in memory of ghk: correlated electrons from intermetallics to organometallics presented at the...

In Memory of GHK: Correlated electrons from intermetallics to

organometallics

Presented at the George Hsing Kwei Memorial Symposium, Los Alamos, June 27, 2006

Corwin H. BoothChemical Sciences Division

Glenn T. Seaborg CenterLawrence Berkeley National

Laboratory

Outline

1. George and me2. George and correlated electrons3. INTRODUCTION: Kondo effect in f-electron

intermetallics4. RESEARCH DIRECTIONS: Possible correlated

electron effects in organometallics

A second career…

• Of George top 50 most-cited papers, nearly 70% came after 1986, 26 in High Tc, 10 in CMR, 7 in mixed valence/heavy fermions and 4 in fullerenes

• During this time, George became a Rietveld jock, solving the 4th most crystal structures using IPNS (I believe, after Jorgensen, Lawson and somebody else)

• George’s mode of operation: facilitator (“vector boson”)— Took a broad view— Put teams of people together— Learned to play to people’s strengths

• George’s interests:— charge, lattice, spin interactions

f-electron intermetallics: History

• Dilute magnetic alloys! The Kondo resistance minimum…

• First observed in 1930’s, not explained until 1964 by Kondo

• CuMn, CuFe, AuFe, etc.• Magnetic properties not very clear

(magnetic impurity, after all), however…

• Requires: a local magnetic moment— dilute magnetic alloys in

transition metals— no need for diluting the 4f and

5f’s!• Unusual ground states develop

near the localized/delocalized boundary: heavy-fermions, superconductivity, non-Fermi liquids

W. J. de Haas, J. de Boer, and G. J. van den Berg, Physica 1, 1115 (1934).

Kondo the focus

RKKY(antiferromagneti

c)(usually)

Kondo(“nonmagnetic”

)

J

f c

Key concepts:

delocalized (conduction) electrons

screen out the local moment

Coupling is so strong it creates a quasi-bound

state (“Kondo singlet”)

Partially localizes conduction electrons, partially delocalizes f-

electronsKondo is a collective, many-body, strongly correlated electron

phenomena…

When is localization/delocalization important?

*adapted from Smith and Kmetko (1983)

4f La Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb LuCe

5f Ac Th Pa Np Pu Am Cm Bk Cf Es Fm Md No LrU

3d Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn

4d Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd

5d Ba La Hf Ta W Re Os Ir Pt Au Hg

Y b 3+

E m p ty S h e ll

P a rtia lly F ille d S h e ll F u llS h e ll

In c re a s in g L o c a liz a tio n

Magnetic behavior in concentrated Kondo

Pauli paramagnet:

(0) ~ B2 ρ(F)

Rajan, PRL 51, 308 (1983).

02

12

J

K

CJT

HT: Curie-Weissparamagnet

LT: Pauliparamagnet

/ 0

T/TK

Valence measurements with x-ray absorption near-edge structure (XANES)

• Energy shift of Yb per valence is ~10 eV.

• singlet binding energy is < 0.1 eV for all these compounds.

creation of a core hole breaks the singlet, giving a probability of Yb3+=nf.

tot(E) = (1-nf) 2+(E)+nf 3+(E)

8930 8935 8940 8945 8950 8955

0.0

0.1

0.2

0.3

(b) YbMgCu4

(E

)t

E (eV)

8930 8935 8940 8945 8950 8955

0.0

0.4

0.8

1.2 (a) YbAuCu4

data two edge fit trivalent component divalent component

cfbfa 1413 ||

4f1

+2p

4f0

Yb LIII edge

Valence measurements with x-ray absorption near-edge structure (XANES)

• Energy shift of Yb per valence is ~10 eV.

• singlet binding energy is < 0.1 eV for all these compounds.

creation of a core hole breaks the singlet, giving a probability of Yb3+=nf.

tot(E) = (1-nf) 2+(E)+nf 3+(E)

8930 8935 8940 8945 8950 8955

0.0

0.1

0.2

0.3

(b) YbMgCu4

(E

)t

E (eV)

8930 8935 8940 8945 8950 8955

0.0

0.4

0.8

1.2 (a) YbAuCu4

data two edge fit trivalent component divalent component

cfbfa 1413 ||

2p

4f0

4f1

Sarrao et al., PRB 59, 6855 (1999)

Some unusual organometallics

Cp*2 Yb (bipy)

Dick Andersen, Jim Boncella, Don Tilley

Cp*2 Yb (terpy)

Kevin John, Dave Morris, Jackie Kiplinger

CePn*2

Dermot O’Hare

Ce(cot)2Many, but not enough

0 50 100 150 200 250 3000.00010

0.00015

0.00020

0.00025

0.00030

0 50 100 150 200 250 3000.0001

0.0002

0.0003

0.0004

0.0005

(e

mu/

mol

Ce)

T (K)

Unusual magnetic properties

Cp*2 Yb (bipy)

Booth et al., PRL 95, 267202 (2005)

Cp*2 Yb (terpy)

Veauthier et al., Inorg. Chem. 44, 5911 (2005).

Ce(cot)2

0 100 200 300 4000.0012

0.0014

0.0016

0.0018

0.0020

0.0022

0 100 200 300 400

0.002

0.003

0.004

0.005

0.006

T (K)

(T

) (e

mu

/mo

l)

Valence tautomerism

“Bullvalene”, aka C10H10

Doering and Roth, Tetrahedron 19, 715 (1963).Wikipedia article on Bullvalene

Valence tautomerism

Ma g

ne t

ic m

o men

t (

B)

This is a chemical equilibrium

C. Pierpont, Coord. Chem. Rev. 216-217, 99 (2001).

semiquinones

XANES example: valence change measurements in valence tautomers

C. Roux et al., Inorg. Chem. 35, 2846 (1996).

Corr

ela

ted e

lect

ron e

ffect

s? M

aybe…

LaB

ute

et

al.,

J. C

hem

. Phys.

116,

3681

(20

02).

8920 8930 8940 8950 89600.0

0.5

1.0

1.5

2.0

2.5

No

rma

lize

d ab

sorp

tion

E (eV)

bipy T=20 K bipy T=150 K bipy T=300 K terpy T=30 K terpy T=150 K terpy T=300 K

Ytterbocenes

8920 8940 8960 8980 90000.0

0.5

1.0

1.5

2.0

2.5

3.0

No

rma

lize

d a

bso

rptio

n

Energy (eV)

Yb(II) Yb-bipy-OMe Yb-bipy Yb-tbut-dad Yb(III)-bipy-I (*)

• From susceptibility, would get 28% and 69% 4f13 in bipy and terpy at RT, with sharp temperature dependence*

• Yb XANES: no observed T-dep in nf from 30-400 K

• Cannot be due to a chemical equilibrium between 4f13 and 4f14 configurations *Veauthier et al., Inorg. Chem. 44, 5911 (2005).

Cerocene and bis(pentalene)cerium

• Temperature-independent paramagnetism (TIP)

• Very low susceptibility: nearly all tetravalent 4f0?

• Ce LIII says more 4f1 character than 4f0 ?!?!?!

• CePn*2 looks same (same electron count)

5700 5720 5740 57600.0

0.5

1.0

1.5

2.0

___2p 4f 1 L 5d

__2p 4f 0 5d

__2p 4f 1 5d

Nor

mal

ized

Abs

orpt

ion

Energy (eV)

Ce(III) Ce(IV) Cerocene

5700 5720 5740 57600.0

0.5

1.0

1.5

2.0

No

rma

lize

d A

bso

rptio

n

Energy (eV)

Cerocene CePn*

2

0 50 100 150 200 250 3000.00010

0.00015

0.00020

0.00025

0.00030

0 50 100 150 200 250 3000.0001

0.0002

0.0003

0.0004

0.0005

(e

mu/

mol

Ce)

T (K)

Ce(cot)2

Booth et al., PRL 95, 267202 (2005)

Metallic bonding in organic compounds

• Benzene discovered in 1825 by Faraday

• Kekulé proposed correct structure in 1858

• Problems include x-ray measurements: 1.53A for typical C-C, 1.34A for typical C=C, and 1.39A for the 6 carbon-carbon bonds in benzene

• Linus Pauling determined the solution: hybrid orbitals!

• Like metals, bonding electrons are delocalized over the ring

• Unlike metals, energy bands are atomic-like (narrow)

A new idea… for 1989!

Ce

Ce(COT)2

Dolg, Fulde and coworkers(1989-1995): intermediate valence state closer to Ce(III) (nf=0.81), but forms a magnetic singlet with the cyclooctatetraene -ligands: a multiconfigurational ground state of 4fe2u

1e2u3 (81%) and 4f0e2u

4 (19%)

The energy difference to the magnetic (triplet) excited state is on the order of 1 eV (TK~11,600 K).

Kondo: a mechanism for f-bonding

Ce

Ce(COT)2

delocalized ’s (HOMO) do the screening of the f-moment

f-shell is close to fully occupied


Ce

Ce(COT)2

delocalized ’s get partially localized

localized f’s get partially delocalized

Kondo in nanoparticles

Schlottmann, PRB 65, 024420 (2001).

HOMO/LUMO Gap,

EFermi

f f

c ’s

Rajan, PRL 51, 308 (1983).

k

E

>>TK<TK

?? “no” says Thimm et al., PRL 82, 2143 (1999)

Cerocene magnetic susceptibility

(0) doesn’t go to zero /TK is smallish

•can use TK~1/0~5000 K

Rajan, PRL 51, 308 (1983).

02

12

J

K

CJT

0 50 100 150 200 250 3000.00010

0.00015

0.00020

0.00025

0.00030

0 50 100 150 200 250 3000.0001

0.0002

0.0003

0.0004

0.0005

(e

mu/

mol

Ce)

T (K)

Booth et al., PRL 95, 267202 (2005)

Ce LIII-edge XANES for f-occupancy

• Ce(III)=Ce[N(Si(CH3)3)2]3

• “Ce(IV)”=(C6H4N2)2Ce[C3H(CH3)2]2 a.k.a Ce(tmtaa)2

• Cerocene f-occupancy nf=0.890.03 from 30-300 K

5700 5720 5740 5760

0.0

0.5

1.0

1.5

2.0

___2p 4f 1 L 5d

__2p 4f 0 5d

__2p 4f 1 5d

Nor

mal

ized

Abs

orpt

ion

Energy (eV)

Cerocene Total fit Fit components

5700 5720 5740 57600.0

0.5

1.0

1.5

2.0

___2p 4f 1 L 5d

__2p 4f 0 5d

__2p 4f 1 5d

Nor

mal

ized

Abs

orpt

ion

Energy (eV)

Ce(III) Ce(IV) Cerocene

Booth et al., PRL 95, 267202 (2005)

Ytterbocenes

-50 0 50 100 150 200 250 300 350 400 4500.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

(em

u/m

ol)

T (K)

Cp*

2Yb(bipy-OMe)

Cp*

2Yb(bipy)

Cp*

2Yb(tbut-dad)

Cp*

2Yb(p-methyl-dad)

8920 8940 8960 8980 90000.0

0.5

1.0

1.5

2.0

2.5

3.0

Nor

mal

ized

abs

orpt

ion

Energy (eV)

Yb(II) Yb-bipy-OMe Yb-bipy Yb-tbut-dad Yb(III)-bipy-I (*)

0 100 200 300 4000.0012

0.0014

0.0016

0.0018

0.0020

0.0022

0 100 200 300 400

0.002

0.003

0.004

0.005

0.006

T (K)

(T

) (e

mu

/mo

l)

TK~1/0=2240 K

TK~4.4Tmax=1670 K

max/0~1.22 (get 1.34)

no observed T-dep in nf from 30-400 K

nf=0.81

Booth et al., PRL 95, 267202 (2005)

The increasingly strange case of Cp*

2Yb(dmb)

0 50 100 150 200 250 3000.000

0.002

0.004

0.006

0.008

0.010

120 140 160 180 200 220 240

0.00050

0.00075

0.00100

0.00125

(em

u/m

ol)

T (K)

T down T up

0 50 100 150 200 250 3000.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

n f

T (K)

T down T up

8920 8930 8940 8950 8960 8970 89800.0

0.5

1.0

1.5

2.0

2.5

Nor

mal

ized

abs

orpt

ion

Energy (eV)

