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Optical properties and Interaction of radiation with matter

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Optical propertiesandInteraction of radiation with matter

S.Nannarone

TASC INFM-CNR & University of Modena

- Elements of Classical description of E.M. field propagation in absorbing/ polarizable media Dielectric function
- Quantum mechanics microscopic treatment of absorption and emission and connection with dielectric function

Physics related to a wide class of Photon-in Photon-out experiments including Absorption, Reflectivity, Diffuse scattering, Luminescence and Fluorescence or radiation-matter interaction

[some experimental arrangements and results, mainly in connection with the BEAR beamline at Elettra

http://new.tasc.infm.it/research/bear/]

- bulk materials the whole space is occupied by matter
- Surfaces matter occupies a semi-space, properties of the vacuum matter interface on top of a semi-infinite bulk
- Interfaces transition region between two different semi-infinite materials

Information [see mainly following lectures]

- Electronic properties full and empty states, valence and core states, localized and delocalized states
- Local atomic geometry /Morphology electronic states – atomic geometry different faces of the same coin

Energy range Visible, Vacuum Ultraviolet, Soft X-rays)

Synchrotron and laboratory sources/LAB

Conceptually Shining light on a system, detecting the products and measuring effects of this interaction

This can be done by Laboratory sources

They cover in principle the whole energy range nowadays covered by synchrotrons (J.A.R.Samson Techniques of vacuum ultraviolet spectroscopy)

- Incandescent sources
- Gas discharge
- X-ray e- bombardment line emission
- Bremsstrahlung continuous emission sources
- Higher harmonic source

Synchrotron and laboratory sources / Synchrotron

- Some well known features
- Collimation
- Intrinsic linear and circular polarization
- Time structure (typically 01-1 ns length, 1 MHz-05GHz repetition rate)
- Continuous spectrum, high energy access to core levels
- Reliable calculability of absolute intensity
- Emission in clean vacuum, no gas or sputtered materials
- High brilliance unprecedented energy resolution
- High brilliance small spot Spectromicroscopy

“The one important complication of synchrotron source is, however, that while laboratory sources are small appendices to the monochromators, in a synchrotron radiation set-up the measuring devices becomes a small appendices to the light source. It is therefore recommendable to make use of synchrotron radiation only when its advantages are really needed.”

C.Kunz, In Optical properties of solids New developments, Ed.B.O.Seraphin, North Holland, 1976

Matter polarizes in presence of an electric field Result is the establishment in the medium of an electric field function of both external and polarization charges

Matter polarizes in presence of a magnetic field Result is the establishment in the medium of a magnetic field function of both external and polarization currents

The presence of fields induce currents

- Mechanisms and peculiarities of polarization and currents induction in presence of an E.M. field
- Scheme to calculate the E.M. field established and propagating in the material
- Basis to understand how this knowledge can be exploited to get information on the microscopic properties of matter

Basic expressions - Charge polarization

and induced currents

Polarization vectors

-Ze-

+Ze

Ze+

e-

_

+

Motion of charge under the effect of the electric field of the E.M. field but in an environment where it is present an E.M. field

Expansion of polarisation

Physical meaningElastic limit the potential is not deformed by the field

Linear and isotropic media

Dielectric function

Permeability function

Formally linear optics implies neglecting terms corresponding to powers of the electric field

Physicallyit meansE.M. forcesnegligible with respect to electron-nuclei coulomb attraction

Nuclear atomic potential is deformed not harmonic (out of the elastic limit) response distortion higher harmonic generation

In very general way

External stimulus

Note is defined as a real quantity

Summary material properties within linear approximation

And

Conduction under a scalar potential – Usual ohmic conduction

Conduction in an e.m. field

Corresponding equations for vacuum case

Wave equation - Vacuum

Vacuum supports the propagation of plane E.M. waves with dispersion / wave vector energy dependence

Wave equation - Matter

Matter supports the propagation of E.M. waves with this dispersion

Formally q is a complex wavevector

Wave vector eigenvalue/dispersion depends on the properties of matter

through (all real quantities)

Absorption

Phase velocity

Real and imaginary parts not independent

Absorption coefficient

Lambert’s law

Complex dielectric constant – Complex wave vector

Supported/propagating E.M. modesdepend on the properties of matter through

The study of modes of the e.m. field supported/propagating in a medium and the related spectroscopical information is the essence of the optical properties of matter

Relation between (r,t), (r,t) (r,t) or (q,) (q, ) (r ) and the properties of matter

1st part Classical scheme / macroscopic picture

2nd part Quantum mechanics / microscopic picture

extension on which the average is made

Note 0 wavevector does not mean lost of dependence on direction anisotropic materials excited close to origin

Unknowns and equations

(real quantities) are the unknowns related with the material properties

(r,t) is close to unity at optical frequencies magnetic effects are small

(not to be confused with magneto-optic effects: i.e. optics in presence of an external magnetic field)

Generally a single spectrum – f.i. absorption – is available from experiment

(An ellipsometric measurement provides real and imaginary parts at the same time.

It is based on the use of polarizers not easily available in an extended energy range)

Real and imaginary parts are related through Kramers – Kronig relations

Sum rules

Kramers – Kronig dispersion relations

Under very general hypothesis including causality and linearity

Models for the dielectric constant / Lorentz oscillator

Mechanical dumped oscillator forced by a local e.m. field Neglecting the magnetic term

e-

Induced dipole

Out of phase – complex/dissipation – polarizability (Lorentzian line shape)

From

Physics

Difference between transverse and longitudinal excitation

EEL spectroscopy

Optical spectroscopy

Non linear Lorentz oscillator

Anarmonic potential

- induced dipole at frequency and 2
- the system is excited by a frequency but oscillates also at frequency 2
- re-emitting both and 2

Lorentz oscillator in a magnetic field 1/2

x and y motions are coupled

Solving for x and y

Larmor frequency

Lorentz oscillator in a magnetic field 2/2

The dielectric function is a tensor

[ Physically lost of symmetry for time reversal ]

Wave equation

Eigenvalue equation

Note≠ 0 in anisotropic media

with

Considering the medium with B||z

Elliptically polarized

Rotation according to n+-n-

Linear polarized

Longitudinal geometry

N+ Right circular polarized wave

N- Left circular polarized wave

Dichroism

Two waves propagating with two different velocities and different absorption

Magneto-optic effects e.g. Faraday and Kerr effects/geometries

are in general tensorial quantities

Scalar medium

Magnetized medium

Longitudinal and transverse dielectric constant 1/2

Any vector field F can be decomposed into two vector fields one of which is irrotational and the other divergenceless

If a field is expanded in plane waves FT is perpendicular to the direction of propagation.

Optics EELS/e- scattering

The description in terms of longitudinal and transverse dielectric function is equivalent to the description in terms of the usual (longitudinal) dielectric function and magnetic permeability. They are both/all real quantities together with conductivity. They combine together to forming the complex dielectric constant defined here.

Propagating waves and excitation modes of matter are two different

manifestation of the same physical situation

Plasmon is a charge oscillation at a frequency defined by the normal modes oscillation produces a field only a field of this kind is able to excite this mode

_

+

- Modes can be transverse or longitudinal in the same meaning of transverse and longitudinal E.M. field
- searching for transverse waves is equivalent to searching for
transverse modes

Searching for modes eigenvectors of

Transverse modes Polaritons

The quantum particles are coupled modes of radiation field and of the elementary excitations of the system, called Polaritons including transverse (opical) phonons, excitons,….

Longitudinal modes

the quantum particles are coupled modes of radiation field and of the elementary excitations of the system: Plasmons, longitudinal opical phonons, longitudinal excitons,….

Polarization waves

Sum rules for the dielectric constant

Examples of sum rules

Of use in experimental spectra interpretation

Macroscopic optical response

Microscopic structure

Transition probability

Ground state HRADIATION + HMATTER perturbed by radiation-matter interaction

- Two approaches
- fully quantum mechanics
- semi classical

- Three processes
- Absorption
- Stimulated emission
- Spontaneous emission

O ° O

O° O °°O

Term neglected for non relativistic particles

System

Radiation

mi,ei

mj,ej

- Interaction Hamiltonian HI
- Effect of the interaction on the states of the unperturbed HR + HI

Matter

Matter Hamiltonian + perturbation Hamiltonian

Problem to be solved

Eigenstate and eigenvector of the matter radiation system in interaction

The solution is found by a perturbative method

- it is assumed here – formally - that the problem in absence of interactions has been solved.
- In practice this can be done with more or less severe approximations.
- The calculation of the electronic properties of the ground state is a special and important topic of the physics of matter

Many particles state

Generally obtained by approximate methods

The effect of perturbation HI on the eigenstates of H0

Obtained by time dependent perturbation theory

The evolution of the state m is obtained calculating the matrix element

System states under perturbation due to

Changes of photon occupation and matter (f.i. electronic) state

It is found that for photon mode k, only

contribute linear terms to matrix elements

+1 photon emission -1 photon absorption

Probability of transition of the system from state

Spontaneous emission

Stimulated emission

Spontaneous emission present only in quantum mechanics treatment

Integrating in time from 0 to infinity for the transition probabilities

per unit time

for

probability of finding the state in a state n, at thermodynamic equilibrium

Dielectric function and microscopic properties

Dissipated

power

Physical meaning Sum of all the absorbing channels at that photon energy

Note dissipation originates from non radiative de-excitation channels

En’

