Femtosecond Photoemission Investigation of Electron Dynamics at ...

Università Cattolica del Sacro Cuore 

Dipartimento di Matematica e Fisica 

Femtosecond Photoemission Investigation 

of Electron Dynamics at Surfaces 

Ph.D. Thesis 

Emanuele Pedersoli 

Brescia, 2006

Scuola di Dottorato di Ricerca in 

Fisica, Astrofisica e Fisica Applicata 

XIX ciclo 

Università degli Studi di Milano 

Facoltà di Scienze Matematiche, Fisiche e Naturali 

Dipartimento di Fisica, Milano (Italy) 

Università Cattolica del Sacro Cuore 

Facoltà di Scienze Matematiche, Fisiche e Naturali 

Dipartimento di Matematica e Fisica, Brescia (Italy) 

(Sede Consorziata) 

Prof. Gianpaolo Bellini (Director) 

Prof. Fulvio Parmigiani (Supervisor)

Contents 

Introduction 1 

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 

I Photocathodes 5 

1 Quantum efficiency of copper photocathodes 7 

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 

1.2 Vectorial photoelectric effect . . . . . . . . . . . . . . . . . . . . . 8 

1.3 Quantum efficiency measurements . . . . . . . . . . . . . . . . . 11 

1.3.1 Experimental setup and sample characterization . . . . . 11 

1.3.2 Experimental results . . . . . . . . . . . . . . . . . . . . . 16 

1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 

1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 

2 Monte Carlo transverse emittance study on Cs 2 Te 21 

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 

2.2 Theoretical model to describe Cs 2 Te . . . . . . . . . . . . . . . . 22 

2.3 Photoemitted electron angular distribution . . . . . . . . . . . . 25 

I

II 

CONTENTS 

2.3.1 Trajectory randomization upon scattering . . . . . . . . . 27 

2.3.2 Quasi elastic scattering . . . . . . . . . . . . . . . . . . . 29 

2.4 Transverse emittance . . . . . . . . . . . . . . . . . . . . . . . . . 34 

2.4.1 Cs 2 Te transverse emittance calculations . . . . . . . . . . 35 

2.4.2 Effects of surface aging and contamination . . . . . . . . . 36 

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 


II Surface and Image Potential States on Noble Metals 39 

3 Phase shift model 41 

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 

3.2 Solution for the phase shift model . . . . . . . . . . . . . . . . . 43 

3.3 Effective mass off the image potential states . . . . . . . . . . . . 45 

3.3.1 Dependence of φ C on the position in the gap . . . . . . . 45 

( ) 

3.3.2 Binding energy ε n k‖ and effective mass m ∗ . . . . . . . 47 

4 Experimental Setup 51 

4.1 Femtosecond pulsed amplified Ti:Sapphire laser system . . . . . . 51 

4.2 Pulse modification and characterization . . . . . . . . . . . . . . 53 

4.3 Ultrahigh vacuum chamber . . . . . . . . . . . . . . . . . . . . . 54 

4.4 Photoemission Measurements . . . . . . . . . . . . . . . . . . . . 57 

5 Spin orbit splitting on Au(111) surface state 61 

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 

5.2 Discussion of experimental results . . . . . . . . . . . . . . . . . . 64 

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 

6 Comparison between theory and experiment on Cu(111) and 

Cu(100) surface electronic states. 69 

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

CONTENTS 

III 

6.2 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . 72 

6.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 

6.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 73 

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 


7 Role of athermal electrons in non-linear photoemission from 

Ag(100) 83 

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 

7.2 Two-photon or three-photon photoemission . . . . . . . . . . . . 85 

7.3 Photoemission autocorrelation . . . . . . . . . . . . . . . . . . . . 89 

7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 

8 Angle resolved photoemission study of image potential states 

and surface state on Cu(111) 95 

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 


8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 

9 Image potential state effective mass variation with hot electrons 

population on Cu(111) 105 

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 

9.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 106 


9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 

Appendix 115 

A Calculations about the phase shift model 115 

A.1 Energy of the image potential state . . . . . . . . . . . . . . . . . 115 

A.1.1 Nearly free electron model . . . . . . . . . . . . . . . . . . 115

IV 

CONTENTS 

A.1.2 Total energy of the image potential state . . . . . . . . . 116 

A.1.3 Energy associated with the imaginary part q . . . . . . . 119 

A.1.4 Non-kinetic part of the energy . . . . . . . . . . . . . . . 119 

A.2 Wavefunction of the image potential state . . . . . . . . . . . . . 120 

A.2.1 Schrödinger’s equation . . . . . . . . . . . . . . . . . . . . 120 

A.2.2 Calculation of the ψ k (r) derivatives . . . . . . . . . . . . 121 

A.2.3 Solution of Schrödinger’s equation . . . . . . . . . . . . . 121 

A.2.4 Phase φ C due to the reflection on the crystal surface . . . 123 

A.3 Hydrogen-like Rydberg series of states . . . . . . . . . . . . . . . 124 

List of Publications 127 

Bibliography 129

Introduction 

Overview 

Ultrafast pulsed lasers have allowed in recent years the investigation of physical 

phenomena in the femtosecond timescale through time resolved techniques. 

In solid state spectroscopy they are a powerful tool to study topics such as 

electron interactions, transiently populated empty band states, ultrafast linear 

and non-linear photoemission phenomena; the generation of short electron 

bunches is another important achievement of fast light sources. Future devices 

will provide X ray femtosecond pulses to implement time resolved diffraction 

and attosecond pulses to probe faster electronic processes. 

Femtosecond pulses are very versatile in photoemission spectroscopy. The 

short duration of the excitation, in photocathodes with fast response, can produce 

short electron bunches. Harmonic or sum frequency generation and parametric 

amplification in non-linear optical crystals allow tunability of the exciting 

photon from the infrared to the ultraviolet range. The high peak intensity of 

short light pulses generates phenomena such as coherent many-photon photoemission 

from occupied states and non-linear photoemission from normally empty 

states populated by relaxation of non-equilibrium excited electrons. 

Impinging with ultrafast laser light on a solid surface it is possible both to 

control some changes in electronic dynamics, by the variation of photon energy, 

1

2 Introduction 

light intensity, direction and polarization, and to probe them, by photoemission 

spectroscopy. In this thesis two main topics were investigated: the first is the 

theoretical and experimental analysis of the femtosecond photoemission yield of 

Cu and Cs 2 Te photocathodes; the second deals with photoemission from surface 

and image potential states on noble metals. 

New generation light sources like free electron lasers are expected, in a few 

years, to supply femtosecond coherent X ray pulses. These devices are based on 

the acceleration of short electron bunches photoemitted by femtosecond laser 

pulses: the quality of the electron beam has to be studied and improved by the 

point of view of time duration, quantum efficiency and velocity distribution. 

The quantum efficiency of copper photocathodes were investigated varying the 

angle of incidence between the light beam and the normal to the sample surface, 

evidencing an enhanced efficiency in p polarization which is not connected to 

absorption increase: this phenomenon is known as vectorial photoelectric effect, 

here explained in terms of non-local conductivity tensor. The thermal emittance 

of photoelectrons from Cs 2 Te thin films was also calculated by a Monte Carlo 

simulation, considering the effects of electron phonon scattering and discussing 

the dependence of thermal emittance and quantum yield on the electron affinity; 

the results highlight the importance of considering the material contamination. 

The investigation of surface Shockley states and image potential states on 

low indexes faces of noble metals allowed to go through several topics concerning 

surface electron dynamics. In particular, image potential states population 

is a two dimensional nearly free electron gas whose internal interactions are 

very sensitive to the laser induced population of the near projected bulk bands: 

different electron behaviors can be caused by laser pulses and revealed by effective 

mass and inverse lifetimes measurements performed through non-linear 

photoemission spectroscopy. Experimental data are compared with theoretical 

calculations and can be qualitatively explained by the phase shift model. 

Shockley surface states were also investigated, obtaining results in agreement 

with literature; the most interesting case is Au(111) whose surface state shows

Outline 3 

a spin orbit splitting due to a spin polarization of the splitted branches of its 

parabolic dispersion. 

Outline 

This thesis is composed of two main parts. 

Part I deals with the characterization of photocathodes that can be used 

as a source of femtosecond electrons bunches: in Chap. 1 experiments on copper 

photocathodes are reported, in which we measured the photoemission total 

yield, finding for p polarized light a quantum efficiency enhancement known as 

vectorial photoelectric effect [1]; Chap. 2 reports the results of Monte Carlo 

calculations on thermal emittance and quantum yield of Cs 2 Te irradiated by a 

light beam [2]. 

In Part II we study the behavior of Shockley surface states and image potential 

states on noble metals. Chap. 3 reports the phase shift model, the theoretical 

description of the image potential states; calculations are shown in App. A. In 

Chap. 4 we show the experimental setup used for experiments described in the 

following. 

The two following chapters describe some measurements confirming theoretical 

calculations: measurements on the spin orbit splitting of the Shockley 

surface states Au(111) are shown in Chap. 5; Chap. 6 reports theoretical calculations 

on Cu(111) and Cu(100) surface electronic states, compared with the 

corresponding experimental results [3]. 

The last three chapters deal with the behavior of image potential states in 

presence of an athermal electrons population due to femtosecond laser pulses 

excitation. Chap. 7 presents time resolved pump & probe experiments studying 

non-linear photoemission from Ag(100) and the influence of athermal hot electrons 

on it [4]. The last two chapters investigate the behavior of the Cu(111) 

image potential state depending on the empty projected bulk bands population 

by hot electrons excited by the laser pulse: Chap. 8 presents two-photon photoe-

4 Introduction 

mission spectroscopy, performed both in resonance with the energy difference 

between Shockley and image potential state and out of resonance [5]; two-color 

photoemission spectroscopy, that allows to control the hot electrons population 

via the pump intensity, and two-photon photoemission are reported in Chap. 9.

Part I 

Photocathodes 

5

Chapter 1 

Quantum efficiency of 

copper photocathodes 

Quantum efficiency measurements of single photon photoemission from a 

Cu(111) single crystal and a Cu polycrystal photocathodes, irradiated by 150 fs 

6.28 eV laser pulses, are reported over a broad range of the incidence angle θ, 

both in s and p polarizations. The maximum quantum efficiency (about 4 × 

10 −4 ) for polycrystalline Cu is obtained in p polarization at an angle of incidence 

θ = 65 ◦ , greater than the angle of maximum light absorption; in fact we observe 

a photoemission enhancement in p polarization which can not be explained in 

terms of optical absorption: this phenomenon is known as vectorial photoelectric 

effect. Issues concerning surface roughness and symmetry considerations are 

addressed; an explanation in terms of non-local conductivity tensor is proposed. 

1.1 Introduction 

Although studied both theoretically and experimentally for more than a 

century, photoemission from solids is still non-completely understood, even for 

7

8 Chapter 1. Quantum efficiency of copper photocathodes 

well known systems such as noble metals photocathodes. Nevertheless, a better 

knowledge of this matter is requested for many theoretical and applicative 

aims, one of which is the advent of the fourth generation free electron laser 

sources [6, 7, 8]. A fundamental issue regards the photocathode material for the 

laser-driven photoinjector devices, to obtain short electron bunches with high 

charge density and low emittance. Metal photocathodes are good candidates, 

having a high reliability, long lifetime and a fast time response between 1 and 

10 fs. However, two major drawbacks limit their usefulness: the small quantum 

efficiency and the high work function, requiring light source in the ultraviolet 

for efficient linear photoemission. 

In this chapter we analyze the variation of the quantum efficiency of copper 

photocathodes as a function of light polarization and incidence angle between 

the normal to the sample surface and the direction of the impinging light beam. 

Results evidence an enhancement of the quantum efficiency for p polarization 

as compared to what would be expected taking into account only the electromagnetic 

absorption process, in accordance with the phenomenon known as 

vectorial photoelectric effect, which can be related to a rapid spatial variation 

of the electric field through the sample surface. 

1.2 Vectorial photoelectric effect 

The quantum efficiency dependence on angle of incidence and light polarization 

is a long standing problem [9, 10, 11, 12, 13] that largely remains to be 

understood. Our data are well fitted by a phenomenological model [11] that 

keeps into account only light absorption, without any explanation in terms of 

microscopic quantum physics. 

If we consider a light beam impinging on a surface sample, we can decompose 

its electric field vector E inside the sample in his components E ‖ = E p‖ + E s 

parallel and E ⊥ = E p⊥ perpendicular to the surface of the sample, where E p 

and E s are the p and s polarized field components respectively, as shown in

1.2. Vectorial photoelectric effect 9 

Figure 1.1: Representation of angles θ and θ t and wave vectors k and k t for 

incident and transmitted light and field components addressed in the text; the 

polarization angle α is also shown. A real index of refraction n is assumed for 

the present figure. 

Fig. 1.1. Calling ε ‖ = ε p‖ + ε s and ε ⊥ = ε p⊥ the electromagnetic energy inside 

the sample due to the light with electric vector E ‖ and E ⊥ respectively, we 

assume that the number of photoemitted electrons is simply proportional to the 

absorbed energy, but with different efficiency due to polarization, and then we 

write the number of photoemitted electrons n f (θ) as a function of the angle of 

incidence as 

n f (θ) = aε ‖ (θ) + bε ⊥ (θ) 

= a[ε ‖ (θ) + (b/a)ε ⊥ (θ)]. 

(1.1) 

The definition of Quantum efficiency Q(θ) can now be given as the ratio 

between the number of photoelectron n f (θ) and the number of photons f impinging 

on the sample surface, resulting 

Q(θ) = n f (θ) 

f 

= a[ε ‖(θ) + (b/a)ε ⊥ (θ)] 

, (1.2) 

f 

that can be normalized dividing by the quantum efficiency at normal incidence,


obtaining 

Q(θ) 

Q(0) = [a/f][ε ‖(θ) + (b/a)ε ⊥ (θ)] 

= ε ‖(θ) 

[a/f]ε ‖ (0) ε ‖ (0) + r ε ⊥(θ) 

ε ‖ (0) , (1.3) 

where a, b and r = b/a are parameters that do not depend on the angle of 

incidence θ, and ε ⊥ (0) = 0. 

We can now deal with two particular conditions of the polarization: with p 

polarized light we can write 

while for s polarization we have E ⊥ = 0 and so 

Q p (θ) 

Q p (0) = ε p‖(θ) 

ε p‖ (0) + r ε p⊥(θ) 

ε p‖ (0) , (1.4) 

Q s (θ) 

Q s (0) = ε s(θ) 

ε s (0) . (1.5) 

The three ratios in the right parts of Eq. (1.4), (1.5) can be calculated from 

the classical theory of electrodynamics [11, 14]: for a metal like Cu, whose complex 

index of refraction is n = 0.98 + 1.49i [15], all non-reflected photons are 

absorbed before penetrating more than about 10 nm into the sample; we can 

identify the electromagnetic energies inside the solid ε ‖ (θ) and ε ⊥ (θ) with the 

total absorptions 1−R p (θ) and 1−R s (θ), where R p (θ) and R s (θ) are the reflectivities 

calculated from the Fresnel laws for p and s polarization respectively. 

Calling θ t the complex angle between the transmitted beam and the normal 

to the surface (see Fig. 1.1) and defining l = (n 2 − sin 2 θ) 1/2 = n cos(θ t ), we 

obtain 

R p (θ) = |n2 cos θ − l| 2 

|n 2 cos θ + l| 2 , R | cos θ − l|2 

s(θ) = 

| cos θ + l| 2 (1.6) 

and the quantum efficiency ratios eventually become 

Q p (θ) 

Q p (0) = 1 − R p(θ) 

1 − R p (0) · |E p‖| 2 + r|E p⊥ | 2 

|E p‖ | 2 + |E p⊥ | 2 , 

Q s (θ) 

Q s (0) = 1 − R s(θ) 

1 − R s (0) . (1.7)

1.3. Quantum efficiency measurements 11 

We note that quantum efficiency of s polarization light is expected to be simply 

proportional to the light absorption governed by Fresnel laws, while in p 

polarization a vectorial photoelectric effect is predicted with a characteristic 

parameter r; in this case, the maximum quantum efficiency is obtained at an 

incident angle θ M different from the pseudo Brewster angle θ B of maximum 

light absorption. 

At normal incidence the electric field polarization is always parallel to the 

surface, allowing to write Q p (0) = Q s (0) = Q(0); in Fig. 1.2 we show, as 

an example for n = 0.98 + 1.49i, r = 10 and Q(0) = 1 × 10 −4 , a graph of the 

calculated value of the quantum efficiency as a function of the angle of incidence 

θ and of the angle of polarization α, defined as drawn in Fig. 1.1. 

1.3 Quantum efficiency measurements 

Quantum efficiency measurements of single photon photoemission from a 

Cu(111) single crystal and a Cu polycrystal photocathodes, irradiated by laser 

beams of 150 fs pulse duration and hν = 6.28 eV photon energy, are reported 

over a broad range of incidence angles, both in s and p polarizations. 

1.3.1 Experimental setup and sample characterization 

The experimental setup is described in Fig. 1.3. 

Our source is an amplified Ti:Sapphire laser supplying pulses with a duration 

less than 150 fs and an average power of 600 mW at 1 kHz repetition rate, giving 

a pulse energy of 600 µJ and a peak power of about 10 10 W. The laser is tuned 

on a near infrared wavelength of 790 nm whose ultraviolet fourth harmonic 

with photon energy hν = 6.28 eV is used in our photoemission experiments. 

This photon is obtained by a double process of second harmonic generation 

in a beta-barium-borate (βBBO) crystal: the first step is in phase matching 

and guarantees a high efficiency; for the doubling of the hν = 3.14 eV second


Figure 1.2: Calculated value of the quantum efficiency as a function of θ and α 

for n = 0.98 + 1.49i, r = 10 and Q(0) = 1 × 10 −4 .


Figure 1.3: Experimental setup for quantum efficiency measurement on copper 

photocathodes.


harmonic photons there is no phase matching allowed, but the small yield of 

fourth harmonic photons requested for photoemission experiments is obtained 

in a thin (200 µm) BBO crystal. After the non-linear optics steps, we obtain 

three collinear beams of different colors: they are spatially dispersed by a MgF 2 

wedge prism, with minimal temporal and pulse front tilt distortions, and the 

beam with the desired photon energy hν = 6.28 eV is selected by a pinhole and 

impinges on the sample in the ultra high vacuum (UHV) chamber through a 

beam splitter and an optical flange. 