T=291 K T=30 K

N

N

Yb

R

R

bipy: R=H

bipy-OMe: R= OCH 3

Cp*2Yb(bipy):

bipy: R=H

bipy-OMe: R=OCH3

dmb:R=Me

The mysteries continue…

0 50 100 150 200 250 3000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Cp*2Yb(dmb)

n f

T (K)

from T(T)/CJ=7/2

from Yb LIII-edge XANES

• Doesn’t appear to be tautomerism…

• If Kondo involved, TK is large enough as to swamp crystal fields

• Could we be thermally occupying a new orbital that also is hybridized strongly with the f hole?

fHOMO

kBTK~Wexp[-f /N(0) Vfc2]

N(0) is density of states at the Fermi level

Wrap up

•George’s contributions to strongly correlated electron systems were broadly based

•For me (and many others) George’s biggest contribution was recognizing a problem and the talents in other people to tackle that problem

• I focussed on f-electron intermetallics

•My work on intermetallics lead me to cerocene and the conjecture that these weird properties of some f-electron organometallics is due to strongly correlated electronic effects

Collaborators and Acknowledgements

Million Daniel, Sang-Wook Han, Evan Werkema,Wayne Lukens, Daniel Kazdan, Dick Andersen (LBNL)

Marc Walter (TU-Kaiserslautern, LBNL)Jon Lawrence (UC Irvine)E. D. Bauer, John Sarrao (Los Alamos National Laboratory)Andrew Ashley, Dermot O’Hare (Oxford)

This research was supported by the Director, Office of Science, Office of Basic Energy Science, Chemical Sciences, Geosciences and Biosciences Division, US Department Energy under Contract Number DE-AC-03-76F00098.

Data were collected on Beamlines 2-3, 10-2 and 11-2 at the Stanford Synchrotron Radiation Laboratory (SSRL), which is operated by the DOE, OBES.

And of course…

George!

Metallocene summary

Attack: Overall lineshape not in the Kondo limit. Impurity model in Kondo limit probably not appropriate HOMO/LUMO gap should become more important for lower TK (?) Better models (for instance, NCA, DFT+correlations, DMFT)

Create molecules with different TK and or HOMO/LUMO gap Explore relationships between the ligand, TK and Other phenomena: valence transitions? U? Np? Pu?

Kondo impurity vs. Anderson lattice

• Kondo investigated the problem of a magnetic impurity in a nonmagnetic host:

• An alternate model was developed by Anderson in 1961 for magnetic alloys, i.e. so-called “concentrated” or “lattice” systems:

,

)0(2k

ckk JnH sS

,

†mix

imp

,band

miximpband

)H.c.(k

k

d

kkk

dcVH

nUnH

nH

HHHH

Kondo Hamiltonian

Anderson Hamiltonian

Kondo=Anderson when d-VU

V2/d=Jc

CeCerocene Ce(COT)2:

Ce

N

N N

N

N N

NN

Ce(Mac)2:

N

N

Yb

R

R

bipy: R=H

bipy-OMe: R= OCH 3

Cp*2Yb(bipy):

bipy: R=H

bipy-OMe: R=OCH3

N

N

Yb

R

Rtbudad: R= C(CH3)3

p-methyldad: R= CH3N

N

Yb

R

Rtbudad: R= C(CH3)3

p-methyldad: R= CH3

Cp*2Yb(dad):

A family of rare-earth metallocenes

Advantages of ytterbium

N

N

Yb

R

R

bipy: R=H

bipy-OMe: R= OCH 3

Cp*2Yb(bipy):

bipy: R=H

bipy-OMe: R=OCH3

Cp*2 Yb (bipy)

N

N

Yb

R

Rtbudad: R= C(CH3)3

p-methyldad: R= CH3N

N

Yb

R

Rtbudad: R= C(CH3)3

p-methyldad: R= CH3

Cp*2Yb(dad):

• Can’t observe the expected temperature dependences if TK is large, like in cerocene!

• Get TK down, need to tune…

• Using Cp* and attached functional groups, can vary the f/ coupling!

The Anderson lattice and the “slow crossover”

0 100 200 3000.000

0.025(e)

YbAgCu4

(c)

(em

u/m

ol)

0.000

0.005

YbTlCu4

(b)

0.000

0.005

YbMgCu4

(a)

0.00

0.05

YbZnCu4

(d)

T(K)

0.000

0.025

YbCdCu4

0.0

0.5

1.0

(ef

f)2 =T/

CJ

0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

J. M. Lawrence et al., Phys. Rev. B, 63:054227, 2001. LBNL-46027

A. L. Cornelius et al., Phys. Rev. Lett., 88:117201, 2002. LBNL-49405

nc

0.52

~2

1.2

0.9

1.6

Local moment in a Pu intermetallic

Sarrao et al., Nature 420, 297 (2002)

=Cel/T~m*~1/TK

=77 mJ/mol·K2

Curie-Weiss behavior above Tc

CeAl2 nanoparticle properties

• Primary Kondo evidence is from ~following:

TK(H)2=TK(0)2+(JgBH)2

• Powder diffraction indicates a ~1.1% volume expansion.

• Gruneisen relation gives TK~2-4K

• The rest is attributed to size effects (generic…)

• Claim is TN is suppressed due to inability to support spin waves over such small distances.

• In addition, TK decreases from ~5K to 0.7K

CePt2+x contracts, rather than expands

Cp*2 Yb (bipy)

8900 8920 8940 8960 8980 9000

-0.2

0.0

0.2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Yb(bipy) Yb(bipy)I

dt)

/dE

E (eV)

Yb L3 XANES Yb(bipy)

Yb(bipy)I

t (

norm

aliz

ed)

• Yb L3 XANES show a divalent component, indicating a valence of 2.80.1

• The valence does not change with temperature from 20 – 300 K

• This result rules out a chemical equilibrium

• Indicates that TK must be at least 750 K

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Bickers, Fig. 14

nf(T

)/n

f(o)

log10 (T/TK)

X-ray absorption spectroscopy

• Main features are single-electron excitations.• Away from edges, energy dependence fits a power

law: AE-3+BE-4 (Victoreen).• Threshold energies increase roughly as Z2.

1 10 100

0.01

0.1

1

10

M

LIII

, LII, L

I

K

Xenon

(cm

-1)

E (keV)From McMaster Tables 1s

filled 3d

continuum

EF

core hole

X-ray absorption spectroscopy (XAS) experimental setup

“white” x-rays from

synchrotron

double-crystal monochromator

collimating slits

ionization detectors

I0I1

I2

beam-stop

LHe cryostatsample

reference sample

• sample absorption is given by

t = log(I1/I0)

• reference absorption is

REF t = log(I2/I1)

• EXAFS (k)=[(k)-0(k)]/0(k)

Fermi gas + interactions = Fermi liquid

• Real system 1023 electrons• 1/r2 interaction is long

range• Compute 1023 interactions

with 1023-1 other fermions

• 1023-1 electrons form a Fermi sea of non-interacting “quasiparticles” excitations

• q.p.’s have same quantum numbers (spin, charge), but a renormalized mass m*

• Amazing fact: FLT describes most pre-1985 behavior in the solid state, including metals, superfluid 3He, BCS superconductivity, quantum Hall liquid state, heavy fermions, mixed valence, Kondo, etc. etc. etc

Fermi gas + interactions = Fermi liquid

• Real system 1023 electrons• 1/r2 interaction is long

range• Compute 1023 interactions

with 1023-1 other fermions

• Ground state properties of a Fermi Liquid: T2

= 0

• C~ T

“Kondo Box”

Liang et al., Nature 2002

• A lot of recent attention on the Kondo effect in quantum dots

• In most of these systems, conduction electrons are injected

• Self-contained “Kondo Box” has been lightly explored:Thimm, Kroha and von Delft, PRL 82, 2143 (1999).



Chen et al., PRL 84, 4990 (2000).


Ce

Ce(COT)2

delocalized ’s (HOMO) do the screening of the f-moment

f-shell is close to fully occupied






Ce

Ce(COT)2

delocalized ’s get partially localized

localized f’s get partially delocalized





Inspiration from the Dark Side?

Cerocenebis-cyclooctatetraene cerium

Ce(COT)2

Ce(C8H8)2

Dolg, Fulde and coworkers(1989-1995): intermediate valence state closer to Ce(III) (nf=0.81), but forms a magnetic singlet with the cyclooctatetraene -ligands

The energy difference to the magnetic (triplet) excited state is on the order of 1 eV (TK~11,600 K).

