Theoretical Spectroscopy: electronic & optical properties...
Transcript of Theoretical Spectroscopy: electronic & optical properties...
Giovanni Onida – Nicola Manini – Guido Fratesi –Lucia Caramella – Elena Molteni –Marco Cazzaniga -
Matteo Ferri – Virginia Carnevali -
Theoretical Spectroscopy:electronic & optical properties of
surfaces, clusters and nanostructures
• Ab-initio: no adjustable parameters – transferable, predictive (but numerically heavy) . Response to probe: one needs to know the ground and excited states
• Several spectroscopies: VIS-UV absorption, Surface optical properties (RAS, SDR, SHG), Circular Dichroism… Also, IR and Raman vibrational spectra.
Optical Spectroscopy
• Non-destructive technique (vs. electrons)
• UHV not needed (ok to measure in situ andin real time during epitaxial growth ex.:MBE, CBE, MOCVD)
• ...sensitivity to the surface?? Measure differential quantities !
(UV photons deeply penetrate into bulk! Surface signal: less than 1 %)
Reflectance Anisotropy Spectroscopy (RAS):
RAS measures the spectral dependence of the quantity:
where are reflectivity intensities for light polarised along the two orthogonal directions parallel to the surface. R0 is the Fresnel reflectivity.
2ii rR =
xr yr
yEr
xEr00 R
RRR
R yx −=
Δ
N.B: r is the complex reflectance, one can also define RAS as thereal part of Δr/r
Image courtesy of Yves Borensztein
ISOTROPIC BULK
polarized light
Isotropy => RAS=0
RAS, SDR: (Isotropy => RAS=0)
SHG: (Inversion symmetry => SHG=0)
CD: (Mirror symmetry => CD=0)
Aminoacids
L-enantiomers are largely dominant ! Chiral photons in the interstellar UV radiation may have induced , by asymmetric photochemical reactions, enantiomeric excess into circumstellar organic compounds prior to their deposition on the Earth.
Origin of L-enantiomeric excess: Nature 266, 567 (1977); Science 283, 1123 (1999).
zwitterionic“neutral”
L-ala_
+
Circular Dichroism Spectra
• Photoemission:
e-hν
Measure EQP = EN – EN-1 = poles of G (Bandstructure)
• STM (I/V):
e-hνe-
• IPE:
P ≠ -i G1hole G1
electron
• Optical absorptionhν
Excitonic effects!
electron+hole: 2 quasiparticles
E,q
e-• El. Energy Loss
P0 = -i G1hole G1
electron Independent Quasiparticles
See e.g. G.O, L. Reining, A. Rubio,Rev. Mod. Phys.74, 601 (2002)
Bulk Silicon
Im [ε] ~ Σλ| Σvc<v|D|c> Aλvc|2 δ (Eλ-ω)
->Mixing of transitions->Modification of excitation energies
ExperimentDFT-LDALDA+GWLDA+GW+BSE
el.-hole interaction: bound excitonlineshape
Im [ε] ~ Σvc |<v|D|c>|2 δ (Ec-Ev-ω)
DFT (RPA) W
DFT+GW P0
BSE P εM
S. Albrecht, L. Reining, R. Del Sole, and G.O.Phys. Rev. Lett. 80, 4510 (1998):
See e.g. G.O, L. Reining, A. Rubio,Rev. Mod. Phys.74, 601 (2002)
Im [ε] ~ Σλ| Σvc<v|D|c> Aλvc|2 δ (Eλ-ω)
->Mixing of transitions->Modification of excitation energies
el.-hole interaction: bound excitonlineshape
Im [ε] ~ Σvc |<v|D|c>|2 δ (Ec-Ev-ω)
DFT (RPA) W
DFT+GW P0
BSE P εM
S. Albrecht, L. Reining, R. Del Sole, and G.O.Phys. Rev. Lett. 80, 4510 (1998):
Solid Argon
V. Olevano (2000)
OPTIMIZEGEOMETRY
• Crystal termination• Structural relaxation• Adatoms of ≠ species{ }
}
}COMPUTEOPTICALPROPERTIES
• Bandstructure needed(filled& empty bands)
• +integrals over the BZ
• Electrons and holes are created simultaneously: Excitonic effects
GW METHOD (Σ=GW) tocompute self-energy effects(perturbative: keep LDA wfc)
e.g.: CAR-PARRINELLO global minimization
DFT-LDA to search for the ground-state geometry;
SLAB METHOD (repeated, for PW)
BETHE-SALPETER EQUATIONfor 2-particles Green’s functions=> dielectric function ε(r,r’,ω)
Moreover...• ε inhomogeneous• ε anisotropic• ε slab ≠ ε surf
{ Three-layers model for ΔR/R(Del Sole ’81, Aspnes ‘71)
Time-Dependent Density Functional Theory (TDDFT):
χ = χ0 + χ0 [ v + fxc ] χHartree
(Local Fields)QP shift
+ e-h attraction
εΜ = 1−lim 4π/q2 χ(q)q→0
Problem ToolPlane-waves ab-initio calculations of the optical properties
Base: Products of KS eigenstates-> eigenvalues equation
We already have W….
