Interband effects and orbital suceptibility of multiband tight-binding models

Transcription

1 Interband effects and orbital suceptibility of multiband tight-binding models Frédéric Piéchon LPS (Orsay) with A. Raoux, J-N. Fuchs and G. Montambaux Orbital suceptibility Berry curvature ky? kx GDR Modmat, Marseille, 2 Avril 2014

2 Preceding talk Orbital suceptibility strongly affected by interband effects beyond the energy spectrum Not a Fermi surface property : need full band knowledge This talk Which intraband quantities characterize interband effects? How these quantities enter in orbital magnetization and orbital suceptibility? ( gauge invariant perturbation theory )

3 Outline Intraband quantities characterizing interband effects Semiclassical picture of interband effects Orbital suceptibility from gauge invariant perturbation Conclusion & Perspectives

4 Intraband quantities characterizing interband effects

5 Characterizing interband effects Eigen band Bloch state : R. Resta Lecture notes 2013, energy band For a N-band tight-binding model, the vectors are the N components eigenstates of the NxN Bloch Hamiltonian matrix Overlapp matrix NxN: but the dependency of originates from interband effects! Are there intraband quantities that give information on interband effects? Yes but only at «second order» : product of two first order interband terms

6 Intraband quantities at second order in interband Effects Intraband metric tensor: Berry curvature : More physical quantities : Inverse mass tensor corrections: Orbital magnetic moment :

7 Rewriting the intraband quantities position operator : velocity operator Using the identity Intraband metric tensor : Inverse mass tensor corrections: Berry curvature vector: Magnetic moment vector: No quantitative studies of metric tensor and inverse mass tensor corrections? This work concentrate on magnetic moment and Berry curvature

8 Time reversal Symmetry For system invariant under time reversal symmetry : Magnetic moment and Berry curvature Dos : Let's now go to examples!

9 2-band tight-binding models on the honeycomb lattice Bloch Hamiltonian matrix t'' t t t' NN hopping t, t' NNNN hoping t'' Onsite potential Pauli matrices

10 Energy band spectrum Semi-metal phases with band touching points ( Graphene Dirac electron t'=t t''=0 Gapped phases =t'-2 t t''=0 Semi-Dirac t'=2t t''=0 =0) «graphene bilayer» t'=t t''=t/2 «Boron-Nitride» t'=t t''=0 =1/2t

11 Magnetic moment and Berry curvature in reciprocal space Not independent quantities! Semi-metal phases Graphene : Dirac semi-dirac «bilayer» Gapped phases B-N No surprise : increasing gap reduces

12 Magnetic moment and curvature Dos Semi Dirac Graphene : Dirac Bilayer BN Magnetic moment independent on gap x100 x100 and by increasing strongly diminished Curvature inverse proportional to gap and strongly depend On the nature of band touching «linear (Dirac), Semi-dirac, Bilayer (quadratic)»

13 3-bands tight-binding models on the Dice lattice Bloch Hamiltonian : A B C Band spectrum : 2 interpenetrating honeycomb lattices Bipartite : (A,C) (B) dispersive bands : (independent on ) flat band : (1/3 of states) Dos similar to 2-band model plus a delta peak at But : all wavefunctions strongly depend on : interband effects!

14 Magnetic moment and Berry curvature (2-band case) Magnetic moment : k-space picture of for all three band looks very much like for the 2-band case : only the quantitative value depend on Berry curvatures : Magnetic moment and Berry curvature have different dependency on Sum rule :

15 Intergrated curvature Graphene : Dirac Semi Dirac BN dependency Bilayer Flat band has a strong effect on curvature. Similar dependency only global scale change

16 Semiclassical picture Of interband effects

17 Textbook's semiclassical equation of motion «minimal Peierls substitution» Slow varying magnetic field and electric field The dispersion relation becomes the effective classical Hamiltonian with the gauge invariant momentum Equations of motion : Limit of validity not clear! Misses low field interband effects!

