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

Cellular-DMFT approach to the electronic
structure of correlated 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 Materials
Kolkata (Calcuta-Calcute) –India – 25 -29 October 2005
Outline
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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 g a Cerium.
Outlook
Concept of Many Body Electronic Structure,
correlated materials.
Effective (DFT-like) single particle spectrum
always consists of delta like peaks
Real excitation spectrum
can be quite different
[  H 0 (k )  ()]G(k , )  1
DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479
(1992). First happy marriage of atomic and band physics.
1
G (k , i ) 
i   k  (i )
Extremize a functional of the local spectra or the local self energy.
Resum all the local graphs using a self consistent impurity model
Reviews: A. Georges G. Kotliar W. Krauth and M.
Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and Dieter
Vollhardt Physics Today 57,(2004)
How good is in practice the local approximation ?
 latt ( k ,  )
 0 ( ) 
1 ( )(cos kx  cos ky )   2 ( )(cos kx.cos ky )  .......
Cellular DMFT [Kotliar
et. al. PRL (2001) ] Test
in 1d Hubbard model
Capone Civelli Sarma
Castellani and Kotliar
PRB 69,195105 (2004) ]
DMFT describes Incoherent and Coherent regimes. M. Rozenberg et. al. Phys. Rev. Lett. 75, 105
(1995)
\
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.
Two paths for ab-initio calculation
of electronic structure of strongly
correlated materials
Crystal structure +Atomic
positions
Model Hamiltonian
Correlation Functions Total
Energies etc.
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
f ( x)VC - 1 ( x, x ')f ( x ') +
ò
ò
2
ò if ( x)y
†
( x )y ( x )
< f ( x ')f ( x ) > - < f ( x ') > < f ( x ) > = W
G = - < y ( x ') y † ( x ) >
1
1
[G,W , M , P ]  TrLn[G0 1  M ]  Tr[G ]  TrLn[VC1  P ]  Tr[ P ]W  Ehartree  [G,W ]
2
2
Introduce Notion of Local Greens functions, Wloc, Gloc
G=Gloc+Gnonloc
.
Ex. Ir>=|R, r> Gloc=G(R r, R r’) dR,R’
[G,W ]
 EDMFT [Gloc ,Wloc , Gnonloc  0,Wnonloc  0]
One can also view

Sum of 2PI graphs
as an approximation to an exact Spectral Density
Functional of Gloc and Wloc.
Order in Perturbation Theory
n=1
Basis
set size.
n=2
Order in PT
l=1
r=1
l=2
r=2
l=lmax
Range of the
clusters
DMFT
r site CDMFT
GW
GW+ first vertex
correction
Conclusions sp systems.
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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.
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Earlier approximations as limiting cases.
 [W  Gloc  ]
cdmft
loc
loc
 t[U G
loc
r  VE]
Spectral Density sdf
loc
loc
xc
dc
Functional
Savrasov Kotliar and Abrahams Nature 410,793 (2001).
LDA+DMFT
lda  dmft[U Edc r lda ; Gloc]
V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and
G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997)
The 3d’s
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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.
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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
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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+DMFT
Energy, eV
Convergence of energy bands in Si using real space local self-energy method.
8
6
4
2
0
-2
-4
-6
-8
-10
-12
-14
Bandwidth
Direct gap
Indirect gap
0
0.5
0.83
1
R=oo
Cutoff Radius R
(after Zein, Savrasov, Kotliar, to appear in condmat 2005)
Conclusions sp systems.
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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


i , j  ,
(tij  d ij )(ci† c j  c†j ci )  U  nini
i
RVB phase diagram of the
Cuprate Superconductors.
Superexchange. G. Kotliar and J. Liu Phys.Rev. B
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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.
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
U /t << 8
8t < < U
or
SC
AF
AF
SC
AF+SC
d
d
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.
A(  0, k )vs k
Ek=t(k)+Re( k ,   0)  
g k = Im( k ,   0)
A( k ,   0) 
gk
g k 2  Ek 2
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.
:
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
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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
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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) .
Revisit with modern DMFT tools. Savrasov and Kotliar
PRL 84,3760 (2000) ……….
J. Lashley et.al.(2004)
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 d Pu to be magnetic with a large moment ( ~5
Bohr) . Experimentally Pu is not magnetic. [Lashley et. al. condmatt 0410634]
oTreating f electrons as core overestimates the volume by 30 %
Total Energy as a function of volume for Pu
W
(ev) vs iw (a.u. 27.2 ev)
(Savrasov, Kotliar, Abrahams, Nature ( 2001)
Non magnetic correlated state of fcc Pu.
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 d-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 d 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)]
F(T,V)=Fphonons
+Finvar
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.
Approach the Mott point from the right Am under
pressureExperimental Equation of State (after Heathman et.al, PRL 2000)
“Soft”
Mott Transition?
“Hard”
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~6B
(Soderlind et.al., PRB 2000).
 Experimentally, Am has non magnetic f6 ground state with
J=0 (7F0)
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.
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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…..
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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