Material Science II. d Electron systems

Transcription

1 Material Science II. d Electron systems 1. Electronic structure of transition-metal ions (June 12) 2. Crystal structure and band structure (June 19) 3. Mott insulators (June 26) 4. Metal-insulator transition (July 3) 5. High-temperature superconductivity (July 10) 6. Spin-related phenomena (July 17)

2 Electronic structure Superconducting gap and pseudogap 5.2 Fe-based high-t c superconductors Electronic structure Superconducting gap and nematic phase

3 1. Electronic structure of transition-metal ions 1.2 Crystal-field splitting Crystal fields from anion ligands (e.g., O 2-, F - ) Octahedral coordination Tetrahedral coordination Giant magneto-resistive material (Perovskite structure) Magnetic semiconductor (Zinc Blende structure) Mn Cu-oxide High-Tc superconductor (Layered perovskite str.) Fe-based superconductor Lowered symmetry

4 4. Metal-insulator transition 4.2 Filling-controlled metal-insulator transition U/W or /W Insulator BANDWIDTH U/W or /W ~ 1 Hole doping Filling-controlled MIT Electron doping Bandwidth-controlled MIT 1 n M. Imada, A. Fujimori and Y. Tokura, Rev. Mod. Phys. 1998

5 /W BANDWIDTH /W ~ 1 n Electron doping Insulator 9 Hole doping Filling-controlled MIT Bandwidth-controlled MIT

6 Electron-doped Nd 2-x Ce x CuO 4 Hole-doped La 2-x Sr x CuO 4 Nd 3+ Ce 4+ La 3+ Sr 2+ e - e - CuO 2 plane CuO 2 plane CuO 2 plane O Cu

7 Electronic structure Superconducting gap and pseudogap 5.2 Fe-based high-t c superconductors Electronic structure Superconducting gap and nematic phase

8 Phase diagram of cuprate superconductors Electron-doped Parent compound Hole-doped Metal AFI: Antiferromagnetic insulator AFM: Antiferromagnetic metal SC: Superconductor

9 small Charge-transfer energy Valence: d p A U I Vacuum level d large small large Transition-metal ions S Non-TM ions large large small small small large

10 Zaanen-Sawatzky-Allen phase diagram Schematic Real materials W p-band metal Charge-transfer insulator Mott-Hubbard insulator d-band metal U Negative charge-transfer energy insulator p-band metal RNiO 3 Charge-transfer insulator Mott-Hubbard insulator W d-band metal

11 Mott-Hubbard-type insulator Charge-transfer-type insulator 3d 3d O 2p W W U 3d ZRS O 2p 3d U ZRS: Zhang-Rice singlet strong p-d hybridization U < U > Gap ~ U - W Gap ~ -W W: Band width U : Atomic Coulomb energy (Coulomb integral) : Charge-transfer enrgy

12 Phase diagram of cuprate superconductors Electron-doped Lightly-dopedHole-doped region Metal AFI: Antiferromagnetic insulator AFM: Antiferromagnetic metal SC: Superconductor

13 3d ZRS O 2p 3d ZRS O 2p (1) U 3d ZRS O 2p 3d Electron doping 3d (9) Parent insulator O Cu 3d ZRS:Zhang-Rice singlet Hole doping Doped electron (d 10 ) Hole d 9 Doped hole (d 9 L)

14 Phase diagram of cuprate superconductors Electron-doped Overdoped region Lightly-dopedHole-doped region Overdoped region Metal AFI: Antiferromagnetic insulator AFM: Antiferromagnetic metal SC: Superconductor

15 2. Crystal structure and band structure 2.3 d-p hybridized bands, d bands d-p band (e.g., d x2-y2 -p x -p y band of CuO 2 plane) O Cu -t pd < 0 Hamiltonian: 3x3 matrix d xy id xy d xy p x d xy p x p x p x d eg 2i t pd sin k x a/2 2i t pd sin k y a/2-2i t pd sin k x a/2 p 0-2i t pd sin k y a/2 0 p

16 2. Crystal structure and band structure 2.3 d-p hybridized bands, d bands d-p band (e.g., d x2-y2 -p x -p y band of CuO 2 plane; Cu (2+x)+ (d 9-x ), x: hole number/cu atom) d band E F p band dx2-y2 p Brillouin zone -t pd < 0 Fermi surfaces k y aa a aa Hamiltonian: 3x3 matrix d eg 2i t pd sin k x a/2 2i t pd sin k y a/2 k -2i t pd sin k x a/2 p 0-2i t pd sin k y a/2 0 p k x

17 2. Crystal structure and band structure 2.3 d-p hybridized bands, d bands Local-density-approximation (LDA) band-structure calculation La/Sr O Cu Energy Cu 3d O 2p L.F. Mattheiss, Phys. Rev. Lett. 1987

18 Fermi surface Band dispersion Hole Hole Electron Hole Hole Band structure: E(k) = -2t(cos k x a+cos k y a) -4t cos k x a cos k y a

19 Specific heat c p / T T 2 Electrical resistivity m m * 1 z b 23 La 2-x Sr x CuO 4 x=0.15 x=0.20 x=0.35 x=0.25 x=0.326 x=0.30 T. Matsuzaki et al., JPSJ 2004 H. Takagi et al., PRB 2001 Overdoped cuprates are correlated Fermi liquids.

20 Electronic structure Superconducting gap and pseudogap 5.2 Fe-based high-t c superconductors Electronic structure Superconducting gap and nematic phase

21 d-wave superconductivity order parameter Superconducting gap 0 (k) Antinode Antinode Node Order parameter (k) = 0 (cosk x a -cosk y a)/2 Superconducting gap Order parameter

22 BCS behavior of d-wave superconductivity Cooper pair formation 0 k Anti-node Fermi surface + -k BCS ( uk vka a k, -k, k u k / v k changes sign 0 ) 0 (ev) Fermi level Superconducting gap Anti-node Fermi k momentum F Bi 2 Sr 2 Ca 2 Cu 3 O Node 2 ( k) ( k k F 2 (k) = 0 (cosk x a -cosk y a) ) 2 k DOS S. Ideta et al., PRL 2010

23 BCS behavior of d-wave superconductivity Density of states 0(0) 4.3k B T c 2 Superconducting gap (T) 2 T c T DOS Electronic specific heat (k) = 0 (cosk x a -cosk y a) Superfluid density = Fermi surface volume/m* C/T T c T

24 Phase diagram of cuprate superconductors Electron-doped Overdoped Underdoped region region Hole-doped Underdoped region Overdoped region Metal AFI: Antiferromagnetic insulator AFM: Antiferromagnetic metal SC: Superconductor

25 Pseudogap behavior of La 2-x Sr x CuO 4 Fermi liquid Superfluid density n s /m* Fermi-liquid system La 1-x Sr x TiO 3 n = 1/qR H ~ 1-x Electron Y. J. Uemura et al. Hole content n=1/er H >0 H. Takagi et al : El.spec.heat coeff. :Pauli susceptibility m* Electronic specific heat coef. M. Oda et al. Band model AFI La 2 CuO 4 d 9 Mott insulator SC Metal Doped hole content x SrTiO 3 d 0 band insulator Doped hole content x LaTiO 3 d 1 Mott insulator

26 Specific heat Pseudo-gap behavior c p / T T 2 Electrical resistivity m* z 1 3 m b x=0.02 La 2-x Sr x CuO 4 x=0.026 decreases! (T)= 0 +AT 2 x=0.035 x=0.05 x=0.06 x=0.075 x=0.10 x=0.15 T. Matsuzaki et al., JPSJ 2004 H. Takagi et al., PRB 2001

27 BCS behavior Pseudogap of d-wave behavior superconductivity Density of states 0(0) 4.3k B T c 2 * Superconducting gap (T) 2* 2 T c T T* DOS Electronic specific heat (k) = 0 (cosk x a -cosk y a) Superfluid density = Fermi surface volume/m* C/T CN/T m* T c T T*

28 d-wave superconductivity order parameter Superconducting gap 0 (k) pseudogap pseudogap Antinode Node Order parameter (k) = 0 (cosk x a -cosk y a)/2 Superconducting gap Order parameter

29 Specific heat Pseudo-gap behavior c p / T T 2 Electrical resistivity m* z 1 3 mb 1 m* z ne nz n x=0.02 x=0.026 La 2-x Sr x CuO 4 decreases! (T)= 0 +AT 2 x=0.035 x=0.05 x=0.06 increases! x=0.075 x=0.10 x=0.15 T. Matsuzaki et al., JPSJ 2004 H. Takagi et al., PRB 2001

30 Proposed mechanism in T c Competing orders Preformed Cooper pairs AFM correlation /AFM SR order Q ~ () Charge order Q ~ () Nematic order Q = ( Time-reversal symmetry breaking Q = ( Proximity to Mott transition Nernst effect Opens PG at E F Opens PG but not necessarily at E F Opens PG but not necessarily at E F Does not open a gap Does not open a gap Opens PG but not necessarily at E F

31 Origin of the pseudogap Preformed Cooper pairs? Nernst effect T T* T CO Y. Wang et al., PRB 2001 T : Nernst temperature O. Cyr-Choinière et al., PRB 2015

32 Electron-hole asymmetry in the pseudogap state. Optimally-doped Bi2201 Particle-hole symmetric Underdoped Bi2212 M. Hashimoto, Z.-X. Shen et al., Nat. Phys. (2010) H.-B. Yang, P. D. Johnson et al., Nature (2008)

33 Origin of the pseudogap Antiferromagnetic fluctuations? Electron-doped Hole-doped Antiferromagn. Q~() Q=() (Nd,Ce) 2 CuO 4 x=0.15 Hot spots Short-range AFM order Bi2212 q Charge fluctuations? Q~() FS: small P. Armitage et al., PRL (2002) A. Koitzsch et al., PRB (2004)

34 Origin of the pseudogap Charge order? YBCO, Hg1201 B. Keimer et al., Nature (2015)

35 Origin of the pseudogap C 4 rotational symmetry breaking? Neutron () AFM peak for YBCO 6.45 STM image and its Fourier transform (QPI) for Bi2212 C 2 : Q = 0 order Pomeranchuk instability of Fermi surface V. Hinkov et al., Science (2008) C.J. Halboth & W. Metzner, PRL (2000) M. J. Lawler, L.S. Davis et al., Nature (2010)

36 Origin of the pseudogap C 4 rotational symmetry breaking? Anisotropy of magnetic susceptibility Anisotropy due to chain Anisotropy due to Cu-O chain Y. Sato, Y. Matsuda et al., Nat. Phys. (2017)

37 Origin of the pseudogap Time-reversal symmetry breaking? Time reversal symmetry breaking due to loop current C.M. Varma and L. Zhu, PRL (2007) Neutron magnetic intensity at Bragg peak of YBCO 6.6 T* Kerr signal in Bi2201 Evidence against TRSB -SR G. J. MacDougall et al., PRL (2008) Gyrotropic order Chiral charge order H. Karapetyan et al., PRL (2014) B. Fraque et al., PRL (2006) R.H. He et al., Science (2011) Birefringence Y. Lubashevsky et al., PRL (2014)

38 Cluster DMFT calculation M. Civelli et al., PRL (2005) s-wave PG hidden Fermion Highly e-h asymmetric pseudogap E F n = 0.95 S. Sakai, M. Imada, Y. Gallais et al., PRL (2013)

39 Proposed mechanism in T c Preformed Cooper pairs Nernst effect No e-hole symmetry (ARPES) Opens PG at E F Competing orders AFM correlation /AFM SR order Q ~ () Charge order Q ~ () Nematic order Q = ( Time-reversal symmetry breaking Q = ( Proximity to Mott transition - NMR, neutron - Hot spots in e-doped cuprates No e-hole symmetry (ARPES) T nem T* Explains polarized INS, Kerr effect, Nodal gap Natural - No clear hot spots in hole-doped HTSC T CO, onset << T* Cannot open a gap Cannot open a gap How to prove it experimentally? Opens PG but not necessarily at E F Opens PG but not necessarily at E F Does not open a gap Does not open a gap Opens PG but not necessarily at E F

40 Electronic structure Superconducting gap and pseudogap 5.2 Fe-based high-t c superconductors Electronic structure Superconducting gap and nematic phase

41 1. Electronic structure of transition-metal ions 1.2 Crystal-field splitting Crystal fields from anion ligands (e.g., O 2-, F - ) Octahedral coordination Tetrahedral coordination Giant magneto-resistive material (Perovskite structure) Magnetic semiconductor (Zinc Blende structure) Mn Cu-oxide High-Tc superconductor (Layered perovskite str.) Fe-based superconductor Lowered symmetry

42 5.1 Fe-based high-t c superconductors Electron doping type LaFeAsO 1-x F x O 2- F - e - Ba 2+ K + e - Hole doping 122 -type Ba 1-x K x Fe 2 As 2 Chemical pressure 122 -type BaFe 2 (As 1-x P x ) 2 FeAs layer 122 -type Ba(Fe 1-x Co x ) 2 As 2 Ba(Fe 1-x Ni x ) 2 As 2 As Fe

43 /W BANDWIDTH /W ~ 1 n Electron doping Insulator 9 Hole doping Filling-controlled MIT Bandwidth-controlled MIT

44 5.1 Fe-based high-t c superconductors /W Insulator BANDWIDTH U/W /W ~ 1 Electron doping Hole doping Filling-controlled n 96 Bandwidth-controlled

45 5.1 Fe-based high-t c superconductors Electron-doped Hole-doped Metal Antiferromagn metal Metal Doped electron x N. Ni et al., PRL (2008) Doped holes x/2 per Fe atom M. Rotter et al., Angew. Chem. Int. Ed. (2008)

46 Electronic structure Superconducting gap and pseudogap 5.2 Fe-based high-t c superconductors Electronic structure Superconducting gap and nematic phase

47 5.1 Fe-based high-t c superconductors small Charge-transfer energy Valence: d p A U I Vacuum level d large small large Transition-metal ions S Non-TM ions large large small small small large

48 5.1 Fe-based high-t c superconductors Zaanen-Sawatzky-Allen phase diagram Schematic Real materials W p-band metal Charge-transfer insulator Mott-Hubbard insulator d-band metal U Negative charge-transfer energy insulator p-band metal RNiO 3 Charge-transfer insulator Mott-Hubbard insulator W d-band metal

49 5.1 Fe-based high-t c superconductors LaOFeAs Hole Fermi surfaces Electron Fermi surfaces La 2-x Sr x CuO 4 Electron Fermi surfaces Hole Fermi surfaces Fe 3d bands Cu 3d-O 2p (d x2-y2 -p x -p y ) antibonding band Energy As 4p bands O 2p bands V. Vildosola et al., PRB (2009) L.F. Mattheiss, PRL (1987)

50 5.1 Fe-based high-t c superconductors 2D Fermi surfaces Fermi surfaces Hole pockets BaFe 2-x Ni x As 2 Electron pockets BaFe 2 As 2 K.Kuroki et al., PRL (2008) Hole FS Electron FS Electron doping F. Ma et al., Front. Phys. China (2010) S. Ideta et al, PRL (2012)

51 5.1 Fe-based high-t c superconductors /W Insulator BANDWIDTH U/W /W ~ 1 Electron doping Hole doping Filling-controlled n 96 Bandwidth-controlled

52 5.1 Fe-based high-t c superconductors Phase diagram Metal Metal

53 5.1 Fe-based high-t c superconductors Fe-As/Se Cu-O Charge-transfer energy (ev) Coulomb energy U d (ev) Transfer integral (pd) (ev) ~2 ~1 ~4 ~7 ~ -0.7 ~ -1.3 Configuration d n Orbitals Parent material AF metal Mott insul. Dimensionality more 3D 2D To induce supercond. d 6+ all d orbitals Carrier doping & Pressure d 9- x 2 -y 2 Carrier doping Cu O La/Sr Fe-As superconductors are more Mott-Hubbard-like

54 Electronic structure Superconducting gap and pseudogap 5.2 Fe-based high-t c superconductors Electronic structure Superconducting gap and nematic phase

55 5.1 Fe-based high-t c superconductors Superconducting gap s + -wave superconductivity Electron FS Hole FS Electron FS - - /a Hole FS /a Ba 1-x K x Fe 2 As 2 /a Order parameter (k) = 0 (cosk x a + cosk y a)/2 H. Ding et al., Europhys. Lett. (2008)

56 5.2 Fe-based high-t c superconductors Horizontal line nodes on hole Fermi surface Loop-like line nodes on electron Fermi surface I. Mazin et al., PRB (2010) K. Suzuki et al., JPSJ (2011) Vertical line nodes P. J. Hirschfeld and D. J. Scalapino, Physics (2010). S. Graser et al., PRB (2010)

57 5.1 Fe-based high-t c superconductors Electronic nematic phase of Fe pnictides Antiferromagnetic -orthorhombic (AFO) phase BaFe 2 (As 1-x P x ) 2 T N =T S AFO phase Q. Huang et al., PRL (2008) S. Kasahara et al., Nature (2013)

58 5.1 Fe-based high-t c superconductors Electronic nematic phase Antiferromagnetic -orthorhombic (AFO) phase T* T N =T S T* T S T N AFO TC T C Doped electron x x(p) Q. Huang et al., PRL (2008) N. Ni et al., PRL (2008) S. Kasahara et al., Nature (2013)

59 5.1 Fe-based high-t c superconductors In-plane resistivity anisotropy in Ba(Fe 1-x Co x ) 2 As 2 T N =T S T* ρ b > ρ a Almost isotropic for non-doped samples b a Residual resistivity Both ρ b and ρ a increase in proportional to the Co concentration (x 0.04) b - a Resistivity anisotropy is due to the anisotropy of impurity scattering. S. Ishida et al., PRL (2013)

60 5.1 Fe-based high-t c superconductors Persistence of the anisotropic band dispersions above T N,S Impurity state around Co Ba(Fe 0.95 Co 0.05 ) 2 As 2 T N T S T* M. Nakajima et al., PRL (2013) M. Yi et al., PNAS (2011) M. P. Allan et al., Nat. Phys. (2013)

61 5.1 Fe-based high-t c superconductors Monolayer FeSe/SrTiO 3 (001) T c ~ 100 K >> 8 K (bulk) Thickness and doping dependences of FeSe using ARPES Gate voltage in-situ measurements of chemically thinned FeSe Ionic liquid J.-F. Ge et al., Nat. Mater. (2014) ARPES T c ~ K Y. Miyata, K. Nakayama et al., Nat. Mater. (2015) S. He, X.J. Zhou et al., Nat. Mater. (2013) J. Shiogai, T. Nojima et al., Nat. Phys. (2016)

62 5.1 Fe-based high-t c superconductors Origin of the T c enhancement in 1ML FeSe? Pressure effect from STO(001)? [cf. A x Fe 2-y Se 2 suspected] No. The same T c for STO (no mismatch) and TiO 2 (large mism) Electron doping from STO? S. N. Rebec, Z.-X. Shen et al., PRL (2017) Yes, but similar effect for MgO! J. Shiogai et al., Nat. Phys. (2016). Enhanced electron-phonon coupling at interface? Phonon replica Distance between FeSe layers? J. J. Lee, D. H. Lee, Z.-X. Shen et al., Nature (2014)

63 5.1 Fe-based high-t c superconductors T c enhancement with distance between FeSe layers A(CHN)-intercalated FeSe (Li0.8Fe0.2)FeSeOH 1M FeSe/STO(001) bulk FeSe T. Noji, Y. Koike et al., Physica C (2014)

64 Electronic structure Superconducting gap and pseudogap 5.2 Fe-based high-t c superconductors Electronic structure Superconducting gap and nematic phase