SEARCH · Search NASA

Results for “Bilayer graphene”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 91 records · Page 5

All sources NASA NTRS DOE OSTI

DOE OSTI2021

Lattice collective modes from a continuum model of magic-angle twisted bilayer graphene

Here, we show that the insulating states of magic-angle twisted bilayer graphene support a series of collective modes corresponding to local particle-hole excitations on triangular lattice sites. Our theory is based on a continuum model of the magic angle flat bands. When the system is insulating at moiré band filling ν = -3, our calculations show that the ground state supports seven low-energy modes that lie well below the charge gap throughout the moiré Brillouin zone, one of which couples strongly to THz photons. The low-energy collective modes are faithfully described by a model with a local SU(8) degree of freedom in each moiré unit cell that we identify as the direct product of spin, valley, and an orbital pseudospin. Apart from spin and valley-wave modes, the collective mode spectrum includes a low-energy intraflavor exciton mode associated with transitions between flat valence and conduction band orbitals.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2022

Gate-defined wires in twisted bilayer graphene: From electrical detection of intervalley coherence to internally engineered Majorana modes

Twisted bilayer graphene (TBG) realizes a highly tunable, strongly interacting system featuring superconductivity and various correlated insulating states. We establish gate-defined wires in TBG with proximity-induced spin-orbit coupling as (i) a tool for revealing the nature of correlated insulators and (ii) a platform for Majorana-based topological qubits. We show that the band structure of a gate-defined wire immersed in an intervalley coherent correlated insulator inherits electrically detectable fingerprints of symmetry breaking native to the latter. Surrounding the wire by a superconducting TBG region on one side and an intervalley coherent correlated insulator on the other further enables the formation of Majorana zero modes—possibly even at zero magnetic field depending on the precise symmetry-breaking order present. Our proposal not only introduces a highly gate-tunable topological qubit medium relying on internally generated proximity effects but can also shed light on the Cooper-pairing mechanism in TBG.

36 MATERIALS SCIENCE↗

DOE OSTI2020

Mechanism for Anomalous Hall Ferromagnetism in Twisted Bilayer Graphene

Motivated by the recent observation of an anomalous Hall effect in twisted bilayer graphene, here we use a lowest Landau level model to understand the origin of the underlying symmetry-broken correlated state. This effective model is rooted in the occurrence of Chern bands which arise due to the coupling between the graphene device and its encapsulating substrate. Our model exhibits a phase transition from a spin-valley polarized insulator to a partial or fully valley unpolarized metal as the bandwidth is increased relative to the interaction strength, consistent with experimental observations. In sharp contrast to standard quantum Hall ferromagnetism, the Chern number structure of the flat bands precludes an instability to an intervalley coherent phase, but allows for an excitonic vortex lattice at large interaction anisotropy.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2022

Magic-Angle Twisted Bilayer Graphene as a Topological Heavy Fermion Problem

Magic-angle (θ=1.05°) twisted bilayer graphene (MATBG) has shown two seemingly contradictory characters: the localization and quantum-dot-like behavior in STM experiments, and delocalization in transport experiments. We construct a model, which naturally captures the two aspects, from the Bistritzer-MacDonald (BM) model in a first principle spirit. A set of local flat-band orbitals (f) centered at the AA-stacking regions are responsible to the localization. A set of extended topological semimetallic conduction bands (c), which are at small energetic separation from the local orbitals, are responsible to the delocalization and transport. The topological flat bands of the BM model appear as a result of the hybridization of f and c electrons. This model then provides a new perspective for the strong correlation physics, which is now described as strongly correlated f electrons coupled to nearly free c electrons—we hence name our model as the topological heavy fermion model. Using this model, we obtain the U(4) and U(4)×U(4) symmetries of Refs. [1–5] as well as the correlated insulator phases and their energies. Simple rules for the ground states and their Chern numbers are derived. Moreover, features such as the large dispersion of the charge ±1 excitations [2,6,7], and the minima of the charge gap at the Γ M point can now, for the first time, be understood both qualitatively and quantitatively in a simple physical picture. Our mapping opens the prospect of using heavy-fermion physics machinery to the superconducting physics of MATBG.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2025

