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.: