1

Dipartimento di Fisica, CNISM, Università di Roma “La Sapienza”

Piazzale Aldo Moro 2, I-00185 Roma 2

Dipartimento di Scienze, Università Roma Tre and CNISM Unità Roma 3

Via della Vasca Navale 84, I-00146 Roma

Abstract Over the past five decades, photoemission has become one of the most popular spectroscopies for investigating matter in its various aggregation states. It finds its application in fields ranging from solid state physics to chemistry, biology and geology, just to mention a few. The number of researchers who need to be introduced to this field is ever increasing, and the spectrum of backgrounds that they possess is very large. Aim of this chapter is to provide for all of them a basic introduction to the models upon which photoemission is based and, by the help of simple examples, to explain what information photoemission can provide.

9.1 Introduction An electron bound to an atom, a molecule or a solid, is characterized by energy, momentum and spin quantum numbers. Among other spectroscopic techniques, photoemission is the best suited tool to study these physical quantities. It is based upon the photoelectric effect, one of the cornerstones on which quantum mechanics description of matter rests. In essence, the photoelectron effect amounts to shining a monochromatic electromagnetic radiation (hν) on a sample and producing free electrons with a well-defined energy spectrum. The Einstein equation connects energy of the quantum of the electromagnetic field (photon) with the MAX

maximum energy of the ejected free electron ( Ee teristic of the sample (work function Φ)

MAX e

E

) through a constant charac-

= hν − Φ . It establishes a close

relationship between sample characteristics and energy spectrum of the ejected electrons and suggests using the photoemission process to build up a wide range of different spectroscopies aimed at studying the electronic structure of matter in its various aggregation states. Such a spectroscopy allows investigation of the occupied electronic states, it gives information on the dielectric (insulator, semiconductor or metal) and chemical state, on the magnetic properties and on the local structure. Although the photoelectron effect is known since over a century [1-4], spectroscopies based upon this phenomenon [5-6] have developed over the past fifty years, mostly driven by progresses in development of monochromatic, bright and tunable light sources [7,8]. Nowadays, photoelectron spectroscopy (PES) has practical applications in many fields of science, such as surface physics and chemistry, material science, nano technologies and still significantly contributes to the understanding of fundamental aspects of physics, chemistry, biology, etc. [9]. The ubiquitous use of photoelectron spectroscopy is testified by the presence of several beam lines devoted to PES in whatever synchrotron radiation source throughout the world. Aim of the present chapter is to establish the fundamental concepts upon which PES relays in order to turn a photoemission experiment into a flexible spectroscopic tool to investigate ground state electronic properties of matter. To this end, we shall make use of the quantum description of the interaction between electromagnetic radiation (EM) and matter outlined in Bertoni’s chapter [10]. A selected set of experiments on quantum objects, starting from simple atoms and ending with solids and surfaces, will provide evidences for discussing values and limitations of the approximations needed in order to turn a photionization process into a spectroscopic tool. This brief introduction to photoelectron spectroscopy with synchrotron radiation, will not rigorously and exhaustively treat the process and the scientific results. The reader will be introduced to the fundamentals of photoemission theory and to the various methods of photoelectron spectroscopy with synchrotron radiation, with a few scientific examples. For in depth description of the photoelectron spectroscopy, we refer to review literature [6,9,11-13].

9.2 Basic concepts The schematics of a modern photoionization spectrometer is shown in Fig. 1. In essence, the experiment amounts to measuring the photoelectron current (Je) as a function of the electron kinetic energy (Ee), the electron ejection angles (θ,φ) and the spin (σ) for each given energy (hν) and polarization vector (ε) of the incident photons. To this end, it is mandatory to analyse in energy, momentum and (possibly) spin the photoelectrons before detecting them and measuring Je. This is usually

achieved by electrostatic spectrometers in conjunction with electron multipliers that allow for detecting single electrons [14].

Fig. 1. Schematics of a photoionization measurement. A monochromatic photon beam (hν) impinges on the sample; photoelectrons emitted within the solid angle Ω defined by the electron detector (Ee) are energy analysed, detected and properly histogrammed as a function of their kinetic energy.

The strong interaction of electrons with matter imposes to perform experiments under vacuum (at least 10-7 mbar) in order to measure a probability distributions of photoelectrons not altered by interaction with the background atmosphere surrounding the sample. As far as photon source is concerned, there are two different types of quasi monochromatic excitation sources that are available under laboratory conditions, namely VUV line spectra of discharge lamps for energies in the range 10-50 eV (UPS), e.g. with rare gases like helium (HeIα=21.218 eV and HeIIα=40.814 eV) and the characteristic lines from the x-ray source (XPS) for which the most commonly used anode materials are aluminium (Al Kα1,2=1486.6 eV) and magnesium (Mg Kα1,2 = 1253.6 eV). Although the line width is small enough for many applications, i.e. few meV for discharge lamps and slightly below 1 eV for x-ray anodes, the use of an additional monochromator can be advantageous for the energy resolution and, more importantly, for suppression of background and satellite intensities. More recently, SR sources have allowed for enhancement of several orders of magnitude in resolution, tunability, wavelength span and polarization control in the UV and X-ray sources (see for example [7]). In the photoemission process, the sample is left in an ionized state when the electron is emitted. Both the sample (atom, molecule or solid) and the emitted electron can be viewed as excited, differing from the fundamental state by the incident photon energy. The photoemitted electron current Je, as measured by an ideal spectrometer resolving energy, angle, and photon and electron polarization

states, is a function of 10 variables Je = f(hν,ε, θ, Φ; Ee, σ, θe,Φe), as shown in Fig. 2:

Fig. 2. Schematic diagram of the photocurrent Fig. 3. Electron mean-free-path in a solid, as a Je and of all the variables. function of photoelectron energy.

In a typical experiment, only a few of these parameters are varied keeping the others either constant or being integrated. For example, in the Energy Distribution Curve (EDC) method, the current Je is measured as a function of the photoelectron energy Ee, while keeping the other variables fixed. In the angular resolved mode, the θe,Φe angles are resolved, otherwise the technique is angular integrated. Moreover, if it is possible to analyze the photoelectron spin state σ, the technique gives information on the magnetic state of the sample. Before entering into the details of the connection between Je and the photoionization cross section, we stress the high surface sensitivity of this technique. In fact, while photons typically penetrate the solid for tens of nm, obviously depending on the material type and photon wavelength, the typical energy of the photoemitted electrons lies in the 5-1500 eV range and their escape depth is limited to a range from a few to a few tens of Å. The escape depth determines the depth from which photoemitted electrons escape from the solid into vacuum without loosing energy. It can be phenomenologically described as an exponential attenuation of the photoemitted current as a function of the depth from which the photoelectron is emitted. The escape depth value is limited by the electron-electron scattering, mostly the excitation of collective modes (plasmons), while the electrons escape into vacuum from the solid. Thus, a fraction of photoelectrons (primary electrons) escape into vacuum without suffering scattering and energy losses, while a large part of photoexcited electrons (secondary electrons) scatter and suffer various energy losses, giving rise to an unstructured background towards low kinetic energy in the photoelectron spectrum, and essentially loosing information. The escape depth, that is directly proportional to the mean free path λ, is a function of the

electron kinetic energy and follows a general behavior, with a minimum value (thus a maximum surface sensitivity) roughly between 10 and 50 eV. There are analytical functions derived for the different materials [15]: as an example the mean free path measured for a series of semiconducting materials [12] is shown in Fig. 3.

9.3 Energy conservation, binding energy and photoelectron energy The photoelectric process can be described as the transition of one electron from an initial occupied state to a final empty free-electron level in vacuum. The photoelectron current Je is a macroscopic quantity directly linked to the photoionization microscopic cross-section through the relation

Je =

∫∆Ω dΩ ∆∫ dE Iρ ( Ω σ ) K e

d A d dEe

s

where ∆Ω and ∆Ee are the electron spec-

Ee

trometer angle and energy acceptances, respectively, I is the incoming photon current, ρA is the sample areal density, Ks the spectrometer efficiency and dΩdσdEe the

(

)

photoionization differential cross section. Hence, to highlight the information contained in the photoelectron current we shall discuss the photoelectric differential cross section. To this end, and for sake of simplicity, let’s take the simplest many electron quantum system we can find in nature: i.e. the He atom. In such an atom two electrons are bound to a doubly charged nucleus in a n=1 ℓ=0 m=0 s= ± ½ quantum state 1s2 whose spectroscopic notation is 1S0. According to treatment of EM interaction with matter [10] and within the validity of the dipole approximation, the photoabsorption cross-section by a quantum object can be written as: 2

r σ = 4π αhν ∑ εˆ • ΨB ∑ ri ΨA δ ( EB − E A − hν ) 2

B

(1)

i

where εˆ is the light polarization, Ψ A is the initial state, Ψ B are the final states of the system and the index i runs over the target electrons. To achieve photoionization of He, one of the two electrons must be promoted to the continuum (Ee>0 with respect to vacuum level), see Fig. 4. Hence, in an experiment where a monochromatic photon energy is larger than the threshold energy (i.e. the minimum energy needed to ionise the system, see Fig. 4) is absorbed by an He atom, the energy conservation δ of equation (1) suggests that electrons are generated with a kinetic energy that satisfies the relation

E B = E A + hν

(2)

Fig. 4. Diagram of the energy levels of the He atom; the arrow represents the photon energy required for the photo ionisation process to take place (hν),

Φ1

and

ε2

are

the

single

electron initial and final states. The label ε2p recalls that dipole selection rules imply a continuum state with ℓ=1.

