Modeling of Nanostructures and Materials

(1)

Modeling of Nanostructures and Materials

Summer Semester 2014 Lecture

Jacek A. Majewski Faculty of Physics, University of Warsaw

E-mail: Jacek.Majewski@fuw.edu.pl

Jacek A. Majewski

Modeling of Nanostructures and Materials

Lecture 8 – April 14, 2014

e-mail: Jacek.Majewski@fuw.edu.pl

• ! Band Gap energies in DFT

• ! Further Developments of DFT methods

Band Gaps in Solids

The DFT & the GW Method

DFT- The Kohn- Sham Method

W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965) System of interacting

electrons with density

!(!r)

System of non-interacting electrons with the same density

!( !r)

“Real” system “Fictitious” or Kohn-Sham reference system

!!

(_{r )}

!

T [ !] T [ !]

S

!!

_S(

!r )== !!( !r )

E[ !! ] == d!r!! !!

_ext

( !r )!!( !r ) ⁺⁺ ^T

^S

[ !! ] ++U [ !! ] ++ E

_x

[ !! ] ++ E

_c

[ !! ]

!! ( !

r ) == !! *

_i

( ! r )

i==1

!

N

! ^!!

ⁱ

⁽ ^{r )} ^!

T

_s

[ !! ] == !! ! 2m

²

# #

_i==1^N

"" d"r ^!!

ⁱ^*

⁽ ^{!r )} ^$ ^$ ^!

²

^!!

ⁱ

⁽ ^{!r )}

unknown!!!

E_x[

!!

] == !! 1

2 d_rd

!

""""

#

i

#

^r'

^! ^!!

ⁱ^*⁽^{r )}

^! ^!!

^j_|⁽_{r !!}

^!

^{r )}

^! ^!! ^!

_{r' |}^*^j⁽^r')

^!

#

j

$$ #

%% && ''

(( )) !!

_i(_r')

!

(2)

!! !

²

2m

"

²

++ !!

_ext

( !r )++!!

_H

( !r )++!!

_x

( !r )++!!

_c

( !r )

##

$$ %% &&

'' ((!!

i

( !r ) == !!

_i

""

_i

( !r )

The Kohn- Sham Method –

The Kohn-Sham Equations

!! ( !

r ) == !! *

_i

( ! r )

i==1

!

N

! ^!!

ⁱ

⁽ ^! ^{r )}

!E [ "]x

!" ^c

!E [ "]

!"

!!_H(! r ) ==!!U

!!"" ⁼⁼ ^d r'!

!!

_|r ""^!!^!⁽^!^r')! r' |

!!_ext(!

r ) == ""e² Z_s

|! r ""!

##_s""! R_n|

$

s,n

$

Schrödinger-like equations with local potential

These equation are nonlinear and must be solved iteratively (self-consistently)

Exact Exchange Method (EXX)

[Optimized Effective Potential (OEPx)]

x x i KS

x i i KS

E E

==

^##

==

" " ^##

!!

^##$$

!!

^##%%

%% ##&& ##$$ ##%% ##&&

KS i i i

T ++

ˆ ==

( !! ) "" ## ""

E

x Ex

!!

!!""

is the first functional derivative of First order perturbation theory determines exactly

Apply chain rule

Ex from

Perturbation theory

Solution:

!!""_i(! r )

!!""_KS(!

r ')== !!^*_j(! r ')!!_i(!

r ')

!!_i!!!!_j

j""ì

#

^""i(!

r )

i

KS i !!""KS K

!!## !!##

!!$$ ==!!"" !!$$ %% !!""^KS K ¹

!!## == ^$$

!!

x

M. Städele et al., Phys. Rev. B 59, 10031 (1999).

LDA & GGA Approximations

E

_xc^GGA

[!!] == d!r !! ^f

^xc

(!!(!r),!!!!(!r))

Generalized Gradient Approximation (GGA)

J. P. Perdew & Y. Wang, Phys. Rev. B 33, 8800 (1986)

f

xc-!constructed to fulfill maximal number of “summation rules”

Exchange-correlation potential can be calculated very easily, since explicit

dependence of E_xc on the density is known. !!

xc

!! E

xc

""

== !!##

Local Density Approximation (LDA):

E

the density is treated locally as constant _xc^LDA

[!!] == d!r !! !!(!r)""

_xc^hom

(!!(!r))

xc

==

x

++

c

hom hom hom

!! !! !!

!(r)

r1

! = !₁ ( )r₁ r

! ₁

The Kohn - Sham Method – One particle energies

HOMO LUMO

! ₁

! ₂

! ^N N

! ₊₊₁ unoccupied occupied

The occupied states are used to calculate one particle density (Aufbau principle) and the total energy

E

^KS_GAP

Kohn-Sham Gap

(3)

The Kohn- Sham Method – The Total Energy

E [ !! ] == !! !2m²

# #

_i==1^N

""

d"r^!!ⁱ^*⁽^{!r )}^$^$^!²^!!ⁱ⁽!r )++U[ !! ] ++ E_x[ !! ] ++ E_c[ !! ] ++ d!r!!

""

_ext(!r )!!⁽^{!r )}

E == !!_i

i==1

"

N

"

^{## 1}₂

%%%%

^d^{r d}^! ^!^r'^$$_|⁽_{r ##}^!^!^{r )}^$$_{r' |}^!⁽^r')^! ⁺⁺^E^x^[^$$^{] ++ E}^c^[^$$^{] ## d}

%%

^r(^! ^&&^x⁽^{r ) ++}^! ^&&^c⁽^!^{r ))}^$$⁽^{r )}^!

"$$$$$$$$$$$$$$$$$$$$$$$$$$$$#$$$$$$$$$$$$$$$$$$$$$$$$$$$$% so-called double counting correction

Sum of the one-particle Kohn-Sham energies

Energy of the reference system differs from the energy of ‘real’ system

!!_i

i==1

!

N

!

⁼⁼ ^""ⁱ| ""!² 2m

"

#

#²++!!_KS(! r )|!!_i

i==1

!

N

!

⁼⁼^T^S^[^""^{] ++ d}

$$

^r^!^!!^KS⁽^{r )}^! ^!!⁽^!^{r )}

The Kohn- Sham Method –

Physical meaning of one particle energies

The Kohn-Sham orbital energy of the highest occupied level is equal to the minus of the ionization energy

,

! max == == !! µ I

!!( !r ) == f

_i

!

i

! ^!!

ⁱ^*

⁽ ^{!r )!!}

ⁱ

⁽ ^{!r )}

i i

E ! f

!! ==

!!

Extension to non-integer occupation numbers

0 f 1 !!

i

!!

Janak theorem (1978)

!!(!

r ) == !!(E_F!!!!_n(! k )

n,!

!

k

!

⁾^!!n*(!

k,! r )!!_n(!

k,! r )

Ca

Energy [eV]

Wave vector !

" ! "#$ % # $ !

! 10

0 2 4 6 8

!

"

Band structure of metals and semiconductors

Band structure of simple metal (Calcium) Fermi energy

In metals

^:

Fermi energy lies in a band Fermi energy must be calculated in each iteration of the self-consistent procedure

N == d³!_r

!

""

₀ ^##⁽^{r ) $}^! ^{$ E}^F

Band structure of metals and semiconductors

GAP GAP GAP

Empty bands

Empty bands Empty bands

Full bands Full bands

Full bands

Wave vector Wave vector

Wave vector

Energy [eV]

Semiconductors Semi-metal

Si Ge

Silicon Germanium Alpha-Tin

Sn

!! ""

In an ideal pure semiconductors at 0 K there are

fully occupied valence bands & completely empty conduction bands separated by the energy gap

Fundamental band gap = Energy difference between the lowest unoccupied state and the highest occupied state Fundamental energy gap can be direct (Ge) or indirect (Si)

Fermi level lies in the energy gap

Insulator – like semiconductor with very flat bands and huge energy gap

(4)

LDA Band Structure Calculations in Semiconductors

-2 -4 0

-6 -8 -10 -12 -14

Energy [eV]

GaAs

Experiment

Pseudopotential theory

Wave vector

!