The ultra high vacuum chamber has a base pressure less than 2×10 −10 mbar 

at room temperature; samples are cleaned by cycles of Ar + sputtering followed 

by annealing at 500 ◦ C. This procedure is continued until the proper value of 

the work function (4.65 eV for the polycrystal and 4.94 eV for the single crystal 

[15]) is measured in photoemission spectra acquired by a time of flight (ToF) 

spectrometer; in these conditions a clear low energy electron diffraction (LEED) 

pattern for the Cu(111) sample is obtained. Photoemission spectra are also 

acquired during total yield measurements in order to measure the sample work 

function and monitor possible onset of sample contaminations and space charge 

effects. 

A more efficient third harmonic conversion, supplying a photon energy hν = 

4.71 eV, is the most usual to study polycrystal photocathodes, but for our 

purposes it is too close to the copper work function. We preferred the fourth 

harmonic photon because its higher energy allows to obtain linear photoemission 

also from samples with higher work function, due to the surface lattice structure 

of the single crystal or to the onset of sample contamination. Moreover, 

ultraviolet short laser pulses induce the removal of oxide contaminants and the 

breaking of chemical bonds on the surface, contributing to maintain the sample 

cleanliness; this effect improves with shorter wavelengths [16, 17]: working with 

a 6.28 eV photon energy should thus help to increase the duty time of machines 

based on Cu photocathodes. 

Several atomic force microscopy (AFM) scans of the samples surface, with


Figure 1.4: Atomic force microscopy images of the two samples’ surfaces. Measured 

route mean squared roughness is 20 nm for the Cu polycrystal and 2 nm 

for the Cu(111) single crystal. 

sizes ranging from 1 × 1 µm 2 to 60 × 60 µm 2 , give values of the root mean 

squared roughness h rms ≃ 20 nm for the Cu polycrystal and h rms ≃ 2 nm for 

the Cu(111) single crystal (see Fig. 1.4). 

The quantum efficiency Q is the ratio between the number of photoemitted 

electrons n f , obtained from the photocurrent measured from the sample 

with a Keithley 6485 Picoammeter, and the number of incident photons f, detected 

measuring on a Tektronix TDS3054B digital oscilloscope the output of a 

Hamamatsu R928 photomultiplier tube. The laser peak intensity on the target, 

regulated by a λ/2 waveplate coupled with a polarizer, is I ≃ 5 × 10 5 W/cm 2 : 

these conditions avoid space charge and guarantee that non-linear photoemission 

is negligible if compared to one-photon photoemission, that is predominant. 

A small part of the impinging beam is directed to the photomultiplier through 

a beam splitter and three mirrors in order to attenuate the light power not to 

saturate the instrument. Due to experimental condition, only p polarized light 

intensity can be measured; s polarized light is renormalized supposing that at


normal incidence there can not be any difference between the two polarizations. 

1.3.2 Experimental results 

The quantum efficiency for linear photoemission in the femtosecond regime 

is measured as a function of the incidence angle θ in the angular range −55 ◦ ≤ 

θ ≤ +80 ◦ , both in s and p polarizations on a Cu polycrystal and a Cu(111) 

single crystal. Results are plotted in Fig. 1.5. 

Experimental data obtained on the two samples are plotted for both p (circles) 

and s (triangles) polarizations and fitted by Eq. (1.7) (lines); dashed 

lines represent the behavior expected taking into account light absorption only 

(r = 1), for p polarization. 

For p polarized light, an enhancement of the quantum efficiency is evident in 

both samples as compared to what would be expected taking into account only 

the electromagnetic absorption process. Data are in good agreement with the 

model described in Sec. 1.2, showing a vectorial photoelectric effect. The fitting 

procedures give the best values of the fitting parameter r of Eq. (1.7): r = 13 for 

the polycrystalline Cu and r = 9 for the Cu(111) single crystal. The maximum 

value of Q does not occur at the pseudo Brewster angle of incidence θ B = 57 ◦ , 

where there is maximum absorption, but is shifted by about 8 ◦ toward grazing 

incidence: the maximum quantum efficiency Q ≃ 4 × 10 −4 , obtained with p 

polarization at θ = 65 ◦ , is four times the value at normal incidence. 

Eq. (1.7) is in agreement with data for s polarization too, showing that in 

this case the photoemission yield n f is simply proportional to the number of 

absorbed photons f (1 − R s (θ)), according to the Fresnel laws. 

1.4 Discussion 

Experimental data exposed in Sec. 1.3 show a behavior in good agreement 

with the vectorial photoelectric effect described in Sec. 1.2.

1.4. Discussion 17 

Figure 1.5: Measurements of quantum efficiency dependence on the angle of 

incidence θ for a Cu polycrystal and a Cu(111) single crystal for p (circles) and 

s (triangles) polarized light. Fits, based on Eq. (1.7), are reported as solid lines. 

The dashed lines are calculated taking into account Fresnel absorption only.


At the light of our data, it is important to investigate the physical mechanisms 

that enhances the photoelectron yield due to E ⊥ compared to the one 

due to E ‖ . 

The crystalline symmetry, important when dealing with polarization dependent 

photoemission, play no role in the present experiment. Both in the Cu(111) 

single crystal, where symmetry considerations could apply, and in the polycrystalline 

Cu, where any contribution related to symmetry is canceled by the random 

orientation of the single crystals domains composing the sample, E ⊥ is 

about 10 times more effective than E ‖ as a cause of the photoemission process. 

Photoemission enhancement due to surface roughness has been recently investigated 

[18, 19, 20]. In the present case surface roughness enhancement can 

be ruled out: the observed vectorial photoelectric effect is comparable on both 

samples, despite their surface roughnesses differ by an order of magnitude, as 

can be seen in the atomic force microscopy scans shown in Fig. 1.4. The comparative 

study of the single crystal Cu(111) and polycrystalline Cu cathodes 

allows to clarify that our experiment is not dependent on sample morphology. 

Therefore, we seek for an explanation in terms of a more general mechanism. 

Solutions of the Maxwell equations on an ideal jellium-vacuum interface for an 

impinging plane electromagnetic wave of frequency ω, evidence an electromagnetic 

field spatially varying on the length scale of about 1 Å on the jellium side 

[21]. The spatially varying electromagnetic field is due to the non-local character 

of the conductivity tensor. This is calculated using free electron-like wave 

functions, so it does not depend on the symmetry of the crystal. The matrix 

element entering the differential cross section for photoemission is composed of 

two terms. The first is the usual electric dipole contribution, the second is due 

to the rapidly varying electric field. The second term prevails for ω < ω p , where 

ω p is the plasma frequency, and leads to an enhancement of the photocurrent 

for the electric field components perpendicular to the sample surface [22, 23]. 

In the present experiment, ħω = 6.28 eV and ħω p ∼ 19 eV [24]. This mechanism 

explains an enhancement of the quantum efficiency for p polarized incident

1.5. Conclusion 19 

radiation while not affecting the results for s polarized light. Furthermore, it 

does not depend on surface roughness or a particular symmetry of the crystal. 

We therefore propose it as the main microscopic mechanism to explain our 

experimental evidences. 

1.5 Conclusion 

Quantum efficiency measurements on Cu photocathodes, irradiated with 

150 fs laser pulses at hν = 6.28 eV, are reported over a broad range of incident 

angles in both s and p polarizations. A quantum efficiency enhancement 

is found for light with electric field perpendicular to the sample’s surface, showing 

a vectorial photoelectric effect. The maximum value of quantum efficiency 

Q ≃ 4 × 10 −4 is four times bigger than the one measured at normal incidence 

and is achieved with p polarized light impinging on the sample at an incidence 

angle of θ = 65 ◦ , greater than the pseudo Brewster angle θ B . 

Investigation of both a Cu(111) single crystal and a Cu polycrystal allows 

us to rule out a microscopic processes based on symmetry considerations and 

surface roughness to explain our data. An explanation in terms of a rapidly 

varying effective field, due to the non-local character of the conductivity tensor, 

is suggested. 

Acknowledgments 

This work was supported by the U.S. Department of Energy, Office of Science 

under Contract No. DE-AC03-76SF00098.

Chapter 2 

Monte Carlo transverse 

emittance study on Cs 2 Te 

The thermal emittance of photoelectrons in Cs 2 Te thin films is investigated 

by a Monte Carlo simulation. The effects of electron-phonon scattering are 

discussed and the thermal emittance calculated for a radiation wavelength of 

265 nm and a spot radius of 1.5 mm, finding a value of ɛ n,rms = 0.56 mrad mm. 

The dependence of ɛ n,rms and the quantum yield on the electron affinity is also 

investigated. The data show the importance of considering aging and contamination 

of the material to assess its emittance. 


High quality electron beams are a fundamental tool in many application areas 

ranging from electron microscopy to accelerator technology. In this context 

the transverse emittance ɛ n,rms of an electron beam at the surface of the photocathode 

is an important parameter: it is a measure of the beam spread in both 

real and momentum spaces and it constitutes the lower limit for the emittance 

21

22 Chapter 2. Monte Carlo transverse emittance study on Cs 2 Te 

that can be generated by an injector. 

Semiconductor photocathodes have attracted attention because of their high 

quantum yield [25]. In particular, Cs 2 Te is a semiconductor compound under 

investigation as a photoemitter under short laser pulse excitation for the 

new generation free electron lasers (FELs) and advanced synchrotron radiation 

sources [6, 7, 8, 26]. Despite the fact that Cs 2 Te has been studied extensively, 

in the past most of the efforts have been devoted to issues such as the energy 

distribution of the photoemitted electrons and the beam dynamics of the photoejected 

electron bunch under high electric field gradients, a typical topic in 

accelerator physics. The issues of the photoemitted electrons’ angular distribution 

and thermal emittance at the cathode have received comparatively little 

attention, nevertheless being a relevant topic both under an applicative and 

fundamental point of view. 

In this work we investigate, via Monte Carlo simulations, the photoemitted 

electron angular distribution and transverse emittance at the surface of a 

Cs 2 Te photocathode under 265 nm radiation wavelength, the fourth harmonic of 

Nd:YAG fundamental wavelength; the quantum yield is also calculated. Attention 

is devoted to the microscopic mechanisms, such as electron-phonon scattering, 

affecting the photoelectrons angular distribution; the effects of aging and 

contamination of the photocathode on thermal emittance and quantum yield 

are also discussed. 

2.2 Theoretical model to describe Cs 2 Te 

The compound Cs 2 Te is a p type semiconductor with a band gap of 3.2 eV 

and an electron affinity of 0.3 eV. A schematic band diagram is illustrated in 

Fig. 2.1. Semiconductor materials are known to be good photoemitters because 

they are good absorbers and photoexcited electrons experience little electronelectron 

scattering. A large ratio between the band gap and the electron affinity 

results in a high quantum efficiency for photons with energy greater than E g +E a

2.2. Theoretical model to describe Cs 2 Te 23 

Figure 2.1: Energy band diagram for Cs 2 Te. The energy gap E g = 3.2 eV separates 

the valence band from the conduction band. The vacuum level is separated 

from the conduction band minimum by the electron affinity E a = 0.3 eV. The 

zero energy reference is taken at the conduction band minimum. The electron 

affinity E a is the potential barrier felt by a photoexcited electron approaching 

the surface. The electron kinetic energy E in inside the material is measured 

with respect to the conduction band minimum. Internal and external emission 

polar angles for a photoemitted electron are also shown: arrows represent the 

momenta of the electron inside and outside the material.


and less than 2E g [27]. 

The Monte Carlo simulation of photoemission from Cs 2 Te follows a scheme 

first introduced to study electron scattering mechanism in metals [28] and successfully 

applied to simulate the quantum yield dependence on photon energy 

in Cs 2 Te [29]. Calculations are based on the three step model for photoemission 

[30], which has proved reliable to interpret photoemission data in many 

materials, including Cs 2 Te [31]. In this model a photoemitted electron is first 

photoexcited inside the material (photoexcitation), secondly it travels toward 

the sample surface (transport) and as a third step it overcomes the surface 

barrier (emission). 

We consider a 30 nm Cs 2 Te polycrystalline layer deposited on a Mo substrate 

and use the microscopic parameters deduced from experimental data taken from 

Ref. [29]. 

We assume that every absorbed photon produces a photoexcited electron. 

The distributions of electron kinetic energy E in , momentum p in and depth z 

from the surface within the material are statistically determined at the beginning 

of the simulation as in reference [31] and are recalculated for every single 

photoexcited electron after each scattering event. The probability for an electron 

to travel without scattering a distance z/ cos θ in in the forward direction to 

the surface is proportional to exp (−z/λ p cos θ in ), where θ in is the angle of the 

electron trajectory inside the sample, defined in Fig. 2.1, and λ p is the electron 

mean free path, assumed to be energy-independent. With the Cs 2 Te parameters 

and a photon energy hν = 4.68 eV, the excited electrons are below the threshold 

for the onset of electron-electron scattering; therefore, only quasi elastic 

electron-phonon scattering is considered. Upon an electron-phonon scattering 

event, the electron energy is reduced by an amount E p = 5 meV [29] and the 

momentum direction is assumed to be randomized over all solid angles with the 

same probability; it is equivalent to consider all values of cos θ in equiprobable. 

When photoexcited electrons reach the surface, they are photoemitted only 

if the kinetic energy associated to their momentum component perpendicular

2.3. Photoemitted electron angular distribution 25 

to the surface is greater than the electron affinity E a , sufficient to overcome 

the surface potential barrier; otherwise electrons are reflected back into the 

semiconductor film. The simulation is iterated until all the initial electrons have 

been eliminated after being photoemitted, crossing the Cs 2 Te/Mo interface or 

having a total energy E in < E a . 

2.3 Photoemitted electron angular distribution 

When electrons are photoemitted from a cathode with photons well above 

threshold, a spread in the transverse and longitudinal components of the electron 

momentum results. 

The emittance of an electron bunch is a measure of its 

spread both in real and momentum space and depends on the spot size, the 

momentum distribution and the angular distribution. The thermal emittance 

is the emittance at the cathode surface, investigated through the photoemitted 

electron angular distribution. 

As mentioned in the previous paragraph, only electrons approaching the 

surface with sufficient energy associated to the perpendicular momentum component 

will cross the semiconductor-vacuum interface. Given the total electron 

energy at the surface E in , an electron will be photoemitted only if the internal 

angle θ in at the surface is lower than a critical angle θ c ≡ arccos( √ E a /E in ). 

The relation between the internal and external polar angles is given by a 

solid-state analog of Snell’s law due to conservation of transverse momentum at 

the interface: 

sin(θ in ) 

sin(θ out ) = √ 

Ein − E a 

E in 

, (2.1) 

where θ in and θ out are the polar angles for the electron trajectories inside and 

outside the sample surface respectively, as shown in Fig. 2.1. 

The calculated distribution of photoemitted electrons is reported in Fig. 2.2. 

The distribution has a maximum for θ in ≃ 34 ◦ and all photoemitted electrons 

are found within an internal cone defined by an angle of value max {θ in } = 63 ◦ ,


Figure 2.2: Angular probability distribution over internal angles θ in of the photoexcited 

electrons that are also photoemitted. The bin angular width is chosen 

equal to 0.36 ◦ . The calculation is performed following 10 5 electrons trajectories 

and assuming a value of E a = 0.3 eV for Cs 2 Te.


where {θ in } denotes the set of values acquired by θ in . A physical model to 

explain this distribution must be considered. 

Assuming a random distribution over polar angles, the probability of finding 

an electron in the range [θ in , θ in + dθ in ] is proportional to sin θ in dθ in ; electrons 

with small θ in have larger probability (proportional to exp (−z/λ p cos θ in )) to 

reach the surface because of the shorter distance |z/ cos θ in | to travel on their 

trajectory. Taking into account both of these mechanisms, integrating over 

the distance z from the surface and assuming that λ p



electrons that are also photoemitted. The assumption of a constant 

θ c = max {θ c } for all the electrons, regardless of their energy, is made. The plots, 

obtained changing the value of E a , map a function proportional to sin(2θ in ). 

The bin angular width is chosen equal to 0.36 ◦ .


max {θ c } as expected. Moreover, changing E a amounts to changing max {θ c }, 

with larger cut off angles associated to lower electron affinity values. 

The photoemitted electron angular distribution with respect to the external 

emission angle θ out is reported, in polar coordinates, in Fig. 2.4 and compared 

with the corresponding angular distribution over internal angles. The distribution 

over the external angles is smeared on the range [0, 90 ◦ ] and is peaked at 

lower polar angles as compared to what would be expected invoking the first 

two mechanisms alone. 

2.3.2 Quasi elastic scattering 

We now investigate the role of electron energy loss, due to quasi elastic scattering, 

on the photoemitted electron angular probability distribution. To this 

purpose, we simulate the angular probability distributions assuming both elastic 

(E p = 0 meV) and quasi elastic (E p = 5 meV) scattering. The distributions are 

equivalent, as can be appreciated in Fig. 2.5. 

Each quasi elastic scattering event reduces the electron energy by 5 meV 

and a photoemitted electron is found to scatter, on average, about 27 times: 

the average photoemitted electron energy loss is about 130 meV. Therefore an 

important issue to be explained is the negligible effect on electrons angular 

distribution of a mean energy loss of about 9% of max {E in }. This energy 

loss mainly affects the high energy part of the photoemitted electron’s internal 

energy distribution. 

To understand the mechanism let us consider an electron with E in ≃ E a . 

Upon scattering, this electron is eliminated from the distribution, because electrons 

with E in < E a cannot be photoemitted, but its energy position can be 

filled with an electron scattered from an higher energy position. This scattering 

driven electron hopping to lower energies depopulates the high energy tail of the 

distribution (compare for instance the electron probability distribution for energies 

above 1.1 eV in the two histograms reported in Fig. 2.6, where no higher


Figure 2.4: Electrons count over internal angles θ in (green graph) and over 

external angles θ out (red graph) of the photoexcited electrons that are also 

photoemitted. The number of photoexcited electrons assumed in the simulation 

is 10 5 . The bin angular width is chosen equal to 0.36 ◦ . The inset shows the angle 

orientation: angles amplitudes are taken clockwise starting from the negative z 

axis.



electrons that are also photoemitted. The bin angular width is chosen 

equal to 0.36 ◦ .


Figure 2.6: Energy probability distribution over internal kinetic energy E in 

of the photoemitted electrons. E in is the electron kinetic energy inside the 

sample, the zero energy reference being at the bottom of the conduction band 

and E a = 0.3 eV. The width of the energy bin is 10 meV. The light blue and 

red histograms show the probability distributions calculated with the scattering 

event supposed elastic (E p = 0 eV) and quasi elastic (E p = 5 meV) respectively.


energy electrons are present to fill the vacancies). This effect is particularly evident 

inspecting the function D(E in ) = ∫ E in 

E a 

[ 

P inel (E ′ in ) − P el(E ′ in ) ] 

dE ′ in , that 

is the integral function of the difference between the two probability distributions 

obtained for the cases with (P inel (E ′ in )) and without (P el(E ′ in )) scattering 

(see Fig. 2.7). The probability difference function is slightly positive for ener- 

Figure 2.7: Graph of the integral function D(E in ) of the difference between the 

two probability distribution reported in Fig. 2.5, 2.6. The dashed line is the 

zero probability line. The width of the energy bin is 10 meV. 

gies below 0.5 eV, indicating that the probability distributions over this range 

are essentially not affected by inelastic scattering. For energies in the range 

comprised between 0.5 and 1.1 eV, D(E in ) becomes negative: inelastic scattering 

increases the number of electrons with kinetic energies in this interval. For 

energies in excess of 1.1 eV, the function D(E in ) increases up to zero value,


attained at E in ≃ 1.4 eV, meaning that electrons added by inelastic scattering 

to the previous energy interval are actually transferred from the 1.1-1.4 eV energy 

interval. This is the mechanism we addressed as inelastic scattering driven 

electron hopping. 

The high energy tail of the probability distribution contributes equally to 

all emission angles and, for this reason, is the portion that least affects the 

photoemitted electron angular distribution. This can be appreciated noting 

that θ c is monotonic with E in and recalling the role of θ c in determining the 

angular electron distribution. We thus conclude that the energy loss due to 

the quasi elastic electron-phonon scattering plays a negligible role in the photoemitted 

electron angular distribution. It is the trajectory randomization of 

electron-phonon scattering that strongly affects the photoemitted electron angular 

distribution via the mechanism explained in the previous subsection. 

2.4 Transverse emittance 

The normalized transverse root mean square emittance ɛ n,rms is a measure 

of the electron beam phase space density that may be deduced by measuring 

the root mean square beam divergence at the cathode for a given beam spot 

size; it is defined as 

ɛ n,rms = 

mc√ 1 σ2 (x)σ 2 (p x ) − σ 2 (xp x ), (2.2) 

where σ 2 (x) ≡ N −1 ∑ N 

i=1 (x i − x) 2 1 , x ≡ N −1 ∑ N 

i=1 x i, m is the electron 

rest mass and c the speed of light; the subscript i indicates quantities referred 

to the i-th photoemitted electron at the surface, whereas N is the number of 

photoemitted electrons. The correlation term σ 2 (xp x ) vanishes at the source, 

since the quantities x and p x are uncorrelated, and x and p x are null, hence 

1 The actual definition should be σ 2 (x) ≡ (N − 1) −1 P N 

i=1 (x i − x) 2 , however, as soon as 

N is big enough, we can consider (N − 1) ≃ N.

2.4. Transverse emittance 35 

Eq. (2.2) reduces to 

ɛ n,rms = 1 mc p x,rms × x rms . (2.3) 

2.4.1 Cs 2 Te transverse emittance calculations 

Figure 2.8: Probability distribution for the normalized transverse momentum 

p x at the cathode for the photoemitted electron. A bin of 5 × 10 −5 is used for 

the normalized transverse momentum. 

The transverse emittance of an electron beam at the photocathode depends 

on the laser beam spot size r, which can be related to x rms , and on the transverse 

momentum distribution p x,rms of the photoemitted electrons. The simulated 

normalized transverse momentum probability distribution at the cathode, calculated 

on the basis of the photoemitted electron angular and energy probability 

distribution obtained in the previous paragraph, is reported in Fig. 2.8. The cal-


culation is performed assuming a wavelength of 265 nm. From the distribution 

we obtain p x,rms /mc = 7.4 × 10 −4 . We assume an hard edge laser spot size of 

radius r = 1.5 × 10 −3 . Making use of Eq. (2.3), we find ɛ n,rms =0.56 mrad mm. 

2.4.2 Effects of surface aging and contamination 

Figure 2.9: Quantum yield (right axis) and normalized transverse root mean 

square emittance at the cathode ɛ n,rms (left axis) as a function of electron 

affinity normalized against the maximum internal electron kinetic energy. 

The reliability of the photocathode characteristics with aging and contamination 

of the surface is one of the key issues in the choice of the photocathode 

material. Surface contamination, occurring during operation of the radio frequency 

gun, affects the material electron affinity: the question arises as to what 

effect this may have on ɛ n,rms and the quantum yield. The results of the simulations 

are reported in Fig. 2.9 against a value of the electron affinity normalized

2.4. Transverse emittance 37 

with respect to the maximum electron kinetic energy E a / (hν − E g ), a quantity 

from now on defined as normalized electron affinity. 

The clean sample corresponds, with the parameters in use, to a normalized 

electron affinity of 0.2, thus showing a quantum yield of 12.5%, a value in good 

agreement with experimental data [32]. The graphs show that the main concern 

is the relatively rapid drop in quantum yield that persists as the electron affinity 

grows in excess of 0.3 eV (normalized electron affinity of 0.2). The decrease 

in the emittance is due to the preferential selection of electrons with forward 

directed momentum. 

Figure 2.10: Normalized transverse root mean square emittance at the cathode 

ɛ n,rms as a function of electron affinity normalized against the maximum electron 

kinetic energy. The empty squares are obtained on the basis of Monte Carlo 

simulation. The full triangles are obtained applying the analytical calculations 

reported in [33] for the case with scattering.


The results of the Monte Carlo simulations are compared in Fig. 2.10 with 

those obtained by applying simple analytical calculations [33]. The difference 

among the Monte Carlo simulation and the analytical calculation is of 

significance for low value of E a / (hν − Eg), and amounts to almost 40% for 

E a = 0.3 eV (normalized electron affinity of 0.2), the value corresponding to 

clean Cs 2 Te. This fact underlines the relevance of statistical approaches, as 

Monte Carlo simulations, when addressing issues related to electron dynamics 

within the material such as thermal emittance at the cathode. 

2.5 Conclusions 

The photoemitted electron angular distribution and the transverse emittance 

at the cathode surface on Cs 2 Te are investigated by Monte Carlo calculations. 

An explanation of the photoemitted electron angular distribution is given in 

terms of electron redistribution over all angles smaller than the proper critical 

angle for each electron. Electron-phonon scattering affects the photoemitted 

electron angular distribution through the randomization of momentum direction, 

whereas an average energy loss of about 0.1 eV per photoemitted electron 

has a negligible effect. The transverse emittance, calculated for an impinging 

radiation at a wavelength of 265 nm and a laser spot size of 1.5 × 10 −3 m, is 

about 0.56 mrad mm. The effect of aging and contamination of the cathode on 

ɛ n,rms and the quantum yield is also investigated. Our results are an improvement 

of those obtained with analytical calculations [33] still in use for thermal 

emittance studies [34], and underline the importance of statistical approaches 

to study photoemission processes where scattering is involved. 


This work was supported by the U.S. Department of Energy, Office of Science 

under Contract No. DE-AC03-76SF00098.

Part II 

Surface and Image Potential 

States on Noble Metals 

39

Chapter 3 

Phase shift model 

Electronic bulk bands calculations can be performed for three dimensional 

infinite crystalline solids; the presence of a surface, breaking the solid symmetry 

in one of the three dimensions of space, allows the rising of some surface related 

features like Shockley surface states and image potential states. 

When a bulk band structure presents an inverted gap in which bands have 

crossed yielding s bands at the top and p bands at the bottom of the gap, the 

symmetry break due to the existence of a surface allows the presence in the 

gap of a surface state whose wave function has a short penetration in the solid, 

exponentially decaying with the distance from the surface. These states took 

their name from Shockley [35] and are well known. 

Another kind of surface states are the image potential states, described in 

Fig. 3.1: they represent a two dimensional free electron gas trapped in front of 

a solid surface when electrons neither can fall into the bulk, for the presence of 

a forbidden bulk band gap at their energy, nor can escape into the vacuum, because 

their presence in front of the surface repels electrons in the solid, creating 

an image charge whose Coulomb potential represents a long range barrier that 

traps electron in the image potential state. These electrons are represented by a 

41

42 Chapter 3. Phase shift model 

standing wave function that is multiply reflected between the crystal barrier and 

the Coulomb image potential barrier: the behavior of these states is described 

in the phase shift model, that we analyze in the following of this chapter. 

Figure 3.1: Description of image potential states. The image charge inside the 

solid creates a Coulomb potential that attracts electrons along the z direction: 

they are trapped in a two dimensional free electron gas. Images are taken from 

Ref. [36]. 


The phase shift model describes the wavefunctions of electrons in the image 

potential states as multiply reflected standing waves between a Coulomb 

boundary, due to an image charge in the solid, and a crystal boundary, due to 

the gap in the bulk bands. In each reflection on the crystal, the wavefunction is 

multiplied by a factor r C e iφ C 

, while a factor r B e iφ B 

is due to each reflection on 

the Coulomb image potential boundary. To obtain a stationary wave we impose 

r B = r C = 1 and φ B + φ C = 2nπ, n ∈ Z. 

Although the definition of a realistic potential matching the periodic lattice 

inside the crystal and the Coulomb-like behavior outside is not trivial and several 

different models were proposed to handle the problem [37, 38, 39, 40, 41], 

all of these potentials have more or less the form described in Fig. 3.2: after

3.2. Solution for the phase shift model 43 

the outermost crystal layer, the bulk potential is continuously matched to an 

external Coulomb potential. 

Figure 3.2: Electric potential at the surface boundary between the bulk solid 

lattice (z < 0) and the Coulomb potential outside the surface (z > 0). 

3.2 Solution for the phase shift model 

In this section we report some well known results of the phase shift model 

[42, 43]; calculations are shown in App. A. We consider the vacuum energy level 

as the origin of the energy axis (E V = 0). 

In a nearly free electron model context, we can consider a band structure 

with a surface state placed in a gap of amplitude 2V g . The surface state’s 

wavefunction exponentially decays in the bulk and, as seen in Eq. (A.32), has 

the form 

ψ(r) = e qz cos(pz + δ) p, q ∈ R, (3.1)


where z is the coordinate on the axis perpendicular to the crystal surface, z < 0 

in the bulk, k = (p − iq)ẑ and δ is a phase we will define in Eq. (3.5); the total 

energy of the state is (Eq. (A.15)) 

Defining 

E = ħ2 k‖ 

2 √ 

2m + ħ2 p 2 

2m − ħ2 q 2 

2m ± Vg 2 − 4 ħ2 p 2 ħ 2 q 2 

2m 2m . (3.2) 

E g = ħ2 p 2 

2m 

and 

ε = E − ħ2 k 2 ‖ 

2m = ħ2 u 2 

2m , (3.3) 

we can identify an energy associated with the imaginary part q of the wavevector 

(Eq. (A.19)) 

ħ 2 q 2 

√ 

2m = − (ε + E g) + Vg 2 + 4εE g (3.4) 

and a non-kinetic part of the energy (Eq. (A.21), (A.22)) 

E − ħ2 k 2 

2m = V ge i2δ , 

sin(2δ) = − ħ2 

2m 2pq/V g. (3.5) 

Matching at the image plane z = z 0 the bulk image potential state wavefunction 

(3.1) for z < z 0 with the standing wave e −iuz + r C e iφ C 

e iuz for z > z 0 

with r C = 1 and imposing the function’s and first derivative’s continuity, we 

obtain (Eq. (A.37)) 

u is defined in Eq. (3.3). 

u tan(φ C /2) = p tan(pz 0 + δ) − q; (3.6) 

The dependence of the phase φ B added by the wavefunction reflection on 

the image potential on the binding energy ε can be calculated analytically or 

numerically for several models of different complexity. The simplest is a reflection 

on a Coulomb potential well, whose solution can be regarded as a useful 

approximation, yielding for the reflection phase the dependence [44] 

(√ ) 

3.4 eV 

φ B = π 

− 1 ; (3.7) 

ε

3.3. Effective mass off the image potential states 45 

imposing the stationarity condition φ B +φ C = 2nπ, n ∈ N according to Sec. A.3, 

we obtain a Rydberg series of hydrogen-like states labeled by the quantum 

number n, whose binding energy is (Eq. (A.42)) 

ε n = 

0.85 eV 

(n + a) 2 , (3.8) 

where we introduced the quantum defect (Eq. (A.41)) 

a = 1 ( 

1 − φ ) 

C 

. (3.9) 

2 π 

3.3 Effective mass off the image potential states 

Image potential states are populated by a two dimensional free electron gas: 

the energy part directly depending on k ‖ are then expected to show a free 

electron parabolic dispersion, yielding for the total energy the expression 

E ( k ‖ 

) 

= εn 

( 

k‖ 

) 

+ 

ħ 2 k 2 ‖ 

2m ; (3.10) 

in this section we want to investigate the dependence of the binding energy 

ε n 

( 

k‖ 

) 

on the parallel momentum k‖ . 

3.3.1 Dependence of φ C on the position in the gap 

The binding energy ε n of the image potential state depends on the quantum 

defect a which is affected by the reflection phase φ C by Eq. (3.8), (3.9); to 

( ) 

predict the values for ε n k‖ we have to investigate the behavior of φC , that 

depends on the position of the vacuum level with respect to the gap in the 

projected bulk bands, as schematically described in Fig. 3.3. 

We are interested in Shockley inverted gaps, in which the upper band is s- 

like and the lower band is p-like. In this case a state in the top of the gap has 

to be matched with even (respect to the atom positions) bulk wavefunctions 

and its phase is φ C = π, while in the bottom the surface state wavefunction


Figure 3.3: Dependence of φ C on the position in the gap and its effect on 

the wavefunction propagation into the bulk. The left panel is reported from 

Ref. [45].


matches an odd function and its phase at the mirror plane is φ C = 0; the phase 

continuously changes from π at the top of the gap to 0 at the bottom. 

Obviously, states whose electron energy is close to any of the gap edges penetrate 

in the bulk for a long depth without decaying, whereas states in the middle 

of the gap vanish faster into the solid. On the contrary, the distance between 

the crystal surface and the image potential state’s wavefunction maximum is 

not dominated by the energy distance between the state and the bulk bands, 

but continuously increases with the decreasing of φ C from the top to the bottom 

of the gap. 

This determines the behavior of the binding energy ε n : if the vacuum level 

is in a position that yields the image potential state near the top of the gap, φ C 

is near to its maximum value π, the quantum defect a is small and the binding 

energy ε n is about 0.85/n eV; if the vacuum level is at the bottom of the gap, 

φ C 

tends to 0, a reaches its maximum value 1/2 and the binding energy ε n 

decreases. 

If we change the sample material from one having an image potential state 

in the top of the gap to several others in which the state is positioned more and 

more down until the gap’s bottom, the values of φ C continuously decreases from 

π to 0 and the binding energy ε 1 of the n = 1 state continuously decreases from 

about 0.85 eV, close to the value obtained for Cu, to 0.85 eV/(1+1/2) 2 = 0.38 eV 

at the bottom of the gap. 

( ) 

3.3.2 Binding energy ε n k‖ and effective mass m 

∗ 

The dispersion of the band edges at the top and at the bottom of the bulk 

gap are different from the free electron and the image potential state dispersion: 

for this reason the energy distance between the image potential states and the 

band edges depends on k ‖ ; this dependence also influences the value of φ C and 

( ) 

consequently of a and ε n k‖ , that then depends on k‖ . 

The case of the n = 1 image potential state is described in Fig. 3.4. The


Figure 3.4: Schematic representation of the reason why the image potential 

state presents an effective mass m ∗ bigger than the free electron mass m.


upper bulk bands disperse less than a free electron: in the point A the state 

and the bands touch each other, φ C = π and the binding energy ε n 

( 

k‖ = k A 

) 

is maximum, as can be seen following the cyan free electron parabola; moving 

to the point k ‖ = k B , φ C decreases and the red free electron parabola passing 

through B has a smaller binding energy ε n 

( 

k‖ = k B 

) 

; in the center of the first 

Brillouin zone, φ C is minimum and the C point stands on the green free electron 

parabola with the smallest binding energy ε n 

( 

k‖ = 0 ) . 

To recover all the points found spanning all the k ‖ range, we draw a blue 

parabola that is flatter than a free electron one: we define an effective mass 

m ∗ > m for which we can rewrite Eq. (3.10) with the approximation 

E ( k ‖ 

) 

= εn + ħ2 k 2 ‖ 

2m ∗ , (3.11) 

where ε n = ε n 

( 

k‖ = 0 ) and the suppression of its dependence on the parallel 

momentum is compensated by the introduction of an effective mass bigger than 

the free electron one.

Chapter 4 

Experimental Setup 

In this chapter we describe the devices and the setup that allowed us to 

perform experiments reported in the following chapters of Part II. 

4.1 Femtosecond pulsed amplified Ti:Sapphire 

laser system 

The pulsed beam for photoemission experiments is supplied by a femtosecond 

amplified Ti:Sapphire laser system. 

The seed of our light source is a Ti:Sapphire oscillator: it provides 130 fs 

pulses with a 76 MHz repetition rate at an average power of about 500 mW; 

the wavelength λ = 790 nm corresponds to a photon energy hν = 1.57 eV. 

This seed is amplified by a second Ti:Sapphire cavity: only one pulse over 

76000, stretched to avoid excessive power concentration that could damage the 

crystal, is trapped into the cavity, thanks to a Pockels cell that changes the 

light’s polarization. The pulse undergoes several passages into the pumped 

crystal to be amplified until it reaches the maximum allowed energy; it is then 

let out by a second polarization rotation in the Pockels cell. The amplified pulse 

51

52 Chapter 4. Experimental Setup 

travels then in a compressor stage. 

The output of this amplified Ti:Sapphire system is a 130 fs pulsed infrared 

(λ = 790 nm, hν = 1.57 eV) laser with a 1 kHz repetition rate at an average 

power of about 500 mW: each pulse carries 500 µJ and the pulse peak power 

can be estimated as the ratio between pulse energy and pulse duration P P eak = 

5 × 10 −4 J/1.3 × 10 −13 s ≃ 4 × 10 9 W. The device is shown in Fig. 4.1. 

Figure 4.1: Femtosecond amplified Ti:Sapphire laser providing λ = 790 nm, 

hν = 1.57 eV infrared 130 fs pulses with a 1 MHz repetition rate at an average 

power of about 500 mW: 500 µJ pulse energy and 4 × 10 9 W pulse peak power.

4.2. Pulse modification and characterization 53 

4.2 Pulse modification and characterization 

The laser beam provided by Ti:Sapphire oscillator and amplifier is modified 

and characterized online. Devices like multilayer or metallic mirrors, lenses, 

λ/2 and λ/4 waveplates, polarizers, prisms allow to control direction, intensity, 

polarization, spot size of the pulsed laser beam. For photoemission purposes, it’s 

also very important to control the light wavelength: tuning the photon energy 

allows us to perform different kinds of experiments, passing, for example, from 

linear to resonant or to indirect non-linear photoemission. 

Tunability is one important feature of femtosecond pulsed sources that allow 

to perform both non-linear optics in uniaxial crystals, to obtain harmonics of 

the fundamental wavelength, and parametric amplification, yielding continuous 

frequency tunability. 

In this work we report experiments in which three kinds of harmonic up 

conversion in beta-barium-borate (βBBO) crystals are used to obtain second 

(λ = 395 nm, hν = 3.14 eV), third (λ = 263 nm, hν = 4.71 eV) and fourth 

(λ = 197 nm, hν = 6.28 eV) harmonic of the fundamental wavelength. Second 

harmonic generation is performed in a type I crystal; the third harmonic is 

obtained by a sum frequency generation between this second harmonic beam and 

the fundamental in a type II crystal; the fourth harmonic is generated frequency 

doubling the second harmonic beam out of phase matching in a 200 µm thin 

βBBO crystal. 

An almost continuous tunability from 1600 nm infrared to 250 nm ultraviolet 

is guaranteed by two parametric amplifiers, shown in Fig. 4.2, and their 

harmonics. 

The first is a traveling-wave optical parametric generator (TOPG) that supplies 

130 fs pulses with an average power of about 30 mW and a wavelength 

tunable in the infrared between 1150 nm and 1500 nm (0.8 eV and 1.1 eV): its 

fourth harmonic spans the range 3.2 eV ≤ hν ≤ 4.4 eV. The second is a noncollinear 

optical parametric amplifier (NOPA) that yields pulses with temporal


Figure 4.2: Parametric amplifiers. Their output and harmonics allow to perform 

photoemission with a photon energy tunability 0.8 eV ≤ hν ≤ 5 eV. 

width down to 20 fs with a tunability between the visible λ = 500 nm to the infrared 

λ = 750 nm ((1.6 eV and 2.5 eV) and an average power of about 10 mW, 

allowing to span the range 3.2 eV ≤ hν ≤ 5 eV with the second harmonic. 

The output wavelength is measured by an online spectrometer. Pulse duration 

and temporal shape can be measured by an online home made fast autocorrelator 

shown in Fig. 4.3. 

4.3 Ultrahigh vacuum chamber 

Experiments are carried out in a ultrahigh vacuum (UHV) chamber system 

shown in Fig. 4.4. A turbomolecular pump keeps a base pressure better than 

3 × 10 −10 mbar at room temperature; the chamber is made in µ-metal, ensuring 

that the residual magnetic field inside is smaller than 10 mG. 

The laser beam enters the chamber through an optical flange to impinge on 

the sample held on a manipulator with 5 degrees of freedom. Photoemitted 

electrons are collected by a time of flight spectrometer (ToF) that measures 

their kinetic energy with a resolution of about 35 meV at E KE = 2 eV. The 

photoemission geometry is described in Fig. 4.5. The angle between the imping-

4.3. Ultrahigh vacuum chamber 55 

Figure 4.3: Home made fast autocorrelator for online pulse characterization.


Figure 4.4: Ultra high vacuum chamber system.

4.4. Photoemission Measurements 57 

ing beam and the trajectory of analyzed electrons is fixed to 30 ◦ , whereas the θ 

angle between the normal to the sample and the position of the analyzer can be 

varied to perform angle resolved photoemission: the geometric dimension of the 

microchannel plate used as the analyzer detector gives an angular resolution of 

about ∆θ = 0.8 ◦ . 

Photoemission measurements described in the following chapters of Part II 

are performed on noble metals single crystals polished with standard optical 

methods to a mirror finish and oriented along the (111) or (100) surface with 

an error of 0.2 ◦ . The sample’s surface is cleaned by cycles of Ar + sputtering 

and subsequent annealing at 500 ◦ C for Au and Cu, 400 ◦ C for Ag: the cleaning 

procedure is carried on until the proper work function for the surface under 

examination is obtained from photoemission spectra. 

Once the sample is clean, a sharp low energy electron diffraction (LEED) 

pattern can be seen, allowing the sample orientation, rotating the φ angle around 

the normal to the surface, for dispersion: along the ΓM for (111) samples, as 

shown in Fig. 4.5, or along the ΓX for (100) samples. 

4.4 Photoemission Measurements 

The detection angle θ between the sample normal and the analyzer axis is 

related to the electron momentum parallel to the surface k ‖ by 

√ 2mEKE 

k ‖ = 

sin θ, (4.1) 

ħ 

where E KE is the kinetic energy of the electrons photoemitted from the investigated 

state. The experimentally observed kinetic energy of the photoelectrons 

is given by 

E KE = hν − E i + E B + V ST , (4.2) 

where hν is the photon energy, E i is the electronic state energy with respect to 

the vacuum level, E B is an external potential of a few tenths of a volt applied


Figure 4.5: Photoemission geometry: varying the φ angle we can chose the direction 

along which to perform the dispersion measurements. The image describes 

the ΓM direction on the LEED pattern of a Cu(111) sample.

4.4. Photoemission Measurements 59 

between the sample and the analyzer to collect the low energy electrons, and 

V ST = Φ S − Φ T oF is a contact potential difference due to the work function 

difference between the sample (Φ S ) and the time of flight spectrometer (Φ T oF = 

4.20 eV). 

Photoemission is measured with a sample to analyzer entrance slit distance 

of 40 mm over a total flight distance of 440 mm. To obtain reliable results, the 

influence of stray fields due to contact potential difference or applied electric 

fields between the sample and the spectrometer entrance slit must be considered. 

The common procedure is to compensate the contact potential difference 

applying a bias voltage. However, the bias voltage compensation is never perfect, 

and the applied decelerating voltage distort the spectra at low electron 

kinetic energy and at non-normal emission angles. For this reason, in the course 

of these experiments, we considered a different approach. In our system the 

spectrometer entrance slit and the sample constitute a parallel plate system 

and the contact potential difference produces an electric field acting only on the 

component of the electron momentum perpendicular to the surface. In this way 

the component of the electron momentum parallel to the surface is conserved, 

while the angular distribution of photoemitted electrons is modified by changing 

the perpendicular electric field. 

We verified that, with an uncompensated contact potential difference, the 

same k ‖ spectral distribution is obtained for different photon energies, implying 

that only the normal component of the electron momentum is modified. This 

amounts to a vertical translation in the kinetic energy versus k ‖ dispersion, 

conserving the curvature of the dispersing electronic states peaks and then their 

effective mass values; spectra deformation must be taken into account only 

for electrons photoemitted at very low kinetic energies (less than 1 eV), for 

which non-normal acceleration due to residual fields is not negligible. In the 

spectra shown in this work the kinetic energy of the photoelectrons include the 

uncompensated contact potential difference.

Chapter 5 

Spin orbit splitting on 

Au(111) surface state 

Linear and two-photon photoemission preliminary studies on Au(111) spin 

orbit splitted Shockley surface state are shown. The splitting is evident in linear 

photoemission spectra, taken in a low kinetic energy region that prevents from 

a correct measurement of the effective mass in the energy versus k ‖ dispersion; 

in two-photon photoemission experiments the expected value of the effective 

mass is measured. To perform future spin polarization dichroism measurements, 

non-linear photoemission has to be performed at low temperature to narrow the 

Shockley surface state linewidth and evidence the splitting. 


The splitting of bulk electronic levels due to spin-orbit coupling is well known. 

In the last decade also the spin-orbit splitting of surface Shockley states was 

discovered and investigated by theory and experiment [46, 47, 48, 49]. 

61

62 Chapter 5. Spin orbit splitting on Au(111) surface state 

According to theory, the E(k ‖ ) dependence on k ‖ = ∣ ∣k ‖ 

∣ ∣ is 

where E B 

E(k ‖ ) = E B + ħ2 k‖ 

2 

2m ± γ SOk ‖ = E B ′ + ħ2 ( 

k‖ ± ∆k ) 2 

, (5.1) 

2m 

is the binding energy at Γ without spin-orbit coupling, m is the 

electron rest mass, γ SO a parameter that represents the coupling intensity; the 

last two terms of Eq. (5.1) are equal if we define ∆k = γ SO m/ħ 2 and E ′ B = 

E B − (ħ∆k)2 

2m 

. The splitted surface state is represented on the E(k ‖) versus k ‖ 

graph by two identical parabolas horizontally shifted by an amount 2∆k; their 

vertexes are shifted of ±∆k from the Γ point. 

Spin-orbit splitting is expected on all (111) faces of noble metals, but its 

intensity is variable due to each solid’s electronic structure: angle resolved photoemission 

can resolve the two splitted features on Au(111) [46, 47, 48, 49] 

for which theory predicts a splitting 2∆k = 0.025 Å −1 [48], whereas for lighter 

noble metals the splitting is too small to be experimentally observed with nowadays 

available techniques (2∆k = 0.0013 Å −1 for Ag(111) [48], even smaller for 

Cu(111) [47]). 

In Fig. 5.1 two examples of spin-orbit splitting calculation for the Au(111) 

surface state are reported from Ref. [48, 49]. Panel a) also reports a comparison 

with angle resolved photoemission data. The dispersion of the splitted surface 

state does not depend on the orientation along a high symmetry direction. 

While the electron population of bulk spin-orbit splitted states is not spin 

polarized, because both up and down spin are allowed on each state, for surface 

states each one of the splitted parabolas is almost completely spin polarized 

[49]: spin depending dichroism could be probed on the Au(111) surface state 

implementing angle resolved photoemission with different light polarizations. 

Since the surface state’s full width at half maximum (FWHM) strongly increases 

with temperature, to study the spin-orbit splitting on this state it is 

necessary to cool down the sample to some tens of K. In this chapter we show 

some preliminary experimental results obtained at room temperature with a 

femtosecond Ti:Sapphire laser source.

5.1. Introduction 63 

Figure 5.1: a) Comparison between calculation (dots) and experiment (gray 

scale plot) on spin-orbit splitted Au(111) surface state, reported from Ref. [48]. 

b) Theoretical description of the E(k ‖ ) dependence on k ‖ (Eq. (5.1)) for the 

spin-orbit splitted Au(111) surface state, reported from Ref. [49]


5.2 Discussion of experimental results 

Angle resolved photoemission measurements were performed on Au(111) surface 

states. The experimental setup is described in Chap. 4. Linear photoemission 

was performed with a photon energy hν = 6.28 eV obtained by two frequency 

doubling steps of the Ti:Sapphire fundamental; two-photon photoemission 

spectra were also acquired impinging with photons of energy hν = 4.28 eV, 

fourth harmonic of the traveling-wave optical parametric generator output. 

In Fig. 5.2, we show the results of direct photoemission. In this conditions 

we can collect a spectrum in some tens of minutes, but the difference hν −Φ S ≃ 

1 eV between the sample work function and the photon energy is small for 

our purposes: electrons leaving the sample have an energy less than 1 eV and 

are then accelerated by the chemical potential difference V ST = Φ S − Φ T oF ≃ 

1 eV between sample and time of flight analyzer. This acceleration is nonnegligible 

and deforms spectra in the low energy region: this is the reason why 

the values of effective mass ratio m ∗ /m = 0.44 ± 0.04 and spin orbit splitting 

2∆k = 0.047 Å −1 are overestimated compared to the values of m ∗ /m = 0.25 

and 2∆k = 0.025 Å −1 found in literature [47, 48, 49]. 

To move the features we are studying to a higher kinetic energy region, we 

can perform non-linear photoemission spectroscopy. In this case our measurements 

are disturbed by the big amount of photoelectrons coming from occupied 

states lying under the surface state: measurement time increases and energy 

resolution gets worse, making it impossible to distinguish the spin-orbit splitting. 

Nevertheless, this configuration allows to measure a value for the effective 

mass ratio m ∗ /m = 0.24 ± 0.02 in agreement with literature. This data are 

shown in Fig. 5.3.

5.2. Discussion of experimental results 65 

Figure 5.2: a) Angular dispersion of the linear photoemission spectra collected 

at hν = 6.28 eV along the ΓM direction of the Brillouin zone. b) Kinetic energy 

versus k ‖ momentum for the spin-orbit splitted Shockley surface state measured 

with linear photoemission on Au(111). A parabolic fit gives an effective mass of 

m ∗ 1/m = 0.45 ± 0.04 and m ∗ 2/m = 0.42 ± 0.04 for the Shockley surface state; the 

two parabolas are horizontally shifted by 2∆k = 0.047 Å −1 . The too high values 

of this results are attributed to a deformation of this region of the photoemission 

spectra, due to the significant acceleration of low energy electrons caused by the 

chemical potential difference between sample and detector.


Figure 5.3: a) Angular dispersion of the two-photon photoemission spectra collected 

at hν = 4.28 eV along the ΓM direction of the Brillouin zone. b) Kinetic 

energy versus k ‖ momentum for the spin-orbit splitted Shockley surface state 

measured with two-photon photoemission on Au(111). A parabolic fit gives an 

effective mass of m ∗ /m = 0.24 ± 0.02 for the Shockley surface state. With this 

photon energy, the studied feature are in a kinetic energy region that allows to 

correctly measure the effective mass, but the loss of resolution prevents us from 

distinguishing the spin orbit splitting.

5.3. Conclusions 67 


These data demonstrate the possibility to observe the spin-orbit spitting 

on the Shockley surface state of Au(111) with femtosecond photoemission. To 

perform further experiments it is necessary to cool down to liquid helium temperature 

to have sharper peaks. Linear photoemission is a good technique to 

see the spin-orbit splitting and can be a good test for sample cleanliness, but for 

quantitative measurements non-linear photoemission is suggested, even though 

a long runtime to perform measurements must be taken into account in these 

conditions.

Chapter 6 

Comparison between theory 

and experiment on Cu(111) 

and Cu(100) surface 

electronic states. 

Recent advances in both the experimental resolution and in the computational 

capabilities motivate new studies of surface properties. In particular, 

a detailed comparison between theoretical and experimental data is expected 

to provide a better insight into surface and image states. In this chapter we 

present a joint effort analyzing such features of the Cu(111) and Cu(100) surfaces. 

The experiments are performed by using both linear and non-linear angle 

resolved photoemission. From the theoretical point of view, we use the Green 

function embedding technique within density functional theory. We account for 

the image states by suitably modifying the effective potential in the Kohn-Sham 

(KS) equation and the generalized boundary conditions on the Green function. 

69

70 Chapter 6.Theory and experiment on copper surface electronic states 

Comprehensive theoretical and experimental results on the effective masses and 

binding energy of the Shockley state and the first image states are reported. 

The theoretical and numerical calculations reported in this chapter were 

performed by the G. P. Brivio and M. I. Trioni group of the Dipartimento di 

Scienza dei Materiali dell’Università degli studi di Milano Bicocca; results are 

published in Ref. [3]. 


When a solid is terminated by a surface, new electronic states are created, 

that are not present in ideal infinite solids: the so called surface states, localized 

at the interface and decaying exponentially into vacuum. If their energy 

lies within a surface projected gap, such states are truly discrete and their wavefunction 

decays exponentially into bulk too; otherwise they display a linewidth 

due to hybridization with the bands, and propagate into the solid. Such surface 

states are important for bond formation in surface reactivity and hence in 

growth kinetics [50]. A particular class of surface states is determined by the 

image potential, which shows a Coulomb tail far enough from the surface [51]. 

Since the probability maxima of the surface states wavefunction are located a 

few Å from the topmost ion layer, those states show an almost free electron 

dispersion as a function of the surface wavevector. Their measurement may also 

allow the determination of the detailed long range shape of the surface potential 

and help in working out losses in photoemission experiments [52, 53]. 

Apart from inverse photoelectron spectroscopy, measurements of image potential 

states are recently performed with two photon photoemission (2PPE). 

This technique, which can also be used for detecting other types of surface state, 

is particularly suitable to probe unoccupied bound states at surfaces, and it can 

be applied to determine their energy, dispersion and lifetime. In fact the energy 

and momentum resolution of two photon photoemission is comparable to 

that achieved by conventional angle resolved photoelectron spectroscopy. The

6.1. Introduction 71 

spectroscopic properties of image potential states are an important and timely 

research topic. For example, studies on the lifetime may help understanding 

the contribution of bulk hybridization [54, 55] and of the many-body effects 

[56]. Nevertheless few reliable and accurate data of surface states, especially of 

image ones, exist; so systematic investigations, even for clean metal surfaces, 

are needed. 

From the theoretical point of view, calculations of surface states within the 

density functional theory (DFT) scheme are reliable only when such states are 

occupied in the system ground state. This rules out the possibility of obtaining 

a density functional theory accurate description of image potential states, also 

considering that the image potential tail cannot be accounted for in any current 

approximation of the exchange-correlation potential. Another difficulty stems 

from the fact that the popular repeated slab geometry does not always allow 

for clearly distinguishing surface from bulk structures and may even suggest 

spurious ones [57, 58]. 

