  ADS  Google Scholar 

  39. Haldar, A., Banerjee, S. & Shenoy, V. B. Higher-dimensional Sachdev-Ye-Kitaev non-Fermi liquids at Lifshitz transitions. Phys. Rev. B 97, 241106 (2018).

    CAS  Article  ADS  Google Scholar 

  40. Gourgout, A. et al. Seebeck coefficient in a cuprate superconductor: particle-hole asymmetry in the strange metal phase and fermi surface transformation in the pseudogap phase. arXiv preprint arXiv:2106.05959 (2021).

  41. Georges, A. & Mravlje, J. Skewed non-fermi liquids and the seebeck effect. arXiv preprint arXiv:2102.13224 (2021).

  42. Kim, K. et al. Tunable moiré bands and strong correlations in small-twist-angle bilayer graphene. Proc. Natl Acad. Sci. 114, 3364–3369 (2017).

    CAS  Article  ADS  Google Scholar 

  43. Naik, M. H. & Jain, M. Ultraflatbands and shear solitons in moire patterns of twisted bilayer transition metal dichalcogenides. Phys. Rev. Lett. 121, 266401 (2018).

    CAS  Article  ADS  Google Scholar 

  44. Plimpton, S. Fast parallel algorithms for short-range molecular dynamics (1993).

  45. https://lammps.sandia.gov

  46. Brenner, D. W. et al. A second-generation reactive empirical bond order (rebo) potential energy expression for hydrocarbons. J. Phys. Cond. Mat. 14, 783 (2002).

    CAS  Article  ADS  Google Scholar 

  47. Wen, M., Carr, S., Fang, S., Kaxiras, E. & Tadmor, E. B. Dihedral-angle-corrected registry-dependent interlayer potential for multilayer graphene structures. Phys. Rev. B 98, 235404 (2018).

    CAS  Article  ADS  Google Scholar 

  48. Slater, J. C. & Koster, G. F. Simplified LCAO method for the periodic potential problem. Phys. Rev. 94, 1498 (1954).

    CAS  Article  ADS  Google Scholar 

Download references

Acknowledgements

The authors thank Nano mission, DST for the financial support. M.J. and S.M. thank the computational facilities in SERC. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, Grant Number JPMXP0112101001, JSPS KAKENHI Grant Numbers JP20H00354 and the CREST(JPMJCR15F3), JST. U.C. acknowledges funding from IISc and SERB (ECR/2017/001566), and H.R.K. from SERB(SB/DF/005/2017). S.B. acknowledges funding from IISc and SERB (ECR/2018/001742).

Author information

Authors and Affiliations

Authors

Contributions

B.G., P.S.M. and M.G. contributed equally to this work. B.G., M.G. and SA.B. fabricated the devices with help from R.S. The transport measurements were performed by B.G., P.S.M. and M.G. with help from A.J. The results were analysed by B.G., P.S.M. and M.G. S.M., M.J., H.R.K. and S.B. provided the presented theory calculations. Hexagonal boron nitride crystals were grown by K.W. and T.T. A.G. and U.C. contributed in the data interpretation, and theoretical understanding of the manuscript. B.G., P.S.M. and A.G. wrote the manuscript with inputs from all authors.

Corresponding authors

Correspondence to Bhaskar Ghawri, Phanibhusan S. Mahapatra, Manjari Garg or Arindam Ghosh.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Source data

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ghawri, B., Mahapatra, P.S., Garg, M. et al. Breakdown of semiclassical description of thermoelectricity in near-magic angle twisted bilayer graphene. Nat Commun 13, 1522 (2022). https://doi.org/10.1038/s41467-022-29198-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1038/s41467-022-29198-4

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing