SC2006c.ppt

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1 1, 2, 1 1 National Institute of Advanced Industrial Science and Technology (AIST) 2 Hakodate National College of Technology

2 Outline 1. Introduction 2. Superconductivity 3. High Temperature Superconductivity 4. Hubbard Model 5. Variational Monte Carlo method 6. Stripes in high-tc cuprates 7. Spin-orbit coupling and Lattice distortion 8. Summary

3 1. Introduction Key words: Physics from U (Coulomb interactions) A possibility of superconductivity Superconductivity from U Competition of AF and SC Incommensurate state Stripes and SC Compete and Collaborate Stripes in the lightly-doped region 300 Electron doped T(K) 200 AF 100 Hole doped AF Pseudo-gap Stripes SC SC Concentration x in Ln M CuO 2-x x 4-y Singular Spectral function

4 Purpose of Theoretical study 1. Origin of the superconductivity Symmetry of Cooper pairs Mechanism of attractive interaction Coulomb interaction U, Exchange interaction J 2. Physics of Anomalous Metallic behavior Inhomogeneous electronic states: stripe Pseudogap phenomena Structural transition LTO, LTT

5 2. Superconductivity 1911 Kamerlingh Onnes Elements that become superconducting Resistivity Superconductive at low temperatures Temperature K Superconductive under pressure

6 k -k k -k BCS

7 0

8 BCS H 1 H = H 0 + H 1 Ψ = c j Ψ j j Ψ j : Slater Ψ 0 : Fermi Ψ = c 0 Ψ + c j Ψ j c 0 1 j 0 c j Ψ 0 H 1 Ψ j < 0 2 ΨHΨ = c j Ψ j H 0 Ψ j + c i c j Ψ i H 1 Ψ j Ψ i H 1 Ψ j < 0 c j > 0 Ψ j c j >0

9 BCS Ψ i = k 1 k 1, k 2 k 2, k 3 k 3,L Ψ j = k' 1 k' 1, k 2 k 2, k 3 k 3,L Ψ i H 1 Ψ j = k' 1 k' 1 V k 1 k 1 < 0 : Ψ = c k1 k 2 L k 1 k 1,k 2 k 2,L Ψ = c k1 c k 2 L Φ = k ( u k + v k k k ) k 1 k 1, k 2 k 2,L BCS ( N N ) 2 N 1 N

Liquid N 2 60 La 40 Liquid H 20 0 Hg Pb 2 NbN Nb 2-x N b Sn x N b Ge 3 3 V Si

10 3. 高温超伝導超伝導臨界温度 160 HgBa2Ca2Cu3Oy Transition temperature (K) (Under Pressure) 140 HgBa2Ca2Cu3Oy TlBaCaCuO Tl 2Ba2Ca2Cu3Oy 120 Bi Sr Ca Cu O y 100 YBa2Cu3O7 80 Liquid N 2 60 La 40 Liquid H 20 0 Hg Pb 2 NbN Nb 2-x N b Sn x N b Ge 3 3 V Si Nb-Al-Ge MgB Sr CuO 1980 La 2-xBaxCuO4 KC Year La2CuO4 YBa2Cu3O7

11 T N T c

12 Cu ε d +U d ε d ε p d x2 -y 2 O p x 1/2 hole level

13 H. Takagi et al. T. Ito et al.

14 LSCO Loram et al., Phys. Rev. Lett. 71, 1740 (1993) Loram et al., Physica C , 498 (1989)

15 YBa 2 Cu 4 O 8 T c = 81K Tc χ Yasuoka et al., Physica B199 (1994)278 T c

16 T= 0 Millis, Moriya 3D C/T χ(q) Ferro. -lnt T -4/3 T 5/3 T -4/3 AF T 1/2 T -3/2 T 3/2 T -3/4 D C/T χ(q) ρ ρ χ s χ s Ferro. T -1/3 (TlnT) -1 T 4/3 χ(q) 3/2 AF -lnt T -1 T T -1

17 4. Hubbard Model Itinerant Electrons Electrons Atoms Mott insulators MnO, FeO, CoO, Mn 3 O 4, Fe 3 O 4, NiO, CuO Insulators due to the Coulomb interaction (Note: Antiferromagnets such as MnO and NiO are not Mott insulators in the strict sense.)

18 On-site Coulomb Interaction Coulomb interaction ε 0 ε 0 +U t U >> t Insulator +U

19 Gap in the Hubbard Model Hartree-Fock theory (Half-filling) AF Gap Δ = Um Δ ~ t e -2πt/U d = 1, 3 1D Hubbard model ~ t e -2π(t/U)1/2 d = 2 U << t U >>t Hubbard gap Δ (16 / π) tue π / (2U ) U Spin-wave velocity 2v s /π = J (4t /π)(1 U /4πt) 4t 2 /U ε Δ v s k v s Δ ~U 0 k ~1/U

20 Cu-O 2 Model and Hubbard Model Cu-O 2 model (d-p model) Zhang-Rice singlet Mixed state of d and p t-j model U >> t Hubbard model + + H = t (c iσ c jσ + h.c.) + J S i S j H = t (c iσ c jσ + h.c.) + U n i ij σ ij ij σ i n i t pd << U d -(ε p ε d ) t pd << ε p ε d ε p ε d << U d ε p ε d ~ 0(t pd )

21 5. Variational Monte Carlo method Gutzwiller ψ G = P G ψ 0 ψ 0 P G = : j ( 1 (1 g)n j n ) j 0 g 1 Gutzwiller weight g weight 1 Coulomb +U

22 Normal state ψ 0 Slater ψ 0 = a l ψ ψ l l l k 1, k 2,, k n j 1, j 2,, j n det D = e ik 1 j 1 eik 1 j 2 L eik 1 j n L e ik n j 1 e ik h j 2 L e ik n j n Slater a l = det D det D

23 ψqψ = a m a n ψ m Qψ n = mn ψ m P m = a 2 m ψqψ = 1 M l a l 2 a n a m m l a m 2 a l 2 a n n a m ψ m Qψ n ψ m Qψ n m = 1,L, M m n Metropolis ψ j ψ n 2 2 R = a n / a j ξ ψ n < ξ ψ ξ: 0 ξ <1 j ψ m Qψ n cpu

24 Ψ s = P N P G ψ ψ BCS BCS = (u k k + v k c + + k c k ) 0 P N Ψ s = P G P N exp v = P k G c u k k k = P G a ij c i k k + + c k + + c j v k u k c k + + c k N / 2 N / 2 a ij = 1 e ik ( R i R j ) V k v k u k Slater BCS N = N i 1, i 2,.. i N/2 ; j 1, j 2,.. j N/2 a i1 j 1 L a i1 j N /2 L a in /2 L a in /2 j N /2 normal state

25 Superconducting state ψ CdS = P G k (u k + v k c k + + c k ) 0 k -k Gutzwiller Projection P G To control the on-site strong correlation Weight g Weight 1 Coulomb +U Parameter 0<g<1 Equivalent to RVB state (Anderson)

26 Superconducting condensation energy SC Condensation energy ΔE SC = Ω n Ω s = T = c (C 0 s T c 0 (S n C n )dt S s )dt C/T Entropy balance T Loram et al. PRL 71, 1740 ( 93) optimally doped YBCO SC Condensation energy ~ 0.2 mev

27 Evaluations in the superconducting state Variational Monte Carlo method 10x10 Hubbard model U= 8 E g /N s T. Nakanishi et al. JPSJ 66, 294 (1997) K. Yamaji et al., Physica C304, 225 (1998) T. Yanagisawa et al., Phys. Rev. B67, (2003) YBCO Δ ΔE/N E cond ~ 0.2meV SC condensation E /N a

28 Condensation Energy for d-p model Condensation energy E cond ~ t dp = 0.56 mev/site 2D d-p model 6x6 and 8x8 δ = t pp = 0.2 SC order parameter Hole density T.Yanagisawa et al., PRB64, ( 01)

29 Superconductivity and Antiferromagnetism Competition Size dependence of SC condensation energy E cond AF SDW(U=6,L=10) SDW(U=6,L=12) SC(U=6,L=10) SC(U=6,L=12) t t' Fig. 1 SC Pure d-wave SC E cond meV Fig. 2 SC E cond (ρ ~.84, U=5, t'=.05) /N s Experiments 0.26 mev/site 0.17~0.26 critical field H c (C/T)

30 6. Stripes in high-tc cuprates Vertical stripes for x > 0.05 Diagonal stripes for x < 0.05 AF coexists with SC? Neutron scattering Vertical SC+Stripes Coexist Diagonal M.Fujita et al. Phys. Rev.B65,064505( 02) S.Wakimoto et al. PRB61, 3699( 00)

31 Vertical Stripes in the under-doped region Vertical stripes: 8 lattice periodicity (Tranquada) Charge rich Charge poor Charge rich 0.0 x = 1/8 VMC 3-band Hubbard E/N Commensurate AF Stripes 8 lattice period t pp T.Y. et al., J.Phys.C14,21( 02)

32 Stripes and Superconductivity Compete and Collaborate SC coexists with stripes (AF) Bogoliubov-de Gennes eq. H ij + F ij F * ji H ji u j λ v j λ = E λ α λ = u i λ a i + v i λ a i + α λ = u i λ a i + v i λ a i + Wave function V λj = v j λ λ u i v i λ λ ( U) λj = u j Nano-scale SC ψ SC = P G P N e λ α λ α λ + 0 P G (U 1 V) ij a + + i a j ij N e / 2 0

33 Diagonal stripes in lightly doped region Diagonal stripes are observed for La 2-x Sr La 2-x Sr La 2-x-y Nd La La La Sr x NiO 4 Sr x CuO 4 Nd y Sr x CuO 4 T(K) Electron doped AF Hole doped AF 100 SC SC Concentration x in Ln M CuO 2-x x 4-y

34 Incommensurability: Comparison with Experiments = x Vertical stripes Diagonal stripes U=8.0 t = -0.2 δ can be explained by 2D Hubbard model.

35 Stripes and Structural transition Structural transitions: Lattice distortions LTT,LTO,LTLO,HTT Stripes: suggested by Incommensurability δ~x N.Ichikawa et al. PRL85, 1738( 00) T HTT LTO LTT LTLO Hole density M.Fujita et al. Phys. Rev.B65,064505( 02)

36 What happens under lattice distortions? 1. Anisotropy of the transfer integrals Anisotropic electronic state vertical stripes Diagonal stripes x< Spin-Orbit Coupling induced from lattice distortions 3. Electron-phonon interaction T vertical HTT LTO LTT LTLO Hole density

37 Anisotropy of the transfer integrals in LTT phase One-band Hubbard model (Miyazaki) X = ty / tx Y LTT structural transitions stabilize stripes.

38 Possible Stripe Structure 1 Mixed phase of LTT and LTLO tx tx Stripes // tilt axis Stabilize stripes ty M. K. Crawford et. al. T HTT LTO LTT LTLO Hole density Hole poor domain Hole rich domain LTLO LTT LTLO LTT

Possible Stripe Structure 2 Charge poor Charge rich Mixed phase of LTT and HTT Stripes perpendicular to tilt axis t x t x Stable H. Oyanagi A. Bianconi M. K. Crawford et.

39 Possible Stripe Structure 2 Charge poor Charge rich Mixed phase of LTT and HTT Stripes perpendicular to tilt axis t x t x Stable H. Oyanagi A. Bianconi M. K. Crawford et. al. t y T HTT LTO LTT ΔE/N LTT-HTT Parallel //stripes Perpendicular Hole density LTLO Oscillation of tilt angles LTT HTT LTT HTT u

40 7. Spin-orbit coupling and Lattice distortion Spin-Orbit Coupling induced by the Lattice distortion Tilting Friedel et al., J.Phys.Chem.Solids 25, 781 (1964) p x (x a / 2, y) H dp d xz (r) = t xz e ik x / 2 a iky / 2 a p y (x, y a / 2) H dp d yz (r) = t yz e 1 H SO = ξ(r)l S d xz (r) H SO d yz (r) = i 2 ξ d yz (r) H SO d xz (r) = i 2 ξ Cu Oxygen d x 2 y 2 (r) H SO d yz (r) = i 2 ξ d x 2 y 2 (r) H SO d xz (r) = 1 2 ξ Effective iξ term for p-p transfer t xz, t yz 0 ~ tilt angle Five orbitals ( ): d x 2 -y 2, d xz, d yz, p x, p y

41 Dispersion in the presence of spin-orbit coupling d-p model Zone boundary One-band effective model H kin = (t ij + icσθ ij )d + iσ d jσ ijσ (Bonesteal et al.,prl68,2684( 92)) = π/100 = 3π/100 E k =-2(cosk x +cosk y ) E k (π/2,π/2) -4-6 ξ = 0.4 (K.Yamaji, JPSJ(1988)) -8 (0,0) (π,0) (π,π) (0,0) k

42 Flux state E(k x,k y ) = ±e i / 4 e ik x + e i / 4 e ik y +e i / 4 e ik x + e i / 4 e ik y Excitation: Dirac fermion E k Linear dispersion Fermi point (half-filling) Small Fermi surface Inhomogeneous d-density wave

43 Pseudo-gap in the density of states Flux state Density of states Pseudo-gap An origin of pseudo-gap N σ (k, ε ) = 1 π Im G σ (k,ε + iδ ) N(ω) = 0 = π/100 = 3π/100 Eigenfunction Hϕ σm (r) = E σm ϕ σm (r) G σ (r, r',iω) = ϕ σm (r) m * ϕ σm (r)ϕ σm (r' ) iω E σm ω/ t T HTT LTO LTT LTLO Hole density

44 Diagonal stripes with Spin-orbit coupling Spin-orbit coupling induces flux. Spin-orbit coupling stabilizes the diagonal stripes x = 0.03 VMC Uniform Diagonal Stripe & d-density wave E/N Diagonal bond-center site-center /π Φ = 4 Vertical

45 d-density wave d-density wave + iδ Q Y(k) = c k +Qσ c kσ Y(k) = cos(k x ) cos(k y ) Q = (π,π) Nayak, Phys. Rev. B62, 4880 ( 00) Chakravarty et al., PRB63, ( 01) Inhomogeneous density wave + iδ Q Y(k) = c k +Qσ c kσ Δ lqs σ = k + c k +lqs σc kσ d-symmetry incommensurate Q s =(π+2πδ,π) vertical Q s =(π+2πδ,π+2πδ) diagonal Stripe

8. Summary ψ CdS = P G k (u k + v k c k + + c k ) 0 0.004 0.

01 1/N a Explanation of spectra Incommensurability Neutron scatterings

46 8. Summary ψ CdS = P G k (u k + v k c k + + c k ) ΔE/N YBCO /N a Explanation of spectra Incommensurability Neutron scatterings Theoretical estimate of SC condensation energy Agreement with Exp. ARPES Measurements underdope overdope E cond ~ 0.2meV

遍歴電子磁性とスピン揺らぎ理論 - 京都大学大学院理学研究科集中講義

遍歴電子磁性とスピン揺らぎ理論 - 京都大学大学院理学研究科集中講義 email: takahash@sci.u-hyogo.ac.jp August 3, 2009 Title of Lecture: SCR Spin Fluctuation Theory 2 / 179 Part I Introduction Introduction Stoner-Wohlfarth Theory Stoner-Wohlfarth Theory Hatree Fock Approximation