Correlation effects in magic-angle twisted bilayer graphene: An auxiliary-field quantum Monte Carlo study

Magic-angle twisted bilayer graphene (MATBG) presents a fascinating platform for investigating the effects of electron interactions in topological flat bands. The Bistritzer-MacDonald (BM) model provides a simplified quantitative description of the flat bands. Introducing long-range Coulomb interactions leads to an interacting BM (IBM) Hamiltonian, a momentum-space continuum description which offers a very natural starting point for many-body studies of MATBG. Accurate and reliable many-body computations in the IBM model are challenging, however, and have been limited mostly to special fillings or smaller lattice sizes. We employ a state-of-the-art auxiliary-field quantum Monte Carlo (AFQMC) method to study the IBM model, which constrains the sign problem to enable accurate treatment of large system sizes. We determine ground-state properties and quantify errors compared to mean-field theory calculations. Our calculations identify correlated metal states and their competition with the insulating Kramers intervalley-coherent state at both half-filling and charge neutrality. Additionally, we investigate one- and three-quarter fillings, and examine the effect of many-body corrections beyond single Slater determinant solutions. We discuss the effect that details of the IBM Hamiltonian have on the results, including different forms of double-counting corrections, and the need to establish and precisely specify many-body Hamiltonians to allow more direct and quantitative comparisons with experiments in MATBG. Published by the American Physical Society 2025

Xiao, Zhi-Yu (ORCID:0000000219531579)↗

DOE OSTI2024

Electron– K -phonon interaction in twisted bilayer graphene

We develop an analytic theory to describe the interaction between electrons and K phonons and study its influence on superconductivity in the bare bands of twisted bilayer graphene (TBG). We find that, due to symmetry and the two-center approximation, only one optical K phonon (160meV) of graphene is responsible for the intervalley electron-phonon interaction. Here, this phonon has recently been found in angular-resolved photoemission spectroscopy to be responsible for replicas of the TBG flat bands. By projecting the interaction to the TBG flat bands, we perform the full symmetry analysis of the phonon-mediated attractive interaction and pairing channels in the Chern basis, and show that several channels are guaranteed to have gapless order parameters. From the linearized gap equations, we find that the highest T c pairing induced by this phonon is a singlet gapped s-wave inter-Chern-band order parameter, followed closely by a gapless nematic d-wave intra-Chern-band order parameter. We justify these results analytically, using the topological heavy-fermion mapping of TBG which has allowed us to obtain an analytic form of a phonon-mediated attractive interaction and to analytically solve the linearized and T = 0 gap equations. For the intra-Chern-band channel, the nematic state with nodes is shown to be stabilized in the chiral flat-band limit. While the flat-band Coulomb interaction can be screened sufficiently enough—around the Van Hove singularities—to allow for electron-phonon based superconductivity, it is unlikely that this effect can be maintained in the lower density of states excitation bands around the correlated insulator states.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