Hedin’s equations in Many-Body Perturbation Theory:
G =Σ = self-energy
W = screened Coulomb int.P = polarization functionΓ = vertex funcion
Base: Products of KS eigenstates-> eigenvalues equation
We already have W….
Hedin’s equations in Many-Body Perturbation Theory:
G =Σ = self-energy
W = screened Coulomb int.P = polarization functionΓ = vertex funcion
Base: Products of KS eigenstates-> eigenvalues equation
We already have W….
Beyond independent QP through Hedin’s eqns:
Base: Products of KS eigenstates-> eigenvalues equation
We already have W….
4-points BSE Kernel:
Beyond independent QP through Hedin’s eqns:
BSE
No approximations (till now…)
BSE
K(1234) = -W(12)δ(13)δ(24)
The GW approximation
GW
BSE
K(1234) = -W(12)δ(13)δ(24)
The GW approximation
GW
εmicro = 1 − vc P
εM =1
−−−−−−−−< ε−1 > Vtot = Vext +Vind = ε−1 Vext
• Adler, Phys. Rev.126 (1962);• Wiser, Phys. Rev.129 (1963)
BSE
K(1234) = -W(12)δ(13)δ(24)
The GW approximation
GW
ε−1 = 1 + vc χ
If no screening (ε = 1) then W=v TDHF!
εmicro = 1 − vc P
εM =1
−−−−−−−−< ε−1 > Vtot = Vext +Vind = ε−1 Vext
• Adler, Phys. Rev.126 (1962);• Wiser, Phys. Rev.129 (1963)
-W(12)δ(13)δ(24) +V(13)δ(12)δ(34) K(1234) = e-h exchangee-h attraction
χ = P0 + P0 [- W + vc ] χ
BSE
K(1234) = -W(12)δ(13)δ(24)
The GW approximation
GW
P = P0 + P0 [- W + vc ] P
εM = 1 − vc P
Local Fields
-W(12)δ(13)δ(24) +V(13)δ(12)δ(34) K(1234) = e-h exchangee-h attraction
v = v + vLR
(Hel + Hhole + Hel-hole ) Aλ = Eλ Aλ
Bethe-Salpeter equation -> eigenvalue equation for aneffective two-particles hamiltonian 2Nx2N (N=nv*nc*nk):
Reduction to an eigenvalue problem
Resonant
Antiresonant
Coupling
(but not the only way)
2Nx2N matrix in transition space (Non Hermitian)
P = P0 + P0 [K] P
Optical properties of a small peptide“83-92” fragment of HIV-protease
10 aminoacids175 atoms 4 species
470 electrons200,000 g-vectors
per state
Different conformations
TDLDA ?
• IP-RPA, x polarization• IP-RPA, y pol• IP-RPA, z pol
Dark transitions!
DOS
Abs. onset is always above 2 eV
Energy gap is ≤ 1 eV(0.6 – 1.0 eV, depending on conformation)
Absorption spectrum
235 (HOMO)
231
225
236 (LUMO)
238
…Charge transfer excitations
244
Absorption
e-hpair
Eabs
E2
Emission
Eem
E4
E3
Groundstate
Cluster geometry
E1
EmissionEmission asas the the timetime--reversalreversal of of absorptionabsorption (LDA, (LDA, GW&BSEGW&BSE))
Excitedstate
Si10H14O
Many-body effects in emission spectra
Emission vsAbsorption(GW+BSE)
a.u.
Redshift between absorption and emission very large
(e-h interaction + Stokes shift)
Probability distribution for finding the electronwhen the hole( ) is fixed
hole
Ma et al. APL, 75 1875 (1999))
S.Ossicini, …, G.O
SiSi1010HH1414OO Emission (VIS)
Absorption
PL peak in the visible, with excitonic nature
Many-body effects in emission spectra
Seen in experiments
S. Ossicini, …, and G.O., Journal ofNanoscience and Nanotechnology 8, 492 (2008)
1900 2000 2100 2200 2300
0
500
1000
Inte
nsity
[cou
nts/
min
]
Raman shift [cm-1]
in situ (RT) exposed to He
RAMAN SPECTRA of linear carbon chains
-86 % -76 % -65 % -56 % -44 %
… having different decay behaviour (when exposed to He).Thanks to the high statistics we could identify several sub-components …
C band
es.: nanostrutture di carbonio
sp-carbon atomic wires
13.05 eV
sp2
sp
L. Ravagnan, N. Manini, G.O, at el., Phys. Rev. Lett. 102, 245502 (2009)
C. Jin, H. Lan, L. Peng,K. Suenaga, and S. Iijima,
Phys. Rev. Lett. 102, 205501 (2009)
L. Chuvilin, J.C. Meyer,G. Algara-Siller and
U. Kaiser, New Journal of Physics11, 083019 (2009)
sp3
8.531 eV
Twisting the wires
sp3
sp3 terminated chainscan be freely twisted.
=> they are notaffected by the
orientation of the terminal groups.
sp2
sp2 terminated chains are torsionally stiff, since a memory of
the orientation of the terminating sp2 carbon propagates along the chain
=> they are affected by the orientation of the terminal groups.
N. Manini and G.O., Phys. Rev. B 81, 127401 (2010)L. Ravagnan, N. Manini, G.O, at el., Phys. Rev. Lett. 102, 245502 (2009)
A
Torsional strain: DFT results
Torsional strain
Twisting the chain the total energy increases
= Non-spin polarized= Spin-polarized
Largely strained chains undergo a magnetic instability.
UV-VIS and Raman spectra change!
C6(CH2)2(TDLSDA)
Optical Absorption:
Thank you
Giovanni Onida – Nicola Manini – Guido Fratesi –Lucia Caramella – Elena Molteni –Marco Cazzaniga -
Matteo Ferri – Virginia Carnevali -
Plane-waves ab-initio calculations of the optical properties of surfaces
SURFACE• Crystal termination• Structural relaxation• Adatoms of ≠ species{ }
}
}OPTICALPROPERTIES
• Bandstructure needed(filled& empty bands)
• +integrals over the BZ
• Electrons and holes are created simultaneously: Excitonic effects
GW METHOD (Σ=GW) tocompute self-energy effects(perturbative: keep LDA wfc)
e.g.: CAR-PARRINELLO global minimization
DFT-LDA to search for the ground-state geometry;
SLAB METHOD (repeated, for PW)
BETHE-SALPETER EQUATIONfor 2-particles Green’s functions=> dielectric function ε(r,r’,ω)
Moreover...• ε inhomogeneous• ε anisotropic• ε slab ≠ ε surf
{ Three-layers model for ΔR/R(Del Sole ’81, Aspnes ‘71)
Plane-waves ab-initio calculations of the optical properties of surfaces
• Crystal termination• Structural relaxation• Adatoms of ≠ species
SURFACE { }
}
}OPTICALPROPERTIES
• Bandstructure needed(filled& empty bands)
• +integrals over the BZ
• Electrons and holes are created simultaneously: Excitonic effects
e.g.: CAR-PARRINELLO global minimization
DFT-LDA to search for the ground-state geometry;
SLAB METHOD (repeated, for PW)
# OF PLANE WAVES # OF EMPTY BANDS
+ CONVERGENCE WITH SLAB THICKNESS AND SEPARATION!
# OF K-POINTS
GW METHOD (Σ=GW) tocompute self-energy effects(perturbative: keep LDA wfc)
BETHE-SALPETER EQUATIONfor 2-particles Green’s functions=> dielectric function ε(r,r’,ω)
Moreover...• ε inhomogeneous• ε anisotropic• ε slab ≠ ε surf
{ Three-layers model for ΔR/R(Del Sole ’81, Aspnes ‘71)
Computational cost
• LDA (electronic ground state, equilibrium geometry, reconstructions): “1”
• Optical properties in DFT-LDA+shift: “1”-”5”
• GW bandstructure calculations “200”
• Bethe-Salpeter equation for excitons “>200”
using: one can reduce everything to matrix elements like: or: 2) < n | rν | m > (length gauge)
1) < n | pν | m > (velocity gauge)
Δε(ω) = εL (ω) − εR (ω) : Measured difference in
molar extinction coefficients for left and right-polarized light.
Circular Dichroism (intrinsic):
electric dipole
For freely rotating molecules, Δε(ω) = Re { Tr [Gμν] }
magnetic dipole
[electric quadrupole only relevant for oriented samples]
• Surface effects (reconstructions, RAS/SDR experiments…)
• Atomic motion (Stokes shift, phonon effects…)
• Large unit cells, disorder ( Numerical bottlenecks)
• …
• Optical excitations are compex by themselves…(MB effects…)
• Real systems add more complexity! e.g.: