Jean-Noël Fuchs Arnaud RaouxGilles Montambaux
Frédéric PiéchonLaboratoire de Physique des Solides, OrsayCNRS, Université Paris-Sud, France
Kaust, March 2018
Orbital Magnetic Susceptibility of Metals and Insulators
Outline :
1. Magnetism of non magnetic materials
2. Revisiting orbital magnetism of free electron
3. in tight-binding models of metals and insulators
4. Orbital susceptibility formula : role of band structure
Outline :
1. Magnetism of non magnetic materials
2. Revisiting orbital magnetism of free electron
3. in tight-binding models of metals and insulators
4. Orbital susceptibility formula : role of band structure
Magnetism basics
Magnetization :
→ spontaneous/permanent magnetization :
→ induced by an external field (susceptibility) :
paramagneticdiamagnetic
Magnetism basics
Magnetization :
→ spontaneous/permanent magnetization :
→ induced by an external field (susceptibility) :
paramagneticdiamagnetic
Spin
Localized magnetic impurities : Langevin paramagnetism
Itinerant electrons in metals : Pauli paramagnetism
Orbital motion of electrons
Atomic (localized) contribution: Larmor diamagnetism
Itinerant electrons in metals : Landau diamagnetism
Periodic Table : paramagnetic and diamagnetic elements
-Diamagnetism discovered by Sebald Justinus Brugmans in 1793 « Bismuth and Antimony repel each other »-Paramagnetism discovered « theoretically » by Faraday in 1845
Periodic Table : paramagnetic and diamagnetic elements
BismuthAntimony
Paramagnetic
Diamagnetic
Carbon
Terbium
Diamagnetism in materials
Superconductor :
perfect diamagnet : the magnetic field is completely expelled (Meissner effect)
Other materials :
Water Bismuth Diamond Graphite Graphite Graphene
Fingerprint of Diamagnetism : Levitation
Levitation of graphiteLevitation of a frog (« water »)
M. Berry and A. Geim, E.J.P. 1997 Ig-Nobel 2000
(ambiant temperature)
Nature 349 p 470 (1991)
Levitation in strong magnetic field with a strong gradient
Why is diamagnetic levitation possible ?
2D free electron gas :Landau diamagnetism, 1930
Levitation of graphite
?
Paramagnetic !
Why is diamagnetic levitation possible ?
Levitation of graphite
Bloch electron gas : Importance of Band structure effects !
2D Dirac electron gas : Mc Clure diamagnetism 1956
Strong diamagnetism at Dirac point !
Orbital susceptibility of tight-binding electrons
square lattice
graphene
?
? ?
Outline :
1. Magnetism of non magnetic materials
2. Revisiting orbital magnetism of free electron
3. in tight-binding models of metals and insulators
4. Orbital susceptibility formula : role of band structure
Classical cyclotron motion
Energy :
Orbital magnetic moment
Is there « classical » orbital magnetism ?
Lorentz force
Classical cyclotron motion : magnetic field scaling
Energy :
Orbital magnetic moment :
Lorentz force is transverse → no work → energy is field independent !
No « classical » orbital magnetism orbital magnetism is a quantum phenomenum !
cyclotron radius and velocity :
Quantum cyclotron motion : Landau quantization
Energy :
Orbital magnetic moment :
cyclotron radius and velocity : « quantum scaling »
Correct calculation but misleading physical picture !
Quantum cyclotron motion of a wave packet
Quantum fluctuations of minimal energy wave packet :
Ehrenfest theoremexactly like classical quantities !
Mean quantum position and velocity :
quantum scaling
The wave packet spreading widthdepends on magnetic field
Quantum cyclotron motion of a wavepacket
Orbital magnetic moment :
Energy :
Quantum fluctuationsprovide a field independentcontribution
Quantum fluctuationsprovide a field dependentcontribution
self rotation of the wavepacket
orbital magnetism originates from quantum fluctuations due to the magnetic field depedent wave packet spreading width !
Outline :
1. Magnetism of non magnetic materials
2. Revisiting orbital magnetism of free electron
3. in tight-binding models of metals and insulators
4. Orbital susceptibility formula : role of band structure
density of states in small magnetic field :
Thermodynamic grand potential:
Spontaneous magnetization :
Susceptibility :
How to calculate orbital magnetic susceptibility
Diamagnetic paramagnetic
First step : Peierls substitution
-Tight-binding models:
Second step : Density of states in magnetic field
1-Exact spectrum in magnetic field :
2-perturbation theory : « linear response »
How to calculate the density of states in magnetic field
Hofstadter Butterfly, 1976
From square to honeycomb : Brikwall lattice
gapped grapheneStaggered square
Square lattice « honeycomb » lattice
Single band square lattice
Numerics of Landau Levels : Hofstadter Butterfly
Hofstadter 1976
square lattice
Metals
paramagnetic Para. & diamagnetic
Square lattice
Graphene
Insulators
Gapped Graphene
Gapped square lattice
susceptibility plateau in the gap ?
*Diamagnetic and paramagnetic
*in metals and in insulators
*Fermi surface and Fermi sea
Orbital susceptibility Pauli susceptibility
*Paramagnetic
*in metals
*Fermi surface
*zero field density of states
Take Home Message
*Diamagnetic and paramagnetic
*in metals and in insulators
*Fermi surface and Fermi sea
Orbital susceptibility
Can we understand orbital susceptibility from zero field band spectrum and wavefunctions ?
Pauli susceptibility
*Paramagnetic
*in metals
*Fermi surface
*zero field density of states
Take Home Message
Outline :
1. Magnetism of non magnetic materials
2. Revisiting orbital magnetism of free electron
3. in tight-binding models of metals and insulators
4. Orbital susceptibility formula : role of band structure
Peierls formula
3737
Peierls 1933
« hessian curvature »
Independent bands approximation
* Fermi surface property (only in metal)* Energy spectrum property
* Diamagnetic and paramagnetic regions compensate
Sum rule :
How good is it ?
From square to honeycomb : Brikwall lattice
gapped grapheneStaggered square
Square lattice « honeycomb » lattice
Single band square lattice
Parabolic band edge → positive inverse mass determinant → diamagnetic
Saddle point → negative inverse mass determinant → paramagnetic
Inverse mass determinant
Graphene
Sum rule :
Landau-Peierls formula fails to reproduce the paramagnetic plateau near Mc Clure peak ?
Staggered square lattice
Landau-Peierls :null in the gapGapped graphene
Para or diamagneticplateau in the gap
Perturbative approaches with interbands effects
Almost free electron limit or low energy modelsRoth (1962) Blount (1964) Fukuyama (1971)Koshino, Ando (2010) Tight-binding models Gomes-santos, Stauber (2011) (graphene)Gao,Niu (2014)Raoux, J-N. Fuchs, G. Montambaux and F.P. (2015)
how many interband contributions ? Fermi surface vs Fermi sea ? Energy spectrum vs wavefunctions ?
Multibands susceptibility formula
Raoux & al (2015)
Bloch Hamiltonian matrix
Interband onlyPeierls+Interband
Greens function matrix
Two bands models orbital susceptibility
Landau-Peierls
Quantum metric
each contribution verifies the sum rule
Interband contributions
Berry curvature
Quantum metric(particle-hole assymetric)
Geometry of Bloch states
modulus phase
Geometry of the phase :
Berry connection
Berry curvature vector
Geometry of the modulus :
quantum metric tensor
Bloch state :
Provost-Valle, 1980
(Fubini-Study metric)
Two-band models
Sublattice pseudospin 1/2
A B
Energy spectrum
Eigen-wave function projector
Geometry of 2-band models in two dimension
Berry curvature (scalar):
Covariant quantum metric (2x2 symmetric matrix):
Anatomy of two-band susceptibility
Landau-Peierls :
Berry curvature contribution :
Fermi sea,dia
Fermi Surf, Para
Quantum metric contributions :
Fermi sea, dia & para
Fermi sea, dia & para
In-gap susceptibility plateau
Dia ParaPara or dia
* Fermi sea contribution only
Staggered square lattice
=
+ +
Landau-Peierls Berry curvature Quantum metric
Gapped graphene
=
+ +
Landau-Peierls Berry curvature Quantum metric
Brikwall lattice
« gapped graphene »
Staggered square
Gapped graphene : lattice vs low energy
Koshino-Ando (2010)
Gapped graphene : lattice vs low energy
Berry curvature
Gapped graphene : lattice vs low energy
Quantum metric
Gapped graphene : lattice vs low energy
Flat band on checkerboard lattice Mielke (1991)
Landau-Peierls : No contribution of the flat band !
Flat band results from destructive interferences
Flat band on checkerboard lattice
Divergent paramagnetic peak near the flat band !
Orbital susceptibility is a subtle quantity !
Single band : Fermi surface effect → only in metals ;Diamagnetic and paramagneticdetermined only by the energy spectrum → Hessian curvature Two bands : Fermi surface+Fermi sea contributions in metal and in insulators (in-gap plateau & flat band)cannot be described by energy spectrum only !interband effects due to Bloch wavefunctions geometric properties Berry curvature Quantum metric
Perspectives : role of spin orbit coupling role of quantum metric tensor & Berry curvature in other magnetic field dependent quantities : transport, plasmon, excitons...
(Peierls 1933)
Supplementary material
Geometric interpretation
Berry connection :
next order in field : Gao, Yang and Niu (2014)
Berry curvature :
Geometric interpretation
Berry connection :
next order in field : Gao, Yang and Niu (2014)
Berry curvature :
Interband susceptibilities :
Geometric interpretation
Thonhauser et al, Xiao et al (2005)Gat, Avron (2003)Magnetization :
Berry connections :
Interband susceptibility :
zero field: first order field corrections :
Geometric interpretation
Thonhauser et al, Xiao et al (2005)Gat, Avron (2003)Magnetization :
Berry connections :
Interband susceptibility :
zero field: first order field corrections :