ASIAN-15 November 2012 1
Electronic Structure Theory
Yesterday, Today and Tomorrow
Richard M. Martin
Department of Physics
University of Illinois at Urbana-Champaign
Department of Applied Physics
Stanford University
Added slide #`17 not in talk:
Includes the good points made by Profs. Marzari and Terakura
ASIAN-15 November 2012 2
Thanks!
For the opportunity to be at this excellent workshop!
Excellent:
Organization – Thank you Committee, Mei-Yin, others
Talks: Content and presentation
Posters: Content and presentation
and many young people!
Only one other workshop like this – Trieste – every two years
(next time, January 2013 – hope many of you will be there!)
Congratulations!
ASIAN-15 November 2012 3
You (we) are working in an
important field
You are the researchers of today and tomorrow
You are the teachers of the students who will make
the advances in the future
It is our opportunity and responsibility to adhere to basic
principles – new problems will emerge, new capabilities,
new knowledge – but basic principles are the guide for
good science
Our field has great impact – more and more it will be an
essential part of research in many fields
Among the greatest intellectual challenges in physics
A bit of history
ASIAN-15 November 2012 4
ASIAN-15 November 2012 5
A long way in 90 years
• L. de Broglie –
Nature 112, 540 (1923).
• E. Schrödinger –
1925, …. • Pauli exclusion Principle - 1925
• Fermi statistics - 1926
• Thomas-Fermi approximation – 1927
• Dirac determinant (Slater 1929)
• First density functional – Dirac – 1928
• Dirac equation – relativistic quantum mechanics - 1928
1900 1920 1940 1960 1980 2000 2020
ASIAN-15 November 2012 6
Quantum Mechanics
Understanding ----- Technology
• Bloch theorem – 1928
• Wigner- Seitz – Quantitative calculation for Na - 1935
• Slater - Bands of Na - 1934 (proposal of APW in 1937)
• Shockly – Bands of NaCl (1935)
• Bardeen - Fermi surface of a metal - 1935
• Invention of the Transistor – 1940’s
– Bardeen – student of Wigner
– Shockley – student of Slater
1900 1920 1940 1960 1980 2000 2020
• Wilson - Implications of band theory - Insulators/metals –1931
• First understanding of semiconductors – 1930’s
ASIAN-15 November 2012 7
Development of the Basic Methods I
• The basic methods
– Slater – Augmented Plane Waves (APW) - 1937
• Not used in practice until 1950’s, 1960’s – electronic computers
– Herring – Orthogonalized Plane Waves (OPW) – 1940
• First realistic bands of a semiconductor – Ge – Herrman, Callaway (1953)
– Hellman, Fermi,– Pseudopotentials - 1930’s
….
1900 1920 1940 1960 1980 2000 2020
• Independent electron approximations
– Assume each electron moves in some effective potential
• Of course there is more recent work
– Pseudopotentials
Kleinman, Phillips – 1950’s, Hamann, Vanderbilt, others – 1980’s
– Korringa (1947) Kohn and Rostocker (1954) KKR Green’s function method
– Andersen – Linearized Muffin Tin Orbitals (LMTO) – 1975
• The full potential “L” methods – LAPW, ….
ASIAN-15 November 2012 8
Development of the Basic Methods II
• The basic advances in many-body theory – 1950’s - 60’s
– Landau – Feyman . . .
– Gell-Mann – Breuckner – Gutzwiller - Hubbard – . . . .
– Bohm – Pines – Baym – Kadanoff – Hedin – . . . .
• Applied mainly to the electron gas
1900 1920 1940 1960 1980 2000 2020
• Many-body methods to treat electron-electron interactions
– Recognized as the key issue since the early days of quantum mechanics
• Hylleraas – numerically exact solution of energy of H2 - 1929
• Bethe – Exact solutions for one-dimensional problems
• Wigner-Seitz – Quantitative study in solid
ASIAN-15 November 2012 9
Wigner- Seitz -- 1933-5 Quantitative calculation for Na
1900 1920 1940 1960 1980 2000 2020
Energy of lowest state
(bottom of the valence band)
Hartree-Fock
(kinetic energy + exchange)
Correlation energy
“This energy will be called the correlation energy”
“The calculation of a wavefunction took about two afternoons,
and five wavefunctions were calculated in the whole, . . ..”
ASIAN-15 November 2012 10
1964-5
• The GW approximation
– Hedin – . .
– Building upon previous work
– “Nothing new” .
.
• Quantum Monte Carlo
– McMillan – 1964
1900 1920 1940 1960 1980 2000 2020
• Three major developments underlie most of our work today
• Density functional theory
• Hohenberg and Kohn, Kohn-Sham
• “Useful”
Ceperely-Alder
Car-Parrinello
Hybertsen-Louie
1985-7
Many-Body Theory I
ASIAN-15 November 2012
• DFT – theory of the many-body interacting-electron system
• Kohn-Sham method - only the ground state
• Extremely successful!
• The basis of much work at this workshop
→ HKS at self-consistent solution
• Provides no way to calculate excited states
• But Time-dependent DFT gives certain excitations
• Stefano Baroni’s talk
11
Many-Body Theory II
ASIAN-15 November 2012
• GW approximation by Hedin
• 1960’s Applied to electron gas
• 1980’s – today (and tomorrow!) -- real materials
• Steve Louie’s talk
• Key idea – screened Coulomb interaction – W(w)
W
G
S =
• Beyond GW – Strong interactions – higher order diagrams
12
13
Many-Body “GW” Calculations • Remarkable agreement with experiment!
Faleev and
van Schilfgaarde
2009
Th
eore
tica
l G
ap
Experimental Gap
13 ASIAN-15 November 2012
14
QMC
• Apologies!
• Benchmarks -- excellent work
• Sign problem --- great intellectual challenge!
14 ASIAN-15 November 2012
15
Electronic structure methods have come a long way . . . Questions:
• What would you do if you want to calculate:
Magnetism -- for example Fe
Transition temperature Tc Magnetic behavior at the center of the earth
Metal-insulator transitions
Possible devices, mantel of the earth
“Strongly correlated” systems
Not well-definedterm But lets look a few cases that show large effects of electron-electron interactions
ASIAN-15 November 2012
While keeping the advantages of methods like DFT and “GW”
Approximations that are remarkably accurate for large classes of materials Better yet: deepening our understanding
16
Ordered magnetization
Magnetism in Fe, Ni
Curie-Weiss law for
thermally disordered spins
χ ~ 1/(T - Tc) χ-1 ~ T - Tc
Magnetization at T=0
given well by DFT
ASIAN-15 November 2012
Both delocalized band-like and localized atom-like behavior
16
17
Magnetism in Fe, Ni
There various ways to approach
the problem
ASIAN-15 November 2012
Added slide not in talk:
Includes the good points made by Profs. Marzari and Terakura
17
A. Use the fact that DFT works rather well for the ground
state moment even though there are errors for electron bands
B. Then one can treat the high T as a disordered array of spins following the approach on
the following slides.
A. Carry out a many-body electron structure calculation that provides a better description
of the bands. Like the GW method it uses Green’s functions – see following slides
B. Treat both ordered and disordered states
C. Much more difficult than the method above. May be less accurate because
of computational limitations.. But it carries over to cases like Ce where electronic
correlations can be much more important. The same method for many problems.
D. An example is the work of Zhang, et al, PRB 84, 140411 (2011).
1
2
D. An example is the work of Liechtenstein, PRL 87, 067205 (2001)
C. Main advantage is the accuracy of DFT for ground state properties. Can include
phonons. But the methods do not work so well for strongly correlated cases
ASIAN-15 November 2012 18
Transition elements and Magnetism
Magnetic moments form on atomic sites – disordered at
high temperature T>Tc
Moments order at a critical temperature to form magnetic
order T<Tc
Ferromagnet
Quandary:
Should this be thought of as a Heisenberg Spin Model –
Or as a periodic system with average spin density as in DFT
No average spin
19
Transition elements and Magnetism Kohn-Sham DFT is a theory of in terms the average density
(spin density for magnetic materials}
Spin dependent DFT
(approximations)
do very well
19 ASIAN-15 November 2012
20
Transition elements and Magnetism Kohn-Sham DFT is a theory of in terms the average density
(spin density for magnetic materials}
No average spin
Up – down = 0 at every point in space
20
Fails to describe fluctuating local moments for T> Tc
ASIAN-15 November 2012
Bear with me
The next three slides are rather dense –
very short summary of a method
But they contain very important ideas
ASIAN-15 November 2012 21
22
Dynamical Mean Field Theory
DMFT
• Classical mean field for a magnet-- Weiss 1907
– Average effective field on a site due to its neighbors
– Thermal fluctuations
• Quantum dynamical mean field
– Metzner, Vollhardt 1989; Georges, Kotliar 1992
– Average quantum field on a site due to its neighbors
– Quantum fluctuations require dynamical mean field even
at T=0 -- Frequency dependent coupling of a site to its
surroundings
No average spin
Fluctuating local moments
22 ASIAN-15 November 2012
23
Dynamical Mean Field Theory
DMFT
• Mean field means neglect of correlation between neighbors
No average spin
Fluctuating local moments
23
Atom surrounded by average
(uncorrelated) neighbors
“Embedded”
S localized to one site –
k-independent
• Solve accurately -- includes diagrams to all orders
• Improved theory – beyond mean field – correlation between sites ASIAN-15 November 2012
24
Dynamical Mean Field Theory
DMFT
• This was a lot of detail!
