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. 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 - PowerPoint PPT PresentationTRANSCRIPT
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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
![Page 4: 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](https://reader031.vdocuments.site/reader031/viewer/2022012303/56814388550346895db00464/html5/thumbnails/4.jpg)
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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