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E.Bascones
EmergenceofQuantumPhasesinNovelMaterials
RamónAguado LeniBascones
MªJoséCalderón
TeoríaySimulacióndeMaterialesInstitutodeCienciadeMaterialesdeMadrid
AlbertoCortijo
E.Bascones
Moreisdifferent
“Theabilitytoreduceeverythingtosimplefundamentallawsdoesnotimplytheabilitytostartfromthoselawsandreconstructtheuniverse.”
“Thebehavioroflargeandcomplexaggregatesofelementaryparticles,itturnsout,isnottobeunderstoodintermsofasimpleextrapolationofthepropertiesofafewparticles.Instead,ateachlevelofcomplexityentirelynewpropertiesappear.”
1972,Anderson
E.Bascones
Emergence
Thearisingofnovelandcoherentstructures,patternsandpropertiesduringtheprocessofself-organizationincomplexsystems
Goldstein,Economist(1999)
E.Bascones
Emergenceincondensedmatterphysics
©Itub©askiitians ©kebes
Solids:brokentranslationalsymmetry Nematicliquidcrystal:Brokenrotationalsymmetry
Temperature
E.Bascones
Emergenceincondensedmatter
Electronsinteractwitheachotherandwiththeenvironment
Collectivemany-bodybehavior(stronglycorrelatedmaterials&mesoscopicsystems)
TounderstandthecomplexbehaviorIsolatethefundamentalprocesses
Morethan35yearsofintensiveresearcheffortandcontinuouslynewsurprisesappear
E.Bascones
Emergenceincondensedmatter
Electrons:itinerantorlocalized
Dimension:Influenceinorderedstatesbutalsoin“normalstate”
Originofmagnetism
Scatteringprocesses
Metallicity
Quantumstates:Magnetism:Ferromagnetism,AntiferromagnetismChargedensitywaves,Superconductivity,Nematicity,andother
Topology:Anewwaytoclassifycondensedmattersystemswithnoveleffects
Pauliexclusionprinciple
Lattice:Symmetries,Frustrated,Bipartite…
[email protected] Andmanyotherissues:interplaywithdisorder…
E.Bascones
Bandtheory&DFT:Ourbasicdescriptionofsolids
Blochstates:electronicbands
-Metalsandinsulators-Dependenceontemperatureofmeasurablequantities(Cv,χ,..)
-DensityFunctionalTheory:Abilitytocalculatethebands
Theelectronmovesintheaverageperiodicpotentialofthesolid
Bandtheory:Basisofourunderstandingofsolids
E.Bascones
Emergenceincondensedmatter
BandsgenerallycalculatedusingDFT
Bandtheoryisbasicallyadescriptionbasedonsingleparticlestates
Whydoesbandtheorywork?
Interactionsarenotsmallandtheelectronsshouldreacttothepresenceofotherelectrons
E.Bascones
Emergenceincondensedmatter
Perturbativeversusnon-perturbativeeffectofinteractions
BandsgenerallycalculatedusingDFT
Fermiliquidtheory
Bandtheoryisbasicallyadescriptionbasedonsingleparticlestates
Metalsandinsulators
Dependenceontemperatureofmeasurablequantities(Cv,χ,..)
Samebehaviorbutwithrenormalizedparameters {
E.Bascones
Emergenceincondensedmatter
Perturbativeversusnon-perturbativeeffectofinteractions
Phasetransitions&symmetrybreaking
BandsgenerallycalculatedusingDFT
Fermiliquidtheory
Bandtheoryisbasicallyadescriptionbasedonsingleparticlestates
E.Bascones
ExamplesofelectronicorderedphasesSpinorder Chargeorder
Thesymmetryoftheatomiclatticeispreserved
Latticetranslationalsymmetrybroken
Latticetranslationalandrotationalsymmetrybroken
Laticerotationalsymmetrybroken
Wignercrystal
ChargestripesStripeantiferromagnet
Neelantiferromagnet
Chargenematic
Chargedensitywave
Spinnematic
E.Bascones
Emergenceincondensedmatter
Perturbativeversusnon-perturbativeeffectofinteractions
Phasetransitions&symmetrybreaking
BandsgenerallycalculatedusingDFT
Fermiliquidtheory
Bandtheoryisbasicallyadescriptionbasedonsingleparticlestates
{ -Fermisurfaceinstabilities(itinerantelectrons)
-Localization(electronicbandsarelost)
-Itinerant+localelectrons
E.Bascones
Emergenceincondensedmatter
Perturbativeversusnon-perturbativeeffectofinteractions
BandsgenerallycalculatedusingDFT
Fermiliquidtheory
Bandtheoryisbasicallyadescriptionbasedonsingleparticlestates
Givenaparticularsystem,shouldweexpectthatbandtheoryworks?
Whatkindofinstabilitiesmaythesystemshow?
E.Bascones
Interactions:localizationandsuperconductivity
Motttransition:amaterialpredictedbybandtheorytobemetallicbecomesaninsulator
duetoelectronicinteractionsevenwithoutsymmetrybreaking
Superconductivity:Belowacriticaltemperatureresistivitygoestozero.
(duetophononsweknow,butprobablyalsoduetomagneticfluctuationsorMottphysics)
E.Bascones
Interactions:localizationandsuperconductivity
Cuprates:hightemperaturesuperconductorsFromanantiferromagneticMottinsulatortoasuperconductorandthenametal
High-temperaturesuperconductor
AntiferromagneticMottinsulator Metal
E.Bascones
Interactions:localizationandsuperconductivity
Cuprates:hightemperaturesuperconductorsFromanantiferromagneticMottinsulatortoasuperconductorandthenametal
Pseudogap:Isitanewstateofmatter?AremanentoftheMottinsulatororanotherbrokensymmetryphase?
E.Bascones
Interactions:localizationandsuperconductivity
Holedopedcuprates:hightemperaturesuperconductors
NatureComms5,5875(2014)
Fig:InnaVishik
Hugedevelopmentsinideasandintheoreticalandexperimentaltechniques
Schmidtetal,NJP13,065014(2011)
E.Bascones
Emergenceincondensedmatter
SCES
SeriousChallengeofEstablishedStandards
Stronglycorrelatedelectronsystems
E.Bascones
Emergenceincondensedmatter
SCES
Stronglycorrelatedelectronsystems
Difficultytounveilthefundamentalpropertiesboththeoretically&experimentally(awayfromcontrolledtheoreticallimits,competingstates,notalwayspossibletodomanyexperiments…)
E.Bascones
Emergenceincondensedmatterfromelectronicinteractions
Stronglycorrelatedelectronsystems:verysensitivetochanges(Pressure,MagneticField,doping…)
Organicmaterial
E.Bascones
Magnetismandsuperconductivityoftenappearclosetoeachother
Cuprates
Hightemperaturesuperconductors
Ironsuperconductors
Theircriticaltemperaturesdonotlookhighbut…
Heavyfermion
Fig:www.supraconductivite.fr
Organicsalt
Fig:Faltelmeier,PRB(2007)
E.Bascones
Dootherphasescompeteorcooperatewithsuperconductivity?Descriptionintermsorlocaloritinerantspins?
Sun et al, Nat. Comm. 12146 (2016)
IronsuperconductorFeSe(nematicstate)
Fig:InnaVishik
Holedopedcuprates(chargeorder)
IronsuperconductorBa-122(quantumcriticality)
Parentcompound:MetalParentcompound:
Insulator
E.Bascones
EmergenceincondensedmatterKondoeffect:
Aminimumintheresistivityofmetalswithmagneticimpuritieswithalotofphysicsbehind
Fig:Kasaietal,(2001)
Onedimension:Dramaticeffectofinteractions:
Luttingerliquids
Fig:Jonpoletal,Science(2009)
E.Bascones
Fig:www.gaia3d.co.uk
Fig:iopscience.iop.org
Fig:www.purdue.edu
Emergenceinmesoscopicsystems
KondoeffectinQuantumDots
Differentphasesputincontact(superconducting&ferromagnetic)
Luttingerliquidbehaviorincarbonnanotubes
E.Bascones
Topologicaland2Dsystems
Newpropertiesinatomiclayers.Evenatthesingleparticlelevel!
Fig:Nanophotonics4,128(2015)
Topologicalinsulators,Weylsemimetals
Fig:Hoffman’[email protected]
E.Bascones
Engineeringnewmaterialsandheterostructures
Fig:Geim&Grigorieva,Nature499,419(2013)
Possibilitytoengineerbandstructuresoralternatematerialswithdifferentfuncionalities(ferromagnetic,superconductor…)
2DLEGO
Moiréheterostructures(hugeunitcells)
Atomiclatticesonsurfaces
©Ponor
Yan&Liljeroth,AdvancesinPhysicsX,4,2019
E.Bascones
Moiréflatbands:anewplatformtostudytheeffectsofcorrelations
Caoetal,Nature556,43(2018)Correlated
insulatingstate
Moiréunitcell>10.000atomsHighlytunablesystem(doping,angle…)
Magicangletwistedbilayergraphene(1.1º)
LETTERRESEARCH
8 2 | N A T U R E | V O L 5 5 6 | 5 A P R I L 2 0 1 8
The emergence of half-filling states is not expected in the absence of interactions between electrons and appears to be correlated with the narrow bandwidth near the first magic angle. In our experiment, sev-eral separate pieces of evidence support the presence of flat bands. First, we measured the temperature dependence of the amplitude of Shubnikov–de Haas oscillations in device D1, from which we extracted the effective mass of the electron, m* (Fig. 3b; see Methods and Extended Data Fig. 3 for analysis). For a Dirac spectrum with eight-fold degeneracy (spin, valley and layer), we expect that ⁎= / πm h n v(8 )2
F2 ,
which scales as 1/vF . The large measured m* near charge neutrality in device D1 indicates a reduction in vF by a factor of 25 compared to monolayer graphene (4 × 104 m s− 1 compared to 106 m s− 1). This large reduction in the Fermi velocity is a characteristic that is expected for flat bands. Second, we analysed the capacitance data of device D2 near the Dirac point (Fig. 3a) and found that vF needs to be reduced to about 0.15v0 for a good fit to the data (Methods, Extended Data Fig. 1b). Third, another direct manifestation of flat bands is the flattening of the con-ductance minimum at charge neutrality above a temperature of 40 K (thermal energy kT = 3.5 meV), as seen in Fig. 3c. Although the con-ductance minimum in monolayer graphene can be observed clearly even near room temperature, it is smeared out in magic-angle TBG when the thermal energy kT becomes comparable to vFkθ/2 ≈ 4 meV—the energy scale that spans the Dirac-like portion of the band (Fig. 1c)24–26.
Owing to the localized nature of the electrons, a plausible explanation for the gapped behaviour at half-filling is the formation of a Mott-like insulator driven by Coulomb interactions between electrons27,28. To this end, we consider a Hubbard model on a triangular lattice, with each site corresponding to a localized region with AA stacking in the moiré pattern (Fig. 1i). In Fig. 3d we show the bandwidth of the E > 0 branch of the low-energy bands for 0.04° < θ < 2° that we calculated numerically using a continuum model of TBG6. The bandwidth W is strongly suppressed near the magic angles. The on-site Coulomb energy U of each site is estimated to be e2/(4π εd), where d is the effective linear
dimension of each site (with the same length scale as the moiré period), ε is the effective dielectric constant including screening and e is the electron charge. Combining ε and the dependence of d on twist angle into a single constant κ, we write U = e2θ/(4π ε0κa), where a = 0.246 nm is the lattice constant of monolayer graphene. In Fig. 3d we plot the on-site energy U versus θ for κ = 4–20. As a reference, κ = 4 if we assume ε = 10ε0 and d is 40% of the moiré wavelength. For a range of possible values of κ it is therefore reasonable that U/W > 1 occurs near the magic angles and results in half-filling Mott-like gaps27. However, the realistic scenario is much more complicated than these simplistic estimates; a complete understanding requires detailed theoretical anal-yses of the interactions responsible for the correlated gaps.
The Shubnikov–de Haas oscillation frequency fSdH (Fig. 3b) also supports the existence of Mott-like correlated gaps at half-filling. Near the charge neutrality point, the oscillation frequency closely follows fSdH = φ0| n| /M where φ0 = h/e is the flux quantum and M = 4 indicates the spin and valley degeneracies. However, at | n| > ns/2, we observe oscillation frequencies that corresponds to straight lines, fSdH = φ0(| n| − ns/2)/M, in which M has a reduced value of 2. Moreover, these lines extrapolate to zero exactly at the densities of the half-filling states, n = ± ns/2. These oscillations point to small Fermi pockets that result from doping the half-filling states, which might originate from charged quasi particles near a Mott-like insulator phase29. The halved degener-acy of the Fermi pockets might be related to the spin–charge separation that is predicted in a Mott insulator29. These results are also supported by Hall measurements at 0.3 K (Extended Data Fig. 4; see Methods for discussion), which show a ‘resetting’ of the Hall densities when the system is electrostatically doped beyond the Mott-like states.
The half-filling states at ± ns/2 are suppressed by the application of a magnetic field. In Fig. 4a, b we show that both insulating phases start to conduct at a perpendicular field of B = 4 T and recover normal conductance by B = 8 T. A similar effect is observed for an in-plane magnetic field (Extended Data Fig. 5d). The insensitivity to field
0
0.05
–1.8 –1.3 0
0.3 K
p-side
p-side
Insulating
Metallic
n-side
n-side
1.7 K
1.0 1.5
0.3 K
1.7 K
Carrier density, n (1012 cm–2)
0 0.1 0.2 0.5 1.0 1.5 2.0 2.5 3.0
T–1 (K–1)
10–2
10–3
10–4
T =
4 K
0
0.05
0.1
–2
Con
duct
ance
, G (m
S)
Con
duct
ance
, G (m
S)
Con
duct
ance
, G (m
S)
Carrier density, n (1012 cm–2)
0
0.05
0.1
–4 0 2 4
–ns
–ns/2 ns/2
–ns/2 ns/2–ns ns
ns
D1
D2
D3
D4
0
D1
D2
D3
D4
a
bc d
T = 1.08°
T = 1.10°
T = 1.12°
T = 1.16°
Figure 2 | Half-filling insulating states in magic-angle TBG. a, Measured conductance G of magic-angle TBG device D1 with θ = 1.08° and T = 0.3 K. The Dirac point is located at n = 0. The lighter-shaded regions are superlattice gaps at carrier density n = ± ns = ± 2.7 × 1012 cm− 2. The darker-shaded regions denote half-filling states at ± ns/2. The inset shows the density locations of half-filling states in the four different devices.
See Methods for a definition of the error bars. b, Minimum conductance values in the p-side (red) and n-side (blue) half-filling states in device D1. The dashed lines are fits of exp[− ∆ /(2kT)] to the data, where ∆ ≈ 0.31 meV is the thermal activation gap. c, d, Temperature-dependent conductance of D1 for temperatures from about 0.3 K (black) to 1.7 K (orange) near the p-side (c) and n-side (d) half-filling states.
© 2018 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
Twocorrelatedinsulatingstates(half-fillingelectron&holebands)
Caoetal,Nature556,80(2018),Nature556,43(2018)
March2018
E.Bascones
Caoetal,Nature556,43(2018)Correlated
insulatingstate
Moiréunitcell>10.000atomsHighlytunablesystem(doping,angle…)
Magicangletwistedbilayergraphene(1.1º)
LETTERRESEARCH
8 2 | N A T U R E | V O L 5 5 6 | 5 A P R I L 2 0 1 8
The emergence of half-filling states is not expected in the absence of interactions between electrons and appears to be correlated with the narrow bandwidth near the first magic angle. In our experiment, sev-eral separate pieces of evidence support the presence of flat bands. First, we measured the temperature dependence of the amplitude of Shubnikov–de Haas oscillations in device D1, from which we extracted the effective mass of the electron, m* (Fig. 3b; see Methods and Extended Data Fig. 3 for analysis). For a Dirac spectrum with eight-fold degeneracy (spin, valley and layer), we expect that ⁎= / πm h n v(8 )2
F2 ,
which scales as 1/vF . The large measured m* near charge neutrality in device D1 indicates a reduction in vF by a factor of 25 compared to monolayer graphene (4 × 104 m s− 1 compared to 106 m s− 1). This large reduction in the Fermi velocity is a characteristic that is expected for flat bands. Second, we analysed the capacitance data of device D2 near the Dirac point (Fig. 3a) and found that vF needs to be reduced to about 0.15v0 for a good fit to the data (Methods, Extended Data Fig. 1b). Third, another direct manifestation of flat bands is the flattening of the con-ductance minimum at charge neutrality above a temperature of 40 K (thermal energy kT = 3.5 meV), as seen in Fig. 3c. Although the con-ductance minimum in monolayer graphene can be observed clearly even near room temperature, it is smeared out in magic-angle TBG when the thermal energy kT becomes comparable to vFkθ/2 ≈ 4 meV—the energy scale that spans the Dirac-like portion of the band (Fig. 1c)24–26.
Owing to the localized nature of the electrons, a plausible explanation for the gapped behaviour at half-filling is the formation of a Mott-like insulator driven by Coulomb interactions between electrons27,28. To this end, we consider a Hubbard model on a triangular lattice, with each site corresponding to a localized region with AA stacking in the moiré pattern (Fig. 1i). In Fig. 3d we show the bandwidth of the E > 0 branch of the low-energy bands for 0.04° < θ < 2° that we calculated numerically using a continuum model of TBG6. The bandwidth W is strongly suppressed near the magic angles. The on-site Coulomb energy U of each site is estimated to be e2/(4π εd), where d is the effective linear
dimension of each site (with the same length scale as the moiré period), ε is the effective dielectric constant including screening and e is the electron charge. Combining ε and the dependence of d on twist angle into a single constant κ, we write U = e2θ/(4π ε0κa), where a = 0.246 nm is the lattice constant of monolayer graphene. In Fig. 3d we plot the on-site energy U versus θ for κ = 4–20. As a reference, κ = 4 if we assume ε = 10ε0 and d is 40% of the moiré wavelength. For a range of possible values of κ it is therefore reasonable that U/W > 1 occurs near the magic angles and results in half-filling Mott-like gaps27. However, the realistic scenario is much more complicated than these simplistic estimates; a complete understanding requires detailed theoretical anal-yses of the interactions responsible for the correlated gaps.
The Shubnikov–de Haas oscillation frequency fSdH (Fig. 3b) also supports the existence of Mott-like correlated gaps at half-filling. Near the charge neutrality point, the oscillation frequency closely follows fSdH = φ0| n| /M where φ0 = h/e is the flux quantum and M = 4 indicates the spin and valley degeneracies. However, at | n| > ns/2, we observe oscillation frequencies that corresponds to straight lines, fSdH = φ0(| n| − ns/2)/M, in which M has a reduced value of 2. Moreover, these lines extrapolate to zero exactly at the densities of the half-filling states, n = ± ns/2. These oscillations point to small Fermi pockets that result from doping the half-filling states, which might originate from charged quasi particles near a Mott-like insulator phase29. The halved degener-acy of the Fermi pockets might be related to the spin–charge separation that is predicted in a Mott insulator29. These results are also supported by Hall measurements at 0.3 K (Extended Data Fig. 4; see Methods for discussion), which show a ‘resetting’ of the Hall densities when the system is electrostatically doped beyond the Mott-like states.
The half-filling states at ± ns/2 are suppressed by the application of a magnetic field. In Fig. 4a, b we show that both insulating phases start to conduct at a perpendicular field of B = 4 T and recover normal conductance by B = 8 T. A similar effect is observed for an in-plane magnetic field (Extended Data Fig. 5d). The insensitivity to field
0
0.05
–1.8 –1.3 0
0.3 K
p-side
p-side
Insulating
Metallic
n-side
n-side
1.7 K
1.0 1.5
0.3 K
1.7 K
Carrier density, n (1012 cm–2)
0 0.1 0.2 0.5 1.0 1.5 2.0 2.5 3.0
T–1 (K–1)
10–2
10–3
10–4
T =
4 K
0
0.05
0.1
–2
Con
duct
ance
, G (m
S)
Con
duct
ance
, G (m
S)
Con
duct
ance
, G (m
S)
Carrier density, n (1012 cm–2)
0
0.05
0.1
–4 0 2 4
–ns
–ns/2 ns/2
–ns/2 ns/2–ns ns
ns
D1
D2
D3
D4
0
D1
D2
D3
D4
a
bc d
T = 1.08°
T = 1.10°
T = 1.12°
T = 1.16°
Figure 2 | Half-filling insulating states in magic-angle TBG. a, Measured conductance G of magic-angle TBG device D1 with θ = 1.08° and T = 0.3 K. The Dirac point is located at n = 0. The lighter-shaded regions are superlattice gaps at carrier density n = ± ns = ± 2.7 × 1012 cm− 2. The darker-shaded regions denote half-filling states at ± ns/2. The inset shows the density locations of half-filling states in the four different devices.
See Methods for a definition of the error bars. b, Minimum conductance values in the p-side (red) and n-side (blue) half-filling states in device D1. The dashed lines are fits of exp[− ∆ /(2kT)] to the data, where ∆ ≈ 0.31 meV is the thermal activation gap. c, d, Temperature-dependent conductance of D1 for temperatures from about 0.3 K (black) to 1.7 K (orange) near the p-side (c) and n-side (d) half-filling states.
© 2018 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
Twocorrelatedinsulatingstates(half-fillingelectron&holebands)Superconductivity
Caoetal,Nature556,80(2018),Nature556,43(2018)
Moiréflatbands:anewplatformtostudytheeffectsofcorrelations
E.Bascones [email protected]
LETTERRESEARCH
8 2 | N A T U R E | V O L 5 5 6 | 5 A P R I L 2 0 1 8
The emergence of half-filling states is not expected in the absence of interactions between electrons and appears to be correlated with the narrow bandwidth near the first magic angle. In our experiment, sev-eral separate pieces of evidence support the presence of flat bands. First, we measured the temperature dependence of the amplitude of Shubnikov–de Haas oscillations in device D1, from which we extracted the effective mass of the electron, m* (Fig. 3b; see Methods and Extended Data Fig. 3 for analysis). For a Dirac spectrum with eight-fold degeneracy (spin, valley and layer), we expect that ⁎= / πm h n v(8 )2
F2 ,
which scales as 1/vF . The large measured m* near charge neutrality in device D1 indicates a reduction in vF by a factor of 25 compared to monolayer graphene (4 × 104 m s− 1 compared to 106 m s− 1). This large reduction in the Fermi velocity is a characteristic that is expected for flat bands. Second, we analysed the capacitance data of device D2 near the Dirac point (Fig. 3a) and found that vF needs to be reduced to about 0.15v0 for a good fit to the data (Methods, Extended Data Fig. 1b). Third, another direct manifestation of flat bands is the flattening of the con-ductance minimum at charge neutrality above a temperature of 40 K (thermal energy kT = 3.5 meV), as seen in Fig. 3c. Although the con-ductance minimum in monolayer graphene can be observed clearly even near room temperature, it is smeared out in magic-angle TBG when the thermal energy kT becomes comparable to vFkθ/2 ≈ 4 meV—the energy scale that spans the Dirac-like portion of the band (Fig. 1c)24–26.
Owing to the localized nature of the electrons, a plausible explanation for the gapped behaviour at half-filling is the formation of a Mott-like insulator driven by Coulomb interactions between electrons27,28. To this end, we consider a Hubbard model on a triangular lattice, with each site corresponding to a localized region with AA stacking in the moiré pattern (Fig. 1i). In Fig. 3d we show the bandwidth of the E > 0 branch of the low-energy bands for 0.04° < θ < 2° that we calculated numerically using a continuum model of TBG6. The bandwidth W is strongly suppressed near the magic angles. The on-site Coulomb energy U of each site is estimated to be e2/(4π εd), where d is the effective linear
dimension of each site (with the same length scale as the moiré period), ε is the effective dielectric constant including screening and e is the electron charge. Combining ε and the dependence of d on twist angle into a single constant κ, we write U = e2θ/(4π ε0κa), where a = 0.246 nm is the lattice constant of monolayer graphene. In Fig. 3d we plot the on-site energy U versus θ for κ = 4–20. As a reference, κ = 4 if we assume ε = 10ε0 and d is 40% of the moiré wavelength. For a range of possible values of κ it is therefore reasonable that U/W > 1 occurs near the magic angles and results in half-filling Mott-like gaps27. However, the realistic scenario is much more complicated than these simplistic estimates; a complete understanding requires detailed theoretical anal-yses of the interactions responsible for the correlated gaps.
The Shubnikov–de Haas oscillation frequency fSdH (Fig. 3b) also supports the existence of Mott-like correlated gaps at half-filling. Near the charge neutrality point, the oscillation frequency closely follows fSdH = φ0| n| /M where φ0 = h/e is the flux quantum and M = 4 indicates the spin and valley degeneracies. However, at | n| > ns/2, we observe oscillation frequencies that corresponds to straight lines, fSdH = φ0(| n| − ns/2)/M, in which M has a reduced value of 2. Moreover, these lines extrapolate to zero exactly at the densities of the half-filling states, n = ± ns/2. These oscillations point to small Fermi pockets that result from doping the half-filling states, which might originate from charged quasi particles near a Mott-like insulator phase29. The halved degener-acy of the Fermi pockets might be related to the spin–charge separation that is predicted in a Mott insulator29. These results are also supported by Hall measurements at 0.3 K (Extended Data Fig. 4; see Methods for discussion), which show a ‘resetting’ of the Hall densities when the system is electrostatically doped beyond the Mott-like states.
The half-filling states at ± ns/2 are suppressed by the application of a magnetic field. In Fig. 4a, b we show that both insulating phases start to conduct at a perpendicular field of B = 4 T and recover normal conductance by B = 8 T. A similar effect is observed for an in-plane magnetic field (Extended Data Fig. 5d). The insensitivity to field
0
0.05
–1.8 –1.3 0
0.3 K
p-side
p-side
Insulating
Metallic
n-side
n-side
1.7 K
1.0 1.5
0.3 K
1.7 K
Carrier density, n (1012 cm–2)
0 0.1 0.2 0.5 1.0 1.5 2.0 2.5 3.0
T–1 (K–1)
10–2
10–3
10–4
T =
4 K
0
0.05
0.1
–2
Con
duct
ance
, G (m
S)
Con
duct
ance
, G (m
S)
Con
duct
ance
, G (m
S)
Carrier density, n (1012 cm–2)
0
0.05
0.1
–4 0 2 4
–ns
–ns/2 ns/2
–ns/2 ns/2–ns ns
ns
D1
D2
D3
D4
0
D1
D2
D3
D4
a
bc d
T = 1.08°
T = 1.10°
T = 1.12°
T = 1.16°
Figure 2 | Half-filling insulating states in magic-angle TBG. a, Measured conductance G of magic-angle TBG device D1 with θ = 1.08° and T = 0.3 K. The Dirac point is located at n = 0. The lighter-shaded regions are superlattice gaps at carrier density n = ± ns = ± 2.7 × 1012 cm− 2. The darker-shaded regions denote half-filling states at ± ns/2. The inset shows the density locations of half-filling states in the four different devices.
See Methods for a definition of the error bars. b, Minimum conductance values in the p-side (red) and n-side (blue) half-filling states in device D1. The dashed lines are fits of exp[− ∆ /(2kT)] to the data, where ∆ ≈ 0.31 meV is the thermal activation gap. c, d, Temperature-dependent conductance of D1 for temperatures from about 0.3 K (black) to 1.7 K (orange) near the p-side (c) and n-side (d) half-filling states.
© 2018 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
March2018
Twocorrelatedinsulatingstates(half-fillingelectron&holebands)Superconductivity
January2019
Correlatedstatesatallintegerfillings
July2021(compressibilitymeasurements)- Correlatedinsulators(integerfilling)
- ChargeDensityWave
- ChernInsulators/IQH
- SymmetrybrokenCherninsulators
- FractionalCherninsulatorsFilling
Caoetal,Nature556,80(2018),Nature556,43(2018) LuetalNature574,653(2019)
Xieetal,arXiv:2107.10854
Moiréflatbandswhere2Dmaterials,correlationsandtopologymeet
E.Bascones [email protected]
Moiréflatbandswhere2Dmaterials,correlationsandtopologymeetTwistingother2DmaterialsOthergraphenebased
twistedheterostructures
Twisteddoublebilayergraphene
Twistedgraphenetrilayer(Moiré3.0)
TrilayerABCgraphene/hBN
Playwiththesubstratealsointwistedheterostructures
(semiconductors,magnetic…)
TwistedbilayerWSe2
TwistedbilayerWSe2/WS2
