Connecting real materials to low-energy effective Hamiltonian: applications of symmetryrespecting

Transcription

1 Connecting real materials to low-energy effective Hamiltonian: applications of symmetryrespecting Wannier functions Wei Ku ( 顧威 ) Brookhaven National Lab & SUNY Stony Brook

2 Outlines Simple illustrations: Orbital order in LaMnO 3 Gapless CDW in TaSe 2 Ferro-orbital order in Fe-SCs Interacting problems: Local excitations Propagation and decay of local excitations Effects of disordered impurities in Fe-SCs: Substitution of Fe: doping or not? Fe vacancy: violation of Luttinger theorem Ru substitution: realization of superdiffusion

3 Energy RG perspective: bifurcation of many-body systems H bare H eff H eff Constraint: no double occupation H eff H eff H eff H eff Other Constraint? Lagrangian: integrate out high-energy sectors vs. Hamiltonian: block diagonalize low-energy sctors

4 Direct mapping to effective Hamiltonian 1 DFT : LDA+U calculate t,u,v,j Wannier Function tight-binding H LDA+U parameters self-consistent Hartree-Fock H HF ab initio full H effective H eff W. Ku, H. Rosner, W.E. Pickett, R.T. Scalettar, PRL 89, (2002). S. White, J. Chem. Phys. 117, 7472 (2002). canonical transformation few-band H eff Team 2 LDA+U+WF=H HF W. Yin, D. Volja, W. Ku, PRL 96, (2006) Applications: CMR manganites HTSC cuprates

5 Orbital Ordering in LaMnO 3 e-e vs. e-l interactions & effective many-body H eff W.-G. Yin et al, PRL 96, (2006) Large O component nature of charge transfer gap Systematic study of e-e vs. e-l interactions in ordering orbitals K H t d d g T Q Q Q U n n E T ' z ij j ' i i i i i eff i i z i ij ' i 2 i i i pseudo-spin: z 1 1 Ti d d d d T d d d d 2 2 x ( ), i ( ). i i i i i i i i.

6 Gapless charge density wave in TaSe 2 R. L. Barnett, A. P., E. Demler, W.-G. Yin, and Wei Ku Phys. Rev. Lett. 96, (2006)

7 Gapless Charge-Density Wave in TaSe 2 commensurate CDW gapless excitations throughout the whole Fermi surface in CDW phase? Fermi surface Nesting vector too large for q CDW? R. Liu, et. al. Phys. Rev. B, 5212 (2000)

8 Conventional gapped CDW picture V q CDW, k kq Fermi surface instability divergent q nesting preferred gap energy gain CDW, 0

9 Local Picture: Low-Energy Wannier Function e g e g a g 3z 2 r 2 (a g ) symmetry near EF, noticeable hybridization with e g WS in one site contains complete information of the full k-space center: a g symmetry; tail: e g symmetry

10 Surprising Hopping Path DFT DFT trn, R n Rn h R n t t t t 1 2,1,2 38(meV) 115(meV) 29(meV) 23(meV) t t 2 1! perfect tight-binding fit to the band structure surprising hopping strength to 2nd nearest neighbors

11 Why t 2 >> t 1? + NN NNN

12 Why t 2 >> t 1? + NN NNN phase interference from e g hybridization tail t t 2 1

13 Gapless Charge-Density Wave in TaSe 2 x 2 -y 2 hybridization t 2 1 decoupling of 3 sublattices minimization of tight-binding H against distortion t one sublattice undistorted gapless band structure R. L. Barnett et al, PRL 96, (2006)

14 Ferro-orbital order & anisotropic magnetic structure in 1111 (&122) Chi-Cheng Lee, Wei-Guo Yin & Wei Ku Phys. Rev. Lett. 103, (2009)

15 Energy resolved, symmetry respecting Wannier function (energy window) Rn km km Rn 1 N 1 km cell km km e ik R U ( k ) mn U km e ( k ) mn Ncell k m ik R x z d xz y x z d yz methods see: W. Ku et al., PRL 89, (2002); W. Yin et al., PRL 96, (2006) y xz xy yz z 2 x 2 -y 2 NM onsite energy (ev) z x 2 -y yz 0.10 xz 0.10 xy 0.34 small crystal field splitting degenerate xz and yz orbital freedom!