Let’s now assume that initial and final states are correctly described in terms of

ΨA = Aˆ φ1φ 2 and ΨB = Aˆ φ1ε 2 where ε2 is a single particle continuum wavefunction and Aˆ is an

antisimmetrized product of fully independent particles: i.e.

antysimmetrization operator. Under these conditions equation 2 applied to the case sketched in Fig. 5 reads

E1s + Ee = E1s + E1s + hν

Ee = E1s + hν Ee = hν − BE1s (24.6eV )

(3) Hence, in the Je distribution a single peak is to be expected whose kinetic energy is directly linked to the unperturbed initial state single particle binding energy (BE). In other words: photoemission spectra should give direct access to spectroscopy of single particle bound states of a multi electron system such as electronic orbitals for atoms, molecules and clusters. A more elaborate concept is needed for photoemission from solids [9]. Is reality as simple as we think? Let’s examine the Je experimental data for Helium (He). For this atom, the independent particle model is expected to be realis-

tic, and the energy level diagram of Fig. 4 can be used to represent both initial and final states. In Fig. 5 it is reported the experimental photoemission spectrum of He as measured with a monochromatic beam of energy 89.5 eV, i.e. the Je(Ee) at a fixed hν. In case of validity of the fully independent particle model, the observed photoelectron spectrum should consist of a single peak, main peak, at the photoelectron energy that satisfies equation 3, the one labeled as n=1 at a binding energy of 24.6 eV, that corresponds to the transition

hν + He(1s 2 ) → He + (1s 1 ) + ε l =1

Fig. 5. The He photoemission spectrum as measured at hν=89.5 eV. The peak labeled n=1 corresponds to the transition hν + He(1s2) -> He+(1s1) + ε l =1, where the label n refers to the principal quantum number of the excited ionic state. From [16]. Copyright Elsevier, reprinted with permission.

Further photoemission peaks appearing in the spectrum labeled as n=2,3,…. and the continuum distribution extending above the He++ threshold energy, cannot be interpreted in terms of independent particle model and to explain these so called satellite structures many-body properties of the systems should be taken into account. Similar satellite structures are observed for all the rare gases [17, 18] and they become more and more relevant as the atomic number increases. We shall see in the following that for targets of increased complexity, such as heavy atoms, molecules, clusters or solids, the photoelectron spectrum is accordingly more complex. To interpret these spectra asks for models more sophisticated than the one outlined in equations 3, and this will be discussed in the following paragraphs.

9.4 Satellite structures To understand the origin of the satellite features appearing in the photoemission spectra we shall use as a case study the He spectrum. Besides the peak associate with the final state 1s1 foreseen by the independent particle approach, a manifold of discrete transitions at larger “apparent” BE followed by a continuum, above the double ionization threshold, is also clearly observed in Fig. 5.

Fig. 6. The energy level diagrams are sketches of the mechanism responsible for generation of the main peak (one electron transition) and satellite structures (two electron transitions, see text for details).

In building up the aforementioned energy conservations (eq. 3), we have implicitly assumed validity of the “frozen core approximation”, i.e. relaxation of the sample upon creation of a hole-state and electron-electron correlation is neglected. To overcome these limitations the system is to be described as formed by N interacting electrons the energy eigenvalues of which are defined according to the Schrödinger equation:

H 0 Ψ A( N ) = E A( N ) Ψ A( N )

(4) ,

in which the initial state Hamiltonian is

H 0 = H 0 (kin) + H 0 (e − n) + H 0 (e − e) + H 0 ( s − o) = N N r r p2 Ze 2 N e 2 N = ∑ i + ∑− + ∑ + ∑ ζ ( r j )l i • s i 1 2m 1 1 i> j r ri ij

(5)

and the N-particle wave function is simply described by a Slater determinant of one electron spin-orbitals for both the initial state

r ΨA( N ) = Â (φ j (r j ,σ j ); ΨR( N −1) )

(6)

with Â an antisymmetrization operator rendering the N particle state compatible ( N −1)

as a minor at N-1 electron of the N electron mawith the Pauli principle, ΨR trix and φj the single particle spin orbital of the j-th initial state from which the photoelectron is extracted. For the final state of the photoionization process

H 0' ΨB( N ) = E B( N ) ΨB( N )

(7)

where H0’ is the final state Hamiltonian (note that in general H0 is not equal to H0’) . Assuming validity of the Sudden Approximation: i.e. the electron in the contin-

r

uum state ε l ,with kinetic energy Ee and momentum K e , (photoelectron) is fully decoupled (does not interact) from the residual (N-1) particles ion, the final N-particle state can be expressed as

ΨB( N ) = Aˆ (ε l ; ΨB( N −1) )

(8)

To appreciate consequences of the many-particle descriptions of the atom on the Je spectrum, let’s write down the photoemission cross section at fixed photon energy and differential in the photoelectron energy and emission angle. It derives directly from the absorption cross section (1) that, in the case of a discrete-to- continuum transition, represents the total probability of photoemission over the 4π solid angle. By replacing expressions (6) and (8) in equation (1) we obtain:

dσ dΩdEe

∝∑ B

r

r

εˆ • ε l r j φ j (r j , σ j ) Ψ B( N −1) Ψ R( N −1) δ ( Ee + E

( N −1) B

2

(9)

− E − hν ) N A

Assuming validity of the Koopman’s theorem, i.e. the target atom electronic structure stays frozen under creation of the hole state, Hamiltonians of initial and final states are identical (H0=H0’) and the monopole matrix element appearing in (9), usually termed Fractional Parentage, is 0 unless the ionic state is equal to the frozen residual (N-1) configuration of the initial state. As a consequence, the photoemission cross section will be different from 0 (Je different from 0) only for the ionic states that correspond to the frozen residues at N-1 of the initial N particles ground state. In this way relaxation and correlations are overlooked, energy conservation reduces to the δ function appearing in (9) and the Koopman’s energy of the photoelectron is directly linked to the binding energy of single particle orbital involved in the photoionization process. Under these circumstances, the only allowed photoemission process for He is: hν + He(1s ) → He (1s ) + ε l =1 that is associated to the most intense peak appearing in Fig. 5 (Main line or adiabatic peak). To explain the faint structures highlighted in Fig. 5, the condition H0=H0’ must be relaxed, thus allowing for a spectrum of possible final ionic states to be associated with removal of an identical single particle spin orbital. There is a manifold of different final states associated to each individual single particle hole state that imply excitation of further electron(s) of the target both to discrete and continuum empty states (shake-up and shake-off satellites respectively). The 2

+

1

monopole matrix element of (9) determines which final states are allowed (monopole selection rules) and with which probability. In the case of He, for instance, the main allowed transitions present in the spectrum are:

hν + He(1s 2 ) → He + (1s1 ) + ε l =1 hν + He(1s 2 ) → He + (2 s1 ) + ε l =1 , He + (3s1 ) + ε l =1 , He + (4 s1 ) + + ε l =1 ,.............He + + + ε l + ε l ' hν + He(1s 2 ) → He + (2 p1 ) + ε l =0 , He + (3 p1 ) + ε l =0 , He + (4 p1 ) + + ε l =0 ,..........He + + + ε l + ε l '

These transitions explain most of the satellite structures observed but the next question is: can Ee still be directly connected to single particle binding energies? It can be shown that sum rules applicable to photionization process imply that: i) the weighted average of all binding energies (main, shakeup and shake off) yields the Koopman’s Theorem binding energy ( BE nlm ), ii) the sum of all intensities is

⎛ ⎝

proportional to the frozen orbital cross section ⎜ εˆ •

r

r

ε l r j φ j (r j , σ j )

2

⎞. ⎟ ⎠

This kind of treatment is very convenient for atoms, molecules and core states of solids, i.e. for all localized electron states. In the case of valence states of solids, i.e. whenever we are dealing with delocalized, continuum electronic state, it is more convenient to formulate the matrix element

r

r

ε l r j φ j (r j , σ j ) ΨB( N −1) ΨR( N −1) r

of (9) as

r

r

r

ε l r j φ j (r j , σ j ) A(k , E )

where A( k , E ) is the so called spectral function of the target that can be related to the single particle Green’s function. In spite of the formal differences, the conclusions that we have reached on the meaning of main and satellite lines of a photoelectron spectrum are valid independently of the state of aggregation of the target. In particular, the richness of satellite structures is directly linked to the degree of electron correlation that affects the initial bound state involved in the photoionization.

