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5.0 引言 5.1 轨道 , 相互作用与自旋 5.2 原子和分子的磁矩 5.3 晶体的磁矩 5.4 晶体的磁各向异性 5.5 习题

5. 　磁性与电子态. 5.0 引言 5.1 轨道 , 相互作用与自旋 5.2 原子和分子的磁矩 5.3 晶体的磁矩 5.4 晶体的磁各向异性 5.5 习题. Outline. Energy bands Spin polarization in crystals Magnetic configuration (FM, AFM, … ) and phase transition Noncollinear magnetism Surface magnetism Orbital quench. Outline. Energy bands

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5.0 引言 5.1 轨道 , 相互作用与自旋 5.2 原子和分子的磁矩 5.3 晶体的磁矩 5.4 晶体的磁各向异性 5.5 习题

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  1. 5.　磁性与电子态 5.0 引言5.1 轨道,相互作用与自旋5.2 原子和分子的磁矩5.3 晶体的磁矩5.4 晶体的磁各向异性5.5 习题

  2. Outline • Energy bands • Spin polarization in crystals • Magnetic configuration (FM, AFM, …) and phase transition • Noncollinear magnetism • Surface magnetism • Orbital quench

  3. Outline • Energy bands • Spin polarization in crystals • Magnetic configuration (FM, AFM, …) and phase transition • Noncollinear magnetism • Surface magnetism • Orbital quench

  4. Symmetry of Atoms and Crystals Isolated Atom: spherical symmetry reduces 3D to 1D Crystal: translational symmetry H(r) = H(r + T) leads to Bloch theorem yi(r)= ykn (r)= e-ikr ukn(r) with ukn(r) = ukn(r + T) Bloch theorem reduces infinite degrees of freedom to integration of finite bands(n) over Brillouin zone(k)

  5. Lattice-Reciprocal-FCC xyz(a1) = 0.5, 0.5, 0 xyz(a2) = 0.5, 0, 0.5 xyz(a3) = 0, 0.5, 0.5 xyz(b1) = 1, 1, -1 xyz(b2) = 1, -1, 1 xyz(b3) = -1, 1, 1 Plotting line is through G-X-W-G-K-L-G G: (b1,b2,b3)=0,0,0 xyz=0,0,0 X: (b1,b2,b3)=0.5,0,0.5 xyz=0,1,0 W: (b1,b2,b3)=0.5,0.25,0.75 xyz=0,1,0.5 G: (b1,b2,b3)=0,0,0 xyz=0,0,0 K: (b1,b2,b3)=0.375,0.375,0.75 xyz=0,0.75,0.75 L: (b1,b2,b3)=0.5,0.5,0.5 xyz=0.5,0.5,0.5 G: (b1,b2,b3)=0,0,0 xyz=0,0,0

  6. Minimization of LDA Energy Kohn-Sham equation: Hykn(r)= eknykn(r) H = -(1/2)Ñ2 + Vc(r) + Vxc(r) Vxc(r) = dExc(r)/dr = Vxc0(r) Basis expansion: yn(r) = Sifi(r) Cin Matrix eigen-problem: Sj Hij Cjn = enSj Sij Cjn Hij = < fi(r)|H|fj(r)> Sij = <fi(r)|fj(r)>

  7. Summation/Integration over StatesJ.D.Pack and H.J.Monkhorst, PRB16, 1748(1977) Fermi energy is determined by Ncell = Sn SsòBZ dk f(Ekns-EF) = Sn SsSR\inGòIBZ dk f(ERkns-EF) Charge density is determined by r(r)= Sn SsSR\inGòIBZ dk f(ERkns-EF) rRkns(r) = Sn SsSR\inGòIBZ dk f(Ekns-EF) rkns(R-1r) Integral replaced by weighted sum over special k points (SKP) òBZ dk = Sk\inSKP w(k)

  8. Total Energy in DFT E(Zm,Rm,r(r)) = Siei + (1/2) Smn ZmZn/(|Rm-Rn|) - (1/2) òdrdr'r(r)r(r')/(|r-r'|) + òdr [Exc(r)-Vxc(r)r(r)]

  9. Augmented Basis The periodic function u is expanded, ukn = SG Ckn(G)FG(r) by augmented planewaves (APW), which is atomic orbitals near atomic core, and augmented by planewave at region far from the cores. With rm = r-Rm, APW is defined as FG(r)=(1/W)1/2 e-iGr outside all MT spheres or FG(r)=SL ALm(G)ulm(rm)YL(rm) in m-MT spheres Advantage: Acceptable basis(<100/atom) and exact over whole space. Disadvantage: Basis depends on k, structure/potential.

  10. LAPW BasisO.K.Anderson PR 12, 3060, 1975 D.D.Koeling et al. J.Phys.F 5, 2041, 1975 Linearized basis FG(r) = (1/W)1/2 e-iGr outside all MT spheres but FG(r) = SL [ ALm(G) ulm(rm) + BLm(G) dulm(rm)/dr ] YL(rm) in m-MT sphere which is both function value continuous and derivative continuous on the MT sphere boundary.

  11. Projector augmented-wave methodP. E. Blochl Phys. Rev. B 50, 17953 (1994) The all-electron wavefunction can be obtained by the pseudo wavefunction, |Y>=|Y> + Si (|fi>-|fi>)<pi|Y> Here |fi> : core states, |fi> : smoothed core states, |pi> : projector localized in the augmentation region and obeys < fi |pi>=dij

  12. Planewave Basis and Pseudopotential The periodic function u is expanded by the planewave basis, ukn(r)=SG Ckn(G)(1/V)1/2 e-iGr Advantage: Basis is structureless and independent of k. Disadvantage: Large basis (>1000/atom) if including cores. Scheme: Pseudopotential makes its nodeless wavefunction identical to the real valence wavefunction beyond r>rc.

  13. Pseudopotential - Norm Conserve Norm conserve and self-consistent: ‘Pseudopotentials that work from H to Pu‘, G.B.Bachelet, D.R.Hamann, and M. Schluter,PRB26, 4199, 1982

  14. This potential is charge state dependent and norm does not conserve. However, it is well suit for plane-wave solid- state calculations, and show promise even for transition metals. Pseudopotential – UltrasoftD.Vanderbilt, PRB 41, 7892 (1990)

  15. Tight Binding Basis (Atomic/Local Orbital) The periodic function u is expanded ukn = Smm Ckn(mm)fkmm(r) by atomic (LCAO),Gaussian (LCGO), and MT.orb. (LMTO) fkmm(r)=(1/N1/2) STe-ik(r-Rm-T))fm(r-Rm-T) Advantage: Minimum basis (10/atom) and exact cores. Disadvantage: Basis depends on k and structure/potential; Poor approximation at far from nuclei.

  16. Slater-Koster Scheme < fkmm(r)|V|fknn(r) > = e-ik(Rm-Rn) ST eikT <fm(r-Rm)|V|fn(r-Rn-T)> < fm(r-Rm)|V|fn(r-Rn-T) > = SM [DlMm(Rn+T-Rm)]* Dl'Mn(Rn+T-Rm) <flM(r-Rm)|V|fl'M(r-Rn-T)> = SM [DlMm(Rn+T-Rm)]* Dl'Mn(Rn+T-Rm) Vll'M(|Rm-Rn-T|)

  17. Slater-Koster Scheme: Canonical TheoryO.K.Andersen et al, PRB17, 1209 (1978) Vll'M(|Rm-Rn-T|) VsssVspsVsds VppsVpppVpdsVpdp VddsVddpVddd VABl'lM(R)=(-1)l+M+1 (l')!(l)!(l+l')![(-1)l+l'VAl'l's VBlls]1/2 [(2l')!(2l)!(l'+M)!(l'-M)!(l+M)!(l-M)!]-1/2 (RAB/R)l+l'+1

  18. Energy Bands of Cu

  19. Experimental Cu Bands

  20. Density of States of Cu

  21. Total Energy vs Cu Lattice Constant

  22. Formation Energy of CuAu Alloy Item Total_energy Cohesive/Formation (Ryd) Calc(eV) Expt(eV) Cu_atom -3304.5211 Cu_xtal -3304.8606 4.62 3.49 Au_atom -38074.2247 Au_xtal -38074.5444 4.35 3.81 Cu_xtal+Au_xtal -41379.4050 CuAu_xtal -41379.4174 0.17 0.11

  23. Outline • Energy bands • Spin polarization in crystals • Magnetic configuration (FM, AFM, …) and phase transition • Noncollinear magnetism • Surface magnetism • Orbital quench

  24. Spin Polarization in Crystals Energy gain in intraatomic exchange E = -(1/2) I si.sj I : energy cost of generating an antiparallel pair of spins E(m) = I [ (N/2+m/2)(N/2-m/2)- (N/2)(N/2)] = -(1/4) I m2

  25. Spin Polarization in Crystals Ef Energy cost of spin polarization of band electrons in a crystal, E = - òdE {EF-e(m/2):EF} n(E)E + ò dE{EF,:EF+e(m/2)} n(E)E = (1/4) m2/n(EF) + O(m4)

  26. Spin Polarization in Crystals Energy gain over cost due to spin polarization E = -(1/4)Im2 I + (1/4)( 1/n(EF)) m2 = -(1/4)(I- 1/n(EF)) m2 Condition for nonvanishing (spontaneous) atomic moment I > 1/ n(EF) (Stoner-Wohlfarth criterion)

  27. Total Energy of FM vs NM bcc Fe

  28. Magnetic Fe Moment in bcc Structure

  29. Slater-Pauling Curve: Experiment vs LSDA

  30. DFT results for Fe: LDA vs GGA J. H. Cho and M. Scheffler, PRB (1996)

  31. Outline • Energy bands • Spin polarization in crystals • Magnetic configuration (FM, AFM, …) and phase transition • Noncollinear magnetism • Surface magnetism • Orbital quench

