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 :