9.4.1 Spin-orbit splitting Besides the aforementioned satellite structures, further splitting of the photoemission peak associated with l ≠ 0 hole states is noticeable. This is particularly evident in the case of the Xe 3d doublet. This is a typical final state effect. In these closed shell atoms the initial state energy doesn’t depend upon spin momentum projections as all orbitals are fully occupied. On the contrary, when a vacancy is created in the orbital identified by n,l,m quantum numbers, the energy of the final

state depends on the spin projection through the term H0(s-o) of the Hamiltonian (5). Hence, it is to be expected that holes in p (l=1) and d (l=2) will generate doublets of final states characterized by quantum number j (j=l+s) equal to 1/2, 3/2 and 3/2, 5/2 respectively.

9.4.2 Multiplet splitting In moving from closed to open shell systems (i.e. quantum systems in which orbitals are partially occupied) complexity of the photoelectron spectrum increases and splitting of peaks not ascribable to spin-orbit interaction is observed. Such effects, usually termed as “multiplet splitting”, were first observed in connection with core photoemission from paramagnetic free molecules and transition metal compounds. An introductory example of multiplet splitting as observed in the photoemission spectra from core O 1s electrons in the paramagnetic free molecules O2 is given in Fig. 7 [19]. The electron configuration of the neutral ground state is:

KK (σ g 2s ) 2 (σ u 2s ) 2 (σ g 2 p) 2 (π u 2 p) 4 (π g 2 p) 2 3 Σ g As shown in Fig. 7, ionization of the K shells leads to two different core hole multiplet states of O+2, one with Σ g - symmetry, the other of Σ g - symmetry. Fig. 7 also shows the XPS spectrum with two separate lines corresponding to these two different final electronic states. 2

4

Fig. 7. O 1s core photoemission spectrum of molecular oxygen as measured with 1487eV photons. The observed multiplet splitting of 1.11eV correspond to the two possible final states outlined in the left side schematics where energetic of the initial, neutral, and two final ionic states is shown [19]. Copyright Institute Of Physics, reprinted with permission.

The transitions observed in the photoelectron spectrum can be schematically illustrated as shown in the diagram of Fig. 7. The difference in transition energy towards the two different final states, 1.11 eV, arises from spin coupling of the

residual 1s core electron with the valence electrons unpaired spins. The relative weight of the two peaks obeys the statistical ratio expected for quartet versus doublet intensity, i.e. 2. It is worth noting that multiplet splittings are observed for both core and valence photoelectron spectra. The valence spectrum of the oxygen molecule, for example, shows very well separated doublet and quartet states. It may also be noted that multiplet splittings are common ingredients in core hole relaxation processes, such as Auger electron spectra, since the final states normally contain two vacancies which may couple in several different ways and give rise to states of different energy.

9.4.3 Chemical shift It is well known that when aggregating atoms to form molecules, clusters or solids, the valence electrons are mostly involved to form bonds while core state remain almost unchanged in their atomic character (i.e. there is a negligible overlap between adjacent atoms core orbitals). Nevertheless, a considerable fraction of photoelectron studies has been primarily involved with the precise measurement of core hole electron binding energies and in particular of what is known as the “chemical shift” suffered by these energies when the chemical environment within which the atom is bound changes. Let’s illustrate this effect with an experimental example. In Fig. 8 it is shown the photoelectron spectrum of the Ethyltrifluoroacetate molecule [20]. In this organic molecule four carbon atoms are present in four different chemical environments.

Fig. 8. C 1s photoelectron spectrum of ethyltrifluoroacetate showing four different lines due to the chemical shift. The spectrum was excited by monochromatized radiation at 340 eV [20]. Copyright Elsevier, reprinted with permission.

Each individual carbon atom gives rise to a different core photoemission peak, thus allowing for investigating the molecular electronic structure with an atomic scale resolution even though this technique has no spatial resolution. Similar differences are observed for all atoms of the periodic table (C 1s 285-300eV, O 1s 530-540eV) and for the same atom bound in different molecules to elements with

different electronegativity ( C CO2 298 eV, C CH4 290.7 eV). The observed difference can be essentially understood in terms of simple electrostatic interactions, when bound to elements with larger electronegativity the binding energy is higher as the screening charge on the atomic site is reduced with respect to the free atom. Vice versa, when bound to elements with smaller electronegativity there is more charge available for screening and e-e-interaction, and the core binding energy accordingly decreases. The chemical shift provides chemically-significant information concerning the initial state electronic structure of the system under study. Both initial (electronegativity) and final (relaxation of neighboring electrons on the hole) state effects are influenced by chemical environment and contribute to determining the chemical shift. Empirically, there seems to be a correlation, although exception exists, between chemical shift and electronegativity of neighbor atoms. For small electronegativity differences ∆χ the core level shift ∆Ε is nearly equal to ∆χ. At larger ∆χ the energy shift saturates, probably due to saturation of the charge transfer. From such an empirical behavior, one may predict core level shift for new ligands, or find out from a measured core level shift which ligand (electronegativity of the ligand) is involved. To accurately calculate chemical shifts, the total “all electron energies” with and without core hole is to be taken into account. In spite of the complexity of this phenomenon, an overall proportionality between binding energy and electronegativity is found in most of the cases. Core energy shift is also associated to changes in coordination number of the same atom within a given aggregate. Clusters give a clear example where atoms bound to the outermost shell (surface) experience a chemical bond different from those bound to the inner shells (bulk) and the chemical shift is accordingly different.

9.5 Molecular photoelectron spectroscopy The next complex quantum system whose electronic structure is to be investigated by PES is a molecule. It is evident that the extra degree of freedom added to the system by the nuclear motion (i.e. molecular vibrations and rotations) will play a role in the related photoelectron spectra. In this case, energy can be transferred from the electromagnetic field to electronic, vibrational and rotational degrees of freedom of the quantum system. In other words, we ask ourselves how vibrations and rotations will influence molecular photoelectron spectra. Even though we start from a ground state configuration, vibrational and rotational excitations are to be observed besides the electronic ones. Let’s take as an example a simple diatomic molecule, the basic principles laid down in this case remain valid for more complex polyatomic molecules though experimental and theoretical treatments will become accordingly more complex.

9.5.1 Vibrational overtones Nitrogen is an archetypal for diatomic molecules and its photoemission spectrum [21] is shown in Fig. 9 as a function of the single-particle binding energy, alongside with the molecular orbital assignments. It shows that the three outermost occupied orbitals give rise to three distinct multiplets.

Fig. 9. Molecular PES of nitrogen excited by HeI radiation. The molecular orbital assignment is reported alongside the spectrum. [21]. Copyright Elsevier, reprinted with permission.

The existence of a fine structure (side bands) for the individual electronic states is ascribable neither to spin-orbit nor to multiplet splitting. This fine structure is due to an effect that we have neglected so far, namely the generation of the ionized molecule in excited vibrational states. Vibrational effects are observed in core photoemission spectra as well, as it is demonstrated by the C 1s photoionisation spectrum in CH4, it displays two overtone peaks displaced by 0.43 and 0.86 eV, respectively, from the carbon main transition [22]. The observed energy splitting is consistent with the vibrational quantum in CH4(C1s-1) while the relative intensities are explained in terms of Frank-Condon factors for transitions to the v=1 and v=2 ionic vibrationally excited states. Vibrational excitation of the initial ground state is not taken into account as the sample was at room temperature (kT = 26 meV) and the nitrogen vibrational quantum is of the order of 400 meV. Vibrational fine structure is usually clearly visible in UPS (typical energy resolution 0.015 eV) but is obscured in XPS (typical energy resolution 1 eV). Note that a resolution of 0.015 eV corresponds to 120 cm-1 (1 eV = 8066 cm-1). This is sufficient to observe a vibrational fine structure but not enough to resolve the rotational fine structure.

To explain the main features of the molecular photoemission spectrum we shall calculate the differential photoemission cross section taking into account that initial and final single particle electronic states are molecular states. For core holes, localization of the electronic state makes atomic like orbitals appropriate for describing both bound and continuum states and an atomic like cross section is appropriate. The situation is more complex for valence delocalised initial states (molecular orbitals) that are described in terms of the usual Born-Oppenhaimer approximation (BO), i.e. as the product of linear combinations of atomic orbitals (LCAO) with independent nuclear motion wavefunctions. Initial and final states should take into account the full geometry of the molecular system and the photoelectron continuum state should account for the interaction with the residual molecular ion, though an atomic continuum model can be assumed valid at high energy and large distance from nuclei. The simplest approximation to continuum wave functions is the Plane Wave (PW) but several other more elaborate models have been successfully applied (OPW, multiple scattering, etc.). To calculate the molecular photoemission cross-section we shall follow Gelius approach for XPS where the photoelectron is described as a PW and initial states are described by LCAO. Furthermore, the BO approximation allows to factorise the WFs in a product of electronic and vibrational WFs (we neglect rotational states as they are usually not resolved in PES and XPS). In other words, the system hamiltonian should now include the nucleus-nucleus interaction

H 0 = H 0 (kin) + H 0 (e − n) + H 0 (e − e) + H 0 ( s − o) + H 0 (n − n) = N r r M e2 Zi Z j pi2 Ze 2 N e 2 N =∑ + ∑− + ∑ + ∑ ζ (rj )li • si + ∑ rij ri i> j 1 2m 1 1 i > j rij

(10)

N

( N ) el

and the WFs read: ΨA, B = ΨA, B (N )

ΨAvib, B . Hence, substituting these initial

and final states WFs in the photoemission cross-section (9), we obtain the molecular expression of the photocurrent Je as calculated under the aforementioned approximations.

dσ dΩdEe

∝∑ B

el el r εˆ • ε l r j ∑ C Aλφ Aλ Ψ B( N −1) Ψ R( N −1) Aλ

2

Ψ Bvib Ψ Avib δ ( Ee + E B( N −1) − E A − hν )

2

(11)

where

∑λ C λφ λ A

A

is the LCAO expression of the single particle molecular obital

A

from which the photoelectron is extracted. In the molecular photoelectron spectrum we shall therefore expect as many peaks as many ionic electron-vibrational states are allowed by monopole and vibrational selection rules and with intensities proportional to product of the relevant Frank-Condon (FC) and FractionalParentage coefficients. The way FC coefficients will work in a molecular photoionization process is sketched in Fig. 10, where the relative probability for exciting final ions in v=1,2,3,….. states is depicted as a function of the molecular equilibrium distance of the final ion. As it gets larger and larger with respect to the ground neutral state equilibrium distance (from panel (a) to (d)), increasingly higher v states are accessed in the final ion, till the fragmentation threshold is reached and excitation (discrete vibrational structure in the photoemission spectrum) as well fragmentation (continuum vibrational structure in the photoemission spectrum) are present.

Fig. 10. Schematics of the dependence of PES features upon change of the molecular equilibrium distance in the final state molecular ion. Minimum change is the leftmost panel, maximum change is the rightmost one.

Multiplet splitting is present in molecular spectra as well. A clear example of the experimental vibrational structure observed in the valence photoionization spectrum of molecular oxygen in conjunction with the aforementioned quartet doublet multiplet splitting is given in the PES of O2 with HeI exciting light [23]. Extensive vibrational structures have been observed also in PES of polyatomic molecules, water is a relevant example. The PES spectrum excited by HeIα radia-

tion at 21.218 eV shows the three outermost valence orbitals whose binding energies have been calculated using a one-electron method (HF) and a many-electron (CI) method. Comparison of the predicted binding energies with the center of mass of the vibrational structure associate to each individual orbital highlights the extreme sensitivity of PES to accurateness of the model adopted in describing the molecular valence orbitals . Sensitivity of PES vibrational overtones to nuclear masses is demonstrated by the spectra of isotope modified water molecules, 18O instead of 16O and D instead of H [24]. Modern molecular PES studies include spectroscopy of transient states and species as well, but this subject is well beyond the scope of this chapter. Rotational fine structure is normally not resolved in photoelectron spectra. At a resolution of about 10 meV, the line profiles may be influenced by the rotational excitations, but the individual lines cannot be resolved. At a resolution below about 5 meV, which can be achieved with modern UPS spectrometers, these influences become clearer and can sometimes be used to draw conclusions about the molecular geometry and the intensity of different rotational branches. For diatomic molecules, and larger molecules containing hydrogen, like H2O or HF, for which the spacing between the individual components of the rotational structure are comparatively large, it may even be possible to observe this fine structure [25].

9.6 Photoelectron angular distributions Till now we have been dealing only with the energy distribution of the photoelectron probability current, but the cross section (9) implies that a distribution in space (ejection angles ) exists as well. We shall discuss this aspect of photoemission having in mind that it has a twofold relevance. On the one side angular distribution of photoelectrons is relevant to highlight fine details of photoionization dynamics (hence they are a stringent test for quantum description of radiationmatter interaction). On the other side, accurate description of this phenomenon provides the basis for various spectroscopies based on diffraction of the photoelectron from surroundings atoms that are aimed at studying local geometrical order in molecules, clusters and solids. Let’s start from the simple atomic case. Starting from equation (9), it can be shown that, for linearly polarized light and for transition to a well defined ionic state (for fixed photon and photoelectron energies that satisfy the energy conservation δ function), the photoemission cross section reads:

r r dσ ∝ ∑ εˆ • ε l r j φ j (r j ,σ j ) ΨB( N −1) ΨR( N −1) dΩ B

2

(12)

Fig. 11. Schematics of an angle resolved photoemission experiment. The linearly polarized light hν photoionises an electron with kinetic energy Ee within the solid angle dΩ in direction (θ,φ) with respect to the electric polarization vector ε. θ and φ are polar and azimuthal angles, respectively.

Hence, we expect the photoelectron current to depend upon direction under which the photoelectron is detected, with respect to polarization vector (see Fig. 11), and characteristics of the single particle orbitals involved, both bound and continuum ones. All molecular and atomic orbitals have a characteristic angular distribution of electrons. This means that the intensity of all final states that we measure in a spectrum will have a certain angular dependence. Upon validity of the approximations of expression (9), the angular distribution of the emitted photoelectrons can be described using one single parameter, the asymmetry parameter β [26]. In practice it means that the photoelectrons are symmetrically distributed about the polarization direction. Furthermore, upon dipole approximation the photoelectron will be ejected with a continuum wavefunction with l = ±1 with respect to the single particle hole quantum state left behind (this is because the incoming photon carries an unitary angular momentum). This gives an angular distribution of the ejected electron that obeys the relation:

σ dσ [1 + βP2 cos(ϑ )] ∝ dΩ 4Π

(13)

where β spans from –1 to 2 and depends on the initial and final state orbitals involved in the ionization process. Expected angular dependence of the atomic photoelectron current for selected β values are reported in Fig. 12.

Fig. 12. Angular distribution of photoionisation in polar coordinates experimental (right panel) and calculated according to (13) for different values of β. The polarisation vector ε is reported as a black bold arrow. Magic angle has been indicated by a black point. Experimental results are relative to He and the best fit to the data is obtained for β=2.

By studying this figure it can be seen that there are four directions in which photocurrents are expected to be independent from β values. This occurs at the so called magic angle of 54.74°. At this angle the photocurrent depends only on the total cross section σ and not on the light polarization or the angular momenta of the photoelectron wavefunction or the symmetry of the initial state. That this description is mostly correct in the case of isolated atoms is demonstrated by the good agreement between experimental results on He (dots) and the theoretical prediction for β=2 (full line) shown in the right panel of Fig. 12. What happens to photoelectron angular distributions when the atom from which the photoelectron is generated is surrounded by neighboring ones (i.e. in molecules, clusters and solids)? The photoelectron wavefunction gets scattered from each individual atom and the photoelectron analyzer collects a photoelectron current that is a coherent superposition of source photoelectron wavefunction and point scattered wavefunctions, as schematically shown in Fig. 13. The angular distribution is not any more a smooth distribution determined by β, it will show two main features: a forward intense peak along atom-atom direction, and a diffraction pattern at all other scattering angles. It can be shown that while in the case of an isolated atoms the measured photoelectron current is proportional to the square modulus of the unperturbed photoelectron wave function, 2

i.e. J e ∝ Φ 0 , in the presence of a scatterer it becomes proportional to the square modulus of the coherent sum of the unperturbed and scattered photoelectron wavefunctions, i.e. J e ∝ Φ 0 + Φ S

2

. If we perform measurement of the

angular distribution of the photoelectrons ejected from a fixed in space molecule, what is observed is schematically shown in Fig. 14 for the case of an aligned

Fig. 13. Schematics of an angle resolved photoemis- Fig. 14. Pictorial view of the photoelecsion experiment from a dimer oriented in space. The tron angular distribution for C ionization linearly polarized light hν photoionises an electron of an aligned in space CO molecule. with kinetic energy Ee. The photoelectron current detected within the solid angle dΩ, in direction (θ,φ) with respect to the electric polarization vector ε, is determined by coherent superposition of wavefunctions directly generated by the photoemission process and elastically diffused by the neighbor atom.

in space CO diatomic molecule. Assuming that a core photoelectron is ejected from the carbon atom, the circle represents an hypotetical polar probability distribution determined by the unperturbed wavefunction, while the waving line represents modulation introduced in the photoelectron angular distribution by interference of the direct and scattered wavefunctions. The intense forward lobe aligned with the molecular axis is to be interpreted as the 0th order peak of the angular diffraction pattern. Intensity maxima appearing at larger angles are higher order diffraction peaks. It is a crucial issue to establish validity for this simple scheme of interpretation of the photoelectron angular distributions, particularly in the case of core ionization, as it will constitute the cornerstone for applications to solids and surfaces, i.e. for the so-called photoelectron diffraction. Forward scattering and diffraction features have been clearly seen in an experiment performed on aligned in space CO molecule [27]. The photoelectrons angular distribution was measured in two modes: with the molecular axis parallel or perpendicular to the light polarization vector ε . In the former case the angular distribution is dominated by the forward 0th order diffraction peak as the intensity of the unperturbed wavefunction is low away from the molecular axis. In the latter case the unperturbed wavefunction is weak along the molecular axis while has maxima perpendicular to it, hence the photoelectron angular distribution is domi-

nated by higher order diffraction peaks. These findings have been confirmed by several other independent experiments on similar or different molecular targets, thus providing a sound background for validity of the aforementioned photoelectron diffraction model.

