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The Alpha to Gamma Transition in Ce: A The Alpha to Gamma Transition in Ce: A Theoretical View From Optical SpectroscopyTheoretical View From Optical Spectroscopy

K. Haule, V. Oudovenko, S. Savrasov, G. Kotliar

DMFT(SUNCA method)

two-band Hubbard model

Bethe lattice, U=4D

IJS Outline

• Some facts about Ce, why is it an interesting material?

• Classical theories explaining Ce volume collapse

• LDA+DMFT Photoemission results

• LDA+DMFT Optics calculation and results

IJS Overview

Electron configuration of Ce

Atom : [Xe]4f25d06s2

Solid or compounds :

trivalent [Xe]4f1(5d6s)3,

tetravalent [Xe]4f0(5d6s)4

The element Ce

γ-α phase transition of Ce

• large volume collapse (~15%)

• loss of local magnetic moment

promotional model promotional model

(Ramirez, Falicov 1971)(Ramirez, Falicov 1971)

IJS Overview

volumes exp. LDA LDA+U 28Å3 24.7Å3

34.4Å3 35.2Å3

•Transition is 1.order•ends with CP very similar to gas-liquid condesation

Various phases :

isostructural phase transition ends in a critical point at (T=600K, P=2GPa)

(fcc) phase

[ magnetic moment

(Curie-Wiess law),

large volume,

stable high-T, low-p]

(fcc) phase

[ loss of magnetic

moment (Pauli-para),

smaller volume,

stable low-T, high-p]

with large

volume collapse

v/v 15

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Mott transition (B. Johansson, 1974):Mott transition (B. Johansson, 1974):

Kondo volume colapse (J.W. Allen, R.M. Martin, 1982):Kondo volume colapse (J.W. Allen, R.M. Martin, 1982):

Classical theories

Hubbard modelHubbard model

Anderson (impurity) modelAnderson (impurity) model

changes and causes Mott tr.changes and causes Mott tr.

changes changes →→ chnange of T chnange of TKK

bath

either constant or

taken from LDA and rescaled

IJS LDA+DMFT

ab initio calculationab initio calculation

is self-consistently determinedis self-consistently determined

contains tcontains tffff and V and Vfdfd hopping hopping

bath for AIMbath for AIM

Kondo volume colapse model resembles DMFT picture:Kondo volume colapse model resembles DMFT picture:

Solution of the Solution of the Anderson impurity modelAnderson impurity model → → Kondo physicsKondo physics

DifferenceDifference: : with DMFT the lattice problem is solved (and therefore with DMFT the lattice problem is solved (and therefore Δ must self-consistently determined) while in KVC Δ is calculated for a fictious impurity (and needs to be rescaled to fit exp.)

IJS LDA+DMFT Formalism

local in localized LMTO base

Impurity problem (14x14):

fermionic bathfermionic bath

mappingmapping

solution AIM

DMFT SCC

IJS Slave particle diagrammatic impurity solvers

NCA

OCA

TCA

Luttinger Ward functional

local (eigen)state - full atomic base

, where

general AIM:

( )

IJS SUNCA vs QMC

two band Hubbard model, Bethe lattice, U=4D

three band Hubbard model,

Bethe lattice, U=5D, T=0.0625D

three band Hubbard model,

Bethe lattice, U=5D, T=0.0625D

IJS LDA and LDA+U

f DOStotal DOSvolumes exp. LDA LDA+U

28Å3 24.7Å3

34.4Å3 35.2Å3

ferromagnetic

IJS LDA+DMFT alpha DOS

TK(exp)=1000-200K

IJS LDA+DMFT gamma DOS

TK(exp)=60-80K

IJS Photoemission&experiment

IJS Thermodynamics of the transition

The impurity level is calculated by the constraint LDA calculation and fixed and is not calculated from the high frequency expansion

of LDA-SCC (Edc is not needed)

Non-self consistent one shot calculation

B. Amadon, S. Biermann, A. Georges, F. Aryasetiawan, cond-mat/0504732

K.Held, A.K.McMahan,R.T. Scalettar, PRL 87,276404(2001)

IJS Optics calculation

double poledouble pole

single pole for ATMOne divergence integrated out!

IJS ATM in short

Analytic tetrahedron method:

Integral is analytic and simple

(combination of logarithms)

IJS Optical conductivity

1eV alpha peak

1eV alpha peak

0.5 eV gamma depletion

0.5 eV gamma depletion

0.33 eV alpha shoulder5K

300K

580K

1160K

*

*

IJS Partial DOS

4f

5d

6s

Z=0.33

IJS Optical conductivity

orbitally resolved "fat" optics for alpha phaseLDA compared to LDA+DMFT

ff contribution to optics <<fd<<ddff contribution to optics <<fd<<dd ff hopping ff hopping very smallvery small,

Kondo resonance mostly due to hybridization with dKondo resonance mostly due to hybridization with d

Mott transition not the right explanation Mott transition not the right explanation

(even if Mott transition is understood in modern sense)

IJS Hybridization pseudogap

peak in f spectra scatters d electronspeak in f spectra scatters d electrons

dd contribution largest, some df

IJS Local spectral function

IJS Bath spectral function

n.k.p.=16x16x16=4096

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d,s,p conducting bands are important for explaining properties of CeKondo peak in low T alpha phase appears due to hybridization with d bandsOptics conductivity has mostly d character Optics shows hybridization pseudogap up to 1eV in alpha phase

and no pseudogap in gamma phaseKVC model better than MT scenario

Conclusion