No average spin
Fluctuating local moments
24
• What is the BIG PICTURE?
• Methods such as Kohn-Sham DFT often work well for ordered
states at T=0 – very hard to generalize to non-zero temperature
Exc Fxc(T) (free energy)
• The same applies to GW – all applications (almost) work at T=0
in ordered states – does not describe fluctuations
• DMFT is the opposite! Works best at high T - includes
fluctuations on a site but ignores correlation between sites
ASIAN-15 November 2012
25
Big Picture
• DMFT is an approximation (more than one approximation)
25
• But is a start toward a new capability
• Strong local interactions --- intra-atomic interactions between
electrons on the same site
• Can include temperature – most natural at high T
• Brings together statistical mechanics with modern electronic
structure theory
• NOT hard to understand the general principles
ASIAN-15 November 2012
26
Note this is scaled!
Actually Tc to high by
~ factor of 2 in Fe
Ordered Magnetization
DMFT in practice
Curie-Weiss law for
thermally disordered spins
χ ~ 1/(T - Tc) χ-1 ~ T - Tc
Lichtenstein, et al.
26 ASIAN-15 November 2012
ASIAN-15 November 2012 27
“Strongly Correlated” Systems
• Atoms with localized electronic states
• Strong local intra-atomic interactions carry over to the solid – Transition metals -- Rare earths
– Open Shells
– Magnetism
– Metal - insulator transitions, Hi-Tc materials
– Catalytic centers
– Transition metal centers in Biological molecules
. . .
ASIAN-15 November 2012 28
Periodic Table
Ce 58
Pr 59
Nd 60
Pm 61
Sm 62
Eu 63
Gd 64
Tb 65
Dy 66
Ho 67
Er 68
Tm 69
Yb 70
Th 90
Pa 91
U 92
Np 93
Pu 94
Am 95
Cm 96
Bk 97
Cf 98
Es 99
Fm 100
Md 101
No 102
Lu 71
Lw 103
Sc 21
Ti 22
V 23
Cr 24
Mn 25
Fe 26
Co 27
Ni 28
Cu 29
Zn 30
Ga 31
Ca 20
K 19
Y 39
Zr 40
Nb 41
Mo 42
Tc 43
Ru 44
Rh 45
Pd 46
Ag 47
Cd 48
In 49
Sr 38
Rb 37
La 57
Hf 72
Ta 73
W 74
Re 75
Os 76
Ir 77
Pt 78
Au 79
Hg 80
Th 81
Ba 56
Cs 55
H 1
C 6
N 7
O 8
F 9
Ne 10
B 5
Be 4
Li 3
Si 14
P 15
S 16
Cl 17
Ar 18
Al 13
Mg 12
Na 11
He 2
Ge 32
As 33
Se 34
Br 35
Kr 36
Sn 50
Sb 51
Te 52
I 53
Xe 54
Pb 82
Bi 83
Po 84
At 85
Rn 86
Ac 89
Ra 88
Fr 87
Transition metals
Lanthanides - Actinides
Localized, Magnetic
Delocalized, Superconducting
Periodic Table of transition elements
(arranged delocalized ---- localized)
Anomalous on the boundary
Original due to J. L. Smith
29 ASIAN-15 November 2012
Transition Elements
Also other localized states!