16 Energy (ev) Energy (ev) Comparing LDA band structures NM in NM 1 st -BZ d xz & d yz most relevant to the low-e Only d yz splits strongly near E F C-AF d xz d yz W Δ d yz more spin polarized ~0.34m B than d xz (~0.15m B ) more different with U=2eV 0.58 vs. 0.23m B orbital symmetry broken D ~ W large (,k)-space involved local picture more suitable Fermi surface nesting not essential SDW less convenient unfolding methods see: Wei Ku et al., PRL 104, (2010)

17 Anti-intuitive hopping parameters <WFs H WFs> Fe1 z 2 x 2 -y 2 yz xz xy Fe2 (Fe4) z (-0.31) (0.00) 0.00 (0.10) 0.00 x 2 -y (-0.31) (0.00) 0.00 (0.42) 0.00 yz (0.00) 0.42 (0.00) (-0.13) (0.23) xz 0.00 (0.10) 0.00 (0.42) (-0.40) (0.00) xy (0.23) (0.00) Fe3 z x 2 -y yz xz xy Fe1 Fe4 Fe2 Fe3 Unusual coupling direction Cubic symmetry broken seriously by As Fe-As phonon modes important Perpendicular hopping direction! Chi-Cheng Lee et al., PRL 103, (2009)

18 C-AF magnetic structure and ferro-orbital order y Fe1 Fe4 x yz xz J 2 J 1x Fe2 Fe3 Strongly anisotropic super-exchange: J 1x > J 2 >> J 1y no competition with G-AF at all! J 1 ~ 2J 2 irrelevant! single orbital Heisenberg model inadequate Orbital polarization and ferro-orbital correlation important rotational symmetry breaking Unusual coupling direction and strong anisotropic hoppings! common staggered OO loses a > b: AF across long bond (rare) J 2 ~ 0.4 J 1x yz xz yz xz ΔE = -2t 2 /U Fe1 Fe2 ΔE = -t 2 /(U -J H ) Fe1 X Fe2 Chi-Cheng Lee et al., PRL 103, (2009)

19 Outlines Simple illustrations: Orbital order in LaMnO 3 Gapless CDW in TaSe 2 Ferro-orbital order in Fe-SCs Interacting problems: Local excitations Propagation and decay of local excitations Effects of disordered impurities in Fe-SCs: Substitution of Fe: doping or not? Fe vacancy: violation of Luttinger theorem Ru substitution: realization of superdiffusion

20 Propagation of local excitations in strongly correlated materials P. Abbamonte et. al., PNAS 105, (2008) C-C Lee et al., Phys. Rev. Lett. 111, (2013)

21 Acknowledgement Funding sources Basic Energy Science, Department of Energy DOE-CMSN Taiwanese Student Fellowship Theoretical works Chi-Cheng Lee (BNL & Tamkang Univ.) Chen-Lin Yeh (BNL & Tamkang Univ.) Hung-Chung Hsueh (Tamkang Univ.) Experiments Peter Abbamonte, Xiaoqian M. Chen, Yu Gan (UIUC) Ben C. Larson (ORNL) John Hill (BNL), Young Cai (BNL) Andrivo Rusydi (NUS, Singapore) Young-June Kim (U. Toronto, Canada) Nozomu Hiraoka (Sping-8, Japna) Chi-Cheng Lee Sinica

22 Local Picture for Strongly Correlated Systems H H V H H 0 local nonlocal V too big for perturbation Maximize the terms in the local part symmetric Wannier Representation defines local Treat local part accurately Add non-local part as modification

23 How to define local in CT-insulators? Periodic symmetry Point group symmetry Simultaneously keep both? How to split the Hilbert space?

