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...

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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

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Outline

• Energy bands • Spin polarization in crystals • Magnetic configuration (FM, AFM, …

) and phase transition• Noncollinear magnetism• Surface magnetism• Orbital quench

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Outline

• Energy bands

• Spin polarization in crystals

• Magnetic configuration (FM, AFM, …) and phase transition

• Noncollinear magnetism

• Surface magnetism

• Orbital quench

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Symmetry of Atoms and CrystalsIsolated Atom: spherical symmetry reduces 3D to 1D

Crystal: translational symmetry H(r) = H(r + T) leads to Bloch theorem

i(r)= kn (r)= e-ikr ukn(r)with

ukn(r) = ukn(r + T)

Bloch theorem reduces infinite degrees of freedomto integration of finite bands(n) over Brillouin zone(k)

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Lattice-Reciprocal-FCCxyz(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 -X-W--K-L-

: (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

: (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

: (b1,b2,b3)=0,0,0 xyz=0,0,0

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Minimization of LDA EnergyKohn-Sham equation: Hkn(r)= knkn(r)

H = -(1/2)2 + Vc(r) + Vxc(r) Vxc(r) = dExc()/d Vxc

()

Basis expansion: n(r) = i i(r) Cin Matrix eigen-problem: j Hij Cjn = n j Sij Cjn Hij = < i(r)|H|j(r)>

Sij = <i(r)|j(r)>

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Summation/Integration over States J.D.Pack and H.J.Monkhorst, PRB16, 1748(1977

)

Fermi energy is determined by Ncell = n BZ dk f(Ekn-EF)

= n R\inG IBZ dk f(ERkn-EF)Charge density is determined by(r)= n R\inG IBZ dk f(ERkn-EF) Rkn(r)

= n R\inG IBZ dk f(Ekn-EF) kn(R-1r) Integral replaced by weighted sum over special k

points (SKP) BZ dk = k\inSKP w(k)

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Total Energy in DFT

E(Z,R,(r)) = i i + (1/2) ZZ/(|R-R|) - (1/2) drdr'(r)(r')/(|r-r'|) + dr [Exc(r)-Vxc(r)(r)]

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Augmented BasisThe periodic function u is expanded,

ukn = G Ckn(G) G(r)by augmented planewaves (APW), which is atomic orbital

s near atomic core, and augmented by planewave at region far from the cores. With r = r-R, APW is defined as

G(r)=(1/)1/2 e-iGr outside all MT spheresor G(r)=L AL(G)ul(r)YL(r) in -MT spheres

Advantage: Acceptable basis(<100/atom) and exact overwhole space. Disadvantage: Basis depends on k, structure/potential.

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LAPW BasisO.K.Anderson PR 12, 3060, 1975

D.D.Koeling et al. J.Phys.F 5, 2041, 1975

Linearized basisG(r) = (1/)1/2 e-iGr outside all MT spheresbut G(r) = L [ AL(G) ul(r) + BL(G) dul(r)/dr ] YL(r) in -MT sphere

which is both function value continuous and derivative continuous on the MT sphere boundary.

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Projector augmented-wave method P. E. Blochl Phys. Rev. B 50, 17953 (1994)

The all-electron wavefunction

can be obtained by the pseudo

wavefunction,

|>=|> + i (|i>-|i>)<pi|>

Here

|i> : core states,

|i> : smoothed core states,

|pi> : projector localized in the

augmentation region and obeys

< i |pi> =ij

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Planewave Basis and Pseudopotential

The periodic function u is expanded by the planewave basis,

ukn(r)=G 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.

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Pseudopotential - Norm ConserveNorm conserve and self-consistent: ‘Pseudopotentials

that work from H to Pu‘, G.B.Bachelet, D.R.Hamann, and M. Schluter,PRB26, 4199, 1982

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Pseudopotential – Ultrasoft D.Vanderbilt, PRB 41, 7892 (1990)

This potential is chargestate dependent and

norm does not conserve.

However, it is well suit for plane-wave solid-state calculations, and show promise even for transition metals.

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Tight Binding Basis (Atomic/Local Orbital)

The periodic function u is expanded ukn = m Ckn(m)km(r)

by atomic (LCAO),Gaussian (LCGO), and MT.orb. (LMTO)

km(r)=(1/N1/2) Te-ik(r-R-T))m(r-R-T)

Advantage: Minimum basis (10/atom) and exact cores.

Disadvantage: Basis depends on k and structure/potential;

Poor approximation at far from nuclei.

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Slater-Koster Scheme

< km(r)|V|kn(r) > = e-ik(R-R) T eikT <m(r-R)|V|n(r-R-T)>

< m(r-R)|V|n(r-R-T) >= M [Dl

Mm(R+T-R)]* Dl'Mn(R+T-R)

<lM(r-R)|V|l'M(r-R-T)> = M [Dl

Mm(R+T-R)]* Dl'Mn(R+T-R)

Vll'M(|R-R-T|)

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Slater-Koster Scheme: Canonical Theory O.K.Andersen et al, PRB17, 1209 (1978)

Vll'M(|R-R-T|)

Vss Vsp Vsd

Vpp Vpp Vpd Vpd

Vdd Vdd Vdd

VAB

l'lM(R)=(-1)l+M+1 (l')!(l)!(l+l')![(-1)l+l'VAl'l'VB

ll]1/2

[(2l')!(2l)!(l'+M)!(l'-M)!(l+M)!(l-M)!]-1/2

(RAB/R)l+l'+1

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Energy Bands of Cu

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Experimental Cu Bands

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Density of States of Cu

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Total Energy vs Cu Lattice Constant

