Interplay between spin, charge, lattice and orbital degrees of

freedom

Lecture notes Les Houches June 2006

George Sawatzky

LSDA+U

V.I. Anisimov et al., PRB 44, 943 (1991)

Simplified version :

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

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

2

3

4

5

6

7

8

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

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•

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

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)

Eskes et al PRL 67, (1991) 1035 Meinders et al PRB 48, (1993) 3916

Exact diagonalization 1D Hubbard Meinders et al, PRB 48, 3916 (1993)

Don’t know of a rigorous Proof of Hubb---t,J (U>>w)

Spin charge separation in 1D

Antiphase Domain wall

Now the charge is free to move

Magnons and spinons in 1D

Magnon S=1

Two spinons

Spinons propagate via J Si+S-

1+1

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-----

Closer to real systems

Oxides

Remember at surfaces U is increased, Madelung is decreased, W is decreased

For divalent cations

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

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---

Zaanen et al prl 55 418

(1985) Anderson impurity ansatz Like DMFT but not self consistantBut also including all multiplet interactions

Kondo resonance

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)

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.

More general multiband model Hamiltonian

•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.

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

Rest comes from bond Polarization involving O 2p and TM 4s states

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.

Note that B and C are only slightly reduced in the solid they do not involve changes in the local charge !!!

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

H. Eskes and G.A. SawatzkyPRL 61, 1415 (1988). Anderson Impurity calculation

Zhang Rice singlet

J. Ghijsen et al

Phys. Rev. B. 42, (1990) 2268. Photoemission spectrum of CuO

Energy below Ef in eV

Example of two cluster calculations to obtain the parameters For a low energy theory ( single band Hubbard or tJ )

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

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)

(Maximize spin)

Crystal and ligand field splittings

Often about 0.5 eVIn Oh symmetry

Eg-O2p hoping is 2 times as large as T2g-O-2p hoping

Often about 1-2eVIn Oxides

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)

Mixed valent system could lead to strange effects Such as spin blockade for charge transport and high thermoelectric powers

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.

If the charge transfer energy gets small we have to Modify the superexchange theory

Anderson 1961

New term