In this chapter, we will pursue a joint experimental and theoretical study of 

Shockley and image potential states of Cu cut along the (111) and (100) surfaces, 

aiming at presenting a more comprehensive set of data than the previous 

works [51, 59]. The photoemission apparatus is described in Chap. 4. Density 

functional calculations are performed within the Green function embedding approach 

[60]. This method, outlined in Sec. 6.2, is particularly suited to describe 

spectral properties. In fact, since it takes into account a truly semi-infinite solid, 

it allows for distinguishing between sharp discrete surface states and resonant 

ones [61]. The image potential is added phenomenologically in the last step of 

the solution of the Kohn-Sham equation [58, 62]. All results are presented and 

discussed in Sec. 6.4.


6.2 Theoretical framework 

The calculations are worked out within the embedding scheme [60]. In particular, 

we adopt the implementation by Ishida [61] for the treatment of realistic 

solid surfaces. We consider a system which is infinite, periodic in the surface 

plane but non-periodic along the surface normal direction z. We define a finite 

region along z, the so called embedded region, determined by the volume in 

which the perturbation due to the surface is well screened. The problem is then 

solved in such a region only. For a metallic Cu substrate, this implies considering 

the two topmost Cu layers on the bulk side, and a vacuum portion extending 

for 12 atomic units from the outermost ion layer. 

The calculation is carried out solving a Kohn-Sham-like equation in the 

density functional theory framework, using a Green function description and a 

full-potential linearized augmented plane wave (FLAPW) method. Additional 

terms, named embedding potentials and acting as a sort of generalized boundary 

conditions, appear in the Kohn-Sham hamiltonian accounting for the presence 

of the semi-infinite substrate [63]. The embedding potential at the vacuum 

region is usually determined analytically from the Green function calculated for 

a constant potential [64]. 

The implementation used in this paper has the additional capability of treating 

image potential states, which are not described correctly in density functional 

theory by the local or semi-local approximations commonly used for the 

evaluation of the exchange and correlation functional. In fact such functionals 

determine an exponential asymptotic decay of the potential outside a surface, 

instead of the correct image-like behavior V (z) ∝ −1/4z. We follow the 

phenomenological approach developed by Nekovee and Inglesfield [62] and just 

applied to Na/Cu(111) in Ref. [58]. It allows one to compute the embedding 

potential at the semi-infinite vacuum taking into account the correct form of 

the asymptotic potential decay. Then, to avoid any discontinuity at the vacuum 

embedding surface, the effective potential V eff inside the outermost part of

6.3. Experiment 73 

the embedded region is obtained by gradually mixing the effective potential of 

the Kohn-Sham equation with the model potential of the form −1/4|z − z 0 | (z 0 

being the image plane position) in the last step of the self-consistent calculation. 

The parameters of the calculation are reported in Ref. [58]. All results are obtained 

in a generalized gradient approximation for the exchange and correlation 

functionals [65]. 

6.3 Experiment 

The photoemission measurements are performed on Cu(111) and Cu(100) 

single crystals. The experimental setup is described in Chap. 4. The second 

and the fourth harmonic of the Ti:sapphire laser system, hν = 3.14 eV and 

hν = 6.28 eV respectively, are used to excite the (111), surface performing 

angle resolved photoemission along the ΓM direction; the fourth harmonic of 

the traveling-wave optical parametric generator, tuned at hν = 4.35 eV, is 

used to excite the (100) surface, measuring the states dispersion along the ΓX 

direction. 

Spectra at normal electron emission are collected with the p polarized laser 

beam at an angle of incidence of 30 ◦ from sample normal. The samples’ work 

functions are 4.95 eV for Cu(111) and 4.6 eV for Cu(100) [15]. 

6.4 Results and discussion 

A complete band structure study of the Cu(111) surface is shown in Fig. 6.1, 

in which a surface plot of the surface states dispersion is presented. Starting 

from the left we observe the boundary of the s band: at about −5 eV from 

E F 

a gap opens up at Γ. At higher energies, roughly in the energy interval 

(−3, −1) eV, we observe the d band structures; then, between −1 eV and 4 eV 

at Γ, we observe a new gap, into which the sharp peaks identify the dispersion 

of the Shockley state, extending up to the first excited state s band above 4 eV.


Here the two small features, with a parabolic-like dispersion, are the first two 

image potential states. 

Figure 6.1: 

surface. 

Surface plot of the band structure calculations for the Cu(111) 

It is interesting to remark that, since those structures are inside a band, 

their width comes out naturally from our embedding calculation because of 

hybridization with bulk surface projected states. On the other hand, the peaks 

of the Shockley surface state, which lies in a gap and is indeed a sharp discrete 

delta function also in our method, have been suitably broadened to be plotted. 

Such important difference would not be displayed by finite volume methods, 

such those based on the (repeated) slab geometry.

6.4. Results and discussion 75 

Figure 6.2: Features of a non-linear photoemission (2PPE) spectrum collected 

at k ‖ = 0 with hν = 3.14 eV for Cu(111) is explained by a schematic description 

of the electronic energy structure of the sample. Features can be ascribed to 

the Shockley surface state, the image potential states, the Fermi level and their 

above threshold replicas.


In Fig. 6.2 we show a schematic explanation of the features in non-linear 

photoemission spectra from Cu(111), collected at k ‖ = 0 with hν = 3.14 eV. 

Calling E F E the kinetic energy of electrons laying on the first (lower energy) 

cyan marked Fermi edge in the spectrum, we can ascribe to the n = 0 Shockley 

surface state the red marked feature at kinetic energy E 0 = E F E − 0.42 eV and 

address as the n = 1 and n = 2 image potential states the peaks located at 

kinetic energies E 1 = E F E + 0.95 eV and E 2 = E F E + 1.6 eV respectively. 

In this experiment on Cu(111), the occupied Shockley surface state is probed 

by two photon photoemission, while the unoccupied image states are populated 

by non-resonant two photon absorption and probed by direct photoemission. 

The addition of one more above threshold photon to each feature explains the 

higher energy replicas shown in Fig. 6.2. Both calculations and measurements 

on the dispersion along ΓM of the Shockley surface state and the n = 1 image 

potential state are shown in Fig. 6.3. 

The occupied Shockley surface state is also probed via direct photoemission 

with a photon energy of 6.28 eV, in the same conditions of the non-linear photoemission 

experiment. The binding energy at the Γ point and the effective mass 

of the Shockley state, measured with both linear and non-linear photoemission, 

are found to be consistent within the experimental error, with a larger variation 

in the effective mass: m ∗ /m = 0.43 ± 0.04 from the non-linear experiment, 

m ∗ /m = 0.48 ± 0.04 from direct photoemission (see Tab. 6.1). 

In Tab. 6.1 and Fig. 6.3 the theoretical and experimental results for the surface 

state of Cu(111) are compared. Note that the lower edge of the excited state 

calculated band has been shifted of 0.4 eV to higher energies. This arbitrary 

shift tries to compensate the well known defect of density functional theory, 

which underestimates the solid gaps. Consequently the n = 1 image potential 

state is a discrete state and not a resonant one as it appears in Fig. 6.1. 

The calculated effective masses underestimate the experimental values by 

about 15%. A possible explanation is that the ab initio calculations describes 

the system in its ground state while in the experiment the system is in a strongly


Figure 6.3: Comparison between the experimental (open circles) and calculated 

(dotted line) dispersion relationships of the surface states of Cu(111). The 

dotted horizontal lines are the Fermi E F and vacuum E V energies, respectively.


Calculated 

Experimental 

System State Energy (eV) m ∗ /m Energy (eV) m ∗ /m 

Cu(111) n = 0 −0.526 0.39 −0.42 ± 0.05 a 0.48 ± 0.04 a 

0.43 ± 0.04 b 

−0.434±0.05[59] . . . 

−0.445±0.05[66] . . . 

n = 1 −0.779 1.10 −0.81 ± 0.05 b 1.28 ± 0.07 b 

−0.82±0.05[51] . . . 

n = 2 −0.226 1.02 −0.18 ± 0.05 b . . . 

−0.27±0.07[51] . . . 

n = 3 −0.107 1.00 . . . . . . 

Cu(100) n = 1 −0.493 0.99 −0.55 ± 0.05 c 1.05 ± 0.07 c 

n = 2 −0.169 1.00 . . . . . . 

n = 3 −0.082 1.00 . . . . . . 

a) This work, hν = 6.28 eV 

b) This work, hν = 3.14 eV 

c) This work, hν = 4.35 eV 

Table 6.1: Key features of the Shockley state (n = 0, for Cu(111) only) and the 

first three image potential states for the clean Cu(111) and Cu(100) surfaces. m 

and m ∗ are the electron and the effective mass, respectively. Energies at the Γ 

point are defined from the Fermi level for the Shockley state and from vacuum 

for the image states, respectively. References to experimental data other than 

those of this paper are given.


excited state. Moreover, since it is known that density functional theory underestimates 

the gap energies in calculating band structures, this is likely to affect 

the parameters of states that are located near the gap edge, like the image potential 

state in Cu(111). However, we point out that the results of this work 

are the most exhaustive ones for the surface states on Cu(111). 

To further test the consistency between experiment and density functional 

calculations, the Cu(100) image potential state effective mass has been measured 

and calculated. For this surface it is well known that there is no Shockley 

state and that the image potential states lie in the middle of a gap [54]. The 

n = 1 image potential state from Cu(100) is measured in the same experimental 

conditions described previously via two photon photoemission with a photon 

energy of 4.35 eV; the results are shown in Fig. 6.4. 

In this case, the calculated effective mass of the n = 1 image potential 

state is in agreement with the experimental value (within experimental errors), 

indicating that the density functional theory failure in reproducing the absolute 

energies do not affect the effective mass of states far from the band gap edges. 

In Fig. 6.5 the theoretical results for the Cu(100) image states are reported. 

Note that the calculation method permits to clearly distinguish the features of 

those states up to n = 7. 


In this chapter we have presented a comprehensive experimental and theoretical 

investigation on surface states of Cu(111) and Cu(100). In particular, 

our work provides a first set of results fully calculated within density functional 

calculation. They are critically compared with new non-linear dispersion 

measurements, binding energies and effective masses. The weakness of density 

functional theory in determining absolute energies is well known. However, going 

beyond the approach of this work is very demanding since it asks for a 

many-body treatment, such the GW one, but developed for a semi infinite solid


Figure 6.4: Measured dispersion of the first image potential state of Cu(100). A 

parabolic fit gives a value m ∗ /m = 1.05±0.07 for the ratio between the effective 

mass and the electron rest mass.


Figure 6.5: Calculated dispersion relationship of the image states of Cu(100).


[55]. 


This work was funded in part by the EU’s 6th Framework Programme 

through the NANOQUANTA Network of Excellence (NMP4-CT-2004-500198).

Chapter 7 

Role of athermal electrons 

in non-linear photoemission 

from Ag(100) 

The non-linear photoelectron spectra obtained by short laser pulses from a 

Ag(100) surface show a high energy electron emission due to an athermal electron 

distribution created by the laser pulse. By comparing the photoemission 

at normal and non-normal emission geometry it is possible to evidence the independence 

of the hot electrons photoemission on the parallel momentum and on 

different final state configurations. A photoemission correlation measurement 

evidences that non-photoelectric effects, as tunneling or thermally assisted photoemission, 

do not contribute to the electron yield. Various theoretical models 

are discussed on the basis of the present data. 

83

84 Chapter 7.Athermal electrons in non-linear photoemission from Ag(100) 


The dynamics of the non-equilibrium electron distribution in solids, induced 

by excitation with ultrashort laser pulses, has been widely investigated in the 

last decade. Time resolved two-photon photoemission has contributed to the 

detailed investigation of the non-equilibrium electron dynamics and has been 

crucial for the understanding of the role played by electron-electron and electronphonon 

scattering in the relaxation processes [67, 68, 56]. In addition, the femtosecond 

excitation of a non-equilibrium electron distribution is a powerful tool 

to study the problem of charge transfer between a metallic substrate and adsorbate 

or polar molecules overlayers, a relevant topic in modern femtochemistry 

[69]. 

Recently, higher order multiphoton photoemission processes have been recognized 

as an important extension of two-photon photoemission technique, increasing 

the energy range of the unoccupied states that can be studied and 

providing information on the photoexcitation mechanisms [18, 70, 71]. Few 

studies have investigated the influence of band structure on this kind of processes, 

especially when above threshold photoemission is involved, a process 

in which the last multiphoton electron transition is entirely above the vacuum 

level. The role played by the non-equilibrium electron distribution in the nonlinear 

photoemission process has not been entirely clarified and the role of the 

intermediate states, especially in cases where no electron bulk band states are 

available, for example when the intermediate state fall in a band gap, is not well 

understood and is a subject of discussion [72]. 

In this chapter, we focus on the non-linear photoemission of electrons excited 

so that the intermediate states are within the extended band gap region along 

the (100) direction of Ag. The photoemission spectra are obtained by exciting 

the electrons by short laser pulses in a regime where the equilibrium distribution 

of the bulk electrons is negligibly perturbed and a small population of excited 

electrons, whose equilibrium electron temperature is not defined, is far from

7.2. Two-photon or three-photon photoemission 85 

equilibrium. By comparing the photoemission at normal and non-normal emission 

geometry it is possible to evidence the independence of the hot electrons 

photoemission on the parallel momentum and from different final and intermediate 

states configurations. A photoemission correlation measurement evidences 

that non-photoelectric effects, as tunneling or thermally assisted photoemission, 

do not contribute to the electron yield. These experimental evidences can be 

rationalized in terms of a non-thermal electron population subject to interactions 

such as phonon scattering [73], inverse bremsstrahlung processes [74] or 

transient exciton formation [75] that quench the dependence of photoelectric 

emission on electron momentum and band structure. 

7.2 Two-photon or three-photon photoemission 

When a metal sample is excited by short light pulses with a photon energy 

smaller than the work function, different non-linear photoelectric processes can 

contribute to the photoemission spectrum. The typical appearance of a nonlinear 

photoemission spectrum, obtained from the experimental setup described 

in Chap. 4, is reported in Fig. 7.1 (black line); light pulses with photon energy 

hν = 3.14 eV, p polarized, were obtained by doubling the amplified Ti:Sapphire 

laser system output. The size of the spot, about 300 × 300 µm 2 , was chosen 

to have a high count rate minimizing space charge effects in the photoemission 

spectra. 

The final state energy of the electrons is measured from the Fermi energy 

E F , the emission angle is 30 ◦ at normal light incidence. The two-photon photoemission 

and three-photon photoemission spectral regions are indicated. A high 

energy electron tail that departs from the two-photon Fermi edge is related to 

the photoemission from a non-equilibrium electron distribution excited in the 

empty bulk states. In this case the photoemission mechanism is a three-photon 

photoemission process [77, 71]. 

A non-normal emission geometry implies that different parallel momenta k ‖


Figure 7.1: Photoemission spectra measured at 30 ◦ emission angle (black line) 

and normal emission angle (gray line). The red dashed line simulates the steplike 

structure expected in the case of coherent photoemission. The projected 

band structure along the ΓX direction of Ag(100) surface Brillouin zone is shown 

(taken from Ref. [76]). A measured work function of 4.3 eV is reported.

7.2. Two-photon or three-photon photoemission 87 

are associated with electrons ejected with different kinetic energies E KE . The 

relationship between k ‖ and the kinetic energy at θ=30 ◦ emission is: 

√ 

k ‖ (Å −1 2mEKE 

) = 

sin θ = 0.256 √ E KE (eV ), (7.1) 

ħ 

where m is the free electron mass. In the projected band structure scheme, 

reported in Fig. 7.1, the dependence of the measured external kinetic energy 

on the parallel momentum is shown. The three-photon photoemission kinetic 

energy region spans a k ‖ interval between 0.3 and 0.5 Å −1 . As can be seen from 

the band structure scheme reported, the spectra span a parallel momentum 

interval where the final states cross the border between the gap and the empty 

bands. The lack of discontinuities in the photoemission spectrum near the 

empty bands crossing suggests a minor role played by the final states in the 

three-photon photoemission process. 

The structure of spectra taken at normal emission (k ‖ = 0), reported in 

gray in Fig. 7.1, do not differ from non-normal emission spectra, except for 

the presence of image potential states (IPS), that can not be photoemitted 

at large emission angles. The excited non-equilibrium electrons populates the 

image potential states by scattering assisted indirect processes, inducing an 

effective mass variation and electric dipole selection rules violation [70]. The 

similar behavior of the three-photon photoemission regions at different k ‖ is the 

signature of the independence of this photoemission process on the availability 

of intermediate states. 

According to the recent literature, three different processes contribute to 

multiphoton transitions in a non-linear photoemission process [78, 79]. Coherent 

transitions, when the intermediate state is a virtual state. Direct transitions, 

when the intermediate state is a real state and there is no change of the 

electron momentum parallel to the surface sample (also referred to as resonant 

transitions). Indirect transitions, when the electron momentum parallel to the 

sample surface is not conserved during the transition from the initial to the 

final state. The high kinetic electron tail detected in the reported spectra can


not be explained by a coherent photoemission because the expected step-like 

structure (see red dashed line in Fig. 7.1) of the spectrum is not reproduced by 

experiments. Moreover, direct resonant dipole transitions between two levels in 

the conduction band are not allowed in the dipole approximation. This follows 

from the energy and quasimomentum conservation [73]. 

A plausible mechanism to explain the high energy electron tail is an indirect 

three-photon photoemission process, mediated by scattering events which 

change the k ‖ momentum of the photoemitted electrons. In this picture the 

scattering mediated photon absorption creates a non-equilibrium electron population 

in the unoccupied bulk states at k ‖ ≠ 0, where available states extend 

up to the vacuum level. A scattering event is also responsible for the electron 

k ‖ momentum exchange necessary to the photoemission process, as shown in 

Fig. 7.1. The scattering mediated absorption completely relaxes the selection 

rules and the dependence on k ‖ , thus explaining the similarity between normal 

emission and non-normal emission spectra. This interpretation is consistent 

with two recent theoretical works that investigate the interaction of the laser 

field with the excited non-equilibrium conduction electrons in metals. Photon 

absorption is attributed to electron collisions with phonons [73] or to an inverse 

bremsstrahlung process [74]. 

Our findings can be explained also by the creation of transient excitonic 

states, through multi photon absorption. It was speculated that the observed 

states in band gaps are caused by an attractive interaction between the photoelectron 

and its localized hole in the d bands, due to the finite time it takes 

the valence band electrons to screen the photohole [80]. Successive theoretical 

models confirmed this possibility [75]. In this case the intermediate electron 

states are given by the transient excitonic levels created by the laser photohole 

production. We note that all the proposed models [73, 74, 75] are based on ab 

initio results, without invoking the so called transport correction [81, 72].

7.3. Photoemission autocorrelation 89 

7.3 Photoemission autocorrelation 

To investigate the origin of the athermal part of the excited electrons, a photoemission 

autocorrelation is used: electron spectra are measured as a function 

of the delay between two identical pulses. Differently from a pump and probe 

experiment, where the electron population is excited (without photoemission) 

by the optical pump and emitted by the probe pulse, in photoemission autocorrelation 

both pulses produce a complete electronic spectrum. The correlation 

of identical pulses via the photoemitted non-linear spectra gives information 

on the relaxation dynamics, the non-linear order of the photoemission process, 

and gives indication on the dependence on non-photoelectric effects, such as 

tunneling processes and thermally assisted photoemission [82]. 

In Fig. 7.2 the spectra are shown as a function of the delay between two 

collinear pulses. Normal light incidence prevents artifacts due to different absorption 

coefficients from the cross polarized beams. It is important to note the 

difference of about 4 order of magnitude in the two-photon and three-photon 

photoemission yields. In order to obtain reasonable electron statistics in the 

three-photon region, the typical duration of each spectrum collection is several 

hours. As a consequence, fluctuations and drifts of the laser intensity result in 

a scattering of the non-linear photoemission intensity, which can not be totally 

compensated by renormalization to the mean incident power. 

Previous pump and probe measurements [83, 84] of the thermalization of the 

non-equilibrium electron population were performed on a polycrystalline metal 

film irradiated by femtosecond laser pulses, with an incident fluence larger than 

1 GW/cm 2 . In this case, the absorption of the electromagnetic field energy resulted 

in a strong perturbation of the equilibrium electron distribution, with an 

electronic temperature increase up to 400 K. On the contrary, our measurements 

are carried out with a laser intensity of about one order of magnitude lower. 

For this reason the equilibrium electron distribution is weakly perturbed after 

the absorption of the laser pulse. The temperature increase of the two-photon


Figure 7.2: The photoemission spectra are plotted versus the delay time between 

pump and probe pulses. The pump and probe measurements are carried out at 

normal incidence and at an angle of 30 ◦ between the detector and the normal 

to the sample surface. Some spectra are omitted for graphical reasons. In the 

inset the two-photon Fermi edge temperature associated to each spectrum is 

reported. The temperature spike seen at 0 delay is attributed to a small space 

charge deformation since this timescale is too short for any energy exchange 

between the electron system and the phonon gas.


photoemission Fermi distribution in the time resolved spectra shown in Fig. 7.2, 

due to energy exchange between the photoexcited electron population and the 

bulk electrons, is at most few tens degrees. The two-photon Fermi edge of the 

spectra at various delays is fitted with a Fermi Dirac distribution convoluted 

with a Gaussian function, to take into account the apparatus resolution. The 

temperature increase of the equilibrium electron distribution, measured when 

the delay exceed the laser pulse duration, is about 20 K. The calculated Fermi 

temperature using the two temperature model is about 345 K after a single laser 

pulse, a value compatible with that measured in this work. 

In Fig. 7.3 the electron yield from the two-photon and three-photon photoemission 

spectral regions is shown as a function of the delay. To measure the 

nonlinearity of the photoemission process in each spectral region, the electron 

yield is reported versus laser fluence I in the inset. The electron yield in the 

two-photon Fermi edge region (1.66 ≤ E KE ≤ 1.88) has a I 2 power law dependence 

on laser fluence, as expected for a two-photon photoemission process. 

The electron yield in the three-photon region (2.18 ≤ E KE ≤ 3.5) has a I 3 dependence 

on fluence, in agreement with the result reported in Ref. [77]. Since 

each region has a different photoemission non-linearity, to compensate for laser 

intensity drifts, we normalized the electron yield using the mean laser intensity 

I 0 to the corresponding power law dependence (I0 2 for two-photon and I0 3 for 

the three-photon photoemission regions). 

The autocorrelation of the photoemission yield from each region shows an 

instantaneous response to the laser excitation on the temporal scale of the pulse 

time width. This result confirms that, within the pulse duration, the laser 

electric field coexists with a non-equilibrium electron gas which is thermalizing 

through electron-electron scattering on the same timescale. An estimate of the 

scattering rate τ of the electron gas averaged over the three-photon photoemission 

region, using the Fermi liquid theory [85] gives a mean scattering time of 

〈τ〉 ≃ 23 fs. This value is compatible with a decay time of the non-equilibrium 

population shorter than the 130 fs pulse time width.


Figure 7.3: The photoemission integrated intensities are plotted versus the delay 

time between the pump and the probe pulses. The pump and probe measurements 

are carried out at normal incidence and at an angle of 30 ◦ between the 

detector and the normal to the sample surface. In the inset the integrated intensities 

for the Fermi (diamonds) and high energy electron (triangles) regions are 

plotted versus the incident laser fluence. In order to estimate the non-linearity 

order n, the data are fitted with a power function.


The ratio between the measured photoemission intensity I 0 at ∆t = 0 and the 

background intensity I ∞ at ∆t = 300 fs have been measured in the two different 

spectral regions. For two-photon photoemission, for which the expected ratio is 

I 0 

= |E pump + E probe | 2 

I ∞ |E pump | 2 + |E probe | 2 = 4 |E pump| 2 

2 

= 2, (7.2) 

2 |E pump | 

where E pump and E probe = E pump are the intensities of the two light beams, we 

measure a value I 0 /I ∞ = 1.85, compatible with a second order process; for a 

three-photon process the measured ratio value is I 0 /I ∞ = 4.6, compatible with 

the value 

I 0 

= |E pump + E probe | 3 

I ∞ |E pump | 3 + |E probe | 3 = 8 |E pump| 3 

3 

= 4, (7.3) 

2 |E pump | 

expected for a third order photoemission process. 

The close correspondence between the non-linearities measured by laser fluence 

versus electron yield and autocorrelation peak to background contrast ratio 

(PBCR) is an indication that temperature assisted processes are negligible. In 

fact, a thermally assisted photoemission should result in a dependence of the 

electron yield on electron temperature. Since the time scale of electron temperature 

relaxation is one picosecond, this could result in a yield increase on 

the wings of the autocorrelation and a modification of the peak to background 

contrast ratio for delays smaller than the electron-lattice relaxation constant 

[82, 86]. 

Ruling out thermally assisted processes is important in view of the proposed 

models to explain short pulse photoemission: both scattering assisted photoemission 

and transient exciton formation are extremely rapid processes, on the 

scale of few tens of femtoseconds, that are decoupled from the electron thermal 

bath. 

The reported results confirm that the non-equilibrium electron population 

created by a laser pulse at very low laser intensities (I ≃ 0.1 GW/cm 2 ) on Ag 

single crystal, perturbs in a negligible way the equilibrium bulk electron system, 

which behaves as a thermal bath. For this reason the recently reported variations


of the properties of the indirectly populated image potential states [70], such 

as a difference in the effective mass, can not be attributed to a perturbation of 

the properties of the equilibrium electron distribution. The physical mechanism 

responsible for these variations has to be found in the population mechanism 

or in the interaction of electrons in the image potential states with the nonequilibrium 

population. 


The photoemission from the athermal electron gas is due to a pure photoelectric 

process where forbidden dipole transition in sp bands are allowed 

through phonon scattering or through intermediate transient state formation 

by excitonic mechanisms. The absence of thermal contribution is confirmed by 

photoemission autocorrelation measurements.

Chapter 8 

Angle resolved 

photoemission study of 

image potential states and 

surface state on Cu(111) 

Angle resolved non-linear photoemission induced by short laser pulses is 

used to investigate both the occupied surface state and the unoccupied image 

potential states on Cu(111). 

The n = 1 image potential state effective mass measured with a non-resonant 

photon energy between the surface state and the image potential state at zero 

parallel momentum is significantly different from the effective mass measured 

at resonance in the same conditions; the surface state effective mass shows no 

variation with photon energy. To explain the modification of the image potential 

state effective mass in the non-resonant excitation, the influence of the high 

energy electron population in the bulk bands near the image potential state 

95

96 Chapter 8.Image potential states and surface state on Cu(111) 

must be taken into account, in agreement with the phase shift model described 

in Chap. 3. 


The study of many-body phenomena at solid surfaces is an important topic of 

present solid state physics that has benefited from the availability of laser sources 

in the femtosecond regime. Image potential states (IPS) on metal surfaces 

[68, 56, 79, 87] are an important class of surface states (SS) in which ultra short 

laser pulses have been effectively used to obtain a detailed understanding of both 

electron excitation and dynamical processes and to investigate the interaction 

of electrons at surfaces with the surrounding environment. The image potential 

states originate by electrons trapped in front of a metal surface when a gap 

of the projected bulk states occurs at energies below the vacuum level: in this 

case a high reflectivity for the electron wavefunction prevents electrons from 

decaying into the bulk and a long range bounding potential is formed by the 

Coulomb interaction between electrons in the vacuum and their image charge 

in the solid. Electrons in image states are a nearly free quasi two dimensional 

electron gas, with a parabolic dispersion of the energy versus parallel momentum 

k ‖ characterized by an effective mass m ∗ close to the free electron mass. The 

study of the m ∗ deviation from the free electron value is an important tool in 

studying the interactions between image potential states electrons and the bulk 

or adsorbates. 

A variation of the image potential states effective mass is usually observed at 

interfaces between dissimilar materials, such as metal-dielectric interfaces, or in 

stepped lattices [88, 69, 89, 90]. Recently, it has been reported the variation of 

the effective mass due to the interaction of the image potential states electrons 

with a non-equilibrium electron distribution created upon femtosecond laser 

pulse absorption on Ag(100) [70, 91]. In this case the variation of the effective 

mass reveals a strong coupling between the electrons that reside outside the


solid with a hot electrons population in the unoccupied bulk states below the 

Fermi level. 

In this chapter, a variation of the effective mass of the n = 1 image potential 

state on Cu(111), related to scattering-assisted indirect population, is 

reported. The effective mass m ∗ has been measured either tuning the photon 

energy in resonance with the energy difference of the transition between the 

occupied Shockley state and the n = 1 image potential state at k ‖ = 0 (see inset 

of Fig. 8.1), either through a non-resonant excitation with a photon energy 

300 meV smaller. A significantly higher image potential state effective mass is 

measured out of resonance, while the Shockley state effective mass is in agreement 

with previously published values and shows no variation with the photon 

energy. 


The photoemission measurements are performed on Cu(111) single crystals 

with the experimental setup described in Chap. 4. The light sources were the 

fourth harmonic of the traveling-wave optical parametric generator, tuned at 

hν = 4.28 eV, and the second harmonic of the non-collinear optical parametric 

amplifier tuned at hν = 4.5 eV. The samples work function is Φ S = 4.93 eV 

[15]. 

In the first part of the experiment, the photon energy hν = 4.45 eV is tuned 

in resonance with the Shockley state - image potential state transition at k ‖ = 0, 

obtaining a single, sharp photoemission peak due to the transition of the electrons 

from the surface state to the image state (Fig. 8.1). The intrinsic linewidth 

of this feature at k ‖ = 0 is about 18 meV, obtained fitting the photoemission 

peak with a Lorentzian convoluted with a Gaussian to take into account the 

experimental resolution. 

In Fig. 8.2 a) the two-photon photoemission spectra, collected at different 

values of the θ angle between the sample normal and the analyzer, are shown.


Figure 8.1: Non-linear photoemission spectrum collected at the center of the 

Brillouin zone with hν = 4.45 eV in p polarization and a fluence of 68 µJ/cm 2 . 

The feature at about 4.55 eV is the image potential state populated at resonance 

from the surface state. It is fitted by a Lorentzian profile convoluted with a 

Gaussian (line). On the energy axis the measured kinetic energy is reported, 

which includes the contact potential difference V ST = 0.75 eV indicated by the 

arrow. In the inset, the projected band structure of Cu(111) is shown: the 

red and the blue arrows indicate the direct and indirect mechanisms of image 

potential state population.


Figure 8.2: a) Angular dispersion of the photoemission spectra collected at hν = 

4.45 eV, p polarization, with a fluence of 68 µJ/cm 2 , along the ΓM direction of 

the Brillouin zone. b) Kinetic energy versus k ‖ momentum for the n = 1 image 

potential state (full circles) and the Shockley surface state (squares) measured 

with two-photon photoemission on Cu(111). A parabolic fit gives an effective 

mass of m ∗ /m = 1.26±0.07 for the image potential state and m ∗ /m = 0.47±0.04 

for the Shockley surface state.


Due to the different k ‖ dispersion of the Shockley and the image potential states, 

two peaks are observed at non-zero parallel momentum: the first, at high kinetic 

energy, is attributed to a two-photon photoemission from the Shockley surface 

state, whereas the peak at lower energy is due to the direct photoemission from 

an image potential state populated by electrons scattered from other parts of 

the Brillouin zone via phonons or ions collisions [70, 91, 73, 74]. 

In Fig. 8.2 b) the dispersion of the kinetic energy versus k ‖ is shown for 

both the Shockley and the image potential states, giving effective masses of 

m ∗ /m = 0.47 ± 0.04 and m ∗ /m = 1.26 ± 0.07 respectively, in agreement with 

those reported in literature on Cu(111) and similar to that reported for Ag(111) 

[92, 43, 42, 93, 94, 47, 95, 96]. 

In Fig. 8.3 a) two-photon photoemission spectra at different angles are shown, 

collected with a photon energy hν = 4.28 eV, about 300 meV smaller than the 

Shockley-image potential states energy separation at zero parallel momentum. 

The spectrum is characterized by two peaks at k ‖ = 0: the feature at low energy 

is attributed to a two-photon photoemission from the Shockley surface state, the 

second feature at high energy is due to a direct photoemission from an already 

populated image potential state, as discussed previously. At a parallel momentum 

of about 0.18 Å −1 , the photon energy is resonant with the energy separation 

between the two states and only one intense and sharp feature appears in the 

spectrum. 

In Fig. 8.3 b) the dispersion of the kinetic energy versus k ‖ is shown, and in 

this case the effective masses are m ∗ /m = 0.46 ± 0.04 for the Shockley surface 

state and m ∗ /m = 1.64 ± 0.07 for the image potential state. While the effective 

mass of the surface state is in agreement with the value measured with hν = 

4.45 eV, the effective mass of the image potential state is quite different from 

the previous result. 

To rationalize this finding, we note that a 4.28 eV photon causes a direct 

transition from the surface state to the image potential state at a k point in the 

Brillouin zone where the image potential state band crosses the unoccupied sp


Figure 8.3: a) Angular dispersion of the photoemission spectra collected at hν = 

4.28 eV, p polarization, with a fluence of 68 µJ/cm 2 , along the ΓM direction 

of the Brillouin zone. b) Kinetic energy versus k ‖ momentum for the n = 1 

image potential state (full circles) and the Shockley surface state (squares). 

A parabolic fit gives an effective mass of m ∗ /m = 1.64 ± 0.07 for the image 

potential state and m ∗ /m = 0.46 ± 0.04 for the Shockley surface state.


bulk bands and hybridizes with them, so that a high non-equilibrium electron 

population is created in the bulk states near the image potential by the laser 

pulses. 

Considering similar cases reported in the theoretical and experimental literature 

[70, 91, 73, 74], it is likely that the scattering interaction between this 

high density non-equilibrium population and the electrons on the image potential 

state is at the origin of the electron effective mass variation due to a 

modification of the effective potential landscape seen by image potential state 

electrons. Note that in the case of resonant excitation at zero parallel momentum 

(hν = 4.45 eV), this effect is quenched by the much higher probability of 

direct transition between the two states, that makes the creation of a high density 

hot electrons population in the bulk bands near the image potential states 

less probable. In both cases an indirect, scattering assisted population mechanism 

of the image potential states is present, but only in the non-resonant case 

a high density population of hot electrons is induced in the bulk bands near the 

image potential state. 

On Ag(100) the image potential state’s effective mass measured via nonresonant 

excitation is reduced by 9% with respect to the free electron mass [70, 

91]; on Cu(111) the effective mass measured in this work in similar conditions 

increases by 30%. While a theoretical model of the wave function response to 

high non-equilibrium electron densities is not available, these differences can 

be phenomenologically justified on the basis of the different band structures 

of the (100) and (111) surfaces of noble metals. While on (100) surface the 

image potential state is located at the center of the gap, on (111) surfaces it 

stands at the top of the energy gap, near the unoccupied bulk derived bands. 

This makes an important difference as far as the excited electron population is 

concerned, because in the (100) case it is created in the whole of the unoccupied 

bulk state above the Fermi level and below the energy gap, while in the (111) 

case the excited electrons are created in the unoccupied bulk band just above 

the energy gap, near the image potential state. Its energy position in the gap


has been recognized to play an important role in the effective mass modification 

in two-photon photoemission spectroscopy (see, for example, the phase shift 

model in Chap. 3 and in Ref. [43, 42]) and the present data suggest that not 

only the image potential state’s energy position, but also the bulk band where 

the hot population is excited influence the effective mass variation. Further 

experiments were performed to investigate the effective mass behavior depending 

on the variation of an hot electrons population of the projected bulk bands near 

the image potential state: results and discussion are reported in the following 

chapter. 


The effective mass of the Cu(111) image state is measured populating the 

state in resonance and off resonance with respect to the Shockley - image potential 

state transition at k ‖ = 0. While the effective mass of the n = 1 image 

potential state in resonance is in agreement with the literature, the effective 

mass measured with an off resonance photon energy results higher than the 

expected value. The observed difference in effective mass is attributed to the 

effect of a high density electron population out of equilibrium excited by the 

laser pulses.

Chapter 9 

Image potential state 

effective mass variation 

with hot electrons 

population on Cu(111) 

The Cu(111) image potential state effective mass is investigated with twophoton 

and two-color angle resolved photoemission spectroscopy. Populating 

with a hν = 4.71 eV the empty projected bulk bands just above the image 

potential states, we change the reflectivity phase φ C at the crystal barrier and 

the image potential state wave function penetration within the crystal: the 

dispersion along the ΓM direction is no longer parabolic and the effective mass 

is continuously increased. 