Symmetric Kondo Lattice States in Doped Strained Twisted Bilayer Graphene

We use the topological heavy fermion (THF) model and its Kondo lattice (KL) formulation to study the possibility of a symmetric Kondo (SK) state in twisted bilayer graphene. Via a large-N approximation, we find a SK state in the KL model at fillings v = 0,±1,±2 where a KL model can be constructed. In the SK state, all symmetries are preserved and the local moments are Kondo screened by the conduction electrons. At the mean-field level of the THF model at v = 0,±1,±2,±3 we also find a similar symmetric state that is adiabatically connected to the symmetric Kondo state. We study the stability of the symmetric state by comparing its energy with the ordered (symmetry-breaking) states found in [H. Hu et al., Phys. Rev. Lett. 131, 026502 (2023)., Z.-D. Song and B. A. Bernevig, Phys. Rev. Lett. 129, 047601 (2022).] and find the ordered states to have lower energy at v = 0,±1,±2. However, moving away from integer fillings by doping the light bands, our mean-field calculations find the energy difference between the ordered state and the symmetric state to be reduced, which suggests the loss of ordering and a tendency toward Kondo screening. In order to include many-body effects beyond the mean-field approximation, we also performed dynamical mean-field theory calculations on the THF model in the nonordered phase. The spin susceptibility follows a Curie behavior at v = 0,±1,±2 down to ~2 K where the onset of screening of the local moment becomes visible. This hints to very low Kondo temperatures at these fillings, in agreement with the outcome of our mean-field calculations. At noninteger filling v = ±0.5,±0.8,±1.2 dynamical mean-field theory shows deviations from a 1/T susceptibility at much higher temperatures, suggesting a more effective screening of local moments with doping. Lastly, we study the effect of a C 3⁢z -rotational-symmetry-breaking strain via mean-field approaches and find that a symmetric phase (that only breaks C 3⁢z symmetry) can be stabilized at sufficiently large strain at v = 0,±1,±2. Our results suggest that a symmetric Kondo phase is strongly suppressed at integer fillings, but could be stabilized either at noninteger fillings or by applying strain.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

Junctions and Superconducting Symmetry in Twisted Bilayer Graphene

Junctions provide a wealth of information on the symmetry of the order parameter of superconductors. Here, we analyze junctions between a scanning tunneling microscope (STM) tip and superconducting twisted bilayer graphene (TBG) and TBG Josephson junctions (JJs). We compare superconducting phases that are even or odd under valley exchange (s- or ƒ-wave). The critical current in mixed (s and ƒ) JJs strongly depends on the angle between the junction and the lattice. In STM-TBG junctions, due to Andreev reflection, the ƒ-wave leads to a prominent peak in subgap conductance, as seen in experiments.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Twisted bilayer graphene. I. Matrix elements, approximations, perturbation theory, and a k · p two-band model

We investigate the twisted bilayer graphene (TBG) model of Bistritzer and MacDonald (BM) [Bistritzer and MacDonald, Proc. Natl. Acad. Sci. 108, 12233 (2011)] to obtain an analytic understanding of its energetics and wave functions needed for many-body calculations. We provide an approximation scheme for the wave functions of the BM model, which first elucidates why the BM K M -point centered original calculation containing only four plane waves provides a good analytical value for the first magic angle (θ M ≈ 1°). The approximation scheme also elucidates why most of the many-body matrix elements in the Coulomb Hamiltonian projected to the active bands can be neglected. By applying our approximation scheme at the first magic angle to a Γ M -point centered model of six plane waves, we analytically understand the reason for the small Γ M -point gap between the active and passive bands in the isotropic limit w 0 = w 1 . Furthermore, we analytically calculate the group velocities of the passive bands in the isotropic limit, and show that they are almost doubly degenerate, even away from the Γ M point, where no symmetry forces them to be. Furthermore, moving away from the Γ M and K M points, we provide an explicit analytical perturbative understanding as to why the TBG bands are flat at the first magic angle, despite the first magic angle is defined by only requiring a vanishing K M -point Dirac velocity. We derive analytically a connected “magic manifold” w 1 = $2\sqrt{1 + w^{2}_{0}}$ $-\sqrt{2 + 3w^2_0}$, on which the bands remain extremely flat as w 0 is tuned between the isotropic (w 0 = w 1 ) and chiral (w 0 = 0) limits. We analytically show why going away from the isotropic limit by making w 0 less (but not larger) than w 1 increases the Γ M -point gap between the active and the passive bands. Finally, by perturbation theory, we provide an analytic Γ M point k ∙ p two-band model that reproduces the TBG band structure and eigenstates within a certain w 0 , w 1 parameter range. Further refinement of this model are discussed, which suggest a possible faithful representation of the TBG bands by a two-band Γ M point k ∙ p model in the full w 0 , w 1 parameter range.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Hyperbolic enhancement of photocurrent patterns in minimally twisted bilayer graphene

