Cellular-DMFT approach to the Cellular-DMFT approach to the electronic structure of correlated electronic structure of correlated solids. solids.

Application to the sp, 3d,4f and 5f electron systems.

Collaborators, N.Zein K. Haule M. Capone M. Civelli O Parcollet, S.Y. Savrasov

G.Kotliar Physics Department and Center for Materials Theory Rutgers University.

and CPHT Ecole Polytechnique , France. Pascal Chair de la Fondation de l’Ecole Normale.

Indo-US conference on Novel and Complex MaterialsKolkata (Calcuta-Calcute) –India – 25 -29 October 2005

Outline

Some introductory comments about Dynamical Mean Field Theory.

sp’s systems. The gap problem in semiconductors.

3d’s Electrons. Superconductivity and the Mott transition. High Tc.

5f’s Mott transition across the actinide series, Plutonium and Americium.

4f’s The Kondo collapse in Cerium. Outlook

Effective (DFT-like) single particle spectrumalways consists of delta like peaks

Real excitation spectrumcan be quite different

Concept of Many Body Electronic Structure, Concept of Many Body Electronic Structure, correlated materials. correlated materials.

0[ ( ) ( )] ( , ) 1H k G k

DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). First happy marriage of atomic and band physics.

Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

1( , )

( )k

G k ii i

Extremize a functional of the local spectra or the local self energy.

Resum all the local graphs using a self consistent impurity model

0

1 2

( , ) ( )

( )(cos cos ) ( )(cos .cos ) .......latt k

kx ky kx ky

How good is in practice the local approximation ?

Cellular DMFT [Kotliar et. al. PRL (2001) ] Test in 1d Hubbard model Capone Civelli Sarma Castellani and Kotliar PRB 69,195105 (2004) ]

T/W

Phase diagram of a Hubbard model at int with partial frustration at integer filling. [no density changes!] Evolution of

the Local Spectra as a function of U,and T.

DMFT describes Incoherent and Coherent regimes. M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)

\

Two paths for ab-initio calculation of electronic structure of strongly correlated materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions

DMFT ideas can be used in both cases.

Functional formulation. Chitra and Kotliar

Phys. Rev. B 62, 12715 (2000) and Phys. Rev.B (2001) . 

1 †1( ) ( , ') ( ') ( ) ( ) ( )

2Cx V x x x i x x xff f y y-+ +òò ò

†( ') ( )G x xy y=- < > ( ') ( ) ( ') ( )x x x x Wff ff< >- < >< >=

Ex. Ir>=|R, > Gloc=G(R, R ’) R,R’’

1 10

1 1[ , , , ] [ ] [ ] [ ] [ ] [ , ]

2 2C hartreeG W M P TrLn G M Tr G TrLn V P Tr P W E G W

Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc .

Sum of 2PI graphs[ , ] [ , , 0, 0]EDMFT loc loc nonloc nonlocG W G W G W

One can also view as an approximation to an exact Spectral Density Functional of Gloc and Wloc.

n=1

Order in Perturbation Theory

Order in PT

Range of the clusters

Basis set size. DMFT

GW

r site CDMFT

l=1

l=2

l=lmax

r=1

r=2

n=2

GW+ first vertex correction

Conclusions sp systems. Not well described by single site DMFT.

But very well described by first principles cdmft with relatively small clusters. [2 or 3 coordination spheres]

Weakly correlated materials. Use cheap impurity solvers.

Fast, self consistent way of getting first principles electronic structure without LDA. Good trends for semiconducting gaps and band withds.

Earlier approximations as limiting cases.

Spectral Density Functional

LDA+DMFT dc lda loc[ E ; ]lda dmft U G

loc loc [ ]cdmft loc locW G

xc loc dc[ ] V Elocsdft U G

V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997)

Savrasov Kotliar and Abrahams Nature 410,793 (2001).

The 3d’s A large number of 3d elements and oxides are

well described by single site DMFT but some materials require 2 CDMFT [ Mott Peierls systems ] , or 4 site cuprate [high Tc superconductors].

CDMFT conceptual tool for formulating a dynamical version of the RVB theory,which removes the conceptual problems of earlier versions and accounts naturally for a large body of experimental observations. [ M. Capone, M. Civelli O. Parcollet and G. Kotliar in preparation ] Civelli et. al. PRL (2005).

Conclusions 5f systems at the Mott boundary. Pu and Am.

Single site DMFT describes well, and even predicted, the total energy of phases, the phonon spectra, the photoemission spectra, of Am and Pu.

Qualitative explanation of mysterious phenomena, such as the negative thermal expansion in delta Pu, the volume contraction in the delta-epsilon transition, the anomalous raise in resistivity as one applies pressure to Am metal, etc…..

Conclusions 4f materials Single site DMFT describes well the

photoemission, total energy, and optical spectra of alpha and gamma cerium.

Analysis of the DMFT results favors (and provides a moder reformulation of) the volume collapse transition.

Application to sp systems. Zein Savrasov and Kotliar (2005). What is the range of the self energy in real solids ? 2nd order PT impurity solver.

Gaps of semiconductors

Band Structure of Si from GW+DMFTBand Structure of Si from GW+DMFT

(after Zein, Savrasov, Kotliar, to appear in condmat 2005)

Cutoff Radius R

Ene

rgy,

eV

-14-12-10

-8-6-4-202468

0 0.5 0.83 1 R=oo

Bandwidth

Direct gap

Indirect gap

Convergence of energy bands in Si using real space local self-energy method.

Conclusions sp systems. Not well described by single site DMFT.

But very well described by first principles cdmft with relatively small clusters. [2 or 3 coordination spheres]

Weakly correlated materials. Use cheap impurity solvers.

Fast, self consistent way of getting first principles electronic structure without LDA. Good trends for semiconducting gaps and band wdiths.

Applications to 3d systems,High Temperature Superconductors. P.W. Anderson, Baskaran Zou and Anderson : connection between Mottness and High Tc. RVB approach

† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

RVB phase diagram of the Cuprate Superconductors. Superexchange. The approach to the

Mott insulator renormalizes the kinetic energy Trvb increases.

The proximity to the Mott insulator reduces T goes to zero.

Superconducting dome. Pseudogap evolves continously into the superconducting state.

G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)

Related approach using wave functions:T. M. Rice group. Zhang et. al. Supercond Scie Tech 1, 36 (1998), Gross Joynt and Rice (1986) M.

Randeria N. Trivedi , A. Paramenkanti PRL 87, 217002 (2001)

Competition of AF and SC

AF

AF+SC

SC

8t U<<

or

AF

SC

U /t << 8

Gap and d-wave order parameter vs doping.

Tunneling spectra

.

.

Low energy inset around the Fermi

level

Follow the “normal state” with doping. Evolution of the spectral function at low frequency.

( 0, )vs k A k

If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

: Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k U=16 t

hole doped

K.M. Shen et.al. 2004

2X2 CDMFT

Approaching the Mott transition: CDMFT Picture

Fermi Surface Breakup. Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state!

D wave gapping of the single particle spectra as the Mott transition is approached.

Similar scenario was encountered in previous study of the kappa organics. O Parcollet G. Biroli and G. Kotliar PRL, 92,

226402. (2004) .

Large Doping

Small Doping

The 3d’s A large number of 3d elements and oxides are

well described by single site DMFT but some materials require 2 CDMFT [ Mott Peierls systems ] , or 4 site cuprate [cuprate superconductors].

CDMFT conceptual tool for formulating a dynamical version of the RVB theory,which removes the conceptual problems of earlier versions and accounts naturally for a large body of experimental observations. [ M. Capone, M. Civelli O. Parcollet and G. Kotliar in preparation ] Civelli et. al. PRL (2005).

5f’s Mott Transition in the Actinide Series

Johansen Phil Mag. 30, 469(1974) .

J. Lashley et.al.(2004)

Revisit with modern DMFT tools. Savrasov and Kotliar PRL 84,3760 (2000) ……….

Pu phases: A. Lawson Los Alamos Science 26, (2000)

oNon magnetic LDA underestimates the volume of fcc Pu by 30%, Negative shear modulus. Bouchet et.al.12, 1723 (2000) .oLSDA predicts Pu to be magnetic with a large moment ( ~5 Bohr) . Experimentally Pu is not magnetic. [Lashley et. al. cond-matt 0410634]oTreating f electrons as core overestimates the volume by 30 %

Total Energy as a function of volume for Total Energy as a function of volume for Pu Pu W (ev) vs (a.u. 27.2 ev)

(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.

iw

Zein Savrasov and Kotliar (2005) Following Aryasetiwan et. al. PRB 70

195104. (2004)

Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

DMFT Phonons in fcc DMFT Phonons in fcc -Pu-Pu

  C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.73 4.78

( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)

Double well structure and Pu Qualitative explanation of negative thermal expansion[Lawson, A. C., Roberts J. A., Martinez, B., and Richardson, J. W., Jr. Phil. Mag. B, 82, 1837,(2002). G. Kotliar J.Low Temp. Physvol.126, 1009 27. (2002)]

Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the

volume expands the insulator and contract the metal.

F(T,V)=Fphonons+Finvar

Approach the Mott point from the right Am under Approach the Mott point from the right Am under pressurepressure

Density functional based electronic structure calculations: Non magnetic LDA/GGA predicts volume 50% off. Magnetic GGA corrects most of error in volume but gives m~6B

(Soderlind et.al., PRB 2000). Experimentally, Am has non magnetic f6 ground state with J=0 (7F0)

Experimental Equation of State (after Heathman et.al, PRL 2000)

Mott Transition?“Soft”

“Hard”

Am equation of state. LDA+DMFT.New acceleration technique for solving DMFT equations S. Savrasov K. Haule G. Kotliar cond-mat. 0507552 (2005)

Resistivity of Am under pressure. J. C. Griveau Rebizant Lander and Kotliar PRL 94, 097002 (2005).

Photoemission spectra using Hubbard I solver [Lichtenstein and Katsnelson, PRB 57, 6884,(1998 ), Svane cond-mat 0508311] and Sunca . [Savrasov Haule and Kotliar cond-mat 0507552] Hubbard bands width is determined by multiplet

splittings.

Photomission Spectra of Am under pressure. Sunca. Onset of mixed valence.

Savrasov Haule Kotliar (2005)

Conclusions 5f systems at the Mott boundary. Pu and Am.

Single site DMFT describes well, and even predicted, the total energy of phases, the phonon spectra, the photoemission spectra, of Am and Pu.

Qualitative explanation of mysterious phenomena, such as the negative thermal expansion in delta Pu, the volume contraction in the delta-epsilon transition, the anomalous raise in resistivity as one applies pressure to Am metal, etc…..

Conclusion CDMFT, method under very active

development. But there is now a clear formulation (and to large extent implementation) as a fully self consistent, controlled many body approach to solids.

It gives good quantitative results for total energies, phonon and photoemission spectra, and transport of materials using elements from all over the periodic table.

Helpful in developing intuition and qualitative insights in correlated electron materials.

Review, G. Kotliar S. Savrasov K. Haule O. Parcollet V. Udovenko and C. Marianetti, submitted to RMP.

Software , and modern programming tools for development and implementation of realistic DMFT http://dmftreview.rutgers.edu