Impurities, defects
Nanostrctures, ……
30
Transition elements and Magnetism Kohn-Sham DFT is a theory of in terms the average density
(spin density for magnetic materials}
Spin dependent DFT
(approximations}
often get the wrong
ground state
Not acceptable
What to do?
30 ASIAN-15 November 2012
ASIAN-15 November 2012 31
DFT + U ---- SIC
• Identify localized state
• Add extra interaction on the localized state
– Depends upon occupation
– U ni nj, i ≠ j
• If orbital j is full, it costs and extra energy U to add an
electron in orbital i
Improves the
approximation
in spin dependent DFT
Often great improvement!
(Marzari, other talk)
32
“Mott Insulator” NiO
Anti-ferromagnetic
order T=0
(Rodl, et al.)
Experiment ~ same
below and above T
DMFT - Parameters
from LDA
(Kunes, et al.)
ASIAN-15 November 2012
33
Metal-insulator transition – V2O3
Ordered Phase
Can explain MI transition Qualitatively
MI transition
Complicated lattice – Mechanisms NOT clear
ASIAN-15 November 2012
34
Ce – anomalous rare earth
Volume collapse – spin fluctuations
g phase
normal rare earth
normal volume
local moments
a phase
anomalous rare earth
Volume collapsed – 15%
“non-magnetic”
fcc
structure
Fermi liquid at low T
Low T critical point
Quanum liquid – gas
Alloy with Tl
Two critical points
No magnetic order
P
T
0
Phase diagram
ASIAN-15 November 2012
Critical point like Classical liquid-gas
ASIAN-15 November 2012 35
Ce – anomalous rare earth
Volume collapse – spin fluctuations
Held, Scalletar, McMahan 2003
g phase
“normal rare earth”
normal volume
local moments
a phase
“anomalous rare earth”
Volume collapsed – 15%
non-magnetic
Experiment
Circle symbols
Non-magnetic
but local moments
(Like Kondo effect)
U
DFT+DMFT -- CeIrIn5 Heavy Fermion Material
Almost the same as DFT
with 4f-states removed
Similar to DFT
including 4-f states
Scaled by 1/100 !
T = 300 K T = 10 K
Choi, H. C., Min, B. I., Shim, J. H., Haule, K. and Kotliar, G. (2011).
Temperature-dependent fermi surface evolution in heavy fermion ceirin5. Cond-
Mat 2011, arXiv:1105.2402v1.
LDA+DMFT, Single-site, CTMC solver
36 ASIAN-15 November 2012
ASIAN-15 November 2012 37
Relation of the methods
• DFT - Kohn-Sham – potential VKS(r ) = VHartree(r) + Vxc(r)
– Static Vxc[n(r)] - functional of the density n(r)
• GW – dynamic self energy S(r,r’,w)
– Calculate with perturbation theory
– Often start with DFT (or extensions of DFT)
– Green’s function G(r,r’,E) = [G0 -1 – S(w)] -1
• DMFT – local dynamical Sii(w)
– Treats local on-site correlations more accurately than GW
– DMFT – a set of techniques – must combine with another method for quantitative calculations
• DFT – rather ad hoc – but captures essential points -- shows the potential
• GW and beyond – Fully first principles methods for strongly correlated systems
ASIAN-15 November 2012 38
Conclusions
Yesterday, Today, and Tomorrow • A long way in ~90 years!
• Electronic Structure is the quintessential many-body problem
of quantum mechanics
– Interacting electrons → real materials and phenomena
• Density functional theory
– Approximate forms have proved to be very successful
– BUT approximations have shortcomings and failures!
• Future – Combinations of DFT, QMC, Many-Body
Perturbation theory, Analytic Theory
– Opportunities and challenges • Strongly correlated systems
• New materials, new phases of matter
• Nanoscale - Biological systems
• Bridging the length and time scales is critical issue
• Future – New discoveries not yet imagined!
ASIAN-15 November 2012 39
Conclusions
Yesterday, Today, and Tomorrow
You are the researchers of today and tomorrow
You are the teachers of the students who will make
the advances in the future
It is our opportunity and responsibility to adhere to basic
principles – new problems will emerge, new capabilities,
new knowledge – but basic principles are the guide for
good science
Our field has great impact – more and more it will be an
essential part of research in many fields
Among the greatest intellectual challenges in physics