Quasi-periodic moiré patterns and their effect on electronic properties of twisted bilayer graphene have been intensely studied. At small twist angle θ, due to atomic reconstruction, the moiré superlattice morphs into a network of narrow domain walls separating micron-scale AB and BA stacking regions. We use scanning probe photocurrent imaging to resolve nanoscale variations of the Seebeck coefficient occurring at these domain walls. The observed features become enhanced in a range of mid-infrared frequencies where the hexagonal boron nitride substrate is optically hyperbolic. Our results illustrate the capabilities of the nano-photocurrent technique for probing nanoscale electronic inhomogeneities in two-dimensional materials.

42 ENGINEERING↗

DOE OSTI2021

Twisted bilayer graphene. II. Stable symmetry anomaly

We show that the entire continuous model of twisted bilayer graphene (TBG) (and not just the two active bands) with particle-hole symmetry is anomalous and hence incompatible with lattice models. Previous works [e.g., Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, Phys. Rev. Lett. 123, 036401 (2019); J. Ahn, S. Park, and B.-J. Yang, Phys. Rev. X 9, 021013 (2019); H. C. Po, L. Zou, T. Senthil, and A. Vishwanath, Phys. Rev. B 99, 195455 (2019); J. Kang and O. Vafek, Phys. Rev. X 8, 031088 (2018); M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi, K. Kuroki, and L. Fu, Phys. Rev. X 8, 031087 (2018); J. Liu, J. Liu, and X. Dai, Phys. Rev. B 99, 155415 (2019); L. Zou, H. C. Po, A. Vishwanath, and T. Senthil, Phys. Rev. B 98, 085435 (2018)] found that the two flatbands in TBG possess a fragile topology protected by the C 2z T symmetry. Song et al. [Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, Phys. Rev. Lett. 123, 036401 (2019)] also pointed out an approximate particle-hole symmetry (P) in the continuous model of TBG. In this paper, we numerically confirm that P is indeed a good approximation for TBG and show that the fragile topology of the two flatbands is enhanced to a P-protected stable topology. This stable topology implies 4l + 2 (l ϵ N) Dirac points between the middle two bands. The P-protected stable topology is robust against arbitrary gap closings between the middle two bands and the other bands. We also show that, remarkably, this P-protected stable topology, as well as the corresponding 4l + 2 Dirac points, cannot be realized in lattice models that preserve both C 2z T and P symmetries. In other words, the continuous model of TBG is anomalous and cannot be realized on lattices. Two other topology related topics, with consequences for the interacting TBG problem, i.e., the choice of Chern band basis in the two flatbands and the perfect metal phase of TBG in the so-called second chiral limit, are also discussed.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Twisted bilayer graphene. III. Interacting Hamiltonian and exact symmetries

We derive the explicit Hamiltonian of twisted bilayer graphene (TBG) with Coulomb interaction projected into the flat bands and study the symmetries of the Hamiltonian. First, we show that all projected TBG Hamiltonians can be written as positive semidefinite Hamiltonians, an example of which was found in work by Kang and Vafek [Phys. Rev. Lett. 122, 246401 (2019)]. We then prove that the interacting TBG Hamiltonian exhibits an exact U(4) symmetry in the exactly flat band (nonchiral-flat) limit. We further define, besides a first chiral limit where the AA stacking hopping is zero, a second chiral limit where the AB/BA stacking hopping is zero. In the first chiral-flat limit (or second chiral-flat limit) with exactly flat bands, the TBG is enhanced to have an exact U(4) × U(4) symmetry, whose generators are different between the two chiral limits. While in the first chiral limit and in the nonchiral case these symmetries have been found in work by Bultinck et al. [Phys. Rev. X 10, 031034 (2020)], for the eight lowest bands, we here prove that they are valid for projection into any 8 n max particle-hole symmetric TBG bands, with n max > 1 being the practical case for small twist angles < 1°. Furthermore, in the first or second chiral-nonflat limit without flat bands, an exact U(4) symmetry still remains. We also elucidate the link between the U(4) symmetry presented here and the similar but different U(4) of Kang and Vafek [Phys. Rev. Lett. 122, 246401 (2019)]. Additionally, we show that our projected Hamiltonian can be viewed as the normal-ordered Coulomb interaction plus a Hartree-Fock term from passive bands, and exhibits a many-body particle-hole symmetry which renders the physics symmetric around charge neutrality. We also provide an efficient parametrization of the interacting Hamiltonian. The existence of two chiral limits with an enlarged symmetry suggests a possible duality of the model yet undiscovered.