24 Symmetric Wannier Functions for CT-Insulators W. Ku et al., Phys. Rev. Lett. 89, (2002). R. L. Barnett et al., Phys. Rev. Lett. 96, (2006). W.-G. Yin et al, Phys. Rev. Lett. 96, (2006). Ni 3 r z 3( y ) x xy yz xz O 2 3( x y 2 ) 2 2 3z r s wave O-p orbitals additional Ni-d orbitals (no double counting of O orbitals) local is now defined by this super-atom

25 Super Atom for Charge Transfer Insulator Ni 3 r z 3( y ) x xy yz xz O 2 3( x y 2 ) H 2 2 3z r H local H nonlocal (exact) (modification) s wave Maximize the contributions of local atom

26 Density response for super atom (eg-t2g=0.65ev) (antibonding-type)

27 Multiplet Splitting Made Possible with MB Hilbert Space GS Ni eg Ni t2g O eg O s Ni eg Ni t2g O eg O s Ni eg Ni t2g O eg O s (0.9484) ( ) (0.2226) T 1g

28 3D Map of Local Excitations in the Mott Gap in NiO 1eV excitation found! very different q-dependence of spectral weight Nozomu Hiraoka et al, Europhys. Lett. 96, (2011)

29 3D Map of Local Excitations in the Mott Gap in NiO B. C. Larson et al.,prl 2007 Detailed comparison: exp vs. TD-LDA+U & cluster model good general agreement with some discrepancies in details

30 How does the local excitations propagate and decay?

31 Propagation of Tightly-Bound Excitons: case study of LiF Tightly bound exciton Charge transfer insulator p-h in different atoms Frenkel or Wannier exciton? Dispersion propagation in space/time Inelastic X-ray scattering Structured spectral weight Clear dispersion at large q! observe fs dynamics P. Abbamonte et. al., PNAS 2008

32 Excitons in LiF as a Frenkel Exciton in a Super Atom ½ x ½ 10 5 y z y x ½ z

33 Matrix Element and Structure in q-space real-space x ½ y z q-space k x k y k z q = 0~1.5 Intensity divided by 2.6 P. Abbamonte et al., PNAS 2008

34 Propagation of tightly bound excitons Treat tightly bound excitons (and other local excitations) as a composite boson. Define its propagation kinetic kernel and full propagator DH L and DH : D T H DH DH TDH via local integrates out all the pair fluctuation in space and encapsulates propagation and decay processes. We then approximate T D 1 L L using unbound exciton propagator H D H ~ D GH D GH L T 0 L Separation of local many-body problem from non-local propagation Many orders of magnitudes cheaper than Bethe-Salpeter equation 0 T C-C Lee et al., Phys. Rev. Lett. 111, (2013)

35 Effective Two-Particle Hopping Define effective two particle kinetic kernel T via D H D H D H TD H L L D D L D T in the basis of local bound pair b L 1 1 T D H D H c c RN Rp Rh and simplify 1 1 D 0 G H L I H L D0 G H I H same location/time 1 1 D0 G H L D0 G H using the empty bubble D0 RN, RN ; t, t G Rp, Rp; t, t G Rh, Rh; t, t D L Rpt Rht T gives hopping of p-h pair in real space dispersion in q-space C-C Lee et al., Phys. Rev. Lett. 111, (2013)

36 Effective Two-Particle Hopping in LiF T() is complex and strongly -dependent to fully account for 1. Landau continuum (integrating out virtual pair breaking processes) exact for E b = 0 2. Lower mobility with stronger p-h binding correct t 2 /E b behavior 3. Renormalization of on-site energy from kinetic energy

37 Effective Two-Particle Hopping Define effective two particle kinetic kernel T via D H D H D H TD H L L D D L D T in the basis of local bound pair b L L 1 1 T D H D H c c RN Rp Rh and simplify 1 1 D 0 G H L I H L D0 G H I H same location/time 1 1 D0 G H L D0 G H using the empty bubble D0 RN, RN ; t, t G Rp, Rp; t, t G Rh, Rh; t, t D L Rpt Rht T gives hopping of p-h pair in real space dispersion in q-space C-C Lee et al., Phys. Rev. Lett. 111, (2013)

38 Exciton Band Structure ReT 1. Full results similar to the diagonalization of exciton exciton 2. Similar dispersion to the F p-bands (same symmetry) C-C Lee et al., Phys. Rev. Lett. 111, (2013)

39 Observation of Multiple Exciton Bands 1. Similar weight in momentum space 2. Similar dispersion 3. Switching of bands (breaks in intensity)

40 Time Evaluation of Charge Fluctuation in LiF exp. 2D map of exciton propagation is in preparation

41 Time Evaluation of Charge Fluctuation in LiF exp. 2D map of exciton propagation is in preparation

42 Outlines Simple illustrations: Orbital order in LaMnO 3 Gapless CDW in TaSe 2 Ferro-orbital order in Fe-SCs Interacting problems: Local excitations Propagation and decay of local excitations Effects of disordered impurities in Fe-SCs: Substitution of Fe: doping or not? Fe vacancy: violation of Luttinger theorem Ru substitution: realization of superdiffusion

43 Treating materials with disordered impurities T. Berlijn, D. Volja, and Wei Ku, PRL 106, (2011) For various applications, see T.S. Herng, et al., Phys. Rev. Lett. 105, (2010) Tom Berlijn, et al., Phys. Rev. Lett. 108, (2012) Tom Berlijn, et al., Phys. Rev. Lett. 109, (2012) L.-M. Wang, et al., Phys. Rev. Lett. 110, (2013)

44 Acknowledgement Funding sources Basic Energy Science, Office of Science, Department of Energy Dmitri Volja MIT Tom Berlijn BNL Limin Wang BNL Collaborators P. Hirschfeld, & H.-P. Chen, U of Florida

45 First-principles methods for disorder Relevance Defects, vacancy, chemical substitution, doping & intercalation Thermal/quantum fluctuation induced spatial inhomogeneity Paramagnetic phase with disordered local moment Different approaches Mean-field potential: virtual crystal approximation Mean-field scattering: coherent potential approximation Perturbation: diagrammatic summation Non-perturbed direct method: configuration average over super cells

46 Mean-field approaches VCA: Virtual Crystal Approximation V virtual crystal = (1-x) V 0 + x V 1 no scattering CPA: Coherent Potential Approximation lack cluster-scattering A. Gonis, Green functions for ordered and disordered systems (1992)

47 Non-local physics: k-dependent broadening same different broadening k

48 Non-local physics: large-sized impurity states Anderson localization, multiple scattering required

49 Non-local physics: short-range order

50 Configuration average over super cell disordered system configuration 1 big super cells configuration N + + P 1 P N <G > P 1 G P N G N Challenge: solving many Hamiltonians with big super cells

51 Configuration average over super cell big super cells (100 ~ 1000 of atoms) +.+ <G > P 1 G P N G N Challenge: solving many Hamiltonians with big super cells

52 Constructing small and accurate effective H Consider H from influence of impurities at x 1,, x N 1. Calculate in same Wannier basis from DFT 2. Calculate D by taking difference of H DFT in Wannier basis 3. Assemble H c for each disorder configuration 4. Solve H and unfold A(k, ) to the normal cell unit 5. Average over configurations with different size & shape... 1, ,, 1 D D N j i x x N i x x x j i i N H H i i j i j i i i x x x x x x x x H H H H D D D D 0,, 0 ;...,, 0, 1 0,,..., 1, 1 D N i i m i m r m r n r r n r r n H n r r n r H r n i N N c c c w k P A w k A,, j i i x x x H H H, 0 &, T. Berlijn, D. Volja, and Wei Ku, PRL 106, (2011)

53 Including Effects in Density Functional Theory two DFT Calculations undoped normal cell 1 impurity in large super cell

54 Energy (ev) Reducing Hilbert Space via Wannier Transformation DFT CoO 2 Wannier functions truncated

55 Energy Energy Obtain Effects of Impurities on H DFT DFT undoped 2 Wannier transformations k DFT 1 impurity Reduced Hilbert space 2 Tight Binding Hamiltonians undoped 1 impurity k x x 0 D i H i H

56 Constructing small and accurate effective H Consider H from influence of impurities at x 1,, x N 1. Calculate in same Wannier basis from DFT 2. Calculate D by taking difference of H DFT in Wannier basis 3. Assemble H c for each disorder configuration 4. Solve H and unfold A(k, ) to the normal cell unit 5. Average over configurations with different size & shape... 1, ,, 1 D D N j i x x N i x x x j i i N H H i i j i j i i i x x x x x x x x H H H H D D D D 0,, 0 ;...,, 0, 1 0,,..., 1, 1 D N i i m i m r m r n r r n r r n H n r r n r H r n i N N c c c w k P A w k A,, j i i x x x H H H, 0 &, T. Berlijn, D. Volja, and Wei Ku, PRL 106, (2011)

57 Substitution & vacancies: Zn 1-x Cu x O 1-y SQUID 300K (O-poor) 300K (O-rich) 5K (O-poor) 300K (O-poor) Q1) Where do the doped electrons reside? Q2) What is the role of oxygen vacancy? T. S. Herng, et. al, PRL 105, (2010)

58 Energy (ev) spin minority spin majority Cu-d V O Cu-d V O A1) Doped electrons reside in Cu orbitals A2) V O mediates Cu moments V O Cu d 9 Cu d 10 T. S. Herng, et. al, PRL 105, (2010)

59 Do transition metal substitution dope carriers in Febased superconductors T. Berlijn & Wei Ku PRL 108, (2012)

60 Do TM substitutions dope carriers in Fe-SC? more valence electrons than Fe expect electron doping

61 Additional Charge Residing in the Substituted Atoms in Fe-SC Theory DFT (density) Co:122 Experiment XANES Co:122 H. Wadati et al, PRL 105 (2010) E. M. Bittar et al, PRL 107 (2011) Substitution does not dope the system (?)

62 12.5% Co Substituted BaFe 2 As 2 a nα 122 Fe 7.22 Co:122 Fe 7.19 x 7/8 Co:122 Co 8.40 x 1/8 BZ E F 3 d k dω An (k,ω) A(k,ω E F ) Co:122 +1e/Co X Γ M Change of density in substituted atoms not doping? Enlarged Fermi surfaces & shift of chemical potential doping?

63 Energy(eV) Example: Does Co/Zn substitution dope Fe-SC? G M X G Z R A Z 0.2 undoped E F G M X G G M X G Fe Co x 7 As +1 e/co Most definitely yes, in the simplest manner, but some carriers lose coherence! Tom Berlijn, et al., PRL 108, (2012)

64 Fe vacancy in K2Fe4Se5 T. Berlijn, P. Hirschfeld, & Wei Ku PRL 109, (2012)

65 A heavily electron doped system? F. Chen et al. Phys. Rev. X 1, (2011)

66 Effective doping with Fe vacancy: Luttinger theorem? Appears to be heavily doped ~ 0.5 e / Fe with disordered Fe vacancy Tom Berlijn, Peter Hirschfeld, and Wei Ku, 109, (2012)

67 Origin of the effective doping with Fe vacancy Need both disorder and the impurity potential Tom Berlijn, Peter Hirschfeld, and Wei Ku, 109, (2012)

