Transcript Document 7311336

Theoretical Modeling of Core Excitation Spectra
Eric L. Shirley (NIST),*
on behalf of
L.X. Benedict, LLNL
J.A. Soininen, U. Helsinki
Z.H. Levine, NIST
J.A. Burnett, NIST
J.J. Rehr, U. Wash., Seattle
S. Dalosto, NIST
H.M. Lawler, U. Wash., Seattle
E.K. Chang
J.A. Soininen, U. Helsinki
J.J. Rehr, U. Wash., Seattle
J.C. Woicik (NIST)
C.S. Hellberg (NRL),
E.K. Chang
(x-ray)
H.M. Lawler, U. Wash., Seattle
(Thz)
(NIR/VIS/UV)
…and others!
*Tel: 301 975 2349 / FAX: 301 975 2950 / email: [email protected]
This work was funded in part by the US Department of Energy
Grant No. DE-FG03-97ER45623 through the Computational
Materials Science Network (CMSN) and the Office of Microelectronics
Programs at NIST and International SEMATECH.
Optical constants through the electromagnetic spectrum:
Various excitations in a solid and associated excitation spectra:
Index of refraction n and
index of absorption k in GaAs:
[taken from Palik, Handbook of Optical
Constants of Solids, Volume I.]
VIS/UV optical properties
TO
phonon
2-phonon...
x-ray
edges
2-phonon
3 phonon
Definition of terms:
D  displaceme nt
E  electric field


D    E  E  4 P, 
P  Pcore  Pval.  Plat.  polarization
  dielectric tensor
Atomic units
If dielectric tensor may be treated as a scalar, one has
  dielectric constant, 1 ,  2 real

  1  i 2  (n  ik ) 2 , n  index of refraction, n  0
k  index of absorption, k  0

Some related quantities:
 ( )  4 k ( )  absorption spectrum ,

Im   L1  2 2 2  loss function,
1   2




q2
S (q, ) 
Im   L1  dynamic structure factor (electroni c)
2
4 ne
Note:
D  D(q,  ), etc.,
q  2 / ,
  2 .
TALK PLAN:
Against the backdrop of the multi-faceted interaction of light and matter, especially in
solids, we shall discuss now-common model (GW/Bethe-Salpeter, a.k.a. GW/BSE) to
calculate effect of electron-electron/electron-hole interactions on optical constants.
This model includes:
(1) self-consistent density-functional calculations to get good one-el.
orbitals and energy bands,
(2) many-body corrections to band energies (here, GW approximation),
(3) a method to solve the coupled el.-hole e.o.m. describe the excited
states resulting from excitation of an electron across the Fermi level.
Sample results include:
* calculation of absorption spectra involving core excitations
* treatment of IXS at a a core edge (F 1s in LiF)
* effects of electron/hole lifetime damping on core/valence spectra
* quadrupolar x-ray transitions in transition metal (TM) oxides
* multiplet effects in TM oxides
* structural determination of ferroelectric TM oxide thin-films
Review Articles:
Bethe-Salpeter calculations, including vs TD-DFT
G. Onida, L. Reining, A. Rubio
Electronic excitations: density-functional vs many-body Green's functions approaches Rev. Mod. Phys., 2002
O. Pulci, M. Marsili, E. Luppi, C. Hogan, V. Garbuio, F. Sottile, R. Magri, and R. Del Sole
Electronic excitations in solids: Density functional and Green's functions theory Phys. Stat. Sol. (b), 2005
M. Palummo, O. Pulci, R. Del Sole, A. Marini, P. Hahn, W.G. Schmidt, and F. Bechstedt
The BSE: a first-principles approach for calculating surface optical spectra J. Phys.: Condens. Matter, 2004
GW (and some GW/BSE) calculations
F. Aryasetiawan and O. Gunnarsson
The GW method Rep. Prog. Phys, 1998
L. Hedin
On correlation effects in electron spectroscopies and the GW approximation J. Phys.: Condens. Matter, 1999
W.G. Aulbur, L. Jönsson, and J.W Wilkins
Quasiparticle Calculations in Solids Solid State Physics, 2000
XAFS
J.J. Rehr and R.C. Albers
Theoretical approaches to x-ray absorption fine structure Rev. Mod. Phys., 2000
Classic
L. Hedin and S. Lundqvist
Effects of Electron-Electron and Electron-Phonon Interactions on the 1-el. States of Solids Solid State Physics, 1969
Describing electron states
Predictive theory needs:
* accurate band structure methods
(Schrödinger equation)
* many-body
corrections to
band energies
this work:
pseudopotential,
plane-wave, +
PAW-type
reconstruction
as needed
Dyson equation:
Electron self-energy
(Hedin’s GW approximation)
electron state energy, wave function
 p2

3

V

V

(
r
)

d
r (r, r; Enk )  nk (r)  Enk  nk (r )


ext
H  nk
 2m

GW self-energy (accounts for many-body
electron-electron interaction effects)
2 2


2m
Energy/momentum domain:
self-energy = convolution of electron propagator,
polarization effects
 electron interacts with its own polarization cloud
GW
LDA
Bethe-Salpeter calculation:
Excited state = linear superposition
of all states produced
by a single electron
excitation.
Eel
In each such electron-hole pair state,
electron in band n,
with crystal momentum k+q.
momentum
hole in [band or core-level] n,
with crystal momentum k,
Call such a state |n n k(q), total
crystal momentum q.
BSE, cont’d.:
In a non-interacting picture, one has
[HE0] |n n k(q) = [ Eel( n , k+q)  Eel ( n, k) ] |n n k(q).
H=He+Hh
The e-h pair states {|n n k(q)} diagonalize Hamiltonian, H.
In an interacting picture, one has
[HE0] |n n k(q) = [ Eel( n , k+q)  Eel ( n, k) ] |n n k(q) +
H=He+Hh+Heh
 n n k V(n n k, nn k) |n n k(q).
This couples {|n n k(q)} states. Stationary states are linear combinations of e-h pair states.
Resulting coupled, el.-hole-pair Schrödinger equation
of motion, i.e., Bethe-Salpeter equation: difficult to
solve, especially within a realistic treatment of a solid.
Maps onto 1-el. eq’n in cases of
core holes without dynamics (e.g., 1s)
WARNING:
This Bethe-Salpeter
treatment is good when
a one-electron/one-hole
description is valid,
and not otherwise.
BSE, cont’d.:
Two parts of electron-hole interaction:
Direct:
VD ( nnk , nnk ; q) 
  d 3 x  n*k  q ( x) nk  q ( x)  d 3 x   n*k ( x) nk  (x) W ( x, x;   0)
Exchange:
nk  q
nk
nk   q
nk 
nk  q
nk
nk   q
nk 
VX (nnk , nnk ; q) 
 2  d 3 x  n*k  q (x) nk  ( x)  d 3 x   n*k (x) nk  q ( x) ( x  x) 1
The attractive direct part is responsible for excitons (bound electron-hole pairs)
and an energetically downward shift in oscillator strength (excitonic effects).
The repulsive exchange part is responsible for plasmons. Including it accounts
for spin degrees of freedom for spin-singlet excitons.
MgO optical constants:
Li halides, Li 1s excitations
(LiH, LiF, LiCl, LiBr, LiI, LiAt*)
J. El. Spect. Rel. Phenom. 137-40: 579 (2004)
Trends on descending through series,
effects of electron-hole interaction
*unstable
measured spectrum (meas.)
calculated spectrum (inter.)
calculated spectrum, omitting
electron-hole interaction (n.i.)
Dynamic structure factor for LiF at
fluorine K edge (x-ray scattering):
Hämäläinen et al., PRB 65, 155111 (2002).
Scattering process:
q, 
qout, Eout
q = qin  qout,
 = Ein  Eout.
Calc.
qin, Ein
d 2
 d 


  F ˆ q I
d d  d  Th F
2
 (EI    EF )
 d 

 S (q,  )
 d  Th
2
1 
 d   q  

  2  Im
 d  Th  4 n    (q,  ) 
Momentum-dependent
spectra betray even-parity
core-hole exciton level.
Meas.
Data collected at
European Synchrotron
Radiation Facility (ESRF)
Damping of electron states
Electron self-energy
for band states in LiF,
versus LDA band energy.
(Self-energy minus
average LDA exchange
and correlation shown;
conduction band minimum
defined as zero.)
F 2p
LDA=independent electron
picture, without effectual
self-energy effects
F 2s
Damping of electron/core states
F 1s edge results in LiF with
and without core-hole,
damping.
Damping/self-energy shifts
realized by probing e-h
Greens function, not at E,
but at E+(E); outstanding
issue: what is self-energy
of e-h pair?
Effective 1-el. e.o.m.
For el-core hole pair:
core-hole effect

 p2


V
(
r
)

V
(
r
)

V
(
r
)

 G (r, r; E ) 
N
H
c
 2m

 d r (r, r; E ) G(r, r; E )  E G(r, r; E )
3
Vc
damping effects
electron-core hole
Bethe-Salpeter equation
What about lifetime
effects in valence spectra?
needs lifetime broadening?
Idea:
average lifetime damping
of electron and hole states at
half the photon energy away
from the mid-gap.
Quadrupole (and dipole) core-excitation spectra:
“Pre-edge” and near-edge excitations in rutile TiO2 XAS
J. El. Spect. Rel. Phenom. 136, 77 (2004)
why does the
spectrum worsen
above here? damping eff.
(E  x) exp[ i(q  x  t )] 
[E  x  i(E  x)(q  x)  ...] exp( it )
quadrupole
dipole
Atomic-basis expansion of Bloch function:
k q
nk q (x)   Anlm
Fl ( x) Ylm (xˆ )
lm
 nk q E  x  i(E  x)(q  x)  ,R 0
k  q *
  ( Anlm
) [ I1 (l , ) A1 (lm, LM ; eˆ ) 
lm
iqI 2 (l , ) A2 (lm, LM ; eˆ , qˆ )] ,
radial integrals
Rutile TiO2
crystal structure
angular integrals
Ti(1s3d)
Ti(1s4p), etc.
note possibility of longit. quadrupole in EELS/IXS
Multiplet effects in 3d L2,3 oxide spectra
Ti4+(1s22s22p63s23p6)Ti4+(1s22s22p53s23p63d1)
* 2p core hole ML,MS degrees of freedom (DOF)
- exchange interaction with electron
- multipole effects
- spin-orbit effects
Atomic view of Ti L2,3 edge in, say, SrTiO3
(taken from de Groot et al., PRB, 1990):
ISOLATED ATOM
H  H av  L  S( p)  L  S(d )  g (i, j )
* excited electron spin DOF
- exchange interaction with hole
- spin-orbit effects
* particles interact
- Coulomb monopole interaction
- Multipole exchange & direct
- Screening by other particles (0.83)
- Charge-transfer (beyond BSE)
* solid-state environment
- crystal field
ATOM + CRYSTAL FIELD
H  H av  L  S( p)  L  S(d )  g (i, j )  H xtalfield
Chicken-egg paradigm: Should we view this as
* crystal field effects in atomic calculation or
* fancier excitonic effects in BSE calculation?
WHICH
CAME
FIRST?
Multiplet effects in 3d-oxide L2,3 spectra:
Example: Ti L2,3 in SrTiO3
J. El. Spect. 144, 1187 (2005)
This work: Bethe-Salpeter solid-state calculation:
Generalize:
 nk(q)   M L M S , nk ; q 
H eff  H BSE  H h  H e  H eh
He 
2m
V
xtal
KS  L  S ( d )
H eh  V (re )  g (i, j )
central part
screened
by RPA
crystal-field
splitting
spin-orbit and
crystal-field
splittings
evident
H h    L  S( p )
p e2
spin-orbit
splitting
Slater-type
integrals,
scaled by  0.83
re-arranged
oscillator
strength
band-induced width;
higher-lying features
included naturally;
C-T satellites absent
Quantifying effects of correlation/multiple excitation
beyond current Bethe-Salpeter treatment
wide band gap (~12 eV)
small band gap (~3 eV)
Results suggest that Coster-Kronig, vibrations
shake-up, charge transfer excitations, etc.
dominate remaining discrepancy (cf. de Groot et al.).
FILM
BULK
Thin-film strontium titanate on Si (100)
Coherent growth of 5 ML film:
compressive strain in a-b plane, c/a stretch,
Leading to AFD cage rot. & FE polarization
80
Local atomic
geometric
in bulk (a)
and film (b)
Ti K and L2,3 spectra (expt=top, theory=bottom)
100
90
cubic SrTiO3
80
q // 100 e // 001
q // 100 e // 010
q // 110 e // 1-10
40
20
Intensity (arb. units)
Intensity (arb. units)
60
70
60
50
40
e out-of-plane
e in-plane
30
20
5 ML SrTiO3
cubic SrTiO3
5 ML SrTiO3
10
0
80
0
100
90
80
cubic SrTiO3
q // 100 e // 001
q // 100 e // 010
q // 110 e // 1-10
40
20
Intensity (arb. units)
Intensity (arb. units)
60
70
50
40
e out-of-plane
30
20
5 ML SrTiO3
cubic SrTiO3
60
e in-plane
5 ML SrTiO3
10
0
0
4960 4962 4964 4966 4968 4970 4972 4974 4976 4978 4980
Photon Energy (eV)
452
454
456
458
460
462
464
Photon Energy (eV)
466
468
470
472
Electron damping (picture from Hedin
Lundqvist 1969 Solid State Physics Review)
Gradual onset
from low-energy
electron-hole
(inter-band)
continuum
05/09
Example: bromellite
Dielectric function betrays
inter-band transitions that
give lowest structure in loss
function.
Main onset and
high-energy tail
from plasmon +
large-q transitions
Plasmon-pole
model misses
electron-hole
continuum
tail broadens
plasmon loss
06/09
Observed requirements of dielectric function:
 Need to go beyond plasmon-pole model!
Proposed model for dielectric function:
2(q,E)
1.) accounting for electron-hole continuum
2.) broadening of plasmon (high-energy tail in 2)
Requires: approximate band gap E0, , occ’d band states
 0 (q, E )  C  ( E  E0 ) [ ( E  E0 )  E ]