2-dimensional systems↗

DOE OSTI2026

Weak localization and universal conductance fluctuations in large-area twisted bilayer graphene

We study diffusive magnetotransport in highly 𝑝-doped large-area twisted bilayer graphene in 1∘, 7∘, 9∘, and 20∘ samples. All samples exhibit weak localization, from which we extract the phase coherence length and intervalley scattering lengths, and from that determine that dephasing is caused by electron-electron scattering and intervalley scattering is caused by point defects. We observe signatures of universal conductance fluctuations in the 9∘ sample, which has high mobility and is near the van Hove singularity. Further improvements in sample quality and applications to large-area moiré materials will open new avenues to observe quantum interference effects.

Talkington, Spenser [University of Pennsylvania]↗

DOE OSTI2023

Critical magnetic fields and electron pairing in magic-angle twisted bilayer graphene

The velocities of the quasiparticles that form Cooper pairs in a superconductor are revealed by the upper critical magnetic field. Here we use this property to assess superconductivity in magic-angle twisted bilayer graphene (MATBG), which has been observed over a range of moiré band-filling, twist angle, and screening environment conditions. An average Fermi velocity can be defined as $v$$^{*}_{F}$ ≡ k B T c ℓ c /ℏ, where T c and ℓ c are the critical temperature and magnetic length, respectively. An advantage of this definition is that $v$$^{*}_{F}$ can be directly extracted from the existing experimental data. Mean-field theory calculations of upper critical fields in model superconductors are consistent with the expectation that Fermi velocities defined in this way are nearly independent of the strength of pairing interaction. Moreover, for fixed strength pairing interaction, minima in $v$$^{*}_{F}$ as a function of band filling are coincident with maxima in T c , as expected from the McMillan formula. Since no association between T c maxima and $v$$^{*}_{F}$ minima is present in MATBG experimental data, we argue that the pairing interaction in MATBG is strongly filling-factor dependent. Furthermore, any theory of MATBG superconductivity must explain this dependence, which is apparently primarily responsible for the observed superconducting domes.

2-dimensional systems↗

DOE OSTI2022

Observation of Reentrant Correlated Insulators and Interaction-Driven Fermi-Surface Reconstructions at One Magnetic Flux Quantum per Moiré Unit Cell in Magic-Angle Twisted Bilayer Graphene

The discovery of flat bands with nontrivial band topology in magic-angle twisted bilayer graphene (MATBG) has provided a unique platform to study strongly correlated phenomena including superconductivity, correlated insulators, Chern insulators, and magnetism. A fundamental feature of the MATBG, so far unexplored, is its high magnetic field Hofstadter spectrum. Here, we report on a detailed magnetotransport study of a MATBG device in external magnetic fields of up to B = 3 1 T , corresponding to one magnetic flux quantum per moiré unit cell Φ 0 . At Φ 0 , we observe reentrant correlated insulators at a flat band filling factors of ν = + 2 and of ν = + 3 , and interaction-driven Fermi-surface reconstructions at other fillings, which are identified by new sets of Landau levels originating from these. These experimental observations are supplemented by theoretical work that predicts a new set of eight well-isolated flat bands at Φ 0 , of comparable band width, but with different topology than in zero field. Overall, our magnetotransport data reveal a qualitatively new Hofstadter spectrum in MATBG, which arises due to the strong electronic correlations in the reentrant flat bands.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2026

Interlayer Exciton Condensates between Second Landau Level Orbitals in Double Bilayer Graphene