68 Effects of disordered impurity potential Tom Berlijn, Peter Hirschfeld, and Wei Ku, 109, (2012)

69 Ru substitution of BaFe2As2 Limin Wang, Tom Berlijn, Chia-Hui Lin, Yan Wang, Peter Hirschfeld, Wei Ku PRL 110, (2013)

70 Temperature-composition phase diagram [R. S. Dhaka et al, PRL 107, ] S. Nandi et al, PRL 104, [N. Katayama, et al, JPSJ 79,113702] [S. Kasahara, et al, PRB ] Ru/Co atoms replace the most essential Fe atom. Superconductivity can survive even with 40% Ru substitution

71 Transport property [F. Rullier-Albenque et al., PRB 81, (2010)] [P. C. Canfield et al., Annu. Rev. Cond. Matt. 27 (2010)] T(K) Ru substituted samples exhibit a residual resistivity comparable to 8% substituted system at a much higher 35% substitution level

72 Experimental (ARPES) controversy Nearly substitution-independent FS [R. S. Dhaka et al, PRL 107, (2011) ]

73 Experimental (ARPES) controversy A crossover from two-dimensional to three dimensional structure [N. Xu et al, PRB 86, ]

74 Band structure for different Ru substitution The bands are heavily smeared.(-0.8ev) x=0.55 is the most disordered one. Fe bands around the Fermi level remain very sharp.

75 Origin of the superdiffusion around the Fermi level diagonal disorder only : strong smearing of the entire d-band off-diagonal disorder only : scattering diminishes near the center of the d-band. combined weak impurity scattering near E F

76 Energy (ev) Origin of the superdiffusion around the Fermi level (a) <A(G,)> (ev -1 ) (b) x=0.38 (c) diagonal disorder G M X G Z R A Z G M X G Z R A Z (e) (d) off-diagonal disorder G M X G Z R A Z k j k ' j ' <kj D k ' j ' > <kj D D k ' j ' > <kj D OD k ' j ' > G 9 0.3GM G 8 0.3GM The insensitivity originates from a coherent interference between on-site and off-site impurity effects.

77 Superdiffusion in 1D model [D. H. Dunlap, K. Kundu, and P. Phillips, PRB 40, (1989)] Single-particle Hamiltonian a correlated disorder ( ) b site-diagonal disorder. c site-off diagonal disorder. Correlations between site and off-diagonal disorder give rise to superdiffusion transport.

78 Fermi surface (2Fe) for different Ru substitution The hole pockets become more three dimensional The electron pockets show a slight substitution dependence.

79 Resolution of the ARPES controversy (a) x=0.02 (b) x=0.21 (c) x=0.36 (d) x=0.55 [R. S. Dhaka et al, PRL 107, ] 1Fe zone This photon polarization couples mostly to the yz orbital. The yz character of our unfolded BS also show little change.

80 Suppression of the long-range order x = 0 x = 0.21 x = 0.38 x = 0.55 x = 0.75 x = 1.0 [R. S. Dhaka et al, PRL 107, ] Larger splitting between the conduction and valence bands carrier reduction The reduced DOS & enhanced 3D nature suppress the longrange orbital/magnetic order

81 Conclusions Symmetry-respecting Wannier function Simple, flexible and efficient basis to describe physics Intuitive microscopic picture + quantitative realism Unfolding 1 st -principles band structures Visualizing each band s coupling to the order parameters and other symmetry breakers Useful for the study of extrinsic (doping, intercalation, vacancy, substitution, & defects) and intrinsic (CDW, SDW, OO, & structure distortion) effects on the electronic structure Direct comparison with ARPES spectrum, built-in matrix elements effects A non-perturbative beyond-mean-field first-principles method for materials with disordered impurities Defects, vacancy, chemical substitution, doping & intercalation Thermal/quantum fluctuation induced spatial inhomogeneity Paramagnetic phase with disordered local moments