N(E) density of states

(Number of states/eV)

Joint Density Of States N(E) N(E’)

En

Matrix elements of position operator

Note that the same result can be obtained by considering the transition probability between quantized states of the matter system under the effect of classical external perturbation of the E.M. field with given by the same expression of

This semi classical approach gives identical results for absorption and stimulated emission probabilities, but does not account for spontaneous emission

Generic light polarization

For radiation polarized along z

[linear polarized light ]

Expressions valid in any central field

The basic step in calculation involves

many particles wavefunctions

Born - Oppenheimer approximation

Nuclear motions separated from electronic motions

One electron description

One electron WF Solution of motion in an average potential generated by all other electrons

Case of crystals

K reduced vector within the Brillouin zone

Crystal states E(k)

Joint Density of States - JDOS

- Interband transitions
- direct/indirect
- Intraband absorption
- Phonon contribution
- Core/localized (e.g. molecular) level absorption

Local field effects - Local (Lorentz) field corrections

Probability of relaxation/decay of excited state as integral on all the spontaneous emission channels of field and matter states

As a consequence the dependence of Im () has to be modified

- Lorentzian broadening
- function substituted for by Lorentzian curve

(e.g. see Lorentz oscillator)

Total/Partial yield measurement of absorption through electron (Secondary, Auger,..) and photon (fluorescence, luminescence,…) yields

- De-excitation spectroscopies
- Fluorescence
- Luminescence XEOL
- Auger electron and photon induced – Selection rules and surface sensitivity

matrix element ~ constant I Density of states and 3

From material filling the whole space to material with boundaries and matter-vacuum interfaces

Reflectivity - Measure of the reflected intensity as a function of incident intensity

Fresnels relations based on boundary conditions of fields link

reflected intensity with dielectric function

Surface plane

Normal to surface

Modellisation of surfaces and interfaces

Multiple boundaries

Small and/or rough objects

Scattered wave

scattering medium

Incident field

Term neglected if dimensions

Inhomogeneous filling of space

Scalar theory of scattering(single Cartesian component)

Defining:Scattering potential

Born approximation

The scattering amplitude is the Fourier transform of the scattering potential

Inverting F(r) n(r)

Classical scheme Introduction of the dielectric function

Microscopic (quantum mechanics) treatment of emission and absorption

Relation between macroscopic dielectric function (measured quantity) and microscopic properties

http://www.gfms.unimore.it/

3

5

x 1012

T

320eV

r = 1 m 0.1% BW

4.5

2

4

3.5

//

1

3

Eoy N/C

2.5

0

2

x 1012

1.5

-1

1

Eoz N/C

0.5

-2

0

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

y (mrad)

-3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

y (mrad)

Source: 3.3 m of arc, 3.1 mx 3.3 m vertical x horizontal

two fields – vertical and horizontal – out of phase of ±/2according to the sign of take off angle (J.Schwinger PR 75(1949)1912)

Electric fields

≈ 103 photons/bunch - bunch duration ≈ 20 ps

320eV

r = 1 m 0.1% BW

Selector fully open:

Zc= 45 mm, Zg = 1 mm

S1=-0.9S2=0.011 S3=-0.068

Ey=0.95Ez=0.04 =-1.4

Ellipticity, =0.04

Polarization selector position:

Zc = 34 mm, Zg = 41 mm

(aperture 4 mm)

S1=-0.97 S2=0.011 S3=0.082

Ey=0.98Ez=0.04 =-1.44

Ellipticity, =0.33

Polarization selector position: Zc = 31 mm, Zg = 31 mm (aperture 14 mm)

S1=-0.77 S2=0.08 S3=-0.57Ey=0.93 Ez=0.31 =1.43

P2

Helicity selector

BPM

GAS CELL

P1

EXIT SLITS

MONO

Source 4 m HxV

Mirrors in sagittal focusing reduction of slope errors effects in the dispersion plane

Intensity monitor

Light spot

Energy range 3- 1600 eV

Energy resolution E/E ≈ 3000 (peak 5000) at vertical slit (typically 30 μm) x 400 μm (variable)Variable

divergence (maximum, variable) 20 m vert x hor

ellipticity variable horizontal/vertical (typically in the range 1.5 – 3.5, Stokes parameters (normalized to the beam intensity) S1 0.5 - 0.6, S2 0 - 0.1, S3 0.75 -0.85 )

helicity variable (typical value for rate of circular polarization P or S3 0.75 – 0.95)

plane-grating-plane mirror monochromator based on the Naletto-Tondello configuration