  32. Collinear Spin Configurations in Layered Structure

  33. Spin Configurations and Lattice DistortionJ.T.Wang(王建涛),et al. APL 79,1507(2001) Layered MnAu

  34. Exchange Integral and Lattice Distortion J.T.Wang(王建涛),et al. APL 79,1507(2001) Layered MnAu

  35. Magnetic Phases of Layered MnAuJ.T.Wang(王建涛),et al. APL 79,1507(2001)

  36. Atomic and Spin Configuration of Layered MnAuJ.T.Wang(王建涛),et al. APL 79,1507(2001) Configuration & Expt & Theo Atomic & B2 & B2 & L10 Vol/atom (A3) & 16.58 & 16.63 & 16.40 a' (A) & 3.18 & 3.184 & a (A) & & & 4.080 c (A) & 3.28 & 3.280 & 3.938 c/a or c/21/2a' & 0.729 & 0.728 & 0.965 Spin & AF4 & AF4 & AF2 Moment (mB) & 4.0 & 3.86 & 3.93 TN (K) & 513 & 528 & 946

  37. Outline • Energy bands • Spin polarization in crystals • Magnetic configuration (FM, AFM, …) and phase transition • Noncollinear magnetism • Surface magnetism • Orbital quench

  38. Non-collinear Magnetism – HamiltonianM. Uhl et al. JMMM 103, 314 (1992) Under LSDA H = [ -(1/2) Ñ2 + V(r(r)) ] + Vm(r(r), m(r)) U sz U The spin-1/2 rotation matrix, cos(q/2) eif/2 sin(q/2) e-if/2 U(r) = ( ) -sin(q/2) eif/2 cos(q/2) e-if/2 depending on the local moment direction (q(r), f(r))

  39. Outline • Energy bands • Spin polarization in crystals • Magnetic configuration (FM, AFM, …) and phase transition • Noncollinear magnetism • Surface magnetism • Orbital quench

  40. Surface Magnetism: Possible Reduction Ef Basic fact for Fe, Co, and Ni spin up band fully filled spin down band > half filled Band narrowing: width ~ Z1/2 Decrease of moment due to band narrowing

  41. Surface Magnetism: Possible Enhance Ef Basic fact for Fe, Co, and Ni spin up band fully filled spin down band > half filled Surface bands are lifted Enhance of moment due to level shift

  42. Surface Dipole and Level Shift

  43. Surface Magnetism Self-consistent calculation gives, Decrease of moment due to band narrowing ^^^^ Enhance of moment due to level shift

  44. Giant Surface Spin Moment LSDA spin moment , and surface core level shift of Ni films LAPW (H.Krakauer et al, 1981 …) System Z m(mB) Shift(eV) Bulk 12 0.561 (100) center 12 0.619 (111) center 12 0.613 (111) surface 9 0.625 0.291 (100) surface 8 0.675 0.354 (111) monolayer 6 0.892 (100) monolayer 4 1.014

  45. Ni Layers on Cu Substrate D.S.Wang(王鼎盛) et al, PRB3, 1340 (1982) Expt. verified: ‘Dead layer’ is dead

  46. Ni Layers on Cu Substrate J.Henk et al, PRB59, 9332 (1999)

  47. Outline • Energy bands • Spin polarization in crystals • Magnetic configuration (FM, AFM, …) and phase transition • Noncollinear magnetism • Surface magnetism • Orbital quench

  48. Vanishing Orbital Moment For any time-inversion invariant system, H = -(1/2)Ñ2 + V(r), and TH = HT, its non-degenerate eigen-states have vanishing orbital moment. Proof: HT|y>=TH|y>=eT|y>, thus, T|y>=|y> -<y|lz|y>=<y|Tlz|y>=<y|lz|y> thus, <y|lz|y>=0 for any z axis i.e., |y>=Sm(Cm|m>+C-m|-m>), and |C-m|=|C-m| Also for evenly occupied degenerate states.

  49. Origin of Orbital Polarization • H includes Zeeman term, –gLSili.Happl • H includes the spin-orbit coupling Hsl = Si [(1/4c2r) ¶V/ ¶r]li (r)×si = Six( r)li (r)×si when there is spin polarization.

  50. Orbital Quench in Magnetic SolidsExample: Fe dimer with 6x2 d electons s* p* d* d p HOMO: p contains two states consisting of |+1> and |-1> orbitals, separated by about x. Lower p has only slightly more |+1> component, and orbital moment is quenched from Hund value (2 mB per atom) to morb = x / D Fe Co Ni Atom Hund rule 2 3 3 Solid Neutron 0.05 0.08 0.05 XMCD 0.09 0.12 0.05 s spin down spin up

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