" # #

L K X

$

1

X₁ X5 X3

#

1

#

1

#

15

#

15

L₁ L₃

L₁

!

1

!

1

! +

3

!

5

$

2

$

1

$

₁

"

1

"

3

"

1

$

Valence bands for GaAs as determined from angle-resolved photoemission

experiments and pseudo-potential theory LDA gives very good description of the occupied s-p valence bands (4s & 4p) in semiconductors

PP LMTO LAPW EXP.

-12.84 -12.85 -12.78 -13.1 X₁ -10.36 -10.49 -10.47 -10.75 X₃ -6.83 -7.06 -6.72 -6.70 X₅ -2.67 -2.83 -2.60 -2.80 L_1p -6.66 -6.94 -6.53 -6.70 Energies [eV] in symmetry points

Various methods of solving Kohn-Sham equations give very similar results

!!1s

Probing the Electronic Structure by Photoemission (ARPES)

Measurements of kinetic energy (and angle) of photo-emitted electrons give valence band energies

ARPES - Energetics of the photoemission process

(i) Optical excitation of the electron in the bulk.

(ii) Travel of the excited electron to the surface.

(iii) Escape of the photoelectron into vacuum.

One-step vs.

Three-step model

E_GAP= E_LUMO - E_HOMO Too small by factor of 2

Si

Band structure of diamond silicon

Energy [eV]

Wave vector

E_GAP

Kohn-Sham Method in LDA (GGA) Approximation Energy Gap of Silicon

KSGap

E == !!

_cbb^KS

"" !!

_vbt^KS

KSGap

E ==!!_N^KS₊₊₁( )N ""!!_N^KS( )N

Kohn-Sham gap

For all semiconductors and insulators, LDA (GGA) give energy gaps that are 40%-70% of experimental gaps

Is the Kohn-Sham gap generally wrong, for description of one particle excitations ? Does the error is caused by the approximation of the functionals ?

“The band gap problem”

Relation of the Kohn-Sham gap to the quasi-particle energy (change of system energy caused by adding a particle) ?

(5)

Calc. band gaps [eV]

Exp. band gaps [eV]

Fundamental Band Gaps

0 1 2 3 4 5 6 0

1 2 3 4 5

6

LDA

EXX

GeSi

GaAs AlAs SiC

GaN

AlN C

Fundamental band gaps in semiconductors:

Local Density Approximation & Exact Exchange

EXX Method leads to Kohn-Sham gaps that agree very well with experiment

Large part of the error in the fundamental gaps is connected to the approximated functionals (LDA, GGA)

P. Rinke et al.

New J. Phys. 7, 126 (2005)

Band Gap of Semiconductors in Exact-Exchange OEP

0 10 20 30 40

0 2 4 6 8 10

EXX LDA expt.

GaAs

! "

2

( )

h [eV]"

L ! X K,U !

GaAs

-14-12 -10-8-6-4-20246

EXX energy [eV] LDA

!Wave vector !

Band structure

Dielectric function

GaAs: electron effective mass: LDA = 0.03m , EXP = 0.07m , EXX = 0.10m ₀ ₀ ₀

Band structure of semiconductors:

Local Density Approximation & Exact Exchange Method

The most pronounced difference between band structure calculated with LDA and EXX methods – rigid shift of the conduction bands Concerning energy differences – LDA should give valuable predictions

theory experiment

biaxial strain e [ % ]_||

energy gap [eV]

!"!

#!"$ !"$

%"&'

%"'!