Here, we present Coulomb-drag measurements on a heterostructure comprising two Bernal-stacked bilayer graphene (BLG) sheets separated by a 2.5 nm hexagonal boron nitride (hBN) spacer in the quantum Hall (QH) regime. Using top and bottom gate control, together with an interlayer bias, we independently tune the two BLG layers into either the lowest (N = 0) or second (N = 1) Landau level (LL) orbital and probe their interlayer QH states. When both layers occupy the N = 0 orbital, we observe both interlayer exciton condensates (ECs) at integer total filling and interlayer fractional QH states, echoing the results in double monolayer graphene. In contrast to previous studies, however, when both BLG layers occupy the N = 1 orbital, we also observe quantized drag signals, signifying an interlayer exciton condensate formed between the second LLs. By tuning the layer degree of freedom, we find that this N = 1 EC state arises only when the N = 1 wave function in each BLG is polarized toward the hBN interface to maximize the interlayer Coulomb interaction.

Electrical conductivity↗

DOE OSTI2021

Moiré commensurability and the quantum anomalous Hall effect in twisted bilayer graphene on hexagonal boron nitride

The quantum anomalous Hall (QAH) effect is sometimes observed in twisted bilayer graphene (tBG) when it is nearly aligned with an encapsulating hexagonal boron nitride (hBN) layer. We propose that the appearance or absence of the QAH effect in individual devices could be related to commensurability between the graphene/graphene and graphene/hBN moiré patterns. We identify a series of points in the (θ GG , θ GBN ) twist-angle space at which the two moiré patterns are commensurate, allowing moiré band theory to be applied, and we show that the band Chern numbers are in this case sensitive to a rigid in-plane hBN displacement. Given this property, we argue that the QAH effect is likely only when (i) the (θ GG , θ GBN ) twist-angle-pair is close enough to a commensurate point that the two moiré patterns yield a supermoiré pattern with a sufficiently long length scale, and (ii) the supermoiré has a percolating topologically nontrivial QAH phase. For twist angles far from commensurability, the hBN layer acts as a source of disorder that can destroy the QAH effect. Our proposal can explain a number of current experimental observations. Finally, further experimental studies that can test this proposal more directly are suggested.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Kondo Lattice Model of Magic-Angle Twisted-Bilayer Graphene: Hund’s Rule, Local-Moment Fluctuations, and Low-Energy Effective Theory

For this work, we apply a generalized Schrieffer-Wolff transformation to the extended Anderson-like topological heavy fermion (THF) model for the magic-angle (θ=1.05°) twisted bilayer graphene (MATBLG) [Phys. Rev. Lett. 129, 047601 (2022)], to obtain its Kondo lattice limit. In this limit localized $\mathcal{f}$ electrons on a triangular lattice interact with topological conduction $\mathcal{c}$ electrons. By solving the exact limit of the THF model, we show that the integer fillings ν=0, ±1, ±2 are controlled by the heavy $\mathcal{f}$ electrons, while ν=±3 is at the border of a phase transition between two $\mathcal{f}$-electron fillings. For ν=0, ±1, ±2, we then calculate the Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions between the $\mathcal{f}$ moments in the full model and analytically prove the SU(4) Hund’s rule for the ground state which maintains that two $\mathcal{f}$ electrons fill the same valley-spin flavor. Our (ferromagnetic interactions in the) spin model dramatically differ from the usual Heisenberg antiferromagnetic interactions expected at strong coupling. We show the ground state in some limits can be found exactly by employing a positive semidefinite “bond-operators” method. We then compute the excitation spectrum of the $\mathcal{f}$ moments in the ordered ground state, prove the stability of the ground state favored by RKKY interactions, and discuss the properties of the Goldstone modes, the (reason for the accidental) degeneracy of (some of) the excitation modes, and the physics of their phase stiffness. We develop a low-energy effective theory for the $\mathcal{f}$ moments and obtain analytic expressions for the dispersion of the collective modes. We discuss the relevance of our results to the spin-entropy experiments in TBG.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