Bulk materials

Surfaces

Interfaces

Absorption

Reflectivity

Fluorescence

Luminescence – XEOL

Diffuse scattering

Experimental arrangements

BEAR (Bending magnet for Emission Absorption Reflectivity) beamline at Elettra

Detection

e- analyser /photodiodes

(2 solid angle)

VIS Luminescencemonochromator

Goniometers

M,A 0.001°

A 0.01°

S 0.05°

C 0.1°

(Positive -Differentially pumped joints)

Sample manipulator

6 degree of freedom

Rotation around beam axis any position of E in the sample frame

See f.i. Mónica Fernández-Perea, Juan I. Larruquert, José A. Aznárez1, José A. Méndez Luca Poletto, Denis Garoli, A. Marco Malvezzi, Angelo Giglia, Stefano Nannarone, JOSA to be published

- ultra-thin deposited films
- buried interface spectroscopy

- Devices of use in spectroscopy

Z

See also posterP III 26

ML : Artificial periodic stack of materials

(Optical technology Band pass mirrors)

BRAGG

At Bragg Z dependent standing e.m. field establishes both inside the structure and at the vacuum-surface interface modulated in amplitude and position

Scanning through

Bragg peak

In energy or angle

Spectroscopy of interfaces

Physics of mirror/Reflection

Si

Mo

Si

Local modulation of excitation

Photoemission, Auger, fluorescence, luminescence etc..

Cr2O3

(6 Å)

Cr

(15 Å)

Sc

(25 Å)

573 eV

X 60

Qualitative analysis

-Opposite behavior of Cr and Sc

- Different chemical states of the buried Sc
- Two signalsfrom oxygen: one bound to Cr at the surface, the second coming from the interface
- Carbon segregation at the interface

Mo

(39.6 Å)

Ru

(15 Å)

Si

(41.2 Å)

Angular scan through the Bragg peak

at 838 eV

X 40

silicide

- Interface morphology
- Calculation of e.m. field inside Ml
- Photoemission was calculated, (Ek= h - EB)
- Minimum position and lineshape depend critically on the morphology profile

Ru

Ru-Si

Mo/Si

61.2 %

58.4 %

Wavelength [nm]

Motivation role of ion kinetic energy and flux during ML growth

ML (P 8 nm, 0.44) Performance- R (10°)

Ion assistance

Ions EK: 5 eV (1st nm), 74 eV > 1nm Controlled activation of surface mobility

Performance differences are to be related to interface quality

Diffuse scattering around the specular beam was measured

KS=Ki + qZ + q//

f.i. Stearns jAP 84,1003

I.f.roughnes produces diffuse scattering around the specular beam

I.f.roughnes can/can not be coherently correleted through the ML

Description on a statistical base, ….fractal properties

Single interface – Autocorrelation function

In plane Fourier transform on q//of potential

KS=Ki + qZ + q//

MLs : S(q) two terms

- incoherent scattering by single interfaces
- correlated/coherent scattering among i.f.
- (interlayer replica of roughness)

See f.i. Stearns JAP, 84, 1003, 1998

Detector

qdetector 0.003 nm-1

At 0.48 nm (13.1 eV)

- scan

Rocking scan

Incident beam

Divergence

qdiv 0.0005 nm-1/m

At 0.48 nm (13.1 eV)

ξ=400 Å

ξ=200 Å

ξ= 300 Å

ξ=120 Å

Correlation function

In plane correlation function - absence of interface correlation

- ξ , correlation length h, fractal dimension/jaggedness

See also poster Borgatti et al. P III 17

Mo/Si

Ion assistance

See also poster Pedio et al. P II 33

Chemisorption morphology - tilt angle & electronic structure

( Concentrating on 1 Mono layer )

- Premise about C22H14/substrates
- He scattering on pentacene deposited by hyperthermalbeams 1ML planar
- Morphology and electronic properties ( delocalization of the p electrons) transport properties highly anisotropic;
- on Metals: nearly planar orientation a condition hindering the formation of an ordered overlayer;
- on semiconductors/oxides: SiO2 standing
- GeS lying.

( By He scattering )

Danışman et al. Phys. Rev. B 72, 085404 (2005)

C3 symmetry

Oblique cell Periodicity (6 x 3) ,

1 monolayer

(At magic angle 54.7°)

Gas phase XAS

(Alagia et al.JChemPhys 122(05)124305)

Redistribution of the oscillator strength in the C1s – LUMO excitation region (1-3 of gas phase)

Tilt angle

LDA calculationsC22H14/Al(100)

Simeoni et al. S.Science 562,43 (2004)

EF

Redistribution of states upon chemisorption

HOMO-LUMO gap increasing

Dichroism/Bond directionality & Tilt angle of the molecule

ε= EV/EH = 0.29

i= 10°

Fit parameter: θ (polar angle of dinamic dipole )

Fit function for Single domain

P