%"''

B C

GaN

CB VB

A B C

!

top A

Chichibu et al.

APL 68, 3766 (1996) Dingle et al.

PRB 4, 1211 (1971)

LDA calculations in wurtzite GaN:

Change of A, B, C exciton energy gaps with biaxial strain

Gives a reference scale to determine strain

in an epitaxially grown sample

(6)

LDA calculation in wurtzite nitrides:

Energy gap deformation potentials for biaxial strain and hydrostatic pressure

Very good agreement with experiment

GaN

-8.0 -22.2 -9.5 -6.1

-15.8 -8.0 dE(C exciton)

de

=

⁰

dE(A exciton) dln V

V0

-8.2

-17.2 -8.0

theory expt.

AlN

dE(A exciton) de

=

⁰

(all data in eV) -8.2

-8.4

Ionisation Potential and Electron Affinity

Ionisation potential ^:

minimal energy to remove an electron

I = E(N ! 1) ! E(N)

Electron affinity:

minimal energy to add an electron

A = E(N) ! E(N + 1)

Ionisation Potential for Atoms

E. Engel in A Primer in DFT, Springer 2003

Ionisation Potential vs. Kohn-Sham HOMO

Kohn-Sham: eigenvalue of the highest occupied

Kohn-Sham level I

_KS

= !"

_N

(N)

For exact density functional – I

_KS

= I = E(N-1) – E(N)

How this relation is fulfilled for approximate functionals ?

Much better for EXX than for LDA !

(7)

EXX versus LDA: Zn and Ga Atoms

Correct asymptotic decay of potential in DFT-EXX

Ionisation Potential - Small Molecules

S. Hamel et al., J. Chem. Phys.

116, 8276 (2002

) Comparison of Kohn-Sham HOMO with experimental values of ionization potential

Very good agreement for EXX (OEPx) !

Band Gaps in Solids

The DFT & the GW Method

Band Gap of Semiconductors

Band gap:

E

_gap

= I ! A = E(N + 1) ! 2E(N) + E(N ! 1) For solids, E(N + 1) and E(N ! 1) cannot be reliable computed in DFT, yet !

In Kohn-Sham the highest occupied state is exact

KS

1

N N

! ( N ) == !! I == E( N ) E( N !! !! )

1

KSN N

!

₊₊

( N ++ == !! ) I

₊₊

== E( N ++ !! ) E( N )

1 1

E

gap

== { E( N ++ !! ) E( N )} { E( N ) E( N !! !! !! )}

1

KS KS

gap N N

E == !

₊₊

( N ++ !! ) ! ( N )

(8)

Band Gap of Semiconductors

Discontinuity Kohn-Sham gap

Band Gap of Semiconductors - Discontinuity in V

_xc

After the addition of an electron into the conduction band (right) the xc potential and the whole band-structure shift up by a quantity !xc . R.W. Godby et al., in A Primer in DFT, Springer 2003

The Quasi-particle Concept

Spectral function

Quasi-particle energies in many-particle theory

Energy of quasi-particle = energy of one-particle excitation = Change of system energy caused by adding a particle to the system

Dyson equation

!² 2m

!!

!²!!!!_ext(! r ) !!!!_H(!

r ) ++ E

!!

""

## $$

%%&& G(! r,!

r ';E) !!

!! d³! r '' (!

r ',! r '';E)

!

"" !

^G(^!^r,^{r ';E) ==}^! ^!!⁽^{r !!}^! ^{r ')}^!

-!self- energy operator (!

r ',! r '';E)

!

- one particle Green’s function G(!

r,! r ';E)

Energies of one-particle excitations = poles of

G( ! r, !

r ';E)

can be complex

Real part – energy of the quasiparticle Imaginary part - Life time

( 1) ( )

tot tot

E N++ !!E N

Difference between total energy of a system with N+1 and N particles L. Hedin & S. Lundquist, Solid State Physics 23, 1 (1969)

(9)

Quasi-particle energies in many-particle theory

!!!² 2m

"!