9.7 Core hole state relaxation We have seen that the main peculiarity of photoemission peaks is the linear dispersion of their characteristic kinetic energy with the photon energy (see equation 3). Inspecting any full energy distribution of photoelectron current, we discover that, besides photoemission peaks, it displays further peaks whose energy is characteristic of the sample but does not change by changing the photon energy. The origin of these peaks can be explained in an elementary way by the energy diagram shown in Fig. 15. The core hole state i (i.e. M+) is created either by direct photoemission (left side panel Fig. 15) or by excitation to the excited state

a (i.e. M*). Obviously, the first process is always possible provided the photon energy exceeds the ionization threshold, while the second (right side panel) only when the photon matches the difference in energy between state

i and a . In

both cases, the i hole state is part of a singly ionized/neutral excited state whose lifetime is finite. This state will decay through either a radiative way (fluorescence) or in a non radiative way i.e. ejecting an electron in the continuum (autoionization). This latter decay channel, originated by electron-electron interaction within the target, is dominant for low Z atoms and is usually termed as autoionization or Auger according to the charge state of the final ion, single or double respectively. Fig. 15. Schematics of the secondary processes leading to electron emission by the decay of a highly excited ionic or neutral system. Usually, the Auger process is associated with the decay of a cation whereas the decay of a neutral system is referred to as autoionization.

It is evident that the energy spectrum of the autoionizing electrons will reflect the target energy structure through the difference between energy of the final and ini-

tial states in as much as the energy of the Auger/autoinozation transition matches the separation energy between

i and a .

9.7.1 The Auger decay in

Auger electrons emission is usually described within a two-step approximation which decay incoherently follows the core hole creation

hυ + A → A + + e p → A + + + e p + e A . The energy distribution of the Auger electrons (eA) is independent from the photon and photoelectron (ep) energies, it will be determined by the difference in energy between the intermediate singly ionized state and the final doubly ionized state E Auger = E A+ − E A+ + . Hence, the Auger energy spectrum will be formed by groups of line transitions, one for each core hole state, having as many components as the multiplet configurations allow for the double hole final state. It is therefore spontaneous to indicate each Auger electron emission with a three letter label. The first letter refers to the orbital involved in the intermediate core hole state creation, the other two to the orbitals that generate the doubly ionized final state. Let’s take the Ne atom as an example. In its ground state configuration the shells K (n=1) and L (n=2) are fully occupied (see leftmost panel in Fig. 16). Fig. 16. Schematics of the KL2L3 Auger process in Ne. Ep and EA are kinetic energy of the photoelectron and Auger electron, respectively.

According to this convention, the Auger transition depicted in the rightmost panel of Fig. 16 is KL2L3. This is not the only possible Auger transition for the K core hole to decay, the main groups of transitions with their multiplet structures are listed in the following table.

Auger Transition KL1L1 KL1L2,3 KL2,3L2,3

Double ion valence configuration 2s0 2p6 2s1 2p5 2s2 2p4

Multiplet Terms 1

S0 P 1, 3P 0, 3P 2, 3P 1 1 S 0, 3P 0, 3P 2, 1D 2 1

The simplest, though coarse, way to calculate individual Auger transition energies is within a simple one-electron model, i.e. ignoring relaxation and final state effects. Within this approximation the multiplet splitting is ignored and the Auger energy can be deduced from the binding energy of the individual orbitals, for instance: EA(KL1L2) = E(K) –E(L1) –E(L2, L1)

(14)

where the latter energy is the binding energy of the L2 orbital computed in presence of a hole in L1. There are several methods to calculate more accurately [28], even ab-initio, Auger transition energies but to describe them is beyond the scope of this chapter. The simplified scheme so far outlined is sufficient for interpreting an atomic Auger spectrum such as the one displayed in Fig. 17.

Fig. 17. Auger spectrum of gaseous Krypton. The energy distribution is recorded as a function of the Auger kinetic energy. The principal Auger lines are indexed according to the usual notation (see text) [29]. Copyright Elsevier, reprinted with permission.

Fig. 17 shows the energy distribution of Auger electrons (N(EA)) from gaseous Krypton as a function of their kinetic energy [29]. The principal transitions can be readily assigned through relation (14) and are indexed according to the aforementioned notation. Multiplet splitting of each individual principal transition is also evident. This effect is better seen, for instance, in the free Zn atom L3M4,5M4,5 spectrum [30]. In Zn, all shells from K to M and the 4s subshell are full, hence a simple two-particles (i.e. two holes in the final doubly ionized state) multiplet structure is appropriate to describe its Auger spectrum. The appropriate spectroscopic terms are: 1S0, 1G4, 3P0,1,2, 1D2, 3F2,3,4. The overall agreement between theory and experiment is rather good, thus demonstrating that, in spite of the complexity, most features of Auger spectra can be predicted and calculated, thus providing a powerful tool for studying correlated behavior of matter as the final state is always a two interacting particle one. For what concerns intensity of the transitions, the two-step model can be invoked to allow us to compute the probability of an Auger transition as the product of those relative to the core hole creation and decay separately. Hence the probability for an Auger transition initiated by a photon that ionizes a system in state

0 to a core hole state i that eventually decays

f is:

to the doubly ionized state

P(0, i, f ) ∝ fε pε A C iε p

2

× iε p D 0

2

(15)

in which C and D are coulomb and dipole operators, respectively. The two step model is appropriate as long as the core hole lifetime is much longer than the coulomb interaction time. On the contrary, the single step model is to be adopted. In this frame the core hole state is an intermediate state for the second order transition that leads to the doubly ionized system and the probability reads:

P (0, i, f ) ∝

∫

fε p ε A C iτ × iτ D 0 E A + E p − Eτ − hν − i Γ 2

2

dτ

(16)

In which τ are virtual intermediate states and Γ is the core hole lifetime broadening. It is therefore evident that the main driving force behind Auger transition is the coulomb interaction among bound electrons of the system. Furthermore, through the final state ger cross-section.

f correlated properties of the system influence the Au-

9.7.2 Resonant Auger and photoemission So far we have discussed core hole de-excitations having a singly charged ion as initial state for the Auger decay. Let’s now consider the case depicted in the rightmost panel of Fig. 18, i.e. the autoionization of a core excited system. In Xray absorption language we are interested in describing the decay channels of the x-ray absorption near edge spectroscopy (XANES) resonances. This is an autoionization process that ends in a singly ionized state, which is a state identical to the one reached by a direct photoionization. Let’s again take Ne as an example. As shown in the schematics of Fig. 18, by absorbing a photon of appropriate energy the 1s electron is promoted to the empty 3p level. Afterwards, the core excited Ne* autoionizes filling up the core hole and ejecting a valence (2p in the example) electron. This process is usually termed as a spectator (the 3p state) Auger decay and the final state, which is identical to the final state of a photoionization transition satellite to the 2p3/2 photoionisation, is termed as a 1 particle 2 hole state.

Fig. 18. Schematics of the resonant Auger process in Ne. the 1s electron is promoted to the empty 3p level. The core excited Ne* autoionises by filling up the core hole and ejecting a valence electron(2p3/2)

In other words, the final ionic state depicted in Fig. 18 can be created either by a resonant spectator Auger decay or by direct photoianization. What happens when the energies of the resonant Auger and the photoelectron coincide? We shall discuss it by the help of a specific result obtained for an Ar atom [31]. In Fig. 19a the lowest ionic states of argon are sketched, that are single hole (1h) states, 3p-1 or 3s-1. At higher energies a multitude of 2h1p states, belonging to the 3p-2nl, 3p-1 3s-1nl, or 3s-2nl manifolds, are found. In direct, nonresonant photoemission, the 1h states are strongly dominant, and the 2h1p states are observed as weak satellites.

Fig. 19. Schematics of the resonant Auger process in Ar. The 2p electron is promoted to the empty 4s level [31]. The core excited Ar* autoionises by filling up the core hole and ejecting a valence electron(3p). See text for details. Copyright American Physical Society, reprinted with permission.

The hν + Ar → Ar (3s,3 p ) nl + e photoemission spectrum was recorded for several different photon energies across the resonance +

−2

−

−1

hν + Ar → Ar * (2 p3 / 2 4s ) (see Fig. 19 panel a). The intensity of the unresolved transitions hν + Ar → Ar (3 p ) 5s,4 f + e is almost zero when the photon is tuned out of resonance (∆Ω = 7.5 eV) while it becomes dominant when tuning the photon on resonance (∆Ω = 0 eV). It is therefore evident that for selected photoionization channels the resonant excitation is dominant over the direct ionisation one. Indeed, panel b of Fig. 19, where intensity of the individual photolines is reported as a function of ħΩ, shows that tuning the photon energy across the core resonance, the photoionisation cross section rises by almost an order of magnitude reaching its maximum for on resonance condition. Observing +

−2

−

the behaviour of the transition hν + Ar → Ar (3 p ) 3d + e as a function of ħΩ, we notice that the cross section “on resonance” is larger than “off resonance”, but the lineshape is also highly asymmetric with respect to the resonance energy, as it is the case for all the other photoemission investigated and reported in Fig. 19b. The origin of this asymmetry is in the double quantum path (see Fig. 19 panel a) that, for a given photon energy, can be followed in going from the ground neutral state to the singly ionised 2h 1p states. The final ionic state is created following the resonant or the direct channel with an identical wavefunction but with a different phase. Upon changing the detuning, the phase associate with the direct +

−2

−

channel changes slowly and monotonically, while the resonant one changes rapidly. As long as amplitude of one channel is much larger than the other, the ionisation cross section reduces to the direct photoemission (equation 12) or to the nonresonant Auger one (equation 16). When the amplitudes of the matrix elements of the two competing channels become comparable they should be coherently added and the probability of the process becomes:

P (0, i, f ) =

∫

f ε A C iτ × iτ D 0 dτ + fε A D 0 E A + E p − Eτ − hν − i Γ 2

2

(17)

The probability of the resonant Auger/photoelectron process is therefore modulated by the interference term resulting from squaring the sum of the two amplitudes. Upon rapid change of the resonant channel phase, interference rapidly switches from constructive to destructive, thus yielding the peculiar lineshapes (Fano profiles) displayed in Fig. 19. In summary, photoemission resonant through an Auger channel provides: a. an effective spectroscopy of many-electron excited states that are otherwise inaccessible to conventional electron spectroscopies; b. a unique way to highlight phase changes in the final state wavefunction that are just not detected by other spectroscopies.