Liuetal,Nature583,221(2020),Parketal,Nature590,249(2021),Chenetal,Nature572,215(2019),Anetal,NanoscaleHorizons9,(2020),JinetalNatureMaterials20,940(2021)
E.Bascones
QuantumPhasesinNovelMaterials
Basicideas&genericconcepts
Differentkindsofphenomena&whytheyhappen
TheoreticaldescriptionManydifferentanalytical
&numericalmethodsandapproaches
Experimentalsignaturesofthequantumstates
Materialsandplatformswherethesequantumphasesappear
E.Bascones
EmergenceofQuantumPhasesinNovelStates:Outlineq Introduction:EmergenceandBasicConcepts(L.Bascones)
q Fermiliquidtheory(L.Bascones)
q Electroniccorrelations:Mott,HundandLuttingerPhysics(L.Bascones)
q Magnetism(M.J.Calderón)
q Superconductivity(M.J.Calderón)
q DiracMaterials(A.Cortijo)q TopologicalInsulatorsandtopologicalsemimetals(A.Cortijo)
q Kondoeffectinmetalsandnanostructures(R.Aguado)q Topologicalsuperconductivity(R.Aguado)
E.Bascones
Basicconcepts:theFermigas• Focusonelectrons.Workatfixedµ
• Electronsarefermions:Fermi-DiracStatistics
Zerotemperature
f(ε)∝e-ε/KBT
Hightemperatures(classicalgas)KBT»µ
f(ε)=1
e(ε-µ)/KBT+1
chemicalpotential
Maxwell-Boltzmanndistributionfunction
Stepfunction
Statesfilledupto
theFermiEnergyεFwhichdefinesaFermisurface
Fig:Wikipedia
energy
E.Bascones
Basicconcepts:theFermigas• Hightemperatures(classicalgas,Maxwell-Boltzmann)
KBT>>µ
• Lowtemperatures(quantumgas,Fermi-Dirac)
KBT<<µ
o Specificheat
Cv=γT Linearintemperature
o Spinsusceptibility
Cv= o Specificheat∂F∂T µ
Cv=independentofT
Freeenergy
χs=∂M∂H
Magnetization
Magneticfield
χs=µB2N(εF)independentofT χs∝
o Spinsusceptibility
Paulisusceptibility
1
TCurielaw
Bohrmagneton Densityofstates
atFermilevel
E.Bascones
Basicconcepts:theFermigas
• Hightemperatures(classicalgas)
• Lowtemperatures(quantumgas)
o Specificheat Cv=γT Linearintemperature
o Spinsusceptibility
Cv=independentofT
χs=µB2N(εF)
independentofT
χs∝
Paulisusceptibility
1
T Curielaw
KBT<<µ KBT>>µ
Differenttemperaturedependenceofmeasurablequantities
inthedegenerate(quantum)andclassicallimits
ConsequenceofPauliexclusionprinciple
Non-interactingfermionsarecorrelatedduetoFermistatistics&Pauliprinciple
E.Bascones
Basicconcepts:theFermigas
Differenttemperaturedependenceofmeasurablequantities
inthedegenerate(quantum)andclassicallimits • Hightemperatures
(classicalgas)
• Lowtemperatures(quantumgas) KBT<<µ
KBT>>µ
FermitemperatureTF=εF/KB
Temperaturebelowwhichquantumeffectsareimportant
E.Bascones
Basicconcepts:theFermigas
Differenttemperaturedependenceofmeasurablequantities
inthedegenerate(quantum)andclassicallimits • Hightemperatures
(classicalgas)
• Lowtemperatures(quantumgas) KBT<<µ
KBT>>µ
FermitemperatureTF=εF/KB
Temperaturebelowwhichquantumeffectsareimportant
Metalsgenericallyhavetobetreatedasdegenerategases
Inmetals εF~3eV
TF~2400K
InTBG&dopedsemiconductors εF~fewmeV
TF~50K
E.Bascones
Basicconcepts:latticeandenergybandsvsthecontinuum
Inasolid:
• periodicpotentialoftheioniclattice
• Lengthscale:thelatticeconstanta
• Momentumconservedmodulus2π/a
• Ingeneral,anisotropicpotential
• EnergybandsfilleduptoεF.Fermisurface
• Effectivemass(bandmass)
Infreespace:Parabolicband Kineticenergy=k2
2m• Isotropic
• Nolengthscale
• Momentumconserved
Fig: Calderón et al, PRB, 80, 094531 (2009) m-1=|∂2ε/∂k2|
E.Bascones
Basicconcepts:latticeandenergybandsvsthecontinuum
Infreespace:Parabolicband Kineticenergy=k2
2m
Fig: Calderón et al, PRB, 80, 094531 (2009)
SomephenomenacanbedescribedintermsofparabolicbandswhichsubstitutetheenergybandsclosetotheFermisurface
Careful!Thisisnotalwayspossible
Someeffectsareintrinsictothelattice
E.Bascones
Basicconcepts:masslessfermionsingrapheneParabolicbands:
Effectivemass(bandmass) m-1=|∂2ε/∂k2|
Fig: Calderón et al, PRB, 80, 094531 (2009)
Continuumapproximation
k22m
Graphene:
LinearbandsclosetoFermilevel
Continuumapproximation
Effectivemass(bandmass)iszero!