"²++!!_ext(! r ) ++!!_H(!

## r )

$$%% &&

''(( us(! r ) ++ d³!

))

r '

* *

⁽^!^r,^{r ')u}^! ^s⁽^{r ') ==}^! ^!!^s^us(! r )

G(_r,

! !

_{r ';E) ==}

!

s

!

^uE ""^s

⁽ ^!

^{r )u}

!!

_s^s^*

±± ⁽

^{r ')}

^!

i

""

It is possible to introduce one particle functions u_s

Green’s function for a non-interacting system

!

(

!

!r,!r') ==V_x^HF(!r,!r') Self-energy operator

!

- independent on energy Self-energy operator

Quasi-particle energies in many-particle theory

( )E

!

introduce functions ^!^!ⁿ^k^!⁽^{r )}^!

!!!² 2m

"!

"²++!!_ext(!r)++!!_H(!r)

##

$$%% &&

''((!!_n_k^!(!r)++ d

))

³!r'

* *

⁽^!r,!r';En!

k)!!_n_k^!(!r') == E_n_k^!!!_n_k^!(!r) Im(E_n_k^!) Re(E_n_k^!)

If one is interested in energies of excitation and not their lifetimes, one can neglect imaginary part of the self-energy operator _{Im( ) == 0}

! !

There exists series expansion for self-energy operator Take the first term

! !

(^r,^!^!r ';E) == i

2!! ^d""e""i""o⁺⁺G(! r,!

r ';E ""!!)

##

^{W (}^r,^!^!^{r ';}^!^!⁾

G(! r,!

r ';E) == !!_n_k^!(! r )!!_n^*_k^!(!

r ') E !! E_n_k^!++io⁺⁺sgn(E_n_k^!!!µµ)

n!

"

k

"

W (! r,!

r ';!!) == d³! r ''""^##1(

$$

^!^r,^{r ',}^! ^!^!⁾_|_{r ''##}^!^e²^!_{r '|}

Inverse of dielectric function

Screened Coulomb potential Self-consistent solution gives energies of single particle excitations

GW - method

Hedin & Lundqist

GW- method

" We make the problem simpler by considering one electron in an effective potential

" The effective potential is the

Coulomb interaction, V, between

the electron and the average of all the others

" We describe the electron’s motion with the Green’s function, G

The GW Method – Green’s Function

" The electron polarizes the system, making effective electron-hole pairs.

" This screens the Coulomb interaction.

" This means that the electron now interacts with a screened

coulomb interaction, W

" In order to make the model better we model the excited states and their interactions.

The GW Method –

Screened Coulomb Interaction

(10)

GW Approximation - Interacting Quasiparticles GW Approximation - Formalism

Quasi-particle energies in many-particle theory Connection to Kohn-Sham energies

!

!_n_k^!(! r ) !!!!_n^KS_k^!(! Self-consistent calculations show that r )

E_n_k^!==!!_n^KS_k^!++ !!_n^KS_k^!(!

r ) |

! !

(E_n_k^!)^""^!!xcKS|!!_n^KS_k^!(! r )

Z_n_k^!== 1 !!""

# #

(E)

""E

E==E_n_k^!

$$

%%

&&

'' ((

))))

!!1

So-called renormalization

E_n_k^!

== !!

_n^KS_k^!

++ ""

_n^KS_k^!

(!r)| (E # #

_n_k^!

) ^$$ ^%%

xcKS

|""

_n^KS_k^!

(!r) Z

_n_k^! Relation between quasi-particle and Kohn-Sham energies

Kohn-Sham orbitals Kohn-Sham energies

GW Approximation for Solids

General improvement of the energy gaps in comparison to DFT-LDA

Aulbur et al.

Solid State Phys. 54 (2000)

(11)

Fundamental band gaps in II-VI semiconductors:

LDA, EXX, and GW calculations

Exp.