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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

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Outline

• Energy bands

• Spin polarization in crystals

• Magnetic configuration (FM, AFM, …) and phase transition

• Noncollinear magnetism

• Surface magnetism

• Orbital quench

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Spin Polarization in Crystals

Energy gain in intraatomic exchange

E = -(1/2) I ij

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

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Spin Polarization in Crystals

Energy cost of spin polarization of band electrons in a crystal,

E = - dE { EF-(m/2):EF n(E)E

+ dE{EF,:EF+(m/2) n(E)E

m2/n(EF) + O(m4)

Ef

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Spin Polarization in Crystals

Energy gain over cost due to spin polarization

E = -(1/4)Im2 I + (1/4)( /n(EF)) m2

= -(1/4)(I- /n(EF)) m2

Condition for nonvanishing (spontaneous) atomic moment

I > 1/ n(EF) (Stoner-Wohlfarth criterion)

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2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4-1.10

-1.05

-1.00

-0.95

-0.90

-0.85

-0.80

NM-FM

Calc. Expt.

Enm: non-magnetic Efm: ferromagnetic

E +

254

0 (R

y/at

om)

Lattice constant (A)

Total Energy of FM vs NM bcc Fe

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2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.40.0

0.5

1.0

1.5

2.0

2.5

3.0

NM=FM

calc. expt.

Expt. m= 2.20B/atom

Mo

me

nt o

f b

cc F

e (

B/a

tom

)

Lattice constants (A)

Magnetic Fe Moment in bcc Structure

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Slater-Pauling Curve: Experiment vs LSDA

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

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Outline

• Energy bands

• Spin polarization in crystals

• Magnetic configuration (FM, AFM, …) and phase transition

• Noncollinear magnetism

• Surface magnetism

• Orbital quench

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Collinear Spin Configurations in Layered Structure

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

Layered MnAu

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

Layered MnAu

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

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Atomic and Spin Configuration of Layered MnAu

J.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 (B) & 4.0 & 3.86 & 3.93

TN (K) & 513 & 528 & 946

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Outline

• Energy bands

• Spin polarization in crystals

• Magnetic configuration (FM, AFM, …) and phase transition

• Noncollinear magnetism

• Surface magnetism

• Orbital quench

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Non-collinear Magnetism – HamiltonianM. Uhl et al. JMMM 103, 314 (1992)

Under LSDA

H = [ -(1/2) 2 + V((r)) ] + Vm((r), m(r)) U z U

The spin-1/2 rotation matrix,

cos(/2) ei/2 sin(/2) e-i/2

U(r) = ( )

-sin(/2) ei/2 cos(/2) e-i/2

depending on the local moment direction ((r), (r)

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Outline

• Energy bands

• Spin polarization in crystals

• Magnetic configuration (FM, AFM, …) and phase transition

• Noncollinear magnetism

• Surface magnetism

• Orbital quench

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Surface Magnetism: Possible Reduction

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

f

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Surface Magnetism: Possible Enhance

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

f

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Surface Dipole and Level Shift

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Surface Magnetism

Self-consistent calculation gives,

Decrease of moment due to

band narrowing

^^^^

Enhance of moment due to

level shift

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Giant Surface Spin MomentLSDA spin moment , and surface core level shift of Ni films

LAPW (H.Krakauer et al, 1981 …)

System 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

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Ni Layers on Cu Substrate D.S.Wang( 王鼎盛 ) et al, PRB3, 1340 (1982)

Expt. verified: ‘Dead layer’ is dead

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

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Outline

• Energy bands

• Spin polarization in crystals

• Magnetic configuration (FM, AFM, …) and phase transition

• Noncollinear magnetism

• Surface magnetism

• Orbital quench

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Vanishing Orbital MomentFor any time-inversion invariant system, H = -(1/2)2 + V(r), and TH = HT,its non-degenerate eigen-states have vanishing orbital mo

ment.

Proof: HT|>=TH|>=T|>, thus, T|>=|> -<|lz|>=<|Tlz|>=<|lz|> thus, <|lz|>=0 for any z axis i.e., |>=m(Cm|m>+C-m|-m>), and |C-m|=|C-m|

Also for evenly occupied degenerate states.

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Origin of Orbital Polarization

1. H includes Zeeman term, –gLili.Happl

2. H includes the spin-orbit coupling

Hsl = i [(1/4c2r) V/r] li (r)i = i r li (r)i

when there is spin polarization.

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Orbital Quench in Magnetic Solids Example: Fe dimer with 6x2 d electons

HOMO: contains two states consisting of |+1> and |-1> orbitals, separated by about

Lower has only slightly more |+1> com

ponent, and orbital moment is quenched from Hund value (2 B per atom) to

orb

Fe Co Ni Atom Hund rule 2 3 3Solid Neutron 0.05 0.08 0.05 XMCD 0.09 0.12 0.05

spin down

spin up

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Orbital Polarization of Ni Clusters

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Ni Cluster Geometry

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Spin Exchange and Orbital Correlation

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Cluster Polarization: Orbital vs Spin X.G.Wan ( 万贤纲 ) et al. PRB 69, 174414 (2004)

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Concluding Remarks

• Electron levels, especially those close to Ef, affect spin polarization strongly.

• Interatomic exchange determines the spin configuration.

• Magnetic phase transition could be well simulated

• Orbital quench/polarization depends on spin-orbit coupling