105

106 Chapter 9.IPS effective mass versus hot electrons population on Cu(111) 


In Chap. 8 we observed that, when populating the image potential state with 

a photon energy hν = 4.5 eV in resonance with the binding energy difference 

at k ‖ = 0 between the Shockley state and the n = 1 image potential state, the 

expected value of m ∗ /m = 1.26 ± 0.07 was obtained; on the contrary, if the 

photon energy is less than the resonance energy between the two states and the 

population of the image potential state at k ‖ = 0 is possible only by scattering 

after a sp bulk band population, the effective mass increases. 

To further investigate this phenomenon, we want to perform different angle 

resolved photoemission experiments in which we populate the empty projected 

bulk bands with several different pump pulse intensities, measuring then the 

behavior of the image potential state effective mass. 

9.2 Experimental Setup 

Two-color photoemission experiments on a Cu(111) single crystal sample 

were performed with the experimental setup described in Chap. 4. A pump 

pulse of photon energy hν = 4.71 eV, third harmonic of the Ti:Sapphire laser 

source, is tuned in intensity to obtain the desired population on the empty 

bulk bands just above the image potential states; a second harmonic probe 

pulse with hν = 3.14 eV extracts photoelectrons from the so populated image 

potential state: the probe light intensity can be tuned to perform photoemission 

avoiding space charge effects. 

The two pulses are produced splitting a hν = 3.14 eV second harmonic beam 

with a beam splitter: the pump beam undergoes a sum frequency generation 

process, to obtain a hν = 4.71 eV third harmonic; the probe beam impinges on 

the sample after a delay line that is set to ensure the temporal coincidence of 

the two beams. 

Another way to perform the same experiment is a one color two-photon


photoemission; in this way we can not tune the hot electron population, because 

the intensity of the laser beam is set to obtain a correct photoemission yield. 

The energy scheme of the experiment is described in Fig. 9.1; the projected 

bulk band structure of Cu(111) is also shown: the violet arrows indicate the 

indirect mechanisms of image potential state population through scattering of 

hot electrons from higher bulk bands, populated by the pump pulse (blue arrow). 

The probe can be a hν = 3.14 eV photon (light blue arrows) for two-color 

photoemission or another hν = 4.71 eV photon (blue arrows) for two-photon 

photoemission. A coherent two-photon process results in the features ascribed to 

the Shockley surface state, whereas an indirect scattering mediated two-photon 

photoemission yields the peaks ascribed to the image potential state. 


The dispersion relation of the kinetic energy E KE (k ‖ ) of electrons photoemitted 

from the n = 1 image potential state as a function of the parallel momentum 

k ‖ is investigated by angle resolved photoemission along the ΓM direction on 

Cu(111). Two-color and two-photon photoemission were performed with different 

pump pulse intensities: in Fig. 9.2 and Fig. 9.3 two-color photoemission 

results are reported, obtained with pump pulses (hν = 4.71 eV) of 1.8×10 10 and 

3.3 × 10 10 photons respectively; in Fig. 9.4 we show two-photon photoemission 

results obtained with a single hν = 4.71 eV pulse of 6.7 × 10 10 photons. 

Dispersions show that hot electron population cause a deviation from the 

free-electron-like parabolic behavior because of the flattening of the curve near 

the Γ point: this causes the increasing of the effective mass that, by definition, 

is measured for small values of k ‖ . This effect grows with the increasing of bulk 

bands population due to higher light intensity of the pump beam; while the 

effective mass of the Shockley surface state is not affected by the empty band 

population, maintaining a value within m ∗ /m = 0.55±0.04, the image potential 

state effective mass increases with the pump photons number till m ∗ /m = 2.2±


Figure 9.1: Features of a k ‖ = 0 two-color photoemission spectrum (hν = 

4.71 and 3.14 eV) for Cu(111) are explained by a comparison with the calculated 

band structure of the sample.


Figure 9.2: a) Angular dispersion of the two-color photoemission spectra collected 

with a 1.8 × 10 10 photons hν = 4.71 eV pump and a hν = 3.14 eV 

probe along the ΓM direction of the Brillouin zone. b) Kinetic energy versus 

k ‖ momentum for the n = 1 image potential state (full circles) and the 

Shockley surface state (squares) measured with two-color photoemission on 

Cu(111). A parabolic fit of data near the Γ point gives an effective mass of 

m ∗ /m = 1.25 ± 0.07 for the image potential state and m ∗ /m = 0.56 ± 0.03 for 

the Shockley surface state.


Figure 9.3: a) Angular dispersion of the two-color photoemission spectra collected 

with a 3.3 × 10 10 photons hν = 4.71 eV pump and a hν = 3.14 eV 

probe along the ΓM direction of the Brillouin zone. b) Kinetic energy versus 

k ‖ momentum for the n = 1 image potential state (full circles) and the 

Shockley surface state (squares) measured with two-color photoemission on 

Cu(111). A parabolic fit of data near the Γ point gives an effective mass of 

m ∗ /m = 1.58 ± 0.05 for the image potential state and m ∗ /m = 0.54 ± 0.02 for 

the Shockley surface state.


Figure 9.4: a) Angular dispersion of the two-photon photoemission spectra collected 

at hν = 4.71 eV along the ΓM direction of the Brillouin zone. The number 

of photons in the pulse is 6.7 × 10 10 . b) Kinetic energy versus k ‖ momentum 

for the n = 1 image potential state (full circles) and the Shockley surface state 

(squares) measured with two-photon photoemission on Cu(111). A parabolic fit 

gives an effective mass of m ∗ /m = 2.2 ± 0.1 for the image potential state and 

m ∗ /m = 0.62 ± 0.03 for the Shockley surface state.


0.1, instead of the expected m ∗ /m = 1.26 ± 0.07. The measured effective mass 

dependence on the pump pulse’s photons number is shown in Fig. 9.5 for both 

Shockley surface state and n = 1 image potential state. 

Figure 9.5: Effective mass dependence on the pump pulse’s photons number. 

While the Shockley surface state dispersion is not affected by the variation in 

the pump intensity, the n = 1 image potential state effective mass increases 

with the empty projected bulk bands population due to the pump pulse. 

An explanation can be given in terms of the phase shift model described in 

Chap. 3. The population of the empty projected bulk bands on the top of the 

gap creates a repulsion for electrons occupying the image potential state, whose 

wave functions are expelled from the bulk: the reflectivity phase φ C at the 

crystal barrier decreases, modifying the quantum defect a and then the binding 

energy; as a last consequence, the effective mass value is also changed.



Performing angle resolved two-color and two-photon photoemission we can 

measure the change of the Cu(111) n = 1 image potential state effective mass, 

in the dispersion along the ΓM direction, as a function of the hot electron population 

of the projected bulk sp bands on the top of the gap; the electron density 

on these normally empty states can be governed tuning the pump beam intensity. 

The image potential state dispersion in these conditions is not parabolic: 

a flattening of the dispersion curve in the region near to Γ increases the effective 

mass value continuously with the bulk bands electron population, up to 

m ∗ /m = 2.2 ± 0.1. This effect can be explained in the phase shift model (see 

Chap. 3) context as a reduction of the reflectivity phase φ C at the crystal barrier, 

due to the repulsion between electrons populating the bulk bands and the 

image potential states, whose wave function is expelled from the bulk.

Appendix A 

Calculations about the 

phase shift model 

In this appendix we show calculations in which we recover some well known 

formulas concerning quantities treated in the phase shift model and that we 

report in Chap 3. 

A.1 Energy of the image potential state 

A.1.1 

Nearly free electron model 

If we consider a system of nearly free electrons [97] subject to a weak periodic 

potential V (r) due to a lattice of ions, we can write 

V (r) = ∑ K 

V K e iK·r , 

(A.1) 

115

116 Appendix A.Calculations about the phase shift model 

where K is any vector of the reciprocal lattice. The wave function of a Block 

level with crystal momentum q 

ψ q (r) = ∑ K 

c q−K e i(q−K)·r 

(A.2) 

has an energy E that depends on the free electron energy 1 

E q = ħ2 q 2 

2m . 

(A.3) 

Focusing on a fixed q for which two vectors of the reciprocal lattice K 1 and K 2 

exist such as 

|E q−K1 − E q−K2 | < V (r) 

|E q−K − E q−Ki | ≫ V (r) for K ≠ K 1 , K 2 and i = 1, 2, 

(A.4) 

we can obtain the energy E annulling the determinant of the matrix 

[ 

ħ 2 k 2 

2m − E V g 

ħ 

V 2 (k−g) 2 

g 2m 

− E 

with V g = V −g = V g . 

] 

for k = q − K 1 and g = K 2 − K 1 , (A.5) 

From (A.2) and (A.5) we note that ψ q (r) = ψ k (r), because of the sum over 

any vector K of the reciprocal lattice. 

A.1.2 

Total energy of the image potential state 

We want to investigate the image potential state that lies in a gap of the projected 

bulk bands, so we consider a wavefunction of energy E whose wavevector 

q has a complex component along the z direction, normal to the solid surface: 

in this case we can write 

k = q − K 1 = k ‖ + k z with k z = p − iq (A.6) 

1 We use the notation q 2 = qx 2 + qy 2 + qz 2 ∈ C.

A.1. Energy of the image potential state 117 

and ψ k (r) becomes a surface wavefunction that exponentially decays in the bulk, 

where z < 0 [43, 42]. As seen in (A.5), its energy is the solution of the equation 

E 2 − ħ2 [ 

k 2 + (k − g) 2] ( ) ħ 

2 2 

[ 

E + k 2 (k − g) 2] − Vg 2 = 0. (A.7) 

2m 

2m 

In the easiest case, with the vector g normal to the solid surface and the only 

g z component different from zero, to fulfill the condition (A.4), we chose k on 

the zone boundary associated with g imposing 

p = g z 

2 

(A.8) 

and obtaining 

R [E q−K1 − E q−K2 ] = R [E k − E g ] = 

= ħ2 

2m R [ k 2 − ( k 2 + g 2 − 2k · g )] = 

= ħ2 

2m R [ 2(p − iq)g z − gz 

2 ] 

= 

[ ( 

= ħ2 

2m R gz 

) ] 

2 

2 − iq g z − gz 

2 = 

= ħ2 

2m R [−2iqg z] = 0; 

(A.9) 

therefore, (A.4) is fulfilled if 

|E q−K1 − E q−K2 | = |−2iqg z | = 2qg z < V (r). (A.10) 

From (A.8), we calculate 

(k − g) 2 = ( k ‖ + k z − g z 

) 2 

= k 

2 

‖ + (p − iq − 2p) 2 = k 2 ‖ + (p + iq)2 (A.11) 

and, developing some terms as 

( 

k 2 + (k − g) 2 = k‖ 2 + (p − iq)2 + k‖ 2 + (p + iq)2 = 2 k‖ 2 + p2 − q 2) 

(A.12)


and 

[ 

[ 

k 2 (k − g) 2 = k‖ 2 + (p − iq)2] k‖ 2 + (p + iq)2] = 

= k‖ 4 + [ k2 ‖ (p + iq) 2 + (p − iq) 2] + (p − iq) 2 (p + iq) 2 = (A.13) 

= k‖ 4 + ( 2k2 ‖ p 2 − q 2) + ( p 2 + q 2) 2 

, 

we can write (A.7) as 

( 

E 2 − 2 ħ2 

k‖ 2 2m 

+ p2 − q 2) E+ 

( ) ħ 

2 2 [ 

+ k‖ 4 2m 

+ ( 2k2 ‖ p 2 − q 2) + ( p 2 + q 2) ] 2 

− Vg 2 = 0 

(A.14) 

to obtain the values of energy 

{ 

( 

( 

E = ħ2 

k‖ 2 2m 

+ p2 − q 2) ) ħ 

2 2 ( 

± 

k‖ 2 2m 

+ p2 − q 2) 2 

+ 

( ) ħ 

2 2 [ 

− k‖ 4 2m 

+ ( 2k2 ‖ p 2 − q 2) + ( p 2 + q 2) ] } 1/2 

2 

+ Vg 2 = 

( 

= ħ2 

2m 

{ ( ħ 

2 

± 

k 2 ‖ + p2 − q 2) ± 

2m 

) 2 [ 

k 4 ‖ + p4 + q 4 + 2k ‖ 

( 

p 2 − q 2) − 2p 2 q 2] + 

( ) ħ 

2 2 

} 

[ 

1/2 

− k‖ 4 2m 

+ ( 2k2 ‖ p 2 − q 2) + p 4 + q 4 + 2p 2 q 2] + Vg 2 = 

= ħ2 k‖ 

2 √ 

2m + ħ2 p 2 

2m − ħ2 q 2 

2m ± Vg 2 − 4 ħ2 p 2 ħ 2 q 2 

2m 2m . 

(A.15) 

We observe that the choice of k on a Bragg plane (A.8) ensures real coefficients 

for (A.14) and then real values for E, once imposed on the discriminant 

the condition 

Vg 2 − 4 ħ2 p 2 ħ 2 q 2 

2m 2m > 0. 

(A.16)

A.1. Energy of the image potential state 119 

A.1.3 

Energy associated with the imaginary part q 

Defining 

E g = ħ2 

2m 

( gz 

) 2 ħ 2 p 2 

= 

2 2m 

and 

ε = E − ħ2 k 2 ‖ 

2m = ħ2 u 2 

2m , 

(A.17) 

we can write 

√ 

ħ 2 q 2 

2m + ε − E g = ± 

( ħ 2 q 2 ) 2 

+ 2 (ε − E g ) ħ2 q 2 

2m 

( ħ 2 q 2 ) 2 

+ 2 (ε + E g ) ħ2 q 2 

2m 

V 2 g − 4E g 

ħ 2 q 2 

2m 

2m + (ε − E g) 2 = Vg 2 ħ 2 q 2 

− 4E g 

2m 

2m + (ε − E g) 2 − V 2 

g = 0 (A.18) 

from which we eventually obtain 

ħ 2 q 2 

√ 

2m = − (ε + E g) + (ε + E g ) 2 + Vg 2 − (ε − E g ) 2 

√ 

= − (ε + E g ) + ε 2 + Eg 2 + 2εE g + Vg 2 − ε 2 − Eg 2 + 2εE g = 

√ 

= − (ε + E g ) + Vg 2 + 4εE g . 

(A.19) 

A.1.4 

Non-kinetic part of the energy 

We can define a non-kinetic part of the energy subtracting from the energy 

E the kinetic contribution ħ 2 k 2 /2m; using (A.15), the absolute value of this


quantity is 

∣ ∣ E − ħ2 k 2 

∣∣∣ 2m ∣ = E − ħ2 

2m 

= 

∣ E − ħ2 

2m 

{ (√ 

and, writing 

= 

= 

√ 

× 

[ 

k 2 ‖ + (p + iq)2]∣ ∣ ∣∣ = 

[ 

k 2 ‖ + p2 − q 2 + 2ipq] ∣ ∣ ∣∣ = 

Vg 2 − 4 ħ2 p 2 

2m 

(√ 

V 2 g − 4 ħ2 p 2 

2m 

V 2 g − 4 ħ2 p 2 

2m 

ħ 2 q 2 

2m + ħ2 

2m 2ipq ) 

× 

ħ 2 q 2 

2m − ħ2 

2m 2ipq ) } 1/2 

= 

ħ 2 q 2 

2m + ( ħ 

2 

2m 

∣ E − ħ2 k 2 

2m = ħ2 k 2 ∣∣∣ 

2m = E − ħ2 k 2 

2m ∣ eiφ = V g e i2δ , 

) 2 

4p 2 q 2 = V g 

(A.20) 

we can calculate 

⎡ ⎤ / ∣ sin(2δ) = I ⎣E − ħ2 k 2 

∣∣∣∣ 

⎦ E − ħ2 k 2 

2m 

2m 

∣ = − ħ2 

2m 2pq/V g. 

(A.21) 

(A.22) 

A.2 Wavefunction of the image potential state 

A.2.1 

Schrödinger’s equation 

The solution (A.2) of the Schrödinger’s equation 

] 

[− ħ2 

2m ∇2 + V (r) ψ k (r) = Eψ k (r), 

(A.23) 

providing (A.4) is fulfilled, using (A.8) and supposing k ‖ = 0 to investigate the 

simplest case, can be written as 

ψ k (r) = ψ q (r) = ∑ K 

c q−K e i(q−K)·r = c k e ik·r + c k−g e i(k−g)·r = 

= c p e i(p−iq)z + c −p e i(−p−iq)z ; 

(A.24)

A.2. Wavefunction of the image potential state 121 

since this is a superposition of two counterpropagating standing waves, we impose 

them to have the same amplitude and choose their coefficients’ modulus 

equal to 1, writing 

ψ k (r) = e qz ( e i(pz+δ+) + e −i(pz+δ− ) ) . 

(A.25) 

A.2.2 

Calculation of the ψ k (r) derivatives 

To verify that ψ k (r) is a solution of(A.23), we have to calculate its derivatives 

along the z direction. The first derivative is 

dψ k (r) 

dz 

= qe qz ( e i(pz+δ+) + e −i(pz+δ− ) ) + 

+ e qz ( ipe i(pz+δ+) − ipe −i(pz+δ− ) ) = 

(A.26) 

the second derivative is 

= e qz [ (q + ip)e i(pz+δ+) + (q − ip)e −i(pz+δ− ) ] ; 

d 2 ψ k (r) 

dz 2 = qe qz [ (q + ip)e i(pz+δ+) + (q − ip)e −i(pz+δ− ) ] + 

+ e qz [ (ipq − p 2 )e i(pz+δ+) + (−ipq − p 2 )e −i(pz+δ− ) ] = 

= −e qz[ (p 2 − 2ipq − q 2 )e i(pz+δ+) + 

(A.27) 

+ (p 2 + 2ipq − q 2 )e −i(pz+δ− ) ] = 

= −e qz [ (p − iq) 2 e i(pz+δ+) + (p + iq) 2 e −i(pz+δ− ) ] . 

A.2.3 

Solution of Schrödinger’s equation 

We want to find out for which values of δ + and δ − the wavefunction ψ k (r) 

(A.25) is solution of the Scrödinger’s equation (A.23). We firstly focus on the 

term 

V (r)ψ k (r) = ∑ V K c h e i(h+K)·r (A.28) 

K,h


which yields significant contributions [97] only for 

K = g, h = k − g or K = −g, h = k, (A.29) 

giving then 

V (r)ψ k (r) = ∑ V K c h e i(h+K)·r = 

K,h 

= V g c k−g e i(k−g)·r e ig·r + V −g c k e ik·r e −ig·r = 

= V g c k−g e ik·r + V −g c k e i(k−g)·r = 

= V g c −p e i(p−iq)z + V g c p e i(−p−iq)z = 

= V g e −iδ− e i(p−iq)z + V g e iδ+ e i(−p−iq)z = 

(A.30) 

= V g e qz ( e i(pz−δ−) + e −i(pz−δ+ ) ) . 

The Scrödinger’s equation (A.23) is written, using (A.21), as 

0 = ħ2 

2m eqz [ (p − iq) 2 e i(pz+δ+) + (p + iq) 2 e −i(pz+δ− ) ] + 

+ V g e qz ( e i(pz−δ−) + e −i(pz−δ+ ) ) + 

( ) 

− Ee qz e i(pz+δ+) + e −i(pz+δ− ) 

= 

{ ] 

= e qz − 

[E − ħ2 

2m (p − iq)2 e i(pz+δ+) − 

} 

+ V g e i(pz−δ−) + V g e −i(pz−δ+ ) 

= 

[E − ħ2 

2m (p + iq)2 ] 

e −i(pz+δ−) + 

{ 

= e qz − [ V g e −i2δ] e i(pz+δ+) − [ V g e i2δ] e −i(pz+δ−) + 

} 

+ V g e i(pz−δ−) + V g e −i(pz−δ+ ) 

= 

{ 

} 

= e qz V g − e i(pz+δ+ −2δ) − e −i(pz+δ−−2δ) + e i(pz−δ−) + e −i(pz−δ+ ) 

= e qz V g 

{ 

e ipz ( e −iδ− − e i(δ+ −2δ) ) + e −ipz ( e iδ+ − e −i(δ− −2δ) ) } 

= 

(A.31)

A.2. Wavefunction of the image potential state 123 

and is fulfilled if δ + = δ − = δ. The wavefunction (A.25) can then be written as 

ψ k (r) = e qz ( e i(pz+δ) + e −i(pz+δ)) = 2e qz cos(pz + δ). 

(A.32) 

A.2.4 

Phase φ C due to the reflection on the crystal surface 

A relation concerning the phase φ C , added to the image potential state 

wavefunction for each reflection on the mirror plane on the crystal surface, 

is obtained imposing the continuous matching of the wave function and first 

derivative in any point of the image plane z = z 0 : we match the bulk wave 

e qz cos(pz + δ) for z < z 0 with the external wave e −iuz + r C e iφ C 

e iuz for z > z 0 , 

whose wavevector u is defined in Eq. (A.17). 

To obtain the matching, we impose r C = 1 and we obtain 

e qz0 cos(pz 0 + δ) = e −iuz0 + e iφ C 

e iuz0 , 

(A.33) 

while from the first derivatives we obtain 

qe qz0 cos(pz 0 + δ) − pe qz0 sin(pz 0 + δ) = −iue −iuz0 + e iφ C 

iue iuz0 

−e qz0 cos(pz 0 + δ) (p tan(pz 0 + δ) − q) = iu ( e iφ C 

e iuz0 − e −iuz0) 

p tan(pz 0 + δ) − q = −iu eiφ C 

e iuz0 − e −iuz0 

e qz0 cos(pz 0 + δ) 

(A.34) 

and, substituting Eq. (A.33) into Eq. (A.34), 

p tan(pz 0 + δ) − q = −iu eiφ C 

e iuz0 − e −iuz0 

e iφ C e 

iuz 0 + e 

−iuz 0 

= −iu eiφ C 

− e −i2uz0 

e iφ C + e 

−i2uz 0 

. 

(A.35) 

Supposing the z = z 0 plane to be characterized by the condition uz 0 = ±π, 

we obtain 

Eq. (A.35) becomes 

e −i2uz0 = e ±i2π = 1 : (A.36) 

u tan(φ C /2) = u sin(φ C/2) 

cos(φ C /2) = −iueiφ C/2 − e −iφ C/2 

e iφ C/2 

+ e −iφ C/2 = −iueiφ C 

− 1 

e iφ C + 1 

= 

= p tan(pz 0 + δ) − q. 

(A.37)


A.3 Hydrogen-like Rydberg series of states 

We interpret the image potential state wavefunction as a multiply reflected 

standing wave between the surface crystal barrier, where the electron can not 

enter because of the gap in the projected bulk bands, and the image charge 

Coulomb potential, that traps electrons with energy less than the vacuum energy 

E V . For each reflection the wavefunction is multiplied by a factor r B e iφ B 

on the 

Coulomb boundary or r C e iφ C 

we impose 

on the crystal surface: to have a standing wave 

r B = r C = 1 and φ B + φ C = 2nπ, n ∈ Z. (A.38) 

When a wavefunction is reflected by a Coulomb potential V (z) = (4πε 0 z) −1 , 

its phase is modified by an amount [44] 

(√ ) 

3.4 eV 

φ B = π 

− 1 ; (A.39) 

ε 

the stationarity condition Eq. (A.38) becomes 

(√ ) 

3.4 eV 

π 

− 1 + φ C = 2nπ, n ∈ Z 

ε 

√ 

3.4 eV 

= 2n + 1 − φ C 

ε n 

π 

√ 

√ 3.4 eV 

εn = 

2n + 1 − φ C /π . 

(A.40) 

In a Shockley inverted gap [35], where the s bands lay at the top and the 

p bands at the bottom of the gap, φ C continuously varies from π in the upper 

edge to 0 in the lower edge: defining the quantum defect 

a = 1 ( 

1 − φ ) 

C 

, (A.41) 

2 π 

ranging from 0 at the top of the gap to 1/2 at the bottom, and imposing n ∈ N 

to obtain a positive value for the square root (A.40) of the binding energy ε n ,

A.3. Hydrogen-like Rydberg series of states 125 

we obtain a series of hydrogen-like Rydberg states with binding energy 

ε n = 

3.4 eV 

(2n + 1 − φ C /π) 2 = 3.4 eV 

0.85 eV 

2 

= 

4 (n + (1 − φ C /π) /2) (n + a) 2 ; (A.42) 

the energy difference between two neighbor states vanishes approaching the 

vacuum energy level as n increases to infinity.

List of Publications 

1. E. Pedersoli, F. Banfi, S. Pagliara, G. Galimberti, G. Ferrini, C. Giannetti 

and F. Parmigiani: Evidence of Vectorial Photoelectric Effect on Copper. 

Appl. Phys. Lett., 87, 081112 (2005). 

2. F. Banfi, G. Ferrini, E. Pedersoli, S. Pagliara, C. Giannetti, G. Galimberti, 

S. Lidia, J. Corlett, B. Ressel, F. Parmigiani: Monte Carlo Transverse 

Emittance Study On Cs 2 Te. JAKoW/eConf, C0508213, 572 (2005). 

3. S. Caravati, G. Butti, G. P. Brivio, M. I. Trioni, S. Pagliara, G. Ferrini, 

G. Galimberti, E. Pedersoli, C. Giannetti and F. Parmigiani: Cu(111) 

and Cu(001) surface electronic states. Comparison between theory and 

experiment. Surf. Sci., 600, 3901 (2006). 

4. C. Giannetti, G. Ferrini, S. Pagliara, G. Galimberti, F. Banfi, E. Pedersoli, 

and F. Parmigiani: On the role of athermal electrons in non-linear 

photoemission from Ag(100). Eur. Phys. J. B, 00337, 0 (2006). 

5. S. Pagliara, G. Ferrini, G. Galimberti, E. Pedersoli, C. Giannetti and 

F. Parmigiani: Angle Resolved Photoemission Study of Image Potential 

States and Surface States on Cu(111). Surf. Sci., 600, 4290 (2006). 

127

Bibliography 

[1] E. Pedersoli et al., Appl. Phys. Lett. 87, 081112 (2005). 

[2] F. Banfi et al., JAKoW / eConf C0508213, 572 (2005). 

[3] S. Caravati et al., Surf. Sci. 600, 3901 (2006). 

[4] C. Gannetti et al., Eur. Phys. J. B 00337, 0 (2006). 

[5] S. Pagliara et al., Surf. Sci. 600, 4290 (2006). 

[6] R. W. Schoenlein et al., Science 287, 5461 (2000). 

[7] R. Neutze et al., Nature 406, 752 (2000). 

[8] H. C. Kapteyn and T. Ditmire, Nature 429, 467 (2002). 

[9] D. W. Juenker, J. P. Waldron, and R. J. Jaccodine, J. Opt. Soc. Am. 54, 

216 (1964). 

[10] R. M. Broudy, Phys. Rev. B 1, 3430 (1970). 

[11] R. M. Broudy, Phys. Rev. B 3, 3641 (1971). 

[12] J. P. Girardeau-Montaut, S. D. Girardeau-Montaut, S. D. Moustaizis, and 

C. Fotakis, Appl. Phys. Lett. 63, 699 (1993). 

129

130 BIBLIOGRAPHY 

[13] T. Srinivasan-Rao, J. Fischer, and T. Tsang, Appl. Phys. Lett. 63, 1596 

(1993). 

[14] H. Y. Fan, Phys. Rev. 68, 43 (1945). 

[15] R. C. Weast and M. J. Astle, Hand Book of Chemistry and Physics (CRC 

Press, Boca Raton, Florida, 1982). 

[16] M. Afif et al., App. Surf. Sci. 96, 469 (1996). 

[17] C. Beleznai, D. Vouagner, J. Girardeau-Montaut, and L. Nanai, App. Surf. 

Sci. 138, 512 (1999). 

[18] G. P. Banfi, G. Ferrini, M. Peloi, and F. Parmigiani, Phys. Rev. B 67, 

035428 (2003). 

[19] J. Zawadzka, D. A. Jaroszynski, J. Carey, and K. Wynne, Appl. Phys. Lett. 

79, 2130 (2001). 

[20] M. Kupersztych and P. Monchicourt, Phys. Rev. Lett. 86, 5180 (2001). 

[21] P. J. Feibelman, Phys. Rev. B 12, 1319 (1975). 

[22] P. J. Feibelman, Phys. Rev. Lett. 34, 1092 (1975). 

[23] H. J. Levinson, E. W. Plummer, and P. J. Feibelman, Phys. Rev. Lett. 43, 

952 (1979). 

[24] D. L. Misell and A. J. Atkins, Philos. Mag. 27, 95 (1973). 

[25] W. E. Spicer and A. Herrera-Gomez, SLAC-PUB- 6306, (1993). 

[26] P. G. O’Shea and H. P. Freund, Science 292, 1853 (2001). 

[27] A. H. Sommer, Photoemissive materials (Robert E. Krieger Pub. Co., New 

York, 1980). 

[28] F. W. R. Stuart and W. E. Spicer, Phys. Rev. 135, A495 (1964).

BIBLIOGRAPHY 131 

[29] G. Ferrini, P. Michelato, and F. Parmigiani, Solid State Comm. 106, 21 

(1998). 

[30] W. E. Spicer, Phys. Rev. 112, 114 (1958). 

[31] R. A. Powell, W. E. Spicer, G. B. Fisher, and P. Gregory, Phys. Rev. B 8, 

3987 (1973). 

[32] G. B. Fisher et al., App. Opt. 12, 799 (1973). 

[33] K. Flottmann, TESLA-FEL-97-01 DESY-HH (1997). 

[34] J. Clendenin and G. A. Mullhollan, SLAC-PUB- 7760, (1998). 

[35] W. Shockley, Phys. Rev. 56, 317 (1939). 

[36] U. Höfer et al., Science 277, 1480 (1997). 

[37] M. W. Cole and M. H. Cohen, Phys. Rev. Lett. 23, 1238 (1969). 

[38] N. Garcia, B. Rehil, K. H. Frank, and A. R. Williams, Phys. Rev. Lett. 54, 

591 (1985). 

[39] M. Ortuño and P. M. Echenique, Phys. Rev. B 34, 5199 (1986). 

[40] N. V. Smith and C. T. Chen, Phys. Rev. B 40, 7565 (1989). 

[41] M. Weinert, S. H. Hulbert, and P. D. Johnson, Phys. Rev. Lett. 55, 2055 

(1985). 

[42] K. Giesen et al., Phys. Rev. B 35, 975 (1987). 

[43] N. V. Smith, Phys. Rev. B 32, 3549 (1985). 

[44] E. G. McRae, Rev. Mod. Phys. 51, 541 (1979). 

[45] Th. Fauster and W. Steinmann, Two-photon photoemission spectroscopy of 

image states, in Photonic Probes of Surfaces (Elsevier, Amsterdam, 1995).


[46] S. LaShell, B. A. McDougall, and E. Jensen, Phys. Rev. Lett. 77, 3419 

(1996). 

[47] F. Reinert et al., Phys. Rev. B 63, 115415 (2001). 

[48] G. Nicolay, F. Reinert, S. Hüfner, and P. Blaha, Phys. Rev. B 65, 033407 

(2001). 

[49] J. Henk, A. Ernst, and P. Bruno, Phys. Rev. B 68, 165416 (2003). 

[50] N. Memmel and R. Bertel, Phys. Rev. Lett. 75, 485 (1995). 

[51] M. Wolf, E. Knoesel, and T. Hertel, Phys. Rev. B 54, R5295 (1996). 

[52] R. Joynt, Science 284, 777 (1999). 

[53] D. L. Mills, Phys. Rev. B 62, 11197 (2000). 

[54] B. Quiniou, V. Bulović, and J. R. M. Osgood, Phys. Rev. B 23, 15890 

(1993). 

[55] G. Fratesi, G. P. Brivio, P. Rinke, and R. W. Godby, Phys. Rev. B 68, 

195404 (2003). 

[56] P. M. Echenique et al., Surf. Sci. Rep. 52, 219 (2004). 

[57] J. M. Carlsson and B. Hellsing, Phys. Rev. B 61, 13973 (2000). 

[58] G. Butti et al., Phys. Rev. B 72, 125405 (2005). 

[59] F. Forster et al., Surf. Sci. 532-535, 160 (2003). 

[60] G. P. Brivio and M. I. Trioni, Rev. Mod. Phys. 71, 231 (1999). 

[61] H. Ishida, Phys. Rev. B 63, 165409 (2001). 

[62] M. Nekovee and J. E. Inglesfield, Progr. Surf. Sci. 50, 149 (1996). 

[63] J. E. Inglesfield, J. Phys. C 14, 3795 (1981).


[64] G. Butti, M. I. Trioni, and H. Ishida, Phys. Rev. B 70, 195425 (2004). 

[65] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 

(1996). 

[66] J. Kliewer et al., Science 288, 1399 (2000). 

[67] R. Haight, Surf. Sci. Rep. 21, 275 (1995). 

[68] H. Petek and S. Ogawa, Prog. Surf. Sci. 56, 239 (1997). 

[69] A. Miller et al., Science 297, 1163 (2002). 

[70] G. Ferrini et al., Phys. Rev. Lett. 92, 256802 (2004). 

[71] F. Bisio et al., Phys. Rev. Lett. 96, 087601 (2006). 

[72] W. Eckart, W.-D. Schöne, and R. Keiling, Appl. Phys. A 71, 529 (2000). 

[73] A. V. Lugovskoy and I. Bray, Phys. Rev. B 60, 3279 (1999). 

[74] B. Rethfeld, A. Kaiser, M. Vicanek, and G. Simon, Phys. Rev. B 65, 214303 

(2002). 

[75] W.-D. Schöne and W. Eckart, Phys. Rev. B 65, 113112 (2002). 

[76] Z. Li and S. Gao, Phys. Rev. B 50, 15394 (1994). 

[77] F. Banfi et al., Phys. Rev. Lett. 94, 037601 (2005). 

[78] S. Ogawa, H. Nagano, and H. Petek, Phys. Rev. B 55, 10869 (1997). 

[79] T. Hertel, E. Knoesel, M. Wolf, and G. Ertl, Phys. Rev. Lett. 76, 535 

(1996). 

[80] J. Cao et al., Phys. Rev. B 56, 1099 (1997). 

[81] E. Knoesel, A. Hotzel, and M. Wolf, Phys. Rev. B 57, 12812 (1998).


[82] J. G. Fujimoto, J. M. Liu, E. P. Ippen, and N. Bloembergen, Phys. Rev. 

Lett. 53, 1387 (1984). 

[83] W. S. Fann, R. Storz, H. W. K. Tom, and J. Bokor, Phys. Rev. Lett. 68, 

2834 (1992). 

[84] W. S. Fann, R. Storz, H. W. K. Tom, and J. Bokor, Phys. Rev. B 46, 13592 

(1992). 

[85] D. Pines and P. Nozieres, The Theory of Quantum Liquids (Benjamin, New 

York, 1969). 

[86] A. Bartoli et al., Phys. Rev. B 56, 1107 (1997). 

[87] M. Wolf, A. Hotzel, E. Knoesel, and D. Velic, Phys. Rev. B 59, 5926 (1999). 

[88] N.-H. Ge et al., Science 279, 202 (1998). 

[89] F. Baumberger et al., Phys. Rev. Lett. 92, 196805 (2004). 

[90] M. Roth et al., Phys. Rev. Lett. 88, 096802 (2002). 

[91] G. Ferrini et al., J. Electron Spectroc. and Relat. Phenom. 144-147, 565 

(2005). 

[92] A. Goldmann, V. Dose, and G. Borstel, Phys. Rev. B 32, 1971 (1985). 

[93] R. W. Schoenlein, J. G. Fujimoto, G. L. Eesley, and T. W. Capehart, Phys. 

Rev. B 43, 4688 (1991). 

[94] D. F. Padowitz, W. R. Merry, R. E. Jordan, and C. B. Harris, Phys. Rev. 

Lett. 69, 3583 (1992). 

[95] M. Weinelt, Appl. Phys. A 71, 493 (2000). 

[96] A. Hotzel, M. Wolf, and J. Gauyaeq, J. Phys. Chem. B 104, 8438 (2000).


[97] N. W. Aschcroft and N. D. Mermin, Solid State Physics (Saunders College, 

Philadelphia, 1976).

Femtosecond Photoemission Investigation of Electron Dynamics at ...

Create successful ePaper yourself

Delete template?

Save as template?