 (    p  1)(   1) 

 M p (q)   dE E p  0 (q, E )  CE0    p 1
(  p)
0


M 1 (q), M 0 (q), M1 (q) 

( z  1)  z( z ) 
E0
E
(q-dependence of
C, ,  is implicit)
C, ,  are known
We obtain
M1(q) from the f-sum rule
M1(q) from the Levine-Louie dielectric function (plus Kramers-Kronig)

M0(q) from
1
1
2
3




d
E

(
r
,
r
;
E
)

|

(
r
,
r
)
|

n
(
r
)

(r  r)
0
1

0
2
08/09
Sample results:
Periclase small-q dielectric function
Other results (periclase loss function,
silicon dielectric function):
Effect on electron self-energy using GW approximation with jellium G, model W
(screened interaction), which is an easy calculation. Inset: improved “ramp-up” of
lifetime damping because of spread of oscillator strength in response function.
 2
e iE G(k  q,   E )W (q, E )
Magnitude of imaginary part
of self-energy:
points=numerical calc (hours)
curves=our model (seconds)
Electron state energy minus conduction band minimum (eV)
Typical plasmon-pole model result
Real part of self-energy:
points=numerical calc (hours)
curves=our model (seconds)
 dE
( , k )  i  d q 
3
So where are we as a community over the whole spectrum? At least this good…
(measured=Palik Handbook; Ikezawa and Ishigame, J. Phys. Soc. Japan, 1981)
Silicon optical constants: x-ray to Thz
n, k
valence
excitations
L1
L2,3
K
sums
differences
phonons
Notes:
NBSE denotes the NIST Bethse-Salpeter Equation (BSE) code(s);
at.=first-principles correction for outer shell absorption asymptote;
For DFPT results, see, for instance, Deinzer and Strauch, Phys. Rev. B, 2004
Closing Remarks
-Present BSE is a robust method to calculate many
excitation spectra
-Vibrational effects being included: work underway
-Stronger correlation effects (greater entanglements
in
(1) ground-state wave function,
(2) excited state wave function beyond el-hole pair
-Generic “bridge” coupling valence BSE code to output of
many plane-wave/pseudopotential codes is in place
-“bridge” coupling core BSE code to output of many
plane-wave/pseudopotentials codes conceived
-with such bridge, codes would be shared with others