electronic structure: yesterday, today and tomorrow

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


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!


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



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


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


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, ….


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


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, . . ..”


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


• 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!


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


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


Both delocalized band-like and localized atom-like behavior

16

17

Magnetism in Fe, Ni

There various ways to approach

the problem


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


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


20



No average spin

Up – down = 0 at every point in space

20

Fails to describe fluctuating local moments for T> Tc


Bear with me

The next three slides are rather dense –

very short summary of a method

But they contain very important ideas


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


23


DMFT

• Mean field means neglect of correlation between neighbors

No average spin


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


DMFT

• This was a lot of detail!

No average spin


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


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


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.



“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

. . .


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


Transition Elements

Also other localized states!

Impurities, defects

Nanostrctures, ……

30



Spin dependent DFT

(approximations}

often get the wrong

ground state

Not acceptable

What to do?



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.)


33

Metal-insulator transition – V2O3

Ordered Phase

Can explain MI transition Qualitatively

MI transition

Complicated lattice – Mechanisms NOT clear


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


Critical point like Classical liquid-gas


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



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


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!


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

electronic structure: yesterday, today and tomorrow

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