2.5

GW EXX

LDA

2.0 2.1 0.8

CdS

Exp.

GW

EXX

LDA

1.7

0.2 1.4 1.3

CdSe

3.0 3.3

Exp.

3.8 1.7

LDA

GW EXX

ZnS

Exp.

GW

EXX

LDA

1.6

0.2 1.3 1.2

CdTe

Energy gaps in eV

LDA gives dramatically too small band gaps LDA + GW – large corrections to LDA gaps - corrected gaps are of order of EXX gaps

EXX + GW – very small correction of order (0.1 - 0.2 eV)

Still some work to do !

P. Rinke et al.

New J. Phys.

7, 126 (2005)

Quasi-particle (GW) Band Gaps

EXX better than LDA basis for quasi-particle calculations

OEPx = EXX

Electronic Structure of Copper in the GW Approximation

Andrea Marini et al.,Phys. Rev. Lett. 88, 016403 (2001)

GW DFT-LDA EXP.

GW Approximation - Merits

Gives accurate band gaps for many materials Allows for calculation of lifetimes

Successfully applied to

bulk materials

surfaces

nanotubes

clusters

defects

defects on surfaces

(12)

A Primer in Density Functional Theory, C. Fiolhais, F. Nogueira and M. Marques, Springer 2003 (ISBN 3-540-03083-2).

“Quasiparticle Calculations in Solids”, W. G. Aulbur and L. Jönsson and J. W. Wilkins,

Solid State Phys. : Advances in Research and Applications 54, 1 (2000).

“Electronic Excitations: Density-Functional Versus Many-Body Green’s Function Approaches”,

G. Onida, L. Reining and A. Rubio, Rev. Mod. Phys. 74, p601 (2002).

“Combining GW calculations with exact-exchange density-functional theory:

An analysis of valence-band photoemission for compound semiconductors”, P. Rinke, A. Qteish, J. Neugebauer, C. Freysoldt and M. Scheffler, New J. Phys. 7, 126 (2005).

Additional reading

DFT (LDA, GGA, EXX) for weakly correlated systems

!! Accuracy of geometries is better than 0.1 A Accuracy of Common DFT implementations

!! Accuracy of calculated energies (relative) is usually better than 0.2 eV

Very often better than 0.01 eV

May we reach so-called chemical accuracy within DFT?

Exact Exchange Kohn-Sham Method – a step in this direction

"!Systematic improvement of existing Kohn-Sham schemes

"!

Computationally very demanding

•!Bulk systems up to now

•!Implementations for larger systems going on Crucial - Better correlation energy functionals

Failures of LSDA for strongly correlated systems

LSDA predicts negative ions (e.g. F^-) to be unstable For strongly correlated systems, LSDA consistently underestimates the tendency to magnetism (e.g., cuprates, NiO)

For strongly fluctuating systems, LSDA consistently overestimates the tendency to magnetism

FeAl : M = 0.7

((

Exp. – paramagnetic) Sr3^Ru2^O7^{: M = 0.6}

_B (Exp. – paramagnetic)

Band Gap Problem

Positions of the cationic d-bands in semiconductors are by 3-4 eV too high in energy

CB p-VB

s-VB d-band (in Cu d-bands are 0.5 eV too high) overestimation of p-d hybridization

-4 -2 0 2 4 -10

-5 0 5 10

SPIN-DOWN

SPIN-UP Cr-3d states Fermi level

Energy [eV]

Density of states [a.u.]

zb-CrAs

Spin-polarized LDA (LSDA) prediction:

zinc-blende CrAs is ferromagnetic

M. Shirai et al., J. Magn. & Magn.

Mater. 177-181, 1383 (1998)

"!

Previously nonexistent

compound

"!

Later thin films

grown by MBE

"!

Curie temperature

larger than 400 K

"!

Magnetic moment = 3

_B

Agreement between

theory and experiment

(13)

EXP LSDA

-3 -2 -1 0 1

Energy [eV]

Intensity [a.u.]

Photoemission spectrum of La

_0.94

Sr

_0.06

TiO

₃

6% hole doping

Exp.: A. Fujimori et al., PRL 69, 1796 (1992)

LSDA: I.A.Nekrasov et al., Euro. Phys.J B 8, 55 (2000)

LDA band structure calculations clearly fail to reproduce the broad band observed in the experiment at energies 1-2 eV below the Fermi energy.

Fermi

energy Spectra are Gauss-broadened (0.3 eV broadening parameter) to simulate the experimental accuracy

Phonon dispersion curves for NiO

Savrasov & Kotliar (2002)

0 2 4 6

8 10 12 16 14 18

Frequency [THz]

X L

LDA EXP

!!

Phonon wave vector

Comparison of LDA results with experiment

LDA overestimates the electronic screening effects by large amount causing

the artificial softening of optical phonons &

lowering of the LO-TO splitting.

LDA overestimates the value of

by a factor of 6.

!!

_"_"

Beyond LDA approach

to correlated electron systems

ˆH == d ""

³

!r ˆ!!

^†

( !r)[!!!! ++V

^ext

(!r)] ^! ^{! (!r)++} ^ˆ

++ 1 2 ""

d³

!rd

³

!r' ^! ^! ^ˆ

^†

⁽ ^{!r) ˆ!!}

^†

⁽ ^!r')!!

^ee

(!r !! !r') ˆ!! (!r) ˆ!! (!r') ˆ

_kin

H

ˆ

_{e e}

H

_!!

ˆH == d ""

³

!r ˆ!!

^†

(!r)[!!!! ++V

_KS

( !r)] ^! ^{! (} ^ˆ ^{!r)++ ˆH}

^corr

ˆ

_LDA

H

!ˆ

!^†(! r ) == ˆc_ilm^†

ilm

!

^!^!ilm(!

r ) _{, ' ',} ^† _{' '}

, ' ',

ˆ_LDA _{ilm jl m} ˆ ˆ_ilm _{jl m}

ilm jl m

H t _!!c c^!! ^!!

!!

==

" "

Expansion of field operators in basis (LMTO, LAPW, ….)

! !

ilm

How to deal with ? H

ˆ

_corr

LDA + local Coulomb correlations

LDA

ˆ ˆ

_LDA

ˆ

_corr^local

ˆ

_corr

ˆ

_res

H H

== ++

H

!!

H

++

H

'' ' '

, , , , ', '

1 ˆ ˆ

2

d

ilm ilm

i l l m m

U

mm^!!

n n

!!

!! !! !!

!!

==

" " ¹ ₂ ^U ^{n n}

^d

⁽

^d

^!! ¹⁾

Ab-initio correlated electron model

H

2

( )

LDA d d

E n

U

== !!

n

!!

Needed basis where interacting orbitals can be identified U can be calculated via constraint LDA:

Hund’s rule coupling can be calculated similarly

LDA + U

: solve HH with Hartree-Fock

LDA + DMFT

: solve HH with Dynamical Mean-Field Theory

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Dynamical Mean-Field Theory

Time

+ -

DMFT

Electron reservoir

0 !! !!""

DMFT replaces the full lattice of atoms and electrons with a single impurity atom imagined to exist in a bath of electrons

DMFT captures the dynamics of electrons on a central atom as it fluctuates among different atomic configurations, shown here as snapshots in time.

DMTS in the simplest case of an s orbital occupying an atom G. Kotliar & D. Vollhardt, Physics Today, March 2004

V V

Dynamical Mean-Field Theory – Basic Mathematical Description

To treat strongly correlated electrons, one has to introduce a frequency resolution for the electron occupancy at a particular lattice site

Green function specifies the probability amplitude to create electron with spin at site i at time and destroy it at the same site at a later time

!! !! '

!!

ˆ ˆ †

( ') ( ) ( ')

i i i

G _!! "" "" ## $$ ## c _!! "" c _!! ""

The dynamicalmean field theory (DMFT) can be used to investigate the full many-body problem of interacting quantum mechanical particles or effective treatments such as the Hubbard model

† ,

ˆ

_ij

ˆ ˆ

_i _j

ˆ ˆ

_i _i

ij i

H t c c

_!! _!!

U n n

!! "" ##

== $ $ ++ $ $

Dynamical Mean-Field Theory – Basic Mathematical Description (2)

, , †0, ,

, ,

ˆ

_AIM

ˆ

_atom ^{bath bath}

(

^bath

. .)

H H

!! "" !! ""

n V c a

!! "" !! ""

h c

!! ""

##

!! ""

== ++ $ $ ++ $ $ ++

The Anderson impurity model

The hybridization function

2

, ,

| | ( )

V_!! _bath

!! "" !! ""

## $ $

$

== $ %%

&&

' '

plays the role of dynamic mean field.

!! "" ( )

has to be determined from the self-consistency condition:

G[!!(!

! )] ==

!

{{ ! ! !! ""[!!(! ! )]!! t

_k^!

}}

"

k

"

^!!1

(( ))

¹

[ ( )] ( ) G [ ( )]

!! "" ## %% "" ## $$ "" ##

^$$

++ ##

Self-energy term

takes on the meaning of a frequency dependent potential

Self-consistent cycle of LDA+DMFA

A functional of both the charge density and the local Green function of the correlated orbital

!![!!,G] == T[!!,G]++ V^ext(! r )!!(!

r )d³! r ++ 1

!!

2 ^!!_|⁽^!r ""^!^{r )}^!!!⁽^{r ')}^! r '| d³!

rd³!

r '++ E_xc[!!,G]

!!

!!( !r) H ˆ _LDA , U ^DMFA n _ilm ^!!( ^!r)

LDA+DMFA – Functional Formulation

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LDA+DMFA – Computational Scheme Photoemission spectrum of La

_0.94

Sr

_0.06

TiO

₃

LSDA

Exp.: A. Fujimori et al., PRL 69, 1796 (1992)

LSDA: I.A.Nekrasov et al., Euro. Phys.J B 8, 55 (2000) EXP

-3 -2 -1 0 1

Energy [eV]

Intensity [a.u.]

Fermi energy

LSDA+DMFT

(U=4.25 eV)

(U=5.0 eV)

LSDA+DMFT

dramatically improves

the photoemission spectrum

Phonon dispersion curves for NiO

0 2 4 8 6 10 12 16 14 18

Frequency [THz]

X L

LDA+DMFT EXP

!!

Phonon wave vector Savrasov & Kotliar (2002)

Results of LDA + DMFT

2 2

|

* 2

| /

LO TO

Z

!! $$ !! % % ""

_#_#

LDA LDA+

DMFT EXP

| Z

*

|

!! _" _"

35.7 2.17

7.2 2.33

5.7, 6.1 2.22 Born effective charge

Accidental agreement with experiment !!

Importance of correlations in lattice dynamics of NiO

Dynamical Mean Field Theory

• DMFT is an intrinsically many body electronic theory.

• It simultaneously handles the atomic and band character of electrons. This is at the heart of correlation physics.

• The approach leads to a non trivial but tractable problem.

• Misses out on spatial correlations. CDMFT can handle them.

• From a curiosity in the early 90’s, it has become now

an indispensable part of the theorists training.

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References:

A. Georges, et al., Rev. Mod. Phys. (1996) T. Maier, et al., Rev. Mod. Phys. (2005) G. Kotliar, et al., Rev. Mod. Phys. (2006) A. Georges, review, cond-mat. (2004) M. Civelli, Rutgers thesis, cond-mat (2007)

Dynamical Mean Field Theory

Thank you !

Modeling of Nanostructures and Materials