9.8 Photoemission from solids: the three-step model We have described the basic principles of the photoemission process from atoms and molecules. It can be described as the transition of one electron from an initial occupied state to a final empty free-electron level in vacuum. In the case of an atom or a molecule, the final state corresponds to the final state of an optical excitation, while in a solid the optically excited electrons, whose photoelectron current can be treated through the Fermi golden rule and the transition probability, must travel towards the surface, go through it, and escape into vacuum as a photoelectron to be detected. In principle, the photoemission process should be treated as a photo-excited electron going into an empty final state in vacuum outside the system as a free electron (one-step model). However, if the system is solid, the photoelectron must acquire velocity in the direction normal to the surface, towards the electron analyzer, following energy and momentum conservation rules: the escaping photoelectron energy must be equal to the sum of the initial level energy and of the photon energy. Moreover, the momentum component parallel to the surface (k//) is conserved, while the translation symmetry normal to the surface is lost, so that k⊥ is not conserved.

A rigorous treatment of these processes in a single coherent step is not straightforward. Furthermore, the photoelectrons can undergo extrinsic scattering processes with the other electrons and intrinsic processes due to the relaxation of the system, once created the photo-hole in the initial state. This is a multi-particle process that can be treated in an approximate framework, through a three-step model, which treats the three processes as independent contributions [32-40]. The three-step model is a better model to interpret the experimental measurements, in particular from the valence band of a solid, which can correlate the measured spectral photoelectron current (Energy Distribution Curve, EDC) to the spectral density of initial valence band states. In this model, the photoemission process can be treated in three sequential independent steps: i) excitation of an electron from a Bloch state of energy Ei to a Bloch state of energy Ei + hν, i.e. electron optical excitation; ii) electron transport through the solid, including inelastic scattering with the other electrons; iii) electron escape from the surface, by traveling from the semi-infinite crystal to vacuum. i) In the first step, the photon with energy hν promotes an electron from an initially occupied Bloch state Φi to a final Bloch state Φf of free-electron out of the crystal, with energy and momentum conservation (also considering the possible presence of a reciprocal lattice vector G). The photoelectron current will be constituted by the sum of the transition probabilities per unit time (wif) weighted by the Fermi-Dirac distribution function f(E) for the initial/final state to be occupied/empty:

Je ∝ ∑if f(Ei) [1 - f(Ef)] wif = ∑if f(Ei) [1 - f(Ef)] Φi ´Hint Φf

2

×

δ [Ekin − (Ef − Φ )] × δ (Ef − Ei − hυ ) = ∑if f(Ei) [1 - f(Ef)] Mif 2 ×

(18)

δ [Ekin − (Ef − Φ )] × δ (Ef − Ei − hυ ) × δ (ki + G − kf )

where Hint is the perturbation hamiltonian and Mif the matrix element for the transition. The δ functions ensure the following conservation laws: of the energy between the initial Bloch state and the final free-electron level, of the energy in the optical excitation process into the crystal, and of the momentum in the extended Brillouin zone. ii) Second step: transmission of the electron through the bulk solid. The traveling photo-electrons can be grouped into two classes, those which reach the solid surface without encountering any scattering and keeping the physical information from within the crystal (primary electrons), and those which undergo multiple inelastic scattering (secondary electrons), mainly due to electron-electron scattering.

The primary electrons are a small fraction of the whole current and can be treated through a coefficient d(Ef, k) representing this fraction of primary electrons with energy Ef and momentum k which reach the surface without suffering inelastic processes. The mean free path λ(Ef, k) between two collisions can be defined as λ(Ef, k) = τ/ħ⎪vg(Ef, k) ⎪ = τ ⎪(∇k, E)Ef,k⎪, where τ is the average time lap between two collisions (assumed as isotropic), and vg is the electron group velocity. Thus, assuming α as the optical absorption coefficient of the solid, based on the probability that the electron can reach the surface without scattering, one can assume

d (Ef , k ) = αλ / (1 + αλ )

(19)

iii) In the third step, the electrons overcoming the surface and escaping into the vacuum must have the normal component of their momentum different from zero. Since they are free electrons, this component can be defined as k⊥ext = 1/ħ√(2mEkin) cos θ, where θ is the angle between the emission direction and the normal to the surface. In this third step, we can define a transmission coefficient T(Ef, kext) as T(Ef, kext) = 0 if k⊥ext < 0 and T(Ef, kext) = x if k⊥ext > 0, with x < 1, since not all the electrons arriving at the surface can overcome it. The transmission function can be explicitly expressed in terms of Bloch functions as T(Ef, kext) = 0 if Ef < EF + Φ and T(Ef, kext) = 1/2 √[1-(EF+ Φ)/Ef] if Ef > EF + Φ. Finally, we can express the photoemission current as the product of these three independent contributions

Je ∝ ∑if f(Ei) [1 - f(Ef)] Mif 2T (Ef , kext )d (Ef , k )δ [Ekin − (Ef − Φ )]

(

× δ (Ef − Ei − hυ )× δ (ki + G − kf )× δ k i// + G // − k //f

)

(20)

where we multiply for a further δ function, ensuring the conservation of the momentum parallel to the surface. In fact, in the definition of the transmission coefficient, we must take into account that for the primary electrons overcoming the surface, only the momentum component parallel to the surface is conserved between the internal and external momentum, because of symmetry considerations. The matrix element of the transition (Mif) is assumed as constant. However, it may smoothly vary as a function of energy, and this must be taken into consideration for quantitative purposes. Furthermore, we can consider matrix elements effects when using polarized radiation. If the photo-electron is excited by polarized radiation from an initial state with definite parity, in order to have nonvanishing matrix elements for the transition, there must be overlap between the polarization direction of the vector potential projected onto the final state, and the

initial state. As an example, if we consider excitation from an even state for reflection with respect to a mirror plane, with the detector positioned in its mirror plane, also the integrand must be even for reflection with respect to the mirror plane, thus the final state must be even as well [41]. A few considerations on the momentum conservation, concerning the angularresolved photoelectron spectroscopy (ARPES). Symmetry considerations impose the conservation of the parallel component of the electron momentum k//int=k//ext, while knorm is not a good quantum number. For a surface or in general for a twodimensional system, the problem of band mapping is solved, insofar an ARPES experiment does map the electron spectral density of states as a function of the angle θ between the surface normal and the electron emission, thus straightly giving the complete E vs. k// bands: // // k int = k ext = 2π / h 2mEkin sin θ

(21)

In order to fully map the band structure in a three-dimensional solid, we should know all the E vs. kint dispersion curves, however an approximate approach brings to a reasonable solution for the dispersion along knorm. The excited state can be approximated as a quasi-free electron Bloch state

E (k ) =

(k 8π m h2 2

norm

2

+ k //

2

)− E

0

(22)

with E0 the bottom of the valence band with respect to the Fermi level; knowing the k// component, we can determine the knorm component as norm k int =

2π h

2mEkin cos 2 θ + E0 + Φ

(23)

Φ is the work function, Eo + Φ is the inner potential which can be estimated [41] either by theoretical methods or it can be inferred by experimentally determining the periodicity of electronic states dispersing along the surface normal, in normal emission, i.e. at emission angle θ=0, thus k//=0.

9.9 Spectral function of interacting electrons in a solid Let us first consider a model based on interacting particles. The photoexcited electrons in a solid experience extrinsic and intrinsic effects while traveling in the solid. When treating a crystalline solid, we can consider the electronic wave function as extended Bloch functions and we can factorize the wave function of the

system with two contributions, the wave function Φk of a Bloch state with wavevector k, and the wavefunction ΨN-1 of an N-1 electron system. This approximation is valid, given the high group velocity of the photo-emitted electron, which almost instantaneously overcomes the surface. In these conditions, the photo-electron does not interact with the remaining system, once excited. The potential of the crystal with N-1 electrons is also considered to change rapidly. Thus, the final state of the system can be written as {ΨNf}= Â {ΨN-1f Φkf}. We choose ΨN-1m as eigenfunction of the Hamiltonian for the N-1 electrons in the final state, and we can decompose the initial state, so that the photoemission current becomes:

J e ∝ ∑if M if2

∑

(

2

m

mim × δ EiN + hυ − EmN −1 − Ekin

)

(24)

where Mif is the matrix element between single particle Bloch states, mim = is the overlap between states of N-1 electrons. If ΨN-1i were an eigenstate of the Hamiltonian of the N-1 electron system, mim would be zero unless i=m; in this latter case the resulting spectrum would be a single peak, like in the case of non-interacting electrons. But the N-1 particle state associated to Φki can be expressed through a destruction operator acting onto the N electron system. Thus, for a correlated system, several elements of the mim overlap will be nonzero. The problem is rather complicated and can be solved within the Green formalism, and one can finally define the single-particle spectral function A(k, ω) as

A(k , ε ) =

1

π

2

Im[G (k , ε )] = ∑m ΨmN −1 ck Ψ i

N

δ (ε + E mN −1 − EiN )

(25)

where A(k,ε) is the sum of an A+ and A- contribution, where ck is the creation (for A+) or annihilation (for A-) operator, respectively. The photoemission current (equation 24), multiplied by the Fermi-Dirac distribution function taking care of the occupation of the single-particle states, can be redefined as Je(k, ω) ∝ Σif⎪Mif⎪2 A(k, Ekin - hν) f(Ekin - hν), and the expression of the photoemission current for non-interacting particles can be obtained through the Green approach as

∑ G (k , ε ) =

m

ΨmN −1 ck ΨiN

ε +E −E N i

N −1 m

2

− iη

=

∑

m

δ im

ε + E − EmN −1 − iη

1 1 = N N −1 ε + E i − E m − iη ε − E k − iη

N i

= (26)

where Ek=EiN-1-EiN is the ionization energy of the Koopman’s theorem, so that we finally obtain A(ε, k)=δ(ε - Ek), i.e. the spectral function relative to a single particle, is represented by a single peak. So far, we have not considered the effects of the multiple interactions on the lineshape. They can be taken into account by introducing a complex quantity, the self-energy Σ(ε, k), containing the effects on the renormalization of the dispersion curves and on the average excitation lifetime. We can re-write the Green function in the E, k space as

G (k , ε ) =

1

ε − Ek − ∑ (ε , k ) m

(27)

with Σ = Σ' + i Σ'', and the single-particle spectral function becomes:

∑ (k , ε ) ''

A(ε , k ) =

1

π ε − E − ' (k , ε ) 2 + '' (k , ε ) 2 ∑ ∑ k

(28)

where the self-energy in general contains contributions due to electronelectron, electron-impurity and electron-phonon scattering. In the case of Σ function independent on k, the single-particle function has the form of a Lorentzian curve, whose width is proportional to the imaginary part of the self-energy. Thus, Σ'' can be experimentally directly obtained from the width of the photoemission peaks, once deconvolved from the experimental resolution contribution. Since the Green function is the linear response to the external perturbation due to the incoming electromagnetic field, its real and imaginary parts are in causal relation through the Kramers-Kronig relation, so that experimentally known Σ'', we can easily determine Σ'. In the following paragraphs, we will present a few examples of studies in which photoemission has given fundamental contributions to the understanding of the physical properties of the investigated systems.

9.10 Energy Distribution Curves (EDC)

9.10.1 Valence band examples (cross-section exploitation and Cooper minima) The EDC mode of photoemission essentially gives a representation of the spectral density of the electronic states of the investigated system, valence band or core-levels, also giving information on its chemical state. In this paragraph we will give a few examples of angle-integrated valence band and core-level data, exploiting the excitation cross-section [42] and the surface sensitivity dependence on the energy. Organic molecules are nowadays attracting scientific and technological attention, due to their potential use for advanced devices at the nano-scale. MetalPhthalocyanines (MPcs, M-C32H16N8) belong to an exemplary class of organic oligomers constituted by a planar aromatic macrocycle (containing C and N atoms) surrounding a central metal atom [43]. MPcs easily self-assemble as ordered planar nano-chains on surfaces [44, 45], and photoemission EDC data can give indispensable hints for experimentally determine the spectral density of states depending on the different species composing the molecule. In Fig. 20, we show the Valence Band (VB) measured at ordered planar thin-films (TF) of FePc and CoPc grown on Au(110). Data are taken with different photon energies, from 21.218 eV to 150 eV, thus obtaining photo-emitted electrons from the same initial states at different kinetic energies. The VB electronic states in this energy range are expected to be mainly associated either to metal-derived d-symmetry states (b2g/eg), or to C and N derived π-states (a1u, highest-occupied molecular-orbital, HOMO) [46], whose relative intensity depends on the photoelectron excitation crosssections [42]. By combining the calculated excitation cross-section at different energies for the C and N 2p-states (composing the π-orbitals) and for the Fe and Co 3d-states, with the experimental data, a correct attribution to the different contributions in the VB has been given, as indicated in the figure [47].

Fig. 20. Valence Band EDCs of FePhthalocyanine and Co-Phthalocyanine thin-films grown on Au(110) [47], taken at different photon energies (left panel). Data adapted from [47]. Copyright American Physical Society, reprinted with permission.

Another example of cross-section exploitation has been used for disentangling the VB contribution of two different metal species. Cobalt nano-chains have been prepared on the regularly stepped vicinal Pt(997) surface, where the Co atoms self-assemble as chains at the step edges close to the (111)-oriented terraces [49]. In order to disentangle the d-contribution due to the Co-3d states from that of the substrate Pt-5d states, the authors exploit the differently varying photo-excitation cross-section of the Pt-5d states with respect to the Co-3d levels. Data taken when the Pt-5d states present the Cooper minimum (see inset in the figure) bring to light the VB features associated to the Co nano-wires, as shown in Fig. 21. We briefly mention that also resonant photoemission exciting VB electrons with photon energies spanning across the core-level of an atomic species, can help in disentangling the VB attribution [50].

Fig. 21. Valence Band EDCs of the clean Pt(997) surface (thin lines) and of Conanowires grown on Pt(997) (dots and thick lines), taken at different photon energies. From Fig. 2 of [49]. Copyright American Physical Society, reprinted with permission.

High-luminosity and high-resolution angular-integrated EDC brought to light experimental measurement of the population of the bottom of the conduction band in a narrow gap semiconductor, i.e. highly n-type doped InAs(110). A tiny amount (hundredths of a monolayer) of alkali atoms (Cs) adosrbed on InAs(110) induce a strong bending of the surface bands with respect to the Fermi level, favoring the occupancy of the previously empty InAs conduction band, as shown in Fig. 22 (left panel) [51]. The charge accumulation region is hundreds of Å deep, so that electronic charge is confined in the direction normal to the surface and free in the surface plane, generating a quasi-free and spatially confined 2D electron gas with quantized eigenstates (right panel). These occupied quantized energy levels are present in the angular-integrated EDCs shown in Fig. 22, in perfect agreement with self-consistent calculations, which results are superimposed to the experimental data [52]. Fig. 22. High-resolution Valence Band and Conduction band EDCs of the InAs(110) surface under tiny electron-doping by exposing to Cs (left panel); experimental density of states in the CB and results of selfconsistent ab-initio calculation of the quantized levels in the confined 2D electron-gas. Reprinted from Fig. 6 (left) and 7 (right) of [51]. Copyright Elsevier BV, reprinted with permission.

9.10.2 Core level examples Use of different photon energies affects the surface sensitivity, as the photoelectron kinetic energy of detection changes for a given excitation (see the "universal" mean free path curve in Fig. 3). It is well known that III-V compound semiconductors when cleaved along the (110) surface, present a relaxation of the first atomic plane as minimum energy configuration, due to the bulk symmetry breaking, maintaining the (1×1) symmetry, causing the surface atoms to present a different position with respect to the bulk-like configuration. This reflects into a modification of the VB structure and, as a consequence, the core-levels experience a surface-core level shift (see also [53] in this same book). In Fig. 23 (upper panel) we show the In-4d core levels at the freshly cleaved InAs(110) surface, taken at two different photon energies (40.814, HeIIα, and 48.372, HeIIβ). Data (red dots)

have been deconvolved with Voigt peaks (Gaussian-Lorentzian shape), and show surface (S, blu lines) and bulk (B, red lines) doublets. It is evident that the surfacerelated In-4d doublet increases its relative intensity upon passing from the HeIIα to the HeIIβ radiation, from about 19 eV to 27 eV kinetic energy, thus increasing the surface sensitivity, as shown in the “universal” curve for the electron inelastic mean free path (lower panel) [15, 54].

Fig. 23. High-resolution In-4d core-levels at freshly cleaved InAs(110), taken with HeIIα and HeIIβ radiation; experimental data (red dotrs), Voigt-profiled fit with surface (S, blu lines) and bulk (B, red lines) doublet components (upper panel); “universal” curve of the electron inelastic mean free path [15] (lower panel); graph from [54].

9.11 Angular Resolved PhotoElectron Spectroscopy (ARPES) 9.11.1 Band structures The ARPES technique does not require any assumption for mapping the band structure if the system under investigation is two-dimensional. A valuable example of 2D systems is given by reconstructed surfaces of semiconductors. In particular, at the Si(111) cleaved surface, angular resolved photoemission gave fundamental hints for the understanding of the (2x1) reconstruction mechanisms. Without entering into historical detail on this surface [55], the Si(111)-(2x1) surface is characterized by formation of chains of Si atoms, Pandey's chain reconstruction model [56], one of the pillars of clean semiconductors' reconstruction mechanisms. The dangling bonds associated to the surface Si atoms form surface states which disperse along the ΓJ direction of the Surface Brillouin Zone (SBZ).

We present in Fig. 24 data from one of the first ARPES experiments showing the dangling-bond dispersing state [57], and one of the most recent, where detail on the formation of isomers on the surface is brought to light thanks to highresolution ARPES on highly n-doped Si crystal [58].

Fig. 24. Dangling-bond surface state dispersion at the Si(111)-(2x1) reconstructed surface along the ΓJ direction of the Surface Brillouin Zone (SBZ). One of the first experimental ARPES dangling-bond dispersion (left panel). Recent high-resolution ARPES dangling-bond dispersion and selected spectra, with the evidence of isomers-related states (center and right panels). From Fig. 2 of [57] (left) and Fig. 1 of [58] (center and right). Copyright American Physical Society, reprinted with permission.

9.11.2 Adsorbates A purely 2D system can also be formed by highly-ordered adsorption of atomic or molecular structures. The substrate effects can be neglected when the adsorbate is weakly bound through van der Waals interactions, like low-temperature physisorbed gases on metals. As first example of highly ordered 2D array, we show the ARPES band structure formed by Xe atoms physisorbed at low temperature on the Cu(110) surface [59], where the noble gas atoms coalesce in an ordered c(2×2) structure with respect to the underlying (1×1) unit mesh. The lowest binding energy electronic states of Xe are the 5p core-levels, energetically separated from the Cu-3d bands, only slightly interacting with the Cu-4s,p bands. These core-levels associated to a compact ordered 2D structure have been shown to form distinct bands [60], as presented in Fig. 25.

Fig. 25. Experimental band structure of the 5p levels of Xe physisorbed in an ordered c(2×2) structure onto the Cu(110) surface. ARPES bands. From Fig. 5 of [60]. Copyright American Physical Society, reprinted with permission.

Aromatic molecule adsorption on surfaces is encountering a new wave of interest in the recent years, due to their use in advanced opto-electronic and magnetic devices at the nano-scale, and to their ability of self-assembling at the nano-scale, by exploitation of naturally patterned surfaces, thus forming a variety of highly ordered 1D or 2D structures. Pentacene (C22H14) molecules self-assemble as long nano-wires onto the Cu(119) vicinal surface. This surface is a regularly stepped surface with (001)oriented 1.2 nm-wide terraces and long-ordered step edges, thus favoring planar adsorption of the long pentacene nano-wires [61-64]. Pentacene interacts with copper, due to an orbital mixing between the pentacene molecular orbitals and the Cu states [65], such as occupied interface states form at completion of the first ordered layer [66]. Upon further pentacene deposition on Cu(119), the molecules keep maintaining their ordered structure forming a highly ordered 2 nm-thich molecular film, whose electronic band dispersion of its HOMO has been probed by normal-emission photoemission, changing the photon energy, thus having access to the HOMO band dispersion along the k⊥ direction of the BZ [63], as shown in Fig. 26. Several other organic molecules adsorbed on surfaces, like rubrene [67], form planar ordered layers and present k⊥ dispersion of the highest-occupied molecular orbital.

Fig. 26. 2-nm thick pentacene film grown on Cu(119). ARPES selection of spectra taken at normal emission and varying the photon energy (left); highestoccupied molecular-orbital (HOMO) band dispersion along k⊥ (right). Reprinted from Figs. 2 and 3 of [63]. Copyright Elsevier BV, reprinted with permission.

9.11.3 Graphite and Graphene as exemplary 2D system The most classical example by far of two-dimensional band-structure is that of graphite, the sequence of slightly interacting graphene layers. Data taken on highly-oriented pyrolitic graphite (HOPG) along different directions of the BZ and the experimentally determined band structure are shown in Fig. 27 [68]: we observe the wide dispersion of the π-bands accompanied by the typical linear behaviour close to the K point and by the higher binding energy σ bands.

Fig. 27. Valence band of graphite (HOPG), stacking of the ARPES spectra as a function of polar angle (left) and experimental band structure (right). From Figs. 1 and 2 of [68]. Copyright American Physical Society, reprinted with permission.

Graphene is one of the most exciting exemplary systems in science of the last few years. In fact, Geim and Novoselov were awarded the 2010 Nobel prize [69] for their discovery, which is of great importance in view of its fundamental properties, strictly depending on the topology of the crystal lattice, and for the wealth of potential applications. Graphene is actually at the spotlight of ongoing research on nanoscale systems, due to its unique electronic, mechanical, and transport properties [70-72]. It is the prototype of a 2D system. Without entering into details, we briefly recall in Fig. 28 the lattice cell (hexagonal lattice with two C atoms per unit cell), its reciprocal lattice, the π and σ bands as calculated by tightbinding approach along the main symmetry directions of the SBZ, and the well known conical dispersion of the π-bands across the K points of the SBZ [73].

Fig. 28. Graphene: real space and reciprocal lattices, tight-binding band-structure along the main symmetry directions, and electronic structure of the π bands, showing the Dirac points where the valence and conduction bands meet at the K points of the Brillouin zone. Reprinted from Fig. 1 of [73]. Copyright Pergamon, reprinted with permission.

In order to study its electronic band structure, it is necessary to prepare graphene as a highly-ordered sheet, as can be done onto a variety of hexagonallyterminated crystalline substrates [73, 74]. In particular, metal surfaces offer a wellsuited support for epitaxial growth [75-83], and the Ir(111) surface has revealed to be an excellent template for minimizing the carbon–substrate interaction [79, 80, 83], while only slightly affecting its proper band structure, leading to a quasi-free standing graphene layer.

The electronic band structure of graphene grown on Ir(111), taken along the main symmetry directions ΓKM of the SBZ, with a zoom on the section of the Dirac cone around the K point, is shown in Fig. 29 [84]. Fig. 29. Graphene band structure along ΓKM and zoom of the Dirac cone around the K point of the SBZ. ARPES data taken with highresolution ARPES and a He discharge source [84].

Actual free-standing suspended exfoliated graphene presents a broader spectral trace [85] with respect to graphene supported on Ir, due to the natural corrugation of actual free-standing graphene, which presents intrinsic ripples in its 2D structure [86]. Graphene can be built layer-by-layer in a highly controlled multi-layer sheet on SiC [74]. Furthermore, the evolution of the band structure in a multi-layer sheet (single layer-by-single layer) can also be followed as a function of the number of layers by ARPES, as shown in Fig. 30, where we report the emerging of several slightly energy-shifted Dirac cone traces at the point K of the SBZ, each trace corresponding to a single graphene sheet, with the slight energy shift due to the π- π interlayer interaction. In the lower panel of the figure, the k⊥ vs. k// bands are reported for a binding energy of 1 eV: for 1 layer there is no dispersion in the direction perpendicular to the plane, as expected, while upon increasing the number of layers, there is a clear intensity modulation due to the layer interaction, eventually evolving in the formation of a dispersing π-band of multi-layer (4 layer) graphene towards graphite.

Fig. 30. Formation of an electronic band, stepwise: from 1-layer (extreme left) to 4-layer (extreme right) graphene band structure along across the Dirac point. From Fig. 2 of [74]. Copyright American Physical Society, reprinted with permission.

Graphene can be easily doped by alkali metal intercalation, producing electron doping and a shift of the Dirac point towards higher binding energies, basically without modifying the graphene band properties, thus maintaining the linear dispersion. However, the band properties can also be heavily modified by more interacting adsorbates, for example by atomic H-adsorption, which causes a distortion of the C bonds, modifying the sp2 hybridization towards an sp3-like condition, with the opening of a band gap [87]. The intriguing properties of graphene are also characterized by many-body interactions slightly affecting the purely linear behaviour of the Dirac cone trace [88]. Finally, deep core-levels are generally expected to be strongly localized and not to present any band dispersion. However, highresolution ARPES on graphene has recently brought to light the dispersion of the C-1s core-level [89], observed as an emission-angle-dependent binding-energy modulation, thus confirming graphene as one of the best examples of 2D system.

9.12 Final considerations and perspectives In this chapter, we introduced the reader to the photoelectron spectroscopy, briefly recalling the basic theoretical concepts and the experimental aspects of the photoemission tecnique. We accompanied the discussion with a few exemplary systems, with photoemission applied from atomic and molecular systems to the solid state, with special emphasys on the angular resolved mode. These examples studied by photoemission demonstrate the powerfulness and suitability of photoelectron spectroscopy in studying and completely determining the electronic properties of a variety of physical systems, ranging from atomic, molecular and solid state physics to chemistry, organic chemistry and low-dimensional hybrid organicinorganic materials.

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