This picture is very closely analogous to the Kondo effect in the heavy-fermion and mixed valence intermetallics…

A molecular magnetic mystery…

Ce

Ce(COT)2Quick history of CEROCENE, Ce(C8H8)2

Cerocene discovered in 1976 (Greco) following uranocene in 1968 (Streitwieser)

NMR, gas phase photoemission, structure consistent with Ce(IV)

Basic behavior of IV metals

High temperature limit: LOCAL MOMENT PARAMAGNET

Integral valence: nf 1 z = 2+nf = 3 Yb 4f13(5d6s)3

Curie Law: CJ/T where CJ = N g2 B2 J(J+1)/ 3 kB J = 7/2 (Yb)

Full moment entropy: S R ln(2J+1)

CROSSOVER at Characteristic temperature TK

Low temperature limit: FERMI LIQUID

Nonintegral valence (nf < 1) Yb 4f14-nf (5d6s)2+nf

Pauli paramagnet: (0) ~ B2 ρ(F)

Linear specific heat: Cv ~ T

= (1/3) 2 ρ(F) kB2

Note: Low T anomalies, relative the AIM,

in χ(T) and C(T)/T.

γ→

χ(0) →

nf →1

← χ ~ CJ/T

Cornelius et al, PRL 88 (2002) 117201

8920 8930 8940 8950 8960 89700.0

0.5

1.0

1.5

2.0

2.5

3.0

No

rma

lize

d A

bsor

ptio

n

E (eV)

py2

bipy(I) (bipy-CO

2Et)I

bipy p-Mebipy bipy-CO

2Me

OMeDAD p-MeDAD tBuDAD terpy

0

2

4

6

8

10

4M (kG

)

0 100 200 300

0

5

10

15

La0.7

Ca0.3

MnO3

TC = 250 K

(m

cm

)

Temperature (K)

M. F. Hundley et al., MRS Proc. 474, 167 (1997).

The colossal magnetoresistance renaissance

Mn3+ Mn4+

Hopping

FM double-exchange

Jahn-Teller distortion

Hund’s rulealigns spins

Hopping

AFM exchange

Mn3+ Mn4+

Hund’s rulealigns spins

eg

t2g

Magnetic energy penalty J

Lattice energy penalty JT

Mn

O La/Ca

0 100 200 3000

10

20MFH/LANL MST-10 11/16/94mr1194fa.org

(m

cm

)

Temperature (K)

-1.0

-0.5

0.0

Bapp

= 5 T

La0.7

Ca0.3

MnO3

/ o (

50 k

Oe)

T (K)

M. F. Hundley et al., MRS Proc. 474, 167 (1997).

0.0 0.2 0.4 0.6 0.8 1.0

-3.5

-3.0

x=0.21 x=0.25 x=0.30

ln(

)

M/M0

0 50 100 150 200 250 300 350

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

x=0.0 x=0.21 x=0.25 x=0.30 x=1.0

2 Mn-

O (

Å2 )

T (K)

2Th+2

FP

2Th

Billinge, DiFrancesco, Kwei, Neumeier, Thompson, PRL 77, 715 (1996).

Booth, Bridges, Kwei, Cornelius, Neumeier, PRL 80, 853 (1998).

The Hamiltonians

• Kondo (1964) wrote the first solvable Hamiltonian for a magnetic impurity in a non-magnetic, metallic host:

)0(2,

Kondo sS ck

kk JnH

• Anderson (1961) wrote the generalized Hamiltonian capable of describing a lattice of “impurities” (“Anderson lattice”):

,

†

,Anderson

)H.c.(k

k

nndk

kk

dcV

UnnH

Phil

Distortion affecting magnetic properties

All that’s left is Kondo:

kBTK~Wexp[-f /N(0) Vfc2]

W is the conduction bandwidth

f is the f-level energy w.r.t. the Fermi level, EF.

N(0) is the electronic density of states at EF.

Vfc is the f-electron/conduction-electron hybridization energy

bonds pf

pffpfp

R

rrV 5

2/15 )(

fd: constant rf: outer f-radiusrp: outer p-radius Rf-d: f-d bond length

tight-binding model:

Harrison and Straub

in memory of ghk: correlated electrons from intermetallics to organometallics presented at the...

Documents