Interplay between spin, charge, lattice and orbital degrees of freedom
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Interplay between spin, charge, lattice and orbital degrees of
freedom
Lecture notes Les Houches June 2006
George Sawatzky
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LSDA+U
V.I. Anisimov et al., PRB 44, 943 (1991)
Simplified version :
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LSDA+U also has no electron correlationSingle Slater det. of Bloch states. No multiplets.
LSDA LSDA+U
Czyzk et l PRB 49, 14211(1994)
LSDA+U antiferromagnetic S=.8 Bohr magnetons, E gap = 1.65 eV
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Num
ber
of h
oles
LDA+U potential correction
SC Hydrogen
a =2.7 ÅU=12eV
LDA+U DOS
0.0 0.2 0.4 0.6 0.8 1.0-8
-7
-6
-5
-4
-3
-2
-1
0
1
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4
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LD
A+
U c
orr
ect
ion
(e
V)
Number of holes
spin up spin down
Problems with LDA+U for metallic systems
Ferromagnetic
Note U gap closes with doping No spectral weight transfer
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Meinders et al, PRB 48, 3916 (1993)
Exact diagonalization in 1DHubbard 10 sites U=10t• U Gap increases with doping•Spectral weight is transfered from upper H band to the lower H band•
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N N
EFPES PES
U
EF
N-1 N-12
EF
N+1 N+12
Mott – Hubbard Spectral weight transfer
Remove one electron Create two addition statesAt low energy
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These particles block 2 or more states
Bosons – block 0 statesFermions – block 1 state
Integral of the low Energy spectral weight For electron addition if Hole doped (left) and Electron removal for eDoped (right side)
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Eskes et al PRL 67, (1991) 1035 Meinders et al PRB 48, (1993) 3916
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Exact diagonalization 1D Hubbard Meinders et al, PRB 48, 3916 (1993)
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Don’t know of a rigorous Proof of Hubb---t,J (U>>w)
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Spin charge separation in 1D
Antiphase Domain wall
Now the charge is free to move
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Magnons and spinons in 1D
Magnon S=1
Two spinons
Spinons propagate via J Si+S-
1+1
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Similar is some sense to the 1D case it is proposed that one has2D rivers of charge separating anti-phase domain walls.Charges can now fluctuate from left to right without costing J
Anisimov, Zaanen ,Andersen, Kivelson,Emery-----
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Closer to real systems
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Oxides
Remember at surfaces U is increased, Madelung is decreased, W is decreased
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For divalent cations
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If the charge transfer gap becomes negative we will get a strange metal
This seems to happen in CrO2 where the O bands cross theFermi driving the system to a half metallic ferromagnet Korotin PRL 80 (1998) 4305
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3 most frequently used methods • Anderson like impurity in a semiconducting host
consisting of full O 2p bands and empty TM 4s bands including all multiplets
Developed for oxides in early 1980’s, Zaanen, Sawtzky, Kotani, Gunnarson,-----
• Cluster exact diagonalization methods. O cluster of the correct symmetry with TM in the center. Again include all multiplets crystal fields etc
Developed for oxides in early 1980’s Fujimori, Sawatzky,Eskes, ------
• Dynamic Mean Field methods, CDMFT, DCA which to date do not include multiplets
Developed in the late 1990’s: Kotliar, George, Vollhart---
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Zaanen et al prl 55 418
(1985) Anderson impurity ansatz Like DMFT but not self consistantBut also including all multiplet interactions
Kondo resonance
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To calculate the gap we calculate the ground state of the system with
n,n-1, and n+1 d electrons Then the gap is
E(Gap)= E(n-1)+E(n+1)-2E(n)
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Two new complications
• d(n) multiplets determined by Slater atomic integrals or Racah parameters A,B,C for d electrons. These determine Hund’s rules and magnetic moments
• d-o(2p) hybridization ( d-p hoping int.) and the o(p)-o(p) hoping ( o 2p band width) determine crystal field splitting, superexchange , super transferred hyperfine fields etc.
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More general multiband model Hamiltonian
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•We usually take U(pp) =0 although it is about 5 eV as Measured with Auger but the O 2p band is usuallu fiull or nearly full.
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Ways to “screen “ or rather reduce Uor F0
U in Cu atom is 18eV in the solid 8eV
In a polarizable medium“Solvation” in chemistry
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Rest comes from bond Polarization involving O 2p and TM 4s states
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As we remove or add d electrons charge moves from O(2p) to or from TM(4s) reducing the d electron Removal energy as well as the d electron addition energy. Reduces U effectively by about 6-8 eV. Recall though that these effects will yield satellites or incoherent Parts to the spectral function at energies corresponding to the O(2p)-TM(4s) energy splitting.
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Note that B and C are only slightly reduced in the solid they do not involve changes in the local charge !!!
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For the N-1 electron states we need d8, d9L, d10L2 where L denotes a hole in O 2p band. The d8 states exhibit multiplets
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H. Eskes and G.A. SawatzkyPRL 61, 1415 (1988). Anderson Impurity calculation
Zhang Rice singlet
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J. Ghijsen et al
Phys. Rev. B. 42, (1990) 2268. Photoemission spectrum of CuO
Energy below Ef in eV
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Example of two cluster calculations to obtain the parameters For a low energy theory ( single band Hubbard or tJ )
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Eskes etal PRB 44,9656, (1991)
0 to 1 hole spectrumOne of the Cu’s is d9The other d10 in the Final state. Bonding Antibonding splitting Measure d-d hoping
1 hole to 2 holes finalState is od9 on both Cu’s Triplet singlet Splitting yields superExchange J
2 holes to 3 holes final state is d9 for both Cu’sPlus a hole on O formingA singlet (ZR) with one of The Cu’s . Splitting in red Yields the ZR-ZR hoping integral as in tJ
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Need multiband models to describe TM compounds
However numerous studies have shown that this can sometimes be reduced to an effective
single band Hubbard model at least for highTc’s BUT ONLY FOR LOW ENERGY
EXCITATIONS E<0.5eV
Macridin et al
Phys. Rev. B 71, 134527 (2005)
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(Maximize spin)
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Crystal and ligand field splittings
Often about 0.5 eVIn Oh symmetry
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Eg-O2p hoping is 2 times as large as T2g-O-2p hoping
Often about 1-2eVIn Oxides
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High Spin – Low Spin transition very common inCo(3+)(d6), as in LaCoO3, not so common in Fe(2+)(d6)Because of the smaller hybridization with O(2p)
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Mixed valent system could lead to strange effects Such as spin blockade for charge transport and high thermoelectric powers
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What would happen if 2Jh <10Dq<3JhIf we remove one electron from d6 we would go fromS=0 in d6 to S=5/2 in d5. The “hole “ would carry a spinOf 5/2 as it moves in the d6 lattice.
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If the charge transfer energy gets small we have to Modify the superexchange theory
Anderson 1961
New term
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