18 Semiclassical equations of motion with interband contributions «improved Peierls substitution» the dispersion relation Rev. Mod. Phys. D. Xiao, M.C. Chang & Q. Niu (2010) R.Resta Lecture notes (2013) becomes the effective classical Hamiltonian «effective Zeeman term» position and gauge invariant momentum that verify anomalous Poisson brackets : are band projected operators Interband effects modify phase space structure Equations of motion «anomalous velocity» Berry curvature is a magnetic field in k-space Validity beyond linear order in external field? (See however arxiv : )

19 Semiclassics for spontaneous orbital magnetization in uniform magnetic field Anomalous Poisson brakets and Zeeman term modify the effective energy spectrum Dos : «reminds linear degeneracy of Landau level» thermodynamic «grand potential» and spontaneous magnetization But ( for time reversal invariant system ) beyond first order in magnetic field : suceptibility? (See however arxiv : )

20 Orbital suceptibility from gauge invariant perturbation

21 Perturbative microscopic approaches A long story R. Peierls : «Peierls substitution in the Hamiltonian». Direct calculation of the partition function. First suceptibility formula. valid for a single band models only Many attempts to improve Peierls formula : Adams, Kjeldaas, Kohn, Hebborn, Sondheimer, Roth, Misra, Kleinman, Wannier... No clear conclusion! 1971-H. Fukyama : Kubo-linear response. Diagrammatic approach. First suceptibility formula with «matrix» Greens functions : contains interband effects. Not valid for tight-binding models G. Gomez-santos & T. Stauber : Fukuyama's method adapted to tight-binding electrons. Can only handle suceptibility but not magnetization. Explicit k-space approach difficult to adapt to inhomogenous and finite systems X. Gonze & J.W.Zwanziger : matrix density approach. Can handle both magnetization and suceptibility. Restricted to zero temperature K.T.Chen & P.A. Lee. thermodynamic approach+ Gauge invariant perturbative calculation of the K-space greens function. Method only skectched. No explicit formula for magnetization or suceptibility! 2012-B. Savoie and P. Briet (mathematical physics). thermodynamic approach+gauge invariant perturbative calculation of the local R-space greens function. Explicit formula for the suceptibility!

22 Gauge invariant perturbation for tight-binding electrons Aim : obtain Zero field lattice greens function : Peierls substitution in the lattice Hamiltonian : Seems to break translation invariance Gauge dependent Peierls phase naive perturbation : Breaks translation invariance + gauge dependent order by order! gauge invariant perturbation : :gauge invariant magnetic flux through the triangular loop

23 Orbital Suceptibility formula (global) generalized Peierls contribution Interband contribution Bloch electron picture Position space picture velocity «inverse mass» One can obtain magnetization formula for both global m and local m(r) and also local suceptibility Relation with magnetic moment and Berry curvature?

24 Orbital Suceptibility formula (global) generalized Peierls contribution Interband contribution Bloch electron picture Position space picture velocity «inverse mass» One can obtain magnetization formula for both global m and local m(r) and also local suceptibility Relation with magnetic moment and Berry curvature?

25 2-band models suceptibility We can obtain various analytical equivalent forms...one example is : generalized Landau-Peierls formula that contains only energy spectrum derivatives but that involved full band knowledge in addition to Fermi surface. : interband term that comes from the first «g^2» contribution to suceptibility formula. Often a small correction. Not a Fermi surface term. Not related to magnetic moment or Berry curvature. : interband term coming from «g^4» contribution. Can be recast as : Involve only magnetic moment and Berry curvature Let's go to examples!

26 2-Band semi-metals semi-dirac Graphene (Dirac) Bilayer (quadratic) Landau-Peierls does not gives the diamagnetic peak LP has diamagnetic peak but with wrong temperature scaling All these semi-metals have a diamagnetic diverging peak. Temperature scaling of the peak depends on the type of band touching.

27 2-Band Gapped phases BN : gapped (Dirac) both Landau-Peierls or Mclure fail to give the correct scale of the diamagnetic plateau in the gap Gap only small corrections to Landau-Peierls formula? Para vs diamagnetism at gap edge?

28 2-Band Gapped phases BN : gapped (Dirac) both Landau-Peierls or Mclure fail to give the correct scale of the diamagnetic plateau in the gap Gap only small correction to Landau-Peierls formula Para vs diamagnetism at gap edge!

29 3-Band cases Dirac electrons semi-dirac electrons Flat band and dependent interband effects induce dia to paramagnetic transition for the peak, the gap plateau and the gap edges. Reversly it induces para to dia transition at larger energy scale.

30 Conclusion Interband effects can be quantified by intraband quantities : magnetic moment, Berry curvature & Mass tensor, Metric tensor Interband effects can be tuned without changing the energy spectrum Interband effects are important for orbital magnetic suceptibility : it provides divergent (dia/para) suceptibility at band touching point, (dia/para) in gap plateau and at gap edges. Depends on the nature of the gap. affect overall shape of suceptibility at large energy scale. Perspectives use gauge invariant perturbation to developp a local picture for magnetization and suceptibility (bulk vs boundary, disorder). role of spin orbit (Z2 insulator) & time reversal symmetry (Chern insulator). Influence of e-e interactions