E~vFk
E.Bascones
Basicconcepts:metalsandinsulators
MetallicityincleansystemsBandscrossingtheFermilevel(finiteDOS)
Fig: Calderón et al, PRB, 80, 094531 (2009)
InsulatingbehaviourincleansystemsBandsbelow
Fermilevelfilled
Fig: Hess & Serene, PRB 59, 15167 (1999) [email protected]
E.Bascones
Spindegeneracy(nosymmetrybreaking):Eachbandcanhold2electronsperunitcell
Evennumberofelectronsperunitcell
Insulating
Metallic(incaseofbandoverlap)
Oddnumberofelectronsperunitcell
Metallic(orsemimetal)
Basicconcepts:metalsandinsulators
E.Bascones
Spindegeneracy:Eachbandcanhold2electronsperunitcell
Evennumberofelectronsperunitcell
Insulating
Metallic(incaseofbandoverlap)
Oddnumberofelectronsperunitcell
Metallic
Basicconcepts:metalsandinsulators
E.Bascones
Basicconcepts:strengthofinteractions
Kineticenergy:Wbandwidth
Strengthofinteractions:UU/W
Interactionsarenotsmallcomparedtokineticenergy
• FocusonelectronsclosetotheFermisurface.
• Coreelectrons+nucleusformtheion
Kineticenergy:tightbindingmodelusingatomicorbitalsasabasis
Largerspreadofwavefunction(lesslocalizedorbitals)
Largeroverlapbetweenorbitalsinneighbouringsites(largerkineticenergy)
Twoelectronsarelesslikelytobefound
veryclose(smallereffectiveinteraction)
Narrowbandsmoresensitivetointeractions
E.Bascones
Basicconcepts:strengthofinteractions
s&pelectronskineticenergymoreimportant
3d:competitionbetweenkineticenergy&interaction
Interactionstrengthdecreasesin4d&overallin5dduetolargerextensionoftheorbital
4foveralland5forbitalsarequitelocalizedonasite.Verysmallkineticenergy.Interactionswin
E.Bascones
Basicconcepts:strengthofinteractionsp-orbitals,extendedwavefunctions(interactionsingraphenenotexpectedtobeveryrelevant)
E.Bascones
Basicconcepts:strengthofinteractionsp-orbitals,extendedwavefunctions(interactionsingraphenenotexpectedtobeveryrelevant)
Flatbandsclosetothemagicangle~1.1º
VerysmallW
E.Bascones
Basicconcepts:strengthofinteractions
Cuprates,manganites,ironsuperconductors
U,W~2-3eV
Twistedbilayergrapheneandothermoiréheterostructures
U,W~1-50meV
E.Bascones
Basicconcepts:strengthofinteractions
Coulombinteraction:e2r
Assume3D,freespace(continuum),letn=Ne/V(nelectronicdensity)
V:VolumeNe:Numberofelectrons
Kineticenergy ∫-∞εFε g(ε) dε ∝n2/3
Interactionenergy ∝<1/r>≅n1/3
Kineticenergy
Interactionenergyn-1/3∝
Carefulinmetals!
DuetoPauliprincipletheimportanceofinteractionsdecreaseswithincreasingdensity
g(ε):densityofstates
Importantinsemiconductors
Validfor1/rinteraction
E.Bascones
Basicconcepts:ScreeningandHubbardmodel
Coulombinteraction:
Longrangecharacter
Interactionislargerwhenelectronsareclosere2
r
KineticEnergy(Tightbinding)
Σijσtijc†iσcjσ +h.c.+UΣjnj↑nj↓+Σi≠jσσ’Uijniσnjσ’
I,jlatticesitesσspin
Interaction(repulsionbetweenelectronsindifferentsites)
Hamiltonianonthelattice(kineticenergy+interactions)
Interaction(repulsionbetweenelectronsinthesamesite,onlydifferentspin)
E.Bascones
Basicconcepts:ScreeningandHubbardmodel
Coulombinteraction:
Longrangecharacter
Interactionislargerwhenelectronsareclosere2
εr
Otherelectronsinthesolid(metal)orgatescreentheCoulombinteraction.Theinteractiondecaysfasterthan1/r
Screeningalsoentersincreasesε
E.Bascones
Basicconcepts:ScreeningandHubbardmodel
Hubbardmodel:latticemodel,1orbitalpersite.Approximation
Interactionsrestrictedtoelectronsinthesamesite
KineticEnergy(Tightbinding)
Σijσtijc†iσcjσ +h.c.+UΣjnj↑nj↓
I,jlatticesitesσspin InteractionEnergy
Worksbetterforlocalizedelectrons
Itlookssimplebutitisnotwellunderstood.Basicinthestudyofcorrelatedelectrons
E.Bascones
Basicconcepts:Summaryq ElectronsinasolidfollowFermi-Diracstatistics.Inatypicalmetalthedegeneracytemperatureismuchhigherthanroomtemperature(inMATBGno).
q
q Parabolicbands(continuum)versuslatticemodels.Linearbands(nomass)
q Bandtheory:metallicorinsulatingcharacteronthebasisofthenumberofelectronsperunitcell.
q U/Wcontrolsstrengthofinteractions
q sandporbitalslesssensitivetointeractions,forbitalsarethemostcorrelated.d-electronsinterplaybetweenkineticenergy&interactions.InTBG&othercorrelatedmoirésinteractionsareimportantduetoverysmallW
q In3Dandwith1/rinthecontinuumlimittheimportanceofinteractionsdecreaseswithincreasingdensity
q ScreeningandHubbardmodel
Cv=γT Linearintemperature χs=µB
2N(εF)independentofT
